Subgroups — #22550250

Order 8128.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14523235	C₂	1	4	yes	maximal, complemented
2	#16861206	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄⋊Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₅₀₈	—	2	2	no	—
6	C₂×D₅₀₈	C₂²	1	2	yes	—
7	#14523235	—	2	8	no	—
8	C₂×C₁₂₇⋊C₈	C₂²	1	2	yes	—
9	C₂×Q₈×C₁₂₇	C₂²	1	2	yes	—
10	C₄⋊C₄×C₁₂₇	—	2	2	no	—
11	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂²×D₁₂₇	—	2	2	no	—
13	D₅₀₈	—	2	4	no	—
14	D₅₀₈	—	4	4	no	—
15	C₁₂₇⋊C₈	D₄	1	4	yes	complemented
16	Q₈×C₁₂₇	—	2	4	no	—
17	C₂×C₄×C₁₂₇	—	2	2	no	—
18	C₂×C₄×C₁₂₇	D₄	1	2	yes	—
19	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
20	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
21	C₂×D₁₂₇	—	4	2	no	—
22	C₂×D₁₂₇	—	4	4	no	—
23	C₄×C₁₂₇	—	4	2	no	—
24	C₄×C₁₂₇	SD₁₆	1	4	yes	complemented
25	C₄×C₁₂₇	C₂×D₄	1	2	yes	—
26	C₂×C₁₂₇	#127	1	1	yes	characteristic
27	C₂×C₁₂₇	C₂×SD₁₆	1	2	yes	—
28	D₁₂₇	—	8	2	no	—
29	C₁₂₇	#492	1	1	yes	characteristic, complemented, Sylow
30	#492	—	127	1	no	maximal, Sylow
31	C₂×SD₁₆	—	127	4	no	—
32	C₄⋊Q₈	D₁₂₇	1	1	yes	characteristic, complemented
33	#127	—	127	1	no	—
34	C₄×C₈	—	127	1	no	—
35	C₂×Q₈	C₂×D₁₂₇	1	2	yes	—
36	C₂×D₄	—	127	2	no	—
37	C₂×D₄	—	254	2	no	—
38	SD₁₆	—	254	8	no	—
39	C₂×C₈	—	127	2	no	—
40	C₄⋊C₄	—	2	2	no	—
41	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
42	C₂³	—	254	2	no	—
43	Q₈	—	2	4	no	—
44	D₄	—	508	4	no	—
45	D₄	—	254	4	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	C₂²⋊D₁₂₇	1	2	yes	—
48	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
49	C₈	—	127	4	no	—
50	C₂²	—	508	2	no	—
51	C₂²	—	508	4	no	—
52	C₂²	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	#14523235	1	4	yes	complemented
55	C₄	D₄×D₁₂₇	1	2	yes	—
56	C₂	—	1016	2	no	—
57	C₂	#16861205	1	1	yes	characteristic
58	C₂	C₂×#14523235	1	2	yes	—

	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	#492
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
1. C₂×#14523235	C₂	semidirect
3. C₄×C₁₂₇⋊C₈	C₂	semidirect
4. C₄⋊Q₈×C₁₂₇	C₂	semidirect
15. C₁₂₇⋊C₈	D₄	semidirect
24. C₄×C₁₂₇	SD₁₆	semidirect
29. C₁₂₇	#492	semidirect
32. C₄⋊Q₈	D₁₂₇	semidirect
54. C₄	#14523235	semidirect