Subgroups — C₃₁×#35139 #21491898

Order 7936.

2150 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#529	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×#125	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2861	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2859	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2664	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2611	C₂	1	4	yes	maximal, complemented
7	C₃₁×#2566	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2565	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2504	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#2503	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#2502	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#2501	C₂	1	1	yes	maximal, characteristic
13	C₃₁×#2413	C₂	1	4	yes	maximal, complemented
14	C₃₁×#2322	C₂	1	4	yes	maximal, complemented
15	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
16	C₃₁×#529	—	2	8	no	—
17	C₂×C₃₁×#126	C₂²	1	2	yes	—
18	C₂×C₃₁×#125	—	2	8	no	—
19	C₂×C₃₁×#125	C₂²	1	2	yes	—
20	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#125	C₂²	1	2	yes	—
22	C₂×C₃₁×#125	C₂²	1	2	yes	—
23	C₂×C₃₁×#125	C₂²	1	2	yes	—
24	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#123	C₂²	1	4	yes	—
26	C₂×C₃₁×#123	C₂²	1	2	yes	—
27	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×#121	C₂²	1	1	yes	characteristic
29	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
30	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
31	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
32	C₂×C₃₁×#117	C₂²	1	2	yes	—
33	C₂×C₃₁×#117	C₂²	1	2	yes	—
34	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
35	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
36	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
37	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
38	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
39	C₃₁×#400	C₂²	1	4	yes	—
40	C₃₁×#399	C₂²	1	4	yes	—
41	C₃₁×#398	C₂²	1	8	yes	—
42	C₃₁×#398	C₂²	1	4	yes	—
43	C₃₁×#397	C₂²	1	4	yes	—
44	C₃₁×#396	C₂²	1	8	yes	—
45	C₃₁×#391	C₂²	1	2	yes	—
46	C₃₁×#391	C₂²	1	2	yes	—
47	C₃₁×#390	C₂²	1	2	yes	—
48	C₃₁×#390	C₂²	1	2	yes	—
49	C₃₁×#389	C₂²	1	8	yes	—
50	C₃₁×#389	C₂²	1	8	yes	—
51	C₃₁×#388	C₂²	1	8	yes	—
52	C₃₁×#388	C₂²	1	8	yes	—
53	C₃₁×#387	C₂²	1	8	yes	—
54	C₃₁×#386	C₂²	1	4	yes	—
55	C₃₁×#385	C₂²	1	8	yes	—
56	C₃₁×#385	C₂²	1	4	yes	—
57	C₃₁×#384	C₂²	1	2	yes	—
58	C₃₁×#384	C₂²	1	2	yes	—
59	C₃₁×#382	C₂²	1	2	yes	—
60	C₃₁×#382	C₂²	1	2	yes	—
61	C₃₁×#381	C₂²	1	4	yes	—
62	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
63	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
64	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
65	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
66	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
67	C₃₁×#376	C₂²	1	2	yes	—
68	C₄³×C₃₁	C₂²	1	2	yes	—
69	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
70	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
71	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
73	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
76	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
77	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
78	C₃₁×#126	—	2	8	no	—
79	C₃₁×#125	—	2	8	no	—
80	C₃₁×#125	—	2	8	no	—
81	C₃₁×#125	—	2	8	no	—
82	C₃₁×#125	—	2	8	no	—
83	C₃₁×#125	—	2	4	no	—
84	C₃₁×#124	—	2	4	no	—
85	C₃₁×#123	—	2	8	no	—
86	C₃₁×#123	—	2	16	no	—
87	C₃₁×C₂²⋊Q₈	—	2	4	no	—
88	C₃₁×#121	—	2	4	no	—
89	C₄×Q₈×C₃₁	—	2	4	no	—
90	C₄×D₄×C₃₁	—	2	8	no	—
91	C₄×D₄×C₃₁	—	2	4	no	—
92	C₃₁×#117	—	2	8	no	—
93	C₃₁×#117	—	2	8	no	—
94	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
95	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
96	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
97	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
98	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
99	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
100	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
101	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
102	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
104	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
105	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
106	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
107	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
108	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
109	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
110	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
111	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
113	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
114	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
115	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
116	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
117	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
118	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
119	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
120	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
121	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
122	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
123	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
124	C₂×C₄²×C₃₁	D₄	1	8	yes	—
125	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
126	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
127	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
128	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
129	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
130	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
131	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
132	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
133	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
134	C₃₁×#95	C₂³	1	8	yes	—
135	C₃₁×#95	C₂³	1	2	yes	—
136	C₃₁×#95	C₂³	1	8	yes	—
137	C₃₁×#95	C₂³	1	2	yes	—
138	C₃₁×#95	C₂³	1	8	yes	—
139	C₃₁×#95	C₂³	1	8	yes	—
140	C₃₁×#95	C₂³	1	4	yes	—
141	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
142	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
143	C₂×Q₈×C₃₁	—	2	2	no	—
144	C₂×Q₈×C₃₁	—	2	4	no	—
145	C₂×D₄×C₃₁	—	2	4	no	—
146	C₂×D₄×C₃₁	—	2	8	no	—
147	C₂×D₄×C₃₁	—	2	2	no	—
148	C₂×D₄×C₃₁	—	2	4	no	—
149	C₂²×C₄×C₃₁	—	2	4	no	—
150	C₂²×C₄×C₃₁	—	2	4	no	—
151	C₂²×C₄×C₃₁	—	2	4	no	—
152	C₂²×C₄×C₃₁	—	2	4	no	—
153	C₂²×C₄×C₃₁	—	2	8	no	—
154	C₂²×C₄×C₃₁	—	2	4	no	—
155	C₂²×C₄×C₃₁	—	4	2	no	—
156	C₂²×C₄×C₃₁	—	4	4	no	—
157	C₂²×C₄×C₃₁	—	4	4	no	—
158	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
159	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
160	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
161	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
162	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
163	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
164	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
165	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
166	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
167	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
168	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
169	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
170	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
171	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
172	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
173	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
174	C₄⋊C₄×C₃₁	—	2	8	no	—
175	C₄⋊C₄×C₃₁	—	2	8	no	—
176	C₄⋊C₄×C₃₁	—	2	4	no	—
177	C₄⋊C₄×C₃₁	—	2	4	no	—
178	C₄⋊C₄×C₃₁	—	2	8	no	—
179	C₄⋊C₄×C₃₁	—	2	8	no	—
180	C₄⋊C₄×C₃₁	—	2	4	no	—
181	C₄⋊C₄×C₃₁	—	2	4	no	—
182	C₄⋊C₄×C₃₁	—	2	4	no	—
183	C₄⋊C₄×C₃₁	—	2	4	no	—
184	C₄⋊C₄×C₃₁	—	2	2	no	—
185	C₄⋊C₄×C₃₁	—	2	4	no	—
186	C₄⋊C₄×C₃₁	—	2	2	no	—
187	C₄⋊C₄×C₃₁	—	2	4	no	—
188	C₄⋊C₄×C₃₁	—	2	8	no	—
189	C₄⋊C₄×C₃₁	—	2	16	no	—
190	C₄⋊C₄×C₃₁	—	4	8	no	—
191	C₂²⋊C₄×C₃₁	—	2	4	no	—
192	C₂²⋊C₄×C₃₁	—	2	4	no	—
193	C₂²⋊C₄×C₃₁	—	2	4	no	—
194	C₂²⋊C₄×C₃₁	—	2	4	no	—
195	C₂²⋊C₄×C₃₁	—	2	4	no	—
196	C₂²⋊C₄×C₃₁	—	2	4	no	—
197	C₂²⋊C₄×C₃₁	—	2	16	no	—
198	C₂²⋊C₄×C₃₁	—	4	8	no	—
199	C₂²⋊C₄×C₃₁	#41	1	8	yes	—
200	C₄²×C₃₁	—	2	4	no	—
201	C₄²×C₃₁	—	2	8	no	—
202	C₄²×C₃₁	—	2	4	no	—
203	C₄²×C₃₁	—	2	4	no	—
204	C₄²×C₃₁	—	4	4	no	—
205	C₄²×C₃₁	C₂×D₄	1	4	yes	—
206	C₄²×C₃₁	C₂×D₄	1	4	yes	—
207	C₄²×C₃₁	C₂×D₄	1	4	yes	—
208	C₄²×C₃₁	C₂×D₄	1	4	yes	—
209	#35139	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
210	C₂³×C₃₁	—	2	2	no	—
211	C₂³×C₃₁	—	2	2	no	—
212	C₂³×C₃₁	—	2	2	no	—
213	C₂³×C₃₁	—	2	4	no	—
214	C₂³×C₃₁	—	2	4	no	—
215	C₂³×C₃₁	—	2	4	no	—
216	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
217	C₂³×C₃₁	C₂×#41	1	4	yes	—
218	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
219	Q₈×C₃₁	—	4	4	no	—
220	D₄×C₃₁	—	4	8	no	—
221	D₄×C₃₁	—	4	4	no	—
222	C₂×C₄×C₃₁	—	2	8	no	—
223	C₂×C₄×C₃₁	—	2	2	no	—
224	C₂×C₄×C₃₁	—	2	2	no	—
225	C₂×C₄×C₃₁	—	2	4	no	—
226	C₂×C₄×C₃₁	—	2	2	no	—
227	C₂×C₄×C₃₁	—	2	2	no	—
228	C₂×C₄×C₃₁	—	2	4	no	—
229	C₂×C₄×C₃₁	—	2	2	no	—
230	C₂×C₄×C₃₁	—	2	8	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	4	no	—
235	C₂×C₄×C₃₁	—	2	4	no	—
236	C₂×C₄×C₃₁	—	2	4	no	—
237	C₂×C₄×C₃₁	—	2	8	no	—
238	C₂×C₄×C₃₁	—	2	4	no	—
239	C₂×C₄×C₃₁	—	2	4	no	—
240	C₂×C₄×C₃₁	—	2	4	no	—
241	C₂×C₄×C₃₁	—	2	4	no	—
242	C₂×C₄×C₃₁	—	2	8	no	—
243	C₂×C₄×C₃₁	—	2	4	no	—
244	C₂×C₄×C₃₁	—	4	4	no	—
245	C₂×C₄×C₃₁	—	4	8	no	—
246	C₂×C₄×C₃₁	—	4	8	no	—
247	C₂×C₄×C₃₁	—	4	8	no	—
248	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
249	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
250	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
251	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
252	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
253	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
254	C₂×C₄×C₃₁	C₂×#41	1	8	yes	—
255	C₂×C₄×C₃₁	2₋¹⁺⁴	1	4	yes	—
256	C₂×#529	C₂×C₃₁	1	1	yes	characteristic
257	C₂²×#125	C₂×C₃₁	1	1	yes	characteristic, complemented
258	#2861	C₂×C₃₁	1	2	yes	complemented
259	#2859	C₂×C₃₁	1	2	yes	complemented
260	#2664	C₂×C₃₁	1	4	yes	complemented
261	#2611	C₂×C₃₁	1	4	yes	complemented
262	#2566	C₂×C₃₁	1	2	yes	complemented
263	#2565	C₂×C₃₁	1	2	yes	complemented
264	#2504	C₂×C₃₁	1	1	yes	characteristic
265	#2503	C₂×C₃₁	1	1	yes	characteristic, complemented
266	#2502	C₂×C₃₁	1	1	yes	characteristic, complemented
267	#2501	C₂×C₃₁	1	1	yes	characteristic
268	#2413	C₂×C₃₁	1	4	yes	complemented
269	#2322	C₂×C₃₁	1	4	yes	complemented
270	C₄×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
271	C₂²×C₃₁	—	2	2	no	—
272	C₂²×C₃₁	—	2	2	no	—
273	C₂²×C₃₁	—	2	8	no	—
274	C₂²×C₃₁	—	4	2	no	—
275	C₂²×C₃₁	—	4	2	no	—
276	C₂²×C₃₁	—	4	2	no	—
277	C₂²×C₃₁	—	4	8	no	—
278	C₂²×C₃₁	#532	1	4	yes	—
279	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
280	C₂²×C₃₁	C₂²×#41	1	2	yes	—
281	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
282	C₂²×C₃₁	#585	1	2	yes	—
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	—	2	2	no	—
288	C₄×C₃₁	—	2	8	no	—
289	C₄×C₃₁	—	4	4	no	—
290	C₄×C₃₁	—	4	2	no	—
291	C₄×C₃₁	—	4	2	no	—
292	C₄×C₃₁	—	4	2	no	—
293	C₄×C₃₁	—	4	4	no	—
294	C₄×C₃₁	—	4	4	no	—
295	C₄×C₃₁	#536	1	4	yes	—
296	#529	—	2	8	no	—
297	C₂×#126	C₂²×C₃₁	1	2	yes	—
298	C₂×#125	—	2	8	no	—
299	C₂×#125	C₂²×C₃₁	1	2	yes	—
300	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
301	C₂×#125	C₂²×C₃₁	1	2	yes	—
302	C₂×#125	C₂²×C₃₁	1	2	yes	—
303	C₂×#125	C₂²×C₃₁	1	2	yes	—
304	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
305	C₂×#123	C₂²×C₃₁	1	4	yes	—
306	C₂×#123	C₂²×C₃₁	1	2	yes	—
307	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
308	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
309	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
310	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
311	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
312	C₂×#117	C₂²×C₃₁	1	2	yes	—
313	C₂×#117	C₂²×C₃₁	1	2	yes	—
314	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
315	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
316	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
317	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
318	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
319	#400	C₂²×C₃₁	1	4	yes	—
320	#399	C₂²×C₃₁	1	4	yes	—
321	#398	C₂²×C₃₁	1	8	yes	—
322	#398	C₂²×C₃₁	1	4	yes	—
323	#397	C₂²×C₃₁	1	4	yes	—
324	#396	C₂²×C₃₁	1	8	yes	—
325	#391	C₂²×C₃₁	1	2	yes	—
326	#391	C₂²×C₃₁	1	2	yes	—
327	#390	C₂²×C₃₁	1	2	yes	—
328	#390	C₂²×C₃₁	1	2	yes	—
329	#389	C₂²×C₃₁	1	8	yes	—
330	#389	C₂²×C₃₁	1	8	yes	—
331	#388	C₂²×C₃₁	1	8	yes	—
332	#388	C₂²×C₃₁	1	8	yes	—
333	#387	C₂²×C₃₁	1	8	yes	—
334	#386	C₂²×C₃₁	1	4	yes	—
335	#385	C₂²×C₃₁	1	8	yes	—
336	#385	C₂²×C₃₁	1	4	yes	—
337	#384	C₂²×C₃₁	1	2	yes	—
338	#384	C₂²×C₃₁	1	2	yes	—
339	#382	C₂²×C₃₁	1	2	yes	—
340	#382	C₂²×C₃₁	1	2	yes	—
341	#381	C₂²×C₃₁	1	4	yes	—
342	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
343	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
344	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
345	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
346	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
347	#376	C₂²×C₃₁	1	2	yes	—
348	C₄³	C₂²×C₃₁	1	2	yes	—
349	C₂×C₃₁	—	2	4	no	—
350	C₂×C₃₁	—	8	2	no	—
351	C₂×C₃₁	C₂×#532	1	1	yes	characteristic
352	C₂×C₃₁	C₂×#536	1	1	yes	characteristic
353	C₂×C₃₁	#3449	1	1	yes	characteristic
354	C₂×C₃₁	#3515	1	4	yes	—
355	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
356	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
357	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
358	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
359	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
360	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
361	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
362	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
363	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
364	#126	—	2	8	no	—
365	#125	—	2	8	no	—
366	#125	—	2	8	no	—
367	#125	—	2	8	no	—
368	#125	—	2	8	no	—
369	#125	—	2	4	no	—
370	#124	—	2	4	no	—
371	#123	—	2	8	no	—
372	#123	—	2	16	no	—
373	C₂²⋊Q₈	—	2	4	no	—
374	#121	—	2	4	no	—
375	C₄×Q₈	—	2	4	no	—
376	C₄×D₄	—	2	8	no	—
377	C₄×D₄	—	2	4	no	—
378	#117	—	2	8	no	—
379	#117	—	2	8	no	—
380	C₂×C₄⋊C₄	—	2	4	no	—
381	C₂×C₄⋊C₄	—	2	4	no	—
382	C₂×C₄⋊C₄	—	2	8	no	—
383	C₂×C₄⋊C₄	—	2	8	no	—
384	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
385	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
386	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
387	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
388	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
389	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
390	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
391	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
392	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
393	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
394	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
395	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
396	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
397	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
398	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
399	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
400	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
401	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
402	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
403	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
404	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
405	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
406	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
407	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
408	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
409	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
410	C₂×C₄²	D₄×C₃₁	1	8	yes	—
411	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
412	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
413	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
414	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
415	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
416	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
417	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
418	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
419	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
420	#95	C₂³×C₃₁	1	4	yes	—
421	#95	C₂³×C₃₁	1	2	yes	—
422	#95	C₂³×C₃₁	1	2	yes	—
423	#95	C₂³×C₃₁	1	8	yes	—
424	#95	C₂³×C₃₁	1	8	yes	—
425	#95	C₂³×C₃₁	1	8	yes	—
426	#95	C₂³×C₃₁	1	8	yes	—
427	C₃₁	#35139	1	1	yes	characteristic, direct product factor, Sylow
428	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
429	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
430	C₂×Q₈	—	2	2	no	—
431	C₂×Q₈	—	2	4	no	—
432	C₂×D₄	—	2	4	no	—
433	C₂×D₄	—	2	8	no	—
434	C₂×D₄	—	2	2	no	—
435	C₂×D₄	—	2	4	no	—
436	C₂²×C₄	—	2	4	no	—
437	C₂²×C₄	—	2	4	no	—
438	C₂²×C₄	—	2	4	no	—
439	C₂²×C₄	—	2	4	no	—
440	C₂²×C₄	—	2	8	no	—
441	C₂²×C₄	—	2	4	no	—
442	C₂²×C₄	—	4	2	no	—
443	C₂²×C₄	—	4	4	no	—
444	C₂²×C₄	—	4	4	no	—
445	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
446	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
447	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
448	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
449	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
450	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
451	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
452	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
453	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
454	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
455	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
456	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
457	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
458	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
459	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
460	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
461	C₄⋊C₄	—	2	8	no	—
462	C₄⋊C₄	—	2	8	no	—
463	C₄⋊C₄	—	2	8	no	—
464	C₄⋊C₄	—	2	8	no	—
465	C₄⋊C₄	—	2	4	no	—
466	C₄⋊C₄	—	2	4	no	—
467	C₄⋊C₄	—	2	4	no	—
468	C₄⋊C₄	—	2	4	no	—
469	C₄⋊C₄	—	2	4	no	—
470	C₄⋊C₄	—	2	4	no	—
471	C₄⋊C₄	—	2	2	no	—
472	C₄⋊C₄	—	2	4	no	—
473	C₄⋊C₄	—	2	2	no	—
474	C₄⋊C₄	—	2	4	no	—
475	C₄⋊C₄	—	2	8	no	—
476	C₄⋊C₄	—	2	16	no	—
477	C₄⋊C₄	—	4	8	no	—
478	C₂²⋊C₄	—	2	4	no	—
479	C₂²⋊C₄	—	2	4	no	—
480	C₂²⋊C₄	—	2	4	no	—
481	C₂²⋊C₄	—	2	4	no	—
482	C₂²⋊C₄	—	2	4	no	—
483	C₂²⋊C₄	—	2	4	no	—
484	C₂²⋊C₄	—	2	16	no	—
485	C₂²⋊C₄	—	4	8	no	—
486	C₂²⋊C₄	#41×C₃₁	1	8	yes	—
487	C₄²	—	2	4	no	—
488	C₄²	—	2	8	no	—
489	C₄²	—	2	4	no	—
490	C₄²	—	2	4	no	—
491	C₄²	—	4	4	no	—
492	C₄²	C₂×D₄×C₃₁	1	4	yes	—
493	C₄²	C₂×D₄×C₃₁	1	4	yes	—
494	C₄²	C₂×D₄×C₃₁	1	4	yes	—
495	C₄²	C₂×D₄×C₃₁	1	4	yes	—
496	C₂³	—	2	2	no	—
497	C₂³	—	2	2	no	—
498	C₂³	—	2	2	no	—
499	C₂³	—	2	4	no	—
500	C₂³	—	2	4	no	—
501	C₂³	—	2	4	no	—
502	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
503	C₂³	C₂×#41×C₃₁	1	4	yes	—
504	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
505	Q₈	—	4	4	no	—
506	D₄	—	4	8	no	—
507	D₄	—	4	4	no	—
508	C₂×C₄	—	2	8	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	4	no	—
512	C₂×C₄	—	2	2	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	4	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	8	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	4	no	—
521	C₂×C₄	—	2	4	no	—
522	C₂×C₄	—	2	4	no	—
523	C₂×C₄	—	2	8	no	—
524	C₂×C₄	—	2	4	no	—
525	C₂×C₄	—	2	4	no	—
526	C₂×C₄	—	2	4	no	—
527	C₂×C₄	—	2	4	no	—
528	C₂×C₄	—	2	8	no	—
529	C₂×C₄	—	2	4	no	—
530	C₂×C₄	—	4	4	no	—
531	C₂×C₄	—	4	8	no	—
532	C₂×C₄	—	4	8	no	—
533	C₂×C₄	—	4	8	no	—
534	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
535	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
536	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
537	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
538	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
539	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
540	C₂×C₄	C₂×#41×C₃₁	1	8	yes	—
541	C₂×C₄	C₃₁×2₋¹⁺⁴	1	4	yes	—
542	C₂²	—	2	2	no	—
543	C₂²	—	2	2	no	—
544	C₂²	—	2	8	no	—
545	C₂²	—	4	2	no	—
546	C₂²	—	4	2	no	—
547	C₂²	—	4	2	no	—
548	C₂²	—	4	8	no	—
549	C₂²	C₃₁×#532	1	4	yes	—
550	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
551	C₂²	C₂²×#41×C₃₁	1	2	yes	—
552	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	2	yes	—
553	C₂²	C₃₁×#585	1	2	yes	—
554	C₄	—	2	2	no	—
555	C₄	—	2	2	no	—
556	C₄	—	2	2	no	—
557	C₄	—	2	2	no	—
558	C₄	—	2	2	no	—
559	C₄	—	2	8	no	—
560	C₄	—	4	4	no	—
561	C₄	—	4	2	no	—
562	C₄	—	4	2	no	—
563	C₄	—	4	2	no	—
564	C₄	—	4	4	no	—
565	C₄	—	4	4	no	—
566	C₄	C₃₁×#536	1	4	yes	—
567	C₂	—	2	4	no	—
568	C₂	—	8	2	no	—
569	C₂	C₂×C₃₁×#532	1	1	yes	characteristic
570	C₂	C₂×C₃₁×#536	1	1	yes	characteristic
571	C₂	C₃₁×#3449	1	1	yes	characteristic
572	C₂	C₃₁×#3515	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂×C₃₁×#121	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃₁×#125	C₂	semidirect
3. C₃₁×#2861	C₂	semidirect
4. C₃₁×#2859	C₂	semidirect
5. C₃₁×#2664	C₂	semidirect
6. C₃₁×#2611	C₂	semidirect
7. C₃₁×#2566	C₂	semidirect
8. C₃₁×#2565	C₂	semidirect
10. C₃₁×#2503	C₂	semidirect
11. C₃₁×#2502	C₂	semidirect
13. C₃₁×#2413	C₂	semidirect
14. C₃₁×#2322	C₂	semidirect
15. C₄×C₃₁×#117	C₂	semidirect
209. #35139	C₃₁	direct (N × H)
257. C₂²×#125	C₂×C₃₁	semidirect
258. #2861	C₂×C₃₁	semidirect
259. #2859	C₂×C₃₁	semidirect
260. #2664	C₂×C₃₁	semidirect
261. #2611	C₂×C₃₁	semidirect
262. #2566	C₂×C₃₁	semidirect
263. #2565	C₂×C₃₁	semidirect
265. #2503	C₂×C₃₁	semidirect
266. #2502	C₂×C₃₁	semidirect
268. #2413	C₂×C₃₁	semidirect
269. #2322	C₂×C₃₁	semidirect
270. C₄×#117	C₂×C₃₁	semidirect
427. C₃₁	#35139	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×#529	2
C₂²×C₃₁×#125	2
C₃₁×#2861	2
C₃₁×#2859	2
C₃₁×#2664	2
C₃₁×#2611	2
C₃₁×#2566	2
C₃₁×#2565	2
C₃₁×#2504	2
C₃₁×#2503	2
C₃₁×#2502	2
C₃₁×#2501	2
C₃₁×#2413	2
C₃₁×#2322	2
C₄×C₃₁×#117	2
#35139	31

Subgroups — C31×#35139 #21491898

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#35139 #21491898