Subgroups — C₃₁×#61943 #21518702

Order 7936.

1650 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3556	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3510	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3400	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3398	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3377	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3371	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3223	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3222	C₂	1	2	yes	maximal
9	C₃₁×#3215	C₂	1	2	yes	maximal, complemented
10	C₃₁×#3127	C₂	1	2	yes	maximal, complemented
11	C₃₁×#3123	C₂	1	2	yes	maximal, complemented
12	C₃₁×#3116	C₂	1	2	yes	maximal, complemented
13	C₃₁×#2987	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#564	—	2	2	no	—
15	C₃₁×#559	C₂²	1	4	yes	—
16	C₃₁×#557	—	2	2	no	—
17	C₃₁×#557	C₂²	1	4	yes	complemented
18	C₃₁×#557	C₂²	1	4	yes	complemented
19	C₃₁×#556	C₂²	1	4	yes	—
20	C₃₁×#551	—	2	4	no	—
21	C₃₁×#542	—	2	2	no	—
22	C₃₁×#541	—	2	4	no	—
23	C₃₁×#541	C₂²	1	2	yes	—
24	C₃₁×#541	C₂²	1	2	yes	—
25	C₃₁×#536	C₂²	1	1	yes	characteristic
26	C₃₁×#535	C₂²	1	1	yes	characteristic
27	C₃₁×#533	C₂²	1	2	yes	—
28	C₃₁×#533	C₂²	1	1	yes	characteristic
29	C₃₁×#529	—	2	2	no	—
30	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
31	C₃₁×#519	C₂²	1	1	yes	characteristic
32	C₃₁×#518	C₂²	1	1	yes	characteristic
33	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
34	C₃₁×#497	C₂²	1	2	yes	—
35	C₃₁×#497	C₂²	1	4	yes	—
36	C₃₁×#497	C₂²	1	2	yes	complemented
37	C₃₁×#494	C₂²	1	2	yes	complemented
38	C₃₁×#492	C₂²	1	2	yes	—
39	C₃₁×#491	C₂²	1	2	yes	complemented
40	C₃₁×#490	C₂²	1	2	yes	—
41	C₃₁×#489	C₂²	1	4	yes	—
42	C₃₁×#488	C₂²	1	2	yes	—
43	C₃₁×#488	C₂²	1	2	yes	complemented
44	C₃₁×#485	C₂²	1	4	yes	—
45	C₃₁×#479	C₂²	1	4	yes	complemented
46	C₃₁×#478	C₂²	1	4	yes	—
47	C₃₁×#477	C₂²	1	2	yes	complemented
48	C₃₁×#476	C₂²	1	2	yes	complemented
49	C₃₁×#464	C₂²	1	4	yes	—
50	C₃₁×#464	C₂²	1	4	yes	—
51	C₃₁×#464	C₂²	1	4	yes	—
52	C₃₁×#464	C₂²	1	4	yes	complemented
53	C₃₁×#462	C₂²	1	2	yes	complemented
54	C₃₁×#462	C₂²	1	4	yes	complemented
55	C₃₁×#462	C₂²	1	4	yes	complemented
56	C₃₁×#461	C₂²	1	4	yes	—
57	C₃₁×#461	C₂²	1	2	yes	complemented
58	C₃₁×#460	C₂²	1	4	yes	—
59	C₃₁×#452	C₂²	1	4	yes	—
60	C₃₁×#451	C₂²	1	4	yes	—
61	C₃₁×#450	C₂²	1	4	yes	—
62	C₃₁×#446	C₂²	1	4	yes	complemented
63	C₃₁×#446	C₂²	1	4	yes	complemented
64	C₃₁×#441	C₂²	1	4	yes	complemented
65	C₃₁×#441	C₂²	1	4	yes	complemented
66	C₃₁×#440	C₂²	1	4	yes	—
67	C₃₁×#440	C₂²	1	4	yes	complemented
68	C₄×Q₁₆×C₃₁	C₂²	1	4	yes	complemented
69	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	—
70	C₃₁×#433	C₂²	1	2	yes	—
71	C₃₁×#433	C₂²	1	2	yes	—
72	C₃₁×#432	C₂²	1	2	yes	—
73	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
74	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
75	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
76	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
77	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
78	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
79	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
80	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
81	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
82	C₃₁×C₄⋊Q₈	—	2	4	no	—
83	C₃₁×C₄⋊Q₈	D₄	1	2	yes	—
84	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
85	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
86	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
87	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
88	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
89	C₃₁×#127	C₂³	1	2	yes	—
90	C₃₁×#126	—	2	4	no	—
91	C₃₁×#126	—	2	4	no	—
92	C₃₁×#126	—	2	4	no	—
93	C₃₁×#126	—	2	4	no	—
94	C₃₁×#125	—	2	4	no	—
95	C₃₁×#125	—	2	4	no	—
96	C₃₁×#125	—	2	4	no	—
97	C₃₁×#125	—	2	4	no	—
98	C₃₁×#125	—	2	4	no	—
99	C₃₁×#125	—	2	4	no	—
100	C₃₁×#125	—	2	4	no	—
101	C₃₁×#125	—	2	2	no	—
102	C₃₁×#125	D₄	1	2	yes	—
103	C₃₁×#125	C₂³	1	2	yes	—
104	C₃₁×#125	C₂³	1	2	yes	—
105	C₃₁×#124	—	2	4	no	—
106	C₃₁×#124	—	2	1	no	—
107	C₃₁×#124	—	2	1	no	—
108	C₃₁×#124	C₂³	1	2	yes	—
109	C₃₁×#124	C₂³	1	4	yes	—
110	C₃₁×#124	C₂³	1	2	yes	—
111	C₃₁×#123	—	2	4	no	—
112	C₃₁×#123	—	2	4	no	—
113	C₃₁×#123	—	2	2	no	—
114	C₃₁×C₂²⋊Q₈	—	2	4	no	—
115	C₃₁×C₂²⋊Q₈	—	2	2	no	—
116	C₃₁×C₂²⋊Q₈	—	2	2	no	—
117	C₃₁×C₂²⋊Q₈	D₄	1	4	yes	—
118	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
119	C₃₁×#121	—	2	2	no	—
120	C₃₁×#120	—	2	2	no	—
121	C₄×Q₈×C₃₁	—	2	4	no	—
122	C₄×Q₈×C₃₁	—	2	2	no	—
123	C₄×Q₈×C₃₁	—	2	4	no	—
124	C₄×Q₈×C₃₁	—	2	2	no	—
125	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
126	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
127	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
128	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
129	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
130	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
131	C₄×D₄×C₃₁	—	2	2	no	—
132	C₄×D₄×C₃₁	—	2	2	no	—
133	C₄×D₄×C₃₁	—	2	4	no	—
134	C₄×D₄×C₃₁	C₂³	1	4	yes	—
135	C₄×D₄×C₃₁	C₂³	1	2	yes	—
136	C₃₁×#117	—	2	2	no	—
137	C₃₁×#117	—	2	4	no	—
138	C₃₁×#117	—	2	2	no	—
139	C₃₁×#117	—	2	1	no	—
140	C₃₁×#117	C₂³	1	1	yes	characteristic
141	C₃₁×#117	C₂³	1	1	yes	characteristic
142	C₃₁×#117	C₂³	1	1	yes	characteristic
143	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
144	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
145	C₂×C₄²×C₃₁	—	2	1	no	—
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	4	yes	—
152	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
153	C₃₁×#103	C₂³	1	4	yes	—
154	C₃₁×#103	C₂³	1	4	yes	—
155	C₃₁×#103	C₂³	1	4	yes	—
156	C₃₁×#103	C₂³	1	4	yes	—
157	C₃₁×#103	C₂³	1	4	yes	—
158	C₃₁×#102	C₂³	1	4	yes	—
159	C₃₁×#102	C₂³	1	4	yes	—
160	C₃₁×#102	C₂³	1	4	yes	—
161	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
162	C₃₁×#97	C₂³	1	4	yes	—
163	C₃₁×#97	C₂³	1	2	yes	—
164	C₃₁×#97	C₂³	1	2	yes	—
165	C₄×C₈×C₃₁	C₂³	1	2	yes	—
166	C₄×C₈×C₃₁	C₂³	1	2	yes	—
167	C₂⁴×C₃₁	—	2	1	no	—
168	#41×C₃₁	—	4	2	no	—
169	C₂×Q₈×C₃₁	—	2	2	no	—
170	C₂×Q₈×C₃₁	—	2	1	no	—
171	C₂×Q₈×C₃₁	—	4	2	no	—
172	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
173	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
174	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
175	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
176	C₂×D₄×C₃₁	—	2	1	no	—
177	C₂×D₄×C₃₁	—	2	1	no	—
178	C₂×D₄×C₃₁	—	2	4	no	—
179	C₂×D₄×C₃₁	—	4	2	no	—
180	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
181	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
182	C₂²×C₄×C₃₁	—	2	1	no	—
183	C₂²×C₄×C₃₁	—	2	1	no	—
184	C₂²×C₄×C₃₁	—	2	2	no	—
185	C₂²×C₄×C₃₁	—	2	4	no	—
186	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
187	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
188	Q₁₆×C₃₁	—	4	4	no	—
189	Q₁₆×C₃₁	—	4	4	no	—
190	Q₁₆×C₃₁	—	4	4	no	—
191	SD₁₆×C₃₁	—	4	4	no	—
192	SD₁₆×C₃₁	—	4	4	no	—
193	SD₁₆×C₃₁	—	4	4	no	—
194	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
195	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
196	C₄⋊C₄×C₃₁	—	2	2	no	—
197	C₄⋊C₄×C₃₁	—	2	4	no	—
198	C₄⋊C₄×C₃₁	—	2	4	no	—
199	C₄⋊C₄×C₃₁	—	2	2	no	—
200	C₄⋊C₄×C₃₁	—	2	2	no	—
201	C₄⋊C₄×C₃₁	—	2	2	no	—
202	C₄⋊C₄×C₃₁	—	2	4	no	—
203	C₄⋊C₄×C₃₁	—	2	4	no	—
204	C₄⋊C₄×C₃₁	—	2	4	no	—
205	C₄⋊C₄×C₃₁	—	2	4	no	—
206	C₄⋊C₄×C₃₁	—	2	4	no	—
207	C₄⋊C₄×C₃₁	—	2	4	no	—
208	C₄⋊C₄×C₃₁	—	2	4	no	—
209	C₄⋊C₄×C₃₁	—	2	4	no	—
210	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
211	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
212	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
213	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
214	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
215	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
216	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
217	C₂²⋊C₄×C₃₁	—	2	1	no	—
218	C₂²⋊C₄×C₃₁	—	2	4	no	—
219	C₂²⋊C₄×C₃₁	—	2	4	no	—
220	C₂²⋊C₄×C₃₁	—	2	4	no	—
221	C₂²⋊C₄×C₃₁	—	2	4	no	—
222	C₂²⋊C₄×C₃₁	—	2	2	no	—
223	C₂²⋊C₄×C₃₁	—	2	4	no	—
224	C₂²⋊C₄×C₃₁	—	2	4	no	—
225	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
226	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
227	C₄²×C₃₁	—	2	2	no	—
228	C₄²×C₃₁	—	2	1	no	—
229	C₄²×C₃₁	—	2	4	no	—
230	C₄²×C₃₁	—	2	4	no	—
231	C₄²×C₃₁	—	2	4	no	—
232	C₄²×C₃₁	—	2	4	no	—
233	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
234	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
235	C₄²×C₃₁	C₂⁴	1	2	yes	—
236	C₄²×C₃₁	C₂⁴	1	2	yes	—
237	C₄²×C₃₁	C₂⁴	1	2	yes	—
238	#61943	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
239	C₂³×C₃₁	—	2	2	no	—
240	C₂³×C₃₁	—	2	1	no	—
241	C₂³×C₃₁	—	4	2	no	—
242	C₂³×C₃₁	—	8	2	no	—
243	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
244	Q₈×C₃₁	—	2	4	no	—
245	Q₈×C₃₁	—	2	4	no	—
246	Q₈×C₃₁	—	2	4	no	—
247	Q₈×C₃₁	—	2	2	no	—
248	Q₈×C₃₁	—	4	4	no	—
249	Q₈×C₃₁	—	4	4	no	—
250	Q₈×C₃₁	—	4	2	no	—
251	D₄×C₃₁	—	2	4	no	—
252	D₄×C₃₁	—	2	2	no	—
253	D₄×C₃₁	—	4	2	no	—
254	D₄×C₃₁	—	4	2	no	—
255	D₄×C₃₁	—	8	4	no	—
256	C₂×C₄×C₃₁	—	2	2	no	—
257	C₂×C₄×C₃₁	—	2	2	no	—
258	C₂×C₄×C₃₁	—	2	4	no	—
259	C₂×C₄×C₃₁	—	2	4	no	—
260	C₂×C₄×C₃₁	—	2	2	no	—
261	C₂×C₄×C₃₁	—	2	1	no	—
262	C₂×C₄×C₃₁	—	4	2	no	—
263	C₂×C₄×C₃₁	—	4	2	no	—
264	C₂×C₄×C₃₁	—	4	2	no	—
265	C₂×C₄×C₃₁	—	8	2	no	—
266	C₂×C₄×C₃₁	—	8	4	no	—
267	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
268	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
269	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
270	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
271	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
272	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
273	C₈×C₃₁	—	2	4	no	—
274	C₈×C₃₁	—	2	4	no	—
275	#3556	C₂×C₃₁	1	1	yes	characteristic, complemented
276	#3510	C₂×C₃₁	1	1	yes	characteristic
277	#3400	C₂×C₃₁	1	4	yes	complemented
278	#3398	C₂×C₃₁	1	4	yes	complemented
279	#3377	C₂×C₃₁	1	4	yes	complemented
280	#3371	C₂×C₃₁	1	4	yes	complemented
281	#3223	C₂×C₃₁	1	2	yes	complemented
282	#3222	C₂×C₃₁	1	2	yes	—
283	#3215	C₂×C₃₁	1	2	yes	complemented
284	#3127	C₂×C₃₁	1	2	yes	complemented
285	#3123	C₂×C₃₁	1	2	yes	complemented
286	#3116	C₂×C₃₁	1	2	yes	complemented
287	#2987	C₂×C₃₁	1	1	yes	characteristic, complemented
288	C₂²×C₃₁	—	2	2	no	—
289	C₂²×C₃₁	—	2	1	no	—
290	C₂²×C₃₁	—	4	2	no	—
291	C₂²×C₃₁	—	4	4	no	—
292	C₂²×C₃₁	—	4	1	no	—
293	C₂²×C₃₁	—	4	2	no	—
294	C₂²×C₃₁	—	16	2	no	—
295	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
296	C₄×C₃₁	—	2	2	no	—
297	C₄×C₃₁	—	2	2	no	—
298	C₄×C₃₁	—	2	4	no	—
299	C₄×C₃₁	—	2	4	no	—
300	C₄×C₃₁	—	2	1	no	—
301	C₄×C₃₁	—	4	2	no	—
302	C₄×C₃₁	—	4	4	no	—
303	C₄×C₃₁	—	4	1	no	—
304	C₄×C₃₁	—	4	2	no	—
305	C₄×C₃₁	—	4	4	no	—
306	C₄×C₃₁	#585	1	2	yes	—
307	#564	—	2	2	no	—
308	#559	C₂²×C₃₁	1	4	yes	—
309	#557	—	2	2	no	—
310	#557	C₂²×C₃₁	1	4	yes	complemented
311	#557	C₂²×C₃₁	1	4	yes	complemented
312	#556	C₂²×C₃₁	1	4	yes	—
313	#551	—	2	4	no	—
314	#542	—	2	2	no	—
315	#541	—	2	4	no	—
316	#541	C₂²×C₃₁	1	2	yes	—
317	#541	C₂²×C₃₁	1	2	yes	—
318	#536	C₂²×C₃₁	1	1	yes	characteristic
319	#535	C₂²×C₃₁	1	1	yes	characteristic
320	#533	C₂²×C₃₁	1	2	yes	—
321	#533	C₂²×C₃₁	1	1	yes	characteristic
322	#529	—	2	2	no	—
323	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
324	#519	C₂²×C₃₁	1	1	yes	characteristic
325	#518	C₂²×C₃₁	1	1	yes	characteristic
326	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
327	#497	C₂²×C₃₁	1	2	yes	—
328	#497	C₂²×C₃₁	1	4	yes	—
329	#497	C₂²×C₃₁	1	2	yes	complemented
330	#494	C₂²×C₃₁	1	2	yes	complemented
331	#492	C₂²×C₃₁	1	2	yes	—
332	#491	C₂²×C₃₁	1	2	yes	complemented
333	#490	C₂²×C₃₁	1	2	yes	—
334	#489	C₂²×C₃₁	1	4	yes	—
335	#488	C₂²×C₃₁	1	2	yes	—
336	#488	C₂²×C₃₁	1	2	yes	complemented
337	#485	C₂²×C₃₁	1	4	yes	—
338	#479	C₂²×C₃₁	1	4	yes	complemented
339	#478	C₂²×C₃₁	1	4	yes	—
340	#477	C₂²×C₃₁	1	2	yes	complemented
341	#476	C₂²×C₃₁	1	2	yes	complemented
342	#464	C₂²×C₃₁	1	4	yes	—
343	#464	C₂²×C₃₁	1	4	yes	—
344	#464	C₂²×C₃₁	1	4	yes	—
345	#464	C₂²×C₃₁	1	4	yes	complemented
346	#462	C₂²×C₃₁	1	2	yes	complemented
347	#462	C₂²×C₃₁	1	4	yes	complemented
348	#462	C₂²×C₃₁	1	4	yes	complemented
349	#461	C₂²×C₃₁	1	4	yes	—
350	#461	C₂²×C₃₁	1	2	yes	complemented
351	#460	C₂²×C₃₁	1	4	yes	—
352	#452	C₂²×C₃₁	1	4	yes	—
353	#451	C₂²×C₃₁	1	4	yes	—
354	#450	C₂²×C₃₁	1	4	yes	—
355	#446	C₂²×C₃₁	1	4	yes	complemented
356	#446	C₂²×C₃₁	1	4	yes	complemented
357	#441	C₂²×C₃₁	1	4	yes	complemented
358	#441	C₂²×C₃₁	1	4	yes	complemented
359	#440	C₂²×C₃₁	1	4	yes	—
360	#440	C₂²×C₃₁	1	4	yes	complemented
361	C₄×Q₁₆	C₂²×C₃₁	1	4	yes	complemented
362	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	—
363	#433	C₂²×C₃₁	1	2	yes	—
364	#433	C₂²×C₃₁	1	2	yes	—
365	#432	C₂²×C₃₁	1	2	yes	—
366	C₂×C₃₁	—	4	1	no	—
367	C₂×C₃₁	—	8	1	no	—
368	C₂×C₃₁	—	8	2	no	—
369	C₂×C₃₁	#3487	1	1	yes	characteristic
370	C₂×C₃₁	#3586	1	2	yes	—
371	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
372	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
373	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
374	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
375	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
376	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
377	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
378	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
379	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
380	C₄⋊Q₈	—	2	4	no	—
381	C₄⋊Q₈	D₄×C₃₁	1	2	yes	—
382	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
383	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
384	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
385	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
386	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
387	#127	C₂³×C₃₁	1	2	yes	—
388	#126	—	2	4	no	—
389	#126	—	2	4	no	—
390	#126	—	2	4	no	—
391	#126	—	2	4	no	—
392	#125	—	2	4	no	—
393	#125	—	2	4	no	—
394	#125	—	2	4	no	—
395	#125	—	2	4	no	—
396	#125	—	2	4	no	—
397	#125	—	2	4	no	—
398	#125	—	2	4	no	—
399	#125	—	2	2	no	—
400	#125	D₄×C₃₁	1	2	yes	—
401	#125	C₂³×C₃₁	1	2	yes	—
402	#125	C₂³×C₃₁	1	2	yes	—
403	#124	—	2	4	no	—
404	#124	—	2	1	no	—
405	#124	—	2	1	no	—
406	#124	C₂³×C₃₁	1	4	yes	—
407	#124	C₂³×C₃₁	1	2	yes	—
408	#124	C₂³×C₃₁	1	2	yes	—
409	#123	—	2	4	no	—
410	#123	—	2	4	no	—
411	#123	—	2	2	no	—
412	C₂²⋊Q₈	—	2	4	no	—
413	C₂²⋊Q₈	—	2	2	no	—
414	C₂²⋊Q₈	—	2	2	no	—
415	C₂²⋊Q₈	D₄×C₃₁	1	4	yes	—
416	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
417	#121	—	2	2	no	—
418	#120	—	2	2	no	—
419	C₄×Q₈	—	2	4	no	—
420	C₄×Q₈	—	2	2	no	—
421	C₄×Q₈	—	2	4	no	—
422	C₄×Q₈	—	2	2	no	—
423	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
424	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
425	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
426	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
427	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
428	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
429	C₄×D₄	—	2	2	no	—
430	C₄×D₄	—	2	2	no	—
431	C₄×D₄	—	2	4	no	—
432	C₄×D₄	C₂³×C₃₁	1	2	yes	—
433	C₄×D₄	C₂³×C₃₁	1	4	yes	—
434	#117	—	2	2	no	—
435	#117	—	2	4	no	—
436	#117	—	2	2	no	—
437	#117	—	2	1	no	—
438	#117	C₂³×C₃₁	1	1	yes	characteristic
439	#117	C₂³×C₃₁	1	1	yes	characteristic
440	#117	C₂³×C₃₁	1	1	yes	characteristic
441	C₂×C₂²⋊C₄	—	2	2	no	—
442	C₂×C₂²⋊C₄	—	2	2	no	—
443	C₂×C₄²	—	2	1	no	—
444	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
445	#107	C₂³×C₃₁	1	4	yes	—
446	#106	C₂³×C₃₁	1	4	yes	—
447	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
448	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
449	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
450	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
451	#103	C₂³×C₃₁	1	4	yes	—
452	#103	C₂³×C₃₁	1	4	yes	—
453	#103	C₂³×C₃₁	1	4	yes	—
454	#103	C₂³×C₃₁	1	4	yes	—
455	#103	C₂³×C₃₁	1	4	yes	—
456	#102	C₂³×C₃₁	1	4	yes	—
457	#102	C₂³×C₃₁	1	4	yes	—
458	#102	C₂³×C₃₁	1	4	yes	—
459	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
460	#97	C₂³×C₃₁	1	4	yes	—
461	#97	C₂³×C₃₁	1	2	yes	—
462	#97	C₂³×C₃₁	1	2	yes	—
463	C₄×C₈	C₂³×C₃₁	1	2	yes	—
464	C₄×C₈	C₂³×C₃₁	1	2	yes	—
465	C₃₁	#61943	1	1	yes	characteristic, direct product factor, Sylow
466	C₂⁴	—	2	1	no	—
467	#41	—	4	2	no	—
468	C₂×Q₈	—	2	2	no	—
469	C₂×Q₈	—	2	1	no	—
470	C₂×Q₈	—	4	2	no	—
471	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
472	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
473	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
474	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
475	C₂×D₄	—	2	1	no	—
476	C₂×D₄	—	2	1	no	—
477	C₂×D₄	—	2	4	no	—
478	C₂×D₄	—	4	2	no	—
479	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
480	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
481	C₂²×C₄	—	2	1	no	—
482	C₂²×C₄	—	2	1	no	—
483	C₂²×C₄	—	2	2	no	—
484	C₂²×C₄	—	2	4	no	—
485	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
486	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
487	Q₁₆	—	4	4	no	—
488	Q₁₆	—	4	4	no	—
489	Q₁₆	—	4	4	no	—
490	SD₁₆	—	4	4	no	—
491	SD₁₆	—	4	4	no	—
492	SD₁₆	—	4	4	no	—
493	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
494	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
495	C₄⋊C₄	—	2	2	no	—
496	C₄⋊C₄	—	2	4	no	—
497	C₄⋊C₄	—	2	4	no	—
498	C₄⋊C₄	—	2	2	no	—
499	C₄⋊C₄	—	2	2	no	—
500	C₄⋊C₄	—	2	2	no	—
501	C₄⋊C₄	—	2	4	no	—
502	C₄⋊C₄	—	2	4	no	—
503	C₄⋊C₄	—	2	4	no	—
504	C₄⋊C₄	—	2	4	no	—
505	C₄⋊C₄	—	2	4	no	—
506	C₄⋊C₄	—	2	4	no	—
507	C₄⋊C₄	—	2	4	no	—
508	C₄⋊C₄	—	2	4	no	—
509	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
510	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
511	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
512	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
513	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
514	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
515	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
516	C₂²⋊C₄	—	2	1	no	—
517	C₂²⋊C₄	—	2	4	no	—
518	C₂²⋊C₄	—	2	4	no	—
519	C₂²⋊C₄	—	2	4	no	—
520	C₂²⋊C₄	—	2	4	no	—
521	C₂²⋊C₄	—	2	2	no	—
522	C₂²⋊C₄	—	2	4	no	—
523	C₂²⋊C₄	—	2	4	no	—
524	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
525	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
526	C₄²	—	2	2	no	—
527	C₄²	—	2	1	no	—
528	C₄²	—	2	4	no	—
529	C₄²	—	2	4	no	—
530	C₄²	—	2	4	no	—
531	C₄²	—	2	4	no	—
532	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
533	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
534	C₄²	C₂⁴×C₃₁	1	2	yes	—
535	C₄²	C₂⁴×C₃₁	1	2	yes	—
536	C₄²	C₂⁴×C₃₁	1	2	yes	—
537	C₂³	—	2	2	no	—
538	C₂³	—	2	1	no	—
539	C₂³	—	4	2	no	—
540	C₂³	—	8	2	no	—
541	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
542	Q₈	—	2	4	no	—
543	Q₈	—	2	4	no	—
544	Q₈	—	2	4	no	—
545	Q₈	—	2	2	no	—
546	Q₈	—	4	4	no	—
547	Q₈	—	4	4	no	—
548	Q₈	—	4	2	no	—
549	D₄	—	2	4	no	—
550	D₄	—	2	2	no	—
551	D₄	—	4	2	no	—
552	D₄	—	4	2	no	—
553	D₄	—	8	4	no	—
554	C₂×C₄	—	2	2	no	—
555	C₂×C₄	—	2	2	no	—
556	C₂×C₄	—	2	4	no	—
557	C₂×C₄	—	2	4	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	1	no	—
560	C₂×C₄	—	4	2	no	—
561	C₂×C₄	—	4	2	no	—
562	C₂×C₄	—	4	2	no	—
563	C₂×C₄	—	8	2	no	—
564	C₂×C₄	—	8	4	no	—
565	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
566	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
567	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
568	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
569	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
570	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
571	C₈	—	2	4	no	—
572	C₈	—	2	4	no	—
573	C₂²	—	2	1	no	—
574	C₂²	—	2	2	no	—
575	C₂²	—	4	1	no	—
576	C₂²	—	4	2	no	—
577	C₂²	—	4	2	no	—
578	C₂²	—	4	4	no	—
579	C₂²	—	16	2	no	—
580	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
581	C₄	—	2	1	no	—
582	C₄	—	2	2	no	—
583	C₄	—	2	2	no	—
584	C₄	—	2	4	no	—
585	C₄	—	2	4	no	—
586	C₄	—	4	1	no	—
587	C₄	—	4	2	no	—
588	C₄	—	4	4	no	—
589	C₄	—	4	2	no	—
590	C₄	—	4	4	no	—
591	C₄	C₃₁×#585	1	2	yes	—
592	C₂	—	4	1	no	—
593	C₂	—	8	2	no	—
594	C₂	—	8	1	no	—
595	C₂	C₃₁×#3487	1	1	yes	characteristic
596	C₂	C₃₁×#3586	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 ◁ C₃₁×#533 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3556	C₂	semidirect
3. C₃₁×#3400	C₂	semidirect
4. C₃₁×#3398	C₂	semidirect
5. C₃₁×#3377	C₂	semidirect
6. C₃₁×#3371	C₂	semidirect
7. C₃₁×#3223	C₂	semidirect
9. C₃₁×#3215	C₂	semidirect
10. C₃₁×#3127	C₂	semidirect
11. C₃₁×#3123	C₂	semidirect
12. C₃₁×#3116	C₂	semidirect
13. C₃₁×#2987	C₂	semidirect
17. C₃₁×#557	C₂²	semidirect
18. C₃₁×#557	C₂²	semidirect
36. C₃₁×#497	C₂²	semidirect
37. C₃₁×#494	C₂²	semidirect
39. C₃₁×#491	C₂²	semidirect
43. C₃₁×#488	C₂²	semidirect
45. C₃₁×#479	C₂²	semidirect
47. C₃₁×#477	C₂²	semidirect
48. C₃₁×#476	C₂²	semidirect
52. C₃₁×#464	C₂²	semidirect
53. C₃₁×#462	C₂²	semidirect
54. C₃₁×#462	C₂²	semidirect
55. C₃₁×#462	C₂²	semidirect
57. C₃₁×#461	C₂²	semidirect
62. C₃₁×#446	C₂²	semidirect
63. C₃₁×#446	C₂²	semidirect
64. C₃₁×#441	C₂²	semidirect
65. C₃₁×#441	C₂²	semidirect
67. C₃₁×#440	C₂²	semidirect
68. C₄×Q₁₆×C₃₁	C₂²	semidirect
238. #61943	C₃₁	direct (N × H)
275. #3556	C₂×C₃₁	semidirect
277. #3400	C₂×C₃₁	semidirect
278. #3398	C₂×C₃₁	semidirect
279. #3377	C₂×C₃₁	semidirect
280. #3371	C₂×C₃₁	semidirect
281. #3223	C₂×C₃₁	semidirect
283. #3215	C₂×C₃₁	semidirect
284. #3127	C₂×C₃₁	semidirect
285. #3123	C₂×C₃₁	semidirect
286. #3116	C₂×C₃₁	semidirect
287. #2987	C₂×C₃₁	semidirect
310. #557	C₂²×C₃₁	semidirect
311. #557	C₂²×C₃₁	semidirect
329. #497	C₂²×C₃₁	semidirect
330. #494	C₂²×C₃₁	semidirect
332. #491	C₂²×C₃₁	semidirect
336. #488	C₂²×C₃₁	semidirect
338. #479	C₂²×C₃₁	semidirect
340. #477	C₂²×C₃₁	semidirect
341. #476	C₂²×C₃₁	semidirect
345. #464	C₂²×C₃₁	semidirect
346. #462	C₂²×C₃₁	semidirect
347. #462	C₂²×C₃₁	semidirect
348. #462	C₂²×C₃₁	semidirect
350. #461	C₂²×C₃₁	semidirect
355. #446	C₂²×C₃₁	semidirect
356. #446	C₂²×C₃₁	semidirect
357. #441	C₂²×C₃₁	semidirect
358. #441	C₂²×C₃₁	semidirect
360. #440	C₂²×C₃₁	semidirect
361. C₄×Q₁₆	C₂²×C₃₁	semidirect
465. C₃₁	#61943	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×#3556	2
C₃₁×#3510	2
C₃₁×#3400	2
C₃₁×#3398	2
C₃₁×#3377	2
C₃₁×#3371	2
C₃₁×#3223	2
C₃₁×#3222	2
C₃₁×#3215	2
C₃₁×#3127	2
C₃₁×#3123	2
C₃₁×#3116	2
C₃₁×#2987	2
#61943	31

Subgroups — C31×#61943 #21518702

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61943 #21518702