Subgroups — #22550163

Order 8128.

108 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	#16861140	C₂	1	2	yes	maximal, complemented
3	C₄⋊(C₁₂₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861129	C₂	1	2	yes	maximal
5	C₄⋊Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₅₀₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₁₂₇	—	2	1	no	—
8	C₂×C₁₂₇⋊C₈	C₂²	1	2	yes	—
9	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
10	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
11	C₂×Q₈×C₁₂₇	—	2	1	no	—
12	C₄⋊C₄×C₁₂₇	—	2	1	no	—
13	C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
14	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂²×D₁₂₇	—	2	1	no	—
16	D₅₀₈	—	2	1	no	—
17	D₅₀₈	Q₈	1	2	yes	complemented
18	C₂×Dic₁₂₇	—	2	1	no	—
19	C₄×D₁₂₇	—	4	2	no	—
20	C₁₂₇⋊C₈	—	2	2	no	—
21	Q₈×C₁₂₇	—	4	2	no	—
22	C₂×C₄×C₁₂₇	—	2	2	no	—
23	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
24	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
25	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
26	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
27	C₂×D₁₂₇	—	2	2	no	—
28	C₂×D₁₂₇	—	4	1	no	—
29	C₂×D₁₂₇	—	4	1	no	—
30	C₄×C₁₂₇	—	2	1	no	—
31	C₄×C₁₂₇	—	4	2	no	—
32	C₄×C₁₂₇	SD₁₆	1	2	yes	—
33	C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
34	C₄×C₁₂₇	#41	1	1	yes	characteristic
35	Dic₁₂₇	—	4	1	no	—
36	C₂×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
37	C₂×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
38	C₂×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
39	D₁₂₇	—	4	2	no	—
40	C₁₂₇	#476	1	1	yes	characteristic, complemented, Sylow
41	#476	—	127	1	no	maximal, Sylow
42	C₄⋊Q₈	D₁₂₇	1	1	yes	characteristic, complemented
43	C₄×D₄	—	127	1	no	—
44	#106	—	127	2	no	—
45	C₄⋊C₈	—	127	1	no	—
46	#102	—	127	2	no	—
47	C₂×Q₈	—	2	1	no	—
48	C₂×D₄	—	127	1	no	—
49	C₂²×C₄	—	254	1	no	—
50	C₂×C₈	—	127	2	no	—
51	C₄⋊C₄	—	2	1	no	—
52	C₄⋊C₄	—	127	1	no	—
53	C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
54	C₂²⋊C₄	—	254	1	no	—
55	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
56	C₂³	—	254	1	no	—
57	Q₈	—	4	2	no	—
58	D₄	—	127	2	no	—
59	D₄	—	254	1	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	508	2	no	—
62	C₂×C₄	—	254	1	no	—
63	C₂×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
64	C₂×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
65	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
66	C₈	—	254	2	no	—
67	C₂²	—	508	1	no	—
68	C₂²	—	508	1	no	—
69	C₂²	—	254	2	no	—
70	C₂²	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	4	2	no	—
73	C₄	—	508	1	no	—
74	C₄	#14523235	1	2	yes	—
75	C₄	Q₈×D₁₂₇	1	1	yes	characteristic
76	C₄	#14523249	1	1	yes	characteristic
77	C₂	—	508	2	no	—
78	C₂	#16861185	1	1	yes	characteristic
79	C₂	C₂×#14523235	1	1	yes	characteristic
80	C₂	#16861235	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	C₄⋊Q₈×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₄
hypercenter	C₄⋊Q₈	D₁₂₇
hypocenter	C₁₂₇	#476
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #16861140	C₂	semidirect
3. C₄⋊(C₁₂₇⋊C₈)	C₂	semidirect
5. C₄⋊Q₈×C₁₂₇	C₂	semidirect
17. D₅₀₈	Q₈	semidirect
40. C₁₂₇	#476	semidirect
42. C₄⋊Q₈	D₁₂₇	semidirect