Subgroups — C₃₁×#33055 #21489814

Order 7936.

922 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#490	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#405	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2629	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2619	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1991	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1957	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1936	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1934	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1895	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1893	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1836	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#121	C₂²	1	2	yes	—
17	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
18	C₃₁×#490	—	2	4	no	—
19	C₂×C₃₁×#102	C₂²	1	2	yes	—
20	C₂×C₃₁×#102	C₂²	1	2	yes	—
21	C₂×C₃₁×#102	C₂²	1	2	yes	—
22	C₂×C₃₁×#102	C₂²	1	2	yes	—
23	C₃₁×#405	—	2	4	no	—
24	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#398	—	2	2	no	—
29	C₃₁×#396	—	2	2	no	—
30	C₃₁×#394	—	2	2	no	—
31	C₃₁×#392	—	2	2	no	—
32	C₃₁×#390	C₂²	1	2	yes	—
33	C₃₁×#389	C₂²	1	2	yes	—
34	C₃₁×#384	C₂²	1	1	yes	characteristic
35	C₃₁×#382	C₂²	1	1	yes	characteristic
36	C₃₁×#380	C₂²	1	2	yes	—
37	C₃₁×#340	C₂²	1	2	yes	—
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#340	C₂²	1	2	yes	—
40	C₃₁×#340	C₂²	1	2	yes	—
41	C₂²×D₄×C₃₁	—	2	2	no	—
42	C₂²×D₄×C₃₁	—	2	2	no	—
43	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
44	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
45	C₂×M₁₆×C₃₁	—	2	2	no	—
46	C₂×M₁₆×C₃₁	—	2	2	no	—
47	C₂×M₁₆×C₃₁	—	2	2	no	—
48	C₂²×C₈×C₃₁	—	2	2	no	—
49	C₂²×C₈×C₃₁	—	2	2	no	—
50	C₂²×C₈×C₃₁	—	2	2	no	—
51	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
55	C₃₁×#127	—	2	2	no	—
56	C₃₁×#127	—	2	2	no	—
57	C₃₁×#125	—	2	2	no	—
58	C₃₁×#125	—	2	2	no	—
59	C₃₁×#121	—	2	4	no	—
60	C₃₁×#121	—	2	4	no	—
61	C₃₁×#117	—	2	2	no	—
62	C₃₁×#117	—	2	2	no	—
63	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
64	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
65	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
66	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
67	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
68	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
69	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
70	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
71	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
72	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
73	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
74	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
75	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
77	C₃₁×#102	—	2	4	no	—
78	C₃₁×#102	—	2	4	no	—
79	C₃₁×#102	—	2	4	no	—
80	C₃₁×#102	—	2	4	no	—
81	C₃₁×#97	—	2	2	no	—
82	C₃₁×#97	D₄	1	4	yes	complemented
83	C₄×C₈×C₃₁	—	2	2	no	—
84	C₄×C₈×C₃₁	—	2	2	no	—
85	C₃₁×#95	—	2	2	no	—
86	C₃₁×#95	—	2	2	no	—
87	C₃₁×#95	—	2	2	no	—
88	C₂⁴×C₃₁	—	2	2	no	—
89	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
90	C₂×D₄×C₃₁	—	2	2	no	—
91	C₂×D₄×C₃₁	—	2	2	no	—
92	C₂×D₄×C₃₁	—	2	2	no	—
93	C₂×D₄×C₃₁	—	2	2	no	—
94	C₂×D₄×C₃₁	—	2	2	no	—
95	C₂×D₄×C₃₁	—	2	2	no	—
96	C₂×D₄×C₃₁	—	4	4	no	—
97	C₂×D₄×C₃₁	—	4	4	no	—
98	C₂×D₄×C₃₁	—	8	2	no	—
99	C₂×D₄×C₃₁	—	8	2	no	—
100	C₂×D₄×C₃₁	—	8	2	no	—
101	C₂×D₄×C₃₁	—	8	2	no	—
102	C₂²×C₄×C₃₁	—	2	1	no	—
103	C₂²×C₄×C₃₁	—	2	1	no	—
104	C₂²×C₄×C₃₁	—	2	1	no	—
105	C₂²×C₄×C₃₁	—	2	1	no	—
106	C₂²×C₄×C₃₁	—	2	2	no	—
107	C₂²×C₄×C₃₁	—	2	2	no	—
108	C₂²×C₄×C₃₁	—	2	2	no	—
109	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₃₁	#41	1	2	yes	—
113	C₂²×C₄×C₃₁	#41	1	2	yes	—
114	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
115	M₁₆×C₃₁	—	2	8	no	—
116	C₂×C₈×C₃₁	—	2	8	no	—
117	C₂×C₈×C₃₁	—	2	1	no	—
118	C₂×C₈×C₃₁	—	2	1	no	—
119	C₂×C₈×C₃₁	—	2	1	no	—
120	C₂×C₈×C₃₁	—	2	1	no	—
121	C₂×C₈×C₃₁	—	2	1	no	—
122	C₂×C₈×C₃₁	—	2	1	no	—
123	C₂×C₈×C₃₁	—	2	1	no	—
124	C₂×C₈×C₃₁	—	2	1	no	—
125	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
126	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
127	C₂×C₈×C₃₁	#41	1	2	yes	—
128	C₂×C₈×C₃₁	#41	1	2	yes	—
129	C₄⋊C₄×C₃₁	—	2	4	no	—
130	C₄⋊C₄×C₃₁	—	2	2	no	—
131	C₄⋊C₄×C₃₁	—	2	2	no	—
132	C₄⋊C₄×C₃₁	—	2	2	no	—
133	C₄⋊C₄×C₃₁	—	2	2	no	—
134	C₄⋊C₄×C₃₁	—	2	2	no	—
135	C₄⋊C₄×C₃₁	—	2	2	no	—
136	C₄⋊C₄×C₃₁	—	4	4	no	—
137	C₄⋊C₄×C₃₁	—	8	2	no	—
138	C₄⋊C₄×C₃₁	—	8	2	no	—
139	C₂²⋊C₄×C₃₁	—	2	4	no	—
140	C₂²⋊C₄×C₃₁	—	4	4	no	—
141	C₂²⋊C₄×C₃₁	—	8	2	no	—
142	C₂²⋊C₄×C₃₁	—	8	2	no	—
143	C₂²⋊C₄×C₃₁	—	8	2	no	—
144	C₄²×C₃₁	—	2	1	no	—
145	C₄²×C₃₁	—	2	1	no	—
146	C₄²×C₃₁	C₂×D₄	1	2	yes	—
147	C₄²×C₃₁	#41	1	2	yes	—
148	#33055	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂³×C₃₁	—	2	1	no	—
150	C₂³×C₃₁	—	2	1	no	—
151	C₂³×C₃₁	—	2	2	no	—
152	C₂³×C₃₁	—	2	2	no	—
153	C₂³×C₃₁	—	4	2	no	—
154	C₂³×C₃₁	—	4	2	no	—
155	C₂³×C₃₁	—	4	2	no	—
156	C₂³×C₃₁	—	4	2	no	—
157	C₂³×C₃₁	—	4	2	no	—
158	C₂³×C₃₁	—	4	2	no	—
159	C₂³×C₃₁	—	4	2	no	—
160	C₂³×C₃₁	#124	1	2	yes	—
161	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
162	D₄×C₃₁	—	4	2	no	—
163	D₄×C₃₁	—	4	2	no	—
164	D₄×C₃₁	—	4	2	no	—
165	D₄×C₃₁	—	4	2	no	—
166	D₄×C₃₁	—	8	8	no	—
167	D₄×C₃₁	—	8	8	no	—
168	C₂×C₄×C₃₁	—	2	1	no	—
169	C₂×C₄×C₃₁	—	2	1	no	—
170	C₂×C₄×C₃₁	—	2	2	no	—
171	C₂×C₄×C₃₁	—	2	2	no	—
172	C₂×C₄×C₃₁	—	2	2	no	—
173	C₂×C₄×C₃₁	—	2	2	no	—
174	C₂×C₄×C₃₁	—	2	2	no	—
175	C₂×C₄×C₃₁	—	2	2	no	—
176	C₂×C₄×C₃₁	—	2	2	no	—
177	C₂×C₄×C₃₁	—	2	2	no	—
178	C₂×C₄×C₃₁	—	2	2	no	—
179	C₂×C₄×C₃₁	—	4	2	no	—
180	C₂×C₄×C₃₁	—	4	2	no	—
181	C₂×C₄×C₃₁	—	4	2	no	—
182	C₂×C₄×C₃₁	—	4	2	no	—
183	C₂×C₄×C₃₁	—	4	2	no	—
184	C₂×C₄×C₃₁	—	4	2	no	—
185	C₂×C₄×C₃₁	—	4	2	no	—
186	C₂×C₄×C₃₁	—	4	2	no	—
187	C₂×C₄×C₃₁	—	4	2	no	—
188	C₂×C₄×C₃₁	#121	1	4	yes	—
189	C₂×C₄×C₃₁	#124	1	2	yes	—
190	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
192	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
193	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
194	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
195	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
196	C₈×C₃₁	—	2	2	no	—
197	C₈×C₃₁	—	2	2	no	—
198	C₈×C₃₁	—	2	2	no	—
199	C₈×C₃₁	—	2	2	no	—
200	C₂×#490	C₂×C₃₁	1	1	yes	characteristic, complemented
201	C₂×#405	C₂×C₃₁	1	1	yes	characteristic, complemented
202	#2629	C₂×C₃₁	1	1	yes	characteristic, complemented
203	#2619	C₂×C₃₁	1	1	yes	characteristic
204	#1991	C₂×C₃₁	1	1	yes	characteristic, complemented
205	#1957	C₂×C₃₁	1	1	yes	characteristic, complemented
206	#1936	C₂×C₃₁	1	2	yes	complemented
207	#1934	C₂×C₃₁	1	2	yes	complemented
208	#1895	C₂×C₃₁	1	2	yes	complemented
209	#1893	C₂×C₃₁	1	2	yes	complemented
210	#1836	C₂×C₃₁	1	1	yes	characteristic, complemented
211	C₂²×C₃₁	—	2	2	no	—
212	C₂²×C₃₁	—	2	4	no	—
213	C₂²×C₃₁	—	4	1	no	—
214	C₂²×C₃₁	—	4	1	no	—
215	C₂²×C₃₁	—	4	2	no	—
216	C₂²×C₃₁	—	8	2	no	—
217	C₂²×C₃₁	—	8	2	no	—
218	C₂²×C₃₁	—	8	2	no	—
219	C₂²×C₃₁	—	8	2	no	—
220	C₂²×C₃₁	—	8	2	no	—
221	C₂²×C₃₁	—	8	2	no	—
222	C₂²×C₃₁	—	8	2	no	—
223	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
224	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
225	C₂²×C₃₁	#532	1	1	yes	characteristic
226	C₂²×C₃₁	#576	1	2	yes	—
227	C₂²×C₃₁	#577	1	2	yes	—
228	C₄×C₃₁	—	2	2	no	—
229	C₄×C₃₁	—	2	4	no	—
230	C₄×C₃₁	—	4	2	no	—
231	C₄×C₃₁	—	8	2	no	—
232	C₄×C₃₁	—	8	2	no	—
233	C₄×C₃₁	—	8	2	no	—
234	C₄×C₃₁	#525	1	1	yes	characteristic
235	C₄×C₃₁	#525	1	1	yes	characteristic
236	C₄×C₃₁	#529	1	1	yes	characteristic
237	C₄×C₃₁	#529	1	1	yes	characteristic
238	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
239	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
240	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
241	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
242	C₂×#121	C₂²×C₃₁	1	2	yes	—
243	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
244	#490	—	2	4	no	—
245	C₂×#102	C₂²×C₃₁	1	2	yes	—
246	C₂×#102	C₂²×C₃₁	1	2	yes	—
247	C₂×#102	C₂²×C₃₁	1	2	yes	—
248	C₂×#102	C₂²×C₃₁	1	2	yes	—
249	#405	—	2	4	no	—
250	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
251	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
252	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
253	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
254	#398	—	2	2	no	—
255	#396	—	2	2	no	—
256	#394	—	2	2	no	—
257	#392	—	2	2	no	—
258	#390	C₂²×C₃₁	1	2	yes	—
259	#389	C₂²×C₃₁	1	2	yes	—
260	#384	C₂²×C₃₁	1	1	yes	characteristic
261	#382	C₂²×C₃₁	1	1	yes	characteristic
262	#380	C₂²×C₃₁	1	2	yes	—
263	#340	C₂²×C₃₁	1	2	yes	—
264	#340	C₂²×C₃₁	1	2	yes	—
265	#340	C₂²×C₃₁	1	2	yes	—
266	#340	C₂²×C₃₁	1	2	yes	—
267	C₂×C₃₁	—	4	2	no	—
268	C₂×C₃₁	—	16	2	no	—
269	C₂×C₃₁	#2850	1	1	yes	characteristic
270	C₂×C₃₁	#3062	1	2	yes	—
271	C₂×C₃₁	#3136	1	2	yes	—
272	C₂×C₃₁	#3174	1	1	yes	characteristic
273	C₂×C₃₁	#3176	1	1	yes	characteristic
274	C₂²×D₄	—	2	2	no	—
275	C₂²×D₄	—	2	2	no	—
276	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
277	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
278	C₂×M₁₆	—	2	2	no	—
279	C₂×M₁₆	—	2	2	no	—
280	C₂×M₁₆	—	2	2	no	—
281	C₂²×C₈	—	2	2	no	—
282	C₂²×C₈	—	2	2	no	—
283	C₂²×C₈	—	2	2	no	—
284	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
285	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
286	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
287	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
288	#127	—	2	2	no	—
289	#127	—	2	2	no	—
290	#125	—	2	2	no	—
291	#125	—	2	2	no	—
292	#121	—	2	4	no	—
293	#121	—	2	4	no	—
294	#117	—	2	2	no	—
295	#117	—	2	2	no	—
296	C₂×C₄⋊C₄	—	2	2	no	—
297	C₂×C₄⋊C₄	—	2	2	no	—
298	C₂×C₄⋊C₄	—	2	2	no	—
299	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
300	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
301	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
302	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
303	C₂×C₂²⋊C₄	—	2	2	no	—
304	C₂×C₂²⋊C₄	—	2	2	no	—
305	C₂×C₂²⋊C₄	—	2	2	no	—
306	C₂×C₂²⋊C₄	—	2	2	no	—
307	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
308	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
309	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
310	#102	—	2	4	no	—
311	#102	—	2	4	no	—
312	#102	—	2	4	no	—
313	#102	—	2	4	no	—
314	#97	—	2	2	no	—
315	#97	D₄×C₃₁	1	4	yes	complemented
316	C₄×C₈	—	2	2	no	—
317	C₄×C₈	—	2	2	no	—
318	#95	—	2	2	no	—
319	#95	—	2	2	no	—
320	#95	—	2	2	no	—
321	C₃₁	#33055	1	1	yes	characteristic, direct product factor, Sylow
322	C₂⁴	—	2	2	no	—
323	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
324	C₂×D₄	—	2	2	no	—
325	C₂×D₄	—	2	2	no	—
326	C₂×D₄	—	2	2	no	—
327	C₂×D₄	—	2	2	no	—
328	C₂×D₄	—	2	2	no	—
329	C₂×D₄	—	2	2	no	—
330	C₂×D₄	—	4	4	no	—
331	C₂×D₄	—	4	4	no	—
332	C₂×D₄	—	8	2	no	—
333	C₂×D₄	—	8	2	no	—
334	C₂×D₄	—	8	2	no	—
335	C₂×D₄	—	8	2	no	—
336	C₂²×C₄	—	2	1	no	—
337	C₂²×C₄	—	2	1	no	—
338	C₂²×C₄	—	2	1	no	—
339	C₂²×C₄	—	2	1	no	—
340	C₂²×C₄	—	2	2	no	—
341	C₂²×C₄	—	2	2	no	—
342	C₂²×C₄	—	2	2	no	—
343	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
344	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
345	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
346	C₂²×C₄	#41×C₃₁	1	2	yes	—
347	C₂²×C₄	#41×C₃₁	1	2	yes	—
348	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
349	M₁₆	—	2	8	no	—
350	C₂×C₈	—	2	8	no	—
351	C₂×C₈	—	2	1	no	—
352	C₂×C₈	—	2	1	no	—
353	C₂×C₈	—	2	1	no	—
354	C₂×C₈	—	2	1	no	—
355	C₂×C₈	—	2	1	no	—
356	C₂×C₈	—	2	1	no	—
357	C₂×C₈	—	2	1	no	—
358	C₂×C₈	—	2	1	no	—
359	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
360	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
361	C₂×C₈	#41×C₃₁	1	2	yes	—
362	C₂×C₈	#41×C₃₁	1	2	yes	—
363	C₄⋊C₄	—	2	4	no	—
364	C₄⋊C₄	—	2	2	no	—
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	—	2	2	no	—
367	C₄⋊C₄	—	2	2	no	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	—	2	2	no	—
370	C₄⋊C₄	—	4	4	no	—
371	C₄⋊C₄	—	8	2	no	—
372	C₄⋊C₄	—	8	2	no	—
373	C₂²⋊C₄	—	2	4	no	—
374	C₂²⋊C₄	—	4	4	no	—
375	C₂²⋊C₄	—	8	2	no	—
376	C₂²⋊C₄	—	8	2	no	—
377	C₂²⋊C₄	—	8	2	no	—
378	C₄²	—	2	1	no	—
379	C₄²	—	2	1	no	—
380	C₄²	C₂×D₄×C₃₁	1	2	yes	—
381	C₄²	#41×C₃₁	1	2	yes	—
382	C₂³	—	2	1	no	—
383	C₂³	—	2	1	no	—
384	C₂³	—	2	2	no	—
385	C₂³	—	2	2	no	—
386	C₂³	—	4	2	no	—
387	C₂³	—	4	2	no	—
388	C₂³	—	4	2	no	—
389	C₂³	—	4	2	no	—
390	C₂³	—	4	2	no	—
391	C₂³	—	4	2	no	—
392	C₂³	—	4	2	no	—
393	C₂³	C₃₁×#124	1	2	yes	—
394	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
395	D₄	—	4	2	no	—
396	D₄	—	4	2	no	—
397	D₄	—	4	2	no	—
398	D₄	—	4	2	no	—
399	D₄	—	8	8	no	—
400	D₄	—	8	8	no	—
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	2	1	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	4	2	no	—
413	C₂×C₄	—	4	2	no	—
414	C₂×C₄	—	4	2	no	—
415	C₂×C₄	—	4	2	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	4	2	no	—
421	C₂×C₄	C₃₁×#121	1	4	yes	—
422	C₂×C₄	C₃₁×#124	1	2	yes	—
423	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
424	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
425	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
426	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
427	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
428	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
429	C₈	—	2	2	no	—
430	C₈	—	2	2	no	—
431	C₈	—	2	2	no	—
432	C₈	—	2	2	no	—
433	C₂²	—	2	2	no	—
434	C₂²	—	2	4	no	—
435	C₂²	—	4	1	no	—
436	C₂²	—	4	1	no	—
437	C₂²	—	4	2	no	—
438	C₂²	—	8	2	no	—
439	C₂²	—	8	2	no	—
440	C₂²	—	8	2	no	—
441	C₂²	—	8	2	no	—
442	C₂²	—	8	2	no	—
443	C₂²	—	8	2	no	—
444	C₂²	—	8	2	no	—
445	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
446	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
447	C₂²	C₃₁×#532	1	1	yes	characteristic
448	C₂²	C₃₁×#576	1	2	yes	—
449	C₂²	C₃₁×#577	1	2	yes	—
450	C₄	—	2	2	no	—
451	C₄	—	2	4	no	—
452	C₄	—	4	2	no	—
453	C₄	—	8	2	no	—
454	C₄	—	8	2	no	—
455	C₄	—	8	2	no	—
456	C₄	C₃₁×#525	1	1	yes	characteristic
457	C₄	C₃₁×#525	1	1	yes	characteristic
458	C₄	C₃₁×#529	1	1	yes	characteristic
459	C₄	C₃₁×#529	1	1	yes	characteristic
460	C₂	—	4	2	no	—
461	C₂	—	16	2	no	—
462	C₂	C₃₁×#2850	1	1	yes	characteristic
463	C₂	C₃₁×#3062	1	2	yes	—
464	C₂	C₃₁×#3136	1	2	yes	—
465	C₂	C₃₁×#3174	1	1	yes	characteristic
466	C₂	C₃₁×#3176	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#2619	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#490	C₂	semidirect
2. C₂×C₃₁×#405	C₂	semidirect
3. C₃₁×#2629	C₂	semidirect
5. C₃₁×#1991	C₂	semidirect
6. C₃₁×#1957	C₂	semidirect
7. C₃₁×#1936	C₂	semidirect
8. C₃₁×#1934	C₂	semidirect
9. C₃₁×#1895	C₂	semidirect
10. C₃₁×#1893	C₂	semidirect
11. C₃₁×#1836	C₂	semidirect
82. C₃₁×#97	D₄	semidirect
148. #33055	C₃₁	direct (N × H)
200. C₂×#490	C₂×C₃₁	semidirect
201. C₂×#405	C₂×C₃₁	semidirect
202. #2629	C₂×C₃₁	semidirect
204. #1991	C₂×C₃₁	semidirect
205. #1957	C₂×C₃₁	semidirect
206. #1936	C₂×C₃₁	semidirect
207. #1934	C₂×C₃₁	semidirect
208. #1895	C₂×C₃₁	semidirect
209. #1893	C₂×C₃₁	semidirect
210. #1836	C₂×C₃₁	semidirect
315. #97	D₄×C₃₁	semidirect
321. C₃₁	#33055	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×#490	2
C₂×C₃₁×#405	2
C₃₁×#2629	2
C₃₁×#2619	2
C₃₁×#1991	2
C₃₁×#1957	2
C₃₁×#1936	2
C₃₁×#1934	2
C₃₁×#1895	2
C₃₁×#1893	2
C₃₁×#1836	2
#33055	31

Subgroups — C31×#33055 #21489814

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33055 #21489814