Subgroups — #44397808

Order 29056.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28408785	C₂	1	2	yes	maximal
2	C₄×C₁₆×C₂₂₇	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₂₂₇⋊C₈)	—	2	2	no	—
4	C₂×C₁₆×C₂₂₇	C₄	1	1	yes	characteristic
5	C₂×C₁₆×C₂₂₇	C₄	1	1	yes	characteristic
6	C₄×C₈×C₂₂₇	C₂²	1	1	yes	characteristic
7	C₂×C₂₂₇⋊C₈	—	4	2	no	—
8	C₂×C₈×C₂₂₇	C₂×C₄	1	1	yes	characteristic
9	C₂×C₈×C₂₂₇	Q₈	1	1	yes	characteristic
10	C₄²×C₂₂₇	D₄	1	1	yes	characteristic
11	C₁₆×C₂₂₇	—	2	1	no	—
12	C₁₆×C₂₂₇	C₈	1	2	yes	complemented
13	C₂₂₇⋊C₈	—	8	2	no	—
14	C₂×C₄×C₂₂₇	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₄×C₂₂₇	D₈	1	1	yes	characteristic
16	C₂×C₄×C₂₂₇	Q₁₆	1	1	yes	characteristic
17	C₈×C₂₂₇	—	2	1	no	—
18	C₈×C₂₂₇	C₂×C₈	1	1	yes	characteristic
19	C₈×C₂₂₇	M₁₆	1	1	yes	characteristic
20	C₂²×C₂₂₇	#107	1	1	yes	characteristic
21	C₄×C₂₂₇	—	2	1	no	—
22	C₄×C₂₂₇	C₄⋊C₈	1	1	yes	characteristic
23	C₄×C₂₂₇	#108	1	1	yes	characteristic
24	C₄×C₂₂₇	SD₃₂	1	2	yes	—
25	C₂×C₂₂₇	#335	1	1	yes	characteristic
26	C₂×C₂₂₇	#367	1	1	yes	characteristic
27	C₂×C₂₂₇	#368	1	1	yes	characteristic
28	C₂₂₇	#1372	1	1	yes	characteristic, complemented, Sylow
29	#1372	—	227	1	no	maximal, Sylow
30	C₄×C₁₆	D₂₂₇	1	1	yes	characteristic
31	#335	—	227	2	no	—
32	C₂×C₁₆	Dic₂₂₇	1	1	yes	characteristic
33	C₂×C₁₆	Dic₂₂₇	1	1	yes	characteristic
34	C₄⋊C₈	—	454	2	no	—
35	C₄×C₈	C₂×D₂₂₇	1	1	yes	characteristic
36	C₂×C₈	—	908	2	no	—
37	C₂×C₈	C₂×Dic₂₂₇	1	1	yes	characteristic
38	C₂×C₈	Dic₄₅₄	1	1	yes	characteristic
39	C₄²	D₉₀₈	1	1	yes	characteristic
40	C₁₆	—	2	1	no	—
41	C₁₆	C₂₂₇⋊C₈	1	2	yes	complemented
42	C₂×C₄	C₄⋊Dic₂₂₇	1	1	yes	characteristic
43	C₂×C₄	D₁₈₁₆	1	1	yes	characteristic
44	C₂×C₄	Dic₉₀₈	1	1	yes	characteristic
45	C₈	—	2	1	no	—
46	C₈	—	1816	2	no	—
47	C₈	C₂×C₂₂₇⋊C₈	1	1	yes	characteristic
48	C₈	#16813868	1	1	yes	characteristic
49	C₂²	#20300222	1	1	yes	characteristic
50	C₄	—	2	1	no	—
51	C₄	C₄⋊(C₂₂₇⋊C₈)	1	1	yes	characteristic
52	C₄	#20300224	1	1	yes	characteristic
53	C₄	#20300233	1	2	yes	—
54	C₂	#28408785	1	1	yes	characteristic
55	C₂	#28408860	1	1	yes	characteristic
56	C₂	#28408862	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₁₈₁₆
commutator	C₈×C₂₂₇	C₂×C₈
Frattini	C₄×C₈	C₂×D₂₂₇
Fitting	C₄×C₁₆×C₂₂₇	C₂
socle, prime radical	C₂²×C₂₂₇	#107
cosocle	C₄×C₈×C₂₂₇	C₂²
hypercenter	C₄×C₁₆	D₂₂₇
hypocenter	C₂₂₇	#1372
LCS	G ▷ C₈×C₂₂₇ ▷ C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
12. C₁₆×C₂₂₇	C₈	semidirect
28. C₂₂₇	#1372	semidirect
41. C₁₆	C₂₂₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#28408785	2
C₄×C₁₆×C₂₂₇	2
#1372	227