Subgroups — #22559803

Order 8576.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16863801	C₂	1	1	yes	maximal, characteristic
2	#16863792	C₂	1	2	yes	maximal
3	#16863748	C₂	1	2	yes	maximal
4	C₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#494×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#14524043	C₂²	1	2	yes	—
7	#14523977	—	2	2	no	—
8	#14523974	—	2	4	no	—
9	#14523966	—	2	4	no	—
10	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×Q₁₆×C₆₇	—	2	2	no	—
13	C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
14	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₃₄	—	2	2	no	—
16	C₂×C₆₇⋊C₈	D₄	1	2	yes	—
17	C₂×C₆₇⋊C₈	D₄	1	2	yes	—
18	C₄⋊Dic₆₇	—	2	2	no	—
19	C₄⋊Dic₆₇	—	2	2	no	—
20	C₂×Q₈×C₆₇	—	2	2	no	—
21	Q₁₆×C₆₇	—	4	4	no	—
22	C₂×C₈×C₆₇	D₄	1	2	yes	—
23	C₄⋊C₄×C₆₇	—	2	2	no	—
24	C₄⋊C₄×C₆₇	—	2	2	no	—
25	C₄²×C₆₇	C₂³	1	1	yes	characteristic
26	Q₈×C₆₇	—	4	4	no	—
27	C₂×C₄×C₆₇	—	4	2	no	—
28	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
31	C₂×Dic₆₇	—	4	2	no	—
32	Dic₁₃₄	—	4	4	no	—
33	C₈×C₆₇	SD₁₆	1	4	yes	—
34	C₆₇⋊C₈	—	2	2	no	—
35	C₆₇⋊C₈	Q₁₆	1	4	yes	complemented
36	C₂²×C₆₇	#127	1	1	yes	characteristic
37	C₄×C₆₇	—	8	2	no	—
38	C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
39	C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
40	C₄×C₆₇	#135	1	2	yes	—
41	Dic₆₇	—	8	2	no	—
42	C₂×C₆₇	#492	1	1	yes	characteristic
43	C₂×C₆₇	#494	1	1	yes	characteristic
44	C₂×C₆₇	#495	1	1	yes	characteristic
45	#1707	—	67	1	no	maximal, Sylow
46	C₆₇	#1707	1	1	yes	characteristic, complemented, Sylow
47	#500	—	67	1	no	—
48	#498	—	67	2	no	—
49	#494	D₆₇	1	1	yes	characteristic
50	#487	—	67	2	no	—
51	C₈²	—	67	1	no	—
52	C₂×Q₁₆	—	2	2	no	—
53	C₄⋊Q₈	—	67	2	no	—
54	C₄⋊Q₈	C₂×D₆₇	1	2	yes	—
55	#107	—	134	2	no	—
56	#106	—	134	4	no	—
57	#103	—	134	4	no	—
58	C₄×C₈	—	67	1	no	—
59	C₄×C₈	—	67	1	no	—
60	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
61	C₂×Q₈	—	2	2	no	—
62	C₂×Q₈	—	134	2	no	—
63	Q₁₆	—	4	4	no	—
64	C₂×C₈	—	67	2	no	—
65	C₂×C₈	—	67	2	no	—
66	C₂×C₈	C₂²⋊D₆₇	1	2	yes	—
67	C₄⋊C₄	—	2	2	no	—
68	C₄⋊C₄	—	2	2	no	—
69	C₄⋊C₄	—	134	2	no	—
70	C₄⋊C₄	—	134	2	no	—
71	C₄²	C₂²×D₆₇	1	1	yes	characteristic
72	Q₈	—	4	4	no	—
73	Q₈	—	268	4	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	268	2	no	—
76	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
77	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
78	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
79	C₈	—	67	4	no	—
80	C₈	—	134	2	no	—
81	C₈	#11761616	1	4	yes	—
82	C₂²	#14524039	1	1	yes	characteristic
83	C₄	—	8	2	no	—
84	C₄	—	536	2	no	—
85	C₄	C₂×#11761616	1	2	yes	—
86	C₄	Q₁₆×D₆₇	1	2	yes	—
87	C₄	#14524066	1	2	yes	—
88	C₂	#16863768	1	1	yes	characteristic
89	C₂	#16863774	1	1	yes	characteristic
90	C₂	#16863781	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14524039
commutator	C₄²×C₆₇	C₂³
Frattini	C₄²	C₂²×D₆₇
Fitting	#494×C₆₇	C₂
socle	C₂²×C₆₇	#127
hypercenter	#494	D₆₇
hypocenter	C₆₇	#1707
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

Complements

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

N	H ≅ G/N	type
35. C₆₇⋊C₈	Q₁₆	semidirect
46. C₆₇	#1707	semidirect