Subgroups — C₃₁×#60879 #21517638

Order 7936.

1654 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3510	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3504	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3244	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3243	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3233	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3232	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3166	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#3160	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#3140	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#488	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#3097	C₂	1	2	yes	maximal, complemented
12	C₃₁×#3096	C₂	1	2	yes	maximal
13	C₃₁×#3082	C₂	1	2	yes	maximal, complemented
14	C₃₁×#3081	C₂	1	2	yes	maximal, complemented
15	C₃₁×#2922	C₂	1	1	yes	maximal, characteristic, complemented
16	C₃₁×#559	—	2	2	no	—
17	C₃₁×#557	—	2	2	no	—
18	C₃₁×#556	—	2	2	no	—
19	C₃₁×#554	—	2	4	no	—
20	C₃₁×#554	—	2	2	no	—
21	C₃₁×#551	—	2	4	no	—
22	C₃₁×#544	C₂²	1	2	yes	—
23	C₃₁×#544	C₂²	1	4	yes	complemented
24	C₃₁×#542	C₂²	1	4	yes	—
25	C₃₁×#536	C₂²	1	1	yes	characteristic
26	C₃₁×#535	C₂²	1	1	yes	characteristic
27	C₃₁×#533	C₂²	1	1	yes	characteristic
28	C₃₁×#533	C₂²	1	1	yes	characteristic
29	C₃₁×#533	C₂²	1	2	yes	complemented
30	C₃₁×#529	C₂²	1	4	yes	—
31	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
32	C₂×C₃₁×#125	C₂²	1	2	yes	complemented
33	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
34	C₃₁×#520	C₂²	1	2	yes	—
35	C₃₁×#519	C₂²	1	1	yes	characteristic
36	C₃₁×#518	C₂²	1	1	yes	characteristic
37	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
38	C₃₁×#501	C₂²	1	4	yes	complemented
39	C₃₁×#500	C₂²	1	2	yes	complemented
40	C₃₁×#499	C₂²	1	2	yes	complemented
41	C₃₁×#499	C₂²	1	2	yes	complemented
42	C₃₁×#499	C₂²	1	4	yes	complemented
43	C₃₁×#498	C₂²	1	2	yes	complemented
44	C₃₁×#489	C₂²	1	4	yes	—
45	C₃₁×#488	C₂²	1	4	yes	—
46	C₃₁×#488	C₂²	1	2	yes	—
47	C₃₁×#488	C₂²	1	2	yes	—
48	C₃₁×#488	C₂²	1	2	yes	—
49	C₃₁×#488	C₂²	1	2	yes	—
50	C₃₁×#485	C₂²	1	4	yes	complemented
51	C₃₁×#485	C₂²	1	4	yes	complemented
52	C₃₁×#484	C₂²	1	4	yes	complemented
53	C₃₁×#483	C₂²	1	4	yes	complemented
54	C₃₁×#482	C₂²	1	4	yes	—
55	C₃₁×#482	C₂²	1	4	yes	—
56	C₃₁×#481	C₂²	1	4	yes	—
57	C₃₁×#480	C₂²	1	4	yes	—
58	C₃₁×#479	C₂²	1	4	yes	complemented
59	C₃₁×#479	C₂²	1	4	yes	complemented
60	C₃₁×#478	C₂²	1	4	yes	—
61	C₃₁×#478	C₂²	1	4	yes	—
62	C₃₁×#477	C₂²	1	4	yes	complemented
63	C₃₁×#476	C₂²	1	4	yes	—
64	C₃₁×#475	C₂²	1	4	yes	complemented
65	C₃₁×#474	C₂²	1	4	yes	—
66	C₃₁×#433	C₂²	1	2	yes	complemented
67	C₃₁×#433	C₂²	1	2	yes	complemented
68	C₃₁×#431	C₂²	1	4	yes	complemented
69	C₃₁×#428	C₂²	1	2	yes	—
70	C₃₁×#428	C₂²	1	2	yes	—
71	C₃₁×#427	C₂²	1	2	yes	—
72	C₃₁×#427	C₂²	1	2	yes	—
73	C₃₁×#424	C₂²	1	2	yes	—
74	C₃₁×#419	C₂²	1	2	yes	—
75	C₃₁×#418	C₂²	1	2	yes	—
76	C₂×C₃₁×#103	C₂²	1	2	yes	—
77	C₂×C₃₁×#102	C₂²	1	2	yes	—
78	C₃₁×#408	C₂²	1	2	yes	—
79	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
80	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
81	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
82	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
83	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
84	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
85	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
86	C₃₁×C₄⋊Q₈	—	2	1	no	—
87	C₃₁×C₄⋊Q₈	—	2	2	no	—
88	C₃₁×C₄⋊Q₈	—	2	4	no	—
89	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
90	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
91	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
92	C₃₁×#127	—	2	1	no	—
93	C₃₁×#126	—	2	4	no	—
94	C₃₁×#126	—	2	4	no	—
95	C₃₁×#125	—	2	2	no	—
96	C₃₁×#125	—	2	4	no	—
97	C₃₁×#125	—	2	2	no	—
98	C₃₁×#125	—	2	4	no	—
99	C₃₁×#125	C₂³	1	4	yes	—
100	C₃₁×#125	C₂³	1	2	yes	—
101	C₃₁×#125	C₂³	1	1	yes	characteristic
102	C₃₁×#125	C₂³	1	1	yes	characteristic
103	C₃₁×#125	C₂³	1	2	yes	—
104	C₃₁×#125	C₂³	1	2	yes	—
105	C₃₁×#124	—	2	1	no	—
106	C₃₁×#124	—	2	1	no	—
107	C₃₁×#124	—	2	4	no	—
108	C₃₁×#124	—	2	2	no	—
109	C₃₁×#124	C₂³	1	2	yes	—
110	C₃₁×#124	C₂³	1	2	yes	—
111	C₃₁×#123	—	2	2	no	—
112	C₃₁×#123	—	2	4	no	—
113	C₃₁×#123	—	2	4	no	—
114	C₃₁×C₂²⋊Q₈	—	2	2	no	—
115	C₃₁×C₂²⋊Q₈	—	2	2	no	—
116	C₃₁×C₂²⋊Q₈	—	2	4	no	—
117	C₃₁×C₂²⋊Q₈	—	2	4	no	—
118	C₃₁×C₂²⋊Q₈	—	2	4	no	—
119	C₃₁×C₂²⋊Q₈	—	2	4	no	—
120	C₃₁×C₂²⋊Q₈	—	2	4	no	—
121	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
122	C₃₁×#121	C₂³	1	4	yes	—
123	C₃₁×#120	—	2	2	no	—
124	C₄×Q₈×C₃₁	—	2	2	no	—
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	4	no	—
129	C₄×Q₈×C₃₁	D₄	1	2	yes	—
130	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
131	C₄×D₄×C₃₁	—	2	4	no	—
132	C₄×D₄×C₃₁	—	2	2	no	—
133	C₄×D₄×C₃₁	—	2	4	no	—
134	C₄×D₄×C₃₁	—	2	2	no	—
135	C₄×D₄×C₃₁	D₄	1	4	yes	—
136	C₄×D₄×C₃₁	D₄	1	2	yes	—
137	C₄×D₄×C₃₁	C₂³	1	4	yes	—
138	C₃₁×#117	—	2	2	no	—
139	C₃₁×#117	—	2	4	no	—
140	C₃₁×#117	C₂³	1	2	yes	—
141	C₃₁×#117	C₂³	1	1	yes	characteristic
142	C₃₁×#117	C₂³	1	1	yes	characteristic
143	C₃₁×#117	C₂³	1	4	yes	—
144	C₃₁×#117	C₂³	1	1	yes	characteristic
145	C₃₁×#117	C₂³	1	2	yes	—
146	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
147	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
148	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
149	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
150	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
151	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
152	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
153	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
154	C₃₁×#107	C₂³	1	2	yes	—
155	C₃₁×#107	C₂³	1	2	yes	—
156	C₃₁×#107	C₂³	1	2	yes	—
157	C₃₁×#107	C₂³	1	2	yes	—
158	C₃₁×#107	C₂³	1	4	yes	—
159	C₃₁×#107	C₂³	1	4	yes	—
160	C₃₁×#106	C₂³	1	2	yes	—
161	C₃₁×#106	C₂³	1	2	yes	—
162	C₃₁×#106	C₂³	1	2	yes	—
163	C₃₁×#106	C₂³	1	2	yes	—
164	C₃₁×#106	C₂³	1	4	yes	—
165	C₃₁×#106	C₂³	1	4	yes	—
166	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
167	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
168	C₃₁×#103	C₂³	1	4	yes	—
169	C₃₁×#103	C₂³	1	4	yes	—
170	C₃₁×#103	C₂³	1	4	yes	—
171	C₃₁×#103	C₂³	1	4	yes	—
172	C₃₁×#102	C₂³	1	4	yes	—
173	C₃₁×#102	C₂³	1	4	yes	—
174	C₃₁×#102	C₂³	1	4	yes	—
175	C₃₁×#102	C₂³	1	4	yes	—
176	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
177	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
178	C₃₁×#97	C₂³	1	2	yes	—
179	C₄×C₈×C₃₁	C₂³	1	2	yes	—
180	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
181	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
182	C₄×C₈×C₃₁	C₂³	1	2	yes	—
183	C₂⁴×C₃₁	—	2	1	no	—
184	#41×C₃₁	—	2	4	no	—
185	C₂×Q₈×C₃₁	—	2	1	no	—
186	C₂×Q₈×C₃₁	—	2	2	no	—
187	C₂×Q₈×C₃₁	—	2	4	no	—
188	C₂×Q₈×C₃₁	—	2	2	no	—
189	C₂×Q₈×C₃₁	—	2	4	no	—
190	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
191	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
192	C₂×D₄×C₃₁	—	2	4	no	—
193	C₂×D₄×C₃₁	—	2	1	no	—
194	C₂×D₄×C₃₁	—	2	4	no	—
195	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
196	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
197	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
198	C₂²×C₄×C₃₁	—	2	1	no	—
199	C₂²×C₄×C₃₁	—	2	4	no	—
200	C₂²×C₄×C₃₁	—	2	2	no	—
201	C₂²×C₄×C₃₁	—	2	4	no	—
202	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
203	C₂²×C₄×C₃₁	C₂×D₄	1	4	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	4	no	—
212	C₄⋊C₄×C₃₁	—	2	2	no	—
213	C₄⋊C₄×C₃₁	—	2	4	no	—
214	C₄⋊C₄×C₃₁	—	2	4	no	—
215	C₄⋊C₄×C₃₁	—	2	4	no	—
216	C₄⋊C₄×C₃₁	—	2	4	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	4	no	—
221	C₄⋊C₄×C₃₁	—	2	2	no	—
222	C₄⋊C₄×C₃₁	—	2	2	no	—
223	C₄⋊C₄×C₃₁	—	2	4	no	—
224	C₄⋊C₄×C₃₁	—	2	2	no	—
225	C₄⋊C₄×C₃₁	—	2	2	no	—
226	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
227	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
228	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
229	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
230	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
231	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
232	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
233	C₂²⋊C₄×C₃₁	—	2	1	no	—
234	C₂²⋊C₄×C₃₁	—	2	1	no	—
235	C₂²⋊C₄×C₃₁	—	2	4	no	—
236	C₂²⋊C₄×C₃₁	—	2	2	no	—
237	C₂²⋊C₄×C₃₁	—	2	4	no	—
238	C₂²⋊C₄×C₃₁	—	2	4	no	—
239	C₂²⋊C₄×C₃₁	—	2	4	no	—
240	C₂²⋊C₄×C₃₁	—	2	4	no	—
241	C₂²⋊C₄×C₃₁	—	2	4	no	—
242	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
243	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
244	C₄²×C₃₁	—	2	4	no	—
245	C₄²×C₃₁	—	2	1	no	—
246	C₄²×C₃₁	—	2	4	no	—
247	C₄²×C₃₁	—	2	2	no	—
248	C₄²×C₃₁	—	2	1	no	—
249	C₄²×C₃₁	C₂×D₄	1	4	yes	—
250	C₄²×C₃₁	C₂×D₄	1	2	yes	—
251	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
252	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
253	C₄²×C₃₁	C₂⁴	1	2	yes	—
254	#60879	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
255	C₂³×C₃₁	—	2	2	no	—
256	C₂³×C₃₁	—	4	2	no	—
257	C₂³×C₃₁	—	8	2	no	—
258	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
259	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
260	Q₈×C₃₁	—	2	2	no	—
261	Q₈×C₃₁	—	2	4	no	—
262	Q₈×C₃₁	—	4	2	no	—
263	Q₈×C₃₁	—	8	4	no	—
264	Q₈×C₃₁	—	8	2	no	—
265	Q₈×C₃₁	—	8	2	no	—
266	D₄×C₃₁	—	2	4	no	—
267	D₄×C₃₁	—	2	2	no	—
268	D₄×C₃₁	—	2	4	no	—
269	D₄×C₃₁	—	4	2	no	—
270	D₄×C₃₁	—	8	4	no	—
271	C₂×C₄×C₃₁	—	2	4	no	—
272	C₂×C₄×C₃₁	—	2	2	no	—
273	C₂×C₄×C₃₁	—	2	2	no	—
274	C₂×C₄×C₃₁	—	2	2	no	—
275	C₂×C₄×C₃₁	—	2	2	no	—
276	C₂×C₄×C₃₁	—	2	4	no	—
277	C₂×C₄×C₃₁	—	2	2	no	—
278	C₂×C₄×C₃₁	—	4	4	no	—
279	C₂×C₄×C₃₁	—	4	2	no	—
280	C₂×C₄×C₃₁	—	4	2	no	—
281	C₂×C₄×C₃₁	—	8	4	no	—
282	C₂×C₄×C₃₁	—	8	2	no	—
283	C₂×C₄×C₃₁	—	8	4	no	—
284	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
285	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
286	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
287	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
288	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
289	C₂×C₄×C₃₁	2₋¹⁺⁴	1	4	yes	—
290	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
291	C₈×C₃₁	—	2	2	no	—
292	C₈×C₃₁	—	2	2	no	—
293	C₈×C₃₁	—	2	4	no	—
294	#3510	C₂×C₃₁	1	1	yes	characteristic
295	#3504	C₂×C₃₁	1	1	yes	characteristic, complemented
296	#3244	C₂×C₃₁	1	4	yes	complemented
297	#3243	C₂×C₃₁	1	4	yes	complemented
298	#3233	C₂×C₃₁	1	4	yes	complemented
299	#3232	C₂×C₃₁	1	4	yes	complemented
300	#3166	C₂×C₃₁	1	1	yes	characteristic, complemented
301	#3160	C₂×C₃₁	1	1	yes	characteristic, complemented
302	#3140	C₂×C₃₁	1	1	yes	characteristic, complemented
303	C₂×#488	C₂×C₃₁	1	1	yes	characteristic, complemented
304	#3097	C₂×C₃₁	1	2	yes	complemented
305	#3096	C₂×C₃₁	1	2	yes	—
306	#3082	C₂×C₃₁	1	2	yes	complemented
307	#3081	C₂×C₃₁	1	2	yes	complemented
308	#2922	C₂×C₃₁	1	1	yes	characteristic, complemented
309	C₂²×C₃₁	—	2	2	no	—
310	C₂²×C₃₁	—	2	4	no	—
311	C₂²×C₃₁	—	2	2	no	—
312	C₂²×C₃₁	—	4	4	no	—
313	C₂²×C₃₁	—	4	2	no	—
314	C₂²×C₃₁	—	16	2	no	—
315	C₂²×C₃₁	#536	1	2	yes	—
316	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
317	C₄×C₃₁	—	2	2	no	—
318	C₄×C₃₁	—	2	4	no	—
319	C₄×C₃₁	—	2	2	no	—
320	C₄×C₃₁	—	2	2	no	—
321	C₄×C₃₁	—	4	2	no	—
322	C₄×C₃₁	—	4	4	no	—
323	C₄×C₃₁	—	4	2	no	—
324	C₄×C₃₁	—	4	2	no	—
325	C₄×C₃₁	—	4	2	no	—
326	C₄×C₃₁	—	4	2	no	—
327	C₄×C₃₁	#536	1	2	yes	—
328	C₄×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
329	#559	—	2	2	no	—
330	#557	—	2	2	no	—
331	#556	—	2	2	no	—
332	#554	—	2	4	no	—
333	#554	—	2	2	no	—
334	#551	—	2	4	no	—
335	#544	C₂²×C₃₁	1	2	yes	—
336	#544	C₂²×C₃₁	1	4	yes	complemented
337	#542	C₂²×C₃₁	1	4	yes	—
338	#536	C₂²×C₃₁	1	1	yes	characteristic
339	#535	C₂²×C₃₁	1	1	yes	characteristic
340	#533	C₂²×C₃₁	1	1	yes	characteristic
341	#533	C₂²×C₃₁	1	1	yes	characteristic
342	#533	C₂²×C₃₁	1	2	yes	complemented
343	#529	C₂²×C₃₁	1	4	yes	—
344	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
345	C₂×#125	C₂²×C₃₁	1	2	yes	complemented
346	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
347	#520	C₂²×C₃₁	1	2	yes	—
348	#519	C₂²×C₃₁	1	1	yes	characteristic
349	#518	C₂²×C₃₁	1	1	yes	characteristic
350	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
351	#501	C₂²×C₃₁	1	4	yes	complemented
352	#500	C₂²×C₃₁	1	2	yes	complemented
353	#499	C₂²×C₃₁	1	2	yes	complemented
354	#499	C₂²×C₃₁	1	2	yes	complemented
355	#499	C₂²×C₃₁	1	4	yes	complemented
356	#498	C₂²×C₃₁	1	2	yes	complemented
357	#489	C₂²×C₃₁	1	4	yes	—
358	#488	C₂²×C₃₁	1	4	yes	—
359	#488	C₂²×C₃₁	1	2	yes	—
360	#488	C₂²×C₃₁	1	2	yes	—
361	#488	C₂²×C₃₁	1	2	yes	—
362	#488	C₂²×C₃₁	1	2	yes	—
363	#485	C₂²×C₃₁	1	4	yes	complemented
364	#485	C₂²×C₃₁	1	4	yes	complemented
365	#484	C₂²×C₃₁	1	4	yes	complemented
366	#483	C₂²×C₃₁	1	4	yes	complemented
367	#482	C₂²×C₃₁	1	4	yes	—
368	#482	C₂²×C₃₁	1	4	yes	—
369	#481	C₂²×C₃₁	1	4	yes	—
370	#480	C₂²×C₃₁	1	4	yes	—
371	#479	C₂²×C₃₁	1	4	yes	complemented
372	#479	C₂²×C₃₁	1	4	yes	complemented
373	#478	C₂²×C₃₁	1	4	yes	—
374	#478	C₂²×C₃₁	1	4	yes	—
375	#477	C₂²×C₃₁	1	4	yes	complemented
376	#476	C₂²×C₃₁	1	4	yes	—
377	#475	C₂²×C₃₁	1	4	yes	complemented
378	#474	C₂²×C₃₁	1	4	yes	—
379	#433	C₂²×C₃₁	1	2	yes	complemented
380	#433	C₂²×C₃₁	1	2	yes	complemented
381	#431	C₂²×C₃₁	1	4	yes	complemented
382	#428	C₂²×C₃₁	1	2	yes	—
383	#428	C₂²×C₃₁	1	2	yes	—
384	#427	C₂²×C₃₁	1	2	yes	—
385	#427	C₂²×C₃₁	1	2	yes	—
386	#424	C₂²×C₃₁	1	2	yes	—
387	#419	C₂²×C₃₁	1	2	yes	—
388	#418	C₂²×C₃₁	1	2	yes	—
389	C₂×#103	C₂²×C₃₁	1	2	yes	—
390	C₂×#102	C₂²×C₃₁	1	2	yes	—
391	#408	C₂²×C₃₁	1	2	yes	—
392	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
393	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
394	C₂×C₃₁	—	2	2	no	—
395	C₂×C₃₁	—	4	2	no	—
396	C₂×C₃₁	—	8	2	no	—
397	C₂×C₃₁	C₂×#536	1	1	yes	characteristic
398	C₂×C₃₁	#3584	1	2	yes	—
399	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
400	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
401	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
402	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
403	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
404	C₄⋊Q₈	—	2	1	no	—
405	C₄⋊Q₈	—	2	2	no	—
406	C₄⋊Q₈	—	2	4	no	—
407	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
408	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
409	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
410	#127	—	2	1	no	—
411	#126	—	2	4	no	—
412	#126	—	2	4	no	—
413	#125	—	2	2	no	—
414	#125	—	2	4	no	—
415	#125	—	2	2	no	—
416	#125	—	2	4	no	—
417	#125	C₂³×C₃₁	1	2	yes	—
418	#125	C₂³×C₃₁	1	2	yes	—
419	#125	C₂³×C₃₁	1	1	yes	characteristic
420	#125	C₂³×C₃₁	1	4	yes	—
421	#125	C₂³×C₃₁	1	2	yes	—
422	#125	C₂³×C₃₁	1	1	yes	characteristic
423	#124	—	2	1	no	—
424	#124	—	2	1	no	—
425	#124	—	2	4	no	—
426	#124	—	2	2	no	—
427	#124	C₂³×C₃₁	1	2	yes	—
428	#124	C₂³×C₃₁	1	2	yes	—
429	#123	—	2	2	no	—
430	#123	—	2	4	no	—
431	#123	—	2	4	no	—
432	C₂²⋊Q₈	—	2	2	no	—
433	C₂²⋊Q₈	—	2	2	no	—
434	C₂²⋊Q₈	—	2	4	no	—
435	C₂²⋊Q₈	—	2	4	no	—
436	C₂²⋊Q₈	—	2	4	no	—
437	C₂²⋊Q₈	—	2	4	no	—
438	C₂²⋊Q₈	—	2	4	no	—
439	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
440	#121	C₂³×C₃₁	1	4	yes	—
441	#120	—	2	2	no	—
442	C₄×Q₈	—	2	2	no	—
443	C₄×Q₈	—	2	2	no	—
444	C₄×Q₈	—	2	2	no	—
445	C₄×Q₈	—	2	2	no	—
446	C₄×Q₈	—	2	4	no	—
447	C₄×Q₈	D₄×C₃₁	1	2	yes	—
448	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
449	C₄×D₄	—	2	4	no	—
450	C₄×D₄	—	2	2	no	—
451	C₄×D₄	—	2	4	no	—
452	C₄×D₄	—	2	2	no	—
453	C₄×D₄	D₄×C₃₁	1	4	yes	—
454	C₄×D₄	D₄×C₃₁	1	2	yes	—
455	C₄×D₄	C₂³×C₃₁	1	4	yes	—
456	#117	—	2	2	no	—
457	#117	—	2	4	no	—
458	#117	C₂³×C₃₁	1	1	yes	characteristic
459	#117	C₂³×C₃₁	1	1	yes	characteristic
460	#117	C₂³×C₃₁	1	2	yes	—
461	#117	C₂³×C₃₁	1	4	yes	—
462	#117	C₂³×C₃₁	1	1	yes	characteristic
463	#117	C₂³×C₃₁	1	2	yes	—
464	C₂×C₄⋊C₄	—	2	2	no	—
465	C₂×C₄⋊C₄	—	2	4	no	—
466	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
467	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
468	C₂×C₂²⋊C₄	—	2	2	no	—
469	C₂×C₂²⋊C₄	—	2	2	no	—
470	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
471	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
472	#107	C₂³×C₃₁	1	2	yes	—
473	#107	C₂³×C₃₁	1	2	yes	—
474	#107	C₂³×C₃₁	1	2	yes	—
475	#107	C₂³×C₃₁	1	2	yes	—
476	#107	C₂³×C₃₁	1	4	yes	—
477	#107	C₂³×C₃₁	1	4	yes	—
478	#106	C₂³×C₃₁	1	2	yes	—
479	#106	C₂³×C₃₁	1	2	yes	—
480	#106	C₂³×C₃₁	1	2	yes	—
481	#106	C₂³×C₃₁	1	2	yes	—
482	#106	C₂³×C₃₁	1	4	yes	—
483	#106	C₂³×C₃₁	1	4	yes	—
484	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
485	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
486	#103	C₂³×C₃₁	1	4	yes	—
487	#103	C₂³×C₃₁	1	4	yes	—
488	#103	C₂³×C₃₁	1	4	yes	—
489	#103	C₂³×C₃₁	1	4	yes	—
490	#102	C₂³×C₃₁	1	4	yes	—
491	#102	C₂³×C₃₁	1	4	yes	—
492	#102	C₂³×C₃₁	1	4	yes	—
493	#102	C₂³×C₃₁	1	4	yes	—
494	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
495	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
496	#97	C₂³×C₃₁	1	2	yes	—
497	C₄×C₈	C₂³×C₃₁	1	2	yes	—
498	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
499	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
500	C₄×C₈	C₂³×C₃₁	1	2	yes	—
501	C₃₁	#60879	1	1	yes	characteristic, direct product factor, Sylow
502	C₂⁴	—	2	1	no	—
503	#41	—	2	4	no	—
504	C₂×Q₈	—	2	1	no	—
505	C₂×Q₈	—	2	2	no	—
506	C₂×Q₈	—	2	4	no	—
507	C₂×Q₈	—	2	2	no	—
508	C₂×Q₈	—	2	4	no	—
509	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
510	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
511	C₂×D₄	—	2	4	no	—
512	C₂×D₄	—	2	1	no	—
513	C₂×D₄	—	2	4	no	—
514	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
515	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
516	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
517	C₂²×C₄	—	2	1	no	—
518	C₂²×C₄	—	2	4	no	—
519	C₂²×C₄	—	2	2	no	—
520	C₂²×C₄	—	2	4	no	—
521	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
522	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
523	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
524	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
525	M₁₆	—	2	4	no	—
526	C₂×C₈	—	2	4	no	—
527	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
528	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
529	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
530	C₄⋊C₄	—	2	4	no	—
531	C₄⋊C₄	—	2	2	no	—
532	C₄⋊C₄	—	2	4	no	—
533	C₄⋊C₄	—	2	4	no	—
534	C₄⋊C₄	—	2	4	no	—
535	C₄⋊C₄	—	2	4	no	—
536	C₄⋊C₄	—	2	2	no	—
537	C₄⋊C₄	—	2	2	no	—
538	C₄⋊C₄	—	2	4	no	—
539	C₄⋊C₄	—	2	4	no	—
540	C₄⋊C₄	—	2	2	no	—
541	C₄⋊C₄	—	2	2	no	—
542	C₄⋊C₄	—	2	4	no	—
543	C₄⋊C₄	—	2	2	no	—
544	C₄⋊C₄	—	2	2	no	—
545	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
546	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
547	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
548	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
549	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
550	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
551	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
552	C₂²⋊C₄	—	2	1	no	—
553	C₂²⋊C₄	—	2	1	no	—
554	C₂²⋊C₄	—	2	4	no	—
555	C₂²⋊C₄	—	2	2	no	—
556	C₂²⋊C₄	—	2	4	no	—
557	C₂²⋊C₄	—	2	4	no	—
558	C₂²⋊C₄	—	2	4	no	—
559	C₂²⋊C₄	—	2	4	no	—
560	C₂²⋊C₄	—	2	4	no	—
561	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
562	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
563	C₄²	—	2	4	no	—
564	C₄²	—	2	1	no	—
565	C₄²	—	2	4	no	—
566	C₄²	—	2	2	no	—
567	C₄²	—	2	1	no	—
568	C₄²	C₂×D₄×C₃₁	1	4	yes	—
569	C₄²	C₂×D₄×C₃₁	1	2	yes	—
570	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
571	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
572	C₄²	C₂⁴×C₃₁	1	2	yes	—
573	C₂³	—	2	2	no	—
574	C₂³	—	4	2	no	—
575	C₂³	—	8	2	no	—
576	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
577	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
578	Q₈	—	2	2	no	—
579	Q₈	—	2	4	no	—
580	Q₈	—	4	2	no	—
581	Q₈	—	8	4	no	—
582	Q₈	—	8	2	no	—
583	Q₈	—	8	2	no	—
584	D₄	—	2	4	no	—
585	D₄	—	2	2	no	—
586	D₄	—	2	4	no	—
587	D₄	—	4	2	no	—
588	D₄	—	8	4	no	—
589	C₂×C₄	—	2	4	no	—
590	C₂×C₄	—	2	2	no	—
591	C₂×C₄	—	2	2	no	—
592	C₂×C₄	—	2	2	no	—
593	C₂×C₄	—	2	2	no	—
594	C₂×C₄	—	2	4	no	—
595	C₂×C₄	—	2	2	no	—
596	C₂×C₄	—	4	4	no	—
597	C₂×C₄	—	4	2	no	—
598	C₂×C₄	—	4	2	no	—
599	C₂×C₄	—	8	4	no	—
600	C₂×C₄	—	8	2	no	—
601	C₂×C₄	—	8	4	no	—
602	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
603	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
604	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
605	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
606	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
607	C₂×C₄	C₃₁×2₋¹⁺⁴	1	4	yes	—
608	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
609	C₈	—	2	2	no	—
610	C₈	—	2	2	no	—
611	C₈	—	2	4	no	—
612	C₂²	—	2	2	no	—
613	C₂²	—	2	2	no	—
614	C₂²	—	2	4	no	—
615	C₂²	—	4	2	no	—
616	C₂²	—	4	4	no	—
617	C₂²	—	16	2	no	—
618	C₂²	C₃₁×#536	1	2	yes	—
619	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
620	C₄	—	2	2	no	—
621	C₄	—	2	2	no	—
622	C₄	—	2	4	no	—
623	C₄	—	2	2	no	—
624	C₄	—	4	2	no	—
625	C₄	—	4	2	no	—
626	C₄	—	4	4	no	—
627	C₄	—	4	2	no	—
628	C₄	—	4	2	no	—
629	C₄	—	4	2	no	—
630	C₄	C₃₁×#536	1	2	yes	—
631	C₄	C₂×C₃₁×2₋¹⁺⁴	1	2	yes	—
632	C₂	—	2	2	no	—
633	C₂	—	4	2	no	—
634	C₂	—	8	2	no	—
635	C₂	C₂×C₃₁×#536	1	1	yes	characteristic
636	C₂	C₃₁×#3584	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₃₁×#535	C₂²
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
2. C₃₁×#3504	C₂	semidirect
3. C₃₁×#3244	C₂	semidirect
4. C₃₁×#3243	C₂	semidirect
5. C₃₁×#3233	C₂	semidirect
6. C₃₁×#3232	C₂	semidirect
7. C₃₁×#3166	C₂	semidirect
8. C₃₁×#3160	C₂	semidirect
9. C₃₁×#3140	C₂	semidirect
10. C₂×C₃₁×#488	C₂	semidirect
11. C₃₁×#3097	C₂	semidirect
13. C₃₁×#3082	C₂	semidirect
14. C₃₁×#3081	C₂	semidirect
15. C₃₁×#2922	C₂	semidirect
23. C₃₁×#544	C₂²	semidirect
29. C₃₁×#533	C₂²	semidirect
32. C₂×C₃₁×#125	C₂²	semidirect
38. C₃₁×#501	C₂²	semidirect
39. C₃₁×#500	C₂²	semidirect
40. C₃₁×#499	C₂²	semidirect
41. C₃₁×#499	C₂²	semidirect
42. C₃₁×#499	C₂²	semidirect
43. C₃₁×#498	C₂²	semidirect
50. C₃₁×#485	C₂²	semidirect
51. C₃₁×#485	C₂²	semidirect
52. C₃₁×#484	C₂²	semidirect
53. C₃₁×#483	C₂²	semidirect
58. C₃₁×#479	C₂²	semidirect
59. C₃₁×#479	C₂²	semidirect
62. C₃₁×#477	C₂²	semidirect
64. C₃₁×#475	C₂²	semidirect
66. C₃₁×#433	C₂²	semidirect
67. C₃₁×#433	C₂²	semidirect
68. C₃₁×#431	C₂²	semidirect
254. #60879	C₃₁	direct (N × H)
295. #3504	C₂×C₃₁	semidirect
296. #3244	C₂×C₃₁	semidirect
297. #3243	C₂×C₃₁	semidirect
298. #3233	C₂×C₃₁	semidirect
299. #3232	C₂×C₃₁	semidirect
300. #3166	C₂×C₃₁	semidirect
301. #3160	C₂×C₃₁	semidirect
302. #3140	C₂×C₃₁	semidirect
303. C₂×#488	C₂×C₃₁	semidirect
304. #3097	C₂×C₃₁	semidirect
306. #3082	C₂×C₃₁	semidirect
307. #3081	C₂×C₃₁	semidirect
308. #2922	C₂×C₃₁	semidirect
336. #544	C₂²×C₃₁	semidirect
342. #533	C₂²×C₃₁	semidirect
345. C₂×#125	C₂²×C₃₁	semidirect
351. #501	C₂²×C₃₁	semidirect
352. #500	C₂²×C₃₁	semidirect
353. #499	C₂²×C₃₁	semidirect
354. #499	C₂²×C₃₁	semidirect
355. #499	C₂²×C₃₁	semidirect
356. #498	C₂²×C₃₁	semidirect
363. #485	C₂²×C₃₁	semidirect
364. #485	C₂²×C₃₁	semidirect
365. #484	C₂²×C₃₁	semidirect
366. #483	C₂²×C₃₁	semidirect
371. #479	C₂²×C₃₁	semidirect
372. #479	C₂²×C₃₁	semidirect
375. #477	C₂²×C₃₁	semidirect
377. #475	C₂²×C₃₁	semidirect
379. #433	C₂²×C₃₁	semidirect
380. #433	C₂²×C₃₁	semidirect
381. #431	C₂²×C₃₁	semidirect
501. C₃₁	#60879	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×#3510	2
C₃₁×#3504	2
C₃₁×#3244	2
C₃₁×#3243	2
C₃₁×#3233	2
C₃₁×#3232	2
C₃₁×#3166	2
C₃₁×#3160	2
C₃₁×#3140	2
C₂×C₃₁×#488	2
C₃₁×#3097	2
C₃₁×#3096	2
C₃₁×#3082	2
C₃₁×#3081	2
C₃₁×#2922	2
#60879	31

Subgroups — C31×#60879 #21517638

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60879 #21517638