Subgroups — C₃₁×#13725 #21470484

Order 7936.

270 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×D₃₂	C₂	1	4	yes	maximal, complemented
2	C₃₁×#2246	C₂	1	2	yes	maximal, complemented
3	C₄×C₃₁×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×D₁₆	—	2	2	no	—
5	C₂×C₃₁×D₁₆	C₂²	1	4	yes	—
6	C₃₁×#493	—	2	2	no	—
7	C₃₁×D₃₂	—	2	8	no	—
8	C₂×C₃₁×C₃₂	C₂²	1	2	yes	—
9	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×D₈×C₃₁	—	2	4	no	—
11	C₂×D₈×C₃₁	—	4	2	no	—
12	C₃₁×#127	—	4	2	no	—
13	C₃₁×D₁₆	—	2	8	no	—
14	C₃₁×D₁₆	—	4	4	no	—
15	C₂×C₁₆×C₃₁	D₄	1	1	yes	characteristic
16	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
17	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
18	C₃₁×C₃₂	D₄	1	4	yes	complemented
19	C₂×D₄×C₃₁	—	4	4	no	—
20	C₂×D₄×C₃₁	—	8	4	no	—
21	D₈×C₃₁	—	4	4	no	—
22	D₈×C₃₁	—	4	4	no	—
23	D₈×C₃₁	—	8	4	no	—
24	C₂×C₈×C₃₁	D₈	1	1	yes	characteristic
25	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
26	C₄²×C₃₁	D₈	1	1	yes	characteristic
27	C₁₆×C₃₁	D₈	1	2	yes	—
28	C₁₆×C₃₁	C₂×D₄	1	2	yes	—
29	#13725	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂³×C₃₁	—	8	4	no	—
31	D₄×C₃₁	—	8	4	no	—
32	D₄×C₃₁	—	8	4	no	—
33	D₄×C₃₁	—	16	8	no	—
34	C₂×C₄×C₃₁	D₁₆	1	2	yes	—
35	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
36	C₈×C₃₁	D₁₆	1	2	yes	—
37	C₈×C₃₁	#127	1	1	yes	characteristic
38	C₈×C₃₁	C₂×D₈	1	1	yes	characteristic
39	C₂×D₃₂	C₂×C₃₁	1	4	yes	complemented
40	#2246	C₂×C₃₁	1	2	yes	complemented
41	C₄×C₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
42	C₂²×C₃₁	—	16	4	no	—
43	C₂²×C₃₁	—	16	8	no	—
44	C₂²×C₃₁	C₂×D₁₆	1	1	yes	characteristic
45	C₄×C₃₁	D₃₂	1	4	yes	complemented
46	C₄×C₃₁	#493	1	1	yes	characteristic
47	C₄×C₃₁	C₂×D₁₆	1	1	yes	characteristic
48	C₂×D₁₆	—	2	2	no	—
49	C₂×D₁₆	C₂²×C₃₁	1	4	yes	—
50	#493	—	2	2	no	—
51	D₃₂	—	2	8	no	—
52	C₂×C₃₂	C₂²×C₃₁	1	2	yes	—
53	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
54	C₂×C₃₁	—	32	4	no	—
55	C₂×C₃₁	#2246	1	1	yes	characteristic
56	C₂×C₃₁	C₂×D₃₂	1	2	yes	—
57	C₂×D₈	—	2	4	no	—
58	C₂×D₈	—	4	2	no	—
59	#127	—	4	2	no	—
60	D₁₆	—	2	8	no	—
61	D₁₆	—	4	4	no	—
62	C₂×C₁₆	D₄×C₃₁	1	1	yes	characteristic
63	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
64	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
65	C₃₂	D₄×C₃₁	1	4	yes	complemented
66	C₃₁	#13725	1	1	yes	characteristic, direct product factor, Sylow
67	C₂×D₄	—	4	4	no	—
68	C₂×D₄	—	8	4	no	—
69	D₈	—	4	4	no	—
70	D₈	—	4	4	no	—
71	D₈	—	8	4	no	—
72	C₂×C₈	D₈×C₃₁	1	1	yes	characteristic
73	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
74	C₄²	D₈×C₃₁	1	1	yes	characteristic
75	C₁₆	D₈×C₃₁	1	2	yes	—
76	C₁₆	C₂×D₄×C₃₁	1	2	yes	—
77	C₂³	—	8	4	no	—
78	D₄	—	8	4	no	—
79	D₄	—	8	4	no	—
80	D₄	—	16	8	no	—
81	C₂×C₄	C₃₁×D₁₆	1	2	yes	—
82	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
83	C₈	C₃₁×D₁₆	1	2	yes	—
84	C₈	C₃₁×#127	1	1	yes	characteristic
85	C₈	C₂×D₈×C₃₁	1	1	yes	characteristic
86	C₂²	—	16	4	no	—
87	C₂²	—	16	8	no	—
88	C₂²	C₂×C₃₁×D₁₆	1	1	yes	characteristic
89	C₄	C₃₁×D₃₂	1	4	yes	complemented
90	C₄	C₃₁×#493	1	1	yes	characteristic
91	C₄	C₂×C₃₁×D₁₆	1	1	yes	characteristic
92	C₂	—	32	4	no	—
93	C₂	C₃₁×#2246	1	1	yes	characteristic
94	C₂	C₂×C₃₁×D₃₂	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₃₁
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄²×C₃₁ ◁ C₄×C₈×C₃₁ ◁ C₄×C₁₆×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×D₃₂	C₂	semidirect
2. C₃₁×#2246	C₂	semidirect
3. C₄×C₃₁×C₃₂	C₂	semidirect
18. C₃₁×C₃₂	D₄	semidirect
29. #13725	C₃₁	direct (N × H)
39. C₂×D₃₂	C₂×C₃₁	semidirect
40. #2246	C₂×C₃₁	semidirect
41. C₄×C₃₂	C₂×C₃₁	semidirect
45. C₄×C₃₁	D₃₂	semidirect
65. C₃₂	D₄×C₃₁	semidirect
66. C₃₁	#13725	direct (N × H)
89. C₄	C₃₁×D₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×D₃₂	2
C₃₁×#2246	2
C₄×C₃₁×C₃₂	2
#13725	31

Subgroups — C31×#13725 #21470484

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13725 #21470484