Subgroups — C₃₁×#7472 #21464231

Order 7936.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×C₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#368	C₂	1	2	yes	maximal
3	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
4	C₂×C₃₁×#108	—	2	2	no	—
5	C₂×C₃₁×C₃₂	—	2	2	no	—
6	C₂×C₃₁×C₃₂	C₄	1	2	yes	—
7	C₃₁×#368	—	2	2	no	—
8	C₃₁×#368	C₄	1	4	yes	—
9	C₂×M₁₆×C₃₁	—	4	2	no	—
10	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
11	C₂×C₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
12	C₂×C₁₆×C₃₁	C₂×C₄	1	2	yes	—
13	C₂×C₁₆×C₃₁	D₄	1	2	yes	—
14	C₂×C₁₆×C₃₁	Q₈	1	1	yes	characteristic
15	C₃₁×#108	—	2	4	no	—
16	C₃₁×#108	—	4	2	no	—
17	C₃₁×C₃₂	—	2	2	no	—
18	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
19	M₁₆×C₃₁	—	4	4	no	—
20	M₁₆×C₃₁	—	8	2	no	—
21	C₂×C₈×C₃₁	—	8	2	no	—
22	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
23	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
24	C₂×C₈×C₃₁	SD₁₆	1	2	yes	—
25	C₂×C₈×C₃₁	Q₁₆	1	1	yes	characteristic
26	C₁₆×C₃₁	C₄²	1	1	yes	characteristic
27	C₁₆×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
28	C₁₆×C₃₁	C₄⋊C₄	1	2	yes	—
29	#7472	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂³×C₃₁	Q₃₂	1	1	yes	characteristic
31	C₂×C₄×C₃₁	#102	1	2	yes	—
32	C₂×C₄×C₃₁	#107	1	1	yes	characteristic
33	C₂×C₄×C₃₁	D₁₆	1	1	yes	characteristic
34	C₂×C₄×C₃₁	SD₃₂	1	2	yes	—
35	C₈×C₃₁	—	8	4	no	—
36	C₈×C₃₁	#95	1	1	yes	characteristic
37	C₈×C₃₁	#103	1	2	yes	—
38	C₈×C₃₁	#106	1	1	yes	characteristic
39	C₂²×C₃₂	C₂×C₃₁	1	1	yes	characteristic
40	C₂×#368	C₂×C₃₁	1	2	yes	—
41	C₂²×C₃₁	—	2	2	no	—
42	C₂²×C₃₁	#358	1	2	yes	—
43	C₂²×C₃₁	#366	1	1	yes	characteristic
44	C₄×C₃₁	#340	1	1	yes	characteristic
45	C₄×C₃₁	#357	1	2	yes	—
46	C₄×C₃₁	#367	1	1	yes	characteristic
47	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
48	C₂×#108	—	2	2	no	—
49	C₂×C₃₂	—	2	2	no	—
50	C₂×C₃₂	C₄×C₃₁	1	2	yes	—
51	#368	—	2	2	no	—
52	#368	C₄×C₃₁	1	4	yes	—
53	C₂×C₃₁	—	2	2	no	—
54	C₂×C₃₁	#1380	1	1	yes	characteristic
55	C₂×C₃₁	#1425	1	2	yes	—
56	C₂×M₁₆	—	4	2	no	—
57	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
58	C₂×C₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
59	C₂×C₁₆	C₂×C₄×C₃₁	1	2	yes	—
60	C₂×C₁₆	D₄×C₃₁	1	2	yes	—
61	C₂×C₁₆	Q₈×C₃₁	1	1	yes	characteristic
62	#108	—	2	4	no	—
63	#108	—	4	2	no	—
64	C₃₂	—	2	2	no	—
65	C₃₁	#7472	1	1	yes	characteristic, direct product factor, Sylow
66	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
67	M₁₆	—	4	4	no	—
68	M₁₆	—	8	2	no	—
69	C₂×C₈	—	8	2	no	—
70	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
71	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
72	C₂×C₈	SD₁₆×C₃₁	1	2	yes	—
73	C₂×C₈	Q₁₆×C₃₁	1	1	yes	characteristic
74	C₁₆	C₄²×C₃₁	1	1	yes	characteristic
75	C₁₆	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
76	C₁₆	C₄⋊C₄×C₃₁	1	2	yes	—
77	C₂³	C₃₁×Q₃₂	1	1	yes	characteristic
78	C₂×C₄	C₃₁×#102	1	2	yes	—
79	C₂×C₄	C₃₁×#107	1	1	yes	characteristic
80	C₂×C₄	C₃₁×D₁₆	1	1	yes	characteristic
81	C₂×C₄	C₃₁×SD₃₂	1	2	yes	—
82	C₈	—	8	4	no	—
83	C₈	C₃₁×#95	1	1	yes	characteristic
84	C₈	C₃₁×#103	1	2	yes	—
85	C₈	C₃₁×#106	1	1	yes	characteristic
86	C₂²	—	2	2	no	—
87	C₂²	C₃₁×#358	1	2	yes	—
88	C₂²	C₃₁×#366	1	1	yes	characteristic
89	C₄	C₃₁×#340	1	1	yes	characteristic
90	C₄	C₃₁×#357	1	2	yes	—
91	C₄	C₃₁×#367	1	1	yes	characteristic
92	C₂	—	2	2	no	—
93	C₂	C₃₁×#1380	1	1	yes	characteristic
94	C₂	C₃₁×#1425	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	D₁₆
commutator	C₁₆	C₄²×C₃₁
Frattini, cosocle	C₂²×C₁₆	C₂²×C₃₁
socle	C₂²×C₃₁	#366
prime radical	C₂³×C₃₁	Q₃₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₂²×C₈×C₃₁ ◁ C₂²×C₁₆×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
29. #7472	C₃₁	direct (N × H)
65. C₃₁	#7472	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×C₃₂	2
C₂×C₃₁×#368	2
#7472	31

Subgroups — C31×#7472 #21464231

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#7472 #21464231