Subgroups — #22550244

Order 8128.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16861201	C₂	1	1	yes	maximal, characteristic
2	#16861133	C₂	1	4	yes	maximal
3	#16861120	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₂₇⋊C₈	C₂²	1	2	yes	—
6	C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
7	Dic₁₂₇⋊C₄	—	2	2	no	—
8	C₂×Q₈×C₁₂₇	C₂²	1	2	yes	—
9	C₄⋊C₄×C₁₂₇	—	2	2	no	—
10	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₂₇	—	2	2	no	—
12	C₁₂₇⋊C₈	—	2	2	no	—
13	Q₈×C₁₂₇	—	2	4	no	—
14	C₂×C₄×C₁₂₇	—	2	2	no	—
15	C₂×C₄×C₁₂₇	D₄	1	2	yes	—
16	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
17	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
18	C₄×C₁₂₇	—	2	2	no	—
19	C₄×C₁₂₇	—	4	2	no	—
20	C₄×C₁₂₇	#41	1	2	yes	—
21	Dic₁₂₇	—	4	2	no	—
22	C₂×C₁₂₇	#124	1	1	yes	characteristic
23	C₂×C₁₂₇	#137	1	2	yes	—
24	C₁₂₇	#491	1	1	yes	characteristic, complemented, Sylow
25	#491	—	127	1	no	maximal, Sylow
26	C₄⋊Q₈	D₁₂₇	1	1	yes	characteristic
27	#125	—	127	1	no	—
28	#103	—	127	4	no	—
29	#97	—	127	1	no	—
30	C₂×Q₈	C₂×D₁₂₇	1	2	yes	—
31	C₂×C₈	—	127	2	no	—
32	C₄⋊C₄	—	2	2	no	—
33	C₄⋊C₄	—	127	2	no	—
34	C₄⋊C₄	—	254	2	no	—
35	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
36	Q₈	—	2	4	no	—
37	C₂×C₄	—	2	2	no	—
38	C₂×C₄	—	254	2	no	—
39	C₂×C₄	C₂²⋊D₁₂₇	1	2	yes	—
40	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
41	C₈	—	254	2	no	—
42	C₂²	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
43	C₄	—	2	2	no	—
44	C₄	—	4	2	no	—
45	C₄	—	508	2	no	—
46	C₄	#14523247	1	2	yes	—
47	C₂	#16861196	1	1	yes	characteristic
48	C₂	#16861240	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	C₄⋊Q₈×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₄
hypercenter	C₄⋊Q₈	D₁₂₇
hypocenter	C₁₂₇	#491
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
24. C₁₂₇	#491	semidirect