Subgroups — C₆₇×#61323 #30740574

Order 17152.

1814 conjugacy classes of subgroups.

The subgroup lattice has 868 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#41×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#556×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3377	C₂	1	4	yes	maximal, complemented
4	C₆₇×#3376	C₂	1	4	yes	maximal, complemented
5	C₆₇×#3369	C₂	1	2	yes	maximal, complemented
6	C₆₇×#3367	C₂	1	2	yes	maximal, complemented
7	C₆₇×#3364	C₂	1	2	yes	maximal, complemented
8	C₆₇×#3360	C₂	1	2	yes	maximal, complemented
9	C₆₇×#3140	C₂	1	2	yes	maximal, complemented
10	C₆₇×#3139	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#3078	C₂	1	2	yes	maximal, complemented
12	C₆₇×#3077	C₂	1	1	yes	maximal, characteristic, complemented
13	C₆₇×#3040	C₂	1	2	yes	maximal
14	C₆₇×#3039	C₂	1	1	yes	maximal, characteristic, complemented
15	Q₈×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₄×#137×C₆₇	C₂	1	2	yes	maximal, complemented
17	C₄×AGL(1,ℤ/8)×C₆₇	C₂	1	1	yes	maximal, characteristic
18	C₂×#137×C₆₇	C₂²	1	2	yes	—
19	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	1	yes	characteristic
20	Q₈²×C₆₇	C₂²	1	1	yes	characteristic, complemented
21	#557×C₆₇	C₂²	1	1	yes	characteristic, complemented
22	#557×C₆₇	C₂²	1	2	yes	complemented
23	#556×C₆₇	—	2	1	no	—
24	#556×C₆₇	—	2	2	no	—
25	#556×C₆₇	—	2	1	no	—
26	#556×C₆₇	C₂²	1	2	yes	—
27	#556×C₆₇	C₂²	1	2	yes	—
28	#556×C₆₇	C₂²	1	4	yes	—
29	#554×C₆₇	—	2	1	no	—
30	#554×C₆₇	—	2	2	no	—
31	#554×C₆₇	C₂²	1	1	yes	characteristic, complemented
32	#554×C₆₇	C₂²	1	2	yes	complemented
33	D₄×Q₈×C₆₇	—	2	1	no	—
34	D₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
35	#533×C₆₇	—	2	1	no	—
36	#533×C₆₇	—	2	2	no	—
37	#533×C₆₇	C₂²	1	1	yes	characteristic
38	#533×C₆₇	C₂²	1	2	yes	—
39	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×#126×C₆₇	—	2	2	no	—
41	C₂×#124×C₆₇	C₂²	1	2	yes	—
42	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
43	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
44	C₄×#41×C₆₇	C₂²	1	2	yes	—
45	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
46	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
47	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
48	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
49	C₂×#117×C₆₇	—	2	1	no	—
50	#491×C₆₇	C₂²	1	4	yes	complemented
51	#489×C₆₇	C₂²	1	4	yes	—
52	#489×C₆₇	C₂²	1	4	yes	—
53	#488×C₆₇	C₂²	1	4	yes	—
54	#488×C₆₇	C₂²	1	4	yes	—
55	#487×C₆₇	C₂²	1	2	yes	complemented
56	#487×C₆₇	C₂²	1	2	yes	complemented
57	#479×C₆₇	C₂²	1	4	yes	complemented
58	#479×C₆₇	C₂²	1	4	yes	complemented
59	#478×C₆₇	C₂²	1	4	yes	—
60	#477×C₆₇	C₂²	1	4	yes	complemented
61	#476×C₆₇	C₂²	1	2	yes	—
62	#475×C₆₇	C₂²	1	4	yes	complemented
63	#474×C₆₇	C₂²	1	2	yes	—
64	#464×C₆₇	C₂²	1	4	yes	—
65	#464×C₆₇	C₂²	1	4	yes	—
66	#463×C₆₇	C₂²	1	4	yes	—
67	#463×C₆₇	C₂²	1	4	yes	—
68	#462×C₆₇	C₂²	1	2	yes	complemented
69	#462×C₆₇	C₂²	1	2	yes	complemented
70	#461×C₆₇	C₂²	1	2	yes	complemented
71	#461×C₆₇	C₂²	1	2	yes	complemented
72	#446×C₆₇	C₂²	1	2	yes	complemented
73	#446×C₆₇	C₂²	1	4	yes	complemented
74	C₈×Q₈×C₆₇	C₂²	1	2	yes	complemented
75	#442×C₆₇	C₂²	1	2	yes	—
76	#441×C₆₇	C₂²	1	4	yes	complemented
77	#440×C₆₇	C₂²	1	2	yes	—
78	#440×C₆₇	C₂²	1	4	yes	complemented
79	C₄×Q₁₆×C₆₇	C₂²	1	4	yes	complemented
80	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
81	C₄×SD₁₆×C₆₇	C₂²	1	4	yes	complemented
82	C₄×D₈×C₆₇	C₂²	1	2	yes	—
83	#428×C₆₇	C₂²	1	2	yes	—
84	#428×C₆₇	C₂²	1	1	yes	characteristic
85	#423×C₆₇	C₂²	1	1	yes	characteristic
86	#423×C₆₇	C₂²	1	2	yes	—
87	#419×C₆₇	C₂²	1	2	yes	—
88	#419×C₆₇	C₂²	1	2	yes	—
89	#419×C₆₇	C₂²	1	2	yes	—
90	#418×C₆₇	C₂²	1	2	yes	—
91	#418×C₆₇	C₂²	1	1	yes	characteristic
92	#417×C₆₇	C₂²	1	1	yes	characteristic
93	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
94	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
95	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
96	C₂²×Q₈×C₆₇	—	2	1	no	—
97	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
98	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
99	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
100	C₂³×C₄×C₆₇	—	2	1	no	—
101	#137×C₆₇	—	2	8	no	—
102	AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
103	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
104	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
105	C₂×SD₁₆×C₆₇	C₂³	1	2	yes	—
106	C₂×D₈×C₆₇	C₂³	1	2	yes	—
107	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
108	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
109	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
110	C₄⋊Q₈×C₆₇	—	2	1	no	—
111	C₄⋊Q₈×C₆₇	—	2	2	no	—
112	C₄⋊Q₈×C₆₇	—	2	1	no	—
113	C₄⋊Q₈×C₆₇	—	2	2	no	—
114	C₄⋊Q₈×C₆₇	—	2	1	no	—
115	C₄⋊Q₈×C₆₇	—	2	1	no	—
116	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
117	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
118	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
119	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
120	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
121	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
122	#126×C₆₇	—	2	4	no	—
123	#126×C₆₇	—	2	4	no	—
124	#126×C₆₇	—	2	4	no	—
125	#126×C₆₇	—	2	4	no	—
126	#125×C₆₇	—	2	2	no	—
127	#125×C₆₇	—	2	2	no	—
128	#125×C₆₇	—	2	2	no	—
129	#125×C₆₇	C₂³	1	2	yes	—
130	#125×C₆₇	C₂³	1	2	yes	—
131	#125×C₆₇	C₂³	1	2	yes	—
132	#124×C₆₇	—	2	2	no	—
133	#124×C₆₇	—	2	2	no	—
134	#124×C₆₇	C₂³	1	4	yes	—
135	#124×C₆₇	C₂³	1	4	yes	—
136	C₂²⋊Q₈×C₆₇	—	2	2	no	—
137	C₂²⋊Q₈×C₆₇	—	2	2	no	—
138	C₂²⋊Q₈×C₆₇	—	2	4	no	—
139	C₂²⋊Q₈×C₆₇	—	2	1	no	—
140	C₂²⋊Q₈×C₆₇	—	2	2	no	—
141	C₂²⋊Q₈×C₆₇	—	2	1	no	—
142	C₂²⋊Q₈×C₆₇	—	2	2	no	—
143	C₂²⋊Q₈×C₆₇	—	2	2	no	—
144	C₂²⋊Q₈×C₆₇	—	2	1	no	—
145	C₂²⋊Q₈×C₆₇	—	2	1	no	—
146	C₂²⋊Q₈×C₆₇	—	2	2	no	—
147	C₂²⋊Q₈×C₆₇	—	2	2	no	—
148	C₂²⋊Q₈×C₆₇	—	2	2	no	—
149	C₂²⋊Q₈×C₆₇	—	2	2	no	—
150	C₄×Q₈×C₆₇	—	2	1	no	—
151	C₄×Q₈×C₆₇	—	2	1	no	—
152	C₄×Q₈×C₆₇	—	2	2	no	—
153	C₄×Q₈×C₆₇	—	2	1	no	—
154	C₄×Q₈×C₆₇	—	2	2	no	—
155	C₄×Q₈×C₆₇	—	2	2	no	—
156	C₄×Q₈×C₆₇	—	2	2	no	—
157	C₄×Q₈×C₆₇	—	2	1	no	—
158	C₄×Q₈×C₆₇	—	2	2	no	—
159	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
160	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
161	C₄×Q₈×C₆₇	D₄	1	2	yes	—
162	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
163	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
164	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
165	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
166	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
167	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
168	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
169	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
170	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
171	C₄×D₄×C₆₇	—	2	1	no	—
172	C₄×D₄×C₆₇	—	2	2	no	—
173	C₄×D₄×C₆₇	—	2	2	no	—
174	C₄×D₄×C₆₇	—	2	1	no	—
175	C₄×D₄×C₆₇	—	2	1	no	—
176	C₄×D₄×C₆₇	—	2	1	no	—
177	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
178	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
179	C₄×D₄×C₆₇	C₂³	1	2	yes	—
180	C₄×D₄×C₆₇	C₂³	1	2	yes	—
181	C₄×D₄×C₆₇	C₂³	1	2	yes	—
182	C₄×D₄×C₆₇	C₂³	1	2	yes	—
183	#117×C₆₇	—	2	2	no	—
184	#117×C₆₇	—	2	4	no	—
185	#117×C₆₇	—	2	2	no	—
186	#117×C₆₇	—	2	1	no	—
187	#117×C₆₇	—	2	2	no	—
188	#117×C₆₇	C₂³	1	1	yes	characteristic
189	#117×C₆₇	C₂³	1	2	yes	—
190	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
191	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
192	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
193	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
194	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
195	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
196	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
197	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
198	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
199	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
200	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
201	C₂×C₄²×C₆₇	—	2	2	no	—
202	C₂×C₄²×C₆₇	—	2	1	no	—
203	C₂×C₄²×C₆₇	—	2	2	no	—
204	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
205	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
206	#107×C₆₇	C₂³	1	4	yes	—
207	#107×C₆₇	C₂³	1	2	yes	—
208	#106×C₆₇	C₂³	1	2	yes	—
209	#106×C₆₇	C₂³	1	4	yes	—
210	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
211	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
212	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
213	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
214	#103×C₆₇	C₂³	1	4	yes	—
215	#103×C₆₇	C₂³	1	4	yes	—
216	#103×C₆₇	C₂³	1	4	yes	—
217	#103×C₆₇	C₂³	1	4	yes	—
218	#103×C₆₇	C₂³	1	4	yes	—
219	#103×C₆₇	C₂³	1	4	yes	—
220	#103×C₆₇	C₂³	1	2	yes	—
221	#102×C₆₇	C₂³	1	4	yes	—
222	#102×C₆₇	C₂³	1	4	yes	—
223	#102×C₆₇	C₂³	1	2	yes	—
224	#102×C₆₇	C₂³	1	2	yes	—
225	#102×C₆₇	C₂³	1	2	yes	—
226	#97×C₆₇	C₂³	1	4	yes	—
227	#97×C₆₇	C₂³	1	2	yes	—
228	C₄×C₈×C₆₇	C₂³	1	2	yes	—
229	C₄×C₈×C₆₇	C₂³	1	4	yes	—
230	C₂⁴×C₆₇	—	2	1	no	—
231	#41×C₆₇	—	2	2	no	—
232	#41×C₆₇	#41	1	4	yes	—
233	C₂×Q₈×C₆₇	—	2	2	no	—
234	C₂×Q₈×C₆₇	—	2	1	no	—
235	C₂×Q₈×C₆₇	—	2	2	no	—
236	C₂×Q₈×C₆₇	—	2	2	no	—
237	C₂×Q₈×C₆₇	—	2	1	no	—
238	C₂×Q₈×C₆₇	—	2	4	no	—
239	C₂×Q₈×C₆₇	—	2	4	no	—
240	C₂×Q₈×C₆₇	—	4	2	no	—
241	C₂×Q₈×C₆₇	—	4	4	no	—
242	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
243	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
244	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
245	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
246	C₂×Q₈×C₆₇	C₂⁴	1	1	yes	characteristic
247	C₂×D₄×C₆₇	—	2	1	no	—
248	C₂×D₄×C₆₇	—	2	1	no	—
249	C₂×D₄×C₆₇	—	2	2	no	—
250	C₂×D₄×C₆₇	—	2	2	no	—
251	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
252	C₂×D₄×C₆₇	C₂⁴	1	1	yes	characteristic
253	C₂²×C₄×C₆₇	—	2	1	no	—
254	C₂²×C₄×C₆₇	—	2	1	no	—
255	C₂²×C₄×C₆₇	—	2	2	no	—
256	C₂²×C₄×C₆₇	—	2	2	no	—
257	C₂²×C₄×C₆₇	—	2	1	no	—
258	C₂²×C₄×C₆₇	—	2	2	no	—
259	C₂²×C₄×C₆₇	—	2	2	no	—
260	C₂²×C₄×C₆₇	—	2	2	no	—
261	C₂²×C₄×C₆₇	—	2	1	no	—
262	C₂²×C₄×C₆₇	—	4	4	no	—
263	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
264	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
265	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
266	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
267	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
268	Q₁₆×C₆₇	—	2	8	no	—
269	SD₁₆×C₆₇	—	2	4	no	—
270	SD₁₆×C₆₇	—	2	8	no	—
271	D₈×C₆₇	—	2	4	no	—
272	M₁₆×C₆₇	—	2	2	no	—
273	M₁₆×C₆₇	—	2	4	no	—
274	M₁₆×C₆₇	—	2	2	no	—
275	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
276	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
277	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
278	C₄⋊C₄×C₆₇	—	2	2	no	—
279	C₄⋊C₄×C₆₇	—	2	2	no	—
280	C₄⋊C₄×C₆₇	—	2	2	no	—
281	C₄⋊C₄×C₆₇	—	2	2	no	—
282	C₄⋊C₄×C₆₇	—	2	2	no	—
283	C₄⋊C₄×C₆₇	—	2	2	no	—
284	C₄⋊C₄×C₆₇	—	2	1	no	—
285	C₄⋊C₄×C₆₇	—	2	4	no	—
286	C₄⋊C₄×C₆₇	—	2	4	no	—
287	C₄⋊C₄×C₆₇	—	2	2	no	—
288	C₄⋊C₄×C₆₇	—	2	2	no	—
289	C₄⋊C₄×C₆₇	—	2	1	no	—
290	C₄⋊C₄×C₆₇	—	2	2	no	—
291	C₄⋊C₄×C₆₇	—	2	2	no	—
292	C₄⋊C₄×C₆₇	—	2	1	no	—
293	C₄⋊C₄×C₆₇	—	2	1	no	—
294	C₄⋊C₄×C₆₇	—	2	2	no	—
295	C₄⋊C₄×C₆₇	—	2	1	no	—
296	C₄⋊C₄×C₆₇	—	2	2	no	—
297	C₄⋊C₄×C₆₇	—	2	2	no	—
298	C₄⋊C₄×C₆₇	—	2	2	no	—
299	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
300	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
301	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
302	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
303	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
304	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
305	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
306	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
307	C₄⋊C₄×C₆₇	C₂⁴	1	1	yes	characteristic
308	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
309	C₄⋊C₄×C₆₇	C₂⁴	1	1	yes	characteristic
310	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
311	C₂²⋊C₄×C₆₇	—	2	1	no	—
312	C₂²⋊C₄×C₆₇	—	2	2	no	—
313	C₂²⋊C₄×C₆₇	—	2	4	no	—
314	C₂²⋊C₄×C₆₇	—	2	4	no	—
315	C₂²⋊C₄×C₆₇	—	2	2	no	—
316	C₂²⋊C₄×C₆₇	—	2	2	no	—
317	C₂²⋊C₄×C₆₇	—	2	4	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	1	no	—
322	C₂²⋊C₄×C₆₇	—	2	1	no	—
323	C₄²×C₆₇	—	2	4	no	—
324	C₄²×C₆₇	—	2	4	no	—
325	C₄²×C₆₇	—	2	1	no	—
326	C₄²×C₆₇	—	2	2	no	—
327	C₄²×C₆₇	—	2	2	no	—
328	C₄²×C₆₇	—	2	1	no	—
329	C₄²×C₆₇	—	2	2	no	—
330	C₄²×C₆₇	—	2	1	no	—
331	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
332	C₄²×C₆₇	C₂×D₄	1	2	yes	—
333	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
334	C₄²×C₆₇	C₂×D₄	1	2	yes	—
335	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
336	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
337	C₄²×C₆₇	C₂⁴	1	2	yes	—
338	C₄²×C₆₇	C₂⁴	1	2	yes	—
339	C₂³×C₆₇	—	2	1	no	—
340	C₂³×C₆₇	—	2	2	no	—
341	C₂³×C₆₇	—	4	2	no	—
342	C₂³×C₆₇	—	4	2	no	—
343	C₂³×C₆₇	—	4	2	no	—
344	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
345	Q₈×C₆₇	—	2	2	no	—
346	Q₈×C₆₇	—	2	2	no	—
347	Q₈×C₆₇	—	2	1	no	—
348	Q₈×C₆₇	—	2	4	no	—
349	Q₈×C₆₇	—	2	4	no	—
350	Q₈×C₆₇	—	4	2	no	—
351	Q₈×C₆₇	—	4	4	no	—
352	Q₈×C₆₇	—	4	4	no	—
353	Q₈×C₆₇	—	4	2	no	—
354	Q₈×C₆₇	—	8	2	no	—
355	Q₈×C₆₇	—	8	1	no	—
356	Q₈×C₆₇	C₂×#41	1	2	yes	—
357	D₄×C₆₇	—	2	4	no	—
358	D₄×C₆₇	—	2	2	no	—
359	D₄×C₆₇	—	2	1	no	—
360	D₄×C₆₇	—	2	2	no	—
361	D₄×C₆₇	—	4	2	no	—
362	D₄×C₆₇	C₂×#41	1	2	yes	—
363	C₂×C₄×C₆₇	—	2	1	no	—
364	C₂×C₄×C₆₇	—	2	4	no	—
365	C₂×C₄×C₆₇	—	2	2	no	—
366	C₂×C₄×C₆₇	—	2	2	no	—
367	C₂×C₄×C₆₇	—	2	4	no	—
368	C₂×C₄×C₆₇	—	2	2	no	—
369	C₂×C₄×C₆₇	—	2	1	no	—
370	C₂×C₄×C₆₇	—	2	2	no	—
371	C₂×C₄×C₆₇	—	2	1	no	—
372	C₂×C₄×C₆₇	—	4	2	no	—
373	C₂×C₄×C₆₇	—	4	1	no	—
374	C₂×C₄×C₆₇	—	4	2	no	—
375	C₂×C₄×C₆₇	—	4	4	no	—
376	C₂×C₄×C₆₇	—	4	4	no	—
377	C₂×C₄×C₆₇	—	4	2	no	—
378	C₂×C₄×C₆₇	—	4	4	no	—
379	C₂×C₄×C₆₇	—	4	4	no	—
380	C₂×C₄×C₆₇	—	4	2	no	—
381	C₂×C₄×C₆₇	—	8	2	no	—
382	C₂×C₄×C₆₇	—	8	1	no	—
383	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
384	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
385	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
386	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
387	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
388	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
389	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
390	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
391	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
392	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
393	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
394	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
395	C₈×C₆₇	—	2	2	no	—
396	C₈×C₆₇	—	2	4	no	—
397	C₈×C₆₇	—	2	2	no	—
398	C₂²×C₆₇	—	2	1	no	—
399	C₂²×C₆₇	—	2	1	no	—
400	C₂²×C₆₇	—	2	2	no	—
401	C₂²×C₆₇	—	4	1	no	—
402	C₂²×C₆₇	—	4	1	no	—
403	C₂²×C₆₇	—	4	2	no	—
404	C₂²×C₆₇	—	4	2	no	—
405	C₂²×C₆₇	—	4	2	no	—
406	C₂²×C₆₇	—	8	4	no	—
407	C₂²×C₆₇	#548	1	2	yes	—
408	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
409	C₄×C₆₇	—	2	2	no	—
410	C₄×C₆₇	—	2	1	no	—
411	C₄×C₆₇	—	2	2	no	—
412	C₄×C₆₇	—	2	1	no	—
413	C₄×C₆₇	—	2	2	no	—
414	C₄×C₆₇	—	2	1	no	—
415	C₄×C₆₇	—	2	2	no	—
416	C₄×C₆₇	—	2	2	no	—
417	C₄×C₆₇	—	4	2	no	—
418	C₄×C₆₇	—	4	1	no	—
419	C₄×C₆₇	—	4	4	no	—
420	C₄×C₆₇	—	4	2	no	—
421	C₄×C₆₇	—	4	2	no	—
422	C₄×C₆₇	—	4	1	no	—
423	C₄×C₆₇	#548	1	2	yes	—
424	C₄×C₆₇	#575	1	2	yes	—
425	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
426	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
427	#61323	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
428	C₂×C₆₇	—	2	2	no	—
429	C₂×C₆₇	—	4	2	no	—
430	C₂×C₆₇	—	8	2	no	—
431	C₂×C₆₇	C₂×#548	1	1	yes	characteristic
432	C₂×C₆₇	C₂×#575	1	1	yes	characteristic
433	C₂×C₆₇	#3586	1	1	yes	characteristic
434	Q₈×#41	C₂×C₆₇	1	1	yes	characteristic, complemented
435	C₂×#556	C₂×C₆₇	1	1	yes	characteristic, complemented
436	#3377	C₂×C₆₇	1	4	yes	complemented
437	#3376	C₂×C₆₇	1	4	yes	complemented
438	#3369	C₂×C₆₇	1	2	yes	complemented
439	#3367	C₂×C₆₇	1	2	yes	complemented
440	#3364	C₂×C₆₇	1	2	yes	complemented
441	#3360	C₂×C₆₇	1	2	yes	complemented
442	#3140	C₂×C₆₇	1	2	yes	complemented
443	#3139	C₂×C₆₇	1	1	yes	characteristic, complemented
444	#3078	C₂×C₆₇	1	2	yes	complemented
445	#3077	C₂×C₆₇	1	1	yes	characteristic, complemented
446	#3040	C₂×C₆₇	1	2	yes	—
447	#3039	C₂×C₆₇	1	1	yes	characteristic, complemented
448	Q₈×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
449	C₄×#137	C₂×C₆₇	1	2	yes	complemented
450	C₄×AGL(1,ℤ/8)	C₂×C₆₇	1	1	yes	characteristic
451	C₆₇	#61323	1	1	yes	characteristic, direct product factor, Sylow
452	C₂×#137	C₂²×C₆₇	1	2	yes	—
453	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	1	yes	characteristic
454	Q₈²	C₂²×C₆₇	1	1	yes	characteristic, complemented
455	#557	C₂²×C₆₇	1	1	yes	characteristic, complemented
456	#557	C₂²×C₆₇	1	2	yes	complemented
457	#556	—	2	1	no	—
458	#556	—	2	2	no	—
459	#556	—	2	1	no	—
460	#556	C₂²×C₆₇	1	2	yes	—
461	#556	C₂²×C₆₇	1	2	yes	—
462	#556	C₂²×C₆₇	1	4	yes	—
463	#554	—	2	1	no	—
464	#554	—	2	2	no	—
465	#554	C₂²×C₆₇	1	1	yes	characteristic, complemented
466	#554	C₂²×C₆₇	1	2	yes	complemented
467	D₄×Q₈	—	2	1	no	—
468	D₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
469	#533	—	2	1	no	—
470	#533	—	2	2	no	—
471	#533	C₂²×C₆₇	1	1	yes	characteristic
472	#533	C₂²×C₆₇	1	2	yes	—
473	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
474	C₂×#126	—	2	2	no	—
475	C₂×#124	C₂²×C₆₇	1	2	yes	—
476	C₂×C₂²⋊Q₈	—	2	1	no	—
477	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
478	C₄×#41	C₂²×C₆₇	1	2	yes	—
479	C₂×C₄×Q₈	—	2	1	no	—
480	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
481	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
482	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
483	C₂×#117	—	2	1	no	—
484	#491	C₂²×C₆₇	1	4	yes	complemented
485	#489	C₂²×C₆₇	1	4	yes	—
486	#489	C₂²×C₆₇	1	4	yes	—
487	#488	C₂²×C₆₇	1	4	yes	—
488	#488	C₂²×C₆₇	1	4	yes	—
489	#487	C₂²×C₆₇	1	2	yes	complemented
490	#487	C₂²×C₆₇	1	2	yes	complemented
491	#479	C₂²×C₆₇	1	4	yes	complemented
492	#479	C₂²×C₆₇	1	4	yes	complemented
493	#478	C₂²×C₆₇	1	4	yes	—
494	#477	C₂²×C₆₇	1	4	yes	complemented
495	#476	C₂²×C₆₇	1	2	yes	—
496	#475	C₂²×C₆₇	1	4	yes	complemented
497	#474	C₂²×C₆₇	1	2	yes	—
498	#464	C₂²×C₆₇	1	4	yes	—
499	#464	C₂²×C₆₇	1	4	yes	—
500	#463	C₂²×C₆₇	1	4	yes	—
501	#463	C₂²×C₆₇	1	4	yes	—
502	#462	C₂²×C₆₇	1	2	yes	complemented
503	#462	C₂²×C₆₇	1	2	yes	complemented
504	#461	C₂²×C₆₇	1	2	yes	complemented
505	#461	C₂²×C₆₇	1	2	yes	complemented
506	#446	C₂²×C₆₇	1	2	yes	complemented
507	#446	C₂²×C₆₇	1	4	yes	complemented
508	C₈×Q₈	C₂²×C₆₇	1	2	yes	complemented
509	#442	C₂²×C₆₇	1	2	yes	—
510	#441	C₂²×C₆₇	1	4	yes	complemented
511	#440	C₂²×C₆₇	1	2	yes	—
512	#440	C₂²×C₆₇	1	4	yes	complemented
513	C₄×Q₁₆	C₂²×C₆₇	1	4	yes	complemented
514	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	—
515	C₄×SD₁₆	C₂²×C₆₇	1	4	yes	complemented
516	C₄×D₈	C₂²×C₆₇	1	2	yes	—
517	#428	C₂²×C₆₇	1	2	yes	—
518	#428	C₂²×C₆₇	1	1	yes	characteristic
519	#423	C₂²×C₆₇	1	1	yes	characteristic
520	#423	C₂²×C₆₇	1	2	yes	—
521	#419	C₂²×C₆₇	1	2	yes	—
522	#419	C₂²×C₆₇	1	2	yes	—
523	#419	C₂²×C₆₇	1	2	yes	—
524	#418	C₂²×C₆₇	1	2	yes	—
525	#418	C₂²×C₆₇	1	1	yes	characteristic
526	#417	C₂²×C₆₇	1	1	yes	characteristic
527	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
528	C₄×M₁₆	C₂²×C₆₇	1	2	yes	—
529	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
530	C₂²×Q₈	—	2	1	no	—
531	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
532	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
533	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
534	C₂³×C₄	—	2	1	no	—
535	#137	—	2	8	no	—
536	AGL(1,ℤ/8)	—	2	4	no	—
537	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
538	C₂×SD₁₆	C₂³×C₆₇	1	2	yes	—
539	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
540	C₂×D₈	C₂³×C₆₇	1	2	yes	—
541	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
542	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
543	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
544	C₄⋊Q₈	—	2	1	no	—
545	C₄⋊Q₈	—	2	2	no	—
546	C₄⋊Q₈	—	2	1	no	—
547	C₄⋊Q₈	—	2	2	no	—
548	C₄⋊Q₈	—	2	1	no	—
549	C₄⋊Q₈	—	2	1	no	—
550	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
551	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
552	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
553	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
554	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
555	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
556	#126	—	2	4	no	—
557	#126	—	2	4	no	—
558	#126	—	2	4	no	—
559	#126	—	2	4	no	—
560	#125	—	2	2	no	—
561	#125	—	2	2	no	—
562	#125	—	2	2	no	—
563	#125	C₂³×C₆₇	1	2	yes	—
564	#125	C₂³×C₆₇	1	2	yes	—
565	#125	C₂³×C₆₇	1	2	yes	—
566	#124	—	2	2	no	—
567	#124	—	2	2	no	—
568	#124	C₂³×C₆₇	1	4	yes	—
569	#124	C₂³×C₆₇	1	4	yes	—
570	C₂²⋊Q₈	—	2	2	no	—
571	C₂²⋊Q₈	—	2	2	no	—
572	C₂²⋊Q₈	—	2	4	no	—
573	C₂²⋊Q₈	—	2	1	no	—
574	C₂²⋊Q₈	—	2	2	no	—
575	C₂²⋊Q₈	—	2	1	no	—
576	C₂²⋊Q₈	—	2	2	no	—
577	C₂²⋊Q₈	—	2	2	no	—
578	C₂²⋊Q₈	—	2	1	no	—
579	C₂²⋊Q₈	—	2	1	no	—
580	C₂²⋊Q₈	—	2	2	no	—
581	C₂²⋊Q₈	—	2	2	no	—
582	C₂²⋊Q₈	—	2	2	no	—
583	C₂²⋊Q₈	—	2	2	no	—
584	C₄×Q₈	—	2	1	no	—
585	C₄×Q₈	—	2	1	no	—
586	C₄×Q₈	—	2	2	no	—
587	C₄×Q₈	—	2	1	no	—
588	C₄×Q₈	—	2	2	no	—
589	C₄×Q₈	—	2	2	no	—
590	C₄×Q₈	—	2	2	no	—
591	C₄×Q₈	—	2	1	no	—
592	C₄×Q₈	—	2	2	no	—
593	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
594	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
595	C₄×Q₈	D₄×C₆₇	1	2	yes	—
596	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
597	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
598	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
599	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
600	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
601	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
602	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
603	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
604	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
605	C₄×D₄	—	2	1	no	—
606	C₄×D₄	—	2	2	no	—
607	C₄×D₄	—	2	2	no	—
608	C₄×D₄	—	2	1	no	—
609	C₄×D₄	—	2	1	no	—
610	C₄×D₄	—	2	1	no	—
611	C₄×D₄	C₂³×C₆₇	1	2	yes	—
612	C₄×D₄	C₂³×C₆₇	1	2	yes	—
613	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
614	C₄×D₄	C₂³×C₆₇	1	2	yes	—
615	C₄×D₄	C₂³×C₆₇	1	2	yes	—
616	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
617	#117	—	2	2	no	—
618	#117	—	2	4	no	—
619	#117	—	2	2	no	—
620	#117	—	2	1	no	—
621	#117	—	2	2	no	—
622	#117	C₂³×C₆₇	1	1	yes	characteristic
623	#117	C₂³×C₆₇	1	2	yes	—
624	C₂×C₄⋊C₄	—	2	1	no	—
625	C₂×C₄⋊C₄	—	2	2	no	—
626	C₂×C₄⋊C₄	—	2	1	no	—
627	C₂×C₄⋊C₄	—	2	1	no	—
628	C₂×C₄⋊C₄	—	2	2	no	—
629	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
630	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
631	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
632	C₂×C₂²⋊C₄	—	2	2	no	—
633	C₂×C₂²⋊C₄	—	2	1	no	—
634	C₂×C₂²⋊C₄	—	2	2	no	—
635	C₂×C₄²	—	2	2	no	—
636	C₂×C₄²	—	2	1	no	—
637	C₂×C₄²	—	2	2	no	—
638	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
639	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
640	#107	C₂³×C₆₇	1	2	yes	—
641	#107	C₂³×C₆₇	1	4	yes	—
642	#106	C₂³×C₆₇	1	2	yes	—
643	#106	C₂³×C₆₇	1	4	yes	—
644	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
645	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
646	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
647	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
648	#103	C₂³×C₆₇	1	2	yes	—
649	#103	C₂³×C₆₇	1	4	yes	—
650	#103	C₂³×C₆₇	1	4	yes	—
651	#103	C₂³×C₆₇	1	4	yes	—
652	#103	C₂³×C₆₇	1	4	yes	—
653	#103	C₂³×C₆₇	1	4	yes	—
654	#103	C₂³×C₆₇	1	4	yes	—
655	#102	C₂³×C₆₇	1	2	yes	—
656	#102	C₂³×C₆₇	1	2	yes	—
657	#102	C₂³×C₆₇	1	2	yes	—
658	#102	C₂³×C₆₇	1	4	yes	—
659	#102	C₂³×C₆₇	1	4	yes	—
660	#97	C₂³×C₆₇	1	2	yes	—
661	#97	C₂³×C₆₇	1	4	yes	—
662	C₄×C₈	C₂³×C₆₇	1	2	yes	—
663	C₄×C₈	C₂³×C₆₇	1	4	yes	—
664	C₂⁴	—	2	1	no	—
665	#41	—	2	2	no	—
666	#41	#41×C₆₇	1	4	yes	—
667	C₂×Q₈	—	2	2	no	—
668	C₂×Q₈	—	2	1	no	—
669	C₂×Q₈	—	2	2	no	—
670	C₂×Q₈	—	2	2	no	—
671	C₂×Q₈	—	2	1	no	—
672	C₂×Q₈	—	2	4	no	—
673	C₂×Q₈	—	2	4	no	—
674	C₂×Q₈	—	4	2	no	—
675	C₂×Q₈	—	4	4	no	—
676	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
677	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
678	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
679	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
680	C₂×Q₈	C₂⁴×C₆₇	1	1	yes	characteristic
681	C₂×D₄	—	2	1	no	—
682	C₂×D₄	—	2	1	no	—
683	C₂×D₄	—	2	2	no	—
684	C₂×D₄	—	2	2	no	—
685	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
686	C₂×D₄	C₂⁴×C₆₇	1	1	yes	characteristic
687	C₂²×C₄	—	2	1	no	—
688	C₂²×C₄	—	2	1	no	—
689	C₂²×C₄	—	2	2	no	—
690	C₂²×C₄	—	2	2	no	—
691	C₂²×C₄	—	2	1	no	—
692	C₂²×C₄	—	2	2	no	—
693	C₂²×C₄	—	2	2	no	—
694	C₂²×C₄	—	2	2	no	—
695	C₂²×C₄	—	2	1	no	—
696	C₂²×C₄	—	4	4	no	—
697	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
698	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
699	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
700	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
701	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
702	Q₁₆	—	2	8	no	—
703	SD₁₆	—	2	4	no	—
704	SD₁₆	—	2	8	no	—
705	D₈	—	2	4	no	—
706	M₁₆	—	2	2	no	—
707	M₁₆	—	2	4	no	—
708	M₁₆	—	2	2	no	—
709	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
710	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
711	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
712	C₄⋊C₄	—	2	2	no	—
713	C₄⋊C₄	—	2	2	no	—
714	C₄⋊C₄	—	2	2	no	—
715	C₄⋊C₄	—	2	2	no	—
716	C₄⋊C₄	—	2	2	no	—
717	C₄⋊C₄	—	2	2	no	—
718	C₄⋊C₄	—	2	1	no	—
719	C₄⋊C₄	—	2	4	no	—
720	C₄⋊C₄	—	2	4	no	—
721	C₄⋊C₄	—	2	2	no	—
722	C₄⋊C₄	—	2	2	no	—
723	C₄⋊C₄	—	2	1	no	—
724	C₄⋊C₄	—	2	2	no	—
725	C₄⋊C₄	—	2	2	no	—
726	C₄⋊C₄	—	2	1	no	—
727	C₄⋊C₄	—	2	1	no	—
728	C₄⋊C₄	—	2	2	no	—
729	C₄⋊C₄	—	2	1	no	—
730	C₄⋊C₄	—	2	2	no	—
731	C₄⋊C₄	—	2	2	no	—
732	C₄⋊C₄	—	2	2	no	—
733	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
734	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
735	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
736	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
737	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
738	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
739	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
740	C₄⋊C₄	C₂⁴×C₆₇	1	1	yes	characteristic
741	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
742	C₄⋊C₄	C₂⁴×C₆₇	1	1	yes	characteristic
743	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
744	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
745	C₂²⋊C₄	—	2	1	no	—
746	C₂²⋊C₄	—	2	2	no	—
747	C₂²⋊C₄	—	2	4	no	—
748	C₂²⋊C₄	—	2	4	no	—
749	C₂²⋊C₄	—	2	2	no	—
750	C₂²⋊C₄	—	2	2	no	—
751	C₂²⋊C₄	—	2	4	no	—
752	C₂²⋊C₄	—	2	4	no	—
753	C₂²⋊C₄	—	2	2	no	—
754	C₂²⋊C₄	—	2	2	no	—
755	C₂²⋊C₄	—	2	1	no	—
756	C₂²⋊C₄	—	2	1	no	—
757	C₄²	—	2	4	no	—
758	C₄²	—	2	4	no	—
759	C₄²	—	2	1	no	—
760	C₄²	—	2	2	no	—
761	C₄²	—	2	2	no	—
762	C₄²	—	2	1	no	—
763	C₄²	—	2	2	no	—
764	C₄²	—	2	1	no	—
765	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
766	C₄²	C₂×D₄×C₆₇	1	2	yes	—
767	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
768	C₄²	C₂×D₄×C₆₇	1	2	yes	—
769	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
770	C₄²	C₂⁴×C₆₇	1	2	yes	—
771	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
772	C₄²	C₂⁴×C₆₇	1	2	yes	—
773	C₂³	—	2	1	no	—
774	C₂³	—	2	2	no	—
775	C₂³	—	4	2	no	—
776	C₂³	—	4	2	no	—
777	C₂³	—	4	2	no	—
778	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
779	Q₈	—	2	2	no	—
780	Q₈	—	2	2	no	—
781	Q₈	—	2	1	no	—
782	Q₈	—	2	4	no	—
783	Q₈	—	2	4	no	—
784	Q₈	—	4	2	no	—
785	Q₈	—	4	4	no	—
786	Q₈	—	4	4	no	—
787	Q₈	—	4	2	no	—
788	Q₈	—	8	2	no	—
789	Q₈	—	8	1	no	—
790	Q₈	C₂×#41×C₆₇	1	2	yes	—
791	D₄	—	2	4	no	—
792	D₄	—	2	2	no	—
793	D₄	—	2	1	no	—
794	D₄	—	2	2	no	—
795	D₄	—	4	2	no	—
796	D₄	C₂×#41×C₆₇	1	2	yes	—
797	C₂×C₄	—	2	1	no	—
798	C₂×C₄	—	2	4	no	—
799	C₂×C₄	—	2	2	no	—
800	C₂×C₄	—	2	2	no	—
801	C₂×C₄	—	2	4	no	—
802	C₂×C₄	—	2	2	no	—
803	C₂×C₄	—	2	1	no	—
804	C₂×C₄	—	2	2	no	—
805	C₂×C₄	—	2	1	no	—
806	C₂×C₄	—	4	2	no	—
807	C₂×C₄	—	4	2	no	—
808	C₂×C₄	—	4	1	no	—
809	C₂×C₄	—	4	4	no	—
810	C₂×C₄	—	4	4	no	—
811	C₂×C₄	—	4	2	no	—
812	C₂×C₄	—	4	4	no	—
813	C₂×C₄	—	4	4	no	—
814	C₂×C₄	—	4	2	no	—
815	C₂×C₄	—	8	2	no	—
816	C₂×C₄	—	8	1	no	—
817	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
818	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
819	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
820	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
821	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
822	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
823	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
824	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
825	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
826	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
827	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
828	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
829	C₈	—	2	2	no	—
830	C₈	—	2	4	no	—
831	C₈	—	2	2	no	—
832	C₂²	—	2	1	no	—
833	C₂²	—	2	1	no	—
834	C₂²	—	2	2	no	—
835	C₂²	—	4	1	no	—
836	C₂²	—	4	1	no	—
837	C₂²	—	4	2	no	—
838	C₂²	—	4	2	no	—
839	C₂²	—	4	2	no	—
840	C₂²	—	8	4	no	—
841	C₂²	#548×C₆₇	1	2	yes	—
842	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
843	C₄	—	2	1	no	—
844	C₄	—	2	2	no	—
845	C₄	—	2	2	no	—
846	C₄	—	2	2	no	—
847	C₄	—	2	1	no	—
848	C₄	—	2	2	no	—
849	C₄	—	2	1	no	—
850	C₄	—	2	2	no	—
851	C₄	—	4	2	no	—
852	C₄	—	4	2	no	—
853	C₄	—	4	1	no	—
854	C₄	—	4	2	no	—
855	C₄	—	4	1	no	—
856	C₄	—	4	4	no	—
857	C₄	#548×C₆₇	1	2	yes	—
858	C₄	#575×C₆₇	1	2	yes	—
859	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
860	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
861	C₂	—	2	2	no	—
862	C₂	—	4	2	no	—
863	C₂	—	8	2	no	—
864	C₂	C₂×#548×C₆₇	1	1	yes	characteristic
865	C₂	C₂×#575×C₆₇	1	1	yes	characteristic
866	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	C₂×AGL(1,ℤ/8)×C₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄×Q₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. Q₈×#41×C₆₇	C₂	semidirect
2. C₂×#556×C₆₇	C₂	semidirect
3. C₆₇×#3377	C₂	semidirect
4. C₆₇×#3376	C₂	semidirect
5. C₆₇×#3369	C₂	semidirect
6. C₆₇×#3367	C₂	semidirect
7. C₆₇×#3364	C₂	semidirect
8. C₆₇×#3360	C₂	semidirect
9. C₆₇×#3140	C₂	semidirect
10. C₆₇×#3139	C₂	semidirect
11. C₆₇×#3078	C₂	semidirect
12. C₆₇×#3077	C₂	semidirect
14. C₆₇×#3039	C₂	semidirect
15. Q₈×M₁₆×C₆₇	C₂	semidirect
16. C₄×#137×C₆₇	C₂	semidirect
20. Q₈²×C₆₇	C₂²	semidirect
21. #557×C₆₇	C₂²	semidirect
22. #557×C₆₇	C₂²	semidirect
31. #554×C₆₇	C₂²	semidirect
32. #554×C₆₇	C₂²	semidirect
34. D₄×Q₈×C₆₇	C₂²	semidirect
50. #491×C₆₇	C₂²	semidirect
55. #487×C₆₇	C₂²	semidirect
56. #487×C₆₇	C₂²	semidirect
57. #479×C₆₇	C₂²	semidirect
58. #479×C₆₇	C₂²	semidirect
60. #477×C₆₇	C₂²	semidirect
62. #475×C₆₇	C₂²	semidirect
68. #462×C₆₇	C₂²	semidirect
69. #462×C₆₇	C₂²	semidirect
70. #461×C₆₇	C₂²	semidirect
71. #461×C₆₇	C₂²	semidirect
72. #446×C₆₇	C₂²	semidirect
73. #446×C₆₇	C₂²	semidirect
74. C₈×Q₈×C₆₇	C₂²	semidirect
76. #441×C₆₇	C₂²	semidirect
78. #440×C₆₇	C₂²	semidirect
79. C₄×Q₁₆×C₆₇	C₂²	semidirect
81. C₄×SD₁₆×C₆₇	C₂²	semidirect
427. #61323	C₆₇	direct (N × H)
434. Q₈×#41	C₂×C₆₇	semidirect
435. C₂×#556	C₂×C₆₇	semidirect
436. #3377	C₂×C₆₇	semidirect
437. #3376	C₂×C₆₇	semidirect
438. #3369	C₂×C₆₇	semidirect
439. #3367	C₂×C₆₇	semidirect
440. #3364	C₂×C₆₇	semidirect
441. #3360	C₂×C₆₇	semidirect
442. #3140	C₂×C₆₇	semidirect
443. #3139	C₂×C₆₇	semidirect
444. #3078	C₂×C₆₇	semidirect
445. #3077	C₂×C₆₇	semidirect
447. #3039	C₂×C₆₇	semidirect
448. Q₈×M₁₆	C₂×C₆₇	semidirect
449. C₄×#137	C₂×C₆₇	semidirect
451. C₆₇	#61323	direct (N × H)
454. Q₈²	C₂²×C₆₇	semidirect
455. #557	C₂²×C₆₇	semidirect
456. #557	C₂²×C₆₇	semidirect
465. #554	C₂²×C₆₇	semidirect
466. #554	C₂²×C₆₇	semidirect
468. D₄×Q₈	C₂²×C₆₇	semidirect
484. #491	C₂²×C₆₇	semidirect
489. #487	C₂²×C₆₇	semidirect
490. #487	C₂²×C₆₇	semidirect
491. #479	C₂²×C₆₇	semidirect
492. #479	C₂²×C₆₇	semidirect
494. #477	C₂²×C₆₇	semidirect
496. #475	C₂²×C₆₇	semidirect
502. #462	C₂²×C₆₇	semidirect
503. #462	C₂²×C₆₇	semidirect
504. #461	C₂²×C₆₇	semidirect
505. #461	C₂²×C₆₇	semidirect
506. #446	C₂²×C₆₇	semidirect
507. #446	C₂²×C₆₇	semidirect
508. C₈×Q₈	C₂²×C₆₇	semidirect
510. #441	C₂²×C₆₇	semidirect
512. #440	C₂²×C₆₇	semidirect
513. C₄×Q₁₆	C₂²×C₆₇	semidirect
515. C₄×SD₁₆	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#41×C₆₇	2
C₂×#556×C₆₇	2
C₆₇×#3377	2
C₆₇×#3376	2
C₆₇×#3369	2
C₆₇×#3367	2
C₆₇×#3364	2
C₆₇×#3360	2
C₆₇×#3140	2
C₆₇×#3139	2
C₆₇×#3078	2
C₆₇×#3077	2
C₆₇×#3040	2
C₆₇×#3039	2
Q₈×M₁₆×C₆₇	2
C₄×#137×C₆₇	2
C₄×AGL(1,ℤ/8)×C₆₇	2
#61323	67

Subgroups — C67×#61323 #30740574

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61323 #30740574