Subgroups — #46800550

Order 33952.

50 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄₂₄₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#22555116	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊C₄×C₁₀₆₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₁₀₆₁	—	2	1	no	—
5	D₄₂₄₄	—	2	1	no	—
6	D₄₂₄₄	C₄	1	2	yes	complemented
7	#22555116	—	2	1	no	—
8	C₂×C₄×C₁₀₆₁	—	2	1	no	—
9	C₂×C₄×C₁₀₆₁	C₂²	1	1	yes	characteristic
10	C₂²×C₁₀₆₁	D₄	1	1	yes	characteristic
11	C₂×D₁₀₆₁	—	2	2	no	—
12	C₂×D₁₀₆₁	—	4	1	no	—
13	C₂×D₁₀₆₁	—	4	1	no	—
14	C₄×C₁₀₆₁	—	4	1	no	—
15	C₄×C₁₀₆₁	C₂×C₄	1	1	yes	characteristic
16	C₄×C₁₀₆₁	D₄	1	1	yes	characteristic
17	C₂×C₁₀₆₁	C₂²⋊C₄	1	1	yes	characteristic
18	C₂×C₁₀₆₁	D₈	1	1	yes	characteristic
19	C₂×C₁₀₆₁	SD₁₆	1	1	yes	characteristic
20	D₁₀₆₁	—	4	2	no	—
21	C₁₀₆₁	#102	1	1	yes	characteristic, complemented, Sylow
22	#102	—	1061	1	no	maximal, Sylow
23	C₂×D₄	—	1061	1	no	—
24	C₂×C₈	—	1061	1	no	—
25	C₄⋊C₄	D₁₀₆₁	1	1	yes	characteristic, complemented
26	C₂³	—	2122	1	no	—
27	D₄	—	1061	2	no	—
28	D₄	—	2122	1	no	—
29	C₂×C₄	—	2	1	no	—
30	C₂×C₄	C₂×D₁₀₆₁	1	1	yes	characteristic
31	C₈	—	2122	1	no	—
32	C₂²	—	2122	2	no	—
33	C₂²	—	4244	1	no	—
34	C₂²	—	4244	1	no	—
35	C₂²	C₂²⋊D₁₀₆₁	1	1	yes	characteristic
36	C₄	—	4	1	no	—
37	C₄	C₄×D₁₀₆₁	1	1	yes	characteristic
38	C₄	D₄₂₄₄	1	1	yes	characteristic
39	C₂	—	4244	2	no	—
40	C₂	#30684093	1	1	yes	characteristic
41	C₂	#30684101	1	1	yes	characteristic
42	C₂	#30684103	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊D₁₀₆₁
commutator	C₄×C₁₀₆₁	C₂×C₄
Frattini	C₂×C₄	C₂×D₁₀₆₁
Fitting	C₄⋊C₄×C₁₀₆₁	C₂
socle	C₂²×C₁₀₆₁	D₄
prime radical	C₂×D₄₂₄₄	C₂
cosocle	C₂×C₄×C₁₀₆₁	C₂²
hypercenter	C₄⋊C₄	D₁₀₆₁
hypocenter	C₁₀₆₁	#102
LCS	G ▷ C₄×C₁₀₆₁ ▷ C₂×C₁₀₆₁ ▷ C₁₀₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×#22555116	C₂	semidirect
3. C₄⋊C₄×C₁₀₆₁	C₂	semidirect
6. D₄₂₄₄	C₄	semidirect
21. C₁₀₆₁	#102	semidirect
25. C₄⋊C₄	D₁₀₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄₂₄₄	2
C₂×#22555116	2
C₄⋊C₄×C₁₀₆₁	2
#102	1061