Subgroups — C₃₁×#24783 #21481542

Order 7936.

984 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#470	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#467	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#435	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₃₁×#106	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2417	C₂	1	2	yes	maximal
6	C₃₁×#2066	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2035	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1971	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1944	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1872	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1809	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
16	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
17	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
19	C₃₁×#470	—	2	4	no	—
20	C₃₁×#467	—	2	4	no	—
21	C₃₁×#435	—	2	4	no	—
22	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#106	—	2	4	no	—
24	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#106	C₂²	1	2	yes	—
26	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
32	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
33	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
34	C₃₁×#398	—	2	2	no	—
35	C₃₁×#396	—	2	2	no	—
36	C₃₁×#391	C₂²	1	2	yes	—
37	C₃₁×#389	C₂²	1	2	yes	—
38	C₃₁×#386	—	2	2	no	—
39	C₃₁×#380	—	2	2	no	—
40	C₃₁×#380	C₂²	1	2	yes	—
41	C₃₁×#340	C₂²	1	2	yes	—
42	C₃₁×#340	C₂²	1	2	yes	—
43	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
44	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
45	C₂³×C₄×C₃₁	—	2	2	no	—
46	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₂×Q₁₆×C₃₁	—	2	2	no	—
49	C₂×Q₁₆×C₃₁	—	2	2	no	—
50	C₂×Q₁₆×C₃₁	—	2	2	no	—
51	C₂×M₁₆×C₃₁	—	2	2	no	—
52	C₂×M₁₆×C₃₁	—	2	2	no	—
53	C₂×M₁₆×C₃₁	—	2	2	no	—
54	C₂²×C₈×C₃₁	—	2	2	no	—
55	C₂²×C₈×C₃₁	—	2	2	no	—
56	C₂²×C₈×C₃₁	D₄	1	4	yes	—
57	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
61	C₃₁×C₂²⋊Q₈	—	2	4	no	—
62	C₃₁×C₂²⋊Q₈	—	2	4	no	—
63	C₃₁×C₂²⋊Q₈	—	2	4	no	—
64	C₃₁×C₂²⋊Q₈	—	4	2	no	—
65	C₄×D₄×C₃₁	—	2	2	no	—
66	C₄×D₄×C₃₁	—	2	2	no	—
67	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
68	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
69	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
70	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
71	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
72	C₂×C₄⋊C₄×C₃₁	—	4	2	no	—
73	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
74	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
75	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
76	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
77	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
78	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
79	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
80	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
81	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
82	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
83	C₃₁×#107	—	2	2	no	—
84	C₃₁×#106	—	2	2	no	—
85	C₃₁×#106	—	2	4	no	—
86	C₃₁×#106	D₄	1	4	yes	complemented
87	C₃₁×C₄⋊C₈	—	2	2	no	—
88	C₃₁×#103	—	2	4	no	—
89	C₃₁×#103	—	2	4	no	—
90	C₃₁×#103	—	4	2	no	—
91	C₃₁×C₂²⋊C₈	—	2	2	no	—
92	C₃₁×C₂²⋊C₈	—	2	2	no	—
93	C₃₁×#97	—	2	2	no	—
94	C₃₁×#95	—	2	2	no	—
95	C₃₁×#95	—	2	2	no	—
96	C₃₁×#95	—	2	2	no	—
97	C₃₁×#95	—	2	2	no	—
98	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
99	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₂×Q₈×C₃₁	—	2	2	no	—
101	C₂×Q₈×C₃₁	—	2	2	no	—
102	C₂×Q₈×C₃₁	—	2	2	no	—
103	C₂×Q₈×C₃₁	—	2	2	no	—
104	C₂×Q₈×C₃₁	—	2	2	no	—
105	C₂×Q₈×C₃₁	—	2	2	no	—
106	C₂×D₄×C₃₁	—	2	2	no	—
107	C₂×D₄×C₃₁	—	2	4	no	—
108	C₂²×C₄×C₃₁	—	2	1	no	—
109	C₂²×C₄×C₃₁	—	2	1	no	—
110	C₂²×C₄×C₃₁	—	2	1	no	—
111	C₂²×C₄×C₃₁	—	2	1	no	—
112	C₂²×C₄×C₃₁	—	2	2	no	—
113	C₂²×C₄×C₃₁	—	2	1	no	—
114	C₂²×C₄×C₃₁	—	2	1	no	—
115	C₂²×C₄×C₃₁	—	2	1	no	—
116	C₂²×C₄×C₃₁	—	2	1	no	—
117	C₂²×C₄×C₃₁	—	2	1	no	—
118	C₂²×C₄×C₃₁	—	2	1	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	—	4	4	no	—
123	C₂²×C₄×C₃₁	—	4	4	no	—
124	C₂²×C₄×C₃₁	—	8	2	no	—
125	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
130	C₂²×C₄×C₃₁	#41	1	2	yes	—
131	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
132	Q₁₆×C₃₁	—	4	4	no	—
133	M₁₆×C₃₁	—	4	4	no	—
134	C₂×C₈×C₃₁	—	2	1	no	—
135	C₂×C₈×C₃₁	—	2	1	no	—
136	C₂×C₈×C₃₁	—	2	8	no	—
137	C₂×C₈×C₃₁	—	2	1	no	—
138	C₂×C₈×C₃₁	—	2	1	no	—
139	C₂×C₈×C₃₁	—	2	1	no	—
140	C₂×C₈×C₃₁	—	2	1	no	—
141	C₂×C₈×C₃₁	—	2	1	no	—
142	C₂×C₈×C₃₁	—	2	1	no	—
143	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
144	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
145	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
146	C₂×C₈×C₃₁	#41	1	2	yes	—
147	C₄⋊C₄×C₃₁	—	2	2	no	—
148	C₄⋊C₄×C₃₁	—	2	2	no	—
149	C₄⋊C₄×C₃₁	—	2	2	no	—
150	C₄⋊C₄×C₃₁	—	2	2	no	—
151	C₄⋊C₄×C₃₁	—	2	2	no	—
152	C₄⋊C₄×C₃₁	—	2	2	no	—
153	C₄⋊C₄×C₃₁	—	4	4	no	—
154	C₄⋊C₄×C₃₁	—	4	4	no	—
155	C₄⋊C₄×C₃₁	—	4	4	no	—
156	C₄⋊C₄×C₃₁	—	8	2	no	—
157	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
158	C₂²⋊C₄×C₃₁	—	2	2	no	—
159	C₂²⋊C₄×C₃₁	—	2	2	no	—
160	C₂²⋊C₄×C₃₁	—	4	4	no	—
161	C₂²⋊C₄×C₃₁	—	4	4	no	—
162	C₂²⋊C₄×C₃₁	—	8	2	no	—
163	C₄²×C₃₁	—	2	1	no	—
164	C₄²×C₃₁	—	2	1	no	—
165	#24783	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
166	C₂³×C₃₁	—	2	1	no	—
167	C₂³×C₃₁	—	2	1	no	—
168	C₂³×C₃₁	—	2	2	no	—
169	C₂³×C₃₁	—	2	1	no	—
170	C₂³×C₃₁	—	2	1	no	—
171	C₂³×C₃₁	—	2	1	no	—
172	C₂³×C₃₁	—	2	2	no	—
173	C₂³×C₃₁	—	2	2	no	—
174	C₂³×C₃₁	#121	1	2	yes	—
175	C₂³×C₃₁	#137	1	4	yes	—
176	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
177	Q₈×C₃₁	—	4	2	no	—
178	Q₈×C₃₁	—	4	2	no	—
179	Q₈×C₃₁	—	4	2	no	—
180	Q₈×C₃₁	—	4	2	no	—
181	D₄×C₃₁	—	4	4	no	—
182	C₂×C₄×C₃₁	—	2	1	no	—
183	C₂×C₄×C₃₁	—	2	1	no	—
184	C₂×C₄×C₃₁	—	2	2	no	—
185	C₂×C₄×C₃₁	—	2	2	no	—
186	C₂×C₄×C₃₁	—	2	2	no	—
187	C₂×C₄×C₃₁	—	2	2	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	4	no	—
192	C₂×C₄×C₃₁	—	2	1	no	—
193	C₂×C₄×C₃₁	—	2	1	no	—
194	C₂×C₄×C₃₁	—	2	2	no	—
195	C₂×C₄×C₃₁	—	2	2	no	—
196	C₂×C₄×C₃₁	—	4	2	no	—
197	C₂×C₄×C₃₁	—	4	2	no	—
198	C₂×C₄×C₃₁	—	4	1	no	—
199	C₂×C₄×C₃₁	—	4	1	no	—
200	C₂×C₄×C₃₁	—	4	1	no	—
201	C₂×C₄×C₃₁	—	4	1	no	—
202	C₂×C₄×C₃₁	—	4	2	no	—
203	C₂×C₄×C₃₁	—	4	2	no	—
204	C₂×C₄×C₃₁	—	4	2	no	—
205	C₂×C₄×C₃₁	—	4	2	no	—
206	C₂×C₄×C₃₁	—	4	2	no	—
207	C₂×C₄×C₃₁	—	4	2	no	—
208	C₂×C₄×C₃₁	—	4	2	no	—
209	C₂×C₄×C₃₁	—	4	2	no	—
210	C₂×C₄×C₃₁	—	4	2	no	—
211	C₂×C₄×C₃₁	—	8	8	no	—
212	C₂×C₄×C₃₁	#121	1	2	yes	—
213	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
215	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
216	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
217	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
218	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
219	C₈×C₃₁	—	2	2	no	—
220	C₈×C₃₁	—	4	1	no	—
221	C₈×C₃₁	—	4	1	no	—
222	C₈×C₃₁	#121	1	4	yes	—
223	C₂×#470	C₂×C₃₁	1	1	yes	characteristic, complemented
224	C₂×#467	C₂×C₃₁	1	1	yes	characteristic, complemented
225	C₂×#435	C₂×C₃₁	1	1	yes	characteristic
226	C₂²×#106	C₂×C₃₁	1	1	yes	characteristic, complemented
227	#2417	C₂×C₃₁	1	2	yes	—
228	#2066	C₂×C₃₁	1	2	yes	complemented
229	#2035	C₂×C₃₁	1	2	yes	complemented
230	#1971	C₂×C₃₁	1	1	yes	characteristic, complemented
231	#1944	C₂×C₃₁	1	1	yes	characteristic, complemented
232	#1872	C₂×C₃₁	1	2	yes	complemented
233	#1809	C₂×C₃₁	1	1	yes	characteristic, complemented
234	C₂²×C₃₁	—	2	4	no	—
235	C₂²×C₃₁	—	2	2	no	—
236	C₂²×C₃₁	—	2	4	no	—
237	C₂²×C₃₁	—	2	2	no	—
238	C₂²×C₃₁	—	4	2	no	—
239	C₂²×C₃₁	—	4	1	no	—
240	C₂²×C₃₁	—	4	2	no	—
241	C₂²×C₃₁	—	4	1	no	—
242	C₂²×C₃₁	—	4	1	no	—
243	C₂²×C₃₁	#470	1	4	yes	complemented
244	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
245	C₂²×C₃₁	D₄²	1	1	yes	characteristic
246	C₂²×C₃₁	#547	1	1	yes	characteristic
247	C₂²×C₃₁	C₂×#137	1	2	yes	—
248	C₂²×C₃₁	#577	1	2	yes	—
249	C₄×C₃₁	—	2	2	no	—
250	C₄×C₃₁	—	4	1	no	—
251	C₄×C₃₁	—	4	4	no	—
252	C₄×C₃₁	—	4	1	no	—
253	C₄×C₃₁	—	4	1	no	—
254	C₄×C₃₁	—	8	2	no	—
255	C₄×C₃₁	—	8	2	no	—
256	C₄×C₃₁	—	8	2	no	—
257	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
258	C₄×C₃₁	#537	1	1	yes	characteristic
259	C₄×C₃₁	#548	1	1	yes	characteristic
260	C₄×C₃₁	#553	1	1	yes	characteristic
261	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
262	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
263	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
264	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
265	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
266	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
267	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
268	#470	—	2	4	no	—
269	#467	—	2	4	no	—
270	#435	—	2	4	no	—
271	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
272	C₂×#106	—	2	4	no	—
273	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
274	C₂×#106	C₂²×C₃₁	1	2	yes	—
275	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
276	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
277	C₂×#103	C₂²×C₃₁	1	2	yes	—
278	C₂×#103	C₂²×C₃₁	1	2	yes	—
279	C₂×#103	C₂²×C₃₁	1	2	yes	—
280	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
281	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
282	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
283	#398	—	2	2	no	—
284	#396	—	2	2	no	—
285	#391	C₂²×C₃₁	1	2	yes	—
286	#389	C₂²×C₃₁	1	2	yes	—
287	#386	—	2	2	no	—
288	#380	—	2	2	no	—
289	#380	C₂²×C₃₁	1	2	yes	—
290	#340	C₂²×C₃₁	1	2	yes	—
291	#340	C₂²×C₃₁	1	2	yes	—
292	C₂×C₃₁	—	2	4	no	—
293	C₂×C₃₁	—	8	1	no	—
294	C₂×C₃₁	#2416	1	1	yes	characteristic
295	C₂×C₃₁	C₂×#470	1	1	yes	characteristic
296	C₂×C₃₁	#3060	1	1	yes	characteristic
297	C₂×C₃₁	#3266	1	2	yes	—
298	C₂×C₃₁	#3344	1	2	yes	—
299	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
300	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
301	C₂³×C₄	—	2	2	no	—
302	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
303	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
304	C₂×Q₁₆	—	2	2	no	—
305	C₂×Q₁₆	—	2	2	no	—
306	C₂×Q₁₆	—	2	2	no	—
307	C₂×M₁₆	—	2	2	no	—
308	C₂×M₁₆	—	2	2	no	—
309	C₂×M₁₆	—	2	2	no	—
310	C₂²×C₈	—	2	2	no	—
311	C₂²×C₈	—	2	2	no	—
312	C₂²×C₈	D₄×C₃₁	1	4	yes	—
313	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
314	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
315	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
316	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
317	C₂²⋊Q₈	—	2	4	no	—
318	C₂²⋊Q₈	—	2	4	no	—
319	C₂²⋊Q₈	—	2	4	no	—
320	C₂²⋊Q₈	—	4	2	no	—
321	C₄×D₄	—	2	2	no	—
322	C₄×D₄	—	2	2	no	—
323	C₂×C₄⋊C₄	—	2	4	no	—
324	C₂×C₄⋊C₄	—	2	2	no	—
325	C₂×C₄⋊C₄	—	2	2	no	—
326	C₂×C₄⋊C₄	—	2	2	no	—
327	C₂×C₄⋊C₄	—	2	2	no	—
328	C₂×C₄⋊C₄	—	4	2	no	—
329	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
330	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
331	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
332	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
333	C₂×C₂²⋊C₄	—	2	2	no	—
334	C₂×C₂²⋊C₄	—	2	2	no	—
335	C₂×C₂²⋊C₄	—	2	2	no	—
336	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
337	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
338	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
339	#107	—	2	2	no	—
340	#106	—	2	2	no	—
341	#106	—	2	4	no	—
342	#106	D₄×C₃₁	1	4	yes	complemented
343	C₄⋊C₈	—	2	2	no	—
344	#103	—	2	4	no	—
345	#103	—	2	4	no	—
346	#103	—	4	2	no	—
347	C₂²⋊C₈	—	2	2	no	—
348	C₂²⋊C₈	—	2	2	no	—
349	#97	—	2	2	no	—
350	#95	—	2	2	no	—
351	#95	—	2	2	no	—
352	#95	—	2	2	no	—
353	#95	—	2	2	no	—
354	C₃₁	#24783	1	1	yes	characteristic, direct product factor, Sylow
355	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
356	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
357	C₂×Q₈	—	2	2	no	—
358	C₂×Q₈	—	2	2	no	—
359	C₂×Q₈	—	2	2	no	—
360	C₂×Q₈	—	2	2	no	—
361	C₂×Q₈	—	2	2	no	—
362	C₂×Q₈	—	2	2	no	—
363	C₂×D₄	—	2	2	no	—
364	C₂×D₄	—	2	4	no	—
365	C₂²×C₄	—	2	1	no	—
366	C₂²×C₄	—	2	1	no	—
367	C₂²×C₄	—	2	1	no	—
368	C₂²×C₄	—	2	1	no	—
369	C₂²×C₄	—	2	2	no	—
370	C₂²×C₄	—	2	1	no	—
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	2	1	no	—
373	C₂²×C₄	—	2	1	no	—
374	C₂²×C₄	—	2	1	no	—
375	C₂²×C₄	—	2	1	no	—
376	C₂²×C₄	—	2	2	no	—
377	C₂²×C₄	—	2	2	no	—
378	C₂²×C₄	—	2	2	no	—
379	C₂²×C₄	—	4	4	no	—
380	C₂²×C₄	—	4	4	no	—
381	C₂²×C₄	—	8	2	no	—
382	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
383	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
384	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
385	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
386	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
387	C₂²×C₄	#41×C₃₁	1	2	yes	—
388	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
389	Q₁₆	—	4	4	no	—
390	M₁₆	—	4	4	no	—
391	C₂×C₈	—	2	1	no	—
392	C₂×C₈	—	2	1	no	—
393	C₂×C₈	—	2	8	no	—
394	C₂×C₈	—	2	1	no	—
395	C₂×C₈	—	2	1	no	—
396	C₂×C₈	—	2	1	no	—
397	C₂×C₈	—	2	1	no	—
398	C₂×C₈	—	2	1	no	—
399	C₂×C₈	—	2	1	no	—
400	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
401	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
402	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
403	C₂×C₈	#41×C₃₁	1	2	yes	—
404	C₄⋊C₄	—	2	2	no	—
405	C₄⋊C₄	—	2	2	no	—
406	C₄⋊C₄	—	2	2	no	—
407	C₄⋊C₄	—	2	2	no	—
408	C₄⋊C₄	—	2	2	no	—
409	C₄⋊C₄	—	2	2	no	—
410	C₄⋊C₄	—	4	4	no	—
411	C₄⋊C₄	—	4	4	no	—
412	C₄⋊C₄	—	4	4	no	—
413	C₄⋊C₄	—	8	2	no	—
414	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
415	C₂²⋊C₄	—	2	2	no	—
416	C₂²⋊C₄	—	2	2	no	—
417	C₂²⋊C₄	—	4	4	no	—
418	C₂²⋊C₄	—	4	4	no	—
419	C₂²⋊C₄	—	8	2	no	—
420	C₄²	—	2	1	no	—
421	C₄²	—	2	1	no	—
422	C₂³	—	2	1	no	—
423	C₂³	—	2	1	no	—
424	C₂³	—	2	2	no	—
425	C₂³	—	2	1	no	—
426	C₂³	—	2	1	no	—
427	C₂³	—	2	1	no	—
428	C₂³	—	2	2	no	—
429	C₂³	—	2	2	no	—
430	C₂³	C₃₁×#121	1	2	yes	—
431	C₂³	C₃₁×#137	1	4	yes	—
432	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
433	Q₈	—	4	2	no	—
434	Q₈	—	4	2	no	—
435	Q₈	—	4	2	no	—
436	Q₈	—	4	2	no	—
437	D₄	—	4	4	no	—
438	C₂×C₄	—	2	1	no	—
439	C₂×C₄	—	2	1	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	1	no	—
445	C₂×C₄	—	2	1	no	—
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	2	4	no	—
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	4	2	no	—
453	C₂×C₄	—	4	2	no	—
454	C₂×C₄	—	4	1	no	—
455	C₂×C₄	—	4	1	no	—
456	C₂×C₄	—	4	1	no	—
457	C₂×C₄	—	4	1	no	—
458	C₂×C₄	—	4	2	no	—
459	C₂×C₄	—	4	2	no	—
460	C₂×C₄	—	4	2	no	—
461	C₂×C₄	—	4	2	no	—
462	C₂×C₄	—	4	2	no	—
463	C₂×C₄	—	4	2	no	—
464	C₂×C₄	—	4	2	no	—
465	C₂×C₄	—	4	2	no	—
466	C₂×C₄	—	4	2	no	—
467	C₂×C₄	—	8	8	no	—
468	C₂×C₄	C₃₁×#121	1	2	yes	—
469	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
470	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
471	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
472	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
473	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
474	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
475	C₈	—	2	2	no	—
476	C₈	—	4	1	no	—
477	C₈	—	4	1	no	—
478	C₈	C₃₁×#121	1	4	yes	—
479	C₂²	—	2	4	no	—
480	C₂²	—	2	2	no	—
481	C₂²	—	2	4	no	—
482	C₂²	—	2	2	no	—
483	C₂²	—	4	2	no	—
484	C₂²	—	4	1	no	—
485	C₂²	—	4	2	no	—
486	C₂²	—	4	1	no	—
487	C₂²	—	4	1	no	—
488	C₂²	C₃₁×#470	1	4	yes	complemented
489	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
490	C₂²	D₄²×C₃₁	1	1	yes	characteristic
491	C₂²	C₃₁×#547	1	1	yes	characteristic
492	C₂²	C₂×C₃₁×#137	1	2	yes	—
493	C₂²	C₃₁×#577	1	2	yes	—
494	C₄	—	2	2	no	—
495	C₄	—	4	1	no	—
496	C₄	—	4	4	no	—
497	C₄	—	4	1	no	—
498	C₄	—	4	1	no	—
499	C₄	—	8	2	no	—
500	C₄	—	8	2	no	—
501	C₄	—	8	2	no	—
502	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
503	C₄	C₃₁×#537	1	1	yes	characteristic
504	C₄	C₃₁×#548	1	1	yes	characteristic
505	C₄	C₃₁×#553	1	1	yes	characteristic
506	C₂	—	2	4	no	—
507	C₂	—	8	1	no	—
508	C₂	C₃₁×#2416	1	1	yes	characteristic
509	C₂	C₂×C₃₁×#470	1	1	yes	characteristic
510	C₂	C₃₁×#3060	1	1	yes	characteristic
511	C₂	C₃₁×#3266	1	2	yes	—
512	C₂	C₃₁×#3344	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#470	C₂	semidirect
2. C₂×C₃₁×#467	C₂	semidirect
4. C₂²×C₃₁×#106	C₂	semidirect
6. C₃₁×#2066	C₂	semidirect
7. C₃₁×#2035	C₂	semidirect
8. C₃₁×#1971	C₂	semidirect
9. C₃₁×#1944	C₂	semidirect
10. C₃₁×#1872	C₂	semidirect
11. C₃₁×#1809	C₂	semidirect
86. C₃₁×#106	D₄	semidirect
165. #24783	C₃₁	direct (N × H)
223. C₂×#470	C₂×C₃₁	semidirect
224. C₂×#467	C₂×C₃₁	semidirect
226. C₂²×#106	C₂×C₃₁	semidirect
228. #2066	C₂×C₃₁	semidirect
229. #2035	C₂×C₃₁	semidirect
230. #1971	C₂×C₃₁	semidirect
231. #1944	C₂×C₃₁	semidirect
232. #1872	C₂×C₃₁	semidirect
233. #1809	C₂×C₃₁	semidirect
243. C₂²×C₃₁	#470	semidirect
342. #106	D₄×C₃₁	semidirect
354. C₃₁	#24783	direct (N × H)
488. C₂²	C₃₁×#470	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#470	2
C₂×C₃₁×#467	2
C₂×C₃₁×#435	2
C₂²×C₃₁×#106	2
C₃₁×#2417	2
C₃₁×#2066	2
C₃₁×#2035	2
C₃₁×#1971	2
C₃₁×#1944	2
C₃₁×#1872	2
C₃₁×#1809	2
#24783	31

Subgroups — C31×#24783 #21481542

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24783 #21481542