Subgroups — #22562489

Order 8576.

128 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	#16863223	C₂	1	4	yes	maximal
3	C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#500×C₆₇	C₂	1	1	yes	maximal, characteristic
5	#14524043	—	2	1	no	—
6	C₂×C₆₇⋊C₁₆	C₂²	1	2	yes	—
7	#14523977	—	2	1	no	—
8	#14523977	C₂²	1	2	yes	—
9	C₄⋊Q₈×C₆₇	—	2	1	no	—
10	#107×C₆₇	—	2	1	no	—
11	#107×C₆₇	C₂²	1	2	yes	—
12	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×Dic₁₃₄	—	4	1	no	—
14	C₄⋊Dic₆₇	—	2	2	no	—
15	C₄⋊Dic₆₇	—	4	1	no	—
16	C₆₇⋊C₁₆	Q₈	1	4	yes	complemented
17	C₂×Q₈×C₆₇	—	4	1	no	—
18	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
19	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₆₇	—	2	2	no	—
21	C₄⋊C₄×C₆₇	—	4	1	no	—
22	C₄²×C₆₇	D₄	1	1	yes	characteristic
23	Q₈×C₆₇	—	8	2	no	—
24	C₂×C₄×C₆₇	—	4	2	no	—
25	C₂×C₄×C₆₇	D₈	1	1	yes	characteristic
26	C₂×C₄×C₆₇	D₈	1	1	yes	characteristic
27	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
28	C₂×Dic₆₇	—	4	2	no	—
29	Dic₁₃₄	—	8	2	no	—
30	C₈×C₆₇	Q₁₆	1	2	yes	—
31	C₈×C₆₇	C₂×Q₈	1	2	yes	—
32	C₂²×C₆₇	C₂×D₈	1	1	yes	characteristic
33	C₄×C₆₇	—	8	2	no	—
34	C₄×C₆₇	SD₃₂	1	2	yes	—
35	C₄×C₆₇	SD₃₂	1	2	yes	—
36	C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
37	C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
38	Dic₆₇	—	8	2	no	—
39	C₂×C₆₇	#500	1	1	yes	characteristic
40	C₂×C₆₇	C₂×SD₃₂	1	1	yes	characteristic
41	C₂×C₆₇	C₂×SD₃₂	1	1	yes	characteristic
42	#2254	—	67	1	no	maximal, Sylow
43	C₆₇	#2254	1	1	yes	characteristic, complemented, Sylow
44	#500	—	67	1	no	—
45	#500	D₆₇	1	1	yes	characteristic
46	#367	—	67	4	no	—
47	C₄×C₁₆	—	67	1	no	—
48	C₄⋊Q₈	—	2	1	no	—
49	C₄⋊Q₈	—	134	1	no	—
50	C₂×C₁₆	—	67	2	no	—
51	#107	—	2	1	no	—
52	#107	—	67	2	no	—
53	#107	—	134	1	no	—
54	#107	C₂×D₆₇	1	2	yes	—
55	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
56	C₂×Q₈	—	4	1	no	—
57	C₂×Q₈	—	268	1	no	—
58	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
59	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
60	C₄⋊C₄	—	2	2	no	—
61	C₄⋊C₄	—	4	1	no	—
62	C₄⋊C₄	—	268	1	no	—
63	C₄⋊C₄	—	134	2	no	—
64	C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
65	C₁₆	—	67	4	no	—
66	Q₈	—	8	2	no	—
67	Q₈	—	536	2	no	—
68	C₂×C₄	—	4	2	no	—
69	C₂×C₄	—	268	2	no	—
70	C₂×C₄	#11761613	1	1	yes	characteristic
71	C₂×C₄	#11761613	1	1	yes	characteristic
72	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
73	C₈	#11761618	1	2	yes	—
74	C₈	Q₈×D₆₇	1	2	yes	—
75	C₂²	C₂×#11761613	1	1	yes	characteristic
76	C₄	—	8	2	no	—
77	C₄	—	536	2	no	—
78	C₄	#14523986	1	2	yes	—
79	C₄	#14523987	1	2	yes	—
80	C₄	#14524041	1	1	yes	characteristic
81	C₄	C₂×#11761618	1	1	yes	characteristic
82	C₂	#16863799	1	1	yes	characteristic
83	C₂	C₂×#14523986	1	1	yes	characteristic
84	C₂	C₂×#14523987	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#11761613
commutator	C₂×C₈×C₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	#500×C₆₇	C₂
socle	C₂²×C₆₇	C₂×D₈
hypercenter	#500	D₆₇
hypocenter	C₆₇	#2254
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

Complements

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

N	H ≅ G/N	type
16. C₆₇⋊C₁₆	Q₈	semidirect
43. C₆₇	#2254	semidirect