Subgroups — #22550624

Order 8128.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16861197	C₂	1	2	yes	maximal, complemented
2	#16861194	C₂	1	4	yes	maximal, complemented
3	#16861187	C₂	1	4	yes	maximal, complemented
4	#16861184	C₂	1	4	yes	maximal, complemented
5	#125×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₂₅₄	C₂²	1	2	yes	—
7	C₂×D₅₀₈	C₂²	1	4	yes	—
8	C₂×D₅₀₈	C₂²	1	2	yes	—
9	C₂×C₄×D₁₂₇	C₂²	1	4	yes	—
10	Dic₁₂₇⋊C₄	C₂²	1	4	yes	—
11	(C₂×D₁₂₇)⋊C₄	C₂²	1	4	yes	—
12	(C₂×D₁₂₇)⋊C₄	C₂²	1	4	yes	—
13	(C₂×D₁₂₇)⋊C₄	C₂²	1	4	yes	—
14	C₄⋊C₄×C₁₂₇	C₂²	1	4	yes	—
15	C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
16	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₂²×D₁₂₇	C₂³	1	4	yes	—
18	Dic₂₅₄	—	4	2	no	—
19	D₅₀₈	—	4	4	no	—
20	D₅₀₈	—	4	2	no	—
21	C₂×Dic₁₂₇	C₂³	1	4	yes	—
22	C₄×D₁₂₇	—	4	4	no	—
23	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₁₂₇	C₂³	1	2	yes	—
25	C₂×C₄×C₁₂₇	C₂³	1	4	yes	—
26	C₂²×C₁₂₇	C₂⁴	1	1	yes	characteristic
27	C₂×D₁₂₇	—	2	4	no	—
28	C₂×D₁₂₇	—	2	4	no	—
29	C₂×D₁₂₇	—	2	4	no	—
30	C₄×C₁₂₇	—	2	4	no	—
31	C₄×C₁₂₇	—	2	2	no	—
32	C₄×C₁₂₇	—	2	1	no	—
33	Dic₁₂₇	—	2	4	no	—
34	C₂×C₁₂₇	H(5,𝔽₂)	1	2	yes	—
35	C₂×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
36	D₁₂₇	—	4	4	no	—
37	C₁₂₇	#562	1	1	yes	characteristic, complemented, Sylow
38	#562	—	127	1	no	maximal, Sylow
39	#125	D₁₂₇	1	1	yes	characteristic, complemented
40	#124	—	127	2	no	—
41	#123	—	127	4	no	—
42	C₂²⋊Q₈	—	127	4	no	—
43	#121	—	127	4	no	—
44	C₂×Q₈	—	127	2	no	—
45	C₂×D₄	—	127	4	no	—
46	C₂×D₄	—	127	2	no	—
47	C₂²×C₄	—	127	4	no	—
48	C₄⋊C₄	—	127	4	no	—
49	C₄⋊C₄	C₂×D₁₂₇	1	4	yes	—
50	C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
51	C₂²⋊C₄	—	127	4	no	—
52	C₂²⋊C₄	—	127	4	no	—
53	C₂²⋊C₄	—	127	4	no	—
54	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
55	C₂³	—	127	4	no	—
56	Q₈	—	508	2	no	—
57	D₄	—	508	4	no	—
58	D₄	—	508	2	no	—
59	C₂×C₄	—	127	4	no	—
60	C₂×C₄	—	508	4	no	—
61	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
63	C₂×C₄	C₂²×D₁₂₇	1	4	yes	—
64	C₂²	—	254	4	no	—
65	C₂²	—	254	4	no	—
66	C₂²	—	254	4	no	—
67	C₂²	C₂³×D₁₂₇	1	1	yes	characteristic
68	C₄	—	2	4	no	—
69	C₄	—	2	2	no	—
70	C₄	—	2	1	no	—
71	C₄	—	254	4	no	—
72	C₂	—	508	4	no	—
73	C₂	#16861255	1	2	yes	—
74	C₂	#16861257	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂⁴
Fitting	#125×C₁₂₇	C₂
hypercenter	#125	D₁₂₇
hypocenter	C₁₂₇	#562
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. #16861197	C₂	semidirect
2. #16861194	C₂	semidirect
3. #16861187	C₂	semidirect
4. #16861184	C₂	semidirect
5. #125×C₁₂₇	C₂	semidirect
37. C₁₂₇	#562	semidirect
39. #125	D₁₂₇	semidirect