Subgroups — #22550605

Order 8128.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16861209	C₂	1	3	yes	maximal
2	#16861200	C₂	1	6	yes	maximal
3	Q₈×Dic₁₂₇	C₂	1	2	yes	maximal
4	C₄×Dic₂₅₄	C₂	1	3	yes	maximal
5	C₄×Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₅₄	C₂²	1	3	yes	—
7	C₄⋊Dic₁₂₇	C₂²	1	6	yes	—
8	C₄⋊Dic₁₂₇	C₂²	1	6	yes	—
9	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
10	Dic₁₂₇⋊C₄	C₂²	1	6	yes	—
11	C₄×Dic₁₂₇	C₂²	1	6	yes	—
12	C₂×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₁₂₇	C₂²	1	3	yes	—
14	C₄²×C₁₂₇	C₂²	1	3	yes	—
15	Dic₂₅₄	—	2	6	no	—
16	C₂×Dic₁₂₇	C₂³	1	2	yes	—
17	C₂×Dic₁₂₇	C₂³	1	6	yes	—
18	Q₈×C₁₂₇	Q₈	1	4	yes	complemented
19	C₂×C₄×C₁₂₇	C₂³	1	3	yes	—
20	C₂×C₄×C₁₂₇	C₂³	1	3	yes	—
21	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
22	C₂²×C₁₂₇	C₂⁴	1	1	yes	characteristic
23	C₄×C₁₂₇	—	2	3	no	—
24	C₄×C₁₂₇	C₂×Q₈	1	6	yes	—
25	C₄×C₁₂₇	#41	1	2	yes	—
26	Dic₁₂₇	—	2	6	no	—
27	Dic₁₂₇	—	2	2	no	—
28	C₂×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
29	C₂×C₁₂₇	C₂×#41	1	1	yes	characteristic
30	C₂×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
31	C₁₂₇	#557	1	1	yes	characteristic, complemented, Sylow
32	#557	—	127	1	no	maximal, Sylow
33	C₄⋊Q₈	—	127	3	no	—
34	#125	—	127	6	no	—
35	C₄×Q₈	—	127	2	no	—
36	C₄×Q₈	—	127	3	no	—
37	C₄×Q₈	D₁₂₇	1	1	yes	characteristic
38	C₂×Q₈	—	127	3	no	—
39	C₂×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
40	C₄⋊C₄	—	127	6	no	—
41	C₄⋊C₄	—	127	1	no	—
42	C₄⋊C₄	—	127	6	no	—
43	C₄⋊C₄	—	127	6	no	—
44	C₄⋊C₄	C₂×D₁₂₇	1	3	yes	—
45	C₄²	—	127	6	no	—
46	C₄²	C₂×D₁₂₇	1	3	yes	—
47	Q₈	—	254	6	no	—
48	Q₈	Dic₂₅₄	1	4	yes	complemented
49	C₂×C₄	—	127	2	no	—
50	C₂×C₄	—	127	6	no	—
51	C₂×C₄	C₂²×D₁₂₇	1	3	yes	—
52	C₂×C₄	C₂²×D₁₂₇	1	3	yes	—
53	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
54	C₂²	C₂³×D₁₂₇	1	1	yes	characteristic
55	C₄	—	2	3	no	—
56	C₄	—	254	6	no	—
57	C₄	—	254	2	no	—
58	C₄	C₂×Dic₂₅₄	1	6	yes	—
59	C₄	#14523249	1	2	yes	—
60	C₂	C₂²×Dic₂₅₄	1	1	yes	characteristic
61	C₂	C₂×#14523249	1	1	yes	characteristic
62	C₂	#16861256	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂⁴
Fitting	C₄×Q₈×C₁₂₇	C₂
hypercenter	C₄×Q₈	D₁₂₇
hypocenter	C₁₂₇	#557
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
18. Q₈×C₁₂₇	Q₈	semidirect
31. C₁₂₇	#557	semidirect
48. Q₈	Dic₂₅₄	semidirect