Subgroups — #22550255

Order 8128.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14523238	C₂	1	4	yes	maximal
2	#16861209	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₂₅₄	C₂²	1	2	yes	—
6	#14523238	—	2	8	no	—
7	C₂×C₁₂₇⋊C₈	C₂²	1	2	yes	—
8	C₄⋊Dic₁₂₇	—	2	2	no	—
9	C₂×Q₈×C₁₂₇	C₂²	1	2	yes	—
10	C₄⋊C₄×C₁₂₇	—	2	2	no	—
11	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
12	Dic₂₅₄	—	2	4	no	—
13	C₂×Dic₁₂₇	—	2	2	no	—
14	C₁₂₇⋊C₈	D₄	1	4	yes	—
15	Q₈×C₁₂₇	—	2	4	no	—
16	C₂×C₄×C₁₂₇	—	2	2	no	—
17	C₂×C₄×C₁₂₇	D₄	1	2	yes	—
18	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
19	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
20	C₄×C₁₂₇	—	4	2	no	—
21	C₄×C₁₂₇	Q₁₆	1	4	yes	complemented
22	C₄×C₁₂₇	C₂×D₄	1	2	yes	—
23	Dic₁₂₇	—	4	2	no	—
24	C₂×C₁₂₇	#127	1	1	yes	characteristic
25	C₂×C₁₂₇	C₂×Q₁₆	1	2	yes	—
26	C₁₂₇	#494	1	1	yes	characteristic, complemented, Sylow
27	#494	—	127	1	no	maximal, Sylow
28	C₂×Q₁₆	—	127	4	no	—
29	C₄⋊Q₈	—	127	1	no	—
30	C₄⋊Q₈	D₁₂₇	1	1	yes	characteristic
31	C₄×C₈	—	127	1	no	—
32	C₂×Q₈	—	127	2	no	—
33	C₂×Q₈	C₂×D₁₂₇	1	2	yes	—
34	Q₁₆	—	254	8	no	—
35	C₂×C₈	—	127	2	no	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄⋊C₄	—	254	2	no	—
38	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
39	Q₈	—	2	4	no	—
40	Q₈	—	254	4	no	—
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	254	2	no	—
43	C₂×C₄	C₂²⋊D₁₂₇	1	2	yes	—
44	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
45	C₈	—	127	4	no	—
46	C₂²	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
47	C₄	—	4	2	no	—
48	C₄	—	508	2	no	—
49	C₄	#14523238	1	4	yes	complemented
50	C₄	D₄×D₁₂₇	1	2	yes	—
51	C₂	#16861205	1	1	yes	characteristic
52	C₂	C₂×#14523238	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	#494
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
21. C₄×C₁₂₇	Q₁₆	semidirect
26. C₁₂₇	#494	semidirect
49. C₄	#14523238	semidirect