Subgroups — C₃₁×#19737 #21476496

Order 7936.

746 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3231	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3228	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3210	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3208	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#3208	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#3205	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1722	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1720	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1717	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1717	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1715	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#1714	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#1708	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×#533	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×#500	C₂²	1	1	yes	characteristic, complemented
22	C₃₁×#498	C₂²	1	1	yes	characteristic, complemented
23	C₃₁×#498	C₂²	1	1	yes	characteristic, complemented
24	C₃₁×#494	C₂²	1	1	yes	characteristic, complemented
25	C₃₁×#493	C₂²	1	1	yes	characteristic
26	C₃₁×#493	C₂²	1	1	yes	characteristic
27	C₃₁×#492	C₂²	1	1	yes	characteristic
28	C₃₁×#492	C₂²	1	1	yes	characteristic
29	C₃₁×#492	C₂²	1	1	yes	characteristic
30	C₃₁×#492	C₂²	1	1	yes	characteristic
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×#487	C₂²	1	1	yes	characteristic, complemented
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
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×#485	—	2	2	no	—
45	C₃₁×#482	—	2	2	no	—
46	C₃₁×#479	—	2	2	no	—
47	C₃₁×#478	—	2	2	no	—
48	C₃₁×#464	—	2	2	no	—
49	C₃₁×#460	—	2	2	no	—
50	C₃₁×#460	—	2	2	no	—
51	C₃₁×#459	—	2	2	no	—
52	C₃₁×#452	—	2	2	no	—
53	C₃₁×#450	—	2	2	no	—
54	C₃₁×#450	—	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	C₂²×D₄×C₃₁	—	2	1	no	—
65	C₂²×D₄×C₃₁	—	2	1	no	—
66	C₂²×D₄×C₃₁	—	2	1	no	—
67	C₂×Q₁₆×C₃₁	—	2	2	no	—
68	C₂×SD₁₆×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	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
76	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
77	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
78	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
79	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
80	C₃₁×#127	—	2	1	no	—
81	C₃₁×#127	—	2	1	no	—
82	C₃₁×#127	—	2	1	no	—
83	C₃₁×#127	C₂³	1	1	yes	characteristic
84	C₃₁×#127	C₂³	1	1	yes	characteristic
85	C₃₁×#127	C₂³	1	1	yes	characteristic
86	C₃₁×#125	—	2	1	no	—
87	C₃₁×#125	—	2	1	no	—
88	C₃₁×#124	—	2	1	no	—
89	C₃₁×C₂²⋊Q₈	—	2	1	no	—
90	C₃₁×C₂²⋊Q₈	—	2	1	no	—
91	C₃₁×C₂²⋊Q₈	—	2	1	no	—
92	C₃₁×C₂²⋊Q₈	—	2	1	no	—
93	C₃₁×#121	—	2	1	no	—
94	C₃₁×#121	—	2	1	no	—
95	C₄×Q₈×C₃₁	—	2	1	no	—
96	C₄×Q₈×C₃₁	—	2	1	no	—
97	C₄×Q₈×C₃₁	—	2	1	no	—
98	C₄×Q₈×C₃₁	—	2	1	no	—
99	C₄×D₄×C₃₁	—	2	1	no	—
100	C₄×D₄×C₃₁	—	2	1	no	—
101	C₃₁×#117	—	2	1	no	—
102	C₃₁×#117	—	2	1	no	—
103	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
104	C₃₁×#107	—	2	2	no	—
105	C₃₁×#107	—	2	2	no	—
106	C₃₁×#106	—	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×#103	—	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	C₂×Q₈×C₃₁	—	2	1	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	—	4	1	no	—
149	C₂×D₄×C₃₁	—	4	1	no	—
150	C₂×D₄×C₃₁	—	4	1	no	—
151	C₂×D₄×C₃₁	—	4	1	no	—
152	C₂×D₄×C₃₁	—	8	2	no	—
153	C₂×D₄×C₃₁	—	8	2	no	—
154	C₂×D₄×C₃₁	—	8	2	no	—
155	C₂²×C₄×C₃₁	—	4	1	no	—
156	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
159	Q₁₆×C₃₁	—	8	2	no	—
160	SD₁₆×C₃₁	—	8	2	no	—
161	SD₁₆×C₃₁	—	8	2	no	—
162	SD₁₆×C₃₁	—	8	2	no	—
163	SD₁₆×C₃₁	—	8	2	no	—
164	SD₁₆×C₃₁	—	8	2	no	—
165	D₈×C₃₁	—	8	2	no	—
166	D₈×C₃₁	—	8	2	no	—
167	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
168	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
169	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
170	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
171	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
172	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
173	C₄⋊C₄×C₃₁	—	2	1	no	—
174	C₄⋊C₄×C₃₁	—	2	1	no	—
175	C₄⋊C₄×C₃₁	—	2	1	no	—
176	C₄⋊C₄×C₃₁	—	2	1	no	—
177	C₄⋊C₄×C₃₁	—	2	1	no	—
178	C₄⋊C₄×C₃₁	—	2	1	no	—
179	C₄⋊C₄×C₃₁	—	2	1	no	—
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	—	4	1	no	—
184	C₄⋊C₄×C₃₁	—	4	1	no	—
185	C₄⋊C₄×C₃₁	—	4	1	no	—
186	C₄⋊C₄×C₃₁	—	4	1	no	—
187	C₂²⋊C₄×C₃₁	—	4	1	no	—
188	C₂²⋊C₄×C₃₁	—	4	1	no	—
189	C₂²⋊C₄×C₃₁	—	4	1	no	—
190	C₂²⋊C₄×C₃₁	—	4	1	no	—
191	C₄²×C₃₁	—	4	1	no	—
192	C₄²×C₃₁	—	4	1	no	—
193	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
194	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
195	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
196	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
197	#19737	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
198	C₂³×C₃₁	—	4	1	no	—
199	C₂³×C₃₁	—	4	1	no	—
200	C₂³×C₃₁	—	4	1	no	—
201	C₂³×C₃₁	—	8	2	no	—
202	C₂³×C₃₁	—	8	2	no	—
203	C₂³×C₃₁	—	8	2	no	—
204	C₂³×C₃₁	#127	1	1	yes	characteristic
205	Q₈×C₃₁	—	4	2	no	—
206	Q₈×C₃₁	—	4	2	no	—
207	Q₈×C₃₁	—	4	2	no	—
208	Q₈×C₃₁	—	4	2	no	—
209	Q₈×C₃₁	—	4	2	no	—
210	D₄×C₃₁	—	4	2	no	—
211	D₄×C₃₁	—	4	2	no	—
212	D₄×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	—	8	2	no	—
220	D₄×C₃₁	—	8	2	no	—
221	D₄×C₃₁	—	8	2	no	—
222	D₄×C₃₁	—	8	2	no	—
223	C₂×C₄×C₃₁	—	2	2	no	—
224	C₂×C₄×C₃₁	—	2	2	no	—
225	C₂×C₄×C₃₁	—	2	2	no	—
226	C₂×C₄×C₃₁	—	4	1	no	—
227	C₂×C₄×C₃₁	—	4	1	no	—
228	C₂×C₄×C₃₁	—	4	1	no	—
229	C₂×C₄×C₃₁	—	4	1	no	—
230	C₂×C₄×C₃₁	—	4	1	no	—
231	C₂×C₄×C₃₁	—	8	2	no	—
232	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
233	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
234	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
235	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
236	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
238	C₈×C₃₁	—	2	2	no	—
239	C₈×C₃₁	—	2	2	no	—
240	C₈×C₃₁	—	2	2	no	—
241	C₈×C₃₁	—	2	2	no	—
242	C₈×C₃₁	—	2	2	no	—
243	C₈×C₃₁	—	2	2	no	—
244	#3231	C₂×C₃₁	1	1	yes	characteristic, complemented
245	#3228	C₂×C₃₁	1	1	yes	characteristic, complemented
246	#3210	C₂×C₃₁	1	1	yes	characteristic, complemented
247	#3208	C₂×C₃₁	1	1	yes	characteristic, complemented
248	#3208	C₂×C₃₁	1	1	yes	characteristic, complemented
249	#3205	C₂×C₃₁	1	1	yes	characteristic
250	#1722	C₂×C₃₁	1	1	yes	characteristic, complemented
251	#1720	C₂×C₃₁	1	1	yes	characteristic, complemented
252	#1717	C₂×C₃₁	1	1	yes	characteristic, complemented
253	#1717	C₂×C₃₁	1	1	yes	characteristic, complemented
254	#1715	C₂×C₃₁	1	1	yes	characteristic, complemented
255	#1714	C₂×C₃₁	1	1	yes	characteristic, complemented
256	#1708	C₂×C₃₁	1	1	yes	characteristic, complemented
257	#1706	C₂×C₃₁	1	1	yes	characteristic, complemented
258	#1454	C₂×C₃₁	1	1	yes	characteristic, complemented
259	C₂²×C₃₁	—	2	1	no	—
260	C₂²×C₃₁	—	2	1	no	—
261	C₂²×C₃₁	—	2	1	no	—
262	C₂²×C₃₁	—	8	1	no	—
263	C₂²×C₃₁	—	8	1	no	—
264	C₂²×C₃₁	—	8	1	no	—
265	C₂²×C₃₁	—	8	1	no	—
266	C₂²×C₃₁	—	8	1	no	—
267	C₂²×C₃₁	—	8	1	no	—
268	C₂²×C₃₁	—	8	1	no	—
269	C₂²×C₃₁	—	8	1	no	—
270	C₂²×C₃₁	—	8	1	no	—
271	C₂²×C₃₁	—	16	4	no	—
272	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
273	C₄×C₃₁	—	2	1	no	—
274	C₄×C₃₁	—	2	1	no	—
275	C₄×C₃₁	—	2	1	no	—
276	C₄×C₃₁	—	8	1	no	—
277	C₄×C₃₁	—	8	1	no	—
278	C₄×C₃₁	—	8	1	no	—
279	C₄×C₃₁	—	8	1	no	—
280	C₄×C₃₁	—	8	1	no	—
281	C₄×C₃₁	#576	1	2	yes	—
282	C₄×C₃₁	#577	1	2	yes	—
283	C₄×C₃₁	#577	1	2	yes	—
284	#533	C₂²×C₃₁	1	1	yes	characteristic
285	#533	C₂²×C₃₁	1	1	yes	characteristic
286	#532	C₂²×C₃₁	1	1	yes	characteristic
287	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
288	#500	C₂²×C₃₁	1	1	yes	characteristic, complemented
289	#500	C₂²×C₃₁	1	1	yes	characteristic, complemented
290	#498	C₂²×C₃₁	1	1	yes	characteristic, complemented
291	#498	C₂²×C₃₁	1	1	yes	characteristic, complemented
292	#494	C₂²×C₃₁	1	1	yes	characteristic, complemented
293	#493	C₂²×C₃₁	1	1	yes	characteristic
294	#493	C₂²×C₃₁	1	1	yes	characteristic
295	#492	C₂²×C₃₁	1	1	yes	characteristic
296	#492	C₂²×C₃₁	1	1	yes	characteristic
297	#492	C₂²×C₃₁	1	1	yes	characteristic
298	#492	C₂²×C₃₁	1	1	yes	characteristic
299	#492	C₂²×C₃₁	1	1	yes	characteristic, complemented
300	#487	C₂²×C₃₁	1	1	yes	characteristic, complemented
301	#487	C₂²×C₃₁	1	1	yes	characteristic, complemented
302	#487	C₂²×C₃₁	1	1	yes	characteristic, complemented
303	#486	C₂²×C₃₁	1	1	yes	characteristic
304	#486	C₂²×C₃₁	1	1	yes	characteristic
305	#486	C₂²×C₃₁	1	1	yes	characteristic
306	#486	C₂²×C₃₁	1	1	yes	characteristic
307	#486	C₂²×C₃₁	1	1	yes	characteristic
308	#486	C₂²×C₃₁	1	1	yes	characteristic
309	#486	C₂²×C₃₁	1	1	yes	characteristic, complemented
310	#486	C₂²×C₃₁	1	1	yes	characteristic, complemented
311	#486	C₂²×C₃₁	1	1	yes	characteristic, complemented
312	#485	—	2	2	no	—
313	#482	—	2	2	no	—
314	#479	—	2	2	no	—
315	#478	—	2	2	no	—
316	#464	—	2	2	no	—
317	#460	—	2	2	no	—
318	#460	—	2	2	no	—
319	#459	—	2	2	no	—
320	#452	—	2	2	no	—
321	#450	—	2	2	no	—
322	#450	—	2	2	no	—
323	#447	—	2	2	no	—
324	#431	C₂²×C₃₁	1	1	yes	characteristic
325	#431	C₂²×C₃₁	1	1	yes	characteristic
326	#431	C₂²×C₃₁	1	1	yes	characteristic
327	#322	C₂²×C₃₁	1	1	yes	characteristic, complemented
328	#322	C₂²×C₃₁	1	1	yes	characteristic, complemented
329	#322	C₂²×C₃₁	1	1	yes	characteristic, complemented
330	C₈²	C₂²×C₃₁	1	1	yes	characteristic, complemented
331	C₂×C₃₁	—	4	1	no	—
332	C₂×C₃₁	—	16	1	no	—
333	C₂×C₃₁	—	16	1	no	—
334	C₂×C₃₁	—	16	1	no	—
335	C₂×C₃₁	#3154	1	1	yes	characteristic
336	C₂×C₃₁	#3154	1	1	yes	characteristic
337	C₂×C₃₁	#3155	1	1	yes	characteristic
338	C₂×#41	—	2	1	no	—
339	C₂²×D₄	—	2	1	no	—
340	C₂²×D₄	—	2	1	no	—
341	C₂²×D₄	—	2	1	no	—
342	C₂×Q₁₆	—	2	2	no	—
343	C₂×SD₁₆	—	2	2	no	—
344	C₂×SD₁₆	—	2	2	no	—
345	C₂×SD₁₆	—	2	2	no	—
346	C₂×SD₁₆	—	2	2	no	—
347	C₂×SD₁₆	—	2	2	no	—
348	C₂×D₈	—	2	2	no	—
349	C₂×D₈	—	2	2	no	—
350	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
351	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
352	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
353	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
354	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
355	#127	—	2	1	no	—
356	#127	—	2	1	no	—
357	#127	—	2	1	no	—
358	#127	C₂³×C₃₁	1	1	yes	characteristic
359	#127	C₂³×C₃₁	1	1	yes	characteristic
360	#127	C₂³×C₃₁	1	1	yes	characteristic
361	#125	—	2	1	no	—
362	#125	—	2	1	no	—
363	#124	—	2	1	no	—
364	C₂²⋊Q₈	—	2	1	no	—
365	C₂²⋊Q₈	—	2	1	no	—
366	C₂²⋊Q₈	—	2	1	no	—
367	C₂²⋊Q₈	—	2	1	no	—
368	#121	—	2	1	no	—
369	#121	—	2	1	no	—
370	C₄×Q₈	—	2	1	no	—
371	C₄×Q₈	—	2	1	no	—
372	C₄×Q₈	—	2	1	no	—
373	C₄×Q₈	—	2	1	no	—
374	C₄×D₄	—	2	1	no	—
375	C₄×D₄	—	2	1	no	—
376	#117	—	2	1	no	—
377	#117	—	2	1	no	—
378	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
379	#107	—	2	2	no	—
380	#107	—	2	2	no	—
381	#106	—	2	2	no	—
382	#106	—	2	2	no	—
383	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
384	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
385	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
386	#103	—	2	2	no	—
387	#103	—	2	2	no	—
388	#103	—	2	2	no	—
389	#102	—	2	2	no	—
390	#102	—	2	2	no	—
391	#102	—	2	2	no	—
392	#102	—	2	2	no	—
393	#102	—	2	2	no	—
394	#102	—	2	2	no	—
395	#102	—	2	2	no	—
396	#102	—	2	2	no	—
397	#102	—	2	2	no	—
398	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
399	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
400	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
401	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
402	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
403	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
404	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
405	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
406	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
407	C₃₁	#19737	1	1	yes	characteristic, direct product factor, Sylow
408	C₂⁴	—	4	1	no	—
409	#41	—	8	2	no	—
410	C₂×Q₈	—	2	1	no	—
411	C₂×Q₈	—	2	1	no	—
412	C₂×Q₈	—	2	1	no	—
413	C₂×Q₈	—	2	1	no	—
414	C₂×Q₈	—	2	1	no	—
415	C₂×D₄	—	2	1	no	—
416	C₂×D₄	—	2	1	no	—
417	C₂×D₄	—	2	1	no	—
418	C₂×D₄	—	2	1	no	—
419	C₂×D₄	—	2	1	no	—
420	C₂×D₄	—	2	1	no	—
421	C₂×D₄	—	2	1	no	—
422	C₂×D₄	—	2	1	no	—
423	C₂×D₄	—	2	1	no	—
424	C₂×D₄	—	4	1	no	—
425	C₂×D₄	—	4	1	no	—
426	C₂×D₄	—	4	1	no	—
427	C₂×D₄	—	4	1	no	—
428	C₂×D₄	—	8	2	no	—
429	C₂×D₄	—	8	2	no	—
430	C₂×D₄	—	8	2	no	—
431	C₂²×C₄	—	4	1	no	—
432	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
433	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
434	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
435	Q₁₆	—	8	2	no	—
436	SD₁₆	—	8	2	no	—
437	SD₁₆	—	8	2	no	—
438	SD₁₆	—	8	2	no	—
439	SD₁₆	—	8	2	no	—
440	SD₁₆	—	8	2	no	—
441	D₈	—	8	2	no	—
442	D₈	—	8	2	no	—
443	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
444	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
445	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
446	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
447	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
448	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
449	C₄⋊C₄	—	2	1	no	—
450	C₄⋊C₄	—	2	1	no	—
451	C₄⋊C₄	—	2	1	no	—
452	C₄⋊C₄	—	2	1	no	—
453	C₄⋊C₄	—	2	1	no	—
454	C₄⋊C₄	—	2	1	no	—
455	C₄⋊C₄	—	2	1	no	—
456	C₄⋊C₄	—	2	1	no	—
457	C₄⋊C₄	—	2	1	no	—
458	C₄⋊C₄	—	2	1	no	—
459	C₄⋊C₄	—	4	1	no	—
460	C₄⋊C₄	—	4	1	no	—
461	C₄⋊C₄	—	4	1	no	—
462	C₄⋊C₄	—	4	1	no	—
463	C₂²⋊C₄	—	4	1	no	—
464	C₂²⋊C₄	—	4	1	no	—
465	C₂²⋊C₄	—	4	1	no	—
466	C₂²⋊C₄	—	4	1	no	—
467	C₄²	—	4	1	no	—
468	C₄²	—	4	1	no	—
469	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
470	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
471	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
472	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
473	C₂³	—	4	1	no	—
474	C₂³	—	4	1	no	—
475	C₂³	—	4	1	no	—
476	C₂³	—	8	2	no	—
477	C₂³	—	8	2	no	—
478	C₂³	—	8	2	no	—
479	C₂³	C₃₁×#127	1	1	yes	characteristic
480	Q₈	—	4	2	no	—
481	Q₈	—	4	2	no	—
482	Q₈	—	4	2	no	—
483	Q₈	—	4	2	no	—
484	Q₈	—	4	2	no	—
485	D₄	—	4	2	no	—
486	D₄	—	4	2	no	—
487	D₄	—	4	2	no	—
488	D₄	—	4	2	no	—
489	D₄	—	4	2	no	—
490	D₄	—	4	2	no	—
491	D₄	—	4	2	no	—
492	D₄	—	4	2	no	—
493	D₄	—	4	2	no	—
494	D₄	—	8	2	no	—
495	D₄	—	8	2	no	—
496	D₄	—	8	2	no	—
497	D₄	—	8	2	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	4	1	no	—
502	C₂×C₄	—	4	1	no	—
503	C₂×C₄	—	4	1	no	—
504	C₂×C₄	—	4	1	no	—
505	C₂×C₄	—	4	1	no	—
506	C₂×C₄	—	8	2	no	—
507	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
508	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
509	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
510	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
511	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
512	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
513	C₈	—	2	2	no	—
514	C₈	—	2	2	no	—
515	C₈	—	2	2	no	—
516	C₈	—	2	2	no	—
517	C₈	—	2	2	no	—
518	C₈	—	2	2	no	—
519	C₂²	—	2	1	no	—
520	C₂²	—	2	1	no	—
521	C₂²	—	2	1	no	—
522	C₂²	—	8	1	no	—
523	C₂²	—	8	1	no	—
524	C₂²	—	8	1	no	—
525	C₂²	—	8	1	no	—
526	C₂²	—	8	1	no	—
527	C₂²	—	8	1	no	—
528	C₂²	—	8	1	no	—
529	C₂²	—	8	1	no	—
530	C₂²	—	8	1	no	—
531	C₂²	—	16	4	no	—
532	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
533	C₄	—	2	1	no	—
534	C₄	—	2	1	no	—
535	C₄	—	2	1	no	—
536	C₄	—	8	1	no	—
537	C₄	—	8	1	no	—
538	C₄	—	8	1	no	—
539	C₄	—	8	1	no	—
540	C₄	—	8	1	no	—
541	C₄	C₃₁×#576	1	2	yes	—
542	C₄	C₃₁×#577	1	2	yes	—
543	C₄	C₃₁×#577	1	2	yes	—
544	C₂	—	4	1	no	—
545	C₂	—	16	1	no	—
546	C₂	—	16	1	no	—
547	C₂	—	16	1	no	—
548	C₂	C₃₁×#3154	1	1	yes	characteristic
549	C₂	C₃₁×#3154	1	1	yes	characteristic
550	C₂	C₃₁×#3155	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×#3231	C₂	semidirect
2. C₃₁×#3228	C₂	semidirect
3. C₃₁×#3210	C₂	semidirect
4. C₃₁×#3208	C₂	semidirect
5. C₃₁×#3208	C₂	semidirect
7. C₃₁×#1722	C₂	semidirect
8. C₃₁×#1720	C₂	semidirect
9. C₃₁×#1717	C₂	semidirect
10. C₃₁×#1717	C₂	semidirect
11. C₃₁×#1715	C₂	semidirect
12. C₃₁×#1714	C₂	semidirect
13. C₃₁×#1708	C₂	semidirect
14. C₃₁×#1706	C₂	semidirect
15. C₃₁×#1454	C₂	semidirect
20. C₃₁×#500	C₂²	semidirect
21. C₃₁×#500	C₂²	semidirect
22. C₃₁×#498	C₂²	semidirect
23. C₃₁×#498	C₂²	semidirect
24. C₃₁×#494	C₂²	semidirect
31. C₃₁×#492	C₂²	semidirect
32. C₃₁×#487	C₂²	semidirect
33. C₃₁×#487	C₂²	semidirect
34. C₃₁×#487	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
197. #19737	C₃₁	direct (N × H)
244. #3231	C₂×C₃₁	semidirect
245. #3228	C₂×C₃₁	semidirect
246. #3210	C₂×C₃₁	semidirect
247. #3208	C₂×C₃₁	semidirect
248. #3208	C₂×C₃₁	semidirect
250. #1722	C₂×C₃₁	semidirect
251. #1720	C₂×C₃₁	semidirect
252. #1717	C₂×C₃₁	semidirect
253. #1717	C₂×C₃₁	semidirect
254. #1715	C₂×C₃₁	semidirect
255. #1714	C₂×C₃₁	semidirect
256. #1708	C₂×C₃₁	semidirect
257. #1706	C₂×C₃₁	semidirect
258. #1454	C₂×C₃₁	semidirect
288. #500	C₂²×C₃₁	semidirect
289. #500	C₂²×C₃₁	semidirect
290. #498	C₂²×C₃₁	semidirect
291. #498	C₂²×C₃₁	semidirect
292. #494	C₂²×C₃₁	semidirect
299. #492	C₂²×C₃₁	semidirect
300. #487	C₂²×C₃₁	semidirect
301. #487	C₂²×C₃₁	semidirect
302. #487	C₂²×C₃₁	semidirect
309. #486	C₂²×C₃₁	semidirect
310. #486	C₂²×C₃₁	semidirect
311. #486	C₂²×C₃₁	semidirect
327. #322	C₂²×C₃₁	semidirect
328. #322	C₂²×C₃₁	semidirect
329. #322	C₂²×C₃₁	semidirect
330. C₈²	C₂²×C₃₁	semidirect
407. C₃₁	#19737	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×#3231	2
C₃₁×#3228	2
C₃₁×#3210	2
C₃₁×#3208	2
C₃₁×#3205	2
C₃₁×#1722	2
C₃₁×#1720	2
C₃₁×#1717	2
C₃₁×#1715	2
C₃₁×#1714	2
C₃₁×#1708	2
C₃₁×#1706	2
C₃₁×#1454	2
#19737	31

Subgroups — C31×#19737 #21476496

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19737 #21476496