Subgroups — #30636766

Order 16256.

112 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	#22549697	C₂	1	1	yes	maximal, characteristic, complemented
3	#364×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₀₁₆	—	2	1	no	—
5	C₂×D₁₀₁₆	—	2	1	no	—
6	#16861206	C₄	1	2	yes	—
7	#16861149	—	2	1	no	—
8	M₃₂×C₁₂₇	—	2	1	no	—
9	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂×D₅₀₈	—	2	2	no	—
11	C₂×D₅₀₈	—	4	2	no	—
12	D₁₀₁₆	—	4	1	no	—
13	D₁₀₁₆	—	4	1	no	—
14	D₁₀₁₆	—	4	1	no	—
15	D₁₀₁₆	—	4	1	no	—
16	C₁₂₇⋊C₁₆	—	4	1	no	—
17	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
18	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
19	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
20	C₁₆×C₁₂₇	—	4	1	no	—
21	C₂²×D₁₂₇	—	4	2	no	—
22	D₅₀₈	—	4	2	no	—
23	D₅₀₈	—	4	2	no	—
24	D₅₀₈	—	8	4	no	—
25	C₂×C₄×C₁₂₇	—	2	1	no	—
26	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₈×C₁₂₇	D₈	1	1	yes	characteristic
28	C₈×C₁₂₇	D₈	1	1	yes	characteristic
29	C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
30	C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
31	C₂²×C₁₂₇	#100	1	1	yes	characteristic
32	C₂×D₁₂₇	—	8	4	no	—
33	C₂×D₁₂₇	—	8	2	no	—
34	C₄×C₁₂₇	—	2	2	no	—
35	C₄×C₁₂₇	#102	1	1	yes	characteristic
36	C₄×C₁₂₇	#102	1	1	yes	characteristic
37	C₂×C₁₂₇	—	2	1	no	—
38	C₂×C₁₂₇	#331	1	1	yes	characteristic
39	D₁₂₇	—	16	2	no	—
40	#1357	—	127	1	no	maximal, Sylow
41	C₁₂₇	#1357	1	1	yes	characteristic, complemented, Sylow
42	#493	—	127	1	no	—
43	#364	—	127	1	no	—
44	#364	D₁₂₇	1	1	yes	characteristic, complemented
45	C₂×D₈	—	254	1	no	—
46	C₂×D₈	—	254	1	no	—
47	#127	—	127	2	no	—
48	M₃₂	—	2	1	no	—
49	M₃₂	—	254	1	no	—
50	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
51	C₂×D₄	—	508	2	no	—
52	C₂×D₄	—	254	2	no	—
53	D₈	—	508	1	no	—
54	D₈	—	508	1	no	—
55	D₈	—	508	1	no	—
56	D₈	—	508	1	no	—
57	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
58	C₂×C₈	D₅₀₈	1	1	yes	characteristic
59	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
60	C₁₆	—	4	1	no	—
61	C₁₆	—	508	1	no	—
62	C₂³	—	508	2	no	—
63	D₄	—	1016	4	no	—
64	D₄	—	508	2	no	—
65	D₄	—	508	2	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
68	C₈	#14523233	1	1	yes	characteristic
69	C₈	D₁₀₁₆	1	1	yes	characteristic
70	C₈	#14523235	1	1	yes	characteristic
71	C₈	#14523237	1	1	yes	characteristic
72	C₂²	—	1016	4	no	—
73	C₂²	—	1016	2	no	—
74	C₂²	#16861125	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	#16861129	1	1	yes	characteristic
77	C₄	#16861131	1	1	yes	characteristic
78	C₂	—	2	1	no	—
79	C₂	—	2032	2	no	—
80	C₂	#22549619	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22549619
commutator	C₄²×C₁₂₇	C₂×C₄
Frattini	C₄×C₈	C₂×D₁₂₇
Fitting	#364×C₁₂₇	C₂
socle	C₂×C₁₂₇	#331
prime radical	#22550253	C₂
cosocle	C₄×C₈×C₁₂₇	C₂²
hypercenter	#364	D₁₂₇
hypocenter	C₁₂₇	#1357
LCS	G ▷ C₄²×C₁₂₇ ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
2. #22549697	C₂	semidirect
3. #364×C₁₂₇	C₂	semidirect
41. C₁₂₇	#1357	semidirect
44. #364	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
#22549697	2
#364×C₁₂₇	2
#1357	127