Subgroups — #22549601

Order 8128.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₂₇⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊(C₁₂₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₁₂₇	C₄	1	1	yes	characteristic
5	C₂²×Dic₁₂₇	C₄	1	1	yes	characteristic
6	C₂×C₁₂₇⋊C₈	—	2	1	no	—
7	Dic₁₂₇⋊C₄	—	2	1	no	—
8	Dic₁₂₇⋊C₄	—	2	1	no	—
9	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂×C₈×C₁₂₇	—	2	1	no	—
11	C₂×Dic₁₂₇	—	2	1	no	—
12	C₂×Dic₁₂₇	—	2	1	no	—
13	C₂×Dic₁₂₇	—	2	1	no	—
14	C₂×Dic₁₂₇	—	4	1	no	—
15	C₂×Dic₁₂₇	C₈	1	2	yes	complemented
16	C₁₂₇⋊C₈	—	4	1	no	—
17	C₂³×C₁₂₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₁₂₇	—	4	1	no	—
19	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
20	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
21	C₈×C₁₂₇	—	4	1	no	—
22	C₂²×C₁₂₇	—	2	1	no	—
23	C₂²×C₁₂₇	—	2	1	no	—
24	C₂²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²×C₁₂₇	C₂×C₈	1	1	yes	characteristic
26	C₂²×C₁₂₇	M₁₆	1	1	yes	characteristic
27	C₄×C₁₂₇	—	2	1	no	—
28	C₄×C₁₂₇	—	2	1	no	—
29	Dic₁₂₇	—	2	2	no	—
30	Dic₁₂₇	—	4	1	no	—
31	C₂×C₁₂₇	—	2	1	no	—
32	C₂×C₁₂₇	—	2	1	no	—
33	C₂×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
34	C₂×C₁₂₇	#99	1	1	yes	characteristic
35	C₂×C₁₂₇	#101	1	1	yes	characteristic
36	C₁₂₇	#324	1	1	yes	characteristic, complemented, Sylow
37	#324	—	127	1	no	maximal, Sylow
38	C₂×C₄⋊C₄	—	127	1	no	—
39	C₂²⋊C₈	—	127	1	no	—
40	C₂²⋊C₈	D₁₂₇	1	1	yes	characteristic
41	C₂²×C₄	—	127	1	no	—
42	C₂²×C₄	—	127	1	no	—
43	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
44	C₂×C₈	—	2	1	no	—
45	C₂×C₈	—	254	1	no	—
46	C₄⋊C₄	—	254	1	no	—
47	C₄⋊C₄	—	254	1	no	—
48	C₂³	C₄×D₁₂₇	1	1	yes	characteristic
49	C₂×C₄	—	4	1	no	—
50	C₂×C₄	—	127	2	no	—
51	C₂×C₄	—	508	1	no	—
52	C₂×C₄	—	254	1	no	—
53	C₂×C₄	—	254	1	no	—
54	C₂×C₄	—	254	1	no	—
55	C₂×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
56	C₂×C₄	D₅₀₈	1	1	yes	characteristic
57	C₈	—	4	1	no	—
58	C₈	—	508	1	no	—
59	C₂²	—	2	1	no	—
60	C₂²	—	2	1	no	—
61	C₂²	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
62	C₂²	C₈×D₁₂₇	1	1	yes	characteristic
63	C₂²	#14523231	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	—	2	1	no	—
66	C₄	—	508	1	no	—
67	C₄	—	254	2	no	—
68	C₂	—	2	1	no	—
69	C₂	—	2	1	no	—
70	C₂	(C₂×D₁₂₇)⋊C₈	1	1	yes	characteristic
71	C₂	#16861123	1	1	yes	characteristic
72	C₂	#16861127	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×D₁₂₇)⋊C₄
commutator	C₂²×C₁₂₇	C₂×C₈
Frattini	C₂²×C₄	C₂×D₁₂₇
Fitting	C₂²⋊C₈×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²⋊C₄
hypercenter	C₂²⋊C₈	D₁₂₇
hypocenter	C₁₂₇	#324
LCS	G ▷ C₂²×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
15. C₂×Dic₁₂₇	C₈	semidirect
36. C₁₂₇	#324	semidirect