Subgroups — #27212963

Order 13504.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₈₄₄	C₂	1	1	yes	maximal, characteristic
2	#20269913	C₂	1	1	yes	maximal, characteristic
3	M₃₂×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₄₂₂	—	2	1	no	—
5	Dic₈₄₄	—	2	1	no	—
6	Dic₈₄₄	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	Dic₄₂₂	—	2	2	no	—
11	Dic₄₂₂	—	4	1	no	—
12	C₂×Dic₂₁₁	—	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	C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
19	C₄×C₂₁₁	D₈	1	1	yes	characteristic
20	Dic₂₁₁	—	4	2	no	—
21	C₂×C₂₁₁	—	2	1	no	—
22	C₂×C₂₁₁	#102	1	1	yes	characteristic
23	C₂₁₁	#362	1	1	yes	characteristic, complemented, Sylow
24	#362	—	211	1	no	maximal, Sylow
25	C₂×Q₁₆	—	211	1	no	—
26	M₃₂	D₂₁₁	1	1	yes	characteristic
27	#108	—	211	1	no	—
28	C₂×Q₈	—	422	1	no	—
29	Q₁₆	—	422	1	no	—
30	Q₁₆	—	211	2	no	—
31	M₁₆	—	422	1	no	—
32	C₂×C₈	C₂×D₂₁₁	1	1	yes	characteristic
33	C₁₆	—	2	1	no	—
34	Q₈	—	844	1	no	—
35	Q₈	—	422	2	no	—
36	C₂×C₄	—	844	1	no	—
37	C₂×C₄	D₈₄₄	1	1	yes	characteristic
38	C₈	—	844	1	no	—
39	C₈	C₄×D₂₁₁	1	1	yes	characteristic
40	C₈	C₂²⋊D₂₁₁	1	1	yes	characteristic
41	C₂²	#16809951	1	1	yes	characteristic
42	C₄	—	844	2	no	—
43	C₄	(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
44	C₄	D₁₆₈₈	1	1	yes	characteristic
45	C₂	—	2	1	no	—
46	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	C₂²×C₂₁₁	SD₁₆
cosocle	C₂×C₈×C₂₁₁	C₂²
hypercenter	M₃₂	D₂₁₁
hypocenter	C₂₁₁	#362
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
23. C₂₁₁	#362	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₈₄₄	2
#20269913	2
M₃₂×C₂₁₁	2
#362	211