Subgroups — #22550602

Order 8128.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16861208	C₂	1	3	yes	maximal
2	#16861201	C₂	1	3	yes	maximal
3	#16861200	C₂	1	3	yes	maximal
4	Q₈×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
5	Q₈×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₂₅₄	C₂	1	3	yes	maximal
7	C₄×Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₅₄	C₂²	1	3	yes	—
9	C₄⋊Dic₁₂₇	C₂²	1	3	yes	—
10	C₄⋊Dic₁₂₇	C₂²	1	3	yes	—
11	C₄⋊Dic₁₂₇	C₂²	1	3	yes	—
12	Dic₁₂₇⋊C₄	C₂²	1	6	yes	—
13	Dic₁₂₇⋊C₄	C₂²	1	3	yes	—
14	Dic₁₂₇⋊C₄	C₂²	1	1	yes	characteristic
15	C₄×Dic₁₂₇	C₂²	1	3	yes	—
16	C₄×Dic₁₂₇	C₂²	1	3	yes	—
17	C₂×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₁₂₇	C₂²	1	3	yes	—
19	C₄²×C₁₂₇	C₂²	1	3	yes	—
20	Dic₂₅₄	—	2	6	no	—
21	C₂×Dic₁₂₇	C₂³	1	1	yes	characteristic
22	C₂×Dic₁₂₇	C₂³	1	3	yes	—
23	C₂×Dic₁₂₇	C₂³	1	3	yes	—
24	C₂×Dic₁₂₇	C₂³	1	1	yes	characteristic
25	Q₈×C₁₂₇	Q₈	1	4	yes	complemented
26	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
27	C₂×C₄×C₁₂₇	C₂³	1	3	yes	—
28	C₂×C₄×C₁₂₇	C₂³	1	3	yes	—
29	C₂²×C₁₂₇	C₂⁴	1	1	yes	characteristic
30	C₄×C₁₂₇	—	2	3	no	—
31	C₄×C₁₂₇	—	2	1	no	—
32	C₄×C₁₂₇	C₂×Q₈	1	6	yes	—
33	Dic₁₂₇	—	2	3	no	—
34	Dic₁₂₇	—	2	1	no	—
35	Dic₁₂₇	—	2	3	no	—
36	Dic₁₂₇	#41	1	2	yes	—
37	C₂×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
38	C₂×C₁₂₇	C₂×#41	1	1	yes	characteristic
39	C₂×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
40	C₁₂₇	#557	1	1	yes	characteristic, complemented, Sylow
41	#557	—	127	1	no	maximal, Sylow
42	C₄⋊Q₈	—	127	3	no	—
43	#125	—	127	3	no	—
44	#125	—	127	3	no	—
45	C₄×Q₈	—	127	1	no	—
46	C₄×Q₈	—	127	1	no	—
47	C₄×Q₈	—	127	3	no	—
48	C₄×Q₈	D₁₂₇	1	1	yes	characteristic
49	C₂×Q₈	—	127	3	no	—
50	C₂×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
51	C₄⋊C₄	—	127	1	no	—
52	C₄⋊C₄	—	127	3	no	—
53	C₄⋊C₄	—	127	6	no	—
54	C₄⋊C₄	—	127	3	no	—
55	C₄⋊C₄	—	127	3	no	—
56	C₄⋊C₄	—	127	3	no	—
57	C₄⋊C₄	C₂×D₁₂₇	1	3	yes	—
58	C₄²	—	127	3	no	—
59	C₄²	—	127	3	no	—
60	C₄²	C₂×D₁₂₇	1	3	yes	—
61	Q₈	—	254	6	no	—
62	Q₈	Dic₂₅₄	1	4	yes	complemented
63	C₂×C₄	—	127	1	no	—
64	C₂×C₄	—	127	1	no	—
65	C₂×C₄	—	127	3	no	—
66	C₂×C₄	—	127	3	no	—
67	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
68	C₂×C₄	C₂²×D₁₂₇	1	3	yes	—
69	C₂×C₄	C₂²×D₁₂₇	1	3	yes	—
70	C₂²	C₂³×D₁₂₇	1	1	yes	characteristic
71	C₄	—	2	3	no	—
72	C₄	—	2	1	no	—
73	C₄	—	127	2	no	—
74	C₄	—	254	3	no	—
75	C₄	—	254	1	no	—
76	C₄	—	254	3	no	—
77	C₄	C₂×Dic₂₅₄	1	6	yes	—
78	C₂	C₂²×Dic₂₅₄	1	1	yes	characteristic
79	C₂	#41×D₁₂₇	1	1	yes	characteristic
80	C₂	#16861257	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
25. Q₈×C₁₂₇	Q₈	semidirect
40. C₁₂₇	#557	semidirect
62. Q₈	Dic₂₅₄	semidirect