Subgroups — #22549616

Order 8128.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16861120	C₂	1	2	yes	maximal, complemented
2	#124×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₂₇⋊C₈	—	2	2	no	—
4	C₂×Q₈×C₁₂₇	C₄	1	1	yes	characteristic
5	C₂×D₄×C₁₂₇	C₄	1	1	yes	characteristic
6	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
7	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₁₂₇⋊C₈	—	2	4	no	—
9	C₂³×C₁₂₇	—	2	1	no	—
10	Q₈×C₁₂₇	—	4	1	no	—
11	D₄×C₁₂₇	—	4	1	no	—
12	C₂×C₄×C₁₂₇	—	2	1	no	—
13	C₂×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₁₂₇	D₄	1	2	yes	—
15	C₂²×C₁₂₇	—	4	1	no	—
16	C₂²×C₁₂₇	—	4	2	no	—
17	C₂²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
18	C₄×C₁₂₇	—	2	1	no	—
19	C₄×C₁₂₇	—	2	2	no	—
20	C₄×C₁₂₇	—	4	1	no	—
21	C₂×C₁₂₇	—	8	1	no	—
22	C₂×C₁₂₇	#100	1	1	yes	characteristic
23	C₂×C₁₂₇	#104	1	2	yes	—
24	C₁₂₇	#329	1	1	yes	characteristic, complemented, Sylow
25	#329	—	127	1	no	maximal, Sylow
26	#124	D₁₂₇	1	1	yes	characteristic
27	#97	—	127	2	no	—
28	C₂×Q₈	Dic₁₂₇	1	1	yes	characteristic
29	C₂×D₄	Dic₁₂₇	1	1	yes	characteristic
30	C₂×C₈	—	254	2	no	—
31	C₂²⋊C₄	—	2	2	no	—
32	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
33	C₂³	—	2	1	no	—
34	Q₈	—	4	1	no	—
35	D₄	—	4	1	no	—
36	C₂×C₄	—	2	1	no	—
37	C₂×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
38	C₂×C₄	C₂²⋊D₁₂₇	1	2	yes	—
39	C₈	—	254	4	no	—
40	C₂²	—	4	1	no	—
41	C₂²	—	4	2	no	—
42	C₂²	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
43	C₄	—	2	1	no	—
44	C₄	—	2	2	no	—
45	C₄	—	4	1	no	—
46	C₂	—	8	1	no	—
47	C₂	#16861126	1	1	yes	characteristic
48	C₂	#16861136	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²⋊Dic₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂×C₄
Frattini	C₄²	C₂×D₁₂₇
Fitting	#124×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²⋊C₄
hypercenter	#124	D₁₂₇
hypocenter	C₁₂₇	#329
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #124

Complements

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

N	H ≅ G/N	type
1. #16861120	C₂	semidirect
24. C₁₂₇	#329	semidirect