Subgroups — C₃₁×#17601 #21474360

Order 7936.

882 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3523	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2984	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2320	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1521	C₂	1	2	yes	maximal
5	C₃₁×#1518	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1515	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1512	C₂	1	2	yes	maximal
8	C₃₁×#1509	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#1506	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1503	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1500	C₂	1	1	yes	maximal, characteristic
12	C₃₁×#557	C₄	1	1	yes	characteristic, complemented
13	C₃₁×#557	C₄	1	1	yes	characteristic, complemented
14	C₃₁×#557	C₄	1	2	yes	complemented
15	C₃₁×#556	C₄	1	1	yes	characteristic, complemented
16	C₃₁×#556	C₄	1	2	yes	complemented
17	C₃₁×#552	—	2	2	no	—
18	C₃₁×#552	—	2	1	no	—
19	D₄×Q₈×C₃₁	—	2	1	no	—
20	D₄×Q₈×C₃₁	C₄	1	1	yes	characteristic, complemented
21	C₃₁×#541	—	2	1	no	—
22	C₃₁×#541	—	2	2	no	—
23	C₃₁×#533	C₂²	1	1	yes	characteristic
24	C₃₁×#533	C₂²	1	2	yes	—
25	C₃₁×#529	C₂²	1	1	yes	characteristic
26	C₃₁×#529	C₂²	1	2	yes	—
27	C₃₁×#519	—	2	1	no	—
28	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#446	—	2	1	no	—
30	C₃₁×#446	—	2	2	no	—
31	C₈×Q₈×C₃₁	—	2	1	no	—
32	C₃₁×#432	C₂²	1	1	yes	characteristic
33	C₃₁×#432	C₂²	1	2	yes	—
34	C₃₁×#431	C₂²	1	1	yes	characteristic
35	C₃₁×#431	C₂²	1	2	yes	—
36	C₃₁×#391	—	2	1	no	—
37	C₃₁×#389	—	2	1	no	—
38	C₃₁×#387	—	2	2	no	—
39	C₃₁×#383	C₂²	1	2	yes	—
40	C₃₁×#382	C₂²	1	2	yes	—
41	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
42	C₃₁×#376	C₂²	1	1	yes	characteristic
43	C₃₁×#328	C₂²	1	1	yes	characteristic
44	C₃₁×#328	C₂²	1	1	yes	characteristic
45	C₃₁×#328	C₂²	1	2	yes	—
46	C₃₁×#328	C₂²	1	2	yes	—
47	C₃₁×#328	C₂²	1	2	yes	—
48	C₃₁×#328	C₂²	1	2	yes	—
49	C₃₁×#328	C₂²	1	2	yes	—
50	C₃₁×#327	C₂²	1	1	yes	characteristic
51	C₃₁×#327	C₂²	1	1	yes	characteristic
52	C₃₁×#327	C₂²	1	2	yes	—
53	C₂²×Q₈×C₃₁	—	2	1	no	—
54	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
55	C₃₁×C₄⋊Q₈	—	2	1	no	—
56	C₃₁×C₄⋊Q₈	—	2	2	no	—
57	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
58	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
59	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
60	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
61	C₃₁×#126	—	2	2	no	—
62	C₃₁×#126	—	2	2	no	—
63	C₃₁×#126	—	2	2	no	—
64	C₃₁×#126	—	2	2	no	—
65	C₃₁×#126	—	2	1	no	—
66	C₃₁×#125	—	2	2	no	—
67	C₃₁×#125	—	2	2	no	—
68	C₃₁×#125	—	2	2	no	—
69	C₃₁×#125	—	2	2	no	—
70	C₃₁×#125	—	2	1	no	—
71	C₃₁×#125	—	2	2	no	—
72	C₃₁×#125	—	2	1	no	—
73	C₃₁×#125	C₂×C₄	1	1	yes	characteristic
74	C₃₁×#125	C₂×C₄	1	2	yes	—
75	C₃₁×#124	C₂×C₄	1	1	yes	characteristic
76	C₃₁×#124	C₂×C₄	1	2	yes	—
77	C₃₁×#123	—	2	1	no	—
78	C₃₁×#123	—	2	2	no	—
79	C₃₁×C₂²⋊Q₈	—	2	1	no	—
80	C₃₁×C₂²⋊Q₈	—	2	2	no	—
81	C₃₁×C₂²⋊Q₈	—	2	1	no	—
82	C₃₁×C₂²⋊Q₈	—	2	2	no	—
83	C₃₁×C₂²⋊Q₈	—	2	2	no	—
84	C₃₁×C₂²⋊Q₈	—	2	1	no	—
85	C₃₁×C₂²⋊Q₈	C₂³	1	1	yes	characteristic
86	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
87	C₃₁×#121	C₂³	1	1	yes	characteristic
88	C₄×Q₈×C₃₁	—	2	1	no	—
89	C₄×Q₈×C₃₁	—	2	2	no	—
90	C₄×Q₈×C₃₁	—	2	1	no	—
91	C₄×Q₈×C₃₁	—	2	1	no	—
92	C₄×Q₈×C₃₁	—	2	2	no	—
93	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
94	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
95	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
96	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
97	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
98	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
99	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
100	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
101	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
102	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
103	C₄×Q₈×C₃₁	D₄	1	2	yes	—
104	C₄×D₄×C₃₁	—	2	1	no	—
105	C₄×D₄×C₃₁	—	2	2	no	—
106	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
107	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
108	C₃₁×#117	—	2	1	no	—
109	C₃₁×#117	—	2	2	no	—
110	C₃₁×#117	—	2	2	no	—
111	C₃₁×#117	—	2	1	no	—
112	C₃₁×#117	—	2	1	no	—
113	C₃₁×#117	—	2	2	no	—
114	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
115	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
116	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
117	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
118	C₂×C₄²×C₃₁	—	2	1	no	—
119	C₂×C₄²×C₃₁	—	2	1	no	—
120	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
122	C₃₁×C₄⋊C₈	—	2	2	no	—
123	C₃₁×C₄⋊C₈	—	2	1	no	—
124	C₃₁×C₄⋊C₈	—	2	1	no	—
125	C₃₁×C₄⋊C₈	—	2	2	no	—
126	C₃₁×C₂²⋊C₈	C₂³	1	1	yes	characteristic
127	C₃₁×C₂²⋊C₈	C₂³	1	1	yes	characteristic
128	C₃₁×C₂²⋊C₈	C₂³	1	2	yes	—
129	C₃₁×#97	—	2	2	no	—
130	C₃₁×#97	—	2	1	no	—
131	C₄×C₈×C₃₁	—	2	1	no	—
132	C₄×C₈×C₃₁	—	2	2	no	—
133	C₃₁×#95	—	2	1	no	—
134	C₃₁×#95	—	2	2	no	—
135	C₃₁×#95	—	2	2	no	—
136	C₃₁×#95	—	2	2	no	—
137	C₃₁×#95	C₂³	1	1	yes	characteristic
138	C₃₁×#95	C₂³	1	1	yes	characteristic
139	C₃₁×#95	C₂³	1	2	yes	—
140	C₂×Q₈×C₃₁	—	2	2	no	—
141	C₂×Q₈×C₃₁	—	2	1	no	—
142	C₂×Q₈×C₃₁	—	4	2	no	—
143	C₂×Q₈×C₃₁	—	4	4	no	—
144	C₂×Q₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
145	C₂×Q₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
146	C₂×Q₈×C₃₁	C₂²⋊C₄	1	2	yes	—
147	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
148	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
149	C₂×D₄×C₃₁	—	2	1	no	—
150	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
151	C₂²×C₄×C₃₁	—	2	2	no	—
152	C₂²×C₄×C₃₁	—	2	1	no	—
153	C₂²×C₄×C₃₁	—	2	1	no	—
154	C₂²×C₄×C₃₁	—	2	1	no	—
155	C₂²×C₄×C₃₁	—	2	2	no	—
156	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
158	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
160	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
161	C₂×C₈×C₃₁	—	2	1	no	—
162	C₂×C₈×C₃₁	—	2	1	no	—
163	C₂×C₈×C₃₁	—	2	2	no	—
164	C₄⋊C₄×C₃₁	—	2	2	no	—
165	C₄⋊C₄×C₃₁	—	2	1	no	—
166	C₄⋊C₄×C₃₁	—	2	1	no	—
167	C₄⋊C₄×C₃₁	—	2	2	no	—
168	C₄⋊C₄×C₃₁	—	2	2	no	—
169	C₄⋊C₄×C₃₁	—	2	1	no	—
170	C₄⋊C₄×C₃₁	—	2	2	no	—
171	C₄⋊C₄×C₃₁	—	2	1	no	—
172	C₄⋊C₄×C₃₁	—	2	1	no	—
173	C₄⋊C₄×C₃₁	—	2	2	no	—
174	C₄⋊C₄×C₃₁	—	2	2	no	—
175	C₄⋊C₄×C₃₁	—	2	2	no	—
176	C₄⋊C₄×C₃₁	—	2	2	no	—
177	C₄⋊C₄×C₃₁	—	2	2	no	—
178	C₄⋊C₄×C₃₁	—	2	1	no	—
179	C₄⋊C₄×C₃₁	—	2	1	no	—
180	C₄⋊C₄×C₃₁	—	2	2	no	—
181	C₄⋊C₄×C₃₁	—	4	2	no	—
182	C₄⋊C₄×C₃₁	—	8	2	no	—
183	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
184	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
185	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
186	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
187	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
188	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
189	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
190	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
191	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
192	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
193	C₂²⋊C₄×C₃₁	—	2	2	no	—
194	C₂²⋊C₄×C₃₁	—	2	1	no	—
195	C₂²⋊C₄×C₃₁	—	2	1	no	—
196	C₂²⋊C₄×C₃₁	—	2	2	no	—
197	C₂²⋊C₄×C₃₁	—	2	1	no	—
198	C₂²⋊C₄×C₃₁	—	2	2	no	—
199	C₂²⋊C₄×C₃₁	—	2	2	no	—
200	C₂²⋊C₄×C₃₁	—	2	1	no	—
201	C₂²⋊C₄×C₃₁	—	2	1	no	—
202	C₄²×C₃₁	—	2	2	no	—
203	C₄²×C₃₁	—	2	2	no	—
204	C₄²×C₃₁	—	2	2	no	—
205	C₄²×C₃₁	—	2	2	no	—
206	C₄²×C₃₁	—	2	1	no	—
207	C₄²×C₃₁	—	4	2	no	—
208	C₄²×C₃₁	—	8	1	no	—
209	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
210	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
211	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
212	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
213	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
214	C₄²×C₃₁	C₂×D₄	1	2	yes	—
215	C₄²×C₃₁	C₂×D₄	1	2	yes	—
216	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
217	#17601	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
218	C₂³×C₃₁	—	2	1	no	—
219	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
220	Q₈×C₃₁	—	2	2	no	—
221	Q₈×C₃₁	—	2	2	no	—
222	Q₈×C₃₁	—	4	2	no	—
223	Q₈×C₃₁	—	4	1	no	—
224	Q₈×C₃₁	—	4	2	no	—
225	Q₈×C₃₁	—	4	4	no	—
226	Q₈×C₃₁	—	4	2	no	—
227	D₄×C₃₁	—	4	1	no	—
228	D₄×C₃₁	—	4	1	no	—
229	D₄×C₃₁	—	4	1	no	—
230	C₂×C₄×C₃₁	—	2	1	no	—
231	C₂×C₄×C₃₁	—	2	2	no	—
232	C₂×C₄×C₃₁	—	2	1	no	—
233	C₂×C₄×C₃₁	—	2	2	no	—
234	C₂×C₄×C₃₁	—	2	2	no	—
235	C₂×C₄×C₃₁	—	2	2	no	—
236	C₂×C₄×C₃₁	—	2	1	no	—
237	C₂×C₄×C₃₁	—	2	1	no	—
238	C₂×C₄×C₃₁	—	2	2	no	—
239	C₂×C₄×C₃₁	—	2	1	no	—
240	C₂×C₄×C₃₁	—	4	2	no	—
241	C₂×C₄×C₃₁	—	4	1	no	—
242	C₂×C₄×C₃₁	—	4	4	no	—
243	C₂×C₄×C₃₁	—	4	2	no	—
244	C₂×C₄×C₃₁	—	4	1	no	—
245	C₂×C₄×C₃₁	—	4	1	no	—
246	C₂×C₄×C₃₁	—	4	2	no	—
247	C₂×C₄×C₃₁	—	4	1	no	—
248	C₂×C₄×C₃₁	—	4	1	no	—
249	C₂×C₄×C₃₁	—	4	2	no	—
250	C₂×C₄×C₃₁	—	4	2	no	—
251	C₂×C₄×C₃₁	—	4	1	no	—
252	C₂×C₄×C₃₁	—	4	1	no	—
253	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
254	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
255	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
256	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
257	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
258	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
259	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
260	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
261	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
262	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
263	C₈×C₃₁	—	4	1	no	—
264	C₈×C₃₁	—	4	1	no	—
265	C₈×C₃₁	—	4	2	no	—
266	#3523	C₂×C₃₁	1	1	yes	characteristic
267	#2984	C₂×C₃₁	1	1	yes	characteristic, complemented
268	#2320	C₂×C₃₁	1	1	yes	characteristic, complemented
269	#1521	C₂×C₃₁	1	2	yes	—
270	#1518	C₂×C₃₁	1	2	yes	complemented
271	#1515	C₂×C₃₁	1	2	yes	complemented
272	#1512	C₂×C₃₁	1	2	yes	—
273	#1509	C₂×C₃₁	1	1	yes	characteristic
274	#1506	C₂×C₃₁	1	1	yes	characteristic, complemented
275	#1503	C₂×C₃₁	1	1	yes	characteristic, complemented
276	#1500	C₂×C₃₁	1	1	yes	characteristic
277	C₂²×C₃₁	—	2	1	no	—
278	C₂²×C₃₁	—	2	1	no	—
279	C₂²×C₃₁	—	4	1	no	—
280	C₂²×C₃₁	—	4	1	no	—
281	C₂²×C₃₁	—	4	1	no	—
282	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
283	C₂²×C₃₁	#577	1	1	yes	characteristic
284	C₂²×C₃₁	#578	1	1	yes	characteristic
285	C₄×C₃₁	—	2	1	no	—
286	C₄×C₃₁	—	2	1	no	—
287	C₄×C₃₁	—	2	1	no	—
288	C₄×C₃₁	—	2	1	no	—
289	C₄×C₃₁	—	2	2	no	—
290	C₄×C₃₁	—	2	2	no	—
291	C₄×C₃₁	—	2	1	no	—
292	C₄×C₃₁	—	2	2	no	—
293	C₄×C₃₁	—	2	2	no	—
294	C₄×C₃₁	—	4	2	no	—
295	C₄×C₃₁	—	4	1	no	—
296	C₄×C₃₁	—	4	1	no	—
297	C₄×C₃₁	—	4	2	no	—
298	C₄×C₃₁	—	4	1	no	—
299	C₄×C₃₁	—	8	1	no	—
300	C₄×C₃₁	—	8	2	no	—
301	C₄×C₃₁	—	8	1	no	—
302	C₄×C₃₁	#421	1	2	yes	—
303	#557	C₄×C₃₁	1	1	yes	characteristic, complemented
304	#557	C₄×C₃₁	1	1	yes	characteristic, complemented
305	#557	C₄×C₃₁	1	2	yes	complemented
306	#556	C₄×C₃₁	1	1	yes	characteristic, complemented
307	#556	C₄×C₃₁	1	2	yes	complemented
308	#552	—	2	2	no	—
309	#552	—	2	1	no	—
310	D₄×Q₈	—	2	1	no	—
311	D₄×Q₈	C₄×C₃₁	1	1	yes	characteristic, complemented
312	#541	—	2	1	no	—
313	#541	—	2	2	no	—
314	#533	C₂²×C₃₁	1	1	yes	characteristic
315	#533	C₂²×C₃₁	1	2	yes	—
316	#529	C₂²×C₃₁	1	1	yes	characteristic
317	#529	C₂²×C₃₁	1	2	yes	—
318	#519	—	2	1	no	—
319	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
320	#446	—	2	1	no	—
321	#446	—	2	2	no	—
322	C₈×Q₈	—	2	1	no	—
323	#432	C₂²×C₃₁	1	1	yes	characteristic
324	#432	C₂²×C₃₁	1	2	yes	—
325	#431	C₂²×C₃₁	1	1	yes	characteristic
326	#431	C₂²×C₃₁	1	2	yes	—
327	#391	—	2	1	no	—
328	#389	—	2	1	no	—
329	#387	—	2	2	no	—
330	#383	C₂²×C₃₁	1	2	yes	—
331	#382	C₂²×C₃₁	1	2	yes	—
332	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
333	#376	C₂²×C₃₁	1	1	yes	characteristic
334	#328	C₂²×C₃₁	1	1	yes	characteristic
335	#328	C₂²×C₃₁	1	1	yes	characteristic
336	#328	C₂²×C₃₁	1	2	yes	—
337	#328	C₂²×C₃₁	1	2	yes	—
338	#328	C₂²×C₃₁	1	2	yes	—
339	#328	C₂²×C₃₁	1	2	yes	—
340	#328	C₂²×C₃₁	1	2	yes	—
341	#327	C₂²×C₃₁	1	1	yes	characteristic
342	#327	C₂²×C₃₁	1	1	yes	characteristic
343	#327	C₂²×C₃₁	1	2	yes	—
344	C₂×C₃₁	—	2	1	no	—
345	C₂×C₃₁	—	2	1	no	—
346	C₂×C₃₁	—	8	1	no	—
347	C₂×C₃₁	#2884	1	1	yes	characteristic
348	C₂×C₃₁	#2898	1	1	yes	characteristic
349	C₂×C₃₁	C₂×#421	1	1	yes	characteristic
350	C₂²×Q₈	—	2	1	no	—
351	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
352	C₄⋊Q₈	—	2	1	no	—
353	C₄⋊Q₈	—	2	2	no	—
354	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
355	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
356	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
357	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
358	#126	—	2	2	no	—
359	#126	—	2	2	no	—
360	#126	—	2	2	no	—
361	#126	—	2	2	no	—
362	#126	—	2	1	no	—
363	#125	—	2	2	no	—
364	#125	—	2	2	no	—
365	#125	—	2	2	no	—
366	#125	—	2	2	no	—
367	#125	—	2	1	no	—
368	#125	—	2	2	no	—
369	#125	—	2	1	no	—
370	#125	C₂×C₄×C₃₁	1	1	yes	characteristic
371	#125	C₂×C₄×C₃₁	1	2	yes	—
372	#124	C₂×C₄×C₃₁	1	1	yes	characteristic
373	#124	C₂×C₄×C₃₁	1	2	yes	—
374	#123	—	2	1	no	—
375	#123	—	2	2	no	—
376	C₂²⋊Q₈	—	2	1	no	—
377	C₂²⋊Q₈	—	2	2	no	—
378	C₂²⋊Q₈	—	2	1	no	—
379	C₂²⋊Q₈	—	2	2	no	—
380	C₂²⋊Q₈	—	2	2	no	—
381	C₂²⋊Q₈	—	2	1	no	—
382	C₂²⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
383	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
384	#121	C₂³×C₃₁	1	1	yes	characteristic
385	C₄×Q₈	—	2	1	no	—
386	C₄×Q₈	—	2	2	no	—
387	C₄×Q₈	—	2	1	no	—
388	C₄×Q₈	—	2	1	no	—
389	C₄×Q₈	—	2	2	no	—
390	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
391	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
392	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
393	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
394	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
395	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
396	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
397	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
398	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
399	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
400	C₄×Q₈	D₄×C₃₁	1	2	yes	—
401	C₄×D₄	—	2	1	no	—
402	C₄×D₄	—	2	2	no	—
403	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
404	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
405	#117	—	2	1	no	—
406	#117	—	2	2	no	—
407	#117	—	2	2	no	—
408	#117	—	2	1	no	—
409	#117	—	2	1	no	—
410	#117	—	2	2	no	—
411	C₂×C₄⋊C₄	—	2	1	no	—
412	C₂×C₄⋊C₄	—	2	2	no	—
413	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
414	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
415	C₂×C₄²	—	2	1	no	—
416	C₂×C₄²	—	2	1	no	—
417	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
418	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
419	C₄⋊C₈	—	2	2	no	—
420	C₄⋊C₈	—	2	1	no	—
421	C₄⋊C₈	—	2	1	no	—
422	C₄⋊C₈	—	2	2	no	—
423	C₂²⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
424	C₂²⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
425	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
426	#97	—	2	2	no	—
427	#97	—	2	1	no	—
428	C₄×C₈	—	2	1	no	—
429	C₄×C₈	—	2	2	no	—
430	#95	—	2	1	no	—
431	#95	—	2	2	no	—
432	#95	—	2	2	no	—
433	#95	—	2	2	no	—
434	#95	C₂³×C₃₁	1	1	yes	characteristic
435	#95	C₂³×C₃₁	1	1	yes	characteristic
436	#95	C₂³×C₃₁	1	2	yes	—
437	C₃₁	#17601	1	1	yes	characteristic, direct product factor, Sylow
438	C₂×Q₈	—	2	2	no	—
439	C₂×Q₈	—	2	1	no	—
440	C₂×Q₈	—	4	2	no	—
441	C₂×Q₈	—	4	4	no	—
442	C₂×Q₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
443	C₂×Q₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
444	C₂×Q₈	C₂²⋊C₄×C₃₁	1	2	yes	—
445	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
446	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
447	C₂×D₄	—	2	1	no	—
448	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
449	C₂²×C₄	—	2	2	no	—
450	C₂²×C₄	—	2	1	no	—
451	C₂²×C₄	—	2	1	no	—
452	C₂²×C₄	—	2	1	no	—
453	C₂²×C₄	—	2	2	no	—
454	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
455	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
456	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
457	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
458	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
459	C₂×C₈	—	2	1	no	—
460	C₂×C₈	—	2	1	no	—
461	C₂×C₈	—	2	2	no	—
462	C₄⋊C₄	—	2	2	no	—
463	C₄⋊C₄	—	2	1	no	—
464	C₄⋊C₄	—	2	1	no	—
465	C₄⋊C₄	—	2	2	no	—
466	C₄⋊C₄	—	2	2	no	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	2	2	no	—
469	C₄⋊C₄	—	2	1	no	—
470	C₄⋊C₄	—	2	1	no	—
471	C₄⋊C₄	—	2	2	no	—
472	C₄⋊C₄	—	2	2	no	—
473	C₄⋊C₄	—	2	2	no	—
474	C₄⋊C₄	—	2	2	no	—
475	C₄⋊C₄	—	2	2	no	—
476	C₄⋊C₄	—	2	1	no	—
477	C₄⋊C₄	—	2	1	no	—
478	C₄⋊C₄	—	2	2	no	—
479	C₄⋊C₄	—	4	2	no	—
480	C₄⋊C₄	—	8	2	no	—
481	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
482	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
483	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
484	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
485	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
486	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
487	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
488	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
489	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
490	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
491	C₂²⋊C₄	—	2	2	no	—
492	C₂²⋊C₄	—	2	1	no	—
493	C₂²⋊C₄	—	2	1	no	—
494	C₂²⋊C₄	—	2	2	no	—
495	C₂²⋊C₄	—	2	1	no	—
496	C₂²⋊C₄	—	2	2	no	—
497	C₂²⋊C₄	—	2	2	no	—
498	C₂²⋊C₄	—	2	1	no	—
499	C₂²⋊C₄	—	2	1	no	—
500	C₄²	—	2	2	no	—
501	C₄²	—	2	2	no	—
502	C₄²	—	2	2	no	—
503	C₄²	—	2	2	no	—
504	C₄²	—	2	1	no	—
505	C₄²	—	4	2	no	—
506	C₄²	—	8	1	no	—
507	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
508	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
509	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
510	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
511	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
512	C₄²	C₂×D₄×C₃₁	1	2	yes	—
513	C₄²	C₂×D₄×C₃₁	1	2	yes	—
514	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
515	C₂³	—	2	1	no	—
516	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
517	Q₈	—	2	2	no	—
518	Q₈	—	2	2	no	—
519	Q₈	—	4	2	no	—
520	Q₈	—	4	1	no	—
521	Q₈	—	4	2	no	—
522	Q₈	—	4	4	no	—
523	Q₈	—	4	2	no	—
524	D₄	—	4	1	no	—
525	D₄	—	4	1	no	—
526	D₄	—	4	1	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	2	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	2	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	4	2	no	—
538	C₂×C₄	—	4	1	no	—
539	C₂×C₄	—	4	4	no	—
540	C₂×C₄	—	4	2	no	—
541	C₂×C₄	—	4	1	no	—
542	C₂×C₄	—	4	1	no	—
543	C₂×C₄	—	4	2	no	—
544	C₂×C₄	—	4	1	no	—
545	C₂×C₄	—	4	1	no	—
546	C₂×C₄	—	4	2	no	—
547	C₂×C₄	—	4	2	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	4	1	no	—
550	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
551	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
552	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
553	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
554	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
555	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
556	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
557	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
558	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
559	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
560	C₈	—	4	1	no	—
561	C₈	—	4	1	no	—
562	C₈	—	4	2	no	—
563	C₂²	—	2	1	no	—
564	C₂²	—	2	1	no	—
565	C₂²	—	4	1	no	—
566	C₂²	—	4	1	no	—
567	C₂²	—	4	1	no	—
568	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
569	C₂²	C₃₁×#577	1	1	yes	characteristic
570	C₂²	C₃₁×#578	1	1	yes	characteristic
571	C₄	—	2	1	no	—
572	C₄	—	2	1	no	—
573	C₄	—	2	1	no	—
574	C₄	—	2	1	no	—
575	C₄	—	2	2	no	—
576	C₄	—	2	2	no	—
577	C₄	—	2	1	no	—
578	C₄	—	2	2	no	—
579	C₄	—	2	2	no	—
580	C₄	—	4	2	no	—
581	C₄	—	4	1	no	—
582	C₄	—	4	1	no	—
583	C₄	—	4	2	no	—
584	C₄	—	4	1	no	—
585	C₄	—	8	1	no	—
586	C₄	—	8	2	no	—
587	C₄	—	8	1	no	—
588	C₄	C₃₁×#421	1	2	yes	—
589	C₂	—	2	1	no	—
590	C₂	—	2	1	no	—
591	C₂	—	8	1	no	—
592	C₂	C₃₁×#2884	1	1	yes	characteristic
593	C₂	C₃₁×#2898	1	1	yes	characteristic
594	C₂	C₂×C₃₁×#421	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#121	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ 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. C₃₁×#2984	C₂	semidirect
3. C₃₁×#2320	C₂	semidirect
5. C₃₁×#1518	C₂	semidirect
6. C₃₁×#1515	C₂	semidirect
9. C₃₁×#1506	C₂	semidirect
10. C₃₁×#1503	C₂	semidirect
12. C₃₁×#557	C₄	semidirect
13. C₃₁×#557	C₄	semidirect
14. C₃₁×#557	C₄	semidirect
15. C₃₁×#556	C₄	semidirect
16. C₃₁×#556	C₄	semidirect
20. D₄×Q₈×C₃₁	C₄	semidirect
217. #17601	C₃₁	direct (N × H)
267. #2984	C₂×C₃₁	semidirect
268. #2320	C₂×C₃₁	semidirect
270. #1518	C₂×C₃₁	semidirect
271. #1515	C₂×C₃₁	semidirect
274. #1506	C₂×C₃₁	semidirect
275. #1503	C₂×C₃₁	semidirect
303. #557	C₄×C₃₁	semidirect
304. #557	C₄×C₃₁	semidirect
305. #557	C₄×C₃₁	semidirect
306. #556	C₄×C₃₁	semidirect
307. #556	C₄×C₃₁	semidirect
311. D₄×Q₈	C₄×C₃₁	semidirect
437. C₃₁	#17601	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×#3523	2
C₃₁×#2984	2
C₃₁×#2320	2
C₃₁×#1521	2
C₃₁×#1518	2
C₃₁×#1515	2
C₃₁×#1512	2
C₃₁×#1509	2
C₃₁×#1506	2
C₃₁×#1503	2
C₃₁×#1500	2
#17601	31

Subgroups — C31×#17601 #21474360

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17601 #21474360