Subgroups — C₂×((C₃₁⋊C₃)⋊C₄)⋊C₅ #72219331

Order 3720.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×(C₃₁⋊C₃)⋊C₅	C₂	1	1	yes	maximal, characteristic
2	((C₃₁⋊C₃)⋊C₄)⋊C₅	C₂	1	2	yes	maximal, direct product factor
3	C₂×Dic₃₁⋊C₅	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂×(C₃₁⋊C₃)⋊C₅	C₄	1	2	yes	complemented
5	C₂×(C₃₁⋊C₃)⋊C₅	C₂²	1	1	yes	characteristic
6	C₂×(C₃₁⋊C₃)⋊C₄	C₅	1	1	yes	maximal, characteristic, complemented, Hall
7	C₂²×ΓL(1,𝔽₃₂)	C₂×C₃	1	1	yes	characteristic
8	Dic₃₁⋊C₅	C₂×C₃	1	2	yes	complemented
9	(C₃₁⋊C₃)⋊C₅	C₂×C₄	1	1	yes	characteristic, complemented, Hall
10	C₂²×C₃₁⋊C₃	C₂×C₅	1	1	yes	characteristic
11	(C₃₁⋊C₃)⋊C₄	C₂×C₅	1	2	yes	complemented
12	C₂×ΓL(1,𝔽₃₂)	C₃×C₄	1	2	yes	complemented
13	C₂×ΓL(1,𝔽₃₂)	C₂²×C₃	1	1	yes	characteristic
14	C₂×Dic₃₁	C₃×C₅	1	1	yes	characteristic, complemented, Hall
15	C₂×C₃₁⋊C₃	C₄×C₅	1	2	yes	complemented
16	C₂×C₃₁⋊C₃	C₂²×C₅	1	1	yes	characteristic
17	ΓL(1,𝔽₃₂)	C₂×C₃×C₄	1	1	yes	characteristic, complemented, Hall
18	C₂²×C₃₁	C₂×C₃×C₅	1	1	yes	characteristic
19	Dic₃₁	C₂×C₃×C₅	1	2	yes	complemented
20	C₂×C₃×C₄×C₅	—	31	1	no	maximal, Hall
21	C₃₁⋊C₃	C₂×C₄×C₅	1	1	yes	characteristic, complemented, Hall
22	C₂×C₃₁	C₃×C₄×C₅	1	2	yes	complemented
23	C₂×C₃₁	C₂²×C₃×C₅	1	1	yes	characteristic
24	C₂²×C₃×C₅	—	31	1	no	—
25	C₃×C₄×C₅	—	31	2	no	—
26	C₂×C₄×C₅	—	31	1	no	Hall
27	C₃₁	C₂×C₃×C₄×C₅	1	1	yes	characteristic, complemented, Sylow
28	C₂×C₃×C₅	—	31	1	no	—
29	C₂×C₃×C₅	—	31	2	no	—
30	C₂×C₃×C₄	—	31	1	no	Hall
31	C₂²×C₅	—	31	1	no	—
32	C₄×C₅	—	31	2	no	—
33	C₃×C₅	—	31	1	no	Hall
34	C₂²×C₃	—	31	1	no	—
35	C₃×C₄	—	31	2	no	—
36	C₂×C₅	—	31	1	no	—
37	C₂×C₅	—	31	2	no	—
38	C₂×C₄	—	31	1	no	Sylow
39	C₂×C₃	—	31	1	no	—
40	C₂×C₃	—	31	2	no	—
41	C₅	—	31	1	no	Sylow
42	C₂²	AGL(1,𝔽₃₁)	1	1	yes	characteristic
43	C₄	—	31	2	no	—
44	C₃	—	31	1	no	Sylow
45	C₂	((C₃₁⋊C₃)⋊C₄)⋊C₅	1	2	yes	direct product factor
46	C₂	C₂×AGL(1,𝔽₃₁)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²	AGL(1,𝔽₃₁)
commutator, hypocenter	C₃₁	C₂×C₃×C₄×C₅
Frattini	C₂	C₂×AGL(1,𝔽₃₁)
Fitting, socle	C₂²×C₃₁	C₂×C₃×C₅
prime radical	C₂²×(C₃₁⋊C₃)⋊C₅	C₂
cosocle	C₂×C₃₁	C₂²×C₃×C₅
LCS	G ▷ C₃₁
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
2. ((C₃₁⋊C₃)⋊C₄)⋊C₅	C₂	direct (N × H)
3. C₂×Dic₃₁⋊C₅	C₃	semidirect
4. C₂×(C₃₁⋊C₃)⋊C₅	C₄	semidirect
6. C₂×(C₃₁⋊C₃)⋊C₄	C₅	semidirect
8. Dic₃₁⋊C₅	C₂×C₃	semidirect
9. (C₃₁⋊C₃)⋊C₅	C₂×C₄	semidirect
11. (C₃₁⋊C₃)⋊C₄	C₂×C₅	semidirect
12. C₂×ΓL(1,𝔽₃₂)	C₃×C₄	semidirect
14. C₂×Dic₃₁	C₃×C₅	semidirect
15. C₂×C₃₁⋊C₃	C₄×C₅	semidirect
17. ΓL(1,𝔽₃₂)	C₂×C₃×C₄	semidirect
19. Dic₃₁	C₂×C₃×C₅	semidirect
21. C₃₁⋊C₃	C₂×C₄×C₅	semidirect
22. C₂×C₃₁	C₃×C₄×C₅	semidirect
27. C₃₁	C₂×C₃×C₄×C₅	semidirect
45. C₂	((C₃₁⋊C₃)⋊C₄)⋊C₅	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×(C₃₁⋊C₃)⋊C₅	2
((C₃₁⋊C₃)⋊C₄)⋊C₅	2
C₂×Dic₃₁⋊C₅	3
C₂×(C₃₁⋊C₃)⋊C₄	5
C₂×C₃×C₄×C₅	31

Subgroups — C2×((C31⋊C3)⋊C4)⋊C5 #72219331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×((C₃₁⋊C₃)⋊C₄)⋊C₅ #72219331