Subgroups — #22549970

Order 8128.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₅₀₈	C₂	1	1	yes	maximal, characteristic
2	#16861174	C₂	1	2	yes	maximal
3	#16861132	C₂	1	2	yes	maximal
4	C₈×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
5	#107×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₅₄	C₂²	1	2	yes	—
7	Dic₅₀₈	C₄	1	4	yes	complemented
8	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
9	Dic₁₂₇⋊C₄	—	2	2	no	—
10	C₄×Dic₁₂₇	—	2	2	no	—
11	C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
14	Dic₂₅₄	—	2	2	no	—
15	Dic₂₅₄	C₂×C₄	1	4	yes	—
16	C₂×Dic₁₂₇	—	2	2	no	—
17	C₂×Dic₁₂₇	D₄	1	1	yes	characteristic
18	C₂×Dic₁₂₇	D₄	1	1	yes	characteristic
19	C₁₂₇⋊C₈	—	2	1	no	—
20	C₂×C₄×C₁₂₇	—	2	2	no	—
21	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
22	C₈×C₁₂₇	C₂×C₄	1	2	yes	—
23	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
24	C₄×C₁₂₇	—	4	2	no	—
25	C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
26	C₄×C₁₂₇	#41	1	1	yes	characteristic
27	Dic₁₂₇	—	2	4	no	—
28	Dic₁₂₇	—	2	1	no	—
29	Dic₁₂₇	Q₁₆	1	2	yes	complemented
30	C₂×C₁₂₇	C₄×D₄	1	1	yes	characteristic
31	C₂×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
32	C₂×C₁₂₇	#135	1	1	yes	characteristic
33	C₁₂₇	C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
34	C₄×Q₁₆	—	127	1	no	maximal, Sylow
35	C₂×Q₁₆	—	127	1	no	—
36	C₄×Q₈	—	127	2	no	—
37	#107	D₁₂₇	1	1	yes	characteristic
38	#103	—	127	2	no	—
39	C₄×C₈	—	127	1	no	—
40	C₂×Q₈	—	127	2	no	—
41	Q₁₆	—	127	4	no	—
42	C₂×C₈	—	127	1	no	—
43	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
44	C₄⋊C₄	—	254	2	no	—
45	C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
46	C₄²	—	127	1	no	—
47	C₄²	—	254	2	no	—
48	Q₈	—	127	4	no	—
49	Q₈	—	254	2	no	—
50	C₂×C₄	—	2	2	no	—
51	C₂×C₄	—	127	1	no	—
52	C₂×C₄	—	127	1	no	—
53	C₂×C₄	—	254	2	no	—
54	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
55	C₈	—	254	1	no	—
56	C₈	C₄×D₁₂₇	1	2	yes	—
57	C₂²	D₄×D₁₂₇	1	1	yes	characteristic
58	C₄	—	4	2	no	—
59	C₄	—	127	2	no	—
60	C₄	—	254	1	no	—
61	C₄	—	254	4	no	—
62	C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
63	C₄	#14523249	1	1	yes	characteristic
64	C₂	#16861171	1	1	yes	characteristic
65	C₂	Q₁₆×D₁₂₇	1	1	yes	characteristic
66	C₂	#16861228	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₂₇
commutator	C₄×C₁₂₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	#107×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₄
hypercenter	#107	D₁₂₇
hypocenter	C₁₂₇	C₄×Q₁₆
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
7. Dic₅₀₈	C₄	semidirect
29. Dic₁₂₇	Q₁₆	semidirect
33. C₁₂₇	C₄×Q₁₆	semidirect