Subgroups — #24924086

Order 10688.

248 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128879	C₂	1	2	yes	maximal, complemented
2	#19128865	C₂	1	6	yes	maximal, complemented
3	#19128854	C₂	1	3	yes	maximal, complemented
4	C₂×Dic₆₆₈	C₂	1	3	yes	maximal, complemented
5	#131×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#14581950	C₂²	1	6	yes	—
7	#14581949	—	2	6	no	—
8	C₂×Dic₃₃₄	C₂²	1	6	yes	—
9	Q₈×D₁₆₇	—	2	2	no	—
10	Dic₆₆₈	C₂²	1	3	yes	—
11	Dic₆₆₈	C₂²	1	6	yes	—
12	#14581939	C₂²	1	6	yes	—
13	D₁₃₃₆	C₂²	1	1	yes	characteristic
14	#41×C₁₆₇	C₂²	1	1	yes	characteristic
15	M₁₆×C₁₆₇	C₂²	1	3	yes	—
16	C₂×C₈×C₁₆₇	C₂²	1	3	yes	—
17	Dic₃₃₄	—	2	6	no	—
18	Dic₃₃₄	C₂³	1	6	yes	—
19	D₆₆₈	C₂³	1	2	yes	—
20	C₂²⋊D₁₆₇	—	2	6	no	—
21	C₂×Dic₁₆₇	—	2	6	no	—
22	C₄×D₁₆₇	—	2	6	no	—
23	Q₈×C₁₆₇	D₄	1	1	yes	characteristic
24	D₄×C₁₆₇	D₄	1	3	yes	—
25	C₂×C₄×C₁₆₇	C₂³	1	3	yes	—
26	C₈×C₁₆₇	C₂³	1	1	yes	characteristic
27	C₈×C₁₆₇	C₂³	1	3	yes	—
28	C₂²×C₁₆₇	C₂×D₄	1	3	yes	—
29	C₂×D₁₆₇	—	2	2	no	—
30	C₄×C₁₆₇	C₂×D₄	1	3	yes	—
31	C₄×C₁₆₇	C₂⁴	1	1	yes	characteristic
32	Dic₁₆₇	—	2	6	no	—
33	C₂×C₁₆₇	—	2	3	no	—
34	C₂×C₁₆₇	C₂²×D₄	1	1	yes	characteristic
35	D₁₆₇	—	4	2	no	—
36	C₁₆₇	#578	1	1	yes	characteristic, complemented, Sylow
37	#578	—	167	1	no	maximal, Sylow
38	2₋¹⁺⁴	—	167	2	no	—
39	#137	—	167	6	no	—
40	#135	—	167	3	no	—
41	C₂×Q₁₆	—	167	3	no	—
42	#131	D₁₆₇	1	1	yes	characteristic, complemented
43	#41	—	334	6	no	—
44	#41	—	167	6	no	—
45	#41	C₂×D₁₆₇	1	1	yes	characteristic
46	C₂×Q₈	—	334	2	no	—
47	C₂×Q₈	—	167	6	no	—
48	Q₁₆	—	167	3	no	—
49	Q₁₆	—	167	6	no	—
50	SD₁₆	—	167	6	no	—
51	D₈	—	167	1	no	—
52	M₁₆	C₂×D₁₆₇	1	3	yes	—
53	C₂×C₈	C₂×D₁₆₇	1	3	yes	—
54	Q₈	—	334	6	no	—
55	Q₈	—	167	6	no	—
56	Q₈	D₆₆₈	1	1	yes	characteristic
57	D₄	—	334	6	no	—
58	D₄	—	167	2	no	—
59	D₄	D₆₆₈	1	3	yes	—
60	C₂×C₄	—	334	6	no	—
61	C₂×C₄	—	334	6	no	—
62	C₂×C₄	C₂²×D₁₆₇	1	3	yes	—
63	C₈	C₂²×D₁₆₇	1	1	yes	characteristic
64	C₈	C₂²×D₁₆₇	1	3	yes	—
65	C₂²	—	334	2	no	—
66	C₂²	C₂×D₆₆₈	1	3	yes	—
67	C₄	—	334	6	no	—
68	C₄	C₂×D₆₆₈	1	3	yes	—
69	C₄	C₂³×D₁₆₇	1	1	yes	characteristic
70	C₂	—	2	3	no	—
71	C₂	—	668	2	no	—
72	C₂	C₂²×D₆₆₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₆₆₈
commutator, cosocle	C₄×C₁₆₇	C₂⁴
Frattini	C₄	C₂³×D₁₆₇
Fitting	#131×C₁₆₇	C₂
socle	C₂×C₁₆₇	C₂²×D₄
hypercenter	#131	D₁₆₇
hypocenter	C₁₆₇	#578
LCS	G ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

Complements

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

N	H ≅ G/N	type
1. #19128879	C₂	semidirect
2. #19128865	C₂	semidirect
3. #19128854	C₂	semidirect
4. C₂×Dic₆₆₈	C₂	semidirect
5. #131×C₁₆₇	C₂	semidirect
36. C₁₆₇	#578	semidirect
42. #131	D₁₆₇	semidirect