Subgroups — #22549603

Order 8128.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₅₀₈	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₅₀₈	C₄	1	1	yes	characteristic
5	C₂×C₄×D₁₂₇	—	2	1	no	—
6	C₂×C₁₂₇⋊C₈	—	2	1	no	—
7	C₄⋊Dic₁₂₇	C₄	1	1	yes	characteristic
8	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
9	C₂×C₈×C₁₂₇	—	2	1	no	—
10	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂²×D₁₂₇	—	2	1	no	—
12	D₅₀₈	—	2	1	no	—
13	D₅₀₈	C₈	1	2	yes	complemented
14	C₂×Dic₁₂₇	—	2	1	no	—
15	C₄×D₁₂₇	—	4	1	no	—
16	C₄×D₁₂₇	—	4	1	no	—
17	C₁₂₇⋊C₈	—	2	2	no	—
18	C₂×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
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	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×D₁₂₇	—	2	2	no	—
24	C₂×D₁₂₇	—	4	1	no	—
25	C₂×D₁₂₇	—	4	1	no	—
26	C₄×C₁₂₇	—	2	1	no	—
27	C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
28	C₄×C₁₂₇	M₁₆	1	1	yes	characteristic
29	C₄×C₁₂₇	D₈	1	1	yes	characteristic
30	C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
31	Dic₁₂₇	—	4	1	no	—
32	C₂×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
33	C₂×C₁₂₇	#102	1	1	yes	characteristic
34	C₂×C₁₂₇	#104	1	1	yes	characteristic
35	D₁₂₇	—	4	2	no	—
36	C₁₂₇	D₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
37	D₄⋊C₈	—	127	1	no	maximal, Sylow
38	C₄×D₄	—	127	1	no	—
39	C₄⋊C₈	D₁₂₇	1	1	yes	characteristic, complemented
40	C₄×C₈	—	127	1	no	—
41	C₂×D₄	—	127	1	no	—
42	C₂²×C₄	—	254	1	no	—
43	C₂×C₈	—	2	1	no	—
44	C₂×C₈	—	254	1	no	—
45	C₄⋊C₄	—	127	1	no	—
46	C₂²⋊C₄	—	254	1	no	—
47	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
48	C₂³	—	254	1	no	—
49	D₄	—	127	2	no	—
50	D₄	—	254	1	no	—
51	C₂×C₄	—	508	1	no	—
52	C₂×C₄	—	508	1	no	—
53	C₂×C₄	—	254	1	no	—
54	C₂×C₄	C₄×D₁₂₇	1	1	yes	characteristic
55	C₂×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
56	C₂×C₄	D₅₀₈	1	1	yes	characteristic
57	C₈	—	4	1	no	—
58	C₈	—	254	2	no	—
59	C₂²	—	508	1	no	—
60	C₂²	—	508	1	no	—
61	C₂²	—	254	2	no	—
62	C₂²	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	—	508	1	no	—
65	C₄	C₈×D₁₂₇	1	1	yes	characteristic
66	C₄	#14523231	1	1	yes	characteristic
67	C₄	#14523233	1	1	yes	characteristic
68	C₄	#14523235	1	1	yes	characteristic
69	C₂	—	508	2	no	—
70	C₂	(C₂×D₁₂₇)⋊C₈	1	1	yes	characteristic
71	C₂	#16861129	1	1	yes	characteristic
72	C₂	#16861135	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₁₂₇
commutator	C₄×C₁₂₇	C₂×C₈
Frattini	C₄²	C₂×D₁₂₇
Fitting	C₄⋊C₈×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²⋊C₄
hypercenter	C₄⋊C₈	D₁₂₇
hypocenter	C₁₂₇	D₄⋊C₈
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₄×C₁₂₇⋊C₈	C₂	semidirect
3. C₄⋊C₈×C₁₂₇	C₂	semidirect
13. D₅₀₈	C₈	semidirect
36. C₁₂₇	D₄⋊C₈	semidirect
39. C₄⋊C₈	D₁₂₇	semidirect