Subgroups — C₃₁×#18603 #21475362

Order 7936.

866 conjugacy classes of subgroups.

The subgroup lattice has 508 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×#3237	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3182	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#3104	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1648	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1640	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1636	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1630	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1553	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#1491	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1485	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×#501	—	2	1	no	—
27	C₃₁×#500	C₂²	1	1	yes	characteristic
28	C₃₁×#492	C₂²	1	1	yes	characteristic
29	C₃₁×#491	—	2	1	no	—
30	C₃₁×#487	C₂²	1	1	yes	characteristic
31	C₃₁×#486	C₂²	1	1	yes	characteristic
32	C₃₁×#485	—	2	2	no	—
33	C₃₁×#484	—	2	1	no	—
34	C₃₁×#477	C₂²	1	2	yes	—
35	C₃₁×#477	C₂²	1	2	yes	—
36	C₃₁×#475	C₂²	1	2	yes	—
37	C₃₁×#468	—	2	2	no	—
38	C₃₁×#462	C₂²	1	2	yes	—
39	C₃₁×#451	—	2	1	no	—
40	C₃₁×#440	—	2	2	no	—
41	C₄×Q₁₆×C₃₁	—	2	2	no	—
42	C₃₁×#432	C₂²	1	1	yes	characteristic
43	C₃₁×#431	C₂²	1	1	yes	characteristic
44	C₄×M₁₆×C₃₁	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×SD₁₆×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×#107	—	2	2	no	—
109	C₃₁×#107	—	2	2	no	—
110	C₃₁×#106	—	2	2	no	—
111	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
112	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
113	C₃₁×#103	—	2	2	no	—
114	C₃₁×#103	—	2	2	no	—
115	C₃₁×#103	—	2	2	no	—
116	C₃₁×#103	—	2	2	no	—
117	C₃₁×#103	—	2	2	no	—
118	C₃₁×#102	—	2	2	no	—
119	C₃₁×C₂²⋊C₈	—	2	1	no	—
120	C₃₁×C₂²⋊C₈	—	2	1	no	—
121	C₃₁×#97	—	2	1	no	—
122	C₃₁×#97	—	2	1	no	—
123	C₄×C₈×C₃₁	—	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	SD₁₆×C₃₁	—	4	4	no	—
148	M₁₆×C₃₁	—	4	2	no	—
149	C₂×C₈×C₃₁	—	2	1	no	—
150	C₂×C₈×C₃₁	—	2	2	no	—
151	C₂×C₈×C₃₁	—	2	1	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	#18603	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	4	no	—
196	Q₈×C₃₁	—	4	2	no	—
197	Q₈×C₃₁	—	4	2	no	—
198	Q₈×C₃₁	—	4	2	no	—
199	Q₈×C₃₁	—	4	2	no	—
200	Q₈×C₃₁	—	8	1	no	—
201	D₄×C₃₁	—	4	2	no	—
202	D₄×C₃₁	—	4	4	no	—
203	D₄×C₃₁	—	4	2	no	—
204	D₄×C₃₁	—	4	2	no	—
205	C₂×C₄×C₃₁	—	2	2	no	—
206	C₂×C₄×C₃₁	—	2	2	no	—
207	C₂×C₄×C₃₁	—	2	2	no	—
208	C₂×C₄×C₃₁	—	2	2	no	—
209	C₂×C₄×C₃₁	—	2	2	no	—
210	C₂×C₄×C₃₁	—	2	2	no	—
211	C₂×C₄×C₃₁	—	2	2	no	—
212	C₂×C₄×C₃₁	—	4	1	no	—
213	C₂×C₄×C₃₁	—	4	4	no	—
214	C₂×C₄×C₃₁	—	4	2	no	—
215	C₂×C₄×C₃₁	—	4	1	no	—
216	C₂×C₄×C₃₁	—	4	1	no	—
217	C₂×C₄×C₃₁	—	8	1	no	—
218	C₂×C₄×C₃₁	—	8	4	no	—
219	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
220	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
221	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
222	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
224	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
225	C₈×C₃₁	—	2	2	no	—
226	C₈×C₃₁	—	4	1	no	—
227	C₈×C₃₁	—	4	2	no	—
228	C₈×C₃₁	—	4	2	no	—
229	#3507	C₂×C₃₁	1	1	yes	characteristic
230	#3237	C₂×C₃₁	1	1	yes	characteristic, complemented
231	#3182	C₂×C₃₁	1	1	yes	characteristic
232	#3104	C₂×C₃₁	1	1	yes	characteristic, complemented
233	#1648	C₂×C₃₁	1	2	yes	complemented
234	#1640	C₂×C₃₁	1	2	yes	complemented
235	#1636	C₂×C₃₁	1	2	yes	complemented
236	#1630	C₂×C₃₁	1	2	yes	complemented
237	#1553	C₂×C₃₁	1	1	yes	characteristic
238	#1491	C₂×C₃₁	1	1	yes	characteristic, complemented
239	#1485	C₂×C₃₁	1	1	yes	characteristic, complemented
240	C₂²×C₃₁	—	2	1	no	—
241	C₂²×C₃₁	—	2	1	no	—
242	C₂²×C₃₁	—	2	1	no	—
243	C₂²×C₃₁	—	4	2	no	—
244	C₂²×C₃₁	—	4	2	no	—
245	C₂²×C₃₁	—	4	2	no	—
246	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
247	C₄×C₃₁	—	2	1	no	—
248	C₄×C₃₁	—	2	1	no	—
249	C₄×C₃₁	—	2	1	no	—
250	C₄×C₃₁	—	2	1	no	—
251	C₄×C₃₁	—	4	2	no	—
252	C₄×C₃₁	—	4	2	no	—
253	C₄×C₃₁	—	4	2	no	—
254	C₄×C₃₁	—	4	2	no	—
255	C₄×C₃₁	—	4	2	no	—
256	C₄×C₃₁	—	8	1	no	—
257	C₄×C₃₁	—	8	1	no	—
258	C₄×C₃₁	#575	1	2	yes	—
259	C₄×C₃₁	#578	1	2	yes	—
260	#559	C₂²×C₃₁	1	2	yes	—
261	#557	C₂²×C₃₁	1	2	yes	—
262	#556	—	2	1	no	—
263	#556	—	2	2	no	—
264	#554	—	2	1	no	—
265	#552	—	2	2	no	—
266	#539	—	2	2	no	—
267	#536	—	2	1	no	—
268	#533	C₂²×C₃₁	1	1	yes	characteristic
269	#533	C₂²×C₃₁	1	1	yes	characteristic
270	#532	C₂²×C₃₁	1	1	yes	characteristic
271	#529	—	2	2	no	—
272	#520	—	2	1	no	—
273	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
274	#501	—	2	1	no	—
275	#500	C₂²×C₃₁	1	1	yes	characteristic
276	#492	C₂²×C₃₁	1	1	yes	characteristic
277	#491	—	2	1	no	—
278	#487	C₂²×C₃₁	1	1	yes	characteristic
279	#486	C₂²×C₃₁	1	1	yes	characteristic
280	#485	—	2	2	no	—
281	#484	—	2	1	no	—
282	#477	C₂²×C₃₁	1	2	yes	—
283	#477	C₂²×C₃₁	1	2	yes	—
284	#475	C₂²×C₃₁	1	2	yes	—
285	#468	—	2	2	no	—
286	#462	C₂²×C₃₁	1	2	yes	—
287	#451	—	2	1	no	—
288	#440	—	2	2	no	—
289	C₄×Q₁₆	—	2	2	no	—
290	#432	C₂²×C₃₁	1	1	yes	characteristic
291	#431	C₂²×C₃₁	1	1	yes	characteristic
292	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
293	#332	C₂²×C₃₁	1	2	yes	—
294	#331	C₂²×C₃₁	1	2	yes	—
295	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
296	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
297	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
298	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
299	C₂×C₃₁	—	4	1	no	—
300	C₂×C₃₁	—	8	2	no	—
301	C₂×C₃₁	#3005	1	1	yes	characteristic
302	C₂×C₃₁	#3011	1	1	yes	characteristic
303	C₂×C₃₁	#3018	1	1	yes	characteristic
304	C₂×#41	—	2	1	no	—
305	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
306	C₂×Q₁₆	—	2	2	no	—
307	C₂×SD₁₆	—	2	2	no	—
308	C₂×M₁₆	—	2	1	no	—
309	C₄⋊Q₈	—	2	2	no	—
310	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
311	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
312	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
313	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
314	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
315	#127	C₂³×C₃₁	1	1	yes	characteristic
316	#126	—	2	4	no	—
317	#126	—	2	4	no	—
318	#125	—	2	2	no	—
319	#125	—	2	4	no	—
320	#125	—	2	1	no	—
321	#125	—	2	1	no	—
322	#124	—	2	2	no	—
323	#124	—	2	2	no	—
324	#124	—	2	1	no	—
325	#123	—	2	4	no	—
326	#123	—	2	4	no	—
327	C₂²⋊Q₈	—	2	2	no	—
328	C₂²⋊Q₈	—	2	2	no	—
329	C₂²⋊Q₈	—	2	4	no	—
330	C₂²⋊Q₈	—	2	1	no	—
331	C₂²⋊Q₈	—	2	1	no	—
332	C₂²⋊Q₈	—	2	1	no	—
333	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
334	#121	—	2	1	no	—
335	#121	D₄×C₃₁	1	2	yes	—
336	C₄×Q₈	—	2	2	no	—
337	C₄×Q₈	—	2	2	no	—
338	C₄×Q₈	—	2	1	no	—
339	C₄×Q₈	—	2	1	no	—
340	C₄×Q₈	—	2	1	no	—
341	C₄×Q₈	—	2	1	no	—
342	C₄×Q₈	—	2	1	no	—
343	C₄×Q₈	D₄×C₃₁	1	2	yes	—
344	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
345	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
346	C₄×D₄	—	2	4	no	—
347	C₄×D₄	—	2	2	no	—
348	C₄×D₄	—	2	2	no	—
349	C₄×D₄	—	2	1	no	—
350	C₄×D₄	D₄×C₃₁	1	2	yes	—
351	#117	—	2	2	no	—
352	#117	—	2	4	no	—
353	#117	—	2	1	no	—
354	#117	—	2	1	no	—
355	C₂×C₄⋊C₄	—	2	2	no	—
356	C₂×C₄⋊C₄	—	2	1	no	—
357	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
358	C₂×C₄²	—	2	1	no	—
359	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
360	#107	—	2	2	no	—
361	#107	—	2	2	no	—
362	#107	—	2	2	no	—
363	#106	—	2	2	no	—
364	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
365	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
366	#103	—	2	2	no	—
367	#103	—	2	2	no	—
368	#103	—	2	2	no	—
369	#103	—	2	2	no	—
370	#103	—	2	2	no	—
371	#102	—	2	2	no	—
372	C₂²⋊C₈	—	2	1	no	—
373	C₂²⋊C₈	—	2	1	no	—
374	#97	—	2	1	no	—
375	#97	—	2	1	no	—
376	C₄×C₈	—	2	1	no	—
377	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
378	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
379	C₃₁	#18603	1	1	yes	characteristic, direct product factor, Sylow
380	#41	—	4	4	no	—
381	C₂×Q₈	—	2	1	no	—
382	C₂×Q₈	—	2	1	no	—
383	C₂×Q₈	—	2	1	no	—
384	C₂×Q₈	—	2	1	no	—
385	C₂×Q₈	—	2	1	no	—
386	C₂×Q₈	—	4	2	no	—
387	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
388	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
389	C₂×D₄	—	2	2	no	—
390	C₂×D₄	—	2	1	no	—
391	C₂×D₄	—	2	1	no	—
392	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
393	C₂²×C₄	—	2	1	no	—
394	C₂²×C₄	—	2	4	no	—
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	2	2	no	—
397	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
398	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
399	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
400	Q₁₆	—	4	4	no	—
401	SD₁₆	—	4	4	no	—
402	M₁₆	—	4	2	no	—
403	C₂×C₈	—	2	1	no	—
404	C₂×C₈	—	2	2	no	—
405	C₂×C₈	—	2	1	no	—
406	C₂×C₈	—	2	2	no	—
407	C₄⋊C₄	—	2	4	no	—
408	C₄⋊C₄	—	2	4	no	—
409	C₄⋊C₄	—	2	4	no	—
410	C₄⋊C₄	—	2	1	no	—
411	C₄⋊C₄	—	2	1	no	—
412	C₄⋊C₄	—	2	2	no	—
413	C₄⋊C₄	—	2	1	no	—
414	C₄⋊C₄	—	2	2	no	—
415	C₄⋊C₄	—	2	1	no	—
416	C₄⋊C₄	—	2	2	no	—
417	C₄⋊C₄	—	2	1	no	—
418	C₄⋊C₄	—	2	2	no	—
419	C₄⋊C₄	—	2	1	no	—
420	C₄⋊C₄	—	2	1	no	—
421	C₄⋊C₄	—	2	1	no	—
422	C₄⋊C₄	—	2	1	no	—
423	C₄⋊C₄	—	4	1	no	—
424	C₄⋊C₄	—	4	1	no	—
425	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
426	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
427	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
428	C₂²⋊C₄	—	2	4	no	—
429	C₂²⋊C₄	—	2	4	no	—
430	C₂²⋊C₄	—	2	4	no	—
431	C₂²⋊C₄	—	2	2	no	—
432	C₂²⋊C₄	—	2	2	no	—
433	C₂²⋊C₄	—	2	2	no	—
434	C₂²⋊C₄	—	4	1	no	—
435	C₄²	—	2	4	no	—
436	C₄²	—	2	2	no	—
437	C₄²	—	2	2	no	—
438	C₄²	—	2	2	no	—
439	C₄²	—	4	1	no	—
440	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
441	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
442	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
443	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
444	C₂³	—	2	2	no	—
445	C₂³	C₃₁×#120	1	1	yes	characteristic
446	Q₈	—	2	2	no	—
447	Q₈	—	2	2	no	—
448	Q₈	—	2	4	no	—
449	Q₈	—	4	2	no	—
450	Q₈	—	4	2	no	—
451	Q₈	—	4	2	no	—
452	Q₈	—	4	2	no	—
453	Q₈	—	8	1	no	—
454	D₄	—	4	2	no	—
455	D₄	—	4	4	no	—
456	D₄	—	4	2	no	—
457	D₄	—	4	2	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	4	1	no	—
466	C₂×C₄	—	4	4	no	—
467	C₂×C₄	—	4	2	no	—
468	C₂×C₄	—	4	1	no	—
469	C₂×C₄	—	4	1	no	—
470	C₂×C₄	—	8	1	no	—
471	C₂×C₄	—	8	4	no	—
472	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
473	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
474	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
475	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
476	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
477	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
478	C₈	—	2	2	no	—
479	C₈	—	4	1	no	—
480	C₈	—	4	2	no	—
481	C₈	—	4	2	no	—
482	C₂²	—	2	1	no	—
483	C₂²	—	2	1	no	—
484	C₂²	—	2	1	no	—
485	C₂²	—	4	2	no	—
486	C₂²	—	4	2	no	—
487	C₂²	—	4	2	no	—
488	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
489	C₄	—	2	1	no	—
490	C₄	—	2	1	no	—
491	C₄	—	2	1	no	—
492	C₄	—	2	1	no	—
493	C₄	—	4	2	no	—
494	C₄	—	4	2	no	—
495	C₄	—	4	2	no	—
496	C₄	—	4	2	no	—
497	C₄	—	4	2	no	—
498	C₄	—	8	1	no	—
499	C₄	—	8	1	no	—
500	C₄	C₃₁×#575	1	2	yes	—
501	C₄	C₃₁×#578	1	2	yes	—
502	C₂	—	4	1	no	—
503	C₂	—	8	2	no	—
504	C₂	C₃₁×#3005	1	1	yes	characteristic
505	C₂	C₃₁×#3011	1	1	yes	characteristic
506	C₂	C₃₁×#3018	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×#3237	C₂	semidirect
4. C₃₁×#3104	C₂	semidirect
5. C₃₁×#1648	C₂	semidirect
6. C₃₁×#1640	C₂	semidirect
7. C₃₁×#1636	C₂	semidirect
8. C₃₁×#1630	C₂	semidirect
10. C₃₁×#1491	C₂	semidirect
11. C₃₁×#1485	C₂	semidirect
190. #18603	C₃₁	direct (N × H)
230. #3237	C₂×C₃₁	semidirect
232. #3104	C₂×C₃₁	semidirect
233. #1648	C₂×C₃₁	semidirect
234. #1640	C₂×C₃₁	semidirect
235. #1636	C₂×C₃₁	semidirect
236. #1630	C₂×C₃₁	semidirect
238. #1491	C₂×C₃₁	semidirect
239. #1485	C₂×C₃₁	semidirect
379. C₃₁	#18603	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×#3237	2
C₃₁×#3182	2
C₃₁×#3104	2
C₃₁×#1648	2
C₃₁×#1640	2
C₃₁×#1636	2
C₃₁×#1630	2
C₃₁×#1553	2
C₃₁×#1491	2
C₃₁×#1485	2
#18603	31

Subgroups — C31×#18603 #21475362

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#18603 #21475362