Subgroups — C₃₁×#8137 #21464896

Order 7936.

354 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#1726	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1450	C₂	1	2	yes	maximal
3	C₃₁×#1275	C₂	1	2	yes	maximal
4	C₃₁×#1274	C₂	1	2	yes	maximal
5	C₃₁×#431	C₂²	1	4	yes	—
6	C₂×C₃₁×#97	C₄	1	2	yes	—
7	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₃₁	C₄	1	2	yes	—
9	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₃₁×#376	C₂²	1	1	yes	characteristic
11	C₃₁×#322	C₄	1	4	yes	complemented
12	C₈²×C₃₁	C₄	1	4	yes	complemented
13	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
14	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
18	C₃₁×C₄⋊C₈	—	2	4	no	—
19	C₃₁×C₂²⋊C₈	—	2	4	no	—
20	C₃₁×#97	—	2	4	no	—
21	C₃₁×#97	—	2	2	no	—
22	C₄×C₈×C₃₁	C₈	1	8	yes	complemented
23	C₄×C₈×C₃₁	C₂×C₄	1	8	yes	—
24	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
26	C₃₁×#95	—	2	2	no	—
27	C₂²×C₄×C₃₁	—	2	2	no	—
28	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₃₁	#41	1	2	yes	—
30	C₂×C₈×C₃₁	—	2	2	no	—
31	C₂×C₈×C₃₁	—	2	2	no	—
32	C₂×C₈×C₃₁	—	2	8	no	—
33	C₂×C₈×C₃₁	C₄²	1	1	yes	characteristic
34	C₂×C₈×C₃₁	C₄²	1	2	yes	—
35	C₂×C₈×C₃₁	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₃₁	C₂×C₈	1	4	yes	—
37	C₂×C₈×C₃₁	C₂×C₈	1	4	yes	—
38	C₂×C₈×C₃₁	M₁₆	1	4	yes	—
39	C₄²×C₃₁	—	2	2	no	—
40	C₄²×C₃₁	C₂×C₈	1	4	yes	—
41	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
42	C₄²×C₃₁	#41	1	2	yes	—
43	#8137	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂³×C₃₁	#117	1	1	yes	characteristic
45	C₂×C₄×C₃₁	—	2	4	no	—
46	C₂×C₄×C₃₁	—	4	2	no	—
47	C₂×C₄×C₃₁	—	4	4	no	—
48	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
49	C₂×C₄×C₃₁	#117	1	4	yes	—
50	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
51	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
52	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
53	C₈×C₃₁	—	2	4	no	—
54	C₈×C₃₁	—	4	8	no	—
55	C₈×C₃₁	C₄×C₈	1	2	yes	—
56	C₈×C₃₁	C₄×C₈	1	2	yes	—
57	C₈×C₃₁	#97	1	4	yes	—
58	#1726	C₂×C₃₁	1	1	yes	characteristic
59	#1450	C₂×C₃₁	1	2	yes	—
60	#1275	C₂×C₃₁	1	2	yes	—
61	#1274	C₂×C₃₁	1	2	yes	—
62	C₂²×C₃₁	—	2	2	no	—
63	C₂²×C₃₁	#376	1	1	yes	characteristic
64	C₂²×C₃₁	#431	1	2	yes	—
65	C₄×C₃₁	—	2	4	no	—
66	C₄×C₃₁	—	4	4	no	—
67	C₄×C₃₁	C₂×C₄×C₈	1	1	yes	characteristic
68	C₄×C₃₁	C₂×#97	1	1	yes	characteristic
69	C₄×C₃₁	#431	1	2	yes	—
70	#431	C₂²×C₃₁	1	4	yes	—
71	C₂×#97	C₄×C₃₁	1	2	yes	—
72	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
73	C₂×C₄×C₈	C₄×C₃₁	1	2	yes	—
74	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
75	#376	C₂²×C₃₁	1	1	yes	characteristic
76	#322	C₄×C₃₁	1	4	yes	complemented
77	C₈²	C₄×C₃₁	1	4	yes	complemented
78	C₂×C₃₁	—	2	2	no	—
79	C₂×C₃₁	#1744	1	1	yes	characteristic
80	C₂×C₃₁	#1747	1	2	yes	—
81	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
82	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
83	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
84	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
85	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
86	C₄⋊C₈	—	2	4	no	—
87	C₂²⋊C₈	—	2	4	no	—
88	#97	—	2	4	no	—
89	#97	—	2	2	no	—
90	C₄×C₈	C₈×C₃₁	1	8	yes	complemented
91	C₄×C₈	C₂×C₄×C₃₁	1	8	yes	—
92	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
93	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
94	#95	—	2	2	no	—
95	C₃₁	#8137	1	1	yes	characteristic, direct product factor, Sylow
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
98	C₂²×C₄	#41×C₃₁	1	2	yes	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	8	no	—
102	C₂×C₈	C₄²×C₃₁	1	1	yes	characteristic
103	C₂×C₈	C₄²×C₃₁	1	1	yes	characteristic
104	C₂×C₈	C₄²×C₃₁	1	2	yes	—
105	C₂×C₈	C₂×C₈×C₃₁	1	4	yes	—
106	C₂×C₈	C₂×C₈×C₃₁	1	4	yes	—
107	C₂×C₈	M₁₆×C₃₁	1	4	yes	—
108	C₄²	—	2	2	no	—
109	C₄²	C₂×C₈×C₃₁	1	4	yes	—
110	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
111	C₄²	#41×C₃₁	1	2	yes	—
112	C₂³	C₃₁×#117	1	1	yes	characteristic
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	4	4	no	—
116	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
117	C₂×C₄	C₃₁×#117	1	4	yes	—
118	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
119	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
120	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
121	C₈	—	2	4	no	—
122	C₈	—	4	8	no	—
123	C₈	C₄×C₈×C₃₁	1	2	yes	—
124	C₈	C₄×C₈×C₃₁	1	2	yes	—
125	C₈	C₃₁×#97	1	4	yes	—
126	C₂²	—	2	2	no	—
127	C₂²	C₃₁×#376	1	1	yes	characteristic
128	C₂²	C₃₁×#431	1	2	yes	—
129	C₄	—	2	4	no	—
130	C₄	—	4	4	no	—
131	C₄	C₂×C₄×C₈×C₃₁	1	1	yes	characteristic
132	C₄	C₂×C₃₁×#97	1	1	yes	characteristic
133	C₄	C₃₁×#431	1	2	yes	—
134	C₂	—	2	2	no	—
135	C₂	C₃₁×#1744	1	1	yes	characteristic
136	C₂	C₃₁×#1747	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₃₁	C₄²
commutator	C₄	C₂×C₄×C₈×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	#376
prime radical	C₂³×C₃₁	#117
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×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
22. C₄×C₈×C₃₁	C₈	semidirect
43. #8137	C₃₁	direct (N × H)
76. #322	C₄×C₃₁	semidirect
77. C₈²	C₄×C₃₁	semidirect
90. C₄×C₈	C₈×C₃₁	semidirect
95. C₃₁	#8137	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×#1726	2
C₃₁×#1450	2
C₃₁×#1275	2
C₃₁×#1274	2
#8137	31

Subgroups — C31×#8137 #21464896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8137 #21464896