Subgroups — C₃₁×#19622 #21476381

Order 7936.

784 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	C₃₁×#3229	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3206	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3167	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#3165	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₁×#498	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#3152	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1716	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1714	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1707	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1705	C₂	1	2	yes	maximal, complemented
11	C₈×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#137	—	2	2	no	—
13	C₃₁×#533	C₂²	1	1	yes	characteristic
14	C₃₁×#532	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₃₁×#501	—	2	2	no	—
18	C₃₁×#500	C₂²	1	2	yes	—
19	C₃₁×#498	—	2	2	no	—
20	C₃₁×#498	C₂²	1	2	yes	—
21	C₃₁×#498	C₂²	1	2	yes	—
22	C₃₁×#498	C₂²	1	2	yes	—
23	C₃₁×#498	C₂²	1	1	yes	characteristic
24	C₃₁×#498	C₂²	1	2	yes	—
25	C₃₁×#497	—	2	2	no	—
26	C₃₁×#494	C₂²	1	2	yes	—
27	C₃₁×#492	C₂²	1	2	yes	—
28	C₃₁×#492	C₂²	1	1	yes	characteristic
29	C₃₁×#492	C₂²	1	2	yes	—
30	C₃₁×#487	C₂²	1	1	yes	characteristic
31	C₃₁×#487	C₂²	1	2	yes	—
32	C₃₁×#487	C₂²	1	2	yes	—
33	C₃₁×#486	C₂²	1	1	yes	characteristic
34	C₃₁×#483	—	2	2	no	—
35	C₃₁×#475	—	2	2	no	—
36	C₃₁×#465	—	2	2	no	—
37	C₃₁×#461	—	2	2	no	—
38	C₃₁×#448	—	2	2	no	—
39	C₄×SD₁₆×C₃₁	—	2	2	no	—
40	C₃₁×#431	C₂²	1	1	yes	characteristic
41	C₃₁×#428	—	2	2	no	—
42	C₂×C₃₁×#106	—	2	2	no	—
43	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
44	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
45	C₃₁×#322	C₂²	1	2	yes	—
46	C₈²×C₃₁	C₂²	1	2	yes	—
47	C₂×#41×C₃₁	—	2	1	no	—
48	C₂²×Q₈×C₃₁	—	2	1	no	—
49	C₂²×Q₈×C₃₁	—	2	1	no	—
50	C₃₁×#137	—	4	8	no	—
51	C₂×Q₁₆×C₃₁	—	2	4	no	—
52	C₂×SD₁₆×C₃₁	—	2	2	no	—
53	C₂×SD₁₆×C₃₁	—	2	4	no	—
54	C₂×SD₁₆×C₃₁	—	2	4	no	—
55	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
56	C₂²×C₈×C₃₁	D₄	1	2	yes	—
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	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
64	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
65	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
66	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
67	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
68	C₃₁×#127	C₂³	1	1	yes	characteristic
69	C₃₁×#125	—	2	1	no	—
70	C₃₁×#124	—	2	1	no	—
71	C₃₁×C₂²⋊Q₈	—	2	1	no	—
72	C₃₁×C₂²⋊Q₈	—	2	1	no	—
73	C₃₁×#121	—	2	1	no	—
74	C₃₁×#121	—	2	1	no	—
75	C₄×Q₈×C₃₁	—	2	1	no	—
76	C₄×Q₈×C₃₁	—	2	1	no	—
77	C₄×D₄×C₃₁	—	2	1	no	—
78	C₄×D₄×C₃₁	—	2	1	no	—
79	C₃₁×#117	—	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×C₄²×C₃₁	C₂³	1	1	yes	characteristic
85	C₃₁×#107	—	2	4	no	—
86	C₃₁×#106	—	2	4	no	—
87	C₃₁×#106	—	2	4	no	—
88	C₃₁×#106	—	2	4	no	—
89	C₃₁×#106	—	2	2	no	—
90	C₃₁×#106	—	2	4	no	—
91	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
92	C₃₁×#103	—	2	2	no	—
93	C₃₁×#103	—	2	4	no	—
94	C₃₁×#103	—	2	4	no	—
95	C₃₁×#102	—	2	2	no	—
96	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
97	C₃₁×#97	D₄	1	2	yes	—
98	C₄×C₈×C₃₁	D₄	1	2	yes	—
99	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
100	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
101	C₄×C₈×C₃₁	C₂³	1	2	yes	—
102	C₄×C₈×C₃₁	C₂³	1	2	yes	—
103	#41×C₃₁	—	4	4	no	—
104	C₂×Q₈×C₃₁	—	2	1	no	—
105	C₂×Q₈×C₃₁	—	2	1	no	—
106	C₂×Q₈×C₃₁	—	2	2	no	—
107	C₂×Q₈×C₃₁	—	2	1	no	—
108	C₂×Q₈×C₃₁	—	2	2	no	—
109	C₂×Q₈×C₃₁	—	4	1	no	—
110	C₂×Q₈×C₃₁	—	4	1	no	—
111	C₂×Q₈×C₃₁	—	4	4	no	—
112	C₂×Q₈×C₃₁	—	8	2	no	—
113	C₂×D₄×C₃₁	—	2	1	no	—
114	C₂×D₄×C₃₁	—	2	1	no	—
115	C₂×D₄×C₃₁	—	2	1	no	—
116	C₂×D₄×C₃₁	—	4	1	no	—
117	C₂²×C₄×C₃₁	—	4	1	no	—
118	C₂²×C₄×C₃₁	—	4	1	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	—	4	8	no	—
124	SD₁₆×C₃₁	—	4	8	no	—
125	SD₁₆×C₃₁	—	4	4	no	—
126	SD₁₆×C₃₁	—	8	4	no	—
127	M₁₆×C₃₁	SD₁₆	1	8	yes	complemented
128	C₂×C₈×C₃₁	—	2	4	no	—
129	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
130	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
131	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
132	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
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	2	no	—
138	C₄⋊C₄×C₃₁	—	2	2	no	—
139	C₄⋊C₄×C₃₁	—	2	1	no	—
140	C₄⋊C₄×C₃₁	—	2	1	no	—
141	C₄⋊C₄×C₃₁	—	2	2	no	—
142	C₄⋊C₄×C₃₁	—	2	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	C₄⋊C₄×C₃₁	—	4	1	no	—
148	C₄⋊C₄×C₃₁	—	4	1	no	—
149	C₂²⋊C₄×C₃₁	—	4	1	no	—
150	C₂²⋊C₄×C₃₁	—	4	1	no	—
151	C₂²⋊C₄×C₃₁	—	4	1	no	—
152	C₄²×C₃₁	—	4	1	no	—
153	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
154	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
155	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
156	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
157	#19622	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₂³×C₃₁	—	4	1	no	—
159	C₂³×C₃₁	#127	1	1	yes	characteristic
160	Q₈×C₃₁	—	4	4	no	—
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	—	8	2	no	—
166	Q₈×C₃₁	—	8	2	no	—
167	D₄×C₃₁	—	4	2	no	—
168	D₄×C₃₁	—	4	2	no	—
169	D₄×C₃₁	—	4	2	no	—
170	D₄×C₃₁	—	8	2	no	—
171	C₂×C₄×C₃₁	—	2	2	no	—
172	C₂×C₄×C₃₁	—	2	2	no	—
173	C₂×C₄×C₃₁	—	4	1	no	—
174	C₂×C₄×C₃₁	—	4	1	no	—
175	C₂×C₄×C₃₁	—	4	1	no	—
176	C₂×C₄×C₃₁	—	4	2	no	—
177	C₂×C₄×C₃₁	—	4	2	no	—
178	C₂×C₄×C₃₁	—	8	2	no	—
179	C₂×C₄×C₃₁	—	8	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	C₂×SD₁₆	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	4	no	—
190	C₈×C₃₁	C₂×SD₁₆	1	8	yes	—
191	C₈×C₃₁	#137	1	4	yes	complemented
192	#3229	C₂×C₃₁	1	1	yes	characteristic, complemented
193	#3206	C₂×C₃₁	1	1	yes	characteristic
194	#3167	C₂×C₃₁	1	1	yes	characteristic
195	#3165	C₂×C₃₁	1	1	yes	characteristic, complemented
196	C₂×#498	C₂×C₃₁	1	1	yes	characteristic, complemented
197	#3152	C₂×C₃₁	1	1	yes	characteristic
198	#1716	C₂×C₃₁	1	2	yes	complemented
199	#1714	C₂×C₃₁	1	2	yes	complemented
200	#1707	C₂×C₃₁	1	2	yes	complemented
201	#1705	C₂×C₃₁	1	2	yes	complemented
202	C₈×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
203	C₂²×C₃₁	—	2	1	no	—
204	C₂²×C₃₁	—	2	1	no	—
205	C₂²×C₃₁	—	8	1	no	—
206	C₂²×C₃₁	—	8	1	no	—
207	C₂²×C₃₁	—	8	1	no	—
208	C₂²×C₃₁	#492	1	2	yes	—
209	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
210	C₄×C₃₁	—	2	1	no	—
211	C₄×C₃₁	—	2	1	no	—
212	C₄×C₃₁	—	8	1	no	—
213	C₄×C₃₁	—	8	2	no	—
214	C₄×C₃₁	—	8	1	no	—
215	C₄×C₃₁	—	8	2	no	—
216	C₄×C₃₁	—	8	1	no	—
217	C₄×C₃₁	#492	1	2	yes	—
218	C₄×C₃₁	C₂²×SD₁₆	1	2	yes	—
219	C₄×C₃₁	C₂×#137	1	2	yes	—
220	C₄×C₃₁	#577	1	2	yes	—
221	C₂×#137	—	2	2	no	—
222	#533	C₂²×C₃₁	1	1	yes	characteristic
223	#532	C₂²×C₃₁	1	1	yes	characteristic
224	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
225	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
226	#501	—	2	2	no	—
227	#500	C₂²×C₃₁	1	2	yes	—
228	#498	—	2	2	no	—
229	#498	C₂²×C₃₁	1	2	yes	—
230	#498	C₂²×C₃₁	1	2	yes	—
231	#498	C₂²×C₃₁	1	2	yes	—
232	#498	C₂²×C₃₁	1	1	yes	characteristic
233	#498	C₂²×C₃₁	1	2	yes	—
234	#497	—	2	2	no	—
235	#494	C₂²×C₃₁	1	2	yes	—
236	#492	C₂²×C₃₁	1	2	yes	—
237	#492	C₂²×C₃₁	1	1	yes	characteristic
238	#492	C₂²×C₃₁	1	2	yes	—
239	#487	C₂²×C₃₁	1	1	yes	characteristic
240	#487	C₂²×C₃₁	1	2	yes	—
241	#487	C₂²×C₃₁	1	2	yes	—
242	#486	C₂²×C₃₁	1	1	yes	characteristic
243	#483	—	2	2	no	—
244	#475	—	2	2	no	—
245	#465	—	2	2	no	—
246	#461	—	2	2	no	—
247	#448	—	2	2	no	—
248	C₄×SD₁₆	—	2	2	no	—
249	#431	C₂²×C₃₁	1	1	yes	characteristic
250	#428	—	2	2	no	—
251	C₂×#106	—	2	2	no	—
252	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
253	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
254	#322	C₂²×C₃₁	1	2	yes	—
255	C₈²	C₂²×C₃₁	1	2	yes	—
256	C₂×C₃₁	—	2	2	no	—
257	C₂×C₃₁	—	16	1	no	—
258	C₂×C₃₁	C₂×#492	1	1	yes	characteristic
259	C₂×C₃₁	#3152	1	1	yes	characteristic
260	C₂×C₃₁	#3154	1	1	yes	characteristic
261	C₂×#41	—	2	1	no	—
262	C₂²×Q₈	—	2	1	no	—
263	C₂²×Q₈	—	2	1	no	—
264	#137	—	4	8	no	—
265	C₂×Q₁₆	—	2	4	no	—
266	C₂×SD₁₆	—	2	4	no	—
267	C₂×SD₁₆	—	2	2	no	—
268	C₂×SD₁₆	—	2	4	no	—
269	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
270	C₂²×C₈	D₄×C₃₁	1	2	yes	—
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	C₂³×C₃₁	1	2	yes	—
278	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
279	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
280	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
281	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
282	#127	C₂³×C₃₁	1	1	yes	characteristic
283	#125	—	2	1	no	—
284	#124	—	2	1	no	—
285	C₂²⋊Q₈	—	2	1	no	—
286	C₂²⋊Q₈	—	2	1	no	—
287	#121	—	2	1	no	—
288	#121	—	2	1	no	—
289	C₄×Q₈	—	2	1	no	—
290	C₄×Q₈	—	2	1	no	—
291	C₄×D₄	—	2	1	no	—
292	C₄×D₄	—	2	1	no	—
293	#117	—	2	1	no	—
294	C₂×C₄⋊C₄	—	2	1	no	—
295	C₂×C₄⋊C₄	—	2	1	no	—
296	C₂×C₄⋊C₄	—	2	1	no	—
297	C₂×C₄⋊C₄	—	2	1	no	—
298	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
299	#107	—	2	4	no	—
300	#106	—	2	4	no	—
301	#106	—	2	4	no	—
302	#106	—	2	4	no	—
303	#106	—	2	2	no	—
304	#106	—	2	4	no	—
305	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
306	#103	—	2	2	no	—
307	#103	—	2	4	no	—
308	#103	—	2	4	no	—
309	#102	—	2	2	no	—
310	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
311	#97	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	2	yes	—
315	C₄×C₈	C₂³×C₃₁	1	2	yes	—
316	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
317	C₃₁	#19622	1	1	yes	characteristic, direct product factor, Sylow
318	#41	—	4	4	no	—
319	C₂×Q₈	—	2	1	no	—
320	C₂×Q₈	—	2	1	no	—
321	C₂×Q₈	—	2	2	no	—
322	C₂×Q₈	—	2	1	no	—
323	C₂×Q₈	—	2	2	no	—
324	C₂×Q₈	—	4	1	no	—
325	C₂×Q₈	—	4	1	no	—
326	C₂×Q₈	—	4	4	no	—
327	C₂×Q₈	—	8	2	no	—
328	C₂×D₄	—	2	1	no	—
329	C₂×D₄	—	2	1	no	—
330	C₂×D₄	—	2	1	no	—
331	C₂×D₄	—	4	1	no	—
332	C₂²×C₄	—	4	1	no	—
333	C₂²×C₄	—	4	1	no	—
334	C₂²×C₄	—	4	1	no	—
335	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
336	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
337	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
338	Q₁₆	—	4	8	no	—
339	SD₁₆	—	4	8	no	—
340	SD₁₆	—	4	4	no	—
341	SD₁₆	—	8	4	no	—
342	M₁₆	SD₁₆×C₃₁	1	8	yes	complemented
343	C₂×C₈	—	2	4	no	—
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	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	1	no	—
352	C₄⋊C₄	—	2	2	no	—
353	C₄⋊C₄	—	2	2	no	—
354	C₄⋊C₄	—	2	1	no	—
355	C₄⋊C₄	—	2	1	no	—
356	C₄⋊C₄	—	2	2	no	—
357	C₄⋊C₄	—	2	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	C₄⋊C₄	—	4	1	no	—
364	C₂²⋊C₄	—	4	1	no	—
365	C₂²⋊C₄	—	4	1	no	—
366	C₂²⋊C₄	—	4	1	no	—
367	C₄²	—	4	1	no	—
368	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
369	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
370	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
371	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
372	C₂³	—	4	1	no	—
373	C₂³	C₃₁×#127	1	1	yes	characteristic
374	Q₈	—	4	4	no	—
375	Q₈	—	4	2	no	—
376	Q₈	—	4	2	no	—
377	Q₈	—	4	4	no	—
378	Q₈	—	4	2	no	—
379	Q₈	—	8	2	no	—
380	Q₈	—	8	2	no	—
381	D₄	—	4	2	no	—
382	D₄	—	4	2	no	—
383	D₄	—	4	2	no	—
384	D₄	—	8	2	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	4	1	no	—
388	C₂×C₄	—	4	1	no	—
389	C₂×C₄	—	4	1	no	—
390	C₂×C₄	—	4	2	no	—
391	C₂×C₄	—	4	2	no	—
392	C₂×C₄	—	8	2	no	—
393	C₂×C₄	—	8	2	no	—
394	C₂×C₄	—	8	2	no	—
395	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
396	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
397	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
398	C₂×C₄	C₂×SD₁₆×C₃₁	1	4	yes	—
399	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
400	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
401	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
402	C₈	—	2	2	no	—
403	C₈	—	2	4	no	—
404	C₈	C₂×SD₁₆×C₃₁	1	8	yes	—
405	C₈	C₃₁×#137	1	4	yes	complemented
406	C₂²	—	2	1	no	—
407	C₂²	—	2	1	no	—
408	C₂²	—	8	1	no	—
409	C₂²	—	8	1	no	—
410	C₂²	—	8	1	no	—
411	C₂²	C₃₁×#492	1	2	yes	—
412	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
413	C₄	—	2	1	no	—
414	C₄	—	2	1	no	—
415	C₄	—	8	1	no	—
416	C₄	—	8	2	no	—
417	C₄	—	8	1	no	—
418	C₄	—	8	2	no	—
419	C₄	—	8	1	no	—
420	C₄	C₃₁×#492	1	2	yes	—
421	C₄	C₂²×SD₁₆×C₃₁	1	2	yes	—
422	C₄	C₂×C₃₁×#137	1	2	yes	—
423	C₄	C₃₁×#577	1	2	yes	—
424	C₂	—	2	2	no	—
425	C₂	—	16	1	no	—
426	C₂	C₂×C₃₁×#492	1	1	yes	characteristic
427	C₂	C₃₁×#3152	1	1	yes	characteristic
428	C₂	C₃₁×#3154	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#532	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×#3229	C₂	semidirect
4. C₃₁×#3165	C₂	semidirect
5. C₂×C₃₁×#498	C₂	semidirect
7. C₃₁×#1716	C₂	semidirect
8. C₃₁×#1714	C₂	semidirect
9. C₃₁×#1707	C₂	semidirect
10. C₃₁×#1705	C₂	semidirect
11. C₈×M₁₆×C₃₁	C₂	semidirect
127. M₁₆×C₃₁	SD₁₆	semidirect
157. #19622	C₃₁	direct (N × H)
191. C₈×C₃₁	#137	semidirect
192. #3229	C₂×C₃₁	semidirect
195. #3165	C₂×C₃₁	semidirect
196. C₂×#498	C₂×C₃₁	semidirect
198. #1716	C₂×C₃₁	semidirect
199. #1714	C₂×C₃₁	semidirect
200. #1707	C₂×C₃₁	semidirect
201. #1705	C₂×C₃₁	semidirect
202. C₈×M₁₆	C₂×C₃₁	semidirect
317. C₃₁	#19622	direct (N × H)
342. M₁₆	SD₁₆×C₃₁	semidirect
405. C₈	C₃₁×#137	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3229	2
C₃₁×#3206	2
C₃₁×#3167	2
C₃₁×#3165	2
C₂×C₃₁×#498	2
C₃₁×#3152	2
C₃₁×#1716	2
C₃₁×#1714	2
C₃₁×#1707	2
C₃₁×#1705	2
C₈×M₁₆×C₃₁	2
#19622	31

Subgroups — C31×#19622 #21476381

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19622 #21476381