Subgroups — C₃₁×#18840 #21475599

Order 7936.

892 conjugacy classes of subgroups.

The subgroup lattice has 538 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, complemented
2	C₃₁×#3254	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3207	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#3112	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1636	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1630	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1626	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1620	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1557	C₂	1	1	yes	maximal, characteristic, complemented
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×#532	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×#493	C₂²	1	1	yes	characteristic
27	C₃₁×#492	C₂²	1	1	yes	characteristic
28	C₃₁×#487	C₂²	1	1	yes	characteristic
29	C₃₁×#486	C₂²	1	1	yes	characteristic
30	C₃₁×#480	—	2	1	no	—
31	C₃₁×#477	C₂²	1	2	yes	—
32	C₃₁×#477	C₂²	1	2	yes	—
33	C₃₁×#474	—	2	1	no	—
34	C₃₁×#468	—	2	2	no	—
35	C₃₁×#463	—	2	2	no	—
36	C₃₁×#461	—	2	1	no	—
37	C₃₁×#460	C₂²	1	2	yes	—
38	C₃₁×#460	C₂²	1	2	yes	—
39	C₃₁×#449	—	2	2	no	—
40	C₃₁×#448	—	2	1	no	—
41	C₃₁×#440	—	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	C₂×#41×C₃₁	—	2	1	no	—
53	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
54	C₂×SD₁₆×C₃₁	—	2	2	no	—
55	C₂×SD₁₆×C₃₁	—	2	2	no	—
56	C₂×SD₁₆×C₃₁	—	2	2	no	—
57	C₂×D₈×C₃₁	—	2	2	no	—
58	C₂×M₁₆×C₃₁	—	2	1	no	—
59	C₃₁×C₄⋊Q₈	—	2	2	no	—
60	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
61	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
62	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
63	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
64	C₃₁×#127	C₂³	1	1	yes	characteristic
65	C₃₁×#127	C₂³	1	1	yes	characteristic
66	C₃₁×#126	—	2	4	no	—
67	C₃₁×#126	—	2	4	no	—
68	C₃₁×#125	—	2	2	no	—
69	C₃₁×#125	—	2	4	no	—
70	C₃₁×#125	—	2	1	no	—
71	C₃₁×#124	—	2	2	no	—
72	C₃₁×#124	—	2	2	no	—
73	C₃₁×#124	—	2	1	no	—
74	C₃₁×#124	—	2	1	no	—
75	C₃₁×#123	—	2	4	no	—
76	C₃₁×#123	—	2	4	no	—
77	C₃₁×C₂²⋊Q₈	—	2	2	no	—
78	C₃₁×C₂²⋊Q₈	—	2	2	no	—
79	C₃₁×C₂²⋊Q₈	—	2	4	no	—
80	C₃₁×C₂²⋊Q₈	—	2	1	no	—
81	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
82	C₃₁×#121	—	2	1	no	—
83	C₃₁×#121	—	2	1	no	—
84	C₃₁×#121	—	2	1	no	—
85	C₃₁×#121	D₄	1	2	yes	—
86	C₄×Q₈×C₃₁	—	2	2	no	—
87	C₄×Q₈×C₃₁	—	2	2	no	—
88	C₄×Q₈×C₃₁	—	2	1	no	—
89	C₄×Q₈×C₃₁	—	2	1	no	—
90	C₄×Q₈×C₃₁	—	2	1	no	—
91	C₄×Q₈×C₃₁	D₄	1	2	yes	—
92	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
93	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
94	C₄×D₄×C₃₁	—	2	4	no	—
95	C₄×D₄×C₃₁	—	2	2	no	—
96	C₄×D₄×C₃₁	—	2	2	no	—
97	C₄×D₄×C₃₁	—	2	1	no	—
98	C₄×D₄×C₃₁	—	2	1	no	—
99	C₄×D₄×C₃₁	—	2	1	no	—
100	C₄×D₄×C₃₁	D₄	1	2	yes	—
101	C₃₁×#117	—	2	2	no	—
102	C₃₁×#117	—	2	4	no	—
103	C₃₁×#117	—	2	1	no	—
104	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
105	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
106	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
107	C₂×C₄²×C₃₁	—	2	1	no	—
108	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
109	C₃₁×#107	—	2	2	no	—
110	C₃₁×#107	—	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×#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	—	4	4	no	—
127	#41×C₃₁	—	8	2	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	—	2	1	no	—
139	C₂×D₄×C₃₁	—	2	1	no	—
140	C₂×D₄×C₃₁	—	2	1	no	—
141	C₂×D₄×C₃₁	—	4	1	no	—
142	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
143	C₂²×C₄×C₃₁	—	2	1	no	—
144	C₂²×C₄×C₃₁	—	2	4	no	—
145	C₂²×C₄×C₃₁	—	2	1	no	—
146	C₂²×C₄×C₃₁	—	2	2	no	—
147	C₂²×C₄×C₃₁	—	4	1	no	—
148	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
150	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
151	SD₁₆×C₃₁	—	4	4	no	—
152	SD₁₆×C₃₁	—	8	2	no	—
153	SD₁₆×C₃₁	—	8	2	no	—
154	D₈×C₃₁	—	8	2	no	—
155	M₁₆×C₃₁	—	8	1	no	—
156	C₂×C₈×C₃₁	—	2	1	no	—
157	C₂×C₈×C₃₁	—	2	1	no	—
158	C₂×C₈×C₃₁	—	2	2	no	—
159	C₂×C₈×C₃₁	—	2	2	no	—
160	C₄⋊C₄×C₃₁	—	2	4	no	—
161	C₄⋊C₄×C₃₁	—	2	4	no	—
162	C₄⋊C₄×C₃₁	—	2	4	no	—
163	C₄⋊C₄×C₃₁	—	2	1	no	—
164	C₄⋊C₄×C₃₁	—	2	1	no	—
165	C₄⋊C₄×C₃₁	—	2	2	no	—
166	C₄⋊C₄×C₃₁	—	2	1	no	—
167	C₄⋊C₄×C₃₁	—	2	2	no	—
168	C₄⋊C₄×C₃₁	—	2	1	no	—
169	C₄⋊C₄×C₃₁	—	2	2	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	C₂²⋊C₄×C₃₁	—	2	4	no	—
178	C₂²⋊C₄×C₃₁	—	2	4	no	—
179	C₂²⋊C₄×C₃₁	—	2	4	no	—
180	C₂²⋊C₄×C₃₁	—	2	2	no	—
181	C₂²⋊C₄×C₃₁	—	2	2	no	—
182	C₂²⋊C₄×C₃₁	—	2	2	no	—
183	C₂²⋊C₄×C₃₁	—	4	1	no	—
184	C₂²⋊C₄×C₃₁	—	4	1	no	—
185	C₄²×C₃₁	—	2	4	no	—
186	C₄²×C₃₁	—	2	2	no	—
187	C₄²×C₃₁	—	2	2	no	—
188	C₄²×C₃₁	—	2	2	no	—
189	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
190	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
191	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
192	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
193	#18840	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
194	C₂³×C₃₁	—	2	2	no	—
195	C₂³×C₃₁	—	4	1	no	—
196	C₂³×C₃₁	#120	1	1	yes	characteristic
197	Q₈×C₃₁	—	2	2	no	—
198	Q₈×C₃₁	—	2	2	no	—
199	Q₈×C₃₁	—	2	2	no	—
200	Q₈×C₃₁	—	2	2	no	—
201	Q₈×C₃₁	—	4	2	no	—
202	Q₈×C₃₁	—	4	2	no	—
203	Q₈×C₃₁	—	4	2	no	—
204	Q₈×C₃₁	—	8	1	no	—
205	D₄×C₃₁	—	4	2	no	—
206	D₄×C₃₁	—	4	2	no	—
207	D₄×C₃₁	—	4	4	no	—
208	D₄×C₃₁	—	4	2	no	—
209	D₄×C₃₁	—	4	2	no	—
210	D₄×C₃₁	—	4	1	no	—
211	D₄×C₃₁	—	4	1	no	—
212	D₄×C₃₁	—	4	1	no	—
213	D₄×C₃₁	—	4	1	no	—
214	D₄×C₃₁	—	8	2	no	—
215	C₂×C₄×C₃₁	—	2	2	no	—
216	C₂×C₄×C₃₁	—	2	1	no	—
217	C₂×C₄×C₃₁	—	2	1	no	—
218	C₂×C₄×C₃₁	—	2	2	no	—
219	C₂×C₄×C₃₁	—	2	2	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	—	4	1	no	—
224	C₂×C₄×C₃₁	—	4	4	no	—
225	C₂×C₄×C₃₁	—	4	2	no	—
226	C₂×C₄×C₃₁	—	4	1	no	—
227	C₂×C₄×C₃₁	—	8	2	no	—
228	C₂×C₄×C₃₁	—	8	1	no	—
229	C₂×C₄×C₃₁	—	8	4	no	—
230	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
231	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
232	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
233	C₂×C₄×C₃₁	C₂²×D₄	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	C₈×C₃₁	—	4	1	no	—
237	C₈×C₃₁	—	4	1	no	—
238	C₈×C₃₁	—	4	2	no	—
239	C₈×C₃₁	—	4	2	no	—
240	#3507	C₂×C₃₁	1	1	yes	characteristic, complemented
241	#3254	C₂×C₃₁	1	1	yes	characteristic, complemented
242	#3207	C₂×C₃₁	1	1	yes	characteristic
243	#3112	C₂×C₃₁	1	1	yes	characteristic, complemented
244	#1636	C₂×C₃₁	1	2	yes	complemented
245	#1630	C₂×C₃₁	1	2	yes	complemented
246	#1626	C₂×C₃₁	1	2	yes	complemented
247	#1620	C₂×C₃₁	1	2	yes	complemented
248	#1557	C₂×C₃₁	1	1	yes	characteristic, complemented
249	#1497	C₂×C₃₁	1	1	yes	characteristic, complemented
250	#1487	C₂×C₃₁	1	1	yes	characteristic, complemented
251	C₂²×C₃₁	—	2	1	no	—
252	C₂²×C₃₁	—	2	1	no	—
253	C₂²×C₃₁	—	2	1	no	—
254	C₂²×C₃₁	—	4	2	no	—
255	C₂²×C₃₁	—	4	2	no	—
256	C₂²×C₃₁	—	4	2	no	—
257	C₂²×C₃₁	—	8	1	no	—
258	C₂²×C₃₁	—	8	1	no	—
259	C₂²×C₃₁	—	8	1	no	—
260	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
261	C₄×C₃₁	—	2	1	no	—
262	C₄×C₃₁	—	2	1	no	—
263	C₄×C₃₁	—	2	1	no	—
264	C₄×C₃₁	—	2	1	no	—
265	C₄×C₃₁	—	4	2	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	—	8	1	no	—
271	C₄×C₃₁	#575	1	2	yes	—
272	C₄×C₃₁	#577	1	1	yes	characteristic
273	C₄×C₃₁	#578	1	1	yes	characteristic
274	#559	C₂²×C₃₁	1	2	yes	—
275	#557	C₂²×C₃₁	1	2	yes	—
276	#556	—	2	1	no	—
277	#556	—	2	2	no	—
278	#554	—	2	1	no	—
279	#552	—	2	2	no	—
280	#539	—	2	2	no	—
281	#536	—	2	1	no	—
282	#533	C₂²×C₃₁	1	1	yes	characteristic
283	#532	C₂²×C₃₁	1	1	yes	characteristic
284	#532	C₂²×C₃₁	1	1	yes	characteristic
285	#529	—	2	2	no	—
286	#520	—	2	1	no	—
287	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
288	#493	C₂²×C₃₁	1	1	yes	characteristic
289	#492	C₂²×C₃₁	1	1	yes	characteristic
290	#487	C₂²×C₃₁	1	1	yes	characteristic
291	#486	C₂²×C₃₁	1	1	yes	characteristic
292	#480	—	2	1	no	—
293	#477	C₂²×C₃₁	1	2	yes	—
294	#477	C₂²×C₃₁	1	2	yes	—
295	#474	—	2	1	no	—
296	#468	—	2	2	no	—
297	#463	—	2	2	no	—
298	#461	—	2	1	no	—
299	#460	C₂²×C₃₁	1	2	yes	—
300	#460	C₂²×C₃₁	1	2	yes	—
301	#449	—	2	2	no	—
302	#448	—	2	1	no	—
303	#440	—	2	2	no	—
304	#432	C₂²×C₃₁	1	1	yes	characteristic
305	#431	C₂²×C₃₁	1	1	yes	characteristic
306	#423	C₂²×C₃₁	1	1	yes	characteristic
307	#332	C₂²×C₃₁	1	2	yes	—
308	#331	C₂²×C₃₁	1	2	yes	—
309	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
310	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
311	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
312	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
313	C₂×C₃₁	—	4	1	no	—
314	C₂×C₃₁	—	8	2	no	—
315	C₂×C₃₁	—	16	1	no	—
316	C₂×C₃₁	#3005	1	1	yes	characteristic
317	C₂×C₃₁	#3013	1	1	yes	characteristic
318	C₂×C₃₁	#3021	1	1	yes	characteristic
319	C₂×#41	—	2	1	no	—
320	C₂×#41	—	2	1	no	—
321	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
322	C₂×SD₁₆	—	2	2	no	—
323	C₂×SD₁₆	—	2	2	no	—
324	C₂×SD₁₆	—	2	2	no	—
325	C₂×D₈	—	2	2	no	—
326	C₂×M₁₆	—	2	1	no	—
327	C₄⋊Q₈	—	2	2	no	—
328	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
329	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
330	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
331	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
332	#127	C₂³×C₃₁	1	1	yes	characteristic
333	#127	C₂³×C₃₁	1	1	yes	characteristic
334	#126	—	2	4	no	—
335	#126	—	2	4	no	—
336	#125	—	2	2	no	—
337	#125	—	2	4	no	—
338	#125	—	2	1	no	—
339	#124	—	2	2	no	—
340	#124	—	2	2	no	—
341	#124	—	2	1	no	—
342	#124	—	2	1	no	—
343	#123	—	2	4	no	—
344	#123	—	2	4	no	—
345	C₂²⋊Q₈	—	2	2	no	—
346	C₂²⋊Q₈	—	2	2	no	—
347	C₂²⋊Q₈	—	2	4	no	—
348	C₂²⋊Q₈	—	2	1	no	—
349	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
350	#121	—	2	1	no	—
351	#121	—	2	1	no	—
352	#121	—	2	1	no	—
353	#121	D₄×C₃₁	1	2	yes	—
354	C₄×Q₈	—	2	2	no	—
355	C₄×Q₈	—	2	2	no	—
356	C₄×Q₈	—	2	1	no	—
357	C₄×Q₈	—	2	1	no	—
358	C₄×Q₈	—	2	1	no	—
359	C₄×Q₈	D₄×C₃₁	1	2	yes	—
360	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
361	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
362	C₄×D₄	—	2	4	no	—
363	C₄×D₄	—	2	2	no	—
364	C₄×D₄	—	2	2	no	—
365	C₄×D₄	—	2	1	no	—
366	C₄×D₄	—	2	1	no	—
367	C₄×D₄	—	2	1	no	—
368	C₄×D₄	D₄×C₃₁	1	2	yes	—
369	#117	—	2	2	no	—
370	#117	—	2	4	no	—
371	#117	—	2	1	no	—
372	C₂×C₄⋊C₄	—	2	2	no	—
373	C₂×C₄⋊C₄	—	2	1	no	—
374	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
375	C₂×C₄²	—	2	1	no	—
376	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
377	#107	—	2	2	no	—
378	#107	—	2	2	no	—
379	C₄⋊C₈	—	2	1	no	—
380	C₄⋊C₈	—	2	1	no	—
381	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
382	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
383	#103	—	2	2	no	—
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	C₂²⋊C₈	—	2	1	no	—
390	C₂²⋊C₈	—	2	1	no	—
391	#97	—	2	1	no	—
392	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
393	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
394	C₃₁	#18840	1	1	yes	characteristic, direct product factor, Sylow
395	#41	—	4	4	no	—
396	#41	—	8	2	no	—
397	C₂×Q₈	—	2	1	no	—
398	C₂×Q₈	—	2	1	no	—
399	C₂×Q₈	—	2	1	no	—
400	C₂×Q₈	—	2	1	no	—
401	C₂×Q₈	—	4	2	no	—
402	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
403	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
404	C₂×D₄	—	2	2	no	—
405	C₂×D₄	—	2	1	no	—
406	C₂×D₄	—	2	1	no	—
407	C₂×D₄	—	2	1	no	—
408	C₂×D₄	—	2	1	no	—
409	C₂×D₄	—	2	1	no	—
410	C₂×D₄	—	4	1	no	—
411	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
412	C₂²×C₄	—	2	1	no	—
413	C₂²×C₄	—	2	4	no	—
414	C₂²×C₄	—	2	1	no	—
415	C₂²×C₄	—	2	2	no	—
416	C₂²×C₄	—	4	1	no	—
417	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
418	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
419	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
420	SD₁₆	—	4	4	no	—
421	SD₁₆	—	8	2	no	—
422	SD₁₆	—	8	2	no	—
423	D₈	—	8	2	no	—
424	M₁₆	—	8	1	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	C₄⋊C₄	—	2	4	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	2	no	—
435	C₄⋊C₄	—	2	1	no	—
436	C₄⋊C₄	—	2	2	no	—
437	C₄⋊C₄	—	2	1	no	—
438	C₄⋊C₄	—	2	2	no	—
439	C₄⋊C₄	—	2	1	no	—
440	C₄⋊C₄	—	2	2	no	—
441	C₄⋊C₄	—	2	1	no	—
442	C₄⋊C₄	—	2	1	no	—
443	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
444	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
445	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
446	C₂²⋊C₄	—	2	4	no	—
447	C₂²⋊C₄	—	2	4	no	—
448	C₂²⋊C₄	—	2	4	no	—
449	C₂²⋊C₄	—	2	2	no	—
450	C₂²⋊C₄	—	2	2	no	—
451	C₂²⋊C₄	—	2	2	no	—
452	C₂²⋊C₄	—	4	1	no	—
453	C₂²⋊C₄	—	4	1	no	—
454	C₄²	—	2	4	no	—
455	C₄²	—	2	2	no	—
456	C₄²	—	2	2	no	—
457	C₄²	—	2	2	no	—
458	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
459	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
460	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
461	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
462	C₂³	—	2	2	no	—
463	C₂³	—	4	1	no	—
464	C₂³	C₃₁×#120	1	1	yes	characteristic
465	Q₈	—	2	2	no	—
466	Q₈	—	2	2	no	—
467	Q₈	—	2	2	no	—
468	Q₈	—	2	2	no	—
469	Q₈	—	4	2	no	—
470	Q₈	—	4	2	no	—
471	Q₈	—	4	2	no	—
472	Q₈	—	8	1	no	—
473	D₄	—	4	2	no	—
474	D₄	—	4	2	no	—
475	D₄	—	4	4	no	—
476	D₄	—	4	2	no	—
477	D₄	—	4	2	no	—
478	D₄	—	4	1	no	—
479	D₄	—	4	1	no	—
480	D₄	—	4	1	no	—
481	D₄	—	4	1	no	—
482	D₄	—	8	2	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	2	1	no	—
485	C₂×C₄	—	2	1	no	—
486	C₂×C₄	—	2	2	no	—
487	C₂×C₄	—	2	2	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	4	1	no	—
492	C₂×C₄	—	4	4	no	—
493	C₂×C₄	—	4	2	no	—
494	C₂×C₄	—	4	1	no	—
495	C₂×C₄	—	8	2	no	—
496	C₂×C₄	—	8	1	no	—
497	C₂×C₄	—	8	4	no	—
498	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
499	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
500	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
501	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
502	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
503	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
504	C₈	—	4	1	no	—
505	C₈	—	4	1	no	—
506	C₈	—	4	2	no	—
507	C₈	—	4	2	no	—
508	C₂²	—	2	1	no	—
509	C₂²	—	2	1	no	—
510	C₂²	—	2	1	no	—
511	C₂²	—	4	2	no	—
512	C₂²	—	4	2	no	—
513	C₂²	—	4	2	no	—
514	C₂²	—	8	1	no	—
515	C₂²	—	8	1	no	—
516	C₂²	—	8	1	no	—
517	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
518	C₄	—	2	1	no	—
519	C₄	—	2	1	no	—
520	C₄	—	2	1	no	—
521	C₄	—	2	1	no	—
522	C₄	—	4	2	no	—
523	C₄	—	4	2	no	—
524	C₄	—	4	2	no	—
525	C₄	—	4	2	no	—
526	C₄	—	4	2	no	—
527	C₄	—	8	1	no	—
528	C₄	C₃₁×#575	1	2	yes	—
529	C₄	C₃₁×#577	1	1	yes	characteristic
530	C₄	C₃₁×#578	1	1	yes	characteristic
531	C₂	—	4	1	no	—
532	C₂	—	8	2	no	—
533	C₂	—	16	1	no	—
534	C₂	C₃₁×#3005	1	1	yes	characteristic
535	C₂	C₃₁×#3013	1	1	yes	characteristic
536	C₂	C₃₁×#3021	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3207	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×#3507	C₂	semidirect
2. C₃₁×#3254	C₂	semidirect
4. C₃₁×#3112	C₂	semidirect
5. C₃₁×#1636	C₂	semidirect
6. C₃₁×#1630	C₂	semidirect
7. C₃₁×#1626	C₂	semidirect
8. C₃₁×#1620	C₂	semidirect
9. C₃₁×#1557	C₂	semidirect
10. C₃₁×#1497	C₂	semidirect
11. C₃₁×#1487	C₂	semidirect
193. #18840	C₃₁	direct (N × H)
240. #3507	C₂×C₃₁	semidirect
241. #3254	C₂×C₃₁	semidirect
243. #3112	C₂×C₃₁	semidirect
244. #1636	C₂×C₃₁	semidirect
245. #1630	C₂×C₃₁	semidirect
246. #1626	C₂×C₃₁	semidirect
247. #1620	C₂×C₃₁	semidirect
248. #1557	C₂×C₃₁	semidirect
249. #1497	C₂×C₃₁	semidirect
250. #1487	C₂×C₃₁	semidirect
394. C₃₁	#18840	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×#3254	2
C₃₁×#3207	2
C₃₁×#3112	2
C₃₁×#1636	2
C₃₁×#1630	2
C₃₁×#1626	2
C₃₁×#1620	2
C₃₁×#1557	2
C₃₁×#1497	2
C₃₁×#1487	2
#18840	31

Subgroups — C31×#18840 #21475599

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#18840 #21475599