Subgroups — #16860423

Order 3968.

1424 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521816	C₂	1	15	yes	maximal, complemented
2	C₂×#11758525	C₂	1	15	yes	maximal, complemented
3	C₃₁×#585	C₂	1	1	yes	maximal, characteristic
4	C₃₁×2₋¹⁺⁴	C₄	1	6	yes	—
5	C₃₁×H(5,𝔽₂)	C₄	1	10	yes	—
6	C₂×#41×C₃₁	C₂²	1	15	yes	—
7	#11758525	C₂²	1	20	yes	complemented
8	#11758525	C₂²	1	60	yes	complemented
9	C₂×#92474	C₂²	1	45	yes	complemented
10	C₂²×C₃₁⋊C₈	C₂²	1	15	yes	complemented
11	#41×C₃₁	C₂×C₄	1	60	yes	—
12	#41×C₃₁	C₂³	1	20	yes	—
13	C₂×Q₈×C₃₁	C₂×C₄	1	15	yes	—
14	C₂×D₄×C₃₁	C₂×C₄	1	45	yes	—
15	C₂²×C₄×C₃₁	C₂³	1	15	yes	—
16	#92474	C₂³	1	60	yes	—
17	C₂×C₃₁⋊C₈	C₂³	1	15	yes	—
18	C₂×C₃₁⋊C₈	C₂³	1	45	yes	—
19	C₂³×C₃₁	C₂²×C₄	1	15	yes	—
20	Q₈×C₃₁	C₂²×C₄	1	20	yes	—
21	D₄×C₃₁	C₂²×C₄	1	60	yes	—
22	C₂×C₄×C₃₁	C₂²×C₄	1	45	yes	—
23	C₂×C₄×C₃₁	C₂⁴	1	15	yes	—
24	C₃₁⋊C₈	C₂⁴	1	1	yes	characteristic
25	C₃₁⋊C₈	C₂⁴	1	15	yes	—
26	#3573	—	31	1	no	maximal, Sylow
27	C₂²×C₃₁	—	4	15	no	—
28	C₂²×C₃₁	C₂³×C₄	1	15	yes	—
29	C₄×C₃₁	C₂³×C₄	1	15	yes	—
30	C₄×C₃₁	C₂⁵	1	1	yes	characteristic
31	#585	D₃₁	1	1	yes	characteristic
32	#568	—	31	15	no	—
33	C₂×#131	—	31	15	no	—
34	C₂×C₃₁	—	2	15	no	—
35	C₂×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
36	2₋¹⁺⁴	Dic₃₁	1	6	yes	—
37	H(5,𝔽₂)	Dic₃₁	1	10	yes	—
38	C₂×#41	C₂×D₃₁	1	15	yes	—
39	#131	—	31	20	no	—
40	#131	—	31	60	no	—
41	C₂×M₁₆	—	31	45	no	—
42	C₂²×C₈	—	31	15	no	—
43	C₃₁	#3573	1	1	yes	characteristic, complemented, Sylow
44	#41	C₂×Dic₃₁	1	60	yes	—
45	#41	C₂²×D₃₁	1	20	yes	—
46	C₂×Q₈	C₂×Dic₃₁	1	15	yes	—
47	C₂×D₄	C₂×Dic₃₁	1	45	yes	—
48	C₂²×C₄	C₂²×D₃₁	1	15	yes	—
49	M₁₆	—	31	60	no	—
50	C₂×C₈	—	31	15	no	—
51	C₂×C₈	—	31	45	no	—
52	C₂³	C₂²×Dic₃₁	1	15	yes	—
53	Q₈	C₂²×Dic₃₁	1	20	yes	—
54	D₄	C₂²×Dic₃₁	1	60	yes	—
55	C₂×C₄	C₂²×Dic₃₁	1	45	yes	—
56	C₂×C₄	C₂³×D₃₁	1	15	yes	—
57	C₈	—	31	1	no	—
58	C₈	—	31	15	no	—
59	C₂²	—	4	15	no	—
60	C₂²	C₂³×Dic₃₁	1	15	yes	—
61	C₄	C₂³×Dic₃₁	1	15	yes	—
62	C₄	C₂⁴×D₃₁	1	1	yes	characteristic
63	C₂	—	2	15	no	—
64	C₂	C₂⁴×Dic₃₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂⁴×D₃₁
commutator, socle	C₂×C₃₁	C₂⁴×C₄
Fitting, prime radical	C₃₁×#585	C₂
cosocle	C₄×C₃₁	C₂⁵
hypercenter	#585	D₃₁
hypocenter	C₃₁	#3573
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₄ ◁ #585

Complements

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

N	H ≅ G/N	type
1. #14521816	C₂	semidirect
2. C₂×#11758525	C₂	semidirect
7. #11758525	C₂²	semidirect
8. #11758525	C₂²	semidirect
9. C₂×#92474	C₂²	semidirect
10. C₂²×C₃₁⋊C₈	C₂²	semidirect
43. C₃₁	#3573	semidirect

Maximal subgroups

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

subgroup	index
#14521816	2
C₂×#11758525	2
C₃₁×#585	2
#3573	31