Subgroups — #27213546

Order 13504.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20269968	C₂	1	2	yes	maximal
2	#20269910	C₂	1	2	yes	maximal
3	#20269909	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₂₁₁	C₂	1	1	yes	maximal, characteristic
5	#106×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
7	C₄⋊Dic₂₁₁	—	2	2	no	—
8	C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
9	Dic₂₁₁⋊C₄	—	2	2	no	—
10	C₄×Dic₂₁₁	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
13	C₂×Dic₂₁₁	—	2	2	no	—
14	C₂×Dic₂₁₁	D₄	1	2	yes	—
15	C₂₁₁⋊C₈	Q₈	1	2	yes	—
16	C₂×C₄×C₂₁₁	—	2	2	no	—
17	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
18	C₈×C₂₁₁	Q₈	1	2	yes	—
19	C₂²×C₂₁₁	C₂×D₄	1	1	yes	characteristic
20	C₄×C₂₁₁	—	4	2	no	—
21	C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
22	C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
23	Dic₂₁₁	—	2	2	no	—
24	Dic₂₁₁	—	4	2	no	—
25	C₂×C₂₁₁	C₄⋊Q₈	1	1	yes	characteristic
26	C₂×C₂₁₁	#135	1	2	yes	—
27	C₂₁₁	#499	1	1	yes	characteristic, complemented, Sylow
28	#499	—	211	1	no	maximal, Sylow
29	#125	—	211	2	no	—
30	#107	—	211	2	no	—
31	#106	—	211	1	no	—
32	#106	D₂₁₁	1	1	yes	characteristic
33	C₄×C₈	—	211	1	no	—
34	C₂×C₈	—	211	1	no	—
35	C₂×C₈	C₂×D₂₁₁	1	1	yes	characteristic
36	C₄⋊C₄	—	422	2	no	—
37	C₄⋊C₄	—	422	2	no	—
38	C₄⋊C₄	—	211	2	no	—
39	C₄⋊C₄	C₂×D₂₁₁	1	2	yes	—
40	C₄²	—	211	1	no	—
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	422	2	no	—
43	C₂×C₄	—	211	2	no	—
44	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
45	C₈	—	211	2	no	—
46	C₈	Dic₄₂₂	1	2	yes	—
47	C₂²	D₄×D₂₁₁	1	1	yes	characteristic
48	C₄	—	4	2	no	—
49	C₄	—	844	2	no	—
50	C₄	—	422	2	no	—
51	C₄	Q₈×D₂₁₁	1	1	yes	characteristic
52	C₄	C₂×Dic₄₂₂	1	1	yes	characteristic
53	C₂	#20269976	1	1	yes	characteristic
54	C₂	#20269998	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₂₁₁
commutator, cosocle	C₂×C₄×C₂₁₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₁₁
Fitting	#106×C₂₁₁	C₂
socle, prime radical	C₂²×C₂₁₁	C₂×D₄
hypercenter	#106	D₂₁₁
hypocenter	C₂₁₁	#499
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

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

Maximal subgroups

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

subgroup	index
#20269968	2
#20269910	2
#20269909	2
C₈×Dic₂₁₁	2
#106×C₂₁₁	2
#499	211