Subgroups — C₃₁×#22954 #21479713

Order 7936.

880 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#446	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1985	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1940	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1940	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1939	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1917	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1916	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1850	C₂	1	1	yes	maximal, characteristic
11	C₃₁×#1850	C₂	1	1	yes	maximal, characteristic
12	C₃₁×#1771	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#1752	C₂	1	1	yes	maximal, characteristic
14	C₈×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
15	C₃₁×#1749	C₂	1	1	yes	maximal, characteristic
16	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×C₂²⋊Q₈	C₄	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#446	—	2	4	no	—
22	C₂×C₃₁×C₄⋊C₈	—	2	4	no	—
23	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
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×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×C₄⋊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×#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×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
39	C₃₁×#391	C₄	1	1	yes	characteristic
40	C₃₁×#389	C₄	1	1	yes	characteristic
41	C₃₁×#387	C₄	1	1	yes	characteristic
42	C₃₁×#387	C₄	1	1	yes	characteristic
43	C₃₁×#385	C₄	1	1	yes	characteristic
44	C₃₁×#385	C₄	1	1	yes	characteristic
45	C₃₁×#380	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×C₄⋊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×Q₈×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	C₂²×C₈×C₃₁	—	2	2	no	—
66	C₂²×C₈×C₃₁	—	2	2	no	—
67	C₂²×C₈×C₃₁	—	2	2	no	—
68	C₂²×C₈×C₃₁	—	2	2	no	—
69	C₂²×C₈×C₃₁	—	4	1	no	—
70	C₂²×C₈×C₃₁	—	4	1	no	—
71	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
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	C₃₁×C₂²⋊Q₈	—	2	4	no	—
80	C₄×Q₈×C₃₁	—	2	2	no	—
81	C₄×Q₈×C₃₁	—	2	2	no	—
82	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
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×C₄²×C₃₁	—	2	2	no	—
96	C₂×C₄²×C₃₁	—	2	2	no	—
97	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
98	C₂×C₄²×C₃₁	C₂×C₄	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	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
105	C₃₁×C₄⋊C₈	—	2	2	no	—
106	C₃₁×C₄⋊C₈	—	2	2	no	—
107	C₃₁×C₄⋊C₈	—	2	2	no	—
108	C₃₁×C₄⋊C₈	—	2	2	no	—
109	C₃₁×C₄⋊C₈	—	2	2	no	—
110	C₃₁×C₄⋊C₈	—	4	1	no	—
111	C₃₁×C₄⋊C₈	—	4	1	no	—
112	C₃₁×C₄⋊C₈	D₄	1	4	yes	—
113	C₃₁×C₂²⋊C₈	—	2	2	no	—
114	C₃₁×C₂²⋊C₈	Q₈	1	4	yes	complemented
115	C₃₁×#97	—	2	2	no	—
116	C₃₁×#97	—	2	2	no	—
117	C₃₁×#97	—	2	2	no	—
118	C₃₁×#97	—	4	1	no	—
119	C₄×C₈×C₃₁	—	2	2	no	—
120	C₄×C₈×C₃₁	—	4	1	no	—
121	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
122	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
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	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
132	C₂×Q₈×C₃₁	—	2	2	no	—
133	C₂×Q₈×C₃₁	—	2	2	no	—
134	C₂×Q₈×C₃₁	—	4	1	no	—
135	C₂²×C₄×C₃₁	—	2	2	no	—
136	C₂²×C₄×C₃₁	—	2	2	no	—
137	C₂²×C₄×C₃₁	—	2	2	no	—
138	C₂²×C₄×C₃₁	—	2	1	no	—
139	C₂²×C₄×C₃₁	—	2	1	no	—
140	C₂²×C₄×C₃₁	—	2	1	no	—
141	C₂²×C₄×C₃₁	—	2	1	no	—
142	C₂²×C₄×C₃₁	—	2	2	no	—
143	C₂²×C₄×C₃₁	—	2	2	no	—
144	C₂²×C₄×C₃₁	—	4	1	no	—
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	1	yes	characteristic
148	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
149	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
150	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
151	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
152	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
153	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
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	C₂×Q₈	1	2	yes	—
159	C₂²×C₄×C₃₁	#41	1	2	yes	—
160	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
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	1	no	—
170	C₂×C₈×C₃₁	—	2	1	no	—
171	C₂×C₈×C₃₁	—	2	1	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	—	4	4	no	—
182	C₂×C₈×C₃₁	—	4	4	no	—
183	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
184	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
185	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
186	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
187	C₄⋊C₄×C₃₁	—	2	2	no	—
188	C₄⋊C₄×C₃₁	—	2	2	no	—
189	C₄⋊C₄×C₃₁	—	2	2	no	—
190	C₄⋊C₄×C₃₁	—	2	2	no	—
191	C₄⋊C₄×C₃₁	—	2	2	no	—
192	C₄⋊C₄×C₃₁	—	2	2	no	—
193	C₄⋊C₄×C₃₁	—	2	2	no	—
194	C₄⋊C₄×C₃₁	—	2	2	no	—
195	C₄⋊C₄×C₃₁	—	4	1	no	—
196	C₂²⋊C₄×C₃₁	—	2	2	no	—
197	C₂²⋊C₄×C₃₁	—	2	2	no	—
198	C₂²⋊C₄×C₃₁	—	2	2	no	—
199	C₂²⋊C₄×C₃₁	M₁₆	1	4	yes	—
200	C₄²×C₃₁	—	2	2	no	—
201	C₄²×C₃₁	—	2	2	no	—
202	C₄²×C₃₁	—	2	1	no	—
203	C₄²×C₃₁	—	2	1	no	—
204	C₄²×C₃₁	—	2	1	no	—
205	C₄²×C₃₁	—	2	1	no	—
206	C₄²×C₃₁	—	2	1	no	—
207	C₄²×C₃₁	—	2	1	no	—
208	C₄²×C₃₁	—	2	1	no	—
209	C₄²×C₃₁	—	2	1	no	—
210	C₄²×C₃₁	—	2	1	no	—
211	C₄²×C₃₁	—	2	1	no	—
212	C₄²×C₃₁	C₂×D₄	1	2	yes	—
213	C₄²×C₃₁	#41	1	2	yes	—
214	#22954	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
215	C₂³×C₃₁	—	2	1	no	—
216	C₂³×C₃₁	—	2	1	no	—
217	C₂³×C₃₁	—	2	1	no	—
218	C₂³×C₃₁	—	2	1	no	—
219	C₂³×C₃₁	C₄×Q₈	1	2	yes	—
220	C₂³×C₃₁	C₂×M₁₆	1	2	yes	—
221	C₂³×C₃₁	#131	1	2	yes	—
222	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
223	Q₈×C₃₁	—	4	4	no	—
224	C₂×C₄×C₃₁	—	2	8	no	—
225	C₂×C₄×C₃₁	—	2	1	no	—
226	C₂×C₄×C₃₁	—	2	1	no	—
227	C₂×C₄×C₃₁	—	2	1	no	—
228	C₂×C₄×C₃₁	—	2	1	no	—
229	C₂×C₄×C₃₁	—	2	1	no	—
230	C₂×C₄×C₃₁	—	2	1	no	—
231	C₂×C₄×C₃₁	—	2	1	no	—
232	C₂×C₄×C₃₁	—	2	1	no	—
233	C₂×C₄×C₃₁	—	2	1	no	—
234	C₂×C₄×C₃₁	—	2	1	no	—
235	C₂×C₄×C₃₁	—	2	1	no	—
236	C₂×C₄×C₃₁	—	2	1	no	—
237	C₂×C₄×C₃₁	—	2	2	no	—
238	C₂×C₄×C₃₁	—	2	2	no	—
239	C₂×C₄×C₃₁	—	2	2	no	—
240	C₂×C₄×C₃₁	—	2	2	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	1	no	—
252	C₂×C₄×C₃₁	—	2	1	no	—
253	C₂×C₄×C₃₁	—	2	1	no	—
254	C₂×C₄×C₃₁	—	2	1	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	—	4	4	no	—
263	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
264	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
265	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
266	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
267	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
268	C₂×C₄×C₃₁	#131	1	2	yes	—
269	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
270	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
271	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
272	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
273	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
274	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
275	C₈×C₃₁	—	2	2	no	—
276	C₈×C₃₁	—	2	2	no	—
277	C₈×C₃₁	—	2	2	no	—
278	C₈×C₃₁	—	2	2	no	—
279	C₈×C₃₁	—	4	1	no	—
280	C₈×C₃₁	—	4	1	no	—
281	C₈×C₃₁	—	4	1	no	—
282	C₈×C₃₁	—	4	1	no	—
283	C₂×#446	C₂×C₃₁	1	1	yes	characteristic, complemented
284	C₂²×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic
285	C₄×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic
286	#1985	C₂×C₃₁	1	1	yes	characteristic, complemented
287	#1940	C₂×C₃₁	1	1	yes	characteristic, complemented
288	#1940	C₂×C₃₁	1	1	yes	characteristic, complemented
289	#1939	C₂×C₃₁	1	1	yes	characteristic, complemented
290	#1917	C₂×C₃₁	1	1	yes	characteristic, complemented
291	#1916	C₂×C₃₁	1	1	yes	characteristic, complemented
292	#1850	C₂×C₃₁	1	1	yes	characteristic
293	#1850	C₂×C₃₁	1	1	yes	characteristic
294	#1771	C₂×C₃₁	1	1	yes	characteristic, complemented
295	#1752	C₂×C₃₁	1	1	yes	characteristic
296	C₈×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic
297	#1749	C₂×C₃₁	1	1	yes	characteristic
298	C₂²×C₃₁	—	2	2	no	—
299	C₂²×C₃₁	—	2	2	no	—
300	C₂²×C₃₁	—	2	2	no	—
301	C₂²×C₃₁	—	2	2	no	—
302	C₂²×C₃₁	—	2	2	no	—
303	C₂²×C₃₁	—	2	2	no	—
304	C₂²×C₃₁	#446	1	4	yes	complemented
305	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
306	C₂²×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
307	C₂²×C₃₁	#520	1	1	yes	characteristic
308	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
309	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
310	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
311	C₂²×C₃₁	#568	1	1	yes	characteristic
312	C₄×C₃₁	—	2	2	no	—
313	C₄×C₃₁	—	2	2	no	—
314	C₄×C₃₁	—	2	2	no	—
315	C₄×C₃₁	—	2	2	no	—
316	C₄×C₃₁	—	2	2	no	—
317	C₄×C₃₁	—	2	2	no	—
318	C₄×C₃₁	—	4	1	no	—
319	C₄×C₃₁	—	4	1	no	—
320	C₄×C₃₁	—	4	1	no	—
321	C₄×C₃₁	—	4	1	no	—
322	C₄×C₃₁	—	4	1	no	—
323	C₄×C₃₁	—	4	1	no	—
324	C₄×C₃₁	—	4	1	no	—
325	C₄×C₃₁	#546	1	1	yes	characteristic
326	C₄×C₃₁	D₄×Q₈	1	1	yes	characteristic
327	C₄×C₃₁	#552	1	1	yes	characteristic
328	C₄×C₃₁	#554	1	1	yes	characteristic
329	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
330	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
331	C₂×C₂²⋊Q₈	C₄×C₃₁	1	1	yes	characteristic
332	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
333	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
334	#446	—	2	4	no	—
335	C₂×C₄⋊C₈	—	2	4	no	—
336	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
337	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
338	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
339	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
340	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
341	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
342	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
343	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
344	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
345	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
346	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
347	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
348	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
349	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
350	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
351	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
352	#391	C₄×C₃₁	1	1	yes	characteristic
353	#389	C₄×C₃₁	1	1	yes	characteristic
354	#387	C₄×C₃₁	1	1	yes	characteristic
355	#387	C₄×C₃₁	1	1	yes	characteristic
356	#385	C₄×C₃₁	1	1	yes	characteristic
357	#385	C₄×C₃₁	1	1	yes	characteristic
358	#380	C₄×C₃₁	1	1	yes	characteristic
359	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
360	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
361	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
362	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
363	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
364	#336	C₂²×C₃₁	1	1	yes	characteristic
365	#336	C₂²×C₃₁	1	1	yes	characteristic
366	#336	C₂²×C₃₁	1	1	yes	characteristic
367	#336	C₂²×C₃₁	1	1	yes	characteristic
368	#336	C₂²×C₃₁	1	1	yes	characteristic
369	#336	C₂²×C₃₁	1	1	yes	characteristic
370	#336	C₂²×C₃₁	1	1	yes	characteristic
371	#336	C₂²×C₃₁	1	1	yes	characteristic
372	#336	C₂²×C₃₁	1	1	yes	characteristic
373	#336	C₂²×C₃₁	1	1	yes	characteristic
374	C₂×C₃₁	—	2	4	no	—
375	C₂×C₃₁	#2360	1	1	yes	characteristic
376	C₂×C₃₁	#2929	1	1	yes	characteristic
377	C₂×C₃₁	D₄×M₁₆	1	1	yes	characteristic
378	C₂×C₃₁	C₂×#446	1	1	yes	characteristic
379	C₂×C₃₁	#2962	1	1	yes	characteristic
380	C₂×C₃₁	#2990	1	1	yes	characteristic
381	C₂×C₃₁	#2992	1	1	yes	characteristic
382	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
383	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
384	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
385	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
386	C₂²×C₈	—	2	2	no	—
387	C₂²×C₈	—	2	2	no	—
388	C₂²×C₈	—	2	2	no	—
389	C₂²×C₈	—	2	2	no	—
390	C₂²×C₈	—	4	1	no	—
391	C₂²×C₈	—	4	1	no	—
392	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
393	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
394	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
395	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
396	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
397	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
398	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
399	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
400	C₂²⋊Q₈	—	2	4	no	—
401	C₄×Q₈	—	2	2	no	—
402	C₄×Q₈	—	2	2	no	—
403	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
404	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
405	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
406	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
407	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
408	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
409	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
410	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
411	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
412	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
413	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
414	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
415	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
416	C₂×C₄²	—	2	2	no	—
417	C₂×C₄²	—	2	2	no	—
418	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
419	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
420	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
421	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
422	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
423	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
424	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
425	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
426	C₄⋊C₈	—	2	2	no	—
427	C₄⋊C₈	—	2	2	no	—
428	C₄⋊C₈	—	2	2	no	—
429	C₄⋊C₈	—	2	2	no	—
430	C₄⋊C₈	—	2	2	no	—
431	C₄⋊C₈	—	4	1	no	—
432	C₄⋊C₈	—	4	1	no	—
433	C₄⋊C₈	D₄×C₃₁	1	4	yes	—
434	C₂²⋊C₈	—	2	2	no	—
435	C₂²⋊C₈	Q₈×C₃₁	1	4	yes	complemented
436	#97	—	2	2	no	—
437	#97	—	2	2	no	—
438	#97	—	2	2	no	—
439	#97	—	4	1	no	—
440	C₄×C₈	—	2	2	no	—
441	C₄×C₈	—	4	1	no	—
442	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
443	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
444	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
445	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
446	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
447	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
448	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
449	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
450	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
451	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
452	C₃₁	#22954	1	1	yes	characteristic, direct product factor, Sylow
453	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
454	C₂×Q₈	—	2	2	no	—
455	C₂×Q₈	—	2	2	no	—
456	C₂×Q₈	—	4	1	no	—
457	C₂²×C₄	—	2	2	no	—
458	C₂²×C₄	—	2	2	no	—
459	C₂²×C₄	—	2	2	no	—
460	C₂²×C₄	—	2	1	no	—
461	C₂²×C₄	—	2	1	no	—
462	C₂²×C₄	—	2	1	no	—
463	C₂²×C₄	—	2	1	no	—
464	C₂²×C₄	—	2	2	no	—
465	C₂²×C₄	—	2	2	no	—
466	C₂²×C₄	—	4	1	no	—
467	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
468	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
469	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
470	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
471	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
472	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
473	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
474	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
475	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
476	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
477	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
478	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
479	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
480	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
481	C₂²×C₄	#41×C₃₁	1	2	yes	—
482	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
483	C₂×C₈	—	2	1	no	—
484	C₂×C₈	—	2	1	no	—
485	C₂×C₈	—	2	1	no	—
486	C₂×C₈	—	2	1	no	—
487	C₂×C₈	—	2	1	no	—
488	C₂×C₈	—	2	1	no	—
489	C₂×C₈	—	2	1	no	—
490	C₂×C₈	—	2	1	no	—
491	C₂×C₈	—	2	1	no	—
492	C₂×C₈	—	2	1	no	—
493	C₂×C₈	—	2	1	no	—
494	C₂×C₈	—	2	1	no	—
495	C₂×C₈	—	2	1	no	—
496	C₂×C₈	—	2	1	no	—
497	C₂×C₈	—	2	1	no	—
498	C₂×C₈	—	2	1	no	—
499	C₂×C₈	—	2	1	no	—
500	C₂×C₈	—	2	1	no	—
501	C₂×C₈	—	2	1	no	—
502	C₂×C₈	—	2	1	no	—
503	C₂×C₈	—	4	4	no	—
504	C₂×C₈	—	4	4	no	—
505	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
506	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
507	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
508	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
509	C₄⋊C₄	—	2	2	no	—
510	C₄⋊C₄	—	2	2	no	—
511	C₄⋊C₄	—	2	2	no	—
512	C₄⋊C₄	—	2	2	no	—
513	C₄⋊C₄	—	2	2	no	—
514	C₄⋊C₄	—	2	2	no	—
515	C₄⋊C₄	—	2	2	no	—
516	C₄⋊C₄	—	2	2	no	—
517	C₄⋊C₄	—	4	1	no	—
518	C₂²⋊C₄	—	2	2	no	—
519	C₂²⋊C₄	—	2	2	no	—
520	C₂²⋊C₄	—	2	2	no	—
521	C₂²⋊C₄	M₁₆×C₃₁	1	4	yes	—
522	C₄²	—	2	2	no	—
523	C₄²	—	2	2	no	—
524	C₄²	—	2	1	no	—
525	C₄²	—	2	1	no	—
526	C₄²	—	2	1	no	—
527	C₄²	—	2	1	no	—
528	C₄²	—	2	1	no	—
529	C₄²	—	2	1	no	—
530	C₄²	—	2	1	no	—
531	C₄²	—	2	1	no	—
532	C₄²	—	2	1	no	—
533	C₄²	—	2	1	no	—
534	C₄²	C₂×D₄×C₃₁	1	2	yes	—
535	C₄²	#41×C₃₁	1	2	yes	—
536	C₂³	—	2	1	no	—
537	C₂³	—	2	1	no	—
538	C₂³	—	2	1	no	—
539	C₂³	—	2	1	no	—
540	C₂³	C₄×Q₈×C₃₁	1	2	yes	—
541	C₂³	C₂×M₁₆×C₃₁	1	2	yes	—
542	C₂³	C₃₁×#131	1	2	yes	—
543	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
544	Q₈	—	4	4	no	—
545	C₂×C₄	—	2	8	no	—
546	C₂×C₄	—	2	1	no	—
547	C₂×C₄	—	2	1	no	—
548	C₂×C₄	—	2	1	no	—
549	C₂×C₄	—	2	1	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	1	no	—
552	C₂×C₄	—	2	1	no	—
553	C₂×C₄	—	2	1	no	—
554	C₂×C₄	—	2	1	no	—
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	2	1	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	1	no	—
563	C₂×C₄	—	2	1	no	—
564	C₂×C₄	—	2	1	no	—
565	C₂×C₄	—	2	1	no	—
566	C₂×C₄	—	2	1	no	—
567	C₂×C₄	—	2	1	no	—
568	C₂×C₄	—	2	1	no	—
569	C₂×C₄	—	2	1	no	—
570	C₂×C₄	—	2	1	no	—
571	C₂×C₄	—	2	1	no	—
572	C₂×C₄	—	2	1	no	—
573	C₂×C₄	—	2	1	no	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	2	1	no	—
577	C₂×C₄	—	2	1	no	—
578	C₂×C₄	—	2	1	no	—
579	C₂×C₄	—	2	1	no	—
580	C₂×C₄	—	2	1	no	—
581	C₂×C₄	—	2	1	no	—
582	C₂×C₄	—	2	1	no	—
583	C₂×C₄	—	4	4	no	—
584	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
585	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
586	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
587	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
588	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
589	C₂×C₄	C₃₁×#131	1	2	yes	—
590	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
591	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
592	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
593	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
594	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
595	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
596	C₈	—	2	2	no	—
597	C₈	—	2	2	no	—
598	C₈	—	2	2	no	—
599	C₈	—	2	2	no	—
600	C₈	—	4	1	no	—
601	C₈	—	4	1	no	—
602	C₈	—	4	1	no	—
603	C₈	—	4	1	no	—
604	C₂²	—	2	2	no	—
605	C₂²	—	2	2	no	—
606	C₂²	—	2	2	no	—
607	C₂²	—	2	2	no	—
608	C₂²	—	2	2	no	—
609	C₂²	—	2	2	no	—
610	C₂²	C₃₁×#446	1	4	yes	complemented
611	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
612	C₂²	C₂×C₄×Q₈×C₃₁	1	1	yes	characteristic
613	C₂²	C₃₁×#520	1	1	yes	characteristic
614	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
615	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
616	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
617	C₂²	C₃₁×#568	1	1	yes	characteristic
618	C₄	—	2	2	no	—
619	C₄	—	2	2	no	—
620	C₄	—	2	2	no	—
621	C₄	—	2	2	no	—
622	C₄	—	2	2	no	—
623	C₄	—	2	2	no	—
624	C₄	—	4	1	no	—
625	C₄	—	4	1	no	—
626	C₄	—	4	1	no	—
627	C₄	—	4	1	no	—
628	C₄	—	4	1	no	—
629	C₄	—	4	1	no	—
630	C₄	—	4	1	no	—
631	C₄	C₃₁×#546	1	1	yes	characteristic
632	C₄	D₄×Q₈×C₃₁	1	1	yes	characteristic
633	C₄	C₃₁×#552	1	1	yes	characteristic
634	C₄	C₃₁×#554	1	1	yes	characteristic
635	C₂	—	2	4	no	—
636	C₂	C₃₁×#2360	1	1	yes	characteristic
637	C₂	C₃₁×#2929	1	1	yes	characteristic
638	C₂	D₄×M₁₆×C₃₁	1	1	yes	characteristic
639	C₂	C₂×C₃₁×#446	1	1	yes	characteristic
640	C₂	C₃₁×#2962	1	1	yes	characteristic
641	C₂	C₃₁×#2990	1	1	yes	characteristic
642	C₂	C₃₁×#2992	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
1. C₂×C₃₁×#446	C₂	semidirect
4. C₃₁×#1985	C₂	semidirect
5. C₃₁×#1940	C₂	semidirect
6. C₃₁×#1940	C₂	semidirect
7. C₃₁×#1939	C₂	semidirect
8. C₃₁×#1917	C₂	semidirect
9. C₃₁×#1916	C₂	semidirect
12. C₃₁×#1771	C₂	semidirect
114. C₃₁×C₂²⋊C₈	Q₈	semidirect
214. #22954	C₃₁	direct (N × H)
283. C₂×#446	C₂×C₃₁	semidirect
286. #1985	C₂×C₃₁	semidirect
287. #1940	C₂×C₃₁	semidirect
288. #1940	C₂×C₃₁	semidirect
289. #1939	C₂×C₃₁	semidirect
290. #1917	C₂×C₃₁	semidirect
291. #1916	C₂×C₃₁	semidirect
294. #1771	C₂×C₃₁	semidirect
304. C₂²×C₃₁	#446	semidirect
435. C₂²⋊C₈	Q₈×C₃₁	semidirect
452. C₃₁	#22954	direct (N × H)
610. C₂²	C₃₁×#446	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃₁×#446	2
C₂²×C₃₁×C₄⋊C₈	2
C₄×C₃₁×C₂²⋊Q₈	2
C₃₁×#1985	2
C₃₁×#1940	2
C₃₁×#1939	2
C₃₁×#1917	2
C₃₁×#1916	2
C₃₁×#1850	2
C₃₁×#1771	2
C₃₁×#1752	2
C₈×C₂²⋊C₄×C₃₁	2
C₃₁×#1749	2
#22954	31

Subgroups — C31×#22954 #21479713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22954 #21479713