Subgroups — C₃₁×#18848 #21475607

Order 7936.

858 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3507	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3255	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3228	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#3113	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1653	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1648	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1640	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1640	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1557	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#1497	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1487	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#559	C₂²	1	2	yes	—
13	C₃₁×#557	C₂²	1	2	yes	—
14	C₃₁×#556	—	2	1	no	—
15	C₃₁×#556	—	2	2	no	—
16	C₃₁×#554	—	2	1	no	—
17	C₃₁×#552	—	2	2	no	—
18	C₃₁×#539	—	2	2	no	—
19	C₃₁×#536	—	2	1	no	—
20	C₃₁×#533	C₂²	1	1	yes	characteristic
21	C₃₁×#533	C₂²	1	1	yes	characteristic
22	C₃₁×#532	C₂²	1	1	yes	characteristic
23	C₃₁×#529	—	2	2	no	—
24	C₃₁×#520	—	2	1	no	—
25	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#500	C₂²	1	1	yes	characteristic
27	C₃₁×#498	C₂²	1	1	yes	characteristic
28	C₃₁×#486	C₂²	1	1	yes	characteristic
29	C₃₁×#486	C₂²	1	1	yes	characteristic
30	C₃₁×#483	—	2	1	no	—
31	C₃₁×#482	—	2	2	no	—
32	C₃₁×#478	—	2	2	no	—
33	C₃₁×#475	—	2	1	no	—
34	C₃₁×#475	C₂²	1	2	yes	—
35	C₃₁×#475	C₂²	1	2	yes	—
36	C₃₁×#470	—	2	2	no	—
37	C₃₁×#462	—	2	1	no	—
38	C₃₁×#462	C₂²	1	2	yes	—
39	C₃₁×#462	C₂²	1	2	yes	—
40	C₃₁×#451	—	2	1	no	—
41	C₃₁×#441	—	2	2	no	—
42	C₃₁×#432	C₂²	1	1	yes	characteristic
43	C₃₁×#431	C₂²	1	1	yes	characteristic
44	C₃₁×#423	C₂²	1	1	yes	characteristic
45	C₃₁×#332	C₂²	1	2	yes	—
46	C₃₁×#331	C₂²	1	2	yes	—
47	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
48	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
49	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
50	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
51	C₂×#41×C₃₁	—	2	1	no	—
52	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
53	C₂×Q₁₆×C₃₁	—	2	2	no	—
54	C₂×Q₁₆×C₃₁	—	2	2	no	—
55	C₂×M₁₆×C₃₁	—	2	1	no	—
56	C₃₁×C₄⋊Q₈	—	2	2	no	—
57	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
58	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
59	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
60	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
61	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
62	C₃₁×#127	C₂³	1	1	yes	characteristic
63	C₃₁×#126	—	2	4	no	—
64	C₃₁×#126	—	2	4	no	—
65	C₃₁×#125	—	2	2	no	—
66	C₃₁×#125	—	2	4	no	—
67	C₃₁×#125	—	2	1	no	—
68	C₃₁×#125	—	2	1	no	—
69	C₃₁×#124	—	2	2	no	—
70	C₃₁×#124	—	2	2	no	—
71	C₃₁×#124	—	2	1	no	—
72	C₃₁×#123	—	2	4	no	—
73	C₃₁×#123	—	2	4	no	—
74	C₃₁×C₂²⋊Q₈	—	2	2	no	—
75	C₃₁×C₂²⋊Q₈	—	2	2	no	—
76	C₃₁×C₂²⋊Q₈	—	2	4	no	—
77	C₃₁×C₂²⋊Q₈	—	2	1	no	—
78	C₃₁×C₂²⋊Q₈	—	2	1	no	—
79	C₃₁×C₂²⋊Q₈	—	2	1	no	—
80	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
81	C₃₁×#121	—	2	1	no	—
82	C₃₁×#121	D₄	1	2	yes	—
83	C₄×Q₈×C₃₁	—	2	2	no	—
84	C₄×Q₈×C₃₁	—	2	2	no	—
85	C₄×Q₈×C₃₁	—	2	1	no	—
86	C₄×Q₈×C₃₁	—	2	1	no	—
87	C₄×Q₈×C₃₁	—	2	1	no	—
88	C₄×Q₈×C₃₁	—	2	1	no	—
89	C₄×Q₈×C₃₁	—	2	1	no	—
90	C₄×Q₈×C₃₁	D₄	1	2	yes	—
91	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
92	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
93	C₄×D₄×C₃₁	—	2	4	no	—
94	C₄×D₄×C₃₁	—	2	2	no	—
95	C₄×D₄×C₃₁	—	2	2	no	—
96	C₄×D₄×C₃₁	—	2	1	no	—
97	C₄×D₄×C₃₁	D₄	1	2	yes	—
98	C₃₁×#117	—	2	2	no	—
99	C₃₁×#117	—	2	4	no	—
100	C₃₁×#117	—	2	1	no	—
101	C₃₁×#117	—	2	1	no	—
102	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
103	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
104	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
105	C₂×C₄²×C₃₁	—	2	1	no	—
106	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
107	C₃₁×#107	—	2	2	no	—
108	C₃₁×#106	—	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×#103	—	2	2	no	—
118	C₃₁×#103	—	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	—	4	4	no	—
127	C₂×Q₈×C₃₁	—	2	1	no	—
128	C₂×Q₈×C₃₁	—	2	1	no	—
129	C₂×Q₈×C₃₁	—	2	1	no	—
130	C₂×Q₈×C₃₁	—	2	1	no	—
131	C₂×Q₈×C₃₁	—	2	1	no	—
132	C₂×Q₈×C₃₁	—	4	2	no	—
133	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
134	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
135	C₂×D₄×C₃₁	—	2	2	no	—
136	C₂×D₄×C₃₁	—	2	1	no	—
137	C₂×D₄×C₃₁	—	2	1	no	—
138	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
139	C₂²×C₄×C₃₁	—	2	1	no	—
140	C₂²×C₄×C₃₁	—	2	4	no	—
141	C₂²×C₄×C₃₁	—	2	1	no	—
142	C₂²×C₄×C₃₁	—	2	2	no	—
143	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
146	Q₁₆×C₃₁	—	4	4	no	—
147	Q₁₆×C₃₁	—	8	2	no	—
148	M₁₆×C₃₁	—	8	1	no	—
149	C₂×C₈×C₃₁	—	2	1	no	—
150	C₂×C₈×C₃₁	—	2	1	no	—
151	C₂×C₈×C₃₁	—	2	2	no	—
152	C₂×C₈×C₃₁	—	2	2	no	—
153	C₄⋊C₄×C₃₁	—	2	4	no	—
154	C₄⋊C₄×C₃₁	—	2	4	no	—
155	C₄⋊C₄×C₃₁	—	2	4	no	—
156	C₄⋊C₄×C₃₁	—	2	1	no	—
157	C₄⋊C₄×C₃₁	—	2	1	no	—
158	C₄⋊C₄×C₃₁	—	2	2	no	—
159	C₄⋊C₄×C₃₁	—	2	1	no	—
160	C₄⋊C₄×C₃₁	—	2	2	no	—
161	C₄⋊C₄×C₃₁	—	2	1	no	—
162	C₄⋊C₄×C₃₁	—	2	2	no	—
163	C₄⋊C₄×C₃₁	—	2	1	no	—
164	C₄⋊C₄×C₃₁	—	2	2	no	—
165	C₄⋊C₄×C₃₁	—	2	1	no	—
166	C₄⋊C₄×C₃₁	—	2	1	no	—
167	C₄⋊C₄×C₃₁	—	2	1	no	—
168	C₄⋊C₄×C₃₁	—	2	1	no	—
169	C₄⋊C₄×C₃₁	—	4	1	no	—
170	C₄⋊C₄×C₃₁	—	4	1	no	—
171	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
172	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
173	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
174	C₂²⋊C₄×C₃₁	—	2	4	no	—
175	C₂²⋊C₄×C₃₁	—	2	4	no	—
176	C₂²⋊C₄×C₃₁	—	2	4	no	—
177	C₂²⋊C₄×C₃₁	—	2	2	no	—
178	C₂²⋊C₄×C₃₁	—	2	2	no	—
179	C₂²⋊C₄×C₃₁	—	2	2	no	—
180	C₂²⋊C₄×C₃₁	—	4	1	no	—
181	C₄²×C₃₁	—	2	4	no	—
182	C₄²×C₃₁	—	2	2	no	—
183	C₄²×C₃₁	—	2	2	no	—
184	C₄²×C₃₁	—	2	2	no	—
185	C₄²×C₃₁	—	4	1	no	—
186	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
187	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
188	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
189	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
190	#18848	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
191	C₂³×C₃₁	—	2	2	no	—
192	C₂³×C₃₁	#120	1	1	yes	characteristic
193	Q₈×C₃₁	—	2	2	no	—
194	Q₈×C₃₁	—	2	2	no	—
195	Q₈×C₃₁	—	2	2	no	—
196	Q₈×C₃₁	—	2	2	no	—
197	Q₈×C₃₁	—	4	2	no	—
198	Q₈×C₃₁	—	4	2	no	—
199	Q₈×C₃₁	—	4	2	no	—
200	Q₈×C₃₁	—	4	1	no	—
201	Q₈×C₃₁	—	4	1	no	—
202	Q₈×C₃₁	—	8	1	no	—
203	D₄×C₃₁	—	4	2	no	—
204	D₄×C₃₁	—	4	4	no	—
205	D₄×C₃₁	—	4	2	no	—
206	D₄×C₃₁	—	4	1	no	—
207	D₄×C₃₁	—	4	1	no	—
208	C₂×C₄×C₃₁	—	2	2	no	—
209	C₂×C₄×C₃₁	—	2	1	no	—
210	C₂×C₄×C₃₁	—	2	1	no	—
211	C₂×C₄×C₃₁	—	2	2	no	—
212	C₂×C₄×C₃₁	—	2	2	no	—
213	C₂×C₄×C₃₁	—	2	2	no	—
214	C₂×C₄×C₃₁	—	2	2	no	—
215	C₂×C₄×C₃₁	—	2	2	no	—
216	C₂×C₄×C₃₁	—	4	1	no	—
217	C₂×C₄×C₃₁	—	4	4	no	—
218	C₂×C₄×C₃₁	—	4	2	no	—
219	C₂×C₄×C₃₁	—	4	1	no	—
220	C₂×C₄×C₃₁	—	4	1	no	—
221	C₂×C₄×C₃₁	—	8	1	no	—
222	C₂×C₄×C₃₁	—	8	4	no	—
223	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
224	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
225	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
226	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
227	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
228	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
229	C₈×C₃₁	—	4	1	no	—
230	C₈×C₃₁	—	4	1	no	—
231	C₈×C₃₁	—	4	2	no	—
232	C₈×C₃₁	—	4	2	no	—
233	#3507	C₂×C₃₁	1	1	yes	characteristic
234	#3255	C₂×C₃₁	1	1	yes	characteristic, complemented
235	#3228	C₂×C₃₁	1	1	yes	characteristic
236	#3113	C₂×C₃₁	1	1	yes	characteristic, complemented
237	#1653	C₂×C₃₁	1	2	yes	complemented
238	#1648	C₂×C₃₁	1	2	yes	complemented
239	#1640	C₂×C₃₁	1	2	yes	complemented
240	#1640	C₂×C₃₁	1	2	yes	complemented
241	#1557	C₂×C₃₁	1	1	yes	characteristic
242	#1497	C₂×C₃₁	1	1	yes	characteristic, complemented
243	#1487	C₂×C₃₁	1	1	yes	characteristic, complemented
244	C₂²×C₃₁	—	2	1	no	—
245	C₂²×C₃₁	—	2	1	no	—
246	C₂²×C₃₁	—	2	1	no	—
247	C₂²×C₃₁	—	4	2	no	—
248	C₂²×C₃₁	—	4	2	no	—
249	C₂²×C₃₁	—	4	2	no	—
250	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
251	C₄×C₃₁	—	2	1	no	—
252	C₄×C₃₁	—	2	1	no	—
253	C₄×C₃₁	—	2	1	no	—
254	C₄×C₃₁	—	2	1	no	—
255	C₄×C₃₁	—	4	2	no	—
256	C₄×C₃₁	—	4	2	no	—
257	C₄×C₃₁	—	4	2	no	—
258	C₄×C₃₁	—	4	2	no	—
259	C₄×C₃₁	—	4	2	no	—
260	C₄×C₃₁	—	8	1	no	—
261	C₄×C₃₁	—	8	1	no	—
262	C₄×C₃₁	#575	1	2	yes	—
263	C₄×C₃₁	#577	1	1	yes	characteristic
264	C₄×C₃₁	#578	1	1	yes	characteristic
265	#559	C₂²×C₃₁	1	2	yes	—
266	#557	C₂²×C₃₁	1	2	yes	—
267	#556	—	2	1	no	—
268	#556	—	2	2	no	—
269	#554	—	2	1	no	—
270	#552	—	2	2	no	—
271	#539	—	2	2	no	—
272	#536	—	2	1	no	—
273	#533	C₂²×C₃₁	1	1	yes	characteristic
274	#533	C₂²×C₃₁	1	1	yes	characteristic
275	#532	C₂²×C₃₁	1	1	yes	characteristic
276	#529	—	2	2	no	—
277	#520	—	2	1	no	—
278	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
279	#500	C₂²×C₃₁	1	1	yes	characteristic
280	#498	C₂²×C₃₁	1	1	yes	characteristic
281	#486	C₂²×C₃₁	1	1	yes	characteristic
282	#486	C₂²×C₃₁	1	1	yes	characteristic
283	#483	—	2	1	no	—
284	#482	—	2	2	no	—
285	#478	—	2	2	no	—
286	#475	—	2	1	no	—
287	#475	C₂²×C₃₁	1	2	yes	—
288	#475	C₂²×C₃₁	1	2	yes	—
289	#470	—	2	2	no	—
290	#462	—	2	1	no	—
291	#462	C₂²×C₃₁	1	2	yes	—
292	#462	C₂²×C₃₁	1	2	yes	—
293	#451	—	2	1	no	—
294	#441	—	2	2	no	—
295	#432	C₂²×C₃₁	1	1	yes	characteristic
296	#431	C₂²×C₃₁	1	1	yes	characteristic
297	#423	C₂²×C₃₁	1	1	yes	characteristic
298	#332	C₂²×C₃₁	1	2	yes	—
299	#331	C₂²×C₃₁	1	2	yes	—
300	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
301	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
302	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
303	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
304	C₂×C₃₁	—	4	1	no	—
305	C₂×C₃₁	—	8	2	no	—
306	C₂×C₃₁	#3005	1	1	yes	characteristic
307	C₂×C₃₁	#3013	1	1	yes	characteristic
308	C₂×C₃₁	#3022	1	1	yes	characteristic
309	C₂×#41	—	2	1	no	—
310	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
311	C₂×Q₁₆	—	2	2	no	—
312	C₂×Q₁₆	—	2	2	no	—
313	C₂×M₁₆	—	2	1	no	—
314	C₄⋊Q₈	—	2	2	no	—
315	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
316	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
317	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
318	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
319	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
320	#127	C₂³×C₃₁	1	1	yes	characteristic
321	#126	—	2	4	no	—
322	#126	—	2	4	no	—
323	#125	—	2	2	no	—
324	#125	—	2	4	no	—
325	#125	—	2	1	no	—
326	#125	—	2	1	no	—
327	#124	—	2	2	no	—
328	#124	—	2	2	no	—
329	#124	—	2	1	no	—
330	#123	—	2	4	no	—
331	#123	—	2	4	no	—
332	C₂²⋊Q₈	—	2	2	no	—
333	C₂²⋊Q₈	—	2	2	no	—
334	C₂²⋊Q₈	—	2	4	no	—
335	C₂²⋊Q₈	—	2	1	no	—
336	C₂²⋊Q₈	—	2	1	no	—
337	C₂²⋊Q₈	—	2	1	no	—
338	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
339	#121	—	2	1	no	—
340	#121	D₄×C₃₁	1	2	yes	—
341	C₄×Q₈	—	2	2	no	—
342	C₄×Q₈	—	2	2	no	—
343	C₄×Q₈	—	2	1	no	—
344	C₄×Q₈	—	2	1	no	—
345	C₄×Q₈	—	2	1	no	—
346	C₄×Q₈	—	2	1	no	—
347	C₄×Q₈	—	2	1	no	—
348	C₄×Q₈	D₄×C₃₁	1	2	yes	—
349	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
350	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
351	C₄×D₄	—	2	4	no	—
352	C₄×D₄	—	2	2	no	—
353	C₄×D₄	—	2	2	no	—
354	C₄×D₄	—	2	1	no	—
355	C₄×D₄	D₄×C₃₁	1	2	yes	—
356	#117	—	2	2	no	—
357	#117	—	2	4	no	—
358	#117	—	2	1	no	—
359	#117	—	2	1	no	—
360	C₂×C₄⋊C₄	—	2	2	no	—
361	C₂×C₄⋊C₄	—	2	1	no	—
362	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
363	C₂×C₄²	—	2	1	no	—
364	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
365	#107	—	2	2	no	—
366	#106	—	2	2	no	—
367	#106	—	2	2	no	—
368	#106	—	2	2	no	—
369	C₄⋊C₈	—	2	1	no	—
370	C₄⋊C₈	—	2	1	no	—
371	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
372	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
373	#103	—	2	2	no	—
374	#103	—	2	2	no	—
375	#103	—	2	2	no	—
376	#103	—	2	2	no	—
377	#102	—	2	2	no	—
378	#102	—	2	2	no	—
379	C₂²⋊C₈	—	2	1	no	—
380	C₂²⋊C₈	—	2	1	no	—
381	#97	—	2	1	no	—
382	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
383	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
384	C₃₁	#18848	1	1	yes	characteristic, direct product factor, Sylow
385	#41	—	4	4	no	—
386	C₂×Q₈	—	2	1	no	—
387	C₂×Q₈	—	2	1	no	—
388	C₂×Q₈	—	2	1	no	—
389	C₂×Q₈	—	2	1	no	—
390	C₂×Q₈	—	2	1	no	—
391	C₂×Q₈	—	4	2	no	—
392	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
393	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
394	C₂×D₄	—	2	2	no	—
395	C₂×D₄	—	2	1	no	—
396	C₂×D₄	—	2	1	no	—
397	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
398	C₂²×C₄	—	2	1	no	—
399	C₂²×C₄	—	2	4	no	—
400	C₂²×C₄	—	2	1	no	—
401	C₂²×C₄	—	2	2	no	—
402	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
403	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
404	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
405	Q₁₆	—	4	4	no	—
406	Q₁₆	—	8	2	no	—
407	M₁₆	—	8	1	no	—
408	C₂×C₈	—	2	1	no	—
409	C₂×C₈	—	2	1	no	—
410	C₂×C₈	—	2	2	no	—
411	C₂×C₈	—	2	2	no	—
412	C₄⋊C₄	—	2	4	no	—
413	C₄⋊C₄	—	2	4	no	—
414	C₄⋊C₄	—	2	4	no	—
415	C₄⋊C₄	—	2	1	no	—
416	C₄⋊C₄	—	2	1	no	—
417	C₄⋊C₄	—	2	2	no	—
418	C₄⋊C₄	—	2	1	no	—
419	C₄⋊C₄	—	2	2	no	—
420	C₄⋊C₄	—	2	1	no	—
421	C₄⋊C₄	—	2	2	no	—
422	C₄⋊C₄	—	2	1	no	—
423	C₄⋊C₄	—	2	2	no	—
424	C₄⋊C₄	—	2	1	no	—
425	C₄⋊C₄	—	2	1	no	—
426	C₄⋊C₄	—	2	1	no	—
427	C₄⋊C₄	—	2	1	no	—
428	C₄⋊C₄	—	4	1	no	—
429	C₄⋊C₄	—	4	1	no	—
430	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
431	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
432	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
433	C₂²⋊C₄	—	2	4	no	—
434	C₂²⋊C₄	—	2	4	no	—
435	C₂²⋊C₄	—	2	4	no	—
436	C₂²⋊C₄	—	2	2	no	—
437	C₂²⋊C₄	—	2	2	no	—
438	C₂²⋊C₄	—	2	2	no	—
439	C₂²⋊C₄	—	4	1	no	—
440	C₄²	—	2	4	no	—
441	C₄²	—	2	2	no	—
442	C₄²	—	2	2	no	—
443	C₄²	—	2	2	no	—
444	C₄²	—	4	1	no	—
445	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
446	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
447	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
448	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
449	C₂³	—	2	2	no	—
450	C₂³	C₃₁×#120	1	1	yes	characteristic
451	Q₈	—	2	2	no	—
452	Q₈	—	2	2	no	—
453	Q₈	—	2	2	no	—
454	Q₈	—	2	2	no	—
455	Q₈	—	4	2	no	—
456	Q₈	—	4	2	no	—
457	Q₈	—	4	2	no	—
458	Q₈	—	4	1	no	—
459	Q₈	—	4	1	no	—
460	Q₈	—	8	1	no	—
461	D₄	—	4	2	no	—
462	D₄	—	4	4	no	—
463	D₄	—	4	2	no	—
464	D₄	—	4	1	no	—
465	D₄	—	4	1	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	2	1	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	4	1	no	—
475	C₂×C₄	—	4	4	no	—
476	C₂×C₄	—	4	2	no	—
477	C₂×C₄	—	4	1	no	—
478	C₂×C₄	—	4	1	no	—
479	C₂×C₄	—	8	1	no	—
480	C₂×C₄	—	8	4	no	—
481	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
482	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
483	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
484	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
485	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
486	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
487	C₈	—	4	1	no	—
488	C₈	—	4	1	no	—
489	C₈	—	4	2	no	—
490	C₈	—	4	2	no	—
491	C₂²	—	2	1	no	—
492	C₂²	—	2	1	no	—
493	C₂²	—	2	1	no	—
494	C₂²	—	4	2	no	—
495	C₂²	—	4	2	no	—
496	C₂²	—	4	2	no	—
497	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
498	C₄	—	2	1	no	—
499	C₄	—	2	1	no	—
500	C₄	—	2	1	no	—
501	C₄	—	2	1	no	—
502	C₄	—	4	2	no	—
503	C₄	—	4	2	no	—
504	C₄	—	4	2	no	—
505	C₄	—	4	2	no	—
506	C₄	—	4	2	no	—
507	C₄	—	8	1	no	—
508	C₄	—	8	1	no	—
509	C₄	C₃₁×#575	1	2	yes	—
510	C₄	C₃₁×#577	1	1	yes	characteristic
511	C₄	C₃₁×#578	1	1	yes	characteristic
512	C₂	—	4	1	no	—
513	C₂	—	8	2	no	—
514	C₂	C₃₁×#3005	1	1	yes	characteristic
515	C₂	C₃₁×#3013	1	1	yes	characteristic
516	C₂	C₃₁×#3022	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#532	C₂²
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
2. C₃₁×#3255	C₂	semidirect
4. C₃₁×#3113	C₂	semidirect
5. C₃₁×#1653	C₂	semidirect
6. C₃₁×#1648	C₂	semidirect
7. C₃₁×#1640	C₂	semidirect
8. C₃₁×#1640	C₂	semidirect
10. C₃₁×#1497	C₂	semidirect
11. C₃₁×#1487	C₂	semidirect
190. #18848	C₃₁	direct (N × H)
234. #3255	C₂×C₃₁	semidirect
236. #3113	C₂×C₃₁	semidirect
237. #1653	C₂×C₃₁	semidirect
238. #1648	C₂×C₃₁	semidirect
239. #1640	C₂×C₃₁	semidirect
240. #1640	C₂×C₃₁	semidirect
242. #1497	C₂×C₃₁	semidirect
243. #1487	C₂×C₃₁	semidirect
384. C₃₁	#18848	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×#3507	2
C₃₁×#3255	2
C₃₁×#3228	2
C₃₁×#3113	2
C₃₁×#1653	2
C₃₁×#1648	2
C₃₁×#1640	2
C₃₁×#1557	2
C₃₁×#1497	2
C₃₁×#1487	2
#18848	31

Subgroups — C31×#18848 #21475607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#18848 #21475607