Subgroups — #24923701

Order 10688.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128854	C₂	1	2	yes	maximal, complemented
2	Dic₁₃₃₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128777	C₂	1	2	yes	maximal, complemented
4	D₂₆₇₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₆×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#14581950	—	2	2	no	—
7	Dic₆₆₈	C₂²	1	2	yes	—
8	#14581939	—	2	2	no	—
9	D₁₃₃₆	C₂²	1	2	yes	—
10	C₂×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
11	C₁₆×C₁₆₇	C₂²	1	1	yes	characteristic
12	C₁₆×C₁₆₇	C₂²	1	1	yes	characteristic
13	Dic₃₃₄	—	2	2	no	—
14	D₆₆₈	—	2	2	no	—
15	C₂²⋊D₁₆₇	—	4	2	no	—
16	C₄×D₁₆₇	—	4	2	no	—
17	C₂×C₄×C₁₆₇	D₄	1	1	yes	characteristic
18	C₈×C₁₆₇	D₄	1	1	yes	characteristic
19	C₈×C₁₆₇	C₂³	1	1	yes	characteristic
20	C₂²×C₁₆₇	D₈	1	1	yes	characteristic
21	C₂×D₁₆₇	—	4	2	no	—
22	C₄×C₁₆₇	D₈	1	1	yes	characteristic
23	C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
24	Dic₁₆₇	—	4	2	no	—
25	C₂×C₁₆₇	—	2	1	no	—
26	C₂×C₁₆₇	C₂×D₈	1	1	yes	characteristic
27	D₁₆₇	—	8	2	no	—
28	C₁₆₇	#508	1	1	yes	characteristic, complemented, Sylow
29	#508	—	167	1	no	maximal, Sylow
30	#135	—	167	2	no	—
31	Q₃₂	—	167	1	no	—
32	SD₃₂	—	167	2	no	—
33	D₁₆	—	167	1	no	—
34	C₂×C₁₆	D₁₆₇	1	1	yes	characteristic, complemented
35	#41	—	334	2	no	—
36	Q₁₆	—	167	2	no	—
37	SD₁₆	—	334	2	no	—
38	D₈	—	167	2	no	—
39	C₂×C₈	C₂×D₁₆₇	1	1	yes	characteristic
40	C₁₆	C₂×D₁₆₇	1	1	yes	characteristic
41	C₁₆	C₂×D₁₆₇	1	1	yes	characteristic
42	Q₈	—	334	2	no	—
43	D₄	—	334	2	no	—
44	D₄	—	668	2	no	—
45	C₂×C₄	—	668	2	no	—
46	C₂×C₄	D₆₆₈	1	1	yes	characteristic
47	C₈	D₆₆₈	1	1	yes	characteristic
48	C₈	C₂²×D₁₆₇	1	1	yes	characteristic
49	C₂²	—	668	2	no	—
50	C₂²	D₁₃₃₆	1	1	yes	characteristic
51	C₄	—	668	2	no	—
52	C₄	D₁₃₃₆	1	1	yes	characteristic
53	C₄	C₂×D₆₆₈	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	1336	2	no	—
56	C₂	C₂×D₁₃₃₆	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₁₃₃₆
commutator, cosocle	C₈×C₁₆₇	C₂³
Frattini	C₈	C₂²×D₁₆₇
Fitting	C₂×C₁₆×C₁₆₇	C₂
socle	C₂×C₁₆₇	C₂×D₈
hypercenter	C₂×C₁₆	D₁₆₇
hypocenter	C₁₆₇	#508
LCS	G ▷ C₈×C₁₆₇ ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
1. #19128854	C₂	semidirect
2. Dic₁₃₃₆	C₂	semidirect
3. #19128777	C₂	semidirect
4. D₂₆₇₂	C₂	semidirect
5. C₂×C₁₆×C₁₆₇	C₂	semidirect
28. C₁₆₇	#508	semidirect
34. C₂×C₁₆	D₁₆₇	semidirect