Subgroups — #27213258

Order 13504.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×Dic₂₁₁	C₂	1	2	yes	maximal, complemented
2	#20269909	C₂	1	1	yes	maximal, characteristic, complemented
3	#20269898	C₂	1	2	yes	maximal, complemented
4	#20269887	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₁₁	—	2	2	no	—
7	C₂×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
9	C₂²⋊Dic₂₁₁	—	2	2	no	—
10	C₄×Dic₂₁₁	C₂²	1	1	yes	characteristic
11	C₂×D₄×C₂₁₁	C₂²	1	2	yes	—
12	D₈×C₂₁₁	C₄	1	4	yes	complemented
13	C₂×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₂×Dic₂₁₁	—	2	2	no	—
15	C₂×Dic₂₁₁	—	4	4	no	—
16	C₂×Dic₂₁₁	D₄	1	2	yes	—
17	C₂₁₁⋊C₈	—	2	1	no	—
18	C₂³×C₂₁₁	—	2	2	no	—
19	D₄×C₂₁₁	—	2	2	no	—
20	D₄×C₂₁₁	C₂×C₄	1	4	yes	complemented
21	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
22	C₈×C₂₁₁	C₂×C₄	1	2	yes	complemented
23	C₂²×C₂₁₁	—	2	4	no	—
24	C₂²×C₂₁₁	—	4	2	no	—
25	C₂²×C₂₁₁	—	4	2	no	—
26	C₂²×C₂₁₁	C₂×D₄	1	1	yes	characteristic
27	C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
28	C₄×C₂₁₁	#41	1	1	yes	characteristic
29	Dic₂₁₁	—	2	2	no	—
30	Dic₂₁₁	—	4	2	no	—
31	C₂×C₂₁₁	—	4	4	no	—
32	C₂×C₂₁₁	C₄×D₄	1	1	yes	characteristic
33	C₂×C₂₁₁	AGL(1,ℤ/8)	1	2	yes	—
34	C₂₁₁	#442	1	1	yes	characteristic, complemented, Sylow
35	#442	—	211	1	no	maximal, Sylow
36	C₂×D₈	D₂₁₁	1	1	yes	characteristic
37	C₄×D₄	—	211	2	no	—
38	#106	—	211	1	no	—
39	#102	—	211	2	no	—
40	#97	—	211	1	no	—
41	C₂×D₄	C₂×D₂₁₁	1	2	yes	—
42	C₂²×C₄	—	422	2	no	—
43	D₈	Dic₂₁₁	1	4	yes	complemented
44	C₂×C₈	—	211	1	no	—
45	C₂×C₈	C₂×D₂₁₁	1	1	yes	characteristic
46	C₄⋊C₄	—	211	2	no	—
47	C₂²⋊C₄	—	422	2	no	—
48	C₄²	—	211	1	no	—
49	C₂³	—	2	2	no	—
50	D₄	—	2	2	no	—
51	D₄	C₂×Dic₂₁₁	1	4	yes	complemented
52	C₂×C₄	—	844	4	no	—
53	C₂×C₄	—	422	2	no	—
54	C₂×C₄	—	211	2	no	—
55	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
56	C₈	—	422	1	no	—
57	C₈	C₂×Dic₂₁₁	1	2	yes	complemented
58	C₂²	—	2	4	no	—
59	C₂²	—	4	2	no	—
60	C₂²	—	4	2	no	—
61	C₂²	D₄×D₂₁₁	1	1	yes	characteristic
62	C₄	—	844	2	no	—
63	C₄	—	422	2	no	—
64	C₄	C₂²×Dic₂₁₁	1	1	yes	characteristic
65	C₄	#16809961	1	1	yes	characteristic
66	C₂	—	4	4	no	—
67	C₂	D₄×Dic₂₁₁	1	1	yes	characteristic
68	C₂	#20270001	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₂₁₁
commutator	C₄×C₂₁₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₁₁
Fitting, prime radical	C₂×D₈×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂×D₄
cosocle	C₂×C₄×C₂₁₁	C₂³
hypercenter	C₂×D₈	D₂₁₁
hypocenter	C₂₁₁	#442
LCS	G ▷ C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. D₄×Dic₂₁₁	C₂	semidirect
2. #20269909	C₂	semidirect
3. #20269898	C₂	semidirect
4. #20269887	C₂	semidirect
12. D₈×C₂₁₁	C₄	semidirect
20. D₄×C₂₁₁	C₂×C₄	semidirect
22. C₈×C₂₁₁	C₂×C₄	semidirect
34. C₂₁₁	#442	semidirect
43. D₈	Dic₂₁₁	semidirect
51. D₄	C₂×Dic₂₁₁	semidirect
57. C₈	C₂×Dic₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
D₄×Dic₂₁₁	2
#20269909	2
#20269898	2
#20269887	2
C₂×D₈×C₂₁₁	2
#442	211