Subgroups — #22550266

Order 8128.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14523235	C₂	1	2	yes	maximal, complemented
2	C₂×#14523233	C₂	1	2	yes	maximal, complemented
3	#16861206	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861120	C₂	1	1	yes	maximal, characteristic, complemented
5	#124×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₅₀₈	—	2	2	no	—
7	C₂×D₅₀₈	C₂²	1	2	yes	—
8	#14523235	—	2	4	no	—
9	#14523233	—	2	4	no	—
10	C₂×C₁₂₇⋊C₈	C₂²	1	2	yes	—
11	C₂×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
14	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂²×D₁₂₇	—	2	2	no	—
16	D₅₀₈	—	2	2	no	—
17	D₅₀₈	—	2	2	no	—
18	D₅₀₈	—	4	4	no	—
19	C₁₂₇⋊C₈	D₄	1	4	yes	complemented
20	C₂³×C₁₂₇	—	2	1	no	—
21	Q₈×C₁₂₇	—	2	2	no	—
22	D₄×C₁₂₇	—	2	2	no	—
23	C₂×C₄×C₁₂₇	—	2	1	no	—
24	C₂×C₄×C₁₂₇	D₄	1	2	yes	—
25	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
26	C₂²×C₁₂₇	—	4	1	no	—
27	C₂²×C₁₂₇	—	4	2	no	—
28	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
29	C₂×D₁₂₇	—	4	2	no	—
30	C₂×D₁₂₇	—	4	4	no	—
31	C₄×C₁₂₇	—	2	2	no	—
32	C₄×C₁₂₇	—	4	1	no	—
33	C₄×C₁₂₇	C₂×D₄	1	2	yes	—
34	C₂×C₁₂₇	—	8	1	no	—
35	C₂×C₁₂₇	#127	1	1	yes	characteristic
36	C₂×C₁₂₇	AGL(1,ℤ/8)	1	2	yes	—
37	D₁₂₇	—	8	2	no	—
38	C₁₂₇	#496	1	1	yes	characteristic, complemented, Sylow
39	#496	—	127	1	no	maximal, Sylow
40	C₂×SD₁₆	—	127	2	no	—
41	C₂×D₈	—	127	2	no	—
42	#127	—	127	1	no	—
43	#124	D₁₂₇	1	1	yes	characteristic, complemented
44	#97	—	127	1	no	—
45	C₂×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
46	C₂×D₄	—	127	2	no	—
47	C₂×D₄	—	254	2	no	—
48	C₂×D₄	C₂×D₁₂₇	1	1	yes	characteristic
49	SD₁₆	—	254	4	no	—
50	D₈	—	254	4	no	—
51	C₂×C₈	—	127	2	no	—
52	C₂²⋊C₄	—	2	2	no	—
53	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
54	C₂³	—	2	1	no	—
55	C₂³	—	254	2	no	—
56	Q₈	—	2	2	no	—
57	D₄	—	2	2	no	—
58	D₄	—	508	4	no	—
59	D₄	—	254	2	no	—
60	D₄	—	254	2	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	C₂²⋊D₁₂₇	1	2	yes	—
63	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
64	C₈	—	127	4	no	—
65	C₂²	—	4	1	no	—
66	C₂²	—	4	2	no	—
67	C₂²	—	508	2	no	—
68	C₂²	—	508	4	no	—
69	C₂²	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	4	1	no	—
72	C₄	D₄×D₁₂₇	1	2	yes	—
73	C₂	—	8	1	no	—
74	C₂	—	1016	2	no	—
75	C₂	#16861205	1	1	yes	characteristic
76	C₂	#16861234	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	#124×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₄
hypercenter	#124	D₁₂₇
hypocenter	C₁₂₇	#496
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #124

Complements

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

N	H ≅ G/N	type
1. C₂×#14523235	C₂	semidirect
2. C₂×#14523233	C₂	semidirect
3. #16861206	C₂	semidirect
4. #16861120	C₂	semidirect
5. #124×C₁₂₇	C₂	semidirect
19. C₁₂₇⋊C₈	D₄	semidirect
38. C₁₂₇	#496	semidirect
43. #124	D₁₂₇	semidirect