Subgroups — #22550713

Order 8128.

810 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16861257	C₂	1	5	yes	maximal, complemented
2	#16861255	C₂	1	10	yes	maximal, complemented
3	C₂×#14523249	C₂	1	5	yes	maximal, complemented
4	#41×D₁₂₇	C₂	1	10	yes	maximal, complemented
5	2₋¹⁺⁴×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#14523249	C₂²	1	10	yes	complemented
7	#14523249	C₂²	1	20	yes	complemented
8	#14523248	C₂²	1	30	yes	complemented
9	#14523247	C₂²	1	10	yes	complemented
10	Q₈×D₁₂₇	C₂²	1	10	yes	complemented
11	C₂×D₅₀₈	C₂²	1	15	yes	complemented
12	D₄×D₁₂₇	C₂²	1	30	yes	complemented
13	C₂×C₄×D₁₂₇	C₂²	1	15	yes	complemented
14	#41×C₁₂₇	C₂²	1	10	yes	complemented
15	C₂×Q₈×C₁₂₇	C₂²	1	5	yes	complemented
16	C₂²×D₁₂₇	C₂³	1	15	yes	—
17	Dic₂₅₄	C₂³	1	10	yes	—
18	D₅₀₈	C₂³	1	30	yes	—
19	C₂²⋊D₁₂₇	C₂³	1	20	yes	—
20	C₂×Dic₁₂₇	C₂³	1	5	yes	—
21	C₄×D₁₂₇	C₂³	1	10	yes	—
22	C₄×D₁₂₇	C₂³	1	30	yes	—
23	Q₈×C₁₂₇	C₂³	1	10	yes	—
24	D₄×C₁₂₇	C₂³	1	10	yes	—
25	C₂×C₄×C₁₂₇	C₂³	1	15	yes	—
26	C₂²×C₁₂₇	C₂⁴	1	5	yes	—
27	C₂×D₁₂₇	—	4	15	no	—
28	C₂×D₁₂₇	C₂⁴	1	10	yes	—
29	C₄×C₁₂₇	C₂⁴	1	10	yes	—
30	Dic₁₂₇	C₂⁴	1	1	yes	characteristic
31	Dic₁₂₇	C₂⁴	1	5	yes	—
32	C₂×C₁₂₇	—	2	5	no	—
33	C₂×C₁₂₇	C₂⁵	1	1	yes	characteristic
34	D₁₂₇	—	2	10	no	—
35	C₁₂₇	#585	1	1	yes	characteristic, complemented, Sylow
36	#585	—	127	1	no	maximal, Sylow
37	2₋¹⁺⁴	—	127	5	no	—
38	2₋¹⁺⁴	D₁₂₇	1	1	yes	characteristic, complemented
39	H(5,𝔽₂)	—	127	10	no	—
40	C₂×#41	—	127	10	no	—
41	C₂×#41	—	127	5	no	—
42	#41	—	127	10	no	—
43	#41	—	127	20	no	—
44	#41	—	127	10	no	—
45	#41	—	127	30	no	—
46	#41	C₂×D₁₂₇	1	10	yes	complemented
47	C₂×Q₈	—	127	10	no	—
48	C₂×Q₈	C₂×D₁₂₇	1	5	yes	complemented
49	C₂×D₄	—	127	30	no	—
50	C₂×D₄	—	127	15	no	—
51	C₂²×C₄	—	127	15	no	—
52	C₂³	—	127	15	no	—
53	Q₈	—	127	10	no	—
54	Q₈	C₂²×D₁₂₇	1	10	yes	—
55	D₄	—	127	20	no	—
56	D₄	—	127	30	no	—
57	D₄	C₂²×D₁₂₇	1	10	yes	—
58	C₂×C₄	—	127	5	no	—
59	C₂×C₄	—	127	10	no	—
60	C₂×C₄	—	127	30	no	—
61	C₂×C₄	C₂²×D₁₂₇	1	15	yes	—
62	C₂²	—	127	10	no	—
63	C₂²	—	508	15	no	—
64	C₂²	C₂³×D₁₂₇	1	5	yes	—
65	C₄	—	127	1	no	—
66	C₄	—	127	5	no	—
67	C₄	C₂³×D₁₂₇	1	10	yes	—
68	C₂	—	2	5	no	—
69	C₂	—	254	10	no	—
70	C₂	C₂⁴×D₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁴×D₁₂₇
commutator, socle	C₂×C₁₂₇	C₂⁵
Fitting	2₋¹⁺⁴×C₁₂₇	C₂
hypercenter	2₋¹⁺⁴	D₁₂₇
hypocenter	C₁₂₇	#585
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ 2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. #16861257	C₂	semidirect
2. #16861255	C₂	semidirect
3. C₂×#14523249	C₂	semidirect
4. #41×D₁₂₇	C₂	semidirect
5. 2₋¹⁺⁴×C₁₂₇	C₂	semidirect
6. #14523249	C₂²	semidirect
7. #14523249	C₂²	semidirect
8. #14523248	C₂²	semidirect
9. #14523247	C₂²	semidirect
10. Q₈×D₁₂₇	C₂²	semidirect
11. C₂×D₅₀₈	C₂²	semidirect
12. D₄×D₁₂₇	C₂²	semidirect
13. C₂×C₄×D₁₂₇	C₂²	semidirect
14. #41×C₁₂₇	C₂²	semidirect
15. C₂×Q₈×C₁₂₇	C₂²	semidirect
35. C₁₂₇	#585	semidirect
38. 2₋¹⁺⁴	D₁₂₇	semidirect
46. #41	C₂×D₁₂₇	semidirect
48. C₂×Q₈	C₂×D₁₂₇	semidirect