Subgroups — C₃₁×#61819 #21518578

Order 7936.

1770 conjugacy classes of subgroups.

The subgroup lattice has 634 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×#554	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3381	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3380	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3361	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3360	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3232	C₂	1	4	yes	maximal, complemented
8	C₃₁×#3230	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#3227	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#3105	C₂	1	4	yes	maximal, complemented
11	C₃₁×#3101	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#3100	C₂	1	1	yes	maximal, characteristic
13	C₃₁×#2982	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#559	C₂²	1	2	yes	complemented
15	C₃₁×#557	C₂²	1	2	yes	complemented
16	C₃₁×#556	—	2	2	no	—
17	C₃₁×#556	C₂²	1	2	yes	complemented
18	C₃₁×#554	—	2	2	no	—
19	C₃₁×#554	—	2	2	no	—
20	C₃₁×#554	C₂²	1	4	yes	—
21	C₃₁×#554	C₂²	1	4	yes	—
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×C₄⋊Q₈	C₂²	1	1	yes	characteristic
29	C₃₁×#529	C₂²	1	4	yes	—
30	C₂×C₃₁×#125	C₂²	1	2	yes	—
31	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
32	C₂×C₃₁×C₂²⋊Q₈	—	2	2	no	—
33	C₃₁×#520	—	2	1	no	—
34	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
35	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
36	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
37	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
38	C₃₁×#500	C₂²	1	2	yes	complemented
39	C₃₁×#499	C₂²	1	8	yes	complemented
40	C₃₁×#498	C₂²	1	2	yes	complemented
41	C₃₁×#488	C₂²	1	8	yes	complemented
42	C₃₁×#487	C₂²	1	2	yes	complemented
43	C₃₁×#486	C₂²	1	2	yes	complemented
44	C₃₁×#484	C₂²	1	4	yes	—
45	C₃₁×#483	C₂²	1	2	yes	—
46	C₃₁×#481	C₂²	1	4	yes	—
47	C₃₁×#480	C₂²	1	2	yes	—
48	C₃₁×#479	C₂²	1	8	yes	complemented
49	C₃₁×#478	C₂²	1	8	yes	—
50	C₃₁×#478	C₂²	1	8	yes	complemented
51	C₃₁×#477	C₂²	1	4	yes	complemented
52	C₃₁×#476	C₂²	1	2	yes	—
53	C₃₁×#476	C₂²	1	4	yes	complemented
54	C₃₁×#474	C₂²	1	2	yes	—
55	C₃₁×#463	C₂²	1	8	yes	—
56	C₃₁×#461	C₂²	1	2	yes	—
57	C₃₁×#459	C₂²	1	2	yes	—
58	C₃₁×#450	C₂²	1	2	yes	—
59	C₃₁×#447	C₂²	1	2	yes	—
60	C₈×Q₈×C₃₁	C₂²	1	4	yes	complemented
61	C₈×Q₈×C₃₁	C₂²	1	4	yes	complemented
62	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	—
63	C₄×SD₁₆×C₃₁	C₂²	1	8	yes	—
64	C₄×D₈×C₃₁	C₂²	1	4	yes	—
65	C₄×D₈×C₃₁	C₂²	1	8	yes	—
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	—	2	1	no	—
71	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
72	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₃₁	—	2	1	no	—
74	C₂³×C₄×C₃₁	—	2	2	no	—
75	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
76	C₂×D₈×C₃₁	C₂³	1	4	yes	—
77	C₃₁×C₄⋊Q₈	—	2	2	no	—
78	C₃₁×C₄⋊Q₈	—	2	1	no	—
79	C₃₁×C₄⋊Q₈	—	2	1	no	—
80	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
81	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
82	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
83	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
84	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
85	C₃₁×#127	C₂³	1	1	yes	characteristic
86	C₃₁×#126	—	2	4	no	—
87	C₃₁×#126	—	2	4	no	—
88	C₃₁×#125	—	2	4	no	—
89	C₃₁×#125	—	2	4	no	—
90	C₃₁×#125	—	2	1	no	—
91	C₃₁×#125	C₂³	1	4	yes	—
92	C₃₁×#125	C₂³	1	8	yes	—
93	C₃₁×#124	—	2	2	no	—
94	C₃₁×#124	—	2	1	no	—
95	C₃₁×#124	C₂³	1	4	yes	—
96	C₃₁×#123	—	2	4	no	—
97	C₃₁×#123	—	2	4	no	—
98	C₃₁×C₂²⋊Q₈	—	2	4	no	—
99	C₃₁×C₂²⋊Q₈	—	2	4	no	—
100	C₃₁×C₂²⋊Q₈	—	2	2	no	—
101	C₃₁×C₂²⋊Q₈	—	2	2	no	—
102	C₃₁×C₂²⋊Q₈	—	2	1	no	—
103	C₃₁×C₂²⋊Q₈	—	2	8	no	—
104	C₃₁×C₂²⋊Q₈	—	2	8	no	—
105	C₃₁×C₂²⋊Q₈	—	2	4	no	—
106	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
107	C₃₁×#121	—	2	1	no	—
108	C₃₁×#121	C₂³	1	2	yes	—
109	C₄×Q₈×C₃₁	—	2	2	no	—
110	C₄×Q₈×C₃₁	—	2	1	no	—
111	C₄×Q₈×C₃₁	D₄	1	2	yes	—
112	C₄×Q₈×C₃₁	D₄	1	2	yes	—
113	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
114	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
115	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
116	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
117	C₄×D₄×C₃₁	—	2	4	no	—
118	C₄×D₄×C₃₁	—	2	2	no	—
119	C₄×D₄×C₃₁	—	2	4	no	—
120	C₄×D₄×C₃₁	—	2	1	no	—
121	C₄×D₄×C₃₁	—	2	1	no	—
122	C₄×D₄×C₃₁	—	2	1	no	—
123	C₄×D₄×C₃₁	C₂³	1	2	yes	—
124	C₄×D₄×C₃₁	C₂³	1	2	yes	—
125	C₄×D₄×C₃₁	C₂³	1	4	yes	—
126	C₄×D₄×C₃₁	C₂³	1	8	yes	—
127	C₄×D₄×C₃₁	C₂³	1	2	yes	—
128	C₃₁×#117	—	2	1	no	—
129	C₃₁×#117	—	2	2	no	—
130	C₃₁×#117	—	2	4	no	—
131	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
132	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
133	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
134	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
135	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
136	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
137	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
138	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
139	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
140	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
141	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
142	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
143	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
144	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
145	C₃₁×#107	C₂³	1	4	yes	—
146	C₃₁×#107	C₂³	1	8	yes	—
147	C₃₁×#106	C₂³	1	4	yes	—
148	C₃₁×#106	C₂³	1	8	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	8	yes	—
154	C₃₁×#102	C₂³	1	4	yes	—
155	C₃₁×#102	C₂³	1	4	yes	—
156	C₃₁×#102	C₂³	1	4	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	2	yes	—
161	C₄×C₈×C₃₁	C₂³	1	8	yes	—
162	C₄×C₈×C₃₁	C₂³	1	2	yes	—
163	C₂⁴×C₃₁	—	2	1	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	2	no	—
169	C₂×Q₈×C₃₁	—	2	2	no	—
170	C₂×Q₈×C₃₁	—	8	2	no	—
171	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
172	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
173	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
174	C₂×D₄×C₃₁	—	2	4	no	—
175	C₂×D₄×C₃₁	—	2	2	no	—
176	C₂×D₄×C₃₁	—	2	1	no	—
177	C₂×D₄×C₃₁	—	2	1	no	—
178	C₂×D₄×C₃₁	—	2	1	no	—
179	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
180	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
181	C₂²×C₄×C₃₁	—	2	1	no	—
182	C₂²×C₄×C₃₁	—	2	2	no	—
183	C₂²×C₄×C₃₁	—	2	2	no	—
184	C₂²×C₄×C₃₁	—	2	2	no	—
185	C₂²×C₄×C₃₁	—	2	2	no	—
186	C₂²×C₄×C₃₁	—	2	8	no	—
187	C₂²×C₄×C₃₁	—	2	4	no	—
188	C₂²×C₄×C₃₁	—	4	4	no	—
189	C₂²×C₄×C₃₁	—	8	4	no	—
190	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
191	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
192	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
193	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
194	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
195	SD₁₆×C₃₁	—	2	8	no	—
196	D₈×C₃₁	—	2	8	no	—
197	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
198	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
199	C₄⋊C₄×C₃₁	—	2	8	no	—
200	C₄⋊C₄×C₃₁	—	2	2	no	—
201	C₄⋊C₄×C₃₁	—	2	2	no	—
202	C₄⋊C₄×C₃₁	—	2	1	no	—
203	C₄⋊C₄×C₃₁	—	2	4	no	—
204	C₄⋊C₄×C₃₁	—	2	8	no	—
205	C₄⋊C₄×C₃₁	—	2	1	no	—
206	C₄⋊C₄×C₃₁	—	2	2	no	—
207	C₄⋊C₄×C₃₁	—	2	1	no	—
208	C₄⋊C₄×C₃₁	—	2	2	no	—
209	C₄⋊C₄×C₃₁	—	2	4	no	—
210	C₄⋊C₄×C₃₁	—	2	1	no	—
211	C₄⋊C₄×C₃₁	—	2	1	no	—
212	C₄⋊C₄×C₃₁	—	2	2	no	—
213	C₄⋊C₄×C₃₁	—	2	4	no	—
214	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
215	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
216	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
217	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
218	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
219	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
220	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
221	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
222	C₂²⋊C₄×C₃₁	—	2	2	no	—
223	C₂²⋊C₄×C₃₁	—	2	2	no	—
224	C₂²⋊C₄×C₃₁	—	2	2	no	—
225	C₂²⋊C₄×C₃₁	—	2	8	no	—
226	C₂²⋊C₄×C₃₁	—	2	2	no	—
227	C₂²⋊C₄×C₃₁	—	2	4	no	—
228	C₂²⋊C₄×C₃₁	—	2	4	no	—
229	C₂²⋊C₄×C₃₁	—	2	2	no	—
230	C₂²⋊C₄×C₃₁	—	2	4	no	—
231	C₄²×C₃₁	—	2	4	no	—
232	C₄²×C₃₁	—	2	2	no	—
233	C₄²×C₃₁	C₂×D₄	1	4	yes	—
234	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
235	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
236	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
237	C₄²×C₃₁	C₂⁴	1	4	yes	—
238	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
239	#61819	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
240	C₂³×C₃₁	—	2	4	no	—
241	C₂³×C₃₁	—	2	2	no	—
242	C₂³×C₃₁	—	2	2	no	—
243	C₂³×C₃₁	—	4	2	no	—
244	C₂³×C₃₁	—	4	2	no	—
245	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
246	Q₈×C₃₁	—	2	4	no	—
247	Q₈×C₃₁	—	2	4	no	—
248	Q₈×C₃₁	—	2	4	no	—
249	Q₈×C₃₁	—	4	2	no	—
250	Q₈×C₃₁	—	8	1	no	—
251	Q₈×C₃₁	—	8	2	no	—
252	Q₈×C₃₁	—	8	2	no	—
253	D₄×C₃₁	—	2	4	no	—
254	D₄×C₃₁	—	2	2	no	—
255	D₄×C₃₁	—	2	4	no	—
256	D₄×C₃₁	—	4	4	no	—
257	D₄×C₃₁	—	4	2	no	—
258	D₄×C₃₁	—	8	1	no	—
259	D₄×C₃₁	2₋¹⁺⁴	1	4	yes	—
260	C₂×C₄×C₃₁	—	2	4	no	—
261	C₂×C₄×C₃₁	—	2	2	no	—
262	C₂×C₄×C₃₁	—	2	2	no	—
263	C₂×C₄×C₃₁	—	2	2	no	—
264	C₂×C₄×C₃₁	—	2	2	no	—
265	C₂×C₄×C₃₁	—	2	2	no	—
266	C₂×C₄×C₃₁	—	2	4	no	—
267	C₂×C₄×C₃₁	—	4	2	no	—
268	C₂×C₄×C₃₁	—	4	1	no	—
269	C₂×C₄×C₃₁	—	4	4	no	—
270	C₂×C₄×C₃₁	—	4	2	no	—
271	C₂×C₄×C₃₁	—	4	4	no	—
272	C₂×C₄×C₃₁	—	4	4	no	—
273	C₂×C₄×C₃₁	—	4	4	no	—
274	C₂×C₄×C₃₁	—	8	4	no	—
275	C₂×C₄×C₃₁	—	8	8	no	—
276	C₂×C₄×C₃₁	#135	1	4	yes	—
277	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
278	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
279	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
280	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
281	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
282	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
283	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
284	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
285	C₈×C₃₁	—	2	4	no	—
286	C₈×C₃₁	—	2	4	no	—
287	#3507	C₂×C₃₁	1	1	yes	characteristic, complemented
288	C₂×#554	C₂×C₃₁	1	1	yes	characteristic, complemented
289	#3381	C₂×C₃₁	1	4	yes	complemented
290	#3380	C₂×C₃₁	1	4	yes	complemented
291	#3361	C₂×C₃₁	1	4	yes	complemented
292	#3360	C₂×C₃₁	1	4	yes	complemented
293	#3232	C₂×C₃₁	1	4	yes	complemented
294	#3230	C₂×C₃₁	1	1	yes	characteristic, complemented
295	#3227	C₂×C₃₁	1	1	yes	characteristic, complemented
296	#3105	C₂×C₃₁	1	4	yes	complemented
297	#3101	C₂×C₃₁	1	1	yes	characteristic, complemented
298	#3100	C₂×C₃₁	1	1	yes	characteristic
299	#2982	C₂×C₃₁	1	1	yes	characteristic, complemented
300	C₂²×C₃₁	—	2	1	no	—
301	C₂²×C₃₁	—	2	1	no	—
302	C₂²×C₃₁	—	2	4	no	—
303	C₂²×C₃₁	—	4	8	no	—
304	C₂²×C₃₁	—	4	2	no	—
305	C₂²×C₃₁	—	4	2	no	—
306	C₂²×C₃₁	—	4	2	no	—
307	C₂²×C₃₁	—	4	2	no	—
308	C₂²×C₃₁	—	4	2	no	—
309	C₂²×C₃₁	C₂×#135	1	2	yes	—
310	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
311	C₄×C₃₁	—	2	4	no	—
312	C₄×C₃₁	—	2	1	no	—
313	C₄×C₃₁	—	2	1	no	—
314	C₄×C₃₁	—	2	1	no	—
315	C₄×C₃₁	—	2	4	no	—
316	C₄×C₃₁	—	4	4	no	—
317	C₄×C₃₁	—	4	2	no	—
318	C₄×C₃₁	—	4	2	no	—
319	C₄×C₃₁	—	4	2	no	—
320	C₄×C₃₁	—	4	2	no	—
321	C₄×C₃₁	C₂×#135	1	2	yes	—
322	C₄×C₃₁	C₂×#135	1	2	yes	—
323	C₄×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
324	#559	C₂²×C₃₁	1	2	yes	complemented
325	#557	C₂²×C₃₁	1	2	yes	complemented
326	#556	—	2	2	no	—
327	#556	C₂²×C₃₁	1	2	yes	complemented
328	#554	—	2	2	no	—
329	#554	—	2	2	no	—
330	#554	C₂²×C₃₁	1	4	yes	—
331	#554	C₂²×C₃₁	1	4	yes	—
332	#554	C₂²×C₃₁	1	2	yes	complemented
333	#552	—	2	2	no	—
334	#539	—	2	2	no	—
335	#536	—	2	1	no	—
336	#533	C₂²×C₃₁	1	1	yes	characteristic
337	#532	C₂²×C₃₁	1	1	yes	characteristic
338	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
339	#529	C₂²×C₃₁	1	4	yes	—
340	C₂×#125	C₂²×C₃₁	1	2	yes	—
341	C₂×C₂²⋊Q₈	—	2	1	no	—
342	C₂×C₂²⋊Q₈	—	2	2	no	—
343	#520	—	2	1	no	—
344	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
345	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
346	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
347	C₂²×C₄⋊C₄	—	2	1	no	—
348	#500	C₂²×C₃₁	1	2	yes	complemented
349	#499	C₂²×C₃₁	1	8	yes	complemented
350	#498	C₂²×C₃₁	1	2	yes	complemented
351	#488	C₂²×C₃₁	1	8	yes	complemented
352	#487	C₂²×C₃₁	1	2	yes	complemented
353	#486	C₂²×C₃₁	1	2	yes	complemented
354	#484	C₂²×C₃₁	1	4	yes	—
355	#483	C₂²×C₃₁	1	2	yes	—
356	#481	C₂²×C₃₁	1	4	yes	—
357	#480	C₂²×C₃₁	1	2	yes	—
358	#479	C₂²×C₃₁	1	8	yes	complemented
359	#478	C₂²×C₃₁	1	8	yes	—
360	#478	C₂²×C₃₁	1	8	yes	complemented
361	#477	C₂²×C₃₁	1	4	yes	complemented
362	#476	C₂²×C₃₁	1	2	yes	—
363	#476	C₂²×C₃₁	1	4	yes	complemented
364	#474	C₂²×C₃₁	1	2	yes	—
365	#463	C₂²×C₃₁	1	8	yes	—
366	#461	C₂²×C₃₁	1	2	yes	—
367	#459	C₂²×C₃₁	1	2	yes	—
368	#450	C₂²×C₃₁	1	2	yes	—
369	#447	C₂²×C₃₁	1	2	yes	—
370	C₈×Q₈	C₂²×C₃₁	1	4	yes	complemented
371	C₈×Q₈	C₂²×C₃₁	1	4	yes	complemented
372	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	—
373	C₄×SD₁₆	C₂²×C₃₁	1	8	yes	—
374	C₄×D₈	C₂²×C₃₁	1	4	yes	—
375	C₄×D₈	C₂²×C₃₁	1	8	yes	—
376	#431	C₂²×C₃₁	1	1	yes	characteristic
377	#431	C₂²×C₃₁	1	1	yes	characteristic
378	#431	C₂²×C₃₁	1	4	yes	—
379	C₂×C₃₁	—	2	2	no	—
380	C₂×C₃₁	—	4	4	no	—
381	C₂×C₃₁	—	8	2	no	—
382	C₂×C₃₁	#3486	1	1	yes	characteristic
383	C₂×C₃₁	C₂²×#135	1	1	yes	characteristic
384	C₂×C₃₁	#3585	1	1	yes	characteristic
385	C₂×#41	—	2	1	no	—
386	C₂²×Q₈	—	2	1	no	—
387	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
388	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
389	C₂³×C₄	—	2	1	no	—
390	C₂³×C₄	—	2	2	no	—
391	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
392	C₂×D₈	C₂³×C₃₁	1	4	yes	—
393	C₄⋊Q₈	—	2	2	no	—
394	C₄⋊Q₈	—	2	1	no	—
395	C₄⋊Q₈	—	2	1	no	—
396	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
397	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
398	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
399	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
400	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
401	#127	C₂³×C₃₁	1	1	yes	characteristic
402	#126	—	2	4	no	—
403	#126	—	2	4	no	—
404	#125	—	2	4	no	—
405	#125	—	2	4	no	—
406	#125	—	2	1	no	—
407	#125	C₂³×C₃₁	1	8	yes	—
408	#125	C₂³×C₃₁	1	4	yes	—
409	#124	—	2	2	no	—
410	#124	—	2	1	no	—
411	#124	C₂³×C₃₁	1	4	yes	—
412	#123	—	2	4	no	—
413	#123	—	2	4	no	—
414	C₂²⋊Q₈	—	2	4	no	—
415	C₂²⋊Q₈	—	2	4	no	—
416	C₂²⋊Q₈	—	2	2	no	—
417	C₂²⋊Q₈	—	2	2	no	—
418	C₂²⋊Q₈	—	2	1	no	—
419	C₂²⋊Q₈	—	2	8	no	—
420	C₂²⋊Q₈	—	2	8	no	—
421	C₂²⋊Q₈	—	2	4	no	—
422	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
423	#121	—	2	1	no	—
424	#121	C₂³×C₃₁	1	2	yes	—
425	C₄×Q₈	—	2	2	no	—
426	C₄×Q₈	—	2	1	no	—
427	C₄×Q₈	D₄×C₃₁	1	2	yes	—
428	C₄×Q₈	D₄×C₃₁	1	2	yes	—
429	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
430	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
431	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
432	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
433	C₄×D₄	—	2	4	no	—
434	C₄×D₄	—	2	2	no	—
435	C₄×D₄	—	2	4	no	—
436	C₄×D₄	—	2	1	no	—
437	C₄×D₄	—	2	1	no	—
438	C₄×D₄	—	2	1	no	—
439	C₄×D₄	C₂³×C₃₁	1	2	yes	—
440	C₄×D₄	C₂³×C₃₁	1	8	yes	—
441	C₄×D₄	C₂³×C₃₁	1	2	yes	—
442	C₄×D₄	C₂³×C₃₁	1	4	yes	—
443	C₄×D₄	C₂³×C₃₁	1	2	yes	—
444	#117	—	2	1	no	—
445	#117	—	2	2	no	—
446	#117	—	2	4	no	—
447	C₂×C₄⋊C₄	—	2	2	no	—
448	C₂×C₄⋊C₄	—	2	1	no	—
449	C₂×C₄⋊C₄	—	2	2	no	—
450	C₂×C₄⋊C₄	—	2	4	no	—
451	C₂×C₄⋊C₄	—	2	4	no	—
452	C₂×C₄⋊C₄	—	2	2	no	—
453	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
454	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
455	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
456	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
457	C₂×C₂²⋊C₄	—	2	1	no	—
458	C₂×C₂²⋊C₄	—	2	2	no	—
459	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
460	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
461	#107	C₂³×C₃₁	1	4	yes	—
462	#107	C₂³×C₃₁	1	8	yes	—
463	#106	C₂³×C₃₁	1	4	yes	—
464	#106	C₂³×C₃₁	1	8	yes	—
465	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
466	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
467	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
468	#103	C₂³×C₃₁	1	4	yes	—
469	#103	C₂³×C₃₁	1	8	yes	—
470	#102	C₂³×C₃₁	1	4	yes	—
471	#102	C₂³×C₃₁	1	4	yes	—
472	#102	C₂³×C₃₁	1	4	yes	—
473	#102	C₂³×C₃₁	1	8	yes	—
474	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
475	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
476	C₄×C₈	C₂³×C₃₁	1	2	yes	—
477	C₄×C₈	C₂³×C₃₁	1	8	yes	—
478	C₄×C₈	C₂³×C₃₁	1	2	yes	—
479	C₃₁	#61819	1	1	yes	characteristic, direct product factor, Sylow
480	C₂⁴	—	2	1	no	—
481	#41	—	4	4	no	—
482	C₂×Q₈	—	2	4	no	—
483	C₂×Q₈	—	2	1	no	—
484	C₂×Q₈	—	2	1	no	—
485	C₂×Q₈	—	2	2	no	—
486	C₂×Q₈	—	2	2	no	—
487	C₂×Q₈	—	8	2	no	—
488	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
489	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
490	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
491	C₂×D₄	—	2	4	no	—
492	C₂×D₄	—	2	2	no	—
493	C₂×D₄	—	2	1	no	—
494	C₂×D₄	—	2	1	no	—
495	C₂×D₄	—	2	1	no	—
496	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
497	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
498	C₂²×C₄	—	2	1	no	—
499	C₂²×C₄	—	2	2	no	—
500	C₂²×C₄	—	2	2	no	—
501	C₂²×C₄	—	2	2	no	—
502	C₂²×C₄	—	2	2	no	—
503	C₂²×C₄	—	2	8	no	—
504	C₂²×C₄	—	2	4	no	—
505	C₂²×C₄	—	4	4	no	—
506	C₂²×C₄	—	8	4	no	—
507	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
508	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
509	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
510	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
511	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
512	SD₁₆	—	2	8	no	—
513	D₈	—	2	8	no	—
514	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
515	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
516	C₄⋊C₄	—	2	8	no	—
517	C₄⋊C₄	—	2	2	no	—
518	C₄⋊C₄	—	2	2	no	—
519	C₄⋊C₄	—	2	1	no	—
520	C₄⋊C₄	—	2	4	no	—
521	C₄⋊C₄	—	2	8	no	—
522	C₄⋊C₄	—	2	1	no	—
523	C₄⋊C₄	—	2	2	no	—
524	C₄⋊C₄	—	2	1	no	—
525	C₄⋊C₄	—	2	2	no	—
526	C₄⋊C₄	—	2	4	no	—
527	C₄⋊C₄	—	2	1	no	—
528	C₄⋊C₄	—	2	1	no	—
529	C₄⋊C₄	—	2	2	no	—
530	C₄⋊C₄	—	2	4	no	—
531	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
532	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
533	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
534	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
535	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
536	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
537	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
538	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
539	C₂²⋊C₄	—	2	2	no	—
540	C₂²⋊C₄	—	2	2	no	—
541	C₂²⋊C₄	—	2	2	no	—
542	C₂²⋊C₄	—	2	8	no	—
543	C₂²⋊C₄	—	2	2	no	—
544	C₂²⋊C₄	—	2	4	no	—
545	C₂²⋊C₄	—	2	4	no	—
546	C₂²⋊C₄	—	2	2	no	—
547	C₂²⋊C₄	—	2	4	no	—
548	C₄²	—	2	4	no	—
549	C₄²	—	2	2	no	—
550	C₄²	C₂×D₄×C₃₁	1	4	yes	—
551	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
552	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
553	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
554	C₄²	C₂⁴×C₃₁	1	4	yes	—
555	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
556	C₂³	—	2	4	no	—
557	C₂³	—	2	2	no	—
558	C₂³	—	2	2	no	—
559	C₂³	—	4	2	no	—
560	C₂³	—	4	2	no	—
561	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
562	Q₈	—	2	4	no	—
563	Q₈	—	2	4	no	—
564	Q₈	—	2	4	no	—
565	Q₈	—	4	2	no	—
566	Q₈	—	8	1	no	—
567	Q₈	—	8	2	no	—
568	Q₈	—	8	2	no	—
569	D₄	—	2	4	no	—
570	D₄	—	2	2	no	—
571	D₄	—	2	4	no	—
572	D₄	—	4	4	no	—
573	D₄	—	4	2	no	—
574	D₄	—	8	1	no	—
575	D₄	C₃₁×2₋¹⁺⁴	1	4	yes	—
576	C₂×C₄	—	2	4	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	2	2	no	—
580	C₂×C₄	—	2	2	no	—
581	C₂×C₄	—	2	2	no	—
582	C₂×C₄	—	2	4	no	—
583	C₂×C₄	—	4	2	no	—
584	C₂×C₄	—	4	1	no	—
585	C₂×C₄	—	4	4	no	—
586	C₂×C₄	—	4	2	no	—
587	C₂×C₄	—	4	4	no	—
588	C₂×C₄	—	4	4	no	—
589	C₂×C₄	—	4	4	no	—
590	C₂×C₄	—	8	4	no	—
591	C₂×C₄	—	8	8	no	—
592	C₂×C₄	C₃₁×#135	1	4	yes	—
593	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
594	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
595	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
596	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
597	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
598	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
599	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
600	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
601	C₈	—	2	4	no	—
602	C₈	—	2	4	no	—
603	C₂²	—	2	1	no	—
604	C₂²	—	2	1	no	—
605	C₂²	—	2	4	no	—
606	C₂²	—	4	8	no	—
607	C₂²	—	4	2	no	—
608	C₂²	—	4	2	no	—
609	C₂²	—	4	2	no	—
610	C₂²	—	4	2	no	—
611	C₂²	—	4	2	no	—
612	C₂²	C₂×C₃₁×#135	1	2	yes	—
613	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
614	C₄	—	2	1	no	—
615	C₄	—	2	4	no	—
616	C₄	—	2	4	no	—
617	C₄	—	2	1	no	—
618	C₄	—	2	1	no	—
619	C₄	—	4	2	no	—
620	C₄	—	4	2	no	—
621	C₄	—	4	2	no	—
622	C₄	—	4	4	no	—
623	C₄	—	4	2	no	—
624	C₄	C₂×C₃₁×#135	1	2	yes	—
625	C₄	C₂×C₃₁×#135	1	2	yes	—
626	C₄	C₂×C₃₁×2₋¹⁺⁴	1	2	yes	—
627	C₂	—	2	2	no	—
628	C₂	—	4	4	no	—
629	C₂	—	8	2	no	—
630	C₂	C₃₁×#3486	1	1	yes	characteristic
631	C₂	C₂²×C₃₁×#135	1	1	yes	characteristic
632	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×#3100	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×#554	C₂	semidirect
3. C₃₁×#3381	C₂	semidirect
4. C₃₁×#3380	C₂	semidirect
5. C₃₁×#3361	C₂	semidirect
6. C₃₁×#3360	C₂	semidirect
7. C₃₁×#3232	C₂	semidirect
8. C₃₁×#3230	C₂	semidirect
9. C₃₁×#3227	C₂	semidirect
10. C₃₁×#3105	C₂	semidirect
11. C₃₁×#3101	C₂	semidirect
13. C₃₁×#2982	C₂	semidirect
14. C₃₁×#559	C₂²	semidirect
15. C₃₁×#557	C₂²	semidirect
17. C₃₁×#556	C₂²	semidirect
22. C₃₁×#554	C₂²	semidirect
38. C₃₁×#500	C₂²	semidirect
39. C₃₁×#499	C₂²	semidirect
40. C₃₁×#498	C₂²	semidirect
41. C₃₁×#488	C₂²	semidirect
42. C₃₁×#487	C₂²	semidirect
43. C₃₁×#486	C₂²	semidirect
48. C₃₁×#479	C₂²	semidirect
50. C₃₁×#478	C₂²	semidirect
51. C₃₁×#477	C₂²	semidirect
53. C₃₁×#476	C₂²	semidirect
60. C₈×Q₈×C₃₁	C₂²	semidirect
61. C₈×Q₈×C₃₁	C₂²	semidirect
239. #61819	C₃₁	direct (N × H)
287. #3507	C₂×C₃₁	semidirect
288. C₂×#554	C₂×C₃₁	semidirect
289. #3381	C₂×C₃₁	semidirect
290. #3380	C₂×C₃₁	semidirect
291. #3361	C₂×C₃₁	semidirect
292. #3360	C₂×C₃₁	semidirect
293. #3232	C₂×C₃₁	semidirect
294. #3230	C₂×C₃₁	semidirect
295. #3227	C₂×C₃₁	semidirect
296. #3105	C₂×C₃₁	semidirect
297. #3101	C₂×C₃₁	semidirect
299. #2982	C₂×C₃₁	semidirect
324. #559	C₂²×C₃₁	semidirect
325. #557	C₂²×C₃₁	semidirect
327. #556	C₂²×C₃₁	semidirect
332. #554	C₂²×C₃₁	semidirect
348. #500	C₂²×C₃₁	semidirect
349. #499	C₂²×C₃₁	semidirect
350. #498	C₂²×C₃₁	semidirect
351. #488	C₂²×C₃₁	semidirect
352. #487	C₂²×C₃₁	semidirect
353. #486	C₂²×C₃₁	semidirect
358. #479	C₂²×C₃₁	semidirect
360. #478	C₂²×C₃₁	semidirect
361. #477	C₂²×C₃₁	semidirect
363. #476	C₂²×C₃₁	semidirect
370. C₈×Q₈	C₂²×C₃₁	semidirect
371. C₈×Q₈	C₂²×C₃₁	semidirect
479. C₃₁	#61819	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×#554	2
C₃₁×#3381	2
C₃₁×#3380	2
C₃₁×#3361	2
C₃₁×#3360	2
C₃₁×#3232	2
C₃₁×#3230	2
C₃₁×#3227	2
C₃₁×#3105	2
C₃₁×#3101	2
C₃₁×#3100	2
C₃₁×#2982	2
#61819	31

Subgroups — C31×#61819 #21518578

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61819 #21518578