Subgroups — #22569367

Order 8576.

364 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16863924	C₂	1	1	yes	maximal, characteristic
2	#16863525	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863520	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863472	C₂	1	1	yes	maximal, characteristic, complemented
5	#16863466	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863465	C₂	1	1	yes	maximal, characteristic, complemented
7	#16863461	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×C₂²⋊D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×C₂²⋊D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#16863457	C₂	1	1	yes	maximal, characteristic, complemented
11	#16863456	C₂	1	1	yes	maximal, characteristic, complemented
12	#16863448	C₂	1	1	yes	maximal, characteristic, complemented
13	#16863421	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄⋊C₈×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
15	#424×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11761611	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₆₇	C₂²	1	1	yes	characteristic
20	#14524038	C₄	1	1	yes	characteristic
21	#14524036	C₄	1	1	yes	characteristic
22	#14524033	C₄	1	1	yes	characteristic
23	#14524029	C₄	1	1	yes	characteristic
24	#14524028	C₄	1	1	yes	characteristic
25	#14524025	C₄	1	1	yes	characteristic
26	#14524021	C₄	1	1	yes	characteristic
27	#14524016	C₄	1	1	yes	characteristic
28	C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
29	C₄×D₂₆₈	C₂²	1	1	yes	characteristic
30	C₄×C₂²⋊D₆₇	C₂²	1	1	yes	characteristic
31	C₄×C₂²⋊D₆₇	C₂²	1	1	yes	characteristic
32	#14524000	C₂²	1	1	yes	characteristic
33	C₄²×D₆₇	C₂²	1	1	yes	characteristic
34	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
35	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
36	Dic₆₇⋊C₈	C₂²	1	1	yes	characteristic
37	Dic₆₇⋊C₈	C₂²	1	1	yes	characteristic
38	Dic₆₇⋊C₈	C₂²	1	1	yes	characteristic
39	Dic₆₇⋊C₈	C₂²	1	1	yes	characteristic
40	C₂²⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
41	C₂²⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
42	(C₂×D₆₇)⋊C₈	C₂²	1	1	yes	characteristic
43	(C₂×D₆₇)⋊C₈	C₂²	1	1	yes	characteristic
44	(C₂×D₆₇)⋊C₈	C₂²	1	1	yes	characteristic
45	(C₂×D₆₇)⋊C₈	C₂²	1	1	yes	characteristic
46	#14523954	C₂²	1	1	yes	characteristic
47	#14523953	C₂²	1	1	yes	characteristic
48	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
49	C₈×Dic₆₇	C₂²	1	1	yes	characteristic
50	C₈×Dic₆₇	C₂²	1	1	yes	characteristic
51	C₈×Dic₆₇	C₂²	1	1	yes	characteristic
52	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
54	#117×C₆₇	C₂²	1	1	yes	characteristic
55	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
57	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
58	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
60	C₂×D₂₆₈	C₂×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₆₇	C₂×C₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₆₇	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
66	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
67	#11761611	—	4	1	no	—
68	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
69	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
71	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
72	C₈×D₆₇	—	4	1	no	—
73	C₈×D₆₇	—	4	1	no	—
74	C₈×D₆₇	—	4	1	no	—
75	C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
76	C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
77	Dic₆₇⋊C₄	C₂×C₄	1	1	yes	characteristic
78	Dic₆₇⋊C₄	C₂×C₄	1	1	yes	characteristic
79	Dic₆₇⋊C₄	C₂×C₄	1	1	yes	characteristic
80	Dic₆₇⋊C₄	C₂×C₄	1	1	yes	characteristic
81	Dic₆₇⋊C₄	C₂×C₄	1	1	yes	characteristic
82	Dic₆₇⋊C₄	C₂×C₄	1	1	yes	characteristic
83	C₂²⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
84	C₂²⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
85	(C₂×D₆₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
86	(C₂×D₆₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
87	(C₂×D₆₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
88	(C₂×D₆₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
89	(C₂×D₆₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
90	(C₂×D₆₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
91	C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
92	C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
93	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
94	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
95	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
96	M₁₆×C₆₇	—	4	1	no	—
97	C₂×C₈×C₆₇	—	4	1	no	—
98	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
103	C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
104	C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
105	C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
106	C₄²×C₆₇	C₂³	1	1	yes	characteristic
107	C₄²×C₆₇	C₂³	1	1	yes	characteristic
108	C₂³×C₆₇	C₂²×C₄	1	1	yes	characteristic
109	C₂²×D₆₇	C₂²×C₄	1	1	yes	characteristic
110	C₂²×D₆₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₆₇	—	2	1	no	—
112	C₂×C₄×C₆₇	—	2	1	no	—
113	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
119	C₂²⋊D₆₇	—	2	2	no	—
120	C₂²⋊D₆₇	—	2	2	no	—
121	C₂×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
122	C₂×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
123	C₂×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
124	C₂×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
125	C₂×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
126	C₂×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
127	D₂₆₈	—	2	2	no	—
128	C₄×D₆₇	—	2	2	no	—
129	C₄×D₆₇	—	2	1	no	—
130	C₄×D₆₇	—	2	1	no	—
131	C₄×D₆₇	—	2	1	no	—
132	C₄×D₆₇	—	2	1	no	—
133	C₄×D₆₇	—	2	2	no	—
134	Dic₁₃₄	—	2	2	no	—
135	C₈×C₆₇	—	2	1	no	—
136	C₈×C₆₇	—	2	1	no	—
137	C₈×C₆₇	—	2	1	no	—
138	C₈×C₆₇	—	2	1	no	—
139	C₆₇⋊C₈	—	2	1	no	—
140	C₆₇⋊C₈	—	2	1	no	—
141	C₆₇⋊C₈	—	2	1	no	—
142	C₆₇⋊C₈	—	2	1	no	—
143	C₂²×C₆₇	—	2	1	no	—
144	C₂²×C₆₇	—	2	1	no	—
145	C₂²×C₆₇	—	2	1	no	—
146	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
147	C₂×D₆₇	—	2	1	no	—
148	C₂×D₆₇	—	2	1	no	—
149	C₂×D₆₇	—	2	1	no	—
150	C₂×D₆₇	—	2	1	no	—
151	C₂×D₆₇	—	2	1	no	—
152	C₂×D₆₇	#131	1	2	yes	—
153	C₄×C₆₇	—	2	1	no	—
154	C₄×C₆₇	—	2	1	no	—
155	C₄×C₆₇	—	2	1	no	—
156	C₄×C₆₇	—	2	1	no	—
157	C₄×C₆₇	—	2	1	no	—
158	C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
159	C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
160	Dic₆₇	—	2	1	no	—
161	Dic₆₇	—	2	1	no	—
162	Dic₆₇	—	2	1	no	—
163	Dic₆₇	—	2	1	no	—
164	Dic₆₇	—	2	1	no	—
165	Dic₆₇	#131	1	2	yes	—
166	C₂×C₆₇	—	4	1	no	—
167	C₂×C₆₇	#520	1	1	yes	characteristic
168	C₂×C₆₇	C₂×#131	1	1	yes	characteristic
169	C₂×C₆₇	C₂×#131	1	1	yes	characteristic
170	D₆₇	—	2	2	no	—
171	D₆₇	—	4	1	no	—
172	#2994	—	67	1	no	maximal, Sylow
173	C₆₇	#2994	1	1	yes	characteristic, complemented, Sylow
174	#529	—	67	1	no	—
175	#446	—	67	1	no	—
176	C₈×Q₈	—	67	1	no	—
177	#436	—	67	1	no	—
178	#435	—	67	1	no	—
179	#435	—	67	1	no	—
180	C₈×D₄	—	67	1	no	—
181	C₈×D₄	—	67	1	no	—
182	C₈×D₄	—	67	1	no	—
183	#433	—	67	1	no	—
184	#433	—	67	1	no	—
185	#431	—	67	1	no	—
186	#424	—	67	1	no	—
187	#424	D₆₇	1	1	yes	characteristic, complemented
188	C₂×C₄⋊C₈	—	67	1	no	—
189	C₂×M₁₆	—	67	1	no	—
190	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
191	C₂²×C₈	—	67	1	no	—
192	C₂²×C₈	—	67	1	no	—
193	C₂²×C₈	—	67	1	no	—
194	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
195	#126	—	67	1	no	—
196	#126	—	67	1	no	—
197	#125	—	67	1	no	—
198	#124	—	67	1	no	—
199	#123	—	67	1	no	—
200	#123	—	67	1	no	—
201	C₂²⋊Q₈	—	67	1	no	—
202	#121	—	67	1	no	—
203	C₄×Q₈	—	67	1	no	—
204	C₄×D₄	—	67	1	no	—
205	C₄×D₄	—	67	1	no	—
206	C₄×D₄	—	67	1	no	—
207	#117	—	67	1	no	—
208	#117	C₂×D₆₇	1	1	yes	characteristic
209	C₂×C₄²	—	67	1	no	—
210	C₄⋊C₈	—	67	1	no	—
211	C₄⋊C₈	—	67	1	no	—
212	C₄⋊C₈	—	67	1	no	—
213	C₄⋊C₈	—	67	1	no	—
214	C₄⋊C₈	—	67	1	no	—
215	C₄⋊C₈	—	67	1	no	—
216	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
217	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
218	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
219	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
220	C₂²⋊C₈	—	67	1	no	—
221	C₂²⋊C₈	—	67	1	no	—
222	C₂²⋊C₈	—	67	1	no	—
223	C₂²⋊C₈	—	67	1	no	—
224	C₂²⋊C₈	—	67	1	no	—
225	C₂²⋊C₈	—	67	1	no	—
226	#97	—	67	1	no	—
227	#97	—	67	1	no	—
228	C₄×C₈	—	67	1	no	—
229	C₄×C₈	—	67	1	no	—
230	C₄×C₈	—	67	1	no	—
231	C₄×C₈	—	67	1	no	—
232	C₂×Q₈	—	67	1	no	—
233	C₂×D₄	—	67	1	no	—
234	C₂×D₄	—	67	1	no	—
235	C₂×D₄	—	67	1	no	—
236	C₂²×C₄	—	67	1	no	—
237	C₂²×C₄	—	67	1	no	—
238	C₂²×C₄	—	67	1	no	—
239	C₂²×C₄	—	67	1	no	—
240	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
241	M₁₆	—	4	1	no	—
242	M₁₆	—	268	1	no	—
243	C₂×C₈	—	4	1	no	—
244	C₂×C₈	—	268	1	no	—
245	C₂×C₈	—	268	1	no	—
246	C₂×C₈	—	268	1	no	—
247	C₂×C₈	—	67	1	no	—
248	C₂×C₈	—	67	1	no	—
249	C₂×C₈	—	67	1	no	—
250	C₂×C₈	—	67	1	no	—
251	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
252	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
253	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
254	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
255	C₄⋊C₄	—	67	1	no	—
256	C₄⋊C₄	—	67	1	no	—
257	C₄⋊C₄	—	67	1	no	—
258	C₄⋊C₄	—	67	1	no	—
259	C₄⋊C₄	—	67	1	no	—
260	C₄⋊C₄	—	67	1	no	—
261	C₄⋊C₄	—	67	1	no	—
262	C₄⋊C₄	—	67	1	no	—
263	C₄⋊C₄	C₄×D₆₇	1	1	yes	characteristic
264	C₄⋊C₄	C₄×D₆₇	1	1	yes	characteristic
265	C₂²⋊C₄	—	67	1	no	—
266	C₂²⋊C₄	—	67	1	no	—
267	C₂²⋊C₄	—	67	1	no	—
268	C₂²⋊C₄	—	67	1	no	—
269	C₂²⋊C₄	—	67	1	no	—
270	C₂²⋊C₄	—	67	1	no	—
271	C₂²⋊C₄	—	67	1	no	—
272	C₂²⋊C₄	—	67	1	no	—
273	C₂²⋊C₄	C₄×D₆₇	1	1	yes	characteristic
274	C₂²⋊C₄	C₄×D₆₇	1	1	yes	characteristic
275	C₄²	—	67	1	no	—
276	C₄²	—	67	1	no	—
277	C₄²	—	67	1	no	—
278	C₄²	—	67	1	no	—
279	C₄²	C₂²×D₆₇	1	1	yes	characteristic
280	C₄²	C₂²×D₆₇	1	1	yes	characteristic
281	C₂³	—	67	1	no	—
282	C₂³	—	67	1	no	—
283	C₂³	C₂×C₄×D₆₇	1	1	yes	characteristic
284	Q₈	—	134	2	no	—
285	D₄	—	134	2	no	—
286	D₄	—	134	2	no	—
287	D₄	—	134	2	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	67	1	no	—
291	C₂×C₄	—	67	1	no	—
292	C₂×C₄	—	67	1	no	—
293	C₂×C₄	—	67	1	no	—
294	C₂×C₄	—	67	1	no	—
295	C₂×C₄	—	67	1	no	—
296	C₂×C₄	—	134	2	no	—
297	C₂×C₄	—	134	2	no	—
298	C₂×C₄	—	134	1	no	—
299	C₂×C₄	—	134	1	no	—
300	C₂×C₄	—	134	1	no	—
301	C₂×C₄	—	134	1	no	—
302	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
303	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
304	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
305	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
306	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	134	1	no	—
313	C₈	—	134	1	no	—
314	C₈	—	134	1	no	—
315	C₈	—	134	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	67	2	no	—
320	C₂²	—	134	1	no	—
321	C₂²	—	134	1	no	—
322	C₂²	—	134	1	no	—
323	C₂²	—	134	1	no	—
324	C₂²	—	134	1	no	—
325	C₂²	C₂²×C₄×D₆₇	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	67	2	no	—
332	C₄	—	134	1	no	—
333	C₄	—	134	1	no	—
334	C₄	—	134	1	no	—
335	C₄	—	134	1	no	—
336	C₄	—	134	1	no	—
337	C₄	#14524088	1	1	yes	characteristic
338	C₄	#14524091	1	1	yes	characteristic
339	C₂	—	4	1	no	—
340	C₂	—	268	1	no	—
341	C₂	—	134	2	no	—
342	C₂	#16863877	1	1	yes	characteristic
343	C₂	#131×D₆₇	1	1	yes	characteristic
344	C₂	#131×D₆₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₄
Fitting	#424×C₆₇	C₂
hypercenter	#424	D₆₇
hypocenter	C₆₇	#2994
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
2. #16863525	C₂	semidirect
3. #16863520	C₂	semidirect
4. #16863472	C₂	semidirect
5. #16863466	C₂	semidirect
6. #16863465	C₂	semidirect
7. #16863461	C₂	semidirect
8. C₈×C₂²⋊D₆₇	C₂	semidirect
9. C₈×C₂²⋊D₆₇	C₂	semidirect
10. #16863457	C₂	semidirect
11. #16863456	C₂	semidirect
12. #16863448	C₂	semidirect
13. #16863421	C₂	semidirect
14. C₄⋊C₈×D₆₇	C₂	semidirect
15. #424×C₆₇	C₂	semidirect
173. C₆₇	#2994	semidirect
187. #424	D₆₇	semidirect