Subgroups — C₃₁×#60878 #21517637

Order 7936.

1666 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3509	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3503	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3242	C₂	1	8	yes	maximal, complemented
4	C₃₁×#3231	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3231	C₂	1	2	yes	maximal, complemented
6	C₃₁×#3230	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3229	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3165	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#3160	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#3139	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#487	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#3097	C₂	1	4	yes	maximal
13	C₃₁×#3082	C₂	1	4	yes	maximal, complemented
14	C₃₁×#2922	C₂	1	1	yes	maximal, characteristic
15	Q₈²×C₃₁	—	2	1	no	—
16	C₃₁×#557	—	2	1	no	—
17	C₃₁×#557	—	2	1	no	—
18	C₃₁×#557	—	2	1	no	—
19	C₃₁×#556	—	2	1	no	—
20	C₃₁×#556	—	2	1	no	—
21	C₃₁×#554	—	2	2	no	—
22	C₃₁×#554	—	2	1	no	—
23	C₃₁×#552	—	2	2	no	—
24	C₃₁×#552	—	2	2	no	—
25	D₄×Q₈×C₃₁	—	2	2	no	—
26	D₄×Q₈×C₃₁	—	2	1	no	—
27	C₃₁×#544	C₂²	1	2	yes	—
28	C₃₁×#544	C₂²	1	4	yes	—
29	C₃₁×#541	C₂²	1	4	yes	—
30	C₃₁×#536	C₂²	1	2	yes	—
31	C₃₁×#533	C₂²	1	2	yes	—
32	C₃₁×#533	C₂²	1	2	yes	—
33	C₃₁×#533	C₂²	1	1	yes	characteristic
34	C₃₁×#533	C₂²	1	1	yes	characteristic
35	C₃₁×#533	C₂²	1	2	yes	—
36	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
37	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
38	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
39	C₃₁×#520	C₂²	1	2	yes	—
40	C₃₁×#519	C₂²	1	2	yes	—
41	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
42	C₃₁×#501	C₂²	1	4	yes	—
43	C₃₁×#500	C₂²	1	1	yes	characteristic
44	C₃₁×#500	C₂²	1	1	yes	characteristic
45	C₃₁×#500	C₂²	1	2	yes	—
46	C₃₁×#499	C₂²	1	4	yes	—
47	C₃₁×#498	C₂²	1	1	yes	characteristic
48	C₃₁×#498	C₂²	1	1	yes	characteristic
49	C₃₁×#498	C₂²	1	2	yes	—
50	C₃₁×#491	C₂²	1	4	yes	—
51	C₃₁×#487	C₂²	1	2	yes	—
52	C₃₁×#487	C₂²	1	2	yes	—
53	C₃₁×#487	C₂²	1	4	yes	—
54	C₃₁×#487	C₂²	1	2	yes	—
55	C₃₁×#487	C₂²	1	2	yes	—
56	C₃₁×#485	C₂²	1	8	yes	—
57	C₃₁×#485	C₂²	1	8	yes	—
58	C₃₁×#484	C₂²	1	8	yes	—
59	C₃₁×#483	C₂²	1	8	yes	—
60	C₃₁×#479	C₂²	1	8	yes	—
61	C₃₁×#479	C₂²	1	8	yes	—
62	C₃₁×#477	C₂²	1	8	yes	—
63	C₃₁×#475	C₂²	1	8	yes	—
64	C₃₁×#433	C₂²	1	4	yes	—
65	C₃₁×#431	C₂²	1	2	yes	—
66	C₃₁×#431	C₂²	1	2	yes	—
67	C₃₁×#428	C₂²	1	4	yes	—
68	C₃₁×#427	C₂²	1	4	yes	—
69	C₃₁×#424	C₂²	1	2	yes	—
70	C₃₁×#419	C₂²	1	4	yes	—
71	C₂×C₃₁×#103	C₂²	1	4	yes	—
72	C₃₁×#408	C₂²	1	2	yes	—
73	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
74	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
75	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
76	C₂²×Q₈×C₃₁	—	2	1	no	—
77	C₂²×Q₈×C₃₁	—	2	2	no	—
78	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
79	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
80	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
81	C₃₁×C₄⋊Q₈	—	2	2	no	—
82	C₃₁×C₄⋊Q₈	—	2	2	no	—
83	C₃₁×C₄⋊Q₈	—	2	4	no	—
84	C₃₁×C₄⋊Q₈	—	2	2	no	—
85	C₃₁×C₄⋊Q₈	—	2	1	no	—
86	C₃₁×C₄⋊Q₈	—	2	2	no	—
87	C₃₁×C₄⋊Q₈	—	2	1	no	—
88	C₃₁×C₄⋊Q₈	—	2	1	no	—
89	C₃₁×C₄⋊Q₈	—	2	1	no	—
90	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
91	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
92	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
93	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
94	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
95	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
96	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
97	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
98	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
99	C₃₁×#126	—	2	4	no	—
100	C₃₁×#126	—	2	4	no	—
101	C₃₁×#125	—	2	4	no	—
102	C₃₁×#125	—	2	4	no	—
103	C₃₁×#125	—	2	2	no	—
104	C₃₁×#125	—	2	4	no	—
105	C₃₁×#125	—	2	2	no	—
106	C₃₁×#125	—	2	2	no	—
107	C₃₁×#125	C₂³	1	4	yes	—
108	C₃₁×#125	C₂³	1	2	yes	—
109	C₃₁×#125	C₂³	1	2	yes	—
110	C₃₁×#124	—	2	2	no	—
111	C₃₁×#123	—	2	4	no	—
112	C₃₁×C₂²⋊Q₈	—	2	1	no	—
113	C₃₁×C₂²⋊Q₈	—	2	2	no	—
114	C₃₁×C₂²⋊Q₈	—	2	1	no	—
115	C₃₁×C₂²⋊Q₈	—	2	1	no	—
116	C₃₁×C₂²⋊Q₈	—	2	2	no	—
117	C₃₁×C₂²⋊Q₈	—	2	2	no	—
118	C₃₁×C₂²⋊Q₈	—	2	1	no	—
119	C₃₁×C₂²⋊Q₈	—	2	2	no	—
120	C₃₁×C₂²⋊Q₈	—	2	4	no	—
121	C₃₁×C₂²⋊Q₈	—	2	4	no	—
122	C₃₁×C₂²⋊Q₈	—	2	4	no	—
123	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
124	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
125	C₄×Q₈×C₃₁	—	2	2	no	—
126	C₄×Q₈×C₃₁	—	2	2	no	—
127	C₄×Q₈×C₃₁	—	2	2	no	—
128	C₄×Q₈×C₃₁	—	2	2	no	—
129	C₄×Q₈×C₃₁	—	2	1	no	—
130	C₄×Q₈×C₃₁	—	2	2	no	—
131	C₄×Q₈×C₃₁	—	2	2	no	—
132	C₄×Q₈×C₃₁	—	2	2	no	—
133	C₄×Q₈×C₃₁	—	2	1	no	—
134	C₄×Q₈×C₃₁	—	2	1	no	—
135	C₄×Q₈×C₃₁	—	2	1	no	—
136	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
137	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
138	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
139	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
140	C₄×D₄×C₃₁	—	2	4	no	—
141	C₄×D₄×C₃₁	—	2	2	no	—
142	C₄×D₄×C₃₁	D₄	1	2	yes	—
143	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
144	C₄×D₄×C₃₁	D₄	1	2	yes	—
145	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
146	C₃₁×#117	—	2	2	no	—
147	C₃₁×#117	—	2	2	no	—
148	C₃₁×#117	—	2	4	no	—
149	C₃₁×#117	C₂³	1	2	yes	—
150	C₃₁×#117	C₂³	1	2	yes	—
151	C₃₁×#117	C₂³	1	4	yes	—
152	C₃₁×#117	C₂³	1	1	yes	characteristic
153	C₃₁×#117	C₂³	1	2	yes	—
154	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
155	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
156	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
157	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
158	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
159	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
160	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
161	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
162	C₃₁×#107	C₂³	1	4	yes	—
163	C₃₁×#107	C₂³	1	4	yes	—
164	C₃₁×#107	C₂³	1	8	yes	—
165	C₃₁×#106	C₂³	1	4	yes	—
166	C₃₁×#106	C₂³	1	4	yes	—
167	C₃₁×#106	C₂³	1	8	yes	—
168	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
169	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
170	C₃₁×#103	C₂³	1	8	yes	—
171	C₃₁×#103	C₂³	1	8	yes	—
172	C₃₁×#103	C₂³	1	8	yes	—
173	C₃₁×#103	C₂³	1	8	yes	—
174	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
175	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
176	C₃₁×#97	C₂³	1	2	yes	—
177	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
178	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
179	C₄×C₈×C₃₁	C₂³	1	2	yes	—
180	C₄×C₈×C₃₁	C₂³	1	2	yes	—
181	#41×C₃₁	—	2	4	no	—
182	C₂×Q₈×C₃₁	—	2	1	no	—
183	C₂×Q₈×C₃₁	—	2	1	no	—
184	C₂×Q₈×C₃₁	—	2	1	no	—
185	C₂×Q₈×C₃₁	—	2	2	no	—
186	C₂×Q₈×C₃₁	—	2	2	no	—
187	C₂×Q₈×C₃₁	—	2	1	no	—
188	C₂×Q₈×C₃₁	—	2	4	no	—
189	C₂×Q₈×C₃₁	—	2	2	no	—
190	C₂×Q₈×C₃₁	—	2	8	no	—
191	C₂×Q₈×C₃₁	—	8	2	no	—
192	C₂×Q₈×C₃₁	—	8	4	no	—
193	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
194	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
195	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
196	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
197	C₂²×C₄×C₃₁	—	2	2	no	—
198	C₂²×C₄×C₃₁	—	2	4	no	—
199	C₂²×C₄×C₃₁	—	2	2	no	—
200	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
201	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
202	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
203	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
204	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
205	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
206	M₁₆×C₃₁	—	2	4	no	—
207	C₂×C₈×C₃₁	—	2	4	no	—
208	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
209	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
210	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
211	C₄⋊C₄×C₃₁	—	2	2	no	—
212	C₄⋊C₄×C₃₁	—	2	1	no	—
213	C₄⋊C₄×C₃₁	—	2	2	no	—
214	C₄⋊C₄×C₃₁	—	2	2	no	—
215	C₄⋊C₄×C₃₁	—	2	4	no	—
216	C₄⋊C₄×C₃₁	—	2	2	no	—
217	C₄⋊C₄×C₃₁	—	2	2	no	—
218	C₄⋊C₄×C₃₁	—	2	2	no	—
219	C₄⋊C₄×C₃₁	—	2	4	no	—
220	C₄⋊C₄×C₃₁	—	2	2	no	—
221	C₄⋊C₄×C₃₁	—	2	1	no	—
222	C₄⋊C₄×C₃₁	—	2	1	no	—
223	C₄⋊C₄×C₃₁	—	2	4	no	—
224	C₄⋊C₄×C₃₁	—	2	4	no	—
225	C₄⋊C₄×C₃₁	—	2	1	no	—
226	C₄⋊C₄×C₃₁	—	2	4	no	—
227	C₄⋊C₄×C₃₁	—	2	2	no	—
228	C₄⋊C₄×C₃₁	—	2	4	no	—
229	C₄⋊C₄×C₃₁	—	2	2	no	—
230	C₄⋊C₄×C₃₁	—	2	2	no	—
231	C₄⋊C₄×C₃₁	—	2	2	no	—
232	C₄⋊C₄×C₃₁	—	2	4	no	—
233	C₄⋊C₄×C₃₁	—	2	4	no	—
234	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
235	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
236	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
237	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
238	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
239	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
240	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
241	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
242	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
243	C₂²⋊C₄×C₃₁	—	2	2	no	—
244	C₂²⋊C₄×C₃₁	—	2	4	no	—
245	C₂²⋊C₄×C₃₁	—	2	4	no	—
246	C₂²⋊C₄×C₃₁	—	2	4	no	—
247	C₂²⋊C₄×C₃₁	—	2	2	no	—
248	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
249	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
250	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
251	C₄²×C₃₁	—	2	4	no	—
252	C₄²×C₃₁	—	2	4	no	—
253	C₄²×C₃₁	—	2	4	no	—
254	C₄²×C₃₁	—	2	2	no	—
255	C₄²×C₃₁	—	2	2	no	—
256	C₄²×C₃₁	C₂×D₄	1	2	yes	—
257	C₄²×C₃₁	C₂×D₄	1	2	yes	—
258	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
259	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
260	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
261	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
262	C₄²×C₃₁	C₂⁴	1	2	yes	—
263	#60878	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
264	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
265	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
266	Q₈×C₃₁	—	2	2	no	—
267	Q₈×C₃₁	—	2	8	no	—
268	Q₈×C₃₁	—	4	4	no	—
269	Q₈×C₃₁	—	4	2	no	—
270	Q₈×C₃₁	—	4	4	no	—
271	Q₈×C₃₁	—	4	2	no	—
272	Q₈×C₃₁	—	4	2	no	—
273	Q₈×C₃₁	—	4	2	no	—
274	Q₈×C₃₁	—	8	4	no	—
275	Q₈×C₃₁	—	8	2	no	—
276	D₄×C₃₁	—	2	4	no	—
277	D₄×C₃₁	—	2	2	no	—
278	C₂×C₄×C₃₁	—	2	2	no	—
279	C₂×C₄×C₃₁	—	2	4	no	—
280	C₂×C₄×C₃₁	—	2	2	no	—
281	C₂×C₄×C₃₁	—	2	4	no	—
282	C₂×C₄×C₃₁	—	2	4	no	—
283	C₂×C₄×C₃₁	—	2	2	no	—
284	C₂×C₄×C₃₁	—	2	2	no	—
285	C₂×C₄×C₃₁	—	2	2	no	—
286	C₂×C₄×C₃₁	—	2	4	no	—
287	C₂×C₄×C₃₁	—	2	4	no	—
288	C₂×C₄×C₃₁	—	2	2	no	—
289	C₂×C₄×C₃₁	—	4	4	no	—
290	C₂×C₄×C₃₁	—	8	2	no	—
291	C₂×C₄×C₃₁	—	8	4	no	—
292	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
293	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
294	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
295	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
296	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
297	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
298	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
299	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
300	C₂×C₄×C₃₁	2₋¹⁺⁴	1	4	yes	—
301	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
302	C₈×C₃₁	—	2	2	no	—
303	C₈×C₃₁	—	2	2	no	—
304	C₈×C₃₁	—	2	4	no	—
305	#3509	C₂×C₃₁	1	1	yes	characteristic
306	#3503	C₂×C₃₁	1	1	yes	characteristic
307	#3242	C₂×C₃₁	1	8	yes	complemented
308	#3231	C₂×C₃₁	1	2	yes	complemented
309	#3231	C₂×C₃₁	1	2	yes	complemented
310	#3230	C₂×C₃₁	1	2	yes	complemented
311	#3229	C₂×C₃₁	1	2	yes	complemented
312	#3165	C₂×C₃₁	1	1	yes	characteristic, complemented
313	#3160	C₂×C₃₁	1	1	yes	characteristic, complemented
314	#3139	C₂×C₃₁	1	1	yes	characteristic, complemented
315	C₂×#487	C₂×C₃₁	1	1	yes	characteristic, complemented
316	#3097	C₂×C₃₁	1	4	yes	—
317	#3082	C₂×C₃₁	1	4	yes	complemented
318	#2922	C₂×C₃₁	1	1	yes	characteristic
319	C₂²×C₃₁	—	2	1	no	—
320	C₂²×C₃₁	—	2	1	no	—
321	C₂²×C₃₁	—	2	2	no	—
322	C₂²×C₃₁	—	2	2	no	—
323	C₂²×C₃₁	—	2	2	no	—
324	C₂²×C₃₁	#536	1	2	yes	—
325	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
326	C₄×C₃₁	—	2	2	no	—
327	C₄×C₃₁	—	2	2	no	—
328	C₄×C₃₁	—	2	1	no	—
329	C₄×C₃₁	—	2	2	no	—
330	C₄×C₃₁	—	2	1	no	—
331	C₄×C₃₁	—	4	4	no	—
332	C₄×C₃₁	—	4	4	no	—
333	C₄×C₃₁	—	4	2	no	—
334	C₄×C₃₁	—	4	2	no	—
335	C₄×C₃₁	—	4	2	no	—
336	C₄×C₃₁	—	4	2	no	—
337	C₄×C₃₁	#536	1	2	yes	—
338	C₄×C₃₁	#577	1	2	yes	—
339	C₄×C₃₁	#578	1	2	yes	—
340	C₄×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
341	Q₈²	—	2	1	no	—
342	#557	—	2	1	no	—
343	#557	—	2	1	no	—
344	#557	—	2	1	no	—
345	#556	—	2	1	no	—
346	#556	—	2	1	no	—
347	#554	—	2	2	no	—
348	#554	—	2	1	no	—
349	#552	—	2	2	no	—
350	#552	—	2	2	no	—
351	D₄×Q₈	—	2	2	no	—
352	D₄×Q₈	—	2	1	no	—
353	#544	C₂²×C₃₁	1	2	yes	—
354	#544	C₂²×C₃₁	1	4	yes	—
355	#541	C₂²×C₃₁	1	4	yes	—
356	#536	C₂²×C₃₁	1	2	yes	—
357	#533	C₂²×C₃₁	1	2	yes	—
358	#533	C₂²×C₃₁	1	2	yes	—
359	#533	C₂²×C₃₁	1	1	yes	characteristic
360	#533	C₂²×C₃₁	1	1	yes	characteristic
361	#533	C₂²×C₃₁	1	2	yes	—
362	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
363	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
364	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
365	#520	C₂²×C₃₁	1	2	yes	—
366	#519	C₂²×C₃₁	1	2	yes	—
367	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
368	#501	C₂²×C₃₁	1	4	yes	—
369	#500	C₂²×C₃₁	1	1	yes	characteristic
370	#500	C₂²×C₃₁	1	1	yes	characteristic
371	#500	C₂²×C₃₁	1	2	yes	—
372	#499	C₂²×C₃₁	1	4	yes	—
373	#498	C₂²×C₃₁	1	1	yes	characteristic
374	#498	C₂²×C₃₁	1	1	yes	characteristic
375	#498	C₂²×C₃₁	1	2	yes	—
376	#491	C₂²×C₃₁	1	4	yes	—
377	#487	C₂²×C₃₁	1	2	yes	—
378	#487	C₂²×C₃₁	1	2	yes	—
379	#487	C₂²×C₃₁	1	4	yes	—
380	#487	C₂²×C₃₁	1	2	yes	—
381	#487	C₂²×C₃₁	1	2	yes	—
382	#485	C₂²×C₃₁	1	8	yes	—
383	#485	C₂²×C₃₁	1	8	yes	—
384	#484	C₂²×C₃₁	1	8	yes	—
385	#483	C₂²×C₃₁	1	8	yes	—
386	#479	C₂²×C₃₁	1	8	yes	—
387	#479	C₂²×C₃₁	1	8	yes	—
388	#477	C₂²×C₃₁	1	8	yes	—
389	#475	C₂²×C₃₁	1	8	yes	—
390	#433	C₂²×C₃₁	1	4	yes	—
391	#431	C₂²×C₃₁	1	2	yes	—
392	#431	C₂²×C₃₁	1	2	yes	—
393	#428	C₂²×C₃₁	1	4	yes	—
394	#427	C₂²×C₃₁	1	4	yes	—
395	#424	C₂²×C₃₁	1	2	yes	—
396	#419	C₂²×C₃₁	1	4	yes	—
397	C₂×#103	C₂²×C₃₁	1	4	yes	—
398	#408	C₂²×C₃₁	1	2	yes	—
399	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
400	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
401	C₂×C₃₁	—	2	2	no	—
402	C₂×C₃₁	—	4	2	no	—
403	C₂×C₃₁	C₂×#536	1	1	yes	characteristic
404	C₂×C₃₁	C₂×#577	1	1	yes	characteristic
405	C₂×C₃₁	C₂×#578	1	1	yes	characteristic
406	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
407	C₂²×Q₈	—	2	1	no	—
408	C₂²×Q₈	—	2	2	no	—
409	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
410	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
411	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
412	C₄⋊Q₈	—	2	2	no	—
413	C₄⋊Q₈	—	2	2	no	—
414	C₄⋊Q₈	—	2	4	no	—
415	C₄⋊Q₈	—	2	2	no	—
416	C₄⋊Q₈	—	2	1	no	—
417	C₄⋊Q₈	—	2	2	no	—
418	C₄⋊Q₈	—	2	1	no	—
419	C₄⋊Q₈	—	2	1	no	—
420	C₄⋊Q₈	—	2	1	no	—
421	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
422	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
423	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
424	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
425	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
426	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
427	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
428	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
429	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
430	#126	—	2	4	no	—
431	#126	—	2	4	no	—
432	#125	—	2	4	no	—
433	#125	—	2	4	no	—
434	#125	—	2	2	no	—
435	#125	—	2	4	no	—
436	#125	—	2	2	no	—
437	#125	—	2	2	no	—
438	#125	C₂³×C₃₁	1	2	yes	—
439	#125	C₂³×C₃₁	1	4	yes	—
440	#125	C₂³×C₃₁	1	2	yes	—
441	#124	—	2	2	no	—
442	#123	—	2	4	no	—
443	C₂²⋊Q₈	—	2	1	no	—
444	C₂²⋊Q₈	—	2	2	no	—
445	C₂²⋊Q₈	—	2	1	no	—
446	C₂²⋊Q₈	—	2	1	no	—
447	C₂²⋊Q₈	—	2	2	no	—
448	C₂²⋊Q₈	—	2	2	no	—
449	C₂²⋊Q₈	—	2	1	no	—
450	C₂²⋊Q₈	—	2	2	no	—
451	C₂²⋊Q₈	—	2	4	no	—
452	C₂²⋊Q₈	—	2	4	no	—
453	C₂²⋊Q₈	—	2	4	no	—
454	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
455	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
456	C₄×Q₈	—	2	2	no	—
457	C₄×Q₈	—	2	2	no	—
458	C₄×Q₈	—	2	2	no	—
459	C₄×Q₈	—	2	2	no	—
460	C₄×Q₈	—	2	1	no	—
461	C₄×Q₈	—	2	2	no	—
462	C₄×Q₈	—	2	2	no	—
463	C₄×Q₈	—	2	2	no	—
464	C₄×Q₈	—	2	1	no	—
465	C₄×Q₈	—	2	1	no	—
466	C₄×Q₈	—	2	1	no	—
467	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
468	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
469	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
470	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
471	C₄×D₄	—	2	4	no	—
472	C₄×D₄	—	2	2	no	—
473	C₄×D₄	D₄×C₃₁	1	2	yes	—
474	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
475	C₄×D₄	D₄×C₃₁	1	2	yes	—
476	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
477	#117	—	2	2	no	—
478	#117	—	2	2	no	—
479	#117	—	2	4	no	—
480	#117	C₂³×C₃₁	1	2	yes	—
481	#117	C₂³×C₃₁	1	2	yes	—
482	#117	C₂³×C₃₁	1	4	yes	—
483	#117	C₂³×C₃₁	1	1	yes	characteristic
484	#117	C₂³×C₃₁	1	2	yes	—
485	C₂×C₄⋊C₄	—	2	1	no	—
486	C₂×C₄⋊C₄	—	2	1	no	—
487	C₂×C₄⋊C₄	—	2	1	no	—
488	C₂×C₄⋊C₄	—	2	2	no	—
489	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
490	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
491	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
492	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
493	#107	C₂³×C₃₁	1	4	yes	—
494	#107	C₂³×C₃₁	1	4	yes	—
495	#107	C₂³×C₃₁	1	8	yes	—
496	#106	C₂³×C₃₁	1	4	yes	—
497	#106	C₂³×C₃₁	1	4	yes	—
498	#106	C₂³×C₃₁	1	8	yes	—
499	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
500	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
501	#103	C₂³×C₃₁	1	8	yes	—
502	#103	C₂³×C₃₁	1	8	yes	—
503	#103	C₂³×C₃₁	1	8	yes	—
504	#103	C₂³×C₃₁	1	8	yes	—
505	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
506	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
507	#97	C₂³×C₃₁	1	2	yes	—
508	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
509	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
510	C₄×C₈	C₂³×C₃₁	1	2	yes	—
511	C₄×C₈	C₂³×C₃₁	1	2	yes	—
512	C₃₁	#60878	1	1	yes	characteristic, direct product factor, Sylow
513	#41	—	2	4	no	—
514	C₂×Q₈	—	2	1	no	—
515	C₂×Q₈	—	2	1	no	—
516	C₂×Q₈	—	2	1	no	—
517	C₂×Q₈	—	2	2	no	—
518	C₂×Q₈	—	2	2	no	—
519	C₂×Q₈	—	2	1	no	—
520	C₂×Q₈	—	2	4	no	—
521	C₂×Q₈	—	2	2	no	—
522	C₂×Q₈	—	2	8	no	—
523	C₂×Q₈	—	8	2	no	—
524	C₂×Q₈	—	8	4	no	—
525	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
526	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
527	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
528	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
529	C₂²×C₄	—	2	2	no	—
530	C₂²×C₄	—	2	4	no	—
531	C₂²×C₄	—	2	2	no	—
532	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
533	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
534	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
535	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
536	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
537	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
538	M₁₆	—	2	4	no	—
539	C₂×C₈	—	2	4	no	—
540	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
541	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
542	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
543	C₄⋊C₄	—	2	2	no	—
544	C₄⋊C₄	—	2	1	no	—
545	C₄⋊C₄	—	2	2	no	—
546	C₄⋊C₄	—	2	2	no	—
547	C₄⋊C₄	—	2	4	no	—
548	C₄⋊C₄	—	2	2	no	—
549	C₄⋊C₄	—	2	2	no	—
550	C₄⋊C₄	—	2	2	no	—
551	C₄⋊C₄	—	2	4	no	—
552	C₄⋊C₄	—	2	2	no	—
553	C₄⋊C₄	—	2	1	no	—
554	C₄⋊C₄	—	2	1	no	—
555	C₄⋊C₄	—	2	4	no	—
556	C₄⋊C₄	—	2	4	no	—
557	C₄⋊C₄	—	2	1	no	—
558	C₄⋊C₄	—	2	4	no	—
559	C₄⋊C₄	—	2	2	no	—
560	C₄⋊C₄	—	2	4	no	—
561	C₄⋊C₄	—	2	2	no	—
562	C₄⋊C₄	—	2	2	no	—
563	C₄⋊C₄	—	2	2	no	—
564	C₄⋊C₄	—	2	4	no	—
565	C₄⋊C₄	—	2	4	no	—
566	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
567	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
568	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
569	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
570	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
571	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
572	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
573	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
574	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
575	C₂²⋊C₄	—	2	2	no	—
576	C₂²⋊C₄	—	2	4	no	—
577	C₂²⋊C₄	—	2	4	no	—
578	C₂²⋊C₄	—	2	4	no	—
579	C₂²⋊C₄	—	2	2	no	—
580	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
581	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
582	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
583	C₄²	—	2	4	no	—
584	C₄²	—	2	4	no	—
585	C₄²	—	2	4	no	—
586	C₄²	—	2	2	no	—
587	C₄²	—	2	2	no	—
588	C₄²	C₂×D₄×C₃₁	1	2	yes	—
589	C₄²	C₂×D₄×C₃₁	1	2	yes	—
590	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
591	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
592	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
593	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
594	C₄²	C₂⁴×C₃₁	1	2	yes	—
595	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
596	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
597	Q₈	—	2	2	no	—
598	Q₈	—	2	8	no	—
599	Q₈	—	4	4	no	—
600	Q₈	—	4	2	no	—
601	Q₈	—	4	4	no	—
602	Q₈	—	4	2	no	—
603	Q₈	—	4	2	no	—
604	Q₈	—	4	2	no	—
605	Q₈	—	8	4	no	—
606	Q₈	—	8	2	no	—
607	D₄	—	2	4	no	—
608	D₄	—	2	2	no	—
609	C₂×C₄	—	2	2	no	—
610	C₂×C₄	—	2	4	no	—
611	C₂×C₄	—	2	2	no	—
612	C₂×C₄	—	2	4	no	—
613	C₂×C₄	—	2	4	no	—
614	C₂×C₄	—	2	2	no	—
615	C₂×C₄	—	2	2	no	—
616	C₂×C₄	—	2	2	no	—
617	C₂×C₄	—	2	4	no	—
618	C₂×C₄	—	2	4	no	—
619	C₂×C₄	—	2	2	no	—
620	C₂×C₄	—	4	4	no	—
621	C₂×C₄	—	8	2	no	—
622	C₂×C₄	—	8	4	no	—
623	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
624	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
625	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
626	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
627	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
628	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
629	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
630	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
631	C₂×C₄	C₃₁×2₋¹⁺⁴	1	4	yes	—
632	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
633	C₈	—	2	2	no	—
634	C₈	—	2	2	no	—
635	C₈	—	2	4	no	—
636	C₂²	—	2	1	no	—
637	C₂²	—	2	1	no	—
638	C₂²	—	2	2	no	—
639	C₂²	—	2	2	no	—
640	C₂²	—	2	2	no	—
641	C₂²	C₃₁×#536	1	2	yes	—
642	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
643	C₄	—	2	2	no	—
644	C₄	—	2	2	no	—
645	C₄	—	2	1	no	—
646	C₄	—	2	2	no	—
647	C₄	—	2	1	no	—
648	C₄	—	4	4	no	—
649	C₄	—	4	4	no	—
650	C₄	—	4	2	no	—
651	C₄	—	4	2	no	—
652	C₄	—	4	2	no	—
653	C₄	—	4	2	no	—
654	C₄	C₃₁×#536	1	2	yes	—
655	C₄	C₃₁×#577	1	2	yes	—
656	C₄	C₃₁×#578	1	2	yes	—
657	C₄	C₂×C₃₁×2₋¹⁺⁴	1	2	yes	—
658	C₂	—	2	2	no	—
659	C₂	—	4	2	no	—
660	C₂	C₂×C₃₁×#536	1	1	yes	characteristic
661	C₂	C₂×C₃₁×#577	1	1	yes	characteristic
662	C₂	C₂×C₃₁×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₂×#41×C₃₁	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×#41×C₃₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₃₁×#3242	C₂	semidirect
4. C₃₁×#3231	C₂	semidirect
5. C₃₁×#3231	C₂	semidirect
6. C₃₁×#3230	C₂	semidirect
7. C₃₁×#3229	C₂	semidirect
8. C₃₁×#3165	C₂	semidirect
9. C₃₁×#3160	C₂	semidirect
10. C₃₁×#3139	C₂	semidirect
11. C₂×C₃₁×#487	C₂	semidirect
13. C₃₁×#3082	C₂	semidirect
263. #60878	C₃₁	direct (N × H)
307. #3242	C₂×C₃₁	semidirect
308. #3231	C₂×C₃₁	semidirect
309. #3231	C₂×C₃₁	semidirect
310. #3230	C₂×C₃₁	semidirect
311. #3229	C₂×C₃₁	semidirect
312. #3165	C₂×C₃₁	semidirect
313. #3160	C₂×C₃₁	semidirect
314. #3139	C₂×C₃₁	semidirect
315. C₂×#487	C₂×C₃₁	semidirect
317. #3082	C₂×C₃₁	semidirect
512. C₃₁	#60878	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×#3509	2
C₃₁×#3503	2
C₃₁×#3242	2
C₃₁×#3231	2
C₃₁×#3230	2
C₃₁×#3229	2
C₃₁×#3165	2
C₃₁×#3160	2
C₃₁×#3139	2
C₂×C₃₁×#487	2
C₃₁×#3097	2
C₃₁×#3082	2
C₃₁×#2922	2
#60878	31

Subgroups — C31×#60878 #21517637

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60878 #21517637