Subgroups — #27212927

Order 13504.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16809945	C₂	1	1	yes	maximal, characteristic
2	#20269917	C₂	1	1	yes	maximal, characteristic, complemented
3	M₃₂×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×D₂₁₁	C₄	1	1	yes	characteristic
5	#16809945	—	2	1	no	—
6	#16809945	—	2	1	no	—
7	C₂×C₂₁₁⋊C₈	C₄	1	1	yes	characteristic
8	C₂₁₁⋊C₁₆	—	2	1	no	—
9	C₂×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
10	C₁₆×C₂₁₁	—	2	1	no	—
11	C₂²×D₂₁₁	C₈	1	1	yes	characteristic
12	C₂×Dic₂₁₁	C₈	1	1	yes	characteristic
13	C₄×D₂₁₁	—	2	1	no	—
14	C₄×D₂₁₁	—	2	1	no	—
15	C₂₁₁⋊C₈	—	2	1	no	—
16	C₂×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
17	C₈×C₂₁₁	D₄	1	1	yes	characteristic
18	C₈×C₂₁₁	D₄	1	1	yes	characteristic
19	C₂²×C₂₁₁	C₂×C₈	1	1	yes	characteristic
20	C₂×D₂₁₁	—	2	1	no	—
21	C₂×D₂₁₁	—	4	1	no	—
22	C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₂₁₁	M₁₆	1	1	yes	characteristic
24	Dic₂₁₁	—	2	1	no	—
25	C₂×C₂₁₁	—	2	1	no	—
26	C₂×C₂₁₁	C₂²⋊C₈	1	1	yes	characteristic
27	D₂₁₁	—	4	1	no	—
28	C₂₁₁	#349	1	1	yes	characteristic, complemented, Sylow
29	#349	—	211	1	no	maximal, Sylow
30	C₂×M₁₆	—	211	1	no	—
31	M₃₂	—	211	1	no	—
32	M₃₂	D₂₁₁	1	1	yes	characteristic, complemented
33	C₂²×C₄	—	211	1	no	—
34	M₁₆	—	422	1	no	—
35	M₁₆	—	422	1	no	—
36	C₂×C₈	—	211	1	no	—
37	C₂×C₈	C₂×D₂₁₁	1	1	yes	characteristic
38	C₁₆	—	2	1	no	—
39	C₁₆	—	422	1	no	—
40	C₂³	—	211	1	no	—
41	C₂×C₄	—	422	1	no	—
42	C₂×C₄	—	422	1	no	—
43	C₂×C₄	—	211	1	no	—
44	C₂×C₄	C₄×D₂₁₁	1	1	yes	characteristic
45	C₈	—	422	1	no	—
46	C₈	C₂²⋊D₂₁₁	1	1	yes	characteristic
47	C₈	D₈₄₄	1	1	yes	characteristic
48	C₂²	—	844	1	no	—
49	C₂²	—	422	1	no	—
50	C₂²	C₈×D₂₁₁	1	1	yes	characteristic
51	C₄	—	422	1	no	—
52	C₄	(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
53	C₄	#16809945	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	844	1	no	—
56	C₂	(C₂×D₂₁₁)⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄	(C₂×D₂₁₁)⋊C₄
commutator	C₂²×C₂₁₁	C₂×C₈
Frattini	C₂×C₈	C₂×D₂₁₁
Fitting	M₃₂×C₂₁₁	C₂
socle	C₂×C₂₁₁	C₂²⋊C₈
prime radical	C₂²×D₂₁₁	C₈
cosocle	C₂×C₈×C₂₁₁	C₂²
hypercenter	M₃₂	D₂₁₁
hypocenter	C₂₁₁	#349
LCS	G ▷ C₂²×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. #20269917	C₂	semidirect
3. M₃₂×C₂₁₁	C₂	semidirect
28. C₂₁₁	#349	semidirect
32. M₃₂	D₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16809945	2
#20269917	2
M₃₂×C₂₁₁	2
#349	211