Subgroups — C₃₁×#19705 #21476464

Order 7936.

746 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	C₃₁×#3230	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3209	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3208	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#3208	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#3160	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×#486	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1722	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1720	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1719	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1714	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1452	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#533	C₂²	1	1	yes	characteristic
13	C₃₁×#533	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
16	C₃₁×#500	C₂²	1	1	yes	characteristic, complemented
17	C₃₁×#500	C₂²	1	1	yes	characteristic, complemented
18	C₃₁×#500	C₂²	1	2	yes	complemented
19	C₃₁×#499	—	2	2	no	—
20	C₃₁×#498	C₂²	1	1	yes	characteristic, complemented
21	C₃₁×#498	C₂²	1	1	yes	characteristic, complemented
22	C₃₁×#494	C₂²	1	2	yes	complemented
23	C₃₁×#492	C₂²	1	2	yes	—
24	C₃₁×#492	C₂²	1	2	yes	—
25	C₃₁×#487	C₂²	1	2	yes	complemented
26	C₃₁×#487	C₂²	1	2	yes	complemented
27	C₃₁×#486	—	2	2	no	—
28	C₃₁×#486	C₂²	1	2	yes	—
29	C₃₁×#486	C₂²	1	2	yes	—
30	C₃₁×#486	C₂²	1	2	yes	—
31	C₃₁×#486	C₂²	1	2	yes	—
32	C₃₁×#484	—	2	2	no	—
33	C₃₁×#477	—	2	2	no	—
34	C₃₁×#462	—	2	2	no	—
35	C₃₁×#460	—	2	2	no	—
36	C₃₁×#460	—	2	2	no	—
37	C₃₁×#451	—	2	2	no	—
38	C₃₁×#450	—	2	2	no	—
39	C₃₁×#450	—	2	2	no	—
40	C₃₁×#431	C₂²	1	1	yes	characteristic
41	C₃₁×#431	C₂²	1	1	yes	characteristic
42	C₃₁×#427	—	2	2	no	—
43	C₂×C₃₁×#102	—	2	2	no	—
44	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
45	C₃₁×#322	C₂²	1	2	yes	complemented
46	C₃₁×#322	C₂²	1	2	yes	complemented
47	C₂²×Q₈×C₃₁	—	2	1	no	—
48	C₂²×D₄×C₃₁	—	2	1	no	—
49	C₂²×D₄×C₃₁	—	2	1	no	—
50	C₂²×D₄×C₃₁	—	2	1	no	—
51	C₂×Q₁₆×C₃₁	—	2	4	no	—
52	C₂×SD₁₆×C₃₁	—	2	4	no	—
53	C₂×SD₁₆×C₃₁	—	2	4	no	—
54	C₂²×C₈×C₃₁	D₄	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	C₂³	1	2	yes	—
59	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
60	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
61	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
62	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
63	C₃₁×#127	—	2	1	no	—
64	C₃₁×#127	—	2	1	no	—
65	C₃₁×#127	—	2	1	no	—
66	C₃₁×#127	C₂³	1	2	yes	—
67	C₃₁×#125	—	2	1	no	—
68	C₃₁×#125	—	2	1	no	—
69	C₃₁×C₂²⋊Q₈	—	2	1	no	—
70	C₃₁×C₂²⋊Q₈	—	2	1	no	—
71	C₃₁×C₂²⋊Q₈	—	2	1	no	—
72	C₃₁×C₂²⋊Q₈	—	2	1	no	—
73	C₄×Q₈×C₃₁	—	2	1	no	—
74	C₄×Q₈×C₃₁	—	2	1	no	—
75	C₄×Q₈×C₃₁	—	2	1	no	—
76	C₄×Q₈×C₃₁	—	2	1	no	—
77	C₃₁×#117	—	2	1	no	—
78	C₃₁×#117	—	2	1	no	—
79	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
80	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
81	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
82	C₃₁×#107	—	2	2	no	—
83	C₃₁×#107	—	2	2	no	—
84	C₃₁×#107	—	2	4	no	—
85	C₃₁×#106	—	2	2	no	—
86	C₃₁×#106	—	2	2	no	—
87	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
88	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
89	C₃₁×#103	—	2	4	no	—
90	C₃₁×#103	—	2	4	no	—
91	C₃₁×#102	—	2	4	no	—
92	C₃₁×#102	—	2	4	no	—
93	C₃₁×#102	—	2	4	no	—
94	C₃₁×#102	—	2	4	no	—
95	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
96	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
97	C₄×C₈×C₃₁	D₄	1	2	yes	—
98	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
99	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
100	C₄×C₈×C₃₁	C₂³	1	2	yes	—
101	C₄×C₈×C₃₁	C₂³	1	2	yes	—
102	C₂⁴×C₃₁	—	4	1	no	—
103	C₂×Q₈×C₃₁	—	2	1	no	—
104	C₂×Q₈×C₃₁	—	2	2	no	—
105	C₂×Q₈×C₃₁	—	2	1	no	—
106	C₂×Q₈×C₃₁	—	2	1	no	—
107	C₂×Q₈×C₃₁	—	2	1	no	—
108	C₂×Q₈×C₃₁	—	4	1	no	—
109	C₂×Q₈×C₃₁	—	8	2	no	—
110	C₂×D₄×C₃₁	—	2	2	no	—
111	C₂×D₄×C₃₁	—	2	2	no	—
112	C₂×D₄×C₃₁	—	2	2	no	—
113	C₂×D₄×C₃₁	—	4	1	no	—
114	C₂×D₄×C₃₁	—	4	1	no	—
115	C₂×D₄×C₃₁	—	4	1	no	—
116	C₂×D₄×C₃₁	—	4	4	no	—
117	C₂×D₄×C₃₁	—	8	2	no	—
118	C₂×D₄×C₃₁	—	8	2	no	—
119	C₂²×C₄×C₃₁	—	4	1	no	—
120	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
123	Q₁₆×C₃₁	—	8	4	no	—
124	SD₁₆×C₃₁	—	8	4	no	—
125	SD₁₆×C₃₁	—	8	4	no	—
126	C₂×C₈×C₃₁	—	2	4	no	—
127	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
128	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
129	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
130	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
131	C₄⋊C₄×C₃₁	—	2	2	no	—
132	C₄⋊C₄×C₃₁	—	2	1	no	—
133	C₄⋊C₄×C₃₁	—	2	1	no	—
134	C₄⋊C₄×C₃₁	—	2	1	no	—
135	C₄⋊C₄×C₃₁	—	2	2	no	—
136	C₄⋊C₄×C₃₁	—	2	1	no	—
137	C₄⋊C₄×C₃₁	—	2	1	no	—
138	C₄⋊C₄×C₃₁	—	2	1	no	—
139	C₄⋊C₄×C₃₁	—	2	1	no	—
140	C₄⋊C₄×C₃₁	—	2	1	no	—
141	C₄⋊C₄×C₃₁	—	4	1	no	—
142	C₄⋊C₄×C₃₁	—	4	1	no	—
143	C₄⋊C₄×C₃₁	—	4	1	no	—
144	C₄⋊C₄×C₃₁	—	4	1	no	—
145	C₄⋊C₄×C₃₁	—	4	1	no	—
146	C₄⋊C₄×C₃₁	—	4	1	no	—
147	C₂²⋊C₄×C₃₁	—	4	1	no	—
148	C₂²⋊C₄×C₃₁	—	4	1	no	—
149	C₄²×C₃₁	—	4	1	no	—
150	C₄²×C₃₁	—	4	1	no	—
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	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
155	#19705	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₂³×C₃₁	—	4	2	no	—
157	C₂³×C₃₁	—	8	2	no	—
158	C₂³×C₃₁	—	8	2	no	—
159	C₂³×C₃₁	—	8	2	no	—
160	C₂³×C₃₁	#127	1	1	yes	characteristic
161	Q₈×C₃₁	—	4	2	no	—
162	Q₈×C₃₁	—	4	2	no	—
163	Q₈×C₃₁	—	4	4	no	—
164	Q₈×C₃₁	—	4	2	no	—
165	Q₈×C₃₁	—	4	2	no	—
166	Q₈×C₃₁	—	8	2	no	—
167	D₄×C₃₁	—	4	4	no	—
168	D₄×C₃₁	—	4	4	no	—
169	D₄×C₃₁	—	4	4	no	—
170	D₄×C₃₁	—	8	2	no	—
171	D₄×C₃₁	—	8	2	no	—
172	D₄×C₃₁	—	8	2	no	—
173	C₂×C₄×C₃₁	—	2	2	no	—
174	C₂×C₄×C₃₁	—	2	2	no	—
175	C₂×C₄×C₃₁	—	4	1	no	—
176	C₂×C₄×C₃₁	—	4	1	no	—
177	C₂×C₄×C₃₁	—	4	1	no	—
178	C₂×C₄×C₃₁	—	4	1	no	—
179	C₂×C₄×C₃₁	—	4	2	no	—
180	C₂×C₄×C₃₁	—	8	2	no	—
181	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
182	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
183	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
184	C₂×C₄×C₃₁	#137	1	4	yes	—
185	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
186	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
187	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
188	C₈×C₃₁	—	2	2	no	—
189	C₈×C₃₁	—	2	2	no	—
190	C₈×C₃₁	—	2	4	no	—
191	C₈×C₃₁	—	2	4	no	—
192	#3230	C₂×C₃₁	1	1	yes	characteristic, complemented
193	#3209	C₂×C₃₁	1	1	yes	characteristic, complemented
194	#3208	C₂×C₃₁	1	1	yes	characteristic
195	#3208	C₂×C₃₁	1	1	yes	characteristic
196	#3160	C₂×C₃₁	1	1	yes	characteristic, complemented
197	C₂×#486	C₂×C₃₁	1	1	yes	characteristic
198	#1722	C₂×C₃₁	1	2	yes	complemented
199	#1720	C₂×C₃₁	1	2	yes	complemented
200	#1719	C₂×C₃₁	1	2	yes	complemented
201	#1714	C₂×C₃₁	1	2	yes	complemented
202	#1452	C₂×C₃₁	1	1	yes	characteristic, complemented
203	C₂²×C₃₁	—	2	1	no	—
204	C₂²×C₃₁	—	2	1	no	—
205	C₂²×C₃₁	—	8	2	no	—
206	C₂²×C₃₁	—	8	2	no	—
207	C₂²×C₃₁	—	8	2	no	—
208	C₂²×C₃₁	—	16	4	no	—
209	C₂²×C₃₁	#497	1	2	yes	—
210	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
211	C₄×C₃₁	—	2	1	no	—
212	C₄×C₃₁	—	2	1	no	—
213	C₄×C₃₁	—	8	1	no	—
214	C₄×C₃₁	—	8	1	no	—
215	C₄×C₃₁	—	8	2	no	—
216	C₄×C₃₁	—	8	1	no	—
217	C₄×C₃₁	—	8	1	no	—
218	C₄×C₃₁	#497	1	2	yes	—
219	C₄×C₃₁	C₂×#137	1	2	yes	—
220	C₄×C₃₁	#576	1	2	yes	—
221	C₄×C₃₁	#577	1	2	yes	—
222	#533	C₂²×C₃₁	1	1	yes	characteristic
223	#533	C₂²×C₃₁	1	1	yes	characteristic
224	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
225	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
226	#500	C₂²×C₃₁	1	1	yes	characteristic, complemented
227	#500	C₂²×C₃₁	1	1	yes	characteristic, complemented
228	#500	C₂²×C₃₁	1	2	yes	complemented
229	#499	—	2	2	no	—
230	#498	C₂²×C₃₁	1	1	yes	characteristic, complemented
231	#498	C₂²×C₃₁	1	1	yes	characteristic, complemented
232	#494	C₂²×C₃₁	1	2	yes	complemented
233	#492	C₂²×C₃₁	1	2	yes	—
234	#492	C₂²×C₃₁	1	2	yes	—
235	#487	C₂²×C₃₁	1	2	yes	complemented
236	#487	C₂²×C₃₁	1	2	yes	complemented
237	#486	—	2	2	no	—
238	#486	C₂²×C₃₁	1	2	yes	—
239	#486	C₂²×C₃₁	1	2	yes	—
240	#486	C₂²×C₃₁	1	2	yes	—
241	#486	C₂²×C₃₁	1	2	yes	—
242	#484	—	2	2	no	—
243	#477	—	2	2	no	—
244	#462	—	2	2	no	—
245	#460	—	2	2	no	—
246	#460	—	2	2	no	—
247	#451	—	2	2	no	—
248	#450	—	2	2	no	—
249	#450	—	2	2	no	—
250	#431	C₂²×C₃₁	1	1	yes	characteristic
251	#431	C₂²×C₃₁	1	1	yes	characteristic
252	#427	—	2	2	no	—
253	C₂×#102	—	2	2	no	—
254	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
255	#322	C₂²×C₃₁	1	2	yes	complemented
256	#322	C₂²×C₃₁	1	2	yes	complemented
257	C₂×C₃₁	—	2	2	no	—
258	C₂×C₃₁	—	16	2	no	—
259	C₂×C₃₁	C₂×#497	1	1	yes	characteristic
260	C₂×C₃₁	#3154	1	1	yes	characteristic
261	C₂×C₃₁	#3155	1	1	yes	characteristic
262	C₂²×Q₈	—	2	1	no	—
263	C₂²×D₄	—	2	1	no	—
264	C₂²×D₄	—	2	1	no	—
265	C₂²×D₄	—	2	1	no	—
266	C₂×Q₁₆	—	2	4	no	—
267	C₂×SD₁₆	—	2	4	no	—
268	C₂×SD₁₆	—	2	4	no	—
269	C₂²×C₈	D₄×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	C₂³×C₃₁	1	2	yes	—
274	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
275	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
276	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
277	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
278	#127	—	2	1	no	—
279	#127	—	2	1	no	—
280	#127	—	2	1	no	—
281	#127	C₂³×C₃₁	1	2	yes	—
282	#125	—	2	1	no	—
283	#125	—	2	1	no	—
284	C₂²⋊Q₈	—	2	1	no	—
285	C₂²⋊Q₈	—	2	1	no	—
286	C₂²⋊Q₈	—	2	1	no	—
287	C₂²⋊Q₈	—	2	1	no	—
288	C₄×Q₈	—	2	1	no	—
289	C₄×Q₈	—	2	1	no	—
290	C₄×Q₈	—	2	1	no	—
291	C₄×Q₈	—	2	1	no	—
292	#117	—	2	1	no	—
293	#117	—	2	1	no	—
294	C₂×C₄⋊C₄	—	2	1	no	—
295	C₂×C₄⋊C₄	—	2	1	no	—
296	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
297	#107	—	2	2	no	—
298	#107	—	2	2	no	—
299	#107	—	2	4	no	—
300	#106	—	2	2	no	—
301	#106	—	2	2	no	—
302	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
303	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
304	#103	—	2	4	no	—
305	#103	—	2	4	no	—
306	#102	—	2	4	no	—
307	#102	—	2	4	no	—
308	#102	—	2	4	no	—
309	#102	—	2	4	no	—
310	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
311	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
312	C₄×C₈	D₄×C₃₁	1	2	yes	—
313	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
314	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
315	C₄×C₈	C₂³×C₃₁	1	2	yes	—
316	C₄×C₈	C₂³×C₃₁	1	2	yes	—
317	C₃₁	#19705	1	1	yes	characteristic, direct product factor, Sylow
318	C₂⁴	—	4	1	no	—
319	C₂×Q₈	—	2	1	no	—
320	C₂×Q₈	—	2	2	no	—
321	C₂×Q₈	—	2	1	no	—
322	C₂×Q₈	—	2	1	no	—
323	C₂×Q₈	—	2	1	no	—
324	C₂×Q₈	—	4	1	no	—
325	C₂×Q₈	—	8	2	no	—
326	C₂×D₄	—	2	2	no	—
327	C₂×D₄	—	2	2	no	—
328	C₂×D₄	—	2	2	no	—
329	C₂×D₄	—	4	1	no	—
330	C₂×D₄	—	4	1	no	—
331	C₂×D₄	—	4	1	no	—
332	C₂×D₄	—	4	4	no	—
333	C₂×D₄	—	8	2	no	—
334	C₂×D₄	—	8	2	no	—
335	C₂²×C₄	—	4	1	no	—
336	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
337	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
338	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
339	Q₁₆	—	8	4	no	—
340	SD₁₆	—	8	4	no	—
341	SD₁₆	—	8	4	no	—
342	C₂×C₈	—	2	4	no	—
343	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
344	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
345	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
346	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
347	C₄⋊C₄	—	2	2	no	—
348	C₄⋊C₄	—	2	1	no	—
349	C₄⋊C₄	—	2	1	no	—
350	C₄⋊C₄	—	2	1	no	—
351	C₄⋊C₄	—	2	2	no	—
352	C₄⋊C₄	—	2	1	no	—
353	C₄⋊C₄	—	2	1	no	—
354	C₄⋊C₄	—	2	1	no	—
355	C₄⋊C₄	—	2	1	no	—
356	C₄⋊C₄	—	2	1	no	—
357	C₄⋊C₄	—	4	1	no	—
358	C₄⋊C₄	—	4	1	no	—
359	C₄⋊C₄	—	4	1	no	—
360	C₄⋊C₄	—	4	1	no	—
361	C₄⋊C₄	—	4	1	no	—
362	C₄⋊C₄	—	4	1	no	—
363	C₂²⋊C₄	—	4	1	no	—
364	C₂²⋊C₄	—	4	1	no	—
365	C₄²	—	4	1	no	—
366	C₄²	—	4	1	no	—
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	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
371	C₂³	—	4	2	no	—
372	C₂³	—	8	2	no	—
373	C₂³	—	8	2	no	—
374	C₂³	—	8	2	no	—
375	C₂³	C₃₁×#127	1	1	yes	characteristic
376	Q₈	—	4	2	no	—
377	Q₈	—	4	2	no	—
378	Q₈	—	4	4	no	—
379	Q₈	—	4	2	no	—
380	Q₈	—	4	2	no	—
381	Q₈	—	8	2	no	—
382	D₄	—	4	4	no	—
383	D₄	—	4	4	no	—
384	D₄	—	4	4	no	—
385	D₄	—	8	2	no	—
386	D₄	—	8	2	no	—
387	D₄	—	8	2	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	4	1	no	—
391	C₂×C₄	—	4	1	no	—
392	C₂×C₄	—	4	1	no	—
393	C₂×C₄	—	4	1	no	—
394	C₂×C₄	—	4	2	no	—
395	C₂×C₄	—	8	2	no	—
396	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
397	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
398	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
399	C₂×C₄	C₃₁×#137	1	4	yes	—
400	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
401	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
402	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
403	C₈	—	2	2	no	—
404	C₈	—	2	2	no	—
405	C₈	—	2	4	no	—
406	C₈	—	2	4	no	—
407	C₂²	—	2	1	no	—
408	C₂²	—	2	1	no	—
409	C₂²	—	8	2	no	—
410	C₂²	—	8	2	no	—
411	C₂²	—	8	2	no	—
412	C₂²	—	16	4	no	—
413	C₂²	C₃₁×#497	1	2	yes	—
414	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
415	C₄	—	2	1	no	—
416	C₄	—	2	1	no	—
417	C₄	—	8	1	no	—
418	C₄	—	8	1	no	—
419	C₄	—	8	2	no	—
420	C₄	—	8	1	no	—
421	C₄	—	8	1	no	—
422	C₄	C₃₁×#497	1	2	yes	—
423	C₄	C₂×C₃₁×#137	1	2	yes	—
424	C₄	C₃₁×#576	1	2	yes	—
425	C₄	C₃₁×#577	1	2	yes	—
426	C₂	—	2	2	no	—
427	C₂	—	16	2	no	—
428	C₂	C₂×C₃₁×#497	1	1	yes	characteristic
429	C₂	C₃₁×#3154	1	1	yes	characteristic
430	C₂	C₃₁×#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ 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. C₃₁×#3230	C₂	semidirect
2. C₃₁×#3209	C₂	semidirect
5. C₃₁×#3160	C₂	semidirect
7. C₃₁×#1722	C₂	semidirect
8. C₃₁×#1720	C₂	semidirect
9. C₃₁×#1719	C₂	semidirect
10. C₃₁×#1714	C₂	semidirect
11. C₃₁×#1452	C₂	semidirect
16. C₃₁×#500	C₂²	semidirect
17. C₃₁×#500	C₂²	semidirect
18. C₃₁×#500	C₂²	semidirect
20. C₃₁×#498	C₂²	semidirect
21. C₃₁×#498	C₂²	semidirect
22. C₃₁×#494	C₂²	semidirect
25. C₃₁×#487	C₂²	semidirect
26. C₃₁×#487	C₂²	semidirect
45. C₃₁×#322	C₂²	semidirect
46. C₃₁×#322	C₂²	semidirect
155. #19705	C₃₁	direct (N × H)
192. #3230	C₂×C₃₁	semidirect
193. #3209	C₂×C₃₁	semidirect
196. #3160	C₂×C₃₁	semidirect
198. #1722	C₂×C₃₁	semidirect
199. #1720	C₂×C₃₁	semidirect
200. #1719	C₂×C₃₁	semidirect
201. #1714	C₂×C₃₁	semidirect
202. #1452	C₂×C₃₁	semidirect
226. #500	C₂²×C₃₁	semidirect
227. #500	C₂²×C₃₁	semidirect
228. #500	C₂²×C₃₁	semidirect
230. #498	C₂²×C₃₁	semidirect
231. #498	C₂²×C₃₁	semidirect
232. #494	C₂²×C₃₁	semidirect
235. #487	C₂²×C₃₁	semidirect
236. #487	C₂²×C₃₁	semidirect
255. #322	C₂²×C₃₁	semidirect
256. #322	C₂²×C₃₁	semidirect
317. C₃₁	#19705	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3230	2
C₃₁×#3209	2
C₃₁×#3208	2
C₃₁×#3160	2
C₂×C₃₁×#486	2
C₃₁×#1722	2
C₃₁×#1720	2
C₃₁×#1719	2
C₃₁×#1714	2
C₃₁×#1452	2
#19705	31

Subgroups — C31×#19705 #21476464

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19705 #21476464