Subgroups — C₃₁×#61822 #21518581

Order 7936.

1694 conjugacy classes of subgroups.

The subgroup lattice has 612 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
2	C₂×C₃₁×#557	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3383	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3381	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3365	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3364	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3232	C₂	1	4	yes	maximal, complemented
8	C₃₁×#3229	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#3226	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#3108	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
13	C₃₁×#2982	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#559	C₂²	1	2	yes	—
15	C₃₁×#557	—	2	2	no	—
16	C₃₁×#557	—	2	2	no	—
17	C₃₁×#557	C₂²	1	4	yes	—
18	C₃₁×#557	C₂²	1	4	yes	—
19	C₃₁×#557	C₂²	1	2	yes	—
20	C₃₁×#556	—	2	2	no	—
21	C₃₁×#556	C₂²	1	2	yes	—
22	C₃₁×#554	C₂²	1	2	yes	—
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	—	2	1	no	—
29	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
30	C₃₁×#529	C₂²	1	4	yes	—
31	C₂×C₃₁×#125	—	2	2	no	—
32	C₂×C₃₁×#125	C₂²	1	2	yes	—
33	C₃₁×#520	—	2	1	no	—
34	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
35	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
36	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
37	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
38	C₃₁×#500	C₂²	1	2	yes	—
39	C₃₁×#499	C₂²	1	8	yes	—
40	C₃₁×#498	C₂²	1	2	yes	—
41	C₃₁×#488	C₂²	1	8	yes	—
42	C₃₁×#487	C₂²	1	2	yes	—
43	C₃₁×#486	C₂²	1	2	yes	—
44	C₃₁×#484	C₂²	1	2	yes	—
45	C₃₁×#483	C₂²	1	4	yes	—
46	C₃₁×#481	C₂²	1	2	yes	—
47	C₃₁×#480	C₂²	1	4	yes	—
48	C₃₁×#479	C₂²	1	8	yes	—
49	C₃₁×#479	C₂²	1	8	yes	—
50	C₃₁×#478	C₂²	1	8	yes	—
51	C₃₁×#477	C₂²	1	2	yes	—
52	C₃₁×#475	C₂²	1	2	yes	—
53	C₃₁×#475	C₂²	1	4	yes	—
54	C₃₁×#474	C₂²	1	4	yes	—
55	C₃₁×#464	C₂²	1	8	yes	—
56	C₃₁×#462	C₂²	1	2	yes	—
57	C₃₁×#460	C₂²	1	2	yes	—
58	C₃₁×#451	C₂²	1	2	yes	—
59	C₃₁×#448	C₂²	1	2	yes	—
60	C₈×Q₈×C₃₁	C₂²	1	4	yes	—
61	C₈×Q₈×C₃₁	C₂²	1	4	yes	—
62	C₄×Q₁₆×C₃₁	C₂²	1	4	yes	—
63	C₄×Q₁₆×C₃₁	C₂²	1	8	yes	—
64	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	—
65	C₄×SD₁₆×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×Q₈×C₃₁	C₂³	1	1	yes	characteristic
73	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
74	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
75	C₃₁×C₄⋊Q₈	—	2	4	no	—
76	C₃₁×C₄⋊Q₈	—	2	4	no	—
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	2	yes	—
82	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
83	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
84	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
85	C₃₁×#127	C₂³	1	1	yes	characteristic
86	C₃₁×#126	—	2	4	no	—
87	C₃₁×#126	—	2	4	no	—
88	C₃₁×#125	—	2	8	no	—
89	C₃₁×#125	—	2	8	no	—
90	C₃₁×#125	—	2	4	no	—
91	C₃₁×#125	—	2	4	no	—
92	C₃₁×#125	—	2	1	no	—
93	C₃₁×#125	C₂³	1	8	yes	—
94	C₃₁×#125	C₂³	1	4	yes	—
95	C₃₁×#124	—	2	2	no	—
96	C₃₁×#124	—	2	1	no	—
97	C₃₁×#124	C₂³	1	4	yes	—
98	C₃₁×#123	—	2	4	no	—
99	C₃₁×#123	—	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	4	no	—
104	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
105	C₃₁×#121	—	2	1	no	—
106	C₃₁×#121	C₂³	1	2	yes	—
107	C₄×Q₈×C₃₁	—	2	4	no	—
108	C₄×Q₈×C₃₁	—	2	4	no	—
109	C₄×Q₈×C₃₁	—	2	2	no	—
110	C₄×Q₈×C₃₁	—	2	4	no	—
111	C₄×Q₈×C₃₁	—	2	1	no	—
112	C₄×Q₈×C₃₁	—	2	1	no	—
113	C₄×Q₈×C₃₁	—	2	1	no	—
114	C₄×Q₈×C₃₁	D₄	1	2	yes	—
115	C₄×Q₈×C₃₁	D₄	1	2	yes	—
116	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
117	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
118	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
119	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
120	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
121	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
122	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
123	C₄×D₄×C₃₁	—	2	4	no	—
124	C₄×D₄×C₃₁	—	2	2	no	—
125	C₄×D₄×C₃₁	—	2	1	no	—
126	C₄×D₄×C₃₁	C₂³	1	2	yes	—
127	C₄×D₄×C₃₁	C₂³	1	4	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	1	no	—
135	C₂×C₄⋊C₄×C₃₁	—	2	2	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	2	yes	—
140	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
141	C₂×C₄²×C₃₁	—	2	2	no	—
142	C₂×C₄²×C₃₁	—	2	1	no	—
143	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
144	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
145	C₃₁×#107	C₂³	1	8	yes	—
146	C₃₁×#107	C₂³	1	4	yes	—
147	C₃₁×#106	C₂³	1	8	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×#103	C₂³	1	8	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	#41×C₃₁	—	4	4	no	—
164	C₂×Q₈×C₃₁	—	2	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	—	2	1	no	—
171	C₂×Q₈×C₃₁	—	4	4	no	—
172	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
173	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
174	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
175	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
176	C₂×D₄×C₃₁	—	2	2	no	—
177	C₂×D₄×C₃₁	—	2	1	no	—
178	C₂×D₄×C₃₁	—	2	1	no	—
179	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
180	C₂²×C₄×C₃₁	—	2	1	no	—
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	4	no	—
185	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
186	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
187	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
188	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
189	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
190	Q₁₆×C₃₁	—	2	8	no	—
191	SD₁₆×C₃₁	—	2	8	no	—
192	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
193	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
194	C₄⋊C₄×C₃₁	—	2	1	no	—
195	C₄⋊C₄×C₃₁	—	2	4	no	—
196	C₄⋊C₄×C₃₁	—	2	8	no	—
197	C₄⋊C₄×C₃₁	—	2	2	no	—
198	C₄⋊C₄×C₃₁	—	2	2	no	—
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	8	no	—
203	C₄⋊C₄×C₃₁	—	2	1	no	—
204	C₄⋊C₄×C₃₁	—	2	2	no	—
205	C₄⋊C₄×C₃₁	—	2	1	no	—
206	C₄⋊C₄×C₃₁	—	2	2	no	—
207	C₄⋊C₄×C₃₁	—	2	4	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	C₂×D₄	1	2	yes	—
213	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
214	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
215	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
216	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
217	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
218	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
219	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
220	C₂²⋊C₄×C₃₁	—	2	2	no	—
221	C₂²⋊C₄×C₃₁	—	2	2	no	—
222	C₂²⋊C₄×C₃₁	—	2	4	no	—
223	C₂²⋊C₄×C₃₁	—	2	4	no	—
224	C₂²⋊C₄×C₃₁	—	2	2	no	—
225	C₂²⋊C₄×C₃₁	—	2	4	no	—
226	C₄²×C₃₁	—	2	8	no	—
227	C₄²×C₃₁	—	2	2	no	—
228	C₄²×C₃₁	—	2	2	no	—
229	C₄²×C₃₁	—	2	4	no	—
230	C₄²×C₃₁	—	2	2	no	—
231	C₄²×C₃₁	C₂×D₄	1	4	yes	—
232	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
233	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
234	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
235	C₄²×C₃₁	C₂⁴	1	4	yes	—
236	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
237	#61822	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
238	C₂³×C₃₁	—	2	2	no	—
239	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
240	Q₈×C₃₁	—	2	4	no	—
241	Q₈×C₃₁	—	2	4	no	—
242	Q₈×C₃₁	—	2	4	no	—
243	Q₈×C₃₁	—	2	2	no	—
244	Q₈×C₃₁	—	2	4	no	—
245	Q₈×C₃₁	—	4	4	no	—
246	Q₈×C₃₁	—	4	4	no	—
247	Q₈×C₃₁	—	4	2	no	—
248	Q₈×C₃₁	—	8	1	no	—
249	Q₈×C₃₁	2₋¹⁺⁴	1	4	yes	—
250	D₄×C₃₁	—	2	4	no	—
251	D₄×C₃₁	—	4	4	no	—
252	D₄×C₃₁	—	4	2	no	—
253	D₄×C₃₁	—	8	1	no	—
254	C₂×C₄×C₃₁	—	2	4	no	—
255	C₂×C₄×C₃₁	—	2	4	no	—
256	C₂×C₄×C₃₁	—	2	2	no	—
257	C₂×C₄×C₃₁	—	2	2	no	—
258	C₂×C₄×C₃₁	—	2	2	no	—
259	C₂×C₄×C₃₁	—	2	2	no	—
260	C₂×C₄×C₃₁	—	2	2	no	—
261	C₂×C₄×C₃₁	—	2	4	no	—
262	C₂×C₄×C₃₁	—	2	2	no	—
263	C₂×C₄×C₃₁	—	4	2	no	—
264	C₂×C₄×C₃₁	—	4	1	no	—
265	C₂×C₄×C₃₁	—	4	2	no	—
266	C₂×C₄×C₃₁	—	4	4	no	—
267	C₂×C₄×C₃₁	—	4	4	no	—
268	C₂×C₄×C₃₁	—	8	4	no	—
269	C₂×C₄×C₃₁	#135	1	4	yes	—
270	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
271	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
272	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
273	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
274	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
275	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
276	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
277	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
278	C₈×C₃₁	—	2	4	no	—
279	C₈×C₃₁	—	2	4	no	—
280	#3507	C₂×C₃₁	1	1	yes	characteristic
281	C₂×#557	C₂×C₃₁	1	1	yes	characteristic, complemented
282	#3383	C₂×C₃₁	1	4	yes	complemented
283	#3381	C₂×C₃₁	1	4	yes	complemented
284	#3365	C₂×C₃₁	1	4	yes	complemented
285	#3364	C₂×C₃₁	1	4	yes	complemented
286	#3232	C₂×C₃₁	1	4	yes	complemented
287	#3229	C₂×C₃₁	1	1	yes	characteristic, complemented
288	#3226	C₂×C₃₁	1	1	yes	characteristic
289	#3108	C₂×C₃₁	1	4	yes	complemented
290	#3104	C₂×C₃₁	1	1	yes	characteristic, complemented
291	#3103	C₂×C₃₁	1	1	yes	characteristic
292	#2982	C₂×C₃₁	1	1	yes	characteristic, complemented
293	C₂²×C₃₁	—	2	1	no	—
294	C₂²×C₃₁	—	2	1	no	—
295	C₂²×C₃₁	—	4	2	no	—
296	C₂²×C₃₁	—	4	2	no	—
297	C₂²×C₃₁	—	4	2	no	—
298	C₂²×C₃₁	C₂×#135	1	2	yes	—
299	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
300	C₄×C₃₁	—	2	4	no	—
301	C₄×C₃₁	—	2	1	no	—
302	C₄×C₃₁	—	2	1	no	—
303	C₄×C₃₁	—	2	4	no	—
304	C₄×C₃₁	—	2	1	no	—
305	C₄×C₃₁	—	2	4	no	—
306	C₄×C₃₁	—	4	2	no	—
307	C₄×C₃₁	—	4	2	no	—
308	C₄×C₃₁	—	4	2	no	—
309	C₄×C₃₁	—	4	2	no	—
310	C₄×C₃₁	—	4	4	no	—
311	C₄×C₃₁	C₂×#135	1	2	yes	—
312	C₄×C₃₁	C₂×#135	1	2	yes	—
313	C₄×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
314	#559	C₂²×C₃₁	1	2	yes	—
315	#557	—	2	2	no	—
316	#557	—	2	2	no	—
317	#557	C₂²×C₃₁	1	4	yes	—
318	#557	C₂²×C₃₁	1	4	yes	—
319	#557	C₂²×C₃₁	1	2	yes	—
320	#556	—	2	2	no	—
321	#556	C₂²×C₃₁	1	2	yes	—
322	#554	C₂²×C₃₁	1	2	yes	—
323	#552	—	2	2	no	—
324	#539	—	2	2	no	—
325	#536	—	2	1	no	—
326	#533	C₂²×C₃₁	1	1	yes	characteristic
327	#532	C₂²×C₃₁	1	1	yes	characteristic
328	C₂×C₄⋊Q₈	—	2	1	no	—
329	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
330	#529	C₂²×C₃₁	1	4	yes	—
331	C₂×#125	—	2	2	no	—
332	C₂×#125	C₂²×C₃₁	1	2	yes	—
333	#520	—	2	1	no	—
334	C₂×C₄×Q₈	—	2	1	no	—
335	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
336	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
337	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
338	#500	C₂²×C₃₁	1	2	yes	—
339	#499	C₂²×C₃₁	1	8	yes	—
340	#498	C₂²×C₃₁	1	2	yes	—
341	#488	C₂²×C₃₁	1	8	yes	—
342	#487	C₂²×C₃₁	1	2	yes	—
343	#486	C₂²×C₃₁	1	2	yes	—
344	#484	C₂²×C₃₁	1	2	yes	—
345	#483	C₂²×C₃₁	1	4	yes	—
346	#481	C₂²×C₃₁	1	2	yes	—
347	#480	C₂²×C₃₁	1	4	yes	—
348	#479	C₂²×C₃₁	1	8	yes	—
349	#479	C₂²×C₃₁	1	8	yes	—
350	#478	C₂²×C₃₁	1	8	yes	—
351	#477	C₂²×C₃₁	1	2	yes	—
352	#475	C₂²×C₃₁	1	2	yes	—
353	#475	C₂²×C₃₁	1	4	yes	—
354	#474	C₂²×C₃₁	1	4	yes	—
355	#464	C₂²×C₃₁	1	8	yes	—
356	#462	C₂²×C₃₁	1	2	yes	—
357	#460	C₂²×C₃₁	1	2	yes	—
358	#451	C₂²×C₃₁	1	2	yes	—
359	#448	C₂²×C₃₁	1	2	yes	—
360	C₈×Q₈	C₂²×C₃₁	1	4	yes	—
361	C₈×Q₈	C₂²×C₃₁	1	4	yes	—
362	C₄×Q₁₆	C₂²×C₃₁	1	4	yes	—
363	C₄×Q₁₆	C₂²×C₃₁	1	8	yes	—
364	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	—
365	C₄×SD₁₆	C₂²×C₃₁	1	8	yes	—
366	#431	C₂²×C₃₁	1	1	yes	characteristic
367	#431	C₂²×C₃₁	1	1	yes	characteristic
368	#431	C₂²×C₃₁	1	4	yes	—
369	C₂×C₃₁	—	2	2	no	—
370	C₂×C₃₁	—	8	2	no	—
371	C₂×C₃₁	#3486	1	1	yes	characteristic
372	C₂×C₃₁	C₂²×#135	1	1	yes	characteristic
373	C₂×C₃₁	#3586	1	1	yes	characteristic
374	C₂×#41	—	2	1	no	—
375	C₂²×Q₈	—	2	1	no	—
376	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
377	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
378	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
379	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
380	C₄⋊Q₈	—	2	4	no	—
381	C₄⋊Q₈	—	2	4	no	—
382	C₄⋊Q₈	—	2	2	no	—
383	C₄⋊Q₈	—	2	1	no	—
384	C₄⋊Q₈	—	2	1	no	—
385	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
386	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
387	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
388	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
389	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
390	#127	C₂³×C₃₁	1	1	yes	characteristic
391	#126	—	2	4	no	—
392	#126	—	2	4	no	—
393	#125	—	2	8	no	—
394	#125	—	2	8	no	—
395	#125	—	2	4	no	—
396	#125	—	2	4	no	—
397	#125	—	2	1	no	—
398	#125	C₂³×C₃₁	1	8	yes	—
399	#125	C₂³×C₃₁	1	4	yes	—
400	#124	—	2	2	no	—
401	#124	—	2	1	no	—
402	#124	C₂³×C₃₁	1	4	yes	—
403	#123	—	2	4	no	—
404	#123	—	2	4	no	—
405	C₂²⋊Q₈	—	2	2	no	—
406	C₂²⋊Q₈	—	2	2	no	—
407	C₂²⋊Q₈	—	2	1	no	—
408	C₂²⋊Q₈	—	2	4	no	—
409	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
410	#121	—	2	1	no	—
411	#121	C₂³×C₃₁	1	2	yes	—
412	C₄×Q₈	—	2	4	no	—
413	C₄×Q₈	—	2	4	no	—
414	C₄×Q₈	—	2	2	no	—
415	C₄×Q₈	—	2	4	no	—
416	C₄×Q₈	—	2	1	no	—
417	C₄×Q₈	—	2	1	no	—
418	C₄×Q₈	—	2	1	no	—
419	C₄×Q₈	D₄×C₃₁	1	2	yes	—
420	C₄×Q₈	D₄×C₃₁	1	2	yes	—
421	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
422	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
423	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
424	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
425	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
426	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
427	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
428	C₄×D₄	—	2	4	no	—
429	C₄×D₄	—	2	2	no	—
430	C₄×D₄	—	2	1	no	—
431	C₄×D₄	C₂³×C₃₁	1	4	yes	—
432	C₄×D₄	C₂³×C₃₁	1	2	yes	—
433	#117	—	2	1	no	—
434	#117	—	2	2	no	—
435	#117	—	2	4	no	—
436	C₂×C₄⋊C₄	—	2	2	no	—
437	C₂×C₄⋊C₄	—	2	1	no	—
438	C₂×C₄⋊C₄	—	2	2	no	—
439	C₂×C₄⋊C₄	—	2	1	no	—
440	C₂×C₄⋊C₄	—	2	2	no	—
441	C₂×C₄⋊C₄	—	2	2	no	—
442	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
443	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
444	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
445	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
446	C₂×C₄²	—	2	2	no	—
447	C₂×C₄²	—	2	1	no	—
448	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
449	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
450	#107	C₂³×C₃₁	1	8	yes	—
451	#107	C₂³×C₃₁	1	4	yes	—
452	#106	C₂³×C₃₁	1	8	yes	—
453	#106	C₂³×C₃₁	1	4	yes	—
454	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
455	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
456	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
457	#103	C₂³×C₃₁	1	4	yes	—
458	#103	C₂³×C₃₁	1	4	yes	—
459	#103	C₂³×C₃₁	1	4	yes	—
460	#103	C₂³×C₃₁	1	8	yes	—
461	#102	C₂³×C₃₁	1	4	yes	—
462	#102	C₂³×C₃₁	1	8	yes	—
463	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
464	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
465	C₄×C₈	C₂³×C₃₁	1	2	yes	—
466	C₄×C₈	C₂³×C₃₁	1	8	yes	—
467	C₄×C₈	C₂³×C₃₁	1	2	yes	—
468	C₃₁	#61822	1	1	yes	characteristic, direct product factor, Sylow
469	#41	—	4	4	no	—
470	C₂×Q₈	—	2	4	no	—
471	C₂×Q₈	—	2	4	no	—
472	C₂×Q₈	—	2	1	no	—
473	C₂×Q₈	—	2	1	no	—
474	C₂×Q₈	—	2	2	no	—
475	C₂×Q₈	—	2	2	no	—
476	C₂×Q₈	—	2	1	no	—
477	C₂×Q₈	—	4	4	no	—
478	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
479	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
480	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
481	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
482	C₂×D₄	—	2	2	no	—
483	C₂×D₄	—	2	1	no	—
484	C₂×D₄	—	2	1	no	—
485	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
486	C₂²×C₄	—	2	1	no	—
487	C₂²×C₄	—	2	1	no	—
488	C₂²×C₄	—	2	2	no	—
489	C₂²×C₄	—	2	2	no	—
490	C₂²×C₄	—	2	4	no	—
491	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
492	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
493	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
494	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
495	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
496	Q₁₆	—	2	8	no	—
497	SD₁₆	—	2	8	no	—
498	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
499	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
500	C₄⋊C₄	—	2	1	no	—
501	C₄⋊C₄	—	2	4	no	—
502	C₄⋊C₄	—	2	8	no	—
503	C₄⋊C₄	—	2	2	no	—
504	C₄⋊C₄	—	2	2	no	—
505	C₄⋊C₄	—	2	8	no	—
506	C₄⋊C₄	—	2	2	no	—
507	C₄⋊C₄	—	2	2	no	—
508	C₄⋊C₄	—	2	8	no	—
509	C₄⋊C₄	—	2	1	no	—
510	C₄⋊C₄	—	2	2	no	—
511	C₄⋊C₄	—	2	1	no	—
512	C₄⋊C₄	—	2	2	no	—
513	C₄⋊C₄	—	2	4	no	—
514	C₄⋊C₄	—	2	1	no	—
515	C₄⋊C₄	—	2	1	no	—
516	C₄⋊C₄	—	2	2	no	—
517	C₄⋊C₄	—	2	4	no	—
518	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
519	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
520	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
521	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
522	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
523	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
524	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
525	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
526	C₂²⋊C₄	—	2	2	no	—
527	C₂²⋊C₄	—	2	2	no	—
528	C₂²⋊C₄	—	2	4	no	—
529	C₂²⋊C₄	—	2	4	no	—
530	C₂²⋊C₄	—	2	2	no	—
531	C₂²⋊C₄	—	2	4	no	—
532	C₄²	—	2	8	no	—
533	C₄²	—	2	2	no	—
534	C₄²	—	2	2	no	—
535	C₄²	—	2	4	no	—
536	C₄²	—	2	2	no	—
537	C₄²	C₂×D₄×C₃₁	1	4	yes	—
538	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
539	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
540	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
541	C₄²	C₂⁴×C₃₁	1	4	yes	—
542	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
543	C₂³	—	2	2	no	—
544	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
545	Q₈	—	2	4	no	—
546	Q₈	—	2	4	no	—
547	Q₈	—	2	4	no	—
548	Q₈	—	2	2	no	—
549	Q₈	—	2	4	no	—
550	Q₈	—	4	4	no	—
551	Q₈	—	4	4	no	—
552	Q₈	—	4	2	no	—
553	Q₈	—	8	1	no	—
554	Q₈	C₃₁×2₋¹⁺⁴	1	4	yes	—
555	D₄	—	2	4	no	—
556	D₄	—	4	4	no	—
557	D₄	—	4	2	no	—
558	D₄	—	8	1	no	—
559	C₂×C₄	—	2	4	no	—
560	C₂×C₄	—	2	4	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	4	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	4	2	no	—
569	C₂×C₄	—	4	1	no	—
570	C₂×C₄	—	4	2	no	—
571	C₂×C₄	—	4	4	no	—
572	C₂×C₄	—	4	4	no	—
573	C₂×C₄	—	8	4	no	—
574	C₂×C₄	C₃₁×#135	1	4	yes	—
575	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
576	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
577	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
578	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
579	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
580	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
581	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
582	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
583	C₈	—	2	4	no	—
584	C₈	—	2	4	no	—
585	C₂²	—	2	1	no	—
586	C₂²	—	2	1	no	—
587	C₂²	—	4	2	no	—
588	C₂²	—	4	2	no	—
589	C₂²	—	4	2	no	—
590	C₂²	C₂×C₃₁×#135	1	2	yes	—
591	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
592	C₄	—	2	4	no	—
593	C₄	—	2	1	no	—
594	C₄	—	2	4	no	—
595	C₄	—	2	4	no	—
596	C₄	—	2	1	no	—
597	C₄	—	2	1	no	—
598	C₄	—	4	2	no	—
599	C₄	—	4	2	no	—
600	C₄	—	4	4	no	—
601	C₄	—	4	2	no	—
602	C₄	—	4	2	no	—
603	C₄	C₂×C₃₁×#135	1	2	yes	—
604	C₄	C₂×C₃₁×#135	1	2	yes	—
605	C₄	C₂×C₃₁×2₋¹⁺⁴	1	2	yes	—
606	C₂	—	2	2	no	—
607	C₂	—	8	2	no	—
608	C₂	C₃₁×#3486	1	1	yes	characteristic
609	C₂	C₂²×C₃₁×#135	1	1	yes	characteristic
610	C₂	C₃₁×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₃₁×#532	C₂²
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
2. C₂×C₃₁×#557	C₂	semidirect
3. C₃₁×#3383	C₂	semidirect
4. C₃₁×#3381	C₂	semidirect
5. C₃₁×#3365	C₂	semidirect
6. C₃₁×#3364	C₂	semidirect
7. C₃₁×#3232	C₂	semidirect
8. C₃₁×#3229	C₂	semidirect
10. C₃₁×#3108	C₂	semidirect
11. C₃₁×#3104	C₂	semidirect
13. C₃₁×#2982	C₂	semidirect
237. #61822	C₃₁	direct (N × H)
281. C₂×#557	C₂×C₃₁	semidirect
282. #3383	C₂×C₃₁	semidirect
283. #3381	C₂×C₃₁	semidirect
284. #3365	C₂×C₃₁	semidirect
285. #3364	C₂×C₃₁	semidirect
286. #3232	C₂×C₃₁	semidirect
287. #3229	C₂×C₃₁	semidirect
289. #3108	C₂×C₃₁	semidirect
290. #3104	C₂×C₃₁	semidirect
292. #2982	C₂×C₃₁	semidirect
468. C₃₁	#61822	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×#557	2
C₃₁×#3383	2
C₃₁×#3381	2
C₃₁×#3365	2
C₃₁×#3364	2
C₃₁×#3232	2
C₃₁×#3229	2
C₃₁×#3226	2
C₃₁×#3108	2
C₃₁×#3104	2
C₃₁×#3103	2
C₃₁×#2982	2
#61822	31

Subgroups — C31×#61822 #21518581

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61822 #21518581