Subgroups — C₃₁×#33057 #21489816

Order 7936.

940 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#489	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2857	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2850	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1993	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1959	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1934	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1933	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1895	C₂	1	2	yes	maximal
10	C₃₁×#1894	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1836	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
13	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
16	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#121	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#489	—	2	4	no	—
20	C₂×C₃₁×#103	C₂²	1	2	yes	—
21	C₂×C₃₁×#103	C₂²	1	2	yes	—
22	C₂×C₃₁×#102	C₂²	1	2	yes	—
23	C₂×C₃₁×#102	C₂²	1	2	yes	—
24	C₄×M₁₆×C₃₁	—	2	4	no	—
25	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
28	C₃₁×#391	—	2	1	no	—
29	C₃₁×#391	C₂²	1	1	yes	characteristic
30	C₃₁×#390	C₂²	1	1	yes	characteristic
31	C₃₁×#389	—	2	1	no	—
32	C₃₁×#389	C₂²	1	2	yes	—
33	C₃₁×#388	—	2	1	no	—
34	C₃₁×#388	—	2	1	no	—
35	C₃₁×#386	—	2	1	no	—
36	C₃₁×#386	—	2	1	no	—
37	C₃₁×#384	C₂²	1	1	yes	characteristic
38	C₃₁×#382	C₂²	1	1	yes	characteristic
39	C₃₁×#380	—	2	1	no	—
40	C₃₁×#380	C₂²	1	2	yes	—
41	C₃₁×#340	C₂²	1	2	yes	—
42	C₃₁×#340	C₂²	1	2	yes	—
43	C₃₁×#340	C₂²	1	2	yes	—
44	C₃₁×#340	C₂²	1	2	yes	—
45	C₂²×Q₈×C₃₁	—	2	1	no	—
46	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₂²×D₄×C₃₁	—	2	1	no	—
48	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
50	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
51	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
52	C₂×M₁₆×C₃₁	—	2	4	no	—
53	C₂×M₁₆×C₃₁	—	2	4	no	—
54	C₂×M₁₆×C₃₁	—	2	4	no	—
55	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
56	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
57	C₃₁×C₄⋊Q₈	—	2	2	no	—
58	C₃₁×C₄⋊Q₈	—	2	2	no	—
59	C₃₁×#124	—	2	2	no	—
60	C₃₁×#124	—	2	2	no	—
61	C₃₁×C₂²⋊Q₈	—	2	4	no	—
62	C₃₁×C₂²⋊Q₈	—	4	4	no	—
63	C₃₁×#121	—	2	4	no	—
64	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
65	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
66	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
67	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
68	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
69	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
70	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
71	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
73	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
74	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
75	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
76	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
77	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
78	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
79	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
80	C₂×C₄²×C₃₁	—	2	2	no	—
81	C₂×C₄²×C₃₁	—	2	2	no	—
82	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
83	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
84	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
85	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
86	C₃₁×#103	—	2	4	no	—
87	C₃₁×#103	—	2	4	no	—
88	C₃₁×#102	—	2	4	no	—
89	C₃₁×#102	—	2	4	no	—
90	C₃₁×#97	—	2	2	no	—
91	C₃₁×#97	D₄	1	4	yes	complemented
92	C₄×C₈×C₃₁	—	2	4	no	—
93	C₃₁×#95	—	2	2	no	—
94	C₃₁×#95	—	2	2	no	—
95	C₃₁×#95	—	2	1	no	—
96	C₃₁×#95	—	2	1	no	—
97	C₃₁×#95	—	2	1	no	—
98	C₃₁×#95	—	2	1	no	—
99	C₂⁴×C₃₁	—	2	1	no	—
100	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₂×Q₈×C₃₁	—	2	1	no	—
102	C₂×Q₈×C₃₁	—	2	2	no	—
103	C₂×Q₈×C₃₁	—	2	2	no	—
104	C₂×Q₈×C₃₁	—	2	1	no	—
105	C₂×Q₈×C₃₁	—	4	2	no	—
106	C₂×Q₈×C₃₁	—	4	2	no	—
107	C₂×Q₈×C₃₁	—	4	2	no	—
108	C₂×D₄×C₃₁	—	2	1	no	—
109	C₂×D₄×C₃₁	—	2	2	no	—
110	C₂×D₄×C₃₁	—	2	2	no	—
111	C₂×D₄×C₃₁	—	2	1	no	—
112	C₂×D₄×C₃₁	—	4	2	no	—
113	C₂×D₄×C₃₁	—	4	2	no	—
114	C₂×D₄×C₃₁	—	4	2	no	—
115	C₂²×C₄×C₃₁	—	2	1	no	—
116	C₂²×C₄×C₃₁	—	2	1	no	—
117	C₂²×C₄×C₃₁	—	2	2	no	—
118	C₂²×C₄×C₃₁	—	2	2	no	—
119	C₂²×C₄×C₃₁	—	2	2	no	—
120	C₂²×C₄×C₃₁	—	2	2	no	—
121	C₂²×C₄×C₃₁	—	2	2	no	—
122	C₂²×C₄×C₃₁	—	2	1	no	—
123	C₂²×C₄×C₃₁	—	2	1	no	—
124	C₂²×C₄×C₃₁	—	2	1	no	—
125	C₂²×C₄×C₃₁	—	2	2	no	—
126	C₂²×C₄×C₃₁	—	2	2	no	—
127	C₂²×C₄×C₃₁	—	4	1	no	—
128	C₂²×C₄×C₃₁	—	4	1	no	—
129	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₃₁	#41	1	4	yes	—
135	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
136	M₁₆×C₃₁	—	2	16	no	—
137	C₂×C₈×C₃₁	—	2	2	no	—
138	C₂×C₈×C₃₁	—	2	2	no	—
139	C₂×C₈×C₃₁	—	2	2	no	—
140	C₂×C₈×C₃₁	—	2	2	no	—
141	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
142	C₂×C₈×C₃₁	#41	1	4	yes	—
143	C₄⋊C₄×C₃₁	—	2	2	no	—
144	C₄⋊C₄×C₃₁	—	2	2	no	—
145	C₄⋊C₄×C₃₁	—	2	2	no	—
146	C₄⋊C₄×C₃₁	—	2	2	no	—
147	C₄⋊C₄×C₃₁	—	2	2	no	—
148	C₄⋊C₄×C₃₁	—	2	2	no	—
149	C₄⋊C₄×C₃₁	—	4	2	no	—
150	C₄⋊C₄×C₃₁	—	4	2	no	—
151	C₄⋊C₄×C₃₁	—	4	2	no	—
152	C₄⋊C₄×C₃₁	—	4	2	no	—
153	C₄⋊C₄×C₃₁	—	4	4	no	—
154	C₂²⋊C₄×C₃₁	—	4	4	no	—
155	C₂²⋊C₄×C₃₁	—	4	2	no	—
156	C₂²⋊C₄×C₃₁	—	4	2	no	—
157	C₂²⋊C₄×C₃₁	—	4	2	no	—
158	C₂²⋊C₄×C₃₁	—	4	2	no	—
159	C₂²⋊C₄×C₃₁	—	4	2	no	—
160	C₂²⋊C₄×C₃₁	—	4	2	no	—
161	C₄²×C₃₁	—	2	2	no	—
162	C₄²×C₃₁	—	2	2	no	—
163	C₄²×C₃₁	—	2	1	no	—
164	C₄²×C₃₁	—	2	1	no	—
165	C₄²×C₃₁	C₂×D₄	1	2	yes	—
166	C₄²×C₃₁	#41	1	2	yes	—
167	#33057	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
168	C₂³×C₃₁	—	2	1	no	—
169	C₂³×C₃₁	—	2	1	no	—
170	C₂³×C₃₁	—	2	1	no	—
171	C₂³×C₃₁	—	2	1	no	—
172	C₂³×C₃₁	—	2	1	no	—
173	C₂³×C₃₁	—	2	1	no	—
174	C₂³×C₃₁	—	4	1	no	—
175	C₂³×C₃₁	—	4	1	no	—
176	C₂³×C₃₁	—	4	1	no	—
177	C₂³×C₃₁	—	4	1	no	—
178	C₂³×C₃₁	—	4	1	no	—
179	C₂³×C₃₁	—	4	1	no	—
180	C₂³×C₃₁	—	4	1	no	—
181	C₂³×C₃₁	#124	1	2	yes	—
182	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
183	Q₈×C₃₁	—	4	2	no	—
184	Q₈×C₃₁	—	4	2	no	—
185	Q₈×C₃₁	—	8	4	no	—
186	D₄×C₃₁	—	4	2	no	—
187	D₄×C₃₁	—	4	2	no	—
188	D₄×C₃₁	—	8	4	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	1	no	—
194	C₂×C₄×C₃₁	—	2	1	no	—
195	C₂×C₄×C₃₁	—	2	1	no	—
196	C₂×C₄×C₃₁	—	2	1	no	—
197	C₂×C₄×C₃₁	—	2	1	no	—
198	C₂×C₄×C₃₁	—	2	1	no	—
199	C₂×C₄×C₃₁	—	2	4	no	—
200	C₂×C₄×C₃₁	—	2	4	no	—
201	C₂×C₄×C₃₁	—	4	4	no	—
202	C₂×C₄×C₃₁	—	4	4	no	—
203	C₂×C₄×C₃₁	—	4	1	no	—
204	C₂×C₄×C₃₁	—	4	2	no	—
205	C₂×C₄×C₃₁	—	4	1	no	—
206	C₂×C₄×C₃₁	—	4	1	no	—
207	C₂×C₄×C₃₁	—	4	1	no	—
208	C₂×C₄×C₃₁	—	4	2	no	—
209	C₂×C₄×C₃₁	—	4	1	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	2	no	—
215	C₂×C₄×C₃₁	—	4	1	no	—
216	C₂×C₄×C₃₁	—	4	1	no	—
217	C₂×C₄×C₃₁	—	4	2	no	—
218	C₂×C₄×C₃₁	#121	1	2	yes	—
219	C₂×C₄×C₃₁	#121	1	2	yes	—
220	C₂×C₄×C₃₁	#124	1	2	yes	—
221	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
222	C₂×C₄×C₃₁	#137	1	2	yes	—
223	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
224	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
225	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
226	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
227	C₈×C₃₁	—	2	4	no	—
228	C₈×C₃₁	—	2	4	no	—
229	C₂×#489	C₂×C₃₁	1	1	yes	characteristic, complemented
230	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
231	#2857	C₂×C₃₁	1	1	yes	characteristic, complemented
232	#2850	C₂×C₃₁	1	1	yes	characteristic
233	#1993	C₂×C₃₁	1	1	yes	characteristic, complemented
234	#1959	C₂×C₃₁	1	1	yes	characteristic, complemented
235	#1934	C₂×C₃₁	1	2	yes	complemented
236	#1933	C₂×C₃₁	1	2	yes	complemented
237	#1895	C₂×C₃₁	1	2	yes	—
238	#1894	C₂×C₃₁	1	2	yes	complemented
239	#1836	C₂×C₃₁	1	1	yes	characteristic, complemented
240	C₂²×C₃₁	—	2	2	no	—
241	C₂²×C₃₁	—	2	2	no	—
242	C₂²×C₃₁	—	2	2	no	—
243	C₂²×C₃₁	—	4	1	no	—
244	C₂²×C₃₁	—	4	1	no	—
245	C₂²×C₃₁	—	4	1	no	—
246	C₂²×C₃₁	—	4	1	no	—
247	C₂²×C₃₁	—	8	1	no	—
248	C₂²×C₃₁	—	8	1	no	—
249	C₂²×C₃₁	—	8	1	no	—
250	C₂²×C₃₁	—	8	1	no	—
251	C₂²×C₃₁	—	8	1	no	—
252	C₂²×C₃₁	—	8	1	no	—
253	C₂²×C₃₁	—	8	1	no	—
254	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
255	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
256	C₂²×C₃₁	#532	1	1	yes	characteristic
257	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
258	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
259	C₂²×C₃₁	#575	1	1	yes	characteristic
260	C₂²×C₃₁	#575	1	1	yes	characteristic
261	C₄×C₃₁	—	2	2	no	—
262	C₄×C₃₁	—	2	2	no	—
263	C₄×C₃₁	—	2	2	no	—
264	C₄×C₃₁	—	2	2	no	—
265	C₄×C₃₁	—	2	2	no	—
266	C₄×C₃₁	—	8	1	no	—
267	C₄×C₃₁	—	8	1	no	—
268	C₄×C₃₁	—	8	1	no	—
269	C₄×C₃₁	—	8	2	no	—
270	C₄×C₃₁	—	8	2	no	—
271	C₄×C₃₁	#525	1	2	yes	—
272	C₄×C₃₁	#529	1	2	yes	—
273	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
274	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
275	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
276	C₂×C₂²⋊Q₈	—	2	1	no	—
277	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
278	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
279	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
280	#489	—	2	4	no	—
281	C₂×#103	C₂²×C₃₁	1	2	yes	—
282	C₂×#103	C₂²×C₃₁	1	2	yes	—
283	C₂×#102	C₂²×C₃₁	1	2	yes	—
284	C₂×#102	C₂²×C₃₁	1	2	yes	—
285	C₄×M₁₆	—	2	4	no	—
286	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
287	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
288	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
289	#391	—	2	1	no	—
290	#391	C₂²×C₃₁	1	1	yes	characteristic
291	#390	C₂²×C₃₁	1	1	yes	characteristic
292	#389	—	2	1	no	—
293	#389	C₂²×C₃₁	1	2	yes	—
294	#388	—	2	1	no	—
295	#388	—	2	1	no	—
296	#386	—	2	1	no	—
297	#386	—	2	1	no	—
298	#384	C₂²×C₃₁	1	1	yes	characteristic
299	#382	C₂²×C₃₁	1	1	yes	characteristic
300	#380	—	2	1	no	—
301	#380	C₂²×C₃₁	1	2	yes	—
302	#340	C₂²×C₃₁	1	2	yes	—
303	#340	C₂²×C₃₁	1	2	yes	—
304	#340	C₂²×C₃₁	1	2	yes	—
305	#340	C₂²×C₃₁	1	2	yes	—
306	C₂×C₃₁	—	4	2	no	—
307	C₂×C₃₁	—	16	1	no	—
308	C₂×C₃₁	#2850	1	1	yes	characteristic
309	C₂×C₃₁	#3055	1	1	yes	characteristic
310	C₂×C₃₁	#3056	1	1	yes	characteristic
311	C₂×C₃₁	#3140	1	1	yes	characteristic
312	C₂×C₃₁	#3141	1	1	yes	characteristic
313	C₂×C₃₁	#3171	1	1	yes	characteristic
314	C₂×C₃₁	#3172	1	1	yes	characteristic
315	C₂²×Q₈	—	2	1	no	—
316	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
317	C₂²×D₄	—	2	1	no	—
318	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
319	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
320	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
321	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
322	C₂×M₁₆	—	2	4	no	—
323	C₂×M₁₆	—	2	4	no	—
324	C₂×M₁₆	—	2	4	no	—
325	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
326	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
327	C₄⋊Q₈	—	2	2	no	—
328	C₄⋊Q₈	—	2	2	no	—
329	#124	—	2	2	no	—
330	#124	—	2	2	no	—
331	C₂²⋊Q₈	—	2	4	no	—
332	C₂²⋊Q₈	—	4	4	no	—
333	#121	—	2	4	no	—
334	C₂×C₄⋊C₄	—	2	1	no	—
335	C₂×C₄⋊C₄	—	2	2	no	—
336	C₂×C₄⋊C₄	—	2	1	no	—
337	C₂×C₄⋊C₄	—	2	1	no	—
338	C₂×C₄⋊C₄	—	2	1	no	—
339	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
340	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
341	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
342	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
343	C₂×C₂²⋊C₄	—	2	2	no	—
344	C₂×C₂²⋊C₄	—	2	1	no	—
345	C₂×C₂²⋊C₄	—	2	1	no	—
346	C₂×C₂²⋊C₄	—	2	1	no	—
347	C₂×C₂²⋊C₄	—	2	1	no	—
348	C₂×C₂²⋊C₄	—	2	1	no	—
349	C₂×C₂²⋊C₄	—	2	1	no	—
350	C₂×C₄²	—	2	2	no	—
351	C₂×C₄²	—	2	2	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	#103	—	2	4	no	—
357	#103	—	2	4	no	—
358	#102	—	2	4	no	—
359	#102	—	2	4	no	—
360	#97	—	2	2	no	—
361	#97	D₄×C₃₁	1	4	yes	complemented
362	C₄×C₈	—	2	4	no	—
363	#95	—	2	2	no	—
364	#95	—	2	2	no	—
365	#95	—	2	1	no	—
366	#95	—	2	1	no	—
367	#95	—	2	1	no	—
368	#95	—	2	1	no	—
369	C₃₁	#33057	1	1	yes	characteristic, direct product factor, Sylow
370	C₂⁴	—	2	1	no	—
371	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
372	C₂×Q₈	—	2	1	no	—
373	C₂×Q₈	—	2	2	no	—
374	C₂×Q₈	—	2	2	no	—
375	C₂×Q₈	—	2	1	no	—
376	C₂×Q₈	—	4	2	no	—
377	C₂×Q₈	—	4	2	no	—
378	C₂×Q₈	—	4	2	no	—
379	C₂×D₄	—	2	1	no	—
380	C₂×D₄	—	2	2	no	—
381	C₂×D₄	—	2	2	no	—
382	C₂×D₄	—	2	1	no	—
383	C₂×D₄	—	4	2	no	—
384	C₂×D₄	—	4	2	no	—
385	C₂×D₄	—	4	2	no	—
386	C₂²×C₄	—	2	1	no	—
387	C₂²×C₄	—	2	1	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	2	no	—
397	C₂²×C₄	—	2	2	no	—
398	C₂²×C₄	—	4	1	no	—
399	C₂²×C₄	—	4	1	no	—
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	C₂×D₄×C₃₁	1	1	yes	characteristic
405	C₂²×C₄	#41×C₃₁	1	4	yes	—
406	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
407	M₁₆	—	2	16	no	—
408	C₂×C₈	—	2	2	no	—
409	C₂×C₈	—	2	2	no	—
410	C₂×C₈	—	2	2	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
413	C₂×C₈	#41×C₃₁	1	4	yes	—
414	C₄⋊C₄	—	2	2	no	—
415	C₄⋊C₄	—	2	2	no	—
416	C₄⋊C₄	—	2	2	no	—
417	C₄⋊C₄	—	2	2	no	—
418	C₄⋊C₄	—	2	2	no	—
419	C₄⋊C₄	—	2	2	no	—
420	C₄⋊C₄	—	4	2	no	—
421	C₄⋊C₄	—	4	2	no	—
422	C₄⋊C₄	—	4	2	no	—
423	C₄⋊C₄	—	4	2	no	—
424	C₄⋊C₄	—	4	4	no	—
425	C₂²⋊C₄	—	4	4	no	—
426	C₂²⋊C₄	—	4	2	no	—
427	C₂²⋊C₄	—	4	2	no	—
428	C₂²⋊C₄	—	4	2	no	—
429	C₂²⋊C₄	—	4	2	no	—
430	C₂²⋊C₄	—	4	2	no	—
431	C₂²⋊C₄	—	4	2	no	—
432	C₄²	—	2	2	no	—
433	C₄²	—	2	2	no	—
434	C₄²	—	2	1	no	—
435	C₄²	—	2	1	no	—
436	C₄²	C₂×D₄×C₃₁	1	2	yes	—
437	C₄²	#41×C₃₁	1	2	yes	—
438	C₂³	—	2	1	no	—
439	C₂³	—	2	1	no	—
440	C₂³	—	2	1	no	—
441	C₂³	—	2	1	no	—
442	C₂³	—	2	1	no	—
443	C₂³	—	2	1	no	—
444	C₂³	—	4	1	no	—
445	C₂³	—	4	1	no	—
446	C₂³	—	4	1	no	—
447	C₂³	—	4	1	no	—
448	C₂³	—	4	1	no	—
449	C₂³	—	4	1	no	—
450	C₂³	—	4	1	no	—
451	C₂³	C₃₁×#124	1	2	yes	—
452	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
453	Q₈	—	4	2	no	—
454	Q₈	—	4	2	no	—
455	Q₈	—	8	4	no	—
456	D₄	—	4	2	no	—
457	D₄	—	4	2	no	—
458	D₄	—	8	4	no	—
459	C₂×C₄	—	2	1	no	—
460	C₂×C₄	—	2	1	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	1	no	—
463	C₂×C₄	—	2	1	no	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	2	1	no	—
466	C₂×C₄	—	2	1	no	—
467	C₂×C₄	—	2	1	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	4	no	—
470	C₂×C₄	—	2	4	no	—
471	C₂×C₄	—	4	4	no	—
472	C₂×C₄	—	4	4	no	—
473	C₂×C₄	—	4	1	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	4	1	no	—
476	C₂×C₄	—	4	1	no	—
477	C₂×C₄	—	4	1	no	—
478	C₂×C₄	—	4	2	no	—
479	C₂×C₄	—	4	1	no	—
480	C₂×C₄	—	4	2	no	—
481	C₂×C₄	—	4	2	no	—
482	C₂×C₄	—	4	1	no	—
483	C₂×C₄	—	4	1	no	—
484	C₂×C₄	—	4	2	no	—
485	C₂×C₄	—	4	1	no	—
486	C₂×C₄	—	4	1	no	—
487	C₂×C₄	—	4	2	no	—
488	C₂×C₄	C₃₁×#121	1	2	yes	—
489	C₂×C₄	C₃₁×#121	1	2	yes	—
490	C₂×C₄	C₃₁×#124	1	2	yes	—
491	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
492	C₂×C₄	C₃₁×#137	1	2	yes	—
493	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
494	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
495	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
496	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
497	C₈	—	2	4	no	—
498	C₈	—	2	4	no	—
499	C₂²	—	2	2	no	—
500	C₂²	—	2	2	no	—
501	C₂²	—	2	2	no	—
502	C₂²	—	4	1	no	—
503	C₂²	—	4	1	no	—
504	C₂²	—	4	1	no	—
505	C₂²	—	4	1	no	—
506	C₂²	—	8	1	no	—
507	C₂²	—	8	1	no	—
508	C₂²	—	8	1	no	—
509	C₂²	—	8	1	no	—
510	C₂²	—	8	1	no	—
511	C₂²	—	8	1	no	—
512	C₂²	—	8	1	no	—
513	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
514	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
515	C₂²	C₃₁×#532	1	1	yes	characteristic
516	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
517	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
518	C₂²	C₃₁×#575	1	1	yes	characteristic
519	C₂²	C₃₁×#575	1	1	yes	characteristic
520	C₄	—	2	2	no	—
521	C₄	—	2	2	no	—
522	C₄	—	2	2	no	—
523	C₄	—	2	2	no	—
524	C₄	—	2	2	no	—
525	C₄	—	8	1	no	—
526	C₄	—	8	1	no	—
527	C₄	—	8	1	no	—
528	C₄	—	8	2	no	—
529	C₄	—	8	2	no	—
530	C₄	C₃₁×#525	1	2	yes	—
531	C₄	C₃₁×#529	1	2	yes	—
532	C₂	—	4	2	no	—
533	C₂	—	16	1	no	—
534	C₂	C₃₁×#2850	1	1	yes	characteristic
535	C₂	C₃₁×#3055	1	1	yes	characteristic
536	C₂	C₃₁×#3056	1	1	yes	characteristic
537	C₂	C₃₁×#3140	1	1	yes	characteristic
538	C₂	C₃₁×#3141	1	1	yes	characteristic
539	C₂	C₃₁×#3171	1	1	yes	characteristic
540	C₂	C₃₁×#3172	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#489	C₂	semidirect
2. C₂×C₄×M₁₆×C₃₁	C₂	semidirect
3. C₃₁×#2857	C₂	semidirect
5. C₃₁×#1993	C₂	semidirect
6. C₃₁×#1959	C₂	semidirect
7. C₃₁×#1934	C₂	semidirect
8. C₃₁×#1933	C₂	semidirect
10. C₃₁×#1894	C₂	semidirect
11. C₃₁×#1836	C₂	semidirect
91. C₃₁×#97	D₄	semidirect
167. #33057	C₃₁	direct (N × H)
229. C₂×#489	C₂×C₃₁	semidirect
230. C₂×C₄×M₁₆	C₂×C₃₁	semidirect
231. #2857	C₂×C₃₁	semidirect
233. #1993	C₂×C₃₁	semidirect
234. #1959	C₂×C₃₁	semidirect
235. #1934	C₂×C₃₁	semidirect
236. #1933	C₂×C₃₁	semidirect
238. #1894	C₂×C₃₁	semidirect
239. #1836	C₂×C₃₁	semidirect
361. #97	D₄×C₃₁	semidirect
369. C₃₁	#33057	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#489	2
C₂×C₄×M₁₆×C₃₁	2
C₃₁×#2857	2
C₃₁×#2850	2
C₃₁×#1993	2
C₃₁×#1959	2
C₃₁×#1934	2
C₃₁×#1933	2
C₃₁×#1895	2
C₃₁×#1894	2
C₃₁×#1836	2
#33057	31

Subgroups — C31×#33057 #21489816

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33057 #21489816