Subgroups — C₃₁×#2176 #16842109

Order 3968.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×Q₃₂	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₁₆×C₃₁	C₂	1	2	yes	maximal
3	C₃₁×#367	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#358	C₂	1	2	yes	maximal
5	C₃₁×#346	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₁₆×C₃₁	C₂²	1	2	yes	—
7	C₄×Q₈×C₃₁	—	2	2	no	—
8	C₃₁×Q₃₂	C₄	1	4	yes	complemented
9	C₂×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#107	C₂²	1	2	yes	—
12	C₃₁×#103	—	2	2	no	—
13	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×Q₈×C₃₁	—	2	2	no	—
15	Q₁₆×C₃₁	—	2	2	no	—
16	Q₁₆×C₃₁	C₂×C₄	1	4	yes	—
17	C₂×C₈×C₃₁	D₄	1	1	yes	characteristic
18	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
19	C₄⋊C₄×C₃₁	—	2	2	no	—
20	C₄⋊C₄×C₃₁	—	4	2	no	—
21	C₄²×C₃₁	—	4	2	no	—
22	C₄²×C₃₁	D₄	1	1	yes	characteristic
23	C₁₆×C₃₁	—	2	1	no	—
24	C₁₆×C₃₁	C₂×C₄	1	2	yes	—
25	Q₈×C₃₁	—	2	4	no	—
26	Q₈×C₃₁	—	4	2	no	—
27	C₂×C₄×C₃₁	—	4	2	no	—
28	C₂×C₄×C₃₁	—	4	2	no	—
29	C₂×C₄×C₃₁	D₈	1	2	yes	—
30	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
31	C₈×C₃₁	—	2	1	no	—
32	C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₃₁	#41	1	1	yes	characteristic
34	#2176	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₂²×C₃₁	C₂×D₈	1	1	yes	characteristic
36	C₄×C₃₁	—	2	2	no	—
37	C₄×C₃₁	—	4	4	no	—
38	C₄×C₃₁	—	8	2	no	—
39	C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
40	C₄×C₃₁	#135	1	1	yes	characteristic
41	C₂×Q₃₂	C₂×C₃₁	1	1	yes	characteristic
42	C₄×Q₁₆	C₂×C₃₁	1	2	yes	—
43	#367	C₂×C₃₁	1	1	yes	characteristic
44	#358	C₂×C₃₁	1	2	yes	—
45	#346	C₂×C₃₁	1	1	yes	characteristic
46	C₂×C₃₁	C₄×D₈	1	1	yes	characteristic
47	C₂×C₃₁	#510	1	2	yes	—
48	C₂×Q₁₆	C₂²×C₃₁	1	2	yes	—
49	C₄×Q₈	—	2	2	no	—
50	Q₃₂	C₄×C₃₁	1	4	yes	complemented
51	C₂×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
52	C₂×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
53	#107	C₂²×C₃₁	1	2	yes	—
54	#103	—	2	2	no	—
55	C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
56	C₃₁	#2176	1	1	yes	characteristic, direct product factor, Sylow
57	C₂×Q₈	—	2	2	no	—
58	Q₁₆	—	2	2	no	—
59	Q₁₆	C₂×C₄×C₃₁	1	4	yes	—
60	C₂×C₈	D₄×C₃₁	1	1	yes	characteristic
61	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
62	C₄⋊C₄	—	2	2	no	—
63	C₄⋊C₄	—	4	2	no	—
64	C₄²	—	4	2	no	—
65	C₄²	D₄×C₃₁	1	1	yes	characteristic
66	C₁₆	—	2	1	no	—
67	C₁₆	C₂×C₄×C₃₁	1	2	yes	—
68	Q₈	—	2	4	no	—
69	Q₈	—	4	2	no	—
70	C₂×C₄	—	4	2	no	—
71	C₂×C₄	—	4	2	no	—
72	C₂×C₄	D₈×C₃₁	1	2	yes	—
73	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
74	C₈	—	2	1	no	—
75	C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
76	C₈	#41×C₃₁	1	1	yes	characteristic
77	C₂²	C₂×D₈×C₃₁	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	4	4	no	—
80	C₄	—	8	2	no	—
81	C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
82	C₄	C₃₁×#135	1	1	yes	characteristic
83	C₂	C₄×D₈×C₃₁	1	1	yes	characteristic
84	C₂	C₃₁×#510	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²×C₃₁	C₂×D₈
commutator	C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₈	C₂³×C₃₁
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄²×C₃₁ ◁ C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₃₁×Q₃₂	C₄	semidirect
34. #2176	C₃₁	direct (N × H)
50. Q₃₂	C₄×C₃₁	semidirect
56. C₃₁	#2176	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×Q₃₂	2
C₄×Q₁₆×C₃₁	2
C₃₁×#367	2
C₃₁×#358	2
C₃₁×#346	2
#2176	31

Subgroups — C31×#2176 #16842109

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#2176 #16842109