Subgroups — C₆₇×#34964 #30714215

Order 17152.

2698 conjugacy classes of subgroups.

The subgroup lattice has 778 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#529×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×#41×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2601	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2592	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2584	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2562	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2540	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#2498	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2430	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2417	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2416	C₂	1	2	yes	maximal, complemented
12	C₆₇×#2396	C₂	1	2	yes	maximal, complemented
13	C₆₇×#2392	C₂	1	2	yes	maximal, complemented
14	C₄×C₂²⋊Q₈×C₆₇	C₂	1	2	yes	maximal, complemented
15	C₄×#121×C₆₇	C₂	1	2	yes	maximal, complemented
16	C₄×#121×C₆₇	C₂	1	4	yes	maximal, complemented
17	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
19	#529×C₆₇	—	2	8	no	—
20	C₂×#126×C₆₇	C₂²	1	2	yes	—
21	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#123×C₆₇	C₂²	1	2	yes	—
24	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
25	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#121×C₆₇	C₂²	1	2	yes	—
27	C₂×#121×C₆₇	C₂²	1	4	yes	—
28	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
29	C₄×#41×C₆₇	—	2	8	no	—
30	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
35	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
36	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
39	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
40	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×#117×C₆₇	C₂²	1	2	yes	—
42	C₂×#117×C₆₇	C₂²	1	2	yes	—
43	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
44	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
45	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
46	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
47	#400×C₆₇	C₂²	1	2	yes	—
48	#399×C₆₇	C₂²	1	2	yes	—
49	#398×C₆₇	C₂²	1	2	yes	—
50	#398×C₆₇	C₂²	1	4	yes	—
51	#397×C₆₇	C₂²	1	2	yes	—
52	#396×C₆₇	C₂²	1	4	yes	—
53	#396×C₆₇	C₂²	1	4	yes	—
54	#395×C₆₇	C₂²	1	2	yes	—
55	#394×C₆₇	C₂²	1	2	yes	—
56	#394×C₆₇	C₂²	1	4	yes	—
57	#393×C₆₇	C₂²	1	2	yes	—
58	#392×C₆₇	C₂²	1	2	yes	—
59	#391×C₆₇	C₂²	1	2	yes	—
60	#390×C₆₇	C₂²	1	4	yes	—
61	#390×C₆₇	C₂²	1	2	yes	—
62	#389×C₆₇	C₂²	1	4	yes	—
63	#389×C₆₇	C₂²	1	2	yes	—
64	#389×C₆₇	C₂²	1	2	yes	—
65	#388×C₆₇	C₂²	1	4	yes	—
66	#388×C₆₇	C₂²	1	4	yes	—
67	#388×C₆₇	C₂²	1	4	yes	—
68	#387×C₆₇	C₂²	1	4	yes	—
69	#386×C₆₇	C₂²	1	4	yes	—
70	#386×C₆₇	C₂²	1	4	yes	—
71	#386×C₆₇	C₂²	1	4	yes	—
72	#385×C₆₇	C₂²	1	4	yes	—
73	#380×C₆₇	C₂²	1	4	yes	—
74	#380×C₆₇	C₂²	1	2	yes	—
75	#380×C₆₇	C₂²	1	2	yes	—
76	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
77	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
78	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
79	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
80	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
81	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
82	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
83	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
84	C₂×#95×C₆₇	C₂²	1	2	yes	—
85	C₂×#41×C₆₇	—	2	8	no	—
86	C₂×#41×C₆₇	—	2	4	no	—
87	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
88	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
89	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
90	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
91	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
92	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
93	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
94	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
95	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
96	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
97	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
98	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
99	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
100	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
101	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
102	#126×C₆₇	—	2	8	no	—
103	#125×C₆₇	—	2	4	no	—
104	#124×C₆₇	—	2	4	no	—
105	#123×C₆₇	—	2	8	no	—
106	C₂²⋊Q₈×C₆₇	—	2	4	no	—
107	C₂²⋊Q₈×C₆₇	—	2	8	no	—
108	#121×C₆₇	—	2	16	no	—
109	#121×C₆₇	—	2	4	no	—
110	#121×C₆₇	—	2	8	no	—
111	C₄×Q₈×C₆₇	—	2	4	no	—
112	C₄×Q₈×C₆₇	—	2	4	no	—
113	C₄×Q₈×C₆₇	—	2	4	no	—
114	C₄×D₄×C₆₇	—	2	8	no	—
115	C₄×D₄×C₆₇	—	2	8	no	—
116	C₄×D₄×C₆₇	—	2	8	no	—
117	C₄×D₄×C₆₇	—	2	4	no	—
118	C₄×D₄×C₆₇	—	2	4	no	—
119	C₄×D₄×C₆₇	—	2	4	no	—
120	C₄×D₄×C₆₇	—	4	8	no	—
121	#117×C₆₇	—	2	4	no	—
122	#117×C₆₇	—	2	8	no	—
123	#117×C₆₇	—	2	8	no	—
124	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
125	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
126	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
127	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
128	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
129	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
130	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
131	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
132	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
133	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
134	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
135	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
136	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
137	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
138	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
139	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
140	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
141	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
142	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
143	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
144	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
145	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
146	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
147	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
148	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
149	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
150	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
151	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
152	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
153	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
154	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
155	C₂×C₄²×C₆₇	—	2	8	no	—
156	C₂×C₄²×C₆₇	—	4	4	no	—
157	C₂×C₄²×C₆₇	D₄	1	8	yes	complemented
158	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
159	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
160	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
161	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
162	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
163	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
164	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
165	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
166	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
167	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
168	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
169	#95×C₆₇	C₂³	1	4	yes	—
170	#95×C₆₇	C₂³	1	2	yes	—
171	#95×C₆₇	C₂³	1	2	yes	—
172	#95×C₆₇	C₂³	1	4	yes	—
173	#95×C₆₇	C₂³	1	4	yes	—
174	#95×C₆₇	C₂³	1	4	yes	—
175	#95×C₆₇	C₂³	1	4	yes	—
176	#95×C₆₇	C₂³	1	4	yes	—
177	#95×C₆₇	C₂³	1	4	yes	—
178	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
179	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
180	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
181	#41×C₆₇	—	2	32	no	—
182	C₂×Q₈×C₆₇	—	2	2	no	—
183	C₂×Q₈×C₆₇	—	2	2	no	—
184	C₂×Q₈×C₆₇	—	2	4	no	—
185	C₂×Q₈×C₆₇	—	2	4	no	—
186	C₂×D₄×C₆₇	—	2	4	no	—
187	C₂×D₄×C₆₇	—	2	4	no	—
188	C₂×D₄×C₆₇	—	2	4	no	—
189	C₂×D₄×C₆₇	—	2	4	no	—
190	C₂×D₄×C₆₇	—	2	8	no	—
191	C₂×D₄×C₆₇	—	2	8	no	—
192	C₂×D₄×C₆₇	—	2	2	no	—
193	C₂×D₄×C₆₇	—	2	2	no	—
194	C₂×D₄×C₆₇	—	2	8	no	—
195	C₂×D₄×C₆₇	—	2	4	no	—
196	C₂×D₄×C₆₇	—	4	4	no	—
197	C₂×D₄×C₆₇	#41	1	8	yes	—
198	C₂²×C₄×C₆₇	—	2	4	no	—
199	C₂²×C₄×C₆₇	—	2	4	no	—
200	C₂²×C₄×C₆₇	—	2	4	no	—
201	C₂²×C₄×C₆₇	—	2	4	no	—
202	C₂²×C₄×C₆₇	—	2	4	no	—
203	C₂²×C₄×C₆₇	—	2	2	no	—
204	C₂²×C₄×C₆₇	—	2	2	no	—
205	C₂²×C₄×C₆₇	—	2	8	no	—
206	C₂²×C₄×C₆₇	—	2	4	no	—
207	C₂²×C₄×C₆₇	—	2	4	no	—
208	C₂²×C₄×C₆₇	—	2	4	no	—
209	C₂²×C₄×C₆₇	—	2	8	no	—
210	C₂²×C₄×C₆₇	—	2	4	no	—
211	C₂²×C₄×C₆₇	—	2	4	no	—
212	C₂²×C₄×C₆₇	—	2	4	no	—
213	C₂²×C₄×C₆₇	—	2	4	no	—
214	C₂²×C₄×C₆₇	—	2	4	no	—
215	C₂²×C₄×C₆₇	—	2	4	no	—
216	C₂²×C₄×C₆₇	—	4	4	no	—
217	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
218	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
219	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
220	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
221	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
222	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
223	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
224	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
225	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
226	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
227	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
228	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
229	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
230	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
231	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
232	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
233	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
234	C₄⋊C₄×C₆₇	—	2	4	no	—
235	C₄⋊C₄×C₆₇	—	2	4	no	—
236	C₄⋊C₄×C₆₇	—	2	2	no	—
237	C₄⋊C₄×C₆₇	—	2	2	no	—
238	C₄⋊C₄×C₆₇	—	2	4	no	—
239	C₄⋊C₄×C₆₇	—	2	4	no	—
240	C₄⋊C₄×C₆₇	—	2	2	no	—
241	C₄⋊C₄×C₆₇	—	2	4	no	—
242	C₄⋊C₄×C₆₇	—	2	2	no	—
243	C₄⋊C₄×C₆₇	—	2	4	no	—
244	C₄⋊C₄×C₆₇	—	2	8	no	—
245	C₄⋊C₄×C₆₇	—	2	4	no	—
246	C₄⋊C₄×C₆₇	—	2	8	no	—
247	C₄⋊C₄×C₆₇	—	2	8	no	—
248	C₄⋊C₄×C₆₇	—	4	4	no	—
249	C₄⋊C₄×C₆₇	#41	1	8	yes	complemented
250	C₂²⋊C₄×C₆₇	—	2	4	no	—
251	C₂²⋊C₄×C₆₇	—	2	4	no	—
252	C₂²⋊C₄×C₆₇	—	2	8	no	—
253	C₂²⋊C₄×C₆₇	—	2	4	no	—
254	C₂²⋊C₄×C₆₇	—	2	4	no	—
255	C₂²⋊C₄×C₆₇	—	2	4	no	—
256	C₂²⋊C₄×C₆₇	—	2	4	no	—
257	C₂²⋊C₄×C₆₇	—	2	4	no	—
258	C₂²⋊C₄×C₆₇	—	2	4	no	—
259	C₂²⋊C₄×C₆₇	—	2	8	no	—
260	C₂²⋊C₄×C₆₇	—	2	8	no	—
261	C₂²⋊C₄×C₆₇	—	2	8	no	—
262	C₂²⋊C₄×C₆₇	—	4	4	no	—
263	C₂²⋊C₄×C₆₇	—	4	4	no	—
264	C₄²×C₆₇	—	2	4	no	—
265	C₄²×C₆₇	—	2	4	no	—
266	C₄²×C₆₇	—	2	4	no	—
267	C₄²×C₆₇	—	2	4	no	—
268	C₄²×C₆₇	—	2	8	no	—
269	C₄²×C₆₇	—	4	2	no	—
270	C₄²×C₆₇	—	4	2	no	—
271	C₄²×C₆₇	C₂×D₄	1	4	yes	—
272	C₄²×C₆₇	C₂×D₄	1	4	yes	—
273	C₄²×C₆₇	C₂×D₄	1	4	yes	—
274	C₄²×C₆₇	C₂×D₄	1	4	yes	—
275	C₂³×C₆₇	—	2	2	no	—
276	C₂³×C₆₇	—	2	1	no	—
277	C₂³×C₆₇	—	2	1	no	—
278	C₂³×C₆₇	—	2	2	no	—
279	C₂³×C₆₇	—	2	2	no	—
280	C₂³×C₆₇	—	2	2	no	—
281	C₂³×C₆₇	—	2	2	no	—
282	C₂³×C₆₇	—	2	2	no	—
283	C₂³×C₆₇	—	2	2	no	—
284	C₂³×C₆₇	—	2	2	no	—
285	C₂³×C₆₇	—	2	4	no	—
286	C₂³×C₆₇	—	2	4	no	—
287	C₂³×C₆₇	—	2	2	no	—
288	C₂³×C₆₇	—	2	2	no	—
289	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
290	C₂³×C₆₇	C₂×#41	1	4	yes	—
291	C₂³×C₆₇	C₂×#41	1	4	yes	—
292	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
293	Q₈×C₆₇	—	2	8	no	—
294	Q₈×C₆₇	—	4	4	no	—
295	D₄×C₆₇	—	2	8	no	—
296	D₄×C₆₇	—	2	16	no	—
297	D₄×C₆₇	—	4	16	no	—
298	D₄×C₆₇	—	4	8	no	—
299	D₄×C₆₇	—	4	4	no	—
300	C₂×C₄×C₆₇	—	2	16	no	—
301	C₂×C₄×C₆₇	—	2	8	no	—
302	C₂×C₄×C₆₇	—	2	2	no	—
303	C₂×C₄×C₆₇	—	2	2	no	—
304	C₂×C₄×C₆₇	—	2	4	no	—
305	C₂×C₄×C₆₇	—	2	2	no	—
306	C₂×C₄×C₆₇	—	2	2	no	—
307	C₂×C₄×C₆₇	—	2	2	no	—
308	C₂×C₄×C₆₇	—	2	4	no	—
309	C₂×C₄×C₆₇	—	2	1	no	—
310	C₂×C₄×C₆₇	—	2	1	no	—
311	C₂×C₄×C₆₇	—	2	8	no	—
312	C₂×C₄×C₆₇	—	2	8	no	—
313	C₂×C₄×C₆₇	—	2	2	no	—
314	C₂×C₄×C₆₇	—	2	2	no	—
315	C₂×C₄×C₆₇	—	2	2	no	—
316	C₂×C₄×C₆₇	—	2	2	no	—
317	C₂×C₄×C₆₇	—	2	2	no	—
318	C₂×C₄×C₆₇	—	2	2	no	—
319	C₂×C₄×C₆₇	—	2	4	no	—
320	C₂×C₄×C₆₇	—	2	4	no	—
321	C₂×C₄×C₆₇	—	2	4	no	—
322	C₂×C₄×C₆₇	—	2	2	no	—
323	C₂×C₄×C₆₇	—	2	4	no	—
324	C₂×C₄×C₆₇	—	2	4	no	—
325	C₂×C₄×C₆₇	—	2	2	no	—
326	C₂×C₄×C₆₇	—	2	2	no	—
327	C₂×C₄×C₆₇	—	2	2	no	—
328	C₂×C₄×C₆₇	—	2	2	no	—
329	C₂×C₄×C₆₇	—	4	16	no	—
330	C₂×C₄×C₆₇	—	4	8	no	—
331	C₂×C₄×C₆₇	—	4	8	no	—
332	C₂×C₄×C₆₇	—	4	8	no	—
333	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
334	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
335	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
336	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
337	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
338	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
339	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
340	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
341	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
342	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
343	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
344	C₂²×C₆₇	—	2	2	no	—
345	C₂²×C₆₇	—	2	2	no	—
346	C₂²×C₆₇	—	2	4	no	—
347	C₂²×C₆₇	—	2	4	no	—
348	C₂²×C₆₇	—	2	4	no	—
349	C₂²×C₆₇	—	2	4	no	—
350	C₂²×C₆₇	—	2	4	no	—
351	C₂²×C₆₇	—	4	2	no	—
352	C₂²×C₆₇	—	4	2	no	—
353	C₂²×C₆₇	—	4	2	no	—
354	C₂²×C₆₇	—	4	2	no	—
355	C₂²×C₆₇	—	4	2	no	—
356	C₂²×C₆₇	—	4	2	no	—
357	C₂²×C₆₇	—	4	2	no	—
358	C₂²×C₆₇	—	4	4	no	—
359	C₂²×C₆₇	—	4	4	no	—
360	C₂²×C₆₇	#525	1	4	yes	—
361	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
362	C₂²×C₆₇	C₂²×#41	1	2	yes	—
363	C₂²×C₆₇	C₂²×#41	1	2	yes	—
364	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
365	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
366	C₄×C₆₇	—	2	2	no	—
367	C₄×C₆₇	—	2	2	no	—
368	C₄×C₆₇	—	2	4	no	—
369	C₄×C₆₇	—	2	2	no	—
370	C₄×C₆₇	—	2	2	no	—
371	C₄×C₆₇	—	2	8	no	—
372	C₄×C₆₇	—	4	2	no	—
373	C₄×C₆₇	—	4	2	no	—
374	C₄×C₆₇	—	4	2	no	—
375	C₄×C₆₇	—	4	2	no	—
376	C₄×C₆₇	—	4	2	no	—
377	C₄×C₆₇	—	4	4	no	—
378	C₄×C₆₇	#532	1	4	yes	—
379	C₄×C₆₇	#537	1	4	yes	—
380	#34964	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
381	C₂×C₆₇	—	2	4	no	—
382	C₂×C₆₇	—	4	4	no	—
383	C₂×C₆₇	—	8	2	no	—
384	C₂×C₆₇	C₂×#525	1	1	yes	characteristic
385	C₂×C₆₇	C₂×#532	1	1	yes	characteristic
386	C₂×C₆₇	C₂×#537	1	1	yes	characteristic
387	C₂×C₆₇	#3481	1	2	yes	—
388	C₂×C₆₇	#3482	1	2	yes	—
389	C₂×#529	C₂×C₆₇	1	1	yes	characteristic, complemented
390	C₂×C₄×#41	C₂×C₆₇	1	1	yes	characteristic, complemented
391	#2601	C₂×C₆₇	1	4	yes	complemented
392	#2592	C₂×C₆₇	1	1	yes	characteristic, complemented
393	#2584	C₂×C₆₇	1	1	yes	characteristic, complemented
394	#2562	C₂×C₆₇	1	1	yes	characteristic, complemented
395	#2540	C₂×C₆₇	1	1	yes	characteristic, complemented
396	#2498	C₂×C₆₇	1	2	yes	complemented
397	#2430	C₂×C₆₇	1	2	yes	complemented
398	#2417	C₂×C₆₇	1	2	yes	complemented
399	#2416	C₂×C₆₇	1	2	yes	complemented
400	#2396	C₂×C₆₇	1	2	yes	complemented
401	#2392	C₂×C₆₇	1	2	yes	complemented
402	C₄×C₂²⋊Q₈	C₂×C₆₇	1	2	yes	complemented
403	C₄×#121	C₂×C₆₇	1	4	yes	complemented
404	C₄×#121	C₂×C₆₇	1	2	yes	complemented
405	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
406	C₆₇	#34964	1	1	yes	characteristic, direct product factor, Sylow
407	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
408	#529	—	2	8	no	—
409	C₂×#126	C₂²×C₆₇	1	2	yes	—
410	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
411	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
412	C₂×#123	C₂²×C₆₇	1	2	yes	—
413	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
414	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
415	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
416	C₂×#121	C₂²×C₆₇	1	4	yes	—
417	C₂×#121	C₂²×C₆₇	1	2	yes	—
418	C₄×#41	—	2	8	no	—
419	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
420	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
421	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
422	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
423	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
424	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
425	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
426	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
427	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
428	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
429	C₂×#117	C₂²×C₆₇	1	2	yes	—
430	C₂×#117	C₂²×C₆₇	1	2	yes	—
431	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
432	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
433	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
434	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
435	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
436	#400	C₂²×C₆₇	1	2	yes	—
437	#399	C₂²×C₆₇	1	2	yes	—
438	#398	C₂²×C₆₇	1	2	yes	—
439	#398	C₂²×C₆₇	1	4	yes	—
440	#397	C₂²×C₆₇	1	2	yes	—
441	#396	C₂²×C₆₇	1	4	yes	—
442	#396	C₂²×C₆₇	1	4	yes	—
443	#395	C₂²×C₆₇	1	2	yes	—
444	#394	C₂²×C₆₇	1	2	yes	—
445	#394	C₂²×C₆₇	1	4	yes	—
446	#393	C₂²×C₆₇	1	2	yes	—
447	#392	C₂²×C₆₇	1	2	yes	—
448	#391	C₂²×C₆₇	1	2	yes	—
449	#390	C₂²×C₆₇	1	2	yes	—
450	#390	C₂²×C₆₇	1	4	yes	—
451	#389	C₂²×C₆₇	1	4	yes	—
452	#389	C₂²×C₆₇	1	2	yes	—
453	#389	C₂²×C₆₇	1	2	yes	—
454	#388	C₂²×C₆₇	1	4	yes	—
455	#388	C₂²×C₆₇	1	4	yes	—
456	#388	C₂²×C₆₇	1	4	yes	—
457	#387	C₂²×C₆₇	1	4	yes	—
458	#386	C₂²×C₆₇	1	4	yes	—
459	#386	C₂²×C₆₇	1	4	yes	—
460	#386	C₂²×C₆₇	1	4	yes	—
461	#385	C₂²×C₆₇	1	4	yes	—
462	#380	C₂²×C₆₇	1	2	yes	—
463	#380	C₂²×C₆₇	1	2	yes	—
464	#380	C₂²×C₆₇	1	4	yes	—
465	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
466	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
467	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
468	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
469	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
470	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
471	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
472	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
473	C₂×#95	C₂²×C₆₇	1	2	yes	—
474	C₂×#41	—	2	8	no	—
475	C₂×#41	—	2	4	no	—
476	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
477	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
478	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
479	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
480	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
481	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
482	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
483	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
484	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
485	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
486	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
487	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
488	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
489	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
490	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
491	#126	—	2	8	no	—
492	#125	—	2	4	no	—
493	#124	—	2	4	no	—
494	#123	—	2	8	no	—
495	C₂²⋊Q₈	—	2	4	no	—
496	C₂²⋊Q₈	—	2	8	no	—
497	#121	—	2	16	no	—
498	#121	—	2	4	no	—
499	#121	—	2	8	no	—
500	C₄×Q₈	—	2	4	no	—
501	C₄×Q₈	—	2	4	no	—
502	C₄×Q₈	—	2	4	no	—
503	C₄×D₄	—	2	8	no	—
504	C₄×D₄	—	2	8	no	—
505	C₄×D₄	—	2	8	no	—
506	C₄×D₄	—	2	4	no	—
507	C₄×D₄	—	2	4	no	—
508	C₄×D₄	—	2	4	no	—
509	C₄×D₄	—	4	8	no	—
510	#117	—	2	4	no	—
511	#117	—	2	8	no	—
512	#117	—	2	8	no	—
513	C₂×C₄⋊C₄	—	2	8	no	—
514	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
515	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
516	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
517	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
518	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
519	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
520	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
521	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
522	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
523	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
524	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
525	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
526	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
527	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
528	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
529	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
530	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
531	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
532	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
533	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
534	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
535	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
536	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
537	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
538	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
539	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
540	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
541	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
542	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
543	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
544	C₂×C₄²	—	2	8	no	—
545	C₂×C₄²	—	4	4	no	—
546	C₂×C₄²	D₄×C₆₇	1	8	yes	complemented
547	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
548	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
549	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
550	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
551	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
552	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
553	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
554	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
555	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
556	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
557	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
558	#95	C₂³×C₆₇	1	4	yes	—
559	#95	C₂³×C₆₇	1	2	yes	—
560	#95	C₂³×C₆₇	1	2	yes	—
561	#95	C₂³×C₆₇	1	4	yes	—
562	#95	C₂³×C₆₇	1	4	yes	—
563	#95	C₂³×C₆₇	1	4	yes	—
564	#95	C₂³×C₆₇	1	4	yes	—
565	#95	C₂³×C₆₇	1	4	yes	—
566	#95	C₂³×C₆₇	1	4	yes	—
567	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
568	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
569	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
570	#41	—	2	32	no	—
571	C₂×Q₈	—	2	2	no	—
572	C₂×Q₈	—	2	2	no	—
573	C₂×Q₈	—	2	4	no	—
574	C₂×Q₈	—	2	4	no	—
575	C₂×D₄	—	2	4	no	—
576	C₂×D₄	—	2	4	no	—
577	C₂×D₄	—	2	4	no	—
578	C₂×D₄	—	2	4	no	—
579	C₂×D₄	—	2	8	no	—
580	C₂×D₄	—	2	8	no	—
581	C₂×D₄	—	2	2	no	—
582	C₂×D₄	—	2	2	no	—
583	C₂×D₄	—	2	8	no	—
584	C₂×D₄	—	2	4	no	—
585	C₂×D₄	—	4	4	no	—
586	C₂×D₄	#41×C₆₇	1	8	yes	—
587	C₂²×C₄	—	2	4	no	—
588	C₂²×C₄	—	2	4	no	—
589	C₂²×C₄	—	2	4	no	—
590	C₂²×C₄	—	2	4	no	—
591	C₂²×C₄	—	2	4	no	—
592	C₂²×C₄	—	2	2	no	—
593	C₂²×C₄	—	2	2	no	—
594	C₂²×C₄	—	2	8	no	—
595	C₂²×C₄	—	2	4	no	—
596	C₂²×C₄	—	2	4	no	—
597	C₂²×C₄	—	2	4	no	—
598	C₂²×C₄	—	2	8	no	—
599	C₂²×C₄	—	2	4	no	—
600	C₂²×C₄	—	2	4	no	—
601	C₂²×C₄	—	2	4	no	—
602	C₂²×C₄	—	2	4	no	—
603	C₂²×C₄	—	2	4	no	—
604	C₂²×C₄	—	2	4	no	—
605	C₂²×C₄	—	4	4	no	—
606	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
607	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
608	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
609	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
610	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
611	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
612	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
613	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
614	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
615	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
616	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
617	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
618	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
619	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
620	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
621	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
622	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
623	C₄⋊C₄	—	2	4	no	—
624	C₄⋊C₄	—	2	4	no	—
625	C₄⋊C₄	—	2	2	no	—
626	C₄⋊C₄	—	2	2	no	—
627	C₄⋊C₄	—	2	4	no	—
628	C₄⋊C₄	—	2	4	no	—
629	C₄⋊C₄	—	2	2	no	—
630	C₄⋊C₄	—	2	4	no	—
631	C₄⋊C₄	—	2	2	no	—
632	C₄⋊C₄	—	2	4	no	—
633	C₄⋊C₄	—	2	8	no	—
634	C₄⋊C₄	—	2	4	no	—
635	C₄⋊C₄	—	2	8	no	—
636	C₄⋊C₄	—	2	8	no	—
637	C₄⋊C₄	—	4	4	no	—
638	C₄⋊C₄	#41×C₆₇	1	8	yes	complemented
639	C₂²⋊C₄	—	2	4	no	—
640	C₂²⋊C₄	—	2	4	no	—
641	C₂²⋊C₄	—	2	8	no	—
642	C₂²⋊C₄	—	2	4	no	—
643	C₂²⋊C₄	—	2	4	no	—
644	C₂²⋊C₄	—	2	4	no	—
645	C₂²⋊C₄	—	2	4	no	—
646	C₂²⋊C₄	—	2	4	no	—
647	C₂²⋊C₄	—	2	4	no	—
648	C₂²⋊C₄	—	2	8	no	—
649	C₂²⋊C₄	—	2	8	no	—
650	C₂²⋊C₄	—	2	8	no	—
651	C₂²⋊C₄	—	4	4	no	—
652	C₂²⋊C₄	—	4	4	no	—
653	C₄²	—	2	4	no	—
654	C₄²	—	2	4	no	—
655	C₄²	—	2	4	no	—
656	C₄²	—	2	4	no	—
657	C₄²	—	2	8	no	—
658	C₄²	—	4	2	no	—
659	C₄²	—	4	2	no	—
660	C₄²	C₂×D₄×C₆₇	1	4	yes	—
661	C₄²	C₂×D₄×C₆₇	1	4	yes	—
662	C₄²	C₂×D₄×C₆₇	1	4	yes	—
663	C₄²	C₂×D₄×C₆₇	1	4	yes	—
664	C₂³	—	2	2	no	—
665	C₂³	—	2	1	no	—
666	C₂³	—	2	1	no	—
667	C₂³	—	2	2	no	—
668	C₂³	—	2	2	no	—
669	C₂³	—	2	2	no	—
670	C₂³	—	2	2	no	—
671	C₂³	—	2	2	no	—
672	C₂³	—	2	2	no	—
673	C₂³	—	2	2	no	—
674	C₂³	—	2	4	no	—
675	C₂³	—	2	4	no	—
676	C₂³	—	2	2	no	—
677	C₂³	—	2	2	no	—
678	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
679	C₂³	C₂×#41×C₆₇	1	4	yes	—
680	C₂³	C₂×#41×C₆₇	1	4	yes	—
681	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
682	Q₈	—	2	8	no	—
683	Q₈	—	4	4	no	—
684	D₄	—	2	8	no	—
685	D₄	—	2	16	no	—
686	D₄	—	4	16	no	—
687	D₄	—	4	8	no	—
688	D₄	—	4	4	no	—
689	C₂×C₄	—	2	16	no	—
690	C₂×C₄	—	2	8	no	—
691	C₂×C₄	—	2	2	no	—
692	C₂×C₄	—	2	2	no	—
693	C₂×C₄	—	2	4	no	—
694	C₂×C₄	—	2	2	no	—
695	C₂×C₄	—	2	2	no	—
696	C₂×C₄	—	2	2	no	—
697	C₂×C₄	—	2	4	no	—
698	C₂×C₄	—	2	1	no	—
699	C₂×C₄	—	2	1	no	—
700	C₂×C₄	—	2	8	no	—
701	C₂×C₄	—	2	8	no	—
702	C₂×C₄	—	2	2	no	—
703	C₂×C₄	—	2	2	no	—
704	C₂×C₄	—	2	2	no	—
705	C₂×C₄	—	2	2	no	—
706	C₂×C₄	—	2	2	no	—
707	C₂×C₄	—	2	2	no	—
708	C₂×C₄	—	2	4	no	—
709	C₂×C₄	—	2	4	no	—
710	C₂×C₄	—	2	4	no	—
711	C₂×C₄	—	2	2	no	—
712	C₂×C₄	—	2	4	no	—
713	C₂×C₄	—	2	4	no	—
714	C₂×C₄	—	2	2	no	—
715	C₂×C₄	—	2	2	no	—
716	C₂×C₄	—	2	2	no	—
717	C₂×C₄	—	2	2	no	—
718	C₂×C₄	—	4	16	no	—
719	C₂×C₄	—	4	8	no	—
720	C₂×C₄	—	4	8	no	—
721	C₂×C₄	—	4	8	no	—
722	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
723	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
724	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
725	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
726	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
727	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
728	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
729	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
730	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
731	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
732	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
733	C₂²	—	2	2	no	—
734	C₂²	—	2	2	no	—
735	C₂²	—	2	4	no	—
736	C₂²	—	2	4	no	—
737	C₂²	—	2	4	no	—
738	C₂²	—	2	4	no	—
739	C₂²	—	2	4	no	—
740	C₂²	—	4	2	no	—
741	C₂²	—	4	2	no	—
742	C₂²	—	4	2	no	—
743	C₂²	—	4	2	no	—
744	C₂²	—	4	2	no	—
745	C₂²	—	4	2	no	—
746	C₂²	—	4	2	no	—
747	C₂²	—	4	4	no	—
748	C₂²	—	4	4	no	—
749	C₂²	#525×C₆₇	1	4	yes	—
750	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
751	C₂²	C₂²×#41×C₆₇	1	2	yes	—
752	C₂²	C₂²×#41×C₆₇	1	2	yes	—
753	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
754	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
755	C₄	—	2	2	no	—
756	C₄	—	2	2	no	—
757	C₄	—	2	4	no	—
758	C₄	—	2	2	no	—
759	C₄	—	2	2	no	—
760	C₄	—	2	8	no	—
761	C₄	—	4	2	no	—
762	C₄	—	4	2	no	—
763	C₄	—	4	2	no	—
764	C₄	—	4	2	no	—
765	C₄	—	4	2	no	—
766	C₄	—	4	4	no	—
767	C₄	#532×C₆₇	1	4	yes	—
768	C₄	#537×C₆₇	1	4	yes	—
769	C₂	—	2	4	no	—
770	C₂	—	4	4	no	—
771	C₂	—	8	2	no	—
772	C₂	C₂×#525×C₆₇	1	1	yes	characteristic
773	C₂	C₂×#532×C₆₇	1	1	yes	characteristic
774	C₂	C₂×#537×C₆₇	1	1	yes	characteristic
775	C₂	C₆₇×#3481	1	2	yes	—
776	C₂	C₆₇×#3482	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
socle	C₂³×C₆₇	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#529×C₆₇	C₂	semidirect
2. C₂×C₄×#41×C₆₇	C₂	semidirect
3. C₆₇×#2601	C₂	semidirect
4. C₆₇×#2592	C₂	semidirect
5. C₆₇×#2584	C₂	semidirect
6. C₆₇×#2562	C₂	semidirect
7. C₆₇×#2540	C₂	semidirect
8. C₆₇×#2498	C₂	semidirect
9. C₆₇×#2430	C₂	semidirect
10. C₆₇×#2417	C₂	semidirect
11. C₆₇×#2416	C₂	semidirect
12. C₆₇×#2396	C₂	semidirect
13. C₆₇×#2392	C₂	semidirect
14. C₄×C₂²⋊Q₈×C₆₇	C₂	semidirect
15. C₄×#121×C₆₇	C₂	semidirect
16. C₄×#121×C₆₇	C₂	semidirect
17. C₂×C₄×C₄⋊C₄×C₆₇	C₂	semidirect
157. C₂×C₄²×C₆₇	D₄	semidirect
249. C₄⋊C₄×C₆₇	#41	semidirect
380. #34964	C₆₇	direct (N × H)
389. C₂×#529	C₂×C₆₇	semidirect
390. C₂×C₄×#41	C₂×C₆₇	semidirect
391. #2601	C₂×C₆₇	semidirect
392. #2592	C₂×C₆₇	semidirect
393. #2584	C₂×C₆₇	semidirect
394. #2562	C₂×C₆₇	semidirect
395. #2540	C₂×C₆₇	semidirect
396. #2498	C₂×C₆₇	semidirect
397. #2430	C₂×C₆₇	semidirect
398. #2417	C₂×C₆₇	semidirect
399. #2416	C₂×C₆₇	semidirect
400. #2396	C₂×C₆₇	semidirect
401. #2392	C₂×C₆₇	semidirect
402. C₄×C₂²⋊Q₈	C₂×C₆₇	semidirect
403. C₄×#121	C₂×C₆₇	semidirect
404. C₄×#121	C₂×C₆₇	semidirect
405. C₂×C₄×C₄⋊C₄	C₂×C₆₇	semidirect
406. C₆₇	#34964	direct (N × H)
546. C₂×C₄²	D₄×C₆₇	semidirect
638. C₄⋊C₄	#41×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#529×C₆₇	2
C₂×C₄×#41×C₆₇	2
C₆₇×#2601	2
C₆₇×#2592	2
C₆₇×#2584	2
C₆₇×#2562	2
C₆₇×#2540	2
C₆₇×#2498	2
C₆₇×#2430	2
C₆₇×#2417	2
C₆₇×#2416	2
C₆₇×#2396	2
C₆₇×#2392	2
C₄×C₂²⋊Q₈×C₆₇	2
C₄×#121×C₆₇	2
C₂×C₄×C₄⋊C₄×C₆₇	2
#34964	67

Subgroups — C67×#34964 #30714215

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34964 #30714215