Subgroups — #22549599

Order 8128.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×(C₂×D₁₂₇)⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊(C₁₂₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₁₂₇	C₄	1	1	yes	characteristic
5	C₂²×Dic₁₂₇	C₄	1	1	yes	characteristic
6	C₂×C₁₂₇⋊C₈	—	2	1	no	—
7	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
8	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
9	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂×C₈×C₁₂₇	—	2	1	no	—
11	C₂²×D₁₂₇	—	2	1	no	—
12	C₂²×D₁₂₇	—	2	1	no	—
13	C₂²×D₁₂₇	—	4	1	no	—
14	C₂²×D₁₂₇	—	4	1	no	—
15	C₂²×D₁₂₇	C₈	1	2	yes	complemented
16	C₂×Dic₁₂₇	—	2	1	no	—
17	C₂×Dic₁₂₇	—	4	1	no	—
18	C₁₂₇⋊C₈	—	4	1	no	—
19	C₂³×C₁₂₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₁₂₇	—	4	1	no	—
21	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
22	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
23	C₈×C₁₂₇	—	4	1	no	—
24	C₂²×C₁₂₇	—	2	1	no	—
25	C₂²×C₁₂₇	—	2	1	no	—
26	C₂²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₁₂₇	C₂×C₈	1	1	yes	characteristic
28	C₂²×C₁₂₇	M₁₆	1	1	yes	characteristic
29	C₂×D₁₂₇	—	2	2	no	—
30	C₂×D₁₂₇	—	4	2	no	—
31	C₂×D₁₂₇	—	4	1	no	—
32	C₂×D₁₂₇	—	4	1	no	—
33	C₂×D₁₂₇	—	4	2	no	—
34	C₄×C₁₂₇	—	2	1	no	—
35	C₄×C₁₂₇	—	2	1	no	—
36	Dic₁₂₇	—	4	1	no	—
37	C₂×C₁₂₇	—	2	1	no	—
38	C₂×C₁₂₇	—	2	1	no	—
39	C₂×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
40	C₂×C₁₂₇	#99	1	1	yes	characteristic
41	C₂×C₁₂₇	#100	1	1	yes	characteristic
42	D₁₂₇	—	4	2	no	—
43	C₁₂₇	#323	1	1	yes	characteristic, complemented, Sylow
44	#323	—	127	1	no	maximal, Sylow
45	C₂×C₂²⋊C₄	—	127	1	no	—
46	C₂²⋊C₈	—	127	1	no	—
47	C₂²⋊C₈	D₁₂₇	1	1	yes	characteristic, complemented
48	C₂⁴	—	127	1	no	—
49	C₂²×C₄	—	127	1	no	—
50	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
51	C₂×C₈	—	2	1	no	—
52	C₂×C₈	—	254	1	no	—
53	C₂²⋊C₄	—	254	1	no	—
54	C₂²⋊C₄	—	254	1	no	—
55	C₂³	—	127	2	no	—
56	C₂³	—	508	1	no	—
57	C₂³	—	508	1	no	—
58	C₂³	—	254	1	no	—
59	C₂³	—	254	1	no	—
60	C₂³	C₄×D₁₂₇	1	1	yes	characteristic
61	C₂×C₄	—	4	1	no	—
62	C₂×C₄	—	508	1	no	—
63	C₂×C₄	—	254	1	no	—
64	C₂×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
65	C₂×C₄	D₅₀₈	1	1	yes	characteristic
66	C₈	—	4	1	no	—
67	C₈	—	508	1	no	—
68	C₂²	—	2	1	no	—
69	C₂²	—	2	1	no	—
70	C₂²	—	508	2	no	—
71	C₂²	—	508	2	no	—
72	C₂²	—	508	1	no	—
73	C₂²	—	508	1	no	—
74	C₂²	—	254	2	no	—
75	C₂²	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
76	C₂²	C₈×D₁₂₇	1	1	yes	characteristic
77	C₂²	#14523231	1	1	yes	characteristic
78	C₄	—	2	1	no	—
79	C₄	—	2	1	no	—
80	C₄	—	508	1	no	—
81	C₂	—	2	1	no	—
82	C₂	—	2	1	no	—
83	C₂	—	508	2	no	—
84	C₂	(C₂×D₁₂₇)⋊C₈	1	1	yes	characteristic
85	C₂	#16861123	1	1	yes	characteristic
86	C₂	#16861125	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×D₁₂₇)⋊C₄
commutator	C₂²×C₁₂₇	C₂×C₈
Frattini	C₂²×C₄	C₂×D₁₂₇
Fitting	C₂²⋊C₈×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²⋊C₄
hypercenter	C₂²⋊C₈	D₁₂₇
hypocenter	C₁₂₇	#323
LCS	G ▷ C₂²×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₂²⋊(C₁₂₇⋊C₈)	C₂	semidirect
3. C₂²⋊C₈×C₁₂₇	C₂	semidirect
15. C₂²×D₁₂₇	C₈	semidirect
43. C₁₂₇	#323	semidirect
47. C₂²⋊C₈	D₁₂₇	semidirect