Subgroups — C₃₁×#23755 #21480514

Order 7936.

880 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×#121	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2285	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2100	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2094	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2084	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2077	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2071	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2034	C₂	1	2	yes	maximal
9	C₄×C₃₁×#106	C₂	1	2	yes	maximal, complemented
10	C₄×C₃₁×#102	C₂	1	2	yes	maximal, complemented
11	C₄×C₃₁×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#121	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₃₁	—	2	1	no	—
14	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#117	—	2	1	no	—
16	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#106	C₂²	1	2	yes	—
18	C₂×C₃₁×#106	C₂²	1	2	yes	—
19	C₂×C₃₁×#102	C₂²	1	2	yes	—
20	C₂×C₃₁×#102	C₂²	1	2	yes	—
21	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
23	C₃₁×#390	C₂²	1	1	yes	characteristic
24	C₃₁×#389	C₂²	1	1	yes	characteristic
25	C₃₁×#388	—	2	1	no	—
26	C₃₁×#386	—	2	1	no	—
27	C₃₁×#384	C₂²	1	1	yes	characteristic
28	C₃₁×#384	C₂²	1	1	yes	characteristic
29	C₃₁×#382	—	2	1	no	—
30	C₃₁×#380	—	2	1	no	—
31	C₃₁×#380	C₂²	1	1	yes	characteristic
32	C₃₁×#380	C₂²	1	2	yes	—
33	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
34	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
35	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
36	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
37	C₃₁×#340	C₂²	1	2	yes	—
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#340	C₂²	1	2	yes	—
40	C₃₁×#340	C₂²	1	2	yes	—
41	C₃₁×#336	C₂²	1	1	yes	characteristic
42	C₃₁×#336	C₂²	1	1	yes	characteristic
43	C₂²×D₄×C₃₁	—	2	1	no	—
44	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₃₁	—	2	1	no	—
46	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
50	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
51	C₃₁×#121	—	2	2	no	—
52	C₃₁×#121	—	2	2	no	—
53	C₄×D₄×C₃₁	—	2	4	no	—
54	C₄×D₄×C₃₁	—	4	4	no	—
55	C₃₁×#117	—	4	4	no	—
56	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
57	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
59	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
60	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
61	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
67	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
69	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
70	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
71	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
72	C₂×C₄²×C₃₁	—	2	4	no	—
73	C₂×C₄²×C₃₁	—	2	2	no	—
74	C₂×C₄²×C₃₁	—	2	1	no	—
75	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
76	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
77	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
79	C₃₁×#106	—	2	4	no	—
80	C₃₁×#106	—	2	4	no	—
81	C₃₁×#102	—	2	4	no	—
82	C₃₁×#102	—	2	4	no	—
83	C₃₁×C₂²⋊C₈	—	2	4	no	—
84	C₄×C₈×C₃₁	—	2	4	no	—
85	C₃₁×#95	—	2	1	no	—
86	C₃₁×#95	—	2	1	no	—
87	C₃₁×#95	—	2	2	no	—
88	C₃₁×#95	—	2	2	no	—
89	C₃₁×#95	—	2	1	no	—
90	C₂⁴×C₃₁	—	2	1	no	—
91	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
92	C₂×D₄×C₃₁	—	2	4	no	—
93	C₂×D₄×C₃₁	—	2	2	no	—
94	C₂×D₄×C₃₁	—	2	2	no	—
95	C₂×D₄×C₃₁	—	2	1	no	—
96	C₂×D₄×C₃₁	—	4	2	no	—
97	C₂×D₄×C₃₁	—	4	2	no	—
98	C₂×D₄×C₃₁	#41	1	2	yes	—
99	C₂²×C₄×C₃₁	—	2	2	no	—
100	C₂²×C₄×C₃₁	—	2	2	no	—
101	C₂²×C₄×C₃₁	—	2	1	no	—
102	C₂²×C₄×C₃₁	—	2	2	no	—
103	C₂²×C₄×C₃₁	—	2	2	no	—
104	C₂²×C₄×C₃₁	—	2	2	no	—
105	C₂²×C₄×C₃₁	—	2	2	no	—
106	C₂²×C₄×C₃₁	—	2	1	no	—
107	C₂²×C₄×C₃₁	—	2	1	no	—
108	C₂²×C₄×C₃₁	—	2	1	no	—
109	C₂²×C₄×C₃₁	—	2	2	no	—
110	C₂²×C₄×C₃₁	—	2	2	no	—
111	C₂²×C₄×C₃₁	—	2	1	no	—
112	C₂²×C₄×C₃₁	—	4	1	no	—
113	C₂²×C₄×C₃₁	—	4	2	no	—
114	C₂²×C₄×C₃₁	—	4	2	no	—
115	C₂²×C₄×C₃₁	—	4	2	no	—
116	C₂²×C₄×C₃₁	SD₁₆	1	4	yes	—
117	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
123	C₂×C₈×C₃₁	—	2	2	no	—
124	C₂×C₈×C₃₁	—	2	2	no	—
125	C₂×C₈×C₃₁	—	2	2	no	—
126	C₂×C₈×C₃₁	—	2	2	no	—
127	C₂×C₈×C₃₁	—	2	2	no	—
128	C₂×C₈×C₃₁	#41	1	4	yes	—
129	C₄⋊C₄×C₃₁	—	2	1	no	—
130	C₄⋊C₄×C₃₁	—	2	2	no	—
131	C₄⋊C₄×C₃₁	—	2	2	no	—
132	C₄⋊C₄×C₃₁	—	2	2	no	—
133	C₄⋊C₄×C₃₁	—	2	2	no	—
134	C₄⋊C₄×C₃₁	—	2	2	no	—
135	C₄⋊C₄×C₃₁	—	2	2	no	—
136	C₄⋊C₄×C₃₁	—	4	2	no	—
137	C₄⋊C₄×C₃₁	—	4	2	no	—
138	C₄⋊C₄×C₃₁	—	4	2	no	—
139	C₄⋊C₄×C₃₁	—	4	2	no	—
140	C₄⋊C₄×C₃₁	—	4	2	no	—
141	C₄⋊C₄×C₃₁	#41	1	2	yes	—
142	C₂²⋊C₄×C₃₁	—	2	4	no	—
143	C₂²⋊C₄×C₃₁	—	4	4	no	—
144	C₂²⋊C₄×C₃₁	—	4	2	no	—
145	C₂²⋊C₄×C₃₁	—	4	2	no	—
146	C₂²⋊C₄×C₃₁	—	4	2	no	—
147	C₂²⋊C₄×C₃₁	—	4	2	no	—
148	C₄²×C₃₁	—	2	2	no	—
149	C₄²×C₃₁	—	2	2	no	—
150	C₄²×C₃₁	—	2	2	no	—
151	C₄²×C₃₁	—	4	2	no	—
152	C₄²×C₃₁	—	4	4	no	—
153	C₄²×C₃₁	#41	1	4	yes	—
154	#23755	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₂³×C₃₁	—	2	1	no	—
156	C₂³×C₃₁	—	2	1	no	—
157	C₂³×C₃₁	—	2	1	no	—
158	C₂³×C₃₁	—	2	1	no	—
159	C₂³×C₃₁	—	2	1	no	—
160	C₂³×C₃₁	—	2	1	no	—
161	C₂³×C₃₁	—	2	2	no	—
162	C₂³×C₃₁	—	4	1	no	—
163	C₂³×C₃₁	—	4	1	no	—
164	C₂³×C₃₁	—	4	1	no	—
165	C₂³×C₃₁	—	4	1	no	—
166	C₂³×C₃₁	—	4	1	no	—
167	C₂³×C₃₁	—	4	1	no	—
168	C₂³×C₃₁	C₂×SD₁₆	1	2	yes	—
169	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
170	D₄×C₃₁	—	2	4	no	—
171	D₄×C₃₁	—	4	8	no	—
172	D₄×C₃₁	—	4	2	no	—
173	C₂×C₄×C₃₁	—	2	1	no	—
174	C₂×C₄×C₃₁	—	2	1	no	—
175	C₂×C₄×C₃₁	—	2	1	no	—
176	C₂×C₄×C₃₁	—	2	1	no	—
177	C₂×C₄×C₃₁	—	2	1	no	—
178	C₂×C₄×C₃₁	—	2	1	no	—
179	C₂×C₄×C₃₁	—	2	8	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	1	no	—
184	C₂×C₄×C₃₁	—	4	4	no	—
185	C₂×C₄×C₃₁	—	4	1	no	—
186	C₂×C₄×C₃₁	—	4	1	no	—
187	C₂×C₄×C₃₁	—	4	2	no	—
188	C₂×C₄×C₃₁	—	4	2	no	—
189	C₂×C₄×C₃₁	—	4	1	no	—
190	C₂×C₄×C₃₁	—	4	2	no	—
191	C₂×C₄×C₃₁	—	4	1	no	—
192	C₂×C₄×C₃₁	—	4	2	no	—
193	C₂×C₄×C₃₁	—	4	1	no	—
194	C₂×C₄×C₃₁	—	4	1	no	—
195	C₂×C₄×C₃₁	—	4	1	no	—
196	C₂×C₄×C₃₁	—	4	2	no	—
197	C₂×C₄×C₃₁	—	4	2	no	—
198	C₂×C₄×C₃₁	—	4	2	no	—
199	C₂×C₄×C₃₁	—	4	1	no	—
200	C₂×C₄×C₃₁	—	4	2	no	—
201	C₂×C₄×C₃₁	—	8	4	no	—
202	C₂×C₄×C₃₁	#123	1	2	yes	—
203	C₂×C₄×C₃₁	#123	1	2	yes	—
204	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
205	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
206	C₂×C₄×C₃₁	#135	1	2	yes	—
207	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
208	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
209	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
210	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
211	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
212	C₈×C₃₁	—	2	4	no	—
213	C₈×C₃₁	—	4	2	no	—
214	C₄×#121	C₂×C₃₁	1	1	yes	characteristic
215	#2285	C₂×C₃₁	1	1	yes	characteristic, complemented
216	#2100	C₂×C₃₁	1	1	yes	characteristic, complemented
217	#2094	C₂×C₃₁	1	1	yes	characteristic, complemented
218	#2084	C₂×C₃₁	1	1	yes	characteristic, complemented
219	#2077	C₂×C₃₁	1	2	yes	complemented
220	#2071	C₂×C₃₁	1	1	yes	characteristic, complemented
221	#2034	C₂×C₃₁	1	2	yes	—
222	C₄×#106	C₂×C₃₁	1	2	yes	complemented
223	C₄×#102	C₂×C₃₁	1	2	yes	complemented
224	C₄×C₂²⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
225	C₂²×C₃₁	—	2	2	no	—
226	C₂²×C₃₁	—	2	2	no	—
227	C₂²×C₃₁	—	2	2	no	—
228	C₂²×C₃₁	—	4	1	no	—
229	C₂²×C₃₁	—	4	1	no	—
230	C₂²×C₃₁	—	4	1	no	—
231	C₂²×C₃₁	—	4	1	no	—
232	C₂²×C₃₁	—	4	2	no	—
233	C₂²×C₃₁	—	4	2	no	—
234	C₂²×C₃₁	—	4	2	no	—
235	C₂²×C₃₁	—	8	1	no	—
236	C₂²×C₃₁	—	8	1	no	—
237	C₂²×C₃₁	—	8	1	no	—
238	C₂²×C₃₁	—	8	1	no	—
239	C₂²×C₃₁	C₂×#123	1	1	yes	characteristic
240	C₂²×C₃₁	#525	1	1	yes	characteristic
241	C₂²×C₃₁	#532	1	1	yes	characteristic
242	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
243	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
244	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
245	C₂²×C₃₁	#575	1	1	yes	characteristic
246	C₄×C₃₁	—	2	2	no	—
247	C₄×C₃₁	—	2	2	no	—
248	C₄×C₃₁	—	2	2	no	—
249	C₄×C₃₁	—	4	2	no	—
250	C₄×C₃₁	—	4	1	no	—
251	C₄×C₃₁	—	8	1	no	—
252	C₄×C₃₁	—	8	2	no	—
253	C₄×C₃₁	—	8	1	no	—
254	C₄×C₃₁	—	8	2	no	—
255	C₄×C₃₁	#481	1	4	yes	—
256	C₄×C₃₁	#529	1	2	yes	—
257	C₄×C₃₁	#529	1	2	yes	—
258	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
259	C₂×C₄×D₄	—	2	1	no	—
260	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
261	C₂×#117	—	2	1	no	—
262	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
263	C₂×#106	C₂²×C₃₁	1	2	yes	—
264	C₂×#106	C₂²×C₃₁	1	2	yes	—
265	C₂×#102	C₂²×C₃₁	1	2	yes	—
266	C₂×#102	C₂²×C₃₁	1	2	yes	—
267	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
268	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
269	#390	C₂²×C₃₁	1	1	yes	characteristic
270	#389	C₂²×C₃₁	1	1	yes	characteristic
271	#388	—	2	1	no	—
272	#386	—	2	1	no	—
273	#384	C₂²×C₃₁	1	1	yes	characteristic
274	#384	C₂²×C₃₁	1	1	yes	characteristic
275	#382	—	2	1	no	—
276	#380	—	2	1	no	—
277	#380	C₂²×C₃₁	1	1	yes	characteristic
278	#380	C₂²×C₃₁	1	2	yes	—
279	C₄×C₄⋊C₄	—	2	1	no	—
280	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
281	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
282	C₄×C₂²⋊C₄	—	2	1	no	—
283	#340	C₂²×C₃₁	1	2	yes	—
284	#340	C₂²×C₃₁	1	2	yes	—
285	#340	C₂²×C₃₁	1	2	yes	—
286	#340	C₂²×C₃₁	1	2	yes	—
287	#336	C₂²×C₃₁	1	1	yes	characteristic
288	#336	C₂²×C₃₁	1	1	yes	characteristic
289	C₂×C₃₁	—	4	2	no	—
290	C₂×C₃₁	—	8	2	no	—
291	C₂×C₃₁	#2395	1	1	yes	characteristic
292	C₂×C₃₁	C₂×#481	1	1	yes	characteristic
293	C₂×C₃₁	#3091	1	1	yes	characteristic
294	C₂×C₃₁	#3101	1	1	yes	characteristic
295	C₂×C₃₁	#3106	1	1	yes	characteristic
296	C₂×C₃₁	#3167	1	1	yes	characteristic
297	C₂×C₃₁	#3171	1	1	yes	characteristic
298	C₂²×D₄	—	2	1	no	—
299	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
300	C₂³×C₄	—	2	1	no	—
301	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
302	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
303	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
304	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
305	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
306	#121	—	2	2	no	—
307	#121	—	2	2	no	—
308	C₄×D₄	—	2	4	no	—
309	C₄×D₄	—	4	4	no	—
310	#117	—	4	4	no	—
311	C₂×C₄⋊C₄	—	2	1	no	—
312	C₂×C₄⋊C₄	—	2	1	no	—
313	C₂×C₄⋊C₄	—	2	1	no	—
314	C₂×C₄⋊C₄	—	2	1	no	—
315	C₂×C₄⋊C₄	—	2	1	no	—
316	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
317	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
318	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
319	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
320	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
321	C₂×C₂²⋊C₄	—	2	1	no	—
322	C₂×C₂²⋊C₄	—	2	2	no	—
323	C₂×C₂²⋊C₄	—	2	1	no	—
324	C₂×C₂²⋊C₄	—	2	1	no	—
325	C₂×C₂²⋊C₄	—	2	1	no	—
326	C₂×C₂²⋊C₄	—	2	1	no	—
327	C₂×C₄²	—	2	4	no	—
328	C₂×C₄²	—	2	2	no	—
329	C₂×C₄²	—	2	1	no	—
330	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
331	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
332	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
333	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
334	#106	—	2	4	no	—
335	#106	—	2	4	no	—
336	#102	—	2	4	no	—
337	#102	—	2	4	no	—
338	C₂²⋊C₈	—	2	4	no	—
339	C₄×C₈	—	2	4	no	—
340	#95	—	2	1	no	—
341	#95	—	2	1	no	—
342	#95	—	2	2	no	—
343	#95	—	2	2	no	—
344	#95	—	2	1	no	—
345	C₃₁	#23755	1	1	yes	characteristic, direct product factor, Sylow
346	C₂⁴	—	2	1	no	—
347	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
348	C₂×D₄	—	2	4	no	—
349	C₂×D₄	—	2	2	no	—
350	C₂×D₄	—	2	2	no	—
351	C₂×D₄	—	2	1	no	—
352	C₂×D₄	—	4	2	no	—
353	C₂×D₄	—	4	2	no	—
354	C₂×D₄	#41×C₃₁	1	2	yes	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	—	2	2	no	—
357	C₂²×C₄	—	2	1	no	—
358	C₂²×C₄	—	2	2	no	—
359	C₂²×C₄	—	2	2	no	—
360	C₂²×C₄	—	2	2	no	—
361	C₂²×C₄	—	2	2	no	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	2	1	no	—
364	C₂²×C₄	—	2	1	no	—
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	—	2	2	no	—
367	C₂²×C₄	—	2	1	no	—
368	C₂²×C₄	—	4	1	no	—
369	C₂²×C₄	—	4	2	no	—
370	C₂²×C₄	—	4	2	no	—
371	C₂²×C₄	—	4	2	no	—
372	C₂²×C₄	SD₁₆×C₃₁	1	4	yes	—
373	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
374	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
375	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
376	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
377	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
378	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
379	C₂×C₈	—	2	2	no	—
380	C₂×C₈	—	2	2	no	—
381	C₂×C₈	—	2	2	no	—
382	C₂×C₈	—	2	2	no	—
383	C₂×C₈	—	2	2	no	—
384	C₂×C₈	#41×C₃₁	1	4	yes	—
385	C₄⋊C₄	—	2	1	no	—
386	C₄⋊C₄	—	2	2	no	—
387	C₄⋊C₄	—	2	2	no	—
388	C₄⋊C₄	—	2	2	no	—
389	C₄⋊C₄	—	2	2	no	—
390	C₄⋊C₄	—	2	2	no	—
391	C₄⋊C₄	—	2	2	no	—
392	C₄⋊C₄	—	4	2	no	—
393	C₄⋊C₄	—	4	2	no	—
394	C₄⋊C₄	—	4	2	no	—
395	C₄⋊C₄	—	4	2	no	—
396	C₄⋊C₄	—	4	2	no	—
397	C₄⋊C₄	#41×C₃₁	1	2	yes	—
398	C₂²⋊C₄	—	2	4	no	—
399	C₂²⋊C₄	—	4	4	no	—
400	C₂²⋊C₄	—	4	2	no	—
401	C₂²⋊C₄	—	4	2	no	—
402	C₂²⋊C₄	—	4	2	no	—
403	C₂²⋊C₄	—	4	2	no	—
404	C₄²	—	2	2	no	—
405	C₄²	—	2	2	no	—
406	C₄²	—	2	2	no	—
407	C₄²	—	4	2	no	—
408	C₄²	—	4	4	no	—
409	C₄²	#41×C₃₁	1	4	yes	—
410	C₂³	—	2	1	no	—
411	C₂³	—	2	1	no	—
412	C₂³	—	2	1	no	—
413	C₂³	—	2	1	no	—
414	C₂³	—	2	1	no	—
415	C₂³	—	2	1	no	—
416	C₂³	—	2	2	no	—
417	C₂³	—	4	1	no	—
418	C₂³	—	4	1	no	—
419	C₂³	—	4	1	no	—
420	C₂³	—	4	1	no	—
421	C₂³	—	4	1	no	—
422	C₂³	—	4	1	no	—
423	C₂³	C₂×SD₁₆×C₃₁	1	2	yes	—
424	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
425	D₄	—	2	4	no	—
426	D₄	—	4	8	no	—
427	D₄	—	4	2	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	1	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	8	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	2	1	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	1	no	—
439	C₂×C₄	—	4	4	no	—
440	C₂×C₄	—	4	1	no	—
441	C₂×C₄	—	4	1	no	—
442	C₂×C₄	—	4	2	no	—
443	C₂×C₄	—	4	2	no	—
444	C₂×C₄	—	4	1	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	4	1	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	4	1	no	—
449	C₂×C₄	—	4	1	no	—
450	C₂×C₄	—	4	1	no	—
451	C₂×C₄	—	4	2	no	—
452	C₂×C₄	—	4	2	no	—
453	C₂×C₄	—	4	2	no	—
454	C₂×C₄	—	4	1	no	—
455	C₂×C₄	—	4	2	no	—
456	C₂×C₄	—	8	4	no	—
457	C₂×C₄	C₃₁×#123	1	2	yes	—
458	C₂×C₄	C₃₁×#123	1	2	yes	—
459	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
460	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
461	C₂×C₄	C₃₁×#135	1	2	yes	—
462	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
463	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
464	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
465	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
466	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
467	C₈	—	2	4	no	—
468	C₈	—	4	2	no	—
469	C₂²	—	2	2	no	—
470	C₂²	—	2	2	no	—
471	C₂²	—	2	2	no	—
472	C₂²	—	4	1	no	—
473	C₂²	—	4	1	no	—
474	C₂²	—	4	1	no	—
475	C₂²	—	4	1	no	—
476	C₂²	—	4	2	no	—
477	C₂²	—	4	2	no	—
478	C₂²	—	4	2	no	—
479	C₂²	—	8	1	no	—
480	C₂²	—	8	1	no	—
481	C₂²	—	8	1	no	—
482	C₂²	—	8	1	no	—
483	C₂²	C₂×C₃₁×#123	1	1	yes	characteristic
484	C₂²	C₃₁×#525	1	1	yes	characteristic
485	C₂²	C₃₁×#532	1	1	yes	characteristic
486	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
487	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
488	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
489	C₂²	C₃₁×#575	1	1	yes	characteristic
490	C₄	—	2	2	no	—
491	C₄	—	2	2	no	—
492	C₄	—	2	2	no	—
493	C₄	—	4	2	no	—
494	C₄	—	4	1	no	—
495	C₄	—	8	1	no	—
496	C₄	—	8	2	no	—
497	C₄	—	8	1	no	—
498	C₄	—	8	2	no	—
499	C₄	C₃₁×#481	1	4	yes	—
500	C₄	C₃₁×#529	1	2	yes	—
501	C₄	C₃₁×#529	1	2	yes	—
502	C₂	—	4	2	no	—
503	C₂	—	8	2	no	—
504	C₂	C₃₁×#2395	1	1	yes	characteristic
505	C₂	C₂×C₃₁×#481	1	1	yes	characteristic
506	C₂	C₃₁×#3091	1	1	yes	characteristic
507	C₂	C₃₁×#3101	1	1	yes	characteristic
508	C₂	C₃₁×#3106	1	1	yes	characteristic
509	C₂	C₃₁×#3167	1	1	yes	characteristic
510	C₂	C₃₁×#3171	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	C₂×C₃₁×#121	C₂²
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. C₃₁×#2285	C₂	semidirect
3. C₃₁×#2100	C₂	semidirect
4. C₃₁×#2094	C₂	semidirect
5. C₃₁×#2084	C₂	semidirect
6. C₃₁×#2077	C₂	semidirect
7. C₃₁×#2071	C₂	semidirect
9. C₄×C₃₁×#106	C₂	semidirect
10. C₄×C₃₁×#102	C₂	semidirect
11. C₄×C₃₁×C₂²⋊C₈	C₂	semidirect
154. #23755	C₃₁	direct (N × H)
215. #2285	C₂×C₃₁	semidirect
216. #2100	C₂×C₃₁	semidirect
217. #2094	C₂×C₃₁	semidirect
218. #2084	C₂×C₃₁	semidirect
219. #2077	C₂×C₃₁	semidirect
220. #2071	C₂×C₃₁	semidirect
222. C₄×#106	C₂×C₃₁	semidirect
223. C₄×#102	C₂×C₃₁	semidirect
224. C₄×C₂²⋊C₈	C₂×C₃₁	semidirect
345. C₃₁	#23755	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₃₁×#121	2
C₃₁×#2285	2
C₃₁×#2100	2
C₃₁×#2094	2
C₃₁×#2084	2
C₃₁×#2077	2
C₃₁×#2071	2
C₃₁×#2034	2
C₄×C₃₁×#106	2
C₄×C₃₁×#102	2
C₄×C₃₁×C₂²⋊C₈	2
#23755	31

Subgroups — C31×#23755 #21480514

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23755 #21480514