Subgroups — #22549683

Order 8128.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₂₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#16861145	C₂	1	1	yes	maximal, characteristic, complemented
3	#135×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	#14523232	—	2	1	no	—
5	C₁₂₇⋊C₁₆	—	2	1	no	—
6	#41×C₁₂₇	—	2	1	no	—
7	Q₁₆×C₁₂₇	C₄	1	1	yes	characteristic
8	SD₁₆×C₁₂₇	—	2	1	no	—
9	D₈×C₁₂₇	C₄	1	1	yes	characteristic
10	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₁₂₇⋊C₈	—	4	1	no	—
12	Q₈×C₁₂₇	—	2	1	no	—
13	D₄×C₁₂₇	—	2	1	no	—
14	D₄×C₁₂₇	—	4	1	no	—
15	C₂×C₄×C₁₂₇	—	4	1	no	—
16	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
17	C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
18	C₈×C₁₂₇	D₄	1	1	yes	characteristic
19	C₂²×C₁₂₇	—	4	1	no	—
20	C₂²×C₁₂₇	SD₁₆	1	1	yes	characteristic
21	C₄×C₁₂₇	—	4	1	no	—
22	C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₁₂₇	D₈	1	1	yes	characteristic
24	C₂×C₁₂₇	—	2	1	no	—
25	C₂×C₁₂₇	—	8	1	no	—
26	C₂×C₁₂₇	#102	1	1	yes	characteristic
27	C₁₂₇	#359	1	1	yes	characteristic, complemented, Sylow
28	#359	—	127	1	no	maximal, Sylow
29	#135	D₁₂₇	1	1	yes	characteristic
30	C₂×C₁₆	—	127	1	no	—
31	#108	—	127	1	no	—
32	#41	—	2	1	no	—
33	Q₁₆	Dic₁₂₇	1	1	yes	characteristic
34	SD₁₆	—	2	1	no	—
35	D₈	Dic₁₂₇	1	1	yes	characteristic
36	M₁₆	—	254	1	no	—
37	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
38	C₁₆	—	254	1	no	—
39	Q₈	—	2	1	no	—
40	D₄	—	2	1	no	—
41	D₄	—	4	1	no	—
42	C₂×C₄	—	4	1	no	—
43	C₂×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
44	C₈	—	508	1	no	—
45	C₈	C₂×Dic₁₂₇	1	1	yes	characteristic
46	C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
47	C₂²	—	4	1	no	—
48	C₂²	#14523236	1	1	yes	characteristic
49	C₄	—	4	1	no	—
50	C₄	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
51	C₄	#14523233	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	8	1	no	—
54	C₂	#16861130	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#14523233
commutator	C₈×C₁₂₇	C₂×C₄
Frattini	C₂×C₈	C₂×D₁₂₇
Fitting	#135×C₁₂₇	C₂
socle	C₂×C₁₂₇	#102
hypercenter	#135	D₁₂₇
hypocenter	C₁₂₇	#359
LCS	G ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #135

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₂₇⋊C₁₆	C₂	semidirect
2. #16861145	C₂	semidirect
27. C₁₂₇	#359	semidirect