Subgroups — #27213543

Order 13504.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20269976	C₂	1	2	yes	maximal
2	#20269909	C₂	1	1	yes	maximal, characteristic
3	#20269908	C₂	1	2	yes	maximal
4	C₈×Dic₂₁₁	C₂	1	1	yes	maximal, characteristic
5	#106×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₂₂	—	2	2	no	—
7	C₂×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
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	Dic₄₂₂	—	4	4	no	—
14	C₂×Dic₂₁₁	—	2	2	no	—
15	C₂×Dic₂₁₁	D₄	1	2	yes	—
16	C₂₁₁⋊C₈	Q₈	1	2	yes	complemented
17	C₂×C₄×C₂₁₁	—	2	2	no	—
18	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
19	C₈×C₂₁₁	Q₈	1	2	yes	complemented
20	C₂²×C₂₁₁	C₂×D₄	1	1	yes	characteristic
21	C₄×C₂₁₁	—	4	2	no	—
22	C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
23	C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
24	Dic₂₁₁	—	4	2	no	—
25	Dic₂₁₁	SD₁₆	1	4	yes	—
26	C₂×C₂₁₁	C₄⋊Q₈	1	1	yes	characteristic
27	C₂×C₂₁₁	C₂×SD₁₆	1	2	yes	—
28	C₂₁₁	#498	1	1	yes	characteristic, complemented, Sylow
29	#498	—	211	1	no	maximal, Sylow
30	C₄⋊Q₈	—	211	2	no	—
31	#106	—	211	1	no	—
32	#106	—	211	2	no	—
33	#106	D₂₁₁	1	1	yes	characteristic
34	C₄×C₈	—	211	1	no	—
35	C₂×Q₈	—	422	2	no	—
36	C₂×C₈	—	211	1	no	—
37	C₂×C₈	C₂×D₂₁₁	1	1	yes	characteristic
38	C₄⋊C₄	—	422	2	no	—
39	C₄⋊C₄	—	211	2	no	—
40	C₄⋊C₄	C₂×D₂₁₁	1	2	yes	—
41	C₄²	—	211	1	no	—
42	Q₈	—	844	4	no	—
43	C₂×C₄	—	2	2	no	—
44	C₂×C₄	—	422	2	no	—
45	C₂×C₄	—	211	2	no	—
46	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
47	C₈	—	211	2	no	—
48	C₈	Dic₄₂₂	1	2	yes	complemented
49	C₂²	D₄×D₂₁₁	1	1	yes	characteristic
50	C₄	—	4	2	no	—
51	C₄	—	844	2	no	—
52	C₄	—	211	4	no	—
53	C₄	Q₈×D₂₁₁	1	1	yes	characteristic
54	C₄	C₂×Dic₄₂₂	1	1	yes	characteristic
55	C₂	#20269976	1	1	yes	characteristic
56	C₂	SD₁₆×D₂₁₁	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	#498
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
16. C₂₁₁⋊C₈	Q₈	semidirect
19. C₈×C₂₁₁	Q₈	semidirect
28. C₂₁₁	#498	semidirect
48. C₈	Dic₄₂₂	semidirect

Maximal subgroups

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

subgroup	index
#20269976	2
#20269909	2
#20269908	2
C₈×Dic₂₁₁	2
#106×C₂₁₁	2
#498	211