Subgroups — #30864166

Order 17152.

674 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560928	C₂	1	1	yes	maximal, characteristic
2	#22560928	C₂	1	1	yes	maximal, characteristic
3	#22560775	C₂	1	2	yes	maximal
4	#22560775	C₂	1	2	yes	maximal
5	#22560326	C₂	1	1	yes	maximal, characteristic
6	#22560084	C₂	1	2	yes	maximal
7	#22560078	C₂	1	2	yes	maximal
8	#22560078	C₂	1	1	yes	maximal, characteristic
9	C₄⋊C₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
10	C₄×#14523954	C₂	1	1	yes	maximal, characteristic
11	C₄×#125×C₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×#14523954	C₂²	1	1	yes	characteristic
17	C₂×#14523954	C₂²	1	1	yes	characteristic
18	C₂×#14523954	C₂²	1	2	yes	—
19	C₂×#14523954	C₂²	1	1	yes	characteristic
20	C₂×#14523954	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
22	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
24	#16863151	C₂²	1	2	yes	—
25	#16863151	C₂²	1	2	yes	—
26	#16863151	C₂²	1	2	yes	—
27	#16863151	C₂²	1	2	yes	—
28	#16863151	C₂²	1	2	yes	—
29	#16863151	C₂²	1	2	yes	—
30	C₂×#125×C₆₇	C₄	1	1	yes	characteristic
31	#389×C₆₇	C₄	1	1	yes	characteristic
32	#389×C₆₇	C₄	1	1	yes	characteristic
33	#387×C₆₇	C₄	1	2	yes	—
34	#387×C₆₇	C₄	1	2	yes	—
35	#382×C₆₇	C₄	1	1	yes	characteristic
36	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
37	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
38	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
39	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
40	C₄³×C₆₇	C₂²	1	1	yes	characteristic
41	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
42	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
43	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
44	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
45	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
46	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
47	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
48	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
49	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
50	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
51	#14523954	—	2	2	no	—
52	#14523954	—	2	2	no	—
53	#14523954	—	2	2	no	—
54	#14523954	—	2	2	no	—
55	#14523954	—	2	4	no	—
56	C₄×C₆₇⋊C₈	—	2	4	no	—
57	C₄×C₆₇⋊C₈	—	2	2	no	—
58	C₄×C₆₇⋊C₈	Q₈	1	4	yes	—
59	#125×C₆₇	—	2	4	no	—
60	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
61	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
62	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
66	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
67	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
73	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
76	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
77	#95×C₆₇	C₂×C₄	1	2	yes	—
78	#95×C₆₇	C₂×C₄	1	2	yes	—
79	#95×C₆₇	C₂×C₄	1	2	yes	—
80	#95×C₆₇	C₂×C₄	1	2	yes	—
81	#95×C₆₇	C₂×C₄	1	2	yes	—
82	#95×C₆₇	C₂×C₄	1	2	yes	—
83	C₂×C₆₇⋊C₈	—	2	1	no	—
84	C₂×C₆₇⋊C₈	—	2	1	no	—
85	C₂×C₆₇⋊C₈	—	2	1	no	—
86	C₂×C₆₇⋊C₈	—	2	1	no	—
87	C₂×C₆₇⋊C₈	—	2	1	no	—
88	C₂×C₆₇⋊C₈	—	2	1	no	—
89	C₂×C₆₇⋊C₈	—	2	2	no	—
90	C₂×C₆₇⋊C₈	—	2	2	no	—
91	C₂×C₆₇⋊C₈	—	2	2	no	—
92	C₂×C₆₇⋊C₈	—	2	2	no	—
93	C₂×C₆₇⋊C₈	—	2	2	no	—
94	C₂×C₆₇⋊C₈	—	2	2	no	—
95	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
96	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
97	C₂×C₆₇⋊C₈	#41	1	2	yes	—
98	C₂×C₆₇⋊C₈	#41	1	2	yes	—
99	C₂×C₆₇⋊C₈	#41	1	2	yes	—
100	C₂×C₆₇⋊C₈	#41	1	2	yes	—
101	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
106	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
107	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
108	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
109	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
110	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
112	C₄⋊C₄×C₆₇	—	2	4	no	—
113	C₄⋊C₄×C₆₇	—	2	4	no	—
114	C₄⋊C₄×C₆₇	—	2	2	no	—
115	C₄⋊C₄×C₆₇	—	2	2	no	—
116	C₄⋊C₄×C₆₇	—	2	2	no	—
117	C₄⋊C₄×C₆₇	—	4	2	no	—
118	C₄⋊C₄×C₆₇	—	4	2	no	—
119	C₄⋊C₄×C₆₇	M₁₆	1	4	yes	—
120	C₄²×C₆₇	—	2	2	no	—
121	C₄²×C₆₇	—	2	2	no	—
122	C₄²×C₆₇	—	2	2	no	—
123	C₄²×C₆₇	—	2	1	no	—
124	C₄²×C₆₇	—	2	1	no	—
125	C₄²×C₆₇	—	2	2	no	—
126	C₄²×C₆₇	—	2	1	no	—
127	C₄²×C₆₇	—	2	1	no	—
128	C₄²×C₆₇	—	2	2	no	—
129	C₄²×C₆₇	—	2	2	no	—
130	C₄²×C₆₇	—	4	1	no	—
131	C₄²×C₆₇	—	4	1	no	—
132	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
133	C₄²×C₆₇	#41	1	2	yes	—
134	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₆₇	—	2	2	no	—
136	C₂×C₄×C₆₇	—	2	1	no	—
137	C₂×C₄×C₆₇	—	2	1	no	—
138	C₂×C₄×C₆₇	—	2	2	no	—
139	C₂×C₄×C₆₇	—	2	2	no	—
140	C₂×C₄×C₆₇	—	2	1	no	—
141	C₂×C₄×C₆₇	—	2	1	no	—
142	C₂×C₄×C₆₇	—	2	2	no	—
143	C₂×C₄×C₆₇	—	2	1	no	—
144	C₂×C₄×C₆₇	—	2	1	no	—
145	C₂×C₄×C₆₇	—	2	2	no	—
146	C₂×C₄×C₆₇	—	2	1	no	—
147	C₂×C₄×C₆₇	—	2	1	no	—
148	C₂×C₄×C₆₇	—	2	2	no	—
149	C₂×C₄×C₆₇	—	2	1	no	—
150	C₂×C₄×C₆₇	—	2	1	no	—
151	C₂×C₄×C₆₇	—	2	2	no	—
152	C₂×C₄×C₆₇	—	2	2	no	—
153	C₂×C₄×C₆₇	—	2	1	no	—
154	C₂×C₄×C₆₇	—	2	1	no	—
155	C₂×C₄×C₆₇	—	2	1	no	—
156	C₂×C₄×C₆₇	—	2	1	no	—
157	C₂×C₄×C₆₇	—	2	2	no	—
158	C₂×C₄×C₆₇	—	2	2	no	—
159	C₂×C₄×C₆₇	—	2	2	no	—
160	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
161	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
162	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
163	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
164	C₂×C₄×C₆₇	#131	1	2	yes	—
165	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
166	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
167	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
168	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
169	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
170	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
171	C₆₇⋊C₈	—	2	2	no	—
172	C₆₇⋊C₈	—	2	2	no	—
173	C₆₇⋊C₈	—	2	2	no	—
174	C₆₇⋊C₈	—	4	2	no	—
175	C₆₇⋊C₈	—	4	2	no	—
176	C₆₇⋊C₈	#125	1	4	yes	complemented
177	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
178	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
179	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
180	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
181	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
182	C₂²×C₆₇	#568	1	1	yes	characteristic
183	C₂²×C₆₇	#568	1	1	yes	characteristic
184	C₄×C₆₇	—	2	2	no	—
185	C₄×C₆₇	—	2	4	no	—
186	C₄×C₆₇	—	4	1	no	—
187	C₄×C₆₇	—	4	1	no	—
188	C₄×C₆₇	—	4	2	no	—
189	C₄×C₆₇	—	4	2	no	—
190	C₄×C₆₇	—	4	1	no	—
191	C₄×C₆₇	—	4	1	no	—
192	C₄×C₆₇	—	4	2	no	—
193	C₄×C₆₇	#446	1	4	yes	—
194	C₄×C₆₇	C₂×#125	1	1	yes	characteristic
195	C₄×C₆₇	#529	1	1	yes	characteristic
196	C₄×C₆₇	#529	1	1	yes	characteristic
197	C₄×C₆₇	#533	1	1	yes	characteristic
198	#21675	—	67	1	no	maximal, Sylow
199	C₂×C₆₇	C₄×#125	1	1	yes	characteristic
200	C₂×C₆₇	C₂×#446	1	1	yes	characteristic
201	C₂×C₆₇	#2961	1	1	yes	characteristic
202	C₂×C₆₇	#2969	1	1	yes	characteristic
203	C₂×C₆₇	#2969	1	1	yes	characteristic
204	C₂×C₆₇	#2970	1	1	yes	characteristic
205	C₂×C₆₇	#2971	1	1	yes	characteristic
206	C₄×#125	D₆₇	1	1	yes	characteristic
207	#1939	—	67	1	no	—
208	#1939	—	67	1	no	—
209	#1917	—	67	2	no	—
210	#1917	—	67	2	no	—
211	#1831	—	67	1	no	—
212	#1771	—	67	2	no	—
213	#1770	—	67	2	no	—
214	#1770	—	67	1	no	—
215	C₈×C₄⋊C₄	—	67	1	no	—
216	C₄×#97	—	67	1	no	—
217	C₆₇	#21675	1	1	yes	characteristic, complemented, Sylow
218	C₂×#125	Dic₆₇	1	1	yes	characteristic
219	C₂×C₄⋊C₈	—	67	1	no	—
220	C₂×C₄⋊C₈	—	67	1	no	—
221	C₂×C₄⋊C₈	—	67	2	no	—
222	C₂×C₄⋊C₈	—	67	2	no	—
223	C₂×#97	—	67	1	no	—
224	C₂×#97	—	67	1	no	—
225	C₂×#97	—	67	1	no	—
226	C₂×#97	—	67	1	no	—
227	C₂×#97	—	67	2	no	—
228	C₂×C₄×C₈	—	67	1	no	—
229	C₂×C₄×C₈	—	67	1	no	—
230	C₂×C₄×C₈	—	67	2	no	—
231	#389	Dic₆₇	1	1	yes	characteristic
232	#389	Dic₆₇	1	1	yes	characteristic
233	#387	Dic₆₇	1	2	yes	—
234	#387	Dic₆₇	1	2	yes	—
235	#382	Dic₆₇	1	1	yes	characteristic
236	C₄×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
237	C₄×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
238	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
239	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
240	C₄³	C₂×D₆₇	1	1	yes	characteristic
241	#336	—	67	2	no	—
242	#336	—	67	2	no	—
243	#336	—	67	2	no	—
244	#336	—	67	2	no	—
245	#336	—	67	2	no	—
246	#336	—	67	2	no	—
247	C₂²×C₈	—	67	1	no	—
248	C₂²×C₈	—	67	1	no	—
249	C₂²×C₈	—	67	1	no	—
250	C₂²×C₈	—	67	1	no	—
251	C₂²×C₈	—	67	2	no	—
252	C₂²×C₈	—	67	2	no	—
253	#125	—	2	4	no	—
254	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
255	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
256	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
257	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
258	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
259	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
260	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
261	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
262	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
263	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
264	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
265	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
266	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
267	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
268	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
269	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
270	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
271	C₄⋊C₈	—	134	4	no	—
272	C₄⋊C₈	—	134	4	no	—
273	C₄⋊C₈	—	134	2	no	—
274	C₄⋊C₈	—	134	2	no	—
275	#97	—	134	2	no	—
276	#97	—	134	4	no	—
277	#97	—	134	2	no	—
278	#97	—	134	2	no	—
279	#97	—	134	2	no	—
280	C₄×C₈	—	67	4	no	—
281	C₄×C₈	—	134	4	no	—
282	C₄×C₈	—	134	2	no	—
283	#95	C₂×Dic₆₇	1	2	yes	—
284	#95	C₂×Dic₆₇	1	2	yes	—
285	#95	C₂×Dic₆₇	1	2	yes	—
286	#95	C₂×Dic₆₇	1	2	yes	—
287	#95	C₂×Dic₆₇	1	2	yes	—
288	#95	C₂×Dic₆₇	1	2	yes	—
289	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
290	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
291	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
292	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
293	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
294	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
295	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
296	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
297	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
298	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
299	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
300	C₂×C₈	—	67	2	no	—
301	C₂×C₈	—	67	2	no	—
302	C₂×C₈	—	67	2	no	—
303	C₂×C₈	—	67	2	no	—
304	C₂×C₈	—	67	2	no	—
305	C₂×C₈	—	67	2	no	—
306	C₂×C₈	—	134	2	no	—
307	C₂×C₈	—	134	2	no	—
308	C₂×C₈	—	134	2	no	—
309	C₂×C₈	—	134	2	no	—
310	C₂×C₈	—	134	2	no	—
311	C₂×C₈	—	134	2	no	—
312	C₂×C₈	—	134	1	no	—
313	C₂×C₈	—	134	1	no	—
314	C₂×C₈	—	134	1	no	—
315	C₂×C₈	—	134	1	no	—
316	C₂×C₈	—	134	1	no	—
317	C₂×C₈	—	134	1	no	—
318	C₄⋊C₄	—	2	4	no	—
319	C₄⋊C₄	—	2	4	no	—
320	C₄⋊C₄	—	2	2	no	—
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	2	2	no	—
323	C₄⋊C₄	—	4	2	no	—
324	C₄⋊C₄	—	4	2	no	—
325	C₄⋊C₄	#11761612	1	4	yes	—
326	C₄²	—	2	2	no	—
327	C₄²	—	2	2	no	—
328	C₄²	—	2	2	no	—
329	C₄²	—	2	1	no	—
330	C₄²	—	2	1	no	—
331	C₄²	—	2	2	no	—
332	C₄²	—	2	1	no	—
333	C₄²	—	2	1	no	—
334	C₄²	—	2	2	no	—
335	C₄²	—	2	2	no	—
336	C₄²	—	4	1	no	—
337	C₄²	—	4	1	no	—
338	C₄²	Q₈×D₆₇	1	2	yes	—
339	C₄²	#11761629	1	2	yes	—
340	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	2	1	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
367	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
368	C₂×C₄	C₂×#11761612	1	4	yes	—
369	C₂×C₄	C₂×#11761612	1	2	yes	—
370	C₂×C₄	#14524050	1	2	yes	—
371	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
372	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
373	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
374	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
375	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
376	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
377	C₈	—	268	2	no	—
378	C₈	—	268	2	no	—
379	C₈	—	67	4	no	—
380	C₈	—	134	2	no	—
381	C₈	—	134	2	no	—
382	C₈	—	134	2	no	—
383	C₂²	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
384	C₂²	#41×Dic₆₇	1	1	yes	characteristic
385	C₂²	#41×Dic₆₇	1	1	yes	characteristic
386	C₂²	C₂²×#11761612	1	1	yes	characteristic
387	C₂²	C₂×#14524050	1	1	yes	characteristic
388	C₂²	#16864163	1	1	yes	characteristic
389	C₂²	#16864163	1	1	yes	characteristic
390	C₄	—	2	4	no	—
391	C₄	—	2	2	no	—
392	C₄	—	4	1	no	—
393	C₄	—	4	1	no	—
394	C₄	—	4	2	no	—
395	C₄	—	4	1	no	—
396	C₄	—	4	1	no	—
397	C₄	—	4	2	no	—
398	C₄	—	4	2	no	—
399	C₄	#16863527	1	4	yes	—
400	C₄	#125×D₆₇	1	1	yes	characteristic
401	C₄	#16863931	1	1	yes	characteristic
402	C₄	#16863931	1	1	yes	characteristic
403	C₄	#16863952	1	1	yes	characteristic
404	C₂	#125×Dic₆₇	1	1	yes	characteristic
405	C₂	C₂×#16863527	1	1	yes	characteristic
406	C₂	#22569150	1	1	yes	characteristic
407	C₂	#22569214	1	1	yes	characteristic
408	C₂	#22569214	1	1	yes	characteristic
409	C₂	#22569223	1	1	yes	characteristic
410	C₂	#22569229	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator, socle, prime radical	C₂³×C₆₇	C₂³×C₄
Fitting	C₄×#125×C₆₇	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄×#125	D₆₇
hypocenter	C₆₇	#21675
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#125

Complements

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

N	H ≅ G/N	type
176. C₆₇⋊C₈	#125	semidirect
217. C₆₇	#21675	semidirect

Maximal subgroups

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

subgroup	index
#22560928	2
#22560775	2
#22560326	2
#22560084	2
#22560078	2
C₄⋊C₄×C₆₇⋊C₈	2
C₄×#14523954	2
C₄×#125×C₆₇	2
#21675	67