Subgroups — #14521884

Order 1984.

810 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	#11758567	C₂	1	6	yes	maximal, complemented
3	#41×D₃₁	C₂	1	6	yes	maximal, complemented
4	#11758561	C₂	1	9	yes	maximal, complemented
5	C₂×#92497	C₂	1	9	yes	maximal, complemented
6	#41×C₃₁	C₂²	1	6	yes	complemented
7	C₂×D₄×C₃₁	C₂²	1	9	yes	complemented
8	C₂×C₂²⋊D₃₁	C₂²	1	18	yes	complemented
9	C₂²×Dic₃₁	C₂²	1	6	yes	complemented
10	#92499	C₂²	1	2	yes	complemented
11	Q₈×D₃₁	C₂²	1	6	yes	complemented
12	#92497	C₂²	1	18	yes	complemented
13	#92497	C₂²	1	36	yes	complemented
14	D₄×D₃₁	C₂²	1	18	yes	complemented
15	#92495	C₂²	1	18	yes	complemented
16	C₂×C₄×D₃₁	C₂²	1	9	yes	complemented
17	C₂×Dic₆₂	C₂²	1	9	yes	complemented
18	C₂³×C₃₁	C₂³	1	6	yes	—
19	C₂²×D₃₁	C₂³	1	9	yes	—
20	Q₈×C₃₁	C₂³	1	2	yes	—
21	D₄×C₃₁	C₂³	1	18	yes	—
22	C₂×C₄×C₃₁	C₂³	1	9	yes	—
23	C₂²⋊D₃₁	C₂³	1	36	yes	—
24	C₂×Dic₃₁	C₂³	1	18	yes	—
25	C₂×Dic₃₁	C₂³	1	9	yes	—
26	D₁₂₄	C₂³	1	6	yes	—
27	C₄×D₃₁	C₂³	1	6	yes	—
28	C₄×D₃₁	C₂³	1	18	yes	—
29	Dic₆₂	C₂³	1	18	yes	—
30	C₂²×C₃₁	—	4	6	no	—
31	C₂²×C₃₁	C₂⁴	1	9	yes	—
32	C₂×D₃₁	—	4	9	no	—
33	C₂×D₃₁	C₂⁴	1	6	yes	—
34	C₄×C₃₁	C₂⁴	1	6	yes	—
35	Dic₃₁	C₂⁴	1	9	yes	—
36	Dic₃₁	C₂⁴	1	1	yes	characteristic
37	#585	—	31	1	no	maximal, Sylow
38	C₂×C₃₁	—	2	9	no	—
39	C₂×C₃₁	C₂⁵	1	1	yes	characteristic
40	D₃₁	—	2	6	no	—
41	2₋¹⁺⁴	—	31	6	no	—
42	H(5,𝔽₂)	—	31	9	no	—
43	H(5,𝔽₂)	D₃₁	1	1	yes	characteristic, complemented
44	C₂×#41	—	31	6	no	—
45	C₂×#41	—	31	9	no	—
46	C₃₁	#585	1	1	yes	characteristic, complemented, Sylow
47	#41	—	31	2	no	—
48	#41	—	31	18	no	—
49	#41	—	31	36	no	—
50	#41	—	31	18	no	—
51	#41	C₂×D₃₁	1	6	yes	complemented
52	C₂×Q₈	—	31	6	no	—
53	C₂×Q₈	—	31	9	no	—
54	C₂×D₄	—	31	18	no	—
55	C₂×D₄	—	31	18	no	—
56	C₂×D₄	C₂×D₃₁	1	9	yes	complemented
57	C₂²×C₄	—	31	6	no	—
58	C₂²×C₄	—	31	9	no	—
59	C₂³	—	31	9	no	—
60	C₂³	C₂²×D₃₁	1	6	yes	—
61	Q₈	—	31	18	no	—
62	Q₈	C₂²×D₃₁	1	2	yes	—
63	D₄	—	31	36	no	—
64	D₄	—	31	6	no	—
65	D₄	C₂²×D₃₁	1	18	yes	—
66	C₂×C₄	—	31	9	no	—
67	C₂×C₄	—	31	18	no	—
68	C₂×C₄	—	31	6	no	—
69	C₂×C₄	—	31	18	no	—
70	C₂×C₄	C₂²×D₃₁	1	9	yes	—
71	C₂²	—	4	6	no	—
72	C₂²	—	31	6	no	—
73	C₂²	—	124	9	no	—
74	C₂²	C₂³×D₃₁	1	9	yes	—
75	C₄	—	31	1	no	—
76	C₄	—	31	9	no	—
77	C₄	C₂³×D₃₁	1	6	yes	—
78	C₂	—	2	9	no	—
79	C₂	—	62	6	no	—
80	C₂	C₂⁴×D₃₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁴×D₃₁
commutator, socle, cosocle	C₂×C₃₁	C₂⁵
Fitting	C₃₁×H(5,𝔽₂)	C₂
hypercenter	H(5,𝔽₂)	D₃₁
hypocenter	C₃₁	#585
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂ ◁ H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₃₁×H(5,𝔽₂)	C₂	semidirect
2. #11758567	C₂	semidirect
3. #41×D₃₁	C₂	semidirect
4. #11758561	C₂	semidirect
5. C₂×#92497	C₂	semidirect
6. #41×C₃₁	C₂²	semidirect
7. C₂×D₄×C₃₁	C₂²	semidirect
8. C₂×C₂²⋊D₃₁	C₂²	semidirect
9. C₂²×Dic₃₁	C₂²	semidirect
10. #92499	C₂²	semidirect
11. Q₈×D₃₁	C₂²	semidirect
12. #92497	C₂²	semidirect
13. #92497	C₂²	semidirect
14. D₄×D₃₁	C₂²	semidirect
15. #92495	C₂²	semidirect
16. C₂×C₄×D₃₁	C₂²	semidirect
17. C₂×Dic₆₂	C₂²	semidirect
43. H(5,𝔽₂)	D₃₁	semidirect
46. C₃₁	#585	semidirect
51. #41	C₂×D₃₁	semidirect
56. C₂×D₄	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×H(5,𝔽₂)	2
#11758567	2
#41×D₃₁	2
#11758561	2
C₂×#92497	2
#585	31