Subgroups — C₃₁×#22906 #21479665

Order 7936.

906 conjugacy classes of subgroups.

The subgroup lattice has 688 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×#431	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₃₁×#123	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1940	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1923	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1923	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1916	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1916	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1870	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#1851	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1851	C₂	1	1	yes	maximal, characteristic, complemented
12	C₈×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#1753	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#1753	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#1728	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	C₂×C₃₁×#123	C₄	1	1	yes	characteristic
20	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
22	C₃₁×#436	—	2	4	no	—
23	C₃₁×#431	—	2	4	no	—
24	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×C₄⋊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	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
32	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
33	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
34	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
35	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
39	C₃₁×#388	C₄	1	1	yes	characteristic
40	C₃₁×#388	C₄	1	1	yes	characteristic
41	C₃₁×#387	C₄	1	1	yes	characteristic
42	C₃₁×#387	C₄	1	1	yes	characteristic
43	C₃₁×#386	C₄	1	1	yes	characteristic
44	C₃₁×#385	C₄	1	1	yes	characteristic
45	C₃₁×#381	C₄	1	1	yes	characteristic
46	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
47	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
48	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
49	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
50	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
51	C₃₁×#336	C₂²	1	1	yes	characteristic
52	C₃₁×#336	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	C₃₁×#336	C₂²	1	1	yes	characteristic
60	C₃₁×#336	C₂²	1	1	yes	characteristic
61	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
62	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
63	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
64	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
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	—	2	2	no	—
70	C₂×M₁₆×C₃₁	—	2	2	no	—
71	C₂×M₁₆×C₃₁	—	2	2	no	—
72	C₂×M₁₆×C₃₁	—	4	1	no	—
73	C₂×M₁₆×C₃₁	—	4	1	no	—
74	C₂×M₁₆×C₃₁	—	4	1	no	—
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	C₃₁×#123	—	2	4	no	—
84	C₄×D₄×C₃₁	—	2	2	no	—
85	C₄×D₄×C₃₁	—	2	2	no	—
86	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
89	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
91	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
92	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
93	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
94	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
95	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
96	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
97	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
98	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
99	C₂×C₄²×C₃₁	—	2	2	no	—
100	C₂×C₄²×C₃₁	—	2	2	no	—
101	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
102	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
103	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
105	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
106	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
107	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
108	C₃₁×C₄⋊C₈	—	2	2	no	—
109	C₃₁×C₄⋊C₈	—	2	2	no	—
110	C₃₁×C₄⋊C₈	—	2	2	no	—
111	C₃₁×C₄⋊C₈	—	2	2	no	—
112	C₃₁×C₄⋊C₈	—	4	1	no	—
113	C₃₁×C₂²⋊C₈	—	2	2	no	—
114	C₃₁×C₂²⋊C₈	—	2	2	no	—
115	C₃₁×C₂²⋊C₈	—	2	2	no	—
116	C₃₁×C₂²⋊C₈	—	4	1	no	—
117	C₃₁×C₂²⋊C₈	D₄	1	4	yes	—
118	C₃₁×#97	—	4	1	no	—
119	C₄×C₈×C₃₁	—	2	2	no	—
120	C₄×C₈×C₃₁	—	2	2	no	—
121	C₄×C₈×C₃₁	—	2	2	no	—
122	C₄×C₈×C₃₁	—	2	2	no	—
123	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
124	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
125	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
126	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
127	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
128	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
129	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
130	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
131	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
132	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
133	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
134	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
135	C₂×D₄×C₃₁	—	2	2	no	—
136	C₂×D₄×C₃₁	—	2	2	no	—
137	C₂×D₄×C₃₁	—	4	1	no	—
138	C₂²×C₄×C₃₁	—	2	2	no	—
139	C₂²×C₄×C₃₁	—	2	2	no	—
140	C₂²×C₄×C₃₁	—	2	2	no	—
141	C₂²×C₄×C₃₁	—	2	1	no	—
142	C₂²×C₄×C₃₁	—	2	1	no	—
143	C₂²×C₄×C₃₁	—	2	1	no	—
144	C₂²×C₄×C₃₁	—	2	1	no	—
145	C₂²×C₄×C₃₁	—	2	2	no	—
146	C₂²×C₄×C₃₁	—	2	2	no	—
147	C₂²×C₄×C₃₁	—	2	1	no	—
148	C₂²×C₄×C₃₁	—	2	1	no	—
149	C₂²×C₄×C₃₁	—	2	1	no	—
150	C₂²×C₄×C₃₁	—	2	1	no	—
151	C₂²×C₄×C₃₁	—	2	1	no	—
152	C₂²×C₄×C₃₁	—	2	1	no	—
153	C₂²×C₄×C₃₁	—	4	1	no	—
154	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
155	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
156	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
158	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
159	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
160	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
161	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
162	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
163	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
164	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
165	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
166	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
167	C₂²×C₄×C₃₁	#41	1	2	yes	—
168	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
169	M₁₆×C₃₁	—	4	4	no	—
170	M₁₆×C₃₁	—	4	4	no	—
171	M₁₆×C₃₁	—	4	4	no	—
172	C₂×C₈×C₃₁	—	2	1	no	—
173	C₂×C₈×C₃₁	—	2	1	no	—
174	C₂×C₈×C₃₁	—	2	1	no	—
175	C₂×C₈×C₃₁	—	2	1	no	—
176	C₂×C₈×C₃₁	—	2	1	no	—
177	C₂×C₈×C₃₁	—	2	1	no	—
178	C₂×C₈×C₃₁	—	2	1	no	—
179	C₂×C₈×C₃₁	—	2	1	no	—
180	C₂×C₈×C₃₁	—	2	1	no	—
181	C₂×C₈×C₃₁	—	2	1	no	—
182	C₂×C₈×C₃₁	—	2	1	no	—
183	C₂×C₈×C₃₁	—	2	1	no	—
184	C₂×C₈×C₃₁	—	2	1	no	—
185	C₂×C₈×C₃₁	—	2	1	no	—
186	C₂×C₈×C₃₁	—	2	1	no	—
187	C₂×C₈×C₃₁	—	2	1	no	—
188	C₂×C₈×C₃₁	—	2	1	no	—
189	C₂×C₈×C₃₁	—	2	1	no	—
190	C₂×C₈×C₃₁	—	2	1	no	—
191	C₂×C₈×C₃₁	—	2	1	no	—
192	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
193	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
194	C₂×C₈×C₃₁	#41	1	2	yes	—
195	C₂×C₈×C₃₁	#41	1	2	yes	—
196	C₄⋊C₄×C₃₁	—	2	2	no	—
197	C₄⋊C₄×C₃₁	—	2	2	no	—
198	C₄⋊C₄×C₃₁	—	2	2	no	—
199	C₄⋊C₄×C₃₁	—	2	2	no	—
200	C₄⋊C₄×C₃₁	M₁₆	1	4	yes	complemented
201	C₂²⋊C₄×C₃₁	—	2	2	no	—
202	C₂²⋊C₄×C₃₁	—	2	2	no	—
203	C₂²⋊C₄×C₃₁	—	2	2	no	—
204	C₂²⋊C₄×C₃₁	—	2	2	no	—
205	C₂²⋊C₄×C₃₁	—	2	2	no	—
206	C₂²⋊C₄×C₃₁	—	2	2	no	—
207	C₂²⋊C₄×C₃₁	—	2	2	no	—
208	C₂²⋊C₄×C₃₁	—	4	1	no	—
209	C₄²×C₃₁	—	2	2	no	—
210	C₄²×C₃₁	—	2	2	no	—
211	C₄²×C₃₁	—	2	1	no	—
212	C₄²×C₃₁	—	2	1	no	—
213	C₄²×C₃₁	—	2	1	no	—
214	C₄²×C₃₁	—	2	1	no	—
215	C₄²×C₃₁	—	2	1	no	—
216	C₄²×C₃₁	—	2	1	no	—
217	C₄²×C₃₁	—	2	1	no	—
218	C₄²×C₃₁	—	2	1	no	—
219	C₄²×C₃₁	—	2	1	no	—
220	C₄²×C₃₁	—	2	1	no	—
221	#22906	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
222	C₂³×C₃₁	—	2	1	no	—
223	C₂³×C₃₁	—	2	1	no	—
224	C₂³×C₃₁	—	2	1	no	—
225	C₂³×C₃₁	—	2	1	no	—
226	C₂³×C₃₁	—	2	1	no	—
227	C₂³×C₃₁	—	2	1	no	—
228	C₂³×C₃₁	—	2	1	no	—
229	C₂³×C₃₁	—	2	1	no	—
230	C₂³×C₃₁	—	2	1	no	—
231	C₂³×C₃₁	—	2	1	no	—
232	C₂³×C₃₁	—	2	1	no	—
233	C₂³×C₃₁	—	2	1	no	—
234	C₂³×C₃₁	—	2	1	no	—
235	C₂³×C₃₁	C₄×D₄	1	2	yes	—
236	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
237	D₄×C₃₁	—	4	4	no	—
238	C₂×C₄×C₃₁	—	2	8	no	—
239	C₂×C₄×C₃₁	—	2	1	no	—
240	C₂×C₄×C₃₁	—	2	1	no	—
241	C₂×C₄×C₃₁	—	2	1	no	—
242	C₂×C₄×C₃₁	—	2	1	no	—
243	C₂×C₄×C₃₁	—	2	1	no	—
244	C₂×C₄×C₃₁	—	2	1	no	—
245	C₂×C₄×C₃₁	—	2	1	no	—
246	C₂×C₄×C₃₁	—	2	1	no	—
247	C₂×C₄×C₃₁	—	2	1	no	—
248	C₂×C₄×C₃₁	—	2	1	no	—
249	C₂×C₄×C₃₁	—	2	1	no	—
250	C₂×C₄×C₃₁	—	2	1	no	—
251	C₂×C₄×C₃₁	—	2	2	no	—
252	C₂×C₄×C₃₁	—	2	2	no	—
253	C₂×C₄×C₃₁	—	2	2	no	—
254	C₂×C₄×C₃₁	—	2	2	no	—
255	C₂×C₄×C₃₁	—	2	1	no	—
256	C₂×C₄×C₃₁	—	2	1	no	—
257	C₂×C₄×C₃₁	—	2	1	no	—
258	C₂×C₄×C₃₁	—	2	1	no	—
259	C₂×C₄×C₃₁	—	2	1	no	—
260	C₂×C₄×C₃₁	—	2	1	no	—
261	C₂×C₄×C₃₁	—	2	1	no	—
262	C₂×C₄×C₃₁	—	2	1	no	—
263	C₂×C₄×C₃₁	—	2	1	no	—
264	C₂×C₄×C₃₁	—	2	1	no	—
265	C₂×C₄×C₃₁	—	2	1	no	—
266	C₂×C₄×C₃₁	—	2	1	no	—
267	C₂×C₄×C₃₁	—	2	1	no	—
268	C₂×C₄×C₃₁	—	2	1	no	—
269	C₂×C₄×C₃₁	—	2	1	no	—
270	C₂×C₄×C₃₁	—	2	1	no	—
271	C₂×C₄×C₃₁	—	2	1	no	—
272	C₂×C₄×C₃₁	—	2	1	no	—
273	C₂×C₄×C₃₁	—	4	4	no	—
274	C₂×C₄×C₃₁	—	4	1	no	—
275	C₂×C₄×C₃₁	—	4	1	no	—
276	C₂×C₄×C₃₁	—	4	1	no	—
277	C₂×C₄×C₃₁	—	4	1	no	—
278	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
279	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
280	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
281	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
282	C₂×C₄×C₃₁	#131	1	2	yes	—
283	C₂×C₄×C₃₁	#131	1	2	yes	—
284	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
285	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
286	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
287	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
288	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
289	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
290	C₈×C₃₁	—	2	2	no	—
291	C₈×C₃₁	—	2	2	no	—
292	C₈×C₃₁	—	2	2	no	—
293	C₈×C₃₁	—	2	2	no	—
294	C₈×C₃₁	—	4	1	no	—
295	C₈×C₃₁	—	4	1	no	—
296	C₈×C₃₁	—	4	1	no	—
297	C₈×C₃₁	—	4	1	no	—
298	C₂×#436	C₂×C₃₁	1	1	yes	characteristic
299	C₂×#431	C₂×C₃₁	1	1	yes	characteristic, complemented
300	C₄×#123	C₂×C₃₁	1	1	yes	characteristic
301	#1940	C₂×C₃₁	1	1	yes	characteristic, complemented
302	#1923	C₂×C₃₁	1	1	yes	characteristic, complemented
303	#1923	C₂×C₃₁	1	1	yes	characteristic, complemented
304	#1916	C₂×C₃₁	1	1	yes	characteristic, complemented
305	#1916	C₂×C₃₁	1	1	yes	characteristic, complemented
306	#1870	C₂×C₃₁	1	1	yes	characteristic
307	#1851	C₂×C₃₁	1	1	yes	characteristic, complemented
308	#1851	C₂×C₃₁	1	1	yes	characteristic, complemented
309	C₈×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
310	#1753	C₂×C₃₁	1	1	yes	characteristic, complemented
311	#1753	C₂×C₃₁	1	1	yes	characteristic, complemented
312	#1728	C₂×C₃₁	1	1	yes	characteristic, complemented
313	C₂²×C₃₁	—	2	2	no	—
314	C₂²×C₃₁	—	2	2	no	—
315	C₂²×C₃₁	—	2	2	no	—
316	C₂²×C₃₁	—	4	1	no	—
317	C₂²×C₃₁	—	4	1	no	—
318	C₂²×C₃₁	—	4	1	no	—
319	C₂²×C₃₁	—	4	1	no	—
320	C₂²×C₃₁	—	4	1	no	—
321	C₂²×C₃₁	—	4	1	no	—
322	C₂²×C₃₁	—	4	1	no	—
323	C₂²×C₃₁	—	4	1	no	—
324	C₂²×C₃₁	—	4	1	no	—
325	C₂²×C₃₁	—	4	1	no	—
326	C₂²×C₃₁	—	4	1	no	—
327	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
328	C₂²×C₃₁	C₄×#41	1	1	yes	characteristic
329	C₂²×C₃₁	#520	1	1	yes	characteristic
330	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
331	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
332	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
333	C₂²×C₃₁	#568	1	1	yes	characteristic
334	C₄×C₃₁	—	2	2	no	—
335	C₄×C₃₁	—	2	2	no	—
336	C₄×C₃₁	—	2	2	no	—
337	C₄×C₃₁	—	2	2	no	—
338	C₄×C₃₁	—	2	2	no	—
339	C₄×C₃₁	—	2	2	no	—
340	C₄×C₃₁	—	4	1	no	—
341	C₄×C₃₁	—	4	1	no	—
342	C₄×C₃₁	—	4	1	no	—
343	C₄×C₃₁	—	4	1	no	—
344	C₄×C₃₁	—	4	1	no	—
345	C₄×C₃₁	—	4	1	no	—
346	C₄×C₃₁	#546	1	1	yes	characteristic
347	C₄×C₃₁	#547	1	1	yes	characteristic
348	C₄×C₃₁	#551	1	1	yes	characteristic
349	C₄×C₃₁	#552	1	1	yes	characteristic
350	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
351	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
352	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
353	C₂×#123	C₄×C₃₁	1	1	yes	characteristic
354	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
355	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
356	#436	—	2	4	no	—
357	#431	—	2	4	no	—
358	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
359	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
360	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
361	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
362	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
363	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
364	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
365	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
366	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
367	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
368	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
369	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
370	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
371	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
372	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
373	#388	C₄×C₃₁	1	1	yes	characteristic
374	#388	C₄×C₃₁	1	1	yes	characteristic
375	#387	C₄×C₃₁	1	1	yes	characteristic
376	#387	C₄×C₃₁	1	1	yes	characteristic
377	#386	C₄×C₃₁	1	1	yes	characteristic
378	#385	C₄×C₃₁	1	1	yes	characteristic
379	#381	C₄×C₃₁	1	1	yes	characteristic
380	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
381	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
382	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
383	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
384	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
385	#336	C₂²×C₃₁	1	1	yes	characteristic
386	#336	C₂²×C₃₁	1	1	yes	characteristic
387	#336	C₂²×C₃₁	1	1	yes	characteristic
388	#336	C₂²×C₃₁	1	1	yes	characteristic
389	#336	C₂²×C₃₁	1	1	yes	characteristic
390	#336	C₂²×C₃₁	1	1	yes	characteristic
391	#336	C₂²×C₃₁	1	1	yes	characteristic
392	#336	C₂²×C₃₁	1	1	yes	characteristic
393	#336	C₂²×C₃₁	1	1	yes	characteristic
394	#336	C₂²×C₃₁	1	1	yes	characteristic
395	C₂×C₃₁	—	4	2	no	—
396	C₂×C₃₁	—	8	1	no	—
397	C₂×C₃₁	#2359	1	1	yes	characteristic
398	C₂×C₃₁	#2929	1	1	yes	characteristic
399	C₂×C₃₁	D₄×M₁₆	1	1	yes	characteristic
400	C₂×C₃₁	#2965	1	1	yes	characteristic
401	C₂×C₃₁	#2969	1	1	yes	characteristic
402	C₂×C₃₁	#2991	1	1	yes	characteristic
403	C₂×C₃₁	#2993	1	1	yes	characteristic
404	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
405	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
406	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
407	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
408	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
409	C₂×M₁₆	—	2	2	no	—
410	C₂×M₁₆	—	2	2	no	—
411	C₂×M₁₆	—	2	2	no	—
412	C₂×M₁₆	—	2	2	no	—
413	C₂×M₁₆	—	2	2	no	—
414	C₂×M₁₆	—	2	2	no	—
415	C₂×M₁₆	—	4	1	no	—
416	C₂×M₁₆	—	4	1	no	—
417	C₂×M₁₆	—	4	1	no	—
418	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
419	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
420	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
421	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
422	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
423	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
424	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
425	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
426	#123	—	2	4	no	—
427	C₄×D₄	—	2	2	no	—
428	C₄×D₄	—	2	2	no	—
429	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
430	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
431	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
432	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
433	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
434	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
435	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
436	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
437	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
438	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
439	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
440	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
441	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
442	C₂×C₄²	—	2	2	no	—
443	C₂×C₄²	—	2	2	no	—
444	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
445	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
446	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
447	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
448	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
449	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
450	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
451	C₄⋊C₈	—	2	2	no	—
452	C₄⋊C₈	—	2	2	no	—
453	C₄⋊C₈	—	2	2	no	—
454	C₄⋊C₈	—	2	2	no	—
455	C₄⋊C₈	—	4	1	no	—
456	C₂²⋊C₈	—	2	2	no	—
457	C₂²⋊C₈	—	2	2	no	—
458	C₂²⋊C₈	—	2	2	no	—
459	C₂²⋊C₈	—	4	1	no	—
460	C₂²⋊C₈	D₄×C₃₁	1	4	yes	—
461	#97	—	4	1	no	—
462	C₄×C₈	—	2	2	no	—
463	C₄×C₈	—	2	2	no	—
464	C₄×C₈	—	2	2	no	—
465	C₄×C₈	—	2	2	no	—
466	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
467	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
468	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
469	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
470	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
471	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
472	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
473	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
474	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
475	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
476	C₃₁	#22906	1	1	yes	characteristic, direct product factor, Sylow
477	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
478	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
479	C₂×D₄	—	2	2	no	—
480	C₂×D₄	—	2	2	no	—
481	C₂×D₄	—	4	1	no	—
482	C₂²×C₄	—	2	2	no	—
483	C₂²×C₄	—	2	2	no	—
484	C₂²×C₄	—	2	2	no	—
485	C₂²×C₄	—	2	1	no	—
486	C₂²×C₄	—	2	1	no	—
487	C₂²×C₄	—	2	1	no	—
488	C₂²×C₄	—	2	1	no	—
489	C₂²×C₄	—	2	2	no	—
490	C₂²×C₄	—	2	2	no	—
491	C₂²×C₄	—	2	1	no	—
492	C₂²×C₄	—	2	1	no	—
493	C₂²×C₄	—	2	1	no	—
494	C₂²×C₄	—	2	1	no	—
495	C₂²×C₄	—	2	1	no	—
496	C₂²×C₄	—	2	1	no	—
497	C₂²×C₄	—	4	1	no	—
498	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
499	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
500	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
501	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
502	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
503	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
504	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
505	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
506	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
507	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
508	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
509	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
510	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
511	C₂²×C₄	#41×C₃₁	1	2	yes	—
512	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
513	M₁₆	—	4	4	no	—
514	M₁₆	—	4	4	no	—
515	M₁₆	—	4	4	no	—
516	C₂×C₈	—	2	1	no	—
517	C₂×C₈	—	2	1	no	—
518	C₂×C₈	—	2	1	no	—
519	C₂×C₈	—	2	1	no	—
520	C₂×C₈	—	2	1	no	—
521	C₂×C₈	—	2	1	no	—
522	C₂×C₈	—	2	1	no	—
523	C₂×C₈	—	2	1	no	—
524	C₂×C₈	—	2	1	no	—
525	C₂×C₈	—	2	1	no	—
526	C₂×C₈	—	2	1	no	—
527	C₂×C₈	—	2	1	no	—
528	C₂×C₈	—	2	1	no	—
529	C₂×C₈	—	2	1	no	—
530	C₂×C₈	—	2	1	no	—
531	C₂×C₈	—	2	1	no	—
532	C₂×C₈	—	2	1	no	—
533	C₂×C₈	—	2	1	no	—
534	C₂×C₈	—	2	1	no	—
535	C₂×C₈	—	2	1	no	—
536	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
537	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
538	C₂×C₈	#41×C₃₁	1	2	yes	—
539	C₂×C₈	#41×C₃₁	1	2	yes	—
540	C₄⋊C₄	—	2	2	no	—
541	C₄⋊C₄	—	2	2	no	—
542	C₄⋊C₄	—	2	2	no	—
543	C₄⋊C₄	—	2	2	no	—
544	C₄⋊C₄	M₁₆×C₃₁	1	4	yes	complemented
545	C₂²⋊C₄	—	2	2	no	—
546	C₂²⋊C₄	—	2	2	no	—
547	C₂²⋊C₄	—	2	2	no	—
548	C₂²⋊C₄	—	2	2	no	—
549	C₂²⋊C₄	—	2	2	no	—
550	C₂²⋊C₄	—	2	2	no	—
551	C₂²⋊C₄	—	2	2	no	—
552	C₂²⋊C₄	—	4	1	no	—
553	C₄²	—	2	2	no	—
554	C₄²	—	2	2	no	—
555	C₄²	—	2	1	no	—
556	C₄²	—	2	1	no	—
557	C₄²	—	2	1	no	—
558	C₄²	—	2	1	no	—
559	C₄²	—	2	1	no	—
560	C₄²	—	2	1	no	—
561	C₄²	—	2	1	no	—
562	C₄²	—	2	1	no	—
563	C₄²	—	2	1	no	—
564	C₄²	—	2	1	no	—
565	C₂³	—	2	1	no	—
566	C₂³	—	2	1	no	—
567	C₂³	—	2	1	no	—
568	C₂³	—	2	1	no	—
569	C₂³	—	2	1	no	—
570	C₂³	—	2	1	no	—
571	C₂³	—	2	1	no	—
572	C₂³	—	2	1	no	—
573	C₂³	—	2	1	no	—
574	C₂³	—	2	1	no	—
575	C₂³	—	2	1	no	—
576	C₂³	—	2	1	no	—
577	C₂³	—	2	1	no	—
578	C₂³	C₄×D₄×C₃₁	1	2	yes	—
579	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
580	D₄	—	4	4	no	—
581	C₂×C₄	—	2	8	no	—
582	C₂×C₄	—	2	1	no	—
583	C₂×C₄	—	2	1	no	—
584	C₂×C₄	—	2	1	no	—
585	C₂×C₄	—	2	1	no	—
586	C₂×C₄	—	2	1	no	—
587	C₂×C₄	—	2	1	no	—
588	C₂×C₄	—	2	1	no	—
589	C₂×C₄	—	2	1	no	—
590	C₂×C₄	—	2	1	no	—
591	C₂×C₄	—	2	1	no	—
592	C₂×C₄	—	2	1	no	—
593	C₂×C₄	—	2	1	no	—
594	C₂×C₄	—	2	2	no	—
595	C₂×C₄	—	2	2	no	—
596	C₂×C₄	—	2	2	no	—
597	C₂×C₄	—	2	2	no	—
598	C₂×C₄	—	2	1	no	—
599	C₂×C₄	—	2	1	no	—
600	C₂×C₄	—	2	1	no	—
601	C₂×C₄	—	2	1	no	—
602	C₂×C₄	—	2	1	no	—
603	C₂×C₄	—	2	1	no	—
604	C₂×C₄	—	2	1	no	—
605	C₂×C₄	—	2	1	no	—
606	C₂×C₄	—	2	1	no	—
607	C₂×C₄	—	2	1	no	—
608	C₂×C₄	—	2	1	no	—
609	C₂×C₄	—	2	1	no	—
610	C₂×C₄	—	2	1	no	—
611	C₂×C₄	—	2	1	no	—
612	C₂×C₄	—	2	1	no	—
613	C₂×C₄	—	2	1	no	—
614	C₂×C₄	—	2	1	no	—
615	C₂×C₄	—	2	1	no	—
616	C₂×C₄	—	4	4	no	—
617	C₂×C₄	—	4	1	no	—
618	C₂×C₄	—	4	1	no	—
619	C₂×C₄	—	4	1	no	—
620	C₂×C₄	—	4	1	no	—
621	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
622	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
623	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
624	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
625	C₂×C₄	C₃₁×#131	1	2	yes	—
626	C₂×C₄	C₃₁×#131	1	2	yes	—
627	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
628	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
629	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
630	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
631	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
632	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
633	C₈	—	2	2	no	—
634	C₈	—	2	2	no	—
635	C₈	—	2	2	no	—
636	C₈	—	2	2	no	—
637	C₈	—	4	1	no	—
638	C₈	—	4	1	no	—
639	C₈	—	4	1	no	—
640	C₈	—	4	1	no	—
641	C₂²	—	2	2	no	—
642	C₂²	—	2	2	no	—
643	C₂²	—	2	2	no	—
644	C₂²	—	4	1	no	—
645	C₂²	—	4	1	no	—
646	C₂²	—	4	1	no	—
647	C₂²	—	4	1	no	—
648	C₂²	—	4	1	no	—
649	C₂²	—	4	1	no	—
650	C₂²	—	4	1	no	—
651	C₂²	—	4	1	no	—
652	C₂²	—	4	1	no	—
653	C₂²	—	4	1	no	—
654	C₂²	—	4	1	no	—
655	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
656	C₂²	C₄×#41×C₃₁	1	1	yes	characteristic
657	C₂²	C₃₁×#520	1	1	yes	characteristic
658	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
659	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
660	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
661	C₂²	C₃₁×#568	1	1	yes	characteristic
662	C₄	—	2	2	no	—
663	C₄	—	2	2	no	—
664	C₄	—	2	2	no	—
665	C₄	—	2	2	no	—
666	C₄	—	2	2	no	—
667	C₄	—	2	2	no	—
668	C₄	—	4	1	no	—
669	C₄	—	4	1	no	—
670	C₄	—	4	1	no	—
671	C₄	—	4	1	no	—
672	C₄	—	4	1	no	—
673	C₄	—	4	1	no	—
674	C₄	C₃₁×#546	1	1	yes	characteristic
675	C₄	C₃₁×#547	1	1	yes	characteristic
676	C₄	C₃₁×#551	1	1	yes	characteristic
677	C₄	C₃₁×#552	1	1	yes	characteristic
678	C₂	—	4	2	no	—
679	C₂	—	8	1	no	—
680	C₂	C₃₁×#2359	1	1	yes	characteristic
681	C₂	C₃₁×#2929	1	1	yes	characteristic
682	C₂	D₄×M₁₆×C₃₁	1	1	yes	characteristic
683	C₂	C₃₁×#2965	1	1	yes	characteristic
684	C₂	C₃₁×#2969	1	1	yes	characteristic
685	C₂	C₃₁×#2991	1	1	yes	characteristic
686	C₂	C₃₁×#2993	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#431	C₂	semidirect
4. C₃₁×#1940	C₂	semidirect
5. C₃₁×#1923	C₂	semidirect
6. C₃₁×#1923	C₂	semidirect
7. C₃₁×#1916	C₂	semidirect
8. C₃₁×#1916	C₂	semidirect
10. C₃₁×#1851	C₂	semidirect
11. C₃₁×#1851	C₂	semidirect
12. C₈×C₄⋊C₄×C₃₁	C₂	semidirect
13. C₃₁×#1753	C₂	semidirect
14. C₃₁×#1753	C₂	semidirect
15. C₃₁×#1728	C₂	semidirect
200. C₄⋊C₄×C₃₁	M₁₆	semidirect
221. #22906	C₃₁	direct (N × H)
299. C₂×#431	C₂×C₃₁	semidirect
301. #1940	C₂×C₃₁	semidirect
302. #1923	C₂×C₃₁	semidirect
303. #1923	C₂×C₃₁	semidirect
304. #1916	C₂×C₃₁	semidirect
305. #1916	C₂×C₃₁	semidirect
307. #1851	C₂×C₃₁	semidirect
308. #1851	C₂×C₃₁	semidirect
309. C₈×C₄⋊C₄	C₂×C₃₁	semidirect
310. #1753	C₂×C₃₁	semidirect
311. #1753	C₂×C₃₁	semidirect
312. #1728	C₂×C₃₁	semidirect
476. C₃₁	#22906	direct (N × H)
544. C₄⋊C₄	M₁₆×C₃₁	semidirect

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×#431	2
C₄×C₃₁×#123	2
C₃₁×#1940	2
C₃₁×#1923	2
C₃₁×#1916	2
C₃₁×#1870	2
C₃₁×#1851	2
C₈×C₄⋊C₄×C₃₁	2
C₃₁×#1753	2
C₃₁×#1728	2
#22906	31

Subgroups — C31×#22906 #21479665

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22906 #21479665