Subgroups — C₆₇×#34602 #30713853

Order 17152.

2294 conjugacy classes of subgroups.

The subgroup lattice has 556 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
2	C₂²×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2637	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2634	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2632	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2627	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2623	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2618	C₂	1	2	yes	maximal
9	C₆₇×#2614	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2399	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2398	C₂	1	2	yes	maximal, complemented
12	C₆₇×#2396	C₂	1	4	yes	maximal, complemented
13	C₄×#123×C₆₇	C₂	1	4	yes	maximal, complemented
14	C₆₇×#2283	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
16	#529×C₆₇	—	2	8	no	—
17	C₂×#126×C₆₇	C₂²	1	2	yes	—
18	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#123×C₆₇	C₂²	1	4	yes	—
21	C₂×#123×C₆₇	C₂²	1	2	yes	—
22	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₆₇	—	2	8	no	—
25	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
27	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
28	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
29	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
31	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×#117×C₆₇	C₂²	1	2	yes	—
33	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
34	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
35	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
36	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
37	#400×C₆₇	C₂²	1	4	yes	—
38	#400×C₆₇	C₂²	1	4	yes	—
39	#399×C₆₇	C₂²	1	4	yes	—
40	#398×C₆₇	C₂²	1	4	yes	—
41	#398×C₆₇	C₂²	1	4	yes	—
42	#398×C₆₇	C₂²	1	8	yes	—
43	#397×C₆₇	C₂²	1	4	yes	—
44	#396×C₆₇	C₂²	1	8	yes	—
45	#395×C₆₇	C₂²	1	4	yes	—
46	#395×C₆₇	C₂²	1	4	yes	—
47	#395×C₆₇	C₂²	1	4	yes	—
48	#394×C₆₇	C₂²	1	4	yes	—
49	#393×C₆₇	C₂²	1	4	yes	—
50	#393×C₆₇	C₂²	1	4	yes	—
51	#388×C₆₇	C₂²	1	8	yes	—
52	#387×C₆₇	C₂²	1	8	yes	—
53	#386×C₆₇	C₂²	1	4	yes	—
54	#385×C₆₇	C₂²	1	4	yes	—
55	#383×C₆₇	C₂²	1	2	yes	—
56	#382×C₆₇	C₂²	1	2	yes	—
57	#381×C₆₇	C₂²	1	2	yes	—
58	#381×C₆₇	C₂²	1	4	yes	—
59	#380×C₆₇	C₂²	1	2	yes	—
60	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
61	C₄×C₄⋊C₄×C₆₇	C₂²	1	8	yes	—
62	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
63	C₄×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	—
64	#376×C₆₇	C₂²	1	2	yes	—
65	C₂²×Q₈×C₆₇	—	2	4	no	—
66	C₂²×Q₈×C₆₇	—	2	8	no	—
67	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
68	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
69	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
70	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
71	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
73	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
76	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
77	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
78	#126×C₆₇	—	2	8	no	—
79	#125×C₆₇	—	2	4	no	—
80	#124×C₆₇	—	2	4	no	—
81	#123×C₆₇	—	2	8	no	—
82	#123×C₆₇	—	2	16	no	—
83	C₂²⋊Q₈×C₆₇	—	2	4	no	—
84	#121×C₆₇	—	2	4	no	—
85	C₄×Q₈×C₆₇	—	2	8	no	—
86	C₄×Q₈×C₆₇	—	2	8	no	—
87	C₄×Q₈×C₆₇	—	2	8	no	—
88	C₄×Q₈×C₆₇	—	2	8	no	—
89	C₄×Q₈×C₆₇	—	2	4	no	—
90	C₄×D₄×C₆₇	—	2	8	no	—
91	C₄×D₄×C₆₇	—	2	4	no	—
92	#117×C₆₇	—	2	8	no	—
93	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
94	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
95	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
96	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
97	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
98	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
99	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
100	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
101	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
102	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
104	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
105	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
106	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
107	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
108	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
109	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
110	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
111	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
112	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
113	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
114	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
115	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
116	C₂×C₄²×C₆₇	—	2	8	no	—
117	C₂×C₄²×C₆₇	D₄	1	8	yes	—
118	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
119	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
120	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
121	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
122	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
123	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
124	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
125	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
126	#95×C₆₇	C₂³	1	4	yes	—
127	#95×C₆₇	C₂³	1	4	yes	—
128	#95×C₆₇	C₂³	1	8	yes	—
129	#95×C₆₇	C₂³	1	4	yes	—
130	#95×C₆₇	C₂³	1	8	yes	—
131	#95×C₆₇	C₂³	1	4	yes	—
132	#95×C₆₇	C₂³	1	8	yes	—
133	#95×C₆₇	C₂³	1	8	yes	—
134	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
135	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
136	C₂×Q₈×C₆₇	—	2	32	no	—
137	C₂×Q₈×C₆₇	—	2	4	no	—
138	C₂×Q₈×C₆₇	—	2	4	no	—
139	C₂×Q₈×C₆₇	—	2	2	no	—
140	C₂×Q₈×C₆₇	—	2	8	no	—
141	C₂×Q₈×C₆₇	—	2	8	no	—
142	C₂×Q₈×C₆₇	—	4	2	no	—
143	C₂×D₄×C₆₇	—	2	4	no	—
144	C₂×D₄×C₆₇	—	2	8	no	—
145	C₂×D₄×C₆₇	—	2	2	no	—
146	C₂×D₄×C₆₇	—	4	2	no	—
147	C₂²×C₄×C₆₇	—	2	4	no	—
148	C₂²×C₄×C₆₇	—	2	4	no	—
149	C₂²×C₄×C₆₇	—	2	4	no	—
150	C₂²×C₄×C₆₇	—	2	4	no	—
151	C₂²×C₄×C₆₇	—	2	4	no	—
152	C₂²×C₄×C₆₇	—	2	4	no	—
153	C₂²×C₄×C₆₇	—	2	8	no	—
154	C₂²×C₄×C₆₇	—	4	2	no	—
155	C₂²×C₄×C₆₇	—	4	4	no	—
156	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
157	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
158	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
159	C₂²×C₄×C₆₇	#41	1	8	yes	—
160	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
161	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
162	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
163	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
164	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
165	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
166	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
167	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
168	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
169	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
170	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
171	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
172	C₄⋊C₄×C₆₇	—	2	8	no	—
173	C₄⋊C₄×C₆₇	—	2	8	no	—
174	C₄⋊C₄×C₆₇	—	2	4	no	—
175	C₄⋊C₄×C₆₇	—	2	4	no	—
176	C₄⋊C₄×C₆₇	—	2	4	no	—
177	C₄⋊C₄×C₆₇	—	2	4	no	—
178	C₄⋊C₄×C₆₇	—	2	2	no	—
179	C₄⋊C₄×C₆₇	—	2	4	no	—
180	C₄⋊C₄×C₆₇	—	2	2	no	—
181	C₄⋊C₄×C₆₇	—	2	4	no	—
182	C₄⋊C₄×C₆₇	—	2	16	no	—
183	C₄⋊C₄×C₆₇	—	4	4	no	—
184	C₄⋊C₄×C₆₇	—	4	8	no	—
185	C₂²⋊C₄×C₆₇	—	2	4	no	—
186	C₂²⋊C₄×C₆₇	—	2	8	no	—
187	C₂²⋊C₄×C₆₇	—	2	4	no	—
188	C₂²⋊C₄×C₆₇	—	2	4	no	—
189	C₂²⋊C₄×C₆₇	—	2	4	no	—
190	C₂²⋊C₄×C₆₇	—	2	4	no	—
191	C₂²⋊C₄×C₆₇	—	2	16	no	—
192	C₂²⋊C₄×C₆₇	—	4	8	no	—
193	C₄²×C₆₇	—	2	8	no	—
194	C₄²×C₆₇	—	2	8	no	—
195	C₄²×C₆₇	—	2	4	no	—
196	C₄²×C₆₇	—	2	8	no	—
197	C₄²×C₆₇	C₂×D₄	1	4	yes	—
198	C₄²×C₆₇	C₂×D₄	1	4	yes	—
199	C₄²×C₆₇	C₂×D₄	1	4	yes	—
200	C₄²×C₆₇	C₂×D₄	1	4	yes	—
201	C₂³×C₆₇	—	2	2	no	—
202	C₂³×C₆₇	—	2	2	no	—
203	C₂³×C₆₇	—	2	2	no	—
204	C₂³×C₆₇	—	2	4	no	—
205	C₂³×C₆₇	—	2	4	no	—
206	C₂³×C₆₇	—	2	4	no	—
207	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
208	C₂³×C₆₇	C₂×#41	1	4	yes	—
209	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
210	Q₈×C₆₇	—	2	16	no	—
211	Q₈×C₆₇	—	2	16	no	—
212	Q₈×C₆₇	—	4	4	no	—
213	D₄×C₆₇	—	4	4	no	—
214	D₄×C₆₇	—	4	8	no	—
215	C₂×C₄×C₆₇	—	2	8	no	—
216	C₂×C₄×C₆₇	—	2	16	no	—
217	C₂×C₄×C₆₇	—	2	4	no	—
218	C₂×C₄×C₆₇	—	2	2	no	—
219	C₂×C₄×C₆₇	—	2	2	no	—
220	C₂×C₄×C₆₇	—	2	4	no	—
221	C₂×C₄×C₆₇	—	2	2	no	—
222	C₂×C₄×C₆₇	—	2	2	no	—
223	C₂×C₄×C₆₇	—	2	2	no	—
224	C₂×C₄×C₆₇	—	2	8	no	—
225	C₂×C₄×C₆₇	—	2	4	no	—
226	C₂×C₄×C₆₇	—	2	2	no	—
227	C₂×C₄×C₆₇	—	2	2	no	—
228	C₂×C₄×C₆₇	—	2	2	no	—
229	C₂×C₄×C₆₇	—	2	8	no	—
230	C₂×C₄×C₆₇	—	2	4	no	—
231	C₂×C₄×C₆₇	—	2	8	no	—
232	C₂×C₄×C₆₇	—	2	4	no	—
233	C₂×C₄×C₆₇	—	2	8	no	—
234	C₂×C₄×C₆₇	—	2	8	no	—
235	C₂×C₄×C₆₇	—	2	4	no	—
236	C₂×C₄×C₆₇	—	4	4	no	—
237	C₂×C₄×C₆₇	—	4	8	no	—
238	C₂×C₄×C₆₇	—	4	8	no	—
239	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
240	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
241	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
242	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
243	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
244	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
245	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
246	C₂×C₄×C₆₇	2₋¹⁺⁴	1	4	yes	—
247	C₂²×C₆₇	—	2	2	no	—
248	C₂²×C₆₇	—	2	2	no	—
249	C₂²×C₆₇	—	2	8	no	—
250	C₂²×C₆₇	—	4	2	no	—
251	C₂²×C₆₇	—	4	2	no	—
252	C₂²×C₆₇	—	4	2	no	—
253	C₂²×C₆₇	—	4	8	no	—
254	C₂²×C₆₇	#525	1	4	yes	—
255	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
256	C₂²×C₆₇	C₂²×#41	1	2	yes	—
257	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
258	C₂²×C₆₇	#585	1	2	yes	—
259	C₄×C₆₇	—	2	2	no	—
260	C₄×C₆₇	—	2	2	no	—
261	C₄×C₆₇	—	2	8	no	—
262	C₄×C₆₇	—	2	2	no	—
263	C₄×C₆₇	—	2	2	no	—
264	C₄×C₆₇	—	2	2	no	—
265	C₄×C₆₇	—	4	2	no	—
266	C₄×C₆₇	—	4	2	no	—
267	C₄×C₆₇	—	4	4	no	—
268	C₄×C₆₇	—	4	2	no	—
269	C₄×C₆₇	—	4	8	no	—
270	C₄×C₆₇	#536	1	4	yes	—
271	#34602	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
272	C₂×C₆₇	—	2	4	no	—
273	C₂×C₆₇	—	8	2	no	—
274	C₂×C₆₇	C₂×#525	1	1	yes	characteristic
275	C₂×C₆₇	C₂×#536	1	1	yes	characteristic
276	C₂×C₆₇	#3449	1	1	yes	characteristic
277	C₂×C₆₇	#3461	1	4	yes	—
278	C₂×#529	C₂×C₆₇	1	1	yes	characteristic
279	C₂²×C₄×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
280	#2637	C₂×C₆₇	1	4	yes	complemented
281	#2634	C₂×C₆₇	1	2	yes	complemented
282	#2632	C₂×C₆₇	1	2	yes	complemented
283	#2627	C₂×C₆₇	1	2	yes	complemented
284	#2623	C₂×C₆₇	1	2	yes	complemented
285	#2618	C₂×C₆₇	1	2	yes	—
286	#2614	C₂×C₆₇	1	2	yes	complemented
287	#2399	C₂×C₆₇	1	2	yes	complemented
288	#2398	C₂×C₆₇	1	2	yes	complemented
289	#2396	C₂×C₆₇	1	4	yes	complemented
290	C₄×#123	C₂×C₆₇	1	4	yes	complemented
291	#2283	C₂×C₆₇	1	1	yes	characteristic, complemented
292	C₆₇	#34602	1	1	yes	characteristic, direct product factor, Sylow
293	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
294	#529	—	2	8	no	—
295	C₂×#126	C₂²×C₆₇	1	2	yes	—
296	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
297	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
298	C₂×#123	C₂²×C₆₇	1	4	yes	—
299	C₂×#123	C₂²×C₆₇	1	2	yes	—
300	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
301	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
302	C₂×C₄×Q₈	—	2	8	no	—
303	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
304	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
305	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
306	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
307	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
308	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
309	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
310	C₂×#117	C₂²×C₆₇	1	2	yes	—
311	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
312	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
313	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
314	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
315	#400	C₂²×C₆₇	1	4	yes	—
316	#400	C₂²×C₆₇	1	4	yes	—
317	#399	C₂²×C₆₇	1	4	yes	—
318	#398	C₂²×C₆₇	1	4	yes	—
319	#398	C₂²×C₆₇	1	4	yes	—
320	#398	C₂²×C₆₇	1	8	yes	—
321	#397	C₂²×C₆₇	1	4	yes	—
322	#396	C₂²×C₆₇	1	8	yes	—
323	#395	C₂²×C₆₇	1	4	yes	—
324	#395	C₂²×C₆₇	1	4	yes	—
325	#395	C₂²×C₆₇	1	4	yes	—
326	#394	C₂²×C₆₇	1	4	yes	—
327	#393	C₂²×C₆₇	1	4	yes	—
328	#393	C₂²×C₆₇	1	4	yes	—
329	#388	C₂²×C₆₇	1	8	yes	—
330	#387	C₂²×C₆₇	1	8	yes	—
331	#386	C₂²×C₆₇	1	4	yes	—
332	#385	C₂²×C₆₇	1	4	yes	—
333	#383	C₂²×C₆₇	1	2	yes	—
334	#382	C₂²×C₆₇	1	2	yes	—
335	#381	C₂²×C₆₇	1	2	yes	—
336	#381	C₂²×C₆₇	1	4	yes	—
337	#380	C₂²×C₆₇	1	2	yes	—
338	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
339	C₄×C₄⋊C₄	C₂²×C₆₇	1	8	yes	—
340	C₄×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	—
341	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
342	#376	C₂²×C₆₇	1	2	yes	—
343	C₂²×Q₈	—	2	4	no	—
344	C₂²×Q₈	—	2	8	no	—
345	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
346	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
347	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
348	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
349	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
350	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
351	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
352	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
353	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
354	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
355	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
356	#126	—	2	8	no	—
357	#125	—	2	4	no	—
358	#124	—	2	4	no	—
359	#123	—	2	8	no	—
360	#123	—	2	16	no	—
361	C₂²⋊Q₈	—	2	4	no	—
362	#121	—	2	4	no	—
363	C₄×Q₈	—	2	8	no	—
364	C₄×Q₈	—	2	8	no	—
365	C₄×Q₈	—	2	8	no	—
366	C₄×Q₈	—	2	8	no	—
367	C₄×Q₈	—	2	4	no	—
368	C₄×D₄	—	2	8	no	—
369	C₄×D₄	—	2	4	no	—
370	#117	—	2	8	no	—
371	C₂×C₄⋊C₄	—	2	8	no	—
372	C₂×C₄⋊C₄	—	2	4	no	—
373	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
374	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
375	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
376	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
377	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
378	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
379	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
380	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
381	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
382	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
383	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
384	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
385	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
386	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
387	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
388	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
389	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
390	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
391	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
392	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
393	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
394	C₂×C₄²	—	2	8	no	—
395	C₂×C₄²	D₄×C₆₇	1	8	yes	—
396	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
397	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
398	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
399	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
400	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
401	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
402	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
403	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
404	#95	C₂³×C₆₇	1	4	yes	—
405	#95	C₂³×C₆₇	1	4	yes	—
406	#95	C₂³×C₆₇	1	8	yes	—
407	#95	C₂³×C₆₇	1	4	yes	—
408	#95	C₂³×C₆₇	1	8	yes	—
409	#95	C₂³×C₆₇	1	4	yes	—
410	#95	C₂³×C₆₇	1	8	yes	—
411	#95	C₂³×C₆₇	1	8	yes	—
412	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
413	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
414	C₂×Q₈	—	2	32	no	—
415	C₂×Q₈	—	2	4	no	—
416	C₂×Q₈	—	2	4	no	—
417	C₂×Q₈	—	2	2	no	—
418	C₂×Q₈	—	2	8	no	—
419	C₂×Q₈	—	2	8	no	—
420	C₂×Q₈	—	4	2	no	—
421	C₂×D₄	—	2	4	no	—
422	C₂×D₄	—	2	8	no	—
423	C₂×D₄	—	2	2	no	—
424	C₂×D₄	—	4	2	no	—
425	C₂²×C₄	—	2	4	no	—
426	C₂²×C₄	—	2	4	no	—
427	C₂²×C₄	—	2	4	no	—
428	C₂²×C₄	—	2	4	no	—
429	C₂²×C₄	—	2	4	no	—
430	C₂²×C₄	—	2	4	no	—
431	C₂²×C₄	—	2	8	no	—
432	C₂²×C₄	—	4	2	no	—
433	C₂²×C₄	—	4	4	no	—
434	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
435	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
436	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
437	C₂²×C₄	#41×C₆₇	1	8	yes	—
438	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
439	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
440	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
441	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
442	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
443	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
444	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
445	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
446	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
447	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
448	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
449	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
450	C₄⋊C₄	—	2	8	no	—
451	C₄⋊C₄	—	2	8	no	—
452	C₄⋊C₄	—	2	4	no	—
453	C₄⋊C₄	—	2	4	no	—
454	C₄⋊C₄	—	2	4	no	—
455	C₄⋊C₄	—	2	4	no	—
456	C₄⋊C₄	—	2	2	no	—
457	C₄⋊C₄	—	2	4	no	—
458	C₄⋊C₄	—	2	2	no	—
459	C₄⋊C₄	—	2	4	no	—
460	C₄⋊C₄	—	2	16	no	—
461	C₄⋊C₄	—	4	4	no	—
462	C₄⋊C₄	—	4	8	no	—
463	C₂²⋊C₄	—	2	4	no	—
464	C₂²⋊C₄	—	2	8	no	—
465	C₂²⋊C₄	—	2	4	no	—
466	C₂²⋊C₄	—	2	4	no	—
467	C₂²⋊C₄	—	2	4	no	—
468	C₂²⋊C₄	—	2	4	no	—
469	C₂²⋊C₄	—	2	16	no	—
470	C₂²⋊C₄	—	4	8	no	—
471	C₄²	—	2	8	no	—
472	C₄²	—	2	8	no	—
473	C₄²	—	2	4	no	—
474	C₄²	—	2	8	no	—
475	C₄²	C₂×D₄×C₆₇	1	4	yes	—
476	C₄²	C₂×D₄×C₆₇	1	4	yes	—
477	C₄²	C₂×D₄×C₆₇	1	4	yes	—
478	C₄²	C₂×D₄×C₆₇	1	4	yes	—
479	C₂³	—	2	2	no	—
480	C₂³	—	2	2	no	—
481	C₂³	—	2	2	no	—
482	C₂³	—	2	4	no	—
483	C₂³	—	2	4	no	—
484	C₂³	—	2	4	no	—
485	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
486	C₂³	C₂×#41×C₆₇	1	4	yes	—
487	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
488	Q₈	—	2	16	no	—
489	Q₈	—	2	16	no	—
490	Q₈	—	4	4	no	—
491	D₄	—	4	4	no	—
492	D₄	—	4	8	no	—
493	C₂×C₄	—	2	8	no	—
494	C₂×C₄	—	2	16	no	—
495	C₂×C₄	—	2	4	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	4	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	2	no	—
502	C₂×C₄	—	2	8	no	—
503	C₂×C₄	—	2	4	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	2	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	8	no	—
508	C₂×C₄	—	2	4	no	—
509	C₂×C₄	—	2	8	no	—
510	C₂×C₄	—	2	4	no	—
511	C₂×C₄	—	2	8	no	—
512	C₂×C₄	—	2	8	no	—
513	C₂×C₄	—	2	4	no	—
514	C₂×C₄	—	4	4	no	—
515	C₂×C₄	—	4	8	no	—
516	C₂×C₄	—	4	8	no	—
517	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
518	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
519	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
520	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
521	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
522	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
523	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
524	C₂×C₄	2₋¹⁺⁴×C₆₇	1	4	yes	—
525	C₂²	—	2	2	no	—
526	C₂²	—	2	2	no	—
527	C₂²	—	2	8	no	—
528	C₂²	—	4	2	no	—
529	C₂²	—	4	2	no	—
530	C₂²	—	4	2	no	—
531	C₂²	—	4	8	no	—
532	C₂²	#525×C₆₇	1	4	yes	—
533	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
534	C₂²	C₂²×#41×C₆₇	1	2	yes	—
535	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	2	yes	—
536	C₂²	#585×C₆₇	1	2	yes	—
537	C₄	—	2	2	no	—
538	C₄	—	2	2	no	—
539	C₄	—	2	8	no	—
540	C₄	—	2	2	no	—
541	C₄	—	2	2	no	—
542	C₄	—	2	2	no	—
543	C₄	—	4	2	no	—
544	C₄	—	4	2	no	—
545	C₄	—	4	4	no	—
546	C₄	—	4	2	no	—
547	C₄	—	4	8	no	—
548	C₄	#536×C₆₇	1	4	yes	—
549	C₂	—	2	4	no	—
550	C₂	—	8	2	no	—
551	C₂	C₂×#525×C₆₇	1	1	yes	characteristic
552	C₂	C₂×#536×C₆₇	1	1	yes	characteristic
553	C₂	C₆₇×#3449	1	1	yes	characteristic
554	C₂	C₆₇×#3461	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
socle	C₂³×C₆₇	C₂⁵
prime radical	C₂×#121×C₆₇	C₂²
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
2. C₂²×C₄×Q₈×C₆₇	C₂	semidirect
3. C₆₇×#2637	C₂	semidirect
4. C₆₇×#2634	C₂	semidirect
5. C₆₇×#2632	C₂	semidirect
6. C₆₇×#2627	C₂	semidirect
7. C₆₇×#2623	C₂	semidirect
9. C₆₇×#2614	C₂	semidirect
10. C₆₇×#2399	C₂	semidirect
11. C₆₇×#2398	C₂	semidirect
12. C₆₇×#2396	C₂	semidirect
13. C₄×#123×C₆₇	C₂	semidirect
14. C₆₇×#2283	C₂	semidirect
271. #34602	C₆₇	direct (N × H)
279. C₂²×C₄×Q₈	C₂×C₆₇	semidirect
280. #2637	C₂×C₆₇	semidirect
281. #2634	C₂×C₆₇	semidirect
282. #2632	C₂×C₆₇	semidirect
283. #2627	C₂×C₆₇	semidirect
284. #2623	C₂×C₆₇	semidirect
286. #2614	C₂×C₆₇	semidirect
287. #2399	C₂×C₆₇	semidirect
288. #2398	C₂×C₆₇	semidirect
289. #2396	C₂×C₆₇	semidirect
290. C₄×#123	C₂×C₆₇	semidirect
291. #2283	C₂×C₆₇	semidirect
292. C₆₇	#34602	direct (N × H)

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
C₂²×C₄×Q₈×C₆₇	2
C₆₇×#2637	2
C₆₇×#2634	2
C₆₇×#2632	2
C₆₇×#2627	2
C₆₇×#2623	2
C₆₇×#2618	2
C₆₇×#2614	2
C₆₇×#2399	2
C₆₇×#2398	2
C₆₇×#2396	2
C₄×#123×C₆₇	2
C₆₇×#2283	2
#34602	67

Subgroups — C67×#34602 #30713853

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34602 #30713853