Subgroups — #22574620

Order 8576.

1002 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16864150	C₂	1	4	yes	maximal, complemented
2	#16864131	C₂	1	2	yes	maximal, complemented
3	Q₈×Dic₁₃₄	C₂	1	2	yes	maximal, complemented
4	#16864120	C₂	1	2	yes	maximal, complemented
5	#16864119	C₂	1	2	yes	maximal, complemented
6	Q₈×D₂₆₈	C₂	1	2	yes	maximal, complemented
7	#16864061	C₂	1	2	yes	maximal, complemented
8	#16864019	C₂	1	4	yes	maximal, complemented
9	#16864017	C₂	1	4	yes	maximal, complemented
10	#16863969	C₂	1	4	yes	maximal, complemented
11	#16863953	C₂	1	1	yes	maximal, characteristic, complemented
12	#16863946	C₂	1	1	yes	maximal, characteristic
13	#533×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#11761628	C₂²	1	2	yes	—
15	C₂×Q₈×D₆₇	C₂²	1	4	yes	—
16	#14524043	C₂²	1	4	yes	—
17	#14524043	C₂²	1	1	yes	characteristic
18	#14524043	C₂²	1	1	yes	characteristic
19	#14524043	C₂²	1	1	yes	characteristic
20	#14524042	C₂²	1	4	yes	—
21	#14524042	C₂²	1	8	yes	—
22	#14524040	C₂²	1	1	yes	characteristic
23	#14524038	C₂²	1	8	yes	—
24	#14524038	C₂²	1	8	yes	—
25	#14524035	C₂²	1	2	yes	—
26	#14524032	C₂²	1	4	yes	—
27	#14524031	C₂²	1	4	yes	—
28	#14524031	C₂²	1	2	yes	—
29	#14524029	C₂²	1	8	yes	—
30	#14524028	C₂²	1	8	yes	—
31	#14524025	C₂²	1	8	yes	—
32	#14524023	C₂²	1	8	yes	—
33	#14524023	C₂²	1	4	yes	—
34	#14524022	C₂²	1	2	yes	—
35	#14524022	C₂²	1	2	yes	—
36	#14524021	C₂²	1	8	yes	—
37	#14524019	C₂²	1	4	yes	—
38	#14524017	C₂²	1	2	yes	—
39	#14524017	C₂²	1	2	yes	—
40	Q₈×Dic₆₇	C₂²	1	4	yes	—
41	C₄×Dic₁₃₄	C₂²	1	4	yes	—
42	C₄×Dic₁₃₄	C₂²	1	2	yes	—
43	C₄×Dic₁₃₄	C₂²	1	2	yes	—
44	C₄×D₂₆₈	C₂²	1	4	yes	—
45	C₄×D₂₆₈	C₂²	1	2	yes	—
46	C₄×D₂₆₈	C₂²	1	2	yes	—
47	#14524005	C₂²	1	4	yes	—
48	#14524002	C₂²	1	2	yes	—
49	#14524001	C₂²	1	4	yes	—
50	C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
51	C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
52	#125×C₆₇	C₂²	1	2	yes	—
53	C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
54	C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
55	C₄×Q₈×C₆₇	C₂²	1	2	yes	—
56	C₄×Q₈×C₆₇	C₂²	1	2	yes	—
57	#117×C₆₇	C₂²	1	2	yes	—
58	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
59	#11761628	—	4	4	no	—
60	C₂×Dic₁₃₄	C₂³	1	2	yes	—
61	C₂×Dic₁₃₄	C₂³	1	2	yes	—
62	C₂×Dic₁₃₄	C₂³	1	8	yes	—
63	C₂×Dic₁₃₄	C₂³	1	2	yes	—
64	Q₈×D₆₇	—	4	8	no	—
65	C₂×D₂₆₈	C₂³	1	2	yes	—
66	C₂×D₂₆₈	C₂³	1	2	yes	—
67	C₂×D₂₆₈	C₂³	1	2	yes	—
68	C₂×C₂²⋊D₆₇	C₂³	1	4	yes	—
69	C₂×C₄×D₆₇	C₂³	1	4	yes	—
70	C₂×C₄×D₆₇	C₂³	1	8	yes	—
71	C₄⋊Dic₆₇	C₂³	1	2	yes	—
72	C₄⋊Dic₆₇	C₂³	1	2	yes	—
73	C₄⋊Dic₆₇	C₂³	1	2	yes	—
74	C₄⋊Dic₆₇	C₂³	1	2	yes	—
75	C₄⋊Dic₆₇	C₂³	1	2	yes	—
76	C₄⋊Dic₆₇	C₂³	1	2	yes	—
77	C₄⋊Dic₆₇	C₂³	1	8	yes	—
78	Dic₆₇⋊C₄	C₂³	1	8	yes	—
79	Dic₆₇⋊C₄	C₂³	1	4	yes	—
80	Dic₆₇⋊C₄	C₂³	1	4	yes	—
81	C₂²⋊Dic₆₇	C₂³	1	4	yes	—
82	(C₂×D₆₇)⋊C₄	C₂³	1	8	yes	—
83	(C₂×D₆₇)⋊C₄	C₂³	1	8	yes	—
84	(C₂×D₆₇)⋊C₄	C₂³	1	4	yes	—
85	(C₂×D₆₇)⋊C₄	C₂³	1	4	yes	—
86	(C₂×D₆₇)⋊C₄	C₂³	1	8	yes	—
87	C₄×Dic₆₇	C₂³	1	8	yes	—
88	C₄×Dic₆₇	C₂³	1	4	yes	—
89	C₂×Q₈×C₆₇	C₂³	1	2	yes	—
90	C₂×Q₈×C₆₇	C₂³	1	2	yes	—
91	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
93	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
94	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
95	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
96	C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
97	C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
98	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
99	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
100	C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
101	C₄²×C₆₇	C₂³	1	1	yes	characteristic
102	C₄²×C₆₇	C₂³	1	1	yes	characteristic
103	C₄²×C₆₇	C₂³	1	1	yes	characteristic
104	C₄²×C₆₇	C₂³	1	1	yes	characteristic
105	C₄²×C₆₇	C₂³	1	4	yes	—
106	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
107	C₂²×D₆₇	C₂⁴	1	4	yes	—
108	Q₈×C₆₇	—	2	4	no	—
109	Q₈×C₆₇	—	2	4	no	—
110	C₂×C₄×C₆₇	—	2	2	no	—
111	C₂×C₄×C₆₇	—	2	2	no	—
112	C₂×C₄×C₆₇	—	2	2	no	—
113	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₂⁴	1	4	yes	—
117	C₂×C₄×C₆₇	C₂⁴	1	4	yes	—
118	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
120	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
121	C₂²⋊D₆₇	—	4	4	no	—
122	C₂×Dic₆₇	C₂⁴	1	4	yes	—
123	C₂×Dic₆₇	C₂⁴	1	4	yes	—
124	C₂×Dic₆₇	C₂⁴	1	4	yes	—
125	D₂₆₈	—	2	4	no	—
126	D₂₆₈	—	2	4	no	—
127	D₂₆₈	—	4	2	no	—
128	C₄×D₆₇	—	2	8	no	—
129	C₄×D₆₇	—	4	8	no	—
130	Dic₁₃₄	—	2	4	no	—
131	Dic₁₃₄	—	2	4	no	—
132	Dic₁₃₄	—	4	8	no	—
133	Dic₁₃₄	—	4	2	no	—
134	C₂²×C₆₇	—	2	1	no	—
135	C₂²×C₆₇	—	2	1	no	—
136	C₂²×C₆₇	—	2	1	no	—
137	C₂²×C₆₇	C₂⁵	1	1	yes	characteristic
138	C₂×D₆₇	—	2	4	no	—
139	C₂×D₆₇	—	2	4	no	—
140	C₂×D₆₇	—	2	4	no	—
141	C₄×C₆₇	—	2	4	no	—
142	C₄×C₆₇	—	2	1	no	—
143	C₄×C₆₇	—	2	1	no	—
144	C₄×C₆₇	—	2	1	no	—
145	C₄×C₆₇	—	2	4	no	—
146	C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
147	C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
148	C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
149	Dic₆₇	—	2	4	no	—
150	Dic₆₇	—	2	4	no	—
151	Dic₆₇	—	2	4	no	—
152	C₂×C₆₇	—	4	1	no	—
153	C₂×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
154	C₂×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
155	C₂×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
156	D₆₇	—	4	4	no	—
157	#3550	—	67	1	no	maximal, Sylow
158	C₆₇	#3550	1	1	yes	characteristic, complemented, Sylow
159	#563	—	67	4	no	—
160	#559	—	67	2	no	—
161	Q₈²	—	67	2	no	—
162	#556	—	67	2	no	—
163	#556	—	67	2	no	—
164	D₄×Q₈	—	67	2	no	—
165	D₄×Q₈	—	67	2	no	—
166	#542	—	67	4	no	—
167	#542	—	67	4	no	—
168	#536	—	67	4	no	—
169	#533	—	67	1	no	—
170	#533	D₆₇	1	1	yes	characteristic, complemented
171	#532	—	67	1	no	—
172	C₂×#41	—	67	2	no	—
173	C₂²×Q₈	—	67	4	no	—
174	C₄⋊Q₈	—	67	8	no	—
175	C₄⋊Q₈	—	67	4	no	—
176	C₄⋊Q₈	—	67	1	no	—
177	C₄⋊Q₈	—	67	1	no	—
178	C₄⋊Q₈	—	67	1	no	—
179	C₄⋊Q₈	—	67	4	no	—
180	C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
181	C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
182	#127	—	67	1	no	—
183	#126	—	67	8	no	—
184	#126	—	67	8	no	—
185	#125	—	67	2	no	—
186	#125	C₂×D₆₇	1	2	yes	—
187	#124	—	67	4	no	—
188	#124	—	67	8	no	—
189	#124	—	67	2	no	—
190	#124	—	67	4	no	—
191	#123	—	67	8	no	—
192	#123	—	67	8	no	—
193	C₂²⋊Q₈	—	67	4	no	—
194	C₂²⋊Q₈	—	67	2	no	—
195	C₂²⋊Q₈	—	67	2	no	—
196	C₂²⋊Q₈	—	67	4	no	—
197	C₂²⋊Q₈	—	67	8	no	—
198	C₂²⋊Q₈	—	67	8	no	—
199	C₂²⋊Q₈	C₂×D₆₇	1	2	yes	—
200	C₂²⋊Q₈	C₂×D₆₇	1	2	yes	—
201	#121	—	67	2	no	—
202	#121	—	67	2	no	—
203	C₄×Q₈	—	67	4	no	—
204	C₄×Q₈	—	67	2	no	—
205	C₄×Q₈	—	67	2	no	—
206	C₄×Q₈	—	67	4	no	—
207	C₄×Q₈	C₂×D₆₇	1	2	yes	—
208	C₄×Q₈	C₂×D₆₇	1	2	yes	—
209	C₄×D₄	—	67	4	no	—
210	C₄×D₄	—	67	2	no	—
211	C₄×D₄	—	67	2	no	—
212	C₄×D₄	—	67	4	no	—
213	#117	—	67	2	no	—
214	#117	—	67	4	no	—
215	#117	C₂×D₆₇	1	2	yes	—
216	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
217	#41	—	268	4	no	—
218	C₂×Q₈	—	268	8	no	—
219	C₂×Q₈	—	67	8	no	—
220	C₂×Q₈	—	67	2	no	—
221	C₂×Q₈	—	67	2	no	—
222	C₂×Q₈	—	67	2	no	—
223	C₂×Q₈	C₂²×D₆₇	1	2	yes	—
224	C₂×Q₈	C₂²×D₆₇	1	2	yes	—
225	C₂×D₄	—	67	4	no	—
226	C₂×D₄	—	67	2	no	—
227	C₂×D₄	—	67	2	no	—
228	C₂×D₄	—	67	2	no	—
229	C₂²×C₄	—	67	8	no	—
230	C₂²×C₄	—	67	4	no	—
231	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
232	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
233	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
234	C₄⋊C₄	—	67	2	no	—
235	C₄⋊C₄	—	67	2	no	—
236	C₄⋊C₄	—	67	2	no	—
237	C₄⋊C₄	—	67	2	no	—
238	C₄⋊C₄	—	67	2	no	—
239	C₄⋊C₄	—	67	2	no	—
240	C₄⋊C₄	—	67	4	no	—
241	C₄⋊C₄	—	67	8	no	—
242	C₄⋊C₄	—	67	8	no	—
243	C₄⋊C₄	—	67	4	no	—
244	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
245	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
246	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
247	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
248	C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
249	C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
250	C₂²⋊C₄	—	67	4	no	—
251	C₂²⋊C₄	—	67	8	no	—
252	C₂²⋊C₄	—	67	4	no	—
253	C₂²⋊C₄	—	67	4	no	—
254	C₂²⋊C₄	—	67	8	no	—
255	C₂²⋊C₄	—	67	8	no	—
256	C₂²⋊C₄	C₂²×D₆₇	1	4	yes	—
257	C₄²	—	67	4	no	—
258	C₄²	—	67	8	no	—
259	C₄²	C₂²×D₆₇	1	1	yes	characteristic
260	C₄²	C₂²×D₆₇	1	1	yes	characteristic
261	C₄²	C₂²×D₆₇	1	1	yes	characteristic
262	C₄²	C₂²×D₆₇	1	1	yes	characteristic
263	C₄²	C₂²×D₆₇	1	4	yes	—
264	C₂³	—	67	4	no	—
265	C₂³	C₂³×D₆₇	1	1	yes	characteristic
266	Q₈	—	2	4	no	—
267	Q₈	—	2	4	no	—
268	Q₈	—	268	8	no	—
269	Q₈	—	268	2	no	—
270	Q₈	—	134	4	no	—
271	Q₈	—	134	4	no	—
272	D₄	—	268	2	no	—
273	D₄	—	268	4	no	—
274	D₄	—	134	4	no	—
275	D₄	—	134	4	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	268	8	no	—
280	C₂×C₄	—	67	4	no	—
281	C₂×C₄	—	67	4	no	—
282	C₂×C₄	—	67	4	no	—
283	C₂×C₄	—	134	8	no	—
284	C₂×C₄	C₂³×D₆₇	1	4	yes	—
285	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
286	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
287	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
288	C₂×C₄	C₂³×D₆₇	1	4	yes	—
289	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
290	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
291	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	134	4	no	—
296	C₂²	—	134	4	no	—
297	C₂²	—	134	4	no	—
298	C₂²	C₂⁴×D₆₇	1	1	yes	characteristic
299	C₄	—	2	1	no	—
300	C₄	—	2	4	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	4	no	—
303	C₄	—	2	1	no	—
304	C₄	—	134	4	no	—
305	C₄	—	134	4	no	—
306	C₄	—	134	4	no	—
307	C₄	#14524089	1	2	yes	—
308	C₄	#14524090	1	2	yes	—
309	C₄	#14524091	1	2	yes	—
310	C₂	—	4	1	no	—
311	C₂	—	268	4	no	—
312	C₂	C₂×#14524089	1	1	yes	characteristic
313	C₂	C₂×#14524090	1	1	yes	characteristic
314	C₂	C₂×#14524091	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₆₇
commutator, socle	C₂²×C₆₇	C₂⁵
Fitting	#533×C₆₇	C₂
hypercenter	#533	D₆₇
hypocenter	C₆₇	#3550
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ #533

Complements

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

N	H ≅ G/N	type
1. #16864150	C₂	semidirect
2. #16864131	C₂	semidirect
3. Q₈×Dic₁₃₄	C₂	semidirect
4. #16864120	C₂	semidirect
5. #16864119	C₂	semidirect
6. Q₈×D₂₆₈	C₂	semidirect
7. #16864061	C₂	semidirect
8. #16864019	C₂	semidirect
9. #16864017	C₂	semidirect
10. #16863969	C₂	semidirect
11. #16863953	C₂	semidirect
13. #533×C₆₇	C₂	semidirect
158. C₆₇	#3550	semidirect
170. #533	D₆₇	semidirect