Subgroups — #27213777

Order 13504.

206 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20269976	C₂	1	4	yes	maximal, complemented
2	#20269968	C₂	1	4	yes	maximal, complemented
3	#20269956	C₂	1	4	yes	maximal
4	#20269936	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₄×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₂₂	C₂²	1	4	yes	—
7	C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
8	C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
9	C₄⋊Dic₂₁₁	C₂²	1	4	yes	—
10	Dic₂₁₁⋊C₄	C₂²	1	8	yes	—
11	C₂²⋊Dic₂₁₁	C₂²	1	4	yes	—
12	C₄×Dic₂₁₁	C₂²	1	4	yes	—
13	C₂²×C₄×C₂₁₁	C₂²	1	2	yes	—
14	C₂²×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₂₁₁	C₂²	1	4	yes	—
16	Dic₄₂₂	—	4	4	no	—
17	C₂×Dic₂₁₁	C₂³	1	4	yes	—
18	C₂×Dic₂₁₁	C₂³	1	4	yes	—
19	C₂³×C₂₁₁	C₂³	1	1	yes	characteristic
20	C₂×C₄×C₂₁₁	—	4	2	no	—
21	C₂×C₄×C₂₁₁	Q₈	1	4	yes	complemented
22	C₂×C₄×C₂₁₁	C₂³	1	4	yes	—
23	C₂×C₄×C₂₁₁	C₂³	1	2	yes	—
24	C₂²×C₂₁₁	—	2	1	no	—
25	C₂²×C₂₁₁	—	2	1	no	—
26	C₂²×C₂₁₁	C₂×Q₈	1	2	yes	—
27	C₂²×C₂₁₁	C₂⁴	1	1	yes	characteristic
28	C₄×C₂₁₁	—	2	4	no	—
29	C₄×C₂₁₁	C₂×Q₈	1	4	yes	—
30	Dic₂₁₁	—	2	4	no	—
31	Dic₂₁₁	—	2	4	no	—
32	C₂×C₂₁₁	—	2	2	no	—
33	C₂×C₂₁₁	C₂²×Q₈	1	1	yes	characteristic
34	C₂×C₂₁₁	H(5,𝔽₂)	1	1	yes	characteristic
35	C₂×C₂₁₁	2₋¹⁺⁴	1	1	yes	characteristic
36	C₂₁₁	#544	1	1	yes	characteristic, complemented, Sylow
37	#544	—	211	1	no	maximal, Sylow
38	C₄⋊Q₈	—	211	4	no	—
39	#125	—	211	4	no	—
40	C₂²⋊Q₈	—	211	4	no	—
41	#117	—	211	2	no	—
42	C₂×C₄⋊C₄	D₂₁₁	1	1	yes	characteristic
43	C₂×Q₈	—	211	4	no	—
44	C₂²×C₄	C₂×D₂₁₁	1	2	yes	—
45	C₂²×C₄	C₂×D₂₁₁	1	1	yes	characteristic
46	C₄⋊C₄	—	211	2	no	—
47	C₄⋊C₄	—	211	2	no	—
48	C₄⋊C₄	—	211	4	no	—
49	C₄⋊C₄	—	211	8	no	—
50	C₄⋊C₄	C₂×D₂₁₁	1	4	yes	—
51	C₂²⋊C₄	—	211	4	no	—
52	C₄²	—	211	4	no	—
53	C₂³	C₂²×D₂₁₁	1	1	yes	characteristic
54	Q₈	—	844	4	no	—
55	C₂×C₄	—	4	2	no	—
56	C₂×C₄	—	211	4	no	—
57	C₂×C₄	—	211	4	no	—
58	C₂×C₄	Dic₄₂₂	1	4	yes	complemented
59	C₂×C₄	C₂²×D₂₁₁	1	4	yes	—
60	C₂×C₄	C₂²×D₂₁₁	1	2	yes	—
61	C₂²	—	2	1	no	—
62	C₂²	—	2	1	no	—
63	C₂²	C₂×Dic₄₂₂	1	2	yes	—
64	C₂²	C₂³×D₂₁₁	1	1	yes	characteristic
65	C₄	—	2	4	no	—
66	C₄	—	422	4	no	—
67	C₄	—	422	4	no	—
68	C₄	C₂×Dic₄₂₂	1	4	yes	—
69	C₂	—	2	2	no	—
70	C₂	C₂²×Dic₄₂₂	1	1	yes	characteristic
71	C₂	#20270022	1	1	yes	characteristic
72	C₂	#20270025	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₁₁
commutator, socle, cosocle	C₂²×C₂₁₁	C₂⁴
Fitting	C₂×C₄⋊C₄×C₂₁₁	C₂
prime radical	C₂³×C₂₁₁	C₂³
hypercenter	C₂×C₄⋊C₄	D₂₁₁
hypocenter	C₂₁₁	#544
LCS	G ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. #20269976	C₂	semidirect
2. #20269968	C₂	semidirect
21. C₂×C₄×C₂₁₁	Q₈	semidirect
36. C₂₁₁	#544	semidirect
58. C₂×C₄	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
#20269968	2
#20269956	2
#20269936	2
C₂×C₄⋊C₄×C₂₁₁	2
#544	211