Subgroups — C₃₁×#18952 #21475711

Order 7936.

880 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3550	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3246	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3228	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3214	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1651	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1650	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1628	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1627	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1552	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1497	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1480	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#563	—	2	2	no	—
13	C₃₁×#559	—	2	1	no	—
14	Q₈²×C₃₁	—	2	1	no	—
15	C₃₁×#556	—	2	1	no	—
16	C₃₁×#556	C₂²	1	2	yes	—
17	D₄×Q₈×C₃₁	—	2	1	no	—
18	D₄×Q₈×C₃₁	C₂²	1	2	yes	—
19	C₃₁×#542	—	2	2	no	—
20	C₃₁×#542	—	2	2	no	—
21	C₃₁×#536	—	2	2	no	—
22	C₃₁×#533	C₂²	1	1	yes	characteristic
23	C₃₁×#533	C₂²	1	1	yes	characteristic
24	C₃₁×#532	C₂²	1	1	yes	characteristic
25	C₃₁×#532	C₂²	1	1	yes	characteristic
26	C₃₁×#500	C₂²	1	1	yes	characteristic
27	C₃₁×#498	C₂²	1	1	yes	characteristic
28	C₃₁×#496	—	2	1	no	—
29	C₃₁×#489	—	2	1	no	—
30	C₃₁×#486	C₂²	1	1	yes	characteristic
31	C₃₁×#486	C₂²	1	1	yes	characteristic
32	C₃₁×#482	—	2	2	no	—
33	C₃₁×#478	—	2	2	no	—
34	C₃₁×#476	—	2	1	no	—
35	C₃₁×#475	C₂²	1	2	yes	—
36	C₃₁×#465	—	2	2	no	—
37	C₃₁×#461	—	2	1	no	—
38	C₃₁×#461	C₂²	1	2	yes	—
39	C₃₁×#460	C₂²	1	2	yes	—
40	C₃₁×#459	C₂²	1	2	yes	—
41	C₃₁×#449	—	2	2	no	—
42	C₃₁×#432	C₂²	1	1	yes	characteristic
43	C₃₁×#431	C₂²	1	1	yes	characteristic
44	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
45	C₃₁×#332	C₂²	1	2	yes	—
46	C₃₁×#332	C₂²	1	2	yes	—
47	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
48	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
49	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
50	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
51	C₂×#41×C₃₁	—	2	1	no	—
52	C₂×#41×C₃₁	—	2	1	no	—
53	C₂²×Q₈×C₃₁	—	2	2	no	—
54	C₂×SD₁₆×C₃₁	—	2	2	no	—
55	C₂×SD₁₆×C₃₁	—	2	2	no	—
56	C₂×D₈×C₃₁	—	2	2	no	—
57	C₂²×C₈×C₃₁	—	2	1	no	—
58	C₃₁×C₄⋊Q₈	—	2	4	no	—
59	C₃₁×C₄⋊Q₈	—	2	2	no	—
60	C₃₁×C₄⋊Q₈	—	2	2	no	—
61	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
62	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
63	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
64	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
65	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
66	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
67	C₃₁×#127	D₄	1	1	yes	characteristic
68	C₃₁×#127	C₂³	1	1	yes	characteristic
69	C₃₁×#126	—	2	4	no	—
70	C₃₁×#126	—	2	4	no	—
71	C₃₁×#125	—	2	1	no	—
72	C₃₁×#125	D₄	1	2	yes	—
73	C₃₁×#124	—	2	4	no	—
74	C₃₁×#124	—	2	2	no	—
75	C₃₁×#124	—	2	2	no	—
76	C₃₁×#124	—	2	1	no	—
77	C₃₁×#124	—	2	1	no	—
78	C₃₁×#123	—	2	4	no	—
79	C₃₁×#123	—	2	4	no	—
80	C₃₁×C₂²⋊Q₈	—	2	4	no	—
81	C₃₁×C₂²⋊Q₈	—	2	2	no	—
82	C₃₁×C₂²⋊Q₈	—	2	1	no	—
83	C₃₁×C₂²⋊Q₈	—	2	1	no	—
84	C₃₁×C₂²⋊Q₈	—	2	1	no	—
85	C₃₁×C₂²⋊Q₈	—	2	2	no	—
86	C₃₁×C₂²⋊Q₈	—	2	4	no	—
87	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
88	C₃₁×#121	—	2	1	no	—
89	C₃₁×#121	—	2	1	no	—
90	C₃₁×#121	—	2	1	no	—
91	C₃₁×#121	D₄	1	2	yes	—
92	C₄×Q₈×C₃₁	—	2	2	no	—
93	C₄×Q₈×C₃₁	—	2	2	no	—
94	C₄×Q₈×C₃₁	—	2	1	no	—
95	C₄×Q₈×C₃₁	—	2	1	no	—
96	C₄×Q₈×C₃₁	—	2	1	no	—
97	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
98	C₄×D₄×C₃₁	—	2	2	no	—
99	C₄×D₄×C₃₁	—	2	2	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	C₂³	1	2	yes	—
104	C₃₁×#117	—	2	1	no	—
105	C₃₁×#117	—	2	2	no	—
106	C₃₁×#117	D₄	1	2	yes	—
107	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
108	C₃₁×#107	—	2	2	no	—
109	C₃₁×#106	—	2	2	no	—
110	C₃₁×#106	—	2	2	no	—
111	C₃₁×C₄⋊C₈	—	2	1	no	—
112	C₃₁×C₄⋊C₈	—	2	1	no	—
113	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
114	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
115	C₃₁×#103	—	2	2	no	—
116	C₃₁×#103	—	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×C₂²⋊C₈	—	2	1	no	—
122	C₃₁×C₂²⋊C₈	—	2	1	no	—
123	C₃₁×#97	—	2	1	no	—
124	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
125	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
126	#41×C₃₁	—	8	1	no	—
127	#41×C₃₁	—	8	1	no	—
128	#41×C₃₁	—	8	2	no	—
129	C₂×Q₈×C₃₁	—	2	4	no	—
130	C₂×Q₈×C₃₁	—	2	1	no	—
131	C₂×Q₈×C₃₁	—	2	1	no	—
132	C₂×Q₈×C₃₁	—	2	1	no	—
133	C₂×Q₈×C₃₁	—	2	1	no	—
134	C₂×Q₈×C₃₁	—	2	1	no	—
135	C₂×Q₈×C₃₁	—	8	2	no	—
136	C₂×Q₈×C₃₁	—	8	2	no	—
137	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
138	C₂×D₄×C₃₁	—	2	2	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	—	4	1	no	—
145	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
146	C₂²×C₄×C₃₁	—	2	4	no	—
147	C₂²×C₄×C₃₁	—	2	2	no	—
148	C₂²×C₄×C₃₁	—	4	1	no	—
149	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
150	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
152	SD₁₆×C₃₁	—	8	2	no	—
153	SD₁₆×C₃₁	—	8	2	no	—
154	D₈×C₃₁	—	8	2	no	—
155	C₂×C₈×C₃₁	—	2	1	no	—
156	C₂×C₈×C₃₁	—	2	1	no	—
157	C₂×C₈×C₃₁	—	2	2	no	—
158	C₂×C₈×C₃₁	—	2	2	no	—
159	C₂×C₈×C₃₁	—	8	1	no	—
160	C₄⋊C₄×C₃₁	—	2	4	no	—
161	C₄⋊C₄×C₃₁	—	2	4	no	—
162	C₄⋊C₄×C₃₁	—	2	1	no	—
163	C₄⋊C₄×C₃₁	—	2	1	no	—
164	C₄⋊C₄×C₃₁	—	2	1	no	—
165	C₄⋊C₄×C₃₁	—	2	1	no	—
166	C₄⋊C₄×C₃₁	—	2	2	no	—
167	C₄⋊C₄×C₃₁	—	2	2	no	—
168	C₄⋊C₄×C₃₁	—	2	2	no	—
169	C₄⋊C₄×C₃₁	—	2	1	no	—
170	C₄⋊C₄×C₃₁	—	2	1	no	—
171	C₄⋊C₄×C₃₁	—	2	2	no	—
172	C₄⋊C₄×C₃₁	—	2	1	no	—
173	C₄⋊C₄×C₃₁	—	2	1	no	—
174	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
175	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
176	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
177	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
178	C₂²⋊C₄×C₃₁	—	2	4	no	—
179	C₂²⋊C₄×C₃₁	—	2	4	no	—
180	C₂²⋊C₄×C₃₁	—	2	4	no	—
181	C₂²⋊C₄×C₃₁	—	2	2	no	—
182	C₂²⋊C₄×C₃₁	—	2	2	no	—
183	C₂²⋊C₄×C₃₁	—	2	2	no	—
184	C₂²⋊C₄×C₃₁	—	2	2	no	—
185	C₂²⋊C₄×C₃₁	—	4	1	no	—
186	C₂²⋊C₄×C₃₁	—	4	1	no	—
187	C₄²×C₃₁	—	2	4	no	—
188	C₄²×C₃₁	—	2	2	no	—
189	C₄²×C₃₁	—	2	2	no	—
190	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
191	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
192	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
193	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
194	#18952	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
195	C₂³×C₃₁	—	2	2	no	—
196	C₂³×C₃₁	—	4	1	no	—
197	C₂³×C₃₁	#120	1	1	yes	characteristic
198	Q₈×C₃₁	—	2	2	no	—
199	Q₈×C₃₁	—	2	2	no	—
200	Q₈×C₃₁	—	4	2	no	—
201	Q₈×C₃₁	—	4	1	no	—
202	Q₈×C₃₁	—	4	2	no	—
203	Q₈×C₃₁	—	4	1	no	—
204	Q₈×C₃₁	—	4	2	no	—
205	Q₈×C₃₁	—	8	1	no	—
206	Q₈×C₃₁	—	8	4	no	—
207	D₄×C₃₁	—	2	2	no	—
208	D₄×C₃₁	—	2	2	no	—
209	D₄×C₃₁	—	4	2	no	—
210	D₄×C₃₁	—	4	2	no	—
211	D₄×C₃₁	—	4	1	no	—
212	D₄×C₃₁	—	4	2	no	—
213	D₄×C₃₁	—	4	1	no	—
214	D₄×C₃₁	—	8	1	no	—
215	D₄×C₃₁	—	8	2	no	—
216	D₄×C₃₁	—	8	2	no	—
217	C₂×C₄×C₃₁	—	2	1	no	—
218	C₂×C₄×C₃₁	—	2	2	no	—
219	C₂×C₄×C₃₁	—	2	1	no	—
220	C₂×C₄×C₃₁	—	2	2	no	—
221	C₂×C₄×C₃₁	—	2	2	no	—
222	C₂×C₄×C₃₁	—	2	2	no	—
223	C₂×C₄×C₃₁	—	2	2	no	—
224	C₂×C₄×C₃₁	—	2	2	no	—
225	C₂×C₄×C₃₁	—	4	1	no	—
226	C₂×C₄×C₃₁	—	4	2	no	—
227	C₂×C₄×C₃₁	—	4	2	no	—
228	C₂×C₄×C₃₁	—	4	1	no	—
229	C₂×C₄×C₃₁	—	8	2	no	—
230	C₂×C₄×C₃₁	—	8	4	no	—
231	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
232	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
233	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
234	C₂×C₄×C₃₁	C₂²×D₄	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	C₈×C₃₁	—	4	1	no	—
238	C₈×C₃₁	—	4	1	no	—
239	C₈×C₃₁	—	4	2	no	—
240	C₈×C₃₁	—	4	2	no	—
241	#3550	C₂×C₃₁	1	1	yes	characteristic, complemented
242	#3246	C₂×C₃₁	1	1	yes	characteristic, complemented
243	#3228	C₂×C₃₁	1	1	yes	characteristic, complemented
244	#3214	C₂×C₃₁	1	1	yes	characteristic
245	#1651	C₂×C₃₁	1	2	yes	complemented
246	#1650	C₂×C₃₁	1	2	yes	complemented
247	#1628	C₂×C₃₁	1	2	yes	complemented
248	#1627	C₂×C₃₁	1	2	yes	complemented
249	#1552	C₂×C₃₁	1	1	yes	characteristic, complemented
250	#1497	C₂×C₃₁	1	1	yes	characteristic, complemented
251	#1480	C₂×C₃₁	1	1	yes	characteristic, complemented
252	C₂²×C₃₁	—	2	1	no	—
253	C₂²×C₃₁	—	2	1	no	—
254	C₂²×C₃₁	—	2	1	no	—
255	C₂²×C₃₁	—	4	2	no	—
256	C₂²×C₃₁	—	4	2	no	—
257	C₂²×C₃₁	—	4	2	no	—
258	C₂²×C₃₁	—	8	1	no	—
259	C₂²×C₃₁	—	8	1	no	—
260	C₂²×C₃₁	—	8	1	no	—
261	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
262	C₄×C₃₁	—	2	1	no	—
263	C₄×C₃₁	—	2	1	no	—
264	C₄×C₃₁	—	2	1	no	—
265	C₄×C₃₁	—	2	1	no	—
266	C₄×C₃₁	—	4	2	no	—
267	C₄×C₃₁	—	4	2	no	—
268	C₄×C₃₁	—	4	2	no	—
269	C₄×C₃₁	—	4	2	no	—
270	C₄×C₃₁	—	4	2	no	—
271	C₄×C₃₁	—	8	1	no	—
272	C₄×C₃₁	#576	1	1	yes	characteristic
273	C₄×C₃₁	#577	1	2	yes	—
274	C₄×C₃₁	#577	1	1	yes	characteristic
275	#563	—	2	2	no	—
276	#559	—	2	1	no	—
277	Q₈²	—	2	1	no	—
278	#556	—	2	1	no	—
279	#556	C₂²×C₃₁	1	2	yes	—
280	D₄×Q₈	—	2	1	no	—
281	D₄×Q₈	C₂²×C₃₁	1	2	yes	—
282	#542	—	2	2	no	—
283	#542	—	2	2	no	—
284	#536	—	2	2	no	—
285	#533	C₂²×C₃₁	1	1	yes	characteristic
286	#533	C₂²×C₃₁	1	1	yes	characteristic
287	#532	C₂²×C₃₁	1	1	yes	characteristic
288	#532	C₂²×C₃₁	1	1	yes	characteristic
289	#500	C₂²×C₃₁	1	1	yes	characteristic
290	#498	C₂²×C₃₁	1	1	yes	characteristic
291	#496	—	2	1	no	—
292	#489	—	2	1	no	—
293	#486	C₂²×C₃₁	1	1	yes	characteristic
294	#486	C₂²×C₃₁	1	1	yes	characteristic
295	#482	—	2	2	no	—
296	#478	—	2	2	no	—
297	#476	—	2	1	no	—
298	#475	C₂²×C₃₁	1	2	yes	—
299	#465	—	2	2	no	—
300	#461	—	2	1	no	—
301	#461	C₂²×C₃₁	1	2	yes	—
302	#460	C₂²×C₃₁	1	2	yes	—
303	#459	C₂²×C₃₁	1	2	yes	—
304	#449	—	2	2	no	—
305	#432	C₂²×C₃₁	1	1	yes	characteristic
306	#431	C₂²×C₃₁	1	1	yes	characteristic
307	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
308	#332	C₂²×C₃₁	1	2	yes	—
309	#332	C₂²×C₃₁	1	2	yes	—
310	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
311	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
312	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
313	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
314	C₂×C₃₁	—	4	1	no	—
315	C₂×C₃₁	—	8	2	no	—
316	C₂×C₃₁	—	16	1	no	—
317	C₂×C₃₁	#3010	1	1	yes	characteristic
318	C₂×C₃₁	#3012	1	1	yes	characteristic
319	C₂×C₃₁	#3020	1	1	yes	characteristic
320	C₂×#41	—	2	1	no	—
321	C₂×#41	—	2	1	no	—
322	C₂²×Q₈	—	2	2	no	—
323	C₂×SD₁₆	—	2	2	no	—
324	C₂×SD₁₆	—	2	2	no	—
325	C₂×D₈	—	2	2	no	—
326	C₂²×C₈	—	2	1	no	—
327	C₄⋊Q₈	—	2	4	no	—
328	C₄⋊Q₈	—	2	2	no	—
329	C₄⋊Q₈	—	2	2	no	—
330	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
331	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
332	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
333	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
334	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
335	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
336	#127	D₄×C₃₁	1	1	yes	characteristic
337	#127	C₂³×C₃₁	1	1	yes	characteristic
338	#126	—	2	4	no	—
339	#126	—	2	4	no	—
340	#125	—	2	1	no	—
341	#125	D₄×C₃₁	1	2	yes	—
342	#124	—	2	4	no	—
343	#124	—	2	2	no	—
344	#124	—	2	2	no	—
345	#124	—	2	1	no	—
346	#124	—	2	1	no	—
347	#123	—	2	4	no	—
348	#123	—	2	4	no	—
349	C₂²⋊Q₈	—	2	4	no	—
350	C₂²⋊Q₈	—	2	2	no	—
351	C₂²⋊Q₈	—	2	1	no	—
352	C₂²⋊Q₈	—	2	1	no	—
353	C₂²⋊Q₈	—	2	1	no	—
354	C₂²⋊Q₈	—	2	2	no	—
355	C₂²⋊Q₈	—	2	4	no	—
356	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
357	#121	—	2	1	no	—
358	#121	—	2	1	no	—
359	#121	—	2	1	no	—
360	#121	D₄×C₃₁	1	2	yes	—
361	C₄×Q₈	—	2	2	no	—
362	C₄×Q₈	—	2	2	no	—
363	C₄×Q₈	—	2	1	no	—
364	C₄×Q₈	—	2	1	no	—
365	C₄×Q₈	—	2	1	no	—
366	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
367	C₄×D₄	—	2	2	no	—
368	C₄×D₄	—	2	2	no	—
369	C₄×D₄	—	2	1	no	—
370	C₄×D₄	—	2	1	no	—
371	C₄×D₄	—	2	1	no	—
372	C₄×D₄	C₂³×C₃₁	1	2	yes	—
373	#117	—	2	1	no	—
374	#117	—	2	2	no	—
375	#117	D₄×C₃₁	1	2	yes	—
376	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
377	#107	—	2	2	no	—
378	#106	—	2	2	no	—
379	#106	—	2	2	no	—
380	C₄⋊C₈	—	2	1	no	—
381	C₄⋊C₈	—	2	1	no	—
382	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
383	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
384	#103	—	2	2	no	—
385	#103	—	2	2	no	—
386	#102	—	2	2	no	—
387	#102	—	2	2	no	—
388	#102	—	2	2	no	—
389	#102	—	2	2	no	—
390	C₂²⋊C₈	—	2	1	no	—
391	C₂²⋊C₈	—	2	1	no	—
392	#97	—	2	1	no	—
393	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
394	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
395	C₃₁	#18952	1	1	yes	characteristic, direct product factor, Sylow
396	#41	—	8	1	no	—
397	#41	—	8	1	no	—
398	#41	—	8	2	no	—
399	C₂×Q₈	—	2	4	no	—
400	C₂×Q₈	—	2	1	no	—
401	C₂×Q₈	—	2	1	no	—
402	C₂×Q₈	—	2	1	no	—
403	C₂×Q₈	—	2	1	no	—
404	C₂×Q₈	—	2	1	no	—
405	C₂×Q₈	—	8	2	no	—
406	C₂×Q₈	—	8	2	no	—
407	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
408	C₂×D₄	—	2	2	no	—
409	C₂×D₄	—	2	1	no	—
410	C₂×D₄	—	2	1	no	—
411	C₂×D₄	—	2	1	no	—
412	C₂×D₄	—	2	1	no	—
413	C₂×D₄	—	2	1	no	—
414	C₂×D₄	—	4	1	no	—
415	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
416	C₂²×C₄	—	2	4	no	—
417	C₂²×C₄	—	2	2	no	—
418	C₂²×C₄	—	4	1	no	—
419	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
420	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
421	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
422	SD₁₆	—	8	2	no	—
423	SD₁₆	—	8	2	no	—
424	D₈	—	8	2	no	—
425	C₂×C₈	—	2	1	no	—
426	C₂×C₈	—	2	1	no	—
427	C₂×C₈	—	2	2	no	—
428	C₂×C₈	—	2	2	no	—
429	C₂×C₈	—	8	1	no	—
430	C₄⋊C₄	—	2	4	no	—
431	C₄⋊C₄	—	2	4	no	—
432	C₄⋊C₄	—	2	1	no	—
433	C₄⋊C₄	—	2	1	no	—
434	C₄⋊C₄	—	2	1	no	—
435	C₄⋊C₄	—	2	1	no	—
436	C₄⋊C₄	—	2	2	no	—
437	C₄⋊C₄	—	2	2	no	—
438	C₄⋊C₄	—	2	2	no	—
439	C₄⋊C₄	—	2	1	no	—
440	C₄⋊C₄	—	2	1	no	—
441	C₄⋊C₄	—	2	2	no	—
442	C₄⋊C₄	—	2	1	no	—
443	C₄⋊C₄	—	2	1	no	—
444	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
445	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
446	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
447	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
448	C₂²⋊C₄	—	2	4	no	—
449	C₂²⋊C₄	—	2	4	no	—
450	C₂²⋊C₄	—	2	4	no	—
451	C₂²⋊C₄	—	2	2	no	—
452	C₂²⋊C₄	—	2	2	no	—
453	C₂²⋊C₄	—	2	2	no	—
454	C₂²⋊C₄	—	2	2	no	—
455	C₂²⋊C₄	—	4	1	no	—
456	C₂²⋊C₄	—	4	1	no	—
457	C₄²	—	2	4	no	—
458	C₄²	—	2	2	no	—
459	C₄²	—	2	2	no	—
460	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
461	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
462	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
463	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
464	C₂³	—	2	2	no	—
465	C₂³	—	4	1	no	—
466	C₂³	C₃₁×#120	1	1	yes	characteristic
467	Q₈	—	2	2	no	—
468	Q₈	—	2	2	no	—
469	Q₈	—	4	2	no	—
470	Q₈	—	4	1	no	—
471	Q₈	—	4	2	no	—
472	Q₈	—	4	1	no	—
473	Q₈	—	4	2	no	—
474	Q₈	—	8	1	no	—
475	Q₈	—	8	4	no	—
476	D₄	—	2	2	no	—
477	D₄	—	2	2	no	—
478	D₄	—	4	2	no	—
479	D₄	—	4	2	no	—
480	D₄	—	4	1	no	—
481	D₄	—	4	2	no	—
482	D₄	—	4	1	no	—
483	D₄	—	8	1	no	—
484	D₄	—	8	2	no	—
485	D₄	—	8	2	no	—
486	C₂×C₄	—	2	1	no	—
487	C₂×C₄	—	2	2	no	—
488	C₂×C₄	—	2	1	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	4	1	no	—
495	C₂×C₄	—	4	2	no	—
496	C₂×C₄	—	4	2	no	—
497	C₂×C₄	—	4	1	no	—
498	C₂×C₄	—	8	2	no	—
499	C₂×C₄	—	8	4	no	—
500	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
501	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
502	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
503	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
504	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
505	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
506	C₈	—	4	1	no	—
507	C₈	—	4	1	no	—
508	C₈	—	4	2	no	—
509	C₈	—	4	2	no	—
510	C₂²	—	2	1	no	—
511	C₂²	—	2	1	no	—
512	C₂²	—	2	1	no	—
513	C₂²	—	4	2	no	—
514	C₂²	—	4	2	no	—
515	C₂²	—	4	2	no	—
516	C₂²	—	8	1	no	—
517	C₂²	—	8	1	no	—
518	C₂²	—	8	1	no	—
519	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
520	C₄	—	2	1	no	—
521	C₄	—	2	1	no	—
522	C₄	—	2	1	no	—
523	C₄	—	2	1	no	—
524	C₄	—	4	2	no	—
525	C₄	—	4	2	no	—
526	C₄	—	4	2	no	—
527	C₄	—	4	2	no	—
528	C₄	—	4	2	no	—
529	C₄	—	8	1	no	—
530	C₄	C₃₁×#576	1	1	yes	characteristic
531	C₄	C₃₁×#577	1	2	yes	—
532	C₄	C₃₁×#577	1	1	yes	characteristic
533	C₂	—	4	1	no	—
534	C₂	—	8	2	no	—
535	C₂	—	16	1	no	—
536	C₂	C₃₁×#3010	1	1	yes	characteristic
537	C₂	C₃₁×#3012	1	1	yes	characteristic
538	C₂	C₃₁×#3020	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3214	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×#3550	C₂	semidirect
2. C₃₁×#3246	C₂	semidirect
3. C₃₁×#3228	C₂	semidirect
5. C₃₁×#1651	C₂	semidirect
6. C₃₁×#1650	C₂	semidirect
7. C₃₁×#1628	C₂	semidirect
8. C₃₁×#1627	C₂	semidirect
9. C₃₁×#1552	C₂	semidirect
10. C₃₁×#1497	C₂	semidirect
11. C₃₁×#1480	C₂	semidirect
194. #18952	C₃₁	direct (N × H)
241. #3550	C₂×C₃₁	semidirect
242. #3246	C₂×C₃₁	semidirect
243. #3228	C₂×C₃₁	semidirect
245. #1651	C₂×C₃₁	semidirect
246. #1650	C₂×C₃₁	semidirect
247. #1628	C₂×C₃₁	semidirect
248. #1627	C₂×C₃₁	semidirect
249. #1552	C₂×C₃₁	semidirect
250. #1497	C₂×C₃₁	semidirect
251. #1480	C₂×C₃₁	semidirect
395. C₃₁	#18952	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×#3550	2
C₃₁×#3246	2
C₃₁×#3228	2
C₃₁×#3214	2
C₃₁×#1651	2
C₃₁×#1650	2
C₃₁×#1628	2
C₃₁×#1627	2
C₃₁×#1552	2
C₃₁×#1497	2
C₃₁×#1480	2
#18952	31

Subgroups — C31×#18952 #21475711

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#18952 #21475711