Subgroups — C₃₁×#23050 #21479809

Order 7936.

880 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#431	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1984	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1939	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1923	C₂	1	2	yes	maximal
7	C₃₁×#1923	C₂	1	2	yes	maximal
8	C₃₁×#1916	C₂	1	2	yes	maximal, complemented
9	C₈×C₄⋊C₄×C₃₁	C₂	1	2	yes	maximal, complemented
10	C₈×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₃₁×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
12	C₂×C₃₁×C₂²⋊Q₈	C₄	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₃₁	—	2	4	no	—
16	C₃₁×#431	—	2	4	no	—
17	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
20	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
22	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#391	C₄	1	1	yes	characteristic
30	C₃₁×#389	C₄	1	1	yes	characteristic
31	C₃₁×#387	C₄	1	2	yes	—
32	C₃₁×#385	C₄	1	2	yes	—
33	C₃₁×#380	C₄	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
35	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
36	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
37	C₃₁×#336	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
43	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
48	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
51	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₃₁×C₂²⋊Q₈	C₈	1	8	yes	complemented
54	C₄×Q₈×C₃₁	—	2	2	no	—
55	C₄×Q₈×C₃₁	—	2	2	no	—
56	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
58	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
60	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
61	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
62	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
63	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
64	C₂×C₄²×C₃₁	—	2	2	no	—
65	C₂×C₄²×C₃₁	—	2	2	no	—
66	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
70	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
71	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
72	C₃₁×C₄⋊C₈	—	2	4	no	—
73	C₃₁×C₄⋊C₈	—	2	4	no	—
74	C₃₁×C₄⋊C₈	—	2	2	no	—
75	C₃₁×C₄⋊C₈	D₄	1	4	yes	—
76	C₃₁×C₂²⋊C₈	—	2	4	no	—
77	C₃₁×C₂²⋊C₈	—	2	2	no	—
78	C₃₁×C₂²⋊C₈	—	2	2	no	—
79	C₄×C₈×C₃₁	—	2	4	no	—
80	C₄×C₈×C₃₁	—	2	4	no	—
81	C₄×C₈×C₃₁	—	2	4	no	—
82	C₄×C₈×C₃₁	—	2	2	no	—
83	C₄×C₈×C₃₁	—	2	2	no	—
84	C₃₁×#95	C₂×C₄	1	2	yes	—
85	C₃₁×#95	C₂×C₄	1	2	yes	—
86	C₃₁×#95	C₂×C₄	1	2	yes	—
87	C₃₁×#95	C₂×C₄	1	2	yes	—
88	C₃₁×#95	C₂×C₄	1	2	yes	—
89	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
90	C₂×Q₈×C₃₁	—	2	2	no	—
91	C₂×Q₈×C₃₁	—	2	2	no	—
92	C₂×Q₈×C₃₁	C₂×C₈	1	4	yes	—
93	C₂²×C₄×C₃₁	—	2	2	no	—
94	C₂²×C₄×C₃₁	—	2	2	no	—
95	C₂²×C₄×C₃₁	—	2	2	no	—
96	C₂²×C₄×C₃₁	—	2	2	no	—
97	C₂²×C₄×C₃₁	—	2	1	no	—
98	C₂²×C₄×C₃₁	—	2	1	no	—
99	C₂²×C₄×C₃₁	—	2	1	no	—
100	C₂²×C₄×C₃₁	—	2	1	no	—
101	C₂²×C₄×C₃₁	—	2	1	no	—
102	C₂²×C₄×C₃₁	—	2	1	no	—
103	C₂²×C₄×C₃₁	—	4	1	no	—
104	C₂²×C₄×C₃₁	C₂×C₈	1	4	yes	—
105	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
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	2	yes	—
111	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
112	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
115	C₂×C₈×C₃₁	—	2	1	no	—
116	C₂×C₈×C₃₁	—	2	2	no	—
117	C₂×C₈×C₃₁	—	2	1	no	—
118	C₂×C₈×C₃₁	—	2	2	no	—
119	C₂×C₈×C₃₁	—	2	1	no	—
120	C₂×C₈×C₃₁	—	2	1	no	—
121	C₂×C₈×C₃₁	—	2	2	no	—
122	C₂×C₈×C₃₁	—	2	1	no	—
123	C₂×C₈×C₃₁	—	2	1	no	—
124	C₂×C₈×C₃₁	—	2	1	no	—
125	C₂×C₈×C₃₁	—	2	1	no	—
126	C₂×C₈×C₃₁	—	2	2	no	—
127	C₂×C₈×C₃₁	—	2	1	no	—
128	C₂×C₈×C₃₁	—	2	2	no	—
129	C₂×C₈×C₃₁	—	2	1	no	—
130	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
131	C₂×C₈×C₃₁	#41	1	2	yes	—
132	C₂×C₈×C₃₁	#41	1	2	yes	—
133	C₄⋊C₄×C₃₁	—	2	4	no	—
134	C₄⋊C₄×C₃₁	—	2	2	no	—
135	C₄⋊C₄×C₃₁	—	2	4	no	—
136	C₄⋊C₄×C₃₁	—	2	2	no	—
137	C₄⋊C₄×C₃₁	C₂×C₈	1	4	yes	—
138	C₄⋊C₄×C₃₁	C₂×C₈	1	8	yes	—
139	C₂²⋊C₄×C₃₁	—	4	2	no	—
140	C₂²⋊C₄×C₃₁	C₂×C₈	1	8	yes	—
141	C₄²×C₃₁	—	2	2	no	—
142	C₄²×C₃₁	—	2	2	no	—
143	C₄²×C₃₁	—	2	1	no	—
144	C₄²×C₃₁	—	2	2	no	—
145	C₄²×C₃₁	—	2	2	no	—
146	C₄²×C₃₁	—	2	2	no	—
147	C₄²×C₃₁	—	2	1	no	—
148	C₄²×C₃₁	—	2	2	no	—
149	C₄²×C₃₁	C₂×D₄	1	2	yes	—
150	C₄²×C₃₁	#41	1	2	yes	—
151	#23050	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂³×C₃₁	—	2	1	no	—
153	C₂³×C₃₁	—	2	1	no	—
154	C₂³×C₃₁	—	2	1	no	—
155	C₂³×C₃₁	—	2	1	no	—
156	C₂³×C₃₁	—	2	1	no	—
157	C₂³×C₃₁	—	2	1	no	—
158	C₂³×C₃₁	C₂²×C₈	1	2	yes	—
159	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
160	Q₈×C₃₁	—	2	8	no	—
161	C₂×C₄×C₃₁	—	2	1	no	—
162	C₂×C₄×C₃₁	—	2	1	no	—
163	C₂×C₄×C₃₁	—	2	2	no	—
164	C₂×C₄×C₃₁	—	2	2	no	—
165	C₂×C₄×C₃₁	—	2	2	no	—
166	C₂×C₄×C₃₁	—	2	1	no	—
167	C₂×C₄×C₃₁	—	2	1	no	—
168	C₂×C₄×C₃₁	—	2	2	no	—
169	C₂×C₄×C₃₁	—	2	1	no	—
170	C₂×C₄×C₃₁	—	2	1	no	—
171	C₂×C₄×C₃₁	—	2	8	no	—
172	C₂×C₄×C₃₁	—	2	1	no	—
173	C₂×C₄×C₃₁	—	2	2	no	—
174	C₂×C₄×C₃₁	—	2	2	no	—
175	C₂×C₄×C₃₁	—	2	1	no	—
176	C₂×C₄×C₃₁	—	2	1	no	—
177	C₂×C₄×C₃₁	—	2	2	no	—
178	C₂×C₄×C₃₁	—	2	2	no	—
179	C₂×C₄×C₃₁	—	2	2	no	—
180	C₂×C₄×C₃₁	—	2	2	no	—
181	C₂×C₄×C₃₁	—	4	4	no	—
182	C₂×C₄×C₃₁	—	4	1	no	—
183	C₂×C₄×C₃₁	—	4	1	no	—
184	C₂×C₄×C₃₁	—	4	1	no	—
185	C₂×C₄×C₃₁	—	4	1	no	—
186	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
187	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
188	C₂×C₄×C₃₁	C₂²×C₈	1	4	yes	—
189	C₂×C₄×C₃₁	C₂²×C₈	1	4	yes	—
190	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
191	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
192	C₂×C₄×C₃₁	#131	1	2	yes	—
193	C₂×C₄×C₃₁	#131	1	2	yes	—
194	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
196	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
197	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
198	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
199	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
200	C₈×C₃₁	—	2	4	no	—
201	C₈×C₃₁	—	2	2	no	—
202	C₈×C₃₁	—	2	2	no	—
203	C₈×C₃₁	—	4	1	no	—
204	C₈×C₃₁	—	4	2	no	—
205	C₈×C₃₁	—	4	1	no	—
206	C₂×C₈×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
207	C₂×#431	C₂×C₃₁	1	1	yes	characteristic
208	C₄×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic
209	#1984	C₂×C₃₁	1	1	yes	characteristic, complemented
210	#1939	C₂×C₃₁	1	1	yes	characteristic, complemented
211	#1923	C₂×C₃₁	1	2	yes	—
212	#1923	C₂×C₃₁	1	2	yes	—
213	#1916	C₂×C₃₁	1	2	yes	complemented
214	C₈×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
215	C₈×C₄⋊C₄	C₂×C₃₁	1	2	yes	complemented
216	C₄×C₂²⋊C₈	C₂×C₃₁	1	1	yes	characteristic
217	C₂²×C₃₁	—	2	2	no	—
218	C₂²×C₃₁	—	2	2	no	—
219	C₂²×C₃₁	—	2	2	no	—
220	C₂²×C₃₁	—	4	1	no	—
221	C₂²×C₃₁	—	4	1	no	—
222	C₂²×C₃₁	—	4	1	no	—
223	C₂²×C₃₁	—	4	1	no	—
224	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
225	C₂²×C₃₁	C₄×#41	1	1	yes	characteristic
226	C₂²×C₃₁	#519	1	1	yes	characteristic
227	C₂²×C₃₁	C₂³×C₈	1	1	yes	characteristic
228	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
229	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
230	C₂²×C₃₁	#568	1	1	yes	characteristic
231	C₄×C₃₁	—	2	2	no	—
232	C₄×C₃₁	—	2	4	no	—
233	C₄×C₃₁	—	2	2	no	—
234	C₄×C₃₁	—	2	2	no	—
235	C₄×C₃₁	—	2	4	no	—
236	C₄×C₃₁	—	4	1	no	—
237	C₄×C₃₁	—	4	2	no	—
238	C₄×C₃₁	—	4	2	no	—
239	C₄×C₃₁	C₈×D₄	1	4	yes	—
240	C₄×C₃₁	#547	1	1	yes	characteristic
241	C₄×C₃₁	#548	1	1	yes	characteristic
242	C₄×C₃₁	#552	1	1	yes	characteristic
243	C₄×C₃₁	#556	1	1	yes	characteristic
244	C₂×C₂²⋊Q₈	C₄×C₃₁	1	1	yes	characteristic
245	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
246	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
247	C₈×Q₈	—	2	4	no	—
248	#431	—	2	4	no	—
249	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
250	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
251	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
252	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
253	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
254	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
255	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
256	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
257	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
258	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
259	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
260	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
261	#391	C₄×C₃₁	1	1	yes	characteristic
262	#389	C₄×C₃₁	1	1	yes	characteristic
263	#387	C₄×C₃₁	1	2	yes	—
264	#385	C₄×C₃₁	1	2	yes	—
265	#380	C₄×C₃₁	1	1	yes	characteristic
266	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
267	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
268	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
269	#336	C₂²×C₃₁	1	1	yes	characteristic
270	#336	C₂²×C₃₁	1	2	yes	—
271	#336	C₂²×C₃₁	1	2	yes	—
272	#336	C₂²×C₃₁	1	2	yes	—
273	#336	C₂²×C₃₁	1	2	yes	—
274	#336	C₂²×C₃₁	1	1	yes	characteristic
275	C₂×C₃₁	—	4	2	no	—
276	C₂×C₃₁	#2362	1	1	yes	characteristic
277	C₂×C₃₁	C₂×C₈×D₄	1	1	yes	characteristic
278	C₂×C₃₁	#2933	1	1	yes	characteristic
279	C₂×C₃₁	C₈×#41	1	1	yes	characteristic
280	C₂×C₃₁	#2969	1	1	yes	characteristic
281	C₂×C₃₁	#2982	1	1	yes	characteristic
282	C₂×C₃₁	#2986	1	1	yes	characteristic
283	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
284	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
285	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
286	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
287	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
288	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
289	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
290	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
291	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
292	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
293	C₂²⋊Q₈	C₈×C₃₁	1	8	yes	complemented
294	C₄×Q₈	—	2	2	no	—
295	C₄×Q₈	—	2	2	no	—
296	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
297	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
298	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
299	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
300	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
301	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
302	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
303	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
304	C₂×C₄²	—	2	2	no	—
305	C₂×C₄²	—	2	2	no	—
306	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
307	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
308	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
309	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
310	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
311	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
312	C₄⋊C₈	—	2	4	no	—
313	C₄⋊C₈	—	2	4	no	—
314	C₄⋊C₈	—	2	2	no	—
315	C₄⋊C₈	D₄×C₃₁	1	4	yes	—
316	C₂²⋊C₈	—	2	4	no	—
317	C₂²⋊C₈	—	2	2	no	—
318	C₂²⋊C₈	—	2	2	no	—
319	C₄×C₈	—	2	4	no	—
320	C₄×C₈	—	2	4	no	—
321	C₄×C₈	—	2	4	no	—
322	C₄×C₈	—	2	2	no	—
323	C₄×C₈	—	2	2	no	—
324	#95	C₂×C₄×C₃₁	1	2	yes	—
325	#95	C₂×C₄×C₃₁	1	2	yes	—
326	#95	C₂×C₄×C₃₁	1	2	yes	—
327	#95	C₂×C₄×C₃₁	1	2	yes	—
328	#95	C₂×C₄×C₃₁	1	2	yes	—
329	C₃₁	#23050	1	1	yes	characteristic, direct product factor, Sylow
330	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
331	C₂×Q₈	—	2	2	no	—
332	C₂×Q₈	—	2	2	no	—
333	C₂×Q₈	C₂×C₈×C₃₁	1	4	yes	—
334	C₂²×C₄	—	2	2	no	—
335	C₂²×C₄	—	2	2	no	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	2	2	no	—
338	C₂²×C₄	—	2	1	no	—
339	C₂²×C₄	—	2	1	no	—
340	C₂²×C₄	—	2	1	no	—
341	C₂²×C₄	—	2	1	no	—
342	C₂²×C₄	—	2	1	no	—
343	C₂²×C₄	—	2	1	no	—
344	C₂²×C₄	—	4	1	no	—
345	C₂²×C₄	C₂×C₈×C₃₁	1	4	yes	—
346	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
347	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
348	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
349	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
350	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
351	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
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	C₂⁴×C₃₁	1	1	yes	characteristic
356	C₂×C₈	—	2	1	no	—
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	—	2	1	no	—
359	C₂×C₈	—	2	2	no	—
360	C₂×C₈	—	2	1	no	—
361	C₂×C₈	—	2	1	no	—
362	C₂×C₈	—	2	2	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	1	no	—
369	C₂×C₈	—	2	2	no	—
370	C₂×C₈	—	2	1	no	—
371	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
372	C₂×C₈	#41×C₃₁	1	2	yes	—
373	C₂×C₈	#41×C₃₁	1	2	yes	—
374	C₄⋊C₄	—	2	4	no	—
375	C₄⋊C₄	—	2	2	no	—
376	C₄⋊C₄	—	2	4	no	—
377	C₄⋊C₄	—	2	2	no	—
378	C₄⋊C₄	C₂×C₈×C₃₁	1	4	yes	—
379	C₄⋊C₄	C₂×C₈×C₃₁	1	8	yes	—
380	C₂²⋊C₄	—	4	2	no	—
381	C₂²⋊C₄	C₂×C₈×C₃₁	1	8	yes	—
382	C₄²	—	2	2	no	—
383	C₄²	—	2	2	no	—
384	C₄²	—	2	1	no	—
385	C₄²	—	2	2	no	—
386	C₄²	—	2	2	no	—
387	C₄²	—	2	2	no	—
388	C₄²	—	2	1	no	—
389	C₄²	—	2	2	no	—
390	C₄²	C₂×D₄×C₃₁	1	2	yes	—
391	C₄²	#41×C₃₁	1	2	yes	—
392	C₂³	—	2	1	no	—
393	C₂³	—	2	1	no	—
394	C₂³	—	2	1	no	—
395	C₂³	—	2	1	no	—
396	C₂³	—	2	1	no	—
397	C₂³	—	2	1	no	—
398	C₂³	C₂²×C₈×C₃₁	1	2	yes	—
399	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
400	Q₈	—	2	8	no	—
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	2	1	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	1	no	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	8	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	1	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	4	4	no	—
422	C₂×C₄	—	4	1	no	—
423	C₂×C₄	—	4	1	no	—
424	C₂×C₄	—	4	1	no	—
425	C₂×C₄	—	4	1	no	—
426	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
427	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
428	C₂×C₄	C₂²×C₈×C₃₁	1	4	yes	—
429	C₂×C₄	C₂²×C₈×C₃₁	1	4	yes	—
430	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
431	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
432	C₂×C₄	C₃₁×#131	1	2	yes	—
433	C₂×C₄	C₃₁×#131	1	2	yes	—
434	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
435	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
436	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
437	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
438	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
439	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
440	C₈	—	2	4	no	—
441	C₈	—	2	2	no	—
442	C₈	—	2	2	no	—
443	C₈	—	4	1	no	—
444	C₈	—	4	2	no	—
445	C₈	—	4	1	no	—
446	C₂²	—	2	2	no	—
447	C₂²	—	2	2	no	—
448	C₂²	—	2	2	no	—
449	C₂²	—	4	1	no	—
450	C₂²	—	4	1	no	—
451	C₂²	—	4	1	no	—
452	C₂²	—	4	1	no	—
453	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
454	C₂²	C₄×#41×C₃₁	1	1	yes	characteristic
455	C₂²	C₃₁×#519	1	1	yes	characteristic
456	C₂²	C₂³×C₈×C₃₁	1	1	yes	characteristic
457	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
458	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
459	C₂²	C₃₁×#568	1	1	yes	characteristic
460	C₄	—	2	2	no	—
461	C₄	—	2	4	no	—
462	C₄	—	2	2	no	—
463	C₄	—	2	2	no	—
464	C₄	—	2	4	no	—
465	C₄	—	4	1	no	—
466	C₄	—	4	2	no	—
467	C₄	—	4	2	no	—
468	C₄	C₈×D₄×C₃₁	1	4	yes	—
469	C₄	C₃₁×#547	1	1	yes	characteristic
470	C₄	C₃₁×#548	1	1	yes	characteristic
471	C₄	C₃₁×#552	1	1	yes	characteristic
472	C₄	C₃₁×#556	1	1	yes	characteristic
473	C₂	—	4	2	no	—
474	C₂	C₃₁×#2362	1	1	yes	characteristic
475	C₂	C₂×C₈×D₄×C₃₁	1	1	yes	characteristic
476	C₂	C₃₁×#2933	1	1	yes	characteristic
477	C₂	C₈×#41×C₃₁	1	1	yes	characteristic
478	C₂	C₃₁×#2969	1	1	yes	characteristic
479	C₂	C₃₁×#2982	1	1	yes	characteristic
480	C₂	C₃₁×#2986	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator	C₂²	C₂³×C₈×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂³×C₄
prime radical	C₂⁴×C₃₁	C₂²×C₄
LCS	G ▷ C₂² ▷ 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×C₈×Q₈×C₃₁	C₂	semidirect
4. C₃₁×#1984	C₂	semidirect
5. C₃₁×#1939	C₂	semidirect
8. C₃₁×#1916	C₂	semidirect
9. C₈×C₄⋊C₄×C₃₁	C₂	semidirect
10. C₈×C₄⋊C₄×C₃₁	C₂	semidirect
53. C₃₁×C₂²⋊Q₈	C₈	semidirect
151. #23050	C₃₁	direct (N × H)
206. C₂×C₈×Q₈	C₂×C₃₁	semidirect
209. #1984	C₂×C₃₁	semidirect
210. #1939	C₂×C₃₁	semidirect
213. #1916	C₂×C₃₁	semidirect
214. C₈×C₄⋊C₄	C₂×C₃₁	semidirect
215. C₈×C₄⋊C₄	C₂×C₃₁	semidirect
293. C₂²⋊Q₈	C₈×C₃₁	semidirect
329. C₃₁	#23050	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×C₈×Q₈×C₃₁	2
C₂×C₃₁×#431	2
C₄×C₃₁×C₂²⋊Q₈	2
C₃₁×#1984	2
C₃₁×#1939	2
C₃₁×#1923	2
C₃₁×#1916	2
C₈×C₄⋊C₄×C₃₁	2
C₄×C₃₁×C₂²⋊C₈	2
#23050	31

Subgroups — C31×#23050 #21479809

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23050 #21479809