Subgroups — C₃₁×#12521 #21469280

Order 7936.

262 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2252	C₂	1	4	yes	maximal
2	C₃₁×#1713	C₂	1	2	yes	maximal
3	C₈×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#500	—	2	2	no	—
5	C₃₁×#500	C₂²	1	4	yes	—
6	C₃₁×#494	—	2	2	no	—
7	C₃₁×#366	—	2	8	no	—
8	C₄×C₁₆×C₃₁	C₂²	1	2	yes	—
9	C₈²×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×Q₁₆×C₃₁	—	4	4	no	—
11	C₃₁×C₄⋊Q₈	—	2	4	no	—
12	C₂×C₁₆×C₃₁	D₄	1	4	yes	—
13	C₃₁×#107	—	2	8	no	—
14	C₃₁×#107	—	4	4	no	—
15	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
16	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
17	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
18	C₂×Q₈×C₃₁	—	4	4	no	—
19	Q₁₆×C₃₁	—	8	8	no	—
20	C₂×C₈×C₃₁	D₈	1	2	yes	—
21	C₂×C₈×C₃₁	D₈	1	2	yes	—
22	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
23	C₄⋊C₄×C₃₁	—	4	4	no	—
24	C₄⋊C₄×C₃₁	—	4	4	no	—
25	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
26	C₁₆×C₃₁	Q₁₆	1	8	yes	complemented
27	#12521	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	Q₈×C₃₁	—	8	8	no	—
29	C₂×C₄×C₃₁	—	8	4	no	—
30	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
31	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
32	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
33	C₈×C₃₁	D₁₆	1	4	yes	—
34	C₈×C₃₁	Q₃₂	1	4	yes	—
35	C₈×C₃₁	C₂×Q₁₆	1	4	yes	—
36	#2252	C₂×C₃₁	1	4	yes	—
37	#1713	C₂×C₃₁	1	2	yes	—
38	C₈×C₁₆	C₂×C₃₁	1	1	yes	characteristic
39	C₂²×C₃₁	#493	1	1	yes	characteristic
40	C₄×C₃₁	—	16	4	no	—
41	C₄×C₃₁	#494	1	2	yes	—
42	C₄×C₃₁	C₂×D₁₆	1	2	yes	—
43	C₄×C₃₁	C₂×Q₃₂	1	2	yes	—
44	#500	—	2	2	no	—
45	#500	C₂²×C₃₁	1	4	yes	—
46	#494	—	2	2	no	—
47	#366	—	2	8	no	—
48	C₄×C₁₆	C₂²×C₃₁	1	2	yes	—
49	C₈²	C₂²×C₃₁	1	1	yes	characteristic
50	C₂×C₃₁	#1713	1	1	yes	characteristic
51	C₂×C₃₁	#2246	1	1	yes	characteristic
52	C₂×C₃₁	#2247	1	1	yes	characteristic
53	C₂×Q₁₆	—	4	4	no	—
54	C₄⋊Q₈	—	2	4	no	—
55	C₂×C₁₆	D₄×C₃₁	1	4	yes	—
56	#107	—	2	8	no	—
57	#107	—	4	4	no	—
58	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
59	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
60	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
61	C₃₁	#12521	1	1	yes	characteristic, direct product factor, Sylow
62	C₂×Q₈	—	4	4	no	—
63	Q₁₆	—	8	8	no	—
64	C₂×C₈	D₈×C₃₁	1	2	yes	—
65	C₂×C₈	D₈×C₃₁	1	2	yes	—
66	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
67	C₄⋊C₄	—	4	4	no	—
68	C₄⋊C₄	—	4	4	no	—
69	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
70	C₁₆	Q₁₆×C₃₁	1	8	yes	complemented
71	Q₈	—	8	8	no	—
72	C₂×C₄	—	8	4	no	—
73	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
74	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
75	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
76	C₈	C₃₁×D₁₆	1	4	yes	—
77	C₈	C₃₁×Q₃₂	1	4	yes	—
78	C₈	C₂×Q₁₆×C₃₁	1	4	yes	—
79	C₂²	C₃₁×#493	1	1	yes	characteristic
80	C₄	—	16	4	no	—
81	C₄	C₃₁×#494	1	2	yes	—
82	C₄	C₂×C₃₁×D₁₆	1	2	yes	—
83	C₄	C₂×C₃₁×Q₃₂	1	2	yes	—
84	C₂	C₃₁×#1713	1	1	yes	characteristic
85	C₂	C₃₁×#2246	1	1	yes	characteristic
86	C₂	C₃₁×#2247	1	1	yes	characteristic

	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
26. C₁₆×C₃₁	Q₁₆	semidirect
27. #12521	C₃₁	direct (N × H)
61. C₃₁	#12521	direct (N × H)
70. C₁₆	Q₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#2252	2
C₃₁×#1713	2
C₈×C₁₆×C₃₁	2
#12521	31

Subgroups — C31×#12521 #21469280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12521 #21469280