Subgroups — #30825133

Order 17152.

616 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569334	C₂	1	1	yes	maximal, characteristic
2	#41×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#22559000	C₂	1	1	yes	maximal, characteristic, complemented
4	#22558997	C₂	1	1	yes	maximal, characteristic, complemented
5	#22558995	C₂	1	2	yes	maximal, complemented
6	#22558992	C₂	1	2	yes	maximal, complemented
7	#22558970	C₂	1	2	yes	maximal, complemented
8	#22558967	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558955	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16863148	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2990	C₂	1	1	yes	maximal, characteristic
12	#16863527	—	2	1	no	—
13	Q₈×C₆₇⋊C₈	C₄	1	1	yes	characteristic
14	Q₈×C₆₇⋊C₈	C₄	1	1	yes	characteristic
15	#16863468	—	2	2	no	—
16	D₄×C₆₇⋊C₈	—	2	1	no	—
17	D₄×C₆₇⋊C₈	C₄	1	2	yes	—
18	D₄×C₆₇⋊C₈	C₄	1	1	yes	characteristic
19	D₄×C₆₇⋊C₈	C₄	1	2	yes	—
20	D₄×C₆₇⋊C₈	C₄	1	1	yes	characteristic
21	#16863453	C₂²	1	2	yes	—
22	#16863449	C₂²	1	1	yes	characteristic
23	#16863449	C₂²	1	2	yes	—
24	#16863449	C₂²	1	1	yes	characteristic
25	#16863420	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
27	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
28	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
29	#16863148	C₂²	1	2	yes	—
30	#16863148	C₂²	1	2	yes	—
31	#16863148	C₂²	1	2	yes	—
32	#16863148	C₂²	1	1	yes	characteristic
33	#16863148	C₂²	1	2	yes	—
34	#16863148	C₂²	1	1	yes	characteristic
35	#16863146	C₂²	1	1	yes	characteristic
36	#16863146	C₂²	1	2	yes	—
37	#16863146	C₂²	1	1	yes	characteristic
38	#16863146	C₂²	1	2	yes	—
39	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
40	#446×C₆₇	—	2	1	no	—
41	#435×C₆₇	—	2	2	no	—
42	C₈×D₄×C₆₇	—	2	1	no	—
43	#432×C₆₇	C₂²	1	2	yes	—
44	#431×C₆₇	C₂²	1	1	yes	characteristic
45	#423×C₆₇	C₂²	1	1	yes	characteristic
46	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
47	C₂×#11761612	—	2	1	no	—
48	C₂²×C₆₇⋊C₈	—	2	2	no	—
49	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
50	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
52	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
54	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
55	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
56	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
57	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
58	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
59	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
60	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
61	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
62	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
63	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
64	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
65	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
66	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
67	#14523954	—	2	2	no	—
68	C₄×C₆₇⋊C₈	—	2	1	no	—
69	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
70	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
71	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
72	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
73	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
74	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
75	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
76	C₂×#41×C₆₇	Q₈	1	1	yes	characteristic
77	C₂×M₁₆×C₆₇	—	2	1	no	—
78	C₂²×C₈×C₆₇	—	2	2	no	—
79	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
80	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
81	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
82	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
83	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
84	C₄×D₄×C₆₇	D₄	1	2	yes	—
85	#117×C₆₇	Q₈	1	1	yes	characteristic
86	#117×C₆₇	Q₈	1	2	yes	—
87	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
88	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
89	C₄⋊C₈×C₆₇	—	2	2	no	—
90	C₄⋊C₈×C₆₇	—	2	1	no	—
91	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
92	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
93	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
94	C₂²⋊C₈×C₆₇	—	2	1	no	—
95	C₂²⋊C₈×C₆₇	—	2	2	no	—
96	#97×C₆₇	—	2	2	no	—
97	C₄×C₈×C₆₇	—	2	1	no	—
98	#11761612	—	8	1	no	—
99	C₂×C₆₇⋊C₈	—	2	4	no	—
100	C₂×C₆₇⋊C₈	—	2	2	no	—
101	C₂×C₆₇⋊C₈	—	2	1	no	—
102	C₂×C₆₇⋊C₈	—	2	1	no	—
103	C₂×C₆₇⋊C₈	—	2	2	no	—
104	C₂×C₆₇⋊C₈	—	4	2	no	—
105	C₂×C₆₇⋊C₈	—	4	1	no	—
106	C₂×C₆₇⋊C₈	—	8	2	no	—
107	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
109	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
111	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
113	#41×C₆₇	—	2	4	no	—
114	C₂×Q₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
115	C₂×D₄×C₆₇	C₄⋊C₄	1	2	yes	—
116	C₂×D₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
117	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
119	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
120	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
122	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
123	M₁₆×C₆₇	—	8	1	no	—
124	C₂×C₈×C₆₇	—	2	1	no	—
125	C₂×C₈×C₆₇	—	2	2	no	—
126	C₂×C₈×C₆₇	—	2	1	no	—
127	C₂×C₈×C₆₇	—	8	2	no	—
128	C₄⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
129	C₄⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
130	C₄⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
131	C₄⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
132	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
133	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
134	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
135	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
136	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
137	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
138	C₄²×C₆₇	C₂×D₄	1	2	yes	—
139	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
140	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
141	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
142	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
144	Q₈×C₆₇	—	2	2	no	—
145	D₄×C₆₇	—	2	4	no	—
146	D₄×C₆₇	#108	1	4	yes	—
147	C₂×C₄×C₆₇	—	2	4	no	—
148	C₂×C₄×C₆₇	—	2	2	no	—
149	C₂×C₄×C₆₇	—	2	1	no	—
150	C₂×C₄×C₆₇	—	2	2	no	—
151	C₂×C₄×C₆₇	—	2	1	no	—
152	C₂×C₄×C₆₇	—	2	2	no	—
153	C₂×C₄×C₆₇	—	2	1	no	—
154	C₂×C₄×C₆₇	—	2	2	no	—
155	C₂×C₄×C₆₇	—	2	1	no	—
156	C₂×C₄×C₆₇	—	2	2	no	—
157	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
158	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
159	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
161	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
162	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
164	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
165	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
166	C₈×C₆₇	—	4	1	no	—
167	C₈×C₆₇	—	4	2	no	—
168	C₈×C₆₇	—	4	1	no	—
169	C₆₇⋊C₈	—	2	2	no	—
170	C₆₇⋊C₈	—	2	1	no	—
171	C₆₇⋊C₈	—	2	1	no	—
172	C₆₇⋊C₈	—	2	2	no	—
173	C₆₇⋊C₈	—	2	1	no	—
174	C₆₇⋊C₈	—	4	1	no	—
175	C₆₇⋊C₈	—	4	2	no	—
176	C₆₇⋊C₈	—	4	1	no	—
177	C₆₇⋊C₈	#131	1	2	yes	—
178	C₂²×C₆₇	—	2	2	no	—
179	C₂²×C₆₇	—	2	2	no	—
180	C₂²×C₆₇	—	2	1	no	—
181	C₂²×C₆₇	—	2	1	no	—
182	C₂²×C₆₇	—	2	1	no	—
183	C₂²×C₆₇	C₂×#108	1	4	yes	—
184	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
185	C₄×C₆₇	—	2	2	no	—
186	C₄×C₆₇	—	2	1	no	—
187	C₄×C₆₇	—	2	2	no	—
188	C₄×C₆₇	—	2	1	no	—
189	C₄×C₆₇	—	2	1	no	—
190	C₄×C₆₇	—	2	2	no	—
191	C₄×C₆₇	C₂×#108	1	2	yes	—
192	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
193	C₄×C₆₇	#568	1	1	yes	characteristic
194	C₄×C₆₇	#576	1	1	yes	characteristic
195	C₄×C₆₇	#578	1	1	yes	characteristic
196	#17836	—	67	1	no	maximal, Sylow
197	C₂×C₆₇	—	2	4	no	—
198	C₂×C₆₇	—	4	1	no	—
199	C₂×C₆₇	#2906	1	1	yes	characteristic
200	C₂×C₆₇	#2913	1	1	yes	characteristic
201	C₂×C₆₇	C₂²×#108	1	1	yes	characteristic
202	#2990	—	67	1	no	—
203	#2990	D₆₇	1	1	yes	characteristic
204	C₈×#41	—	67	1	no	—
205	#1574	—	67	1	no	—
206	#1574	—	67	1	no	—
207	#1573	—	67	2	no	—
208	#1573	—	67	2	no	—
209	#1568	—	67	2	no	—
210	#1567	—	67	1	no	—
211	#1564	—	67	1	no	—
212	C₂×#335	—	67	2	no	—
213	C₆₇	#17836	1	1	yes	characteristic, complemented, Sylow
214	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
215	#446	—	2	1	no	—
216	#446	—	134	1	no	—
217	C₈×Q₈	—	67	1	no	—
218	C₈×Q₈	—	67	1	no	—
219	#435	—	2	2	no	—
220	#435	—	134	2	no	—
221	C₈×D₄	—	2	1	no	—
222	C₈×D₄	—	67	2	no	—
223	C₈×D₄	—	67	1	no	—
224	C₈×D₄	—	67	2	no	—
225	C₈×D₄	—	67	1	no	—
226	C₈×D₄	—	134	1	no	—
227	#432	—	67	2	no	—
228	#432	C₂×D₆₇	1	2	yes	—
229	#431	—	67	2	no	—
230	#431	—	67	1	no	—
231	#431	—	67	1	no	—
232	#431	C₂×D₆₇	1	1	yes	characteristic
233	#423	—	67	1	no	—
234	#423	C₂×D₆₇	1	1	yes	characteristic
235	C₂×C₄⋊C₈	—	67	2	no	—
236	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
237	C₂×C₄×C₈	—	67	2	no	—
238	C₂×C₄×C₈	—	67	1	no	—
239	#335	—	67	1	no	—
240	#335	—	67	2	no	—
241	#335	—	67	2	no	—
242	#335	—	67	2	no	—
243	#335	—	67	1	no	—
244	#335	—	67	2	no	—
245	#334	—	67	1	no	—
246	#334	—	67	1	no	—
247	#334	—	67	2	no	—
248	#334	—	67	2	no	—
249	C₂×#41	Dic₁₃₄	1	1	yes	characteristic
250	C₂×M₁₆	—	2	1	no	—
251	C₂×M₁₆	—	134	1	no	—
252	C₂²×C₈	—	2	2	no	—
253	C₂²×C₈	—	67	2	no	—
254	C₂²×C₈	—	67	1	no	—
255	C₂²×C₈	—	67	2	no	—
256	C₂²×C₈	—	67	1	no	—
257	C₂²×C₈	—	134	2	no	—
258	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
259	C₄×Q₈	C₂²⋊D₆₇	1	1	yes	characteristic
260	C₄×D₄	C₄×D₆₇	1	2	yes	—
261	C₄×D₄	C₄×D₆₇	1	1	yes	characteristic
262	C₄×D₄	C₂²⋊D₆₇	1	1	yes	characteristic
263	C₄×D₄	C₂²⋊D₆₇	1	2	yes	—
264	#117	Dic₁₃₄	1	1	yes	characteristic
265	#117	Dic₁₃₄	1	2	yes	—
266	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
267	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
268	C₄⋊C₈	—	2	2	no	—
269	C₄⋊C₈	—	2	1	no	—
270	C₄⋊C₈	—	67	2	no	—
271	C₄⋊C₈	—	67	2	no	—
272	C₄⋊C₈	—	67	1	no	—
273	C₄⋊C₈	—	67	1	no	—
274	C₄⋊C₈	—	67	1	no	—
275	C₄⋊C₈	—	67	1	no	—
276	C₄⋊C₈	—	67	2	no	—
277	C₄⋊C₈	—	134	2	no	—
278	C₄⋊C₈	—	134	1	no	—
279	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
280	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
281	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
282	C₂²⋊C₈	—	2	1	no	—
283	C₂²⋊C₈	—	2	2	no	—
284	C₂²⋊C₈	—	67	2	no	—
285	C₂²⋊C₈	—	67	2	no	—
286	C₂²⋊C₈	—	67	2	no	—
287	C₂²⋊C₈	—	134	1	no	—
288	C₂²⋊C₈	—	134	2	no	—
289	#97	—	2	2	no	—
290	#97	—	134	2	no	—
291	C₄×C₈	—	2	1	no	—
292	C₄×C₈	—	67	2	no	—
293	C₄×C₈	—	67	1	no	—
294	C₄×C₈	—	67	2	no	—
295	C₄×C₈	—	67	1	no	—
296	C₄×C₈	—	67	1	no	—
297	C₄×C₈	—	67	2	no	—
298	C₄×C₈	—	67	1	no	—
299	C₄×C₈	—	134	1	no	—
300	#41	—	2	4	no	—
301	C₂×Q₈	Dic₆₇⋊C₄	1	1	yes	characteristic
302	C₂×D₄	Dic₆₇⋊C₄	1	2	yes	—
303	C₂×D₄	Dic₆₇⋊C₄	1	1	yes	characteristic
304	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
305	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
306	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
307	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
308	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
309	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
310	M₁₆	—	8	1	no	—
311	M₁₆	—	536	1	no	—
312	C₂×C₈	—	2	1	no	—
313	C₂×C₈	—	2	1	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	8	2	no	—
316	C₂×C₈	—	268	2	no	—
317	C₂×C₈	—	268	1	no	—
318	C₂×C₈	—	536	2	no	—
319	C₂×C₈	—	67	1	no	—
320	C₂×C₈	—	67	1	no	—
321	C₂×C₈	—	67	2	no	—
322	C₂×C₈	—	67	1	no	—
323	C₂×C₈	—	67	2	no	—
324	C₂×C₈	—	67	1	no	—
325	C₂×C₈	—	134	1	no	—
326	C₂×C₈	—	134	1	no	—
327	C₂×C₈	—	134	2	no	—
328	C₂×C₈	—	134	4	no	—
329	C₂×C₈	—	134	2	no	—
330	C₄⋊C₄	Dic₆₇⋊C₄	1	2	yes	—
331	C₄⋊C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
332	C₄⋊C₄	Dic₆₇⋊C₄	1	2	yes	—
333	C₄⋊C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
334	C₂²⋊C₄	Dic₆₇⋊C₄	1	2	yes	—
335	C₂²⋊C₄	Dic₆₇⋊C₄	1	2	yes	—
336	C₂²⋊C₄	Dic₆₇⋊C₄	1	2	yes	—
337	C₄²	C₂×C₄×D₆₇	1	2	yes	—
338	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
339	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
340	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
341	C₄²	C₂×Dic₁₃₄	1	2	yes	—
342	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
343	C₄²	C₂³×D₆₇	1	1	yes	characteristic
344	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
345	C₂³	C₂×Dic₆₇⋊C₄	1	2	yes	—
346	Q₈	—	2	2	no	—
347	D₄	—	2	4	no	—
348	D₄	#14523978	1	4	yes	—
349	C₂×C₄	—	2	4	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	1	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
360	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
361	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
362	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
363	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
364	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
365	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
366	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
367	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
368	C₈	—	4	1	no	—
369	C₈	—	4	2	no	—
370	C₈	—	4	1	no	—
371	C₈	—	268	1	no	—
372	C₈	—	268	2	no	—
373	C₈	—	268	1	no	—
374	C₈	—	67	2	no	—
375	C₈	—	134	2	no	—
376	C₈	—	134	1	no	—
377	C₈	—	134	1	no	—
378	C₈	—	134	2	no	—
379	C₈	—	134	1	no	—
380	C₂²	—	2	2	no	—
381	C₂²	—	2	2	no	—
382	C₂²	—	2	1	no	—
383	C₂²	—	2	1	no	—
384	C₂²	—	2	1	no	—
385	C₂²	C₂×#14523978	1	4	yes	—
386	C₂²	C₂²×Dic₆₇⋊C₄	1	1	yes	characteristic
387	C₄	—	2	2	no	—
388	C₄	—	2	1	no	—
389	C₄	—	2	2	no	—
390	C₄	—	2	1	no	—
391	C₄	—	2	1	no	—
392	C₄	—	2	2	no	—
393	C₄	C₂×#14523978	1	2	yes	—
394	C₄	#131×D₆₇	1	1	yes	characteristic
395	C₄	#16864161	1	1	yes	characteristic
396	C₄	#16864199	1	1	yes	characteristic
397	C₄	#16864211	1	1	yes	characteristic
398	C₂	—	2	4	no	—
399	C₂	—	4	1	no	—
400	C₂	#22568775	1	1	yes	characteristic
401	C₂	#22568807	1	1	yes	characteristic
402	C₂	C₂²×#14523978	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#2990	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₆₇	Q₈
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#2990	D₆₇
hypocenter	C₆₇	#17836
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ #2990

Complements

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

N	H ≅ G/N	type
3. #22559000	C₂	semidirect
4. #22558997	C₂	semidirect
5. #22558995	C₂	semidirect
6. #22558992	C₂	semidirect
7. #22558970	C₂	semidirect
8. #22558967	C₂	semidirect
9. #22558955	C₂	semidirect
10. C₂×#16863148	C₂	semidirect
213. C₆₇	#17836	semidirect

Maximal subgroups

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

subgroup	index
#22569334	2
#41×C₆₇⋊C₈	2
#22559000	2
#22558997	2
#22558995	2
#22558992	2
#22558970	2
#22558967	2
#22558955	2
C₂×#16863148	2
C₆₇×#2990	2
#17836	67