Subgroups — C₃₁×#24785 #21481544

Order 7936.

940 conjugacy classes of subgroups.

The subgroup lattice has 508 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	C₂×C₃₁×#427	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2415	C₂	1	2	yes	maximal
6	C₃₁×#2068	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2036	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×#1873	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	C₂×C₃₁×#117	C₂²	1	2	yes	—
19	C₃₁×#470	—	2	4	no	—
20	C₃₁×#467	—	2	4	no	—
21	C₃₁×#435	—	2	4	no	—
22	C₃₁×#427	—	2	4	no	—
23	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#107	C₂²	1	2	yes	—
25	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
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×#397	—	2	2	no	—
35	C₃₁×#394	—	2	2	no	—
36	C₃₁×#391	C₂²	1	2	yes	—
37	C₃₁×#389	—	2	2	no	—
38	C₃₁×#389	C₂²	1	2	yes	—
39	C₃₁×#388	—	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	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
47	C₂×Q₁₆×C₃₁	—	2	2	no	—
48	C₂×Q₁₆×C₃₁	—	2	2	no	—
49	C₂×Q₁₆×C₃₁	—	2	2	no	—
50	C₂×M₁₆×C₃₁	—	2	2	no	—
51	C₂×M₁₆×C₃₁	—	2	2	no	—
52	C₂×M₁₆×C₃₁	—	2	2	no	—
53	C₂²×C₈×C₃₁	—	2	2	no	—
54	C₂²×C₈×C₃₁	—	2	2	no	—
55	C₂²×C₈×C₃₁	D₄	1	4	yes	—
56	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
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	C₃₁×C₂²⋊Q₈	—	2	4	no	—
61	C₃₁×C₂²⋊Q₈	—	2	4	no	—
62	C₃₁×C₂²⋊Q₈	—	2	4	no	—
63	C₃₁×C₂²⋊Q₈	—	4	2	no	—
64	C₄×D₄×C₃₁	—	2	2	no	—
65	C₄×D₄×C₃₁	—	2	2	no	—
66	C₃₁×#117	—	2	4	no	—
67	C₃₁×#117	—	4	2	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	D₄	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
73	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
74	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
75	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
76	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
77	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
78	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
79	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
80	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
81	C₂×C₄²×C₃₁	—	2	2	no	—
82	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
83	C₃₁×#107	—	2	2	no	—
84	C₃₁×#107	—	2	4	no	—
85	C₃₁×#106	—	2	2	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	—	4	2	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	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
127	C₂²×C₄×C₃₁	#41	1	2	yes	—
128	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
129	Q₁₆×C₃₁	—	4	4	no	—
130	M₁₆×C₃₁	—	4	4	no	—
131	C₂×C₈×C₃₁	—	2	1	no	—
132	C₂×C₈×C₃₁	—	2	1	no	—
133	C₂×C₈×C₃₁	—	2	8	no	—
134	C₂×C₈×C₃₁	—	2	1	no	—
135	C₂×C₈×C₃₁	—	2	1	no	—
136	C₂×C₈×C₃₁	—	2	1	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	C₂×D₄	1	2	yes	—
141	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
142	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
143	C₂×C₈×C₃₁	#41	1	2	yes	—
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	—	2	2	no	—
150	C₄⋊C₄×C₃₁	—	4	4	no	—
151	C₄⋊C₄×C₃₁	—	4	4	no	—
152	C₄⋊C₄×C₃₁	—	4	4	no	—
153	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
154	C₂²⋊C₄×C₃₁	—	2	2	no	—
155	C₂²⋊C₄×C₃₁	—	2	2	no	—
156	C₂²⋊C₄×C₃₁	—	4	4	no	—
157	C₂²⋊C₄×C₃₁	—	4	4	no	—
158	C₂²⋊C₄×C₃₁	—	4	4	no	—
159	C₂²⋊C₄×C₃₁	—	4	4	no	—
160	C₄²×C₃₁	—	2	1	no	—
161	C₄²×C₃₁	—	2	1	no	—
162	C₄²×C₃₁	—	4	4	no	—
163	#24785	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
164	C₂³×C₃₁	—	2	1	no	—
165	C₂³×C₃₁	—	2	1	no	—
166	C₂³×C₃₁	—	2	2	no	—
167	C₂³×C₃₁	—	2	2	no	—
168	C₂³×C₃₁	—	2	1	no	—
169	C₂³×C₃₁	—	2	1	no	—
170	C₂³×C₃₁	—	2	1	no	—
171	C₂³×C₃₁	—	2	2	no	—
172	C₂³×C₃₁	—	2	2	no	—
173	C₂³×C₃₁	#121	1	2	yes	—
174	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
175	Q₈×C₃₁	—	4	2	no	—
176	Q₈×C₃₁	—	4	2	no	—
177	Q₈×C₃₁	—	4	2	no	—
178	Q₈×C₃₁	—	4	2	no	—
179	D₄×C₃₁	—	4	4	no	—
180	C₂×C₄×C₃₁	—	2	1	no	—
181	C₂×C₄×C₃₁	—	2	1	no	—
182	C₂×C₄×C₃₁	—	2	2	no	—
183	C₂×C₄×C₃₁	—	2	2	no	—
184	C₂×C₄×C₃₁	—	2	2	no	—
185	C₂×C₄×C₃₁	—	2	2	no	—
186	C₂×C₄×C₃₁	—	2	1	no	—
187	C₂×C₄×C₃₁	—	2	1	no	—
188	C₂×C₄×C₃₁	—	2	1	no	—
189	C₂×C₄×C₃₁	—	2	4	no	—
190	C₂×C₄×C₃₁	—	2	1	no	—
191	C₂×C₄×C₃₁	—	2	1	no	—
192	C₂×C₄×C₃₁	—	2	2	no	—
193	C₂×C₄×C₃₁	—	2	2	no	—
194	C₂×C₄×C₃₁	—	4	2	no	—
195	C₂×C₄×C₃₁	—	4	2	no	—
196	C₂×C₄×C₃₁	—	4	1	no	—
197	C₂×C₄×C₃₁	—	4	1	no	—
198	C₂×C₄×C₃₁	—	4	1	no	—
199	C₂×C₄×C₃₁	—	4	1	no	—
200	C₂×C₄×C₃₁	—	4	2	no	—
201	C₂×C₄×C₃₁	—	4	2	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	#121	1	2	yes	—
210	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
211	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
212	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
213	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
214	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
215	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
216	C₈×C₃₁	—	2	2	no	—
217	C₈×C₃₁	—	4	1	no	—
218	C₈×C₃₁	—	4	1	no	—
219	C₈×C₃₁	#121	1	4	yes	—
220	C₂×#470	C₂×C₃₁	1	1	yes	characteristic, complemented
221	C₂×#467	C₂×C₃₁	1	1	yes	characteristic, complemented
222	C₂×#435	C₂×C₃₁	1	1	yes	characteristic
223	C₂×#427	C₂×C₃₁	1	1	yes	characteristic, complemented
224	#2415	C₂×C₃₁	1	2	yes	—
225	#2068	C₂×C₃₁	1	2	yes	complemented
226	#2036	C₂×C₃₁	1	2	yes	complemented
227	#1971	C₂×C₃₁	1	1	yes	characteristic, complemented
228	#1944	C₂×C₃₁	1	1	yes	characteristic, complemented
229	#1873	C₂×C₃₁	1	2	yes	complemented
230	#1809	C₂×C₃₁	1	1	yes	characteristic, complemented
231	C₂²×C₃₁	—	2	2	no	—
232	C₂²×C₃₁	—	2	2	no	—
233	C₂²×C₃₁	—	2	2	no	—
234	C₂²×C₃₁	—	4	2	no	—
235	C₂²×C₃₁	—	4	2	no	—
236	C₂²×C₃₁	—	4	2	no	—
237	C₂²×C₃₁	—	4	1	no	—
238	C₂²×C₃₁	—	4	2	no	—
239	C₂²×C₃₁	—	4	1	no	—
240	C₂²×C₃₁	—	4	1	no	—
241	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
242	C₂²×C₃₁	D₄²	1	1	yes	characteristic
243	C₂²×C₃₁	#547	1	1	yes	characteristic
244	C₂²×C₃₁	#575	1	2	yes	—
245	C₂²×C₃₁	#578	1	2	yes	—
246	C₄×C₃₁	—	2	2	no	—
247	C₄×C₃₁	—	4	1	no	—
248	C₄×C₃₁	—	4	4	no	—
249	C₄×C₃₁	—	4	1	no	—
250	C₄×C₃₁	—	4	1	no	—
251	C₄×C₃₁	—	8	2	no	—
252	C₄×C₃₁	—	8	2	no	—
253	C₄×C₃₁	—	8	2	no	—
254	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
255	C₄×C₃₁	#537	1	1	yes	characteristic
256	C₄×C₃₁	#548	1	1	yes	characteristic
257	C₄×C₃₁	#553	1	1	yes	characteristic
258	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
259	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
260	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
261	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
262	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
263	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
264	C₂×#117	C₂²×C₃₁	1	2	yes	—
265	#470	—	2	4	no	—
266	#467	—	2	4	no	—
267	#435	—	2	4	no	—
268	#427	—	2	4	no	—
269	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
270	C₂×#107	C₂²×C₃₁	1	2	yes	—
271	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
272	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
273	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
274	C₂×#103	C₂²×C₃₁	1	2	yes	—
275	C₂×#103	C₂²×C₃₁	1	2	yes	—
276	C₂×#103	C₂²×C₃₁	1	2	yes	—
277	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
278	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
279	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
280	#397	—	2	2	no	—
281	#394	—	2	2	no	—
282	#391	C₂²×C₃₁	1	2	yes	—
283	#389	—	2	2	no	—
284	#389	C₂²×C₃₁	1	2	yes	—
285	#388	—	2	2	no	—
286	#380	C₂²×C₃₁	1	2	yes	—
287	#340	C₂²×C₃₁	1	2	yes	—
288	#340	C₂²×C₃₁	1	2	yes	—
289	C₂×C₃₁	—	4	2	no	—
290	C₂×C₃₁	—	8	1	no	—
291	C₂×C₃₁	#2416	1	1	yes	characteristic
292	C₂×C₃₁	#3054	1	1	yes	characteristic
293	C₂×C₃₁	#3059	1	1	yes	characteristic
294	C₂×C₃₁	#3269	1	2	yes	—
295	C₂×C₃₁	#3349	1	2	yes	—
296	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
297	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
298	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
299	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
300	C₂×Q₁₆	—	2	2	no	—
301	C₂×Q₁₆	—	2	2	no	—
302	C₂×Q₁₆	—	2	2	no	—
303	C₂×M₁₆	—	2	2	no	—
304	C₂×M₁₆	—	2	2	no	—
305	C₂×M₁₆	—	2	2	no	—
306	C₂²×C₈	—	2	2	no	—
307	C₂²×C₈	—	2	2	no	—
308	C₂²×C₈	D₄×C₃₁	1	4	yes	—
309	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
310	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
311	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
312	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
313	C₂²⋊Q₈	—	2	4	no	—
314	C₂²⋊Q₈	—	2	4	no	—
315	C₂²⋊Q₈	—	2	4	no	—
316	C₂²⋊Q₈	—	4	2	no	—
317	C₄×D₄	—	2	2	no	—
318	C₄×D₄	—	2	2	no	—
319	#117	—	2	4	no	—
320	#117	—	4	2	no	—
321	C₂×C₄⋊C₄	—	2	2	no	—
322	C₂×C₄⋊C₄	—	2	2	no	—
323	C₂×C₄⋊C₄	—	2	2	no	—
324	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
325	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
326	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
327	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
328	C₂×C₂²⋊C₄	—	2	2	no	—
329	C₂×C₂²⋊C₄	—	2	2	no	—
330	C₂×C₂²⋊C₄	—	2	2	no	—
331	C₂×C₂²⋊C₄	—	2	2	no	—
332	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
333	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
334	C₂×C₄²	—	2	2	no	—
335	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
336	#107	—	2	2	no	—
337	#107	—	2	4	no	—
338	#106	—	2	2	no	—
339	#106	D₄×C₃₁	1	4	yes	complemented
340	C₄⋊C₈	—	2	2	no	—
341	#103	—	2	4	no	—
342	#103	—	2	4	no	—
343	#103	—	4	2	no	—
344	C₂²⋊C₈	—	2	2	no	—
345	C₂²⋊C₈	—	2	2	no	—
346	#97	—	2	2	no	—
347	#95	—	2	2	no	—
348	#95	—	2	2	no	—
349	#95	—	2	2	no	—
350	#95	—	2	2	no	—
351	C₃₁	#24785	1	1	yes	characteristic, direct product factor, Sylow
352	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
353	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
354	C₂×Q₈	—	2	2	no	—
355	C₂×Q₈	—	2	2	no	—
356	C₂×Q₈	—	2	2	no	—
357	C₂×Q₈	—	2	2	no	—
358	C₂×Q₈	—	2	2	no	—
359	C₂×Q₈	—	2	2	no	—
360	C₂×D₄	—	2	2	no	—
361	C₂×D₄	—	4	2	no	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	2	1	no	—
364	C₂²×C₄	—	2	1	no	—
365	C₂²×C₄	—	2	1	no	—
366	C₂²×C₄	—	2	2	no	—
367	C₂²×C₄	—	2	1	no	—
368	C₂²×C₄	—	2	1	no	—
369	C₂²×C₄	—	2	1	no	—
370	C₂²×C₄	—	2	1	no	—
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	2	1	no	—
373	C₂²×C₄	—	2	2	no	—
374	C₂²×C₄	—	2	2	no	—
375	C₂²×C₄	—	2	2	no	—
376	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
377	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
378	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
379	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
380	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
381	C₂²×C₄	#41×C₃₁	1	2	yes	—
382	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
383	Q₁₆	—	4	4	no	—
384	M₁₆	—	4	4	no	—
385	C₂×C₈	—	2	1	no	—
386	C₂×C₈	—	2	1	no	—
387	C₂×C₈	—	2	8	no	—
388	C₂×C₈	—	2	1	no	—
389	C₂×C₈	—	2	1	no	—
390	C₂×C₈	—	2	1	no	—
391	C₂×C₈	—	2	1	no	—
392	C₂×C₈	—	2	1	no	—
393	C₂×C₈	—	2	1	no	—
394	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
395	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
396	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
397	C₂×C₈	#41×C₃₁	1	2	yes	—
398	C₄⋊C₄	—	2	2	no	—
399	C₄⋊C₄	—	2	2	no	—
400	C₄⋊C₄	—	2	2	no	—
401	C₄⋊C₄	—	2	2	no	—
402	C₄⋊C₄	—	2	2	no	—
403	C₄⋊C₄	—	2	2	no	—
404	C₄⋊C₄	—	4	4	no	—
405	C₄⋊C₄	—	4	4	no	—
406	C₄⋊C₄	—	4	4	no	—
407	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
408	C₂²⋊C₄	—	2	2	no	—
409	C₂²⋊C₄	—	2	2	no	—
410	C₂²⋊C₄	—	4	4	no	—
411	C₂²⋊C₄	—	4	4	no	—
412	C₂²⋊C₄	—	4	4	no	—
413	C₂²⋊C₄	—	4	4	no	—
414	C₄²	—	2	1	no	—
415	C₄²	—	2	1	no	—
416	C₄²	—	4	4	no	—
417	C₂³	—	2	1	no	—
418	C₂³	—	2	1	no	—
419	C₂³	—	2	2	no	—
420	C₂³	—	2	2	no	—
421	C₂³	—	2	1	no	—
422	C₂³	—	2	1	no	—
423	C₂³	—	2	1	no	—
424	C₂³	—	2	2	no	—
425	C₂³	—	2	2	no	—
426	C₂³	C₃₁×#121	1	2	yes	—
427	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
428	Q₈	—	4	2	no	—
429	Q₈	—	4	2	no	—
430	Q₈	—	4	2	no	—
431	Q₈	—	4	2	no	—
432	D₄	—	4	4	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	1	no	—
440	C₂×C₄	—	2	1	no	—
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	2	4	no	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	1	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	4	2	no	—
449	C₂×C₄	—	4	1	no	—
450	C₂×C₄	—	4	1	no	—
451	C₂×C₄	—	4	1	no	—
452	C₂×C₄	—	4	1	no	—
453	C₂×C₄	—	4	2	no	—
454	C₂×C₄	—	4	2	no	—
455	C₂×C₄	—	4	2	no	—
456	C₂×C₄	—	4	2	no	—
457	C₂×C₄	—	4	2	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	C₃₁×#121	1	2	yes	—
463	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
464	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
465	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
466	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
467	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
468	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
469	C₈	—	2	2	no	—
470	C₈	—	4	1	no	—
471	C₈	—	4	1	no	—
472	C₈	C₃₁×#121	1	4	yes	—
473	C₂²	—	2	2	no	—
474	C₂²	—	2	2	no	—
475	C₂²	—	2	2	no	—
476	C₂²	—	4	2	no	—
477	C₂²	—	4	2	no	—
478	C₂²	—	4	2	no	—
479	C₂²	—	4	1	no	—
480	C₂²	—	4	2	no	—
481	C₂²	—	4	1	no	—
482	C₂²	—	4	1	no	—
483	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
484	C₂²	D₄²×C₃₁	1	1	yes	characteristic
485	C₂²	C₃₁×#547	1	1	yes	characteristic
486	C₂²	C₃₁×#575	1	2	yes	—
487	C₂²	C₃₁×#578	1	2	yes	—
488	C₄	—	2	2	no	—
489	C₄	—	4	1	no	—
490	C₄	—	4	4	no	—
491	C₄	—	4	1	no	—
492	C₄	—	4	1	no	—
493	C₄	—	8	2	no	—
494	C₄	—	8	2	no	—
495	C₄	—	8	2	no	—
496	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
497	C₄	C₃₁×#537	1	1	yes	characteristic
498	C₄	C₃₁×#548	1	1	yes	characteristic
499	C₄	C₃₁×#553	1	1	yes	characteristic
500	C₂	—	4	2	no	—
501	C₂	—	8	1	no	—
502	C₂	C₃₁×#2416	1	1	yes	characteristic
503	C₂	C₃₁×#3054	1	1	yes	characteristic
504	C₂	C₃₁×#3059	1	1	yes	characteristic
505	C₂	C₃₁×#3269	1	2	yes	—
506	C₂	C₃₁×#3349	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. C₂×C₃₁×#427	C₂	semidirect
6. C₃₁×#2068	C₂	semidirect
7. C₃₁×#2036	C₂	semidirect
8. C₃₁×#1971	C₂	semidirect
9. C₃₁×#1944	C₂	semidirect
10. C₃₁×#1873	C₂	semidirect
11. C₃₁×#1809	C₂	semidirect
86. C₃₁×#106	D₄	semidirect
163. #24785	C₃₁	direct (N × H)
220. C₂×#470	C₂×C₃₁	semidirect
221. C₂×#467	C₂×C₃₁	semidirect
223. C₂×#427	C₂×C₃₁	semidirect
225. #2068	C₂×C₃₁	semidirect
226. #2036	C₂×C₃₁	semidirect
227. #1971	C₂×C₃₁	semidirect
228. #1944	C₂×C₃₁	semidirect
229. #1873	C₂×C₃₁	semidirect
230. #1809	C₂×C₃₁	semidirect
339. #106	D₄×C₃₁	semidirect
351. C₃₁	#24785	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×#470	2
C₂×C₃₁×#467	2
C₂×C₃₁×#435	2
C₂×C₃₁×#427	2
C₃₁×#2415	2
C₃₁×#2068	2
C₃₁×#2036	2
C₃₁×#1971	2
C₃₁×#1944	2
C₃₁×#1873	2
C₃₁×#1809	2
#24785	31

Subgroups — C31×#24785 #21481544

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24785 #21481544