Subgroups — C₃₁×#33061 #21489820

Order 7936.

770 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#487	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3079	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3039	C₂	1	1	yes	maximal, characteristic
4	C₄×C₃₁×#131	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1997	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1966	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1930	C₂	1	4	yes	maximal, complemented
8	C₃₁×#1908	C₂	1	4	yes	maximal
9	C₃₁×#1837	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#137	C₂²	1	2	yes	—
11	C₂×C₃₁×#131	C₂²	1	2	yes	—
12	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#487	—	2	1	no	—
16	C₃₁×#487	—	2	1	no	—
17	C₃₁×#487	—	2	2	no	—
18	C₃₁×#477	—	2	1	no	—
19	C₃₁×#476	—	2	1	no	—
20	C₃₁×#475	—	2	1	no	—
21	C₃₁×#474	—	2	1	no	—
22	C₃₁×#462	—	2	1	no	—
23	C₃₁×#462	—	2	1	no	—
24	C₃₁×#461	—	2	1	no	—
25	C₃₁×#461	—	2	1	no	—
26	C₃₁×#428	C₂²	1	2	yes	—
27	C₃₁×#423	C₂²	1	1	yes	characteristic
28	C₃₁×#423	C₂²	1	1	yes	characteristic
29	C₃₁×#419	C₂²	1	2	yes	—
30	C₃₁×#417	C₂²	1	2	yes	—
31	C₂×C₃₁×#103	C₂²	1	4	yes	—
32	C₂×C₃₁×#101	C₂²	1	4	yes	—
33	C₃₁×#405	—	2	2	no	—
34	C₃₁×#405	C₂²	1	2	yes	—
35	C₄×M₁₆×C₃₁	—	2	1	no	—
36	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₃₁	—	2	1	no	—
38	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
39	C₃₁×#343	C₂²	1	4	yes	—
40	C₃₁×#341	C₂²	1	4	yes	—
41	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
42	C₂²×Q₈×C₃₁	—	2	1	no	—
43	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
44	C₃₁×#137	—	2	8	no	—
45	C₂×Q₁₆×C₃₁	—	2	2	no	—
46	C₂×SD₁₆×C₃₁	—	2	2	no	—
47	C₃₁×#131	—	2	4	no	—
48	C₃₁×#131	—	2	4	no	—
49	C₂×M₁₆×C₃₁	—	2	2	no	—
50	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
51	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
52	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
53	C₂²×C₈×C₃₁	—	2	2	no	—
54	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
55	C₃₁×C₄⋊Q₈	—	2	1	no	—
56	C₃₁×C₄⋊Q₈	—	2	1	no	—
57	C₃₁×C₄⋊Q₈	—	2	1	no	—
58	C₃₁×C₄⋊Q₈	—	2	1	no	—
59	C₃₁×C₄⋊Q₈	—	2	1	no	—
60	C₃₁×C₄⋊Q₈	—	2	1	no	—
61	C₃₁×C₄⋊Q₈	—	2	1	no	—
62	C₃₁×C₄⋊Q₈	—	2	1	no	—
63	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
64	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
65	C₄×D₄×C₃₁	—	2	1	no	—
66	C₄×D₄×C₃₁	—	2	1	no	—
67	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
68	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
69	C₃₁×#117	—	2	1	no	—
70	C₃₁×#117	C₂³	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
72	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
73	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
74	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
75	C₂×C₄²×C₃₁	—	2	1	no	—
76	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
77	C₃₁×#107	—	2	2	no	—
78	C₃₁×#106	—	2	2	no	—
79	C₃₁×C₄⋊C₈	—	2	1	no	—
80	C₃₁×C₄⋊C₈	—	2	1	no	—
81	C₃₁×C₄⋊C₈	—	2	1	no	—
82	C₃₁×C₄⋊C₈	—	2	1	no	—
83	C₃₁×#103	—	2	2	no	—
84	C₃₁×#103	—	2	4	no	—
85	C₃₁×#103	—	2	4	no	—
86	C₃₁×#103	—	2	2	no	—
87	C₃₁×#103	—	2	2	no	—
88	C₃₁×#102	—	2	2	no	—
89	C₃₁×#101	—	2	8	no	—
90	C₃₁×#97	—	2	2	no	—
91	C₃₁×#97	—	2	1	no	—
92	C₄×C₈×C₃₁	—	2	2	no	—
93	C₄×C₈×C₃₁	—	2	1	no	—
94	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
95	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
96	C₄×C₈×C₃₁	D₄	1	2	yes	—
97	#41×C₃₁	—	2	2	no	—
98	#41×C₃₁	#41	1	4	yes	—
99	C₂×Q₈×C₃₁	—	2	2	no	—
100	C₂×Q₈×C₃₁	—	2	2	no	—
101	C₂×Q₈×C₃₁	—	2	4	no	—
102	C₂×Q₈×C₃₁	—	2	4	no	—
103	C₂×Q₈×C₃₁	—	4	1	no	—
104	C₂×Q₈×C₃₁	—	4	1	no	—
105	C₂×Q₈×C₃₁	—	8	2	no	—
106	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂×D₄×C₃₁	—	2	1	no	—
108	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₃₁	—	2	1	no	—
110	C₂²×C₄×C₃₁	—	2	1	no	—
111	C₂²×C₄×C₃₁	—	2	1	no	—
112	C₂²×C₄×C₃₁	—	2	2	no	—
113	C₂²×C₄×C₃₁	—	2	2	no	—
114	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
115	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
116	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
117	Q₁₆×C₃₁	—	4	4	no	—
118	SD₁₆×C₃₁	—	4	4	no	—
119	M₁₆×C₃₁	—	2	8	no	—
120	M₁₆×C₃₁	—	2	2	no	—
121	M₁₆×C₃₁	—	2	4	no	—
122	M₁₆×C₃₁	—	2	4	no	—
123	M₁₆×C₃₁	#41	1	4	yes	—
124	C₂×C₈×C₃₁	—	2	8	no	—
125	C₂×C₈×C₃₁	—	2	2	no	—
126	C₂×C₈×C₃₁	—	2	2	no	—
127	C₂×C₈×C₃₁	—	2	2	no	—
128	C₂×C₈×C₃₁	—	2	2	no	—
129	C₂×C₈×C₃₁	—	2	2	no	—
130	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
131	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
132	C₄⋊C₄×C₃₁	—	2	1	no	—
133	C₄⋊C₄×C₃₁	—	2	1	no	—
134	C₄⋊C₄×C₃₁	—	2	2	no	—
135	C₄⋊C₄×C₃₁	—	2	2	no	—
136	C₄⋊C₄×C₃₁	—	4	1	no	—
137	C₄⋊C₄×C₃₁	—	4	1	no	—
138	C₄⋊C₄×C₃₁	—	4	1	no	—
139	C₄⋊C₄×C₃₁	—	4	1	no	—
140	C₄⋊C₄×C₃₁	—	4	1	no	—
141	C₄⋊C₄×C₃₁	—	4	1	no	—
142	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
143	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
144	C₂²⋊C₄×C₃₁	—	2	1	no	—
145	C₂²⋊C₄×C₃₁	—	2	1	no	—
146	C₂²⋊C₄×C₃₁	—	2	1	no	—
147	C₄²×C₃₁	—	2	1	no	—
148	C₄²×C₃₁	—	2	1	no	—
149	C₄²×C₃₁	—	2	1	no	—
150	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
151	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
152	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
153	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
154	#33061	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₂³×C₃₁	—	2	1	no	—
156	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
157	Q₈×C₃₁	—	2	1	no	—
158	Q₈×C₃₁	—	4	4	no	—
159	Q₈×C₃₁	—	4	2	no	—
160	Q₈×C₃₁	—	4	2	no	—
161	Q₈×C₃₁	—	8	2	no	—
162	Q₈×C₃₁	—	8	2	no	—
163	Q₈×C₃₁	#124	1	2	yes	—
164	D₄×C₃₁	—	2	4	no	—
165	D₄×C₃₁	—	2	1	no	—
166	D₄×C₃₁	#124	1	2	yes	—
167	C₂×C₄×C₃₁	—	2	1	no	—
168	C₂×C₄×C₃₁	—	2	4	no	—
169	C₂×C₄×C₃₁	—	2	2	no	—
170	C₂×C₄×C₃₁	—	2	2	no	—
171	C₂×C₄×C₃₁	—	2	1	no	—
172	C₂×C₄×C₃₁	—	2	1	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	—	4	4	no	—
178	C₂×C₄×C₃₁	—	8	2	no	—
179	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
180	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
181	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
182	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
183	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
184	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
186	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
187	C₈×C₃₁	—	2	2	no	—
188	C₈×C₃₁	—	2	4	no	—
189	C₈×C₃₁	—	2	2	no	—
190	C₈×C₃₁	—	4	2	no	—
191	C₈×C₃₁	—	4	2	no	—
192	C₂×#487	C₂×C₃₁	1	1	yes	characteristic, complemented
193	#3079	C₂×C₃₁	1	1	yes	characteristic
194	#3039	C₂×C₃₁	1	1	yes	characteristic
195	C₄×#131	C₂×C₃₁	1	1	yes	characteristic
196	#1997	C₂×C₃₁	1	1	yes	characteristic, complemented
197	#1966	C₂×C₃₁	1	1	yes	characteristic, complemented
198	#1930	C₂×C₃₁	1	4	yes	complemented
199	#1908	C₂×C₃₁	1	4	yes	—
200	#1837	C₂×C₃₁	1	1	yes	characteristic, complemented
201	C₂²×C₃₁	—	2	1	no	—
202	C₂²×C₃₁	—	2	1	no	—
203	C₂²×C₃₁	—	2	2	no	—
204	C₂²×C₃₁	—	4	1	no	—
205	C₂²×C₃₁	—	4	1	no	—
206	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
207	C₂²×C₃₁	#529	1	2	yes	—
208	C₄×C₃₁	—	2	2	no	—
209	C₄×C₃₁	—	2	1	no	—
210	C₄×C₃₁	—	2	1	no	—
211	C₄×C₃₁	—	4	1	no	—
212	C₄×C₃₁	—	4	1	no	—
213	C₄×C₃₁	—	8	2	no	—
214	C₄×C₃₁	—	8	2	no	—
215	C₄×C₃₁	#471	1	2	yes	—
216	C₄×C₃₁	#473	1	2	yes	—
217	C₄×C₃₁	#525	1	2	yes	—
218	C₄×C₃₁	C₂×#124	1	1	yes	characteristic
219	C₄×C₃₁	#532	1	1	yes	characteristic
220	C₂×#137	C₂²×C₃₁	1	2	yes	—
221	C₂×#131	C₂²×C₃₁	1	2	yes	—
222	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
223	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
224	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
225	#487	—	2	1	no	—
226	#487	—	2	1	no	—
227	#487	—	2	2	no	—
228	#477	—	2	1	no	—
229	#476	—	2	1	no	—
230	#475	—	2	1	no	—
231	#474	—	2	1	no	—
232	#462	—	2	1	no	—
233	#462	—	2	1	no	—
234	#461	—	2	1	no	—
235	#461	—	2	1	no	—
236	#428	C₂²×C₃₁	1	2	yes	—
237	#423	C₂²×C₃₁	1	1	yes	characteristic
238	#423	C₂²×C₃₁	1	1	yes	characteristic
239	#419	C₂²×C₃₁	1	2	yes	—
240	#417	C₂²×C₃₁	1	2	yes	—
241	C₂×#103	C₂²×C₃₁	1	4	yes	—
242	C₂×#101	C₂²×C₃₁	1	4	yes	—
243	#405	—	2	2	no	—
244	#405	C₂²×C₃₁	1	2	yes	—
245	C₄×M₁₆	—	2	1	no	—
246	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
247	C₂×C₄×C₈	—	2	1	no	—
248	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
249	#343	C₂²×C₃₁	1	4	yes	—
250	#341	C₂²×C₃₁	1	4	yes	—
251	C₂×C₃₁	—	2	2	no	—
252	C₂×C₃₁	—	4	2	no	—
253	C₂×C₃₁	#2850	1	1	yes	characteristic
254	C₂×C₃₁	C₂×#471	1	1	yes	characteristic
255	C₂×C₃₁	C₂×#473	1	1	yes	characteristic
256	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
257	C₂²×Q₈	—	2	1	no	—
258	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
259	#137	—	2	8	no	—
260	C₂×Q₁₆	—	2	2	no	—
261	C₂×SD₁₆	—	2	2	no	—
262	#131	—	2	4	no	—
263	#131	—	2	4	no	—
264	C₂×M₁₆	—	2	2	no	—
265	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
266	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
267	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
268	C₂²×C₈	—	2	2	no	—
269	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
270	C₄⋊Q₈	—	2	1	no	—
271	C₄⋊Q₈	—	2	1	no	—
272	C₄⋊Q₈	—	2	1	no	—
273	C₄⋊Q₈	—	2	1	no	—
274	C₄⋊Q₈	—	2	1	no	—
275	C₄⋊Q₈	—	2	1	no	—
276	C₄⋊Q₈	—	2	1	no	—
277	C₄⋊Q₈	—	2	1	no	—
278	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
279	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
280	C₄×D₄	—	2	1	no	—
281	C₄×D₄	—	2	1	no	—
282	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
283	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
284	#117	—	2	1	no	—
285	#117	C₂³×C₃₁	1	1	yes	characteristic
286	C₂×C₄⋊C₄	—	2	1	no	—
287	C₂×C₄⋊C₄	—	2	1	no	—
288	C₂×C₄⋊C₄	—	2	1	no	—
289	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
290	C₂×C₄²	—	2	1	no	—
291	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
292	#107	—	2	2	no	—
293	#106	—	2	2	no	—
294	C₄⋊C₈	—	2	1	no	—
295	C₄⋊C₈	—	2	1	no	—
296	C₄⋊C₈	—	2	1	no	—
297	C₄⋊C₈	—	2	1	no	—
298	#103	—	2	2	no	—
299	#103	—	2	4	no	—
300	#103	—	2	4	no	—
301	#103	—	2	2	no	—
302	#103	—	2	2	no	—
303	#102	—	2	2	no	—
304	#101	—	2	8	no	—
305	#97	—	2	2	no	—
306	#97	—	2	1	no	—
307	C₄×C₈	—	2	2	no	—
308	C₄×C₈	—	2	1	no	—
309	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
310	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
311	C₄×C₈	D₄×C₃₁	1	2	yes	—
312	C₃₁	#33061	1	1	yes	characteristic, direct product factor, Sylow
313	#41	—	2	2	no	—
314	#41	#41×C₃₁	1	4	yes	—
315	C₂×Q₈	—	2	2	no	—
316	C₂×Q₈	—	2	2	no	—
317	C₂×Q₈	—	2	4	no	—
318	C₂×Q₈	—	2	4	no	—
319	C₂×Q₈	—	4	1	no	—
320	C₂×Q₈	—	4	1	no	—
321	C₂×Q₈	—	8	2	no	—
322	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
323	C₂×D₄	—	2	1	no	—
324	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	—	2	1	no	—
327	C₂²×C₄	—	2	1	no	—
328	C₂²×C₄	—	2	2	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
331	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
332	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
333	Q₁₆	—	4	4	no	—
334	SD₁₆	—	4	4	no	—
335	M₁₆	—	2	8	no	—
336	M₁₆	—	2	2	no	—
337	M₁₆	—	2	4	no	—
338	M₁₆	—	2	4	no	—
339	M₁₆	#41×C₃₁	1	4	yes	—
340	C₂×C₈	—	2	8	no	—
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	—	2	2	no	—
343	C₂×C₈	—	2	2	no	—
344	C₂×C₈	—	2	2	no	—
345	C₂×C₈	—	2	2	no	—
346	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
347	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
348	C₄⋊C₄	—	2	1	no	—
349	C₄⋊C₄	—	2	1	no	—
350	C₄⋊C₄	—	2	2	no	—
351	C₄⋊C₄	—	2	2	no	—
352	C₄⋊C₄	—	4	1	no	—
353	C₄⋊C₄	—	4	1	no	—
354	C₄⋊C₄	—	4	1	no	—
355	C₄⋊C₄	—	4	1	no	—
356	C₄⋊C₄	—	4	1	no	—
357	C₄⋊C₄	—	4	1	no	—
358	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
359	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
360	C₂²⋊C₄	—	2	1	no	—
361	C₂²⋊C₄	—	2	1	no	—
362	C₂²⋊C₄	—	2	1	no	—
363	C₄²	—	2	1	no	—
364	C₄²	—	2	1	no	—
365	C₄²	—	2	1	no	—
366	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
367	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
368	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
369	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
370	C₂³	—	2	1	no	—
371	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
372	Q₈	—	2	1	no	—
373	Q₈	—	4	4	no	—
374	Q₈	—	4	2	no	—
375	Q₈	—	4	2	no	—
376	Q₈	—	8	2	no	—
377	Q₈	—	8	2	no	—
378	Q₈	C₃₁×#124	1	2	yes	—
379	D₄	—	2	4	no	—
380	D₄	—	2	1	no	—
381	D₄	C₃₁×#124	1	2	yes	—
382	C₂×C₄	—	2	1	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	4	no	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	2	1	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	—	4	2	no	—
391	C₂×C₄	—	4	2	no	—
392	C₂×C₄	—	4	4	no	—
393	C₂×C₄	—	8	2	no	—
394	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
395	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
396	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
397	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
398	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
399	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
400	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
401	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
402	C₈	—	2	2	no	—
403	C₈	—	2	4	no	—
404	C₈	—	2	2	no	—
405	C₈	—	4	2	no	—
406	C₈	—	4	2	no	—
407	C₂²	—	2	1	no	—
408	C₂²	—	2	1	no	—
409	C₂²	—	2	2	no	—
410	C₂²	—	4	1	no	—
411	C₂²	—	4	1	no	—
412	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
413	C₂²	C₃₁×#529	1	2	yes	—
414	C₄	—	2	2	no	—
415	C₄	—	2	1	no	—
416	C₄	—	2	1	no	—
417	C₄	—	4	1	no	—
418	C₄	—	4	1	no	—
419	C₄	—	8	2	no	—
420	C₄	—	8	2	no	—
421	C₄	C₃₁×#471	1	2	yes	—
422	C₄	C₃₁×#473	1	2	yes	—
423	C₄	C₃₁×#525	1	2	yes	—
424	C₄	C₂×C₃₁×#124	1	1	yes	characteristic
425	C₄	C₃₁×#532	1	1	yes	characteristic
426	C₂	—	2	2	no	—
427	C₂	—	4	2	no	—
428	C₂	C₃₁×#2850	1	1	yes	characteristic
429	C₂	C₂×C₃₁×#471	1	1	yes	characteristic
430	C₂	C₂×C₃₁×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×#41×C₃₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#487	C₂	semidirect
5. C₃₁×#1997	C₂	semidirect
6. C₃₁×#1966	C₂	semidirect
7. C₃₁×#1930	C₂	semidirect
9. C₃₁×#1837	C₂	semidirect
154. #33061	C₃₁	direct (N × H)
192. C₂×#487	C₂×C₃₁	semidirect
196. #1997	C₂×C₃₁	semidirect
197. #1966	C₂×C₃₁	semidirect
198. #1930	C₂×C₃₁	semidirect
200. #1837	C₂×C₃₁	semidirect
312. C₃₁	#33061	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×#487	2
C₃₁×#3079	2
C₃₁×#3039	2
C₄×C₃₁×#131	2
C₃₁×#1997	2
C₃₁×#1966	2
C₃₁×#1930	2
C₃₁×#1908	2
C₃₁×#1837	2
#33061	31

Subgroups — C31×#33061 #21489820

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33061 #21489820