Subgroups — C₃₁×#23571 #21480330

Order 7936.

1156 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
3	C₄×C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2057	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2043	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2036	C₂	1	2	yes	maximal
7	C₃₁×#2024	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2016	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#2001	C₂	1	2	yes	maximal
10	C₄×C₃₁×#103	C₂	1	2	yes	maximal, complemented
11	C₄×C₃₁×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
13	C₃₁×#533	—	2	8	no	—
14	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#125	—	2	1	no	—
17	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
18	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
19	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
21	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
23	C₂×C₃₁×#117	—	2	1	no	—
24	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
25	C₄×Q₁₆×C₃₁	—	2	8	no	—
26	C₂×C₃₁×#107	C₂²	1	2	yes	—
27	C₂×C₃₁×#103	C₂²	1	2	yes	—
28	C₂×C₃₁×#103	C₂²	1	2	yes	—
29	C₂×C₃₁×#103	C₂²	1	2	yes	—
30	C₂×C₃₁×#103	C₂²	1	2	yes	—
31	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
33	C₃₁×#391	C₂²	1	1	yes	characteristic
34	C₃₁×#389	C₂²	1	1	yes	characteristic
35	C₃₁×#387	—	2	1	no	—
36	C₃₁×#385	—	2	1	no	—
37	C₃₁×#380	C₂²	1	1	yes	characteristic
38	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
39	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
40	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
41	C₃₁×#340	C₂²	1	2	yes	—
42	C₃₁×#340	C₂²	1	2	yes	—
43	C₃₁×#336	C₂²	1	1	yes	characteristic
44	C₃₁×#336	C₂²	1	1	yes	characteristic
45	C₂²×Q₈×C₃₁	—	2	1	no	—
46	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
47	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
49	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
50	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
51	C₂×Q₁₆×C₃₁	—	2	4	no	—
52	C₂×Q₁₆×C₃₁	—	2	4	no	—
53	C₂×Q₁₆×C₃₁	—	2	4	no	—
54	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
55	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
56	C₃₁×C₄⋊Q₈	—	2	4	no	—
57	C₃₁×C₄⋊Q₈	—	2	4	no	—
58	C₃₁×#125	—	2	8	no	—
59	C₃₁×C₂²⋊Q₈	—	2	8	no	—
60	C₃₁×C₂²⋊Q₈	—	2	4	no	—
61	C₃₁×C₂²⋊Q₈	—	2	2	no	—
62	C₃₁×C₂²⋊Q₈	—	2	4	no	—
63	C₃₁×C₂²⋊Q₈	—	2	2	no	—
64	C₄×Q₈×C₃₁	—	2	8	no	—
65	C₄×Q₈×C₃₁	—	2	4	no	—
66	C₄×Q₈×C₃₁	—	2	4	no	—
67	C₄×Q₈×C₃₁	D₄	1	8	yes	—
68	C₃₁×#117	—	2	8	no	—
69	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
70	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
71	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
72	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
73	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
74	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
75	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
76	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
77	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
80	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
81	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
82	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
83	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
84	C₂×C₄²×C₃₁	—	2	2	no	—
85	C₂×C₄²×C₃₁	—	2	4	no	—
86	C₂×C₄²×C₃₁	—	2	1	no	—
87	C₂×C₄²×C₃₁	—	2	1	no	—
88	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
89	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
90	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
91	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
92	C₃₁×#107	—	2	4	no	—
93	C₃₁×#103	—	2	4	no	—
94	C₃₁×#103	—	2	4	no	—
95	C₃₁×#103	—	2	4	no	—
96	C₃₁×#103	—	2	4	no	—
97	C₃₁×C₂²⋊C₈	—	2	4	no	—
98	C₄×C₈×C₃₁	—	2	4	no	—
99	C₃₁×#95	—	2	1	no	—
100	C₃₁×#95	—	2	1	no	—
101	C₃₁×#95	—	2	1	no	—
102	C₃₁×#95	—	2	1	no	—
103	C₃₁×#95	—	2	1	no	—
104	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂×Q₈×C₃₁	—	2	4	no	—
106	C₂×Q₈×C₃₁	—	2	2	no	—
107	C₂×Q₈×C₃₁	—	2	1	no	—
108	C₂×Q₈×C₃₁	—	2	2	no	—
109	C₂×Q₈×C₃₁	—	2	2	no	—
110	C₂×Q₈×C₃₁	—	2	2	no	—
111	C₂×Q₈×C₃₁	—	2	2	no	—
112	C₂×Q₈×C₃₁	—	2	2	no	—
113	C₂×Q₈×C₃₁	—	2	2	no	—
114	C₂×Q₈×C₃₁	—	4	2	no	—
115	C₂×Q₈×C₃₁	—	4	2	no	—
116	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
117	C₂×Q₈×C₃₁	#41	1	2	yes	—
118	C₂²×C₄×C₃₁	—	2	2	no	—
119	C₂²×C₄×C₃₁	—	2	2	no	—
120	C₂²×C₄×C₃₁	—	2	1	no	—
121	C₂²×C₄×C₃₁	—	2	1	no	—
122	C₂²×C₄×C₃₁	—	2	2	no	—
123	C₂²×C₄×C₃₁	—	2	2	no	—
124	C₂²×C₄×C₃₁	—	2	2	no	—
125	C₂²×C₄×C₃₁	—	2	2	no	—
126	C₂²×C₄×C₃₁	—	2	2	no	—
127	C₂²×C₄×C₃₁	—	2	2	no	—
128	C₂²×C₄×C₃₁	—	2	1	no	—
129	C₂²×C₄×C₃₁	—	2	1	no	—
130	C₂²×C₄×C₃₁	—	2	1	no	—
131	C₂²×C₄×C₃₁	—	2	1	no	—
132	C₂²×C₄×C₃₁	—	4	1	no	—
133	C₂²×C₄×C₃₁	Q₁₆	1	4	yes	complemented
134	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
140	Q₁₆×C₃₁	—	2	16	no	—
141	C₂×C₈×C₃₁	—	2	2	no	—
142	C₂×C₈×C₃₁	—	2	2	no	—
143	C₂×C₈×C₃₁	—	2	2	no	—
144	C₂×C₈×C₃₁	—	2	2	no	—
145	C₂×C₈×C₃₁	—	2	2	no	—
146	C₂×C₈×C₃₁	#41	1	4	yes	—
147	C₄⋊C₄×C₃₁	—	2	4	no	—
148	C₄⋊C₄×C₃₁	—	2	4	no	—
149	C₄⋊C₄×C₃₁	—	2	8	no	—
150	C₄⋊C₄×C₃₁	—	2	4	no	—
151	C₄⋊C₄×C₃₁	—	2	4	no	—
152	C₄⋊C₄×C₃₁	—	2	8	no	—
153	C₄⋊C₄×C₃₁	—	2	2	no	—
154	C₄⋊C₄×C₃₁	—	2	1	no	—
155	C₄⋊C₄×C₃₁	—	2	2	no	—
156	C₄⋊C₄×C₃₁	—	2	2	no	—
157	C₄⋊C₄×C₃₁	—	4	2	no	—
158	C₄⋊C₄×C₃₁	—	4	2	no	—
159	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
160	C₄⋊C₄×C₃₁	#41	1	2	yes	—
161	C₂²⋊C₄×C₃₁	—	2	8	no	—
162	C₂²⋊C₄×C₃₁	—	2	4	no	—
163	C₂²⋊C₄×C₃₁	—	4	2	no	—
164	C₄²×C₃₁	—	2	2	no	—
165	C₄²×C₃₁	—	2	2	no	—
166	C₄²×C₃₁	—	2	8	no	—
167	C₄²×C₃₁	—	2	2	no	—
168	C₄²×C₃₁	—	4	2	no	—
169	C₄²×C₃₁	—	4	2	no	—
170	C₄²×C₃₁	C₂×D₄	1	4	yes	—
171	#23571	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
172	C₂³×C₃₁	—	2	1	no	—
173	C₂³×C₃₁	—	2	1	no	—
174	C₂³×C₃₁	—	2	1	no	—
175	C₂³×C₃₁	—	2	1	no	—
176	C₂³×C₃₁	—	2	1	no	—
177	C₂³×C₃₁	—	2	1	no	—
178	C₂³×C₃₁	C₂×Q₁₆	1	2	yes	—
179	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
180	Q₈×C₃₁	—	2	4	no	—
181	Q₈×C₃₁	—	2	4	no	—
182	Q₈×C₃₁	—	2	4	no	—
183	Q₈×C₃₁	—	4	8	no	—
184	Q₈×C₃₁	—	4	2	no	—
185	Q₈×C₃₁	—	4	2	no	—
186	Q₈×C₃₁	—	4	2	no	—
187	C₂×C₄×C₃₁	—	2	1	no	—
188	C₂×C₄×C₃₁	—	2	1	no	—
189	C₂×C₄×C₃₁	—	2	1	no	—
190	C₂×C₄×C₃₁	—	2	1	no	—
191	C₂×C₄×C₃₁	—	2	1	no	—
192	C₂×C₄×C₃₁	—	2	1	no	—
193	C₂×C₄×C₃₁	—	2	8	no	—
194	C₂×C₄×C₃₁	—	2	1	no	—
195	C₂×C₄×C₃₁	—	2	4	no	—
196	C₂×C₄×C₃₁	—	2	4	no	—
197	C₂×C₄×C₃₁	—	2	2	no	—
198	C₂×C₄×C₃₁	—	2	2	no	—
199	C₂×C₄×C₃₁	—	2	1	no	—
200	C₂×C₄×C₃₁	—	2	1	no	—
201	C₂×C₄×C₃₁	—	4	4	no	—
202	C₂×C₄×C₃₁	—	4	1	no	—
203	C₂×C₄×C₃₁	—	4	2	no	—
204	C₂×C₄×C₃₁	—	4	1	no	—
205	C₂×C₄×C₃₁	—	4	2	no	—
206	C₂×C₄×C₃₁	—	4	1	no	—
207	C₂×C₄×C₃₁	—	4	2	no	—
208	C₂×C₄×C₃₁	—	4	1	no	—
209	C₂×C₄×C₃₁	—	4	2	no	—
210	C₂×C₄×C₃₁	—	4	2	no	—
211	C₂×C₄×C₃₁	—	4	2	no	—
212	C₂×C₄×C₃₁	—	4	1	no	—
213	C₂×C₄×C₃₁	—	4	1	no	—
214	C₂×C₄×C₃₁	—	4	1	no	—
215	C₂×C₄×C₃₁	—	4	1	no	—
216	C₂×C₄×C₃₁	—	4	1	no	—
217	C₂×C₄×C₃₁	—	4	1	no	—
218	C₂×C₄×C₃₁	#120	1	2	yes	—
219	C₂×C₄×C₃₁	#120	1	2	yes	—
220	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
221	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
222	C₂×C₄×C₃₁	#135	1	2	yes	—
223	C₂×C₄×C₃₁	#137	1	2	yes	—
224	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
225	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
226	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
227	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
228	C₈×C₃₁	—	2	4	no	—
229	C₈×C₃₁	—	4	2	no	—
230	C₂×#533	C₂×C₃₁	1	1	yes	characteristic
231	C₂×C₄×Q₁₆	C₂×C₃₁	1	2	yes	complemented
232	C₄×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic
233	#2057	C₂×C₃₁	1	1	yes	characteristic, complemented
234	#2043	C₂×C₃₁	1	1	yes	characteristic, complemented
235	#2036	C₂×C₃₁	1	2	yes	—
236	#2024	C₂×C₃₁	1	1	yes	characteristic, complemented
237	#2016	C₂×C₃₁	1	1	yes	characteristic, complemented
238	#2001	C₂×C₃₁	1	2	yes	—
239	C₄×#103	C₂×C₃₁	1	2	yes	complemented
240	C₄×C₂²⋊C₈	C₂×C₃₁	1	1	yes	characteristic
241	C₂²×C₃₁	—	2	2	no	—
242	C₂²×C₃₁	—	2	2	no	—
243	C₂²×C₃₁	—	2	2	no	—
244	C₂²×C₃₁	—	4	1	no	—
245	C₂²×C₃₁	—	4	1	no	—
246	C₂²×C₃₁	—	4	1	no	—
247	C₂²×C₃₁	—	4	1	no	—
248	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
249	C₂²×C₃₁	#525	1	1	yes	characteristic
250	C₂²×C₃₁	#532	1	1	yes	characteristic
251	C₂²×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
252	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
253	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
254	C₂²×C₃₁	#575	1	1	yes	characteristic
255	C₄×C₃₁	—	2	2	no	—
256	C₄×C₃₁	—	2	2	no	—
257	C₄×C₃₁	—	2	2	no	—
258	C₄×C₃₁	—	4	2	no	—
259	C₄×C₃₁	—	4	1	no	—
260	C₄×C₃₁	—	4	4	no	—
261	C₄×C₃₁	—	4	2	no	—
262	C₄×C₃₁	—	4	4	no	—
263	C₄×C₃₁	—	8	1	no	—
264	C₄×C₃₁	—	8	1	no	—
265	C₄×C₃₁	#451	1	4	yes	—
266	C₄×C₃₁	#525	1	2	yes	—
267	C₄×C₃₁	#532	1	2	yes	—
268	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
269	#533	—	2	8	no	—
270	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
271	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
272	C₂×#125	—	2	1	no	—
273	C₂×C₂²⋊Q₈	—	2	1	no	—
274	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
275	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
276	C₂×C₄×Q₈	—	2	1	no	—
277	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
278	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
279	C₂×#117	—	2	1	no	—
280	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
281	C₄×Q₁₆	—	2	8	no	—
282	C₂×#107	C₂²×C₃₁	1	2	yes	—
283	C₂×#103	C₂²×C₃₁	1	2	yes	—
284	C₂×#103	C₂²×C₃₁	1	2	yes	—
285	C₂×#103	C₂²×C₃₁	1	2	yes	—
286	C₂×#103	C₂²×C₃₁	1	2	yes	—
287	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
288	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
289	#391	C₂²×C₃₁	1	1	yes	characteristic
290	#389	C₂²×C₃₁	1	1	yes	characteristic
291	#387	—	2	1	no	—
292	#385	—	2	1	no	—
293	#380	C₂²×C₃₁	1	1	yes	characteristic
294	C₄×C₄⋊C₄	—	2	1	no	—
295	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
296	C₄×C₂²⋊C₄	—	2	1	no	—
297	#340	C₂²×C₃₁	1	2	yes	—
298	#340	C₂²×C₃₁	1	2	yes	—
299	#336	C₂²×C₃₁	1	1	yes	characteristic
300	#336	C₂²×C₃₁	1	1	yes	characteristic
301	C₂×C₃₁	—	4	2	no	—
302	C₂×C₃₁	#2388	1	1	yes	characteristic
303	C₂×C₃₁	C₂×#451	1	1	yes	characteristic
304	C₂×C₃₁	#3002	1	1	yes	characteristic
305	C₂×C₃₁	#3104	1	1	yes	characteristic
306	C₂×C₃₁	#3107	1	1	yes	characteristic
307	C₂×C₃₁	#3170	1	1	yes	characteristic
308	C₂×C₃₁	#3172	1	1	yes	characteristic
309	C₂²×Q₈	—	2	1	no	—
310	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
311	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
312	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
313	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
314	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
315	C₂×Q₁₆	—	2	4	no	—
316	C₂×Q₁₆	—	2	4	no	—
317	C₂×Q₁₆	—	2	4	no	—
318	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
319	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
320	C₄⋊Q₈	—	2	4	no	—
321	C₄⋊Q₈	—	2	4	no	—
322	#125	—	2	8	no	—
323	C₂²⋊Q₈	—	2	8	no	—
324	C₂²⋊Q₈	—	2	4	no	—
325	C₂²⋊Q₈	—	2	2	no	—
326	C₂²⋊Q₈	—	2	4	no	—
327	C₂²⋊Q₈	—	2	2	no	—
328	C₄×Q₈	—	2	8	no	—
329	C₄×Q₈	—	2	4	no	—
330	C₄×Q₈	—	2	4	no	—
331	C₄×Q₈	D₄×C₃₁	1	8	yes	—
332	#117	—	2	8	no	—
333	C₂×C₄⋊C₄	—	2	1	no	—
334	C₂×C₄⋊C₄	—	2	1	no	—
335	C₂×C₄⋊C₄	—	2	1	no	—
336	C₂×C₄⋊C₄	—	2	2	no	—
337	C₂×C₄⋊C₄	—	2	2	no	—
338	C₂×C₄⋊C₄	—	2	1	no	—
339	C₂×C₄⋊C₄	—	2	1	no	—
340	C₂×C₄⋊C₄	—	2	1	no	—
341	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
342	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
343	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
344	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
345	C₂×C₂²⋊C₄	—	2	2	no	—
346	C₂×C₂²⋊C₄	—	2	1	no	—
347	C₂×C₂²⋊C₄	—	2	1	no	—
348	C₂×C₄²	—	2	2	no	—
349	C₂×C₄²	—	2	4	no	—
350	C₂×C₄²	—	2	1	no	—
351	C₂×C₄²	—	2	1	no	—
352	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
353	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
354	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
355	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
356	#107	—	2	4	no	—
357	#103	—	2	4	no	—
358	#103	—	2	4	no	—
359	#103	—	2	4	no	—
360	#103	—	2	4	no	—
361	C₂²⋊C₈	—	2	4	no	—
362	C₄×C₈	—	2	4	no	—
363	#95	—	2	1	no	—
364	#95	—	2	1	no	—
365	#95	—	2	1	no	—
366	#95	—	2	1	no	—
367	#95	—	2	1	no	—
368	C₃₁	#23571	1	1	yes	characteristic, direct product factor, Sylow
369	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
370	C₂×Q₈	—	2	4	no	—
371	C₂×Q₈	—	2	2	no	—
372	C₂×Q₈	—	2	1	no	—
373	C₂×Q₈	—	2	2	no	—
374	C₂×Q₈	—	2	2	no	—
375	C₂×Q₈	—	2	2	no	—
376	C₂×Q₈	—	2	2	no	—
377	C₂×Q₈	—	2	2	no	—
378	C₂×Q₈	—	2	2	no	—
379	C₂×Q₈	—	4	2	no	—
380	C₂×Q₈	—	4	2	no	—
381	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
382	C₂×Q₈	#41×C₃₁	1	2	yes	—
383	C₂²×C₄	—	2	2	no	—
384	C₂²×C₄	—	2	2	no	—
385	C₂²×C₄	—	2	1	no	—
386	C₂²×C₄	—	2	1	no	—
387	C₂²×C₄	—	2	2	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	—	2	2	no	—
391	C₂²×C₄	—	2	2	no	—
392	C₂²×C₄	—	2	2	no	—
393	C₂²×C₄	—	2	1	no	—
394	C₂²×C₄	—	2	1	no	—
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	2	1	no	—
397	C₂²×C₄	—	4	1	no	—
398	C₂²×C₄	Q₁₆×C₃₁	1	4	yes	complemented
399	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
400	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
401	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
402	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
403	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
404	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
405	Q₁₆	—	2	16	no	—
406	C₂×C₈	—	2	2	no	—
407	C₂×C₈	—	2	2	no	—
408	C₂×C₈	—	2	2	no	—
409	C₂×C₈	—	2	2	no	—
410	C₂×C₈	—	2	2	no	—
411	C₂×C₈	#41×C₃₁	1	4	yes	—
412	C₄⋊C₄	—	2	4	no	—
413	C₄⋊C₄	—	2	4	no	—
414	C₄⋊C₄	—	2	8	no	—
415	C₄⋊C₄	—	2	4	no	—
416	C₄⋊C₄	—	2	4	no	—
417	C₄⋊C₄	—	2	8	no	—
418	C₄⋊C₄	—	2	2	no	—
419	C₄⋊C₄	—	2	1	no	—
420	C₄⋊C₄	—	2	2	no	—
421	C₄⋊C₄	—	2	2	no	—
422	C₄⋊C₄	—	4	2	no	—
423	C₄⋊C₄	—	4	2	no	—
424	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
425	C₄⋊C₄	#41×C₃₁	1	2	yes	—
426	C₂²⋊C₄	—	2	8	no	—
427	C₂²⋊C₄	—	2	4	no	—
428	C₂²⋊C₄	—	4	2	no	—
429	C₄²	—	2	2	no	—
430	C₄²	—	2	2	no	—
431	C₄²	—	2	8	no	—
432	C₄²	—	2	2	no	—
433	C₄²	—	4	2	no	—
434	C₄²	—	4	2	no	—
435	C₄²	C₂×D₄×C₃₁	1	4	yes	—
436	C₂³	—	2	1	no	—
437	C₂³	—	2	1	no	—
438	C₂³	—	2	1	no	—
439	C₂³	—	2	1	no	—
440	C₂³	—	2	1	no	—
441	C₂³	—	2	1	no	—
442	C₂³	C₂×Q₁₆×C₃₁	1	2	yes	—
443	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
444	Q₈	—	2	4	no	—
445	Q₈	—	2	4	no	—
446	Q₈	—	2	4	no	—
447	Q₈	—	4	8	no	—
448	Q₈	—	4	2	no	—
449	Q₈	—	4	2	no	—
450	Q₈	—	4	2	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	1	no	—
454	C₂×C₄	—	2	1	no	—
455	C₂×C₄	—	2	1	no	—
456	C₂×C₄	—	2	1	no	—
457	C₂×C₄	—	2	8	no	—
458	C₂×C₄	—	2	1	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	2	4	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	1	no	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	4	4	no	—
466	C₂×C₄	—	4	1	no	—
467	C₂×C₄	—	4	2	no	—
468	C₂×C₄	—	4	1	no	—
469	C₂×C₄	—	4	2	no	—
470	C₂×C₄	—	4	1	no	—
471	C₂×C₄	—	4	2	no	—
472	C₂×C₄	—	4	1	no	—
473	C₂×C₄	—	4	2	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	4	2	no	—
476	C₂×C₄	—	4	1	no	—
477	C₂×C₄	—	4	1	no	—
478	C₂×C₄	—	4	1	no	—
479	C₂×C₄	—	4	1	no	—
480	C₂×C₄	—	4	1	no	—
481	C₂×C₄	—	4	1	no	—
482	C₂×C₄	C₃₁×#120	1	2	yes	—
483	C₂×C₄	C₃₁×#120	1	2	yes	—
484	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
485	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
486	C₂×C₄	C₃₁×#135	1	2	yes	—
487	C₂×C₄	C₃₁×#137	1	2	yes	—
488	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
489	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
490	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
491	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
492	C₈	—	2	4	no	—
493	C₈	—	4	2	no	—
494	C₂²	—	2	2	no	—
495	C₂²	—	2	2	no	—
496	C₂²	—	2	2	no	—
497	C₂²	—	4	1	no	—
498	C₂²	—	4	1	no	—
499	C₂²	—	4	1	no	—
500	C₂²	—	4	1	no	—
501	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
502	C₂²	C₃₁×#525	1	1	yes	characteristic
503	C₂²	C₃₁×#532	1	1	yes	characteristic
504	C₂²	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
505	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
506	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
507	C₂²	C₃₁×#575	1	1	yes	characteristic
508	C₄	—	2	2	no	—
509	C₄	—	2	2	no	—
510	C₄	—	2	2	no	—
511	C₄	—	4	2	no	—
512	C₄	—	4	1	no	—
513	C₄	—	4	4	no	—
514	C₄	—	4	2	no	—
515	C₄	—	4	4	no	—
516	C₄	—	8	1	no	—
517	C₄	—	8	1	no	—
518	C₄	C₃₁×#451	1	4	yes	—
519	C₄	C₃₁×#525	1	2	yes	—
520	C₄	C₃₁×#532	1	2	yes	—
521	C₂	—	4	2	no	—
522	C₂	C₃₁×#2388	1	1	yes	characteristic
523	C₂	C₂×C₃₁×#451	1	1	yes	characteristic
524	C₂	C₃₁×#3002	1	1	yes	characteristic
525	C₂	C₃₁×#3104	1	1	yes	characteristic
526	C₂	C₃₁×#3107	1	1	yes	characteristic
527	C₂	C₃₁×#3170	1	1	yes	characteristic
528	C₂	C₃₁×#3172	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂²×D₄
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Q₁₆×C₃₁	C₂	semidirect
4. C₃₁×#2057	C₂	semidirect
5. C₃₁×#2043	C₂	semidirect
7. C₃₁×#2024	C₂	semidirect
8. C₃₁×#2016	C₂	semidirect
10. C₄×C₃₁×#103	C₂	semidirect
133. C₂²×C₄×C₃₁	Q₁₆	semidirect
171. #23571	C₃₁	direct (N × H)
231. C₂×C₄×Q₁₆	C₂×C₃₁	semidirect
233. #2057	C₂×C₃₁	semidirect
234. #2043	C₂×C₃₁	semidirect
236. #2024	C₂×C₃₁	semidirect
237. #2016	C₂×C₃₁	semidirect
239. C₄×#103	C₂×C₃₁	semidirect
368. C₃₁	#23571	direct (N × H)
398. C₂²×C₄	Q₁₆×C₃₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃₁×#533	2
C₂×C₄×Q₁₆×C₃₁	2
C₄×C₃₁×C₂²⋊Q₈	2
C₃₁×#2057	2
C₃₁×#2043	2
C₃₁×#2036	2
C₃₁×#2024	2
C₃₁×#2016	2
C₃₁×#2001	2
C₄×C₃₁×#103	2
C₄×C₃₁×C₂²⋊C₈	2
#23571	31

Subgroups — C31×#23571 #21480330

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23571 #21480330