Subgroups — C₃₁×#11409 #21468168

Order 7936.

284 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2162	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₁×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₈×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1312	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1312	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1311	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×M₃₂	C₄	1	2	yes	—
9	C₃₁×#431	C₄	1	1	yes	characteristic
10	C₄×M₁₆×C₃₁	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₃₁×C₄⋊C₁₆	C₄	1	2	yes	—
13	C₃₁×C₂²⋊C₁₆	C₄	1	2	yes	—
14	C₃₁×#346	C₄	1	2	yes	—
15	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
16	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
17	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
18	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#322	C₄	1	1	yes	characteristic
20	C₃₁×#322	C₄	1	1	yes	characteristic
21	C₈²×C₃₁	C₂²	1	1	yes	characteristic
22	C₈²×C₃₁	C₂²	1	1	yes	characteristic
23	C₂×M₁₆×C₃₁	C₈	1	2	yes	—
24	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₃₁×M₃₂	—	2	4	no	—
28	C₂×C₁₆×C₃₁	C₂×C₄	1	2	yes	—
29	C₂×C₁₆×C₃₁	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₃₁	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₃₁	C₂×C₄	1	2	yes	—
32	C₃₁×C₄⋊C₈	C₈	1	2	yes	—
33	C₃₁×C₂²⋊C₈	C₈	1	2	yes	—
34	C₃₁×#97	C₈	1	2	yes	—
35	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₃₁	C₄²	1	1	yes	characteristic
44	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
46	M₁₆×C₃₁	—	2	4	no	—
47	C₂×C₈×C₃₁	—	2	2	no	—
48	C₂×C₈×C₃₁	—	2	2	no	—
49	C₂×C₈×C₃₁	C₄²	1	1	yes	characteristic
50	C₂×C₈×C₃₁	C₄²	1	1	yes	characteristic
51	C₂×C₈×C₃₁	C₂×C₈	1	2	yes	—
52	C₂×C₈×C₃₁	C₂×C₈	1	2	yes	—
53	C₂×C₈×C₃₁	C₂×C₈	1	2	yes	—
54	C₂×C₈×C₃₁	C₂×C₈	1	2	yes	—
55	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₃₁	C₄²	1	1	yes	characteristic
58	C₄²×C₃₁	C₂×C₈	1	1	yes	characteristic
59	C₄²×C₃₁	C₂×C₈	1	1	yes	characteristic
60	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
61	C₁₆×C₃₁	—	2	2	no	—
62	C₁₆×C₃₁	—	2	2	no	—
63	C₁₆×C₃₁	—	2	2	no	—
64	C₁₆×C₃₁	M₁₆	1	4	yes	complemented
65	#11409	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂³×C₃₁	C₄×C₈	1	1	yes	characteristic
67	C₂×C₄×C₃₁	—	2	1	no	—
68	C₂×C₄×C₃₁	—	2	1	no	—
69	C₂×C₄×C₃₁	—	2	2	no	—
70	C₂×C₄×C₃₁	—	2	2	no	—
71	C₂×C₄×C₃₁	C₄×C₈	1	1	yes	characteristic
72	C₂×C₄×C₃₁	C₄×C₈	1	1	yes	characteristic
73	C₂×C₄×C₃₁	C₄×C₈	1	1	yes	characteristic
74	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
75	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
77	C₈×C₃₁	—	2	1	no	—
78	C₈×C₃₁	—	2	1	no	—
79	C₈×C₃₁	—	2	2	no	—
80	C₈×C₃₁	—	2	2	no	—
81	C₈×C₃₁	—	2	2	no	—
82	C₈×C₃₁	M₃₂	1	4	yes	complemented
83	C₈×C₃₁	C₂×M₁₆	1	2	yes	—
84	C₈×C₃₁	#131	1	2	yes	—
85	#2162	C₂×C₃₁	1	1	yes	characteristic
86	C₄×M₃₂	C₂×C₃₁	1	1	yes	characteristic
87	C₈×M₁₆	C₂×C₃₁	1	1	yes	characteristic
88	#1312	C₂×C₃₁	1	1	yes	characteristic, complemented
89	#1312	C₂×C₃₁	1	1	yes	characteristic, complemented
90	#1311	C₂×C₃₁	1	1	yes	characteristic, complemented
91	C₈×C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
92	C₂²×C₃₁	—	2	1	no	—
93	C₂²×C₃₁	—	2	1	no	—
94	C₂²×C₃₁	—	2	1	no	—
95	C₂²×C₃₁	C₂×C₄×C₈	1	1	yes	characteristic
96	C₄×C₃₁	—	2	1	no	—
97	C₄×C₃₁	—	2	1	no	—
98	C₄×C₃₁	—	2	1	no	—
99	C₄×C₃₁	C₄×M₁₆	1	1	yes	characteristic
100	C₄×C₃₁	#405	1	1	yes	characteristic
101	C₄×C₃₁	C₂×M₃₂	1	2	yes	—
102	C₄×C₃₁	#504	1	2	yes	—
103	C₂×M₃₂	C₄×C₃₁	1	2	yes	—
104	#431	C₄×C₃₁	1	1	yes	characteristic
105	C₄×M₁₆	C₄×C₃₁	1	1	yes	characteristic
106	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
107	C₄⋊C₁₆	C₄×C₃₁	1	2	yes	—
108	C₂²⋊C₁₆	C₄×C₃₁	1	2	yes	—
109	#346	C₄×C₃₁	1	2	yes	—
110	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
111	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
112	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
113	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
114	#322	C₄×C₃₁	1	1	yes	characteristic
115	#322	C₄×C₃₁	1	1	yes	characteristic
116	C₈²	C₂²×C₃₁	1	1	yes	characteristic
117	C₈²	C₂²×C₃₁	1	1	yes	characteristic
118	C₂×C₃₁	—	4	1	no	—
119	C₂×C₃₁	C₈×M₁₆	1	1	yes	characteristic
120	C₂×C₃₁	C₄×M₃₂	1	1	yes	characteristic
121	C₂×C₃₁	#2108	1	1	yes	characteristic
122	C₂×M₁₆	C₈×C₃₁	1	2	yes	—
123	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
124	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
125	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
126	M₃₂	—	2	4	no	—
127	C₂×C₁₆	C₂×C₄×C₃₁	1	2	yes	—
128	C₂×C₁₆	C₂×C₄×C₃₁	1	2	yes	—
129	C₂×C₁₆	C₂×C₄×C₃₁	1	2	yes	—
130	C₂×C₁₆	C₂×C₄×C₃₁	1	2	yes	—
131	C₄⋊C₈	C₈×C₃₁	1	2	yes	—
132	C₂²⋊C₈	C₈×C₃₁	1	2	yes	—
133	#97	C₈×C₃₁	1	2	yes	—
134	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
135	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
136	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
137	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
138	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
139	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
140	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
141	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
142	C₃₁	#11409	1	1	yes	characteristic, direct product factor, Sylow
143	C₂²×C₄	C₄²×C₃₁	1	1	yes	characteristic
144	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
146	M₁₆	—	2	4	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	C₄²×C₃₁	1	1	yes	characteristic
150	C₂×C₈	C₄²×C₃₁	1	1	yes	characteristic
151	C₂×C₈	C₂×C₈×C₃₁	1	2	yes	—
152	C₂×C₈	C₂×C₈×C₃₁	1	2	yes	—
153	C₂×C₈	C₂×C₈×C₃₁	1	2	yes	—
154	C₂×C₈	C₂×C₈×C₃₁	1	2	yes	—
155	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
156	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
157	C₄²	C₄²×C₃₁	1	1	yes	characteristic
158	C₄²	C₂×C₈×C₃₁	1	1	yes	characteristic
159	C₄²	C₂×C₈×C₃₁	1	1	yes	characteristic
160	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
161	C₁₆	—	2	2	no	—
162	C₁₆	—	2	2	no	—
163	C₁₆	—	2	2	no	—
164	C₁₆	M₁₆×C₃₁	1	4	yes	complemented
165	C₂³	C₄×C₈×C₃₁	1	1	yes	characteristic
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	C₄×C₈×C₃₁	1	1	yes	characteristic
171	C₂×C₄	C₄×C₈×C₃₁	1	1	yes	characteristic
172	C₂×C₄	C₄×C₈×C₃₁	1	1	yes	characteristic
173	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
174	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
175	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
176	C₈	—	2	1	no	—
177	C₈	—	2	1	no	—
178	C₈	—	2	2	no	—
179	C₈	—	2	2	no	—
180	C₈	—	2	2	no	—
181	C₈	C₃₁×M₃₂	1	4	yes	complemented
182	C₈	C₂×M₁₆×C₃₁	1	2	yes	—
183	C₈	C₃₁×#131	1	2	yes	—
184	C₂²	—	2	1	no	—
185	C₂²	—	2	1	no	—
186	C₂²	—	2	1	no	—
187	C₂²	C₂×C₄×C₈×C₃₁	1	1	yes	characteristic
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	—	2	1	no	—
191	C₄	C₄×M₁₆×C₃₁	1	1	yes	characteristic
192	C₄	C₃₁×#405	1	1	yes	characteristic
193	C₄	C₂×C₃₁×M₃₂	1	2	yes	—
194	C₄	C₃₁×#504	1	2	yes	—
195	C₂	—	4	1	no	—
196	C₂	C₈×M₁₆×C₃₁	1	1	yes	characteristic
197	C₂	C₄×C₃₁×M₃₂	1	1	yes	characteristic
198	C₂	C₃₁×#2108	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1312	C₂	semidirect
5. C₃₁×#1312	C₂	semidirect
6. C₃₁×#1311	C₂	semidirect
7. C₈×C₁₆×C₃₁	C₂	semidirect
64. C₁₆×C₃₁	M₁₆	semidirect
65. #11409	C₃₁	direct (N × H)
82. C₈×C₃₁	M₃₂	semidirect
88. #1312	C₂×C₃₁	semidirect
89. #1312	C₂×C₃₁	semidirect
90. #1311	C₂×C₃₁	semidirect
91. C₈×C₁₆	C₂×C₃₁	semidirect
142. C₃₁	#11409	direct (N × H)
164. C₁₆	M₁₆×C₃₁	semidirect
181. C₈	C₃₁×M₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#2162	2
C₄×C₃₁×M₃₂	2
C₈×M₁₆×C₃₁	2
C₃₁×#1312	2
C₃₁×#1311	2
C₈×C₁₆×C₃₁	2
#11409	31

Subgroups — C31×#11409 #21468168

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11409 #21468168