Subgroups — #22550233

Order 8128.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16861197	C₂	1	1	yes	maximal, characteristic
2	#16861132	C₂	1	2	yes	maximal, complemented
3	#16861129	C₂	1	2	yes	maximal
4	C₄×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#125×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₂₅₄	C₂²	1	1	yes	characteristic
7	C₂×D₅₀₈	C₂²	1	1	yes	characteristic
8	C₂×C₁₂₇⋊C₈	C₂²	1	2	yes	—
9	(C₂×D₁₂₇)⋊C₄	—	2	2	no	—
10	C₄⋊C₄×C₁₂₇	—	2	2	no	—
11	C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
12	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂²×D₁₂₇	—	2	1	no	—
14	Dic₂₅₄	—	2	2	no	—
15	D₅₀₈	—	2	2	no	—
16	C₂×Dic₁₂₇	—	2	1	no	—
17	C₁₂₇⋊C₈	—	2	2	no	—
18	C₂×C₄×C₁₂₇	—	2	2	no	—
19	C₂×C₄×C₁₂₇	D₄	1	2	yes	—
20	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
21	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
22	C₂×D₁₂₇	—	4	1	no	—
23	C₂×D₁₂₇	—	4	2	no	—
24	C₄×C₁₂₇	—	2	2	no	—
25	C₄×C₁₂₇	—	4	2	no	—
26	C₄×C₁₂₇	#41	1	2	yes	—
27	Dic₁₂₇	—	4	1	no	—
28	C₂×C₁₂₇	#124	1	1	yes	characteristic
29	C₂×C₁₂₇	#135	1	2	yes	—
30	D₁₂₇	—	8	1	no	—
31	C₁₂₇	#488	1	1	yes	characteristic, complemented, Sylow
32	#488	—	127	1	no	maximal, Sylow
33	#125	D₁₂₇	1	1	yes	characteristic, complemented
34	#124	—	127	1	no	—
35	#103	—	127	2	no	—
36	#102	—	127	2	no	—
37	C₄×C₈	—	127	1	no	—
38	C₂×Q₈	—	127	1	no	—
39	C₂×D₄	—	127	1	no	—
40	C₂×C₈	—	127	2	no	—
41	C₄⋊C₄	—	2	2	no	—
42	C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
43	C₂²⋊C₄	—	254	2	no	—
44	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
45	C₂³	—	254	1	no	—
46	Q₈	—	254	2	no	—
47	D₄	—	254	2	no	—
48	C₂×C₄	—	2	2	no	—
49	C₂×C₄	—	254	1	no	—
50	C₂×C₄	C₂²⋊D₁₂₇	1	2	yes	—
51	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
52	C₈	—	254	2	no	—
53	C₂²	—	508	1	no	—
54	C₂²	—	508	2	no	—
55	C₂²	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	—	4	2	no	—
58	C₄	—	508	1	no	—
59	C₄	#14523249	1	2	yes	—
60	C₂	—	1016	1	no	—
61	C₂	#16861198	1	1	yes	characteristic
62	C₂	#16861229	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	#125×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₄
hypercenter	#125	D₁₂₇
hypocenter	C₁₂₇	#488
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
2. #16861132	C₂	semidirect
4. C₄×C₁₂₇⋊C₈	C₂	semidirect
5. #125×C₁₂₇	C₂	semidirect
31. C₁₂₇	#488	semidirect
33. #125	D₁₂₇	semidirect