Subgroups — #22550612

Order 8128.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16861196	C₂	1	4	yes	maximal, complemented
2	#16861190	C₂	1	4	yes	maximal, complemented
3	C₄×Dic₂₅₄	C₂	1	2	yes	maximal, complemented
4	D₄×Dic₁₂₇	C₂	1	4	yes	maximal, complemented
5	#127×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₅₄	C₂²	1	2	yes	—
7	C₂²×Dic₁₂₇	C₂²	1	4	yes	—
8	C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
9	Dic₁₂₇⋊C₄	C₂²	1	4	yes	—
10	C₂²⋊Dic₁₂₇	C₂²	1	4	yes	—
11	C₂²⋊Dic₁₂₇	C₂²	1	8	yes	—
12	C₄×Dic₁₂₇	C₂²	1	4	yes	—
13	C₂×D₄×C₁₂₇	C₂²	1	4	yes	—
14	C₂×D₄×C₁₂₇	C₂²	1	2	yes	—
15	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
16	Dic₂₅₄	—	2	4	no	—
17	C₂×Dic₁₂₇	—	4	4	no	—
18	C₂×Dic₁₂₇	C₂³	1	4	yes	—
19	C₂×Dic₁₂₇	C₂³	1	4	yes	—
20	C₂³×C₁₂₇	C₂³	1	4	yes	—
21	D₄×C₁₂₇	—	2	8	no	—
22	D₄×C₁₂₇	—	4	2	no	—
23	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₁₂₇	C₂³	1	2	yes	—
25	C₂²×C₁₂₇	—	2	8	no	—
26	C₂²×C₁₂₇	—	2	4	no	—
27	C₂²×C₁₂₇	C₂⁴	1	1	yes	characteristic
28	C₄×C₁₂₇	—	2	1	no	—
29	C₄×C₁₂₇	#41	1	4	yes	—
30	Dic₁₂₇	—	2	4	no	—
31	Dic₁₂₇	—	2	4	no	—
32	C₂×C₁₂₇	—	4	4	no	—
33	C₂×C₁₂₇	C₂×#41	1	2	yes	—
34	C₂×C₁₂₇	H(5,𝔽₂)	1	1	yes	characteristic
35	C₁₂₇	#559	1	1	yes	characteristic, complemented, Sylow
36	#559	—	127	1	no	maximal, Sylow
37	#127	D₁₂₇	1	1	yes	characteristic
38	#124	—	127	4	no	—
39	#123	—	127	4	no	—
40	C₄×Q₈	—	127	2	no	—
41	C₄×D₄	—	127	4	no	—
42	C₂×Q₈	—	127	2	no	—
43	C₂×D₄	C₂×D₁₂₇	1	4	yes	—
44	C₂×D₄	C₂×D₁₂₇	1	2	yes	—
45	C₂²×C₄	—	127	4	no	—
46	C₄⋊C₄	—	127	4	no	—
47	C₄⋊C₄	—	127	2	no	—
48	C₂²⋊C₄	—	127	8	no	—
49	C₂²⋊C₄	—	127	4	no	—
50	C₄²	—	127	4	no	—
51	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
52	C₂³	C₂²×D₁₂₇	1	4	yes	—
53	Q₈	—	254	4	no	—
54	D₄	—	2	8	no	—
55	D₄	—	4	2	no	—
56	C₂×C₄	—	127	4	no	—
57	C₂×C₄	—	127	4	no	—
58	C₂×C₄	—	508	4	no	—
59	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
60	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
61	C₂²	—	2	8	no	—
62	C₂²	—	2	4	no	—
63	C₂²	C₂³×D₁₂₇	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	—	254	4	no	—
66	C₄	—	254	4	no	—
67	C₄	#14523247	1	4	yes	—
68	C₂	—	4	4	no	—
69	C₂	C₂×#14523247	1	2	yes	—
70	C₂	#16861254	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂⁴
Fitting	#127×C₁₂₇	C₂
hypercenter	#127	D₁₂₇
hypocenter	C₁₂₇	#559
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ #127

Complements

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

N	H ≅ G/N	type
1. #16861196	C₂	semidirect
2. #16861190	C₂	semidirect
3. C₄×Dic₂₅₄	C₂	semidirect
4. D₄×Dic₁₂₇	C₂	semidirect
35. C₁₂₇	#559	semidirect