Subgroups — C₃₁×#33141 #21489900

Order 7936.

810 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#470	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#405	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2625	C₂	1	2	yes	maximal
4	C₃₁×#1971	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1969	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1943	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1941	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1935	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1933	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1892	C₂	1	2	yes	maximal
11	C₃₁×#1809	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
16	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
17	C₃₁×#470	—	2	4	no	—
18	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#103	C₂²	1	2	yes	—
22	C₂×C₃₁×#103	C₂²	1	2	yes	—
23	C₂×C₃₁×#103	C₂²	1	2	yes	—
24	C₃₁×#405	—	2	4	no	—
25	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#400	—	2	2	no	—
30	C₃₁×#398	—	2	2	no	—
31	C₃₁×#397	—	2	2	no	—
32	C₃₁×#397	—	2	2	no	—
33	C₃₁×#391	C₂²	1	2	yes	—
34	C₃₁×#391	C₂²	1	2	yes	—
35	C₃₁×#389	C₂²	1	2	yes	—
36	C₃₁×#389	C₂²	1	2	yes	—
37	C₃₁×#380	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×Q₈×C₃₁	C₂³	1	2	yes	—
42	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
43	C₂×Q₁₆×C₃₁	—	2	2	no	—
44	C₂×Q₁₆×C₃₁	—	2	2	no	—
45	C₂×Q₁₆×C₃₁	—	2	2	no	—
46	C₂×M₁₆×C₃₁	—	2	2	no	—
47	C₂×M₁₆×C₃₁	—	2	2	no	—
48	C₂×M₁₆×C₃₁	D₄	1	4	yes	—
49	C₂²×C₈×C₃₁	—	2	2	no	—
50	C₂²×C₈×C₃₁	—	2	2	no	—
51	C₂²×C₈×C₃₁	—	2	2	no	—
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	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
56	C₃₁×C₂²⋊Q₈	—	2	4	no	—
57	C₃₁×C₂²⋊Q₈	—	2	4	no	—
58	C₃₁×#117	—	2	2	no	—
59	C₃₁×#117	—	2	2	no	—
60	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
61	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
62	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
63	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
64	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
67	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
69	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
70	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
72	C₃₁×#107	—	2	2	no	—
73	C₃₁×#106	—	2	2	no	—
74	C₃₁×#106	—	2	2	no	—
75	C₃₁×#103	—	2	4	no	—
76	C₃₁×#103	—	2	4	no	—
77	C₃₁×#103	—	2	4	no	—
78	C₃₁×#97	—	2	2	no	—
79	C₃₁×#97	—	2	2	no	—
80	C₄×C₈×C₃₁	—	2	2	no	—
81	C₄×C₈×C₃₁	—	2	2	no	—
82	C₃₁×#95	—	2	2	no	—
83	C₃₁×#95	—	2	2	no	—
84	C₃₁×#95	—	2	2	no	—
85	C₃₁×#95	—	2	2	no	—
86	C₃₁×#95	—	2	2	no	—
87	C₃₁×#95	—	2	2	no	—
88	C₃₁×#95	—	2	2	no	—
89	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
90	C₂×Q₈×C₃₁	—	2	2	no	—
91	C₂×Q₈×C₃₁	—	2	2	no	—
92	C₂×Q₈×C₃₁	—	2	2	no	—
93	C₂×Q₈×C₃₁	—	2	2	no	—
94	C₂×Q₈×C₃₁	—	2	2	no	—
95	C₂×Q₈×C₃₁	—	2	2	no	—
96	C₂²×C₄×C₃₁	—	2	1	no	—
97	C₂²×C₄×C₃₁	—	2	1	no	—
98	C₂²×C₄×C₃₁	—	2	1	no	—
99	C₂²×C₄×C₃₁	—	2	1	no	—
100	C₂²×C₄×C₃₁	—	2	2	no	—
101	C₂²×C₄×C₃₁	—	2	2	no	—
102	C₂²×C₄×C₃₁	—	2	2	no	—
103	C₂²×C₄×C₃₁	—	2	2	no	—
104	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
105	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
108	C₂²×C₄×C₃₁	#41	1	2	yes	—
109	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
110	Q₁₆×C₃₁	—	4	4	no	—
111	M₁₆×C₃₁	—	2	8	no	—
112	C₂×C₈×C₃₁	—	2	8	no	—
113	C₂×C₈×C₃₁	—	2	1	no	—
114	C₂×C₈×C₃₁	—	2	1	no	—
115	C₂×C₈×C₃₁	—	2	1	no	—
116	C₂×C₈×C₃₁	—	2	1	no	—
117	C₂×C₈×C₃₁	—	2	1	no	—
118	C₂×C₈×C₃₁	—	2	1	no	—
119	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
120	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
121	C₂×C₈×C₃₁	#41	1	2	yes	—
122	C₂×C₈×C₃₁	#41	1	2	yes	—
123	C₂×C₈×C₃₁	#41	1	2	yes	—
124	C₂×C₈×C₃₁	#41	1	2	yes	—
125	C₄⋊C₄×C₃₁	—	2	4	no	—
126	C₄⋊C₄×C₃₁	—	2	2	no	—
127	C₄⋊C₄×C₃₁	—	2	2	no	—
128	C₄⋊C₄×C₃₁	—	2	2	no	—
129	C₄⋊C₄×C₃₁	—	2	2	no	—
130	C₄⋊C₄×C₃₁	—	2	2	no	—
131	C₄⋊C₄×C₃₁	—	2	2	no	—
132	C₄⋊C₄×C₃₁	—	4	4	no	—
133	C₄⋊C₄×C₃₁	—	8	2	no	—
134	C₄⋊C₄×C₃₁	—	8	2	no	—
135	C₂²⋊C₄×C₃₁	—	2	4	no	—
136	C₂²⋊C₄×C₃₁	—	4	4	no	—
137	C₂²⋊C₄×C₃₁	—	8	2	no	—
138	C₄²×C₃₁	—	2	1	no	—
139	C₄²×C₃₁	—	2	1	no	—
140	C₄²×C₃₁	—	2	1	no	—
141	C₄²×C₃₁	—	2	1	no	—
142	#33141	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂³×C₃₁	—	2	1	no	—
144	C₂³×C₃₁	—	2	1	no	—
145	C₂³×C₃₁	—	2	2	no	—
146	C₂³×C₃₁	—	2	2	no	—
147	C₂³×C₃₁	#121	1	2	yes	—
148	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
149	Q₈×C₃₁	—	4	2	no	—
150	Q₈×C₃₁	—	4	2	no	—
151	Q₈×C₃₁	—	4	2	no	—
152	Q₈×C₃₁	—	4	2	no	—
153	C₂×C₄×C₃₁	—	2	1	no	—
154	C₂×C₄×C₃₁	—	2	1	no	—
155	C₂×C₄×C₃₁	—	2	2	no	—
156	C₂×C₄×C₃₁	—	2	2	no	—
157	C₂×C₄×C₃₁	—	2	2	no	—
158	C₂×C₄×C₃₁	—	2	2	no	—
159	C₂×C₄×C₃₁	—	2	2	no	—
160	C₂×C₄×C₃₁	—	2	2	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	2	2	no	—
163	C₂×C₄×C₃₁	—	2	2	no	—
164	C₂×C₄×C₃₁	—	2	2	no	—
165	C₂×C₄×C₃₁	—	4	2	no	—
166	C₂×C₄×C₃₁	—	4	2	no	—
167	C₂×C₄×C₃₁	—	4	2	no	—
168	C₂×C₄×C₃₁	—	4	2	no	—
169	C₂×C₄×C₃₁	—	4	2	no	—
170	C₂×C₄×C₃₁	—	4	2	no	—
171	C₂×C₄×C₃₁	—	4	2	no	—
172	C₂×C₄×C₃₁	—	4	2	no	—
173	C₂×C₄×C₃₁	—	4	2	no	—
174	C₂×C₄×C₃₁	—	4	2	no	—
175	C₂×C₄×C₃₁	—	4	2	no	—
176	C₂×C₄×C₃₁	—	4	2	no	—
177	C₂×C₄×C₃₁	#121	1	2	yes	—
178	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
179	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
180	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
181	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
182	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
183	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
184	C₈×C₃₁	—	2	2	no	—
185	C₈×C₃₁	—	2	2	no	—
186	C₈×C₃₁	—	2	2	no	—
187	C₈×C₃₁	#123	1	4	yes	—
188	C₂×#470	C₂×C₃₁	1	1	yes	characteristic
189	C₂×#405	C₂×C₃₁	1	1	yes	characteristic
190	#2625	C₂×C₃₁	1	2	yes	—
191	#1971	C₂×C₃₁	1	1	yes	characteristic, complemented
192	#1969	C₂×C₃₁	1	1	yes	characteristic, complemented
193	#1943	C₂×C₃₁	1	1	yes	characteristic, complemented
194	#1941	C₂×C₃₁	1	1	yes	characteristic, complemented
195	#1935	C₂×C₃₁	1	2	yes	complemented
196	#1933	C₂×C₃₁	1	2	yes	complemented
197	#1892	C₂×C₃₁	1	2	yes	—
198	#1809	C₂×C₃₁	1	1	yes	characteristic
199	C₂²×C₃₁	—	2	2	no	—
200	C₂²×C₃₁	—	2	4	no	—
201	C₂²×C₃₁	—	4	1	no	—
202	C₂²×C₃₁	—	4	1	no	—
203	C₂²×C₃₁	—	4	2	no	—
204	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
205	C₂²×C₃₁	#532	1	1	yes	characteristic
206	C₂²×C₃₁	#532	1	1	yes	characteristic
207	C₂²×C₃₁	#577	1	2	yes	—
208	C₂²×C₃₁	#578	1	2	yes	—
209	C₄×C₃₁	—	2	2	no	—
210	C₄×C₃₁	—	4	2	no	—
211	C₄×C₃₁	—	4	2	no	—
212	C₄×C₃₁	—	8	2	no	—
213	C₄×C₃₁	—	8	2	no	—
214	C₄×C₃₁	—	8	2	no	—
215	C₄×C₃₁	—	8	2	no	—
216	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
217	C₄×C₃₁	#525	1	1	yes	characteristic
218	C₄×C₃₁	#529	1	1	yes	characteristic
219	C₄×C₃₁	#529	1	1	yes	characteristic
220	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
221	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
222	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
223	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
224	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
225	#470	—	2	4	no	—
226	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
227	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
228	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
229	C₂×#103	C₂²×C₃₁	1	2	yes	—
230	C₂×#103	C₂²×C₃₁	1	2	yes	—
231	C₂×#103	C₂²×C₃₁	1	2	yes	—
232	#405	—	2	4	no	—
233	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
234	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
235	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
236	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
237	#400	—	2	2	no	—
238	#398	—	2	2	no	—
239	#397	—	2	2	no	—
240	#397	—	2	2	no	—
241	#391	C₂²×C₃₁	1	2	yes	—
242	#391	C₂²×C₃₁	1	2	yes	—
243	#389	C₂²×C₃₁	1	2	yes	—
244	#389	C₂²×C₃₁	1	2	yes	—
245	#380	C₂²×C₃₁	1	2	yes	—
246	#340	C₂²×C₃₁	1	2	yes	—
247	#340	C₂²×C₃₁	1	2	yes	—
248	#340	C₂²×C₃₁	1	2	yes	—
249	C₂×C₃₁	—	4	2	no	—
250	C₂×C₃₁	#2854	1	1	yes	characteristic
251	C₂×C₃₁	#3063	1	2	yes	—
252	C₂×C₃₁	#3173	1	1	yes	characteristic
253	C₂×C₃₁	#3177	1	1	yes	characteristic
254	C₂×C₃₁	#3178	1	2	yes	—
255	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
256	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
257	C₂×Q₁₆	—	2	2	no	—
258	C₂×Q₁₆	—	2	2	no	—
259	C₂×Q₁₆	—	2	2	no	—
260	C₂×M₁₆	—	2	2	no	—
261	C₂×M₁₆	—	2	2	no	—
262	C₂×M₁₆	D₄×C₃₁	1	4	yes	—
263	C₂²×C₈	—	2	2	no	—
264	C₂²×C₈	—	2	2	no	—
265	C₂²×C₈	—	2	2	no	—
266	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
267	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
268	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
269	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
270	C₂²⋊Q₈	—	2	4	no	—
271	C₂²⋊Q₈	—	2	4	no	—
272	#117	—	2	2	no	—
273	#117	—	2	2	no	—
274	C₂×C₄⋊C₄	—	2	2	no	—
275	C₂×C₄⋊C₄	—	2	2	no	—
276	C₂×C₄⋊C₄	—	2	2	no	—
277	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
278	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
279	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
280	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
281	C₂×C₂²⋊C₄	—	2	2	no	—
282	C₂×C₂²⋊C₄	—	2	2	no	—
283	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
284	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
285	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
286	#107	—	2	2	no	—
287	#106	—	2	2	no	—
288	#106	—	2	2	no	—
289	#103	—	2	4	no	—
290	#103	—	2	4	no	—
291	#103	—	2	4	no	—
292	#97	—	2	2	no	—
293	#97	—	2	2	no	—
294	C₄×C₈	—	2	2	no	—
295	C₄×C₈	—	2	2	no	—
296	#95	—	2	2	no	—
297	#95	—	2	2	no	—
298	#95	—	2	2	no	—
299	#95	—	2	2	no	—
300	#95	—	2	2	no	—
301	#95	—	2	2	no	—
302	#95	—	2	2	no	—
303	C₃₁	#33141	1	1	yes	characteristic, direct product factor, Sylow
304	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
305	C₂×Q₈	—	2	2	no	—
306	C₂×Q₈	—	2	2	no	—
307	C₂×Q₈	—	2	2	no	—
308	C₂×Q₈	—	2	2	no	—
309	C₂×Q₈	—	2	2	no	—
310	C₂×Q₈	—	2	2	no	—
311	C₂²×C₄	—	2	1	no	—
312	C₂²×C₄	—	2	1	no	—
313	C₂²×C₄	—	2	1	no	—
314	C₂²×C₄	—	2	1	no	—
315	C₂²×C₄	—	2	2	no	—
316	C₂²×C₄	—	2	2	no	—
317	C₂²×C₄	—	2	2	no	—
318	C₂²×C₄	—	2	2	no	—
319	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
320	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
321	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
322	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
323	C₂²×C₄	#41×C₃₁	1	2	yes	—
324	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
325	Q₁₆	—	4	4	no	—
326	M₁₆	—	2	8	no	—
327	C₂×C₈	—	2	8	no	—
328	C₂×C₈	—	2	1	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	—	2	1	no	—
332	C₂×C₈	—	2	1	no	—
333	C₂×C₈	—	2	1	no	—
334	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
335	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
336	C₂×C₈	#41×C₃₁	1	2	yes	—
337	C₂×C₈	#41×C₃₁	1	2	yes	—
338	C₂×C₈	#41×C₃₁	1	2	yes	—
339	C₂×C₈	#41×C₃₁	1	2	yes	—
340	C₄⋊C₄	—	2	4	no	—
341	C₄⋊C₄	—	2	2	no	—
342	C₄⋊C₄	—	2	2	no	—
343	C₄⋊C₄	—	2	2	no	—
344	C₄⋊C₄	—	2	2	no	—
345	C₄⋊C₄	—	2	2	no	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	4	4	no	—
348	C₄⋊C₄	—	8	2	no	—
349	C₄⋊C₄	—	8	2	no	—
350	C₂²⋊C₄	—	2	4	no	—
351	C₂²⋊C₄	—	4	4	no	—
352	C₂²⋊C₄	—	8	2	no	—
353	C₄²	—	2	1	no	—
354	C₄²	—	2	1	no	—
355	C₄²	—	2	1	no	—
356	C₄²	—	2	1	no	—
357	C₂³	—	2	1	no	—
358	C₂³	—	2	1	no	—
359	C₂³	—	2	2	no	—
360	C₂³	—	2	2	no	—
361	C₂³	C₃₁×#121	1	2	yes	—
362	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
363	Q₈	—	4	2	no	—
364	Q₈	—	4	2	no	—
365	Q₈	—	4	2	no	—
366	Q₈	—	4	2	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	4	2	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	—	4	2	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	—	4	2	no	—
391	C₂×C₄	C₃₁×#121	1	2	yes	—
392	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
393	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
394	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
395	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
396	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
397	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
398	C₈	—	2	2	no	—
399	C₈	—	2	2	no	—
400	C₈	—	2	2	no	—
401	C₈	C₃₁×#123	1	4	yes	—
402	C₂²	—	2	2	no	—
403	C₂²	—	2	4	no	—
404	C₂²	—	4	1	no	—
405	C₂²	—	4	1	no	—
406	C₂²	—	4	2	no	—
407	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
408	C₂²	C₃₁×#532	1	1	yes	characteristic
409	C₂²	C₃₁×#532	1	1	yes	characteristic
410	C₂²	C₃₁×#577	1	2	yes	—
411	C₂²	C₃₁×#578	1	2	yes	—
412	C₄	—	2	2	no	—
413	C₄	—	4	2	no	—
414	C₄	—	4	2	no	—
415	C₄	—	8	2	no	—
416	C₄	—	8	2	no	—
417	C₄	—	8	2	no	—
418	C₄	—	8	2	no	—
419	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
420	C₄	C₃₁×#525	1	1	yes	characteristic
421	C₄	C₃₁×#529	1	1	yes	characteristic
422	C₄	C₃₁×#529	1	1	yes	characteristic
423	C₂	—	4	2	no	—
424	C₂	C₃₁×#2854	1	1	yes	characteristic
425	C₂	C₃₁×#3063	1	2	yes	—
426	C₂	C₃₁×#3173	1	1	yes	characteristic
427	C₂	C₃₁×#3177	1	1	yes	characteristic
428	C₂	C₃₁×#3178	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂⁴×C₃₁	C₂×D₄
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
4. C₃₁×#1971	C₂	semidirect
5. C₃₁×#1969	C₂	semidirect
6. C₃₁×#1943	C₂	semidirect
7. C₃₁×#1941	C₂	semidirect
8. C₃₁×#1935	C₂	semidirect
9. C₃₁×#1933	C₂	semidirect
142. #33141	C₃₁	direct (N × H)
191. #1971	C₂×C₃₁	semidirect
192. #1969	C₂×C₃₁	semidirect
193. #1943	C₂×C₃₁	semidirect
194. #1941	C₂×C₃₁	semidirect
195. #1935	C₂×C₃₁	semidirect
196. #1933	C₂×C₃₁	semidirect
303. C₃₁	#33141	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×#470	2
C₂×C₃₁×#405	2
C₃₁×#2625	2
C₃₁×#1971	2
C₃₁×#1969	2
C₃₁×#1943	2
C₃₁×#1941	2
C₃₁×#1935	2
C₃₁×#1933	2
C₃₁×#1892	2
C₃₁×#1809	2
#33141	31

Subgroups — C31×#33141 #21489900

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33141 #21489900