Subgroups — C₃₁×#22651 #21479410

Order 7936.

992 conjugacy classes of subgroups.

The subgroup lattice has 490 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×C₈×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#424	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2293	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1939	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1939	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1851	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1850	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1842	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1798	C₂	1	2	yes	maximal, complemented
11	C₄×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
13	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#117	C₂²	1	2	yes	—
16	C₃₁×#436	—	2	4	no	—
17	C₈×D₄×C₃₁	—	2	4	no	—
18	C₃₁×#424	—	2	8	no	—
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
24	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
25	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
26	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
28	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
31	C₃₁×#390	C₄	1	1	yes	characteristic
32	C₃₁×#390	C₄	1	1	yes	characteristic
33	C₃₁×#389	C₄	1	1	yes	characteristic
34	C₃₁×#389	C₄	1	1	yes	characteristic
35	C₃₁×#388	C₄	1	2	yes	—
36	C₃₁×#388	C₄	1	2	yes	—
37	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
38	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
39	C₄³×C₃₁	C₂²	1	1	yes	characteristic
40	C₃₁×#336	C₂²	1	2	yes	—
41	C₃₁×#336	C₂²	1	2	yes	—
42	C₃₁×#336	C₂²	1	1	yes	characteristic
43	C₃₁×#336	C₂²	1	1	yes	characteristic
44	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
46	C₂×M₁₆×C₃₁	—	2	4	no	—
47	C₂×M₁₆×C₃₁	—	2	4	no	—
48	C₂×M₁₆×C₃₁	—	4	2	no	—
49	C₂²×C₈×C₃₁	—	2	4	no	—
50	C₂²×C₈×C₃₁	—	2	4	no	—
51	C₂²×C₈×C₃₁	—	4	2	no	—
52	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
55	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
56	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
58	C₄×D₄×C₃₁	D₄	1	4	yes	—
59	C₄×D₄×C₃₁	Q₈	1	4	yes	—
60	C₃₁×#117	—	2	4	no	—
61	C₃₁×#117	—	2	4	no	—
62	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
63	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
64	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
67	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
68	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
69	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
70	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
71	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
72	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
77	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
80	C₃₁×C₄⋊C₈	—	2	4	no	—
81	C₃₁×C₄⋊C₈	—	2	4	no	—
82	C₃₁×C₄⋊C₈	—	2	4	no	—
83	C₃₁×C₄⋊C₈	—	2	2	no	—
84	C₃₁×C₄⋊C₈	—	2	2	no	—
85	C₃₁×C₄⋊C₈	—	2	2	no	—
86	C₃₁×C₄⋊C₈	—	4	2	no	—
87	C₃₁×C₄⋊C₈	D₄	1	4	yes	complemented
88	C₃₁×C₂²⋊C₈	—	2	4	no	—
89	C₃₁×C₂²⋊C₈	—	2	4	no	—
90	C₄×C₈×C₃₁	—	2	2	no	—
91	C₄×C₈×C₃₁	—	2	2	no	—
92	C₃₁×#95	C₂×C₄	1	2	yes	—
93	C₃₁×#95	C₂×C₄	1	2	yes	—
94	C₃₁×#95	C₂×C₄	1	2	yes	—
95	C₂⁴×C₃₁	C₂²×C₄	1	2	yes	—
96	C₂×D₄×C₃₁	—	2	2	no	—
97	C₂×D₄×C₃₁	—	2	2	no	—
98	C₂×D₄×C₃₁	C₄⋊C₄	1	4	yes	—
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	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
105	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
106	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
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	C₂×D₄	1	4	yes	—
113	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
114	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
115	M₁₆×C₃₁	—	4	8	no	—
116	C₂×C₈×C₃₁	—	2	1	no	—
117	C₂×C₈×C₃₁	—	2	1	no	—
118	C₂×C₈×C₃₁	—	2	1	no	—
119	C₂×C₈×C₃₁	—	2	1	no	—
120	C₂×C₈×C₃₁	—	2	1	no	—
121	C₂×C₈×C₃₁	—	2	1	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	1	no	—
127	C₂×C₈×C₃₁	—	2	1	no	—
128	C₂×C₈×C₃₁	—	2	2	no	—
129	C₂×C₈×C₃₁	—	2	2	no	—
130	C₂×C₈×C₃₁	—	2	1	no	—
131	C₂×C₈×C₃₁	—	2	1	no	—
132	C₂×C₈×C₃₁	—	4	8	no	—
133	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
134	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
135	C₄⋊C₄×C₃₁	—	2	4	no	—
136	C₄⋊C₄×C₃₁	—	2	4	no	—
137	C₄⋊C₄×C₃₁	—	2	2	no	—
138	C₄⋊C₄×C₃₁	—	2	2	no	—
139	C₄⋊C₄×C₃₁	C₄⋊C₄	1	4	yes	—
140	C₂²⋊C₄×C₃₁	—	2	4	no	—
141	C₂²⋊C₄×C₃₁	—	2	4	no	—
142	C₂²⋊C₄×C₃₁	—	2	4	no	—
143	C₂²⋊C₄×C₃₁	—	4	2	no	—
144	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	8	yes	—
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	1	no	—
151	C₄²×C₃₁	—	2	1	no	—
152	C₄²×C₃₁	—	4	1	no	—
153	C₄²×C₃₁	C₂×D₄	1	2	yes	—
154	C₄²×C₃₁	C₂×D₄	1	2	yes	—
155	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
156	C₄²×C₃₁	#41	1	2	yes	—
157	#22651	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₂³×C₃₁	—	2	2	no	—
159	C₂³×C₃₁	—	2	2	no	—
160	C₂³×C₃₁	—	2	2	no	—
161	C₂³×C₃₁	—	2	2	no	—
162	C₂³×C₃₁	—	2	2	no	—
163	C₂³×C₃₁	—	2	2	no	—
164	C₂³×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
165	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
166	D₄×C₃₁	—	2	8	no	—
167	C₂×C₄×C₃₁	—	2	2	no	—
168	C₂×C₄×C₃₁	—	2	1	no	—
169	C₂×C₄×C₃₁	—	2	2	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	4	no	—
178	C₂×C₄×C₃₁	—	2	4	no	—
179	C₂×C₄×C₃₁	—	2	4	no	—
180	C₂×C₄×C₃₁	—	2	4	no	—
181	C₂×C₄×C₃₁	—	2	2	no	—
182	C₂×C₄×C₃₁	—	2	2	no	—
183	C₂×C₄×C₃₁	—	2	1	no	—
184	C₂×C₄×C₃₁	—	2	1	no	—
185	C₂×C₄×C₃₁	—	2	2	no	—
186	C₂×C₄×C₃₁	—	2	2	no	—
187	C₂×C₄×C₃₁	—	2	2	no	—
188	C₂×C₄×C₃₁	—	2	1	no	—
189	C₂×C₄×C₃₁	—	2	2	no	—
190	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
191	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
192	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
193	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
194	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
195	C₂×C₄×C₃₁	#131	1	2	yes	—
196	C₂×C₄×C₃₁	#131	1	2	yes	—
197	C₂×C₄×C₃₁	#131	1	2	yes	—
198	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
199	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
200	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
201	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
202	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
203	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
204	C₈×C₃₁	—	2	2	no	—
205	C₈×C₃₁	—	2	2	no	—
206	C₈×C₃₁	—	4	2	no	—
207	C₈×C₃₁	—	4	2	no	—
208	C₈×C₃₁	—	4	1	no	—
209	C₈×C₃₁	—	4	1	no	—
210	C₂×#436	C₂×C₃₁	1	1	yes	characteristic
211	C₂×C₈×D₄	C₂×C₃₁	1	1	yes	characteristic
212	C₂×#424	C₂×C₃₁	1	2	yes	complemented
213	#2293	C₂×C₃₁	1	1	yes	characteristic
214	#1939	C₂×C₃₁	1	1	yes	characteristic, complemented
215	#1939	C₂×C₃₁	1	1	yes	characteristic, complemented
216	#1851	C₂×C₃₁	1	2	yes	complemented
217	#1850	C₂×C₃₁	1	2	yes	complemented
218	#1842	C₂×C₃₁	1	1	yes	characteristic, complemented
219	#1798	C₂×C₃₁	1	2	yes	complemented
220	C₄×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
221	C₂²×C₃₁	—	2	4	no	—
222	C₂²×C₃₁	—	2	4	no	—
223	C₂²×C₃₁	—	2	4	no	—
224	C₂²×C₃₁	—	4	2	no	—
225	C₂²×C₃₁	—	4	2	no	—
226	C₂²×C₃₁	—	4	2	no	—
227	C₂²×C₃₁	—	4	2	no	—
228	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
229	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
230	C₂²×C₃₁	#520	1	1	yes	characteristic
231	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
232	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
233	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
234	C₂²×C₃₁	#568	1	1	yes	characteristic
235	C₄×C₃₁	—	2	4	no	—
236	C₄×C₃₁	—	2	2	no	—
237	C₄×C₃₁	—	2	2	no	—
238	C₄×C₃₁	—	2	2	no	—
239	C₄×C₃₁	—	2	2	no	—
240	C₄×C₃₁	—	4	2	no	—
241	C₄×C₃₁	—	4	2	no	—
242	C₄×C₃₁	—	4	1	no	—
243	C₄×C₃₁	#424	1	4	yes	complemented
244	C₄×C₃₁	D₄²	1	1	yes	characteristic
245	C₄×C₃₁	#547	1	1	yes	characteristic
246	C₄×C₃₁	D₄×Q₈	1	1	yes	characteristic
247	C₄×C₃₁	#554	1	1	yes	characteristic
248	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
249	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
250	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
251	C₂×#117	C₂²×C₃₁	1	2	yes	—
252	#436	—	2	4	no	—
253	C₈×D₄	—	2	4	no	—
254	#424	—	2	8	no	—
255	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
256	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
257	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
258	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
259	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
260	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
261	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
262	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
263	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
264	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
265	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
266	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
267	#390	C₄×C₃₁	1	1	yes	characteristic
268	#390	C₄×C₃₁	1	1	yes	characteristic
269	#389	C₄×C₃₁	1	1	yes	characteristic
270	#389	C₄×C₃₁	1	1	yes	characteristic
271	#388	C₄×C₃₁	1	2	yes	—
272	#388	C₄×C₃₁	1	2	yes	—
273	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
274	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
275	C₄³	C₂²×C₃₁	1	1	yes	characteristic
276	#336	C₂²×C₃₁	1	2	yes	—
277	#336	C₂²×C₃₁	1	2	yes	—
278	#336	C₂²×C₃₁	1	1	yes	characteristic
279	#336	C₂²×C₃₁	1	1	yes	characteristic
280	C₂×C₃₁	—	4	4	no	—
281	C₂×C₃₁	D₄×C₄⋊C₄	1	1	yes	characteristic
282	C₂×C₃₁	C₂×#424	1	1	yes	characteristic
283	C₂×C₃₁	#2906	1	1	yes	characteristic
284	C₂×C₃₁	#2931	1	1	yes	characteristic
285	C₂×C₃₁	#2935	1	1	yes	characteristic
286	C₂×C₃₁	#2993	1	1	yes	characteristic
287	C₂×C₃₁	#2994	1	1	yes	characteristic
288	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
289	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
290	C₂×M₁₆	—	2	4	no	—
291	C₂×M₁₆	—	2	4	no	—
292	C₂×M₁₆	—	4	2	no	—
293	C₂²×C₈	—	2	4	no	—
294	C₂²×C₈	—	2	4	no	—
295	C₂²×C₈	—	4	2	no	—
296	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
297	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
298	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
299	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
300	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
301	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
302	C₄×D₄	D₄×C₃₁	1	4	yes	—
303	C₄×D₄	Q₈×C₃₁	1	4	yes	—
304	#117	—	2	4	no	—
305	#117	—	2	4	no	—
306	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
307	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
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	2	yes	—
312	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
313	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
314	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
315	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
316	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
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	C₂³×C₃₁	1	2	yes	—
321	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
322	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
323	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
324	C₄⋊C₈	—	2	4	no	—
325	C₄⋊C₈	—	2	4	no	—
326	C₄⋊C₈	—	2	4	no	—
327	C₄⋊C₈	—	2	2	no	—
328	C₄⋊C₈	—	2	2	no	—
329	C₄⋊C₈	—	2	2	no	—
330	C₄⋊C₈	—	4	2	no	—
331	C₄⋊C₈	D₄×C₃₁	1	4	yes	complemented
332	C₂²⋊C₈	—	2	4	no	—
333	C₂²⋊C₈	—	2	4	no	—
334	C₄×C₈	—	2	2	no	—
335	C₄×C₈	—	2	2	no	—
336	#95	C₂×C₄×C₃₁	1	2	yes	—
337	#95	C₂×C₄×C₃₁	1	2	yes	—
338	#95	C₂×C₄×C₃₁	1	2	yes	—
339	C₃₁	#22651	1	1	yes	characteristic, direct product factor, Sylow
340	C₂⁴	C₂²×C₄×C₃₁	1	2	yes	—
341	C₂×D₄	—	2	2	no	—
342	C₂×D₄	—	2	2	no	—
343	C₂×D₄	C₄⋊C₄×C₃₁	1	4	yes	—
344	C₂²×C₄	—	2	2	no	—
345	C₂²×C₄	—	2	2	no	—
346	C₂²×C₄	—	2	2	no	—
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
349	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
350	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
351	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
352	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
353	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
354	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
355	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
356	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
357	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
358	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
359	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
360	M₁₆	—	4	8	no	—
361	C₂×C₈	—	2	1	no	—
362	C₂×C₈	—	2	1	no	—
363	C₂×C₈	—	2	1	no	—
364	C₂×C₈	—	2	1	no	—
365	C₂×C₈	—	2	1	no	—
366	C₂×C₈	—	2	1	no	—
367	C₂×C₈	—	2	2	no	—
368	C₂×C₈	—	2	2	no	—
369	C₂×C₈	—	2	2	no	—
370	C₂×C₈	—	2	2	no	—
371	C₂×C₈	—	2	1	no	—
372	C₂×C₈	—	2	1	no	—
373	C₂×C₈	—	2	2	no	—
374	C₂×C₈	—	2	2	no	—
375	C₂×C₈	—	2	1	no	—
376	C₂×C₈	—	2	1	no	—
377	C₂×C₈	—	4	8	no	—
378	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
379	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
380	C₄⋊C₄	—	2	4	no	—
381	C₄⋊C₄	—	2	4	no	—
382	C₄⋊C₄	—	2	2	no	—
383	C₄⋊C₄	—	2	2	no	—
384	C₄⋊C₄	C₄⋊C₄×C₃₁	1	4	yes	—
385	C₂²⋊C₄	—	2	4	no	—
386	C₂²⋊C₄	—	2	4	no	—
387	C₂²⋊C₄	—	2	4	no	—
388	C₂²⋊C₄	—	4	2	no	—
389	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	8	yes	—
390	C₄²	—	2	2	no	—
391	C₄²	—	2	2	no	—
392	C₄²	—	2	2	no	—
393	C₄²	—	2	2	no	—
394	C₄²	—	2	2	no	—
395	C₄²	—	2	1	no	—
396	C₄²	—	2	1	no	—
397	C₄²	—	4	1	no	—
398	C₄²	C₂×D₄×C₃₁	1	2	yes	—
399	C₄²	C₂×D₄×C₃₁	1	2	yes	—
400	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
401	C₄²	#41×C₃₁	1	2	yes	—
402	C₂³	—	2	2	no	—
403	C₂³	—	2	2	no	—
404	C₂³	—	2	2	no	—
405	C₂³	—	2	2	no	—
406	C₂³	—	2	2	no	—
407	C₂³	—	2	2	no	—
408	C₂³	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
409	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
410	D₄	—	2	8	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	1	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	1	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	4	no	—
422	C₂×C₄	—	2	4	no	—
423	C₂×C₄	—	2	4	no	—
424	C₂×C₄	—	2	4	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
435	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
436	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
437	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
438	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
439	C₂×C₄	C₃₁×#131	1	2	yes	—
440	C₂×C₄	C₃₁×#131	1	2	yes	—
441	C₂×C₄	C₃₁×#131	1	2	yes	—
442	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
443	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
444	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
445	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
446	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
447	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
448	C₈	—	2	2	no	—
449	C₈	—	2	2	no	—
450	C₈	—	4	2	no	—
451	C₈	—	4	2	no	—
452	C₈	—	4	1	no	—
453	C₈	—	4	1	no	—
454	C₂²	—	2	4	no	—
455	C₂²	—	2	4	no	—
456	C₂²	—	2	4	no	—
457	C₂²	—	4	2	no	—
458	C₂²	—	4	2	no	—
459	C₂²	—	4	2	no	—
460	C₂²	—	4	2	no	—
461	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
462	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
463	C₂²	C₃₁×#520	1	1	yes	characteristic
464	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
465	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
466	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
467	C₂²	C₃₁×#568	1	1	yes	characteristic
468	C₄	—	2	4	no	—
469	C₄	—	2	2	no	—
470	C₄	—	2	2	no	—
471	C₄	—	2	2	no	—
472	C₄	—	2	2	no	—
473	C₄	—	4	2	no	—
474	C₄	—	4	2	no	—
475	C₄	—	4	1	no	—
476	C₄	C₃₁×#424	1	4	yes	complemented
477	C₄	D₄²×C₃₁	1	1	yes	characteristic
478	C₄	C₃₁×#547	1	1	yes	characteristic
479	C₄	D₄×Q₈×C₃₁	1	1	yes	characteristic
480	C₄	C₃₁×#554	1	1	yes	characteristic
481	C₂	—	4	4	no	—
482	C₂	D₄×C₄⋊C₄×C₃₁	1	1	yes	characteristic
483	C₂	C₂×C₃₁×#424	1	1	yes	characteristic
484	C₂	C₃₁×#2906	1	1	yes	characteristic
485	C₂	C₃₁×#2931	1	1	yes	characteristic
486	C₂	C₃₁×#2935	1	1	yes	characteristic
487	C₂	C₃₁×#2993	1	1	yes	characteristic
488	C₂	C₃₁×#2994	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
3. C₂×C₃₁×#424	C₂	semidirect
5. C₃₁×#1939	C₂	semidirect
6. C₃₁×#1939	C₂	semidirect
7. C₃₁×#1851	C₂	semidirect
8. C₃₁×#1850	C₂	semidirect
9. C₃₁×#1842	C₂	semidirect
10. C₃₁×#1798	C₂	semidirect
11. C₄×C₃₁×C₄⋊C₈	C₂	semidirect
87. C₃₁×C₄⋊C₈	D₄	semidirect
157. #22651	C₃₁	direct (N × H)
212. C₂×#424	C₂×C₃₁	semidirect
214. #1939	C₂×C₃₁	semidirect
215. #1939	C₂×C₃₁	semidirect
216. #1851	C₂×C₃₁	semidirect
217. #1850	C₂×C₃₁	semidirect
218. #1842	C₂×C₃₁	semidirect
219. #1798	C₂×C₃₁	semidirect
220. C₄×C₄⋊C₈	C₂×C₃₁	semidirect
243. C₄×C₃₁	#424	semidirect
331. C₄⋊C₈	D₄×C₃₁	semidirect
339. C₃₁	#22651	direct (N × H)
476. C₄	C₃₁×#424	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×C₈×D₄×C₃₁	2
C₂×C₃₁×#424	2
C₃₁×#2293	2
C₃₁×#1939	2
C₃₁×#1851	2
C₃₁×#1850	2
C₃₁×#1842	2
C₃₁×#1798	2
C₄×C₃₁×C₄⋊C₈	2
#22651	31

Subgroups — C31×#22651 #21479410

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22651 #21479410