Subgroups — #14521886

Order 1984.

810 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
2	#11758566	C₂	1	10	yes	maximal, complemented
3	#41×D₃₁	C₂	1	10	yes	maximal, complemented
4	#11758564	C₂	1	5	yes	maximal, complemented
5	C₂×#92499	C₂	1	5	yes	maximal, complemented
6	#41×C₃₁	C₂²	1	10	yes	complemented
7	C₂×Q₈×C₃₁	C₂²	1	5	yes	complemented
8	#92499	C₂²	1	10	yes	complemented
9	#92499	C₂²	1	20	yes	complemented
10	Q₈×D₃₁	C₂²	1	10	yes	complemented
11	#92497	C₂²	1	10	yes	complemented
12	D₄×D₃₁	C₂²	1	30	yes	complemented
13	#92495	C₂²	1	30	yes	complemented
14	C₂×D₁₂₄	C₂²	1	15	yes	complemented
15	C₂×C₄×D₃₁	C₂²	1	15	yes	complemented
16	C₂²×D₃₁	C₂³	1	15	yes	—
17	Q₈×C₃₁	C₂³	1	10	yes	—
18	D₄×C₃₁	C₂³	1	10	yes	—
19	C₂×C₄×C₃₁	C₂³	1	15	yes	—
20	C₂²⋊D₃₁	C₂³	1	20	yes	—
21	C₂×Dic₃₁	C₂³	1	5	yes	—
22	D₁₂₄	C₂³	1	30	yes	—
23	C₄×D₃₁	C₂³	1	10	yes	—
24	C₄×D₃₁	C₂³	1	30	yes	—
25	Dic₆₂	C₂³	1	10	yes	—
26	C₂²×C₃₁	C₂⁴	1	5	yes	—
27	C₂×D₃₁	—	4	15	no	—
28	C₂×D₃₁	C₂⁴	1	10	yes	—
29	C₄×C₃₁	C₂⁴	1	10	yes	—
30	Dic₃₁	C₂⁴	1	5	yes	—
31	Dic₃₁	C₂⁴	1	1	yes	characteristic
32	#585	—	31	1	no	maximal, Sylow
33	C₂×C₃₁	—	2	5	no	—
34	C₂×C₃₁	C₂⁵	1	1	yes	characteristic
35	D₃₁	—	2	10	no	—
36	2₋¹⁺⁴	—	31	5	no	—
37	2₋¹⁺⁴	D₃₁	1	1	yes	characteristic, complemented
38	H(5,𝔽₂)	—	31	10	no	—
39	C₂×#41	—	31	10	no	—
40	C₂×#41	—	31	5	no	—
41	C₃₁	#585	1	1	yes	characteristic, complemented, Sylow
42	#41	—	31	10	no	—
43	#41	—	31	20	no	—
44	#41	—	31	10	no	—
45	#41	—	31	30	no	—
46	#41	C₂×D₃₁	1	10	yes	complemented
47	C₂×Q₈	—	31	10	no	—
48	C₂×Q₈	C₂×D₃₁	1	5	yes	complemented
49	C₂×D₄	—	31	30	no	—
50	C₂×D₄	—	31	15	no	—
51	C₂²×C₄	—	31	15	no	—
52	C₂³	—	31	15	no	—
53	Q₈	—	31	10	no	—
54	Q₈	C₂²×D₃₁	1	10	yes	—
55	D₄	—	31	20	no	—
56	D₄	—	31	30	no	—
57	D₄	C₂²×D₃₁	1	10	yes	—
58	C₂×C₄	—	31	5	no	—
59	C₂×C₄	—	31	10	no	—
60	C₂×C₄	—	31	30	no	—
61	C₂×C₄	C₂²×D₃₁	1	15	yes	—
62	C₂²	—	31	10	no	—
63	C₂²	—	124	15	no	—
64	C₂²	C₂³×D₃₁	1	5	yes	—
65	C₄	—	31	1	no	—
66	C₄	—	31	5	no	—
67	C₄	C₂³×D₃₁	1	10	yes	—
68	C₂	—	2	5	no	—
69	C₂	—	62	10	no	—
70	C₂	C₂⁴×D₃₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁴×D₃₁
commutator, socle, cosocle	C₂×C₃₁	C₂⁵
Fitting	C₃₁×2₋¹⁺⁴	C₂
hypercenter	2₋¹⁺⁴	D₃₁
hypocenter	C₃₁	#585
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂ ◁ 2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. C₃₁×2₋¹⁺⁴	C₂	semidirect
2. #11758566	C₂	semidirect
3. #41×D₃₁	C₂	semidirect
4. #11758564	C₂	semidirect
5. C₂×#92499	C₂	semidirect
6. #41×C₃₁	C₂²	semidirect
7. C₂×Q₈×C₃₁	C₂²	semidirect
8. #92499	C₂²	semidirect
9. #92499	C₂²	semidirect
10. Q₈×D₃₁	C₂²	semidirect
11. #92497	C₂²	semidirect
12. D₄×D₃₁	C₂²	semidirect
13. #92495	C₂²	semidirect
14. C₂×D₁₂₄	C₂²	semidirect
15. C₂×C₄×D₃₁	C₂²	semidirect
37. 2₋¹⁺⁴	D₃₁	semidirect
41. C₃₁	#585	semidirect
46. #41	C₂×D₃₁	semidirect
48. C₂×Q₈	C₂×D₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×2₋¹⁺⁴	2
#11758566	2
#41×D₃₁	2
#11758564	2
C₂×#92499	2
#585	31