Subgroups — C₃₁×#20490 #21477249

Order 7936.

1138 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#436	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#435	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#405	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2290	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1771	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1768	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1753	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1753	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1753	C₂	1	1	yes	maximal, characteristic, complemented
12	C₈×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#1750	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×C₃₁×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#1726	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
17	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
18	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
22	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
23	C₃₁×#436	C₄	1	8	yes	complemented
24	C₃₁×#435	C₄	1	8	yes	complemented
25	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
31	C₃₁×#405	—	2	4	no	—
32	C₄×M₁₆×C₃₁	—	2	4	no	—
33	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
34	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
35	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
36	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
37	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
41	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
42	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
43	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
44	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
45	C₃₁×#388	C₄	1	2	yes	—
46	C₃₁×#387	C₄	1	2	yes	—
47	C₃₁×#386	C₄	1	2	yes	—
48	C₃₁×#385	C₄	1	2	yes	—
49	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
50	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
51	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
52	C₃₁×#376	C₂²	1	1	yes	characteristic
53	C₃₁×#336	C₂²	1	1	yes	characteristic
54	C₃₁×#336	C₂²	1	1	yes	characteristic
55	C₃₁×#336	C₂²	1	1	yes	characteristic
56	C₃₁×#336	C₂²	1	1	yes	characteristic
57	C₃₁×#336	C₂²	1	1	yes	characteristic
58	C₃₁×#336	C₂²	1	1	yes	characteristic
59	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
60	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
61	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×M₁₆×C₃₁	—	2	2	no	—
64	C₂×M₁₆×C₃₁	—	2	2	no	—
65	C₂×M₁₆×C₃₁	—	2	2	no	—
66	C₂×M₁₆×C₃₁	—	2	2	no	—
67	C₂×M₁₆×C₃₁	—	2	2	no	—
68	C₂×M₁₆×C₃₁	—	2	2	no	—
69	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
70	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
71	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	complemented
72	C₂²×C₈×C₃₁	—	2	2	no	—
73	C₂²×C₈×C₃₁	—	2	2	no	—
74	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	complemented
75	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
78	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
79	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
80	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
81	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
82	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
83	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
84	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
85	C₃₁×#117	—	2	2	no	—
86	C₃₁×#117	—	2	2	no	—
87	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
88	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
89	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
90	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
91	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
92	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
93	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
94	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
95	C₂×C₄²×C₃₁	—	2	2	no	—
96	C₂×C₄²×C₃₁	—	2	2	no	—
97	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
98	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
99	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
100	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
102	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
103	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
104	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
105	C₃₁×C₂²⋊C₈	C₂×C₄	1	4	yes	—
106	C₃₁×C₂²⋊C₈	C₂×C₄	1	4	yes	—
107	C₃₁×C₂²⋊C₈	C₂×C₄	1	4	yes	complemented
108	C₃₁×C₂²⋊C₈	C₂×C₄	1	4	yes	complemented
109	C₃₁×#97	—	2	2	no	—
110	C₃₁×#97	—	2	2	no	—
111	C₃₁×#97	—	2	2	no	—
112	C₃₁×#97	C₂×C₄	1	4	yes	complemented
113	C₃₁×#97	D₄	1	4	yes	complemented
114	C₄×C₈×C₃₁	—	2	2	no	—
115	C₄×C₈×C₃₁	—	2	2	no	—
116	C₄×C₈×C₃₁	—	2	2	no	—
117	C₄×C₈×C₃₁	—	2	2	no	—
118	C₄×C₈×C₃₁	—	2	2	no	—
119	C₄×C₈×C₃₁	—	2	2	no	—
120	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	complemented
121	C₃₁×#95	C₂×C₄	1	2	yes	—
122	C₃₁×#95	C₂×C₄	1	2	yes	—
123	C₃₁×#95	C₂×C₄	1	2	yes	—
124	C₃₁×#95	C₂×C₄	1	2	yes	—
125	C₃₁×#95	C₂×C₄	1	2	yes	—
126	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
127	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
128	C₂×D₄×C₃₁	—	4	2	no	—
129	C₂×D₄×C₃₁	C₄²	1	4	yes	—
130	C₂²×C₄×C₃₁	—	2	1	no	—
131	C₂²×C₄×C₃₁	—	2	1	no	—
132	C₂²×C₄×C₃₁	—	2	1	no	—
133	C₂²×C₄×C₃₁	—	2	1	no	—
134	C₂²×C₄×C₃₁	—	2	1	no	—
135	C₂²×C₄×C₃₁	—	2	1	no	—
136	C₂²×C₄×C₃₁	—	2	1	no	—
137	C₂²×C₄×C₃₁	—	2	1	no	—
138	C₂²×C₄×C₃₁	—	2	4	no	—
139	C₂²×C₄×C₃₁	—	2	4	no	—
140	C₂²×C₄×C₃₁	—	4	2	no	—
141	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
142	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
143	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
144	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
145	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
146	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
147	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
148	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
149	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
150	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
151	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
152	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
153	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
154	M₁₆×C₃₁	—	2	8	no	—
155	M₁₆×C₃₁	—	2	8	no	—
156	M₁₆×C₃₁	—	2	8	no	—
157	C₂×C₈×C₃₁	—	2	1	no	—
158	C₂×C₈×C₃₁	—	2	1	no	—
159	C₂×C₈×C₃₁	—	2	1	no	—
160	C₂×C₈×C₃₁	—	2	1	no	—
161	C₂×C₈×C₃₁	—	2	1	no	—
162	C₂×C₈×C₃₁	—	2	1	no	—
163	C₂×C₈×C₃₁	—	2	1	no	—
164	C₂×C₈×C₃₁	—	2	1	no	—
165	C₂×C₈×C₃₁	—	2	1	no	—
166	C₂×C₈×C₃₁	—	2	1	no	—
167	C₂×C₈×C₃₁	—	2	1	no	—
168	C₂×C₈×C₃₁	—	2	1	no	—
169	C₂×C₈×C₃₁	—	2	8	no	—
170	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
171	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
172	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
173	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
174	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
175	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
176	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
177	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
178	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
179	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
180	C₂×C₈×C₃₁	#41	1	2	yes	—
181	C₂×C₈×C₃₁	#41	1	2	yes	—
182	C₄⋊C₄×C₃₁	—	2	4	no	—
183	C₄⋊C₄×C₃₁	—	2	4	no	—
184	C₄⋊C₄×C₃₁	C₄²	1	4	yes	—
185	C₂²⋊C₄×C₃₁	—	2	4	no	—
186	C₂²⋊C₄×C₃₁	—	2	4	no	—
187	C₂²⋊C₄×C₃₁	—	4	2	no	—
188	C₂²⋊C₄×C₃₁	C₄²	1	4	yes	—
189	C₂²⋊C₄×C₃₁	C₄²	1	4	yes	—
190	C₄²×C₃₁	—	2	4	no	—
191	C₄²×C₃₁	—	2	1	no	—
192	C₄²×C₃₁	—	2	1	no	—
193	C₄²×C₃₁	—	2	1	no	—
194	C₄²×C₃₁	—	2	1	no	—
195	C₄²×C₃₁	—	2	1	no	—
196	C₄²×C₃₁	—	2	1	no	—
197	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
198	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
199	C₄²×C₃₁	C₂×D₄	1	2	yes	—
200	C₄²×C₃₁	#41	1	2	yes	—
201	#20490	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
202	C₂³×C₃₁	—	2	1	no	—
203	C₂³×C₃₁	—	2	1	no	—
204	C₂³×C₃₁	—	2	1	no	—
205	C₂³×C₃₁	—	2	1	no	—
206	C₂³×C₃₁	—	2	2	no	—
207	C₂³×C₃₁	—	2	2	no	—
208	C₂³×C₃₁	—	2	2	no	—
209	C₂³×C₃₁	—	2	2	no	—
210	C₂³×C₃₁	C₂×C₄²	1	2	yes	—
211	C₂³×C₃₁	C₂×C₄²	1	2	yes	—
212	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
213	D₄×C₃₁	—	4	4	no	—
214	C₂×C₄×C₃₁	—	2	2	no	—
215	C₂×C₄×C₃₁	—	2	1	no	—
216	C₂×C₄×C₃₁	—	2	1	no	—
217	C₂×C₄×C₃₁	—	2	1	no	—
218	C₂×C₄×C₃₁	—	2	1	no	—
219	C₂×C₄×C₃₁	—	2	2	no	—
220	C₂×C₄×C₃₁	—	2	2	no	—
221	C₂×C₄×C₃₁	—	2	2	no	—
222	C₂×C₄×C₃₁	—	2	2	no	—
223	C₂×C₄×C₃₁	—	2	2	no	—
224	C₂×C₄×C₃₁	—	2	2	no	—
225	C₂×C₄×C₃₁	—	2	2	no	—
226	C₂×C₄×C₃₁	—	2	2	no	—
227	C₂×C₄×C₃₁	—	2	2	no	—
228	C₂×C₄×C₃₁	—	2	2	no	—
229	C₂×C₄×C₃₁	—	2	2	no	—
230	C₂×C₄×C₃₁	—	2	2	no	—
231	C₂×C₄×C₃₁	—	2	2	no	—
232	C₂×C₄×C₃₁	—	2	2	no	—
233	C₂×C₄×C₃₁	—	2	2	no	—
234	C₂×C₄×C₃₁	—	2	2	no	—
235	C₂×C₄×C₃₁	—	2	2	no	—
236	C₂×C₄×C₃₁	—	2	2	no	—
237	C₂×C₄×C₃₁	—	2	2	no	—
238	C₂×C₄×C₃₁	—	4	8	no	—
239	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
240	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
241	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
242	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
243	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
244	C₂×C₄×C₃₁	#131	1	4	yes	—
245	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
246	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
247	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
248	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
249	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
250	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
251	C₈×C₃₁	—	2	2	no	—
252	C₈×C₃₁	—	2	2	no	—
253	C₈×C₃₁	—	2	2	no	—
254	C₈×C₃₁	—	2	2	no	—
255	C₈×C₃₁	—	2	2	no	—
256	C₈×C₃₁	—	2	2	no	—
257	C₈×C₃₁	C₄×D₄	1	4	yes	—
258	C₈×C₃₁	C₄×D₄	1	4	yes	—
259	C₂×#436	C₂×C₃₁	1	1	yes	characteristic
260	C₂×#435	C₂×C₃₁	1	1	yes	characteristic
261	C₂×#405	C₂×C₃₁	1	1	yes	characteristic, complemented
262	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
263	#2290	C₂×C₃₁	1	1	yes	characteristic
264	#1771	C₂×C₃₁	1	1	yes	characteristic, complemented
265	C₈×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
266	#1768	C₂×C₃₁	1	1	yes	characteristic, complemented
267	#1753	C₂×C₃₁	1	1	yes	characteristic, complemented
268	#1753	C₂×C₃₁	1	1	yes	characteristic, complemented
269	#1753	C₂×C₃₁	1	1	yes	characteristic, complemented
270	C₈×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
271	#1750	C₂×C₃₁	1	1	yes	characteristic, complemented
272	C₄×C₂²⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
273	#1726	C₂×C₃₁	1	1	yes	characteristic, complemented
274	C₂²×C₃₁	—	2	2	no	—
275	C₂²×C₃₁	—	2	2	no	—
276	C₂²×C₃₁	—	2	2	no	—
277	C₂²×C₃₁	—	2	2	no	—
278	C₂²×C₃₁	—	2	2	no	—
279	C₂²×C₃₁	—	2	2	no	—
280	C₂²×C₃₁	—	4	2	no	—
281	C₂²×C₃₁	—	4	2	no	—
282	C₂²×C₃₁	—	4	2	no	—
283	C₂²×C₃₁	—	4	2	no	—
284	C₂²×C₃₁	C₂²×C₄²	1	1	yes	characteristic
285	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
286	C₂²×C₃₁	C₄×#41	1	1	yes	characteristic
287	C₂²×C₃₁	C₂×#131	1	2	yes	—
288	C₂²×C₃₁	#568	1	2	yes	—
289	C₄×C₃₁	—	2	2	no	—
290	C₄×C₃₁	—	2	2	no	—
291	C₄×C₃₁	—	2	4	no	—
292	C₄×C₃₁	—	2	4	no	—
293	C₄×C₃₁	—	2	2	no	—
294	C₄×C₃₁	—	2	2	no	—
295	C₄×C₃₁	—	4	2	no	—
296	C₄×C₃₁	—	4	2	no	—
297	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
298	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
299	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
300	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
301	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
302	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
303	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
304	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
305	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
306	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
307	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
308	#436	C₄×C₃₁	1	8	yes	complemented
309	#435	C₄×C₃₁	1	8	yes	complemented
310	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
311	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
312	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
313	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
314	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
315	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
316	#405	—	2	4	no	—
317	C₄×M₁₆	—	2	4	no	—
318	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
319	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
320	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
321	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
322	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
323	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
324	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
325	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
326	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
327	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
328	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
329	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
330	#388	C₄×C₃₁	1	2	yes	—
331	#387	C₄×C₃₁	1	2	yes	—
332	#386	C₄×C₃₁	1	2	yes	—
333	#385	C₄×C₃₁	1	2	yes	—
334	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
335	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
336	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
337	#376	C₂²×C₃₁	1	1	yes	characteristic
338	#336	C₂²×C₃₁	1	1	yes	characteristic
339	#336	C₂²×C₃₁	1	1	yes	characteristic
340	#336	C₂²×C₃₁	1	1	yes	characteristic
341	#336	C₂²×C₃₁	1	1	yes	characteristic
342	#336	C₂²×C₃₁	1	1	yes	characteristic
343	#336	C₂²×C₃₁	1	1	yes	characteristic
344	C₂×C₃₁	—	4	2	no	—
345	C₂×C₃₁	—	4	2	no	—
346	C₂×C₃₁	C₄²×D₄	1	1	yes	characteristic
347	C₂×C₃₁	#2873	1	1	yes	characteristic
348	C₂×C₃₁	C₄×#131	1	1	yes	characteristic
349	C₂×C₃₁	#2933	1	2	yes	—
350	C₂×C₃₁	#2969	1	2	yes	—
351	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
352	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
353	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
354	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
355	C₂×M₁₆	—	2	2	no	—
356	C₂×M₁₆	—	2	2	no	—
357	C₂×M₁₆	—	2	2	no	—
358	C₂×M₁₆	—	2	2	no	—
359	C₂×M₁₆	—	2	2	no	—
360	C₂×M₁₆	—	2	2	no	—
361	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
362	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
363	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	complemented
364	C₂²×C₈	—	2	2	no	—
365	C₂²×C₈	—	2	2	no	—
366	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	complemented
367	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
368	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
369	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
370	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
371	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
372	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
373	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
374	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
375	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
376	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
377	#117	—	2	2	no	—
378	#117	—	2	2	no	—
379	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
380	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
381	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
382	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
383	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
384	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
385	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
386	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
387	C₂×C₄²	—	2	2	no	—
388	C₂×C₄²	—	2	2	no	—
389	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
390	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
391	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
392	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
393	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
394	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
395	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	complemented
396	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	complemented
397	C₂²⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
398	C₂²⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
399	C₂²⋊C₈	C₂×C₄×C₃₁	1	4	yes	complemented
400	C₂²⋊C₈	C₂×C₄×C₃₁	1	4	yes	complemented
401	#97	—	2	2	no	—
402	#97	—	2	2	no	—
403	#97	—	2	2	no	—
404	#97	C₂×C₄×C₃₁	1	4	yes	complemented
405	#97	D₄×C₃₁	1	4	yes	complemented
406	C₄×C₈	—	2	2	no	—
407	C₄×C₈	—	2	2	no	—
408	C₄×C₈	—	2	2	no	—
409	C₄×C₈	—	2	2	no	—
410	C₄×C₈	—	2	2	no	—
411	C₄×C₈	—	2	2	no	—
412	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	complemented
413	#95	C₂×C₄×C₃₁	1	2	yes	—
414	#95	C₂×C₄×C₃₁	1	2	yes	—
415	#95	C₂×C₄×C₃₁	1	2	yes	—
416	#95	C₂×C₄×C₃₁	1	2	yes	—
417	#95	C₂×C₄×C₃₁	1	2	yes	—
418	C₃₁	#20490	1	1	yes	characteristic, direct product factor, Sylow
419	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
420	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
421	C₂×D₄	—	4	2	no	—
422	C₂×D₄	C₄²×C₃₁	1	4	yes	—
423	C₂²×C₄	—	2	1	no	—
424	C₂²×C₄	—	2	1	no	—
425	C₂²×C₄	—	2	1	no	—
426	C₂²×C₄	—	2	1	no	—
427	C₂²×C₄	—	2	1	no	—
428	C₂²×C₄	—	2	1	no	—
429	C₂²×C₄	—	2	1	no	—
430	C₂²×C₄	—	2	1	no	—
431	C₂²×C₄	—	2	4	no	—
432	C₂²×C₄	—	2	4	no	—
433	C₂²×C₄	—	4	2	no	—
434	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
435	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
436	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
437	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
438	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
439	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
440	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
441	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
442	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
443	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
444	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
445	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
446	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
447	M₁₆	—	2	8	no	—
448	M₁₆	—	2	8	no	—
449	M₁₆	—	2	8	no	—
450	C₂×C₈	—	2	1	no	—
451	C₂×C₈	—	2	1	no	—
452	C₂×C₈	—	2	1	no	—
453	C₂×C₈	—	2	1	no	—
454	C₂×C₈	—	2	1	no	—
455	C₂×C₈	—	2	1	no	—
456	C₂×C₈	—	2	1	no	—
457	C₂×C₈	—	2	1	no	—
458	C₂×C₈	—	2	1	no	—
459	C₂×C₈	—	2	1	no	—
460	C₂×C₈	—	2	1	no	—
461	C₂×C₈	—	2	1	no	—
462	C₂×C₈	—	2	8	no	—
463	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
464	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
465	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
466	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
467	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
468	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
469	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
470	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
471	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
472	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
473	C₂×C₈	#41×C₃₁	1	2	yes	—
474	C₂×C₈	#41×C₃₁	1	2	yes	—
475	C₄⋊C₄	—	2	4	no	—
476	C₄⋊C₄	—	2	4	no	—
477	C₄⋊C₄	C₄²×C₃₁	1	4	yes	—
478	C₂²⋊C₄	—	2	4	no	—
479	C₂²⋊C₄	—	2	4	no	—
480	C₂²⋊C₄	—	4	2	no	—
481	C₂²⋊C₄	C₄²×C₃₁	1	4	yes	—
482	C₂²⋊C₄	C₄²×C₃₁	1	4	yes	—
483	C₄²	—	2	4	no	—
484	C₄²	—	2	1	no	—
485	C₄²	—	2	1	no	—
486	C₄²	—	2	1	no	—
487	C₄²	—	2	1	no	—
488	C₄²	—	2	1	no	—
489	C₄²	—	2	1	no	—
490	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
491	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
492	C₄²	C₂×D₄×C₃₁	1	2	yes	—
493	C₄²	#41×C₃₁	1	2	yes	—
494	C₂³	—	2	1	no	—
495	C₂³	—	2	1	no	—
496	C₂³	—	2	1	no	—
497	C₂³	—	2	1	no	—
498	C₂³	—	2	2	no	—
499	C₂³	—	2	2	no	—
500	C₂³	—	2	2	no	—
501	C₂³	—	2	2	no	—
502	C₂³	C₂×C₄²×C₃₁	1	2	yes	—
503	C₂³	C₂×C₄²×C₃₁	1	2	yes	—
504	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
505	D₄	—	4	4	no	—
506	C₂×C₄	—	2	2	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	2	no	—
514	C₂×C₄	—	2	2	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	2	no	—
521	C₂×C₄	—	2	2	no	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	2	2	no	—
524	C₂×C₄	—	2	2	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	2	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	2	2	no	—
529	C₂×C₄	—	2	2	no	—
530	C₂×C₄	—	4	8	no	—
531	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
532	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
533	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
534	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
535	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
536	C₂×C₄	C₃₁×#131	1	4	yes	—
537	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
538	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
539	C₂×C₄	C₂²×D₄×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	2	no	—
544	C₈	—	2	2	no	—
545	C₈	—	2	2	no	—
546	C₈	—	2	2	no	—
547	C₈	—	2	2	no	—
548	C₈	—	2	2	no	—
549	C₈	C₄×D₄×C₃₁	1	4	yes	—
550	C₈	C₄×D₄×C₃₁	1	4	yes	—
551	C₂²	—	2	2	no	—
552	C₂²	—	2	2	no	—
553	C₂²	—	2	2	no	—
554	C₂²	—	2	2	no	—
555	C₂²	—	2	2	no	—
556	C₂²	—	2	2	no	—
557	C₂²	—	4	2	no	—
558	C₂²	—	4	2	no	—
559	C₂²	—	4	2	no	—
560	C₂²	—	4	2	no	—
561	C₂²	C₂²×C₄²×C₃₁	1	1	yes	characteristic
562	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
563	C₂²	C₄×#41×C₃₁	1	1	yes	characteristic
564	C₂²	C₂×C₃₁×#131	1	2	yes	—
565	C₂²	C₃₁×#568	1	2	yes	—
566	C₄	—	2	2	no	—
567	C₄	—	2	2	no	—
568	C₄	—	2	4	no	—
569	C₄	—	2	4	no	—
570	C₄	—	2	2	no	—
571	C₄	—	2	2	no	—
572	C₄	—	4	2	no	—
573	C₄	—	4	2	no	—
574	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
575	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
576	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
577	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
578	C₂	—	4	2	no	—
579	C₂	—	4	2	no	—
580	C₂	C₄²×D₄×C₃₁	1	1	yes	characteristic
581	C₂	C₃₁×#2873	1	1	yes	characteristic
582	C₂	C₄×C₃₁×#131	1	1	yes	characteristic
583	C₂	C₃₁×#2933	1	2	yes	—
584	C₂	C₃₁×#2969	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator	C₂²	C₂²×C₄²×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂³×C₄
prime radical	C₂²×D₄×C₃₁	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₃₁×#405	C₂	semidirect
4. C₂×C₄×M₁₆×C₃₁	C₂	semidirect
6. C₃₁×#1771	C₂	semidirect
7. C₈×C₄⋊C₄×C₃₁	C₂	semidirect
8. C₃₁×#1768	C₂	semidirect
9. C₃₁×#1753	C₂	semidirect
10. C₃₁×#1753	C₂	semidirect
11. C₃₁×#1753	C₂	semidirect
12. C₈×C₂²⋊C₄×C₃₁	C₂	semidirect
13. C₃₁×#1750	C₂	semidirect
14. C₄×C₃₁×C₂²⋊C₈	C₂	semidirect
15. C₃₁×#1726	C₂	semidirect
23. C₃₁×#436	C₄	semidirect
24. C₃₁×#435	C₄	semidirect
71. C₂×M₁₆×C₃₁	C₂×C₄	semidirect
74. C₂²×C₈×C₃₁	C₂×C₄	semidirect
103. C₃₁×C₄⋊C₈	C₂×C₄	semidirect
104. C₃₁×C₄⋊C₈	C₂×C₄	semidirect
107. C₃₁×C₂²⋊C₈	C₂×C₄	semidirect
108. C₃₁×C₂²⋊C₈	C₂×C₄	semidirect
112. C₃₁×#97	C₂×C₄	semidirect
113. C₃₁×#97	D₄	semidirect
120. C₄×C₈×C₃₁	C₂×C₄	semidirect
201. #20490	C₃₁	direct (N × H)
261. C₂×#405	C₂×C₃₁	semidirect
262. C₂×C₄×M₁₆	C₂×C₃₁	semidirect
264. #1771	C₂×C₃₁	semidirect
265. C₈×C₄⋊C₄	C₂×C₃₁	semidirect
266. #1768	C₂×C₃₁	semidirect
267. #1753	C₂×C₃₁	semidirect
268. #1753	C₂×C₃₁	semidirect
269. #1753	C₂×C₃₁	semidirect
270. C₈×C₂²⋊C₄	C₂×C₃₁	semidirect
271. #1750	C₂×C₃₁	semidirect
272. C₄×C₂²⋊C₈	C₂×C₃₁	semidirect
273. #1726	C₂×C₃₁	semidirect
308. #436	C₄×C₃₁	semidirect
309. #435	C₄×C₃₁	semidirect
363. C₂×M₁₆	C₂×C₄×C₃₁	semidirect
366. C₂²×C₈	C₂×C₄×C₃₁	semidirect
395. C₄⋊C₈	C₂×C₄×C₃₁	semidirect
396. C₄⋊C₈	C₂×C₄×C₃₁	semidirect
399. C₂²⋊C₈	C₂×C₄×C₃₁	semidirect
400. C₂²⋊C₈	C₂×C₄×C₃₁	semidirect
404. #97	C₂×C₄×C₃₁	semidirect
405. #97	D₄×C₃₁	semidirect
412. C₄×C₈	C₂×C₄×C₃₁	semidirect
418. C₃₁	#20490	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×#436	2
C₂×C₃₁×#435	2
C₂×C₃₁×#405	2
C₂×C₄×M₁₆×C₃₁	2
C₃₁×#2290	2
C₃₁×#1771	2
C₈×C₄⋊C₄×C₃₁	2
C₃₁×#1768	2
C₃₁×#1753	2
C₈×C₂²⋊C₄×C₃₁	2
C₃₁×#1750	2
C₄×C₃₁×C₂²⋊C₈	2
C₃₁×#1726	2
#20490	31

Subgroups — C31×#20490 #21477249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20490 #21477249