Subgroups — #27165770

Order 12832.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20264366	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₈₀₂	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₁₆₀₄	C₂	1	2	yes	maximal, complemented
4	#20264355	C₂	1	2	yes	maximal, complemented
5	M₁₆×C₄₀₁	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₈₀₂	—	2	1	no	—
7	Dic₈₀₂	C₂²	1	2	yes	—
8	Dic₈₀₂	C₂²	1	1	yes	characteristic
9	D₁₆₀₄	C₂²	1	1	yes	characteristic
10	C₂²⋊D₄₀₁	—	2	1	no	—
11	C₂×Dic₄₀₁	—	2	1	no	—
12	C₄×D₄₀₁	—	2	1	no	—
13	C₂×C₄×C₄₀₁	C₂²	1	1	yes	characteristic
14	C₈×C₄₀₁	C₂²	1	2	yes	—
15	C₂²×C₄₀₁	D₄	1	1	yes	characteristic
16	C₂×D₄₀₁	—	2	1	no	—
17	C₄×C₄₀₁	D₄	1	1	yes	characteristic
18	C₄×C₄₀₁	C₂³	1	1	yes	characteristic
19	Dic₄₀₁	—	2	1	no	—
20	Dic₄₀₁	—	2	2	no	—
21	C₂×C₄₀₁	—	2	1	no	—
22	C₂×C₄₀₁	C₂×D₄	1	1	yes	characteristic
23	D₄₀₁	—	4	1	no	—
24	C₄₀₁	#137	1	1	yes	characteristic, complemented, Sylow
25	#137	—	401	1	no	maximal, Sylow
26	#41	—	401	1	no	—
27	C₂×Q₈	—	401	1	no	—
28	Q₁₆	—	401	2	no	—
29	SD₁₆	—	401	2	no	—
30	M₁₆	D₄₀₁	1	1	yes	characteristic, complemented
31	Q₈	—	802	1	no	—
32	Q₈	—	401	2	no	—
33	Q₈	—	401	1	no	—
34	D₄	—	802	1	no	—
35	D₄	—	401	1	no	—
36	C₂×C₄	—	802	1	no	—
37	C₂×C₄	—	802	1	no	—
38	C₂×C₄	C₂×D₄₀₁	1	1	yes	characteristic
39	C₈	C₂×D₄₀₁	1	2	yes	—
40	C₂²	—	802	1	no	—
41	C₂²	D₁₆₀₄	1	1	yes	characteristic
42	C₄	—	802	1	no	—
43	C₄	—	802	2	no	—
44	C₄	D₁₆₀₄	1	1	yes	characteristic
45	C₄	C₂²×D₄₀₁	1	1	yes	characteristic
46	C₂	—	2	1	no	—
47	C₂	—	1604	1	no	—
48	C₂	C₂×D₁₆₀₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₁₆₀₄
commutator, cosocle	C₄×C₄₀₁	C₂³
Frattini	C₄	C₂²×D₄₀₁
Fitting	M₁₆×C₄₀₁	C₂
socle	C₂×C₄₀₁	C₂×D₄
prime radical	#20264366	C₂
hypercenter	M₁₆	D₄₀₁
hypocenter	C₄₀₁	#137
LCS	G ▷ C₄×C₄₀₁ ▷ C₂×C₄₀₁ ▷ C₄₀₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₈₀₂	C₂	semidirect
3. Dic₁₆₀₄	C₂	semidirect
4. #20264355	C₂	semidirect
5. M₁₆×C₄₀₁	C₂	semidirect
24. C₄₀₁	#137	semidirect
30. M₁₆	D₄₀₁	semidirect

Maximal subgroups

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

subgroup	index
#20264366	2
C₂×Dic₈₀₂	2
Dic₁₆₀₄	2
#20264355	2
M₁₆×C₄₀₁	2
#137	401