Subgroups — C₃₁×#62624 #21519383

Order 7936.

2038 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3573	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3407	C₂	1	15	yes	maximal, complemented
3	C₂×C₃₁×#504	C₂	1	15	yes	maximal, complemented
4	C₃₁×#585	C₄	1	1	yes	characteristic
5	C₃₁×#568	C₄	1	15	yes	—
6	C₂×C₃₁×#131	C₂²	1	15	yes	—
7	C₃₁×#504	C₂²	1	20	yes	complemented
8	C₃₁×#504	C₂²	1	60	yes	complemented
9	C₂×C₃₁×M₃₂	C₂²	1	45	yes	complemented
10	C₂²×C₁₆×C₃₁	C₂²	1	15	yes	complemented
11	C₃₁×2₋¹⁺⁴	C₈	1	6	yes	—
12	C₃₁×H(5,𝔽₂)	C₈	1	10	yes	—
13	C₂×#41×C₃₁	C₂×C₄	1	15	yes	—
14	C₃₁×#131	C₂×C₄	1	60	yes	—
15	C₃₁×#131	C₂³	1	20	yes	—
16	C₂×M₁₆×C₃₁	C₂×C₄	1	45	yes	—
17	C₂²×C₈×C₃₁	C₂³	1	15	yes	—
18	C₃₁×M₃₂	C₂³	1	60	yes	—
19	C₂×C₁₆×C₃₁	C₂³	1	45	yes	—
20	C₂×C₁₆×C₃₁	C₂³	1	15	yes	—
21	#41×C₃₁	C₂×C₈	1	60	yes	—
22	#41×C₃₁	C₂²×C₄	1	20	yes	—
23	C₂×Q₈×C₃₁	C₂×C₈	1	15	yes	—
24	C₂×D₄×C₃₁	C₂×C₈	1	45	yes	—
25	C₂²×C₄×C₃₁	C₂²×C₄	1	15	yes	—
26	M₁₆×C₃₁	C₂²×C₄	1	60	yes	—
27	C₂×C₈×C₃₁	C₂²×C₄	1	45	yes	—
28	C₂×C₈×C₃₁	C₂⁴	1	15	yes	—
29	C₁₆×C₃₁	C₂⁴	1	15	yes	—
30	C₁₆×C₃₁	C₂⁴	1	1	yes	characteristic
31	#62624	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂³×C₃₁	C₂²×C₈	1	15	yes	—
33	Q₈×C₃₁	C₂²×C₈	1	20	yes	—
34	D₄×C₃₁	C₂²×C₈	1	60	yes	—
35	C₂×C₄×C₃₁	C₂²×C₈	1	45	yes	—
36	C₂×C₄×C₃₁	C₂³×C₄	1	15	yes	—
37	C₈×C₃₁	C₂³×C₄	1	15	yes	—
38	C₈×C₃₁	C₂⁵	1	1	yes	characteristic
39	#3573	C₂×C₃₁	1	1	yes	characteristic
40	#3407	C₂×C₃₁	1	15	yes	complemented
41	C₂×#504	C₂×C₃₁	1	15	yes	complemented
42	C₂²×C₃₁	—	4	15	no	—
43	C₂²×C₃₁	C₂³×C₈	1	15	yes	—
44	C₄×C₃₁	C₂³×C₈	1	15	yes	—
45	C₄×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
46	#585	C₄×C₃₁	1	1	yes	characteristic
47	#568	C₄×C₃₁	1	15	yes	—
48	C₂×#131	C₂²×C₃₁	1	15	yes	—
49	#504	C₂²×C₃₁	1	20	yes	complemented
50	#504	C₂²×C₃₁	1	60	yes	complemented
51	C₂×M₃₂	C₂²×C₃₁	1	45	yes	complemented
52	C₂²×C₁₆	C₂²×C₃₁	1	15	yes	complemented
53	C₂×C₃₁	—	2	15	no	—
54	C₂×C₃₁	C₂⁴×C₈	1	1	yes	characteristic
55	2₋¹⁺⁴	C₈×C₃₁	1	6	yes	—
56	H(5,𝔽₂)	C₈×C₃₁	1	10	yes	—
57	C₂×#41	C₂×C₄×C₃₁	1	15	yes	—
58	#131	C₂×C₄×C₃₁	1	60	yes	—
59	#131	C₂³×C₃₁	1	20	yes	—
60	C₂×M₁₆	C₂×C₄×C₃₁	1	45	yes	—
61	C₂²×C₈	C₂³×C₃₁	1	15	yes	—
62	M₃₂	C₂³×C₃₁	1	60	yes	—
63	C₂×C₁₆	C₂³×C₃₁	1	15	yes	—
64	C₂×C₁₆	C₂³×C₃₁	1	45	yes	—
65	C₃₁	#62624	1	1	yes	characteristic, direct product factor, Sylow
66	#41	C₂×C₈×C₃₁	1	60	yes	—
67	#41	C₂²×C₄×C₃₁	1	20	yes	—
68	C₂×Q₈	C₂×C₈×C₃₁	1	15	yes	—
69	C₂×D₄	C₂×C₈×C₃₁	1	45	yes	—
70	C₂²×C₄	C₂²×C₄×C₃₁	1	15	yes	—
71	M₁₆	C₂²×C₄×C₃₁	1	60	yes	—
72	C₂×C₈	C₂²×C₄×C₃₁	1	45	yes	—
73	C₂×C₈	C₂⁴×C₃₁	1	15	yes	—
74	C₁₆	C₂⁴×C₃₁	1	15	yes	—
75	C₁₆	C₂⁴×C₃₁	1	1	yes	characteristic
76	C₂³	C₂²×C₈×C₃₁	1	15	yes	—
77	Q₈	C₂²×C₈×C₃₁	1	20	yes	—
78	D₄	C₂²×C₈×C₃₁	1	60	yes	—
79	C₂×C₄	C₂²×C₈×C₃₁	1	45	yes	—
80	C₂×C₄	C₂³×C₄×C₃₁	1	15	yes	—
81	C₈	C₂³×C₄×C₃₁	1	15	yes	—
82	C₈	C₂⁵×C₃₁	1	1	yes	characteristic
83	C₂²	—	4	15	no	—
84	C₂²	C₂³×C₈×C₃₁	1	15	yes	—
85	C₄	C₂³×C₈×C₃₁	1	15	yes	—
86	C₄	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
87	C₂	—	2	15	no	—
88	C₂	C₂⁴×C₈×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆×C₃₁	C₂⁴
commutator	C₂	C₂⁴×C₈×C₃₁
Frattini, cosocle	C₈	C₂⁵×C₃₁
socle	C₂×C₃₁	C₂⁴×C₈
prime radical	C₃₁×#585	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#3407	C₂	semidirect
3. C₂×C₃₁×#504	C₂	semidirect
7. C₃₁×#504	C₂²	semidirect
8. C₃₁×#504	C₂²	semidirect
9. C₂×C₃₁×M₃₂	C₂²	semidirect
10. C₂²×C₁₆×C₃₁	C₂²	semidirect
31. #62624	C₃₁	direct (N × H)
40. #3407	C₂×C₃₁	semidirect
41. C₂×#504	C₂×C₃₁	semidirect
49. #504	C₂²×C₃₁	semidirect
50. #504	C₂²×C₃₁	semidirect
51. C₂×M₃₂	C₂²×C₃₁	semidirect
52. C₂²×C₁₆	C₂²×C₃₁	semidirect
65. C₃₁	#62624	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3573	2
C₃₁×#3407	2
C₂×C₃₁×#504	2
#62624	31

Subgroups — C31×#62624 #21519383

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#62624 #21519383