Subgroups — #27213519

Order 13504.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16809951	C₂	1	4	yes	maximal, complemented
2	#20269977	C₂	1	1	yes	maximal, characteristic, complemented
3	#20269974	C₂	1	1	yes	maximal, characteristic
4	C₄×C₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Dic₄₂₂	C₂²	1	2	yes	—
6	C₂×D₈₄₄	—	2	2	no	—
7	C₂×D₈₄₄	C₂²	1	2	yes	—
8	#16809951	—	2	8	no	—
9	C₄⋊Dic₂₁₁	—	2	2	no	—
10	C₂×C₈×C₂₁₁	C₂²	1	2	yes	—
11	C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
12	C₂²×D₂₁₁	—	2	2	no	—
13	Dic₄₂₂	—	2	4	no	—
14	D₈₄₄	—	2	4	no	—
15	D₈₄₄	—	4	4	no	—
16	C₂×Dic₂₁₁	—	2	2	no	—
17	C₂×C₄×C₂₁₁	D₄	1	2	yes	—
18	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
19	C₈×C₂₁₁	D₄	1	4	yes	complemented
20	C₂²×C₂₁₁	C₂×D₄	1	1	yes	characteristic
21	C₂×D₂₁₁	—	4	2	no	—
22	C₂×D₂₁₁	—	4	4	no	—
23	C₄×C₂₁₁	SD₁₆	1	4	yes	complemented
24	C₄×C₂₁₁	C₂×D₄	1	2	yes	—
25	Dic₂₁₁	—	4	2	no	—
26	C₂×C₂₁₁	#127	1	1	yes	characteristic
27	C₂×C₂₁₁	C₂×SD₁₆	1	2	yes	—
28	D₂₁₁	—	8	2	no	—
29	C₂₁₁	#492	1	1	yes	characteristic, complemented, Sylow
30	#492	—	211	1	no	maximal, Sylow
31	C₂×SD₁₆	—	211	4	no	—
32	C₄⋊Q₈	—	211	1	no	—
33	#127	—	211	1	no	—
34	C₄×C₈	D₂₁₁	1	1	yes	characteristic, complemented
35	C₂×Q₈	—	211	2	no	—
36	C₂×D₄	—	422	2	no	—
37	C₂×D₄	—	211	2	no	—
38	SD₁₆	—	422	8	no	—
39	C₂×C₈	C₂×D₂₁₁	1	2	yes	—
40	C₄⋊C₄	—	422	2	no	—
41	C₄²	C₂×D₂₁₁	1	1	yes	characteristic
42	C₂³	—	422	2	no	—
43	Q₈	—	422	4	no	—
44	D₄	—	844	4	no	—
45	D₄	—	422	4	no	—
46	C₂×C₄	—	422	2	no	—
47	C₂×C₄	D₈₄₄	1	2	yes	—
48	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
49	C₈	D₈₄₄	1	4	yes	complemented
50	C₂²	—	844	2	no	—
51	C₂²	—	844	4	no	—
52	C₂²	C₂×D₈₄₄	1	1	yes	characteristic
53	C₄	—	844	2	no	—
54	C₄	#16809951	1	4	yes	complemented
55	C₄	C₂×D₈₄₄	1	2	yes	—
56	C₂	—	1688	2	no	—
57	C₂	#20269974	1	1	yes	characteristic
58	C₂	C₂×#16809951	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈₄₄
commutator, cosocle	C₂×C₄×C₂₁₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₁₁
Fitting	C₄×C₈×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂×D₄
prime radical	#20269974	C₂
hypercenter	C₄×C₈	D₂₁₁
hypocenter	C₂₁₁	#492
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#16809951	C₂	semidirect
2. #20269977	C₂	semidirect
4. C₄×C₈×C₂₁₁	C₂	semidirect
19. C₈×C₂₁₁	D₄	semidirect
23. C₄×C₂₁₁	SD₁₆	semidirect
29. C₂₁₁	#492	semidirect
34. C₄×C₈	D₂₁₁	semidirect
49. C₈	D₈₄₄	semidirect
54. C₄	#16809951	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16809951	2
#20269977	2
#20269974	2
C₄×C₈×C₂₁₁	2
#492	211