Subgroups — C₃₁×#19628 #21476387

Order 7936.

708 conjugacy classes of subgroups.

The subgroup lattice has 422 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×#3226	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3167	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#3141	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#3139	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×#487	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1719	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1718	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1711	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1710	C₂	1	2	yes	maximal, complemented
11	C₈×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#533	C₂²	1	1	yes	characteristic
13	C₃₁×#532	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₃₁×#500	C₂²	1	2	yes	—
17	C₃₁×#500	C₂²	1	2	yes	—
18	C₃₁×#498	C₂²	1	1	yes	characteristic
19	C₃₁×#492	C₂²	1	1	yes	characteristic
20	C₃₁×#491	—	2	2	no	—
21	C₃₁×#489	—	2	2	no	—
22	C₃₁×#487	—	2	2	no	—
23	C₃₁×#487	C₂²	1	2	yes	—
24	C₃₁×#487	C₂²	1	2	yes	—
25	C₃₁×#487	C₂²	1	2	yes	—
26	C₃₁×#487	C₂²	1	2	yes	—
27	C₃₁×#487	C₂²	1	2	yes	—
28	C₃₁×#487	C₂²	1	1	yes	characteristic
29	C₃₁×#487	C₂²	1	2	yes	—
30	C₃₁×#486	C₂²	1	2	yes	—
31	C₃₁×#486	C₂²	1	1	yes	characteristic
32	C₃₁×#486	C₂²	1	2	yes	—
33	C₃₁×#484	—	2	2	no	—
34	C₃₁×#480	—	2	2	no	—
35	C₃₁×#477	—	2	2	no	—
36	C₃₁×#474	—	2	2	no	—
37	C₃₁×#465	—	2	2	no	—
38	C₄×SD₁₆×C₃₁	—	2	2	no	—
39	C₃₁×#431	C₂²	1	1	yes	characteristic
40	C₃₁×#419	—	2	2	no	—
41	C₃₁×#417	—	2	2	no	—
42	C₂×C₃₁×#103	—	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	C₂×SD₁₆×C₃₁	—	2	2	no	—
51	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
52	C₂²×C₈×C₃₁	D₄	1	2	yes	—
53	C₃₁×C₄⋊Q₈	—	2	1	no	—
54	C₃₁×C₄⋊Q₈	—	2	1	no	—
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	C₂³	1	1	yes	characteristic
60	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
61	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
62	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
63	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
64	C₃₁×#127	C₂³	1	1	yes	characteristic
65	C₃₁×#125	—	2	1	no	—
66	C₃₁×#124	—	2	1	no	—
67	C₃₁×C₂²⋊Q₈	—	2	1	no	—
68	C₃₁×C₂²⋊Q₈	—	2	1	no	—
69	C₃₁×#121	—	2	1	no	—
70	C₃₁×#121	—	2	1	no	—
71	C₄×Q₈×C₃₁	—	2	1	no	—
72	C₄×Q₈×C₃₁	—	2	1	no	—
73	C₄×D₄×C₃₁	—	2	1	no	—
74	C₄×D₄×C₃₁	—	2	1	no	—
75	C₃₁×#117	—	2	1	no	—
76	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
77	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
78	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
79	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
80	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
81	C₃₁×#107	—	2	4	no	—
82	C₃₁×#107	—	2	4	no	—
83	C₃₁×#106	—	2	2	no	—
84	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
85	C₃₁×#103	—	2	4	no	—
86	C₃₁×#103	—	2	4	no	—
87	C₃₁×#103	—	2	4	no	—
88	C₃₁×#103	—	2	4	no	—
89	C₃₁×#103	—	2	4	no	—
90	C₃₁×#103	—	2	2	no	—
91	C₃₁×#103	—	2	4	no	—
92	C₃₁×#102	—	2	4	no	—
93	C₃₁×#102	—	2	2	no	—
94	C₃₁×#102	—	2	4	no	—
95	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
96	C₃₁×#97	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	#41×C₃₁	—	4	4	no	—
103	C₂×Q₈×C₃₁	—	2	1	no	—
104	C₂×Q₈×C₃₁	—	2	1	no	—
105	C₂×Q₈×C₃₁	—	2	2	no	—
106	C₂×Q₈×C₃₁	—	2	2	no	—
107	C₂×Q₈×C₃₁	—	2	1	no	—
108	C₂×Q₈×C₃₁	—	4	4	no	—
109	C₂×Q₈×C₃₁	—	4	1	no	—
110	C₂×Q₈×C₃₁	—	4	1	no	—
111	C₂×Q₈×C₃₁	—	4	4	no	—
112	C₂×D₄×C₃₁	—	2	1	no	—
113	C₂×D₄×C₃₁	—	2	1	no	—
114	C₂×D₄×C₃₁	—	2	1	no	—
115	C₂×D₄×C₃₁	—	4	1	no	—
116	C₂²×C₄×C₃₁	—	4	1	no	—
117	C₂²×C₄×C₃₁	—	4	1	no	—
118	C₂²×C₄×C₃₁	—	4	1	no	—
119	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
122	SD₁₆×C₃₁	—	4	4	no	—
123	M₁₆×C₃₁	—	2	4	no	—
124	C₂×C₈×C₃₁	—	4	2	no	—
125	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
126	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
127	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
128	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
129	C₄⋊C₄×C₃₁	—	2	2	no	—
130	C₄⋊C₄×C₃₁	—	2	1	no	—
131	C₄⋊C₄×C₃₁	—	2	2	no	—
132	C₄⋊C₄×C₃₁	—	2	2	no	—
133	C₄⋊C₄×C₃₁	—	2	1	no	—
134	C₄⋊C₄×C₃₁	—	2	1	no	—
135	C₄⋊C₄×C₃₁	—	2	1	no	—
136	C₄⋊C₄×C₃₁	—	2	2	no	—
137	C₄⋊C₄×C₃₁	—	2	1	no	—
138	C₄⋊C₄×C₃₁	—	2	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	—	4	1	no	—
143	C₄⋊C₄×C₃₁	—	4	1	no	—
144	C₄⋊C₄×C₃₁	—	4	1	no	—
145	C₂²⋊C₄×C₃₁	—	4	1	no	—
146	C₂²⋊C₄×C₃₁	—	4	1	no	—
147	C₂²⋊C₄×C₃₁	—	4	1	no	—
148	C₄²×C₃₁	—	4	1	no	—
149	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
150	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
151	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
152	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
153	#19628	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₂³×C₃₁	—	4	1	no	—
155	C₂³×C₃₁	#127	1	1	yes	characteristic
156	Q₈×C₃₁	—	4	4	no	—
157	Q₈×C₃₁	—	4	2	no	—
158	Q₈×C₃₁	—	4	2	no	—
159	Q₈×C₃₁	—	4	4	no	—
160	Q₈×C₃₁	—	4	2	no	—
161	Q₈×C₃₁	—	8	2	no	—
162	Q₈×C₃₁	—	8	2	no	—
163	D₄×C₃₁	—	4	2	no	—
164	D₄×C₃₁	—	4	2	no	—
165	D₄×C₃₁	—	4	2	no	—
166	D₄×C₃₁	—	8	2	no	—
167	C₂×C₄×C₃₁	—	2	2	no	—
168	C₂×C₄×C₃₁	—	2	2	no	—
169	C₂×C₄×C₃₁	—	4	1	no	—
170	C₂×C₄×C₃₁	—	4	1	no	—
171	C₂×C₄×C₃₁	—	4	1	no	—
172	C₂×C₄×C₃₁	—	4	2	no	—
173	C₂×C₄×C₃₁	—	4	2	no	—
174	C₂×C₄×C₃₁	—	8	2	no	—
175	C₂×C₄×C₃₁	—	8	2	no	—
176	C₂×C₄×C₃₁	—	8	2	no	—
177	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
178	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
179	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
180	C₂×C₄×C₃₁	#135	1	4	yes	—
181	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
182	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
183	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
184	C₈×C₃₁	—	2	4	no	—
185	C₈×C₃₁	—	2	2	no	—
186	C₈×C₃₁	—	2	2	no	—
187	C₈×C₃₁	—	2	4	no	—
188	#3230	C₂×C₃₁	1	1	yes	characteristic, complemented
189	#3226	C₂×C₃₁	1	1	yes	characteristic
190	#3167	C₂×C₃₁	1	1	yes	characteristic
191	#3141	C₂×C₃₁	1	1	yes	characteristic
192	#3139	C₂×C₃₁	1	1	yes	characteristic, complemented
193	C₂×#487	C₂×C₃₁	1	1	yes	characteristic, complemented
194	#1719	C₂×C₃₁	1	2	yes	complemented
195	#1718	C₂×C₃₁	1	2	yes	complemented
196	#1711	C₂×C₃₁	1	2	yes	complemented
197	#1710	C₂×C₃₁	1	2	yes	complemented
198	C₈×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
199	C₂²×C₃₁	—	2	1	no	—
200	C₂²×C₃₁	—	2	1	no	—
201	C₂²×C₃₁	—	8	1	no	—
202	C₂²×C₃₁	—	8	1	no	—
203	C₂²×C₃₁	—	8	1	no	—
204	C₂²×C₃₁	#495	1	2	yes	—
205	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
206	C₄×C₃₁	—	2	1	no	—
207	C₄×C₃₁	—	2	1	no	—
208	C₄×C₃₁	—	8	1	no	—
209	C₄×C₃₁	—	8	2	no	—
210	C₄×C₃₁	—	8	2	no	—
211	C₄×C₃₁	—	8	1	no	—
212	C₄×C₃₁	—	8	1	no	—
213	C₄×C₃₁	#495	1	2	yes	—
214	C₄×C₃₁	C₂×#135	1	2	yes	—
215	C₄×C₃₁	#575	1	2	yes	—
216	C₄×C₃₁	#578	1	2	yes	—
217	#533	C₂²×C₃₁	1	1	yes	characteristic
218	#532	C₂²×C₃₁	1	1	yes	characteristic
219	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
220	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
221	#500	C₂²×C₃₁	1	2	yes	—
222	#500	C₂²×C₃₁	1	2	yes	—
223	#498	C₂²×C₃₁	1	1	yes	characteristic
224	#492	C₂²×C₃₁	1	1	yes	characteristic
225	#491	—	2	2	no	—
226	#489	—	2	2	no	—
227	#487	—	2	2	no	—
228	#487	C₂²×C₃₁	1	2	yes	—
229	#487	C₂²×C₃₁	1	2	yes	—
230	#487	C₂²×C₃₁	1	2	yes	—
231	#487	C₂²×C₃₁	1	2	yes	—
232	#487	C₂²×C₃₁	1	2	yes	—
233	#487	C₂²×C₃₁	1	1	yes	characteristic
234	#487	C₂²×C₃₁	1	2	yes	—
235	#486	C₂²×C₃₁	1	2	yes	—
236	#486	C₂²×C₃₁	1	1	yes	characteristic
237	#486	C₂²×C₃₁	1	2	yes	—
238	#484	—	2	2	no	—
239	#480	—	2	2	no	—
240	#477	—	2	2	no	—
241	#474	—	2	2	no	—
242	#465	—	2	2	no	—
243	C₄×SD₁₆	—	2	2	no	—
244	#431	C₂²×C₃₁	1	1	yes	characteristic
245	#419	—	2	2	no	—
246	#417	—	2	2	no	—
247	C₂×#103	—	2	2	no	—
248	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
249	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
250	#322	C₂²×C₃₁	1	2	yes	—
251	C₈²	C₂²×C₃₁	1	2	yes	—
252	C₂×C₃₁	—	2	2	no	—
253	C₂×C₃₁	—	16	1	no	—
254	C₂×C₃₁	C₂×#495	1	1	yes	characteristic
255	C₂×C₃₁	#3153	1	1	yes	characteristic
256	C₂×C₃₁	#3156	1	1	yes	characteristic
257	C₂×#41	—	2	1	no	—
258	C₂²×Q₈	—	2	1	no	—
259	C₂²×Q₈	—	2	1	no	—
260	C₂×SD₁₆	—	2	2	no	—
261	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
262	C₂²×C₈	D₄×C₃₁	1	2	yes	—
263	C₄⋊Q₈	—	2	1	no	—
264	C₄⋊Q₈	—	2	1	no	—
265	C₄⋊Q₈	—	2	1	no	—
266	C₄⋊Q₈	—	2	1	no	—
267	C₄⋊Q₈	—	2	1	no	—
268	C₄⋊Q₈	—	2	1	no	—
269	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
270	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
271	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
272	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
273	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
274	#127	C₂³×C₃₁	1	1	yes	characteristic
275	#125	—	2	1	no	—
276	#124	—	2	1	no	—
277	C₂²⋊Q₈	—	2	1	no	—
278	C₂²⋊Q₈	—	2	1	no	—
279	#121	—	2	1	no	—
280	#121	—	2	1	no	—
281	C₄×Q₈	—	2	1	no	—
282	C₄×Q₈	—	2	1	no	—
283	C₄×D₄	—	2	1	no	—
284	C₄×D₄	—	2	1	no	—
285	#117	—	2	1	no	—
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	—	2	1	no	—
290	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
291	#107	—	2	4	no	—
292	#107	—	2	4	no	—
293	#106	—	2	2	no	—
294	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
295	#103	—	2	4	no	—
296	#103	—	2	4	no	—
297	#103	—	2	4	no	—
298	#103	—	2	4	no	—
299	#103	—	2	4	no	—
300	#103	—	2	2	no	—
301	#103	—	2	4	no	—
302	#102	—	2	4	no	—
303	#102	—	2	2	no	—
304	#102	—	2	4	no	—
305	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
306	#97	D₄×C₃₁	1	2	yes	—
307	C₄×C₈	D₄×C₃₁	1	2	yes	—
308	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
309	C₄×C₈	C₂³×C₃₁	1	2	yes	—
310	C₄×C₈	C₂³×C₃₁	1	2	yes	—
311	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
312	C₃₁	#19628	1	1	yes	characteristic, direct product factor, Sylow
313	#41	—	4	4	no	—
314	C₂×Q₈	—	2	1	no	—
315	C₂×Q₈	—	2	1	no	—
316	C₂×Q₈	—	2	2	no	—
317	C₂×Q₈	—	2	2	no	—
318	C₂×Q₈	—	2	1	no	—
319	C₂×Q₈	—	4	4	no	—
320	C₂×Q₈	—	4	1	no	—
321	C₂×Q₈	—	4	1	no	—
322	C₂×Q₈	—	4	4	no	—
323	C₂×D₄	—	2	1	no	—
324	C₂×D₄	—	2	1	no	—
325	C₂×D₄	—	2	1	no	—
326	C₂×D₄	—	4	1	no	—
327	C₂²×C₄	—	4	1	no	—
328	C₂²×C₄	—	4	1	no	—
329	C₂²×C₄	—	4	1	no	—
330	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
331	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
332	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
333	SD₁₆	—	4	4	no	—
334	M₁₆	—	2	4	no	—
335	C₂×C₈	—	4	2	no	—
336	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
337	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
338	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
339	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	2	1	no	—
342	C₄⋊C₄	—	2	2	no	—
343	C₄⋊C₄	—	2	2	no	—
344	C₄⋊C₄	—	2	1	no	—
345	C₄⋊C₄	—	2	1	no	—
346	C₄⋊C₄	—	2	1	no	—
347	C₄⋊C₄	—	2	2	no	—
348	C₄⋊C₄	—	2	1	no	—
349	C₄⋊C₄	—	2	1	no	—
350	C₄⋊C₄	—	4	1	no	—
351	C₄⋊C₄	—	4	1	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	C₂²⋊C₄	—	4	1	no	—
357	C₂²⋊C₄	—	4	1	no	—
358	C₂²⋊C₄	—	4	1	no	—
359	C₄²	—	4	1	no	—
360	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
361	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
362	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
363	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
364	C₂³	—	4	1	no	—
365	C₂³	C₃₁×#127	1	1	yes	characteristic
366	Q₈	—	4	4	no	—
367	Q₈	—	4	2	no	—
368	Q₈	—	4	2	no	—
369	Q₈	—	4	4	no	—
370	Q₈	—	4	2	no	—
371	Q₈	—	8	2	no	—
372	Q₈	—	8	2	no	—
373	D₄	—	4	2	no	—
374	D₄	—	4	2	no	—
375	D₄	—	4	2	no	—
376	D₄	—	8	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	4	1	no	—
380	C₂×C₄	—	4	1	no	—
381	C₂×C₄	—	4	1	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	8	2	no	—
385	C₂×C₄	—	8	2	no	—
386	C₂×C₄	—	8	2	no	—
387	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
388	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
389	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
390	C₂×C₄	C₃₁×#135	1	4	yes	—
391	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
392	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
393	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
394	C₈	—	2	4	no	—
395	C₈	—	2	2	no	—
396	C₈	—	2	2	no	—
397	C₈	—	2	4	no	—
398	C₂²	—	2	1	no	—
399	C₂²	—	2	1	no	—
400	C₂²	—	8	1	no	—
401	C₂²	—	8	1	no	—
402	C₂²	—	8	1	no	—
403	C₂²	C₃₁×#495	1	2	yes	—
404	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
405	C₄	—	2	1	no	—
406	C₄	—	2	1	no	—
407	C₄	—	8	1	no	—
408	C₄	—	8	2	no	—
409	C₄	—	8	2	no	—
410	C₄	—	8	1	no	—
411	C₄	—	8	1	no	—
412	C₄	C₃₁×#495	1	2	yes	—
413	C₄	C₂×C₃₁×#135	1	2	yes	—
414	C₄	C₃₁×#575	1	2	yes	—
415	C₄	C₃₁×#578	1	2	yes	—
416	C₂	—	2	2	no	—
417	C₂	—	16	1	no	—
418	C₂	C₂×C₃₁×#495	1	1	yes	characteristic
419	C₂	C₃₁×#3153	1	1	yes	characteristic
420	C₂	C₃₁×#3156	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×#3230	C₂	semidirect
5. C₃₁×#3139	C₂	semidirect
6. C₂×C₃₁×#487	C₂	semidirect
7. C₃₁×#1719	C₂	semidirect
8. C₃₁×#1718	C₂	semidirect
9. C₃₁×#1711	C₂	semidirect
10. C₃₁×#1710	C₂	semidirect
11. C₈×M₁₆×C₃₁	C₂	semidirect
153. #19628	C₃₁	direct (N × H)
188. #3230	C₂×C₃₁	semidirect
192. #3139	C₂×C₃₁	semidirect
193. C₂×#487	C₂×C₃₁	semidirect
194. #1719	C₂×C₃₁	semidirect
195. #1718	C₂×C₃₁	semidirect
196. #1711	C₂×C₃₁	semidirect
197. #1710	C₂×C₃₁	semidirect
198. C₈×M₁₆	C₂×C₃₁	semidirect
312. C₃₁	#19628	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×#3226	2
C₃₁×#3167	2
C₃₁×#3141	2
C₃₁×#3139	2
C₂×C₃₁×#487	2
C₃₁×#1719	2
C₃₁×#1718	2
C₃₁×#1711	2
C₃₁×#1710	2
C₈×M₁₆×C₃₁	2
#19628	31

Subgroups — C31×#19628 #21476387

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19628 #21476387