Subgroups — #27213548

Order 13504.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20269969	C₂	1	2	yes	maximal
2	#20269911	C₂	1	2	yes	maximal
3	#20269909	C₂	1	2	yes	maximal
4	C₄×C₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
5	C₄⋊Dic₂₁₁	C₂²	1	4	yes	—
6	Dic₂₁₁⋊C₄	—	2	4	no	—
7	C₂×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
8	C₂×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
9	C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
10	C₂×Dic₂₁₁	—	2	4	no	—
11	C₂×C₄×C₂₁₁	D₄	1	2	yes	—
12	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
13	C₈×C₂₁₁	Q₈	1	2	yes	—
14	C₈×C₂₁₁	Q₈	1	2	yes	—
15	C₂²×C₂₁₁	C₂×D₄	1	1	yes	characteristic
16	C₄×C₂₁₁	—	2	2	no	—
17	C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
18	C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
19	Dic₂₁₁	—	4	4	no	—
20	C₂×C₂₁₁	C₄⋊Q₈	1	1	yes	characteristic
21	C₂×C₂₁₁	#135	1	2	yes	—
22	C₂₁₁	#499	1	1	yes	characteristic, complemented, Sylow
23	#499	—	211	1	no	maximal, Sylow
24	#125	—	211	2	no	—
25	#107	—	211	2	no	—
26	#106	—	211	2	no	—
27	C₄×C₈	D₂₁₁	1	1	yes	characteristic
28	C₂×C₈	C₂×D₂₁₁	1	1	yes	characteristic
29	C₂×C₈	C₂×D₂₁₁	1	1	yes	characteristic
30	C₄⋊C₄	—	422	4	no	—
31	C₄⋊C₄	—	211	4	no	—
32	C₄²	C₂×D₂₁₁	1	1	yes	characteristic
33	C₂×C₄	—	422	4	no	—
34	C₂×C₄	D₈₄₄	1	2	yes	—
35	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
36	C₈	Dic₄₂₂	1	2	yes	—
37	C₈	Dic₄₂₂	1	2	yes	—
38	C₂²	C₂×D₈₄₄	1	1	yes	characteristic
39	C₄	—	2	2	no	—
40	C₄	—	844	4	no	—
41	C₄	C₂×Dic₄₂₂	1	1	yes	characteristic
42	C₄	C₂×Dic₄₂₂	1	1	yes	characteristic
43	C₂	#20269977	1	1	yes	characteristic
44	C₂	#20269999	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈₄₄
commutator, cosocle	C₂×C₄×C₂₁₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₁₁
Fitting	C₄×C₈×C₂₁₁	C₂
socle, prime radical	C₂²×C₂₁₁	C₂×D₄
hypercenter	C₄×C₈	D₂₁₁
hypocenter	C₂₁₁	#499
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
22. C₂₁₁	#499	semidirect

Maximal subgroups

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

subgroup	index
#20269969	2
#20269911	2
#20269909	2
C₄×C₈×C₂₁₁	2
#499	211