Subgroups — #30825503

Order 17152.

726 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569250	C₂	1	1	yes	maximal, characteristic, complemented
2	#22569233	C₂	1	1	yes	maximal, characteristic, complemented
3	#22559074	C₂	1	4	yes	maximal, complemented
4	#22559039	C₂	1	4	yes	maximal, complemented
5	#22558957	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558600	C₂	1	1	yes	maximal, characteristic, complemented
7	#22558600	C₂	1	1	yes	maximal, characteristic, complemented
8	#22558501	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic
10	#16863474	—	2	2	no	—
11	#16863474	C₂²	1	4	yes	complemented
12	D₄×C₆₇⋊C₈	—	2	2	no	—
13	D₄×C₆₇⋊C₈	C₂²	1	4	yes	complemented
14	#16863449	C₂²	1	1	yes	characteristic
15	#16863449	C₂²	1	1	yes	characteristic
16	#16863449	C₂²	1	1	yes	characteristic
17	#16863449	C₂²	1	1	yes	characteristic
18	#16863360	—	2	2	no	—
19	C₂×C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
20	#16863149	C₂²	1	2	yes	complemented
21	#16863147	C₂²	1	2	yes	complemented
22	D₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	complemented
23	D₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	complemented
24	#16863118	C₂²	1	2	yes	complemented
25	#16863118	C₂²	1	2	yes	complemented
26	C₂²×D₈×C₆₇	C₄	1	1	yes	characteristic
27	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
29	C₄×D₈×C₆₇	C₄	1	2	yes	—
30	C₄×D₈×C₆₇	C₄	1	2	yes	—
31	C₄×D₈×C₆₇	C₂²	1	2	yes	—
32	C₄×D₈×C₆₇	C₂²	1	2	yes	—
33	C₂×#107×C₆₇	C₄	1	1	yes	characteristic
34	C₂×#102×C₆₇	C₄	1	1	yes	characteristic
35	C₂×#102×C₆₇	C₄	1	1	yes	characteristic
36	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×#11761612	—	2	4	no	—
38	C₂²×C₆₇⋊C₈	—	2	4	no	—
39	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
40	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
41	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
42	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
43	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
44	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
45	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
46	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
47	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
48	C₂²⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
49	C₄×C₆₇⋊C₈	—	2	1	no	—
50	C₄×C₆₇⋊C₈	—	2	1	no	—
51	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
52	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
53	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂³×C₄×C₆₇	—	2	1	no	—
56	C₂³×C₄×C₆₇	—	2	1	no	—
57	C₂×D₈×C₆₇	—	2	4	no	—
58	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
59	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
60	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
62	C₄×D₄×C₆₇	—	2	1	no	—
63	C₄×D₄×C₆₇	—	2	1	no	—
64	C₄×D₄×C₆₇	—	2	1	no	—
65	C₄×D₄×C₆₇	—	2	1	no	—
66	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
67	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
68	C₄×D₄×C₆₇	C₂³	1	2	yes	—
69	C₄×D₄×C₆₇	C₂³	1	2	yes	—
70	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
72	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
73	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
74	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
75	#107×C₆₇	C₂×C₄	1	2	yes	—
76	#107×C₆₇	C₂×C₄	1	2	yes	—
77	#102×C₆₇	C₂×C₄	1	2	yes	—
78	#102×C₆₇	C₂×C₄	1	2	yes	—
79	#102×C₆₇	C₂×C₄	1	2	yes	—
80	#102×C₆₇	C₂×C₄	1	2	yes	—
81	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
82	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
83	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
85	#11761612	—	8	4	no	—
86	C₂×C₆₇⋊C₈	—	2	2	no	—
87	C₂×C₆₇⋊C₈	—	2	2	no	—
88	C₂×C₆₇⋊C₈	—	2	2	no	—
89	C₂×C₆₇⋊C₈	—	8	4	no	—
90	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
91	C₂⁴×C₆₇	—	2	1	no	—
92	C₂⁴×C₆₇	—	2	1	no	—
93	C₂×D₄×C₆₇	—	2	1	no	—
94	C₂×D₄×C₆₇	—	2	1	no	—
95	C₂×D₄×C₆₇	—	2	2	no	—
96	C₂×D₄×C₆₇	—	2	2	no	—
97	C₂×D₄×C₆₇	—	2	2	no	—
98	C₂×D₄×C₆₇	M₁₆	1	4	yes	complemented
99	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
100	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₆₇	—	2	1	no	—
102	C₂²×C₄×C₆₇	—	2	1	no	—
103	C₂²×C₄×C₆₇	—	2	2	no	—
104	C₂²×C₄×C₆₇	—	2	2	no	—
105	C₂²×C₄×C₆₇	—	2	2	no	—
106	C₂²×C₄×C₆₇	—	2	2	no	—
107	C₂²×C₄×C₆₇	—	8	2	no	—
108	C₂²×C₄×C₆₇	—	8	2	no	—
109	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
110	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
112	D₈×C₆₇	—	2	4	no	—
113	D₈×C₆₇	—	2	4	no	—
114	C₂×C₈×C₆₇	—	2	2	no	—
115	C₂×C₈×C₆₇	—	2	2	no	—
116	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
120	C₄⋊C₄×C₆₇	—	2	1	no	—
121	C₄⋊C₄×C₆₇	—	2	1	no	—
122	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
123	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
124	C₂²⋊C₄×C₆₇	—	2	2	no	—
125	C₂²⋊C₄×C₆₇	—	2	2	no	—
126	C₂²⋊C₄×C₆₇	—	2	2	no	—
127	C₂²⋊C₄×C₆₇	—	2	2	no	—
128	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
129	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₆₇	#41	1	1	yes	characteristic
131	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
132	C₂³×C₆₇	—	2	2	no	—
133	C₂³×C₆₇	—	2	4	no	—
134	C₂³×C₆₇	—	2	2	no	—
135	C₂³×C₆₇	—	4	2	no	—
136	C₂³×C₆₇	—	8	1	no	—
137	C₂³×C₆₇	—	8	1	no	—
138	C₂³×C₆₇	—	8	1	no	—
139	C₂³×C₆₇	—	8	1	no	—
140	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
141	D₄×C₆₇	—	2	2	no	—
142	D₄×C₆₇	—	2	2	no	—
143	D₄×C₆₇	—	2	2	no	—
144	D₄×C₆₇	—	2	4	no	—
145	D₄×C₆₇	—	4	2	no	—
146	D₄×C₆₇	C₂×M₁₆	1	4	yes	—
147	C₂×C₄×C₆₇	—	2	2	no	—
148	C₂×C₄×C₆₇	—	2	2	no	—
149	C₂×C₄×C₆₇	—	2	2	no	—
150	C₂×C₄×C₆₇	—	4	1	no	—
151	C₂×C₄×C₆₇	—	4	4	no	—
152	C₂×C₄×C₆₇	—	4	4	no	—
153	C₂×C₄×C₆₇	—	8	2	no	—
154	C₂×C₄×C₆₇	—	8	2	no	—
155	C₂×C₄×C₆₇	—	8	1	no	—
156	C₂×C₄×C₆₇	—	8	1	no	—
157	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
158	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
159	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
160	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
161	C₂×C₄×C₆₇	#131	1	2	yes	—
162	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
163	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
165	C₈×C₆₇	—	2	1	no	—
166	C₈×C₆₇	—	2	1	no	—
167	C₈×C₆₇	—	2	1	no	—
168	C₈×C₆₇	—	2	1	no	—
169	C₆₇⋊C₈	—	2	4	no	—
170	C₆₇⋊C₈	—	4	2	no	—
171	C₆₇⋊C₈	—	4	2	no	—
172	C₆₇⋊C₈	—	4	2	no	—
173	C₂²×C₆₇	—	2	1	no	—
174	C₂²×C₆₇	—	2	1	no	—
175	C₂²×C₆₇	—	2	4	no	—
176	C₂²×C₆₇	—	4	2	no	—
177	C₂²×C₆₇	—	4	2	no	—
178	C₂²×C₆₇	—	4	2	no	—
179	C₂²×C₆₇	—	4	2	no	—
180	C₂²×C₆₇	—	4	2	no	—
181	C₂²×C₆₇	—	8	4	no	—
182	C₂²×C₆₇	—	8	4	no	—
183	C₂²×C₆₇	#435	1	2	yes	—
184	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
185	C₄×C₆₇	—	2	1	no	—
186	C₄×C₆₇	—	2	1	no	—
187	C₄×C₆₇	—	2	1	no	—
188	C₄×C₆₇	—	4	2	no	—
189	C₄×C₆₇	—	4	2	no	—
190	C₄×C₆₇	#435	1	2	yes	—
191	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
192	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
193	C₄×C₆₇	#576	1	2	yes	—
194	#17877	—	67	1	no	maximal, Sylow
195	C₂×C₆₇	—	2	2	no	—
196	C₂×C₆₇	—	4	4	no	—
197	C₂×C₆₇	—	8	2	no	—
198	C₂×C₆₇	C₂×#435	1	1	yes	characteristic
199	C₂×C₆₇	#2948	1	1	yes	characteristic
200	C₂×C₆₇	#2957	1	1	yes	characteristic
201	#2975	—	67	1	no	—
202	#2972	—	67	1	no	—
203	C₂×C₄×D₈	D₆₇	1	1	yes	characteristic
204	#1586	—	67	4	no	—
205	#1581	—	67	4	no	—
206	#1564	—	67	1	no	—
207	#1478	—	67	1	no	—
208	#1478	—	67	1	no	—
209	#1452	—	67	1	no	—
210	C₆₇	#17877	1	1	yes	characteristic, complemented, Sylow
211	C₂²×D₈	Dic₆₇	1	1	yes	characteristic
212	C₂×C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
213	C₂×C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
214	C₄×D₈	Dic₆₇	1	2	yes	—
215	C₄×D₈	Dic₆₇	1	2	yes	—
216	C₄×D₈	C₂×D₆₇	1	2	yes	—
217	C₄×D₈	C₂×D₆₇	1	2	yes	—
218	#436	—	67	4	no	—
219	#436	—	134	2	no	—
220	C₈×D₄	—	67	4	no	—
221	C₈×D₄	—	134	2	no	—
222	#431	—	67	1	no	—
223	#431	—	67	1	no	—
224	#431	—	67	1	no	—
225	#431	—	67	1	no	—
226	C₂×#107	Dic₆₇	1	1	yes	characteristic
227	C₂×#102	Dic₆₇	1	1	yes	characteristic
228	C₂×#102	Dic₆₇	1	1	yes	characteristic
229	#407	—	134	2	no	—
230	C₂×C₂²⋊C₈	—	134	2	no	—
231	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
232	#335	—	67	2	no	—
233	#334	—	67	2	no	—
234	D₄⋊C₈	—	67	4	no	—
235	D₄⋊C₈	—	67	4	no	—
236	#322	—	67	2	no	—
237	#322	—	67	2	no	—
238	C₂²×D₄	C₂×Dic₆₇	1	1	yes	characteristic
239	C₂²×D₄	C₂×Dic₆₇	1	1	yes	characteristic
240	C₂³×C₄	—	2	1	no	—
241	C₂³×C₄	—	2	1	no	—
242	C₂×D₈	—	2	4	no	—
243	C₂×D₈	C₂×Dic₆₇	1	2	yes	—
244	C₂×D₈	C₂×Dic₆₇	1	2	yes	—
245	C₂×M₁₆	—	134	4	no	—
246	C₂²×C₈	—	134	4	no	—
247	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
248	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
249	C₄×D₄	—	2	1	no	—
250	C₄×D₄	—	2	1	no	—
251	C₄×D₄	—	2	1	no	—
252	C₄×D₄	—	2	1	no	—
253	C₄×D₄	C₂×Dic₆₇	1	2	yes	—
254	C₄×D₄	C₂×Dic₆₇	1	2	yes	—
255	C₄×D₄	C₂²×D₆₇	1	2	yes	—
256	C₄×D₄	C₂²×D₆₇	1	2	yes	—
257	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
258	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
259	C₂×C₂²⋊C₄	—	2	1	no	—
260	C₂×C₂²⋊C₄	—	2	1	no	—
261	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
262	#107	C₂×Dic₆₇	1	2	yes	—
263	#107	C₂×Dic₆₇	1	2	yes	—
264	C₄⋊C₈	—	67	2	no	—
265	C₄⋊C₈	—	67	2	no	—
266	C₄⋊C₈	—	67	2	no	—
267	C₄⋊C₈	—	134	1	no	—
268	#102	C₂×Dic₆₇	1	2	yes	—
269	#102	C₂×Dic₆₇	1	2	yes	—
270	#102	C₂×Dic₆₇	1	2	yes	—
271	#102	C₂×Dic₆₇	1	2	yes	—
272	C₂²⋊C₈	—	67	2	no	—
273	C₂²⋊C₈	—	134	4	no	—
274	C₂²⋊C₈	—	134	4	no	—
275	C₂²⋊C₈	—	134	1	no	—
276	C₂²⋊C₈	—	134	1	no	—
277	C₂²⋊C₈	—	134	1	no	—
278	C₄×C₈	—	67	2	no	—
279	C₄×C₈	—	67	2	no	—
280	C₄×C₈	—	134	1	no	—
281	C₄×C₈	—	134	1	no	—
282	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
283	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
284	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
285	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
286	C₂⁴	—	2	1	no	—
287	C₂⁴	—	2	1	no	—
288	C₂×D₄	—	2	1	no	—
289	C₂×D₄	—	2	1	no	—
290	C₂×D₄	—	2	2	no	—
291	C₂×D₄	—	2	2	no	—
292	C₂×D₄	—	2	2	no	—
293	C₂×D₄	#11761612	1	4	yes	complemented
294	C₂×D₄	C₂²×Dic₆₇	1	2	yes	—
295	C₂×D₄	C₂²×Dic₆₇	1	2	yes	—
296	C₂²×C₄	—	2	1	no	—
297	C₂²×C₄	—	2	1	no	—
298	C₂²×C₄	—	2	2	no	—
299	C₂²×C₄	—	2	2	no	—
300	C₂²×C₄	—	2	2	no	—
301	C₂²×C₄	—	2	2	no	—
302	C₂²×C₄	—	8	2	no	—
303	C₂²×C₄	—	8	2	no	—
304	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
305	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
306	C₂²×C₄	#11761627	1	1	yes	characteristic
307	D₈	—	2	4	no	—
308	D₈	—	2	4	no	—
309	M₁₆	—	536	4	no	—
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	536	4	no	—
313	C₂×C₈	—	67	4	no	—
314	C₂×C₈	—	134	2	no	—
315	C₂×C₈	—	134	2	no	—
316	C₂×C₈	—	134	2	no	—
317	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
318	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
319	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
320	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
321	C₄⋊C₄	—	2	1	no	—
322	C₄⋊C₄	—	2	1	no	—
323	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
324	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
325	C₂²⋊C₄	—	2	2	no	—
326	C₂²⋊C₄	—	2	2	no	—
327	C₂²⋊C₄	—	2	2	no	—
328	C₂²⋊C₄	—	2	2	no	—
329	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
330	C₄²	D₄×D₆₇	1	1	yes	characteristic
331	C₄²	#11761627	1	1	yes	characteristic
332	C₄²	C₂³×D₆₇	1	1	yes	characteristic
333	C₂³	—	2	2	no	—
334	C₂³	—	2	4	no	—
335	C₂³	—	2	2	no	—
336	C₂³	—	4	2	no	—
337	C₂³	—	8	1	no	—
338	C₂³	—	8	1	no	—
339	C₂³	—	8	1	no	—
340	C₂³	—	8	1	no	—
341	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
342	D₄	—	2	2	no	—
343	D₄	—	2	2	no	—
344	D₄	—	2	2	no	—
345	D₄	—	2	4	no	—
346	D₄	—	4	2	no	—
347	D₄	C₂×#11761612	1	4	yes	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	4	1	no	—
352	C₂×C₄	—	4	4	no	—
353	C₂×C₄	—	4	4	no	—
354	C₂×C₄	—	8	2	no	—
355	C₂×C₄	—	8	2	no	—
356	C₂×C₄	—	8	1	no	—
357	C₂×C₄	—	8	1	no	—
358	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
359	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
360	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
361	C₂×C₄	C₂×#11761612	1	2	yes	—
362	C₂×C₄	#14524050	1	2	yes	—
363	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
364	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
365	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
366	C₈	—	2	1	no	—
367	C₈	—	2	1	no	—
368	C₈	—	2	1	no	—
369	C₈	—	2	1	no	—
370	C₈	—	268	2	no	—
371	C₈	—	268	2	no	—
372	C₈	—	268	2	no	—
373	C₈	—	134	4	no	—
374	C₂²	—	2	1	no	—
375	C₂²	—	2	1	no	—
376	C₂²	—	2	4	no	—
377	C₂²	—	4	2	no	—
378	C₂²	—	4	2	no	—
379	C₂²	—	4	2	no	—
380	C₂²	—	4	2	no	—
381	C₂²	—	4	2	no	—
382	C₂²	—	8	4	no	—
383	C₂²	—	8	4	no	—
384	C₂²	#16863468	1	2	yes	—
385	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
386	C₄	—	2	1	no	—
387	C₄	—	2	1	no	—
388	C₄	—	2	1	no	—
389	C₄	—	4	2	no	—
390	C₄	—	4	2	no	—
391	C₄	#16863468	1	2	yes	—
392	C₄	C₂²×#11761612	1	1	yes	characteristic
393	C₄	C₂×#14524050	1	1	yes	characteristic
394	C₄	#16864197	1	2	yes	—
395	C₂	—	2	2	no	—
396	C₂	—	4	4	no	—
397	C₂	—	8	2	no	—
398	C₂	C₂×#16863468	1	1	yes	characteristic
399	C₂	#22569061	1	1	yes	characteristic
400	C₂	#22569126	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×C₄×D₈×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂²×D₈×C₆₇	C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₂×C₄×D₈	D₆₇
hypocenter	C₆₇	#17877
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. #22569250	C₂	semidirect
2. #22569233	C₂	semidirect
3. #22559074	C₂	semidirect
4. #22559039	C₂	semidirect
5. #22558957	C₂	semidirect
6. #22558600	C₂	semidirect
7. #22558600	C₂	semidirect
8. #22558501	C₂	semidirect
11. #16863474	C₂²	semidirect
13. D₄×C₆₇⋊C₈	C₂²	semidirect
20. #16863149	C₂²	semidirect
21. #16863147	C₂²	semidirect
22. D₄⋊(C₆₇⋊C₈)	C₂²	semidirect
23. D₄⋊(C₆₇⋊C₈)	C₂²	semidirect
24. #16863118	C₂²	semidirect
25. #16863118	C₂²	semidirect
98. C₂×D₄×C₆₇	M₁₆	semidirect
210. C₆₇	#17877	semidirect
293. C₂×D₄	#11761612	semidirect

Maximal subgroups

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

subgroup	index
#22569250	2
#22569233	2
#22559074	2
#22559039	2
#22558957	2
#22558600	2
#22558501	2
C₂×C₄×D₈×C₆₇	2
#17877	67