Subgroups — #31008943

Order 17152.

590 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568631	C₂	1	1	yes	maximal, characteristic
2	#22562016	C₂	1	1	yes	maximal, characteristic, complemented
3	#22562015	C₂	1	1	yes	maximal, characteristic, complemented
4	#22562015	C₂	1	1	yes	maximal, characteristic, complemented
5	#22562010	C₂	1	1	yes	maximal, characteristic, complemented
6	#22562009	C₂	1	1	yes	maximal, characteristic, complemented
7	#22562006	C₂	1	1	yes	maximal, characteristic, complemented
8	#22562005	C₂	1	1	yes	maximal, characteristic, complemented
9	#22561963	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄⋊C₁₆×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#22561821	C₂	1	1	yes	maximal, characteristic
12	#22561821	C₂	1	1	yes	maximal, characteristic
13	#22561807	C₂	1	1	yes	maximal, characteristic, complemented
14	#22561804	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₇×#2164	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14524051	C₄	1	1	yes	characteristic
17	C₂×#14524049	C₂²	1	1	yes	characteristic
18	#16863864	C₄	1	1	yes	characteristic
19	C₂×#14523982	C₂²	1	1	yes	characteristic
20	C₂×#14523982	C₂²	1	1	yes	characteristic
21	C₂×C₁₆×D₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₁₆×D₆₇	C₂²	1	1	yes	characteristic
23	#16863353	C₄	1	1	yes	characteristic
24	#16863353	C₄	1	1	yes	characteristic
25	#16863352	C₄	1	1	yes	characteristic
26	#16863350	C₂²	1	1	yes	characteristic
27	#16863350	C₂²	1	1	yes	characteristic
28	#16863348	C₄	1	1	yes	characteristic
29	#16863348	C₄	1	1	yes	characteristic
30	M₁₆×Dic₆₇	C₄	1	1	yes	characteristic
31	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
32	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
33	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
34	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	1	yes	characteristic
35	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	1	yes	characteristic
36	Dic₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
37	Dic₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
38	Dic₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
39	Dic₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
40	C₂²⋊(C₆₇⋊C₁₆)	C₂²	1	1	yes	characteristic
41	C₂²⋊(C₆₇⋊C₁₆)	C₂²	1	1	yes	characteristic
42	(C₂×D₆₇)⋊C₁₆	C₂²	1	1	yes	characteristic
43	(C₂×D₆₇)⋊C₁₆	C₂²	1	1	yes	characteristic
44	(C₂×D₆₇)⋊C₁₆	C₂²	1	1	yes	characteristic
45	(C₂×D₆₇)⋊C₁₆	C₂²	1	1	yes	characteristic
46	#16863172	C₂²	1	1	yes	characteristic
47	#16863171	C₂²	1	1	yes	characteristic
48	#16863171	C₂²	1	1	yes	characteristic
49	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
50	C₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
51	C₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
52	#405×C₆₇	C₂²	1	1	yes	characteristic
53	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
54	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
55	C₂²⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
56	C₂²⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
57	#346×C₆₇	C₂²	1	1	yes	characteristic
58	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
59	C₂×#11761628	C₂×C₄	1	1	yes	characteristic
60	#14524051	—	2	4	no	—
61	#14524049	—	2	2	no	—
62	#14524049	—	2	2	no	—
63	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
64	C₂×#11761611	C₂×C₄	1	1	yes	characteristic
65	C₂×#11761611	C₂×C₄	1	1	yes	characteristic
66	C₂×#11761611	C₂×C₄	1	1	yes	characteristic
67	C₂×#11761611	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
69	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
70	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
71	C₂×C₈×D₆₇	C₂³	1	1	yes	characteristic
72	C₂×C₈×D₆₇	C₂³	1	1	yes	characteristic
73	#14524008	C₈	1	1	yes	characteristic
74	#14524008	C₈	1	1	yes	characteristic
75	#14524005	C₈	1	1	yes	characteristic
76	#14524005	C₈	1	1	yes	characteristic
77	(C₂²⋊D₆₇)⋊C₄	C₈	1	1	yes	characteristic
78	(C₂²⋊D₆₇)⋊C₄	C₈	1	1	yes	characteristic
79	(C₂²⋊D₆₇)⋊C₄	C₈	1	1	yes	characteristic
80	(C₂²⋊D₆₇)⋊C₄	C₈	1	1	yes	characteristic
81	#14524000	C₂×C₄	1	1	yes	characteristic
82	#14524000	C₂×C₄	1	1	yes	characteristic
83	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
84	C₄²×D₆₇	C₂×C₄	1	1	yes	characteristic
85	C₄²×D₆₇	C₂×C₄	1	1	yes	characteristic
86	#14523982	—	2	2	no	—
87	#14523982	—	2	2	no	—
88	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
89	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
90	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
91	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
92	C₁₆×D₆₇	—	2	2	no	—
93	C₁₆×D₆₇	—	2	2	no	—
94	#14523954	C₂×C₄	1	1	yes	characteristic
95	#14523954	C₂×C₄	1	1	yes	characteristic
96	#14523953	C₂×C₄	1	1	yes	characteristic
97	#14523953	C₂×C₄	1	1	yes	characteristic
98	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
99	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
100	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
101	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
102	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
103	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
104	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
105	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
106	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
107	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
108	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
109	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
110	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
111	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
112	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
113	#97×C₆₇	C₂×C₄	1	1	yes	characteristic
114	#97×C₆₇	C₂×C₄	1	1	yes	characteristic
115	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
116	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
117	#11761628	—	2	2	no	—
118	#11761628	—	2	2	no	—
119	C₂×Dic₁₃₄	C₂×C₈	1	1	yes	characteristic
120	C₂×D₂₆₈	C₂×C₈	1	1	yes	characteristic
121	C₂×C₂²⋊D₆₇	C₂×C₈	1	1	yes	characteristic
122	C₂×C₂²⋊D₆₇	C₂×C₈	1	1	yes	characteristic
123	C₂²×Dic₆₇	C₂×C₈	1	1	yes	characteristic
124	C₂²×Dic₆₇	C₂×C₈	1	1	yes	characteristic
125	C₂×C₄×D₆₇	C₂×C₈	1	1	yes	characteristic
126	C₂×C₄×D₆₇	C₂×C₈	1	1	yes	characteristic
127	C₂×C₄×D₆₇	C₂×C₈	1	1	yes	characteristic
128	C₂×C₄×D₆₇	C₂×C₈	1	1	yes	characteristic
129	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
131	#11761612	—	2	2	no	—
132	#11761611	—	2	2	no	—
133	#11761611	—	2	2	no	—
134	#11761611	—	2	2	no	—
135	#11761611	M₁₆	1	4	yes	—
136	C₂×C₆₇⋊C₈	—	2	2	no	—
137	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
138	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
139	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
140	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
141	C₈×D₆₇	—	2	2	no	—
142	C₈×D₆₇	—	2	2	no	—
143	C₈×D₆₇	—	2	1	no	—
144	C₈×D₆₇	—	2	1	no	—
145	C₈×D₆₇	#41	1	2	yes	—
146	C₈×D₆₇	#41	1	2	yes	—
147	Dic₆₇⋊C₄	C₂×C₈	1	1	yes	characteristic
148	Dic₆₇⋊C₄	C₂×C₈	1	1	yes	characteristic
149	Dic₆₇⋊C₄	C₂×C₈	1	1	yes	characteristic
150	Dic₆₇⋊C₄	C₂×C₈	1	1	yes	characteristic
151	(C₂×D₆₇)⋊C₄	C₂×C₈	1	1	yes	characteristic
152	(C₂×D₆₇)⋊C₄	C₂×C₈	1	1	yes	characteristic
153	(C₂×D₆₇)⋊C₄	C₂×C₈	1	1	yes	characteristic
154	(C₂×D₆₇)⋊C₄	C₂×C₈	1	1	yes	characteristic
155	C₄×Dic₆₇	C₂×C₈	1	1	yes	characteristic
156	C₄×Dic₆₇	C₂×C₈	1	1	yes	characteristic
157	C₄×Dic₆₇	C₂×C₈	1	1	yes	characteristic
158	C₄×Dic₆₇	C₂×C₈	1	1	yes	characteristic
159	C₄×Dic₆₇	C₂×C₈	1	1	yes	characteristic
160	C₄×Dic₆₇	C₂×C₈	1	1	yes	characteristic
161	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
162	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
163	C₆₇⋊C₁₆	—	2	1	no	—
164	C₆₇⋊C₁₆	—	2	1	no	—
165	C₆₇⋊C₁₆	—	2	1	no	—
166	C₆₇⋊C₁₆	—	2	1	no	—
167	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
168	M₁₆×C₆₇	—	2	2	no	—
169	C₂×C₈×C₆₇	—	2	1	no	—
170	C₂×C₈×C₆₇	—	2	1	no	—
171	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
172	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
173	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
174	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
175	C₄⋊C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
176	C₄⋊C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
177	C₂²⋊C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
178	C₂²⋊C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
179	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
180	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
181	C₁₆×C₆₇	—	2	1	no	—
182	C₁₆×C₆₇	—	2	1	no	—
183	C₁₆×C₆₇	—	2	1	no	—
184	C₁₆×C₆₇	—	2	1	no	—
185	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
186	C₂²×D₆₇	C₂²×C₈	1	1	yes	characteristic
187	C₂²×D₆₇	C₂²×C₈	1	1	yes	characteristic
188	C₂×C₄×C₆₇	—	2	1	no	—
189	C₂×C₄×C₆₇	—	2	1	no	—
190	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
191	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
192	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
193	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
194	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
195	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
196	C₂²⋊D₆₇	—	2	2	no	—
197	C₂²⋊D₆₇	—	2	2	no	—
198	C₂×Dic₆₇	—	2	2	no	—
199	C₂×Dic₆₇	—	4	1	no	—
200	C₂×Dic₆₇	C₂²×C₈	1	1	yes	characteristic
201	C₂×Dic₆₇	C₂²×C₈	1	1	yes	characteristic
202	C₂×Dic₆₇	C₂²×C₈	1	1	yes	characteristic
203	C₂×Dic₆₇	C₂²×C₈	1	1	yes	characteristic
204	C₂×Dic₆₇	C₂²×C₈	1	1	yes	characteristic
205	C₂×Dic₆₇	C₂²×C₈	1	1	yes	characteristic
206	D₂₆₈	—	2	2	no	—
207	C₄×D₆₇	—	2	1	no	—
208	C₄×D₆₇	—	2	1	no	—
209	C₄×D₆₇	—	2	2	no	—
210	C₄×D₆₇	—	2	2	no	—
211	C₄×D₆₇	—	4	1	no	—
212	C₄×D₆₇	—	4	1	no	—
213	C₄×D₆₇	#117	1	2	yes	—
214	C₄×D₆₇	C₂×M₁₆	1	2	yes	—
215	Dic₁₃₄	—	2	2	no	—
216	C₈×C₆₇	—	2	1	no	—
217	C₈×C₆₇	—	2	1	no	—
218	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
219	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
220	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
221	C₆₇⋊C₈	—	2	1	no	—
222	C₆₇⋊C₈	—	2	1	no	—
223	C₆₇⋊C₈	#117	1	2	yes	—
224	C₆₇⋊C₈	C₂×M₁₆	1	2	yes	—
225	C₂²×C₆₇	—	2	1	no	—
226	C₂²×C₆₇	—	2	1	no	—
227	C₂²×C₆₇	—	2	1	no	—
228	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
229	C₂×D₆₇	—	2	1	no	—
230	C₂×D₆₇	—	2	1	no	—
231	C₂×D₆₇	—	2	1	no	—
232	C₂×D₆₇	—	2	1	no	—
233	C₂×D₆₇	—	2	1	no	—
234	C₂×D₆₇	#431	1	2	yes	—
235	C₄×C₆₇	—	2	1	no	—
236	C₄×C₆₇	—	2	1	no	—
237	C₄×C₆₇	—	2	1	no	—
238	C₄×C₆₇	—	2	1	no	—
239	C₄×C₆₇	—	2	1	no	—
240	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
241	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
242	Dic₆₇	—	2	1	no	—
243	Dic₆₇	—	2	1	no	—
244	Dic₆₇	—	2	1	no	—
245	Dic₆₇	—	2	1	no	—
246	Dic₆₇	#431	1	2	yes	—
247	Dic₆₇	#504	1	2	yes	—
248	#33423	—	67	1	no	maximal, Sylow
249	C₂×C₆₇	—	4	1	no	—
250	C₂×C₆₇	C₂×#431	1	1	yes	characteristic
251	C₂×C₆₇	C₂×#504	1	1	yes	characteristic
252	C₂×C₆₇	#3407	1	1	yes	characteristic
253	D₆₇	—	2	2	no	—
254	D₆₇	—	4	1	no	—
255	C₄×#131	—	67	1	no	—
256	#2164	—	67	1	no	—
257	#2164	—	67	1	no	—
258	#2164	—	67	1	no	—
259	#2164	D₆₇	1	1	yes	characteristic, complemented
260	#2163	—	67	1	no	—
261	#2163	—	67	1	no	—
262	#2162	—	67	1	no	—
263	#2162	—	67	1	no	—
264	#2150	—	67	1	no	—
265	C₂×C₄⋊C₁₆	—	67	1	no	—
266	#2113	—	67	1	no	—
267	#2113	—	67	1	no	—
268	#2108	—	67	1	no	—
269	#2108	—	67	1	no	—
270	C₆₇	#33423	1	1	yes	characteristic, complemented, Sylow
271	C₂×#131	—	67	1	no	—
272	C₂×#131	—	67	1	no	—
273	C₄×#41	—	67	1	no	—
274	C₂×M₃₂	—	67	1	no	—
275	C₂×M₃₂	—	67	1	no	—
276	C₂²×C₁₆	—	67	1	no	—
277	C₂²×C₁₆	—	67	1	no	—
278	#405	—	67	1	no	—
279	#405	—	67	1	no	—
280	#405	—	67	1	no	—
281	#405	—	67	1	no	—
282	#405	—	67	1	no	—
283	#405	C₂×D₆₇	1	1	yes	characteristic
284	C₄×M₁₆	—	67	1	no	—
285	C₄×M₁₆	—	67	1	no	—
286	C₄×M₁₆	—	67	1	no	—
287	C₂×C₄×C₈	—	67	1	no	—
288	C₂×C₄×C₈	—	67	1	no	—
289	C₂×C₄×C₈	—	67	1	no	—
290	C₄⋊C₁₆	—	67	1	no	—
291	C₄⋊C₁₆	—	67	1	no	—
292	C₄⋊C₁₆	—	67	1	no	—
293	C₄⋊C₁₆	—	67	1	no	—
294	C₄⋊C₁₆	—	67	1	no	—
295	C₄⋊C₁₆	—	67	1	no	—
296	C₄⋊C₁₆	C₂×D₆₇	1	1	yes	characteristic
297	C₄⋊C₁₆	C₂×D₆₇	1	1	yes	characteristic
298	C₂²⋊C₁₆	—	67	1	no	—
299	C₂²⋊C₁₆	—	67	1	no	—
300	C₂²⋊C₁₆	—	67	1	no	—
301	C₂²⋊C₁₆	—	67	1	no	—
302	C₂²⋊C₁₆	—	67	1	no	—
303	C₂²⋊C₁₆	—	67	1	no	—
304	C₂²⋊C₁₆	C₂×D₆₇	1	1	yes	characteristic
305	C₂²⋊C₁₆	C₂×D₆₇	1	1	yes	characteristic
306	#346	—	67	1	no	—
307	#346	—	67	1	no	—
308	#346	—	67	1	no	—
309	#346	C₂×D₆₇	1	1	yes	characteristic
310	C₄×C₁₆	—	67	1	no	—
311	C₄×C₁₆	—	67	1	no	—
312	C₄×C₁₆	—	67	1	no	—
313	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
314	C₂×#41	—	67	1	no	—
315	#131	—	134	2	no	—
316	#131	—	134	2	no	—
317	#131	—	134	4	no	—
318	C₂×M₁₆	—	67	1	no	—
319	C₂×M₁₆	—	67	1	no	—
320	C₂×M₁₆	—	67	1	no	—
321	C₂×M₁₆	—	67	1	no	—
322	C₂×M₁₆	—	67	1	no	—
323	C₂×M₁₆	C₄×D₆₇	1	1	yes	characteristic
324	C₂²×C₈	—	67	1	no	—
325	C₂²×C₈	—	67	1	no	—
326	C₂²×C₈	—	67	1	no	—
327	C₂²×C₈	—	67	1	no	—
328	C₂²×C₈	—	67	1	no	—
329	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
330	C₄×Q₈	—	67	1	no	—
331	C₄×Q₈	—	67	1	no	—
332	C₄×D₄	—	67	1	no	—
333	C₄×D₄	—	67	1	no	—
334	C₄×D₄	—	67	1	no	—
335	C₄×D₄	—	67	1	no	—
336	C₄×D₄	—	67	1	no	—
337	C₄×D₄	—	67	1	no	—
338	#117	—	67	1	no	—
339	#117	—	67	1	no	—
340	#117	C₄×D₆₇	1	1	yes	characteristic
341	C₂×C₄²	—	67	1	no	—
342	C₂×C₄²	—	67	1	no	—
343	C₂×C₄²	—	67	1	no	—
344	M₃₂	—	134	2	no	—
345	M₃₂	—	134	2	no	—
346	C₂×C₁₆	—	67	1	no	—
347	C₂×C₁₆	—	67	1	no	—
348	C₂×C₁₆	—	67	1	no	—
349	C₂×C₁₆	—	67	1	no	—
350	C₂×C₁₆	—	134	2	no	—
351	C₂×C₁₆	—	134	2	no	—
352	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
353	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
354	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
355	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
356	#97	—	67	1	no	—
357	#97	—	67	1	no	—
358	#97	—	67	1	no	—
359	#97	—	67	1	no	—
360	#97	C₄×D₆₇	1	1	yes	characteristic
361	#97	C₄×D₆₇	1	1	yes	characteristic
362	C₄×C₈	—	67	1	no	—
363	C₄×C₈	—	67	1	no	—
364	C₄×C₈	—	67	1	no	—
365	C₄×C₈	—	67	1	no	—
366	C₄×C₈	—	67	1	no	—
367	C₄×C₈	—	67	1	no	—
368	C₄×C₈	—	67	1	no	—
369	C₄×C₈	—	67	1	no	—
370	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
371	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
372	#41	—	134	2	no	—
373	#41	—	134	2	no	—
374	C₂×Q₈	—	67	1	no	—
375	C₂×D₄	—	67	1	no	—
376	C₂×D₄	—	67	1	no	—
377	C₂×D₄	—	67	1	no	—
378	C₂²×C₄	—	67	1	no	—
379	C₂²×C₄	—	67	1	no	—
380	C₂²×C₄	—	67	1	no	—
381	C₂²×C₄	—	67	1	no	—
382	C₂²×C₄	—	67	1	no	—
383	C₂²×C₄	—	67	1	no	—
384	C₂²×C₄	—	67	1	no	—
385	C₂²×C₄	—	67	1	no	—
386	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
387	M₁₆	—	2	2	no	—
388	M₁₆	—	67	4	no	—
389	M₁₆	—	134	2	no	—
390	M₁₆	—	134	2	no	—
391	M₁₆	—	134	2	no	—
392	M₁₆	—	134	2	no	—
393	C₂×C₈	—	2	1	no	—
394	C₂×C₈	—	2	1	no	—
395	C₂×C₈	—	67	1	no	—
396	C₂×C₈	—	67	1	no	—
397	C₂×C₈	—	67	2	no	—
398	C₂×C₈	—	67	2	no	—
399	C₂×C₈	—	67	1	no	—
400	C₂×C₈	—	67	1	no	—
401	C₂×C₈	—	134	2	no	—
402	C₂×C₈	—	134	2	no	—
403	C₂×C₈	—	134	2	no	—
404	C₂×C₈	—	134	1	no	—
405	C₂×C₈	—	134	1	no	—
406	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
407	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
408	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
409	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
410	C₄⋊C₄	—	67	1	no	—
411	C₄⋊C₄	—	67	1	no	—
412	C₄⋊C₄	—	67	1	no	—
413	C₄⋊C₄	—	67	1	no	—
414	C₄⋊C₄	C₈×D₆₇	1	1	yes	characteristic
415	C₄⋊C₄	C₈×D₆₇	1	1	yes	characteristic
416	C₂²⋊C₄	—	67	1	no	—
417	C₂²⋊C₄	—	67	1	no	—
418	C₂²⋊C₄	—	67	1	no	—
419	C₂²⋊C₄	—	67	1	no	—
420	C₂²⋊C₄	C₈×D₆₇	1	1	yes	characteristic
421	C₂²⋊C₄	C₈×D₆₇	1	1	yes	characteristic
422	C₄²	—	67	1	no	—
423	C₄²	—	67	1	no	—
424	C₄²	—	67	1	no	—
425	C₄²	—	67	1	no	—
426	C₄²	—	67	1	no	—
427	C₄²	—	67	1	no	—
428	C₄²	—	67	1	no	—
429	C₄²	—	67	1	no	—
430	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
431	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
432	C₁₆	—	2	1	no	—
433	C₁₆	—	2	1	no	—
434	C₁₆	—	2	1	no	—
435	C₁₆	—	2	1	no	—
436	C₁₆	—	134	1	no	—
437	C₁₆	—	134	1	no	—
438	C₁₆	—	134	1	no	—
439	C₁₆	—	134	1	no	—
440	C₂³	—	67	1	no	—
441	C₂³	—	67	1	no	—
442	C₂³	C₂×C₈×D₆₇	1	1	yes	characteristic
443	Q₈	—	134	2	no	—
444	D₄	—	134	2	no	—
445	D₄	—	134	2	no	—
446	D₄	—	134	2	no	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	268	1	no	—
450	C₂×C₄	—	268	1	no	—
451	C₂×C₄	—	268	1	no	—
452	C₂×C₄	—	67	1	no	—
453	C₂×C₄	—	67	1	no	—
454	C₂×C₄	—	67	2	no	—
455	C₂×C₄	—	67	2	no	—
456	C₂×C₄	—	67	1	no	—
457	C₂×C₄	—	67	1	no	—
458	C₂×C₄	—	67	1	no	—
459	C₂×C₄	—	67	1	no	—
460	C₂×C₄	—	134	2	no	—
461	C₂×C₄	—	134	2	no	—
462	C₂×C₄	—	134	2	no	—
463	C₂×C₄	—	134	1	no	—
464	C₂×C₄	—	134	1	no	—
465	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
466	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
467	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
468	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
469	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
470	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
471	C₈	—	2	1	no	—
472	C₈	—	2	1	no	—
473	C₈	—	67	2	no	—
474	C₈	—	67	2	no	—
475	C₈	—	134	1	no	—
476	C₈	—	134	1	no	—
477	C₈	M₁₆×D₆₇	1	2	yes	—
478	C₈	#41×D₆₇	1	1	yes	characteristic
479	C₈	#41×D₆₇	1	1	yes	characteristic
480	C₂²	—	2	1	no	—
481	C₂²	—	2	1	no	—
482	C₂²	—	2	1	no	—
483	C₂²	—	67	2	no	—
484	C₂²	—	134	1	no	—
485	C₂²	—	134	1	no	—
486	C₂²	—	134	1	no	—
487	C₂²	—	134	1	no	—
488	C₂²	—	134	1	no	—
489	C₂²	C₂²×C₈×D₆₇	1	1	yes	characteristic
490	C₄	—	2	1	no	—
491	C₄	—	2	1	no	—
492	C₄	—	2	1	no	—
493	C₄	—	2	1	no	—
494	C₄	—	2	1	no	—
495	C₄	—	67	2	no	—
496	C₄	—	67	2	no	—
497	C₄	—	134	1	no	—
498	C₄	—	134	1	no	—
499	C₄	—	134	1	no	—
500	C₄	—	134	1	no	—
501	C₄	#117×D₆₇	1	1	yes	characteristic
502	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
503	C₂	—	4	1	no	—
504	C₂	—	268	1	no	—
505	C₂	—	134	2	no	—
506	C₂	#431×D₆₇	1	1	yes	characteristic
507	C₂	#504×D₆₇	1	1	yes	characteristic
508	C₂	#22573310	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₈
Fitting	C₆₇×#2164	C₂
prime radical	C₂×#11761628	C₂×C₄
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	#2164	D₆₇
hypocenter	C₆₇	#33423
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ #2164

Complements

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

N	H ≅ G/N	type
2. #22562016	C₂	semidirect
3. #22562015	C₂	semidirect
4. #22562015	C₂	semidirect
5. #22562010	C₂	semidirect
6. #22562009	C₂	semidirect
7. #22562006	C₂	semidirect
8. #22562005	C₂	semidirect
9. #22561963	C₂	semidirect
10. C₄⋊C₁₆×D₆₇	C₂	semidirect
13. #22561807	C₂	semidirect
14. #22561804	C₂	semidirect
15. C₆₇×#2164	C₂	semidirect
259. #2164	D₆₇	semidirect
270. C₆₇	#33423	semidirect

Maximal subgroups

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

subgroup	index
#22568631	2
#22562016	2
#22562015	2
#22562010	2
#22562009	2
#22562006	2
#22562005	2
#22561963	2
C₄⋊C₁₆×D₆₇	2
#22561821	2
#22561807	2
#22561804	2
C₆₇×#2164	2
#33423	67