Subgroups — C₃₁×#12516 #21469275

Order 7936.

310 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2247	C₂	1	4	yes	maximal
2	C₃₁×#1713	C₂	1	2	yes	maximal
3	C₈×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×Q₃₂	—	2	8	no	—
5	C₃₁×#500	—	2	4	no	—
6	C₃₁×#494	C₂²	1	4	yes	—
7	C₄×C₁₆×C₃₁	C₂²	1	2	yes	—
8	C₈²×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×Q₁₆×C₃₁	—	2	8	no	—
10	C₃₁×C₄⋊Q₈	—	2	4	no	—
11	C₃₁×Q₃₂	—	4	16	no	—
12	C₂×C₁₆×C₃₁	D₄	1	4	yes	—
13	C₃₁×#107	—	4	8	no	—
14	C₄×C₈×C₃₁	D₄	1	2	yes	—
15	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
16	C₂×Q₈×C₃₁	—	4	4	no	—
17	Q₁₆×C₃₁	—	4	16	no	—
18	C₂×C₈×C₃₁	D₈	1	4	yes	—
19	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
20	C₄⋊C₄×C₃₁	—	4	8	no	—
21	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
22	C₁₆×C₃₁	D₈	1	8	yes	—
23	#12516	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	Q₈×C₃₁	—	8	8	no	—
25	C₂×C₄×C₃₁	—	8	4	no	—
26	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
27	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
28	C₈×C₃₁	Q₃₂	1	8	yes	complemented
29	C₈×C₃₁	C₂×D₈	1	4	yes	—
30	#2247	C₂×C₃₁	1	4	yes	—
31	#1713	C₂×C₃₁	1	2	yes	—
32	C₈×C₁₆	C₂×C₃₁	1	1	yes	characteristic
33	C₂²×C₃₁	#493	1	1	yes	characteristic
34	C₄×C₃₁	—	16	4	no	—
35	C₄×C₃₁	#493	1	2	yes	—
36	C₄×C₃₁	C₂×Q₃₂	1	4	yes	—
37	C₂×Q₃₂	—	2	8	no	—
38	#500	—	2	4	no	—
39	#494	C₂²×C₃₁	1	4	yes	—
40	C₄×C₁₆	C₂²×C₃₁	1	2	yes	—
41	C₈²	C₂²×C₃₁	1	1	yes	characteristic
42	C₂×C₃₁	#1712	1	1	yes	characteristic
43	C₂×C₃₁	#2247	1	2	yes	—
44	C₂×Q₁₆	—	2	8	no	—
45	C₄⋊Q₈	—	2	4	no	—
46	Q₃₂	—	4	16	no	—
47	C₂×C₁₆	D₄×C₃₁	1	4	yes	—
48	#107	—	4	8	no	—
49	C₄×C₈	D₄×C₃₁	1	2	yes	—
50	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
51	C₃₁	#12516	1	1	yes	characteristic, direct product factor, Sylow
52	C₂×Q₈	—	4	4	no	—
53	Q₁₆	—	4	16	no	—
54	C₂×C₈	D₈×C₃₁	1	4	yes	—
55	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
56	C₄⋊C₄	—	4	8	no	—
57	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
58	C₁₆	D₈×C₃₁	1	8	yes	—
59	Q₈	—	8	8	no	—
60	C₂×C₄	—	8	4	no	—
61	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
62	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
63	C₈	C₃₁×Q₃₂	1	8	yes	complemented
64	C₈	C₂×D₈×C₃₁	1	4	yes	—
65	C₂²	C₃₁×#493	1	1	yes	characteristic
66	C₄	—	16	4	no	—
67	C₄	C₃₁×#493	1	2	yes	—
68	C₄	C₂×C₃₁×Q₃₂	1	4	yes	—
69	C₂	C₃₁×#1712	1	1	yes	characteristic
70	C₂	C₃₁×#2247	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²×C₃₁	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₃₁
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄²×C₃₁ ◁ C₈²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
23. #12516	C₃₁	direct (N × H)
28. C₈×C₃₁	Q₃₂	semidirect
51. C₃₁	#12516	direct (N × H)
63. C₈	C₃₁×Q₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#2247	2
C₃₁×#1713	2
C₈×C₁₆×C₃₁	2
#12516	31

Subgroups — C31×#12516 #21469275

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12516 #21469275