Subgroups — C₃₁×#21643 #21478402

Order 7936.

818 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#488	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2330	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2326	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1992	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1959	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1936	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1935	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1934	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#1933	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1834	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1763	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#1762	C₂	1	1	yes	maximal, characteristic
13	C₄×C₃₁×#103	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×C₃₁×#102	C₂	1	1	yes	maximal, characteristic
15	C₃₁×#1726	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
20	C₃₁×#488	C₄	1	8	yes	complemented
21	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
35	C₃₁×#391	C₂²	1	1	yes	characteristic
36	C₃₁×#390	C₂²	1	1	yes	characteristic
37	C₃₁×#389	—	2	1	no	—
38	C₃₁×#389	—	2	1	no	—
39	C₃₁×#389	C₂²	1	1	yes	characteristic
40	C₃₁×#389	C₂²	1	1	yes	characteristic
41	C₃₁×#388	—	2	1	no	—
42	C₃₁×#388	—	2	1	no	—
43	C₃₁×#387	—	2	1	no	—
44	C₃₁×#387	—	2	1	no	—
45	C₃₁×#385	—	2	1	no	—
46	C₃₁×#385	—	2	1	no	—
47	C₃₁×#384	C₂²	1	1	yes	characteristic
48	C₃₁×#384	C₂²	1	1	yes	characteristic
49	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
50	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
51	C₃₁×#376	C₂²	1	1	yes	characteristic
52	C₃₁×#340	C₂²	1	1	yes	characteristic
53	C₃₁×#340	C₂²	1	1	yes	characteristic
54	C₃₁×#340	C₂²	1	1	yes	characteristic
55	C₃₁×#340	C₂²	1	1	yes	characteristic
56	C₃₁×#340	C₂²	1	1	yes	characteristic
57	C₃₁×#340	C₂²	1	1	yes	characteristic
58	C₃₁×#340	C₂²	1	1	yes	characteristic
59	C₃₁×#340	C₂²	1	1	yes	characteristic
60	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
61	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₃₁	—	2	1	no	—
63	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
67	C₃₁×#125	—	2	2	no	—
68	C₃₁×#125	C₂×C₄	1	4	yes	complemented
69	C₃₁×#124	—	2	2	no	—
70	C₃₁×#124	C₂×C₄	1	4	yes	—
71	C₄×Q₈×C₃₁	—	2	2	no	—
72	C₄×Q₈×C₃₁	—	2	2	no	—
73	C₄×D₄×C₃₁	—	2	2	no	—
74	C₄×D₄×C₃₁	—	2	2	no	—
75	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
76	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
77	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
78	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
79	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
80	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
81	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
82	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
83	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
84	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
89	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
90	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
91	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
92	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
93	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
94	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
95	C₂×C₄²×C₃₁	—	2	1	no	—
96	C₂×C₄²×C₃₁	—	2	1	no	—
97	C₂×C₄²×C₃₁	—	2	1	no	—
98	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
99	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
100	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
101	C₃₁×#103	—	2	2	no	—
102	C₃₁×#103	—	2	2	no	—
103	C₃₁×#103	C₂×C₄	1	4	yes	complemented
104	C₃₁×#103	C₂×C₄	1	4	yes	complemented
105	C₃₁×#102	—	2	2	no	—
106	C₃₁×#102	—	2	2	no	—
107	C₃₁×#102	C₂×C₄	1	4	yes	—
108	C₃₁×#102	C₂×C₄	1	4	yes	—
109	C₃₁×#97	—	2	2	no	—
110	C₃₁×#97	—	2	2	no	—
111	C₃₁×#97	—	2	2	no	—
112	C₄×C₈×C₃₁	—	2	2	no	—
113	C₄×C₈×C₃₁	—	2	2	no	—
114	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	complemented
115	C₃₁×#95	—	2	1	no	—
116	C₃₁×#95	—	2	1	no	—
117	C₃₁×#95	—	2	1	no	—
118	C₃₁×#95	—	2	1	no	—
119	C₃₁×#95	—	2	1	no	—
120	C₃₁×#95	—	2	1	no	—
121	C₃₁×#95	D₄	1	1	yes	characteristic
122	C₃₁×#95	D₄	1	1	yes	characteristic
123	C₃₁×#95	D₄	1	1	yes	characteristic
124	C₃₁×#95	D₄	1	1	yes	characteristic
125	C₂⁴×C₃₁	—	2	1	no	—
126	C₂×Q₈×C₃₁	—	2	1	no	—
127	C₂×Q₈×C₃₁	—	2	1	no	—
128	C₂×Q₈×C₃₁	—	2	1	no	—
129	C₂×Q₈×C₃₁	—	2	1	no	—
130	C₂×Q₈×C₃₁	—	2	1	no	—
131	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
132	C₂×D₄×C₃₁	—	2	1	no	—
133	C₂×D₄×C₃₁	—	2	1	no	—
134	C₂×D₄×C₃₁	—	2	1	no	—
135	C₂×D₄×C₃₁	—	2	1	no	—
136	C₂×D₄×C₃₁	—	2	1	no	—
137	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
138	C₂²×C₄×C₃₁	—	2	1	no	—
139	C₂²×C₄×C₃₁	—	2	1	no	—
140	C₂²×C₄×C₃₁	—	2	1	no	—
141	C₂²×C₄×C₃₁	—	2	1	no	—
142	C₂²×C₄×C₃₁	—	2	1	no	—
143	C₂²×C₄×C₃₁	—	2	1	no	—
144	C₂²×C₄×C₃₁	—	2	1	no	—
145	C₂²×C₄×C₃₁	—	4	2	no	—
146	C₂²×C₄×C₃₁	—	8	1	no	—
147	C₂²×C₄×C₃₁	—	8	1	no	—
148	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
150	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
155	C₂×C₈×C₃₁	—	2	1	no	—
156	C₂×C₈×C₃₁	—	2	1	no	—
157	C₂×C₈×C₃₁	—	2	1	no	—
158	C₂×C₈×C₃₁	—	2	1	no	—
159	C₂×C₈×C₃₁	—	2	1	no	—
160	C₂×C₈×C₃₁	—	2	1	no	—
161	C₂×C₈×C₃₁	—	2	1	no	—
162	C₂×C₈×C₃₁	—	2	1	no	—
163	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
164	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
165	C₂×C₈×C₃₁	#41	1	2	yes	—
166	C₂×C₈×C₃₁	#41	1	2	yes	—
167	C₄⋊C₄×C₃₁	—	2	4	no	—
168	C₄⋊C₄×C₃₁	—	2	4	no	—
169	C₄⋊C₄×C₃₁	—	2	1	no	—
170	C₄⋊C₄×C₃₁	—	2	1	no	—
171	C₄⋊C₄×C₃₁	—	2	1	no	—
172	C₄⋊C₄×C₃₁	—	2	1	no	—
173	C₄⋊C₄×C₃₁	—	2	1	no	—
174	C₄⋊C₄×C₃₁	—	2	1	no	—
175	C₄⋊C₄×C₃₁	—	2	1	no	—
176	C₄⋊C₄×C₃₁	—	2	1	no	—
177	C₄⋊C₄×C₃₁	—	2	1	no	—
178	C₄⋊C₄×C₃₁	—	2	1	no	—
179	C₄⋊C₄×C₃₁	—	4	2	no	—
180	C₄⋊C₄×C₃₁	—	8	1	no	—
181	C₄⋊C₄×C₃₁	—	8	1	no	—
182	C₄⋊C₄×C₃₁	—	8	1	no	—
183	C₄⋊C₄×C₃₁	—	8	1	no	—
184	C₄⋊C₄×C₃₁	—	8	1	no	—
185	C₄⋊C₄×C₃₁	—	8	1	no	—
186	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
187	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
188	C₂²⋊C₄×C₃₁	—	2	4	no	—
189	C₂²⋊C₄×C₃₁	—	2	4	no	—
190	C₂²⋊C₄×C₃₁	—	4	2	no	—
191	C₂²⋊C₄×C₃₁	—	8	1	no	—
192	C₂²⋊C₄×C₃₁	—	8	1	no	—
193	C₄²×C₃₁	—	2	1	no	—
194	C₄²×C₃₁	—	2	1	no	—
195	C₄²×C₃₁	—	4	2	no	—
196	C₄²×C₃₁	—	8	1	no	—
197	C₄²×C₃₁	—	8	1	no	—
198	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
199	C₄²×C₃₁	#41	1	2	yes	—
200	C₄²×C₃₁	#41	1	2	yes	—
201	C₄²×C₃₁	#41	1	2	yes	—
202	#21643	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
203	C₂³×C₃₁	—	2	2	no	—
204	C₂³×C₃₁	—	4	1	no	—
205	C₂³×C₃₁	—	4	1	no	—
206	C₂³×C₃₁	—	4	1	no	—
207	C₂³×C₃₁	—	4	1	no	—
208	C₂³×C₃₁	—	4	1	no	—
209	C₂³×C₃₁	—	4	1	no	—
210	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
211	Q₈×C₃₁	—	2	2	no	—
212	Q₈×C₃₁	—	2	2	no	—
213	Q₈×C₃₁	—	4	1	no	—
214	Q₈×C₃₁	—	4	1	no	—
215	D₄×C₃₁	—	2	2	no	—
216	D₄×C₃₁	—	2	2	no	—
217	D₄×C₃₁	—	4	1	no	—
218	D₄×C₃₁	—	4	1	no	—
219	C₂×C₄×C₃₁	—	2	1	no	—
220	C₂×C₄×C₃₁	—	2	1	no	—
221	C₂×C₄×C₃₁	—	2	1	no	—
222	C₂×C₄×C₃₁	—	2	1	no	—
223	C₂×C₄×C₃₁	—	2	1	no	—
224	C₂×C₄×C₃₁	—	2	1	no	—
225	C₂×C₄×C₃₁	—	2	1	no	—
226	C₂×C₄×C₃₁	—	2	1	no	—
227	C₂×C₄×C₃₁	—	2	2	no	—
228	C₂×C₄×C₃₁	—	2	2	no	—
229	C₂×C₄×C₃₁	—	2	2	no	—
230	C₂×C₄×C₃₁	—	2	1	no	—
231	C₂×C₄×C₃₁	—	2	1	no	—
232	C₂×C₄×C₃₁	—	2	1	no	—
233	C₂×C₄×C₃₁	—	2	1	no	—
234	C₂×C₄×C₃₁	—	2	1	no	—
235	C₂×C₄×C₃₁	—	2	1	no	—
236	C₂×C₄×C₃₁	—	4	1	no	—
237	C₂×C₄×C₃₁	—	4	1	no	—
238	C₂×C₄×C₃₁	—	4	1	no	—
239	C₂×C₄×C₃₁	—	4	1	no	—
240	C₂×C₄×C₃₁	—	4	1	no	—
241	C₂×C₄×C₃₁	—	4	1	no	—
242	C₂×C₄×C₃₁	—	4	1	no	—
243	C₂×C₄×C₃₁	—	4	1	no	—
244	C₂×C₄×C₃₁	—	4	1	no	—
245	C₂×C₄×C₃₁	—	4	1	no	—
246	C₂×C₄×C₃₁	—	4	1	no	—
247	C₂×C₄×C₃₁	—	4	1	no	—
248	C₂×C₄×C₃₁	—	4	1	no	—
249	C₂×C₄×C₃₁	—	4	1	no	—
250	C₂×C₄×C₃₁	—	4	1	no	—
251	C₂×C₄×C₃₁	—	4	1	no	—
252	C₂×C₄×C₃₁	—	4	1	no	—
253	C₂×C₄×C₃₁	—	4	1	no	—
254	C₂×C₄×C₃₁	—	8	4	no	—
255	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
256	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
257	C₂×C₄×C₃₁	#124	1	2	yes	—
258	C₂×C₄×C₃₁	#124	1	2	yes	—
259	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
260	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
261	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
262	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
263	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
264	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
265	C₈×C₃₁	—	2	2	no	—
266	C₈×C₃₁	—	2	2	no	—
267	C₈×C₃₁	—	2	2	no	—
268	C₈×C₃₁	—	2	2	no	—
269	C₂×#488	C₂×C₃₁	1	1	yes	characteristic
270	#2330	C₂×C₃₁	1	1	yes	characteristic, complemented
271	#2326	C₂×C₃₁	1	1	yes	characteristic
272	#1992	C₂×C₃₁	1	1	yes	characteristic, complemented
273	#1959	C₂×C₃₁	1	1	yes	characteristic
274	#1936	C₂×C₃₁	1	1	yes	characteristic
275	#1935	C₂×C₃₁	1	1	yes	characteristic, complemented
276	#1934	C₂×C₃₁	1	1	yes	characteristic
277	#1933	C₂×C₃₁	1	1	yes	characteristic, complemented
278	#1834	C₂×C₃₁	1	1	yes	characteristic, complemented
279	#1763	C₂×C₃₁	1	1	yes	characteristic, complemented
280	#1762	C₂×C₃₁	1	1	yes	characteristic
281	C₄×#103	C₂×C₃₁	1	1	yes	characteristic, complemented
282	C₄×#102	C₂×C₃₁	1	1	yes	characteristic
283	#1726	C₂×C₃₁	1	1	yes	characteristic, complemented
284	C₂²×C₃₁	—	4	2	no	—
285	C₂²×C₃₁	—	4	2	no	—
286	C₂²×C₃₁	—	4	2	no	—
287	C₂²×C₃₁	—	8	1	no	—
288	C₂²×C₃₁	—	8	1	no	—
289	C₂²×C₃₁	—	8	1	no	—
290	C₂²×C₃₁	—	8	1	no	—
291	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
292	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
293	C₂²×C₃₁	#532	1	1	yes	characteristic
294	C₂²×C₃₁	#576	1	1	yes	characteristic
295	C₂²×C₃₁	#577	1	1	yes	characteristic
296	C₂²×C₃₁	#577	1	1	yes	characteristic
297	C₂²×C₃₁	#578	1	1	yes	characteristic
298	C₄×C₃₁	—	2	2	no	—
299	C₄×C₃₁	—	2	2	no	—
300	C₄×C₃₁	—	2	2	no	—
301	C₄×C₃₁	—	2	2	no	—
302	C₄×C₃₁	—	4	1	no	—
303	C₄×C₃₁	—	4	1	no	—
304	C₄×C₃₁	—	4	2	no	—
305	C₄×C₃₁	—	4	2	no	—
306	C₄×C₃₁	—	4	2	no	—
307	C₄×C₃₁	—	8	1	no	—
308	C₄×C₃₁	—	8	1	no	—
309	C₄×C₃₁	—	8	1	no	—
310	C₄×C₃₁	—	8	1	no	—
311	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
312	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
313	C₄×C₃₁	#529	1	1	yes	characteristic
314	C₄×C₃₁	#529	1	1	yes	characteristic
315	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
316	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
317	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
318	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
319	#488	C₄×C₃₁	1	8	yes	complemented
320	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
321	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
322	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
323	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
324	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
325	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
326	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
327	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
328	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
329	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
330	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
331	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
332	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
333	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
334	#391	C₂²×C₃₁	1	1	yes	characteristic
335	#390	C₂²×C₃₁	1	1	yes	characteristic
336	#389	—	2	1	no	—
337	#389	—	2	1	no	—
338	#389	C₂²×C₃₁	1	1	yes	characteristic
339	#389	C₂²×C₃₁	1	1	yes	characteristic
340	#388	—	2	1	no	—
341	#388	—	2	1	no	—
342	#387	—	2	1	no	—
343	#387	—	2	1	no	—
344	#385	—	2	1	no	—
345	#385	—	2	1	no	—
346	#384	C₂²×C₃₁	1	1	yes	characteristic
347	#384	C₂²×C₃₁	1	1	yes	characteristic
348	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
349	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
350	#376	C₂²×C₃₁	1	1	yes	characteristic
351	#340	C₂²×C₃₁	1	1	yes	characteristic
352	#340	C₂²×C₃₁	1	1	yes	characteristic
353	#340	C₂²×C₃₁	1	1	yes	characteristic
354	#340	C₂²×C₃₁	1	1	yes	characteristic
355	#340	C₂²×C₃₁	1	1	yes	characteristic
356	#340	C₂²×C₃₁	1	1	yes	characteristic
357	#340	C₂²×C₃₁	1	1	yes	characteristic
358	#340	C₂²×C₃₁	1	1	yes	characteristic
359	C₂×C₃₁	—	8	2	no	—
360	C₂×C₃₁	C₄×#124	1	1	yes	characteristic
361	C₂×C₃₁	#2951	1	1	yes	characteristic
362	C₂×C₃₁	#2952	1	1	yes	characteristic
363	C₂×C₃₁	#3136	1	1	yes	characteristic
364	C₂×C₃₁	#3137	1	1	yes	characteristic
365	C₂×C₃₁	#3178	1	1	yes	characteristic
366	C₂×C₃₁	#3179	1	1	yes	characteristic
367	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
368	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
369	C₂³×C₄	—	2	1	no	—
370	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
371	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
372	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
373	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
374	#125	—	2	2	no	—
375	#125	C₂×C₄×C₃₁	1	4	yes	complemented
376	#124	—	2	2	no	—
377	#124	C₂×C₄×C₃₁	1	4	yes	—
378	C₄×Q₈	—	2	2	no	—
379	C₄×Q₈	—	2	2	no	—
380	C₄×D₄	—	2	2	no	—
381	C₄×D₄	—	2	2	no	—
382	C₂×C₄⋊C₄	—	2	1	no	—
383	C₂×C₄⋊C₄	—	2	1	no	—
384	C₂×C₄⋊C₄	—	2	1	no	—
385	C₂×C₄⋊C₄	—	2	1	no	—
386	C₂×C₄⋊C₄	—	2	1	no	—
387	C₂×C₄⋊C₄	—	2	1	no	—
388	C₂×C₄⋊C₄	—	2	1	no	—
389	C₂×C₄⋊C₄	—	2	1	no	—
390	C₂×C₄⋊C₄	—	2	1	no	—
391	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
392	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
393	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
394	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
395	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
396	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
397	C₂×C₂²⋊C₄	—	2	1	no	—
398	C₂×C₂²⋊C₄	—	2	1	no	—
399	C₂×C₂²⋊C₄	—	2	1	no	—
400	C₂×C₂²⋊C₄	—	2	1	no	—
401	C₂×C₂²⋊C₄	—	2	1	no	—
402	C₂×C₄²	—	2	1	no	—
403	C₂×C₄²	—	2	1	no	—
404	C₂×C₄²	—	2	1	no	—
405	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
406	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
407	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
408	#103	—	2	2	no	—
409	#103	—	2	2	no	—
410	#103	C₂×C₄×C₃₁	1	4	yes	complemented
411	#103	C₂×C₄×C₃₁	1	4	yes	complemented
412	#102	—	2	2	no	—
413	#102	—	2	2	no	—
414	#102	C₂×C₄×C₃₁	1	4	yes	—
415	#102	C₂×C₄×C₃₁	1	4	yes	—
416	#97	—	2	2	no	—
417	#97	—	2	2	no	—
418	#97	—	2	2	no	—
419	C₄×C₈	—	2	2	no	—
420	C₄×C₈	—	2	2	no	—
421	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	complemented
422	#95	—	2	1	no	—
423	#95	—	2	1	no	—
424	#95	—	2	1	no	—
425	#95	—	2	1	no	—
426	#95	—	2	1	no	—
427	#95	—	2	1	no	—
428	#95	D₄×C₃₁	1	1	yes	characteristic
429	#95	D₄×C₃₁	1	1	yes	characteristic
430	#95	D₄×C₃₁	1	1	yes	characteristic
431	#95	D₄×C₃₁	1	1	yes	characteristic
432	C₃₁	#21643	1	1	yes	characteristic, direct product factor, Sylow
433	C₂⁴	—	2	1	no	—
434	C₂×Q₈	—	2	1	no	—
435	C₂×Q₈	—	2	1	no	—
436	C₂×Q₈	—	2	1	no	—
437	C₂×Q₈	—	2	1	no	—
438	C₂×Q₈	—	2	1	no	—
439	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
440	C₂×D₄	—	2	1	no	—
441	C₂×D₄	—	2	1	no	—
442	C₂×D₄	—	2	1	no	—
443	C₂×D₄	—	2	1	no	—
444	C₂×D₄	—	2	1	no	—
445	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
446	C₂²×C₄	—	2	1	no	—
447	C₂²×C₄	—	2	1	no	—
448	C₂²×C₄	—	2	1	no	—
449	C₂²×C₄	—	2	1	no	—
450	C₂²×C₄	—	2	1	no	—
451	C₂²×C₄	—	2	1	no	—
452	C₂²×C₄	—	2	1	no	—
453	C₂²×C₄	—	4	2	no	—
454	C₂²×C₄	—	8	1	no	—
455	C₂²×C₄	—	8	1	no	—
456	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
457	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
458	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
459	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
460	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
461	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
462	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
463	C₂×C₈	—	2	1	no	—
464	C₂×C₈	—	2	1	no	—
465	C₂×C₈	—	2	1	no	—
466	C₂×C₈	—	2	1	no	—
467	C₂×C₈	—	2	1	no	—
468	C₂×C₈	—	2	1	no	—
469	C₂×C₈	—	2	1	no	—
470	C₂×C₈	—	2	1	no	—
471	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
472	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
473	C₂×C₈	#41×C₃₁	1	2	yes	—
474	C₂×C₈	#41×C₃₁	1	2	yes	—
475	C₄⋊C₄	—	2	4	no	—
476	C₄⋊C₄	—	2	4	no	—
477	C₄⋊C₄	—	2	1	no	—
478	C₄⋊C₄	—	2	1	no	—
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	—	2	1	no	—
481	C₄⋊C₄	—	2	1	no	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	—	2	1	no	—
484	C₄⋊C₄	—	2	1	no	—
485	C₄⋊C₄	—	2	1	no	—
486	C₄⋊C₄	—	2	1	no	—
487	C₄⋊C₄	—	4	2	no	—
488	C₄⋊C₄	—	8	1	no	—
489	C₄⋊C₄	—	8	1	no	—
490	C₄⋊C₄	—	8	1	no	—
491	C₄⋊C₄	—	8	1	no	—
492	C₄⋊C₄	—	8	1	no	—
493	C₄⋊C₄	—	8	1	no	—
494	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
495	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
496	C₂²⋊C₄	—	2	4	no	—
497	C₂²⋊C₄	—	2	4	no	—
498	C₂²⋊C₄	—	4	2	no	—
499	C₂²⋊C₄	—	8	1	no	—
500	C₂²⋊C₄	—	8	1	no	—
501	C₄²	—	2	1	no	—
502	C₄²	—	2	1	no	—
503	C₄²	—	4	2	no	—
504	C₄²	—	8	1	no	—
505	C₄²	—	8	1	no	—
506	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
507	C₄²	#41×C₃₁	1	2	yes	—
508	C₄²	#41×C₃₁	1	2	yes	—
509	C₄²	#41×C₃₁	1	2	yes	—
510	C₂³	—	2	2	no	—
511	C₂³	—	4	1	no	—
512	C₂³	—	4	1	no	—
513	C₂³	—	4	1	no	—
514	C₂³	—	4	1	no	—
515	C₂³	—	4	1	no	—
516	C₂³	—	4	1	no	—
517	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
518	Q₈	—	2	2	no	—
519	Q₈	—	2	2	no	—
520	Q₈	—	4	1	no	—
521	Q₈	—	4	1	no	—
522	D₄	—	2	2	no	—
523	D₄	—	2	2	no	—
524	D₄	—	4	1	no	—
525	D₄	—	4	1	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	2	1	no	—
541	C₂×C₄	—	2	1	no	—
542	C₂×C₄	—	2	1	no	—
543	C₂×C₄	—	4	1	no	—
544	C₂×C₄	—	4	1	no	—
545	C₂×C₄	—	4	1	no	—
546	C₂×C₄	—	4	1	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	4	1	no	—
550	C₂×C₄	—	4	1	no	—
551	C₂×C₄	—	4	1	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	4	1	no	—
556	C₂×C₄	—	4	1	no	—
557	C₂×C₄	—	4	1	no	—
558	C₂×C₄	—	4	1	no	—
559	C₂×C₄	—	4	1	no	—
560	C₂×C₄	—	4	1	no	—
561	C₂×C₄	—	8	4	no	—
562	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
563	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
564	C₂×C₄	C₃₁×#124	1	2	yes	—
565	C₂×C₄	C₃₁×#124	1	2	yes	—
566	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
567	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
568	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
569	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
570	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
571	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
572	C₈	—	2	2	no	—
573	C₈	—	2	2	no	—
574	C₈	—	2	2	no	—
575	C₈	—	2	2	no	—
576	C₂²	—	4	2	no	—
577	C₂²	—	4	2	no	—
578	C₂²	—	4	2	no	—
579	C₂²	—	8	1	no	—
580	C₂²	—	8	1	no	—
581	C₂²	—	8	1	no	—
582	C₂²	—	8	1	no	—
583	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
584	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
585	C₂²	C₃₁×#532	1	1	yes	characteristic
586	C₂²	C₃₁×#576	1	1	yes	characteristic
587	C₂²	C₃₁×#577	1	1	yes	characteristic
588	C₂²	C₃₁×#577	1	1	yes	characteristic
589	C₂²	C₃₁×#578	1	1	yes	characteristic
590	C₄	—	2	2	no	—
591	C₄	—	2	2	no	—
592	C₄	—	2	2	no	—
593	C₄	—	2	2	no	—
594	C₄	—	4	1	no	—
595	C₄	—	4	1	no	—
596	C₄	—	4	2	no	—
597	C₄	—	4	2	no	—
598	C₄	—	4	2	no	—
599	C₄	—	8	1	no	—
600	C₄	—	8	1	no	—
601	C₄	—	8	1	no	—
602	C₄	—	8	1	no	—
603	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
604	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
605	C₄	C₃₁×#529	1	1	yes	characteristic
606	C₄	C₃₁×#529	1	1	yes	characteristic
607	C₂	—	8	2	no	—
608	C₂	C₄×C₃₁×#124	1	1	yes	characteristic
609	C₂	C₃₁×#2951	1	1	yes	characteristic
610	C₂	C₃₁×#2952	1	1	yes	characteristic
611	C₂	C₃₁×#3136	1	1	yes	characteristic
612	C₂	C₃₁×#3137	1	1	yes	characteristic
613	C₂	C₃₁×#3178	1	1	yes	characteristic
614	C₂	C₃₁×#3179	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₃₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2330	C₂	semidirect
4. C₃₁×#1992	C₂	semidirect
7. C₃₁×#1935	C₂	semidirect
9. C₃₁×#1933	C₂	semidirect
10. C₃₁×#1834	C₂	semidirect
11. C₃₁×#1763	C₂	semidirect
13. C₄×C₃₁×#103	C₂	semidirect
15. C₃₁×#1726	C₂	semidirect
20. C₃₁×#488	C₄	semidirect
68. C₃₁×#125	C₂×C₄	semidirect
103. C₃₁×#103	C₂×C₄	semidirect
104. C₃₁×#103	C₂×C₄	semidirect
114. C₄×C₈×C₃₁	C₂×C₄	semidirect
202. #21643	C₃₁	direct (N × H)
270. #2330	C₂×C₃₁	semidirect
272. #1992	C₂×C₃₁	semidirect
275. #1935	C₂×C₃₁	semidirect
277. #1933	C₂×C₃₁	semidirect
278. #1834	C₂×C₃₁	semidirect
279. #1763	C₂×C₃₁	semidirect
281. C₄×#103	C₂×C₃₁	semidirect
283. #1726	C₂×C₃₁	semidirect
319. #488	C₄×C₃₁	semidirect
375. #125	C₂×C₄×C₃₁	semidirect
410. #103	C₂×C₄×C₃₁	semidirect
411. #103	C₂×C₄×C₃₁	semidirect
421. C₄×C₈	C₂×C₄×C₃₁	semidirect
432. C₃₁	#21643	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×C₃₁×#488	2
C₃₁×#2330	2
C₃₁×#2326	2
C₃₁×#1992	2
C₃₁×#1959	2
C₃₁×#1936	2
C₃₁×#1935	2
C₃₁×#1934	2
C₃₁×#1933	2
C₃₁×#1834	2
C₃₁×#1763	2
C₃₁×#1762	2
C₄×C₃₁×#103	2
C₄×C₃₁×#102	2
C₃₁×#1726	2
#21643	31

Subgroups — C31×#21643 #21478402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21643 #21478402