Subgroups — C₃₁×#24747 #21481506

Order 7936.

968 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#464	C₂	1	2	yes	maximal, complemented
2	C₂×C₃₁×#460	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#441	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2548	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#2351	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2065	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2042	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1961	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#1940	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1880	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1805	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
14	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#124	C₂²	1	2	yes	—
16	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
18	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#464	—	2	8	no	—
20	C₃₁×#460	—	2	4	no	—
21	C₃₁×#441	—	2	4	no	—
22	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
25	C₂×C₃₁×#103	C₂²	1	2	yes	—
26	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×#102	C₂²	1	2	yes	—
29	C₂×C₃₁×#102	C₂²	1	2	yes	—
30	C₂×C₃₁×#102	C₂²	1	2	yes	—
31	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
32	C₃₁×#394	—	2	1	no	—
33	C₃₁×#394	—	2	1	no	—
34	C₃₁×#390	—	2	2	no	—
35	C₃₁×#390	C₂²	1	2	yes	—
36	C₃₁×#389	—	2	2	no	—
37	C₃₁×#389	C₂²	1	2	yes	—
38	C₃₁×#382	—	2	1	no	—
39	C₃₁×#382	C₂²	1	1	yes	characteristic
40	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
41	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
42	C₃₁×#340	C₂²	1	2	yes	—
43	C₃₁×#336	C₂²	1	2	yes	—
44	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₂²×D₄×C₃₁	—	2	1	no	—
47	C₂²×D₄×C₃₁	—	2	1	no	—
48	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
49	C₂×Q₁₆×C₃₁	—	2	2	no	—
50	C₂×Q₁₆×C₃₁	—	2	2	no	—
51	C₂×Q₁₆×C₃₁	D₄	1	4	yes	complemented
52	C₂×SD₁₆×C₃₁	—	2	4	no	—
53	C₂×SD₁₆×C₃₁	—	2	4	no	—
54	C₂×SD₁₆×C₃₁	—	2	4	no	—
55	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
57	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
58	C₃₁×#127	—	2	2	no	—
59	C₃₁×#127	—	2	2	no	—
60	C₃₁×#124	—	2	4	no	—
61	C₃₁×#124	—	4	2	no	—
62	C₄×Q₈×C₃₁	—	2	2	no	—
63	C₄×Q₈×C₃₁	—	2	4	no	—
64	C₄×Q₈×C₃₁	—	2	2	no	—
65	C₄×Q₈×C₃₁	—	2	2	no	—
66	C₄×Q₈×C₃₁	—	4	2	no	—
67	C₄×Q₈×C₃₁	D₄	1	4	yes	—
68	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
69	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
70	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
71	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
72	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
73	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
74	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
75	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
76	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
77	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
78	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
79	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
80	C₂×C₄²×C₃₁	—	2	1	no	—
81	C₂×C₄²×C₃₁	—	2	2	no	—
82	C₂×C₄²×C₃₁	—	2	1	no	—
83	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
84	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
85	C₃₁×#106	—	2	2	no	—
86	C₃₁×C₄⋊C₈	—	2	4	no	—
87	C₃₁×C₄⋊C₈	—	2	2	no	—
88	C₃₁×#103	—	2	2	no	—
89	C₃₁×#103	—	2	4	no	—
90	C₃₁×#103	—	2	2	no	—
91	C₃₁×#102	—	2	4	no	—
92	C₃₁×#102	—	2	4	no	—
93	C₃₁×#102	—	4	2	no	—
94	C₃₁×#97	—	2	2	no	—
95	C₃₁×#95	—	2	1	no	—
96	C₃₁×#95	—	2	2	no	—
97	C₃₁×#95	—	2	1	no	—
98	C₃₁×#95	D₄	1	1	yes	characteristic
99	C₃₁×#95	D₄	1	1	yes	characteristic
100	C₂⁴×C₃₁	—	2	2	no	—
101	C₂×Q₈×C₃₁	—	2	1	no	—
102	C₂×Q₈×C₃₁	—	2	1	no	—
103	C₂×Q₈×C₃₁	—	2	1	no	—
104	C₂×Q₈×C₃₁	—	2	1	no	—
105	C₂×Q₈×C₃₁	—	2	1	no	—
106	C₂×Q₈×C₃₁	—	2	1	no	—
107	C₂×Q₈×C₃₁	—	2	1	no	—
108	C₂×Q₈×C₃₁	—	2	1	no	—
109	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
110	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
111	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
112	C₂×Q₈×C₃₁	#41	1	2	yes	—
113	C₂×D₄×C₃₁	—	2	2	no	—
114	C₂×D₄×C₃₁	—	2	2	no	—
115	C₂×D₄×C₃₁	—	2	2	no	—
116	C₂×D₄×C₃₁	—	2	2	no	—
117	C₂×D₄×C₃₁	—	2	2	no	—
118	C₂×D₄×C₃₁	—	2	2	no	—
119	C₂×D₄×C₃₁	—	4	4	no	—
120	C₂×D₄×C₃₁	—	4	2	no	—
121	C₂×D₄×C₃₁	—	4	2	no	—
122	C₂×D₄×C₃₁	—	8	2	no	—
123	C₂²×C₄×C₃₁	—	2	1	no	—
124	C₂²×C₄×C₃₁	—	2	1	no	—
125	C₂²×C₄×C₃₁	—	2	1	no	—
126	C₂²×C₄×C₃₁	—	2	2	no	—
127	C₂²×C₄×C₃₁	—	2	1	no	—
128	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
129	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
130	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
133	Q₁₆×C₃₁	—	2	8	no	—
134	SD₁₆×C₃₁	—	4	8	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	2	no	—
139	C₂×C₈×C₃₁	—	2	2	no	—
140	C₂×C₈×C₃₁	—	2	2	no	—
141	C₂×C₈×C₃₁	—	2	1	no	—
142	C₂×C₈×C₃₁	—	2	1	no	—
143	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
144	C₄⋊C₄×C₃₁	—	2	4	no	—
145	C₄⋊C₄×C₃₁	—	2	1	no	—
146	C₄⋊C₄×C₃₁	—	2	2	no	—
147	C₄⋊C₄×C₃₁	—	2	1	no	—
148	C₄⋊C₄×C₃₁	—	2	1	no	—
149	C₄⋊C₄×C₃₁	—	2	2	no	—
150	C₄⋊C₄×C₃₁	—	4	2	no	—
151	C₄⋊C₄×C₃₁	—	4	4	no	—
152	C₄⋊C₄×C₃₁	—	4	4	no	—
153	C₄⋊C₄×C₃₁	—	4	2	no	—
154	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
155	C₂²⋊C₄×C₃₁	—	4	4	no	—
156	C₂²⋊C₄×C₃₁	—	4	4	no	—
157	C₂²⋊C₄×C₃₁	—	8	2	no	—
158	C₂²⋊C₄×C₃₁	—	8	2	no	—
159	C₄²×C₃₁	—	2	4	no	—
160	C₄²×C₃₁	—	2	2	no	—
161	C₄²×C₃₁	—	2	2	no	—
162	C₄²×C₃₁	—	4	4	no	—
163	C₄²×C₃₁	—	4	2	no	—
164	C₄²×C₃₁	C₂×D₄	1	2	yes	—
165	C₄²×C₃₁	#41	1	2	yes	—
166	#24747	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
167	C₂³×C₃₁	—	4	2	no	—
168	C₂³×C₃₁	—	4	2	no	—
169	C₂³×C₃₁	—	4	2	no	—
170	C₂³×C₃₁	—	4	2	no	—
171	C₂³×C₃₁	—	4	2	no	—
172	C₂³×C₃₁	—	4	2	no	—
173	C₂³×C₃₁	—	4	2	no	—
174	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
175	Q₈×C₃₁	—	2	2	no	—
176	Q₈×C₃₁	—	2	2	no	—
177	Q₈×C₃₁	—	2	2	no	—
178	Q₈×C₃₁	—	2	2	no	—
179	Q₈×C₃₁	—	2	2	no	—
180	Q₈×C₃₁	—	4	1	no	—
181	Q₈×C₃₁	—	4	1	no	—
182	Q₈×C₃₁	#121	1	4	yes	—
183	D₄×C₃₁	—	4	2	no	—
184	D₄×C₃₁	—	4	2	no	—
185	D₄×C₃₁	—	4	2	no	—
186	D₄×C₃₁	—	4	2	no	—
187	D₄×C₃₁	—	8	4	no	—
188	D₄×C₃₁	—	8	4	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	—	4	1	no	—
203	C₂×C₄×C₃₁	—	4	1	no	—
204	C₂×C₄×C₃₁	—	4	2	no	—
205	C₂×C₄×C₃₁	—	4	1	no	—
206	C₂×C₄×C₃₁	—	4	2	no	—
207	C₂×C₄×C₃₁	—	4	2	no	—
208	C₂×C₄×C₃₁	—	4	2	no	—
209	C₂×C₄×C₃₁	—	4	2	no	—
210	C₂×C₄×C₃₁	#121	1	2	yes	—
211	C₂×C₄×C₃₁	#121	1	2	yes	—
212	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
213	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
215	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
216	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
217	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
218	C₈×C₃₁	—	2	2	no	—
219	C₈×C₃₁	—	4	2	no	—
220	C₈×C₃₁	—	4	1	no	—
221	C₂×#464	C₂×C₃₁	1	2	yes	complemented
222	C₂×#460	C₂×C₃₁	1	1	yes	characteristic
223	C₂×#441	C₂×C₃₁	1	1	yes	characteristic, complemented
224	#2548	C₂×C₃₁	1	1	yes	characteristic
225	#2351	C₂×C₃₁	1	1	yes	characteristic, complemented
226	#2065	C₂×C₃₁	1	2	yes	complemented
227	#2042	C₂×C₃₁	1	2	yes	complemented
228	#1961	C₂×C₃₁	1	1	yes	characteristic
229	#1940	C₂×C₃₁	1	1	yes	characteristic, complemented
230	#1880	C₂×C₃₁	1	2	yes	complemented
231	#1805	C₂×C₃₁	1	1	yes	characteristic, complemented
232	C₂²×C₃₁	—	8	2	no	—
233	C₂²×C₃₁	—	8	2	no	—
234	C₂²×C₃₁	—	8	2	no	—
235	C₂²×C₃₁	—	8	2	no	—
236	C₂²×C₃₁	—	8	2	no	—
237	C₂²×C₃₁	—	8	2	no	—
238	C₂²×C₃₁	—	8	2	no	—
239	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
240	C₂²×C₃₁	D₄²	1	1	yes	characteristic
241	C₂²×C₃₁	#548	1	1	yes	characteristic
242	C₂²×C₃₁	#575	1	2	yes	—
243	C₂²×C₃₁	#577	1	2	yes	—
244	C₄×C₃₁	—	2	4	no	—
245	C₄×C₃₁	—	2	2	no	—
246	C₄×C₃₁	—	2	2	no	—
247	C₄×C₃₁	—	4	2	no	—
248	C₄×C₃₁	—	4	2	no	—
249	C₄×C₃₁	—	4	2	no	—
250	C₄×C₃₁	—	4	2	no	—
251	C₄×C₃₁	—	8	2	no	—
252	C₄×C₃₁	—	8	1	no	—
253	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
254	C₄×C₃₁	#537	1	1	yes	characteristic
255	C₄×C₃₁	#547	1	1	yes	characteristic
256	C₄×C₃₁	#553	1	1	yes	characteristic
257	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
258	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
259	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
260	C₂×#124	C₂²×C₃₁	1	2	yes	—
261	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
262	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
263	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
264	#464	—	2	8	no	—
265	#460	—	2	4	no	—
266	#441	—	2	4	no	—
267	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
268	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
269	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
270	C₂×#103	C₂²×C₃₁	1	2	yes	—
271	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
272	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
273	C₂×#102	C₂²×C₃₁	1	2	yes	—
274	C₂×#102	C₂²×C₃₁	1	2	yes	—
275	C₂×#102	C₂²×C₃₁	1	2	yes	—
276	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
277	#394	—	2	1	no	—
278	#394	—	2	1	no	—
279	#390	—	2	2	no	—
280	#390	C₂²×C₃₁	1	2	yes	—
281	#389	—	2	2	no	—
282	#389	C₂²×C₃₁	1	2	yes	—
283	#382	—	2	1	no	—
284	#382	C₂²×C₃₁	1	1	yes	characteristic
285	C₄×C₄⋊C₄	—	2	1	no	—
286	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
287	#340	C₂²×C₃₁	1	2	yes	—
288	#336	C₂²×C₃₁	1	2	yes	—
289	C₂×C₃₁	—	16	2	no	—
290	C₂×C₃₁	#2416	1	1	yes	characteristic
291	C₂×C₃₁	#3040	1	1	yes	characteristic
292	C₂×C₃₁	#3043	1	1	yes	characteristic
293	C₂×C₃₁	#3271	1	2	yes	—
294	C₂×C₃₁	#3371	1	2	yes	—
295	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
296	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
297	C₂²×D₄	—	2	1	no	—
298	C₂²×D₄	—	2	1	no	—
299	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
300	C₂×Q₁₆	—	2	2	no	—
301	C₂×Q₁₆	—	2	2	no	—
302	C₂×Q₁₆	D₄×C₃₁	1	4	yes	complemented
303	C₂×SD₁₆	—	2	4	no	—
304	C₂×SD₁₆	—	2	4	no	—
305	C₂×SD₁₆	—	2	4	no	—
306	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
307	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
308	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
309	#127	—	2	2	no	—
310	#127	—	2	2	no	—
311	#124	—	2	4	no	—
312	#124	—	4	2	no	—
313	C₄×Q₈	—	2	2	no	—
314	C₄×Q₈	—	2	4	no	—
315	C₄×Q₈	—	2	2	no	—
316	C₄×Q₈	—	2	2	no	—
317	C₄×Q₈	—	4	2	no	—
318	C₄×Q₈	D₄×C₃₁	1	4	yes	—
319	C₂×C₄⋊C₄	—	2	1	no	—
320	C₂×C₄⋊C₄	—	2	2	no	—
321	C₂×C₄⋊C₄	—	2	1	no	—
322	C₂×C₄⋊C₄	—	2	2	no	—
323	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
324	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
325	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
326	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
327	C₂×C₂²⋊C₄	—	2	2	no	—
328	C₂×C₂²⋊C₄	—	2	2	no	—
329	C₂×C₂²⋊C₄	—	2	2	no	—
330	C₂×C₂²⋊C₄	—	2	2	no	—
331	C₂×C₄²	—	2	1	no	—
332	C₂×C₄²	—	2	2	no	—
333	C₂×C₄²	—	2	1	no	—
334	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
335	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
336	#106	—	2	2	no	—
337	C₄⋊C₈	—	2	4	no	—
338	C₄⋊C₈	—	2	2	no	—
339	#103	—	2	2	no	—
340	#103	—	2	4	no	—
341	#103	—	2	2	no	—
342	#102	—	2	4	no	—
343	#102	—	2	4	no	—
344	#102	—	4	2	no	—
345	#97	—	2	2	no	—
346	#95	—	2	1	no	—
347	#95	—	2	2	no	—
348	#95	—	2	1	no	—
349	#95	D₄×C₃₁	1	1	yes	characteristic
350	#95	D₄×C₃₁	1	1	yes	characteristic
351	C₃₁	#24747	1	1	yes	characteristic, direct product factor, Sylow
352	C₂⁴	—	2	2	no	—
353	C₂×Q₈	—	2	1	no	—
354	C₂×Q₈	—	2	1	no	—
355	C₂×Q₈	—	2	1	no	—
356	C₂×Q₈	—	2	1	no	—
357	C₂×Q₈	—	2	1	no	—
358	C₂×Q₈	—	2	1	no	—
359	C₂×Q₈	—	2	1	no	—
360	C₂×Q₈	—	2	1	no	—
361	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
362	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
363	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
364	C₂×Q₈	#41×C₃₁	1	2	yes	—
365	C₂×D₄	—	2	2	no	—
366	C₂×D₄	—	2	2	no	—
367	C₂×D₄	—	2	2	no	—
368	C₂×D₄	—	2	2	no	—
369	C₂×D₄	—	2	2	no	—
370	C₂×D₄	—	2	2	no	—
371	C₂×D₄	—	4	4	no	—
372	C₂×D₄	—	4	2	no	—
373	C₂×D₄	—	4	2	no	—
374	C₂×D₄	—	8	2	no	—
375	C₂²×C₄	—	2	1	no	—
376	C₂²×C₄	—	2	1	no	—
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	2	2	no	—
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
381	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
382	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
383	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
384	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
385	Q₁₆	—	2	8	no	—
386	SD₁₆	—	4	8	no	—
387	C₂×C₈	—	2	1	no	—
388	C₂×C₈	—	2	1	no	—
389	C₂×C₈	—	2	1	no	—
390	C₂×C₈	—	2	2	no	—
391	C₂×C₈	—	2	2	no	—
392	C₂×C₈	—	2	2	no	—
393	C₂×C₈	—	2	1	no	—
394	C₂×C₈	—	2	1	no	—
395	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
396	C₄⋊C₄	—	2	4	no	—
397	C₄⋊C₄	—	2	1	no	—
398	C₄⋊C₄	—	2	2	no	—
399	C₄⋊C₄	—	2	1	no	—
400	C₄⋊C₄	—	2	1	no	—
401	C₄⋊C₄	—	2	2	no	—
402	C₄⋊C₄	—	4	2	no	—
403	C₄⋊C₄	—	4	4	no	—
404	C₄⋊C₄	—	4	4	no	—
405	C₄⋊C₄	—	4	2	no	—
406	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
407	C₂²⋊C₄	—	4	4	no	—
408	C₂²⋊C₄	—	4	4	no	—
409	C₂²⋊C₄	—	8	2	no	—
410	C₂²⋊C₄	—	8	2	no	—
411	C₄²	—	2	4	no	—
412	C₄²	—	2	2	no	—
413	C₄²	—	2	2	no	—
414	C₄²	—	4	4	no	—
415	C₄²	—	4	2	no	—
416	C₄²	C₂×D₄×C₃₁	1	2	yes	—
417	C₄²	#41×C₃₁	1	2	yes	—
418	C₂³	—	4	2	no	—
419	C₂³	—	4	2	no	—
420	C₂³	—	4	2	no	—
421	C₂³	—	4	2	no	—
422	C₂³	—	4	2	no	—
423	C₂³	—	4	2	no	—
424	C₂³	—	4	2	no	—
425	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
426	Q₈	—	2	2	no	—
427	Q₈	—	2	2	no	—
428	Q₈	—	2	2	no	—
429	Q₈	—	2	2	no	—
430	Q₈	—	2	2	no	—
431	Q₈	—	4	1	no	—
432	Q₈	—	4	1	no	—
433	Q₈	C₃₁×#121	1	4	yes	—
434	D₄	—	4	2	no	—
435	D₄	—	4	2	no	—
436	D₄	—	4	2	no	—
437	D₄	—	4	2	no	—
438	D₄	—	8	4	no	—
439	D₄	—	8	4	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	2	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	—	4	1	no	—
454	C₂×C₄	—	4	1	no	—
455	C₂×C₄	—	4	2	no	—
456	C₂×C₄	—	4	1	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	—	4	2	no	—
459	C₂×C₄	—	4	2	no	—
460	C₂×C₄	—	4	2	no	—
461	C₂×C₄	C₃₁×#121	1	2	yes	—
462	C₂×C₄	C₃₁×#121	1	2	yes	—
463	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
464	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
465	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
466	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
467	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
468	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
469	C₈	—	2	2	no	—
470	C₈	—	4	2	no	—
471	C₈	—	4	1	no	—
472	C₂²	—	8	2	no	—
473	C₂²	—	8	2	no	—
474	C₂²	—	8	2	no	—
475	C₂²	—	8	2	no	—
476	C₂²	—	8	2	no	—
477	C₂²	—	8	2	no	—
478	C₂²	—	8	2	no	—
479	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
480	C₂²	D₄²×C₃₁	1	1	yes	characteristic
481	C₂²	C₃₁×#548	1	1	yes	characteristic
482	C₂²	C₃₁×#575	1	2	yes	—
483	C₂²	C₃₁×#577	1	2	yes	—
484	C₄	—	2	4	no	—
485	C₄	—	2	2	no	—
486	C₄	—	2	2	no	—
487	C₄	—	4	2	no	—
488	C₄	—	4	2	no	—
489	C₄	—	4	2	no	—
490	C₄	—	4	2	no	—
491	C₄	—	8	2	no	—
492	C₄	—	8	1	no	—
493	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
494	C₄	C₃₁×#537	1	1	yes	characteristic
495	C₄	C₃₁×#547	1	1	yes	characteristic
496	C₄	C₃₁×#553	1	1	yes	characteristic
497	C₂	—	16	2	no	—
498	C₂	C₃₁×#2416	1	1	yes	characteristic
499	C₂	C₃₁×#3040	1	1	yes	characteristic
500	C₂	C₃₁×#3043	1	1	yes	characteristic
501	C₂	C₃₁×#3271	1	2	yes	—
502	C₂	C₃₁×#3371	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄×C₄⋊C₄×C₃₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₃₁×#464	C₂	semidirect
3. C₂×C₃₁×#441	C₂	semidirect
5. C₃₁×#2351	C₂	semidirect
6. C₃₁×#2065	C₂	semidirect
7. C₃₁×#2042	C₂	semidirect
9. C₃₁×#1940	C₂	semidirect
10. C₃₁×#1880	C₂	semidirect
11. C₃₁×#1805	C₂	semidirect
51. C₂×Q₁₆×C₃₁	D₄	semidirect
166. #24747	C₃₁	direct (N × H)
221. C₂×#464	C₂×C₃₁	semidirect
223. C₂×#441	C₂×C₃₁	semidirect
225. #2351	C₂×C₃₁	semidirect
226. #2065	C₂×C₃₁	semidirect
227. #2042	C₂×C₃₁	semidirect
229. #1940	C₂×C₃₁	semidirect
230. #1880	C₂×C₃₁	semidirect
231. #1805	C₂×C₃₁	semidirect
302. C₂×Q₁₆	D₄×C₃₁	semidirect
351. C₃₁	#24747	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×#464	2
C₂×C₃₁×#460	2
C₂×C₃₁×#441	2
C₃₁×#2548	2
C₃₁×#2351	2
C₃₁×#2065	2
C₃₁×#2042	2
C₃₁×#1961	2
C₃₁×#1940	2
C₃₁×#1880	2
C₃₁×#1805	2
#24747	31

Subgroups — C31×#24747 #21481506

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24747 #21481506