Subgroups — #23752483

Order 9664.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#18012767	C₂	1	2	yes	maximal, complemented
2	C₂×D₁₂₀₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#18012745	C₂	1	1	yes	maximal, characteristic, complemented
4	#18012655	C₂	1	2	yes	maximal, complemented
5	#108×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#14559004	C₂²	1	1	yes	characteristic
7	C₂×D₆₀₄	C₂²	1	2	yes	—
8	#14558993	—	2	2	no	—
9	D₁₂₀₈	—	2	2	no	—
10	D₁₂₀₈	—	2	1	no	—
11	D₁₂₀₈	—	2	1	no	—
12	#14558988	C₂²	1	1	yes	characteristic
13	#14558987	—	2	1	no	—
14	C₈×D₁₅₁	—	2	1	no	—
15	M₁₆×C₁₅₁	C₂²	1	2	yes	—
16	C₂×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
17	C₂²×D₁₅₁	—	2	2	no	—
18	Dic₃₀₂	D₄	1	1	yes	characteristic
19	D₆₀₄	—	2	2	no	—
20	D₆₀₄	—	2	2	no	—
21	D₆₀₄	D₄	1	1	yes	characteristic
22	C₂²⋊D₁₅₁	—	2	1	no	—
23	C₄×D₁₅₁	—	2	1	no	—
24	C₁₅₁⋊C₈	—	2	1	no	—
25	C₂×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
26	C₈×C₁₅₁	—	2	2	no	—
27	C₈×C₁₅₁	D₄	1	1	yes	characteristic
28	C₈×C₁₅₁	D₄	1	1	yes	characteristic
29	C₂²×C₁₅₁	#41	1	1	yes	characteristic
30	C₂×D₁₅₁	—	2	1	no	—
31	C₂×D₁₅₁	—	4	2	no	—
32	C₂×D₁₅₁	—	8	2	no	—
33	C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
34	C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
35	Dic₁₅₁	—	2	1	no	—
36	C₂×C₁₅₁	—	2	1	no	—
37	C₂×C₁₅₁	#121	1	1	yes	characteristic
38	D₁₅₁	—	4	1	no	—
39	D₁₅₁	—	8	2	no	—
40	C₁₅₁	#472	1	1	yes	characteristic, complemented, Sylow
41	#472	—	151	1	no	maximal, Sylow
42	AGL(1,ℤ/8)	—	151	2	no	—
43	C₂×D₈	—	151	1	no	—
44	#131	—	151	1	no	—
45	#108	D₁₅₁	1	1	yes	characteristic, complemented
46	#100	—	151	2	no	—
47	#41	—	151	1	no	—
48	C₂×D₄	—	151	2	no	—
49	SD₁₆	—	302	2	no	—
50	D₈	—	302	2	no	—
51	D₈	—	302	1	no	—
52	D₈	—	302	1	no	—
53	M₁₆	—	302	1	no	—
54	M₁₆	—	151	1	no	—
55	M₁₆	C₂×D₁₅₁	1	2	yes	—
56	C₂×C₈	—	302	1	no	—
57	C₂×C₈	C₂×D₁₅₁	1	1	yes	characteristic
58	C₂³	—	302	2	no	—
59	Q₈	—	151	1	no	—
60	D₄	—	302	1	no	—
61	D₄	—	302	2	no	—
62	D₄	—	302	2	no	—
63	D₄	—	151	1	no	—
64	C₂×C₄	—	302	1	no	—
65	C₂×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
66	C₈	—	2	2	no	—
67	C₈	—	302	1	no	—
68	C₈	D₆₀₄	1	1	yes	characteristic
69	C₈	D₆₀₄	1	1	yes	characteristic
70	C₂²	—	302	1	no	—
71	C₂²	—	604	2	no	—
72	C₂²	—	1208	2	no	—
73	C₂²	#14559005	1	1	yes	characteristic
74	C₄	—	302	1	no	—
75	C₄	D₄×D₁₅₁	1	1	yes	characteristic
76	C₄	C₂×D₆₀₄	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	—	604	1	no	—
79	C₂	—	1208	2	no	—
80	C₂	#18012714	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#18012714
commutator, cosocle	C₂×C₄×C₁₅₁	C₂³
Frattini	C₂×C₄	C₂²×D₁₅₁
Fitting	#108×C₁₅₁	C₂
socle	C₂×C₁₅₁	#121
hypercenter	#108	D₁₅₁
hypocenter	C₁₅₁	#472
LCS	G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #108

Complements

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

N	H ≅ G/N	type
1. #18012767	C₂	semidirect
2. C₂×D₁₂₀₈	C₂	semidirect
3. #18012745	C₂	semidirect
4. #18012655	C₂	semidirect
5. #108×C₁₅₁	C₂	semidirect
40. C₁₅₁	#472	semidirect
45. #108	D₁₅₁	semidirect