Subgroups — C₃₁×#10305 #21467064

Order 7936.

802 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×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#492	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#486	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1990	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1957	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1957	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1273	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₁₆×C₃₁	—	2	2	no	—
9	C₂²×D₈×C₃₁	—	2	2	no	—
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
13	C₃₁×#493	—	2	2	no	—
14	C₃₁×#493	—	2	2	no	—
15	C₃₁×#492	—	2	2	no	—
16	C₃₁×#492	—	2	2	no	—
17	C₃₁×#486	—	2	2	no	—
18	C₃₁×#486	—	2	2	no	—
19	C₂×C₃₁×#102	—	2	2	no	—
20	C₂×C₃₁×#102	—	2	2	no	—
21	C₂×C₃₁×#102	—	2	2	no	—
22	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₃₁×#384	C₂²	1	1	yes	characteristic
26	C₃₁×#340	—	2	2	no	—
27	C₂²×Q₈×C₃₁	—	2	1	no	—
28	C₂²×D₄×C₃₁	—	2	1	no	—
29	C₂²×D₄×C₃₁	—	2	1	no	—
30	C₂²×D₄×C₃₁	—	2	1	no	—
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	8	no	—
35	C₂×SD₁₆×C₃₁	—	4	4	no	—
36	C₂×SD₁₆×C₃₁	—	4	4	no	—
37	C₂×D₈×C₃₁	—	2	8	no	—
38	C₂×D₈×C₃₁	—	4	4	no	—
39	C₂×D₈×C₃₁	—	4	4	no	—
40	C₂²×C₈×C₃₁	D₄	1	2	yes	—
41	C₂²×C₈×C₃₁	D₄	1	2	yes	—
42	C₂²×C₈×C₃₁	D₄	1	2	yes	—
43	C₃₁×C₄⋊Q₈	—	2	1	no	—
44	C₃₁×C₄⋊Q₈	—	2	1	no	—
45	C₃₁×C₄⋊Q₈	—	2	1	no	—
46	C₃₁×C₄⋊Q₈	D₄	1	2	yes	—
47	C₃₁×#127	—	2	1	no	—
48	C₃₁×#127	—	2	1	no	—
49	C₃₁×#127	—	2	1	no	—
50	C₃₁×#127	—	2	1	no	—
51	C₃₁×#127	—	2	1	no	—
52	C₃₁×#127	—	2	1	no	—
53	C₃₁×#127	D₄	1	2	yes	—
54	C₃₁×#127	D₄	1	2	yes	—
55	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
56	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
57	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
59	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
60	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
61	C₃₁×#102	—	2	8	no	—
62	C₃₁×#102	—	4	4	no	—
63	C₃₁×#102	—	4	4	no	—
64	C₄×C₈×C₃₁	—	2	1	no	—
65	C₄×C₈×C₃₁	—	2	1	no	—
66	C₄×C₈×C₃₁	—	2	1	no	—
67	C₄×C₈×C₃₁	—	2	1	no	—
68	C₄×C₈×C₃₁	—	2	1	no	—
69	C₄×C₈×C₃₁	—	2	1	no	—
70	C₂⁴×C₃₁	—	4	1	no	—
71	C₂⁴×C₃₁	—	4	1	no	—
72	C₂×Q₈×C₃₁	—	2	2	no	—
73	C₂×Q₈×C₃₁	—	4	1	no	—
74	C₂×Q₈×C₃₁	—	4	2	no	—
75	C₂×Q₈×C₃₁	—	4	2	no	—
76	C₂×D₄×C₃₁	—	2	2	no	—
77	C₂×D₄×C₃₁	—	2	2	no	—
78	C₂×D₄×C₃₁	—	2	2	no	—
79	C₂×D₄×C₃₁	—	2	2	no	—
80	C₂×D₄×C₃₁	—	2	2	no	—
81	C₂×D₄×C₃₁	—	2	2	no	—
82	C₂×D₄×C₃₁	—	4	1	no	—
83	C₂×D₄×C₃₁	—	4	1	no	—
84	C₂×D₄×C₃₁	—	4	1	no	—
85	C₂×D₄×C₃₁	—	4	1	no	—
86	C₂×D₄×C₃₁	—	4	1	no	—
87	C₂×D₄×C₃₁	—	4	1	no	—
88	C₂×D₄×C₃₁	—	4	2	no	—
89	C₂×D₄×C₃₁	—	4	2	no	—
90	C₂×D₄×C₃₁	—	4	2	no	—
91	C₂×D₄×C₃₁	—	4	2	no	—
92	C₂×D₄×C₃₁	—	4	2	no	—
93	C₂×D₄×C₃₁	—	4	2	no	—
94	C₂×D₄×C₃₁	—	4	2	no	—
95	C₂×D₄×C₃₁	—	4	2	no	—
96	C₂×D₄×C₃₁	—	4	2	no	—
97	C₂×D₄×C₃₁	—	4	2	no	—
98	C₂×D₄×C₃₁	—	4	2	no	—
99	C₂×D₄×C₃₁	—	4	2	no	—
100	C₂²×C₄×C₃₁	—	4	1	no	—
101	C₂²×C₄×C₃₁	—	4	1	no	—
102	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
105	SD₁₆×C₃₁	—	4	16	no	—
106	D₈×C₃₁	—	4	16	no	—
107	C₂×C₈×C₃₁	—	2	2	no	—
108	C₂×C₈×C₃₁	—	2	2	no	—
109	C₂×C₈×C₃₁	—	2	2	no	—
110	C₂×C₈×C₃₁	—	2	2	no	—
111	C₂×C₈×C₃₁	—	2	2	no	—
112	C₂×C₈×C₃₁	—	2	2	no	—
113	C₂×C₈×C₃₁	—	2	2	no	—
114	C₂×C₈×C₃₁	D₈	1	4	yes	complemented
115	C₂×C₈×C₃₁	SD₁₆	1	4	yes	complemented
116	C₄⋊C₄×C₃₁	—	2	2	no	—
117	C₄⋊C₄×C₃₁	—	2	2	no	—
118	C₄⋊C₄×C₃₁	—	4	1	no	—
119	C₄⋊C₄×C₃₁	—	4	1	no	—
120	C₄⋊C₄×C₃₁	—	4	2	no	—
121	C₄⋊C₄×C₃₁	—	4	2	no	—
122	C₄⋊C₄×C₃₁	—	4	2	no	—
123	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₃₁	#41	1	1	yes	characteristic
127	#10305	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂³×C₃₁	—	4	2	no	—
129	C₂³×C₃₁	—	4	2	no	—
130	C₂³×C₃₁	—	8	1	no	—
131	C₂³×C₃₁	—	8	1	no	—
132	C₂³×C₃₁	—	8	1	no	—
133	C₂³×C₃₁	—	8	1	no	—
134	C₂³×C₃₁	—	8	1	no	—
135	C₂³×C₃₁	—	8	1	no	—
136	C₂³×C₃₁	—	8	1	no	—
137	C₂³×C₃₁	—	8	1	no	—
138	C₂³×C₃₁	—	8	1	no	—
139	C₂³×C₃₁	—	8	1	no	—
140	C₂³×C₃₁	—	8	1	no	—
141	C₂³×C₃₁	—	8	1	no	—
142	C₂³×C₃₁	#127	1	1	yes	characteristic
143	Q₈×C₃₁	—	4	4	no	—
144	Q₈×C₃₁	—	8	2	no	—
145	D₄×C₃₁	—	4	4	no	—
146	D₄×C₃₁	—	4	4	no	—
147	D₄×C₃₁	—	4	4	no	—
148	D₄×C₃₁	—	4	4	no	—
149	D₄×C₃₁	—	4	4	no	—
150	D₄×C₃₁	—	4	4	no	—
151	D₄×C₃₁	—	8	2	no	—
152	D₄×C₃₁	—	8	2	no	—
153	D₄×C₃₁	—	8	2	no	—
154	D₄×C₃₁	—	8	2	no	—
155	D₄×C₃₁	—	8	2	no	—
156	D₄×C₃₁	—	8	2	no	—
157	C₂×C₄×C₃₁	—	4	2	no	—
158	C₂×C₄×C₃₁	—	8	1	no	—
159	C₂×C₄×C₃₁	—	8	1	no	—
160	C₂×C₄×C₃₁	—	8	1	no	—
161	C₂×C₄×C₃₁	—	8	1	no	—
162	C₂×C₄×C₃₁	—	8	1	no	—
163	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
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	C₂×D₈	1	2	yes	—
170	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
171	C₂×C₄×C₃₁	#135	1	2	yes	—
172	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
173	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
174	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
175	C₈×C₃₁	—	2	4	no	—
176	C₈×C₃₁	—	2	4	no	—
177	C₈×C₃₁	—	4	2	no	—
178	C₂×#493	C₂×C₃₁	1	1	yes	characteristic
179	C₂×#492	C₂×C₃₁	1	1	yes	characteristic, complemented
180	C₂×#486	C₂×C₃₁	1	1	yes	characteristic, complemented
181	#1990	C₂×C₃₁	1	1	yes	characteristic, complemented
182	#1957	C₂×C₃₁	1	1	yes	characteristic, complemented
183	#1957	C₂×C₃₁	1	1	yes	characteristic, complemented
184	#1273	C₂×C₃₁	1	1	yes	characteristic, complemented
185	C₂²×C₃₁	—	8	2	no	—
186	C₂²×C₃₁	—	8	2	no	—
187	C₂²×C₃₁	—	8	2	no	—
188	C₂²×C₃₁	—	8	2	no	—
189	C₂²×C₃₁	—	8	2	no	—
190	C₂²×C₃₁	—	8	2	no	—
191	C₂²×C₃₁	—	16	1	no	—
192	C₂²×C₃₁	—	16	1	no	—
193	C₂²×C₃₁	—	16	1	no	—
194	C₂²×C₃₁	—	16	1	no	—
195	C₂²×C₃₁	—	16	1	no	—
196	C₂²×C₃₁	—	16	1	no	—
197	C₂²×C₃₁	—	16	1	no	—
198	C₂²×C₃₁	—	16	1	no	—
199	C₂²×C₃₁	#392	1	1	yes	characteristic
200	C₂²×C₃₁	#492	1	1	yes	characteristic
201	C₂²×C₃₁	#493	1	1	yes	characteristic
202	C₂²×C₃₁	#495	1	1	yes	characteristic
203	C₂²×C₃₁	#496	1	1	yes	characteristic
204	C₂²×C₃₁	#496	1	1	yes	characteristic
205	C₂²×C₃₁	#496	1	1	yes	characteristic
206	C₄×C₃₁	—	8	2	no	—
207	C₄×C₃₁	—	16	1	no	—
208	C₄×C₃₁	#447	1	2	yes	—
209	C₄×C₃₁	#449	1	2	yes	—
210	C₄×C₃₁	#450	1	2	yes	—
211	C₄×C₃₁	#459	1	2	yes	—
212	C₄×C₃₁	#460	1	2	yes	—
213	C₄×C₃₁	#463	1	2	yes	—
214	C₂²×SD₁₆	—	2	2	no	—
215	C₂²×D₈	—	2	2	no	—
216	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
217	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
218	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
219	#493	—	2	2	no	—
220	#493	—	2	2	no	—
221	#492	—	2	2	no	—
222	#492	—	2	2	no	—
223	#486	—	2	2	no	—
224	#486	—	2	2	no	—
225	C₂×#102	—	2	2	no	—
226	C₂×#102	—	2	2	no	—
227	C₂×#102	—	2	2	no	—
228	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
229	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
230	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
231	#384	C₂²×C₃₁	1	1	yes	characteristic
232	#340	—	2	2	no	—
233	C₂×C₃₁	—	16	2	no	—
234	C₂×C₃₁	—	16	2	no	—
235	C₂×C₃₁	#1706	1	1	yes	characteristic
236	C₂×C₃₁	#1715	1	1	yes	characteristic
237	C₂×C₃₁	#1717	1	1	yes	characteristic
238	C₂×C₃₁	#1721	1	1	yes	characteristic
239	C₂×C₃₁	#2011	1	1	yes	characteristic
240	C₂×C₃₁	#2012	1	1	yes	characteristic
241	C₂×C₃₁	#2014	1	1	yes	characteristic
242	C₂²×Q₈	—	2	1	no	—
243	C₂²×D₄	—	2	1	no	—
244	C₂²×D₄	—	2	1	no	—
245	C₂²×D₄	—	2	1	no	—
246	C₂²×D₄	—	2	1	no	—
247	C₂²×D₄	—	2	1	no	—
248	C₂²×D₄	—	2	1	no	—
249	C₂×SD₁₆	—	2	8	no	—
250	C₂×SD₁₆	—	4	4	no	—
251	C₂×SD₁₆	—	4	4	no	—
252	C₂×D₈	—	2	8	no	—
253	C₂×D₈	—	4	4	no	—
254	C₂×D₈	—	4	4	no	—
255	C₂²×C₈	D₄×C₃₁	1	2	yes	—
256	C₂²×C₈	D₄×C₃₁	1	2	yes	—
257	C₂²×C₈	D₄×C₃₁	1	2	yes	—
258	C₄⋊Q₈	—	2	1	no	—
259	C₄⋊Q₈	—	2	1	no	—
260	C₄⋊Q₈	—	2	1	no	—
261	C₄⋊Q₈	D₄×C₃₁	1	2	yes	—
262	#127	—	2	1	no	—
263	#127	—	2	1	no	—
264	#127	—	2	1	no	—
265	#127	—	2	1	no	—
266	#127	—	2	1	no	—
267	#127	—	2	1	no	—
268	#127	D₄×C₃₁	1	2	yes	—
269	#127	D₄×C₃₁	1	2	yes	—
270	C₂×C₄⋊C₄	—	2	1	no	—
271	C₂×C₄⋊C₄	—	2	1	no	—
272	C₂×C₄⋊C₄	—	2	1	no	—
273	C₂×C₄⋊C₄	—	2	1	no	—
274	C₂×C₄⋊C₄	—	2	1	no	—
275	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
276	#102	—	2	8	no	—
277	#102	—	4	4	no	—
278	#102	—	4	4	no	—
279	C₄×C₈	—	2	1	no	—
280	C₄×C₈	—	2	1	no	—
281	C₄×C₈	—	2	1	no	—
282	C₄×C₈	—	2	1	no	—
283	C₄×C₈	—	2	1	no	—
284	C₄×C₈	—	2	1	no	—
285	C₃₁	#10305	1	1	yes	characteristic, direct product factor, Sylow
286	C₂⁴	—	4	1	no	—
287	C₂⁴	—	4	1	no	—
288	C₂×Q₈	—	2	2	no	—
289	C₂×Q₈	—	4	1	no	—
290	C₂×Q₈	—	4	2	no	—
291	C₂×Q₈	—	4	2	no	—
292	C₂×D₄	—	2	2	no	—
293	C₂×D₄	—	2	2	no	—
294	C₂×D₄	—	2	2	no	—
295	C₂×D₄	—	2	2	no	—
296	C₂×D₄	—	2	2	no	—
297	C₂×D₄	—	2	2	no	—
298	C₂×D₄	—	4	1	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	1	no	—
304	C₂×D₄	—	4	2	no	—
305	C₂×D₄	—	4	2	no	—
306	C₂×D₄	—	4	2	no	—
307	C₂×D₄	—	4	2	no	—
308	C₂×D₄	—	4	2	no	—
309	C₂×D₄	—	4	2	no	—
310	C₂×D₄	—	4	2	no	—
311	C₂×D₄	—	4	2	no	—
312	C₂×D₄	—	4	2	no	—
313	C₂×D₄	—	4	2	no	—
314	C₂×D₄	—	4	2	no	—
315	C₂×D₄	—	4	2	no	—
316	C₂²×C₄	—	4	1	no	—
317	C₂²×C₄	—	4	1	no	—
318	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
319	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
320	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
321	SD₁₆	—	4	16	no	—
322	D₈	—	4	16	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	2	2	no	—
325	C₂×C₈	—	2	2	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	D₈×C₃₁	1	4	yes	complemented
331	C₂×C₈	SD₁₆×C₃₁	1	4	yes	complemented
332	C₄⋊C₄	—	2	2	no	—
333	C₄⋊C₄	—	2	2	no	—
334	C₄⋊C₄	—	4	1	no	—
335	C₄⋊C₄	—	4	1	no	—
336	C₄⋊C₄	—	4	2	no	—
337	C₄⋊C₄	—	4	2	no	—
338	C₄⋊C₄	—	4	2	no	—
339	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
340	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
341	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
342	C₄²	#41×C₃₁	1	1	yes	characteristic
343	C₂³	—	4	2	no	—
344	C₂³	—	4	2	no	—
345	C₂³	—	8	1	no	—
346	C₂³	—	8	1	no	—
347	C₂³	—	8	1	no	—
348	C₂³	—	8	1	no	—
349	C₂³	—	8	1	no	—
350	C₂³	—	8	1	no	—
351	C₂³	—	8	1	no	—
352	C₂³	—	8	1	no	—
353	C₂³	—	8	1	no	—
354	C₂³	—	8	1	no	—
355	C₂³	—	8	1	no	—
356	C₂³	—	8	1	no	—
357	C₂³	C₃₁×#127	1	1	yes	characteristic
358	Q₈	—	4	4	no	—
359	Q₈	—	8	2	no	—
360	D₄	—	4	4	no	—
361	D₄	—	4	4	no	—
362	D₄	—	4	4	no	—
363	D₄	—	4	4	no	—
364	D₄	—	4	4	no	—
365	D₄	—	4	4	no	—
366	D₄	—	8	2	no	—
367	D₄	—	8	2	no	—
368	D₄	—	8	2	no	—
369	D₄	—	8	2	no	—
370	D₄	—	8	2	no	—
371	D₄	—	8	2	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	—	8	1	no	—
374	C₂×C₄	—	8	1	no	—
375	C₂×C₄	—	8	1	no	—
376	C₂×C₄	—	8	1	no	—
377	C₂×C₄	—	8	1	no	—
378	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
379	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
380	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
381	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
382	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
383	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
384	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
385	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
386	C₂×C₄	C₃₁×#135	1	2	yes	—
387	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
388	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
389	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
390	C₈	—	2	4	no	—
391	C₈	—	2	4	no	—
392	C₈	—	4	2	no	—
393	C₂²	—	8	2	no	—
394	C₂²	—	8	2	no	—
395	C₂²	—	8	2	no	—
396	C₂²	—	8	2	no	—
397	C₂²	—	8	2	no	—
398	C₂²	—	8	2	no	—
399	C₂²	—	16	1	no	—
400	C₂²	—	16	1	no	—
401	C₂²	—	16	1	no	—
402	C₂²	—	16	1	no	—
403	C₂²	—	16	1	no	—
404	C₂²	—	16	1	no	—
405	C₂²	—	16	1	no	—
406	C₂²	—	16	1	no	—
407	C₂²	C₃₁×#392	1	1	yes	characteristic
408	C₂²	C₃₁×#492	1	1	yes	characteristic
409	C₂²	C₃₁×#493	1	1	yes	characteristic
410	C₂²	C₃₁×#495	1	1	yes	characteristic
411	C₂²	C₃₁×#496	1	1	yes	characteristic
412	C₂²	C₃₁×#496	1	1	yes	characteristic
413	C₂²	C₃₁×#496	1	1	yes	characteristic
414	C₄	—	8	2	no	—
415	C₄	—	16	1	no	—
416	C₄	C₃₁×#447	1	2	yes	—
417	C₄	C₃₁×#449	1	2	yes	—
418	C₄	C₃₁×#450	1	2	yes	—
419	C₄	C₃₁×#459	1	2	yes	—
420	C₄	C₃₁×#460	1	2	yes	—
421	C₄	C₃₁×#463	1	2	yes	—
422	C₂	—	16	2	no	—
423	C₂	—	16	2	no	—
424	C₂	C₃₁×#1706	1	1	yes	characteristic
425	C₂	C₃₁×#1715	1	1	yes	characteristic
426	C₂	C₃₁×#1717	1	1	yes	characteristic
427	C₂	C₃₁×#1721	1	1	yes	characteristic
428	C₂	C₃₁×#2011	1	1	yes	characteristic
429	C₂	C₃₁×#2012	1	1	yes	characteristic
430	C₂	C₃₁×#2014	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₂×C₃₁×#493	C₂
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
2. C₂×C₃₁×#492	C₂	semidirect
3. C₂×C₃₁×#486	C₂	semidirect
4. C₃₁×#1990	C₂	semidirect
5. C₃₁×#1957	C₂	semidirect
6. C₃₁×#1957	C₂	semidirect
7. C₃₁×#1273	C₂	semidirect
114. C₂×C₈×C₃₁	D₈	semidirect
115. C₂×C₈×C₃₁	SD₁₆	semidirect
127. #10305	C₃₁	direct (N × H)
179. C₂×#492	C₂×C₃₁	semidirect
180. C₂×#486	C₂×C₃₁	semidirect
181. #1990	C₂×C₃₁	semidirect
182. #1957	C₂×C₃₁	semidirect
183. #1957	C₂×C₃₁	semidirect
184. #1273	C₂×C₃₁	semidirect
285. C₃₁	#10305	direct (N × H)
330. C₂×C₈	D₈×C₃₁	semidirect
331. C₂×C₈	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×#493	2
C₂×C₃₁×#492	2
C₂×C₃₁×#486	2
C₃₁×#1990	2
C₃₁×#1957	2
C₃₁×#1273	2
#10305	31

Subgroups — C31×#10305 #21467064

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#10305 #21467064