Subgroups — #22549611

Order 8128.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16861183	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇⋊#95	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₅₀₈	C₄	1	1	yes	characteristic, complemented
5	C₂×C₂²⋊D₁₂₇	—	2	1	no	—
6	C₂²×Dic₁₂₇	—	2	1	no	—
7	C₄⋊Dic₁₂₇	C₄	1	1	yes	characteristic, complemented
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	D₅₀₈	—	4	1	no	—
13	C₂²⋊D₁₂₇	—	4	1	no	—
14	C₂²⋊D₁₂₇	—	4	1	no	—
15	C₂×Dic₁₂₇	—	2	2	no	—
16	C₂×Dic₁₂₇	—	2	1	no	—
17	C₂×Dic₁₂₇	—	4	1	no	—
18	C₂×Dic₁₂₇	—	4	1	no	—
19	C₂³×C₁₂₇	D₄	1	1	yes	characteristic
20	C₂×C₄×C₁₂₇	—	4	1	no	—
21	C₂×C₄×C₁₂₇	C₂×C₄	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	D₈	1	1	yes	characteristic
28	C₂²×C₁₂₇	SD₁₆	1	1	yes	characteristic
29	C₂×D₁₂₇	—	4	1	no	—
30	C₂×D₁₂₇	—	4	1	no	—
31	C₂×D₁₂₇	—	4	1	no	—
32	C₄×C₁₂₇	—	2	1	no	—
33	C₄×C₁₂₇	—	2	1	no	—
34	Dic₁₂₇	—	4	2	no	—
35	Dic₁₂₇	—	4	1	no	—
36	C₂×C₁₂₇	—	2	1	no	—
37	C₂×C₁₂₇	—	2	1	no	—
38	C₂×C₁₂₇	#99	1	1	yes	characteristic
39	C₂×C₁₂₇	#102	1	1	yes	characteristic
40	C₂×C₁₂₇	#104	1	1	yes	characteristic
41	D₁₂₇	—	8	1	no	—
42	C₁₂₇	#327	1	1	yes	characteristic, complemented, Sylow
43	#327	—	127	1	no	maximal, Sylow
44	#121	—	127	1	no	—
45	C₂²⋊C₈	D₁₂₇	1	1	yes	characteristic, complemented
46	#95	—	127	1	no	—
47	C₂×D₄	—	127	1	no	—
48	C₂×D₄	—	254	1	no	—
49	C₂²×C₄	—	254	1	no	—
50	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
51	C₂×C₈	—	2	1	no	—
52	C₄⋊C₄	—	127	1	no	—
53	C₂²⋊C₄	—	254	1	no	—
54	C₂³	—	254	1	no	—
55	C₂³	D₅₀₈	1	1	yes	characteristic
56	D₄	—	508	1	no	—
57	D₄	—	508	1	no	—
58	D₄	—	508	1	no	—
59	C₂×C₄	—	4	1	no	—
60	C₂×C₄	—	508	1	no	—
61	C₂×C₄	—	508	1	no	—
62	C₂×C₄	—	254	2	no	—
63	C₂×C₄	—	254	1	no	—
64	C₂×C₄	C₄×D₁₂₇	1	1	yes	characteristic
65	C₂×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
66	C₈	—	4	1	no	—
67	C₂²	—	2	1	no	—
68	C₂²	—	2	1	no	—
69	C₂²	—	508	1	no	—
70	C₂²	—	508	1	no	—
71	C₂²	—	508	1	no	—
72	C₂²	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
73	C₂²	D₁₀₁₆	1	1	yes	characteristic
74	C₂²	#14523237	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	2	1	no	—
77	C₄	—	508	2	no	—
78	C₄	—	508	1	no	—
79	C₂	—	2	1	no	—
80	C₂	—	2	1	no	—
81	C₂	—	1016	1	no	—
82	C₂	#16861123	1	1	yes	characteristic
83	C₂	#16861131	1	1	yes	characteristic
84	C₂	#16861135	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×D₁₂₇)⋊C₄
commutator	C₂×C₄×C₁₂₇	C₂×C₄
Frattini	C₂²×C₄	C₂×D₁₂₇
Fitting	C₂²⋊C₈×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²⋊C₄
hypercenter	C₂²⋊C₈	D₁₂₇
hypocenter	C₁₂₇	#327
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂²×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. C₁₂₇⋊#95	C₂	semidirect
3. C₂²⋊C₈×C₁₂₇	C₂	semidirect
4. C₂×D₅₀₈	C₄	semidirect
7. C₄⋊Dic₁₂₇	C₄	semidirect
42. C₁₂₇	#327	semidirect
45. C₂²⋊C₈	D₁₂₇	semidirect