Subgroups — #27212953

Order 13504.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₁₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#20269913	C₂	1	1	yes	maximal, characteristic, complemented
3	#135×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	#16809946	—	2	1	no	—
5	C₂₁₁⋊C₁₆	—	2	1	no	—
6	#41×C₂₁₁	—	2	1	no	—
7	Q₁₆×C₂₁₁	C₄	1	1	yes	characteristic
8	SD₁₆×C₂₁₁	—	2	1	no	—
9	D₈×C₂₁₁	C₄	1	1	yes	characteristic
10	C₂×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
11	C₂₁₁⋊C₈	—	4	1	no	—
12	Q₈×C₂₁₁	—	2	1	no	—
13	D₄×C₂₁₁	—	2	1	no	—
14	D₄×C₂₁₁	—	4	1	no	—
15	C₂×C₄×C₂₁₁	—	4	1	no	—
16	C₂×C₄×C₂₁₁	D₄	1	1	yes	characteristic
17	C₈×C₂₁₁	C₂×C₄	1	1	yes	characteristic
18	C₈×C₂₁₁	D₄	1	1	yes	characteristic
19	C₂²×C₂₁₁	—	4	1	no	—
20	C₂²×C₂₁₁	SD₁₆	1	1	yes	characteristic
21	C₄×C₂₁₁	—	4	1	no	—
22	C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₂₁₁	D₈	1	1	yes	characteristic
24	C₂×C₂₁₁	—	2	1	no	—
25	C₂×C₂₁₁	—	8	1	no	—
26	C₂×C₂₁₁	#102	1	1	yes	characteristic
27	C₂₁₁	#359	1	1	yes	characteristic, complemented, Sylow
28	#359	—	211	1	no	maximal, Sylow
29	#135	D₂₁₁	1	1	yes	characteristic
30	C₂×C₁₆	—	211	1	no	—
31	#108	—	211	1	no	—
32	#41	—	2	1	no	—
33	Q₁₆	Dic₂₁₁	1	1	yes	characteristic
34	SD₁₆	—	2	1	no	—
35	D₈	Dic₂₁₁	1	1	yes	characteristic
36	M₁₆	—	422	1	no	—
37	C₂×C₈	C₂×D₂₁₁	1	1	yes	characteristic
38	C₁₆	—	422	1	no	—
39	Q₈	—	2	1	no	—
40	D₄	—	2	1	no	—
41	D₄	—	4	1	no	—
42	C₂×C₄	—	4	1	no	—
43	C₂×C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
44	C₈	—	844	1	no	—
45	C₈	C₂×Dic₂₁₁	1	1	yes	characteristic
46	C₈	C₂²⋊D₂₁₁	1	1	yes	characteristic
47	C₂²	—	4	1	no	—
48	C₂²	#16809950	1	1	yes	characteristic
49	C₄	—	4	1	no	—
50	C₄	C₂²⋊Dic₂₁₁	1	1	yes	characteristic
51	C₄	#16809947	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	8	1	no	—
54	C₂	#20269898	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16809947
commutator	C₈×C₂₁₁	C₂×C₄
Frattini	C₂×C₈	C₂×D₂₁₁
Fitting, prime radical	#135×C₂₁₁	C₂
socle	C₂×C₂₁₁	#102
cosocle	C₂×C₈×C₂₁₁	C₂²
hypercenter	#135	D₂₁₁
hypocenter	C₂₁₁	#359
LCS	G ▷ C₈×C₂₁₁ ▷ C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #135

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₁₁⋊C₁₆	C₂	semidirect
2. #20269913	C₂	semidirect
27. C₂₁₁	#359	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₁₁⋊C₁₆	2
#20269913	2
#135×C₂₁₁	2
#359	211