Subgroups — C₃₁×#61885 #21518644

Order 7936.

1814 conjugacy classes of subgroups.

The subgroup lattice has 644 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	C₂×C₃₁×#559	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3396	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3394	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3365	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3361	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3211	C₂	1	4	yes	maximal, complemented
8	C₃₁×#3208	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#3205	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#3105	C₂	1	4	yes	maximal, complemented
11	C₃₁×#3104	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#3103	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#2982	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#559	—	2	2	no	—
15	C₃₁×#559	—	2	2	no	—
16	C₃₁×#559	C₂²	1	4	yes	—
17	C₃₁×#559	C₂²	1	4	yes	—
18	C₃₁×#559	C₂²	1	2	yes	complemented
19	C₃₁×#557	C₂²	1	2	yes	complemented
20	C₃₁×#556	—	2	2	no	—
21	C₃₁×#556	C₂²	1	2	yes	complemented
22	C₃₁×#554	C₂²	1	2	yes	complemented
23	C₃₁×#552	—	2	2	no	—
24	C₃₁×#539	—	2	2	no	—
25	C₃₁×#536	—	2	1	no	—
26	C₃₁×#533	C₂²	1	1	yes	characteristic
27	C₃₁×#532	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
29	C₃₁×#529	C₂²	1	4	yes	—
30	C₂×C₃₁×#124	—	2	1	no	—
31	C₂×C₃₁×#124	C₂²	1	2	yes	—
32	C₂×C₃₁×#123	—	2	2	no	—
33	C₃₁×#520	—	2	1	no	—
34	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
35	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
36	C₂×C₄×D₄×C₃₁	—	2	1	no	—
37	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
38	C₃₁×#495	C₂²	1	8	yes	complemented
39	C₃₁×#493	C₂²	1	2	yes	—
40	C₃₁×#492	C₂²	1	2	yes	—
41	C₃₁×#488	C₂²	1	8	yes	complemented
42	C₃₁×#486	C₂²	1	2	yes	—
43	C₃₁×#486	C₂²	1	2	yes	—
44	C₃₁×#484	C₂²	1	2	yes	—
45	C₃₁×#481	C₂²	1	2	yes	—
46	C₃₁×#478	C₂²	1	8	yes	complemented
47	C₃₁×#477	C₂²	1	2	yes	complemented
48	C₃₁×#475	C₂²	1	2	yes	complemented
49	C₃₁×#464	C₂²	1	8	yes	complemented
50	C₃₁×#463	C₂²	1	8	yes	complemented
51	C₃₁×#463	C₂²	1	8	yes	complemented
52	C₃₁×#462	C₂²	1	2	yes	complemented
53	C₃₁×#460	C₂²	1	4	yes	—
54	C₃₁×#460	C₂²	1	2	yes	complemented
55	C₃₁×#459	C₂²	1	4	yes	—
56	C₃₁×#451	C₂²	1	2	yes	—
57	C₃₁×#450	C₂²	1	4	yes	—
58	C₃₁×#448	C₂²	1	2	yes	—
59	C₃₁×#447	C₂²	1	4	yes	—
60	C₈×Q₈×C₃₁	C₂²	1	4	yes	complemented
61	C₈×Q₈×C₃₁	C₂²	1	4	yes	complemented
62	C₄×Q₁₆×C₃₁	C₂²	1	4	yes	complemented
63	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	complemented
64	C₄×SD₁₆×C₃₁	C₂²	1	8	yes	complemented
65	C₄×D₈×C₃₁	C₂²	1	8	yes	complemented
66	C₃₁×#431	C₂²	1	1	yes	characteristic
67	C₃₁×#431	C₂²	1	1	yes	characteristic
68	C₃₁×#431	C₂²	1	4	yes	—
69	C₂×#41×C₃₁	—	2	1	no	—
70	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
71	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
72	C₂²×D₄×C₃₁	—	2	1	no	—
73	C₂²×D₄×C₃₁	—	2	1	no	—
74	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
75	C₂³×C₄×C₃₁	—	2	2	no	—
76	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
77	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
78	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
79	C₂×D₈×C₃₁	C₂³	1	8	yes	—
80	C₃₁×C₄⋊Q₈	—	2	2	no	—
81	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
82	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
83	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
84	C₃₁×#127	—	2	1	no	—
85	C₃₁×#127	—	2	1	no	—
86	C₃₁×#127	C₂³	1	2	yes	—
87	C₃₁×#127	C₂³	1	2	yes	—
88	C₃₁×#127	C₂³	1	1	yes	characteristic
89	C₃₁×#126	—	2	4	no	—
90	C₃₁×#126	—	2	4	no	—
91	C₃₁×#125	—	2	4	no	—
92	C₃₁×#125	—	2	1	no	—
93	C₃₁×#125	C₂³	1	4	yes	—
94	C₃₁×#124	—	2	4	no	—
95	C₃₁×#124	—	2	4	no	—
96	C₃₁×#124	—	2	2	no	—
97	C₃₁×#124	—	2	4	no	—
98	C₃₁×#124	—	2	1	no	—
99	C₃₁×#124	C₂³	1	4	yes	—
100	C₃₁×#124	C₂³	1	8	yes	—
101	C₃₁×#123	—	2	4	no	—
102	C₃₁×#123	—	2	8	no	—
103	C₃₁×#123	—	2	8	no	—
104	C₃₁×#123	—	2	4	no	—
105	C₃₁×C₂²⋊Q₈	—	2	2	no	—
106	C₃₁×C₂²⋊Q₈	—	2	2	no	—
107	C₃₁×C₂²⋊Q₈	—	2	1	no	—
108	C₃₁×C₂²⋊Q₈	—	2	4	no	—
109	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
110	C₃₁×#121	—	2	1	no	—
111	C₃₁×#121	C₂³	1	2	yes	—
112	C₄×Q₈×C₃₁	—	2	2	no	—
113	C₄×Q₈×C₃₁	—	2	1	no	—
114	C₄×Q₈×C₃₁	—	2	1	no	—
115	C₄×Q₈×C₃₁	—	2	1	no	—
116	C₄×Q₈×C₃₁	D₄	1	2	yes	—
117	C₄×Q₈×C₃₁	D₄	1	2	yes	—
118	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
119	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
120	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
121	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
122	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
123	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
124	C₄×D₄×C₃₁	—	2	4	no	—
125	C₄×D₄×C₃₁	—	2	4	no	—
126	C₄×D₄×C₃₁	—	2	4	no	—
127	C₄×D₄×C₃₁	—	2	2	no	—
128	C₄×D₄×C₃₁	—	2	4	no	—
129	C₄×D₄×C₃₁	—	2	1	no	—
130	C₄×D₄×C₃₁	C₂³	1	4	yes	—
131	C₄×D₄×C₃₁	C₂³	1	8	yes	—
132	C₄×D₄×C₃₁	C₂³	1	2	yes	—
133	C₃₁×#117	—	2	1	no	—
134	C₃₁×#117	—	2	2	no	—
135	C₃₁×#117	—	2	4	no	—
136	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
137	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
138	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
139	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
140	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
141	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
142	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
143	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
144	C₂×C₄²×C₃₁	—	2	1	no	—
145	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
146	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
147	C₃₁×#107	C₂³	1	4	yes	—
148	C₃₁×#106	C₂³	1	4	yes	—
149	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
150	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
151	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
152	C₃₁×#103	C₂³	1	4	yes	—
153	C₃₁×#103	C₂³	1	4	yes	—
154	C₃₁×#103	C₂³	1	4	yes	—
155	C₃₁×#102	C₂³	1	4	yes	—
156	C₃₁×#102	C₂³	1	8	yes	—
157	C₃₁×#102	C₂³	1	8	yes	—
158	C₃₁×C₂²⋊C₈	C₂³	1	2	yes	—
159	C₃₁×C₂²⋊C₈	C₂³	1	2	yes	—
160	C₄×C₈×C₃₁	C₂³	1	8	yes	—
161	C₄×C₈×C₃₁	C₂³	1	2	yes	—
162	C₄×C₈×C₃₁	C₂³	1	2	yes	—
163	C₂⁴×C₃₁	—	2	2	no	—
164	#41×C₃₁	—	4	4	no	—
165	C₂×Q₈×C₃₁	—	2	4	no	—
166	C₂×Q₈×C₃₁	—	2	1	no	—
167	C₂×Q₈×C₃₁	—	2	1	no	—
168	C₂×Q₈×C₃₁	—	2	1	no	—
169	C₂×Q₈×C₃₁	—	4	2	no	—
170	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
171	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
172	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
173	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
174	C₂×D₄×C₃₁	—	2	2	no	—
175	C₂×D₄×C₃₁	—	2	1	no	—
176	C₂×D₄×C₃₁	—	2	2	no	—
177	C₂×D₄×C₃₁	—	2	2	no	—
178	C₂×D₄×C₃₁	—	2	2	no	—
179	C₂×D₄×C₃₁	—	2	1	no	—
180	C₂×D₄×C₃₁	—	2	2	no	—
181	C₂×D₄×C₃₁	—	2	2	no	—
182	C₂×D₄×C₃₁	—	4	4	no	—
183	C₂×D₄×C₃₁	—	4	4	no	—
184	C₂×D₄×C₃₁	—	8	2	no	—
185	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
186	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
187	C₂²×C₄×C₃₁	—	2	1	no	—
188	C₂²×C₄×C₃₁	—	2	1	no	—
189	C₂²×C₄×C₃₁	—	2	2	no	—
190	C₂²×C₄×C₃₁	—	2	4	no	—
191	C₂²×C₄×C₃₁	—	2	8	no	—
192	C₂²×C₄×C₃₁	—	8	4	no	—
193	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
194	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
195	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
196	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
197	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
198	Q₁₆×C₃₁	—	4	4	no	—
199	SD₁₆×C₃₁	—	4	4	no	—
200	SD₁₆×C₃₁	—	4	8	no	—
201	D₈×C₃₁	—	4	8	no	—
202	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
203	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
204	C₄⋊C₄×C₃₁	—	2	1	no	—
205	C₄⋊C₄×C₃₁	—	2	4	no	—
206	C₄⋊C₄×C₃₁	—	2	2	no	—
207	C₄⋊C₄×C₃₁	—	2	2	no	—
208	C₄⋊C₄×C₃₁	—	2	1	no	—
209	C₄⋊C₄×C₃₁	—	2	1	no	—
210	C₄⋊C₄×C₃₁	—	2	2	no	—
211	C₄⋊C₄×C₃₁	—	2	4	no	—
212	C₄⋊C₄×C₃₁	—	2	1	no	—
213	C₄⋊C₄×C₃₁	—	2	1	no	—
214	C₄⋊C₄×C₃₁	—	2	2	no	—
215	C₄⋊C₄×C₃₁	—	2	4	no	—
216	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
217	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
218	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
219	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
220	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
221	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
222	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
223	C₂²⋊C₄×C₃₁	—	2	2	no	—
224	C₂²⋊C₄×C₃₁	—	2	8	no	—
225	C₂²⋊C₄×C₃₁	—	2	8	no	—
226	C₂²⋊C₄×C₃₁	—	2	8	no	—
227	C₂²⋊C₄×C₃₁	—	2	2	no	—
228	C₂²⋊C₄×C₃₁	—	2	4	no	—
229	C₂²⋊C₄×C₃₁	—	2	4	no	—
230	C₂²⋊C₄×C₃₁	—	2	2	no	—
231	C₂²⋊C₄×C₃₁	—	2	4	no	—
232	C₄²×C₃₁	—	2	2	no	—
233	C₄²×C₃₁	—	2	2	no	—
234	C₄²×C₃₁	—	2	4	no	—
235	C₄²×C₃₁	—	2	2	no	—
236	C₄²×C₃₁	C₂×D₄	1	4	yes	—
237	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
238	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
239	C₄²×C₃₁	C₂⁴	1	4	yes	—
240	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
241	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
242	#61885	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
243	C₂³×C₃₁	—	2	2	no	—
244	C₂³×C₃₁	—	2	4	no	—
245	C₂³×C₃₁	—	4	4	no	—
246	C₂³×C₃₁	—	4	4	no	—
247	C₂³×C₃₁	—	4	4	no	—
248	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
249	Q₈×C₃₁	—	2	4	no	—
250	Q₈×C₃₁	—	2	4	no	—
251	Q₈×C₃₁	—	2	4	no	—
252	Q₈×C₃₁	—	2	2	no	—
253	Q₈×C₃₁	—	2	2	no	—
254	Q₈×C₃₁	—	4	2	no	—
255	Q₈×C₃₁	—	4	2	no	—
256	Q₈×C₃₁	—	8	1	no	—
257	D₄×C₃₁	—	2	4	no	—
258	D₄×C₃₁	—	2	8	no	—
259	D₄×C₃₁	—	4	4	no	—
260	D₄×C₃₁	—	4	4	no	—
261	D₄×C₃₁	—	4	4	no	—
262	D₄×C₃₁	—	4	4	no	—
263	D₄×C₃₁	—	4	2	no	—
264	D₄×C₃₁	—	8	1	no	—
265	D₄×C₃₁	—	8	2	no	—
266	D₄×C₃₁	—	8	2	no	—
267	C₂×C₄×C₃₁	—	2	4	no	—
268	C₂×C₄×C₃₁	—	2	2	no	—
269	C₂×C₄×C₃₁	—	2	2	no	—
270	C₂×C₄×C₃₁	—	2	2	no	—
271	C₂×C₄×C₃₁	—	2	2	no	—
272	C₂×C₄×C₃₁	—	2	2	no	—
273	C₂×C₄×C₃₁	—	2	2	no	—
274	C₂×C₄×C₃₁	—	4	2	no	—
275	C₂×C₄×C₃₁	—	4	1	no	—
276	C₂×C₄×C₃₁	—	4	2	no	—
277	C₂×C₄×C₃₁	—	4	2	no	—
278	C₂×C₄×C₃₁	—	4	4	no	—
279	C₂×C₄×C₃₁	—	8	4	no	—
280	C₂×C₄×C₃₁	—	8	8	no	—
281	C₂×C₄×C₃₁	#135	1	4	yes	—
282	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
283	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
284	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
285	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
286	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
287	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
288	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
289	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
290	C₈×C₃₁	—	2	4	no	—
291	C₈×C₃₁	—	2	4	no	—
292	#3507	C₂×C₃₁	1	1	yes	characteristic, complemented
293	C₂×#559	C₂×C₃₁	1	1	yes	characteristic, complemented
294	#3396	C₂×C₃₁	1	4	yes	complemented
295	#3394	C₂×C₃₁	1	4	yes	complemented
296	#3365	C₂×C₃₁	1	4	yes	complemented
297	#3361	C₂×C₃₁	1	4	yes	complemented
298	#3211	C₂×C₃₁	1	4	yes	complemented
299	#3208	C₂×C₃₁	1	1	yes	characteristic, complemented
300	#3205	C₂×C₃₁	1	1	yes	characteristic
301	#3105	C₂×C₃₁	1	4	yes	complemented
302	#3104	C₂×C₃₁	1	1	yes	characteristic, complemented
303	#3103	C₂×C₃₁	1	1	yes	characteristic, complemented
304	#2982	C₂×C₃₁	1	1	yes	characteristic, complemented
305	C₂²×C₃₁	—	2	1	no	—
306	C₂²×C₃₁	—	2	1	no	—
307	C₂²×C₃₁	—	4	4	no	—
308	C₂²×C₃₁	—	4	2	no	—
309	C₂²×C₃₁	—	4	4	no	—
310	C₂²×C₃₁	—	4	2	no	—
311	C₂²×C₃₁	—	4	2	no	—
312	C₂²×C₃₁	—	4	4	no	—
313	C₂²×C₃₁	—	8	8	no	—
314	C₂²×C₃₁	C₂×#135	1	2	yes	—
315	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
316	C₄×C₃₁	—	2	4	no	—
317	C₄×C₃₁	—	2	1	no	—
318	C₄×C₃₁	—	2	1	no	—
319	C₄×C₃₁	—	2	1	no	—
320	C₄×C₃₁	—	2	4	no	—
321	C₄×C₃₁	—	4	2	no	—
322	C₄×C₃₁	—	4	2	no	—
323	C₄×C₃₁	—	4	2	no	—
324	C₄×C₃₁	—	4	2	no	—
325	C₄×C₃₁	—	4	2	no	—
326	C₄×C₃₁	C₂×#135	1	2	yes	—
327	C₄×C₃₁	C₂×#135	1	2	yes	—
328	C₄×C₃₁	#585	1	2	yes	—
329	#559	—	2	2	no	—
330	#559	—	2	2	no	—
331	#559	C₂²×C₃₁	1	4	yes	—
332	#559	C₂²×C₃₁	1	4	yes	—
333	#559	C₂²×C₃₁	1	2	yes	complemented
334	#557	C₂²×C₃₁	1	2	yes	complemented
335	#556	—	2	2	no	—
336	#556	C₂²×C₃₁	1	2	yes	complemented
337	#554	C₂²×C₃₁	1	2	yes	complemented
338	#552	—	2	2	no	—
339	#539	—	2	2	no	—
340	#536	—	2	1	no	—
341	#533	C₂²×C₃₁	1	1	yes	characteristic
342	#532	C₂²×C₃₁	1	1	yes	characteristic
343	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
344	#529	C₂²×C₃₁	1	4	yes	—
345	C₂×#124	—	2	1	no	—
346	C₂×#124	C₂²×C₃₁	1	2	yes	—
347	C₂×#123	—	2	2	no	—
348	#520	—	2	1	no	—
349	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
350	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
351	C₂×C₄×D₄	—	2	1	no	—
352	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
353	#495	C₂²×C₃₁	1	8	yes	complemented
354	#493	C₂²×C₃₁	1	2	yes	—
355	#492	C₂²×C₃₁	1	2	yes	—
356	#488	C₂²×C₃₁	1	8	yes	complemented
357	#486	C₂²×C₃₁	1	2	yes	—
358	#486	C₂²×C₃₁	1	2	yes	—
359	#484	C₂²×C₃₁	1	2	yes	—
360	#481	C₂²×C₃₁	1	2	yes	—
361	#478	C₂²×C₃₁	1	8	yes	complemented
362	#477	C₂²×C₃₁	1	2	yes	complemented
363	#475	C₂²×C₃₁	1	2	yes	complemented
364	#464	C₂²×C₃₁	1	8	yes	complemented
365	#463	C₂²×C₃₁	1	8	yes	complemented
366	#463	C₂²×C₃₁	1	8	yes	complemented
367	#462	C₂²×C₃₁	1	2	yes	complemented
368	#460	C₂²×C₃₁	1	4	yes	—
369	#460	C₂²×C₃₁	1	2	yes	complemented
370	#459	C₂²×C₃₁	1	4	yes	—
371	#451	C₂²×C₃₁	1	2	yes	—
372	#450	C₂²×C₃₁	1	4	yes	—
373	#448	C₂²×C₃₁	1	2	yes	—
374	#447	C₂²×C₃₁	1	4	yes	—
375	C₈×Q₈	C₂²×C₃₁	1	4	yes	complemented
376	C₈×Q₈	C₂²×C₃₁	1	4	yes	complemented
377	C₄×Q₁₆	C₂²×C₃₁	1	4	yes	complemented
378	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	complemented
379	C₄×SD₁₆	C₂²×C₃₁	1	8	yes	complemented
380	C₄×D₈	C₂²×C₃₁	1	8	yes	complemented
381	#431	C₂²×C₃₁	1	1	yes	characteristic
382	#431	C₂²×C₃₁	1	1	yes	characteristic
383	#431	C₂²×C₃₁	1	4	yes	—
384	C₂×C₃₁	—	2	2	no	—
385	C₂×C₃₁	—	8	2	no	—
386	C₂×C₃₁	—	8	4	no	—
387	C₂×C₃₁	#3487	1	1	yes	characteristic
388	C₂×C₃₁	C₂²×#135	1	1	yes	characteristic
389	C₂×C₃₁	#3585	1	1	yes	characteristic
390	C₂×#41	—	2	1	no	—
391	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
392	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
393	C₂²×D₄	—	2	1	no	—
394	C₂²×D₄	—	2	1	no	—
395	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
396	C₂³×C₄	—	2	2	no	—
397	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
398	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
399	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
400	C₂×D₈	C₂³×C₃₁	1	8	yes	—
401	C₄⋊Q₈	—	2	2	no	—
402	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
403	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
404	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
405	#127	—	2	1	no	—
406	#127	—	2	1	no	—
407	#127	C₂³×C₃₁	1	2	yes	—
408	#127	C₂³×C₃₁	1	2	yes	—
409	#127	C₂³×C₃₁	1	1	yes	characteristic
410	#126	—	2	4	no	—
411	#126	—	2	4	no	—
412	#125	—	2	4	no	—
413	#125	—	2	1	no	—
414	#125	C₂³×C₃₁	1	4	yes	—
415	#124	—	2	4	no	—
416	#124	—	2	4	no	—
417	#124	—	2	2	no	—
418	#124	—	2	4	no	—
419	#124	—	2	1	no	—
420	#124	C₂³×C₃₁	1	8	yes	—
421	#124	C₂³×C₃₁	1	4	yes	—
422	#123	—	2	4	no	—
423	#123	—	2	8	no	—
424	#123	—	2	8	no	—
425	#123	—	2	4	no	—
426	C₂²⋊Q₈	—	2	2	no	—
427	C₂²⋊Q₈	—	2	2	no	—
428	C₂²⋊Q₈	—	2	1	no	—
429	C₂²⋊Q₈	—	2	4	no	—
430	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
431	#121	—	2	1	no	—
432	#121	C₂³×C₃₁	1	2	yes	—
433	C₄×Q₈	—	2	2	no	—
434	C₄×Q₈	—	2	1	no	—
435	C₄×Q₈	—	2	1	no	—
436	C₄×Q₈	—	2	1	no	—
437	C₄×Q₈	D₄×C₃₁	1	2	yes	—
438	C₄×Q₈	D₄×C₃₁	1	2	yes	—
439	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
440	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
441	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
442	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
443	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
444	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
445	C₄×D₄	—	2	4	no	—
446	C₄×D₄	—	2	4	no	—
447	C₄×D₄	—	2	4	no	—
448	C₄×D₄	—	2	2	no	—
449	C₄×D₄	—	2	4	no	—
450	C₄×D₄	—	2	1	no	—
451	C₄×D₄	C₂³×C₃₁	1	8	yes	—
452	C₄×D₄	C₂³×C₃₁	1	4	yes	—
453	C₄×D₄	C₂³×C₃₁	1	2	yes	—
454	#117	—	2	1	no	—
455	#117	—	2	2	no	—
456	#117	—	2	4	no	—
457	C₂×C₄⋊C₄	—	2	1	no	—
458	C₂×C₄⋊C₄	—	2	2	no	—
459	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
460	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
461	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
462	C₂×C₂²⋊C₄	—	2	2	no	—
463	C₂×C₂²⋊C₄	—	2	2	no	—
464	C₂×C₂²⋊C₄	—	2	2	no	—
465	C₂×C₄²	—	2	1	no	—
466	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
467	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
468	#107	C₂³×C₃₁	1	4	yes	—
469	#106	C₂³×C₃₁	1	4	yes	—
470	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
471	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
472	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
473	#103	C₂³×C₃₁	1	4	yes	—
474	#103	C₂³×C₃₁	1	4	yes	—
475	#103	C₂³×C₃₁	1	4	yes	—
476	#102	C₂³×C₃₁	1	4	yes	—
477	#102	C₂³×C₃₁	1	8	yes	—
478	#102	C₂³×C₃₁	1	8	yes	—
479	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
480	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
481	C₄×C₈	C₂³×C₃₁	1	8	yes	—
482	C₄×C₈	C₂³×C₃₁	1	2	yes	—
483	C₄×C₈	C₂³×C₃₁	1	2	yes	—
484	C₃₁	#61885	1	1	yes	characteristic, direct product factor, Sylow
485	C₂⁴	—	2	2	no	—
486	#41	—	4	4	no	—
487	C₂×Q₈	—	2	4	no	—
488	C₂×Q₈	—	2	1	no	—
489	C₂×Q₈	—	2	1	no	—
490	C₂×Q₈	—	2	1	no	—
491	C₂×Q₈	—	4	2	no	—
492	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
493	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
494	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
495	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
496	C₂×D₄	—	2	2	no	—
497	C₂×D₄	—	2	1	no	—
498	C₂×D₄	—	2	2	no	—
499	C₂×D₄	—	2	2	no	—
500	C₂×D₄	—	2	2	no	—
501	C₂×D₄	—	2	1	no	—
502	C₂×D₄	—	2	2	no	—
503	C₂×D₄	—	2	2	no	—
504	C₂×D₄	—	4	4	no	—
505	C₂×D₄	—	4	4	no	—
506	C₂×D₄	—	8	2	no	—
507	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
508	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
509	C₂²×C₄	—	2	1	no	—
510	C₂²×C₄	—	2	1	no	—
511	C₂²×C₄	—	2	2	no	—
512	C₂²×C₄	—	2	4	no	—
513	C₂²×C₄	—	2	8	no	—
514	C₂²×C₄	—	8	4	no	—
515	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
516	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
517	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
518	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
519	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
520	Q₁₆	—	4	4	no	—
521	SD₁₆	—	4	4	no	—
522	SD₁₆	—	4	8	no	—
523	D₈	—	4	8	no	—
524	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
525	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
526	C₄⋊C₄	—	2	1	no	—
527	C₄⋊C₄	—	2	4	no	—
528	C₄⋊C₄	—	2	2	no	—
529	C₄⋊C₄	—	2	2	no	—
530	C₄⋊C₄	—	2	1	no	—
531	C₄⋊C₄	—	2	1	no	—
532	C₄⋊C₄	—	2	2	no	—
533	C₄⋊C₄	—	2	4	no	—
534	C₄⋊C₄	—	2	1	no	—
535	C₄⋊C₄	—	2	1	no	—
536	C₄⋊C₄	—	2	2	no	—
537	C₄⋊C₄	—	2	4	no	—
538	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
539	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
540	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
541	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
542	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
543	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
544	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
545	C₂²⋊C₄	—	2	2	no	—
546	C₂²⋊C₄	—	2	8	no	—
547	C₂²⋊C₄	—	2	8	no	—
548	C₂²⋊C₄	—	2	8	no	—
549	C₂²⋊C₄	—	2	2	no	—
550	C₂²⋊C₄	—	2	4	no	—
551	C₂²⋊C₄	—	2	4	no	—
552	C₂²⋊C₄	—	2	2	no	—
553	C₂²⋊C₄	—	2	4	no	—
554	C₄²	—	2	2	no	—
555	C₄²	—	2	2	no	—
556	C₄²	—	2	4	no	—
557	C₄²	—	2	2	no	—
558	C₄²	C₂×D₄×C₃₁	1	4	yes	—
559	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
560	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
561	C₄²	C₂⁴×C₃₁	1	4	yes	—
562	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
563	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
564	C₂³	—	2	2	no	—
565	C₂³	—	2	4	no	—
566	C₂³	—	4	4	no	—
567	C₂³	—	4	4	no	—
568	C₂³	—	4	4	no	—
569	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
570	Q₈	—	2	4	no	—
571	Q₈	—	2	4	no	—
572	Q₈	—	2	4	no	—
573	Q₈	—	2	2	no	—
574	Q₈	—	2	2	no	—
575	Q₈	—	4	2	no	—
576	Q₈	—	4	2	no	—
577	Q₈	—	8	1	no	—
578	D₄	—	2	4	no	—
579	D₄	—	2	8	no	—
580	D₄	—	4	4	no	—
581	D₄	—	4	4	no	—
582	D₄	—	4	4	no	—
583	D₄	—	4	4	no	—
584	D₄	—	4	2	no	—
585	D₄	—	8	1	no	—
586	D₄	—	8	2	no	—
587	D₄	—	8	2	no	—
588	C₂×C₄	—	2	4	no	—
589	C₂×C₄	—	2	2	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	2	no	—
595	C₂×C₄	—	4	2	no	—
596	C₂×C₄	—	4	1	no	—
597	C₂×C₄	—	4	2	no	—
598	C₂×C₄	—	4	2	no	—
599	C₂×C₄	—	4	4	no	—
600	C₂×C₄	—	8	4	no	—
601	C₂×C₄	—	8	8	no	—
602	C₂×C₄	C₃₁×#135	1	4	yes	—
603	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
604	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
605	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
606	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
607	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
608	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
609	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
610	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
611	C₈	—	2	4	no	—
612	C₈	—	2	4	no	—
613	C₂²	—	2	1	no	—
614	C₂²	—	2	1	no	—
615	C₂²	—	4	4	no	—
616	C₂²	—	4	2	no	—
617	C₂²	—	4	2	no	—
618	C₂²	—	4	4	no	—
619	C₂²	—	4	4	no	—
620	C₂²	—	4	2	no	—
621	C₂²	—	8	8	no	—
622	C₂²	C₂×C₃₁×#135	1	2	yes	—
623	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
624	C₄	—	2	1	no	—
625	C₄	—	2	4	no	—
626	C₄	—	2	4	no	—
627	C₄	—	2	1	no	—
628	C₄	—	2	1	no	—
629	C₄	—	4	2	no	—
630	C₄	—	4	2	no	—
631	C₄	—	4	2	no	—
632	C₄	—	4	2	no	—
633	C₄	—	4	2	no	—
634	C₄	C₂×C₃₁×#135	1	2	yes	—
635	C₄	C₂×C₃₁×#135	1	2	yes	—
636	C₄	C₃₁×#585	1	2	yes	—
637	C₂	—	2	2	no	—
638	C₂	—	8	4	no	—
639	C₂	—	8	2	no	—
640	C₂	C₃₁×#3487	1	1	yes	characteristic
641	C₂	C₂²×C₃₁×#135	1	1	yes	characteristic
642	C₂	C₃₁×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₃₁×#3205	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3507	C₂	semidirect
2. C₂×C₃₁×#559	C₂	semidirect
3. C₃₁×#3396	C₂	semidirect
4. C₃₁×#3394	C₂	semidirect
5. C₃₁×#3365	C₂	semidirect
6. C₃₁×#3361	C₂	semidirect
7. C₃₁×#3211	C₂	semidirect
8. C₃₁×#3208	C₂	semidirect
10. C₃₁×#3105	C₂	semidirect
11. C₃₁×#3104	C₂	semidirect
12. C₃₁×#3103	C₂	semidirect
13. C₃₁×#2982	C₂	semidirect
18. C₃₁×#559	C₂²	semidirect
19. C₃₁×#557	C₂²	semidirect
21. C₃₁×#556	C₂²	semidirect
22. C₃₁×#554	C₂²	semidirect
38. C₃₁×#495	C₂²	semidirect
41. C₃₁×#488	C₂²	semidirect
46. C₃₁×#478	C₂²	semidirect
47. C₃₁×#477	C₂²	semidirect
48. C₃₁×#475	C₂²	semidirect
49. C₃₁×#464	C₂²	semidirect
50. C₃₁×#463	C₂²	semidirect
51. C₃₁×#463	C₂²	semidirect
52. C₃₁×#462	C₂²	semidirect
54. C₃₁×#460	C₂²	semidirect
60. C₈×Q₈×C₃₁	C₂²	semidirect
61. C₈×Q₈×C₃₁	C₂²	semidirect
62. C₄×Q₁₆×C₃₁	C₂²	semidirect
63. C₄×SD₁₆×C₃₁	C₂²	semidirect
64. C₄×SD₁₆×C₃₁	C₂²	semidirect
65. C₄×D₈×C₃₁	C₂²	semidirect
242. #61885	C₃₁	direct (N × H)
292. #3507	C₂×C₃₁	semidirect
293. C₂×#559	C₂×C₃₁	semidirect
294. #3396	C₂×C₃₁	semidirect
295. #3394	C₂×C₃₁	semidirect
296. #3365	C₂×C₃₁	semidirect
297. #3361	C₂×C₃₁	semidirect
298. #3211	C₂×C₃₁	semidirect
299. #3208	C₂×C₃₁	semidirect
301. #3105	C₂×C₃₁	semidirect
302. #3104	C₂×C₃₁	semidirect
303. #3103	C₂×C₃₁	semidirect
304. #2982	C₂×C₃₁	semidirect
333. #559	C₂²×C₃₁	semidirect
334. #557	C₂²×C₃₁	semidirect
336. #556	C₂²×C₃₁	semidirect
337. #554	C₂²×C₃₁	semidirect
353. #495	C₂²×C₃₁	semidirect
356. #488	C₂²×C₃₁	semidirect
361. #478	C₂²×C₃₁	semidirect
362. #477	C₂²×C₃₁	semidirect
363. #475	C₂²×C₃₁	semidirect
364. #464	C₂²×C₃₁	semidirect
365. #463	C₂²×C₃₁	semidirect
366. #463	C₂²×C₃₁	semidirect
367. #462	C₂²×C₃₁	semidirect
369. #460	C₂²×C₃₁	semidirect
375. C₈×Q₈	C₂²×C₃₁	semidirect
376. C₈×Q₈	C₂²×C₃₁	semidirect
377. C₄×Q₁₆	C₂²×C₃₁	semidirect
378. C₄×SD₁₆	C₂²×C₃₁	semidirect
379. C₄×SD₁₆	C₂²×C₃₁	semidirect
380. C₄×D₈	C₂²×C₃₁	semidirect
484. C₃₁	#61885	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
C₂×C₃₁×#559	2
C₃₁×#3396	2
C₃₁×#3394	2
C₃₁×#3365	2
C₃₁×#3361	2
C₃₁×#3211	2
C₃₁×#3208	2
C₃₁×#3205	2
C₃₁×#3105	2
C₃₁×#3104	2
C₃₁×#3103	2
C₃₁×#2982	2
#61885	31

Subgroups — C31×#61885 #21518644

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61885 #21518644