Subgroups — C₃₁×#3407 #16843340

Order 3968.

360 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#131	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#504	C₂	1	8	yes	maximal, complemented
3	C₂×C₃₁×M₃₂	C₂	1	6	yes	maximal, complemented
4	C₂×#41×C₃₁	C₄	1	1	yes	characteristic
5	C₃₁×#131	C₄	1	4	yes	—
6	C₃₁×#131	C₂²	1	4	yes	—
7	C₂×M₁₆×C₃₁	C₄	1	3	yes	—
8	C₂²×C₈×C₃₁	C₂²	1	3	yes	—
9	C₃₁×M₃₂	C₂²	1	4	yes	—
10	C₃₁×M₃₂	C₂²	1	12	yes	complemented
11	C₂×C₁₆×C₃₁	C₂²	1	12	yes	complemented
12	#41×C₃₁	C₈	1	4	yes	—
13	#41×C₃₁	C₂×C₄	1	4	yes	—
14	C₂×Q₈×C₃₁	C₈	1	1	yes	characteristic
15	C₂×D₄×C₃₁	C₈	1	3	yes	—
16	C₂²×C₄×C₃₁	C₂×C₄	1	3	yes	—
17	M₁₆×C₃₁	C₂×C₄	1	12	yes	—
18	C₂×C₈×C₃₁	C₂×C₄	1	3	yes	—
19	C₂×C₈×C₃₁	C₂×C₄	1	6	yes	—
20	C₂×C₈×C₃₁	C₂³	1	6	yes	—
21	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
22	C₁₆×C₃₁	C₂³	1	8	yes	—
23	C₂³×C₃₁	C₂×C₈	1	3	yes	—
24	Q₈×C₃₁	C₂×C₈	1	4	yes	—
25	D₄×C₃₁	C₂×C₈	1	12	yes	—
26	C₂×C₄×C₃₁	C₂×C₈	1	3	yes	—
27	C₂×C₄×C₃₁	C₂×C₈	1	6	yes	—
28	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₃₁	C₂²×C₄	1	6	yes	—
30	C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
31	C₈×C₃₁	C₂²×C₄	1	6	yes	—
32	C₈×C₃₁	C₂⁴	1	1	yes	characteristic
33	#3407	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂²×C₃₁	—	4	3	no	—
35	C₂²×C₃₁	C₂²×C₈	1	1	yes	characteristic
36	C₂²×C₃₁	C₂²×C₈	1	6	yes	—
37	C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
38	C₄×C₃₁	C₂²×C₈	1	6	yes	—
39	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
40	C₂×#131	C₂×C₃₁	1	1	yes	characteristic
41	#504	C₂×C₃₁	1	8	yes	complemented
42	C₂×M₃₂	C₂×C₃₁	1	6	yes	complemented
43	C₂×C₃₁	—	2	1	no	—
44	C₂×C₃₁	—	2	6	no	—
45	C₂×C₃₁	C₂³×C₈	1	1	yes	characteristic
46	C₂×#41	C₄×C₃₁	1	1	yes	characteristic
47	#131	C₄×C₃₁	1	4	yes	—
48	#131	C₂²×C₃₁	1	4	yes	—
49	C₂×M₁₆	C₄×C₃₁	1	3	yes	—
50	C₂²×C₈	C₂²×C₃₁	1	3	yes	—
51	M₃₂	C₂²×C₃₁	1	4	yes	—
52	M₃₂	C₂²×C₃₁	1	12	yes	complemented
53	C₂×C₁₆	C₂²×C₃₁	1	12	yes	complemented
54	C₃₁	#3407	1	1	yes	characteristic, direct product factor, Sylow
55	#41	C₈×C₃₁	1	4	yes	—
56	#41	C₂×C₄×C₃₁	1	4	yes	—
57	C₂×Q₈	C₈×C₃₁	1	1	yes	characteristic
58	C₂×D₄	C₈×C₃₁	1	3	yes	—
59	C₂²×C₄	C₂×C₄×C₃₁	1	3	yes	—
60	M₁₆	C₂×C₄×C₃₁	1	12	yes	—
61	C₂×C₈	C₂×C₄×C₃₁	1	3	yes	—
62	C₂×C₈	C₂×C₄×C₃₁	1	6	yes	—
63	C₂×C₈	C₂³×C₃₁	1	6	yes	—
64	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
65	C₁₆	C₂³×C₃₁	1	8	yes	—
66	C₂³	C₂×C₈×C₃₁	1	3	yes	—
67	Q₈	C₂×C₈×C₃₁	1	4	yes	—
68	D₄	C₂×C₈×C₃₁	1	12	yes	—
69	C₂×C₄	C₂×C₈×C₃₁	1	3	yes	—
70	C₂×C₄	C₂×C₈×C₃₁	1	6	yes	—
71	C₂×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
72	C₂×C₄	C₂²×C₄×C₃₁	1	6	yes	—
73	C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
74	C₈	C₂²×C₄×C₃₁	1	6	yes	—
75	C₈	C₂⁴×C₃₁	1	1	yes	characteristic
76	C₂²	—	4	3	no	—
77	C₂²	C₂²×C₈×C₃₁	1	1	yes	characteristic
78	C₂²	C₂²×C₈×C₃₁	1	6	yes	—
79	C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
80	C₄	C₂²×C₈×C₃₁	1	6	yes	—
81	C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
82	C₂	—	2	1	no	—
83	C₂	—	2	6	no	—
84	C₂	C₂³×C₈×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₃₁	C₂⁴
commutator	C₂	C₂³×C₈×C₃₁
Frattini, cosocle	C₈	C₂⁴×C₃₁
socle	C₂×C₃₁	C₂³×C₈
prime radical	C₂×#41×C₃₁	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#504	C₂	semidirect
3. C₂×C₃₁×M₃₂	C₂	semidirect
10. C₃₁×M₃₂	C₂²	semidirect
11. C₂×C₁₆×C₃₁	C₂²	semidirect
33. #3407	C₃₁	direct (N × H)
41. #504	C₂×C₃₁	semidirect
42. C₂×M₃₂	C₂×C₃₁	semidirect
52. M₃₂	C₂²×C₃₁	semidirect
53. C₂×C₁₆	C₂²×C₃₁	semidirect
54. C₃₁	#3407	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×#131	2
C₃₁×#504	2
C₂×C₃₁×M₃₂	2
#3407	31

Subgroups — C31×#3407 #16843340

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3407 #16843340