Subgroups — C₃₁×#10049 #21466808

Order 7936.

714 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#492	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#460	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2567	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1982	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃₁×#333	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1279	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₁₆×C₃₁	—	2	2	no	—
9	C₂²×SD₁₆×C₃₁	—	2	1	no	—
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#492	—	2	2	no	—
14	C₃₁×#492	—	2	2	no	—
15	C₃₁×#460	—	2	2	no	—
16	C₃₁×#460	—	2	2	no	—
17	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#102	—	2	1	no	—
20	C₂×C₃₁×#102	—	2	2	no	—
21	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₃₁×#395	—	2	2	no	—
23	C₃₁×#391	—	2	1	no	—
24	C₃₁×#389	—	2	2	no	—
25	C₃₁×#384	—	2	1	no	—
26	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#333	—	2	2	no	—
28	C₃₁×Q₈⋊C₈	—	2	2	no	—
29	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
30	C₂²×Q₈×C₃₁	D₄	1	2	yes	—
31	C₂²×D₄×C₃₁	—	2	1	no	—
32	C₂²×D₄×C₃₁	—	2	1	no	—
33	C₂²×D₄×C₃₁	—	2	1	no	—
34	C₂×SD₁₆×C₃₁	—	2	4	no	—
35	C₂×SD₁₆×C₃₁	—	2	8	no	—
36	C₂×SD₁₆×C₃₁	—	4	8	no	—
37	C₂×SD₁₆×C₃₁	—	4	2	no	—
38	C₂×SD₁₆×C₃₁	—	4	2	no	—
39	C₂²×C₈×C₃₁	—	2	1	no	—
40	C₂²×C₈×C₃₁	—	2	1	no	—
41	C₂²×C₈×C₃₁	—	2	1	no	—
42	C₃₁×C₄⋊Q₈	—	2	1	no	—
43	C₃₁×C₄⋊Q₈	—	2	1	no	—
44	C₃₁×C₄⋊Q₈	—	2	1	no	—
45	C₃₁×C₄⋊Q₈	—	2	1	no	—
46	C₃₁×#127	—	2	1	no	—
47	C₃₁×#127	—	2	1	no	—
48	C₃₁×#127	—	2	1	no	—
49	C₃₁×#127	D₄	1	2	yes	—
50	C₄×Q₈×C₃₁	—	2	1	no	—
51	C₄×Q₈×C₃₁	—	2	1	no	—
52	C₄×Q₈×C₃₁	—	2	1	no	—
53	C₄×Q₈×C₃₁	—	2	1	no	—
54	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
55	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
56	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
57	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
58	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
60	C₂×C₄²×C₃₁	—	2	1	no	—
61	C₂×C₄²×C₃₁	—	2	1	no	—
62	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
63	C₃₁×C₄⋊C₈	—	2	1	no	—
64	C₃₁×C₄⋊C₈	—	2	1	no	—
65	C₃₁×C₄⋊C₈	—	2	1	no	—
66	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
67	C₃₁×#102	—	2	4	no	—
68	C₃₁×#102	—	4	4	no	—
69	C₄×C₈×C₃₁	—	2	1	no	—
70	C₄×C₈×C₃₁	D₄	1	2	yes	—
71	C₃₁×#95	—	2	4	no	—
72	C₃₁×#95	—	2	1	no	—
73	C₂⁴×C₃₁	—	4	1	no	—
74	C₂×Q₈×C₃₁	—	2	1	no	—
75	C₂×Q₈×C₃₁	—	2	1	no	—
76	C₂×Q₈×C₃₁	—	2	2	no	—
77	C₂×Q₈×C₃₁	—	2	1	no	—
78	C₂×Q₈×C₃₁	—	2	2	no	—
79	C₂×Q₈×C₃₁	—	2	1	no	—
80	C₂×Q₈×C₃₁	—	2	1	no	—
81	C₂×Q₈×C₃₁	—	4	4	no	—
82	C₂×Q₈×C₃₁	SD₁₆	1	2	yes	complemented
83	C₂×D₄×C₃₁	—	2	2	no	—
84	C₂×D₄×C₃₁	—	2	2	no	—
85	C₂×D₄×C₃₁	—	2	2	no	—
86	C₂×D₄×C₃₁	—	4	4	no	—
87	C₂×D₄×C₃₁	—	4	1	no	—
88	C₂×D₄×C₃₁	—	4	1	no	—
89	C₂×D₄×C₃₁	—	4	1	no	—
90	C₂×D₄×C₃₁	—	4	2	no	—
91	C₂×D₄×C₃₁	—	4	1	no	—
92	C₂×D₄×C₃₁	—	4	1	no	—
93	C₂×D₄×C₃₁	—	4	1	no	—
94	C₂×D₄×C₃₁	—	4	1	no	—
95	C₂×D₄×C₃₁	—	4	2	no	—
96	C₂²×C₄×C₃₁	—	2	1	no	—
97	C₂²×C₄×C₃₁	—	2	2	no	—
98	C₂²×C₄×C₃₁	—	2	2	no	—
99	C₂²×C₄×C₃₁	—	2	1	no	—
100	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
103	SD₁₆×C₃₁	—	4	16	no	—
104	SD₁₆×C₃₁	—	4	8	no	—
105	C₂×C₈×C₃₁	—	2	4	no	—
106	C₂×C₈×C₃₁	—	2	2	no	—
107	C₂×C₈×C₃₁	—	2	2	no	—
108	C₂×C₈×C₃₁	—	4	1	no	—
109	C₂×C₈×C₃₁	—	4	1	no	—
110	C₂×C₈×C₃₁	—	4	1	no	—
111	C₂×C₈×C₃₁	—	4	2	no	—
112	C₄⋊C₄×C₃₁	—	2	2	no	—
113	C₄⋊C₄×C₃₁	—	2	1	no	—
114	C₄⋊C₄×C₃₁	—	2	1	no	—
115	C₄⋊C₄×C₃₁	—	4	1	no	—
116	C₄⋊C₄×C₃₁	—	4	1	no	—
117	C₄⋊C₄×C₃₁	—	4	1	no	—
118	C₄⋊C₄×C₃₁	—	4	1	no	—
119	C₄⋊C₄×C₃₁	—	4	1	no	—
120	C₄⋊C₄×C₃₁	—	4	1	no	—
121	C₄⋊C₄×C₃₁	—	8	4	no	—
122	C₄⋊C₄×C₃₁	SD₁₆	1	4	yes	complemented
123	C₄²×C₃₁	—	4	1	no	—
124	C₄²×C₃₁	—	4	1	no	—
125	C₄²×C₃₁	—	4	1	no	—
126	C₄²×C₃₁	—	4	1	no	—
127	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
129	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₃₁	#41	1	1	yes	characteristic
131	#10049	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂³×C₃₁	—	4	2	no	—
133	C₂³×C₃₁	—	8	1	no	—
134	C₂³×C₃₁	—	8	2	no	—
135	C₂³×C₃₁	—	8	1	no	—
136	C₂³×C₃₁	—	8	2	no	—
137	C₂³×C₃₁	#120	1	1	yes	characteristic
138	Q₈×C₃₁	—	2	2	no	—
139	Q₈×C₃₁	—	2	2	no	—
140	Q₈×C₃₁	—	4	4	no	—
141	Q₈×C₃₁	—	4	1	no	—
142	Q₈×C₃₁	—	4	1	no	—
143	Q₈×C₃₁	—	8	2	no	—
144	D₄×C₃₁	—	4	4	no	—
145	D₄×C₃₁	—	4	2	no	—
146	D₄×C₃₁	—	4	2	no	—
147	D₄×C₃₁	—	4	4	no	—
148	D₄×C₃₁	—	8	2	no	—
149	D₄×C₃₁	—	8	1	no	—
150	D₄×C₃₁	—	8	1	no	—
151	D₄×C₃₁	—	8	1	no	—
152	D₄×C₃₁	—	8	1	no	—
153	C₂×C₄×C₃₁	—	2	2	no	—
154	C₂×C₄×C₃₁	—	2	4	no	—
155	C₂×C₄×C₃₁	—	2	2	no	—
156	C₂×C₄×C₃₁	—	4	1	no	—
157	C₂×C₄×C₃₁	—	4	2	no	—
158	C₂×C₄×C₃₁	—	4	1	no	—
159	C₂×C₄×C₃₁	—	4	2	no	—
160	C₂×C₄×C₃₁	—	4	1	no	—
161	C₂×C₄×C₃₁	—	4	2	no	—
162	C₂×C₄×C₃₁	—	4	2	no	—
163	C₂×C₄×C₃₁	—	4	4	no	—
164	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
165	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
166	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
167	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
168	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
169	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
170	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
171	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
172	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
173	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
174	C₂×C₄×C₃₁	#137	1	1	yes	characteristic
175	C₂×C₄×C₃₁	#137	1	2	yes	—
176	C₈×C₃₁	—	2	4	no	—
177	C₈×C₃₁	—	4	2	no	—
178	C₈×C₃₁	—	8	1	no	—
179	C₂×#492	C₂×C₃₁	1	1	yes	characteristic
180	C₂×#460	C₂×C₃₁	1	1	yes	characteristic
181	#2567	C₂×C₃₁	1	1	yes	characteristic, complemented
182	#1982	C₂×C₃₁	1	1	yes	characteristic
183	C₂×#333	C₂×C₃₁	1	1	yes	characteristic, complemented
184	C₂×Q₈⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
185	#1279	C₂×C₃₁	1	1	yes	characteristic, complemented
186	C₂²×C₃₁	—	8	2	no	—
187	C₂²×C₃₁	—	8	2	no	—
188	C₂²×C₃₁	—	8	2	no	—
189	C₂²×C₃₁	—	16	2	no	—
190	C₂²×C₃₁	—	16	2	no	—
191	C₂²×C₃₁	#392	1	1	yes	characteristic
192	C₂²×C₃₁	#448	1	2	yes	—
193	C₂²×C₃₁	#448	1	1	yes	characteristic
194	C₂²×C₃₁	#449	1	1	yes	characteristic
195	C₂²×C₃₁	#453	1	2	yes	—
196	C₄×C₃₁	—	2	1	no	—
197	C₄×C₃₁	—	2	1	no	—
198	C₄×C₃₁	—	4	4	no	—
199	C₄×C₃₁	—	4	2	no	—
200	C₄×C₃₁	—	8	1	no	—
201	C₄×C₃₁	—	8	2	no	—
202	C₄×C₃₁	#450	1	1	yes	characteristic
203	C₄×C₃₁	#452	1	1	yes	characteristic
204	C₄×C₃₁	#461	1	2	yes	—
205	C₄×C₃₁	#465	1	1	yes	characteristic
206	C₄×C₃₁	#468	1	1	yes	characteristic
207	C₄×C₃₁	#492	1	1	yes	characteristic
208	C₄×C₃₁	#497	1	1	yes	characteristic
209	C₂²×SD₁₆	—	2	2	no	—
210	C₂²×SD₁₆	—	2	1	no	—
211	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
212	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
213	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
214	#492	—	2	2	no	—
215	#492	—	2	2	no	—
216	#460	—	2	2	no	—
217	#460	—	2	2	no	—
218	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
219	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
220	C₂×#102	—	2	1	no	—
221	C₂×#102	—	2	2	no	—
222	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
223	#395	—	2	2	no	—
224	#391	—	2	1	no	—
225	#389	—	2	2	no	—
226	#384	—	2	1	no	—
227	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
228	#333	—	2	2	no	—
229	Q₈⋊C₈	—	2	2	no	—
230	C₂×C₃₁	—	16	2	no	—
231	C₂×C₃₁	#1622	1	2	yes	—
232	C₂×C₃₁	#1623	1	2	yes	—
233	C₂×C₃₁	#2000	1	1	yes	characteristic
234	C₂×C₃₁	#2002	1	1	yes	characteristic
235	C₂×C₃₁	#2013	1	1	yes	characteristic
236	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
237	C₂²×Q₈	D₄×C₃₁	1	2	yes	—
238	C₂²×D₄	—	2	1	no	—
239	C₂²×D₄	—	2	1	no	—
240	C₂²×D₄	—	2	1	no	—
241	C₂×SD₁₆	—	2	4	no	—
242	C₂×SD₁₆	—	2	8	no	—
243	C₂×SD₁₆	—	4	8	no	—
244	C₂×SD₁₆	—	4	2	no	—
245	C₂×SD₁₆	—	4	2	no	—
246	C₂²×C₈	—	2	1	no	—
247	C₂²×C₈	—	2	1	no	—
248	C₂²×C₈	—	2	1	no	—
249	C₄⋊Q₈	—	2	1	no	—
250	C₄⋊Q₈	—	2	1	no	—
251	C₄⋊Q₈	—	2	1	no	—
252	C₄⋊Q₈	—	2	1	no	—
253	#127	—	2	1	no	—
254	#127	—	2	1	no	—
255	#127	—	2	1	no	—
256	#127	D₄×C₃₁	1	2	yes	—
257	C₄×Q₈	—	2	1	no	—
258	C₄×Q₈	—	2	1	no	—
259	C₄×Q₈	—	2	1	no	—
260	C₄×Q₈	—	2	1	no	—
261	C₂×C₄⋊C₄	—	2	1	no	—
262	C₂×C₄⋊C₄	—	2	1	no	—
263	C₂×C₄⋊C₄	—	2	1	no	—
264	C₂×C₄⋊C₄	—	2	4	no	—
265	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
266	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
267	C₂×C₄²	—	2	1	no	—
268	C₂×C₄²	—	2	1	no	—
269	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
270	C₄⋊C₈	—	2	1	no	—
271	C₄⋊C₈	—	2	1	no	—
272	C₄⋊C₈	—	2	1	no	—
273	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
274	#102	—	2	4	no	—
275	#102	—	4	4	no	—
276	C₄×C₈	—	2	1	no	—
277	C₄×C₈	D₄×C₃₁	1	2	yes	—
278	#95	—	2	4	no	—
279	#95	—	2	1	no	—
280	C₃₁	#10049	1	1	yes	characteristic, direct product factor, Sylow
281	C₂⁴	—	4	1	no	—
282	C₂×Q₈	—	2	1	no	—
283	C₂×Q₈	—	2	1	no	—
284	C₂×Q₈	—	2	2	no	—
285	C₂×Q₈	—	2	1	no	—
286	C₂×Q₈	—	2	2	no	—
287	C₂×Q₈	—	2	1	no	—
288	C₂×Q₈	—	2	1	no	—
289	C₂×Q₈	—	4	4	no	—
290	C₂×Q₈	SD₁₆×C₃₁	1	2	yes	complemented
291	C₂×D₄	—	2	2	no	—
292	C₂×D₄	—	2	2	no	—
293	C₂×D₄	—	2	2	no	—
294	C₂×D₄	—	4	4	no	—
295	C₂×D₄	—	4	1	no	—
296	C₂×D₄	—	4	1	no	—
297	C₂×D₄	—	4	1	no	—
298	C₂×D₄	—	4	2	no	—
299	C₂×D₄	—	4	1	no	—
300	C₂×D₄	—	4	1	no	—
301	C₂×D₄	—	4	1	no	—
302	C₂×D₄	—	4	1	no	—
303	C₂×D₄	—	4	2	no	—
304	C₂²×C₄	—	2	1	no	—
305	C₂²×C₄	—	2	2	no	—
306	C₂²×C₄	—	2	2	no	—
307	C₂²×C₄	—	2	1	no	—
308	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
309	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
310	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
311	SD₁₆	—	4	16	no	—
312	SD₁₆	—	4	8	no	—
313	C₂×C₈	—	2	4	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	4	1	no	—
317	C₂×C₈	—	4	1	no	—
318	C₂×C₈	—	4	1	no	—
319	C₂×C₈	—	4	2	no	—
320	C₄⋊C₄	—	2	2	no	—
321	C₄⋊C₄	—	2	1	no	—
322	C₄⋊C₄	—	2	1	no	—
323	C₄⋊C₄	—	4	1	no	—
324	C₄⋊C₄	—	4	1	no	—
325	C₄⋊C₄	—	4	1	no	—
326	C₄⋊C₄	—	4	1	no	—
327	C₄⋊C₄	—	4	1	no	—
328	C₄⋊C₄	—	4	1	no	—
329	C₄⋊C₄	—	8	4	no	—
330	C₄⋊C₄	SD₁₆×C₃₁	1	4	yes	complemented
331	C₄²	—	4	1	no	—
332	C₄²	—	4	1	no	—
333	C₄²	—	4	1	no	—
334	C₄²	—	4	1	no	—
335	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
336	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
337	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
338	C₄²	#41×C₃₁	1	1	yes	characteristic
339	C₂³	—	4	2	no	—
340	C₂³	—	8	1	no	—
341	C₂³	—	8	2	no	—
342	C₂³	—	8	1	no	—
343	C₂³	—	8	2	no	—
344	C₂³	C₃₁×#120	1	1	yes	characteristic
345	Q₈	—	2	2	no	—
346	Q₈	—	2	2	no	—
347	Q₈	—	4	4	no	—
348	Q₈	—	4	1	no	—
349	Q₈	—	4	1	no	—
350	Q₈	—	8	2	no	—
351	D₄	—	4	4	no	—
352	D₄	—	4	2	no	—
353	D₄	—	4	2	no	—
354	D₄	—	4	4	no	—
355	D₄	—	8	2	no	—
356	D₄	—	8	1	no	—
357	D₄	—	8	1	no	—
358	D₄	—	8	1	no	—
359	D₄	—	8	1	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	4	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	4	1	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	4	1	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	4	4	no	—
369	C₂×C₄	—	4	1	no	—
370	C₂×C₄	—	4	2	no	—
371	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
372	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
373	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
374	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
375	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
376	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
377	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
378	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
379	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
380	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
381	C₂×C₄	C₃₁×#137	1	1	yes	characteristic
382	C₂×C₄	C₃₁×#137	1	2	yes	—
383	C₈	—	2	4	no	—
384	C₈	—	4	2	no	—
385	C₈	—	8	1	no	—
386	C₂²	—	8	2	no	—
387	C₂²	—	8	2	no	—
388	C₂²	—	8	2	no	—
389	C₂²	—	16	2	no	—
390	C₂²	—	16	2	no	—
391	C₂²	C₃₁×#392	1	1	yes	characteristic
392	C₂²	C₃₁×#448	1	2	yes	—
393	C₂²	C₃₁×#448	1	1	yes	characteristic
394	C₂²	C₃₁×#449	1	1	yes	characteristic
395	C₂²	C₃₁×#453	1	2	yes	—
396	C₄	—	2	1	no	—
397	C₄	—	2	1	no	—
398	C₄	—	4	4	no	—
399	C₄	—	4	2	no	—
400	C₄	—	8	2	no	—
401	C₄	—	8	1	no	—
402	C₄	C₃₁×#450	1	1	yes	characteristic
403	C₄	C₃₁×#452	1	1	yes	characteristic
404	C₄	C₃₁×#461	1	2	yes	—
405	C₄	C₃₁×#465	1	1	yes	characteristic
406	C₄	C₃₁×#468	1	1	yes	characteristic
407	C₄	C₃₁×#492	1	1	yes	characteristic
408	C₄	C₃₁×#497	1	1	yes	characteristic
409	C₂	—	16	2	no	—
410	C₂	C₃₁×#1622	1	2	yes	—
411	C₂	C₃₁×#1623	1	2	yes	—
412	C₂	C₃₁×#2000	1	1	yes	characteristic
413	C₂	C₃₁×#2002	1	1	yes	characteristic
414	C₂	C₃₁×#2013	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#2567	C₂	semidirect
5. C₂×C₃₁×#333	C₂	semidirect
6. C₂×C₃₁×Q₈⋊C₈	C₂	semidirect
7. C₃₁×#1279	C₂	semidirect
82. C₂×Q₈×C₃₁	SD₁₆	semidirect
122. C₄⋊C₄×C₃₁	SD₁₆	semidirect
131. #10049	C₃₁	direct (N × H)
181. #2567	C₂×C₃₁	semidirect
183. C₂×#333	C₂×C₃₁	semidirect
184. C₂×Q₈⋊C₈	C₂×C₃₁	semidirect
185. #1279	C₂×C₃₁	semidirect
280. C₃₁	#10049	direct (N × H)
290. C₂×Q₈	SD₁₆×C₃₁	semidirect
330. C₄⋊C₄	SD₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#492	2
C₂×C₃₁×#460	2
C₃₁×#2567	2
C₃₁×#1982	2
C₂×C₃₁×#333	2
C₂×C₃₁×Q₈⋊C₈	2
C₃₁×#1279	2
#10049	31

Subgroups — C31×#10049 #21466808

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#10049 #21466808