Subgroups — #30638344

Order 16256.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550253	C₂	1	1	yes	maximal, characteristic, complemented
2	#22550252	C₂	1	4	yes	maximal, complemented
3	C₈×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#493×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₀₁₆	—	2	2	no	—
6	C₂×#14523233	—	2	8	no	—
7	#16861206	C₂²	1	2	yes	—
8	C₄×C₁₂₇⋊C₈	C₂²	1	2	yes	—
9	C₂×D₈×C₁₂₇	—	2	2	no	—
10	#127×C₁₂₇	C₂²	1	2	yes	—
11	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂×D₅₀₈	—	2	4	no	—
13	C₂×D₅₀₈	—	2	2	no	—
14	D₁₀₁₆	—	4	4	no	—
15	#14523233	—	4	16	no	—
16	C₂×C₁₂₇⋊C₈	D₄	1	4	yes	—
17	C₂×D₄×C₁₂₇	—	2	4	no	—
18	C₂×D₄×C₁₂₇	—	2	2	no	—
19	D₈×C₁₂₇	—	4	4	no	—
20	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
21	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
22	C₂²×D₁₂₇	—	4	2	no	—
23	D₅₀₈	—	4	8	no	—
24	D₅₀₈	—	4	4	no	—
25	C₁₂₇⋊C₈	D₈	1	8	yes	complemented
26	C₂³×C₁₂₇	—	4	2	no	—
27	D₄×C₁₂₇	—	4	4	no	—
28	D₄×C₁₂₇	—	4	8	no	—
29	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
30	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
31	C₈×C₁₂₇	D₈	1	4	yes	complemented
32	C₂²×C₁₂₇	—	8	4	no	—
33	C₂²×C₁₂₇	—	8	2	no	—
34	C₂²×C₁₂₇	#127	1	1	yes	characteristic
35	C₂×D₁₂₇	—	8	4	no	—
36	C₂×D₁₂₇	—	8	2	no	—
37	C₄×C₁₂₇	C₂×D₈	1	4	yes	—
38	C₄×C₁₂₇	C₂×D₈	1	2	yes	—
39	C₂×C₁₂₇	—	16	2	no	—
40	C₂×C₁₂₇	#493	1	2	yes	—
41	C₂×C₁₂₇	#493	1	1	yes	characteristic
42	D₁₂₇	—	16	2	no	—
43	#1712	—	127	1	no	maximal, Sylow
44	C₁₂₇	#1712	1	1	yes	characteristic, complemented, Sylow
45	#493	—	127	4	no	—
46	#493	—	127	1	no	—
47	#493	D₁₂₇	1	1	yes	characteristic, complemented
48	C₈²	—	127	1	no	—
49	C₂×D₈	—	2	2	no	—
50	C₂×D₈	—	254	8	no	—
51	C₂×D₈	—	254	2	no	—
52	#127	—	127	2	no	—
53	#127	C₂×D₁₂₇	1	2	yes	—
54	C₄×C₈	—	127	2	no	—
55	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
56	C₂×D₄	—	2	4	no	—
57	C₂×D₄	—	2	2	no	—
58	C₂×D₄	—	254	4	no	—
59	C₂×D₄	—	254	2	no	—
60	D₈	—	4	4	no	—
61	D₈	—	508	16	no	—
62	D₈	—	508	4	no	—
63	C₂×C₈	—	127	4	no	—
64	C₂×C₈	C₂²⋊D₁₂₇	1	2	yes	—
65	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
66	C₂³	—	4	2	no	—
67	C₂³	—	508	2	no	—
68	D₄	—	4	8	no	—
69	D₄	—	4	4	no	—
70	D₄	—	508	8	no	—
71	D₄	—	508	4	no	—
72	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
73	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
74	C₈	—	127	8	no	—
75	C₈	#14523233	1	4	yes	complemented
76	C₂²	—	8	4	no	—
77	C₂²	—	8	2	no	—
78	C₂²	—	1016	4	no	—
79	C₂²	—	1016	2	no	—
80	C₂²	#16861205	1	1	yes	characteristic
81	C₄	D₈×D₁₂₇	1	4	yes	—
82	C₄	C₂×#14523233	1	2	yes	—
83	C₂	—	16	2	no	—
84	C₂	—	2032	2	no	—
85	C₂	#22550251	1	2	yes	—
86	C₂	#22550252	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16861205
commutator, cosocle	C₄²×C₁₂₇	C₂³
Frattini	C₄²	C₂²×D₁₂₇
Fitting	#493×C₁₂₇	C₂
socle	C₂²×C₁₂₇	#127
hypercenter	#493	D₁₂₇
hypocenter	C₁₂₇	#1712
LCS	G ▷ C₄²×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

Complements

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

N	H ≅ G/N	type
1. #22550253	C₂	semidirect
2. #22550252	C₂	semidirect
3. C₈×C₁₂₇⋊C₈	C₂	semidirect
4. #493×C₁₂₇	C₂	semidirect
25. C₁₂₇⋊C₈	D₈	semidirect
31. C₈×C₁₂₇	D₈	semidirect
44. C₁₂₇	#1712	semidirect
47. #493	D₁₂₇	semidirect
75. C₈	#14523233	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22550253	2
#22550252	2
C₈×C₁₂₇⋊C₈	2
#493×C₁₂₇	2
#1712	127