Subgroups — C₃₁×#21668 #21478427

Order 7936.

702 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2401	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1924	C₂	1	3	yes	maximal
3	C₃₁×#1917	C₂	1	3	yes	maximal, complemented
4	C₃₁×#1840	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×C₄⋊C₄×C₃₁	C₂	1	3	yes	maximal, complemented
6	C₈×C₂²⋊C₄×C₃₁	C₂	1	3	yes	maximal
7	C₃₁×#1726	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×C₄⋊C₈	C₂²	1	3	yes	—
10	C₂×C₃₁×C₂²⋊C₈	C₂²	1	3	yes	—
11	C₂×C₃₁×#97	C₂²	1	3	yes	—
12	C₂×C₄×C₈×C₃₁	C₂²	1	3	yes	—
13	C₂×C₄×C₈×C₃₁	C₂²	1	3	yes	—
14	C₃₁×#401	C₄	1	1	yes	characteristic
15	C₃₁×#400	C₄	1	3	yes	—
16	C₃₁×#397	C₄	1	3	yes	—
17	C₃₁×#396	C₄	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₃₁	C₂²	1	3	yes	—
19	C₄×C₂²⋊C₄×C₃₁	C₂²	1	3	yes	—
20	C₃₁×#376	C₂²	1	1	yes	characteristic
21	C₃₁×#336	C₂²	1	3	yes	—
22	C₃₁×#336	C₂²	1	3	yes	—
23	C₃₁×#336	C₂²	1	3	yes	—
24	C₃₁×#336	C₂²	1	3	yes	—
25	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₃₁	—	2	6	no	—
27	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
28	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
30	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	3	yes	—
32	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	3	yes	—
33	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	3	yes	—
34	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	3	yes	—
35	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
36	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
37	C₃₁×C₄⋊C₈	—	2	6	no	—
38	C₃₁×C₂²⋊C₈	—	2	6	no	—
39	C₃₁×#97	—	2	6	no	—
40	C₄×C₈×C₃₁	—	2	6	no	—
41	C₄×C₈×C₃₁	—	2	6	no	—
42	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
43	C₃₁×#95	C₂×C₄	1	3	yes	—
44	C₃₁×#95	C₂×C₄	1	3	yes	—
45	C₃₁×#95	C₂×C₄	1	3	yes	—
46	C₃₁×#95	C₂×C₄	1	3	yes	—
47	C₃₁×#95	C₂×C₄	1	3	yes	—
48	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₃₁	—	2	3	no	—
50	C₂²×C₄×C₃₁	—	2	3	no	—
51	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₃₁	C₂²×C₄	1	3	yes	—
53	C₂²×C₄×C₃₁	C₂²×C₄	1	3	yes	—
54	C₂²×C₄×C₃₁	C₂²×C₄	1	3	yes	—
55	C₂²×C₄×C₃₁	C₂²×C₄	1	3	yes	—
56	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
57	C₂×C₈×C₃₁	—	2	3	no	—
58	C₂×C₈×C₃₁	—	2	3	no	—
59	C₂×C₈×C₃₁	—	2	3	no	—
60	C₂×C₈×C₃₁	—	2	3	no	—
61	C₂×C₈×C₃₁	—	2	3	no	—
62	C₂×C₈×C₃₁	—	2	3	no	—
63	C₂×C₈×C₃₁	—	4	4	no	—
64	C₂×C₈×C₃₁	#41	1	6	yes	—
65	C₂×C₈×C₃₁	#41	1	6	yes	—
66	C₄⋊C₄×C₃₁	—	2	6	no	—
67	C₄⋊C₄×C₃₁	—	2	6	no	—
68	C₂²⋊C₄×C₃₁	—	2	6	no	—
69	C₂²⋊C₄×C₃₁	—	2	6	no	—
70	C₄²×C₃₁	—	2	3	no	—
71	C₄²×C₃₁	—	2	3	no	—
72	C₄²×C₃₁	—	2	3	no	—
73	C₄²×C₃₁	—	2	3	no	—
74	#21668	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂³×C₃₁	—	2	3	no	—
76	C₂³×C₃₁	—	2	3	no	—
77	C₂³×C₃₁	#131	1	2	yes	—
78	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
79	C₂×C₄×C₃₁	—	2	3	no	—
80	C₂×C₄×C₃₁	—	2	3	no	—
81	C₂×C₄×C₃₁	—	2	3	no	—
82	C₂×C₄×C₃₁	—	2	3	no	—
83	C₂×C₄×C₃₁	—	2	3	no	—
84	C₂×C₄×C₃₁	—	2	3	no	—
85	C₂×C₄×C₃₁	—	2	3	no	—
86	C₂×C₄×C₃₁	—	2	3	no	—
87	C₂×C₄×C₃₁	—	2	3	no	—
88	C₂×C₄×C₃₁	—	2	3	no	—
89	C₂×C₄×C₃₁	—	2	3	no	—
90	C₂×C₄×C₃₁	—	2	3	no	—
91	C₂×C₄×C₃₁	—	4	1	no	—
92	C₂×C₄×C₃₁	—	4	3	no	—
93	C₂×C₄×C₃₁	#131	1	6	yes	—
94	C₂×C₄×C₃₁	C₂×#41	1	3	yes	—
95	C₂×C₄×C₃₁	C₂×#41	1	3	yes	—
96	C₈×C₃₁	—	2	6	no	—
97	C₈×C₃₁	—	4	1	no	—
98	C₈×C₃₁	—	4	3	no	—
99	C₈×C₃₁	#126	1	4	yes	—
100	#2401	C₂×C₃₁	1	1	yes	characteristic
101	#1924	C₂×C₃₁	1	3	yes	—
102	#1917	C₂×C₃₁	1	3	yes	complemented
103	#1840	C₂×C₃₁	1	1	yes	characteristic, complemented
104	C₈×C₄⋊C₄	C₂×C₃₁	1	3	yes	complemented
105	C₈×C₂²⋊C₄	C₂×C₃₁	1	3	yes	—
106	#1726	C₂×C₃₁	1	1	yes	characteristic, complemented
107	C₂²×C₃₁	—	2	6	no	—
108	C₂²×C₃₁	—	4	1	no	—
109	C₂²×C₃₁	—	4	3	no	—
110	C₂²×C₃₁	C₄×#41	1	3	yes	—
111	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
112	C₂²×C₃₁	C₂×#131	1	3	yes	—
113	C₄×C₃₁	—	2	6	no	—
114	C₄×C₃₁	—	4	1	no	—
115	C₄×C₃₁	—	4	3	no	—
116	C₄×C₃₁	—	4	3	no	—
117	C₄×C₃₁	—	4	3	no	—
118	C₄×C₃₁	C₂×#126	1	1	yes	characteristic
119	C₄×C₃₁	#529	1	3	yes	—
120	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
121	C₂×C₄⋊C₈	C₂²×C₃₁	1	3	yes	—
122	C₂×C₂²⋊C₈	C₂²×C₃₁	1	3	yes	—
123	C₂×#97	C₂²×C₃₁	1	3	yes	—
124	C₂×C₄×C₈	C₂²×C₃₁	1	3	yes	—
125	C₂×C₄×C₈	C₂²×C₃₁	1	3	yes	—
126	#401	C₄×C₃₁	1	1	yes	characteristic
127	#400	C₄×C₃₁	1	3	yes	—
128	#397	C₄×C₃₁	1	3	yes	—
129	#396	C₄×C₃₁	1	1	yes	characteristic
130	C₄×C₄⋊C₄	C₂²×C₃₁	1	3	yes	—
131	C₄×C₂²⋊C₄	C₂²×C₃₁	1	3	yes	—
132	#376	C₂²×C₃₁	1	1	yes	characteristic
133	#336	C₂²×C₃₁	1	3	yes	—
134	#336	C₂²×C₃₁	1	3	yes	—
135	#336	C₂²×C₃₁	1	3	yes	—
136	#336	C₂²×C₃₁	1	3	yes	—
137	C₂×C₃₁	—	4	2	no	—
138	C₂×C₃₁	C₄×#126	1	1	yes	characteristic
139	C₂×C₃₁	#2966	1	3	yes	—
140	C₂×C₃₁	#2966	1	3	yes	—
141	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
142	C₂²×C₈	—	2	6	no	—
143	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
144	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
145	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
146	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
147	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	3	yes	—
148	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	3	yes	—
149	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	3	yes	—
150	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	3	yes	—
151	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
152	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
153	C₄⋊C₈	—	2	6	no	—
154	C₂²⋊C₈	—	2	6	no	—
155	#97	—	2	6	no	—
156	C₄×C₈	—	2	6	no	—
157	C₄×C₈	—	2	6	no	—
158	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
159	#95	C₂×C₄×C₃₁	1	3	yes	—
160	#95	C₂×C₄×C₃₁	1	3	yes	—
161	#95	C₂×C₄×C₃₁	1	3	yes	—
162	#95	C₂×C₄×C₃₁	1	3	yes	—
163	#95	C₂×C₄×C₃₁	1	3	yes	—
164	C₃₁	#21668	1	1	yes	characteristic, direct product factor, Sylow
165	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
166	C₂²×C₄	—	2	3	no	—
167	C₂²×C₄	—	2	3	no	—
168	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
169	C₂²×C₄	C₂²×C₄×C₃₁	1	3	yes	—
170	C₂²×C₄	C₂²×C₄×C₃₁	1	3	yes	—
171	C₂²×C₄	C₂²×C₄×C₃₁	1	3	yes	—
172	C₂²×C₄	C₂²×C₄×C₃₁	1	3	yes	—
173	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
174	C₂×C₈	—	2	3	no	—
175	C₂×C₈	—	2	3	no	—
176	C₂×C₈	—	2	3	no	—
177	C₂×C₈	—	2	3	no	—
178	C₂×C₈	—	2	3	no	—
179	C₂×C₈	—	2	3	no	—
180	C₂×C₈	—	4	4	no	—
181	C₂×C₈	#41×C₃₁	1	6	yes	—
182	C₂×C₈	#41×C₃₁	1	6	yes	—
183	C₄⋊C₄	—	2	6	no	—
184	C₄⋊C₄	—	2	6	no	—
185	C₂²⋊C₄	—	2	6	no	—
186	C₂²⋊C₄	—	2	6	no	—
187	C₄²	—	2	3	no	—
188	C₄²	—	2	3	no	—
189	C₄²	—	2	3	no	—
190	C₄²	—	2	3	no	—
191	C₂³	—	2	3	no	—
192	C₂³	—	2	3	no	—
193	C₂³	C₃₁×#131	1	2	yes	—
194	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
195	C₂×C₄	—	2	3	no	—
196	C₂×C₄	—	2	3	no	—
197	C₂×C₄	—	2	3	no	—
198	C₂×C₄	—	2	3	no	—
199	C₂×C₄	—	2	3	no	—
200	C₂×C₄	—	2	3	no	—
201	C₂×C₄	—	2	3	no	—
202	C₂×C₄	—	2	3	no	—
203	C₂×C₄	—	2	3	no	—
204	C₂×C₄	—	2	3	no	—
205	C₂×C₄	—	2	3	no	—
206	C₂×C₄	—	2	3	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	—	4	3	no	—
209	C₂×C₄	C₃₁×#131	1	6	yes	—
210	C₂×C₄	C₂×#41×C₃₁	1	3	yes	—
211	C₂×C₄	C₂×#41×C₃₁	1	3	yes	—
212	C₈	—	2	6	no	—
213	C₈	—	4	1	no	—
214	C₈	—	4	3	no	—
215	C₈	C₃₁×#126	1	4	yes	—
216	C₂²	—	2	6	no	—
217	C₂²	—	4	3	no	—
218	C₂²	—	4	1	no	—
219	C₂²	C₄×#41×C₃₁	1	3	yes	—
220	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
221	C₂²	C₂×C₃₁×#131	1	3	yes	—
222	C₄	—	2	6	no	—
223	C₄	—	4	3	no	—
224	C₄	—	4	3	no	—
225	C₄	—	4	3	no	—
226	C₄	—	4	1	no	—
227	C₄	C₂×C₃₁×#126	1	1	yes	characteristic
228	C₄	C₃₁×#529	1	3	yes	—
229	C₂	—	4	2	no	—
230	C₂	C₄×C₃₁×#126	1	1	yes	characteristic
231	C₂	C₃₁×#2966	1	3	yes	—
232	C₂	C₃₁×#2966	1	3	yes	—

	subgroup	G/H
center	C₂²×C₈×C₃₁	C₂³
commutator	C₂³	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂³×C₄
prime radical	C₂⁴×C₃₁	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1917	C₂	semidirect
4. C₃₁×#1840	C₂	semidirect
5. C₈×C₄⋊C₄×C₃₁	C₂	semidirect
7. C₃₁×#1726	C₂	semidirect
74. #21668	C₃₁	direct (N × H)
102. #1917	C₂×C₃₁	semidirect
103. #1840	C₂×C₃₁	semidirect
104. C₈×C₄⋊C₄	C₂×C₃₁	semidirect
106. #1726	C₂×C₃₁	semidirect
164. C₃₁	#21668	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×#2401	2
C₃₁×#1924	2
C₃₁×#1917	2
C₃₁×#1840	2
C₈×C₄⋊C₄×C₃₁	2
C₈×C₂²⋊C₄×C₃₁	2
C₃₁×#1726	2
#21668	31

Subgroups — C31×#21668 #21478427

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21668 #21478427