Subgroups — #30636792

Order 16256.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550278	C₂	1	1	yes	maximal, characteristic
2	#22549697	C₂	1	1	yes	maximal, characteristic
3	#364×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	#16861201	C₄	1	2	yes	—
5	#16861149	—	2	1	no	—
6	#16861143	—	2	1	no	—
7	#16861141	—	2	1	no	—
8	M₃₂×C₁₂₇	—	2	1	no	—
9	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₁₂₇	—	2	2	no	—
11	Dic₁₂₇⋊C₄	—	4	2	no	—
12	C₁₂₇⋊C₁₆	—	4	1	no	—
13	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
14	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
15	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₁₂₇	—	4	1	no	—
17	C₂×Dic₁₂₇	—	4	2	no	—
18	C₂×C₄×C₁₂₇	—	2	1	no	—
19	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
20	C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
21	C₈×C₁₂₇	SD₁₆	1	1	yes	characteristic
22	C₈×C₁₂₇	Q₁₆	1	1	yes	characteristic
23	C₈×C₁₂₇	Q₁₆	1	1	yes	characteristic
24	C₂²×C₁₂₇	#100	1	1	yes	characteristic
25	C₄×C₁₂₇	—	4	1	no	—
26	C₄×C₁₂₇	#103	1	1	yes	characteristic
27	C₄×C₁₂₇	#103	1	1	yes	characteristic
28	Dic₁₂₇	—	8	2	no	—
29	C₂×C₁₂₇	—	2	1	no	—
30	C₂×C₁₂₇	#333	1	1	yes	characteristic
31	#1366	—	127	1	no	maximal, Sylow
32	C₁₂₇	#1366	1	1	yes	characteristic, complemented, Sylow
33	#499	—	127	1	no	—
34	#364	—	127	1	no	—
35	#364	D₁₂₇	1	1	yes	characteristic
36	#125	—	127	2	no	—
37	M₃₂	—	2	1	no	—
38	M₃₂	—	254	1	no	—
39	#107	—	254	1	no	—
40	#106	—	254	1	no	—
41	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
42	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
43	C₂×C₈	D₅₀₈	1	1	yes	characteristic
44	C₄⋊C₄	—	508	2	no	—
45	C₄⋊C₄	—	254	2	no	—
46	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
47	C₁₆	—	4	1	no	—
48	C₁₆	—	508	1	no	—
49	C₂×C₄	—	2	1	no	—
50	C₂×C₄	—	508	2	no	—
51	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
52	C₈	#14523236	1	1	yes	characteristic
53	C₈	#14523237	1	1	yes	characteristic
54	C₈	#14523238	1	1	yes	characteristic
55	C₈	Dic₅₀₈	1	1	yes	characteristic
56	C₂²	#16861125	1	1	yes	characteristic
57	C₄	—	4	1	no	—
58	C₄	—	1016	2	no	—
59	C₄	#16861132	1	1	yes	characteristic
60	C₄	#16861134	1	1	yes	characteristic
61	C₂	—	2	1	no	—
62	C₂	#22549624	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22549624
commutator	C₄²×C₁₂₇	C₂×C₄
Frattini	C₄×C₈	C₂×D₁₂₇
Fitting	#364×C₁₂₇	C₂
socle	C₂×C₁₂₇	#333
prime radical	C₂²×C₁₂₇	#100
cosocle	C₄×C₈×C₁₂₇	C₂²
hypercenter	#364	D₁₂₇
hypocenter	C₁₂₇	#1366
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
32. C₁₂₇	#1366	semidirect

Maximal subgroups

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

subgroup	index
#22550278	2
#22549697	2
#364×C₁₂₇	2
#1366	127