Subgroups — #22549619

Order 8128.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16861206	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	—	2	1	no	—
5	C₂×D₅₀₈	—	2	1	no	—
6	C₂×D₅₀₈	C₄	1	2	yes	—
7	C₂×C₁₂₇⋊C₈	—	2	1	no	—
8	C₂×C₈×C₁₂₇	—	2	1	no	—
9	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂²×D₁₂₇	—	2	2	no	—
11	D₅₀₈	—	4	2	no	—
12	D₅₀₈	—	4	1	no	—
13	D₅₀₈	—	4	1	no	—
14	D₅₀₈	—	4	1	no	—
15	D₅₀₈	—	4	1	no	—
16	C₁₂₇⋊C₈	—	4	1	no	—
17	C₂×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
19	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
20	C₈×C₁₂₇	—	4	1	no	—
21	C₂²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×D₁₂₇	—	4	2	no	—
23	C₂×D₁₂₇	—	4	2	no	—
24	C₂×D₁₂₇	—	4	2	no	—
25	C₄×C₁₂₇	—	2	1	no	—
26	C₄×C₁₂₇	D₈	1	1	yes	characteristic
27	C₄×C₁₂₇	D₈	1	1	yes	characteristic
28	C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
29	C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
30	C₂×C₁₂₇	#100	1	1	yes	characteristic
31	C₂×C₁₂₇	#102	1	1	yes	characteristic
32	C₂×C₁₂₇	#102	1	1	yes	characteristic
33	D₁₂₇	—	8	2	no	—
34	C₁₂₇	#331	1	1	yes	characteristic, complemented, Sylow
35	#331	—	127	1	no	maximal, Sylow
36	#127	—	127	1	no	—
37	C₄⋊C₈	—	127	1	no	—
38	C₄⋊C₈	D₁₂₇	1	1	yes	characteristic, complemented
39	C₂×D₄	—	127	2	no	—
40	C₂×D₄	—	254	1	no	—
41	C₂×D₄	—	254	1	no	—
42	C₂×C₈	—	2	1	no	—
43	C₂×C₈	—	254	1	no	—
44	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
45	C₂³	—	254	2	no	—
46	D₄	—	508	2	no	—
47	D₄	—	508	1	no	—
48	D₄	—	508	1	no	—
49	D₄	—	508	1	no	—
50	D₄	—	508	1	no	—
51	C₂×C₄	C₄×D₁₂₇	1	1	yes	characteristic
52	C₂×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
53	C₂×C₄	D₅₀₈	1	1	yes	characteristic
54	C₈	—	4	1	no	—
55	C₈	—	508	1	no	—
56	C₂²	—	508	2	no	—
57	C₂²	—	508	2	no	—
58	C₂²	—	508	2	no	—
59	C₂²	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
60	C₄	—	2	1	no	—
61	C₄	#14523233	1	1	yes	characteristic
62	C₄	D₁₀₁₆	1	1	yes	characteristic
63	C₄	#14523235	1	1	yes	characteristic
64	C₄	#14523237	1	1	yes	characteristic
65	C₂	—	1016	2	no	—
66	C₂	#16861125	1	1	yes	characteristic
67	C₂	#16861129	1	1	yes	characteristic
68	C₂	#16861131	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	#331
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
2. C₄⋊(C₁₂₇⋊C₈)	C₂	semidirect
3. C₄⋊C₈×C₁₂₇	C₂	semidirect
34. C₁₂₇	#331	semidirect
38. C₄⋊C₈	D₁₂₇	semidirect