Subgroups — #22549637

Order 8128.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16861168	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₁₂₇	—	2	1	no	—
5	C₄⋊Dic₁₂₇	C₄	1	2	yes	complemented
6	C₂²⋊Dic₁₂₇	—	2	1	no	—
7	C₄×Dic₁₂₇	—	2	1	no	—
8	C₄×Dic₁₂₇	—	2	1	no	—
9	C₄×Dic₁₂₇	C₄	1	2	yes	complemented
10	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
11	M₁₆×C₁₂₇	—	2	1	no	—
12	M₁₆×C₁₂₇	C₄	1	2	yes	complemented
13	C₂×C₈×C₁₂₇	—	2	1	no	—
14	C₂×Dic₁₂₇	—	2	4	no	—
15	C₂×Dic₁₂₇	—	2	2	no	—
16	C₂×Dic₁₂₇	—	2	2	no	—
17	C₂³×C₁₂₇	D₄	1	1	yes	characteristic
18	C₂×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
22	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
23	C₂×C₄×C₁₂₇	Q₈	1	1	yes	characteristic
24	C₈×C₁₂₇	—	2	2	no	—
25	C₂²×C₁₂₇	—	2	2	no	—
26	C₂²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
28	C₂²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
29	C₄×C₁₂₇	C₄²	1	1	yes	characteristic
30	C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
31	C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
32	C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
33	Dic₁₂₇	—	2	4	no	—
34	Dic₁₂₇	—	4	2	no	—
35	C₂×C₁₂₇	—	2	1	no	—
36	C₂×C₁₂₇	—	2	1	no	—
37	C₂×C₁₂₇	#95	1	1	yes	characteristic
38	C₂×C₁₂₇	#104	1	2	yes	—
39	C₁₂₇	#339	1	1	yes	characteristic, complemented, Sylow
40	#339	—	127	1	no	maximal, Sylow
41	C₂×M₁₆	D₁₂₇	1	1	yes	characteristic
42	#117	—	127	1	no	—
43	C₂×C₄²	—	127	1	no	—
44	C₂²×C₄	—	254	1	no	—
45	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
46	M₁₆	—	2	1	no	—
47	M₁₆	Dic₁₂₇	1	2	yes	complemented
48	C₂×C₈	—	2	1	no	—
49	C₄⋊C₄	—	127	2	no	—
50	C₂²⋊C₄	—	254	1	no	—
51	C₄²	—	127	2	no	—
52	C₄²	—	254	1	no	—
53	C₄²	—	254	1	no	—
54	C₂³	D₅₀₈	1	1	yes	characteristic
55	C₂×C₄	—	254	2	no	—
56	C₂×C₄	—	254	4	no	—
57	C₂×C₄	—	254	2	no	—
58	C₂×C₄	C₄×D₁₂₇	1	1	yes	characteristic
59	C₂×C₄	C₄×D₁₂₇	1	1	yes	characteristic
60	C₂×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
61	C₂×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
62	C₂×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
63	C₂×C₄	Dic₂₅₄	1	1	yes	characteristic
64	C₈	—	2	2	no	—
65	C₂²	—	2	2	no	—
66	C₂²	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
67	C₂²	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
68	C₂²	C₄⋊Dic₁₂₇	1	1	yes	characteristic
69	C₄	—	508	2	no	—
70	C₄	—	254	4	no	—
71	C₄	C₄×Dic₁₂₇	1	1	yes	characteristic
72	C₄	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
73	C₄	Dic₁₂₇⋊C₄	1	1	yes	characteristic
74	C₄	Dic₁₂₇⋊C₄	1	1	yes	characteristic
75	C₂	—	2	1	no	—
76	C₂	—	2	1	no	—
77	C₂	C₁₂₇⋊#95	1	1	yes	characteristic
78	C₂	#16861135	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₁₂₇
commutator	C₄×C₁₂₇	C₄²
Frattini	C₂²×C₄	C₂×D₁₂₇
Fitting	C₂×M₁₆×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²⋊C₄
hypercenter	C₂×M₁₆	D₁₂₇
hypocenter	C₁₂₇	#339
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
5. C₄⋊Dic₁₂₇	C₄	semidirect
9. C₄×Dic₁₂₇	C₄	semidirect
12. M₁₆×C₁₂₇	C₄	semidirect
39. C₁₂₇	#339	semidirect
47. M₁₆	Dic₁₂₇	semidirect