Subgroups — C₃₁×#24812 #21481571

Order 7936.

942 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×#476	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2563	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2085	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2065	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1990	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1939	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1879	C₂	1	2	yes	maximal
10	C₃₁×#1804	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1802	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
16	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
18	C₃₁×#476	—	2	4	no	—
19	C₄×D₈×C₃₁	—	2	4	no	—
20	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#106	C₂²	1	2	yes	—
22	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
24	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#102	C₂²	1	2	yes	—
26	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×#102	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#393	—	2	2	no	—
30	C₃₁×#390	C₂²	1	2	yes	—
31	C₃₁×#389	C₂²	1	2	yes	—
32	C₃₁×#385	—	2	2	no	—
33	C₃₁×#384	C₂²	1	1	yes	characteristic
34	C₃₁×#382	C₂²	1	2	yes	—
35	C₃₁×#380	—	2	1	no	—
36	C₃₁×#380	—	2	1	no	—
37	C₃₁×#380	—	2	1	no	—
38	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
39	C₃₁×#340	C₂²	1	2	yes	—
40	C₃₁×#340	C₂²	1	2	yes	—
41	C₃₁×#340	C₂²	1	2	yes	—
42	C₃₁×#336	C₂²	1	2	yes	—
43	C₂²×Q₈×C₃₁	—	2	1	no	—
44	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
45	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₃₁	—	2	1	no	—
47	C₂³×C₄×C₃₁	—	2	1	no	—
48	C₂³×C₄×C₃₁	—	2	2	no	—
49	C₂×D₈×C₃₁	—	2	2	no	—
50	C₂×D₈×C₃₁	—	2	2	no	—
51	C₂×D₈×C₃₁	Q₈	1	4	yes	complemented
52	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
54	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
55	C₃₁×C₄⋊Q₈	—	2	2	no	—
56	C₃₁×C₄⋊Q₈	—	2	2	no	—
57	C₄×D₄×C₃₁	—	2	4	no	—
58	C₄×D₄×C₃₁	—	2	2	no	—
59	C₄×D₄×C₃₁	—	2	2	no	—
60	C₄×D₄×C₃₁	—	2	2	no	—
61	C₄×D₄×C₃₁	—	4	2	no	—
62	C₄×D₄×C₃₁	D₄	1	4	yes	—
63	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
64	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
65	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
66	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
67	C₂×C₄⋊C₄×C₃₁	—	4	4	no	—
68	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
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	2	yes	—
73	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
74	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
75	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
76	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
77	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
78	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
79	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
80	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
81	C₃₁×#107	—	2	2	no	—
82	C₃₁×#106	—	2	4	no	—
83	C₃₁×C₄⋊C₈	—	2	2	no	—
84	C₃₁×C₄⋊C₈	—	4	2	no	—
85	C₃₁×#102	—	2	2	no	—
86	C₃₁×#102	—	2	4	no	—
87	C₃₁×#102	—	2	2	no	—
88	C₃₁×#102	—	4	2	no	—
89	C₄×C₈×C₃₁	—	2	2	no	—
90	C₃₁×#95	—	2	2	no	—
91	C₃₁×#95	—	2	2	no	—
92	C₃₁×#95	—	2	2	no	—
93	C₃₁×#95	D₄	1	2	yes	—
94	C₂⁴×C₃₁	—	2	1	no	—
95	C₂⁴×C₃₁	—	2	1	no	—
96	C₂×Q₈×C₃₁	—	4	2	no	—
97	C₂×Q₈×C₃₁	—	8	1	no	—
98	C₂×Q₈×C₃₁	—	8	1	no	—
99	C₂×D₄×C₃₁	—	2	1	no	—
100	C₂×D₄×C₃₁	—	2	1	no	—
101	C₂×D₄×C₃₁	—	2	1	no	—
102	C₂×D₄×C₃₁	—	2	1	no	—
103	C₂×D₄×C₃₁	—	2	1	no	—
104	C₂×D₄×C₃₁	—	2	1	no	—
105	C₂×D₄×C₃₁	—	2	1	no	—
106	C₂×D₄×C₃₁	—	2	1	no	—
107	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
108	C₂×D₄×C₃₁	C₂×Q₈	1	2	yes	—
109	C₂×D₄×C₃₁	C₂×Q₈	1	2	yes	—
110	C₂×D₄×C₃₁	#41	1	2	yes	—
111	C₂²×C₄×C₃₁	—	2	1	no	—
112	C₂²×C₄×C₃₁	—	2	4	no	—
113	C₂²×C₄×C₃₁	—	2	1	no	—
114	C₂²×C₄×C₃₁	—	2	2	no	—
115	C₂²×C₄×C₃₁	—	2	2	no	—
116	C₂²×C₄×C₃₁	—	4	4	no	—
117	C₂²×C₄×C₃₁	—	4	4	no	—
118	C₂²×C₄×C₃₁	—	4	2	no	—
119	C₂²×C₄×C₃₁	—	4	2	no	—
120	C₂²×C₄×C₃₁	—	4	2	no	—
121	C₂²×C₄×C₃₁	—	4	2	no	—
122	C₂²×C₄×C₃₁	—	4	2	no	—
123	C₂²×C₄×C₃₁	—	8	2	no	—
124	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
125	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
126	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
129	D₈×C₃₁	—	2	8	no	—
130	C₂×C₈×C₃₁	—	2	1	no	—
131	C₂×C₈×C₃₁	—	2	1	no	—
132	C₂×C₈×C₃₁	—	2	1	no	—
133	C₂×C₈×C₃₁	—	2	2	no	—
134	C₂×C₈×C₃₁	—	2	2	no	—
135	C₂×C₈×C₃₁	—	2	1	no	—
136	C₂×C₈×C₃₁	—	2	1	no	—
137	C₂×C₈×C₃₁	—	2	2	no	—
138	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
139	C₄⋊C₄×C₃₁	—	2	2	no	—
140	C₄⋊C₄×C₃₁	—	2	2	no	—
141	C₄⋊C₄×C₃₁	—	2	1	no	—
142	C₄⋊C₄×C₃₁	—	2	2	no	—
143	C₄⋊C₄×C₃₁	—	2	1	no	—
144	C₄⋊C₄×C₃₁	—	2	1	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	—	4	4	no	—
149	C₄⋊C₄×C₃₁	—	4	2	no	—
150	C₄⋊C₄×C₃₁	—	4	2	no	—
151	C₄⋊C₄×C₃₁	—	8	1	no	—
152	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
153	C₂²⋊C₄×C₃₁	—	2	4	no	—
154	C₂²⋊C₄×C₃₁	—	4	4	no	—
155	C₂²⋊C₄×C₃₁	—	4	2	no	—
156	C₂²⋊C₄×C₃₁	—	8	2	no	—
157	C₂²⋊C₄×C₃₁	—	8	2	no	—
158	C₄²×C₃₁	—	2	2	no	—
159	C₄²×C₃₁	—	2	2	no	—
160	C₄²×C₃₁	C₂×D₄	1	2	yes	—
161	C₄²×C₃₁	#41	1	2	yes	—
162	#24812	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₂³×C₃₁	—	2	2	no	—
164	C₂³×C₃₁	—	2	2	no	—
165	C₂³×C₃₁	—	2	2	no	—
166	C₂³×C₃₁	—	2	2	no	—
167	C₂³×C₃₁	—	4	1	no	—
168	C₂³×C₃₁	—	4	1	no	—
169	C₂³×C₃₁	—	4	1	no	—
170	C₂³×C₃₁	—	4	1	no	—
171	C₂³×C₃₁	—	4	1	no	—
172	C₂³×C₃₁	—	4	1	no	—
173	C₂³×C₃₁	—	4	1	no	—
174	C₂³×C₃₁	—	4	1	no	—
175	C₂³×C₃₁	—	4	1	no	—
176	C₂³×C₃₁	—	4	1	no	—
177	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
178	Q₈×C₃₁	—	8	4	no	—
179	D₄×C₃₁	—	2	2	no	—
180	D₄×C₃₁	—	2	2	no	—
181	D₄×C₃₁	—	2	2	no	—
182	D₄×C₃₁	—	2	2	no	—
183	D₄×C₃₁	—	2	2	no	—
184	D₄×C₃₁	—	4	1	no	—
185	D₄×C₃₁	—	4	1	no	—
186	D₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
187	C₂×C₄×C₃₁	—	2	1	no	—
188	C₂×C₄×C₃₁	—	2	1	no	—
189	C₂×C₄×C₃₁	—	2	2	no	—
190	C₂×C₄×C₃₁	—	2	2	no	—
191	C₂×C₄×C₃₁	—	2	2	no	—
192	C₂×C₄×C₃₁	—	2	2	no	—
193	C₂×C₄×C₃₁	—	2	2	no	—
194	C₂×C₄×C₃₁	—	2	2	no	—
195	C₂×C₄×C₃₁	—	2	2	no	—
196	C₂×C₄×C₃₁	—	4	1	no	—
197	C₂×C₄×C₃₁	—	4	1	no	—
198	C₂×C₄×C₃₁	—	4	8	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	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	2	no	—
207	C₂×C₄×C₃₁	—	4	2	no	—
208	C₂×C₄×C₃₁	—	8	8	no	—
209	C₂×C₄×C₃₁	—	8	4	no	—
210	C₂×C₄×C₃₁	#121	1	2	yes	—
211	C₂×C₄×C₃₁	#121	1	2	yes	—
212	C₂×C₄×C₃₁	#135	1	2	yes	—
213	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
214	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
215	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
216	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
217	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
218	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
219	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
220	C₈×C₃₁	—	2	2	no	—
221	C₈×C₃₁	—	4	1	no	—
222	C₈×C₃₁	—	4	2	no	—
223	C₂×#476	C₂×C₃₁	1	1	yes	characteristic, complemented
224	C₂×C₄×D₈	C₂×C₃₁	1	1	yes	characteristic
225	#2563	C₂×C₃₁	1	1	yes	characteristic, complemented
226	D₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
227	#2085	C₂×C₃₁	1	2	yes	complemented
228	#2065	C₂×C₃₁	1	2	yes	complemented
229	#1990	C₂×C₃₁	1	1	yes	characteristic, complemented
230	#1939	C₂×C₃₁	1	1	yes	characteristic, complemented
231	#1879	C₂×C₃₁	1	2	yes	—
232	#1804	C₂×C₃₁	1	2	yes	complemented
233	#1802	C₂×C₃₁	1	1	yes	characteristic, complemented
234	C₂²×C₃₁	—	2	4	no	—
235	C₂²×C₃₁	—	4	2	no	—
236	C₂²×C₃₁	—	4	2	no	—
237	C₂²×C₃₁	—	4	2	no	—
238	C₂²×C₃₁	—	4	2	no	—
239	C₂²×C₃₁	—	4	2	no	—
240	C₂²×C₃₁	—	4	2	no	—
241	C₂²×C₃₁	—	4	2	no	—
242	C₂²×C₃₁	—	4	2	no	—
243	C₂²×C₃₁	—	4	2	no	—
244	C₂²×C₃₁	—	8	1	no	—
245	C₂²×C₃₁	—	8	1	no	—
246	C₂²×C₃₁	—	8	1	no	—
247	C₂²×C₃₁	—	8	1	no	—
248	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
249	C₂²×C₃₁	#546	1	1	yes	characteristic
250	C₂²×C₃₁	D₄×Q₈	1	1	yes	characteristic
251	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
252	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
253	C₂²×C₃₁	#576	1	1	yes	characteristic
254	C₂²×C₃₁	#577	1	1	yes	characteristic
255	C₄×C₃₁	—	2	2	no	—
256	C₄×C₃₁	—	4	2	no	—
257	C₄×C₃₁	—	4	2	no	—
258	C₄×C₃₁	—	4	2	no	—
259	C₄×C₃₁	—	8	1	no	—
260	C₄×C₃₁	—	8	2	no	—
261	C₄×C₃₁	—	8	2	no	—
262	C₄×C₃₁	#463	1	4	yes	—
263	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
264	C₄×C₃₁	#537	1	1	yes	characteristic
265	C₄×C₃₁	#552	1	1	yes	characteristic
266	C₄×C₃₁	#554	1	1	yes	characteristic
267	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
268	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
269	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
270	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
271	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
272	C₂²×C₄⋊C₄	—	2	1	no	—
273	#476	—	2	4	no	—
274	C₄×D₈	—	2	4	no	—
275	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
276	C₂×#106	C₂²×C₃₁	1	2	yes	—
277	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
278	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
279	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
280	C₂×#102	C₂²×C₃₁	1	2	yes	—
281	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
282	C₂×#102	C₂²×C₃₁	1	2	yes	—
283	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
284	#393	—	2	2	no	—
285	#390	C₂²×C₃₁	1	2	yes	—
286	#389	C₂²×C₃₁	1	2	yes	—
287	#385	—	2	2	no	—
288	#384	C₂²×C₃₁	1	1	yes	characteristic
289	#382	C₂²×C₃₁	1	2	yes	—
290	#380	—	2	1	no	—
291	#380	—	2	1	no	—
292	#380	—	2	1	no	—
293	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
294	#340	C₂²×C₃₁	1	2	yes	—
295	#340	C₂²×C₃₁	1	2	yes	—
296	#340	C₂²×C₃₁	1	2	yes	—
297	#336	C₂²×C₃₁	1	2	yes	—
298	C₂×C₃₁	—	4	4	no	—
299	C₂×C₃₁	—	8	2	no	—
300	C₂×C₃₁	#2417	1	1	yes	characteristic
301	C₂×C₃₁	C₂×#463	1	1	yes	characteristic
302	C₂×C₃₁	#3045	1	1	yes	characteristic
303	C₂×C₃₁	#3303	1	1	yes	characteristic
304	C₂×C₃₁	#3325	1	1	yes	characteristic
305	C₂×C₃₁	#3380	1	1	yes	characteristic
306	C₂×C₃₁	#3384	1	1	yes	characteristic
307	C₂²×Q₈	—	2	1	no	—
308	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
309	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
310	C₂³×C₄	—	2	1	no	—
311	C₂³×C₄	—	2	1	no	—
312	C₂³×C₄	—	2	2	no	—
313	C₂×D₈	—	2	2	no	—
314	C₂×D₈	—	2	2	no	—
315	C₂×D₈	Q₈×C₃₁	1	4	yes	complemented
316	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
317	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
318	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
319	C₄⋊Q₈	—	2	2	no	—
320	C₄⋊Q₈	—	2	2	no	—
321	C₄×D₄	—	2	4	no	—
322	C₄×D₄	—	2	2	no	—
323	C₄×D₄	—	2	2	no	—
324	C₄×D₄	—	2	2	no	—
325	C₄×D₄	—	4	2	no	—
326	C₄×D₄	D₄×C₃₁	1	4	yes	—
327	C₂×C₄⋊C₄	—	2	1	no	—
328	C₂×C₄⋊C₄	—	2	2	no	—
329	C₂×C₄⋊C₄	—	2	1	no	—
330	C₂×C₄⋊C₄	—	2	1	no	—
331	C₂×C₄⋊C₄	—	4	4	no	—
332	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
333	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
334	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
335	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
336	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
337	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
338	C₂×C₂²⋊C₄	—	2	1	no	—
339	C₂×C₂²⋊C₄	—	2	2	no	—
340	C₂×C₂²⋊C₄	—	2	1	no	—
341	C₂×C₂²⋊C₄	—	2	2	no	—
342	C₂×C₂²⋊C₄	—	2	2	no	—
343	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
344	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
345	#107	—	2	2	no	—
346	#106	—	2	4	no	—
347	C₄⋊C₈	—	2	2	no	—
348	C₄⋊C₈	—	4	2	no	—
349	#102	—	2	2	no	—
350	#102	—	2	4	no	—
351	#102	—	2	2	no	—
352	#102	—	4	2	no	—
353	C₄×C₈	—	2	2	no	—
354	#95	—	2	2	no	—
355	#95	—	2	2	no	—
356	#95	—	2	2	no	—
357	#95	D₄×C₃₁	1	2	yes	—
358	C₃₁	#24812	1	1	yes	characteristic, direct product factor, Sylow
359	C₂⁴	—	2	1	no	—
360	C₂⁴	—	2	1	no	—
361	C₂×Q₈	—	4	2	no	—
362	C₂×Q₈	—	8	1	no	—
363	C₂×Q₈	—	8	1	no	—
364	C₂×D₄	—	2	1	no	—
365	C₂×D₄	—	2	1	no	—
366	C₂×D₄	—	2	1	no	—
367	C₂×D₄	—	2	1	no	—
368	C₂×D₄	—	2	1	no	—
369	C₂×D₄	—	2	1	no	—
370	C₂×D₄	—	2	1	no	—
371	C₂×D₄	—	2	1	no	—
372	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
373	C₂×D₄	C₂×Q₈×C₃₁	1	2	yes	—
374	C₂×D₄	C₂×Q₈×C₃₁	1	2	yes	—
375	C₂×D₄	#41×C₃₁	1	2	yes	—
376	C₂²×C₄	—	2	1	no	—
377	C₂²×C₄	—	2	4	no	—
378	C₂²×C₄	—	2	1	no	—
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	4	4	no	—
382	C₂²×C₄	—	4	4	no	—
383	C₂²×C₄	—	4	2	no	—
384	C₂²×C₄	—	4	2	no	—
385	C₂²×C₄	—	4	2	no	—
386	C₂²×C₄	—	4	2	no	—
387	C₂²×C₄	—	4	2	no	—
388	C₂²×C₄	—	8	2	no	—
389	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
390	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
391	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
392	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
393	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
394	D₈	—	2	8	no	—
395	C₂×C₈	—	2	1	no	—
396	C₂×C₈	—	2	1	no	—
397	C₂×C₈	—	2	1	no	—
398	C₂×C₈	—	2	2	no	—
399	C₂×C₈	—	2	2	no	—
400	C₂×C₈	—	2	1	no	—
401	C₂×C₈	—	2	2	no	—
402	C₂×C₈	—	2	1	no	—
403	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
404	C₄⋊C₄	—	2	2	no	—
405	C₄⋊C₄	—	2	2	no	—
406	C₄⋊C₄	—	2	1	no	—
407	C₄⋊C₄	—	2	2	no	—
408	C₄⋊C₄	—	2	1	no	—
409	C₄⋊C₄	—	2	1	no	—
410	C₄⋊C₄	—	2	2	no	—
411	C₄⋊C₄	—	2	2	no	—
412	C₄⋊C₄	—	2	2	no	—
413	C₄⋊C₄	—	4	4	no	—
414	C₄⋊C₄	—	4	2	no	—
415	C₄⋊C₄	—	4	2	no	—
416	C₄⋊C₄	—	8	1	no	—
417	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
418	C₂²⋊C₄	—	2	4	no	—
419	C₂²⋊C₄	—	4	4	no	—
420	C₂²⋊C₄	—	4	2	no	—
421	C₂²⋊C₄	—	8	2	no	—
422	C₂²⋊C₄	—	8	2	no	—
423	C₄²	—	2	2	no	—
424	C₄²	—	2	2	no	—
425	C₄²	C₂×D₄×C₃₁	1	2	yes	—
426	C₄²	#41×C₃₁	1	2	yes	—
427	C₂³	—	2	2	no	—
428	C₂³	—	2	2	no	—
429	C₂³	—	2	2	no	—
430	C₂³	—	2	2	no	—
431	C₂³	—	4	1	no	—
432	C₂³	—	4	1	no	—
433	C₂³	—	4	1	no	—
434	C₂³	—	4	1	no	—
435	C₂³	—	4	1	no	—
436	C₂³	—	4	1	no	—
437	C₂³	—	4	1	no	—
438	C₂³	—	4	1	no	—
439	C₂³	—	4	1	no	—
440	C₂³	—	4	1	no	—
441	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
442	Q₈	—	8	4	no	—
443	D₄	—	2	2	no	—
444	D₄	—	2	2	no	—
445	D₄	—	2	2	no	—
446	D₄	—	2	2	no	—
447	D₄	—	2	2	no	—
448	D₄	—	4	1	no	—
449	D₄	—	4	1	no	—
450	D₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	4	1	no	—
461	C₂×C₄	—	4	1	no	—
462	C₂×C₄	—	4	8	no	—
463	C₂×C₄	—	4	1	no	—
464	C₂×C₄	—	4	2	no	—
465	C₂×C₄	—	4	2	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	2	no	—
471	C₂×C₄	—	4	2	no	—
472	C₂×C₄	—	8	8	no	—
473	C₂×C₄	—	8	4	no	—
474	C₂×C₄	C₃₁×#121	1	2	yes	—
475	C₂×C₄	C₃₁×#121	1	2	yes	—
476	C₂×C₄	C₃₁×#135	1	2	yes	—
477	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
478	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
479	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
480	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
481	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
482	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
483	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
484	C₈	—	2	2	no	—
485	C₈	—	4	1	no	—
486	C₈	—	4	2	no	—
487	C₂²	—	2	4	no	—
488	C₂²	—	4	2	no	—
489	C₂²	—	4	2	no	—
490	C₂²	—	4	2	no	—
491	C₂²	—	4	2	no	—
492	C₂²	—	4	2	no	—
493	C₂²	—	4	2	no	—
494	C₂²	—	4	2	no	—
495	C₂²	—	4	2	no	—
496	C₂²	—	4	2	no	—
497	C₂²	—	8	1	no	—
498	C₂²	—	8	1	no	—
499	C₂²	—	8	1	no	—
500	C₂²	—	8	1	no	—
501	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
502	C₂²	C₃₁×#546	1	1	yes	characteristic
503	C₂²	D₄×Q₈×C₃₁	1	1	yes	characteristic
504	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
505	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
506	C₂²	C₃₁×#576	1	1	yes	characteristic
507	C₂²	C₃₁×#577	1	1	yes	characteristic
508	C₄	—	2	2	no	—
509	C₄	—	4	2	no	—
510	C₄	—	4	2	no	—
511	C₄	—	4	2	no	—
512	C₄	—	8	1	no	—
513	C₄	—	8	2	no	—
514	C₄	—	8	2	no	—
515	C₄	C₃₁×#463	1	4	yes	—
516	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
517	C₄	C₃₁×#537	1	1	yes	characteristic
518	C₄	C₃₁×#552	1	1	yes	characteristic
519	C₄	C₃₁×#554	1	1	yes	characteristic
520	C₂	—	4	4	no	—
521	C₂	—	8	2	no	—
522	C₂	C₃₁×#2417	1	1	yes	characteristic
523	C₂	C₂×C₃₁×#463	1	1	yes	characteristic
524	C₂	C₃₁×#3045	1	1	yes	characteristic
525	C₂	C₃₁×#3303	1	1	yes	characteristic
526	C₂	C₃₁×#3325	1	1	yes	characteristic
527	C₂	C₃₁×#3380	1	1	yes	characteristic
528	C₂	C₃₁×#3384	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₈×C₃₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄×C₄⋊C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#476	C₂	semidirect
3. C₃₁×#2563	C₂	semidirect
4. D₄×C₄⋊C₄×C₃₁	C₂	semidirect
5. C₃₁×#2085	C₂	semidirect
6. C₃₁×#2065	C₂	semidirect
7. C₃₁×#1990	C₂	semidirect
8. C₃₁×#1939	C₂	semidirect
10. C₃₁×#1804	C₂	semidirect
11. C₃₁×#1802	C₂	semidirect
51. C₂×D₈×C₃₁	Q₈	semidirect
162. #24812	C₃₁	direct (N × H)
223. C₂×#476	C₂×C₃₁	semidirect
225. #2563	C₂×C₃₁	semidirect
226. D₄×C₄⋊C₄	C₂×C₃₁	semidirect
227. #2085	C₂×C₃₁	semidirect
228. #2065	C₂×C₃₁	semidirect
229. #1990	C₂×C₃₁	semidirect
230. #1939	C₂×C₃₁	semidirect
232. #1804	C₂×C₃₁	semidirect
233. #1802	C₂×C₃₁	semidirect
315. C₂×D₈	Q₈×C₃₁	semidirect
358. C₃₁	#24812	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×#476	2
C₂×C₄×D₈×C₃₁	2
C₃₁×#2563	2
D₄×C₄⋊C₄×C₃₁	2
C₃₁×#2085	2
C₃₁×#2065	2
C₃₁×#1990	2
C₃₁×#1939	2
C₃₁×#1879	2
C₃₁×#1804	2
C₃₁×#1802	2
#24812	31

Subgroups — C31×#24812 #21481571

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24812 #21481571