Subgroups — #22549938

Order 8128.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16861166	C₂	1	1	yes	maximal, characteristic
2	Dic₁₂₇⋊C₈	C₂	1	2	yes	maximal, complemented
3	(C₂×D₁₂₇)⋊C₈	C₂	1	2	yes	maximal
4	C₄×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#97×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
7	C₂×C₁₂₇⋊C₈	C₂²	1	2	yes	—
8	Dic₁₂₇⋊C₄	C₄	1	2	yes	—
9	(C₂×D₁₂₇)⋊C₄	C₄	1	2	yes	—
10	C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₁₂₇	C₂²	1	2	yes	—
12	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂²×D₁₂₇	C₂×C₄	1	1	yes	characteristic
14	C₂×Dic₁₂₇	C₂×C₄	1	2	yes	—
15	C₂×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
16	C₄×D₁₂₇	—	2	2	no	—
17	C₁₂₇⋊C₈	—	2	2	no	—
18	C₂×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
20	C₈×C₁₂₇	—	2	2	no	—
21	C₂²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
22	C₂×D₁₂₇	—	2	1	no	—
23	C₂×D₁₂₇	—	2	2	no	—
24	C₄×C₁₂₇	—	2	2	no	—
25	C₄×C₁₂₇	#41	1	2	yes	—
26	Dic₁₂₇	—	2	1	no	—
27	Dic₁₂₇	—	2	2	no	—
28	C₂×C₁₂₇	#117	1	1	yes	characteristic
29	C₂×C₁₂₇	#131	1	2	yes	—
30	D₁₂₇	—	4	1	no	—
31	C₁₂₇	#433	1	1	yes	characteristic, complemented, Sylow
32	#433	—	127	1	no	maximal, Sylow
33	#117	—	127	1	no	—
34	C₄⋊C₈	—	127	2	no	—
35	C₂²⋊C₈	—	127	2	no	—
36	#97	D₁₂₇	1	1	yes	characteristic, complemented
37	C₄×C₈	—	127	1	no	—
38	C₂²×C₄	—	127	1	no	—
39	C₂×C₈	—	127	2	no	—
40	C₂×C₈	C₂×D₁₂₇	1	2	yes	—
41	C₄⋊C₄	—	127	2	no	—
42	C₂²⋊C₄	—	127	2	no	—
43	C₄²	—	127	1	no	—
44	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
45	C₂³	—	127	1	no	—
46	C₂×C₄	—	127	2	no	—
47	C₂×C₄	—	127	1	no	—
48	C₂×C₄	—	254	2	no	—
49	C₂×C₄	C₄×D₁₂₇	1	2	yes	—
50	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
51	C₈	—	2	2	no	—
52	C₈	—	254	2	no	—
53	C₂²	—	254	2	no	—
54	C₂²	—	254	1	no	—
55	C₂²	C₂×C₄×D₁₂₇	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	—	254	2	no	—
58	C₄	—	254	1	no	—
59	C₄	#14523248	1	2	yes	—
60	C₂	—	508	1	no	—
61	C₂	#16861166	1	1	yes	characteristic
62	C₂	#16861217	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂²×C₄
Fitting	#97×C₁₂₇	C₂
hypercenter	#97	D₁₂₇
hypocenter	C₁₂₇	#433
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ #97

Complements

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

N	H ≅ G/N	type
2. Dic₁₂₇⋊C₈	C₂	semidirect
4. C₄×C₁₂₇⋊C₈	C₂	semidirect
5. #97×C₁₂₇	C₂	semidirect
31. C₁₂₇	#433	semidirect
36. #97	D₁₂₇	semidirect