Subgroups — C₃₁×#1722 #16841655

Order 3968.

198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#500	C₂	1	2	yes	maximal, complemented
2	C₃₁×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#492	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#486	C₂	1	2	yes	maximal
5	C₃₁×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₁₆×C₃₁	—	2	2	no	—
7	C₂×SD₁₆×C₃₁	—	2	2	no	—
8	C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₃₁×#127	C₂²	1	1	yes	characteristic
11	C₃₁×#107	—	2	4	no	—
12	C₃₁×#102	—	2	4	no	—
13	C₄×C₈×C₃₁	C₂²	1	2	yes	—
14	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₃₁	—	2	1	no	—
16	C₂×Q₈×C₃₁	—	2	2	no	—
17	C₂×D₄×C₃₁	—	2	2	no	—
18	C₂×D₄×C₃₁	—	2	1	no	—
19	Q₁₆×C₃₁	—	4	4	no	—
20	SD₁₆×C₃₁	—	4	4	no	—
21	C₂×C₈×C₃₁	D₄	1	4	yes	—
22	C₂×C₈×C₃₁	D₄	1	2	yes	—
23	C₄⋊C₄×C₃₁	—	2	2	no	—
24	C₄⋊C₄×C₃₁	—	2	2	no	—
25	C₄⋊C₄×C₃₁	—	2	2	no	—
26	C₄²×C₃₁	C₂³	1	1	yes	characteristic
27	C₂³×C₃₁	—	4	1	no	—
28	Q₈×C₃₁	—	4	2	no	—
29	Q₈×C₃₁	—	4	4	no	—
30	D₄×C₃₁	—	4	2	no	—
31	D₄×C₃₁	—	4	4	no	—
32	C₂×C₄×C₃₁	—	4	2	no	—
33	C₂×C₄×C₃₁	—	4	1	no	—
34	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
36	C₈×C₃₁	—	2	4	no	—
37	C₈×C₃₁	D₈	1	4	yes	—
38	#1722	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₂²×C₃₁	—	8	1	no	—
40	C₂²×C₃₁	—	8	2	no	—
41	C₂²×C₃₁	#127	1	1	yes	characteristic
42	C₄×C₃₁	—	8	1	no	—
43	C₄×C₃₁	—	8	2	no	—
44	C₄×C₃₁	C₂×D₈	1	2	yes	—
45	C₄×C₃₁	#137	1	4	yes	—
46	#500	C₂×C₃₁	1	2	yes	complemented
47	#494	C₂×C₃₁	1	1	yes	characteristic, complemented
48	#492	C₂×C₃₁	1	1	yes	characteristic
49	#486	C₂×C₃₁	1	2	yes	—
50	#322	C₂×C₃₁	1	1	yes	characteristic, complemented
51	C₂×C₃₁	—	16	1	no	—
52	C₂×C₃₁	#493	1	1	yes	characteristic
53	C₂×C₃₁	#497	1	2	yes	—
54	C₂×Q₁₆	—	2	2	no	—
55	C₂×SD₁₆	—	2	2	no	—
56	C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
57	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
58	#127	C₂²×C₃₁	1	1	yes	characteristic
59	#107	—	2	4	no	—
60	#102	—	2	4	no	—
61	C₄×C₈	C₂²×C₃₁	1	2	yes	—
62	C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
63	C₃₁	#1722	1	1	yes	characteristic, direct product factor, Sylow
64	C₂×Q₈	—	2	1	no	—
65	C₂×Q₈	—	2	2	no	—
66	C₂×D₄	—	2	2	no	—
67	C₂×D₄	—	2	1	no	—
68	Q₁₆	—	4	4	no	—
69	SD₁₆	—	4	4	no	—
70	C₂×C₈	D₄×C₃₁	1	4	yes	—
71	C₂×C₈	D₄×C₃₁	1	2	yes	—
72	C₄⋊C₄	—	2	2	no	—
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	—	2	2	no	—
75	C₄²	C₂³×C₃₁	1	1	yes	characteristic
76	C₂³	—	4	1	no	—
77	Q₈	—	4	2	no	—
78	Q₈	—	4	4	no	—
79	D₄	—	4	2	no	—
80	D₄	—	4	4	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	—	4	1	no	—
83	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
84	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
85	C₈	—	2	4	no	—
86	C₈	D₈×C₃₁	1	4	yes	—
87	C₂²	—	8	1	no	—
88	C₂²	—	8	2	no	—
89	C₂²	C₃₁×#127	1	1	yes	characteristic
90	C₄	—	8	1	no	—
91	C₄	—	8	2	no	—
92	C₄	C₂×D₈×C₃₁	1	2	yes	—
93	C₄	C₃₁×#137	1	4	yes	—
94	C₂	—	16	1	no	—
95	C₂	C₃₁×#493	1	1	yes	characteristic
96	C₂	C₃₁×#497	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	#127
commutator, Frattini, cosocle	C₄²	C₂³×C₃₁
prime radical	C₃₁×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#500	C₂	semidirect
2. C₃₁×#494	C₂	semidirect
5. C₃₁×#322	C₂	semidirect
38. #1722	C₃₁	direct (N × H)
46. #500	C₂×C₃₁	semidirect
47. #494	C₂×C₃₁	semidirect
50. #322	C₂×C₃₁	semidirect
63. C₃₁	#1722	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×#500	2
C₃₁×#494	2
C₃₁×#492	2
C₃₁×#486	2
C₃₁×#322	2
#1722	31

Subgroups — C31×#1722 #16841655

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#1722 #16841655