Subgroups — C₃₁×#7841 #21464600

Order 7936.

574 conjugacy classes of subgroups.

The subgroup lattice has 118 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#1744	C₂	1	2	yes	maximal
2	C₄²×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#322	C₂	1	2	yes	maximal
4	C₂×C₈²×C₃₁	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₃₁	C₄	1	4	yes	—
6	C₂×C₄×C₈×C₃₁	C₂²	1	4	yes	—
7	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₄³×C₃₁	C₂²	1	1	yes	characteristic
10	C₃₁×#336	C₄	1	8	yes	—
11	C₃₁×#322	C₄	1	8	yes	complemented
12	C₈²×C₃₁	C₄	1	8	yes	complemented
13	C₂²×C₈×C₃₁	C₂×C₄	1	8	yes	—
14	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
15	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
16	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
17	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
19	C₄×C₈×C₃₁	C₈	1	16	yes	complemented
20	C₄×C₈×C₃₁	C₂×C₄	1	8	yes	—
21	C₄×C₈×C₃₁	C₂×C₄	1	8	yes	—
22	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
23	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₃₁	C₄²	1	4	yes	—
25	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
27	C₂×C₈×C₃₁	—	2	16	no	—
28	C₂×C₈×C₃₁	C₄²	1	16	yes	—
29	C₂×C₈×C₃₁	C₄²	1	4	yes	—
30	C₂×C₈×C₃₁	C₄²	1	4	yes	—
31	C₂×C₈×C₃₁	C₂×C₈	1	8	yes	—
32	C₂×C₈×C₃₁	C₂×C₈	1	8	yes	—
33	C₄²×C₃₁	—	2	8	no	—
34	C₄²×C₃₁	C₂×C₈	1	8	yes	—
35	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
36	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
37	#7841	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₂³×C₃₁	C₂×C₄²	1	1	yes	characteristic
39	C₂×C₄×C₃₁	—	2	8	no	—
40	C₂×C₄×C₃₁	C₄×C₈	1	8	yes	—
41	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
42	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
43	C₂×C₄×C₃₁	C₂×C₄²	1	4	yes	—
44	C₂×C₄×C₃₁	C₂²×C₈	1	4	yes	—
45	C₂×C₄×C₃₁	#131	1	8	yes	—
46	C₈×C₃₁	—	2	16	no	—
47	C₈×C₃₁	C₄×C₈	1	8	yes	—
48	C₈×C₃₁	C₄×C₈	1	8	yes	—
49	#1744	C₂×C₃₁	1	2	yes	—
50	C₄²×C₈	C₂×C₃₁	1	1	yes	characteristic
51	C₂×#322	C₂×C₃₁	1	2	yes	—
52	C₂×C₈²	C₂×C₃₁	1	2	yes	—
53	C₂²×C₃₁	C₄³	1	1	yes	characteristic
54	C₂²×C₃₁	C₂×C₄×C₈	1	2	yes	—
55	C₂²×C₃₁	#405	1	4	yes	—
56	C₄×C₃₁	—	2	8	no	—
57	C₄×C₃₁	C₂×C₄×C₈	1	2	yes	—
58	C₄×C₃₁	C₂×C₄×C₈	1	2	yes	—
59	C₄×C₃₁	#405	1	8	yes	—
60	C₂×C₄×C₈	C₄×C₃₁	1	4	yes	—
61	C₂×C₄×C₈	C₂²×C₃₁	1	4	yes	—
62	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
63	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
64	C₄³	C₂²×C₃₁	1	1	yes	characteristic
65	#336	C₄×C₃₁	1	8	yes	—
66	#322	C₄×C₃₁	1	8	yes	complemented
67	C₈²	C₄×C₃₁	1	8	yes	complemented
68	C₂×C₃₁	#1450	1	4	yes	—
69	C₂×C₃₁	C₄²×C₈	1	1	yes	characteristic
70	C₂×C₃₁	#1726	1	2	yes	—
71	C₂²×C₈	C₂×C₄×C₃₁	1	8	yes	—
72	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
73	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
74	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
75	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
76	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
77	C₄×C₈	C₈×C₃₁	1	16	yes	complemented
78	C₄×C₈	C₂×C₄×C₃₁	1	8	yes	—
79	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
80	C₄×C₈	C₂×C₄×C₃₁	1	8	yes	—
81	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
82	C₃₁	#7841	1	1	yes	characteristic, direct product factor, Sylow
83	C₂²×C₄	C₄²×C₃₁	1	4	yes	—
84	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
85	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
86	C₂×C₈	—	2	16	no	—
87	C₂×C₈	C₄²×C₃₁	1	16	yes	—
88	C₂×C₈	C₄²×C₃₁	1	4	yes	—
89	C₂×C₈	C₄²×C₃₁	1	4	yes	—
90	C₂×C₈	C₂×C₈×C₃₁	1	8	yes	—
91	C₂×C₈	C₂×C₈×C₃₁	1	8	yes	—
92	C₄²	—	2	8	no	—
93	C₄²	C₂×C₈×C₃₁	1	8	yes	—
94	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
95	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
96	C₂³	C₂×C₄²×C₃₁	1	1	yes	characteristic
97	C₂×C₄	—	2	8	no	—
98	C₂×C₄	C₄×C₈×C₃₁	1	8	yes	—
99	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
100	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
101	C₂×C₄	C₂×C₄²×C₃₁	1	4	yes	—
102	C₂×C₄	C₂²×C₈×C₃₁	1	4	yes	—
103	C₂×C₄	C₃₁×#131	1	8	yes	—
104	C₈	—	2	16	no	—
105	C₈	C₄×C₈×C₃₁	1	8	yes	—
106	C₈	C₄×C₈×C₃₁	1	8	yes	—
107	C₂²	C₄³×C₃₁	1	1	yes	characteristic
108	C₂²	C₂×C₄×C₈×C₃₁	1	2	yes	—
109	C₂²	C₃₁×#405	1	4	yes	—
110	C₄	—	2	8	no	—
111	C₄	C₂×C₄×C₈×C₃₁	1	2	yes	—
112	C₄	C₂×C₄×C₈×C₃₁	1	2	yes	—
113	C₄	C₃₁×#405	1	8	yes	—
114	C₂	C₃₁×#1450	1	4	yes	—
115	C₂	C₄²×C₈×C₃₁	1	1	yes	characteristic
116	C₂	C₃₁×#1726	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
11. C₃₁×#322	C₄	semidirect
12. C₈²×C₃₁	C₄	semidirect
19. C₄×C₈×C₃₁	C₈	semidirect
37. #7841	C₃₁	direct (N × H)
66. #322	C₄×C₃₁	semidirect
67. C₈²	C₄×C₃₁	semidirect
77. C₄×C₈	C₈×C₃₁	semidirect
82. C₃₁	#7841	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×#1744	2
C₄²×C₈×C₃₁	2
C₂×C₃₁×#322	2
C₂×C₈²×C₃₁	2
#7841	31

Subgroups — C31×#7841 #21464600

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#7841 #21464600