Subgroups — #22549631

Order 8128.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₂₇⋊C₈	C₂	1	2	yes	maximal
2	C₂×C₄²×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₂₇⋊C₈	—	2	2	no	—
4	C₂×C₁₂₇⋊C₈	—	2	2	no	—
5	C₂×C₁₂₇⋊C₈	C₄	1	4	yes	complemented
6	C₂²×C₄×C₁₂₇	C₄	1	2	yes	—
7	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₄²×C₁₂₇	—	2	2	no	—
9	C₁₂₇⋊C₈	—	2	4	no	—
10	C₂³×C₁₂₇	C₂×C₄	1	1	yes	characteristic
11	C₂×C₄×C₁₂₇	—	2	2	no	—
12	C₂×C₄×C₁₂₇	—	2	2	no	—
13	C₂×C₄×C₁₂₇	C₈	1	4	yes	complemented
14	C₂×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
15	C₂×C₄×C₁₂₇	D₄	1	2	yes	—
16	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
17	C₂×C₄×C₁₂₇	Q₈	1	1	yes	characteristic
18	C₂²×C₁₂₇	C₄²	1	1	yes	characteristic
19	C₂²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
20	C₂²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
21	C₂²×C₁₂₇	C₂×C₈	1	2	yes	—
22	C₂²×C₁₂₇	M₁₆	1	2	yes	—
23	C₄×C₁₂₇	—	2	4	no	—
24	C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
25	C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
26	C₂×C₁₂₇	#95	1	1	yes	characteristic
27	C₂×C₁₂₇	C₄×C₈	1	1	yes	characteristic
28	C₂×C₁₂₇	#97	1	1	yes	characteristic
29	C₂×C₁₂₇	C₂²⋊C₈	1	2	yes	—
30	C₂×C₁₂₇	C₄⋊C₈	1	2	yes	—
31	C₁₂₇	#336	1	1	yes	characteristic, complemented, Sylow
32	#336	—	127	1	no	maximal, Sylow
33	C₂²×C₈	—	127	2	no	—
34	C₂×C₄²	D₁₂₇	1	1	yes	characteristic
35	C₂²×C₄	Dic₁₂₇	1	2	yes	—
36	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
37	C₂×C₈	—	127	4	no	—
38	C₂×C₈	—	254	2	no	—
39	C₂×C₈	—	254	2	no	—
40	C₄²	—	2	2	no	—
41	C₂³	C₂×Dic₁₂₇	1	1	yes	characteristic
42	C₂×C₄	—	2	2	no	—
43	C₂×C₄	—	2	2	no	—
44	C₂×C₄	C₁₂₇⋊C₈	1	4	yes	complemented
45	C₂×C₄	C₄×D₁₂₇	1	2	yes	—
46	C₂×C₄	C₂²⋊D₁₂₇	1	2	yes	—
47	C₂×C₄	D₅₀₈	1	1	yes	characteristic
48	C₂×C₄	Dic₂₅₄	1	1	yes	characteristic
49	C₈	—	254	4	no	—
50	C₂²	C₄×Dic₁₂₇	1	1	yes	characteristic
51	C₂²	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
52	C₂²	C₄⋊Dic₁₂₇	1	1	yes	characteristic
53	C₂²	C₂×C₁₂₇⋊C₈	1	2	yes	—
54	C₂²	#14523232	1	2	yes	—
55	C₄	—	2	4	no	—
56	C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
57	C₄	Dic₁₂₇⋊C₄	1	2	yes	—
58	C₂	C₁₂₇⋊#95	1	1	yes	characteristic
59	C₂	C₄×C₁₂₇⋊C₈	1	1	yes	characteristic
60	C₂	#16861120	1	1	yes	characteristic
61	C₂	C₂²⋊(C₁₂₇⋊C₈)	1	2	yes	—
62	C₂	C₄⋊(C₁₂₇⋊C₈)	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂×D₁₂₇
commutator	C₂×C₁₂₇	C₄×C₈
Fitting	C₂×C₄²×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂×C₄
hypercenter	C₂×C₄²	D₁₂₇
hypocenter	C₁₂₇	#336
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
5. C₂×C₁₂₇⋊C₈	C₄	semidirect
13. C₂×C₄×C₁₂₇	C₈	semidirect
31. C₁₂₇	#336	semidirect
44. C₂×C₄	C₁₂₇⋊C₈	semidirect