Subgroups — #22574106

Order 8576.

1040 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16864131	C₂	1	4	yes	maximal, complemented
2	#16864118	C₂	1	4	yes	maximal, complemented
3	#16864070	C₂	1	8	yes	maximal, complemented
4	#16864018	C₂	1	4	yes	maximal, complemented
5	#16863970	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863965	C₂	1	1	yes	maximal, characteristic
7	#16863949	C₂	1	1	yes	maximal, characteristic, complemented
8	#16863933	C₂	1	4	yes	maximal, complemented
9	C₂×#14524032	C₂	1	1	yes	maximal, characteristic, complemented
10	#16863870	C₂	1	1	yes	maximal, characteristic, complemented
11	#16863864	C₂	1	1	yes	maximal, characteristic, complemented
12	#519×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#11761628	C₂²	1	1	yes	characteristic
14	C₂²×D₂₆₈	C₂²	1	1	yes	characteristic
15	#14524042	C₂²	1	2	yes	complemented
16	#14524041	C₂²	1	2	yes	complemented
17	#14524040	C₂²	1	2	yes	—
18	#14524038	C₂²	1	8	yes	complemented
19	#14524037	C₂²	1	8	yes	complemented
20	#14524034	C₂²	1	2	yes	complemented
21	#14524032	C₂²	1	2	yes	—
22	#14524032	C₂²	1	4	yes	—
23	#14524032	C₂²	1	2	yes	—
24	#14524032	C₂²	1	2	yes	complemented
25	#14524031	C₂²	1	2	yes	—
26	#14524031	C₂²	1	8	yes	complemented
27	#14524028	C₂²	1	8	yes	complemented
28	#14524028	C₂²	1	8	yes	complemented
29	#14524026	C₂²	1	4	yes	—
30	#14524021	C₂²	1	8	yes	complemented
31	#14524020	C₂²	1	4	yes	—
32	#14524019	C₂²	1	4	yes	complemented
33	#14524018	C₂²	1	4	yes	—
34	#14524013	C₂²	1	4	yes	—
35	C₄×Dic₁₃₄	C₂²	1	4	yes	complemented
36	#14524008	C₂²	1	2	yes	complemented
37	C₄×D₂₆₈	C₂²	1	4	yes	—
38	C₄×D₂₆₈	C₂²	1	8	yes	complemented
39	#14524005	C₂²	1	4	yes	—
40	#14524005	C₂²	1	2	yes	complemented
41	(C₂²⋊D₆₇)⋊C₄	C₂²	1	4	yes	—
42	#14524002	C₂²	1	1	yes	characteristic
43	#14524000	C₂²	1	2	yes	complemented
44	#14524000	C₂²	1	8	yes	complemented
45	C₂×(C₂×D₆₇)⋊C₄	C₂²	1	4	yes	—
46	C₂²⋊C₄×D₆₇	C₂²	1	4	yes	—
47	C₂×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
48	C₄²×D₆₇	C₂²	1	2	yes	complemented
49	C₂²×Q₈×C₆₇	C₂²	1	1	yes	characteristic
50	C₄×Q₈×C₆₇	C₂²	1	4	yes	complemented
51	C₄×Q₈×C₆₇	C₂²	1	4	yes	complemented
52	#117×C₆₇	C₂²	1	4	yes	—
53	#117×C₆₇	C₂²	1	1	yes	characteristic
54	#117×C₆₇	C₂²	1	1	yes	characteristic
55	C₂³×D₆₇	C₂³	1	2	yes	—
56	#11761628	—	2	4	no	—
57	C₂×Dic₁₃₄	C₂³	1	1	yes	characteristic
58	C₂×Dic₁₃₄	C₂³	1	4	yes	—
59	C₂×D₂₆₈	—	2	4	no	—
60	C₂×D₂₆₈	C₂³	1	2	yes	—
61	C₂×D₂₆₈	C₂³	1	1	yes	characteristic
62	C₂×D₂₆₈	C₂³	1	2	yes	—
63	C₂×D₂₆₈	C₂³	1	8	yes	—
64	C₂×C₂²⋊D₆₇	C₂³	1	2	yes	—
65	C₂²×Dic₆₇	C₂³	1	2	yes	—
66	C₂×C₄×D₆₇	C₂³	1	2	yes	—
67	C₂×C₄×D₆₇	C₂³	1	4	yes	—
68	C₂×C₄×D₆₇	C₂³	1	8	yes	—
69	C₄⋊Dic₆₇	C₂³	1	2	yes	—
70	C₄⋊Dic₆₇	C₂³	1	4	yes	—
71	Dic₆₇⋊C₄	C₂³	1	4	yes	—
72	Dic₆₇⋊C₄	C₂³	1	4	yes	—
73	Dic₆₇⋊C₄	C₂³	1	8	yes	—
74	Dic₆₇⋊C₄	C₂³	1	4	yes	—
75	C₂²⋊Dic₆₇	C₂³	1	2	yes	—
76	(C₂×D₆₇)⋊C₄	C₂³	1	8	yes	—
77	(C₂×D₆₇)⋊C₄	C₂³	1	8	yes	—
78	(C₂×D₆₇)⋊C₄	C₂³	1	8	yes	—
79	(C₂×D₆₇)⋊C₄	C₂³	1	4	yes	—
80	(C₂×D₆₇)⋊C₄	C₂³	1	8	yes	—
81	(C₂×D₆₇)⋊C₄	C₂³	1	8	yes	—
82	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
83	C₄×Dic₆₇	C₂³	1	4	yes	—
84	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
85	C₄×Dic₆₇	C₂³	1	2	yes	—
86	C₄×Dic₆₇	C₂³	1	2	yes	—
87	C₂×Q₈×C₆₇	—	2	4	no	—
88	C₂×Q₈×C₆₇	C₂³	1	2	yes	—
89	C₂×Q₈×C₆₇	C₂³	1	2	yes	—
90	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
91	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
92	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
93	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
94	C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
95	C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
96	C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
97	C₄²×C₆₇	C₂³	1	2	yes	—
98	C₄²×C₆₇	C₂³	1	8	yes	—
99	C₄²×C₆₇	C₂³	1	2	yes	—
100	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
101	C₂²×D₆₇	—	2	4	no	—
102	C₂²×D₆₇	—	4	4	no	—
103	C₂²×D₆₇	C₂⁴	1	4	yes	—
104	C₂²×D₆₇	C₂⁴	1	2	yes	—
105	Q₈×C₆₇	—	2	4	no	—
106	Q₈×C₆₇	—	2	4	no	—
107	C₂×C₄×C₆₇	—	2	4	no	—
108	C₂×C₄×C₆₇	#41	1	4	yes	—
109	C₂×C₄×C₆₇	C₂⁴	1	4	yes	—
110	C₂×C₄×C₆₇	C₂⁴	1	4	yes	—
111	C₂×C₄×C₆₇	C₂⁴	1	4	yes	—
112	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
114	C₂²⋊D₆₇	—	2	4	no	—
115	C₂×Dic₆₇	—	4	2	no	—
116	C₂×Dic₆₇	C₂⁴	1	2	yes	—
117	C₂×Dic₆₇	C₂⁴	1	2	yes	—
118	C₂×Dic₆₇	C₂⁴	1	2	yes	—
119	C₂×Dic₆₇	C₂⁴	1	4	yes	—
120	D₂₆₈	—	2	4	no	—
121	D₂₆₈	—	2	4	no	—
122	D₂₆₈	—	2	2	no	—
123	D₂₆₈	—	4	8	no	—
124	C₄×D₆₇	—	2	4	no	—
125	C₄×D₆₇	—	4	8	no	—
126	C₄×D₆₇	—	4	4	no	—
127	Dic₁₃₄	—	2	2	no	—
128	Dic₁₃₄	—	4	4	no	—
129	C₂²×C₆₇	—	2	2	no	—
130	C₂²×C₆₇	C₂×#41	1	2	yes	—
131	C₂²×C₆₇	C₂⁵	1	1	yes	characteristic
132	C₂×D₆₇	—	2	2	no	—
133	C₂×D₆₇	—	2	4	no	—
134	C₂×D₆₇	—	2	4	no	—
135	C₂×D₆₇	—	2	8	no	—
136	C₂×D₆₇	—	8	4	no	—
137	C₄×C₆₇	—	2	4	no	—
138	C₄×C₆₇	—	2	4	no	—
139	C₄×C₆₇	—	2	4	no	—
140	C₄×C₆₇	C₂×#41	1	2	yes	—
141	C₄×C₆₇	C₂×#41	1	2	yes	—
142	Dic₆₇	—	2	2	no	—
143	Dic₆₇	—	2	2	no	—
144	Dic₆₇	—	2	4	no	—
145	Dic₆₇	—	2	2	no	—
146	C₂×C₆₇	—	2	2	no	—
147	C₂×C₆₇	C₂²×#41	1	1	yes	characteristic
148	C₂×C₆₇	#585	1	2	yes	—
149	D₆₇	—	4	2	no	—
150	D₆₇	—	4	4	no	—
151	#3510	—	67	1	no	maximal, Sylow
152	C₆₇	#3510	1	1	yes	characteristic, complemented, Sylow
153	#559	—	67	4	no	—
154	#556	—	67	4	no	—
155	#551	—	67	8	no	—
156	#542	—	67	4	no	—
157	#536	—	67	1	no	—
158	#535	—	67	1	no	—
159	#533	—	67	1	no	—
160	#529	—	67	4	no	—
161	C₂×#124	—	67	1	no	—
162	#519	D₆₇	1	1	yes	characteristic, complemented
163	#518	—	67	1	no	—
164	C₄×#41	—	67	1	no	—
165	C₂×#41	—	67	1	no	—
166	C₂²×Q₈	C₂×D₆₇	1	1	yes	characteristic
167	C₂²×D₄	—	67	1	no	—
168	C₄⋊Q₈	—	67	2	no	—
169	C₄⋊Q₈	—	67	2	no	—
170	#127	—	67	2	no	—
171	#126	—	67	8	no	—
172	#126	—	67	8	no	—
173	#125	—	67	2	no	—
174	#124	—	67	2	no	—
175	#124	—	67	2	no	—
176	#124	—	67	4	no	—
177	#124	—	67	2	no	—
178	#124	—	67	2	no	—
179	#124	—	67	8	no	—
180	#123	—	67	4	no	—
181	#123	—	67	8	no	—
182	#123	—	67	8	no	—
183	C₂²⋊Q₈	—	67	4	no	—
184	C₂²⋊Q₈	—	67	8	no	—
185	C₂²⋊Q₈	—	67	4	no	—
186	#121	—	67	4	no	—
187	#120	—	67	4	no	—
188	C₄×Q₈	—	67	2	no	—
189	C₄×Q₈	—	67	4	no	—
190	C₄×Q₈	C₂×D₆₇	1	4	yes	complemented
191	C₄×Q₈	C₂×D₆₇	1	4	yes	complemented
192	C₄×D₄	—	67	2	no	—
193	C₄×D₄	—	67	4	no	—
194	C₄×D₄	—	67	4	no	—
195	C₄×D₄	—	67	4	no	—
196	C₄×D₄	—	67	8	no	—
197	#117	—	67	1	no	—
198	#117	—	67	2	no	—
199	#117	—	67	8	no	—
200	#117	C₂×D₆₇	1	1	yes	characteristic
201	#117	C₂×D₆₇	1	1	yes	characteristic
202	#117	C₂×D₆₇	1	4	yes	—
203	C₂×C₂²⋊C₄	—	67	4	no	—
204	C₂×C₂²⋊C₄	—	67	4	no	—
205	C₂×C₄²	—	67	1	no	—
206	C₂×C₄²	—	67	2	no	—
207	C₂⁴	—	67	2	no	—
208	#41	—	134	4	no	—
209	C₂×Q₈	—	2	4	no	—
210	C₂×Q₈	—	67	4	no	—
211	C₂×Q₈	—	67	1	no	—
212	C₂×Q₈	C₂²×D₆₇	1	2	yes	—
213	C₂×Q₈	C₂²×D₆₇	1	2	yes	—
214	C₂×D₄	—	67	2	no	—
215	C₂×D₄	—	67	8	no	—
216	C₂×D₄	—	67	2	no	—
217	C₂×D₄	—	67	2	no	—
218	C₂×D₄	—	67	1	no	—
219	C₂×D₄	—	134	4	no	—
220	C₂²×C₄	—	67	2	no	—
221	C₂²×C₄	—	67	4	no	—
222	C₂²×C₄	—	67	8	no	—
223	C₂²×C₄	—	67	2	no	—
224	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
225	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
226	C₄⋊C₄	—	67	4	no	—
227	C₄⋊C₄	—	67	2	no	—
228	C₄⋊C₄	—	67	4	no	—
229	C₄⋊C₄	—	67	8	no	—
230	C₄⋊C₄	—	67	4	no	—
231	C₄⋊C₄	—	67	4	no	—
232	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
233	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
234	C₄⋊C₄	C₂²×D₆₇	1	8	yes	—
235	C₂²⋊C₄	—	67	2	no	—
236	C₂²⋊C₄	—	67	8	no	—
237	C₂²⋊C₄	—	67	8	no	—
238	C₂²⋊C₄	—	67	8	no	—
239	C₂²⋊C₄	—	67	4	no	—
240	C₂²⋊C₄	—	67	8	no	—
241	C₂²⋊C₄	—	67	8	no	—
242	C₂²⋊C₄	C₂²×D₆₇	1	2	yes	—
243	C₂²⋊C₄	C₂²×D₆₇	1	2	yes	—
244	C₄²	—	67	1	no	—
245	C₄²	—	67	4	no	—
246	C₄²	—	67	1	no	—
247	C₄²	—	67	2	no	—
248	C₄²	—	67	2	no	—
249	C₄²	C₂²×D₆₇	1	2	yes	—
250	C₄²	C₂²×D₆₇	1	8	yes	—
251	C₄²	C₂²×D₆₇	1	2	yes	—
252	C₂³	—	268	4	no	—
253	C₂³	—	67	4	no	—
254	C₂³	—	67	2	no	—
255	C₂³	—	134	4	no	—
256	C₂³	C₂³×D₆₇	1	1	yes	characteristic
257	Q₈	—	2	4	no	—
258	Q₈	—	2	4	no	—
259	Q₈	—	268	4	no	—
260	Q₈	—	134	2	no	—
261	D₄	—	268	8	no	—
262	D₄	—	134	4	no	—
263	D₄	—	134	4	no	—
264	D₄	—	134	4	no	—
265	D₄	—	134	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	268	4	no	—
268	C₂×C₄	—	268	8	no	—
269	C₂×C₄	—	268	2	no	—
270	C₂×C₄	—	67	2	no	—
271	C₂×C₄	—	67	2	no	—
272	C₂×C₄	—	67	2	no	—
273	C₂×C₄	—	67	4	no	—
274	C₂×C₄	—	134	4	no	—
275	C₂×C₄	#11761628	1	4	yes	—
276	C₂×C₄	C₂³×D₆₇	1	4	yes	—
277	C₂×C₄	C₂³×D₆₇	1	4	yes	—
278	C₂×C₄	C₂³×D₆₇	1	4	yes	—
279	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
280	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
281	C₂²	—	2	2	no	—
282	C₂²	—	536	4	no	—
283	C₂²	—	134	2	no	—
284	C₂²	—	134	4	no	—
285	C₂²	—	134	4	no	—
286	C₂²	—	134	8	no	—
287	C₂²	C₂×#11761628	1	2	yes	—
288	C₂²	C₂⁴×D₆₇	1	1	yes	characteristic
289	C₄	—	2	4	no	—
290	C₄	—	2	4	no	—
291	C₄	—	2	4	no	—
292	C₄	—	134	2	no	—
293	C₄	—	134	2	no	—
294	C₄	—	134	4	no	—
295	C₄	—	134	2	no	—
296	C₄	C₂×#11761628	1	2	yes	—
297	C₄	C₂×#11761628	1	2	yes	—
298	C₂	—	2	2	no	—
299	C₂	—	268	4	no	—
300	C₂	—	268	2	no	—
301	C₂	C₂²×#11761628	1	1	yes	characteristic
302	C₂	#16864233	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₆₇
commutator, socle	C₂²×C₆₇	C₂⁵
Fitting	#519×C₆₇	C₂
hypercenter	#519	D₆₇
hypocenter	C₆₇	#3510
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ #519

Complements

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

N	H ≅ G/N	type
1. #16864131	C₂	semidirect
2. #16864118	C₂	semidirect
3. #16864070	C₂	semidirect
4. #16864018	C₂	semidirect
5. #16863970	C₂	semidirect
7. #16863949	C₂	semidirect
8. #16863933	C₂	semidirect
9. C₂×#14524032	C₂	semidirect
10. #16863870	C₂	semidirect
11. #16863864	C₂	semidirect
12. #519×C₆₇	C₂	semidirect
15. #14524042	C₂²	semidirect
16. #14524041	C₂²	semidirect
18. #14524038	C₂²	semidirect
19. #14524037	C₂²	semidirect
20. #14524034	C₂²	semidirect
24. #14524032	C₂²	semidirect
26. #14524031	C₂²	semidirect
27. #14524028	C₂²	semidirect
28. #14524028	C₂²	semidirect
30. #14524021	C₂²	semidirect
32. #14524019	C₂²	semidirect
35. C₄×Dic₁₃₄	C₂²	semidirect
36. #14524008	C₂²	semidirect
38. C₄×D₂₆₈	C₂²	semidirect
40. #14524005	C₂²	semidirect
43. #14524000	C₂²	semidirect
44. #14524000	C₂²	semidirect
48. C₄²×D₆₇	C₂²	semidirect
50. C₄×Q₈×C₆₇	C₂²	semidirect
51. C₄×Q₈×C₆₇	C₂²	semidirect
152. C₆₇	#3510	semidirect
162. #519	D₆₇	semidirect
190. C₄×Q₈	C₂×D₆₇	semidirect
191. C₄×Q₈	C₂×D₆₇	semidirect