Subgroups — C₃₁×#20365 #21477124

Order 7936.

1054 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆×C₃₁	C₂	1	2	yes	maximal
2	C₂×C₃₁×#376	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1746	C₂	1	8	yes	maximal, complemented
4	C₃₁×#1728	C₂	1	4	yes	maximal
5	C₂²×M₁₆×C₃₁	C₂²	1	4	yes	—
6	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
7	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
8	C₄×M₁₆×C₃₁	C₄	1	16	yes	complemented
9	C₂×C₃₁×#97	C₂²	1	4	yes	—
10	C₂×C₄×C₈×C₃₁	C₂²	1	4	yes	—
11	C₃₁×#376	C₄	1	4	yes	—
12	C₃₁×#376	C₂²	1	4	yes	—
13	C₂×C₃₁×#95	C₄	1	4	yes	—
14	C₃₁×#336	C₂²	1	16	yes	—
15	C₂³×C₄×C₃₁	C₂×C₄	1	4	yes	—
16	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
17	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
18	C₂×M₁₆×C₃₁	—	2	16	no	—
19	C₂×M₁₆×C₃₁	C₂×C₄	1	16	yes	—
20	C₂²×C₈×C₃₁	C₂³	1	8	yes	—
21	C₂×C₄²×C₃₁	—	2	8	no	—
22	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
23	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₃₁	C₂×C₄	1	8	yes	—
25	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
26	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
27	C₃₁×#97	C₂×C₄	1	16	yes	complemented
28	C₄×C₈×C₃₁	C₂×C₄	1	16	yes	complemented
29	C₃₁×#95	C₂×C₄	1	16	yes	—
30	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₃₁	—	2	8	no	—
32	C₂²×C₄×C₃₁	—	4	4	no	—
33	C₂²×C₄×C₃₁	—	4	8	no	—
34	C₂²×C₄×C₃₁	C₄²	1	8	yes	—
35	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
36	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
37	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
39	C₂²×C₄×C₃₁	#41	1	8	yes	—
40	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
41	M₁₆×C₃₁	—	2	32	no	—
42	C₂×C₈×C₃₁	—	2	16	no	—
43	C₂×C₈×C₃₁	C₂²×C₄	1	16	yes	—
44	C₄²×C₃₁	—	2	8	no	—
45	C₄²×C₃₁	—	2	8	no	—
46	C₄²×C₃₁	C₄²	1	8	yes	—
47	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
48	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
49	#20365	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂³×C₃₁	—	2	4	no	—
51	C₂³×C₃₁	C₂×C₄²	1	2	yes	—
52	C₂³×C₃₁	#117	1	4	yes	—
53	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
54	C₂×C₄×C₃₁	—	2	8	no	—
55	C₂×C₄×C₃₁	—	2	8	no	—
56	C₂×C₄×C₃₁	—	2	16	no	—
57	C₂×C₄×C₃₁	—	4	8	no	—
58	C₂×C₄×C₃₁	—	4	16	no	—
59	C₂×C₄×C₃₁	C₂×C₄²	1	8	yes	—
60	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
61	C₂×C₄×C₃₁	#117	1	16	yes	—
62	C₂×C₄×C₃₁	#117	1	4	yes	—
63	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
64	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
65	C₈×C₃₁	—	2	16	no	—
66	C₂×C₄×M₁₆	C₂×C₃₁	1	2	yes	—
67	C₂×#376	C₂×C₃₁	1	1	yes	characteristic
68	#1746	C₂×C₃₁	1	8	yes	complemented
69	#1728	C₂×C₃₁	1	4	yes	—
70	C₂²×C₃₁	—	2	4	no	—
71	C₂²×C₃₁	—	2	8	no	—
72	C₂²×C₃₁	#376	1	4	yes	—
73	C₂²×C₃₁	C₂²×C₄²	1	1	yes	characteristic
74	C₂²×C₃₁	C₂×#117	1	2	yes	—
75	C₂²×C₃₁	#568	1	4	yes	—
76	C₄×C₃₁	—	2	8	no	—
77	C₄×C₃₁	—	4	8	no	—
78	C₄×C₃₁	#376	1	4	yes	—
79	C₄×C₃₁	C₂×#117	1	4	yes	—
80	C₂²×M₁₆	C₂²×C₃₁	1	4	yes	—
81	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
82	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
83	C₄×M₁₆	C₄×C₃₁	1	16	yes	complemented
84	C₂×#97	C₂²×C₃₁	1	4	yes	—
85	C₂×C₄×C₈	C₂²×C₃₁	1	4	yes	—
86	#376	C₄×C₃₁	1	4	yes	—
87	#376	C₂²×C₃₁	1	4	yes	—
88	C₂×#95	C₄×C₃₁	1	4	yes	—
89	#336	C₂²×C₃₁	1	16	yes	—
90	C₂×C₃₁	—	2	4	no	—
91	C₂×C₃₁	C₂×#376	1	1	yes	characteristic
92	C₂×C₃₁	#2873	1	2	yes	—
93	C₂×C₃₁	#2921	1	4	yes	—
94	C₂³×C₄	C₂×C₄×C₃₁	1	4	yes	—
95	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
96	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
97	C₂×M₁₆	—	2	16	no	—
98	C₂×M₁₆	C₂×C₄×C₃₁	1	16	yes	—
99	C₂²×C₈	C₂³×C₃₁	1	8	yes	—
100	C₂×C₄²	—	2	8	no	—
101	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
102	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
103	C₂×C₄²	C₂×C₄×C₃₁	1	8	yes	—
104	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
105	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
106	#97	C₂×C₄×C₃₁	1	16	yes	complemented
107	C₄×C₈	C₂×C₄×C₃₁	1	16	yes	complemented
108	#95	C₂×C₄×C₃₁	1	16	yes	—
109	C₃₁	#20365	1	1	yes	characteristic, direct product factor, Sylow
110	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
111	C₂²×C₄	—	2	8	no	—
112	C₂²×C₄	—	4	4	no	—
113	C₂²×C₄	—	4	8	no	—
114	C₂²×C₄	C₄²×C₃₁	1	8	yes	—
115	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
116	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
117	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
118	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
119	C₂²×C₄	#41×C₃₁	1	8	yes	—
120	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
121	M₁₆	—	2	32	no	—
122	C₂×C₈	—	2	16	no	—
123	C₂×C₈	C₂²×C₄×C₃₁	1	16	yes	—
124	C₄²	—	2	8	no	—
125	C₄²	—	2	8	no	—
126	C₄²	C₄²×C₃₁	1	8	yes	—
127	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
128	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
129	C₂³	—	2	4	no	—
130	C₂³	C₂×C₄²×C₃₁	1	2	yes	—
131	C₂³	C₃₁×#117	1	4	yes	—
132	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
133	C₂×C₄	—	2	8	no	—
134	C₂×C₄	—	2	8	no	—
135	C₂×C₄	—	2	16	no	—
136	C₂×C₄	—	4	8	no	—
137	C₂×C₄	—	4	16	no	—
138	C₂×C₄	C₂×C₄²×C₃₁	1	8	yes	—
139	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
140	C₂×C₄	C₃₁×#117	1	16	yes	—
141	C₂×C₄	C₃₁×#117	1	4	yes	—
142	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
143	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
144	C₈	—	2	16	no	—
145	C₂²	—	2	4	no	—
146	C₂²	—	2	8	no	—
147	C₂²	C₃₁×#376	1	4	yes	—
148	C₂²	C₂²×C₄²×C₃₁	1	1	yes	characteristic
149	C₂²	C₂×C₃₁×#117	1	2	yes	—
150	C₂²	C₃₁×#568	1	4	yes	—
151	C₄	—	2	8	no	—
152	C₄	—	4	8	no	—
153	C₄	C₃₁×#376	1	4	yes	—
154	C₄	C₂×C₃₁×#117	1	4	yes	—
155	C₂	—	2	4	no	—
156	C₂	C₂×C₃₁×#376	1	1	yes	characteristic
157	C₂	C₃₁×#2873	1	2	yes	—
158	C₂	C₃₁×#2921	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator	C₂²	C₂²×C₄²×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂³×C₄
prime radical	C₂⁴×C₃₁	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1746	C₂	semidirect
8. C₄×M₁₆×C₃₁	C₄	semidirect
27. C₃₁×#97	C₂×C₄	semidirect
28. C₄×C₈×C₃₁	C₂×C₄	semidirect
49. #20365	C₃₁	direct (N × H)
68. #1746	C₂×C₃₁	semidirect
83. C₄×M₁₆	C₄×C₃₁	semidirect
106. #97	C₂×C₄×C₃₁	semidirect
107. C₄×C₈	C₂×C₄×C₃₁	semidirect
109. C₃₁	#20365	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×M₁₆×C₃₁	2
C₂×C₃₁×#376	2
C₃₁×#1746	2
C₃₁×#1728	2
#20365	31

Subgroups — C31×#20365 #21477124

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20365 #21477124