Subgroups — #22549886

Order 8128.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₂₅₄	C₂	1	1	yes	maximal, characteristic
2	#16861168	C₂	1	1	yes	maximal, characteristic
3	#16861134	C₂	1	4	yes	maximal, complemented
4	C₂×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₂₅₄	—	2	2	no	—
6	C₂×Dic₂₅₄	—	2	1	no	—
7	C₂×Dic₂₅₄	C₄	1	4	yes	complemented
8	C₂×Dic₂₅₄	C₂²	1	2	yes	—
9	C₂²×Dic₁₂₇	—	2	1	no	—
10	C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
11	C₂²⋊Dic₁₂₇	—	2	1	no	—
12	C₄×Dic₁₂₇	—	2	1	no	—
13	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
14	M₁₆×C₁₂₇	—	2	2	no	—
15	C₂×C₈×C₁₂₇	C₂²	1	2	yes	—
16	Dic₂₅₄	—	2	4	no	—
17	Dic₂₅₄	—	2	2	no	—
18	Dic₂₅₄	C₂×C₄	1	4	yes	—
19	C₂×Dic₁₂₇	—	2	4	no	—
20	C₂×Dic₁₂₇	—	2	2	no	—
21	C₂×Dic₁₂₇	—	2	2	no	—
22	C₂³×C₁₂₇	D₄	1	1	yes	characteristic
23	C₂×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
24	C₂×C₄×C₁₂₇	D₄	1	2	yes	—
25	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
26	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
27	C₈×C₁₂₇	—	2	2	no	—
28	C₂²×C₁₂₇	—	2	2	no	—
29	C₂²×C₁₂₇	C₂²⋊C₄	1	2	yes	—
30	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
31	C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
32	C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
33	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
34	Dic₁₂₇	—	2	4	no	—
35	Dic₁₂₇	—	4	2	no	—
36	C₂×C₁₂₇	—	2	2	no	—
37	C₂×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₁₂₇	#137	1	2	yes	—
39	C₁₂₇	#419	1	1	yes	characteristic, complemented, Sylow
40	#419	—	127	1	no	maximal, Sylow
41	C₂²×Q₈	—	127	1	no	—
42	C₂×M₁₆	D₁₂₇	1	1	yes	characteristic
43	#117	—	127	1	no	—
44	#103	—	127	4	no	—
45	C₂×Q₈	—	127	4	no	—
46	C₂×Q₈	—	127	2	no	—
47	C₂×Q₈	—	254	2	no	—
48	C₂×Q₈	—	254	1	no	—
49	C₂²×C₄	—	254	1	no	—
50	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
51	M₁₆	—	2	2	no	—
52	C₂×C₈	C₂×D₁₂₇	1	2	yes	—
53	C₄⋊C₄	—	127	2	no	—
54	C₂²⋊C₄	—	254	1	no	—
55	C₄²	—	254	1	no	—
56	C₂³	D₅₀₈	1	1	yes	characteristic
57	Q₈	—	127	4	no	—
58	Q₈	—	254	4	no	—
59	Q₈	—	254	2	no	—
60	C₂×C₄	—	254	2	no	—
61	C₂×C₄	—	254	4	no	—
62	C₂×C₄	—	254	2	no	—
63	C₂×C₄	C₄×D₁₂₇	1	2	yes	—
64	C₂×C₄	C₂²⋊D₁₂₇	1	2	yes	—
65	C₂×C₄	D₅₀₈	1	1	yes	characteristic
66	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
67	C₈	—	2	2	no	—
68	C₂²	—	2	2	no	—
69	C₂²	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
70	C₂²	C₂×D₅₀₈	1	1	yes	characteristic
71	C₄	—	508	2	no	—
72	C₄	—	254	4	no	—
73	C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
74	C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
75	C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
76	C₂	—	2	2	no	—
77	C₂	C₂×(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
78	C₂	#16861242	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₅₀₈
commutator	C₄×C₁₂₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	C₂×M₁₆×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₄
hypercenter	C₂×M₁₆	D₁₂₇
hypocenter	C₁₂₇	#419
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #16861134	C₂	semidirect
7. C₂×Dic₂₅₄	C₄	semidirect
39. C₁₂₇	#419	semidirect