Subgroups — #27212960

Order 13504.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₆₈₈	C₂	1	1	yes	maximal, characteristic
2	#20269913	C₂	1	1	yes	maximal, characteristic, complemented
3	M₃₂×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₈₄₄	—	2	1	no	—
5	D₁₆₈₈	—	2	1	no	—
6	D₁₆₈₈	C₄	1	2	yes	—
7	#16809946	—	2	1	no	—
8	C₂×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
9	C₁₆×C₂₁₁	—	2	1	no	—
10	C₂²×D₂₁₁	—	4	1	no	—
11	D₈₄₄	—	2	2	no	—
12	D₈₄₄	—	4	1	no	—
13	C₂₁₁⋊C₈	—	4	1	no	—
14	C₂×C₄×C₂₁₁	D₄	1	1	yes	characteristic
15	C₈×C₂₁₁	C₂×C₄	1	1	yes	characteristic
16	C₈×C₂₁₁	D₄	1	1	yes	characteristic
17	C₂²×C₂₁₁	SD₁₆	1	1	yes	characteristic
18	C₂×D₂₁₁	—	4	2	no	—
19	C₂×D₂₁₁	—	8	2	no	—
20	C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₂₁₁	D₈	1	1	yes	characteristic
22	C₂×C₂₁₁	—	2	1	no	—
23	C₂×C₂₁₁	#102	1	1	yes	characteristic
24	D₂₁₁	—	8	2	no	—
25	C₂₁₁	#361	1	1	yes	characteristic, complemented, Sylow
26	#361	—	211	1	no	maximal, Sylow
27	C₂×D₈	—	211	1	no	—
28	M₃₂	D₂₁₁	1	1	yes	characteristic, complemented
29	#108	—	211	1	no	—
30	C₂×D₄	—	422	1	no	—
31	D₈	—	422	1	no	—
32	D₈	—	211	2	no	—
33	M₁₆	—	422	1	no	—
34	C₂×C₈	C₂×D₂₁₁	1	1	yes	characteristic
35	C₁₆	—	2	1	no	—
36	C₂³	—	844	1	no	—
37	D₄	—	844	1	no	—
38	D₄	—	422	2	no	—
39	C₂×C₄	D₈₄₄	1	1	yes	characteristic
40	C₈	—	844	1	no	—
41	C₈	C₄×D₂₁₁	1	1	yes	characteristic
42	C₈	C₂²⋊D₂₁₁	1	1	yes	characteristic
43	C₂²	—	844	2	no	—
44	C₂²	—	1688	2	no	—
45	C₂²	#16809951	1	1	yes	characteristic
46	C₄	(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
47	C₄	D₁₆₈₈	1	1	yes	characteristic
48	C₂	—	2	1	no	—
49	C₂	—	1688	2	no	—
50	C₂	#20269899	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20269899
commutator	C₈×C₂₁₁	C₂×C₄
Frattini	C₂×C₈	C₂×D₂₁₁
Fitting	M₃₂×C₂₁₁	C₂
socle	C₂×C₂₁₁	#102
prime radical	C₂×D₁₆₈₈	C₂
cosocle	C₂×C₈×C₂₁₁	C₂²
hypercenter	M₃₂	D₂₁₁
hypocenter	C₂₁₁	#361
LCS	G ▷ C₈×C₂₁₁ ▷ C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. #20269913	C₂	semidirect
3. M₃₂×C₂₁₁	C₂	semidirect
25. C₂₁₁	#361	semidirect
28. 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×D₁₆₈₈	2
#20269913	2
M₃₂×C₂₁₁	2
#361	211