Subgroups — #22549624

Order 8128.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16861209	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₁₂₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₂₅₄	C₄	1	2	yes	—
5	C₂×C₁₂₇⋊C₈	—	2	1	no	—
6	C₄⋊Dic₁₂₇	—	2	1	no	—
7	C₄⋊Dic₁₂₇	—	2	1	no	—
8	C₂×C₈×C₁₂₇	—	2	1	no	—
9	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
10	Dic₂₅₄	—	4	2	no	—
11	C₂×Dic₁₂₇	—	2	2	no	—
12	C₁₂₇⋊C₈	—	4	1	no	—
13	C₂×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
15	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
16	C₈×C₁₂₇	—	4	1	no	—
17	C₂²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
18	C₄×C₁₂₇	—	2	1	no	—
19	C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
20	C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
21	C₄×C₁₂₇	Q₁₆	1	1	yes	characteristic
22	C₄×C₁₂₇	Q₁₆	1	1	yes	characteristic
23	Dic₁₂₇	—	4	2	no	—
24	C₂×C₁₂₇	#100	1	1	yes	characteristic
25	C₂×C₁₂₇	#103	1	1	yes	characteristic
26	C₂×C₁₂₇	#103	1	1	yes	characteristic
27	C₁₂₇	#333	1	1	yes	characteristic, complemented, Sylow
28	#333	—	127	1	no	maximal, Sylow
29	C₄⋊Q₈	—	127	1	no	—
30	C₄⋊C₈	—	127	1	no	—
31	C₄⋊C₈	D₁₂₇	1	1	yes	characteristic
32	C₂×Q₈	—	127	2	no	—
33	C₂×C₈	—	2	1	no	—
34	C₂×C₈	—	254	1	no	—
35	C₄⋊C₄	—	254	1	no	—
36	C₄⋊C₄	—	254	1	no	—
37	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
38	Q₈	—	508	2	no	—
39	C₂×C₄	—	254	2	no	—
40	C₂×C₄	C₄×D₁₂₇	1	1	yes	characteristic
41	C₂×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
42	C₂×C₄	D₅₀₈	1	1	yes	characteristic
43	C₈	—	4	1	no	—
44	C₈	—	508	1	no	—
45	C₂²	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
46	C₄	—	2	1	no	—
47	C₄	—	508	2	no	—
48	C₄	#14523236	1	1	yes	characteristic
49	C₄	#14523237	1	1	yes	characteristic
50	C₄	#14523238	1	1	yes	characteristic
51	C₄	Dic₅₀₈	1	1	yes	characteristic
52	C₂	#16861125	1	1	yes	characteristic
53	C₂	#16861132	1	1	yes	characteristic
54	C₂	#16861134	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×D₁₂₇)⋊C₄
commutator	C₂×C₄×C₁₂₇	C₂×C₄
Frattini	C₄²	C₂×D₁₂₇
Fitting	C₄⋊C₈×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²⋊C₄
hypercenter	C₄⋊C₈	D₁₂₇
hypocenter	C₁₂₇	#333
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
27. C₁₂₇	#333	semidirect