Subgroups — C₃₁×#21980 #21478739

Order 7936.

774 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#433	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#424	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2298	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1940	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1940	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1939	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1939	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1851	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#1851	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#1850	C₂	1	1	yes	maximal, characteristic
11	C₃₁×#1850	C₂	1	1	yes	maximal, characteristic
12	C₃₁×#1843	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#1842	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#1832	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#1831	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
19	C₃₁×#433	—	2	4	no	—
20	C₃₁×#424	—	2	4	no	—
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	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×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
36	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
37	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
39	C₃₁×#388	C₄	1	1	yes	characteristic
40	C₃₁×#388	C₄	1	1	yes	characteristic
41	C₃₁×#388	C₄	1	1	yes	characteristic
42	C₃₁×#388	C₄	1	1	yes	characteristic
43	C₃₁×#387	C₄	1	1	yes	characteristic
44	C₃₁×#387	C₄	1	1	yes	characteristic
45	C₃₁×#387	C₄	1	1	yes	characteristic
46	C₃₁×#387	C₄	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×#376	C₂²	1	1	yes	characteristic
52	C₄³×C₃₁	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	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
62	C₂×M₁₆×C₃₁	—	2	2	no	—
63	C₂×M₁₆×C₃₁	—	4	1	no	—
64	C₂×M₁₆×C₃₁	—	4	1	no	—
65	C₂²×C₈×C₃₁	—	2	2	no	—
66	C₂²×C₈×C₃₁	—	4	1	no	—
67	C₂²×C₈×C₃₁	—	4	1	no	—
68	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
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	C₃₁×#117	D₄	1	4	yes	—
77	C₃₁×#117	Q₈	1	4	yes	—
78	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
84	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
85	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
86	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
87	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
88	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
89	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
90	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
91	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
92	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
93	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
94	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
95	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
96	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
97	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
98	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
99	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
100	C₃₁×C₄⋊C₈	—	2	2	no	—
101	C₃₁×C₄⋊C₈	—	2	2	no	—
102	C₃₁×C₄⋊C₈	—	2	2	no	—
103	C₃₁×C₄⋊C₈	—	2	2	no	—
104	C₃₁×C₄⋊C₈	—	2	2	no	—
105	C₃₁×C₄⋊C₈	—	2	2	no	—
106	C₃₁×C₄⋊C₈	—	4	1	no	—
107	C₃₁×C₄⋊C₈	—	4	1	no	—
108	C₃₁×C₄⋊C₈	—	4	1	no	—
109	C₃₁×C₄⋊C₈	—	4	1	no	—
110	C₃₁×C₄⋊C₈	—	4	1	no	—
111	C₃₁×C₄⋊C₈	—	4	1	no	—
112	C₃₁×C₄⋊C₈	—	4	1	no	—
113	C₃₁×C₄⋊C₈	—	4	1	no	—
114	C₃₁×C₂²⋊C₈	—	2	2	no	—
115	C₃₁×C₂²⋊C₈	—	2	2	no	—
116	C₃₁×#97	—	2	2	no	—
117	C₄×C₈×C₃₁	—	2	2	no	—
118	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
119	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
120	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
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	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
131	C₂²×C₄×C₃₁	—	2	1	no	—
132	C₂²×C₄×C₃₁	—	2	1	no	—
133	C₂²×C₄×C₃₁	—	2	1	no	—
134	C₂²×C₄×C₃₁	—	2	1	no	—
135	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
136	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
137	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
138	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
139	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
140	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
141	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
142	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
143	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
144	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
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	C₂×D₄	1	2	yes	—
149	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
150	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
151	M₁₆×C₃₁	—	4	4	no	—
152	C₂×C₈×C₃₁	—	2	1	no	—
153	C₂×C₈×C₃₁	—	2	1	no	—
154	C₂×C₈×C₃₁	—	2	1	no	—
155	C₂×C₈×C₃₁	—	2	1	no	—
156	C₂×C₈×C₃₁	—	2	1	no	—
157	C₂×C₈×C₃₁	—	2	1	no	—
158	C₂×C₈×C₃₁	—	2	1	no	—
159	C₂×C₈×C₃₁	—	2	1	no	—
160	C₂×C₈×C₃₁	—	2	1	no	—
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	—	4	4	no	—
177	C₄⋊C₄×C₃₁	—	2	2	no	—
178	C₄⋊C₄×C₃₁	—	2	2	no	—
179	C₄⋊C₄×C₃₁	—	4	1	no	—
180	C₄⋊C₄×C₃₁	—	4	1	no	—
181	C₄⋊C₄×C₃₁	C₄⋊C₄	1	4	yes	—
182	C₄⋊C₄×C₃₁	C₄⋊C₄	1	4	yes	—
183	C₂²⋊C₄×C₃₁	—	2	2	no	—
184	C₂²⋊C₄×C₃₁	—	2	2	no	—
185	C₂²⋊C₄×C₃₁	—	4	1	no	—
186	C₂²⋊C₄×C₃₁	—	4	1	no	—
187	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	4	yes	—
188	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	4	yes	—
189	C₄²×C₃₁	—	2	2	no	—
190	C₄²×C₃₁	—	2	2	no	—
191	C₄²×C₃₁	—	2	1	no	—
192	C₄²×C₃₁	—	2	1	no	—
193	C₄²×C₃₁	—	2	1	no	—
194	C₄²×C₃₁	—	2	1	no	—
195	C₄²×C₃₁	—	2	1	no	—
196	C₄²×C₃₁	—	2	1	no	—
197	C₄²×C₃₁	—	2	1	no	—
198	C₄²×C₃₁	—	2	1	no	—
199	C₄²×C₃₁	—	4	1	no	—
200	C₄²×C₃₁	—	4	1	no	—
201	C₄²×C₃₁	C₂×D₄	1	2	yes	—
202	C₄²×C₃₁	C₂×D₄	1	2	yes	—
203	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
204	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
205	#21980	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
206	C₂³×C₃₁	—	2	1	no	—
207	C₂³×C₃₁	—	2	1	no	—
208	C₂³×C₃₁	—	2	1	no	—
209	C₂³×C₃₁	—	2	1	no	—
210	C₂³×C₃₁	—	2	1	no	—
211	C₂³×C₃₁	—	2	1	no	—
212	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
213	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
214	C₂×C₄×C₃₁	—	2	1	no	—
215	C₂×C₄×C₃₁	—	2	1	no	—
216	C₂×C₄×C₃₁	—	2	1	no	—
217	C₂×C₄×C₃₁	—	2	1	no	—
218	C₂×C₄×C₃₁	—	2	1	no	—
219	C₂×C₄×C₃₁	—	2	1	no	—
220	C₂×C₄×C₃₁	—	2	1	no	—
221	C₂×C₄×C₃₁	—	2	1	no	—
222	C₂×C₄×C₃₁	—	2	1	no	—
223	C₂×C₄×C₃₁	—	2	1	no	—
224	C₂×C₄×C₃₁	—	2	1	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	2	no	—
229	C₂×C₄×C₃₁	—	2	2	no	—
230	C₂×C₄×C₃₁	—	2	2	no	—
231	C₂×C₄×C₃₁	—	2	2	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	1	no	—
238	C₂×C₄×C₃₁	—	2	1	no	—
239	C₂×C₄×C₃₁	—	2	1	no	—
240	C₂×C₄×C₃₁	—	2	1	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	C₂×C₄⋊C₄	1	2	yes	—
251	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
252	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
253	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
254	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
255	C₂×C₄×C₃₁	#131	1	2	yes	—
256	C₂×C₄×C₃₁	#131	1	2	yes	—
257	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
258	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
259	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
260	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
261	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
262	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
263	C₈×C₃₁	—	4	1	no	—
264	C₈×C₃₁	—	4	1	no	—
265	C₈×C₃₁	—	4	1	no	—
266	C₈×C₃₁	—	4	1	no	—
267	C₈×C₃₁	—	4	1	no	—
268	C₈×C₃₁	—	4	1	no	—
269	C₈×C₃₁	—	4	1	no	—
270	C₈×C₃₁	—	4	1	no	—
271	C₂×#433	C₂×C₃₁	1	1	yes	characteristic
272	C₂×#424	C₂×C₃₁	1	1	yes	characteristic
273	#2298	C₂×C₃₁	1	1	yes	characteristic
274	#1940	C₂×C₃₁	1	1	yes	characteristic, complemented
275	#1940	C₂×C₃₁	1	1	yes	characteristic, complemented
276	#1939	C₂×C₃₁	1	1	yes	characteristic, complemented
277	#1939	C₂×C₃₁	1	1	yes	characteristic, complemented
278	#1851	C₂×C₃₁	1	1	yes	characteristic
279	#1851	C₂×C₃₁	1	1	yes	characteristic
280	#1850	C₂×C₃₁	1	1	yes	characteristic
281	#1850	C₂×C₃₁	1	1	yes	characteristic
282	#1843	C₂×C₃₁	1	1	yes	characteristic, complemented
283	#1842	C₂×C₃₁	1	1	yes	characteristic, complemented
284	#1832	C₂×C₃₁	1	1	yes	characteristic, complemented
285	#1831	C₂×C₃₁	1	1	yes	characteristic, complemented
286	C₂²×C₃₁	—	2	2	no	—
287	C₂²×C₃₁	—	2	2	no	—
288	C₂²×C₃₁	—	2	2	no	—
289	C₂²×C₃₁	—	4	1	no	—
290	C₂²×C₃₁	—	4	1	no	—
291	C₂²×C₃₁	—	4	1	no	—
292	C₂²×C₃₁	—	4	1	no	—
293	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
294	C₂²×C₃₁	#520	1	1	yes	characteristic
295	C₂²×C₃₁	#520	1	1	yes	characteristic
296	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
297	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
298	C₂²×C₃₁	#568	1	1	yes	characteristic
299	C₂²×C₃₁	#568	1	1	yes	characteristic
300	C₄×C₃₁	—	2	2	no	—
301	C₄×C₃₁	—	2	2	no	—
302	C₄×C₃₁	—	2	2	no	—
303	C₄×C₃₁	—	2	2	no	—
304	C₄×C₃₁	—	2	2	no	—
305	C₄×C₃₁	—	2	2	no	—
306	C₄×C₃₁	—	2	2	no	—
307	C₄×C₃₁	—	4	1	no	—
308	C₄×C₃₁	—	4	1	no	—
309	C₄×C₃₁	—	4	1	no	—
310	C₄×C₃₁	—	4	1	no	—
311	C₄×C₃₁	—	4	1	no	—
312	C₄×C₃₁	—	4	1	no	—
313	C₄×C₃₁	#535	1	1	yes	characteristic
314	C₄×C₃₁	#536	1	1	yes	characteristic
315	C₄×C₃₁	#544	1	1	yes	characteristic
316	C₄×C₃₁	#544	1	1	yes	characteristic
317	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
318	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
319	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
320	#433	—	2	4	no	—
321	#424	—	2	4	no	—
322	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
323	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
324	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
325	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
326	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
327	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
328	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
329	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
330	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
331	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
332	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
333	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
334	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
335	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
336	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
337	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
338	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
339	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
340	#388	C₄×C₃₁	1	1	yes	characteristic
341	#388	C₄×C₃₁	1	1	yes	characteristic
342	#388	C₄×C₃₁	1	1	yes	characteristic
343	#388	C₄×C₃₁	1	1	yes	characteristic
344	#387	C₄×C₃₁	1	1	yes	characteristic
345	#387	C₄×C₃₁	1	1	yes	characteristic
346	#387	C₄×C₃₁	1	1	yes	characteristic
347	#387	C₄×C₃₁	1	1	yes	characteristic
348	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
349	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
350	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
351	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
352	#376	C₂²×C₃₁	1	1	yes	characteristic
353	C₄³	C₂²×C₃₁	1	1	yes	characteristic
354	#336	C₂²×C₃₁	1	1	yes	characteristic
355	#336	C₂²×C₃₁	1	1	yes	characteristic
356	#336	C₂²×C₃₁	1	1	yes	characteristic
357	#336	C₂²×C₃₁	1	1	yes	characteristic
358	#336	C₂²×C₃₁	1	1	yes	characteristic
359	#336	C₂²×C₃₁	1	1	yes	characteristic
360	#336	C₂²×C₃₁	1	1	yes	characteristic
361	#336	C₂²×C₃₁	1	1	yes	characteristic
362	C₂×C₃₁	—	4	2	no	—
363	C₂×C₃₁	#2317	1	1	yes	characteristic
364	C₂×C₃₁	#2906	1	1	yes	characteristic
365	C₂×C₃₁	#2906	1	1	yes	characteristic
366	C₂×C₃₁	#2992	1	1	yes	characteristic
367	C₂×C₃₁	#2993	1	1	yes	characteristic
368	C₂×C₃₁	#2994	1	1	yes	characteristic
369	C₂×C₃₁	#2995	1	1	yes	characteristic
370	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
371	C₂×M₁₆	—	2	2	no	—
372	C₂×M₁₆	—	4	1	no	—
373	C₂×M₁₆	—	4	1	no	—
374	C₂²×C₈	—	2	2	no	—
375	C₂²×C₈	—	4	1	no	—
376	C₂²×C₈	—	4	1	no	—
377	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
378	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
379	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
380	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
381	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
382	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
383	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
384	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
385	#117	D₄×C₃₁	1	4	yes	—
386	#117	Q₈×C₃₁	1	4	yes	—
387	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
388	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
389	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
390	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
391	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
392	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
393	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
394	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
395	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
396	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
397	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
398	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
399	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
400	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
401	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
402	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
403	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
404	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
405	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
406	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
407	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
408	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
409	C₄⋊C₈	—	2	2	no	—
410	C₄⋊C₈	—	2	2	no	—
411	C₄⋊C₈	—	2	2	no	—
412	C₄⋊C₈	—	2	2	no	—
413	C₄⋊C₈	—	2	2	no	—
414	C₄⋊C₈	—	2	2	no	—
415	C₄⋊C₈	—	4	1	no	—
416	C₄⋊C₈	—	4	1	no	—
417	C₄⋊C₈	—	4	1	no	—
418	C₄⋊C₈	—	4	1	no	—
419	C₄⋊C₈	—	4	1	no	—
420	C₄⋊C₈	—	4	1	no	—
421	C₄⋊C₈	—	4	1	no	—
422	C₄⋊C₈	—	4	1	no	—
423	C₂²⋊C₈	—	2	2	no	—
424	C₂²⋊C₈	—	2	2	no	—
425	#97	—	2	2	no	—
426	C₄×C₈	—	2	2	no	—
427	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
428	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
429	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
430	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
431	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
432	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
433	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
434	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
435	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
436	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
437	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
438	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
439	C₃₁	#21980	1	1	yes	characteristic, direct product factor, Sylow
440	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
441	C₂²×C₄	—	2	1	no	—
442	C₂²×C₄	—	2	1	no	—
443	C₂²×C₄	—	2	1	no	—
444	C₂²×C₄	—	2	1	no	—
445	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
446	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
447	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
448	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
449	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
450	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
451	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
452	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
453	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
454	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
455	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
456	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
457	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
458	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
459	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
460	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
461	M₁₆	—	4	4	no	—
462	C₂×C₈	—	2	1	no	—
463	C₂×C₈	—	2	1	no	—
464	C₂×C₈	—	2	1	no	—
465	C₂×C₈	—	2	1	no	—
466	C₂×C₈	—	2	1	no	—
467	C₂×C₈	—	2	1	no	—
468	C₂×C₈	—	2	1	no	—
469	C₂×C₈	—	2	1	no	—
470	C₂×C₈	—	2	1	no	—
471	C₂×C₈	—	2	1	no	—
472	C₂×C₈	—	2	1	no	—
473	C₂×C₈	—	2	1	no	—
474	C₂×C₈	—	2	1	no	—
475	C₂×C₈	—	2	1	no	—
476	C₂×C₈	—	2	1	no	—
477	C₂×C₈	—	2	1	no	—
478	C₂×C₈	—	2	1	no	—
479	C₂×C₈	—	2	1	no	—
480	C₂×C₈	—	2	1	no	—
481	C₂×C₈	—	2	1	no	—
482	C₂×C₈	—	2	1	no	—
483	C₂×C₈	—	2	1	no	—
484	C₂×C₈	—	2	1	no	—
485	C₂×C₈	—	2	1	no	—
486	C₂×C₈	—	4	4	no	—
487	C₄⋊C₄	—	2	2	no	—
488	C₄⋊C₄	—	2	2	no	—
489	C₄⋊C₄	—	4	1	no	—
490	C₄⋊C₄	—	4	1	no	—
491	C₄⋊C₄	C₄⋊C₄×C₃₁	1	4	yes	—
492	C₄⋊C₄	C₄⋊C₄×C₃₁	1	4	yes	—
493	C₂²⋊C₄	—	2	2	no	—
494	C₂²⋊C₄	—	2	2	no	—
495	C₂²⋊C₄	—	4	1	no	—
496	C₂²⋊C₄	—	4	1	no	—
497	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	4	yes	—
498	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	4	yes	—
499	C₄²	—	2	2	no	—
500	C₄²	—	2	2	no	—
501	C₄²	—	2	1	no	—
502	C₄²	—	2	1	no	—
503	C₄²	—	2	1	no	—
504	C₄²	—	2	1	no	—
505	C₄²	—	2	1	no	—
506	C₄²	—	2	1	no	—
507	C₄²	—	2	1	no	—
508	C₄²	—	2	1	no	—
509	C₄²	—	4	1	no	—
510	C₄²	—	4	1	no	—
511	C₄²	C₂×D₄×C₃₁	1	2	yes	—
512	C₄²	C₂×D₄×C₃₁	1	2	yes	—
513	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
514	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
515	C₂³	—	2	1	no	—
516	C₂³	—	2	1	no	—
517	C₂³	—	2	1	no	—
518	C₂³	—	2	1	no	—
519	C₂³	—	2	1	no	—
520	C₂³	—	2	1	no	—
521	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
522	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	1	no	—
525	C₂×C₄	—	2	1	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	1	no	—
542	C₂×C₄	—	2	1	no	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	2	1	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	1	no	—
559	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
560	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
561	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
562	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
563	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
564	C₂×C₄	C₃₁×#131	1	2	yes	—
565	C₂×C₄	C₃₁×#131	1	2	yes	—
566	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
567	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
568	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
569	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
570	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
571	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
572	C₈	—	4	1	no	—
573	C₈	—	4	1	no	—
574	C₈	—	4	1	no	—
575	C₈	—	4	1	no	—
576	C₈	—	4	1	no	—
577	C₈	—	4	1	no	—
578	C₈	—	4	1	no	—
579	C₈	—	4	1	no	—
580	C₂²	—	2	2	no	—
581	C₂²	—	2	2	no	—
582	C₂²	—	2	2	no	—
583	C₂²	—	4	1	no	—
584	C₂²	—	4	1	no	—
585	C₂²	—	4	1	no	—
586	C₂²	—	4	1	no	—
587	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
588	C₂²	C₃₁×#520	1	1	yes	characteristic
589	C₂²	C₃₁×#520	1	1	yes	characteristic
590	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
591	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
592	C₂²	C₃₁×#568	1	1	yes	characteristic
593	C₂²	C₃₁×#568	1	1	yes	characteristic
594	C₄	—	2	2	no	—
595	C₄	—	2	2	no	—
596	C₄	—	2	2	no	—
597	C₄	—	2	2	no	—
598	C₄	—	2	2	no	—
599	C₄	—	2	2	no	—
600	C₄	—	2	2	no	—
601	C₄	—	4	1	no	—
602	C₄	—	4	1	no	—
603	C₄	—	4	1	no	—
604	C₄	—	4	1	no	—
605	C₄	—	4	1	no	—
606	C₄	—	4	1	no	—
607	C₄	C₃₁×#535	1	1	yes	characteristic
608	C₄	C₃₁×#536	1	1	yes	characteristic
609	C₄	C₃₁×#544	1	1	yes	characteristic
610	C₄	C₃₁×#544	1	1	yes	characteristic
611	C₂	—	4	2	no	—
612	C₂	C₃₁×#2317	1	1	yes	characteristic
613	C₂	C₃₁×#2906	1	1	yes	characteristic
614	C₂	C₃₁×#2906	1	1	yes	characteristic
615	C₂	C₃₁×#2992	1	1	yes	characteristic
616	C₂	C₃₁×#2993	1	1	yes	characteristic
617	C₂	C₃₁×#2994	1	1	yes	characteristic
618	C₂	C₃₁×#2995	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×#1940	C₂	semidirect
5. C₃₁×#1940	C₂	semidirect
6. C₃₁×#1939	C₂	semidirect
7. C₃₁×#1939	C₂	semidirect
12. C₃₁×#1843	C₂	semidirect
13. C₃₁×#1842	C₂	semidirect
14. C₃₁×#1832	C₂	semidirect
15. C₃₁×#1831	C₂	semidirect
205. #21980	C₃₁	direct (N × H)
274. #1940	C₂×C₃₁	semidirect
275. #1940	C₂×C₃₁	semidirect
276. #1939	C₂×C₃₁	semidirect
277. #1939	C₂×C₃₁	semidirect
282. #1843	C₂×C₃₁	semidirect
283. #1842	C₂×C₃₁	semidirect
284. #1832	C₂×C₃₁	semidirect
285. #1831	C₂×C₃₁	semidirect
439. C₃₁	#21980	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×#433	2
C₂×C₃₁×#424	2
C₃₁×#2298	2
C₃₁×#1940	2
C₃₁×#1939	2
C₃₁×#1851	2
C₃₁×#1850	2
C₃₁×#1843	2
C₃₁×#1842	2
C₃₁×#1832	2
C₃₁×#1831	2
#21980	31

Subgroups — C31×#21980 #21478739

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21980 #21478739