Subgroups — #22562491

Order 8576.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16863801	C₂	1	2	yes	maximal
2	#16863224	C₂	1	4	yes	maximal
3	C₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
4	#14524043	—	2	2	no	—
5	#14523977	—	2	2	no	—
6	#14523977	C₂²	1	4	yes	—
7	C₂×C₁₆×C₆₇	C₂²	1	2	yes	—
8	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×Dic₁₃₄	—	4	2	no	—
10	C₄⋊Dic₆₇	—	2	4	no	—
11	C₄⋊Dic₆₇	—	4	2	no	—
12	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
13	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
14	C₄²×C₆₇	D₄	1	1	yes	characteristic
15	C₁₆×C₆₇	Q₈	1	4	yes	complemented
16	C₂×C₄×C₆₇	D₈	1	2	yes	—
17	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
18	C₂×Dic₆₇	—	4	4	no	—
19	Dic₁₃₄	—	8	4	no	—
20	C₈×C₆₇	Q₁₆	1	2	yes	—
21	C₈×C₆₇	C₂×Q₈	1	2	yes	—
22	C₂²×C₆₇	C₂×D₈	1	1	yes	characteristic
23	C₄×C₆₇	SD₃₂	1	4	yes	—
24	C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
25	C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
26	Dic₆₇	—	8	4	no	—
27	C₂×C₆₇	#500	1	1	yes	characteristic
28	C₂×C₆₇	C₂×SD₃₂	1	2	yes	—
29	#2254	—	67	1	no	maximal, Sylow
30	C₆₇	#2254	1	1	yes	characteristic, complemented, Sylow
31	#500	—	67	2	no	—
32	#367	—	67	4	no	—
33	C₄×C₁₆	D₆₇	1	1	yes	characteristic
34	C₄⋊Q₈	—	134	2	no	—
35	C₂×C₁₆	C₂×D₆₇	1	2	yes	—
36	#107	—	67	4	no	—
37	#107	—	134	2	no	—
38	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
39	C₂×Q₈	—	268	2	no	—
40	C₂×C₈	D₂₆₈	1	1	yes	characteristic
41	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
42	C₄⋊C₄	—	268	2	no	—
43	C₄⋊C₄	—	134	4	no	—
44	C₄²	D₂₆₈	1	1	yes	characteristic
45	C₁₆	Dic₁₃₄	1	4	yes	complemented
46	Q₈	—	536	4	no	—
47	C₂×C₄	—	268	4	no	—
48	C₂×C₄	D₅₃₆	1	2	yes	—
49	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
50	C₈	Dic₂₆₈	1	2	yes	—
51	C₈	C₂×Dic₁₃₄	1	2	yes	—
52	C₂²	C₂×D₅₃₆	1	1	yes	characteristic
53	C₄	—	536	4	no	—
54	C₄	#14523988	1	4	yes	—
55	C₄	#14524043	1	1	yes	characteristic
56	C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
57	C₂	#16863801	1	1	yes	characteristic
58	C₂	C₂×#14523988	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₅₃₆
commutator	C₂×C₈×C₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	C₄×C₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂×D₈
hypercenter	C₄×C₁₆	D₆₇
hypocenter	C₆₇	#2254
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
15. C₁₆×C₆₇	Q₈	semidirect
30. C₆₇	#2254	semidirect
45. C₁₆	Dic₁₃₄	semidirect