Subgroups — C₃₁×#12542 #21469301

Order 7936.

350 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2110	C₂	1	3	yes	maximal
2	C₃₁×#2108	C₂	1	3	yes	maximal
3	C₃₁×#1747	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×M₃₂	C₂²	1	3	yes	—
5	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
6	C₃₁×#405	C₂²	1	3	yes	—
7	C₃₁×#347	C₄	1	12	yes	complemented
8	C₃₁×#346	C₄	1	6	yes	complemented
9	C₄×C₁₆×C₃₁	C₄	1	6	yes	complemented
10	C₃₁×#338	C₄	1	1	yes	characteristic
11	C₃₁×H(3,ℤ/4)	C₄	1	3	yes	—
12	C₂×M₁₆×C₃₁	C₂×C₄	1	3	yes	—
13	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
14	C₃₁×#117	C₂×C₄	1	3	yes	—
15	C₃₁×M₃₂	C₂×C₄	1	12	yes	—
16	C₂×C₁₆×C₃₁	C₂×C₄	1	3	yes	—
17	C₂×C₁₆×C₃₁	C₂×C₄	1	6	yes	—
18	C₂×C₁₆×C₃₁	C₂×C₄	1	3	yes	—
19	C₃₁×#97	C₂×C₄	1	6	yes	—
20	C₄×C₈×C₃₁	C₂×C₄	1	6	yes	—
21	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
22	M₁₆×C₃₁	—	2	6	no	—
23	C₂×C₈×C₃₁	C₄²	1	6	yes	—
24	C₂×C₈×C₃₁	C₂²×C₄	1	3	yes	—
25	C₂×C₈×C₃₁	C₂²×C₄	1	3	yes	—
26	C₄⋊C₄×C₃₁	—	2	3	no	—
27	C₂²⋊C₄×C₃₁	—	2	3	no	—
28	C₄²×C₃₁	C₄²	1	6	yes	—
29	C₁₆×C₃₁	C₄²	1	3	yes	—
30	C₁₆×C₃₁	C₄²	1	12	yes	—
31	C₁₆×C₃₁	C₄²	1	1	yes	characteristic
32	#12542	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂³×C₃₁	#131	1	1	yes	characteristic
34	C₂×C₄×C₃₁	—	2	6	no	—
35	C₂×C₄×C₃₁	C₂×C₄²	1	3	yes	—
36	C₂×C₄×C₃₁	#131	1	3	yes	—
37	C₈×C₃₁	—	2	6	no	—
38	C₈×C₃₁	C₂×C₄²	1	3	yes	—
39	C₈×C₃₁	C₂×C₄²	1	1	yes	characteristic
40	#2110	C₂×C₃₁	1	3	yes	—
41	#2108	C₂×C₃₁	1	3	yes	—
42	#1747	C₂×C₃₁	1	1	yes	characteristic
43	C₂²×C₃₁	—	4	1	no	—
44	C₂²×C₃₁	#405	1	3	yes	—
45	C₄×C₃₁	—	4	6	no	—
46	C₄×C₃₁	C₄³	1	1	yes	characteristic
47	C₄×C₃₁	#405	1	3	yes	—
48	C₂×M₃₂	C₂²×C₃₁	1	3	yes	—
49	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
50	#405	C₂²×C₃₁	1	3	yes	—
51	#347	C₄×C₃₁	1	12	yes	complemented
52	#346	C₄×C₃₁	1	6	yes	complemented
53	C₄×C₁₆	C₄×C₃₁	1	6	yes	complemented
54	#338	C₄×C₃₁	1	1	yes	characteristic
55	H(3,ℤ/4)	C₄×C₃₁	1	3	yes	—
56	C₂×C₃₁	—	2	3	no	—
57	C₂×C₃₁	#1726	1	1	yes	characteristic
58	C₂×M₁₆	C₂×C₄×C₃₁	1	3	yes	—
59	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
60	#117	C₂×C₄×C₃₁	1	3	yes	—
61	M₃₂	C₂×C₄×C₃₁	1	12	yes	—
62	C₂×C₁₆	C₂×C₄×C₃₁	1	3	yes	—
63	C₂×C₁₆	C₂×C₄×C₃₁	1	6	yes	—
64	C₂×C₁₆	C₂×C₄×C₃₁	1	3	yes	—
65	#97	C₂×C₄×C₃₁	1	6	yes	—
66	C₄×C₈	C₂×C₄×C₃₁	1	6	yes	—
67	C₃₁	#12542	1	1	yes	characteristic, direct product factor, Sylow
68	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
69	M₁₆	—	2	6	no	—
70	C₂×C₈	C₄²×C₃₁	1	6	yes	—
71	C₂×C₈	C₂²×C₄×C₃₁	1	3	yes	—
72	C₂×C₈	C₂²×C₄×C₃₁	1	3	yes	—
73	C₄⋊C₄	—	2	3	no	—
74	C₂²⋊C₄	—	2	3	no	—
75	C₄²	C₄²×C₃₁	1	6	yes	—
76	C₁₆	C₄²×C₃₁	1	3	yes	—
77	C₁₆	C₄²×C₃₁	1	12	yes	—
78	C₁₆	C₄²×C₃₁	1	1	yes	characteristic
79	C₂³	C₃₁×#131	1	1	yes	characteristic
80	C₂×C₄	—	2	6	no	—
81	C₂×C₄	C₂×C₄²×C₃₁	1	3	yes	—
82	C₂×C₄	C₃₁×#131	1	3	yes	—
83	C₈	—	2	6	no	—
84	C₈	C₂×C₄²×C₃₁	1	3	yes	—
85	C₈	C₂×C₄²×C₃₁	1	1	yes	characteristic
86	C₂²	—	4	1	no	—
87	C₂²	C₃₁×#405	1	3	yes	—
88	C₄	—	4	6	no	—
89	C₄	C₄³×C₃₁	1	1	yes	characteristic
90	C₄	C₃₁×#405	1	3	yes	—
91	C₂	—	2	3	no	—
92	C₂	C₃₁×#1726	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆×C₃₁	C₄²
commutator	C₄	C₄³×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
socle	C₂×C₃₁	#1726
prime radical	C₂³×C₃₁	#131
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₁₆×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
7. C₃₁×#347	C₄	semidirect
8. C₃₁×#346	C₄	semidirect
9. C₄×C₁₆×C₃₁	C₄	semidirect
32. #12542	C₃₁	direct (N × H)
51. #347	C₄×C₃₁	semidirect
52. #346	C₄×C₃₁	semidirect
53. C₄×C₁₆	C₄×C₃₁	semidirect
67. C₃₁	#12542	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×#2110	2
C₃₁×#2108	2
C₃₁×#1747	2
#12542	31

Subgroups — C31×#12542 #21469301

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12542 #21469301