Subgroups — #30640973

Order 16256.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550253	C₂	1	1	yes	maximal, characteristic
2	#22549676	C₂	1	4	yes	maximal, complemented
3	#22549652	C₂	1	1	yes	maximal, characteristic, complemented
4	#499×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₀₁₆	—	2	1	no	—
6	C₂×D₁₀₁₆	C₂²	1	2	yes	—
7	#16861206	—	2	1	no	—
8	C₂×C₁₂₇⋊C₁₆	C₂²	1	2	yes	—
9	#125×C₁₂₇	—	2	1	no	—
10	#107×C₁₂₇	C₂²	1	2	yes	—
11	#106×C₁₂₇	—	2	1	no	—
12	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×D₅₀₈	—	2	2	no	—
14	C₂×D₅₀₈	—	4	2	no	—
15	D₁₀₁₆	—	2	4	no	—
16	D₁₀₁₆	—	4	2	no	—
17	C₁₂₇⋊C₁₆	—	2	2	no	—
18	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
19	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₁₂₇	—	2	2	no	—
21	C₄⋊C₄×C₁₂₇	—	4	2	no	—
22	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
23	C₂²×D₁₂₇	—	4	2	no	—
24	D₅₀₈	—	4	2	no	—
25	D₅₀₈	—	4	2	no	—
26	D₅₀₈	—	8	4	no	—
27	C₂×C₄×C₁₂₇	—	4	2	no	—
28	C₂×C₄×C₁₂₇	D₈	1	2	yes	—
29	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
30	C₈×C₁₂₇	SD₁₆	1	2	yes	—
31	C₈×C₁₂₇	#41	1	2	yes	—
32	C₂²×C₁₂₇	C₂×D₈	1	1	yes	characteristic
33	C₂×D₁₂₇	—	8	2	no	—
34	C₂×D₁₂₇	—	8	4	no	—
35	C₄×C₁₂₇	—	2	2	no	—
36	C₄×C₁₂₇	—	8	2	no	—
37	C₄×C₁₂₇	#124	1	1	yes	characteristic
38	C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
39	C₂×C₁₂₇	#486	1	1	yes	characteristic
40	C₂×C₁₂₇	#509	1	2	yes	—
41	D₁₂₇	—	16	2	no	—
42	#2244	—	127	1	no	maximal, Sylow
43	C₁₂₇	#2244	1	1	yes	characteristic, complemented, Sylow
44	#499	D₁₂₇	1	1	yes	characteristic, complemented
45	#493	—	127	1	no	—
46	#357	—	127	4	no	—
47	#346	—	127	1	no	—
48	C₂×D₈	—	127	2	no	—
49	C₂×D₈	—	254	1	no	—
50	#127	—	254	1	no	—
51	#125	—	2	1	no	—
52	C₂×C₁₆	—	127	2	no	—
53	#107	C₂×D₁₂₇	1	2	yes	—
54	#106	—	2	1	no	—
55	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
56	C₂×D₄	—	508	2	no	—
57	C₂×D₄	—	254	2	no	—
58	D₈	—	508	2	no	—
59	D₈	—	254	4	no	—
60	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
61	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
62	C₄⋊C₄	—	2	2	no	—
63	C₄⋊C₄	—	4	2	no	—
64	C₄²	C₂²⋊D₁₂₇	1	1	yes	characteristic
65	C₁₆	—	254	2	no	—
66	C₂³	—	508	2	no	—
67	D₄	—	1016	4	no	—
68	D₄	—	508	2	no	—
69	D₄	—	508	2	no	—
70	C₂×C₄	—	4	2	no	—
71	C₂×C₄	#14523233	1	2	yes	—
72	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
73	C₈	#14523235	1	2	yes	—
74	C₈	#14523249	1	2	yes	—
75	C₂²	—	1016	4	no	—
76	C₂²	—	1016	2	no	—
77	C₂²	C₂×#14523233	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	8	2	no	—
80	C₄	#16861198	1	1	yes	characteristic
81	C₄	C₂×#14523235	1	1	yes	characteristic
82	C₂	—	2032	2	no	—
83	C₂	#22550224	1	1	yes	characteristic
84	C₂	#22550311	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#14523233
commutator, cosocle	C₂×C₈×C₁₂₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	#499×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₈
prime radical	#22550253	C₂
hypercenter	#499	D₁₂₇
hypocenter	C₁₂₇	#2244
LCS	G ▷ C₂×C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #499

Complements

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

N	H ≅ G/N	type
2. #22549676	C₂	semidirect
3. #22549652	C₂	semidirect
4. #499×C₁₂₇	C₂	semidirect
43. C₁₂₇	#2244	semidirect
44. #499	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550253	2
#22549676	2
#22549652	2
#499×C₁₂₇	2
#2244	127