Subgroups — C₆₇×#61574 #30740825

Order 17152.

2094 conjugacy classes of subgroups.

The subgroup lattice has 908 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3505	C₂	1	1	yes	maximal, characteristic
2	C₂×D₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3320	C₂	1	4	yes	maximal, complemented
4	C₆₇×#3312	C₂	1	4	yes	maximal, complemented
5	C₆₇×#3270	C₂	1	4	yes	maximal, complemented
6	C₆₇×#3266	C₂	1	4	yes	maximal, complemented
7	C₆₇×#3218	C₂	1	2	yes	maximal, complemented
8	C₆₇×#3196	C₂	1	2	yes	maximal, complemented
9	C₆₇×#3189	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#3187	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#3117	C₂	1	2	yes	maximal, complemented
12	C₆₇×#3004	C₂	1	2	yes	maximal, complemented
13	C₂×#451×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#448×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₇×#2973	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
18	C₂×#137×C₆₇	C₂²	1	4	yes	complemented
19	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	complemented
20	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	complemented
21	#553×C₆₇	—	2	2	no	—
22	#553×C₆₇	C₂²	1	2	yes	—
23	#552×C₆₇	C₂²	1	2	yes	—
24	#551×C₆₇	—	2	2	no	—
25	D₄×Q₈×C₆₇	—	2	2	no	—
26	D₄×Q₈×C₆₇	—	2	2	no	—
27	D₄×Q₈×C₆₇	C₂²	1	4	yes	—
28	D₄×Q₈×C₆₇	C₂²	1	4	yes	complemented
29	#548×C₆₇	C₂²	1	2	yes	—
30	#546×C₆₇	C₂²	1	2	yes	—
31	#539×C₆₇	—	2	1	no	—
32	#538×C₆₇	—	2	1	no	—
33	#534×C₆₇	—	2	1	no	—
34	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
35	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
36	#529×C₆₇	C₂²	1	2	yes	—
37	#525×C₆₇	C₂²	1	2	yes	—
38	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
39	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
40	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
42	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
43	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
44	#518×C₆₇	—	2	1	no	—
45	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
46	C₂×C₄×D₄×C₆₇	—	2	1	no	—
47	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
48	#497×C₆₇	C₂²	1	4	yes	complemented
49	#491×C₆₇	C₂²	1	4	yes	complemented
50	#485×C₆₇	C₂²	1	4	yes	complemented
51	#484×C₆₇	C₂²	1	2	yes	complemented
52	#484×C₆₇	C₂²	1	2	yes	complemented
53	#483×C₆₇	C₂²	1	2	yes	complemented
54	#481×C₆₇	C₂²	1	2	yes	—
55	#479×C₆₇	C₂²	1	4	yes	complemented
56	#470×C₆₇	C₂²	1	4	yes	complemented
57	#468×C₆₇	C₂²	1	4	yes	complemented
58	#467×C₆₇	C₂²	1	4	yes	complemented
59	#465×C₆₇	C₂²	1	4	yes	complemented
60	#464×C₆₇	C₂²	1	4	yes	complemented
61	#462×C₆₇	C₂²	1	4	yes	complemented
62	#461×C₆₇	C₂²	1	4	yes	complemented
63	#452×C₆₇	C₂²	1	4	yes	complemented
64	#451×C₆₇	C₂²	1	4	yes	complemented
65	#451×C₆₇	C₂²	1	2	yes	complemented
66	#451×C₆₇	C₂²	1	4	yes	complemented
67	#451×C₆₇	C₂²	1	2	yes	complemented
68	#451×C₆₇	C₂²	1	2	yes	complemented
69	#451×C₆₇	C₂²	1	2	yes	complemented
70	#450×C₆₇	C₂²	1	2	yes	complemented
71	#448×C₆₇	C₂²	1	2	yes	—
72	#448×C₆₇	C₂²	1	2	yes	—
73	#448×C₆₇	C₂²	1	2	yes	—
74	#448×C₆₇	C₂²	1	4	yes	complemented
75	#448×C₆₇	C₂²	1	4	yes	complemented
76	#441×C₆₇	C₂²	1	4	yes	complemented
77	#440×C₆₇	C₂²	1	4	yes	complemented
78	#435×C₆₇	C₂²	1	4	yes	—
79	#435×C₆₇	C₂²	1	4	yes	complemented
80	#432×C₆₇	C₂²	1	2	yes	complemented
81	#419×C₆₇	C₂²	1	4	yes	complemented
82	C₂×#103×C₆₇	C₂²	1	2	yes	complemented
83	C₂×#103×C₆₇	C₂²	1	2	yes	complemented
84	#407×C₆₇	C₂²	1	2	yes	complemented
85	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
86	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
87	C₂×#41×C₆₇	C₂³	1	2	yes	—
88	C₂²×Q₈×C₆₇	—	2	1	no	—
89	C₂²×Q₈×C₆₇	—	2	1	no	—
90	C₂²×Q₈×C₆₇	—	2	1	no	—
91	C₂²×Q₈×C₆₇	—	2	1	no	—
92	C₂²×Q₈×C₆₇	—	2	2	no	—
93	C₂²×Q₈×C₆₇	—	4	4	no	—
94	C₂²×Q₈×C₆₇	—	4	4	no	—
95	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
96	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
97	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
98	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
99	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
100	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
101	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
102	C₂²×D₄×C₆₇	—	2	1	no	—
103	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
104	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
105	C₂³×C₄×C₆₇	—	2	1	no	—
106	C₂³×C₄×C₆₇	—	2	1	no	—
107	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
108	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
109	#137×C₆₇	—	4	8	no	—
110	C₂×Q₁₆×C₆₇	—	4	4	no	—
111	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
112	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
113	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
114	C₂×SD₁₆×C₆₇	—	4	4	no	—
115	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
116	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
117	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
118	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
119	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
120	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
121	C₄⋊Q₈×C₆₇	—	2	4	no	—
122	C₄⋊Q₈×C₆₇	—	2	4	no	—
123	C₄⋊Q₈×C₆₇	—	2	2	no	—
124	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
125	#126×C₆₇	—	2	2	no	—
126	#126×C₆₇	—	2	2	no	—
127	#126×C₆₇	—	2	2	no	—
128	#125×C₆₇	—	2	2	no	—
129	#125×C₆₇	C₂³	1	2	yes	—
130	#124×C₆₇	—	2	2	no	—
131	#124×C₆₇	C₂³	1	2	yes	—
132	#123×C₆₇	—	2	2	no	—
133	#123×C₆₇	—	2	2	no	—
134	#123×C₆₇	—	2	2	no	—
135	#123×C₆₇	—	2	2	no	—
136	#123×C₆₇	—	2	2	no	—
137	C₂²⋊Q₈×C₆₇	—	2	1	no	—
138	C₂²⋊Q₈×C₆₇	—	2	1	no	—
139	C₂²⋊Q₈×C₆₇	—	2	1	no	—
140	C₂²⋊Q₈×C₆₇	—	2	1	no	—
141	C₂²⋊Q₈×C₆₇	—	2	2	no	—
142	C₂²⋊Q₈×C₆₇	—	2	2	no	—
143	C₂²⋊Q₈×C₆₇	—	2	4	no	—
144	C₂²⋊Q₈×C₆₇	—	2	4	no	—
145	C₂²⋊Q₈×C₆₇	—	2	4	no	—
146	C₂²⋊Q₈×C₆₇	—	2	4	no	—
147	C₂²⋊Q₈×C₆₇	C₂³	1	2	yes	—
148	C₂²⋊Q₈×C₆₇	C₂³	1	2	yes	—
149	C₂²⋊Q₈×C₆₇	C₂³	1	2	yes	—
150	C₂²⋊Q₈×C₆₇	C₂³	1	2	yes	—
151	C₂²⋊Q₈×C₆₇	C₂³	1	2	yes	—
152	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
153	#121×C₆₇	—	2	2	no	—
154	#121×C₆₇	—	2	2	no	—
155	#121×C₆₇	—	2	1	no	—
156	#121×C₆₇	—	2	1	no	—
157	#121×C₆₇	—	2	2	no	—
158	#121×C₆₇	C₂³	1	2	yes	—
159	#121×C₆₇	C₂³	1	2	yes	—
160	#121×C₆₇	C₂³	1	2	yes	—
161	#120×C₆₇	—	2	2	no	—
162	#120×C₆₇	—	2	2	no	—
163	C₄×Q₈×C₆₇	—	2	2	no	—
164	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
165	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
166	C₄×D₄×C₆₇	—	2	4	no	—
167	C₄×D₄×C₆₇	—	2	4	no	—
168	C₄×D₄×C₆₇	—	2	2	no	—
169	C₄×D₄×C₆₇	—	2	2	no	—
170	C₄×D₄×C₆₇	—	2	2	no	—
171	C₄×D₄×C₆₇	—	2	2	no	—
172	C₄×D₄×C₆₇	—	2	2	no	—
173	C₄×D₄×C₆₇	D₄	1	2	yes	—
174	C₄×D₄×C₆₇	D₄	1	2	yes	—
175	C₄×D₄×C₆₇	C₂³	1	2	yes	—
176	C₄×D₄×C₆₇	C₂³	1	2	yes	—
177	C₄×D₄×C₆₇	C₂³	1	2	yes	—
178	#117×C₆₇	—	2	2	no	—
179	#117×C₆₇	—	2	2	no	—
180	#117×C₆₇	C₂³	1	2	yes	—
181	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
182	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
183	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
184	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
185	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
186	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
187	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
188	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
189	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
190	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
191	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
192	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
193	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
194	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
195	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
196	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
197	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
198	C₂×C₄²×C₆₇	—	2	1	no	—
199	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
200	#107×C₆₇	C₂³	1	4	yes	—
201	#106×C₆₇	C₂³	1	4	yes	—
202	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
203	#103×C₆₇	C₂³	1	4	yes	—
204	#103×C₆₇	C₂³	1	4	yes	—
205	#103×C₆₇	C₂³	1	4	yes	—
206	#103×C₆₇	C₂³	1	4	yes	—
207	#103×C₆₇	C₂³	1	4	yes	—
208	#103×C₆₇	C₂³	1	4	yes	—
209	#103×C₆₇	C₂³	1	4	yes	—
210	#102×C₆₇	C₂³	1	4	yes	—
211	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
212	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
213	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
214	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
215	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
216	#97×C₆₇	C₂³	1	4	yes	—
217	C₂⁴×C₆₇	—	2	1	no	—
218	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
219	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
220	#41×C₆₇	—	4	4	no	—
221	C₂×Q₈×C₆₇	—	2	4	no	—
222	C₂×Q₈×C₆₇	—	2	1	no	—
223	C₂×Q₈×C₆₇	—	2	4	no	—
224	C₂×Q₈×C₆₇	—	2	2	no	—
225	C₂×Q₈×C₆₇	—	2	2	no	—
226	C₂×Q₈×C₆₇	—	2	1	no	—
227	C₂×Q₈×C₆₇	—	2	1	no	—
228	C₂×Q₈×C₆₇	—	2	1	no	—
229	C₂×Q₈×C₆₇	—	2	4	no	—
230	C₂×Q₈×C₆₇	—	2	4	no	—
231	C₂×Q₈×C₆₇	—	2	8	no	—
232	C₂×Q₈×C₆₇	—	4	2	no	—
233	C₂×Q₈×C₆₇	—	4	2	no	—
234	C₂×Q₈×C₆₇	—	4	4	no	—
235	C₂×Q₈×C₆₇	—	4	2	no	—
236	C₂×Q₈×C₆₇	—	4	2	no	—
237	C₂×Q₈×C₆₇	—	4	4	no	—
238	C₂×Q₈×C₆₇	—	4	4	no	—
239	C₂×Q₈×C₆₇	—	4	2	no	—
240	C₂×Q₈×C₆₇	—	4	4	no	—
241	C₂×Q₈×C₆₇	—	4	4	no	—
242	C₂×Q₈×C₆₇	—	4	8	no	—
243	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
244	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
245	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
246	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
247	C₂×D₄×C₆₇	—	2	4	no	—
248	C₂×D₄×C₆₇	—	2	2	no	—
249	C₂×D₄×C₆₇	—	2	2	no	—
250	C₂×D₄×C₆₇	—	2	2	no	—
251	C₂×D₄×C₆₇	—	2	2	no	—
252	C₂×D₄×C₆₇	—	2	1	no	—
253	C₂×D₄×C₆₇	—	4	2	no	—
254	C₂×D₄×C₆₇	—	4	4	no	—
255	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
256	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
257	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
258	C₂²×C₄×C₆₇	—	2	1	no	—
259	C₂²×C₄×C₆₇	—	2	1	no	—
260	C₂²×C₄×C₆₇	—	2	1	no	—
261	C₂²×C₄×C₆₇	—	2	1	no	—
262	C₂²×C₄×C₆₇	—	2	1	no	—
263	C₂²×C₄×C₆₇	—	2	4	no	—
264	C₂²×C₄×C₆₇	—	2	2	no	—
265	C₂²×C₄×C₆₇	—	2	2	no	—
266	C₂²×C₄×C₆₇	—	2	1	no	—
267	C₂²×C₄×C₆₇	—	2	1	no	—
268	C₂²×C₄×C₆₇	—	2	2	no	—
269	C₂²×C₄×C₆₇	—	2	2	no	—
270	C₂²×C₄×C₆₇	—	4	2	no	—
271	C₂²×C₄×C₆₇	—	4	2	no	—
272	C₂²×C₄×C₆₇	—	4	4	no	—
273	C₂²×C₄×C₆₇	—	4	4	no	—
274	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
275	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
276	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
277	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
278	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
279	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
280	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
281	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
282	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
283	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
284	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
285	Q₁₆×C₆₇	—	2	8	no	—
286	Q₁₆×C₆₇	—	4	4	no	—
287	Q₁₆×C₆₇	—	4	4	no	—
288	SD₁₆×C₆₇	—	2	8	no	—
289	SD₁₆×C₆₇	—	4	4	no	—
290	SD₁₆×C₆₇	—	4	4	no	—
291	M₁₆×C₆₇	—	4	4	no	—
292	C₂×C₈×C₆₇	—	4	2	no	—
293	C₂×C₈×C₆₇	—	4	2	no	—
294	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
295	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
296	C₄⋊C₄×C₆₇	—	2	2	no	—
297	C₄⋊C₄×C₆₇	—	2	4	no	—
298	C₄⋊C₄×C₆₇	—	2	4	no	—
299	C₄⋊C₄×C₆₇	—	2	2	no	—
300	C₄⋊C₄×C₆₇	—	2	4	no	—
301	C₄⋊C₄×C₆₇	—	2	4	no	—
302	C₄⋊C₄×C₆₇	—	2	2	no	—
303	C₄⋊C₄×C₆₇	—	2	2	no	—
304	C₄⋊C₄×C₆₇	—	2	1	no	—
305	C₄⋊C₄×C₆₇	—	2	1	no	—
306	C₄⋊C₄×C₆₇	—	2	1	no	—
307	C₄⋊C₄×C₆₇	—	2	2	no	—
308	C₄⋊C₄×C₆₇	—	2	2	no	—
309	C₄⋊C₄×C₆₇	—	2	2	no	—
310	C₄⋊C₄×C₆₇	—	2	2	no	—
311	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
312	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
313	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
314	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
315	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
316	C₂²⋊C₄×C₆₇	—	2	2	no	—
317	C₂²⋊C₄×C₆₇	—	2	2	no	—
318	C₂²⋊C₄×C₆₇	—	2	4	no	—
319	C₂²⋊C₄×C₆₇	—	2	2	no	—
320	C₂²⋊C₄×C₆₇	—	2	2	no	—
321	C₂²⋊C₄×C₆₇	—	2	2	no	—
322	C₂²⋊C₄×C₆₇	—	2	2	no	—
323	C₂²⋊C₄×C₆₇	—	2	2	no	—
324	C₂²⋊C₄×C₆₇	—	2	2	no	—
325	C₂²⋊C₄×C₆₇	—	2	2	no	—
326	C₂²⋊C₄×C₆₇	—	2	2	no	—
327	C₂²⋊C₄×C₆₇	—	2	2	no	—
328	C₂²⋊C₄×C₆₇	—	2	2	no	—
329	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
330	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
331	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
332	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
333	C₄²×C₆₇	—	2	4	no	—
334	C₄²×C₆₇	—	2	2	no	—
335	C₄²×C₆₇	—	2	2	no	—
336	C₄²×C₆₇	C₂×D₄	1	2	yes	—
337	C₄²×C₆₇	C₂⁴	1	2	yes	—
338	C₂³×C₆₇	—	2	2	no	—
339	C₂³×C₆₇	—	2	2	no	—
340	C₂³×C₆₇	—	4	1	no	—
341	C₂³×C₆₇	—	4	1	no	—
342	C₂³×C₆₇	—	4	1	no	—
343	C₂³×C₆₇	—	4	1	no	—
344	C₂³×C₆₇	—	4	2	no	—
345	C₂³×C₆₇	—	8	1	no	—
346	C₂³×C₆₇	—	8	1	no	—
347	C₂³×C₆₇	#137	1	4	yes	complemented
348	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
349	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
350	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
351	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
352	Q₈×C₆₇	—	2	4	no	—
353	Q₈×C₆₇	—	2	2	no	—
354	Q₈×C₆₇	—	2	2	no	—
355	Q₈×C₆₇	—	2	4	no	—
356	Q₈×C₆₇	—	2	4	no	—
357	Q₈×C₆₇	—	2	2	no	—
358	Q₈×C₆₇	—	4	8	no	—
359	Q₈×C₆₇	—	4	4	no	—
360	Q₈×C₆₇	—	4	4	no	—
361	Q₈×C₆₇	—	4	2	no	—
362	Q₈×C₆₇	—	4	2	no	—
363	Q₈×C₆₇	—	8	1	no	—
364	Q₈×C₆₇	H(5,𝔽₂)	1	4	yes	—
365	D₄×C₆₇	—	2	4	no	—
366	D₄×C₆₇	—	2	4	no	—
367	D₄×C₆₇	—	2	4	no	—
368	D₄×C₆₇	—	4	4	no	—
369	D₄×C₆₇	—	4	4	no	—
370	D₄×C₆₇	—	8	1	no	—
371	D₄×C₆₇	—	8	2	no	—
372	D₄×C₆₇	—	8	2	no	—
373	C₂×C₄×C₆₇	—	2	4	no	—
374	C₂×C₄×C₆₇	—	2	2	no	—
375	C₂×C₄×C₆₇	—	2	2	no	—
376	C₂×C₄×C₆₇	—	2	2	no	—
377	C₂×C₄×C₆₇	—	2	2	no	—
378	C₂×C₄×C₆₇	—	2	4	no	—
379	C₂×C₄×C₆₇	—	2	2	no	—
380	C₂×C₄×C₆₇	—	2	2	no	—
381	C₂×C₄×C₆₇	—	2	2	no	—
382	C₂×C₄×C₆₇	—	2	2	no	—
383	C₂×C₄×C₆₇	—	2	2	no	—
384	C₂×C₄×C₆₇	—	2	2	no	—
385	C₂×C₄×C₆₇	—	2	2	no	—
386	C₂×C₄×C₆₇	—	4	2	no	—
387	C₂×C₄×C₆₇	—	4	1	no	—
388	C₂×C₄×C₆₇	—	4	1	no	—
389	C₂×C₄×C₆₇	—	4	1	no	—
390	C₂×C₄×C₆₇	—	4	1	no	—
391	C₂×C₄×C₆₇	—	4	1	no	—
392	C₂×C₄×C₆₇	—	4	4	no	—
393	C₂×C₄×C₆₇	—	4	4	no	—
394	C₂×C₄×C₆₇	—	4	8	no	—
395	C₂×C₄×C₆₇	—	4	2	no	—
396	C₂×C₄×C₆₇	—	4	2	no	—
397	C₂×C₄×C₆₇	—	4	2	no	—
398	C₂×C₄×C₆₇	—	4	2	no	—
399	C₂×C₄×C₆₇	—	8	2	no	—
400	C₂×C₄×C₆₇	—	8	2	no	—
401	C₂×C₄×C₆₇	—	8	1	no	—
402	C₂×C₄×C₆₇	—	8	1	no	—
403	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
404	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
405	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
406	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
407	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
408	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
409	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
410	C₈×C₆₇	—	2	4	no	—
411	C₈×C₆₇	—	2	4	no	—
412	C₂²×C₆₇	—	2	1	no	—
413	C₂²×C₆₇	—	2	1	no	—
414	C₂²×C₆₇	—	2	1	no	—
415	C₂²×C₆₇	—	2	1	no	—
416	C₂²×C₆₇	—	2	1	no	—
417	C₂²×C₆₇	—	2	1	no	—
418	C₂²×C₆₇	—	2	2	no	—
419	C₂²×C₆₇	—	2	2	no	—
420	C₂²×C₆₇	—	2	2	no	—
421	C₂²×C₆₇	—	4	4	no	—
422	C₂²×C₆₇	—	4	4	no	—
423	C₂²×C₆₇	—	4	2	no	—
424	C₂²×C₆₇	—	4	2	no	—
425	C₂²×C₆₇	—	4	2	no	—
426	C₂²×C₆₇	—	16	2	no	—
427	C₂²×C₆₇	C₂×#137	1	2	yes	—
428	C₂²×C₆₇	C₂×#137	1	2	yes	—
429	C₂²×C₆₇	C₂×#137	1	2	yes	—
430	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
431	C₄×C₆₇	—	2	2	no	—
432	C₄×C₆₇	—	2	4	no	—
433	C₄×C₆₇	—	2	1	no	—
434	C₄×C₆₇	—	2	2	no	—
435	C₄×C₆₇	—	2	1	no	—
436	C₄×C₆₇	—	2	1	no	—
437	C₄×C₆₇	—	2	2	no	—
438	C₄×C₆₇	—	4	2	no	—
439	C₄×C₆₇	—	4	2	no	—
440	C₄×C₆₇	—	4	2	no	—
441	C₄×C₆₇	—	4	2	no	—
442	C₄×C₆₇	—	4	2	no	—
443	C₄×C₆₇	—	4	2	no	—
444	C₄×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
445	#61574	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
446	C₂×C₆₇	—	2	2	no	—
447	C₂×C₆₇	—	2	2	no	—
448	C₂×C₆₇	—	2	2	no	—
449	C₂×C₆₇	—	4	2	no	—
450	C₂×C₆₇	—	8	2	no	—
451	C₂×C₆₇	#3484	1	1	yes	characteristic
452	C₂×C₆₇	C₂²×#137	1	1	yes	characteristic
453	C₂×C₆₇	#3586	1	1	yes	characteristic
454	#3505	C₂×C₆₇	1	1	yes	characteristic
455	C₂×D₄×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
456	#3320	C₂×C₆₇	1	4	yes	complemented
457	#3312	C₂×C₆₇	1	4	yes	complemented
458	#3270	C₂×C₆₇	1	4	yes	complemented
459	#3266	C₂×C₆₇	1	4	yes	complemented
460	#3218	C₂×C₆₇	1	2	yes	complemented
461	#3196	C₂×C₆₇	1	2	yes	complemented
462	#3189	C₂×C₆₇	1	1	yes	characteristic, complemented
463	#3187	C₂×C₆₇	1	1	yes	characteristic, complemented
464	#3117	C₂×C₆₇	1	2	yes	complemented
465	#3004	C₂×C₆₇	1	2	yes	complemented
466	C₂×#451	C₂×C₆₇	1	1	yes	characteristic, complemented
467	C₂×#448	C₂×C₆₇	1	1	yes	characteristic, complemented
468	#2973	C₂×C₆₇	1	1	yes	characteristic, complemented
469	C₆₇	#61574	1	1	yes	characteristic, direct product factor, Sylow
470	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
471	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
472	C₂×#137	C₂²×C₆₇	1	4	yes	complemented
473	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	complemented
474	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	complemented
475	#553	—	2	2	no	—
476	#553	C₂²×C₆₇	1	2	yes	—
477	#552	C₂²×C₆₇	1	2	yes	—
478	#551	—	2	2	no	—
479	D₄×Q₈	—	2	2	no	—
480	D₄×Q₈	—	2	2	no	—
481	D₄×Q₈	C₂²×C₆₇	1	4	yes	—
482	D₄×Q₈	C₂²×C₆₇	1	4	yes	complemented
483	#548	C₂²×C₆₇	1	2	yes	—
484	#546	C₂²×C₆₇	1	2	yes	—
485	#539	—	2	1	no	—
486	#538	—	2	1	no	—
487	#534	—	2	1	no	—
488	C₂×C₄⋊Q₈	—	2	1	no	—
489	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
490	#529	C₂²×C₆₇	1	2	yes	—
491	#525	C₂²×C₆₇	1	2	yes	—
492	C₂×C₂²⋊Q₈	—	2	1	no	—
493	C₂×C₂²⋊Q₈	—	2	1	no	—
494	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
495	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
496	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
497	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
498	#518	—	2	1	no	—
499	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
500	C₂×C₄×D₄	—	2	1	no	—
501	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
502	#497	C₂²×C₆₇	1	4	yes	complemented
503	#491	C₂²×C₆₇	1	4	yes	complemented
504	#485	C₂²×C₆₇	1	4	yes	complemented
505	#484	C₂²×C₆₇	1	2	yes	complemented
506	#484	C₂²×C₆₇	1	2	yes	complemented
507	#483	C₂²×C₆₇	1	2	yes	complemented
508	#481	C₂²×C₆₇	1	2	yes	—
509	#479	C₂²×C₆₇	1	4	yes	complemented
510	#470	C₂²×C₆₇	1	4	yes	complemented
511	#468	C₂²×C₆₇	1	4	yes	complemented
512	#467	C₂²×C₆₇	1	4	yes	complemented
513	#465	C₂²×C₆₇	1	4	yes	complemented
514	#464	C₂²×C₆₇	1	4	yes	complemented
515	#462	C₂²×C₆₇	1	4	yes	complemented
516	#461	C₂²×C₆₇	1	4	yes	complemented
517	#452	C₂²×C₆₇	1	4	yes	complemented
518	#451	C₂²×C₆₇	1	2	yes	complemented
519	#451	C₂²×C₆₇	1	2	yes	complemented
520	#451	C₂²×C₆₇	1	2	yes	complemented
521	#451	C₂²×C₆₇	1	2	yes	complemented
522	#451	C₂²×C₆₇	1	4	yes	complemented
523	#451	C₂²×C₆₇	1	4	yes	complemented
524	#450	C₂²×C₆₇	1	2	yes	complemented
525	#448	C₂²×C₆₇	1	2	yes	—
526	#448	C₂²×C₆₇	1	2	yes	—
527	#448	C₂²×C₆₇	1	2	yes	—
528	#448	C₂²×C₆₇	1	4	yes	complemented
529	#448	C₂²×C₆₇	1	4	yes	complemented
530	#441	C₂²×C₆₇	1	4	yes	complemented
531	#440	C₂²×C₆₇	1	4	yes	complemented
532	#435	C₂²×C₆₇	1	4	yes	—
533	#435	C₂²×C₆₇	1	4	yes	complemented
534	#432	C₂²×C₆₇	1	2	yes	complemented
535	#419	C₂²×C₆₇	1	4	yes	complemented
536	C₂×#103	C₂²×C₆₇	1	2	yes	complemented
537	C₂×#103	C₂²×C₆₇	1	2	yes	complemented
538	#407	C₂²×C₆₇	1	2	yes	complemented
539	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
540	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
541	C₂×#41	C₂³×C₆₇	1	2	yes	—
542	C₂²×Q₈	—	2	1	no	—
543	C₂²×Q₈	—	2	1	no	—
544	C₂²×Q₈	—	2	1	no	—
545	C₂²×Q₈	—	2	1	no	—
546	C₂²×Q₈	—	2	2	no	—
547	C₂²×Q₈	—	4	4	no	—
548	C₂²×Q₈	—	4	4	no	—
549	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
550	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
551	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
552	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
553	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
554	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
555	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
556	C₂²×D₄	—	2	1	no	—
557	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
558	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
559	C₂³×C₄	—	2	1	no	—
560	C₂³×C₄	—	2	1	no	—
561	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
562	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
563	#137	—	4	8	no	—
564	C₂×Q₁₆	—	4	4	no	—
565	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
566	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
567	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
568	C₂×SD₁₆	—	4	4	no	—
569	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
570	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
571	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
572	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
573	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
574	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
575	C₄⋊Q₈	—	2	4	no	—
576	C₄⋊Q₈	—	2	4	no	—
577	C₄⋊Q₈	—	2	2	no	—
578	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
579	#126	—	2	2	no	—
580	#126	—	2	2	no	—
581	#126	—	2	2	no	—
582	#125	—	2	2	no	—
583	#125	C₂³×C₆₇	1	2	yes	—
584	#124	—	2	2	no	—
585	#124	C₂³×C₆₇	1	2	yes	—
586	#123	—	2	2	no	—
587	#123	—	2	2	no	—
588	#123	—	2	2	no	—
589	#123	—	2	2	no	—
590	#123	—	2	2	no	—
591	C₂²⋊Q₈	—	2	1	no	—
592	C₂²⋊Q₈	—	2	1	no	—
593	C₂²⋊Q₈	—	2	1	no	—
594	C₂²⋊Q₈	—	2	1	no	—
595	C₂²⋊Q₈	—	2	2	no	—
596	C₂²⋊Q₈	—	2	2	no	—
597	C₂²⋊Q₈	—	2	4	no	—
598	C₂²⋊Q₈	—	2	4	no	—
599	C₂²⋊Q₈	—	2	4	no	—
600	C₂²⋊Q₈	—	2	4	no	—
601	C₂²⋊Q₈	C₂³×C₆₇	1	2	yes	—
602	C₂²⋊Q₈	C₂³×C₆₇	1	2	yes	—
603	C₂²⋊Q₈	C₂³×C₆₇	1	2	yes	—
604	C₂²⋊Q₈	C₂³×C₆₇	1	2	yes	—
605	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
606	C₂²⋊Q₈	C₂³×C₆₇	1	2	yes	—
607	#121	—	2	2	no	—
608	#121	—	2	2	no	—
609	#121	—	2	1	no	—
610	#121	—	2	1	no	—
611	#121	—	2	2	no	—
612	#121	C₂³×C₆₇	1	2	yes	—
613	#121	C₂³×C₆₇	1	2	yes	—
614	#121	C₂³×C₆₇	1	2	yes	—
615	#120	—	2	2	no	—
616	#120	—	2	2	no	—
617	C₄×Q₈	—	2	2	no	—
618	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
619	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
620	C₄×D₄	—	2	4	no	—
621	C₄×D₄	—	2	4	no	—
622	C₄×D₄	—	2	2	no	—
623	C₄×D₄	—	2	2	no	—
624	C₄×D₄	—	2	2	no	—
625	C₄×D₄	—	2	2	no	—
626	C₄×D₄	—	2	2	no	—
627	C₄×D₄	D₄×C₆₇	1	2	yes	—
628	C₄×D₄	D₄×C₆₇	1	2	yes	—
629	C₄×D₄	C₂³×C₆₇	1	2	yes	—
630	C₄×D₄	C₂³×C₆₇	1	2	yes	—
631	C₄×D₄	C₂³×C₆₇	1	2	yes	—
632	#117	—	2	2	no	—
633	#117	—	2	2	no	—
634	#117	C₂³×C₆₇	1	2	yes	—
635	C₂×C₄⋊C₄	—	2	1	no	—
636	C₂×C₄⋊C₄	—	2	1	no	—
637	C₂×C₄⋊C₄	—	2	1	no	—
638	C₂×C₄⋊C₄	—	2	1	no	—
639	C₂×C₄⋊C₄	—	2	1	no	—
640	C₂×C₄⋊C₄	—	2	1	no	—
641	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
642	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
643	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
644	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
645	C₂×C₂²⋊C₄	—	2	1	no	—
646	C₂×C₂²⋊C₄	—	2	1	no	—
647	C₂×C₂²⋊C₄	—	2	1	no	—
648	C₂×C₂²⋊C₄	—	2	1	no	—
649	C₂×C₂²⋊C₄	—	2	2	no	—
650	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
651	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
652	C₂×C₄²	—	2	1	no	—
653	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
654	#107	C₂³×C₆₇	1	4	yes	—
655	#106	C₂³×C₆₇	1	4	yes	—
656	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
657	#103	C₂³×C₆₇	1	4	yes	—
658	#103	C₂³×C₆₇	1	4	yes	—
659	#103	C₂³×C₆₇	1	4	yes	—
660	#103	C₂³×C₆₇	1	4	yes	—
661	#103	C₂³×C₆₇	1	4	yes	—
662	#103	C₂³×C₆₇	1	4	yes	—
663	#103	C₂³×C₆₇	1	4	yes	—
664	#102	C₂³×C₆₇	1	4	yes	—
665	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
666	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
667	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
668	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
669	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
670	#97	C₂³×C₆₇	1	4	yes	—
671	C₂⁴	—	2	1	no	—
672	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
673	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
674	#41	—	4	4	no	—
675	C₂×Q₈	—	2	4	no	—
676	C₂×Q₈	—	2	1	no	—
677	C₂×Q₈	—	2	4	no	—
678	C₂×Q₈	—	2	2	no	—
679	C₂×Q₈	—	2	2	no	—
680	C₂×Q₈	—	2	1	no	—
681	C₂×Q₈	—	2	1	no	—
682	C₂×Q₈	—	2	1	no	—
683	C₂×Q₈	—	2	4	no	—
684	C₂×Q₈	—	2	4	no	—
685	C₂×Q₈	—	2	8	no	—
686	C₂×Q₈	—	4	2	no	—
687	C₂×Q₈	—	4	2	no	—
688	C₂×Q₈	—	4	4	no	—
689	C₂×Q₈	—	4	2	no	—
690	C₂×Q₈	—	4	2	no	—
691	C₂×Q₈	—	4	4	no	—
692	C₂×Q₈	—	4	4	no	—
693	C₂×Q₈	—	4	2	no	—
694	C₂×Q₈	—	4	4	no	—
695	C₂×Q₈	—	4	4	no	—
696	C₂×Q₈	—	4	8	no	—
697	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
698	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
699	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
700	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
701	C₂×D₄	—	2	4	no	—
702	C₂×D₄	—	2	2	no	—
703	C₂×D₄	—	2	2	no	—
704	C₂×D₄	—	2	2	no	—
705	C₂×D₄	—	2	2	no	—
706	C₂×D₄	—	2	1	no	—
707	C₂×D₄	—	4	2	no	—
708	C₂×D₄	—	4	4	no	—
709	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
710	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
711	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
712	C₂²×C₄	—	2	1	no	—
713	C₂²×C₄	—	2	1	no	—
714	C₂²×C₄	—	2	1	no	—
715	C₂²×C₄	—	2	1	no	—
716	C₂²×C₄	—	2	1	no	—
717	C₂²×C₄	—	2	4	no	—
718	C₂²×C₄	—	2	2	no	—
719	C₂²×C₄	—	2	2	no	—
720	C₂²×C₄	—	2	1	no	—
721	C₂²×C₄	—	2	1	no	—
722	C₂²×C₄	—	2	2	no	—
723	C₂²×C₄	—	2	2	no	—
724	C₂²×C₄	—	4	2	no	—
725	C₂²×C₄	—	4	2	no	—
726	C₂²×C₄	—	4	4	no	—
727	C₂²×C₄	—	4	4	no	—
728	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
729	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
730	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
731	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
732	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
733	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
734	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
735	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
736	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
737	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
738	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
739	Q₁₆	—	2	8	no	—
740	Q₁₆	—	4	4	no	—
741	Q₁₆	—	4	4	no	—
742	SD₁₆	—	2	8	no	—
743	SD₁₆	—	4	4	no	—
744	SD₁₆	—	4	4	no	—
745	M₁₆	—	4	4	no	—
746	C₂×C₈	—	4	2	no	—
747	C₂×C₈	—	4	2	no	—
748	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
749	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
750	C₄⋊C₄	—	2	2	no	—
751	C₄⋊C₄	—	2	4	no	—
752	C₄⋊C₄	—	2	4	no	—
753	C₄⋊C₄	—	2	2	no	—
754	C₄⋊C₄	—	2	4	no	—
755	C₄⋊C₄	—	2	4	no	—
756	C₄⋊C₄	—	2	2	no	—
757	C₄⋊C₄	—	2	2	no	—
758	C₄⋊C₄	—	2	1	no	—
759	C₄⋊C₄	—	2	1	no	—
760	C₄⋊C₄	—	2	1	no	—
761	C₄⋊C₄	—	2	2	no	—
762	C₄⋊C₄	—	2	2	no	—
763	C₄⋊C₄	—	2	2	no	—
764	C₄⋊C₄	—	2	2	no	—
765	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
766	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
767	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
768	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
769	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
770	C₂²⋊C₄	—	2	2	no	—
771	C₂²⋊C₄	—	2	2	no	—
772	C₂²⋊C₄	—	2	4	no	—
773	C₂²⋊C₄	—	2	2	no	—
774	C₂²⋊C₄	—	2	2	no	—
775	C₂²⋊C₄	—	2	2	no	—
776	C₂²⋊C₄	—	2	2	no	—
777	C₂²⋊C₄	—	2	2	no	—
778	C₂²⋊C₄	—	2	2	no	—
779	C₂²⋊C₄	—	2	2	no	—
780	C₂²⋊C₄	—	2	2	no	—
781	C₂²⋊C₄	—	2	2	no	—
782	C₂²⋊C₄	—	2	2	no	—
783	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
784	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
785	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
786	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
787	C₄²	—	2	4	no	—
788	C₄²	—	2	2	no	—
789	C₄²	—	2	2	no	—
790	C₄²	C₂×D₄×C₆₇	1	2	yes	—
791	C₄²	C₂⁴×C₆₇	1	2	yes	—
792	C₂³	—	2	2	no	—
793	C₂³	—	2	2	no	—
794	C₂³	—	4	1	no	—
795	C₂³	—	4	1	no	—
796	C₂³	—	4	1	no	—
797	C₂³	—	4	1	no	—
798	C₂³	—	4	2	no	—
799	C₂³	—	8	1	no	—
800	C₂³	—	8	1	no	—
801	C₂³	#137×C₆₇	1	4	yes	complemented
802	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
803	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
804	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
805	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
806	Q₈	—	2	4	no	—
807	Q₈	—	2	2	no	—
808	Q₈	—	2	2	no	—
809	Q₈	—	2	4	no	—
810	Q₈	—	2	4	no	—
811	Q₈	—	2	2	no	—
812	Q₈	—	4	8	no	—
813	Q₈	—	4	4	no	—
814	Q₈	—	4	4	no	—
815	Q₈	—	4	2	no	—
816	Q₈	—	4	2	no	—
817	Q₈	—	8	1	no	—
818	Q₈	H(5,𝔽₂)×C₆₇	1	4	yes	—
819	D₄	—	2	4	no	—
820	D₄	—	2	4	no	—
821	D₄	—	2	4	no	—
822	D₄	—	4	4	no	—
823	D₄	—	4	4	no	—
824	D₄	—	8	1	no	—
825	D₄	—	8	2	no	—
826	D₄	—	8	2	no	—
827	C₂×C₄	—	2	4	no	—
828	C₂×C₄	—	2	2	no	—
829	C₂×C₄	—	2	2	no	—
830	C₂×C₄	—	2	2	no	—
831	C₂×C₄	—	2	2	no	—
832	C₂×C₄	—	2	4	no	—
833	C₂×C₄	—	2	2	no	—
834	C₂×C₄	—	2	2	no	—
835	C₂×C₄	—	2	2	no	—
836	C₂×C₄	—	2	2	no	—
837	C₂×C₄	—	2	2	no	—
838	C₂×C₄	—	2	2	no	—
839	C₂×C₄	—	2	2	no	—
840	C₂×C₄	—	4	2	no	—
841	C₂×C₄	—	4	1	no	—
842	C₂×C₄	—	4	1	no	—
843	C₂×C₄	—	4	1	no	—
844	C₂×C₄	—	4	1	no	—
845	C₂×C₄	—	4	1	no	—
846	C₂×C₄	—	4	4	no	—
847	C₂×C₄	—	4	4	no	—
848	C₂×C₄	—	4	8	no	—
849	C₂×C₄	—	4	2	no	—
850	C₂×C₄	—	4	2	no	—
851	C₂×C₄	—	4	2	no	—
852	C₂×C₄	—	4	2	no	—
853	C₂×C₄	—	8	2	no	—
854	C₂×C₄	—	8	2	no	—
855	C₂×C₄	—	8	1	no	—
856	C₂×C₄	—	8	1	no	—
857	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
858	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
859	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
860	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
861	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
862	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
863	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
864	C₈	—	2	4	no	—
865	C₈	—	2	4	no	—
866	C₂²	—	2	1	no	—
867	C₂²	—	2	1	no	—
868	C₂²	—	2	1	no	—
869	C₂²	—	2	1	no	—
870	C₂²	—	2	1	no	—
871	C₂²	—	2	1	no	—
872	C₂²	—	2	2	no	—
873	C₂²	—	2	2	no	—
874	C₂²	—	2	2	no	—
875	C₂²	—	4	4	no	—
876	C₂²	—	4	4	no	—
877	C₂²	—	4	2	no	—
878	C₂²	—	4	2	no	—
879	C₂²	—	4	2	no	—
880	C₂²	—	16	2	no	—
881	C₂²	C₂×#137×C₆₇	1	2	yes	—
882	C₂²	C₂×#137×C₆₇	1	2	yes	—
883	C₂²	C₂×#137×C₆₇	1	2	yes	—
884	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
885	C₄	—	2	2	no	—
886	C₄	—	2	2	no	—
887	C₄	—	2	4	no	—
888	C₄	—	2	1	no	—
889	C₄	—	2	2	no	—
890	C₄	—	2	1	no	—
891	C₄	—	2	1	no	—
892	C₄	—	4	2	no	—
893	C₄	—	4	2	no	—
894	C₄	—	4	2	no	—
895	C₄	—	4	2	no	—
896	C₄	—	4	2	no	—
897	C₄	—	4	2	no	—
898	C₄	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
899	C₂	—	2	2	no	—
900	C₂	—	2	2	no	—
901	C₂	—	2	2	no	—
902	C₂	—	4	2	no	—
903	C₂	—	8	2	no	—
904	C₂	C₆₇×#3484	1	1	yes	characteristic
905	C₂	C₂²×#137×C₆₇	1	1	yes	characteristic
906	C₂	C₆₇×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₆₇×#3505	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄×D₄×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×D₄×Q₈×C₆₇	C₂	semidirect
3. C₆₇×#3320	C₂	semidirect
4. C₆₇×#3312	C₂	semidirect
5. C₆₇×#3270	C₂	semidirect
6. C₆₇×#3266	C₂	semidirect
7. C₆₇×#3218	C₂	semidirect
8. C₆₇×#3196	C₂	semidirect
9. C₆₇×#3189	C₂	semidirect
10. C₆₇×#3187	C₂	semidirect
11. C₆₇×#3117	C₂	semidirect
12. C₆₇×#3004	C₂	semidirect
13. C₂×#451×C₆₇	C₂	semidirect
14. C₂×#448×C₆₇	C₂	semidirect
15. C₆₇×#2973	C₂	semidirect
17. C₂³×Q₈×C₆₇	C₂²	semidirect
18. C₂×#137×C₆₇	C₂²	semidirect
19. C₂²×Q₁₆×C₆₇	C₂²	semidirect
20. C₂²×SD₁₆×C₆₇	C₂²	semidirect
28. D₄×Q₈×C₆₇	C₂²	semidirect
35. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
42. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
45. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
48. #497×C₆₇	C₂²	semidirect
49. #491×C₆₇	C₂²	semidirect
50. #485×C₆₇	C₂²	semidirect
51. #484×C₆₇	C₂²	semidirect
52. #484×C₆₇	C₂²	semidirect
53. #483×C₆₇	C₂²	semidirect
55. #479×C₆₇	C₂²	semidirect
56. #470×C₆₇	C₂²	semidirect
57. #468×C₆₇	C₂²	semidirect
58. #467×C₆₇	C₂²	semidirect
59. #465×C₆₇	C₂²	semidirect
60. #464×C₆₇	C₂²	semidirect
61. #462×C₆₇	C₂²	semidirect
62. #461×C₆₇	C₂²	semidirect
63. #452×C₆₇	C₂²	semidirect
64. #451×C₆₇	C₂²	semidirect
65. #451×C₆₇	C₂²	semidirect
66. #451×C₆₇	C₂²	semidirect
67. #451×C₆₇	C₂²	semidirect
68. #451×C₆₇	C₂²	semidirect
69. #451×C₆₇	C₂²	semidirect
70. #450×C₆₇	C₂²	semidirect
74. #448×C₆₇	C₂²	semidirect
75. #448×C₆₇	C₂²	semidirect
76. #441×C₆₇	C₂²	semidirect
77. #440×C₆₇	C₂²	semidirect
79. #435×C₆₇	C₂²	semidirect
80. #432×C₆₇	C₂²	semidirect
81. #419×C₆₇	C₂²	semidirect
82. C₂×#103×C₆₇	C₂²	semidirect
83. C₂×#103×C₆₇	C₂²	semidirect
84. #407×C₆₇	C₂²	semidirect
347. C₂³×C₆₇	#137	semidirect
445. #61574	C₆₇	direct (N × H)
455. C₂×D₄×Q₈	C₂×C₆₇	semidirect
456. #3320	C₂×C₆₇	semidirect
457. #3312	C₂×C₆₇	semidirect
458. #3270	C₂×C₆₇	semidirect
459. #3266	C₂×C₆₇	semidirect
460. #3218	C₂×C₆₇	semidirect
461. #3196	C₂×C₆₇	semidirect
462. #3189	C₂×C₆₇	semidirect
463. #3187	C₂×C₆₇	semidirect
464. #3117	C₂×C₆₇	semidirect
465. #3004	C₂×C₆₇	semidirect
466. C₂×#451	C₂×C₆₇	semidirect
467. C₂×#448	C₂×C₆₇	semidirect
468. #2973	C₂×C₆₇	semidirect
469. C₆₇	#61574	direct (N × H)
471. C₂³×Q₈	C₂²×C₆₇	semidirect
472. C₂×#137	C₂²×C₆₇	semidirect
473. C₂²×Q₁₆	C₂²×C₆₇	semidirect
474. C₂²×SD₁₆	C₂²×C₆₇	semidirect
482. D₄×Q₈	C₂²×C₆₇	semidirect
489. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
496. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
499. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
502. #497	C₂²×C₆₇	semidirect
503. #491	C₂²×C₆₇	semidirect
504. #485	C₂²×C₆₇	semidirect
505. #484	C₂²×C₆₇	semidirect
506. #484	C₂²×C₆₇	semidirect
507. #483	C₂²×C₆₇	semidirect
509. #479	C₂²×C₆₇	semidirect
510. #470	C₂²×C₆₇	semidirect
511. #468	C₂²×C₆₇	semidirect
512. #467	C₂²×C₆₇	semidirect
513. #465	C₂²×C₆₇	semidirect
514. #464	C₂²×C₆₇	semidirect
515. #462	C₂²×C₆₇	semidirect
516. #461	C₂²×C₆₇	semidirect
517. #452	C₂²×C₆₇	semidirect
518. #451	C₂²×C₆₇	semidirect
519. #451	C₂²×C₆₇	semidirect
520. #451	C₂²×C₆₇	semidirect
521. #451	C₂²×C₆₇	semidirect
522. #451	C₂²×C₆₇	semidirect
523. #451	C₂²×C₆₇	semidirect
524. #450	C₂²×C₆₇	semidirect
528. #448	C₂²×C₆₇	semidirect
529. #448	C₂²×C₆₇	semidirect
530. #441	C₂²×C₆₇	semidirect
531. #440	C₂²×C₆₇	semidirect
533. #435	C₂²×C₆₇	semidirect
534. #432	C₂²×C₆₇	semidirect
535. #419	C₂²×C₆₇	semidirect
536. C₂×#103	C₂²×C₆₇	semidirect
537. C₂×#103	C₂²×C₆₇	semidirect
538. #407	C₂²×C₆₇	semidirect
801. C₂³	#137×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₆₇×#3505	2
C₂×D₄×Q₈×C₆₇	2
C₆₇×#3320	2
C₆₇×#3312	2
C₆₇×#3270	2
C₆₇×#3266	2
C₆₇×#3218	2
C₆₇×#3196	2
C₆₇×#3189	2
C₆₇×#3187	2
C₆₇×#3117	2
C₆₇×#3004	2
C₂×#451×C₆₇	2
C₂×#448×C₆₇	2
C₆₇×#2973	2
#61574	67

Subgroups — C67×#61574 #30740825

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61574 #30740825