Subgroups — #24923080

Order 10688.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128854	C₂	1	1	yes	maximal, characteristic
2	#19128764	C₂	1	1	yes	maximal, characteristic, complemented
3	M₃₂×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#14581950	—	2	1	no	—
5	Dic₆₆₈	C₄	1	1	yes	characteristic, complemented
6	#14581939	—	2	1	no	—
7	D₁₃₃₆	C₄	1	1	yes	characteristic, complemented
8	C₄⋊Dic₁₆₇	—	2	1	no	—
9	C₂×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
10	C₁₆×C₁₆₇	—	2	1	no	—
11	Dic₃₃₄	—	2	1	no	—
12	D₆₆₈	—	2	1	no	—
13	C₂²⋊D₁₆₇	—	4	1	no	—
14	C₂×Dic₁₆₇	—	4	1	no	—
15	C₄×D₁₆₇	—	4	1	no	—
16	C₂×C₄×C₁₆₇	D₄	1	1	yes	characteristic
17	C₈×C₁₆₇	C₂×C₄	1	1	yes	characteristic
18	C₈×C₁₆₇	D₄	1	1	yes	characteristic
19	C₂²×C₁₆₇	D₈	1	1	yes	characteristic
20	C₂×D₁₆₇	—	4	1	no	—
21	C₄×C₁₆₇	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₁₆₇	SD₁₆	1	1	yes	characteristic
23	Dic₁₆₇	—	4	1	no	—
24	Dic₁₆₇	—	8	1	no	—
25	C₂×C₁₆₇	—	2	1	no	—
26	C₂×C₁₆₇	#102	1	1	yes	characteristic
27	D₁₆₇	—	8	1	no	—
28	C₁₆₇	#360	1	1	yes	characteristic, complemented, Sylow
29	#360	—	167	1	no	maximal, Sylow
30	#135	—	167	1	no	—
31	M₃₂	D₁₆₇	1	1	yes	characteristic, complemented
32	#106	—	167	1	no	—
33	#41	—	334	1	no	—
34	Q₁₆	—	167	1	no	—
35	SD₁₆	—	334	1	no	—
36	D₈	—	167	1	no	—
37	C₂×C₈	C₂×D₁₆₇	1	1	yes	characteristic
38	C₄⋊C₄	—	334	1	no	—
39	C₁₆	—	2	1	no	—
40	Q₈	—	334	1	no	—
41	D₄	—	334	1	no	—
42	D₄	—	668	1	no	—
43	C₂×C₄	—	668	1	no	—
44	C₂×C₄	—	668	1	no	—
45	C₂×C₄	D₆₆₈	1	1	yes	characteristic
46	C₈	C₄×D₁₆₇	1	1	yes	characteristic
47	C₈	C₂²⋊D₁₆₇	1	1	yes	characteristic
48	C₂²	—	668	1	no	—
49	C₂²	D₁₃₃₆	1	1	yes	characteristic
50	C₄	—	1336	1	no	—
51	C₄	—	668	1	no	—
52	C₄	(C₂×D₁₆₇)⋊C₄	1	1	yes	characteristic
53	C₄	#14581939	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	1336	1	no	—
56	C₂	#19128754	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#19128754
commutator	C₈×C₁₆₇	C₂×C₄
Frattini	C₂×C₈	C₂×D₁₆₇
Fitting	M₃₂×C₁₆₇	C₂
socle	C₂×C₁₆₇	#102
prime radical	#19128854	C₂
cosocle	C₂×C₈×C₁₆₇	C₂²
hypercenter	M₃₂	D₁₆₇
hypocenter	C₁₆₇	#360
LCS	G ▷ C₈×C₁₆₇ ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. #19128764	C₂	semidirect
3. M₃₂×C₁₆₇	C₂	semidirect
5. Dic₆₆₈	C₄	semidirect
7. D₁₃₃₆	C₄	semidirect
28. C₁₆₇	#360	semidirect
31. M₃₂	D₁₆₇	semidirect