Subgroups — C₃₁×#61914 #21518673

Order 7936.

1752 conjugacy classes of subgroups.

The subgroup lattice has 748 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3548	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3507	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3399	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3397	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3392	C₂	1	2	yes	maximal, complemented
6	C₃₁×#3376	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3372	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3361	C₂	1	2	yes	maximal, complemented
9	C₃₁×#3221	C₂	1	4	yes	maximal, complemented
10	C₃₁×#3214	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#3207	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#3119	C₂	1	4	yes	maximal, complemented
13	C₃₁×#3112	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#3103	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#2985	C₂	1	1	yes	maximal, characteristic, complemented
16	C₃₁×#562	—	2	2	no	—
17	C₃₁×#559	C₂²	1	2	yes	—
18	C₃₁×#557	C₂²	1	2	yes	—
19	C₃₁×#556	—	2	1	no	—
20	C₃₁×#556	C₂²	1	4	yes	—
21	C₃₁×#554	—	2	1	no	—
22	C₃₁×#554	C₂²	1	2	yes	—
23	C₃₁×#554	C₂²	1	2	yes	—
24	C₃₁×#552	—	2	2	no	—
25	C₃₁×#550	C₂²	1	2	yes	—
26	C₃₁×#550	C₂²	1	2	yes	—
27	C₃₁×#547	—	2	1	no	—
28	C₃₁×#547	—	2	1	no	—
29	C₃₁×#540	C₂²	1	4	yes	—
30	C₃₁×#539	—	2	2	no	—
31	C₃₁×#537	—	2	2	no	—
32	C₃₁×#537	—	2	2	no	—
33	C₃₁×#536	C₂²	1	2	yes	—
34	C₃₁×#533	C₂²	1	1	yes	characteristic
35	C₃₁×#532	C₂²	1	1	yes	characteristic
36	C₃₁×#532	C₂²	1	1	yes	characteristic
37	C₃₁×#532	C₂²	1	1	yes	characteristic
38	C₃₁×#529	—	2	2	no	—
39	C₃₁×#520	C₂²	1	2	yes	—
40	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
41	C₃₁×#496	C₂²	1	2	yes	—
42	C₃₁×#496	C₂²	1	4	yes	—
43	C₃₁×#493	C₂²	1	1	yes	characteristic
44	C₃₁×#492	C₂²	1	1	yes	characteristic
45	C₃₁×#492	C₂²	1	4	yes	—
46	C₃₁×#491	C₂²	1	4	yes	—
47	C₃₁×#489	C₂²	1	2	yes	—
48	C₃₁×#488	C₂²	1	4	yes	—
49	C₃₁×#487	C₂²	1	1	yes	characteristic
50	C₃₁×#486	C₂²	1	1	yes	characteristic
51	C₃₁×#481	C₂²	1	2	yes	—
52	C₃₁×#480	C₂²	1	2	yes	—
53	C₃₁×#479	C₂²	1	4	yes	—
54	C₃₁×#478	C₂²	1	4	yes	—
55	C₃₁×#477	C₂²	1	2	yes	—
56	C₃₁×#475	C₂²	1	2	yes	—
57	C₃₁×#463	C₂²	1	4	yes	—
58	C₃₁×#463	C₂²	1	4	yes	—
59	C₃₁×#463	C₂²	1	4	yes	—
60	C₃₁×#461	C₂²	1	2	yes	—
61	C₃₁×#461	C₂²	1	4	yes	—
62	C₃₁×#461	C₂²	1	4	yes	—
63	C₃₁×#460	C₂²	1	2	yes	—
64	C₃₁×#460	C₂²	1	2	yes	—
65	C₃₁×#460	C₂²	1	4	yes	—
66	C₃₁×#459	C₂²	1	2	yes	—
67	C₃₁×#459	C₂²	1	4	yes	—
68	C₃₁×#449	C₂²	1	4	yes	—
69	C₃₁×#449	C₂²	1	4	yes	—
70	C₃₁×#448	C₂²	1	2	yes	—
71	C₃₁×#448	C₂²	1	2	yes	—
72	C₃₁×#446	C₂²	1	2	yes	—
73	C₃₁×#446	C₂²	1	4	yes	—
74	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
75	C₃₁×#442	C₂²	1	4	yes	—
76	C₃₁×#440	C₂²	1	4	yes	—
77	C₃₁×#440	C₂²	1	4	yes	—
78	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
79	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
80	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	—
81	C₄×D₈×C₃₁	C₂²	1	4	yes	—
82	C₃₁×#433	C₂²	1	4	yes	—
83	C₃₁×#432	C₂²	1	1	yes	characteristic
84	C₃₁×#431	C₂²	1	1	yes	characteristic
85	C₂×#41×C₃₁	—	2	2	no	—
86	C₂×#41×C₃₁	—	2	2	no	—
87	C₂×#41×C₃₁	—	2	1	no	—
88	C₂×#41×C₃₁	—	2	1	no	—
89	C₂×#41×C₃₁	C₂³	1	2	yes	—
90	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
91	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
92	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
93	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
94	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
95	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
96	C₂×D₈×C₃₁	C₂³	1	4	yes	—
97	C₂×D₈×C₃₁	C₂³	1	4	yes	—
98	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
99	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
100	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
101	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
102	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
103	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
104	C₃₁×#127	C₂³	1	1	yes	characteristic
105	C₃₁×#127	C₂³	1	4	yes	—
106	C₃₁×#127	C₂³	1	1	yes	characteristic
107	C₃₁×#127	C₂³	1	1	yes	characteristic
108	C₃₁×#126	—	2	4	no	—
109	C₃₁×#126	—	2	4	no	—
110	C₃₁×#125	—	2	2	no	—
111	C₃₁×#125	—	2	4	no	—
112	C₃₁×#125	D₄	1	2	yes	—
113	C₃₁×#125	C₂³	1	4	yes	—
114	C₃₁×#124	—	2	2	no	—
115	C₃₁×#124	—	2	1	no	—
116	C₃₁×#124	—	2	1	no	—
117	C₃₁×#124	C₂³	1	4	yes	—
118	C₃₁×#124	C₂³	1	2	yes	—
119	C₃₁×#123	—	2	4	no	—
120	C₃₁×#123	—	2	4	no	—
121	C₃₁×#123	—	2	4	no	—
122	C₃₁×#123	—	2	4	no	—
123	C₃₁×C₂²⋊Q₈	—	2	2	no	—
124	C₃₁×C₂²⋊Q₈	—	2	2	no	—
125	C₃₁×C₂²⋊Q₈	—	2	4	no	—
126	C₃₁×C₂²⋊Q₈	—	2	2	no	—
127	C₃₁×C₂²⋊Q₈	—	2	2	no	—
128	C₃₁×C₂²⋊Q₈	—	2	4	no	—
129	C₃₁×C₂²⋊Q₈	—	2	4	no	—
130	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
131	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
132	C₃₁×#121	—	2	1	no	—
133	C₃₁×#121	—	2	1	no	—
134	C₃₁×#121	—	2	1	no	—
135	C₃₁×#121	—	2	1	no	—
136	C₃₁×#121	—	2	2	no	—
137	C₃₁×#121	—	2	2	no	—
138	C₃₁×#121	—	2	4	no	—
139	C₃₁×#121	—	2	4	no	—
140	C₃₁×#121	—	2	4	no	—
141	C₃₁×#121	—	2	4	no	—
142	C₃₁×#121	C₂³	1	2	yes	—
143	C₃₁×#121	C₂³	1	2	yes	—
144	C₄×Q₈×C₃₁	—	2	2	no	—
145	C₄×Q₈×C₃₁	—	2	1	no	—
146	C₄×Q₈×C₃₁	—	2	1	no	—
147	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
148	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
149	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
150	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
151	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
152	C₄×D₄×C₃₁	—	2	2	no	—
153	C₄×D₄×C₃₁	—	2	4	no	—
154	C₄×D₄×C₃₁	—	2	1	no	—
155	C₄×D₄×C₃₁	—	2	1	no	—
156	C₄×D₄×C₃₁	—	2	1	no	—
157	C₄×D₄×C₃₁	—	2	1	no	—
158	C₄×D₄×C₃₁	C₂³	1	4	yes	—
159	C₄×D₄×C₃₁	C₂³	1	4	yes	—
160	C₄×D₄×C₃₁	C₂³	1	2	yes	—
161	C₄×D₄×C₃₁	C₂³	1	4	yes	—
162	C₄×D₄×C₃₁	C₂³	1	2	yes	—
163	C₃₁×#117	—	2	2	no	—
164	C₃₁×#117	—	2	4	no	—
165	C₃₁×#117	C₂³	1	2	yes	—
166	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
167	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
168	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
169	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
170	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
171	C₂×C₄²×C₃₁	—	2	1	no	—
172	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
173	C₃₁×#107	C₂³	1	4	yes	—
174	C₃₁×#106	C₂³	1	2	yes	—
175	C₃₁×#106	C₂³	1	2	yes	—
176	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
177	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
178	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
179	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
180	C₃₁×#103	C₂³	1	2	yes	—
181	C₃₁×#103	C₂³	1	2	yes	—
182	C₃₁×#103	C₂³	1	4	yes	—
183	C₃₁×#103	C₂³	1	4	yes	—
184	C₃₁×#103	C₂³	1	4	yes	—
185	C₃₁×#102	C₂³	1	2	yes	—
186	C₃₁×#102	C₂³	1	4	yes	—
187	C₃₁×#102	C₂³	1	2	yes	—
188	C₃₁×#102	C₂³	1	4	yes	—
189	C₃₁×#102	C₂³	1	4	yes	—
190	C₃₁×C₂²⋊C₈	C₂³	1	2	yes	—
191	C₃₁×C₂²⋊C₈	C₂³	1	2	yes	—
192	C₃₁×#97	C₂³	1	2	yes	—
193	C₃₁×#97	C₂³	1	4	yes	—
194	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
195	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
196	C₄×C₈×C₃₁	C₂³	1	4	yes	—
197	#41×C₃₁	—	4	4	no	—
198	#41×C₃₁	—	8	2	no	—
199	#41×C₃₁	—	8	2	no	—
200	#41×C₃₁	—	8	4	no	—
201	#41×C₃₁	—	8	4	no	—
202	C₂×Q₈×C₃₁	—	2	1	no	—
203	C₂×Q₈×C₃₁	—	2	1	no	—
204	C₂×Q₈×C₃₁	—	2	1	no	—
205	C₂×Q₈×C₃₁	—	4	2	no	—
206	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
207	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
208	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
209	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
210	C₂×D₄×C₃₁	—	2	2	no	—
211	C₂×D₄×C₃₁	—	2	2	no	—
212	C₂×D₄×C₃₁	—	2	2	no	—
213	C₂×D₄×C₃₁	—	2	1	no	—
214	C₂×D₄×C₃₁	—	2	1	no	—
215	C₂×D₄×C₃₁	—	2	1	no	—
216	C₂×D₄×C₃₁	—	2	1	no	—
217	C₂×D₄×C₃₁	—	2	4	no	—
218	C₂×D₄×C₃₁	—	2	4	no	—
219	C₂×D₄×C₃₁	—	2	1	no	—
220	C₂×D₄×C₃₁	—	2	1	no	—
221	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
222	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
223	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
224	C₂²×C₄×C₃₁	—	2	1	no	—
225	C₂²×C₄×C₃₁	—	2	1	no	—
226	C₂²×C₄×C₃₁	—	2	2	no	—
227	C₂²×C₄×C₃₁	—	2	2	no	—
228	C₂²×C₄×C₃₁	—	2	2	no	—
229	C₂²×C₄×C₃₁	—	2	4	no	—
230	C₂²×C₄×C₃₁	—	2	4	no	—
231	C₂²×C₄×C₃₁	—	2	4	no	—
232	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
233	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
234	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
235	SD₁₆×C₃₁	—	4	2	no	—
236	SD₁₆×C₃₁	—	4	4	no	—
237	SD₁₆×C₃₁	—	4	4	no	—
238	SD₁₆×C₃₁	—	4	2	no	—
239	SD₁₆×C₃₁	—	4	4	no	—
240	D₈×C₃₁	—	4	4	no	—
241	D₈×C₃₁	—	4	4	no	—
242	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
243	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
244	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
245	C₄⋊C₄×C₃₁	—	2	1	no	—
246	C₄⋊C₄×C₃₁	—	2	1	no	—
247	C₄⋊C₄×C₃₁	—	2	1	no	—
248	C₄⋊C₄×C₃₁	—	2	4	no	—
249	C₄⋊C₄×C₃₁	—	2	2	no	—
250	C₄⋊C₄×C₃₁	—	2	2	no	—
251	C₄⋊C₄×C₃₁	—	2	4	no	—
252	C₄⋊C₄×C₃₁	—	2	4	no	—
253	C₄⋊C₄×C₃₁	—	2	4	no	—
254	C₄⋊C₄×C₃₁	—	2	1	no	—
255	C₄⋊C₄×C₃₁	—	2	1	no	—
256	C₄⋊C₄×C₃₁	—	2	4	no	—
257	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
258	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
259	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
260	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
261	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
262	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
263	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
264	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
265	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
266	C₂²⋊C₄×C₃₁	—	2	2	no	—
267	C₂²⋊C₄×C₃₁	—	2	4	no	—
268	C₂²⋊C₄×C₃₁	—	2	4	no	—
269	C₂²⋊C₄×C₃₁	—	2	2	no	—
270	C₂²⋊C₄×C₃₁	—	2	2	no	—
271	C₂²⋊C₄×C₃₁	—	2	2	no	—
272	C₂²⋊C₄×C₃₁	—	2	4	no	—
273	C₂²⋊C₄×C₃₁	—	2	4	no	—
274	C₂²⋊C₄×C₃₁	—	2	4	no	—
275	C₂²⋊C₄×C₃₁	—	2	2	no	—
276	C₂²⋊C₄×C₃₁	—	2	2	no	—
277	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
278	C₄²×C₃₁	—	2	2	no	—
279	C₄²×C₃₁	—	2	2	no	—
280	C₄²×C₃₁	—	2	4	no	—
281	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
282	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
283	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
284	C₄²×C₃₁	C₂⁴	1	4	yes	—
285	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
286	#61914	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
287	C₂³×C₃₁	—	2	2	no	—
288	C₂³×C₃₁	—	2	2	no	—
289	C₂³×C₃₁	—	2	2	no	—
290	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
291	Q₈×C₃₁	—	2	4	no	—
292	Q₈×C₃₁	—	2	2	no	—
293	Q₈×C₃₁	—	2	2	no	—
294	Q₈×C₃₁	—	2	4	no	—
295	Q₈×C₃₁	—	4	2	no	—
296	Q₈×C₃₁	—	4	2	no	—
297	Q₈×C₃₁	—	8	1	no	—
298	Q₈×C₃₁	—	8	1	no	—
299	D₄×C₃₁	—	2	2	no	—
300	D₄×C₃₁	—	2	4	no	—
301	D₄×C₃₁	—	2	4	no	—
302	D₄×C₃₁	—	2	2	no	—
303	D₄×C₃₁	—	4	2	no	—
304	D₄×C₃₁	—	4	4	no	—
305	D₄×C₃₁	—	4	2	no	—
306	D₄×C₃₁	—	4	2	no	—
307	D₄×C₃₁	—	8	1	no	—
308	D₄×C₃₁	—	8	1	no	—
309	D₄×C₃₁	—	8	4	no	—
310	D₄×C₃₁	—	8	1	no	—
311	D₄×C₃₁	—	8	4	no	—
312	D₄×C₃₁	—	8	2	no	—
313	D₄×C₃₁	—	8	2	no	—
314	C₂×C₄×C₃₁	—	2	2	no	—
315	C₂×C₄×C₃₁	—	2	2	no	—
316	C₂×C₄×C₃₁	—	2	2	no	—
317	C₂×C₄×C₃₁	—	2	2	no	—
318	C₂×C₄×C₃₁	—	2	2	no	—
319	C₂×C₄×C₃₁	—	2	2	no	—
320	C₂×C₄×C₃₁	—	2	2	no	—
321	C₂×C₄×C₃₁	—	4	1	no	—
322	C₂×C₄×C₃₁	—	4	4	no	—
323	C₂×C₄×C₃₁	—	4	4	no	—
324	C₂×C₄×C₃₁	—	4	2	no	—
325	C₂×C₄×C₃₁	—	4	4	no	—
326	C₂×C₄×C₃₁	—	8	4	no	—
327	C₂×C₄×C₃₁	—	8	4	no	—
328	C₂×C₄×C₃₁	—	8	4	no	—
329	C₂×C₄×C₃₁	—	8	1	no	—
330	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
331	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
332	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
333	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
334	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
335	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
336	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
337	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
338	C₈×C₃₁	—	2	2	no	—
339	C₈×C₃₁	—	2	4	no	—
340	C₈×C₃₁	—	2	2	no	—
341	#3548	C₂×C₃₁	1	1	yes	characteristic, complemented
342	#3507	C₂×C₃₁	1	1	yes	characteristic, complemented
343	#3399	C₂×C₃₁	1	2	yes	complemented
344	#3397	C₂×C₃₁	1	4	yes	complemented
345	#3392	C₂×C₃₁	1	2	yes	complemented
346	#3376	C₂×C₃₁	1	4	yes	complemented
347	#3372	C₂×C₃₁	1	2	yes	complemented
348	#3361	C₂×C₃₁	1	2	yes	complemented
349	#3221	C₂×C₃₁	1	4	yes	complemented
350	#3214	C₂×C₃₁	1	1	yes	characteristic, complemented
351	#3207	C₂×C₃₁	1	1	yes	characteristic, complemented
352	#3119	C₂×C₃₁	1	4	yes	complemented
353	#3112	C₂×C₃₁	1	1	yes	characteristic, complemented
354	#3103	C₂×C₃₁	1	1	yes	characteristic, complemented
355	#2985	C₂×C₃₁	1	1	yes	characteristic, complemented
356	C₂²×C₃₁	—	2	1	no	—
357	C₂²×C₃₁	—	2	1	no	—
358	C₂²×C₃₁	—	2	1	no	—
359	C₂²×C₃₁	—	4	2	no	—
360	C₂²×C₃₁	—	4	2	no	—
361	C₂²×C₃₁	—	4	2	no	—
362	C₂²×C₃₁	—	4	2	no	—
363	C₂²×C₃₁	—	4	2	no	—
364	C₂²×C₃₁	—	4	2	no	—
365	C₂²×C₃₁	—	4	2	no	—
366	C₂²×C₃₁	—	4	2	no	—
367	C₂²×C₃₁	—	4	2	no	—
368	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
369	C₄×C₃₁	—	2	1	no	—
370	C₄×C₃₁	—	2	1	no	—
371	C₄×C₃₁	—	2	4	no	—
372	C₄×C₃₁	—	2	1	no	—
373	C₄×C₃₁	—	2	1	no	—
374	C₄×C₃₁	—	2	4	no	—
375	C₄×C₃₁	—	4	2	no	—
376	C₄×C₃₁	—	4	2	no	—
377	C₄×C₃₁	—	4	2	no	—
378	C₄×C₃₁	—	4	2	no	—
379	C₄×C₃₁	—	4	2	no	—
380	C₄×C₃₁	#577	1	2	yes	—
381	C₄×C₃₁	#585	1	2	yes	—
382	#562	—	2	2	no	—
383	#559	C₂²×C₃₁	1	2	yes	—
384	#557	C₂²×C₃₁	1	2	yes	—
385	#556	—	2	1	no	—
386	#556	C₂²×C₃₁	1	4	yes	—
387	#554	—	2	1	no	—
388	#554	C₂²×C₃₁	1	2	yes	—
389	#554	C₂²×C₃₁	1	2	yes	—
390	#552	—	2	2	no	—
391	#550	C₂²×C₃₁	1	2	yes	—
392	#550	C₂²×C₃₁	1	2	yes	—
393	#547	—	2	1	no	—
394	#547	—	2	1	no	—
395	#540	C₂²×C₃₁	1	4	yes	—
396	#539	—	2	2	no	—
397	#537	—	2	2	no	—
398	#537	—	2	2	no	—
399	#536	C₂²×C₃₁	1	2	yes	—
400	#533	C₂²×C₃₁	1	1	yes	characteristic
401	#532	C₂²×C₃₁	1	1	yes	characteristic
402	#532	C₂²×C₃₁	1	1	yes	characteristic
403	#532	C₂²×C₃₁	1	1	yes	characteristic
404	#529	—	2	2	no	—
405	#520	C₂²×C₃₁	1	2	yes	—
406	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
407	#496	C₂²×C₃₁	1	2	yes	—
408	#496	C₂²×C₃₁	1	4	yes	—
409	#493	C₂²×C₃₁	1	1	yes	characteristic
410	#492	C₂²×C₃₁	1	1	yes	characteristic
411	#492	C₂²×C₃₁	1	4	yes	—
412	#491	C₂²×C₃₁	1	4	yes	—
413	#489	C₂²×C₃₁	1	2	yes	—
414	#488	C₂²×C₃₁	1	4	yes	—
415	#487	C₂²×C₃₁	1	1	yes	characteristic
416	#486	C₂²×C₃₁	1	1	yes	characteristic
417	#481	C₂²×C₃₁	1	2	yes	—
418	#480	C₂²×C₃₁	1	2	yes	—
419	#479	C₂²×C₃₁	1	4	yes	—
420	#478	C₂²×C₃₁	1	4	yes	—
421	#477	C₂²×C₃₁	1	2	yes	—
422	#475	C₂²×C₃₁	1	2	yes	—
423	#463	C₂²×C₃₁	1	4	yes	—
424	#463	C₂²×C₃₁	1	4	yes	—
425	#463	C₂²×C₃₁	1	4	yes	—
426	#461	C₂²×C₃₁	1	2	yes	—
427	#461	C₂²×C₃₁	1	4	yes	—
428	#461	C₂²×C₃₁	1	4	yes	—
429	#460	C₂²×C₃₁	1	2	yes	—
430	#460	C₂²×C₃₁	1	2	yes	—
431	#460	C₂²×C₃₁	1	4	yes	—
432	#459	C₂²×C₃₁	1	2	yes	—
433	#459	C₂²×C₃₁	1	4	yes	—
434	#449	C₂²×C₃₁	1	4	yes	—
435	#449	C₂²×C₃₁	1	4	yes	—
436	#448	C₂²×C₃₁	1	2	yes	—
437	#448	C₂²×C₃₁	1	2	yes	—
438	#446	C₂²×C₃₁	1	2	yes	—
439	#446	C₂²×C₃₁	1	4	yes	—
440	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
441	#442	C₂²×C₃₁	1	4	yes	—
442	#440	C₂²×C₃₁	1	4	yes	—
443	#440	C₂²×C₃₁	1	4	yes	—
444	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
445	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
446	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	—
447	C₄×D₈	C₂²×C₃₁	1	4	yes	—
448	#433	C₂²×C₃₁	1	4	yes	—
449	#432	C₂²×C₃₁	1	1	yes	characteristic
450	#431	C₂²×C₃₁	1	1	yes	characteristic
451	C₂×C₃₁	—	4	1	no	—
452	C₂×C₃₁	—	8	2	no	—
453	C₂×C₃₁	—	8	2	no	—
454	C₂×C₃₁	—	8	2	no	—
455	C₂×C₃₁	#3487	1	1	yes	characteristic
456	C₂×C₃₁	C₂×#577	1	1	yes	characteristic
457	C₂×C₃₁	#3585	1	1	yes	characteristic
458	C₂×#41	—	2	2	no	—
459	C₂×#41	—	2	2	no	—
460	C₂×#41	—	2	1	no	—
461	C₂×#41	—	2	1	no	—
462	C₂×#41	C₂³×C₃₁	1	2	yes	—
463	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
464	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
465	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
466	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
467	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
468	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
469	C₂×D₈	C₂³×C₃₁	1	4	yes	—
470	C₂×D₈	C₂³×C₃₁	1	4	yes	—
471	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
472	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
473	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
474	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
475	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
476	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
477	#127	C₂³×C₃₁	1	1	yes	characteristic
478	#127	C₂³×C₃₁	1	1	yes	characteristic
479	#127	C₂³×C₃₁	1	1	yes	characteristic
480	#127	C₂³×C₃₁	1	4	yes	—
481	#126	—	2	4	no	—
482	#126	—	2	4	no	—
483	#125	—	2	2	no	—
484	#125	—	2	4	no	—
485	#125	D₄×C₃₁	1	2	yes	—
486	#125	C₂³×C₃₁	1	4	yes	—
487	#124	—	2	2	no	—
488	#124	—	2	1	no	—
489	#124	—	2	1	no	—
490	#124	C₂³×C₃₁	1	2	yes	—
491	#124	C₂³×C₃₁	1	4	yes	—
492	#123	—	2	4	no	—
493	#123	—	2	4	no	—
494	#123	—	2	4	no	—
495	#123	—	2	4	no	—
496	C₂²⋊Q₈	—	2	2	no	—
497	C₂²⋊Q₈	—	2	2	no	—
498	C₂²⋊Q₈	—	2	4	no	—
499	C₂²⋊Q₈	—	2	2	no	—
500	C₂²⋊Q₈	—	2	2	no	—
501	C₂²⋊Q₈	—	2	4	no	—
502	C₂²⋊Q₈	—	2	4	no	—
503	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
504	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
505	#121	—	2	1	no	—
506	#121	—	2	1	no	—
507	#121	—	2	1	no	—
508	#121	—	2	1	no	—
509	#121	—	2	2	no	—
510	#121	—	2	2	no	—
511	#121	—	2	4	no	—
512	#121	—	2	4	no	—
513	#121	—	2	4	no	—
514	#121	—	2	4	no	—
515	#121	C₂³×C₃₁	1	2	yes	—
516	#121	C₂³×C₃₁	1	2	yes	—
517	C₄×Q₈	—	2	2	no	—
518	C₄×Q₈	—	2	1	no	—
519	C₄×Q₈	—	2	1	no	—
520	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
521	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
522	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
523	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
524	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
525	C₄×D₄	—	2	2	no	—
526	C₄×D₄	—	2	4	no	—
527	C₄×D₄	—	2	1	no	—
528	C₄×D₄	—	2	1	no	—
529	C₄×D₄	—	2	1	no	—
530	C₄×D₄	—	2	1	no	—
531	C₄×D₄	C₂³×C₃₁	1	4	yes	—
532	C₄×D₄	C₂³×C₃₁	1	2	yes	—
533	C₄×D₄	C₂³×C₃₁	1	2	yes	—
534	C₄×D₄	C₂³×C₃₁	1	4	yes	—
535	C₄×D₄	C₂³×C₃₁	1	4	yes	—
536	#117	—	2	2	no	—
537	#117	—	2	4	no	—
538	#117	C₂³×C₃₁	1	2	yes	—
539	C₂×C₄⋊C₄	—	2	1	no	—
540	C₂×C₄⋊C₄	—	2	2	no	—
541	C₂×C₄⋊C₄	—	2	2	no	—
542	C₂×C₄⋊C₄	—	2	2	no	—
543	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
544	C₂×C₄²	—	2	1	no	—
545	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
546	#107	C₂³×C₃₁	1	4	yes	—
547	#106	C₂³×C₃₁	1	2	yes	—
548	#106	C₂³×C₃₁	1	2	yes	—
549	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
550	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
551	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
552	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
553	#103	C₂³×C₃₁	1	2	yes	—
554	#103	C₂³×C₃₁	1	2	yes	—
555	#103	C₂³×C₃₁	1	4	yes	—
556	#103	C₂³×C₃₁	1	4	yes	—
557	#103	C₂³×C₃₁	1	4	yes	—
558	#102	C₂³×C₃₁	1	2	yes	—
559	#102	C₂³×C₃₁	1	4	yes	—
560	#102	C₂³×C₃₁	1	2	yes	—
561	#102	C₂³×C₃₁	1	4	yes	—
562	#102	C₂³×C₃₁	1	4	yes	—
563	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
564	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
565	#97	C₂³×C₃₁	1	2	yes	—
566	#97	C₂³×C₃₁	1	4	yes	—
567	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
568	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
569	C₄×C₈	C₂³×C₃₁	1	4	yes	—
570	C₃₁	#61914	1	1	yes	characteristic, direct product factor, Sylow
571	#41	—	4	4	no	—
572	#41	—	8	2	no	—
573	#41	—	8	2	no	—
574	#41	—	8	4	no	—
575	#41	—	8	4	no	—
576	C₂×Q₈	—	2	1	no	—
577	C₂×Q₈	—	2	1	no	—
578	C₂×Q₈	—	2	1	no	—
579	C₂×Q₈	—	4	2	no	—
580	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
581	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
582	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
583	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
584	C₂×D₄	—	2	2	no	—
585	C₂×D₄	—	2	2	no	—
586	C₂×D₄	—	2	2	no	—
587	C₂×D₄	—	2	1	no	—
588	C₂×D₄	—	2	1	no	—
589	C₂×D₄	—	2	1	no	—
590	C₂×D₄	—	2	1	no	—
591	C₂×D₄	—	2	4	no	—
592	C₂×D₄	—	2	4	no	—
593	C₂×D₄	—	2	1	no	—
594	C₂×D₄	—	2	1	no	—
595	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
596	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
597	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
598	C₂²×C₄	—	2	1	no	—
599	C₂²×C₄	—	2	1	no	—
600	C₂²×C₄	—	2	2	no	—
601	C₂²×C₄	—	2	2	no	—
602	C₂²×C₄	—	2	2	no	—
603	C₂²×C₄	—	2	4	no	—
604	C₂²×C₄	—	2	4	no	—
605	C₂²×C₄	—	2	4	no	—
606	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
607	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
608	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
609	SD₁₆	—	4	2	no	—
610	SD₁₆	—	4	4	no	—
611	SD₁₆	—	4	4	no	—
612	SD₁₆	—	4	2	no	—
613	SD₁₆	—	4	4	no	—
614	D₈	—	4	4	no	—
615	D₈	—	4	4	no	—
616	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
617	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
618	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
619	C₄⋊C₄	—	2	1	no	—
620	C₄⋊C₄	—	2	1	no	—
621	C₄⋊C₄	—	2	1	no	—
622	C₄⋊C₄	—	2	4	no	—
623	C₄⋊C₄	—	2	2	no	—
624	C₄⋊C₄	—	2	2	no	—
625	C₄⋊C₄	—	2	4	no	—
626	C₄⋊C₄	—	2	4	no	—
627	C₄⋊C₄	—	2	4	no	—
628	C₄⋊C₄	—	2	1	no	—
629	C₄⋊C₄	—	2	1	no	—
630	C₄⋊C₄	—	2	4	no	—
631	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
632	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
633	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
634	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
635	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
636	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
637	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
638	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
639	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
640	C₂²⋊C₄	—	2	2	no	—
641	C₂²⋊C₄	—	2	4	no	—
642	C₂²⋊C₄	—	2	4	no	—
643	C₂²⋊C₄	—	2	2	no	—
644	C₂²⋊C₄	—	2	2	no	—
645	C₂²⋊C₄	—	2	2	no	—
646	C₂²⋊C₄	—	2	4	no	—
647	C₂²⋊C₄	—	2	4	no	—
648	C₂²⋊C₄	—	2	4	no	—
649	C₂²⋊C₄	—	2	2	no	—
650	C₂²⋊C₄	—	2	2	no	—
651	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
652	C₄²	—	2	2	no	—
653	C₄²	—	2	2	no	—
654	C₄²	—	2	4	no	—
655	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
656	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
657	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
658	C₄²	C₂⁴×C₃₁	1	4	yes	—
659	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
660	C₂³	—	2	2	no	—
661	C₂³	—	2	2	no	—
662	C₂³	—	2	2	no	—
663	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
664	Q₈	—	2	4	no	—
665	Q₈	—	2	2	no	—
666	Q₈	—	2	2	no	—
667	Q₈	—	2	4	no	—
668	Q₈	—	4	2	no	—
669	Q₈	—	4	2	no	—
670	Q₈	—	8	1	no	—
671	Q₈	—	8	1	no	—
672	D₄	—	2	2	no	—
673	D₄	—	2	4	no	—
674	D₄	—	2	4	no	—
675	D₄	—	2	2	no	—
676	D₄	—	4	2	no	—
677	D₄	—	4	4	no	—
678	D₄	—	4	2	no	—
679	D₄	—	4	2	no	—
680	D₄	—	8	1	no	—
681	D₄	—	8	1	no	—
682	D₄	—	8	4	no	—
683	D₄	—	8	1	no	—
684	D₄	—	8	4	no	—
685	D₄	—	8	2	no	—
686	D₄	—	8	2	no	—
687	C₂×C₄	—	2	2	no	—
688	C₂×C₄	—	2	2	no	—
689	C₂×C₄	—	2	2	no	—
690	C₂×C₄	—	2	2	no	—
691	C₂×C₄	—	2	2	no	—
692	C₂×C₄	—	2	2	no	—
693	C₂×C₄	—	2	2	no	—
694	C₂×C₄	—	4	1	no	—
695	C₂×C₄	—	4	4	no	—
696	C₂×C₄	—	4	4	no	—
697	C₂×C₄	—	4	2	no	—
698	C₂×C₄	—	4	4	no	—
699	C₂×C₄	—	8	4	no	—
700	C₂×C₄	—	8	4	no	—
701	C₂×C₄	—	8	4	no	—
702	C₂×C₄	—	8	1	no	—
703	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
704	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
705	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
706	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
707	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
708	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
709	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
710	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
711	C₈	—	2	2	no	—
712	C₈	—	2	4	no	—
713	C₈	—	2	2	no	—
714	C₂²	—	2	1	no	—
715	C₂²	—	2	1	no	—
716	C₂²	—	2	1	no	—
717	C₂²	—	4	2	no	—
718	C₂²	—	4	2	no	—
719	C₂²	—	4	2	no	—
720	C₂²	—	4	2	no	—
721	C₂²	—	4	2	no	—
722	C₂²	—	4	2	no	—
723	C₂²	—	4	2	no	—
724	C₂²	—	4	2	no	—
725	C₂²	—	4	2	no	—
726	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
727	C₄	—	2	1	no	—
728	C₄	—	2	1	no	—
729	C₄	—	2	4	no	—
730	C₄	—	2	1	no	—
731	C₄	—	2	1	no	—
732	C₄	—	2	4	no	—
733	C₄	—	4	2	no	—
734	C₄	—	4	2	no	—
735	C₄	—	4	2	no	—
736	C₄	—	4	2	no	—
737	C₄	—	4	2	no	—
738	C₄	C₃₁×#577	1	2	yes	—
739	C₄	C₃₁×#585	1	2	yes	—
740	C₂	—	4	1	no	—
741	C₂	—	8	2	no	—
742	C₂	—	8	2	no	—
743	C₂	—	8	2	no	—
744	C₂	C₃₁×#3487	1	1	yes	characteristic
745	C₂	C₂×C₃₁×#577	1	1	yes	characteristic
746	C₂	C₃₁×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
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×#3548	C₂	semidirect
2. C₃₁×#3507	C₂	semidirect
3. C₃₁×#3399	C₂	semidirect
4. C₃₁×#3397	C₂	semidirect
5. C₃₁×#3392	C₂	semidirect
6. C₃₁×#3376	C₂	semidirect
7. C₃₁×#3372	C₂	semidirect
8. C₃₁×#3361	C₂	semidirect
9. C₃₁×#3221	C₂	semidirect
10. C₃₁×#3214	C₂	semidirect
11. C₃₁×#3207	C₂	semidirect
12. C₃₁×#3119	C₂	semidirect
13. C₃₁×#3112	C₂	semidirect
14. C₃₁×#3103	C₂	semidirect
15. C₃₁×#2985	C₂	semidirect
286. #61914	C₃₁	direct (N × H)
341. #3548	C₂×C₃₁	semidirect
342. #3507	C₂×C₃₁	semidirect
343. #3399	C₂×C₃₁	semidirect
344. #3397	C₂×C₃₁	semidirect
345. #3392	C₂×C₃₁	semidirect
346. #3376	C₂×C₃₁	semidirect
347. #3372	C₂×C₃₁	semidirect
348. #3361	C₂×C₃₁	semidirect
349. #3221	C₂×C₃₁	semidirect
350. #3214	C₂×C₃₁	semidirect
351. #3207	C₂×C₃₁	semidirect
352. #3119	C₂×C₃₁	semidirect
353. #3112	C₂×C₃₁	semidirect
354. #3103	C₂×C₃₁	semidirect
355. #2985	C₂×C₃₁	semidirect
570. C₃₁	#61914	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×#3548	2
C₃₁×#3507	2
C₃₁×#3399	2
C₃₁×#3397	2
C₃₁×#3392	2
C₃₁×#3376	2
C₃₁×#3372	2
C₃₁×#3361	2
C₃₁×#3221	2
C₃₁×#3214	2
C₃₁×#3207	2
C₃₁×#3119	2
C₃₁×#3112	2
C₃₁×#3103	2
C₃₁×#2985	2
#61914	31

Subgroups — C31×#61914 #21518673

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61914 #21518673