Subgroups — C₃₁×#21435 #21478194

Order 7936.

790 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×#117	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1924	C₂	1	2	yes	maximal
3	C₃₁×#1923	C₂	1	2	yes	maximal
4	C₃₁×#1917	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1916	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1767	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1767	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1749	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#1749	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#1745	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₃₁×#97	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#117	C₄	1	1	yes	characteristic
13	C₂×C₃₁×#117	C₄	1	1	yes	characteristic
14	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
18	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
19	C₂×C₃₁×#97	C₂²	1	2	yes	—
20	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
29	C₄×C₂²⋊C₄×C₃₁	C₄	1	1	yes	characteristic
30	C₄×C₂²⋊C₄×C₃₁	C₄	1	1	yes	characteristic
31	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
33	C₃₁×#376	C₄	1	1	yes	characteristic
34	C₃₁×#376	C₄	1	1	yes	characteristic
35	C₄³×C₃₁	C₂²	1	1	yes	characteristic
36	C₄³×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#336	C₂²	1	2	yes	—
38	C₃₁×#336	C₂²	1	2	yes	—
39	C₃₁×#336	C₂²	1	2	yes	—
40	C₃₁×#336	C₂²	1	2	yes	—
41	C₃₁×#336	C₂²	1	2	yes	—
42	C₃₁×#336	C₂²	1	2	yes	—
43	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
47	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
48	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
49	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
50	C₃₁×#117	—	2	4	no	—
51	C₃₁×#117	—	2	4	no	—
52	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
56	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
57	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
58	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
59	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₃₁	—	2	2	no	—
61	C₂×C₄²×C₃₁	—	2	2	no	—
62	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
78	C₃₁×C₄⋊C₈	—	2	4	no	—
79	C₃₁×C₄⋊C₈	—	2	4	no	—
80	C₃₁×C₂²⋊C₈	—	2	4	no	—
81	C₃₁×C₂²⋊C₈	—	2	4	no	—
82	C₃₁×#97	—	2	4	no	—
83	C₃₁×#97	—	2	2	no	—
84	C₃₁×#97	—	2	2	no	—
85	C₃₁×#97	—	2	2	no	—
86	C₃₁×#97	—	2	2	no	—
87	C₄×C₈×C₃₁	—	2	4	no	—
88	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
89	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
90	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
91	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
92	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
93	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
94	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
95	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
96	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₃₁	—	2	2	no	—
98	C₂²×C₄×C₃₁	—	2	2	no	—
99	C₂²×C₄×C₃₁	—	2	2	no	—
100	C₂²×C₄×C₃₁	—	2	2	no	—
101	C₂²×C₄×C₃₁	—	2	2	no	—
102	C₂²×C₄×C₃₁	—	2	2	no	—
103	C₂²×C₄×C₃₁	—	2	1	no	—
104	C₂²×C₄×C₃₁	—	2	1	no	—
105	C₂²×C₄×C₃₁	—	4	1	no	—
106	C₂²×C₄×C₃₁	—	4	1	no	—
107	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
109	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
110	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
115	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
117	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
120	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
121	C₂×C₈×C₃₁	—	2	2	no	—
122	C₂×C₈×C₃₁	—	2	2	no	—
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	—	2	2	no	—
129	C₂×C₈×C₃₁	—	2	2	no	—
130	C₂×C₈×C₃₁	—	2	2	no	—
131	C₂×C₈×C₃₁	#41	1	4	yes	—
132	C₂×C₈×C₃₁	#41	1	4	yes	—
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	M₁₆	1	4	yes	—
137	C₂²⋊C₄×C₃₁	—	2	2	no	—
138	C₂²⋊C₄×C₃₁	—	2	2	no	—
139	C₂²⋊C₄×C₃₁	—	2	2	no	—
140	C₂²⋊C₄×C₃₁	M₁₆	1	4	yes	—
141	C₄²×C₃₁	—	2	2	no	—
142	C₄²×C₃₁	—	2	2	no	—
143	C₄²×C₃₁	—	2	2	no	—
144	C₄²×C₃₁	—	2	2	no	—
145	C₄²×C₃₁	—	2	2	no	—
146	C₄²×C₃₁	—	2	2	no	—
147	C₄²×C₃₁	—	2	2	no	—
148	C₄²×C₃₁	—	2	2	no	—
149	C₄²×C₃₁	—	2	2	no	—
150	C₄²×C₃₁	—	2	2	no	—
151	C₄²×C₃₁	—	2	1	no	—
152	C₄²×C₃₁	—	2	1	no	—
153	C₄²×C₃₁	—	4	1	no	—
154	C₄²×C₃₁	—	4	1	no	—
155	C₄²×C₃₁	—	4	1	no	—
156	C₄²×C₃₁	—	4	1	no	—
157	C₄²×C₃₁	#41	1	2	yes	—
158	C₄²×C₃₁	#41	1	2	yes	—
159	#21435	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₂³×C₃₁	—	2	1	no	—
161	C₂³×C₃₁	—	2	1	no	—
162	C₂³×C₃₁	—	2	1	no	—
163	C₂³×C₃₁	—	2	1	no	—
164	C₂³×C₃₁	—	2	1	no	—
165	C₂³×C₃₁	—	2	1	no	—
166	C₂³×C₃₁	C₂×M₁₆	1	2	yes	—
167	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
168	C₂×C₄×C₃₁	—	2	8	no	—
169	C₂×C₄×C₃₁	—	2	1	no	—
170	C₂×C₄×C₃₁	—	2	1	no	—
171	C₂×C₄×C₃₁	—	2	1	no	—
172	C₂×C₄×C₃₁	—	2	1	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	1	no	—
180	C₂×C₄×C₃₁	—	2	1	no	—
181	C₂×C₄×C₃₁	—	2	1	no	—
182	C₂×C₄×C₃₁	—	2	1	no	—
183	C₂×C₄×C₃₁	—	2	1	no	—
184	C₂×C₄×C₃₁	—	2	1	no	—
185	C₂×C₄×C₃₁	—	2	1	no	—
186	C₂×C₄×C₃₁	—	2	1	no	—
187	C₂×C₄×C₃₁	—	2	1	no	—
188	C₂×C₄×C₃₁	—	2	1	no	—
189	C₂×C₄×C₃₁	—	2	1	no	—
190	C₂×C₄×C₃₁	—	2	1	no	—
191	C₂×C₄×C₃₁	—	2	1	no	—
192	C₂×C₄×C₃₁	—	2	1	no	—
193	C₂×C₄×C₃₁	—	2	1	no	—
194	C₂×C₄×C₃₁	—	2	1	no	—
195	C₂×C₄×C₃₁	—	2	1	no	—
196	C₂×C₄×C₃₁	—	2	1	no	—
197	C₂×C₄×C₃₁	—	2	1	no	—
198	C₂×C₄×C₃₁	—	2	1	no	—
199	C₂×C₄×C₃₁	—	2	1	no	—
200	C₂×C₄×C₃₁	—	2	1	no	—
201	C₂×C₄×C₃₁	—	4	4	no	—
202	C₂×C₄×C₃₁	—	4	2	no	—
203	C₂×C₄×C₃₁	—	4	2	no	—
204	C₂×C₄×C₃₁	#117	1	2	yes	—
205	C₂×C₄×C₃₁	#117	1	2	yes	—
206	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
207	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
208	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
209	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
210	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
211	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
212	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
213	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
214	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
215	C₈×C₃₁	—	2	4	no	—
216	C₈×C₃₁	—	2	4	no	—
217	C₈×C₃₁	—	4	2	no	—
218	C₈×C₃₁	—	4	2	no	—
219	C₄×#117	C₂×C₃₁	1	1	yes	characteristic
220	#1924	C₂×C₃₁	1	2	yes	—
221	#1923	C₂×C₃₁	1	2	yes	—
222	#1917	C₂×C₃₁	1	2	yes	complemented
223	#1916	C₂×C₃₁	1	2	yes	complemented
224	#1767	C₂×C₃₁	1	1	yes	characteristic, complemented
225	#1767	C₂×C₃₁	1	1	yes	characteristic, complemented
226	#1749	C₂×C₃₁	1	1	yes	characteristic
227	#1749	C₂×C₃₁	1	1	yes	characteristic
228	#1745	C₂×C₃₁	1	1	yes	characteristic, complemented
229	C₄×#97	C₂×C₃₁	1	1	yes	characteristic, complemented
230	C₂²×C₃₁	—	2	2	no	—
231	C₂²×C₃₁	—	2	2	no	—
232	C₂²×C₃₁	—	2	2	no	—
233	C₂²×C₃₁	—	4	1	no	—
234	C₂²×C₃₁	—	4	1	no	—
235	C₂²×C₃₁	—	4	1	no	—
236	C₂²×C₃₁	—	4	1	no	—
237	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
238	C₂²×C₃₁	C₄×#41	1	1	yes	characteristic
239	C₂²×C₃₁	C₄×#41	1	1	yes	characteristic
240	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
241	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
242	C₂²×C₃₁	#568	1	1	yes	characteristic
243	C₂²×C₃₁	#568	1	1	yes	characteristic
244	C₄×C₃₁	—	2	2	no	—
245	C₄×C₃₁	—	2	2	no	—
246	C₄×C₃₁	—	2	2	no	—
247	C₄×C₃₁	—	2	2	no	—
248	C₄×C₃₁	—	2	2	no	—
249	C₄×C₃₁	—	2	2	no	—
250	C₄×C₃₁	—	2	2	no	—
251	C₄×C₃₁	—	4	1	no	—
252	C₄×C₃₁	—	4	1	no	—
253	C₄×C₃₁	—	4	1	no	—
254	C₄×C₃₁	—	4	1	no	—
255	C₄×C₃₁	—	4	1	no	—
256	C₄×C₃₁	—	4	1	no	—
257	C₄×C₃₁	#529	1	2	yes	—
258	C₄×C₃₁	#529	1	2	yes	—
259	C₂×#117	C₄×C₃₁	1	1	yes	characteristic
260	C₂×#117	C₄×C₃₁	1	1	yes	characteristic
261	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
262	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
263	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
264	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
265	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
266	C₂×#97	C₂²×C₃₁	1	2	yes	—
267	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
268	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
269	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
270	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
271	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
272	C₄×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
273	C₄×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
274	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
275	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
276	C₄×C₂²⋊C₄	C₄×C₃₁	1	1	yes	characteristic
277	C₄×C₂²⋊C₄	C₄×C₃₁	1	1	yes	characteristic
278	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
279	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
280	#376	C₄×C₃₁	1	1	yes	characteristic
281	#376	C₄×C₃₁	1	1	yes	characteristic
282	C₄³	C₂²×C₃₁	1	1	yes	characteristic
283	C₄³	C₂²×C₃₁	1	1	yes	characteristic
284	#336	C₂²×C₃₁	1	2	yes	—
285	#336	C₂²×C₃₁	1	2	yes	—
286	#336	C₂²×C₃₁	1	2	yes	—
287	#336	C₂²×C₃₁	1	2	yes	—
288	#336	C₂²×C₃₁	1	2	yes	—
289	#336	C₂²×C₃₁	1	2	yes	—
290	C₂×C₃₁	—	4	2	no	—
291	C₂×C₃₁	#2298	1	1	yes	characteristic
292	C₂×C₃₁	#2920	1	1	yes	characteristic
293	C₂×C₃₁	#2921	1	1	yes	characteristic
294	C₂×C₃₁	#2970	1	1	yes	characteristic
295	C₂×C₃₁	#2970	1	1	yes	characteristic
296	C₂×C₃₁	#2970	1	1	yes	characteristic
297	C₂×C₃₁	#2971	1	1	yes	characteristic
298	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
299	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
300	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
301	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
302	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
303	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
304	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
305	#117	—	2	4	no	—
306	#117	—	2	4	no	—
307	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
308	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
309	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
310	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
311	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
312	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
313	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
314	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
315	C₂×C₄²	—	2	2	no	—
316	C₂×C₄²	—	2	2	no	—
317	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
318	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
319	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
320	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
321	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
322	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
323	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
324	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
325	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
326	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
327	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
328	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
329	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
330	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
331	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
332	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
333	C₄⋊C₈	—	2	4	no	—
334	C₄⋊C₈	—	2	4	no	—
335	C₂²⋊C₈	—	2	4	no	—
336	C₂²⋊C₈	—	2	4	no	—
337	#97	—	2	4	no	—
338	#97	—	2	2	no	—
339	#97	—	2	2	no	—
340	#97	—	2	2	no	—
341	#97	—	2	2	no	—
342	C₄×C₈	—	2	4	no	—
343	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
344	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
345	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
346	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
347	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
348	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
349	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
350	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
351	C₃₁	#21435	1	1	yes	characteristic, direct product factor, Sylow
352	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
353	C₂²×C₄	—	2	2	no	—
354	C₂²×C₄	—	2	2	no	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	—	2	2	no	—
357	C₂²×C₄	—	2	2	no	—
358	C₂²×C₄	—	2	2	no	—
359	C₂²×C₄	—	2	1	no	—
360	C₂²×C₄	—	2	1	no	—
361	C₂²×C₄	—	4	1	no	—
362	C₂²×C₄	—	4	1	no	—
363	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
364	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
365	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
366	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
367	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
368	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
369	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
370	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
371	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
372	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
373	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
374	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
375	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
376	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	—	2	2	no	—
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	—	2	2	no	—
385	C₂×C₈	—	2	2	no	—
386	C₂×C₈	—	2	2	no	—
387	C₂×C₈	#41×C₃₁	1	4	yes	—
388	C₂×C₈	#41×C₃₁	1	4	yes	—
389	C₄⋊C₄	—	2	2	no	—
390	C₄⋊C₄	—	2	2	no	—
391	C₄⋊C₄	—	2	2	no	—
392	C₄⋊C₄	M₁₆×C₃₁	1	4	yes	—
393	C₂²⋊C₄	—	2	2	no	—
394	C₂²⋊C₄	—	2	2	no	—
395	C₂²⋊C₄	—	2	2	no	—
396	C₂²⋊C₄	M₁₆×C₃₁	1	4	yes	—
397	C₄²	—	2	2	no	—
398	C₄²	—	2	2	no	—
399	C₄²	—	2	2	no	—
400	C₄²	—	2	2	no	—
401	C₄²	—	2	2	no	—
402	C₄²	—	2	2	no	—
403	C₄²	—	2	2	no	—
404	C₄²	—	2	2	no	—
405	C₄²	—	2	2	no	—
406	C₄²	—	2	2	no	—
407	C₄²	—	2	1	no	—
408	C₄²	—	2	1	no	—
409	C₄²	—	4	1	no	—
410	C₄²	—	4	1	no	—
411	C₄²	—	4	1	no	—
412	C₄²	—	4	1	no	—
413	C₄²	#41×C₃₁	1	2	yes	—
414	C₄²	#41×C₃₁	1	2	yes	—
415	C₂³	—	2	1	no	—
416	C₂³	—	2	1	no	—
417	C₂³	—	2	1	no	—
418	C₂³	—	2	1	no	—
419	C₂³	—	2	1	no	—
420	C₂³	—	2	1	no	—
421	C₂³	C₂×M₁₆×C₃₁	1	2	yes	—
422	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
423	C₂×C₄	—	2	8	no	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	1	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	1	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	2	1	no	—
437	C₂×C₄	—	2	1	no	—
438	C₂×C₄	—	2	1	no	—
439	C₂×C₄	—	2	1	no	—
440	C₂×C₄	—	2	1	no	—
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	2	1	no	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	1	no	—
445	C₂×C₄	—	2	1	no	—
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	1	no	—
454	C₂×C₄	—	2	1	no	—
455	C₂×C₄	—	2	1	no	—
456	C₂×C₄	—	4	4	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	—	4	2	no	—
459	C₂×C₄	C₃₁×#117	1	2	yes	—
460	C₂×C₄	C₃₁×#117	1	2	yes	—
461	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
462	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
463	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
464	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
465	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
466	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
467	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
468	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
469	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
470	C₈	—	2	4	no	—
471	C₈	—	2	4	no	—
472	C₈	—	4	2	no	—
473	C₈	—	4	2	no	—
474	C₂²	—	2	2	no	—
475	C₂²	—	2	2	no	—
476	C₂²	—	2	2	no	—
477	C₂²	—	4	1	no	—
478	C₂²	—	4	1	no	—
479	C₂²	—	4	1	no	—
480	C₂²	—	4	1	no	—
481	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
482	C₂²	C₄×#41×C₃₁	1	1	yes	characteristic
483	C₂²	C₄×#41×C₃₁	1	1	yes	characteristic
484	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
485	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
486	C₂²	C₃₁×#568	1	1	yes	characteristic
487	C₂²	C₃₁×#568	1	1	yes	characteristic
488	C₄	—	2	2	no	—
489	C₄	—	2	2	no	—
490	C₄	—	2	2	no	—
491	C₄	—	2	2	no	—
492	C₄	—	2	2	no	—
493	C₄	—	2	2	no	—
494	C₄	—	2	2	no	—
495	C₄	—	4	1	no	—
496	C₄	—	4	1	no	—
497	C₄	—	4	1	no	—
498	C₄	—	4	1	no	—
499	C₄	—	4	1	no	—
500	C₄	—	4	1	no	—
501	C₄	C₃₁×#529	1	2	yes	—
502	C₄	C₃₁×#529	1	2	yes	—
503	C₂	—	4	2	no	—
504	C₂	C₃₁×#2298	1	1	yes	characteristic
505	C₂	C₃₁×#2920	1	1	yes	characteristic
506	C₂	C₃₁×#2921	1	1	yes	characteristic
507	C₂	C₃₁×#2970	1	1	yes	characteristic
508	C₂	C₃₁×#2970	1	1	yes	characteristic
509	C₂	C₃₁×#2970	1	1	yes	characteristic
510	C₂	C₃₁×#2971	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	C₂⁴×C₃₁	C₂²×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
4. C₃₁×#1917	C₂	semidirect
5. C₃₁×#1916	C₂	semidirect
6. C₃₁×#1767	C₂	semidirect
7. C₃₁×#1767	C₂	semidirect
10. C₃₁×#1745	C₂	semidirect
11. C₄×C₃₁×#97	C₂	semidirect
159. #21435	C₃₁	direct (N × H)
222. #1917	C₂×C₃₁	semidirect
223. #1916	C₂×C₃₁	semidirect
224. #1767	C₂×C₃₁	semidirect
225. #1767	C₂×C₃₁	semidirect
228. #1745	C₂×C₃₁	semidirect
229. C₄×#97	C₂×C₃₁	semidirect
351. C₃₁	#21435	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×#117	2
C₃₁×#1924	2
C₃₁×#1923	2
C₃₁×#1917	2
C₃₁×#1916	2
C₃₁×#1767	2
C₃₁×#1749	2
C₃₁×#1745	2
C₄×C₃₁×#97	2
#21435	31

Subgroups — C31×#21435 #21478194

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21435 #21478194