Subgroups — C₃₁×#20478 #21477237

Order 7936.

926 conjugacy classes of subgroups.

The subgroup lattice has 512 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×#405	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2298	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1771	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1771	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1770	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1753	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#1753	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#1752	C₂	1	1	yes	maximal, characteristic
11	C₈×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
12	C₃₁×#1746	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#1744	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#1726	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×C₃₁×#97	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	C₄	1	8	yes	complemented
20	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
24	C₃₁×#405	C₄	1	8	yes	complemented
25	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
35	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₃₁	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×#388	C₄	1	2	yes	—
40	C₃₁×#388	C₄	1	2	yes	—
41	C₃₁×#387	C₄	1	2	yes	—
42	C₃₁×#387	C₄	1	2	yes	—
43	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
44	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
45	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
46	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
47	C₃₁×#376	C₂²	1	1	yes	characteristic
48	C₄³×C₃₁	C₂²	1	1	yes	characteristic
49	C₃₁×#336	C₂²	1	1	yes	characteristic
50	C₃₁×#336	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	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
58	C₂×M₁₆×C₃₁	—	2	2	no	—
59	C₂×M₁₆×C₃₁	—	2	2	no	—
60	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
61	C₂²×C₈×C₃₁	—	2	2	no	—
62	C₂²×C₈×C₃₁	—	2	2	no	—
63	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
64	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
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	C₃₁×#117	C₂×C₄	1	4	yes	—
73	C₃₁×#117	C₂×C₄	1	4	yes	—
74	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
75	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
76	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
78	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
79	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
80	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
81	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
82	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
83	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
84	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
85	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
86	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
87	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
88	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
89	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
90	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
91	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
92	C₃₁×C₂²⋊C₈	C₂×C₄	1	4	yes	—
93	C₃₁×C₂²⋊C₈	C₂×C₄	1	4	yes	—
94	C₃₁×#97	—	2	2	no	—
95	C₃₁×#97	—	2	2	no	—
96	C₃₁×#97	—	2	2	no	—
97	C₃₁×#97	—	2	2	no	—
98	C₃₁×#97	C₂×C₄	1	4	yes	—
99	C₃₁×#97	C₂×C₄	1	4	yes	—
100	C₃₁×#97	C₂×C₄	1	4	yes	—
101	C₄×C₈×C₃₁	—	2	2	no	—
102	C₄×C₈×C₃₁	—	2	2	no	—
103	C₄×C₈×C₃₁	—	2	2	no	—
104	C₄×C₈×C₃₁	—	2	2	no	—
105	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
106	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
107	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
108	C₃₁×#95	C₂×C₄	1	2	yes	—
109	C₃₁×#95	C₂×C₄	1	2	yes	—
110	C₃₁×#95	C₂×C₄	1	2	yes	—
111	C₃₁×#95	C₂×C₄	1	2	yes	—
112	C₃₁×#95	C₂×C₄	1	2	yes	—
113	C₃₁×#95	C₂×C₄	1	2	yes	—
114	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
115	C₂²×C₄×C₃₁	—	2	1	no	—
116	C₂²×C₄×C₃₁	—	2	1	no	—
117	C₂²×C₄×C₃₁	—	2	1	no	—
118	C₂²×C₄×C₃₁	—	2	1	no	—
119	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
120	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
121	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
122	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
123	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
124	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
125	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
126	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
127	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
128	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
129	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
130	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
131	M₁₆×C₃₁	—	2	8	no	—
132	C₂×C₈×C₃₁	—	2	1	no	—
133	C₂×C₈×C₃₁	—	2	1	no	—
134	C₂×C₈×C₃₁	—	2	1	no	—
135	C₂×C₈×C₃₁	—	2	1	no	—
136	C₂×C₈×C₃₁	—	2	1	no	—
137	C₂×C₈×C₃₁	—	2	1	no	—
138	C₂×C₈×C₃₁	—	2	1	no	—
139	C₂×C₈×C₃₁	—	2	1	no	—
140	C₂×C₈×C₃₁	—	2	1	no	—
141	C₂×C₈×C₃₁	—	2	1	no	—
142	C₂×C₈×C₃₁	—	2	1	no	—
143	C₂×C₈×C₃₁	—	2	1	no	—
144	C₂×C₈×C₃₁	—	2	8	no	—
145	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
146	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
147	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
148	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
149	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
150	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
151	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
152	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
153	C₂×C₈×C₃₁	#41	1	2	yes	—
154	C₂×C₈×C₃₁	#41	1	2	yes	—
155	C₂×C₈×C₃₁	#41	1	2	yes	—
156	C₂×C₈×C₃₁	#41	1	2	yes	—
157	C₄⋊C₄×C₃₁	—	2	4	no	—
158	C₄⋊C₄×C₃₁	—	4	2	no	—
159	C₄⋊C₄×C₃₁	C₄²	1	4	yes	—
160	C₄⋊C₄×C₃₁	C₄²	1	4	yes	—
161	C₂²⋊C₄×C₃₁	—	2	4	no	—
162	C₂²⋊C₄×C₃₁	—	4	2	no	—
163	C₂²⋊C₄×C₃₁	C₄²	1	4	yes	—
164	C₂²⋊C₄×C₃₁	C₄²	1	4	yes	—
165	C₄²×C₃₁	—	2	4	no	—
166	C₄²×C₃₁	—	2	1	no	—
167	C₄²×C₃₁	—	2	1	no	—
168	C₄²×C₃₁	—	2	1	no	—
169	C₄²×C₃₁	—	2	1	no	—
170	C₄²×C₃₁	—	2	1	no	—
171	C₄²×C₃₁	—	2	1	no	—
172	C₄²×C₃₁	—	2	1	no	—
173	C₄²×C₃₁	—	2	1	no	—
174	C₄²×C₃₁	—	4	2	no	—
175	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
176	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
177	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
178	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
179	#20478	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
180	C₂³×C₃₁	—	2	1	no	—
181	C₂³×C₃₁	—	2	1	no	—
182	C₂³×C₃₁	—	2	2	no	—
183	C₂³×C₃₁	—	2	2	no	—
184	C₂³×C₃₁	C₂×C₄²	1	2	yes	—
185	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
186	C₂×C₄×C₃₁	—	2	2	no	—
187	C₂×C₄×C₃₁	—	2	1	no	—
188	C₂×C₄×C₃₁	—	2	1	no	—
189	C₂×C₄×C₃₁	—	2	2	no	—
190	C₂×C₄×C₃₁	—	2	2	no	—
191	C₂×C₄×C₃₁	—	2	2	no	—
192	C₂×C₄×C₃₁	—	2	2	no	—
193	C₂×C₄×C₃₁	—	2	2	no	—
194	C₂×C₄×C₃₁	—	2	2	no	—
195	C₂×C₄×C₃₁	—	2	2	no	—
196	C₂×C₄×C₃₁	—	2	2	no	—
197	C₂×C₄×C₃₁	—	2	2	no	—
198	C₂×C₄×C₃₁	—	2	2	no	—
199	C₂×C₄×C₃₁	—	2	2	no	—
200	C₂×C₄×C₃₁	—	2	2	no	—
201	C₂×C₄×C₃₁	—	2	2	no	—
202	C₂×C₄×C₃₁	—	2	2	no	—
203	C₂×C₄×C₃₁	—	2	2	no	—
204	C₂×C₄×C₃₁	—	2	2	no	—
205	C₂×C₄×C₃₁	—	2	2	no	—
206	C₂×C₄×C₃₁	—	2	2	no	—
207	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
208	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
209	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
210	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
211	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
212	C₂×C₄×C₃₁	#131	1	4	yes	—
213	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
214	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
215	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
216	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
217	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
218	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
219	C₈×C₃₁	—	2	2	no	—
220	C₈×C₃₁	—	2	2	no	—
221	C₈×C₃₁	—	2	2	no	—
222	C₈×C₃₁	—	2	2	no	—
223	C₈×C₃₁	—	2	2	no	—
224	C₈×C₃₁	—	2	2	no	—
225	C₈×C₃₁	#117	1	4	yes	—
226	C₈×C₃₁	#117	1	4	yes	—
227	C₂×#433	C₂×C₃₁	1	1	yes	characteristic
228	C₂×#405	C₂×C₃₁	1	1	yes	characteristic
229	#2298	C₂×C₃₁	1	1	yes	characteristic
230	#1771	C₂×C₃₁	1	1	yes	characteristic, complemented
231	#1771	C₂×C₃₁	1	1	yes	characteristic, complemented
232	#1770	C₂×C₃₁	1	1	yes	characteristic, complemented
233	C₈×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
234	#1753	C₂×C₃₁	1	1	yes	characteristic
235	#1753	C₂×C₃₁	1	1	yes	characteristic
236	#1752	C₂×C₃₁	1	1	yes	characteristic
237	C₈×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic
238	#1746	C₂×C₃₁	1	1	yes	characteristic, complemented
239	#1744	C₂×C₃₁	1	1	yes	characteristic, complemented
240	#1726	C₂×C₃₁	1	1	yes	characteristic, complemented
241	C₄×#97	C₂×C₃₁	1	1	yes	characteristic, complemented
242	C₂²×C₃₁	—	2	2	no	—
243	C₂²×C₃₁	—	2	2	no	—
244	C₂²×C₃₁	—	2	2	no	—
245	C₂²×C₃₁	—	4	2	no	—
246	C₂²×C₃₁	—	4	2	no	—
247	C₂²×C₃₁	C₂²×C₄²	1	1	yes	characteristic
248	C₂²×C₃₁	C₄×#41	1	1	yes	characteristic
249	C₂²×C₃₁	C₄×#41	1	1	yes	characteristic
250	C₂²×C₃₁	C₂×#131	1	2	yes	—
251	C₂²×C₃₁	#568	1	2	yes	—
252	C₄×C₃₁	—	2	2	no	—
253	C₄×C₃₁	—	2	2	no	—
254	C₄×C₃₁	—	2	2	no	—
255	C₄×C₃₁	—	2	2	no	—
256	C₄×C₃₁	—	2	4	no	—
257	C₄×C₃₁	—	2	2	no	—
258	C₄×C₃₁	—	4	2	no	—
259	C₄×C₃₁	—	4	2	no	—
260	C₄×C₃₁	—	4	2	no	—
261	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
262	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
263	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
264	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
265	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
266	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
267	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
268	#433	C₄×C₃₁	1	8	yes	complemented
269	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
270	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
271	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
272	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
273	#405	C₄×C₃₁	1	8	yes	complemented
274	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
275	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
276	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
277	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
278	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
279	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
280	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
281	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
282	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
283	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
284	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
285	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
286	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
287	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
288	#388	C₄×C₃₁	1	2	yes	—
289	#388	C₄×C₃₁	1	2	yes	—
290	#387	C₄×C₃₁	1	2	yes	—
291	#387	C₄×C₃₁	1	2	yes	—
292	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
293	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
294	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
295	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
296	#376	C₂²×C₃₁	1	1	yes	characteristic
297	C₄³	C₂²×C₃₁	1	1	yes	characteristic
298	#336	C₂²×C₃₁	1	1	yes	characteristic
299	#336	C₂²×C₃₁	1	1	yes	characteristic
300	#336	C₂²×C₃₁	1	1	yes	characteristic
301	#336	C₂²×C₃₁	1	1	yes	characteristic
302	#336	C₂²×C₃₁	1	1	yes	characteristic
303	#336	C₂²×C₃₁	1	1	yes	characteristic
304	#336	C₂²×C₃₁	1	1	yes	characteristic
305	#336	C₂²×C₃₁	1	1	yes	characteristic
306	C₂×C₃₁	—	4	2	no	—
307	C₂×C₃₁	C₄×#117	1	1	yes	characteristic
308	C₂×C₃₁	C₄×#131	1	1	yes	characteristic
309	C₂×C₃₁	#2875	1	1	yes	characteristic
310	C₂×C₃₁	#2968	1	2	yes	—
311	C₂×C₃₁	#2969	1	2	yes	—
312	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
313	C₂×M₁₆	—	2	2	no	—
314	C₂×M₁₆	—	2	2	no	—
315	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
316	C₂²×C₈	—	2	2	no	—
317	C₂²×C₈	—	2	2	no	—
318	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
319	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
320	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
321	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
322	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
323	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
324	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
325	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
326	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
327	#117	C₂×C₄×C₃₁	1	4	yes	—
328	#117	C₂×C₄×C₃₁	1	4	yes	—
329	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
330	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
331	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
332	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
333	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
334	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
335	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
336	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
337	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
338	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
339	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
340	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
341	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
342	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
343	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
344	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
345	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
346	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
347	C₂²⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
348	C₂²⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
349	#97	—	2	2	no	—
350	#97	—	2	2	no	—
351	#97	—	2	2	no	—
352	#97	—	2	2	no	—
353	#97	C₂×C₄×C₃₁	1	4	yes	—
354	#97	C₂×C₄×C₃₁	1	4	yes	—
355	#97	C₂×C₄×C₃₁	1	4	yes	—
356	C₄×C₈	—	2	2	no	—
357	C₄×C₈	—	2	2	no	—
358	C₄×C₈	—	2	2	no	—
359	C₄×C₈	—	2	2	no	—
360	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
361	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
362	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
363	#95	C₂×C₄×C₃₁	1	2	yes	—
364	#95	C₂×C₄×C₃₁	1	2	yes	—
365	#95	C₂×C₄×C₃₁	1	2	yes	—
366	#95	C₂×C₄×C₃₁	1	2	yes	—
367	#95	C₂×C₄×C₃₁	1	2	yes	—
368	#95	C₂×C₄×C₃₁	1	2	yes	—
369	C₃₁	#20478	1	1	yes	characteristic, direct product factor, Sylow
370	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	2	1	no	—
373	C₂²×C₄	—	2	1	no	—
374	C₂²×C₄	—	2	1	no	—
375	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
376	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
377	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
378	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
379	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
380	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
381	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
382	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
383	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
384	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
385	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
386	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
387	M₁₆	—	2	8	no	—
388	C₂×C₈	—	2	1	no	—
389	C₂×C₈	—	2	1	no	—
390	C₂×C₈	—	2	1	no	—
391	C₂×C₈	—	2	1	no	—
392	C₂×C₈	—	2	1	no	—
393	C₂×C₈	—	2	1	no	—
394	C₂×C₈	—	2	1	no	—
395	C₂×C₈	—	2	1	no	—
396	C₂×C₈	—	2	1	no	—
397	C₂×C₈	—	2	1	no	—
398	C₂×C₈	—	2	1	no	—
399	C₂×C₈	—	2	1	no	—
400	C₂×C₈	—	2	8	no	—
401	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
402	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
403	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
404	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
405	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
406	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
407	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
408	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
409	C₂×C₈	#41×C₃₁	1	2	yes	—
410	C₂×C₈	#41×C₃₁	1	2	yes	—
411	C₂×C₈	#41×C₃₁	1	2	yes	—
412	C₂×C₈	#41×C₃₁	1	2	yes	—
413	C₄⋊C₄	—	2	4	no	—
414	C₄⋊C₄	—	4	2	no	—
415	C₄⋊C₄	C₄²×C₃₁	1	4	yes	—
416	C₄⋊C₄	C₄²×C₃₁	1	4	yes	—
417	C₂²⋊C₄	—	2	4	no	—
418	C₂²⋊C₄	—	4	2	no	—
419	C₂²⋊C₄	C₄²×C₃₁	1	4	yes	—
420	C₂²⋊C₄	C₄²×C₃₁	1	4	yes	—
421	C₄²	—	2	4	no	—
422	C₄²	—	2	1	no	—
423	C₄²	—	2	1	no	—
424	C₄²	—	2	1	no	—
425	C₄²	—	2	1	no	—
426	C₄²	—	2	1	no	—
427	C₄²	—	2	1	no	—
428	C₄²	—	2	1	no	—
429	C₄²	—	2	1	no	—
430	C₄²	—	4	2	no	—
431	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
432	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
433	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
434	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
435	C₂³	—	2	1	no	—
436	C₂³	—	2	1	no	—
437	C₂³	—	2	2	no	—
438	C₂³	—	2	2	no	—
439	C₂³	C₂×C₄²×C₃₁	1	2	yes	—
440	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	1	no	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
463	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
464	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
465	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
466	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
467	C₂×C₄	C₃₁×#131	1	4	yes	—
468	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
469	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
470	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
471	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
472	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
473	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
474	C₈	—	2	2	no	—
475	C₈	—	2	2	no	—
476	C₈	—	2	2	no	—
477	C₈	—	2	2	no	—
478	C₈	—	2	2	no	—
479	C₈	—	2	2	no	—
480	C₈	C₃₁×#117	1	4	yes	—
481	C₈	C₃₁×#117	1	4	yes	—
482	C₂²	—	2	2	no	—
483	C₂²	—	2	2	no	—
484	C₂²	—	2	2	no	—
485	C₂²	—	4	2	no	—
486	C₂²	—	4	2	no	—
487	C₂²	C₂²×C₄²×C₃₁	1	1	yes	characteristic
488	C₂²	C₄×#41×C₃₁	1	1	yes	characteristic
489	C₂²	C₄×#41×C₃₁	1	1	yes	characteristic
490	C₂²	C₂×C₃₁×#131	1	2	yes	—
491	C₂²	C₃₁×#568	1	2	yes	—
492	C₄	—	2	2	no	—
493	C₄	—	2	2	no	—
494	C₄	—	2	2	no	—
495	C₄	—	2	2	no	—
496	C₄	—	2	4	no	—
497	C₄	—	2	2	no	—
498	C₄	—	4	2	no	—
499	C₄	—	4	2	no	—
500	C₄	—	4	2	no	—
501	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
502	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
503	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
504	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
505	C₂	—	4	2	no	—
506	C₂	C₄×C₃₁×#117	1	1	yes	characteristic
507	C₂	C₄×C₃₁×#131	1	1	yes	characteristic
508	C₂	C₃₁×#2875	1	1	yes	characteristic
509	C₂	C₃₁×#2968	1	2	yes	—
510	C₂	C₃₁×#2969	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator	C₂²	C₂²×C₄²×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
4. C₃₁×#1771	C₂	semidirect
5. C₃₁×#1771	C₂	semidirect
6. C₃₁×#1770	C₂	semidirect
7. C₈×C₄⋊C₄×C₃₁	C₂	semidirect
12. C₃₁×#1746	C₂	semidirect
13. C₃₁×#1744	C₂	semidirect
14. C₃₁×#1726	C₂	semidirect
15. C₄×C₃₁×#97	C₂	semidirect
19. C₃₁×#433	C₄	semidirect
24. C₃₁×#405	C₄	semidirect
179. #20478	C₃₁	direct (N × H)
230. #1771	C₂×C₃₁	semidirect
231. #1771	C₂×C₃₁	semidirect
232. #1770	C₂×C₃₁	semidirect
233. C₈×C₄⋊C₄	C₂×C₃₁	semidirect
238. #1746	C₂×C₃₁	semidirect
239. #1744	C₂×C₃₁	semidirect
240. #1726	C₂×C₃₁	semidirect
241. C₄×#97	C₂×C₃₁	semidirect
268. #433	C₄×C₃₁	semidirect
273. #405	C₄×C₃₁	semidirect
369. C₃₁	#20478	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×#405	2
C₃₁×#2298	2
C₃₁×#1771	2
C₃₁×#1770	2
C₈×C₄⋊C₄×C₃₁	2
C₃₁×#1753	2
C₃₁×#1752	2
C₈×C₂²⋊C₄×C₃₁	2
C₃₁×#1746	2
C₃₁×#1744	2
C₃₁×#1726	2
C₄×C₃₁×#97	2
#20478	31

Subgroups — C31×#20478 #21477237

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20478 #21477237