Subgroups — C₃₁×#33418 #21490177

Order 7936.

682 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2163	C₂	1	4	yes	maximal, complemented
3	C₃₁×#2162	C₂	1	4	yes	maximal, complemented
4	C₃₁×#2150	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2112	C₂	1	2	yes	maximal
6	C₄×C₃₁×M₃₂	C₂	1	2	yes	maximal, complemented
7	C₂³×C₈×C₃₁	C₄	1	1	yes	characteristic
8	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₃₁	C₄	1	1	yes	characteristic
10	C₂×C₃₁×M₃₂	C₂²	1	4	yes	—
11	C₂×C₄×C₈×C₃₁	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₃₁	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₃₁	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₃₁	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
19	C₃₁×C₄⋊C₁₆	C₂²	1	8	yes	complemented
20	C₃₁×C₂²⋊C₁₆	C₂²	1	8	yes	—
21	C₃₁×#346	C₂²	1	4	yes	complemented
22	C₄×C₁₆×C₃₁	C₂²	1	4	yes	complemented
23	C₂³×C₄×C₃₁	C₈	1	2	yes	—
24	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₃₁	—	2	4	no	—
26	C₂²×C₈×C₃₁	—	2	4	no	—
27	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
34	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₃₁	C₈	1	2	yes	—
36	C₂×C₄²×C₃₁	C₈	1	2	yes	—
37	C₂×C₄²×C₃₁	C₈	1	2	yes	—
38	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
42	C₃₁×M₃₂	—	2	8	no	—
43	C₂×C₁₆×C₃₁	C₂³	1	8	yes	—
44	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
50	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₄×C₈×C₃₁	C₂³	1	2	yes	—
53	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₂⁴×C₃₁	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₃₁	—	2	8	no	—
56	C₂²×C₄×C₃₁	—	2	2	no	—
57	C₂²×C₄×C₃₁	—	2	2	no	—
58	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	C₂×C₈	1	2	yes	—
60	C₂²×C₄×C₃₁	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₃₁	C₂×C₈	1	2	yes	—
62	C₂²×C₄×C₃₁	C₂×C₈	1	2	yes	—
63	C₂²×C₄×C₃₁	C₂×C₈	1	2	yes	—
64	C₂²×C₄×C₃₁	C₂×C₈	1	2	yes	—
65	C₂²×C₄×C₃₁	C₂×C₈	1	2	yes	—
66	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₃₁	—	2	2	no	—
69	C₂×C₈×C₃₁	—	2	2	no	—
70	C₂×C₈×C₃₁	—	2	4	no	—
71	C₂×C₈×C₃₁	—	2	4	no	—
72	C₂×C₈×C₃₁	—	2	4	no	—
73	C₂×C₈×C₃₁	—	2	4	no	—
74	C₂×C₈×C₃₁	M₁₆	1	4	yes	—
75	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₃₁	#41	1	4	yes	—
81	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
82	C₄²×C₃₁	C₂×C₈	1	2	yes	—
83	C₄²×C₃₁	C₂×C₈	1	2	yes	—
84	C₄²×C₃₁	C₂×C₈	1	2	yes	—
85	C₄²×C₃₁	C₂×C₈	1	2	yes	—
86	C₄²×C₃₁	C₂×C₈	1	1	yes	characteristic
87	C₄²×C₃₁	C₂×C₈	1	2	yes	—
88	C₄²×C₃₁	C₂×C₈	1	1	yes	characteristic
89	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
91	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
92	C₁₆×C₃₁	—	2	8	no	—
93	#33418	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₃₁	—	2	2	no	—
95	C₂³×C₃₁	—	2	4	no	—
96	C₂³×C₃₁	C₂²×C₈	1	1	yes	characteristic
97	C₂³×C₃₁	C₂²×C₈	1	2	yes	—
98	C₂×C₄×C₃₁	—	2	2	no	—
99	C₂×C₄×C₃₁	—	2	4	no	—
100	C₂×C₄×C₃₁	—	2	2	no	—
101	C₂×C₄×C₃₁	—	2	2	no	—
102	C₂×C₄×C₃₁	—	2	4	no	—
103	C₂×C₄×C₃₁	—	2	4	no	—
104	C₂×C₄×C₃₁	—	2	4	no	—
105	C₂×C₄×C₃₁	—	2	4	no	—
106	C₂×C₄×C₃₁	—	2	4	no	—
107	C₂×C₄×C₃₁	M₃₂	1	8	yes	complemented
108	C₂×C₄×C₃₁	#117	1	2	yes	—
109	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
110	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
111	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
112	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
114	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
115	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
116	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
117	C₈×C₃₁	—	2	2	no	—
118	C₈×C₃₁	—	2	2	no	—
119	C₈×C₃₁	#117	1	2	yes	—
120	C₈×C₃₁	C₂×M₁₆	1	2	yes	—
121	C₈×C₃₁	C₂×M₁₆	1	2	yes	—
122	C₈×C₃₁	C₂×#41	1	2	yes	—
123	C₂²×C₄×C₈	C₂×C₃₁	1	1	yes	characteristic
124	#2163	C₂×C₃₁	1	4	yes	complemented
125	#2162	C₂×C₃₁	1	4	yes	complemented
126	#2150	C₂×C₃₁	1	2	yes	complemented
127	#2112	C₂×C₃₁	1	2	yes	—
128	C₄×M₃₂	C₂×C₃₁	1	2	yes	complemented
129	C₂²×C₃₁	—	2	8	no	—
130	C₂²×C₃₁	—	2	2	no	—
131	C₂²×C₃₁	—	2	2	no	—
132	C₂²×C₃₁	—	2	2	no	—
133	C₂²×C₃₁	#431	1	2	yes	—
134	C₂²×C₃₁	C₂×M₃₂	1	4	yes	—
135	C₂²×C₃₁	C₂³×C₈	1	1	yes	characteristic
136	C₄×C₃₁	—	2	2	no	—
137	C₄×C₃₁	—	2	2	no	—
138	C₄×C₃₁	—	2	2	no	—
139	C₄×C₃₁	#431	1	2	yes	—
140	C₄×C₃₁	C₂×M₃₂	1	4	yes	—
141	C₄×C₃₁	C₂×M₃₂	1	4	yes	—
142	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
143	C₄×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
144	C₂³×C₈	C₄×C₃₁	1	1	yes	characteristic
145	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
146	C₂²×C₄²	C₄×C₃₁	1	1	yes	characteristic
147	C₂×M₃₂	C₂²×C₃₁	1	4	yes	—
148	C₂×C₄×C₈	C₄×C₃₁	1	1	yes	characteristic
149	C₂×C₄×C₈	C₄×C₃₁	1	2	yes	—
150	C₂×C₄×C₈	C₄×C₃₁	1	2	yes	—
151	C₂×C₄×C₈	C₄×C₃₁	1	1	yes	characteristic
152	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
154	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
155	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
156	C₄⋊C₁₆	C₂²×C₃₁	1	8	yes	complemented
157	C₂²⋊C₁₆	C₂²×C₃₁	1	8	yes	—
158	#346	C₂²×C₃₁	1	4	yes	complemented
159	C₄×C₁₆	C₂²×C₃₁	1	4	yes	complemented
160	C₂×C₃₁	—	2	2	no	—
161	C₂×C₃₁	—	2	4	no	—
162	C₂×C₃₁	C₂×#431	1	1	yes	characteristic
163	C₂×C₃₁	C₂²×M₃₂	1	2	yes	—
164	C₂³×C₄	C₈×C₃₁	1	2	yes	—
165	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
166	C₂²×C₈	—	2	4	no	—
167	C₂²×C₈	—	2	4	no	—
168	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
169	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
170	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
171	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
172	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
173	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
174	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
175	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
176	C₂×C₄²	C₈×C₃₁	1	2	yes	—
177	C₂×C₄²	C₈×C₃₁	1	2	yes	—
178	C₂×C₄²	C₈×C₃₁	1	2	yes	—
179	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
180	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
181	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
182	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
183	M₃₂	—	2	8	no	—
184	C₂×C₁₆	C₂³×C₃₁	1	8	yes	—
185	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
186	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
187	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
188	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
189	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
190	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
191	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
192	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
193	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
194	C₄×C₈	C₂³×C₃₁	1	2	yes	—
195	C₃₁	#33418	1	1	yes	characteristic, direct product factor, Sylow
196	C₂⁴	C₂×C₈×C₃₁	1	1	yes	characteristic
197	C₂²×C₄	—	2	8	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₈×C₃₁	1	2	yes	—
202	C₂²×C₄	C₂×C₈×C₃₁	1	2	yes	—
203	C₂²×C₄	C₂×C₈×C₃₁	1	2	yes	—
204	C₂²×C₄	C₂×C₈×C₃₁	1	2	yes	—
205	C₂²×C₄	C₂×C₈×C₃₁	1	2	yes	—
206	C₂²×C₄	C₂×C₈×C₃₁	1	2	yes	—
207	C₂²×C₄	C₂×C₈×C₃₁	1	2	yes	—
208	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
209	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	M₁₆×C₃₁	1	4	yes	—
217	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
218	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
219	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
220	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
221	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
222	C₂×C₈	#41×C₃₁	1	4	yes	—
223	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
224	C₄²	C₂×C₈×C₃₁	1	2	yes	—
225	C₄²	C₂×C₈×C₃₁	1	2	yes	—
226	C₄²	C₂×C₈×C₃₁	1	2	yes	—
227	C₄²	C₂×C₈×C₃₁	1	2	yes	—
228	C₄²	C₂×C₈×C₃₁	1	1	yes	characteristic
229	C₄²	C₂×C₈×C₃₁	1	2	yes	—
230	C₄²	C₂×C₈×C₃₁	1	1	yes	characteristic
231	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
232	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
233	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
234	C₁₆	—	2	8	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	2	4	no	—
237	C₂³	C₂²×C₈×C₃₁	1	1	yes	characteristic
238	C₂³	C₂²×C₈×C₃₁	1	2	yes	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	C₃₁×M₃₂	1	8	yes	complemented
249	C₂×C₄	C₃₁×#117	1	2	yes	—
250	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
251	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
252	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
253	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
254	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
255	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
256	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
257	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
258	C₈	—	2	2	no	—
259	C₈	—	2	2	no	—
260	C₈	C₃₁×#117	1	2	yes	—
261	C₈	C₂×M₁₆×C₃₁	1	2	yes	—
262	C₈	C₂×M₁₆×C₃₁	1	2	yes	—
263	C₈	C₂×#41×C₃₁	1	2	yes	—
264	C₂²	—	2	8	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	C₃₁×#431	1	2	yes	—
269	C₂²	C₂×C₃₁×M₃₂	1	4	yes	—
270	C₂²	C₂³×C₈×C₃₁	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	C₃₁×#431	1	2	yes	—
275	C₄	C₂×C₃₁×M₃₂	1	4	yes	—
276	C₄	C₂×C₃₁×M₃₂	1	4	yes	—
277	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
278	C₄	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
279	C₂	—	2	2	no	—
280	C₂	—	2	4	no	—
281	C₂	C₂×C₃₁×#431	1	1	yes	characteristic
282	C₂	C₂²×C₃₁×M₃₂	1	2	yes	—

	subgroup	G/H
center	C₄×C₈×C₃₁	C₂³
commutator	C₂²	C₂³×C₈×C₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂³×C₈
prime radical	C₂⁴×C₃₁	C₂×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
2. C₃₁×#2163	C₂	semidirect
3. C₃₁×#2162	C₂	semidirect
4. C₃₁×#2150	C₂	semidirect
6. C₄×C₃₁×M₃₂	C₂	semidirect
19. C₃₁×C₄⋊C₁₆	C₂²	semidirect
21. C₃₁×#346	C₂²	semidirect
22. C₄×C₁₆×C₃₁	C₂²	semidirect
93. #33418	C₃₁	direct (N × H)
107. C₂×C₄×C₃₁	M₃₂	semidirect
124. #2163	C₂×C₃₁	semidirect
125. #2162	C₂×C₃₁	semidirect
126. #2150	C₂×C₃₁	semidirect
128. C₄×M₃₂	C₂×C₃₁	semidirect
156. C₄⋊C₁₆	C₂²×C₃₁	semidirect
158. #346	C₂²×C₃₁	semidirect
159. C₄×C₁₆	C₂²×C₃₁	semidirect
195. C₃₁	#33418	direct (N × H)
248. C₂×C₄	C₃₁×M₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₈×C₃₁	2
C₃₁×#2163	2
C₃₁×#2162	2
C₃₁×#2150	2
C₃₁×#2112	2
C₄×C₃₁×M₃₂	2
#33418	31

Subgroups — C31×#33418 #21490177

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33418 #21490177