Subgroups — C₃₁×#12514 #21469273

Order 7936.

246 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2241	C₂	1	4	yes	maximal
2	C₃₁×#1713	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1710	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1311	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#500	—	2	2	no	—
6	C₃₁×#500	C₂²	1	2	yes	—
7	C₃₁×#494	C₂²	1	2	yes	—
8	C₃₁×#487	—	2	2	no	—
9	C₃₁×#358	—	2	8	no	—
10	C₄×C₁₆×C₃₁	C₂²	1	2	yes	—
11	C₈²×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×Q₁₆×C₃₁	—	2	4	no	—
13	C₃₁×C₄⋊Q₈	—	2	2	no	—
14	C₃₁×C₄⋊Q₈	—	2	2	no	—
15	C₂×C₁₆×C₃₁	D₄	1	4	yes	—
16	C₃₁×#107	—	2	4	no	—
17	C₃₁×#107	—	4	4	no	—
18	C₃₁×#103	—	4	4	no	—
19	C₄×C₈×C₃₁	D₄	1	2	yes	—
20	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
21	C₂×Q₈×C₃₁	—	4	2	no	—
22	C₂×Q₈×C₃₁	—	4	2	no	—
23	Q₁₆×C₃₁	—	4	8	no	—
24	C₂×C₈×C₃₁	D₈	1	4	yes	—
25	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
26	C₄⋊C₄×C₃₁	—	4	2	no	—
27	C₄⋊C₄×C₃₁	—	4	2	no	—
28	C₄⋊C₄×C₃₁	—	4	4	no	—
29	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
30	C₁₆×C₃₁	—	2	4	no	—
31	#12514	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	Q₈×C₃₁	—	8	4	no	—
33	Q₈×C₃₁	—	8	4	no	—
34	C₂×C₄×C₃₁	—	8	2	no	—
35	C₂×C₄×C₃₁	—	8	2	no	—
36	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
37	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
38	C₈×C₃₁	—	2	4	no	—
39	C₈×C₃₁	#135	1	4	yes	—
40	#2241	C₂×C₃₁	1	4	yes	—
41	#1713	C₂×C₃₁	1	1	yes	characteristic
42	#1710	C₂×C₃₁	1	1	yes	characteristic
43	#1311	C₂×C₃₁	1	1	yes	characteristic
44	C₂²×C₃₁	#493	1	1	yes	characteristic
45	C₄×C₃₁	—	16	2	no	—
46	C₄×C₃₁	—	16	2	no	—
47	C₄×C₃₁	#495	1	2	yes	—
48	C₄×C₃₁	#510	1	4	yes	—
49	#500	—	2	2	no	—
50	#500	C₂²×C₃₁	1	2	yes	—
51	#494	C₂²×C₃₁	1	2	yes	—
52	#487	—	2	2	no	—
53	#358	—	2	8	no	—
54	C₄×C₁₆	C₂²×C₃₁	1	2	yes	—
55	C₈²	C₂²×C₃₁	1	1	yes	characteristic
56	C₂×C₃₁	#1709	1	1	yes	characteristic
57	C₂×C₃₁	#2251	1	2	yes	—
58	C₂×Q₁₆	—	2	4	no	—
59	C₄⋊Q₈	—	2	2	no	—
60	C₄⋊Q₈	—	2	2	no	—
61	C₂×C₁₆	D₄×C₃₁	1	4	yes	—
62	#107	—	2	4	no	—
63	#107	—	4	4	no	—
64	#103	—	4	4	no	—
65	C₄×C₈	D₄×C₃₁	1	2	yes	—
66	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
67	C₃₁	#12514	1	1	yes	characteristic, direct product factor, Sylow
68	C₂×Q₈	—	4	2	no	—
69	C₂×Q₈	—	4	2	no	—
70	Q₁₆	—	4	8	no	—
71	C₂×C₈	D₈×C₃₁	1	4	yes	—
72	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
73	C₄⋊C₄	—	4	2	no	—
74	C₄⋊C₄	—	4	2	no	—
75	C₄⋊C₄	—	4	4	no	—
76	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
77	C₁₆	—	2	4	no	—
78	Q₈	—	8	4	no	—
79	Q₈	—	8	4	no	—
80	C₂×C₄	—	8	2	no	—
81	C₂×C₄	—	8	2	no	—
82	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
83	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
84	C₈	—	2	4	no	—
85	C₈	C₃₁×#135	1	4	yes	—
86	C₂²	C₃₁×#493	1	1	yes	characteristic
87	C₄	—	16	2	no	—
88	C₄	—	16	2	no	—
89	C₄	C₃₁×#495	1	2	yes	—
90	C₄	C₃₁×#510	1	4	yes	—
91	C₂	C₃₁×#1709	1	1	yes	characteristic
92	C₂	C₃₁×#2251	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
31. #12514	C₃₁	direct (N × H)
67. C₃₁	#12514	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×#2241	2
C₃₁×#1713	2
C₃₁×#1710	2
C₃₁×#1311	2
#12514	31

Subgroups — C31×#12514 #21469273

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12514 #21469273