Subgroups — C₃₁×#12539 #21469298

Order 7936.

478 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#346	C₂	1	3	yes	maximal
2	C₂×C₄×C₁₆×C₃₁	C₂	1	3	yes	maximal
3	C₃₁×#1726	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₆×C₃₁	C₂²	1	3	yes	—
5	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
6	C₂×C₃₁×#97	C₄	1	3	yes	—
7	C₂×C₄×C₈×C₃₁	C₂²	1	3	yes	—
8	C₃₁×#376	C₄	1	1	yes	characteristic
9	C₃₁×#346	C₄	1	12	yes	complemented
10	C₄×C₁₆×C₃₁	C₄	1	12	yes	complemented
11	C₂²×C₈×C₃₁	C₂×C₄	1	3	yes	—
12	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
13	C₂×C₄²×C₃₁	C₂×C₄	1	3	yes	—
14	C₂×C₁₆×C₃₁	C₂×C₄	1	6	yes	—
15	C₂×C₁₆×C₃₁	C₂×C₄	1	12	yes	—
16	C₂×C₁₆×C₃₁	C₂×C₄	1	6	yes	—
17	C₃₁×#97	C₈	1	12	yes	—
18	C₄×C₈×C₃₁	C₂×C₄	1	6	yes	—
19	C₄×C₈×C₃₁	C₂×C₄	1	6	yes	—
20	C₃₁×#95	C₈	1	4	yes	—
21	C₂²×C₄×C₃₁	C₂×C₈	1	6	yes	—
22	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
23	C₂×C₈×C₃₁	C₄²	1	6	yes	—
24	C₂×C₈×C₃₁	C₂×C₈	1	12	yes	—
25	C₂×C₈×C₃₁	C₂²×C₄	1	3	yes	—
26	C₂×C₈×C₃₁	C₂²×C₄	1	3	yes	—
27	C₄²×C₃₁	C₄²	1	6	yes	—
28	C₄²×C₃₁	C₂×C₈	1	6	yes	—
29	C₁₆×C₃₁	C₄²	1	12	yes	—
30	C₁₆×C₃₁	C₄²	1	4	yes	—
31	#12539	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂³×C₃₁	C₂²×C₈	1	1	yes	characteristic
33	C₂×C₄×C₃₁	—	2	12	no	—
34	C₂×C₄×C₃₁	C₄×C₈	1	12	yes	—
35	C₂×C₄×C₃₁	C₂×C₄²	1	3	yes	—
36	C₂×C₄×C₃₁	C₂²×C₈	1	3	yes	—
37	C₈×C₃₁	C₄×C₈	1	12	yes	—
38	C₈×C₃₁	C₂×C₄²	1	3	yes	—
39	C₈×C₃₁	C₂×C₄²	1	1	yes	characteristic
40	C₂×#346	C₂×C₃₁	1	3	yes	—
41	C₂×C₄×C₁₆	C₂×C₃₁	1	3	yes	—
42	#1726	C₂×C₃₁	1	1	yes	characteristic
43	C₂²×C₃₁	C₂×C₄×C₈	1	3	yes	—
44	C₂²×C₃₁	#504	1	4	yes	—
45	C₄×C₃₁	—	2	12	no	—
46	C₄×C₃₁	C₄³	1	1	yes	characteristic
47	C₄×C₃₁	C₂×C₄×C₈	1	3	yes	—
48	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
49	C₂²×C₁₆	C₂²×C₃₁	1	3	yes	—
50	C₂×#97	C₄×C₃₁	1	3	yes	—
51	C₂×C₄×C₈	C₂²×C₃₁	1	3	yes	—
52	#376	C₄×C₃₁	1	1	yes	characteristic
53	#346	C₄×C₃₁	1	12	yes	complemented
54	C₄×C₁₆	C₄×C₃₁	1	12	yes	complemented
55	C₂×C₃₁	C₄²×C₈	1	1	yes	characteristic
56	C₂×C₃₁	#2108	1	6	yes	—
57	C₂²×C₈	C₂×C₄×C₃₁	1	3	yes	—
58	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
59	C₂×C₄²	C₂×C₄×C₃₁	1	3	yes	—
60	C₂×C₁₆	C₂×C₄×C₃₁	1	6	yes	—
61	C₂×C₁₆	C₂×C₄×C₃₁	1	6	yes	—
62	C₂×C₁₆	C₂×C₄×C₃₁	1	12	yes	—
63	#97	C₈×C₃₁	1	12	yes	—
64	C₄×C₈	C₂×C₄×C₃₁	1	6	yes	—
65	C₄×C₈	C₂×C₄×C₃₁	1	6	yes	—
66	#95	C₈×C₃₁	1	4	yes	—
67	C₃₁	#12539	1	1	yes	characteristic, direct product factor, Sylow
68	C₂²×C₄	C₂×C₈×C₃₁	1	6	yes	—
69	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
70	C₂×C₈	C₄²×C₃₁	1	6	yes	—
71	C₂×C₈	C₂×C₈×C₃₁	1	12	yes	—
72	C₂×C₈	C₂²×C₄×C₃₁	1	3	yes	—
73	C₂×C₈	C₂²×C₄×C₃₁	1	3	yes	—
74	C₄²	C₄²×C₃₁	1	6	yes	—
75	C₄²	C₂×C₈×C₃₁	1	6	yes	—
76	C₁₆	C₄²×C₃₁	1	4	yes	—
77	C₁₆	C₄²×C₃₁	1	12	yes	—
78	C₂³	C₂²×C₈×C₃₁	1	1	yes	characteristic
79	C₂×C₄	—	2	12	no	—
80	C₂×C₄	C₄×C₈×C₃₁	1	12	yes	—
81	C₂×C₄	C₂×C₄²×C₃₁	1	3	yes	—
82	C₂×C₄	C₂²×C₈×C₃₁	1	3	yes	—
83	C₈	C₄×C₈×C₃₁	1	12	yes	—
84	C₈	C₂×C₄²×C₃₁	1	1	yes	characteristic
85	C₈	C₂×C₄²×C₃₁	1	3	yes	—
86	C₂²	C₂×C₄×C₈×C₃₁	1	3	yes	—
87	C₂²	C₃₁×#504	1	4	yes	—
88	C₄	—	2	12	no	—
89	C₄	C₄³×C₃₁	1	1	yes	characteristic
90	C₄	C₂×C₄×C₈×C₃₁	1	3	yes	—
91	C₂	C₄²×C₈×C₃₁	1	1	yes	characteristic
92	C₂	C₃₁×#2108	1	6	yes	—

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

Complements

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

N	H ≅ G/N	type
9. C₃₁×#346	C₄	semidirect
10. C₄×C₁₆×C₃₁	C₄	semidirect
31. #12539	C₃₁	direct (N × H)
53. #346	C₄×C₃₁	semidirect
54. C₄×C₁₆	C₄×C₃₁	semidirect
67. C₃₁	#12539	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#346	2
C₂×C₄×C₁₆×C₃₁	2
C₃₁×#1726	2
#12539	31

Subgroups — C31×#12539 #21469298

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12539 #21469298