Subgroups — #27213503

Order 13504.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20269965	C₂	1	1	yes	maximal, characteristic
2	#20269900	C₂	1	2	yes	maximal, complemented
3	#20269897	C₂	1	2	yes	maximal
4	C₄×C₂₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#125×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₄₂₂	C₂²	1	1	yes	characteristic
7	C₂×D₈₄₄	C₂²	1	1	yes	characteristic
8	C₂×C₂₁₁⋊C₈	C₂²	1	2	yes	—
9	(C₂×D₂₁₁)⋊C₄	—	2	2	no	—
10	C₄⋊C₄×C₂₁₁	—	2	2	no	—
11	C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
12	C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
13	C₂²×D₂₁₁	—	2	1	no	—
14	Dic₄₂₂	—	2	2	no	—
15	D₈₄₄	—	2	2	no	—
16	C₂×Dic₂₁₁	—	2	1	no	—
17	C₂₁₁⋊C₈	—	2	2	no	—
18	C₂×C₄×C₂₁₁	—	2	2	no	—
19	C₂×C₄×C₂₁₁	D₄	1	2	yes	—
20	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
21	C₂²×C₂₁₁	C₂×D₄	1	1	yes	characteristic
22	C₂×D₂₁₁	—	4	1	no	—
23	C₂×D₂₁₁	—	4	2	no	—
24	C₄×C₂₁₁	—	2	2	no	—
25	C₄×C₂₁₁	—	4	2	no	—
26	C₄×C₂₁₁	#41	1	2	yes	—
27	Dic₂₁₁	—	4	1	no	—
28	C₂×C₂₁₁	#124	1	1	yes	characteristic
29	C₂×C₂₁₁	#135	1	2	yes	—
30	D₂₁₁	—	8	1	no	—
31	C₂₁₁	#488	1	1	yes	characteristic, complemented, Sylow
32	#488	—	211	1	no	maximal, Sylow
33	#125	D₂₁₁	1	1	yes	characteristic, complemented
34	#124	—	211	1	no	—
35	#103	—	211	2	no	—
36	#102	—	211	2	no	—
37	C₄×C₈	—	211	1	no	—
38	C₂×Q₈	—	211	1	no	—
39	C₂×D₄	—	211	1	no	—
40	C₂×C₈	—	211	2	no	—
41	C₄⋊C₄	—	2	2	no	—
42	C₄⋊C₄	C₂×D₂₁₁	1	2	yes	—
43	C₂²⋊C₄	—	422	2	no	—
44	C₄²	C₂×D₂₁₁	1	1	yes	characteristic
45	C₂³	—	422	1	no	—
46	Q₈	—	422	2	no	—
47	D₄	—	422	2	no	—
48	C₂×C₄	—	2	2	no	—
49	C₂×C₄	—	422	1	no	—
50	C₂×C₄	C₂²⋊D₂₁₁	1	2	yes	—
51	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
52	C₈	—	422	2	no	—
53	C₂²	—	844	1	no	—
54	C₂²	—	844	2	no	—
55	C₂²	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	—	4	2	no	—
58	C₄	—	844	1	no	—
59	C₄	#16809963	1	2	yes	—
60	C₂	—	1688	1	no	—
61	C₂	#20269966	1	1	yes	characteristic
62	C₂	#20269997	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₁₁
commutator, cosocle	C₂×C₄×C₂₁₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₁₁
Fitting	#125×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂×D₄
prime radical	C₂×D₈₄₄	C₂²
hypercenter	#125	D₂₁₁
hypocenter	C₂₁₁	#488
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
2. #20269900	C₂	semidirect
4. C₄×C₂₁₁⋊C₈	C₂	semidirect
5. #125×C₂₁₁	C₂	semidirect
31. C₂₁₁	#488	semidirect
33. #125	D₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
#20269965	2
#20269900	2
#20269897	2
C₄×C₂₁₁⋊C₈	2
#125×C₂₁₁	2
#488	211