Subgroups — C₃₁×#21503 #21478262

Order 7936.

1260 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#459	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#442	C₂	1	2	yes	maximal, complemented
3	C₄×C₃₁×#127	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄²×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1982	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1948	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1926	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1879	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1802	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1767	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1762	C₂	1	2	yes	maximal, complemented
12	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
13	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₃₁	—	2	1	no	—
15	C₂×C₄×D₄×C₃₁	—	2	1	no	—
16	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
18	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₃₁	—	2	1	no	—
21	C₃₁×#459	C₄	1	8	yes	complemented
22	C₃₁×#442	—	2	8	no	—
23	C₂×C₃₁×#106	C₂²	1	2	yes	—
24	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#102	C₂²	1	2	yes	—
27	C₂×C₃₁×#102	C₂²	1	2	yes	—
28	C₂×C₃₁×#102	C₂²	1	2	yes	—
29	C₂×C₃₁×#102	C₂²	1	2	yes	—
30	C₂×C₃₁×#97	C₂²	1	2	yes	—
31	C₃₁×#390	—	2	1	no	—
32	C₃₁×#390	—	2	1	no	—
33	C₃₁×#390	C₂²	1	2	yes	—
34	C₃₁×#384	C₂²	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
36	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
37	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
38	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
39	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
40	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
41	C₄³×C₃₁	C₂²	1	1	yes	characteristic
42	C₃₁×#340	C₂²	1	2	yes	—
43	C₃₁×#340	C₂²	1	2	yes	—
44	C₃₁×#336	C₂²	1	2	yes	—
45	C₂²×D₄×C₃₁	—	2	1	no	—
46	C₂²×D₄×C₃₁	—	2	1	no	—
47	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
48	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₃₁	—	2	1	no	—
50	C₂³×C₄×C₃₁	—	2	1	no	—
51	C₂³×C₄×C₃₁	—	2	1	no	—
52	C₂³×C₄×C₃₁	—	2	2	no	—
53	C₂×D₈×C₃₁	—	2	4	no	—
54	C₂×D₈×C₃₁	—	2	4	no	—
55	C₂×D₈×C₃₁	C₂×C₄	1	8	yes	complemented
56	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
57	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
58	C₃₁×#127	—	2	2	no	—
59	C₃₁×#127	C₂×C₄	1	4	yes	complemented
60	C₄×D₄×C₃₁	—	2	2	no	—
61	C₄×D₄×C₃₁	—	2	2	no	—
62	C₄×D₄×C₃₁	—	2	4	no	—
63	C₄×D₄×C₃₁	—	2	2	no	—
64	C₄×D₄×C₃₁	—	2	2	no	—
65	C₄×D₄×C₃₁	—	2	4	no	—
66	C₄×D₄×C₃₁	—	4	4	no	—
67	C₄×D₄×C₃₁	—	4	4	no	—
68	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	complemented
69	C₄×D₄×C₃₁	D₄	1	4	yes	complemented
70	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
71	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
72	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
73	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
74	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
75	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
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	2	no	—
80	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
81	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
82	C₂×C₄²×C₃₁	—	2	1	no	—
83	C₂×C₄²×C₃₁	—	2	1	no	—
84	C₂×C₄²×C₃₁	—	2	1	no	—
85	C₂×C₄²×C₃₁	—	4	4	no	—
86	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
87	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
88	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
89	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
90	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
91	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
92	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
93	C₃₁×#106	—	2	4	no	—
94	C₃₁×C₄⋊C₈	—	2	2	no	—
95	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
96	C₃₁×#102	—	2	4	no	—
97	C₃₁×#102	—	2	4	no	—
98	C₃₁×#102	—	2	4	no	—
99	C₃₁×#102	C₂×C₄	1	8	yes	complemented
100	C₃₁×#97	—	2	4	no	—
101	C₃₁×#95	—	2	1	no	—
102	C₃₁×#95	—	2	1	no	—
103	C₃₁×#95	—	2	1	no	—
104	C₂⁴×C₃₁	—	2	1	no	—
105	C₂⁴×C₃₁	—	2	2	no	—
106	C₂×D₄×C₃₁	—	2	4	no	—
107	C₂×D₄×C₃₁	—	2	4	no	—
108	C₂×D₄×C₃₁	—	2	2	no	—
109	C₂×D₄×C₃₁	—	2	1	no	—
110	C₂×D₄×C₃₁	—	2	1	no	—
111	C₂×D₄×C₃₁	—	2	1	no	—
112	C₂×D₄×C₃₁	—	2	2	no	—
113	C₂×D₄×C₃₁	—	2	2	no	—
114	C₂×D₄×C₃₁	—	2	2	no	—
115	C₂×D₄×C₃₁	—	2	2	no	—
116	C₂×D₄×C₃₁	—	4	2	no	—
117	C₂×D₄×C₃₁	—	4	2	no	—
118	C₂×D₄×C₃₁	—	4	2	no	—
119	C₂×D₄×C₃₁	—	4	2	no	—
120	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
121	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
122	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
123	C₂×D₄×C₃₁	#41	1	2	yes	—
124	C₂²×C₄×C₃₁	—	2	4	no	—
125	C₂²×C₄×C₃₁	—	2	1	no	—
126	C₂²×C₄×C₃₁	—	2	1	no	—
127	C₂²×C₄×C₃₁	—	2	1	no	—
128	C₂²×C₄×C₃₁	—	2	4	no	—
129	C₂²×C₄×C₃₁	—	2	4	no	—
130	C₂²×C₄×C₃₁	—	2	2	no	—
131	C₂²×C₄×C₃₁	—	4	2	no	—
132	C₂²×C₄×C₃₁	—	4	2	no	—
133	C₂²×C₄×C₃₁	—	4	2	no	—
134	C₂²×C₄×C₃₁	—	4	4	no	—
135	C₂²×C₄×C₃₁	—	4	4	no	—
136	C₂²×C₄×C₃₁	—	4	2	no	—
137	C₂²×C₄×C₃₁	—	4	2	no	—
138	C₂²×C₄×C₃₁	—	4	2	no	—
139	C₂²×C₄×C₃₁	—	4	4	no	—
140	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
142	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
147	D₈×C₃₁	—	2	16	no	—
148	C₂×C₈×C₃₁	—	2	2	no	—
149	C₂×C₈×C₃₁	—	2	2	no	—
150	C₂×C₈×C₃₁	—	2	2	no	—
151	C₂×C₈×C₃₁	—	2	2	no	—
152	C₂×C₈×C₃₁	—	2	2	no	—
153	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
154	C₄⋊C₄×C₃₁	—	2	1	no	—
155	C₄⋊C₄×C₃₁	—	2	1	no	—
156	C₄⋊C₄×C₃₁	—	2	1	no	—
157	C₄⋊C₄×C₃₁	—	2	2	no	—
158	C₄⋊C₄×C₃₁	—	2	2	no	—
159	C₄⋊C₄×C₃₁	—	4	2	no	—
160	C₄⋊C₄×C₃₁	—	4	2	no	—
161	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
162	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
163	C₄⋊C₄×C₃₁	#41	1	2	yes	—
164	C₂²⋊C₄×C₃₁	—	2	4	no	—
165	C₂²⋊C₄×C₃₁	—	2	4	no	—
166	C₂²⋊C₄×C₃₁	—	2	4	no	—
167	C₂²⋊C₄×C₃₁	—	4	4	no	—
168	C₂²⋊C₄×C₃₁	—	4	4	no	—
169	C₂²⋊C₄×C₃₁	—	4	4	no	—
170	C₄²×C₃₁	—	2	2	no	—
171	C₄²×C₃₁	—	2	2	no	—
172	C₄²×C₃₁	—	2	2	no	—
173	C₄²×C₃₁	—	2	1	no	—
174	C₄²×C₃₁	—	2	1	no	—
175	C₄²×C₃₁	—	4	1	no	—
176	C₄²×C₃₁	—	4	2	no	—
177	C₄²×C₃₁	—	4	2	no	—
178	C₄²×C₃₁	—	4	2	no	—
179	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
180	C₄²×C₃₁	C₂×D₄	1	2	yes	—
181	C₄²×C₃₁	#41	1	2	yes	—
182	C₄²×C₃₁	#41	1	2	yes	—
183	#21503	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
184	C₂³×C₃₁	—	2	2	no	—
185	C₂³×C₃₁	—	2	2	no	—
186	C₂³×C₃₁	—	2	2	no	—
187	C₂³×C₃₁	—	2	4	no	—
188	C₂³×C₃₁	—	4	1	no	—
189	C₂³×C₃₁	—	4	1	no	—
190	C₂³×C₃₁	—	4	1	no	—
191	C₂³×C₃₁	—	4	1	no	—
192	C₂³×C₃₁	—	4	2	no	—
193	C₂³×C₃₁	—	4	2	no	—
194	C₂³×C₃₁	—	4	2	no	—
195	C₂³×C₃₁	—	4	2	no	—
196	C₂³×C₃₁	—	4	2	no	—
197	C₂³×C₃₁	—	4	2	no	—
198	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
199	D₄×C₃₁	—	2	4	no	—
200	D₄×C₃₁	—	2	2	no	—
201	D₄×C₃₁	—	2	4	no	—
202	D₄×C₃₁	—	2	2	no	—
203	D₄×C₃₁	—	2	2	no	—
204	D₄×C₃₁	—	4	8	no	—
205	D₄×C₃₁	—	4	2	no	—
206	D₄×C₃₁	—	4	2	no	—
207	D₄×C₃₁	—	4	8	no	—
208	D₄×C₃₁	C₄×D₄	1	4	yes	complemented
209	C₂×C₄×C₃₁	—	2	1	no	—
210	C₂×C₄×C₃₁	—	2	1	no	—
211	C₂×C₄×C₃₁	—	2	1	no	—
212	C₂×C₄×C₃₁	—	2	1	no	—
213	C₂×C₄×C₃₁	—	2	1	no	—
214	C₂×C₄×C₃₁	—	2	1	no	—
215	C₂×C₄×C₃₁	—	2	1	no	—
216	C₂×C₄×C₃₁	—	2	1	no	—
217	C₂×C₄×C₃₁	—	2	2	no	—
218	C₂×C₄×C₃₁	—	2	2	no	—
219	C₂×C₄×C₃₁	—	2	1	no	—
220	C₂×C₄×C₃₁	—	2	2	no	—
221	C₂×C₄×C₃₁	—	2	1	no	—
222	C₂×C₄×C₃₁	—	2	1	no	—
223	C₂×C₄×C₃₁	—	4	8	no	—
224	C₂×C₄×C₃₁	—	4	8	no	—
225	C₂×C₄×C₃₁	—	4	1	no	—
226	C₂×C₄×C₃₁	—	4	1	no	—
227	C₂×C₄×C₃₁	—	4	1	no	—
228	C₂×C₄×C₃₁	—	4	1	no	—
229	C₂×C₄×C₃₁	—	4	1	no	—
230	C₂×C₄×C₃₁	—	4	1	no	—
231	C₂×C₄×C₃₁	—	4	8	no	—
232	C₂×C₄×C₃₁	—	4	2	no	—
233	C₂×C₄×C₃₁	—	4	2	no	—
234	C₂×C₄×C₃₁	—	4	2	no	—
235	C₂×C₄×C₃₁	—	8	8	no	—
236	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
237	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
238	C₂×C₄×C₃₁	#121	1	2	yes	—
239	C₂×C₄×C₃₁	#121	1	2	yes	—
240	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
241	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
242	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
243	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
244	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
245	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
246	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
247	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
248	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
249	C₈×C₃₁	—	2	4	no	—
250	C₈×C₃₁	—	4	2	no	—
251	C₂×#459	C₂×C₃₁	1	1	yes	characteristic
252	C₂×#442	C₂×C₃₁	1	2	yes	complemented
253	C₄×#127	C₂×C₃₁	1	1	yes	characteristic, complemented
254	C₄²×D₄	C₂×C₃₁	1	1	yes	characteristic, complemented
255	#1982	C₂×C₃₁	1	1	yes	characteristic, complemented
256	#1948	C₂×C₃₁	1	1	yes	characteristic, complemented
257	#1926	C₂×C₃₁	1	2	yes	complemented
258	#1879	C₂×C₃₁	1	2	yes	complemented
259	#1802	C₂×C₃₁	1	1	yes	characteristic, complemented
260	#1767	C₂×C₃₁	1	1	yes	characteristic, complemented
261	#1762	C₂×C₃₁	1	2	yes	complemented
262	C₂²×C₃₁	—	2	4	no	—
263	C₂²×C₃₁	—	4	2	no	—
264	C₂²×C₃₁	—	4	2	no	—
265	C₂²×C₃₁	—	4	2	no	—
266	C₂²×C₃₁	—	4	2	no	—
267	C₂²×C₃₁	—	4	2	no	—
268	C₂²×C₃₁	—	4	2	no	—
269	C₂²×C₃₁	—	4	4	no	—
270	C₂²×C₃₁	—	4	4	no	—
271	C₂²×C₃₁	—	4	4	no	—
272	C₂²×C₃₁	—	8	2	no	—
273	C₂²×C₃₁	—	8	2	no	—
274	C₂²×C₃₁	—	8	2	no	—
275	C₂²×C₃₁	—	8	2	no	—
276	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
277	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
278	C₂²×C₃₁	#525	1	1	yes	characteristic
279	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
280	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
281	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
282	C₂²×C₃₁	#575	1	1	yes	characteristic
283	C₄×C₃₁	—	2	2	no	—
284	C₄×C₃₁	—	2	2	no	—
285	C₄×C₃₁	—	2	2	no	—
286	C₄×C₃₁	—	2	2	no	—
287	C₄×C₃₁	—	4	2	no	—
288	C₄×C₃₁	—	4	1	no	—
289	C₄×C₃₁	—	4	2	no	—
290	C₄×C₃₁	—	4	2	no	—
291	C₄×C₃₁	—	8	2	no	—
292	C₄×C₃₁	#442	1	4	yes	complemented
293	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
294	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
295	C₄×C₃₁	#529	1	1	yes	characteristic
296	C₄×C₃₁	#532	1	1	yes	characteristic
297	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
298	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
299	C₂×C₄×D₄	—	2	1	no	—
300	C₂×C₄×D₄	—	2	1	no	—
301	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
302	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
303	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
304	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
305	C₂²×C₄²	—	2	1	no	—
306	#459	C₄×C₃₁	1	8	yes	complemented
307	#442	—	2	8	no	—
308	C₂×#106	C₂²×C₃₁	1	2	yes	—
309	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
310	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
311	C₂×#102	C₂²×C₃₁	1	2	yes	—
312	C₂×#102	C₂²×C₃₁	1	2	yes	—
313	C₂×#102	C₂²×C₃₁	1	2	yes	—
314	C₂×#102	C₂²×C₃₁	1	2	yes	—
315	C₂×#97	C₂²×C₃₁	1	2	yes	—
316	#390	—	2	1	no	—
317	#390	—	2	1	no	—
318	#390	C₂²×C₃₁	1	2	yes	—
319	#384	C₂²×C₃₁	1	1	yes	characteristic
320	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
321	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
322	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
323	C₄×C₂²⋊C₄	—	2	1	no	—
324	C₄×C₂²⋊C₄	—	2	1	no	—
325	C₄×C₂²⋊C₄	—	2	1	no	—
326	C₄³	C₂²×C₃₁	1	1	yes	characteristic
327	#340	C₂²×C₃₁	1	2	yes	—
328	#340	C₂²×C₃₁	1	2	yes	—
329	#336	C₂²×C₃₁	1	2	yes	—
330	C₂×C₃₁	—	4	4	no	—
331	C₂×C₃₁	—	8	4	no	—
332	C₂×C₃₁	C₄×#121	1	1	yes	characteristic
333	C₂×C₃₁	C₂×#442	1	1	yes	characteristic
334	C₂×C₃₁	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
335	C₂×C₃₁	#3036	1	1	yes	characteristic
336	C₂×C₃₁	#3038	1	1	yes	characteristic
337	C₂×C₃₁	#3109	1	1	yes	characteristic
338	C₂×C₃₁	#3115	1	1	yes	characteristic
339	C₂²×D₄	—	2	1	no	—
340	C₂²×D₄	—	2	1	no	—
341	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
342	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
343	C₂³×C₄	—	2	1	no	—
344	C₂³×C₄	—	2	1	no	—
345	C₂³×C₄	—	2	1	no	—
346	C₂³×C₄	—	2	2	no	—
347	C₂×D₈	—	2	4	no	—
348	C₂×D₈	—	2	4	no	—
349	C₂×D₈	C₂×C₄×C₃₁	1	8	yes	complemented
350	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
351	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
352	#127	—	2	2	no	—
353	#127	C₂×C₄×C₃₁	1	4	yes	complemented
354	C₄×D₄	—	2	2	no	—
355	C₄×D₄	—	2	2	no	—
356	C₄×D₄	—	2	4	no	—
357	C₄×D₄	—	2	2	no	—
358	C₄×D₄	—	2	2	no	—
359	C₄×D₄	—	2	4	no	—
360	C₄×D₄	—	4	4	no	—
361	C₄×D₄	—	4	4	no	—
362	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	complemented
363	C₄×D₄	D₄×C₃₁	1	4	yes	complemented
364	C₂×C₄⋊C₄	—	2	1	no	—
365	C₂×C₄⋊C₄	—	2	1	no	—
366	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
367	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
368	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
369	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
370	C₂×C₂²⋊C₄	—	2	1	no	—
371	C₂×C₂²⋊C₄	—	2	1	no	—
372	C₂×C₂²⋊C₄	—	2	1	no	—
373	C₂×C₂²⋊C₄	—	2	2	no	—
374	C₂×C₂²⋊C₄	—	2	2	no	—
375	C₂×C₂²⋊C₄	—	2	2	no	—
376	C₂×C₄²	—	2	1	no	—
377	C₂×C₄²	—	2	1	no	—
378	C₂×C₄²	—	2	1	no	—
379	C₂×C₄²	—	4	4	no	—
380	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
381	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
382	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
383	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
384	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
385	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
386	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
387	#106	—	2	4	no	—
388	C₄⋊C₈	—	2	2	no	—
389	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	complemented
390	#102	—	2	4	no	—
391	#102	—	2	4	no	—
392	#102	—	2	4	no	—
393	#102	C₂×C₄×C₃₁	1	8	yes	complemented
394	#97	—	2	4	no	—
395	#95	—	2	1	no	—
396	#95	—	2	1	no	—
397	#95	—	2	1	no	—
398	C₃₁	#21503	1	1	yes	characteristic, direct product factor, Sylow
399	C₂⁴	—	2	1	no	—
400	C₂⁴	—	2	2	no	—
401	C₂×D₄	—	2	4	no	—
402	C₂×D₄	—	2	4	no	—
403	C₂×D₄	—	2	2	no	—
404	C₂×D₄	—	2	1	no	—
405	C₂×D₄	—	2	1	no	—
406	C₂×D₄	—	2	1	no	—
407	C₂×D₄	—	2	2	no	—
408	C₂×D₄	—	2	2	no	—
409	C₂×D₄	—	2	2	no	—
410	C₂×D₄	—	2	2	no	—
411	C₂×D₄	—	4	2	no	—
412	C₂×D₄	—	4	2	no	—
413	C₂×D₄	—	4	2	no	—
414	C₂×D₄	—	4	2	no	—
415	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
416	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
417	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
418	C₂×D₄	#41×C₃₁	1	2	yes	—
419	C₂²×C₄	—	2	4	no	—
420	C₂²×C₄	—	2	1	no	—
421	C₂²×C₄	—	2	1	no	—
422	C₂²×C₄	—	2	1	no	—
423	C₂²×C₄	—	2	4	no	—
424	C₂²×C₄	—	2	4	no	—
425	C₂²×C₄	—	2	2	no	—
426	C₂²×C₄	—	4	2	no	—
427	C₂²×C₄	—	4	2	no	—
428	C₂²×C₄	—	4	2	no	—
429	C₂²×C₄	—	4	4	no	—
430	C₂²×C₄	—	4	4	no	—
431	C₂²×C₄	—	4	2	no	—
432	C₂²×C₄	—	4	2	no	—
433	C₂²×C₄	—	4	2	no	—
434	C₂²×C₄	—	4	4	no	—
435	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
436	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
437	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
438	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
439	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
440	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
441	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
442	D₈	—	2	16	no	—
443	C₂×C₈	—	2	2	no	—
444	C₂×C₈	—	2	2	no	—
445	C₂×C₈	—	2	2	no	—
446	C₂×C₈	—	2	2	no	—
447	C₂×C₈	—	2	2	no	—
448	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
449	C₄⋊C₄	—	2	1	no	—
450	C₄⋊C₄	—	2	1	no	—
451	C₄⋊C₄	—	2	1	no	—
452	C₄⋊C₄	—	2	2	no	—
453	C₄⋊C₄	—	2	2	no	—
454	C₄⋊C₄	—	4	2	no	—
455	C₄⋊C₄	—	4	2	no	—
456	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
457	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
458	C₄⋊C₄	#41×C₃₁	1	2	yes	—
459	C₂²⋊C₄	—	2	4	no	—
460	C₂²⋊C₄	—	2	4	no	—
461	C₂²⋊C₄	—	2	4	no	—
462	C₂²⋊C₄	—	4	4	no	—
463	C₂²⋊C₄	—	4	4	no	—
464	C₂²⋊C₄	—	4	4	no	—
465	C₄²	—	2	2	no	—
466	C₄²	—	2	2	no	—
467	C₄²	—	2	2	no	—
468	C₄²	—	2	1	no	—
469	C₄²	—	2	1	no	—
470	C₄²	—	4	1	no	—
471	C₄²	—	4	2	no	—
472	C₄²	—	4	2	no	—
473	C₄²	—	4	2	no	—
474	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
475	C₄²	C₂×D₄×C₃₁	1	2	yes	—
476	C₄²	#41×C₃₁	1	2	yes	—
477	C₄²	#41×C₃₁	1	2	yes	—
478	C₂³	—	2	2	no	—
479	C₂³	—	2	2	no	—
480	C₂³	—	2	2	no	—
481	C₂³	—	2	4	no	—
482	C₂³	—	4	1	no	—
483	C₂³	—	4	1	no	—
484	C₂³	—	4	1	no	—
485	C₂³	—	4	1	no	—
486	C₂³	—	4	2	no	—
487	C₂³	—	4	2	no	—
488	C₂³	—	4	2	no	—
489	C₂³	—	4	2	no	—
490	C₂³	—	4	2	no	—
491	C₂³	—	4	2	no	—
492	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
493	D₄	—	2	4	no	—
494	D₄	—	2	2	no	—
495	D₄	—	2	4	no	—
496	D₄	—	2	2	no	—
497	D₄	—	2	2	no	—
498	D₄	—	4	8	no	—
499	D₄	—	4	8	no	—
500	D₄	—	4	2	no	—
501	D₄	—	4	2	no	—
502	D₄	C₄×D₄×C₃₁	1	4	yes	complemented
503	C₂×C₄	—	2	1	no	—
504	C₂×C₄	—	2	1	no	—
505	C₂×C₄	—	2	1	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	1	no	—
508	C₂×C₄	—	2	1	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	2	2	no	—
512	C₂×C₄	—	2	2	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	2	no	—
515	C₂×C₄	—	2	1	no	—
516	C₂×C₄	—	2	1	no	—
517	C₂×C₄	—	4	8	no	—
518	C₂×C₄	—	4	8	no	—
519	C₂×C₄	—	4	1	no	—
520	C₂×C₄	—	4	1	no	—
521	C₂×C₄	—	4	1	no	—
522	C₂×C₄	—	4	1	no	—
523	C₂×C₄	—	4	1	no	—
524	C₂×C₄	—	4	1	no	—
525	C₂×C₄	—	4	8	no	—
526	C₂×C₄	—	4	2	no	—
527	C₂×C₄	—	4	2	no	—
528	C₂×C₄	—	4	2	no	—
529	C₂×C₄	—	8	8	no	—
530	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
531	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
532	C₂×C₄	C₃₁×#121	1	2	yes	—
533	C₂×C₄	C₃₁×#121	1	2	yes	—
534	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
535	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
536	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
537	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
538	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
539	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
540	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
541	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
542	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
543	C₈	—	2	4	no	—
544	C₈	—	4	2	no	—
545	C₂²	—	2	4	no	—
546	C₂²	—	4	2	no	—
547	C₂²	—	4	2	no	—
548	C₂²	—	4	2	no	—
549	C₂²	—	4	2	no	—
550	C₂²	—	4	2	no	—
551	C₂²	—	4	2	no	—
552	C₂²	—	4	4	no	—
553	C₂²	—	4	4	no	—
554	C₂²	—	4	4	no	—
555	C₂²	—	8	2	no	—
556	C₂²	—	8	2	no	—
557	C₂²	—	8	2	no	—
558	C₂²	—	8	2	no	—
559	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
560	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
561	C₂²	C₃₁×#525	1	1	yes	characteristic
562	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
563	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
564	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
565	C₂²	C₃₁×#575	1	1	yes	characteristic
566	C₄	—	2	2	no	—
567	C₄	—	2	2	no	—
568	C₄	—	2	2	no	—
569	C₄	—	2	2	no	—
570	C₄	—	4	2	no	—
571	C₄	—	4	1	no	—
572	C₄	—	4	2	no	—
573	C₄	—	4	2	no	—
574	C₄	—	8	2	no	—
575	C₄	C₃₁×#442	1	4	yes	complemented
576	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
577	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
578	C₄	C₃₁×#529	1	1	yes	characteristic
579	C₄	C₃₁×#532	1	1	yes	characteristic
580	C₂	—	4	4	no	—
581	C₂	—	8	4	no	—
582	C₂	C₄×C₃₁×#121	1	1	yes	characteristic
583	C₂	C₂×C₃₁×#442	1	1	yes	characteristic
584	C₂	C₄×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
585	C₂	C₃₁×#3036	1	1	yes	characteristic
586	C₂	C₃₁×#3038	1	1	yes	characteristic
587	C₂	C₃₁×#3109	1	1	yes	characteristic
588	C₂	C₃₁×#3115	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#459	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#442	C₂	semidirect
3. C₄×C₃₁×#127	C₂	semidirect
4. C₄²×D₄×C₃₁	C₂	semidirect
5. C₃₁×#1982	C₂	semidirect
6. C₃₁×#1948	C₂	semidirect
7. C₃₁×#1926	C₂	semidirect
8. C₃₁×#1879	C₂	semidirect
9. C₃₁×#1802	C₂	semidirect
10. C₃₁×#1767	C₂	semidirect
11. C₃₁×#1762	C₂	semidirect
21. C₃₁×#459	C₄	semidirect
55. C₂×D₈×C₃₁	C₂×C₄	semidirect
59. C₃₁×#127	C₂×C₄	semidirect
68. C₄×D₄×C₃₁	C₂×C₄	semidirect
69. C₄×D₄×C₃₁	D₄	semidirect
95. C₃₁×C₄⋊C₈	C₂×C₄	semidirect
99. C₃₁×#102	C₂×C₄	semidirect
183. #21503	C₃₁	direct (N × H)
208. D₄×C₃₁	C₄×D₄	semidirect
252. C₂×#442	C₂×C₃₁	semidirect
253. C₄×#127	C₂×C₃₁	semidirect
254. C₄²×D₄	C₂×C₃₁	semidirect
255. #1982	C₂×C₃₁	semidirect
256. #1948	C₂×C₃₁	semidirect
257. #1926	C₂×C₃₁	semidirect
258. #1879	C₂×C₃₁	semidirect
259. #1802	C₂×C₃₁	semidirect
260. #1767	C₂×C₃₁	semidirect
261. #1762	C₂×C₃₁	semidirect
292. C₄×C₃₁	#442	semidirect
306. #459	C₄×C₃₁	semidirect
349. C₂×D₈	C₂×C₄×C₃₁	semidirect
353. #127	C₂×C₄×C₃₁	semidirect
362. C₄×D₄	C₂×C₄×C₃₁	semidirect
363. C₄×D₄	D₄×C₃₁	semidirect
389. C₄⋊C₈	C₂×C₄×C₃₁	semidirect
393. #102	C₂×C₄×C₃₁	semidirect
398. C₃₁	#21503	direct (N × H)
502. D₄	C₄×D₄×C₃₁	semidirect
575. C₄	C₃₁×#442	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#459	2
C₂×C₃₁×#442	2
C₄×C₃₁×#127	2
C₄²×D₄×C₃₁	2
C₃₁×#1982	2
C₃₁×#1948	2
C₃₁×#1926	2
C₃₁×#1879	2
C₃₁×#1802	2
C₃₁×#1767	2
C₃₁×#1762	2
#21503	31

Subgroups — C31×#21503 #21478262

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21503 #21478262