Subgroups — C₂×C₃₁×#1312 #21468159

Order 7936.

350 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	C₂×C₄×C₁₆×C₃₁	C₂	1	2	yes	maximal
2	C₂×C₈²×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1312	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₁₆×C₃₁	C₄	1	4	yes	—
5	C₂×C₄×C₈×C₃₁	C₄	1	1	yes	characteristic
6	C₂×C₄×C₈×C₃₁	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₄×C₁₆×C₃₁	C₄	1	4	yes	—
9	C₄×C₁₆×C₃₁	C₂²	1	4	yes	—
10	C₈²×C₃₁	C₄	1	2	yes	—
11	C₈²×C₃₁	C₂²	1	2	yes	—
12	C₂²×C₈×C₃₁	C₈	1	2	yes	—
13	C₂²×C₈×C₃₁	C₈	1	2	yes	—
14	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
16	C₂×C₁₆×C₃₁	—	2	8	no	—
17	C₂×C₁₆×C₃₁	C₂×C₄	1	8	yes	—
18	C₄×C₈×C₃₁	C₈	1	2	yes	—
19	C₄×C₈×C₃₁	C₈	1	2	yes	—
20	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₃₁	C₄²	1	1	yes	characteristic
26	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
27	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
28	C₂×C₈×C₃₁	—	2	4	no	—
29	C₂×C₈×C₃₁	C₁₆	1	8	yes	complemented
30	C₂×C₈×C₃₁	C₄²	1	2	yes	—
31	C₂×C₈×C₃₁	C₂×C₈	1	4	yes	—
32	C₂×C₈×C₃₁	C₂×C₈	1	4	yes	—
33	C₂×C₈×C₃₁	M₁₆	1	8	yes	—
34	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
35	C₄²×C₃₁	C₄²	1	1	yes	characteristic
36	C₄²×C₃₁	C₂×C₈	1	1	yes	characteristic
37	C₄²×C₃₁	C₂×C₈	1	1	yes	characteristic
38	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
39	C₁₆×C₃₁	—	2	8	no	—
40	C₂×#1312	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₂³×C₃₁	C₄×C₈	1	1	yes	characteristic
42	C₂×C₄×C₃₁	C₄×C₈	1	1	yes	characteristic
43	C₂×C₄×C₃₁	C₄×C₈	1	1	yes	characteristic
44	C₂×C₄×C₃₁	C₄×C₈	1	1	yes	characteristic
45	C₂×C₄×C₃₁	#97	1	4	yes	—
46	C₂×C₄×C₃₁	C₂×C₁₆	1	4	yes	—
47	C₂×C₄×C₃₁	M₃₂	1	4	yes	—
48	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
49	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
50	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
51	C₈×C₃₁	—	2	4	no	—
52	C₈×C₃₁	#97	1	4	yes	—
53	C₈×C₃₁	C₂×C₁₆	1	8	yes	complemented
54	C₈×C₃₁	C₂×M₁₆	1	4	yes	—
55	C₂×C₄×C₁₆	C₂×C₃₁	1	2	yes	—
56	C₂×C₈²	C₂×C₃₁	1	1	yes	characteristic
57	#1312	C₂×C₃₁	1	4	yes	direct product factor
58	C₂²×C₃₁	#322	1	2	yes	—
59	C₂²×C₃₁	C₄×C₁₆	1	2	yes	—
60	C₂²×C₃₁	#346	1	2	yes	—
61	C₂²×C₃₁	C₂×C₄×C₈	1	1	yes	characteristic
62	C₄×C₃₁	#322	1	2	yes	—
63	C₄×C₃₁	C₄×C₁₆	1	2	yes	—
64	C₄×C₃₁	#346	1	2	yes	—
65	C₄×C₃₁	C₂×#97	1	2	yes	—
66	C₄×C₃₁	C₂²×C₁₆	1	2	yes	—
67	C₄×C₃₁	C₂×M₃₂	1	2	yes	—
68	C₂²×C₁₆	C₄×C₃₁	1	4	yes	—
69	C₂×C₄×C₈	C₄×C₃₁	1	1	yes	characteristic
70	C₂×C₄×C₈	C₄×C₃₁	1	1	yes	characteristic
71	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
72	C₄×C₁₆	C₄×C₃₁	1	4	yes	—
73	C₄×C₁₆	C₂²×C₃₁	1	4	yes	—
74	C₈²	C₄×C₃₁	1	2	yes	—
75	C₈²	C₂²×C₃₁	1	2	yes	—
76	C₂×C₃₁	#1312	1	4	yes	direct product factor
77	C₂×C₃₁	C₂×#322	1	1	yes	characteristic
78	C₂×C₃₁	C₂×C₄×C₁₆	1	1	yes	characteristic
79	C₂×C₃₁	C₂×#346	1	1	yes	characteristic
80	C₂²×C₈	C₈×C₃₁	1	2	yes	—
81	C₂²×C₈	C₈×C₃₁	1	2	yes	—
82	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
83	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
84	C₂×C₁₆	—	2	8	no	—
85	C₂×C₁₆	C₂×C₄×C₃₁	1	8	yes	—
86	C₄×C₈	C₈×C₃₁	1	2	yes	—
87	C₄×C₈	C₈×C₃₁	1	2	yes	—
88	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
89	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
90	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
91	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
92	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
93	C₃₁	C₂×#1312	1	1	yes	characteristic, direct product factor, Sylow
94	C₂²×C₄	C₄²×C₃₁	1	1	yes	characteristic
95	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
96	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
97	C₂×C₈	—	2	4	no	—
98	C₂×C₈	C₁₆×C₃₁	1	8	yes	complemented
99	C₂×C₈	C₄²×C₃₁	1	2	yes	—
100	C₂×C₈	C₂×C₈×C₃₁	1	4	yes	—
101	C₂×C₈	C₂×C₈×C₃₁	1	4	yes	—
102	C₂×C₈	M₁₆×C₃₁	1	8	yes	—
103	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
104	C₄²	C₄²×C₃₁	1	1	yes	characteristic
105	C₄²	C₂×C₈×C₃₁	1	1	yes	characteristic
106	C₄²	C₂×C₈×C₃₁	1	1	yes	characteristic
107	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
108	C₁₆	—	2	8	no	—
109	C₂³	C₄×C₈×C₃₁	1	1	yes	characteristic
110	C₂×C₄	C₄×C₈×C₃₁	1	1	yes	characteristic
111	C₂×C₄	C₄×C₈×C₃₁	1	1	yes	characteristic
112	C₂×C₄	C₄×C₈×C₃₁	1	1	yes	characteristic
113	C₂×C₄	C₃₁×#97	1	4	yes	—
114	C₂×C₄	C₂×C₁₆×C₃₁	1	4	yes	—
115	C₂×C₄	C₃₁×M₃₂	1	4	yes	—
116	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
117	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
118	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
119	C₈	—	2	4	no	—
120	C₈	C₃₁×#97	1	4	yes	—
121	C₈	C₂×C₁₆×C₃₁	1	8	yes	complemented
122	C₈	C₂×M₁₆×C₃₁	1	4	yes	—
123	C₂²	C₃₁×#322	1	2	yes	—
124	C₂²	C₄×C₁₆×C₃₁	1	2	yes	—
125	C₂²	C₃₁×#346	1	2	yes	—
126	C₂²	C₂×C₄×C₈×C₃₁	1	1	yes	characteristic
127	C₄	C₃₁×#322	1	2	yes	—
128	C₄	C₄×C₁₆×C₃₁	1	2	yes	—
129	C₄	C₃₁×#346	1	2	yes	—
130	C₄	C₂×C₃₁×#97	1	2	yes	—
131	C₄	C₂²×C₁₆×C₃₁	1	2	yes	—
132	C₄	C₂×C₃₁×M₃₂	1	2	yes	—
133	C₂	C₃₁×#1312	1	4	yes	direct product factor
134	C₂	C₂×C₃₁×#322	1	1	yes	characteristic
135	C₂	C₂×C₄×C₁₆×C₃₁	1	1	yes	characteristic
136	C₂	C₂×C₃₁×#346	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₈×C₃₁	C₂²
commutator	C₂	C₂×C₄×C₁₆×C₃₁
Frattini, cosocle	C₄×C₈	C₂³×C₃₁
socle, prime radical	C₂³×C₃₁	C₄×C₈
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
3. C₃₁×#1312	C₂	direct (N × H)
29. C₂×C₈×C₃₁	C₁₆	semidirect
40. C₂×#1312	C₃₁	direct (N × H)
53. C₈×C₃₁	C₂×C₁₆	semidirect
57. #1312	C₂×C₃₁	direct (N × H)
76. C₂×C₃₁	#1312	direct (N × H)
93. C₃₁	C₂×#1312	direct (N × H)
98. C₂×C₈	C₁₆×C₃₁	semidirect
121. C₈	C₂×C₁₆×C₃₁	semidirect
133. C₂	C₃₁×#1312	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×C₄×C₁₆×C₃₁	2
C₂×C₈²×C₃₁	2
C₃₁×#1312	2
C₂×#1312	31

Subgroups — C2×C31×#1312 #21468159

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1312 #21468159