Subgroups — #22562490

Order 8576.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16863800	C₂	1	2	yes	maximal
2	#16863224	C₂	1	1	yes	maximal, characteristic
3	#16863223	C₂	1	2	yes	maximal
4	C₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic
5	#367×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#14524042	—	2	2	no	—
7	C₂×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	#14523977	C₂²	1	2	yes	—
9	#14523976	—	2	2	no	—
10	C₈×Dic₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
12	#107×C₆₇	C₂²	1	2	yes	—
13	C₂×Dic₁₃₄	—	4	2	no	—
14	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
15	C₄⋊Dic₆₇	—	2	2	no	—
16	Dic₆₇⋊C₄	—	4	2	no	—
17	C₄×Dic₆₇	D₄	1	1	yes	characteristic
18	C₆₇⋊C₁₆	Q₈	1	2	yes	complemented
19	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₆₇	—	2	2	no	—
21	C₁₆×C₆₇	Q₈	1	2	yes	complemented
22	C₂×C₄×C₆₇	—	4	2	no	—
23	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
24	C₂×Dic₆₇	—	4	2	no	—
25	C₂×Dic₆₇	D₈	1	2	yes	—
26	Dic₁₃₄	—	8	4	no	—
27	C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
28	C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
29	C₆₇⋊C₈	Q₁₆	1	2	yes	—
30	C₂²×C₆₇	C₂×D₈	1	1	yes	characteristic
31	C₄×C₆₇	—	8	2	no	—
32	C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
33	C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
34	Dic₆₇	—	8	2	no	—
35	Dic₆₇	SD₃₂	1	4	yes	—
36	C₂×C₆₇	#500	1	1	yes	characteristic
37	C₂×C₆₇	C₂×SD₃₂	1	2	yes	—
38	#2254	—	67	1	no	maximal, Sylow
39	C₆₇	#2254	1	1	yes	characteristic, complemented, Sylow
40	#500	—	67	2	no	—
41	#367	—	67	1	no	—
42	#367	—	67	2	no	—
43	#367	D₆₇	1	1	yes	characteristic
44	C₄×C₁₆	—	67	1	no	—
45	C₄⋊Q₈	—	134	2	no	—
46	C₂×C₁₆	—	67	1	no	—
47	C₂×C₁₆	C₂×D₆₇	1	1	yes	characteristic
48	#107	—	67	2	no	—
49	#107	—	134	2	no	—
50	#107	C₂×D₆₇	1	2	yes	—
51	C₄×C₈	—	67	1	no	—
52	C₂×Q₈	—	268	2	no	—
53	C₂×C₈	—	67	1	no	—
54	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
55	C₄⋊C₄	—	2	2	no	—
56	C₄⋊C₄	—	268	2	no	—
57	C₄⋊C₄	—	134	2	no	—
58	C₄²	—	67	1	no	—
59	C₁₆	—	67	2	no	—
60	C₁₆	Dic₁₃₄	1	2	yes	complemented
61	Q₈	—	536	4	no	—
62	C₂×C₄	—	4	2	no	—
63	C₂×C₄	—	268	2	no	—
64	C₂×C₄	—	67	2	no	—
65	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
66	C₈	—	67	2	no	—
67	C₈	Q₈×D₆₇	1	1	yes	characteristic
68	C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
69	C₂²	D₈×D₆₇	1	1	yes	characteristic
70	C₄	—	8	2	no	—
71	C₄	—	536	2	no	—
72	C₄	—	67	4	no	—
73	C₄	#14524042	1	1	yes	characteristic
74	C₄	Q₁₆×D₆₇	1	1	yes	characteristic
75	C₂	#16863800	1	1	yes	characteristic
76	C₂	SD₃₂×D₆₇	1	2	yes	—

	subgroup	G/H
center	C₂²	D₈×D₆₇
commutator	C₂×C₈×C₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	#367×C₆₇	C₂
socle	C₂²×C₆₇	C₂×D₈
hypercenter	#367	D₆₇
hypocenter	C₆₇	#2254
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

Complements

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

N	H ≅ G/N	type
18. C₆₇⋊C₁₆	Q₈	semidirect
21. C₁₆×C₆₇	Q₈	semidirect
39. C₆₇	#2254	semidirect
60. C₁₆	Dic₁₃₄	semidirect