Subgroups — C₃₁×#19697 #21476456

Order 7936.

820 conjugacy classes of subgroups.

The subgroup lattice has 452 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×#3226	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3208	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3205	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃₁×#492	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#3131	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1719	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1717	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1715	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	C₂²×SD₁₆×C₃₁	—	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×#127	C₂²	1	1	yes	characteristic
17	C₃₁×#500	C₂²	1	2	yes	complemented
18	C₃₁×#498	C₂²	1	2	yes	complemented
19	C₃₁×#493	C₂²	1	2	yes	—
20	C₃₁×#492	—	2	2	no	—
21	C₃₁×#492	C₂²	1	2	yes	—
22	C₃₁×#492	C₂²	1	2	yes	—
23	C₃₁×#492	C₂²	1	2	yes	—
24	C₃₁×#488	—	2	2	no	—
25	C₃₁×#487	C₂²	1	1	yes	characteristic, complemented
26	C₃₁×#487	C₂²	1	1	yes	characteristic, complemented
27	C₃₁×#487	C₂²	1	2	yes	complemented
28	C₃₁×#486	C₂²	1	2	yes	—
29	C₃₁×#486	C₂²	1	2	yes	—
30	C₃₁×#486	C₂²	1	1	yes	characteristic, complemented
31	C₃₁×#486	C₂²	1	1	yes	characteristic, complemented
32	C₃₁×#486	C₂²	1	2	yes	complemented
33	C₃₁×#483	—	2	2	no	—
34	C₃₁×#480	—	2	2	no	—
35	C₃₁×#475	—	2	2	no	—
36	C₃₁×#474	—	2	2	no	—
37	C₃₁×#460	—	2	2	no	—
38	C₃₁×#459	—	2	2	no	—
39	C₃₁×#450	—	2	2	no	—
40	C₃₁×#447	—	2	2	no	—
41	C₃₁×#431	C₂²	1	1	yes	characteristic
42	C₃₁×#431	C₂²	1	1	yes	characteristic
43	C₃₁×#416	—	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	C₂×#41×C₃₁	—	2	1	no	—
48	C₂²×Q₈×C₃₁	—	2	1	no	—
49	C₂²×D₄×C₃₁	—	2	1	no	—
50	C₂²×D₄×C₃₁	—	2	1	no	—
51	C₂²×D₄×C₃₁	—	2	1	no	—
52	C₂×SD₁₆×C₃₁	—	2	4	no	—
53	C₂×SD₁₆×C₃₁	—	2	4	no	—
54	C₂×SD₁₆×C₃₁	—	2	4	no	—
55	C₂×SD₁₆×C₃₁	—	4	8	no	—
56	C₂×D₈×C₃₁	—	2	4	no	—
57	C₂²×C₈×C₃₁	D₄	1	2	yes	—
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	C₂³	1	1	yes	characteristic
62	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
63	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
64	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
65	C₃₁×#127	—	2	1	no	—
66	C₃₁×#127	—	2	1	no	—
67	C₃₁×#127	—	2	1	no	—
68	C₃₁×#127	C₂³	1	2	yes	—
69	C₃₁×#127	C₂³	1	1	yes	characteristic
70	C₃₁×#125	—	2	1	no	—
71	C₃₁×#124	—	2	1	no	—
72	C₃₁×C₂²⋊Q₈	—	2	1	no	—
73	C₃₁×C₂²⋊Q₈	—	2	1	no	—
74	C₃₁×#121	—	2	1	no	—
75	C₃₁×#121	—	2	1	no	—
76	C₄×Q₈×C₃₁	—	2	1	no	—
77	C₄×Q₈×C₃₁	—	2	1	no	—
78	C₄×D₄×C₃₁	—	2	1	no	—
79	C₄×D₄×C₃₁	—	2	1	no	—
80	C₃₁×#117	—	2	1	no	—
81	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
82	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
83	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
84	C₃₁×#107	—	2	4	no	—
85	C₃₁×#106	—	2	4	no	—
86	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
87	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
88	C₃₁×#103	—	2	2	no	—
89	C₃₁×#103	—	2	2	no	—
90	C₃₁×#103	—	2	4	no	—
91	C₃₁×#102	—	2	2	no	—
92	C₃₁×#102	—	2	2	no	—
93	C₃₁×#102	—	2	4	no	—
94	C₃₁×#102	—	2	4	no	—
95	C₃₁×#102	—	2	4	no	—
96	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
97	C₃₁×C₂²⋊C₈	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	C₂⁴×C₃₁	—	4	1	no	—
104	#41×C₃₁	—	4	4	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	—	2	2	no	—
109	C₂×Q₈×C₃₁	—	4	1	no	—
110	C₂×Q₈×C₃₁	—	4	4	no	—
111	C₂×D₄×C₃₁	—	2	1	no	—
112	C₂×D₄×C₃₁	—	2	1	no	—
113	C₂×D₄×C₃₁	—	2	2	no	—
114	C₂×D₄×C₃₁	—	2	2	no	—
115	C₂×D₄×C₃₁	—	2	1	no	—
116	C₂×D₄×C₃₁	—	2	2	no	—
117	C₂×D₄×C₃₁	—	4	1	no	—
118	C₂×D₄×C₃₁	—	4	1	no	—
119	C₂×D₄×C₃₁	—	4	1	no	—
120	C₂×D₄×C₃₁	—	4	1	no	—
121	C₂×D₄×C₃₁	—	4	4	no	—
122	C₂×D₄×C₃₁	—	8	2	no	—
123	C₂×D₄×C₃₁	—	8	2	no	—
124	C₂²×C₄×C₃₁	—	4	1	no	—
125	C₂²×C₄×C₃₁	—	4	1	no	—
126	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
129	SD₁₆×C₃₁	—	4	8	no	—
130	SD₁₆×C₃₁	—	4	8	no	—
131	SD₁₆×C₃₁	—	8	4	no	—
132	D₈×C₃₁	—	8	4	no	—
133	C₂×C₈×C₃₁	—	2	4	no	—
134	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
135	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
136	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
137	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
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	—	2	1	no	—
142	C₄⋊C₄×C₃₁	—	2	2	no	—
143	C₄⋊C₄×C₃₁	—	2	2	no	—
144	C₄⋊C₄×C₃₁	—	2	1	no	—
145	C₄⋊C₄×C₃₁	—	2	1	no	—
146	C₄⋊C₄×C₃₁	—	4	1	no	—
147	C₄⋊C₄×C₃₁	—	4	1	no	—
148	C₄⋊C₄×C₃₁	—	4	1	no	—
149	C₄⋊C₄×C₃₁	—	4	1	no	—
150	C₂²⋊C₄×C₃₁	—	4	1	no	—
151	C₂²⋊C₄×C₃₁	—	4	1	no	—
152	C₂²⋊C₄×C₃₁	—	4	1	no	—
153	C₄²×C₃₁	—	4	1	no	—
154	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
155	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
156	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
157	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
158	#19697	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
159	C₂³×C₃₁	—	4	1	no	—
160	C₂³×C₃₁	—	4	2	no	—
161	C₂³×C₃₁	—	8	2	no	—
162	C₂³×C₃₁	—	8	2	no	—
163	C₂³×C₃₁	—	8	2	no	—
164	C₂³×C₃₁	#127	1	1	yes	characteristic
165	Q₈×C₃₁	—	4	4	no	—
166	Q₈×C₃₁	—	4	2	no	—
167	Q₈×C₃₁	—	4	2	no	—
168	Q₈×C₃₁	—	4	2	no	—
169	Q₈×C₃₁	—	8	2	no	—
170	D₄×C₃₁	—	4	4	no	—
171	D₄×C₃₁	—	4	2	no	—
172	D₄×C₃₁	—	4	2	no	—
173	D₄×C₃₁	—	4	4	no	—
174	D₄×C₃₁	—	4	4	no	—
175	D₄×C₃₁	—	4	2	no	—
176	D₄×C₃₁	—	8	2	no	—
177	D₄×C₃₁	—	8	2	no	—
178	D₄×C₃₁	—	8	2	no	—
179	D₄×C₃₁	—	8	2	no	—
180	C₂×C₄×C₃₁	—	2	2	no	—
181	C₂×C₄×C₃₁	—	2	2	no	—
182	C₂×C₄×C₃₁	—	4	1	no	—
183	C₂×C₄×C₃₁	—	4	1	no	—
184	C₂×C₄×C₃₁	—	4	1	no	—
185	C₂×C₄×C₃₁	—	4	2	no	—
186	C₂×C₄×C₃₁	—	8	2	no	—
187	C₂×C₄×C₃₁	—	8	2	no	—
188	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
189	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
190	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
191	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
192	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
193	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
195	C₈×C₃₁	—	2	2	no	—
196	C₈×C₃₁	—	2	2	no	—
197	C₈×C₃₁	—	2	4	no	—
198	C₈×C₃₁	—	2	4	no	—
199	#3229	C₂×C₃₁	1	1	yes	characteristic, complemented
200	#3226	C₂×C₃₁	1	1	yes	characteristic, complemented
201	#3208	C₂×C₃₁	1	1	yes	characteristic, complemented
202	#3205	C₂×C₃₁	1	1	yes	characteristic
203	C₂×#492	C₂×C₃₁	1	1	yes	characteristic, complemented
204	#3131	C₂×C₃₁	1	1	yes	characteristic, complemented
205	#1719	C₂×C₃₁	1	2	yes	complemented
206	#1717	C₂×C₃₁	1	2	yes	complemented
207	#1715	C₂×C₃₁	1	2	yes	complemented
208	#1714	C₂×C₃₁	1	2	yes	complemented
209	#1452	C₂×C₃₁	1	1	yes	characteristic, complemented
210	C₂²×C₃₁	—	2	1	no	—
211	C₂²×C₃₁	—	2	1	no	—
212	C₂²×C₃₁	—	8	1	no	—
213	C₂²×C₃₁	—	8	2	no	—
214	C₂²×C₃₁	—	8	2	no	—
215	C₂²×C₃₁	—	8	2	no	—
216	C₂²×C₃₁	—	8	1	no	—
217	C₂²×C₃₁	—	8	1	no	—
218	C₂²×C₃₁	—	16	4	no	—
219	C₂²×C₃₁	#496	1	2	yes	—
220	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
221	C₄×C₃₁	—	2	1	no	—
222	C₄×C₃₁	—	2	1	no	—
223	C₄×C₃₁	—	8	1	no	—
224	C₄×C₃₁	—	8	2	no	—
225	C₄×C₃₁	—	8	1	no	—
226	C₄×C₃₁	—	8	1	no	—
227	C₄×C₃₁	#496	1	2	yes	—
228	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
229	C₄×C₃₁	#577	1	2	yes	—
230	C₄×C₃₁	#577	1	2	yes	—
231	C₂²×SD₁₆	—	2	2	no	—
232	#533	C₂²×C₃₁	1	1	yes	characteristic
233	#532	C₂²×C₃₁	1	1	yes	characteristic
234	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
235	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
236	#500	C₂²×C₃₁	1	2	yes	complemented
237	#498	C₂²×C₃₁	1	2	yes	complemented
238	#493	C₂²×C₃₁	1	2	yes	—
239	#492	—	2	2	no	—
240	#492	C₂²×C₃₁	1	2	yes	—
241	#492	C₂²×C₃₁	1	2	yes	—
242	#492	C₂²×C₃₁	1	2	yes	—
243	#488	—	2	2	no	—
244	#487	C₂²×C₃₁	1	1	yes	characteristic, complemented
245	#487	C₂²×C₃₁	1	1	yes	characteristic, complemented
246	#487	C₂²×C₃₁	1	2	yes	complemented
247	#486	C₂²×C₃₁	1	2	yes	—
248	#486	C₂²×C₃₁	1	2	yes	—
249	#486	C₂²×C₃₁	1	1	yes	characteristic, complemented
250	#486	C₂²×C₃₁	1	1	yes	characteristic, complemented
251	#486	C₂²×C₃₁	1	2	yes	complemented
252	#483	—	2	2	no	—
253	#480	—	2	2	no	—
254	#475	—	2	2	no	—
255	#474	—	2	2	no	—
256	#460	—	2	2	no	—
257	#459	—	2	2	no	—
258	#450	—	2	2	no	—
259	#447	—	2	2	no	—
260	#431	C₂²×C₃₁	1	1	yes	characteristic
261	#431	C₂²×C₃₁	1	1	yes	characteristic
262	#416	—	2	2	no	—
263	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
264	#322	C₂²×C₃₁	1	2	yes	complemented
265	#322	C₂²×C₃₁	1	2	yes	complemented
266	C₂×C₃₁	—	2	2	no	—
267	C₂×C₃₁	—	16	1	no	—
268	C₂×C₃₁	—	16	2	no	—
269	C₂×C₃₁	C₂×#496	1	1	yes	characteristic
270	C₂×C₃₁	#3154	1	1	yes	characteristic
271	C₂×C₃₁	#3154	1	1	yes	characteristic
272	C₂×#41	—	2	1	no	—
273	C₂²×Q₈	—	2	1	no	—
274	C₂²×D₄	—	2	1	no	—
275	C₂²×D₄	—	2	1	no	—
276	C₂²×D₄	—	2	1	no	—
277	C₂×SD₁₆	—	2	4	no	—
278	C₂×SD₁₆	—	2	4	no	—
279	C₂×SD₁₆	—	2	4	no	—
280	C₂×SD₁₆	—	4	8	no	—
281	C₂×D₈	—	2	4	no	—
282	C₂²×C₈	D₄×C₃₁	1	2	yes	—
283	C₄⋊Q₈	—	2	1	no	—
284	C₄⋊Q₈	—	2	1	no	—
285	C₄⋊Q₈	—	2	1	no	—
286	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
287	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
288	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
289	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
290	#127	—	2	1	no	—
291	#127	—	2	1	no	—
292	#127	—	2	1	no	—
293	#127	C₂³×C₃₁	1	2	yes	—
294	#127	C₂³×C₃₁	1	1	yes	characteristic
295	#125	—	2	1	no	—
296	#124	—	2	1	no	—
297	C₂²⋊Q₈	—	2	1	no	—
298	C₂²⋊Q₈	—	2	1	no	—
299	#121	—	2	1	no	—
300	#121	—	2	1	no	—
301	C₄×Q₈	—	2	1	no	—
302	C₄×Q₈	—	2	1	no	—
303	C₄×D₄	—	2	1	no	—
304	C₄×D₄	—	2	1	no	—
305	#117	—	2	1	no	—
306	C₂×C₄⋊C₄	—	2	1	no	—
307	C₂×C₄⋊C₄	—	2	1	no	—
308	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
309	#107	—	2	4	no	—
310	#106	—	2	4	no	—
311	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
312	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
313	#103	—	2	2	no	—
314	#103	—	2	2	no	—
315	#103	—	2	4	no	—
316	#102	—	2	2	no	—
317	#102	—	2	2	no	—
318	#102	—	2	4	no	—
319	#102	—	2	4	no	—
320	#102	—	2	4	no	—
321	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
322	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
323	C₄×C₈	D₄×C₃₁	1	2	yes	—
324	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
325	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
326	C₄×C₈	C₂³×C₃₁	1	2	yes	—
327	C₄×C₈	C₂³×C₃₁	1	2	yes	—
328	C₃₁	#19697	1	1	yes	characteristic, direct product factor, Sylow
329	C₂⁴	—	4	1	no	—
330	#41	—	4	4	no	—
331	C₂×Q₈	—	2	1	no	—
332	C₂×Q₈	—	2	1	no	—
333	C₂×Q₈	—	2	1	no	—
334	C₂×Q₈	—	2	2	no	—
335	C₂×Q₈	—	4	1	no	—
336	C₂×Q₈	—	4	4	no	—
337	C₂×D₄	—	2	1	no	—
338	C₂×D₄	—	2	1	no	—
339	C₂×D₄	—	2	2	no	—
340	C₂×D₄	—	2	2	no	—
341	C₂×D₄	—	2	1	no	—
342	C₂×D₄	—	2	2	no	—
343	C₂×D₄	—	4	1	no	—
344	C₂×D₄	—	4	1	no	—
345	C₂×D₄	—	4	1	no	—
346	C₂×D₄	—	4	1	no	—
347	C₂×D₄	—	4	4	no	—
348	C₂×D₄	—	8	2	no	—
349	C₂×D₄	—	8	2	no	—
350	C₂²×C₄	—	4	1	no	—
351	C₂²×C₄	—	4	1	no	—
352	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
353	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
354	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
355	SD₁₆	—	4	8	no	—
356	SD₁₆	—	4	8	no	—
357	SD₁₆	—	8	4	no	—
358	D₈	—	8	4	no	—
359	C₂×C₈	—	2	4	no	—
360	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
361	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
362	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
363	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
364	C₄⋊C₄	—	2	1	no	—
365	C₄⋊C₄	—	2	1	no	—
366	C₄⋊C₄	—	2	1	no	—
367	C₄⋊C₄	—	2	1	no	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	—	2	2	no	—
370	C₄⋊C₄	—	2	1	no	—
371	C₄⋊C₄	—	2	1	no	—
372	C₄⋊C₄	—	4	1	no	—
373	C₄⋊C₄	—	4	1	no	—
374	C₄⋊C₄	—	4	1	no	—
375	C₄⋊C₄	—	4	1	no	—
376	C₂²⋊C₄	—	4	1	no	—
377	C₂²⋊C₄	—	4	1	no	—
378	C₂²⋊C₄	—	4	1	no	—
379	C₄²	—	4	1	no	—
380	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
381	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
382	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
383	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
384	C₂³	—	4	1	no	—
385	C₂³	—	4	2	no	—
386	C₂³	—	8	2	no	—
387	C₂³	—	8	2	no	—
388	C₂³	—	8	2	no	—
389	C₂³	C₃₁×#127	1	1	yes	characteristic
390	Q₈	—	4	4	no	—
391	Q₈	—	4	2	no	—
392	Q₈	—	4	2	no	—
393	Q₈	—	4	2	no	—
394	Q₈	—	8	2	no	—
395	D₄	—	4	4	no	—
396	D₄	—	4	2	no	—
397	D₄	—	4	2	no	—
398	D₄	—	4	4	no	—
399	D₄	—	4	4	no	—
400	D₄	—	4	2	no	—
401	D₄	—	8	2	no	—
402	D₄	—	8	2	no	—
403	D₄	—	8	2	no	—
404	D₄	—	8	2	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	4	1	no	—
409	C₂×C₄	—	4	1	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	—	8	2	no	—
412	C₂×C₄	—	8	2	no	—
413	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
414	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
415	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
416	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
417	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
418	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
419	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
420	C₈	—	2	2	no	—
421	C₈	—	2	2	no	—
422	C₈	—	2	4	no	—
423	C₈	—	2	4	no	—
424	C₂²	—	2	1	no	—
425	C₂²	—	2	1	no	—
426	C₂²	—	8	1	no	—
427	C₂²	—	8	2	no	—
428	C₂²	—	8	2	no	—
429	C₂²	—	8	2	no	—
430	C₂²	—	8	1	no	—
431	C₂²	—	8	1	no	—
432	C₂²	—	16	4	no	—
433	C₂²	C₃₁×#496	1	2	yes	—
434	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
435	C₄	—	2	1	no	—
436	C₄	—	2	1	no	—
437	C₄	—	8	1	no	—
438	C₄	—	8	2	no	—
439	C₄	—	8	1	no	—
440	C₄	—	8	1	no	—
441	C₄	C₃₁×#496	1	2	yes	—
442	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
443	C₄	C₃₁×#577	1	2	yes	—
444	C₄	C₃₁×#577	1	2	yes	—
445	C₂	—	2	2	no	—
446	C₂	—	16	2	no	—
447	C₂	—	16	1	no	—
448	C₂	C₂×C₃₁×#496	1	1	yes	characteristic
449	C₂	C₃₁×#3154	1	1	yes	characteristic
450	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×#3205	C₂
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
2. C₃₁×#3226	C₂	semidirect
3. C₃₁×#3208	C₂	semidirect
5. C₂×C₃₁×#492	C₂	semidirect
6. C₃₁×#3131	C₂	semidirect
7. C₃₁×#1719	C₂	semidirect
8. C₃₁×#1717	C₂	semidirect
9. C₃₁×#1715	C₂	semidirect
10. C₃₁×#1714	C₂	semidirect
11. C₃₁×#1452	C₂	semidirect
17. C₃₁×#500	C₂²	semidirect
18. C₃₁×#498	C₂²	semidirect
25. C₃₁×#487	C₂²	semidirect
26. C₃₁×#487	C₂²	semidirect
27. C₃₁×#487	C₂²	semidirect
30. C₃₁×#486	C₂²	semidirect
31. C₃₁×#486	C₂²	semidirect
32. C₃₁×#486	C₂²	semidirect
45. C₃₁×#322	C₂²	semidirect
46. C₃₁×#322	C₂²	semidirect
158. #19697	C₃₁	direct (N × H)
199. #3229	C₂×C₃₁	semidirect
200. #3226	C₂×C₃₁	semidirect
201. #3208	C₂×C₃₁	semidirect
203. C₂×#492	C₂×C₃₁	semidirect
204. #3131	C₂×C₃₁	semidirect
205. #1719	C₂×C₃₁	semidirect
206. #1717	C₂×C₃₁	semidirect
207. #1715	C₂×C₃₁	semidirect
208. #1714	C₂×C₃₁	semidirect
209. #1452	C₂×C₃₁	semidirect
236. #500	C₂²×C₃₁	semidirect
237. #498	C₂²×C₃₁	semidirect
244. #487	C₂²×C₃₁	semidirect
245. #487	C₂²×C₃₁	semidirect
246. #487	C₂²×C₃₁	semidirect
249. #486	C₂²×C₃₁	semidirect
250. #486	C₂²×C₃₁	semidirect
251. #486	C₂²×C₃₁	semidirect
264. #322	C₂²×C₃₁	semidirect
265. #322	C₂²×C₃₁	semidirect
328. C₃₁	#19697	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×#3229	2
C₃₁×#3226	2
C₃₁×#3208	2
C₃₁×#3205	2
C₂×C₃₁×#492	2
C₃₁×#3131	2
C₃₁×#1719	2
C₃₁×#1717	2
C₃₁×#1715	2
C₃₁×#1714	2
C₃₁×#1452	2
#19697	31

Subgroups — C31×#19697 #21476456

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19697 #21476456