Subgroups — C₃₁×#19735 #21476494

Order 7936.

780 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3228	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3210	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3208	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3207	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#3205	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#3205	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1721	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1721	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1720	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1719	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1717	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#1715	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#1709	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#1706	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#1454	C₂	1	1	yes	maximal, characteristic, complemented
16	C₃₁×#533	C₂²	1	1	yes	characteristic
17	C₃₁×#532	C₂²	1	1	yes	characteristic
18	C₃₁×#532	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
20	C₃₁×#500	C₂²	1	1	yes	characteristic, complemented
21	C₃₁×#498	C₂²	1	1	yes	characteristic, complemented
22	C₃₁×#494	C₂²	1	1	yes	characteristic, complemented
23	C₃₁×#493	C₂²	1	1	yes	characteristic
24	C₃₁×#493	C₂²	1	1	yes	characteristic
25	C₃₁×#493	C₂²	1	1	yes	characteristic
26	C₃₁×#493	C₂²	1	1	yes	characteristic
27	C₃₁×#493	C₂²	1	1	yes	characteristic, complemented
28	C₃₁×#492	C₂²	1	1	yes	characteristic
29	C₃₁×#492	C₂²	1	1	yes	characteristic
30	C₃₁×#492	C₂²	1	1	yes	characteristic, complemented
31	C₃₁×#492	C₂²	1	1	yes	characteristic, complemented
32	C₃₁×#487	C₂²	1	1	yes	characteristic, complemented
33	C₃₁×#487	C₂²	1	1	yes	characteristic, complemented
34	C₃₁×#486	C₂²	1	1	yes	characteristic
35	C₃₁×#486	C₂²	1	1	yes	characteristic
36	C₃₁×#486	C₂²	1	1	yes	characteristic
37	C₃₁×#486	C₂²	1	1	yes	characteristic
38	C₃₁×#486	C₂²	1	1	yes	characteristic
39	C₃₁×#486	C₂²	1	1	yes	characteristic
40	C₃₁×#486	C₂²	1	1	yes	characteristic, complemented
41	C₃₁×#486	C₂²	1	1	yes	characteristic, complemented
42	C₃₁×#486	C₂²	1	1	yes	characteristic, complemented
43	C₃₁×#486	C₂²	1	1	yes	characteristic, complemented
44	C₃₁×#482	—	2	2	no	—
45	C₃₁×#478	—	2	2	no	—
46	C₃₁×#464	—	2	2	no	—
47	C₃₁×#463	—	2	2	no	—
48	C₃₁×#460	—	2	2	no	—
49	C₃₁×#459	—	2	2	no	—
50	C₃₁×#459	—	2	2	no	—
51	C₃₁×#452	—	2	2	no	—
52	C₃₁×#450	—	2	2	no	—
53	C₃₁×#449	—	2	2	no	—
54	C₃₁×#447	—	2	2	no	—
55	C₃₁×#447	—	2	2	no	—
56	C₃₁×#431	C₂²	1	1	yes	characteristic
57	C₃₁×#431	C₂²	1	1	yes	characteristic
58	C₃₁×#431	C₂²	1	1	yes	characteristic
59	C₃₁×#322	C₂²	1	1	yes	characteristic, complemented
60	C₃₁×#322	C₂²	1	1	yes	characteristic, complemented
61	C₃₁×#322	C₂²	1	1	yes	characteristic, complemented
62	C₈²×C₃₁	C₂²	1	1	yes	characteristic, complemented
63	C₂×#41×C₃₁	—	2	1	no	—
64	C₂×#41×C₃₁	—	2	1	no	—
65	C₂²×D₄×C₃₁	—	2	1	no	—
66	C₂²×D₄×C₃₁	—	2	1	no	—
67	C₂²×D₄×C₃₁	—	2	1	no	—
68	C₂×Q₁₆×C₃₁	—	2	2	no	—
69	C₂×SD₁₆×C₃₁	—	2	2	no	—
70	C₂×SD₁₆×C₃₁	—	2	2	no	—
71	C₂×SD₁₆×C₃₁	—	2	2	no	—
72	C₂×SD₁₆×C₃₁	—	2	2	no	—
73	C₂×D₈×C₃₁	—	2	2	no	—
74	C₂×D₈×C₃₁	—	2	2	no	—
75	C₂×D₈×C₃₁	—	2	2	no	—
76	C₂×D₈×C₃₁	—	2	2	no	—
77	C₂×D₈×C₃₁	—	2	2	no	—
78	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
79	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
80	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
81	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
82	C₃₁×#127	—	2	1	no	—
83	C₃₁×#127	—	2	1	no	—
84	C₃₁×#127	—	2	1	no	—
85	C₃₁×#127	C₂³	1	1	yes	characteristic
86	C₃₁×#127	C₂³	1	1	yes	characteristic
87	C₃₁×#127	C₂³	1	1	yes	characteristic
88	C₃₁×#127	C₂³	1	1	yes	characteristic
89	C₃₁×#125	—	2	1	no	—
90	C₃₁×#124	—	2	1	no	—
91	C₃₁×#124	—	2	1	no	—
92	C₃₁×C₂²⋊Q₈	—	2	1	no	—
93	C₃₁×C₂²⋊Q₈	—	2	1	no	—
94	C₃₁×#121	—	2	1	no	—
95	C₃₁×#121	—	2	1	no	—
96	C₃₁×#121	—	2	1	no	—
97	C₃₁×#121	—	2	1	no	—
98	C₄×Q₈×C₃₁	—	2	1	no	—
99	C₄×Q₈×C₃₁	—	2	1	no	—
100	C₄×D₄×C₃₁	—	2	1	no	—
101	C₄×D₄×C₃₁	—	2	1	no	—
102	C₄×D₄×C₃₁	—	2	1	no	—
103	C₄×D₄×C₃₁	—	2	1	no	—
104	C₃₁×#117	—	2	1	no	—
105	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
106	C₃₁×#107	—	2	2	no	—
107	C₃₁×#106	—	2	2	no	—
108	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
109	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
110	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
111	C₃₁×#103	—	2	2	no	—
112	C₃₁×#103	—	2	2	no	—
113	C₃₁×#102	—	2	2	no	—
114	C₃₁×#102	—	2	2	no	—
115	C₃₁×#102	—	2	2	no	—
116	C₃₁×#102	—	2	2	no	—
117	C₃₁×#102	—	2	2	no	—
118	C₃₁×#102	—	2	2	no	—
119	C₃₁×#102	—	2	2	no	—
120	C₃₁×#102	—	2	2	no	—
121	C₃₁×#102	—	2	2	no	—
122	C₃₁×#102	—	2	2	no	—
123	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
124	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
125	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
126	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
127	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
128	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
129	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
130	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
131	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
132	C₂⁴×C₃₁	—	4	1	no	—
133	#41×C₃₁	—	8	2	no	—
134	#41×C₃₁	—	8	2	no	—
135	C₂×Q₈×C₃₁	—	2	1	no	—
136	C₂×Q₈×C₃₁	—	2	1	no	—
137	C₂×Q₈×C₃₁	—	2	1	no	—
138	C₂×Q₈×C₃₁	—	2	1	no	—
139	C₂×D₄×C₃₁	—	2	1	no	—
140	C₂×D₄×C₃₁	—	2	1	no	—
141	C₂×D₄×C₃₁	—	2	1	no	—
142	C₂×D₄×C₃₁	—	2	1	no	—
143	C₂×D₄×C₃₁	—	2	1	no	—
144	C₂×D₄×C₃₁	—	2	1	no	—
145	C₂×D₄×C₃₁	—	2	1	no	—
146	C₂×D₄×C₃₁	—	2	1	no	—
147	C₂×D₄×C₃₁	—	2	1	no	—
148	C₂×D₄×C₃₁	—	2	1	no	—
149	C₂×D₄×C₃₁	—	2	1	no	—
150	C₂×D₄×C₃₁	—	2	1	no	—
151	C₂×D₄×C₃₁	—	4	1	no	—
152	C₂×D₄×C₃₁	—	4	1	no	—
153	C₂×D₄×C₃₁	—	4	1	no	—
154	C₂×D₄×C₃₁	—	4	1	no	—
155	C₂×D₄×C₃₁	—	4	1	no	—
156	C₂×D₄×C₃₁	—	8	2	no	—
157	C₂×D₄×C₃₁	—	8	2	no	—
158	C₂×D₄×C₃₁	—	8	2	no	—
159	C₂²×C₄×C₃₁	—	4	1	no	—
160	C₂²×C₄×C₃₁	—	4	1	no	—
161	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
164	Q₁₆×C₃₁	—	8	2	no	—
165	SD₁₆×C₃₁	—	8	2	no	—
166	SD₁₆×C₃₁	—	8	2	no	—
167	SD₁₆×C₃₁	—	8	2	no	—
168	SD₁₆×C₃₁	—	8	2	no	—
169	D₈×C₃₁	—	8	2	no	—
170	D₈×C₃₁	—	8	2	no	—
171	D₈×C₃₁	—	8	2	no	—
172	D₈×C₃₁	—	8	2	no	—
173	D₈×C₃₁	—	8	2	no	—
174	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
175	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
176	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
177	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
178	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
179	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
180	C₄⋊C₄×C₃₁	—	2	1	no	—
181	C₄⋊C₄×C₃₁	—	2	1	no	—
182	C₄⋊C₄×C₃₁	—	2	1	no	—
183	C₄⋊C₄×C₃₁	—	2	1	no	—
184	C₄⋊C₄×C₃₁	—	2	1	no	—
185	C₄⋊C₄×C₃₁	—	2	1	no	—
186	C₄⋊C₄×C₃₁	—	2	1	no	—
187	C₄⋊C₄×C₃₁	—	2	1	no	—
188	C₄⋊C₄×C₃₁	—	4	1	no	—
189	C₄⋊C₄×C₃₁	—	4	1	no	—
190	C₂²⋊C₄×C₃₁	—	4	1	no	—
191	C₂²⋊C₄×C₃₁	—	4	1	no	—
192	C₂²⋊C₄×C₃₁	—	4	1	no	—
193	C₂²⋊C₄×C₃₁	—	4	1	no	—
194	C₂²⋊C₄×C₃₁	—	4	1	no	—
195	C₄²×C₃₁	—	4	1	no	—
196	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
197	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
198	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
199	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
200	#19735	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
201	C₂³×C₃₁	—	4	1	no	—
202	C₂³×C₃₁	—	4	1	no	—
203	C₂³×C₃₁	—	4	1	no	—
204	C₂³×C₃₁	—	4	1	no	—
205	C₂³×C₃₁	—	8	2	no	—
206	C₂³×C₃₁	—	8	2	no	—
207	C₂³×C₃₁	—	8	2	no	—
208	C₂³×C₃₁	#127	1	1	yes	characteristic
209	Q₈×C₃₁	—	4	2	no	—
210	Q₈×C₃₁	—	4	2	no	—
211	Q₈×C₃₁	—	4	2	no	—
212	Q₈×C₃₁	—	4	2	no	—
213	D₄×C₃₁	—	4	2	no	—
214	D₄×C₃₁	—	4	2	no	—
215	D₄×C₃₁	—	4	2	no	—
216	D₄×C₃₁	—	4	2	no	—
217	D₄×C₃₁	—	4	2	no	—
218	D₄×C₃₁	—	4	2	no	—
219	D₄×C₃₁	—	4	2	no	—
220	D₄×C₃₁	—	4	2	no	—
221	D₄×C₃₁	—	4	2	no	—
222	D₄×C₃₁	—	4	2	no	—
223	D₄×C₃₁	—	4	2	no	—
224	D₄×C₃₁	—	4	2	no	—
225	D₄×C₃₁	—	8	2	no	—
226	D₄×C₃₁	—	8	2	no	—
227	D₄×C₃₁	—	8	2	no	—
228	D₄×C₃₁	—	8	2	no	—
229	D₄×C₃₁	—	8	2	no	—
230	C₂×C₄×C₃₁	—	2	2	no	—
231	C₂×C₄×C₃₁	—	2	2	no	—
232	C₂×C₄×C₃₁	—	2	2	no	—
233	C₂×C₄×C₃₁	—	4	1	no	—
234	C₂×C₄×C₃₁	—	4	1	no	—
235	C₂×C₄×C₃₁	—	4	1	no	—
236	C₂×C₄×C₃₁	—	4	1	no	—
237	C₂×C₄×C₃₁	—	8	2	no	—
238	C₂×C₄×C₃₁	—	8	2	no	—
239	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
240	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
241	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
242	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
243	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
244	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
245	C₈×C₃₁	—	2	2	no	—
246	C₈×C₃₁	—	2	2	no	—
247	C₈×C₃₁	—	2	2	no	—
248	C₈×C₃₁	—	2	2	no	—
249	C₈×C₃₁	—	2	2	no	—
250	C₈×C₃₁	—	2	2	no	—
251	#3228	C₂×C₃₁	1	1	yes	characteristic, complemented
252	#3210	C₂×C₃₁	1	1	yes	characteristic, complemented
253	#3208	C₂×C₃₁	1	1	yes	characteristic, complemented
254	#3207	C₂×C₃₁	1	1	yes	characteristic, complemented
255	#3205	C₂×C₃₁	1	1	yes	characteristic, complemented
256	#3205	C₂×C₃₁	1	1	yes	characteristic, complemented
257	#1721	C₂×C₃₁	1	1	yes	characteristic, complemented
258	#1721	C₂×C₃₁	1	1	yes	characteristic, complemented
259	#1720	C₂×C₃₁	1	1	yes	characteristic, complemented
260	#1719	C₂×C₃₁	1	1	yes	characteristic, complemented
261	#1717	C₂×C₃₁	1	1	yes	characteristic, complemented
262	#1715	C₂×C₃₁	1	1	yes	characteristic, complemented
263	#1709	C₂×C₃₁	1	1	yes	characteristic, complemented
264	#1706	C₂×C₃₁	1	1	yes	characteristic, complemented
265	#1454	C₂×C₃₁	1	1	yes	characteristic, complemented
266	C₂²×C₃₁	—	2	1	no	—
267	C₂²×C₃₁	—	2	1	no	—
268	C₂²×C₃₁	—	2	1	no	—
269	C₂²×C₃₁	—	8	1	no	—
270	C₂²×C₃₁	—	8	1	no	—
271	C₂²×C₃₁	—	8	1	no	—
272	C₂²×C₃₁	—	8	1	no	—
273	C₂²×C₃₁	—	8	1	no	—
274	C₂²×C₃₁	—	8	1	no	—
275	C₂²×C₃₁	—	8	1	no	—
276	C₂²×C₃₁	—	8	1	no	—
277	C₂²×C₃₁	—	8	1	no	—
278	C₂²×C₃₁	—	8	1	no	—
279	C₂²×C₃₁	—	8	1	no	—
280	C₂²×C₃₁	—	8	1	no	—
281	C₂²×C₃₁	—	16	4	no	—
282	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
283	C₄×C₃₁	—	2	1	no	—
284	C₄×C₃₁	—	2	1	no	—
285	C₄×C₃₁	—	2	1	no	—
286	C₄×C₃₁	—	8	1	no	—
287	C₄×C₃₁	—	8	1	no	—
288	C₄×C₃₁	—	8	1	no	—
289	C₄×C₃₁	—	8	1	no	—
290	C₄×C₃₁	#576	1	2	yes	—
291	C₄×C₃₁	#576	1	2	yes	—
292	C₄×C₃₁	#577	1	2	yes	—
293	#533	C₂²×C₃₁	1	1	yes	characteristic
294	#532	C₂²×C₃₁	1	1	yes	characteristic
295	#532	C₂²×C₃₁	1	1	yes	characteristic
296	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
297	#500	C₂²×C₃₁	1	1	yes	characteristic, complemented
298	#498	C₂²×C₃₁	1	1	yes	characteristic, complemented
299	#494	C₂²×C₃₁	1	1	yes	characteristic, complemented
300	#493	C₂²×C₃₁	1	1	yes	characteristic
301	#493	C₂²×C₃₁	1	1	yes	characteristic
302	#493	C₂²×C₃₁	1	1	yes	characteristic
303	#493	C₂²×C₃₁	1	1	yes	characteristic
304	#493	C₂²×C₃₁	1	1	yes	characteristic, complemented
305	#492	C₂²×C₃₁	1	1	yes	characteristic
306	#492	C₂²×C₃₁	1	1	yes	characteristic
307	#492	C₂²×C₃₁	1	1	yes	characteristic, complemented
308	#492	C₂²×C₃₁	1	1	yes	characteristic, complemented
309	#487	C₂²×C₃₁	1	1	yes	characteristic, complemented
310	#487	C₂²×C₃₁	1	1	yes	characteristic, complemented
311	#486	C₂²×C₃₁	1	1	yes	characteristic
312	#486	C₂²×C₃₁	1	1	yes	characteristic
313	#486	C₂²×C₃₁	1	1	yes	characteristic
314	#486	C₂²×C₃₁	1	1	yes	characteristic
315	#486	C₂²×C₃₁	1	1	yes	characteristic
316	#486	C₂²×C₃₁	1	1	yes	characteristic
317	#486	C₂²×C₃₁	1	1	yes	characteristic, complemented
318	#486	C₂²×C₃₁	1	1	yes	characteristic, complemented
319	#486	C₂²×C₃₁	1	1	yes	characteristic, complemented
320	#486	C₂²×C₃₁	1	1	yes	characteristic, complemented
321	#482	—	2	2	no	—
322	#478	—	2	2	no	—
323	#464	—	2	2	no	—
324	#463	—	2	2	no	—
325	#460	—	2	2	no	—
326	#459	—	2	2	no	—
327	#459	—	2	2	no	—
328	#452	—	2	2	no	—
329	#450	—	2	2	no	—
330	#449	—	2	2	no	—
331	#447	—	2	2	no	—
332	#447	—	2	2	no	—
333	#431	C₂²×C₃₁	1	1	yes	characteristic
334	#431	C₂²×C₃₁	1	1	yes	characteristic
335	#431	C₂²×C₃₁	1	1	yes	characteristic
336	#322	C₂²×C₃₁	1	1	yes	characteristic, complemented
337	#322	C₂²×C₃₁	1	1	yes	characteristic, complemented
338	#322	C₂²×C₃₁	1	1	yes	characteristic, complemented
339	C₈²	C₂²×C₃₁	1	1	yes	characteristic, complemented
340	C₂×C₃₁	—	4	1	no	—
341	C₂×C₃₁	—	16	1	no	—
342	C₂×C₃₁	—	16	1	no	—
343	C₂×C₃₁	—	16	1	no	—
344	C₂×C₃₁	—	16	1	no	—
345	C₂×C₃₁	#3154	1	1	yes	characteristic
346	C₂×C₃₁	#3155	1	1	yes	characteristic
347	C₂×C₃₁	#3155	1	1	yes	characteristic
348	C₂×#41	—	2	1	no	—
349	C₂×#41	—	2	1	no	—
350	C₂²×D₄	—	2	1	no	—
351	C₂²×D₄	—	2	1	no	—
352	C₂²×D₄	—	2	1	no	—
353	C₂×Q₁₆	—	2	2	no	—
354	C₂×SD₁₆	—	2	2	no	—
355	C₂×SD₁₆	—	2	2	no	—
356	C₂×SD₁₆	—	2	2	no	—
357	C₂×SD₁₆	—	2	2	no	—
358	C₂×D₈	—	2	2	no	—
359	C₂×D₈	—	2	2	no	—
360	C₂×D₈	—	2	2	no	—
361	C₂×D₈	—	2	2	no	—
362	C₂×D₈	—	2	2	no	—
363	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
364	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
365	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
366	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
367	#127	—	2	1	no	—
368	#127	—	2	1	no	—
369	#127	—	2	1	no	—
370	#127	C₂³×C₃₁	1	1	yes	characteristic
371	#127	C₂³×C₃₁	1	1	yes	characteristic
372	#127	C₂³×C₃₁	1	1	yes	characteristic
373	#127	C₂³×C₃₁	1	1	yes	characteristic
374	#125	—	2	1	no	—
375	#124	—	2	1	no	—
376	#124	—	2	1	no	—
377	C₂²⋊Q₈	—	2	1	no	—
378	C₂²⋊Q₈	—	2	1	no	—
379	#121	—	2	1	no	—
380	#121	—	2	1	no	—
381	#121	—	2	1	no	—
382	#121	—	2	1	no	—
383	C₄×Q₈	—	2	1	no	—
384	C₄×Q₈	—	2	1	no	—
385	C₄×D₄	—	2	1	no	—
386	C₄×D₄	—	2	1	no	—
387	C₄×D₄	—	2	1	no	—
388	C₄×D₄	—	2	1	no	—
389	#117	—	2	1	no	—
390	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
391	#107	—	2	2	no	—
392	#106	—	2	2	no	—
393	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
394	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
395	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
396	#103	—	2	2	no	—
397	#103	—	2	2	no	—
398	#102	—	2	2	no	—
399	#102	—	2	2	no	—
400	#102	—	2	2	no	—
401	#102	—	2	2	no	—
402	#102	—	2	2	no	—
403	#102	—	2	2	no	—
404	#102	—	2	2	no	—
405	#102	—	2	2	no	—
406	#102	—	2	2	no	—
407	#102	—	2	2	no	—
408	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
409	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
410	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
411	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
412	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
413	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
414	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
415	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
416	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
417	C₃₁	#19735	1	1	yes	characteristic, direct product factor, Sylow
418	C₂⁴	—	4	1	no	—
419	#41	—	8	2	no	—
420	#41	—	8	2	no	—
421	C₂×Q₈	—	2	1	no	—
422	C₂×Q₈	—	2	1	no	—
423	C₂×Q₈	—	2	1	no	—
424	C₂×Q₈	—	2	1	no	—
425	C₂×D₄	—	2	1	no	—
426	C₂×D₄	—	2	1	no	—
427	C₂×D₄	—	2	1	no	—
428	C₂×D₄	—	2	1	no	—
429	C₂×D₄	—	2	1	no	—
430	C₂×D₄	—	2	1	no	—
431	C₂×D₄	—	2	1	no	—
432	C₂×D₄	—	2	1	no	—
433	C₂×D₄	—	2	1	no	—
434	C₂×D₄	—	2	1	no	—
435	C₂×D₄	—	2	1	no	—
436	C₂×D₄	—	2	1	no	—
437	C₂×D₄	—	4	1	no	—
438	C₂×D₄	—	4	1	no	—
439	C₂×D₄	—	4	1	no	—
440	C₂×D₄	—	4	1	no	—
441	C₂×D₄	—	4	1	no	—
442	C₂×D₄	—	8	2	no	—
443	C₂×D₄	—	8	2	no	—
444	C₂×D₄	—	8	2	no	—
445	C₂²×C₄	—	4	1	no	—
446	C₂²×C₄	—	4	1	no	—
447	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
448	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
449	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
450	Q₁₆	—	8	2	no	—
451	SD₁₆	—	8	2	no	—
452	SD₁₆	—	8	2	no	—
453	SD₁₆	—	8	2	no	—
454	SD₁₆	—	8	2	no	—
455	D₈	—	8	2	no	—
456	D₈	—	8	2	no	—
457	D₈	—	8	2	no	—
458	D₈	—	8	2	no	—
459	D₈	—	8	2	no	—
460	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
461	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
462	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
463	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
464	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
465	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
466	C₄⋊C₄	—	2	1	no	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	—	2	1	no	—
470	C₄⋊C₄	—	2	1	no	—
471	C₄⋊C₄	—	2	1	no	—
472	C₄⋊C₄	—	2	1	no	—
473	C₄⋊C₄	—	2	1	no	—
474	C₄⋊C₄	—	4	1	no	—
475	C₄⋊C₄	—	4	1	no	—
476	C₂²⋊C₄	—	4	1	no	—
477	C₂²⋊C₄	—	4	1	no	—
478	C₂²⋊C₄	—	4	1	no	—
479	C₂²⋊C₄	—	4	1	no	—
480	C₂²⋊C₄	—	4	1	no	—
481	C₄²	—	4	1	no	—
482	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
483	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
484	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
485	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
486	C₂³	—	4	1	no	—
487	C₂³	—	4	1	no	—
488	C₂³	—	4	1	no	—
489	C₂³	—	4	1	no	—
490	C₂³	—	8	2	no	—
491	C₂³	—	8	2	no	—
492	C₂³	—	8	2	no	—
493	C₂³	C₃₁×#127	1	1	yes	characteristic
494	Q₈	—	4	2	no	—
495	Q₈	—	4	2	no	—
496	Q₈	—	4	2	no	—
497	Q₈	—	4	2	no	—
498	D₄	—	4	2	no	—
499	D₄	—	4	2	no	—
500	D₄	—	4	2	no	—
501	D₄	—	4	2	no	—
502	D₄	—	4	2	no	—
503	D₄	—	4	2	no	—
504	D₄	—	4	2	no	—
505	D₄	—	4	2	no	—
506	D₄	—	4	2	no	—
507	D₄	—	4	2	no	—
508	D₄	—	4	2	no	—
509	D₄	—	4	2	no	—
510	D₄	—	8	2	no	—
511	D₄	—	8	2	no	—
512	D₄	—	8	2	no	—
513	D₄	—	8	2	no	—
514	D₄	—	8	2	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	4	1	no	—
519	C₂×C₄	—	4	1	no	—
520	C₂×C₄	—	4	1	no	—
521	C₂×C₄	—	4	1	no	—
522	C₂×C₄	—	8	2	no	—
523	C₂×C₄	—	8	2	no	—
524	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
525	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
526	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
527	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
528	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
529	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
530	C₈	—	2	2	no	—
531	C₈	—	2	2	no	—
532	C₈	—	2	2	no	—
533	C₈	—	2	2	no	—
534	C₈	—	2	2	no	—
535	C₈	—	2	2	no	—
536	C₂²	—	2	1	no	—
537	C₂²	—	2	1	no	—
538	C₂²	—	2	1	no	—
539	C₂²	—	8	1	no	—
540	C₂²	—	8	1	no	—
541	C₂²	—	8	1	no	—
542	C₂²	—	8	1	no	—
543	C₂²	—	8	1	no	—
544	C₂²	—	8	1	no	—
545	C₂²	—	8	1	no	—
546	C₂²	—	8	1	no	—
547	C₂²	—	8	1	no	—
548	C₂²	—	8	1	no	—
549	C₂²	—	8	1	no	—
550	C₂²	—	8	1	no	—
551	C₂²	—	16	4	no	—
552	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
553	C₄	—	2	1	no	—
554	C₄	—	2	1	no	—
555	C₄	—	2	1	no	—
556	C₄	—	8	1	no	—
557	C₄	—	8	1	no	—
558	C₄	—	8	1	no	—
559	C₄	—	8	1	no	—
560	C₄	C₃₁×#576	1	2	yes	—
561	C₄	C₃₁×#576	1	2	yes	—
562	C₄	C₃₁×#577	1	2	yes	—
563	C₂	—	4	1	no	—
564	C₂	—	16	1	no	—
565	C₂	—	16	1	no	—
566	C₂	—	16	1	no	—
567	C₂	—	16	1	no	—
568	C₂	C₃₁×#3154	1	1	yes	characteristic
569	C₂	C₃₁×#3155	1	1	yes	characteristic
570	C₂	C₃₁×#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
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×#3228	C₂	semidirect
2. C₃₁×#3210	C₂	semidirect
3. C₃₁×#3208	C₂	semidirect
4. C₃₁×#3207	C₂	semidirect
5. C₃₁×#3205	C₂	semidirect
6. C₃₁×#3205	C₂	semidirect
7. C₃₁×#1721	C₂	semidirect
8. C₃₁×#1721	C₂	semidirect
9. C₃₁×#1720	C₂	semidirect
10. C₃₁×#1719	C₂	semidirect
11. C₃₁×#1717	C₂	semidirect
12. C₃₁×#1715	C₂	semidirect
13. C₃₁×#1709	C₂	semidirect
14. C₃₁×#1706	C₂	semidirect
15. C₃₁×#1454	C₂	semidirect
20. C₃₁×#500	C₂²	semidirect
21. C₃₁×#498	C₂²	semidirect
22. C₃₁×#494	C₂²	semidirect
27. C₃₁×#493	C₂²	semidirect
30. C₃₁×#492	C₂²	semidirect
31. C₃₁×#492	C₂²	semidirect
32. C₃₁×#487	C₂²	semidirect
33. C₃₁×#487	C₂²	semidirect
40. C₃₁×#486	C₂²	semidirect
41. C₃₁×#486	C₂²	semidirect
42. C₃₁×#486	C₂²	semidirect
43. C₃₁×#486	C₂²	semidirect
59. C₃₁×#322	C₂²	semidirect
60. C₃₁×#322	C₂²	semidirect
61. C₃₁×#322	C₂²	semidirect
62. C₈²×C₃₁	C₂²	semidirect
200. #19735	C₃₁	direct (N × H)
251. #3228	C₂×C₃₁	semidirect
252. #3210	C₂×C₃₁	semidirect
253. #3208	C₂×C₃₁	semidirect
254. #3207	C₂×C₃₁	semidirect
255. #3205	C₂×C₃₁	semidirect
256. #3205	C₂×C₃₁	semidirect
257. #1721	C₂×C₃₁	semidirect
258. #1721	C₂×C₃₁	semidirect
259. #1720	C₂×C₃₁	semidirect
260. #1719	C₂×C₃₁	semidirect
261. #1717	C₂×C₃₁	semidirect
262. #1715	C₂×C₃₁	semidirect
263. #1709	C₂×C₃₁	semidirect
264. #1706	C₂×C₃₁	semidirect
265. #1454	C₂×C₃₁	semidirect
297. #500	C₂²×C₃₁	semidirect
298. #498	C₂²×C₃₁	semidirect
299. #494	C₂²×C₃₁	semidirect
304. #493	C₂²×C₃₁	semidirect
307. #492	C₂²×C₃₁	semidirect
308. #492	C₂²×C₃₁	semidirect
309. #487	C₂²×C₃₁	semidirect
310. #487	C₂²×C₃₁	semidirect
317. #486	C₂²×C₃₁	semidirect
318. #486	C₂²×C₃₁	semidirect
319. #486	C₂²×C₃₁	semidirect
320. #486	C₂²×C₃₁	semidirect
336. #322	C₂²×C₃₁	semidirect
337. #322	C₂²×C₃₁	semidirect
338. #322	C₂²×C₃₁	semidirect
339. C₈²	C₂²×C₃₁	semidirect
417. C₃₁	#19735	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×#3228	2
C₃₁×#3210	2
C₃₁×#3208	2
C₃₁×#3207	2
C₃₁×#3205	2
C₃₁×#1721	2
C₃₁×#1720	2
C₃₁×#1719	2
C₃₁×#1717	2
C₃₁×#1715	2
C₃₁×#1709	2
C₃₁×#1706	2
C₃₁×#1454	2
#19735	31

Subgroups — C31×#19735 #21476494

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19735 #21476494