Subgroups — C₂×#22575820 #31772633

Order 17312.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575820	C₂	1	4	yes	maximal, direct product factor
2	C₂×#16864513	C₂	1	2	yes	maximal, complemented
3	C₂²×C₄×C₅₄₁	C₂	1	1	yes	maximal, characteristic
4	#16864513	C₂²	1	4	yes	complemented
5	C₂³×C₅₄₁	C₄	1	1	yes	characteristic
6	C₂×C₄×C₅₄₁	C₄	1	2	yes	—
7	C₂×C₄×C₅₄₁	C₄	1	1	yes	characteristic
8	C₂×C₄×C₅₄₁	C₂²	1	2	yes	—
9	C₂×C₄×C₅₄₁	C₂²	1	1	yes	characteristic
10	C₂²×C₅₄₁	—	2	2	no	—
11	C₂²×C₅₄₁	C₂×C₄	1	2	yes	—
12	C₂²×C₅₄₁	C₂×C₄	1	1	yes	characteristic
13	C₄×C₅₄₁	C₂×C₄	1	2	yes	—
14	C₄×C₅₄₁	C₂×C₄	1	1	yes	characteristic
15	C₄×C₅₄₁	C₂³	1	1	yes	characteristic
16	C₂×C₅₄₁	—	2	2	no	—
17	C₂×C₅₄₁	M₁₆	1	2	yes	complemented
18	C₂×C₅₄₁	C₂²×C₄	1	1	yes	characteristic
19	C₅₄₁	C₂×M₁₆	1	1	yes	characteristic, complemented, Sylow
20	C₂×M₁₆	—	541	1	no	maximal, Sylow
21	C₂²×C₄	D₅₄₁	1	1	yes	characteristic
22	M₁₆	—	541	4	no	—
23	C₂×C₈	—	541	2	no	—
24	C₂³	Dic₅₄₁	1	1	yes	characteristic
25	C₂×C₄	Dic₅₄₁	1	1	yes	characteristic
26	C₂×C₄	Dic₅₄₁	1	2	yes	—
27	C₂×C₄	C₂×D₅₄₁	1	2	yes	—
28	C₂×C₄	C₂×D₅₄₁	1	1	yes	characteristic
29	C₈	—	541	4	no	—
30	C₂²	—	2	2	no	—
31	C₂²	C₂×Dic₅₄₁	1	2	yes	—
32	C₂²	C₂×Dic₅₄₁	1	1	yes	characteristic
33	C₄	C₂×Dic₅₄₁	1	2	yes	—
34	C₄	C₂×Dic₅₄₁	1	1	yes	characteristic
35	C₄	C₂²×D₅₄₁	1	1	yes	characteristic
36	C₂	—	2	2	no	—
37	C₂	#22575820	1	2	yes	direct product factor
38	C₂	C₂²×Dic₅₄₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₅₄₁
commutator	C₂×C₅₄₁	C₂²×C₄
Frattini	C₄	C₂²×D₅₄₁
Fitting	C₂²×C₄×C₅₄₁	C₂
socle	C₂²×C₅₄₁	C₂×C₄
prime radical	C₂³×C₅₄₁	C₄
cosocle	C₄×C₅₄₁	C₂³
hypercenter	C₂²×C₄	D₅₄₁
hypocenter	C₅₄₁	C₂×M₁₆
LCS	G ▷ C₂×C₅₄₁ ▷ C₅₄₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. #22575820	C₂	direct (N × H)
2. C₂×#16864513	C₂	semidirect
4. #16864513	C₂²	semidirect
17. C₂×C₅₄₁	M₁₆	semidirect
19. C₅₄₁	C₂×M₁₆	semidirect
37. C₂	#22575820	direct (N × H)

Maximal subgroups

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

subgroup	index
#22575820	2
C₂×#16864513	2
C₂²×C₄×C₅₄₁	2
C₂×M₁₆	541

Subgroups — C2×#22575820 #31772633

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22575820 #31772633