Subgroups — C₃₁×#24729 #21481488

Order 7936.

946 conjugacy classes of subgroups.

The subgroup lattice has 520 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
2	C₂×C₃₁×#462	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2547	C₂	1	1	yes	maximal, characteristic
5	Q₈×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2066	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2042	C₂	1	2	yes	maximal
8	C₃₁×#1956	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1939	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1878	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1802	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
13	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#124	C₂²	1	2	yes	—
16	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
17	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#464	—	2	8	no	—
20	C₃₁×#462	—	2	4	no	—
21	C₄×SD₁₆×C₃₁	—	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	2	yes	—
27	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×#103	C₂²	1	2	yes	—
29	C₂×C₃₁×#102	C₂²	1	2	yes	—
30	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
32	C₃₁×#397	—	2	2	no	—
33	C₃₁×#391	C₂²	1	2	yes	—
34	C₃₁×#389	—	2	2	no	—
35	C₃₁×#389	C₂²	1	2	yes	—
36	C₃₁×#384	—	2	1	no	—
37	C₃₁×#384	C₂²	1	1	yes	characteristic
38	C₃₁×#380	—	2	1	no	—
39	C₃₁×#380	—	2	1	no	—
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	2	yes	—
45	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₃₁	—	2	1	no	—
48	C₂×Q₁₆×C₃₁	—	2	4	no	—
49	C₂×Q₁₆×C₃₁	—	2	4	no	—
50	C₂×Q₁₆×C₃₁	—	2	4	no	—
51	C₂×SD₁₆×C₃₁	—	2	2	no	—
52	C₂×SD₁₆×C₃₁	—	2	2	no	—
53	C₂×SD₁₆×C₃₁	D₄	1	4	yes	—
54	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
56	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
57	C₃₁×C₄⋊Q₈	—	2	2	no	—
58	C₃₁×C₄⋊Q₈	—	2	2	no	—
59	C₃₁×#124	—	2	4	no	—
60	C₃₁×#124	—	4	2	no	—
61	C₄×Q₈×C₃₁	—	2	2	no	—
62	C₄×Q₈×C₃₁	—	2	4	no	—
63	C₄×Q₈×C₃₁	—	4	2	no	—
64	C₄×Q₈×C₃₁	D₄	1	4	yes	—
65	C₄×D₄×C₃₁	—	2	2	no	—
66	C₄×D₄×C₃₁	—	2	2	no	—
67	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
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	1	no	—
72	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
73	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
74	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
75	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
76	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
77	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
79	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
80	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
81	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
82	C₂×C₄²×C₃₁	—	2	1	no	—
83	C₂×C₄²×C₃₁	—	2	2	no	—
84	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
85	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
86	C₃₁×#106	—	2	2	no	—
87	C₃₁×C₄⋊C₈	—	2	4	no	—
88	C₃₁×C₄⋊C₈	—	2	2	no	—
89	C₃₁×#103	—	2	2	no	—
90	C₃₁×#103	—	2	4	no	—
91	C₃₁×#103	—	2	4	no	—
92	C₃₁×#103	—	4	2	no	—
93	C₃₁×#102	—	2	4	no	—
94	C₃₁×#102	—	2	2	no	—
95	C₄×C₈×C₃₁	—	2	2	no	—
96	C₃₁×#95	—	2	2	no	—
97	C₃₁×#95	—	2	2	no	—
98	C₃₁×#95	—	2	2	no	—
99	C₃₁×#95	D₄	1	2	yes	—
100	C₂⁴×C₃₁	—	2	1	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	2	no	—
105	C₂×Q₈×C₃₁	—	2	2	no	—
106	C₂×Q₈×C₃₁	—	2	2	no	—
107	C₂×Q₈×C₃₁	—	2	2	no	—
108	C₂×Q₈×C₃₁	—	2	2	no	—
109	C₂×Q₈×C₃₁	—	2	2	no	—
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	1	no	—
114	C₂×D₄×C₃₁	—	2	1	no	—
115	C₂×D₄×C₃₁	—	2	1	no	—
116	C₂×D₄×C₃₁	—	2	1	no	—
117	C₂×D₄×C₃₁	—	2	1	no	—
118	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
119	C₂²×C₄×C₃₁	—	2	1	no	—
120	C₂²×C₄×C₃₁	—	2	1	no	—
121	C₂²×C₄×C₃₁	—	2	1	no	—
122	C₂²×C₄×C₃₁	—	2	2	no	—
123	C₂²×C₄×C₃₁	—	2	2	no	—
124	C₂²×C₄×C₃₁	—	4	2	no	—
125	C₂²×C₄×C₃₁	—	4	2	no	—
126	C₂²×C₄×C₃₁	—	4	2	no	—
127	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
128	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
129	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
132	Q₁₆×C₃₁	—	4	8	no	—
133	SD₁₆×C₃₁	—	2	8	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	2	no	—
138	C₂×C₈×C₃₁	—	2	2	no	—
139	C₂×C₈×C₃₁	—	2	2	no	—
140	C₂×C₈×C₃₁	—	2	1	no	—
141	C₂×C₈×C₃₁	—	2	1	no	—
142	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
143	C₄⋊C₄×C₃₁	—	2	4	no	—
144	C₄⋊C₄×C₃₁	—	2	2	no	—
145	C₄⋊C₄×C₃₁	—	2	2	no	—
146	C₄⋊C₄×C₃₁	—	2	1	no	—
147	C₄⋊C₄×C₃₁	—	2	2	no	—
148	C₄⋊C₄×C₃₁	—	2	1	no	—
149	C₄⋊C₄×C₃₁	—	2	1	no	—
150	C₄⋊C₄×C₃₁	—	2	2	no	—
151	C₄⋊C₄×C₃₁	—	4	4	no	—
152	C₄⋊C₄×C₃₁	—	4	2	no	—
153	C₄⋊C₄×C₃₁	—	4	2	no	—
154	C₄⋊C₄×C₃₁	—	4	2	no	—
155	C₄⋊C₄×C₃₁	—	8	1	no	—
156	C₄⋊C₄×C₃₁	—	8	2	no	—
157	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
158	C₂²⋊C₄×C₃₁	—	4	4	no	—
159	C₂²⋊C₄×C₃₁	—	4	4	no	—
160	C₂²⋊C₄×C₃₁	—	4	2	no	—
161	C₄²×C₃₁	—	2	4	no	—
162	C₄²×C₃₁	—	2	2	no	—
163	C₄²×C₃₁	—	2	2	no	—
164	C₄²×C₃₁	—	4	4	no	—
165	C₄²×C₃₁	C₂×D₄	1	2	yes	—
166	C₄²×C₃₁	#41	1	2	yes	—
167	#24729	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
168	C₂³×C₃₁	—	2	2	no	—
169	C₂³×C₃₁	—	4	1	no	—
170	C₂³×C₃₁	—	4	1	no	—
171	C₂³×C₃₁	—	4	1	no	—
172	C₂³×C₃₁	—	4	1	no	—
173	C₂³×C₃₁	—	4	1	no	—
174	C₂³×C₃₁	—	4	1	no	—
175	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
176	Q₈×C₃₁	—	2	2	no	—
177	Q₈×C₃₁	—	2	2	no	—
178	Q₈×C₃₁	—	2	2	no	—
179	Q₈×C₃₁	—	4	2	no	—
180	Q₈×C₃₁	—	4	2	no	—
181	Q₈×C₃₁	—	4	2	no	—
182	Q₈×C₃₁	—	4	2	no	—
183	Q₈×C₃₁	#121	1	4	yes	—
184	D₄×C₃₁	—	2	2	no	—
185	D₄×C₃₁	—	2	2	no	—
186	D₄×C₃₁	—	4	1	no	—
187	D₄×C₃₁	—	4	1	no	—
188	C₂×C₄×C₃₁	—	2	2	no	—
189	C₂×C₄×C₃₁	—	2	2	no	—
190	C₂×C₄×C₃₁	—	2	2	no	—
191	C₂×C₄×C₃₁	—	2	1	no	—
192	C₂×C₄×C₃₁	—	2	1	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	—	4	1	no	—
201	C₂×C₄×C₃₁	—	4	1	no	—
202	C₂×C₄×C₃₁	—	4	1	no	—
203	C₂×C₄×C₃₁	—	4	2	no	—
204	C₂×C₄×C₃₁	—	4	2	no	—
205	C₂×C₄×C₃₁	—	4	2	no	—
206	C₂×C₄×C₃₁	—	4	2	no	—
207	C₂×C₄×C₃₁	—	4	2	no	—
208	C₂×C₄×C₃₁	—	4	1	no	—
209	C₂×C₄×C₃₁	—	4	1	no	—
210	C₂×C₄×C₃₁	—	4	2	no	—
211	C₂×C₄×C₃₁	—	8	4	no	—
212	C₂×C₄×C₃₁	#121	1	2	yes	—
213	C₂×C₄×C₃₁	#121	1	2	yes	—
214	C₂×C₄×C₃₁	#135	1	2	yes	—
215	C₂×C₄×C₃₁	#137	1	2	yes	—
216	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
217	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
218	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
219	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
220	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
221	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
222	C₈×C₃₁	—	2	2	no	—
223	C₈×C₃₁	—	4	2	no	—
224	C₈×C₃₁	—	4	1	no	—
225	C₂×#464	C₂×C₃₁	1	2	yes	—
226	C₂×#462	C₂×C₃₁	1	1	yes	characteristic, complemented
227	C₂×C₄×SD₁₆	C₂×C₃₁	1	1	yes	characteristic
228	#2547	C₂×C₃₁	1	1	yes	characteristic
229	Q₈×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
230	#2066	C₂×C₃₁	1	2	yes	complemented
231	#2042	C₂×C₃₁	1	2	yes	—
232	#1956	C₂×C₃₁	1	1	yes	characteristic, complemented
233	#1939	C₂×C₃₁	1	1	yes	characteristic, complemented
234	#1878	C₂×C₃₁	1	2	yes	complemented
235	#1802	C₂×C₃₁	1	1	yes	characteristic
236	C₂²×C₃₁	—	4	2	no	—
237	C₂²×C₃₁	—	4	2	no	—
238	C₂²×C₃₁	—	4	2	no	—
239	C₂²×C₃₁	—	8	1	no	—
240	C₂²×C₃₁	—	8	1	no	—
241	C₂²×C₃₁	—	8	1	no	—
242	C₂²×C₃₁	—	8	1	no	—
243	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
244	C₂²×C₃₁	D₄²	1	1	yes	characteristic
245	C₂²×C₃₁	#548	1	1	yes	characteristic
246	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
247	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
248	C₂²×C₃₁	#577	1	1	yes	characteristic
249	C₂²×C₃₁	#578	1	1	yes	characteristic
250	C₄×C₃₁	—	2	4	no	—
251	C₄×C₃₁	—	2	2	no	—
252	C₄×C₃₁	—	4	2	no	—
253	C₄×C₃₁	—	4	2	no	—
254	C₄×C₃₁	—	4	2	no	—
255	C₄×C₃₁	—	8	2	no	—
256	C₄×C₃₁	—	8	2	no	—
257	C₄×C₃₁	—	8	1	no	—
258	C₄×C₃₁	#464	1	4	yes	complemented
259	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
260	C₄×C₃₁	#537	1	1	yes	characteristic
261	C₄×C₃₁	#547	1	1	yes	characteristic
262	C₄×C₃₁	#553	1	1	yes	characteristic
263	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
264	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
265	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
266	C₂×#124	C₂²×C₃₁	1	2	yes	—
267	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
268	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
269	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
270	#464	—	2	8	no	—
271	#462	—	2	4	no	—
272	C₄×SD₁₆	—	2	4	no	—
273	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
274	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
275	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
276	C₂×#103	C₂²×C₃₁	1	2	yes	—
277	C₂×#103	C₂²×C₃₁	1	2	yes	—
278	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
279	C₂×#103	C₂²×C₃₁	1	2	yes	—
280	C₂×#102	C₂²×C₃₁	1	2	yes	—
281	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
282	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
283	#397	—	2	2	no	—
284	#391	C₂²×C₃₁	1	2	yes	—
285	#389	—	2	2	no	—
286	#389	C₂²×C₃₁	1	2	yes	—
287	#384	—	2	1	no	—
288	#384	C₂²×C₃₁	1	1	yes	characteristic
289	#380	—	2	1	no	—
290	#380	—	2	1	no	—
291	C₄×C₄⋊C₄	—	2	1	no	—
292	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
293	#340	C₂²×C₃₁	1	2	yes	—
294	#336	C₂²×C₃₁	1	2	yes	—
295	C₂×C₃₁	—	8	2	no	—
296	C₂×C₃₁	#2416	1	1	yes	characteristic
297	C₂×C₃₁	C₂×#464	1	1	yes	characteristic
298	C₂×C₃₁	#3046	1	1	yes	characteristic
299	C₂×C₃₁	#3276	1	1	yes	characteristic
300	C₂×C₃₁	#3284	1	1	yes	characteristic
301	C₂×C₃₁	#3364	1	1	yes	characteristic
302	C₂×C₃₁	#3369	1	1	yes	characteristic
303	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
304	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
305	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
306	C₂³×C₄	—	2	1	no	—
307	C₂×Q₁₆	—	2	4	no	—
308	C₂×Q₁₆	—	2	4	no	—
309	C₂×Q₁₆	—	2	4	no	—
310	C₂×SD₁₆	—	2	2	no	—
311	C₂×SD₁₆	—	2	2	no	—
312	C₂×SD₁₆	D₄×C₃₁	1	4	yes	—
313	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
314	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
315	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
316	C₄⋊Q₈	—	2	2	no	—
317	C₄⋊Q₈	—	2	2	no	—
318	#124	—	2	4	no	—
319	#124	—	4	2	no	—
320	C₄×Q₈	—	2	2	no	—
321	C₄×Q₈	—	2	4	no	—
322	C₄×Q₈	—	4	2	no	—
323	C₄×Q₈	D₄×C₃₁	1	4	yes	—
324	C₄×D₄	—	2	2	no	—
325	C₄×D₄	—	2	2	no	—
326	C₂×C₄⋊C₄	—	2	1	no	—
327	C₂×C₄⋊C₄	—	2	2	no	—
328	C₂×C₄⋊C₄	—	2	1	no	—
329	C₂×C₄⋊C₄	—	2	1	no	—
330	C₂×C₄⋊C₄	—	2	1	no	—
331	C₂×C₄⋊C₄	—	2	2	no	—
332	C₂×C₄⋊C₄	—	2	1	no	—
333	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
334	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
335	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
336	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
337	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
338	C₂×C₂²⋊C₄	—	2	1	no	—
339	C₂×C₂²⋊C₄	—	2	2	no	—
340	C₂×C₂²⋊C₄	—	2	2	no	—
341	C₂×C₄²	—	2	1	no	—
342	C₂×C₄²	—	2	2	no	—
343	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
344	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
345	#106	—	2	2	no	—
346	C₄⋊C₈	—	2	4	no	—
347	C₄⋊C₈	—	2	2	no	—
348	#103	—	2	2	no	—
349	#103	—	2	4	no	—
350	#103	—	2	4	no	—
351	#103	—	4	2	no	—
352	#102	—	2	4	no	—
353	#102	—	2	2	no	—
354	C₄×C₈	—	2	2	no	—
355	#95	—	2	2	no	—
356	#95	—	2	2	no	—
357	#95	—	2	2	no	—
358	#95	D₄×C₃₁	1	2	yes	—
359	C₃₁	#24729	1	1	yes	characteristic, direct product factor, Sylow
360	C₂⁴	—	2	1	no	—
361	C₂×Q₈	—	2	1	no	—
362	C₂×Q₈	—	2	1	no	—
363	C₂×Q₈	—	2	1	no	—
364	C₂×Q₈	—	2	2	no	—
365	C₂×Q₈	—	2	2	no	—
366	C₂×Q₈	—	2	2	no	—
367	C₂×Q₈	—	2	2	no	—
368	C₂×Q₈	—	2	2	no	—
369	C₂×Q₈	—	2	2	no	—
370	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
371	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
372	C₂×Q₈	#41×C₃₁	1	2	yes	—
373	C₂×D₄	—	2	1	no	—
374	C₂×D₄	—	2	1	no	—
375	C₂×D₄	—	2	1	no	—
376	C₂×D₄	—	2	1	no	—
377	C₂×D₄	—	2	1	no	—
378	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	—	2	1	no	—
381	C₂²×C₄	—	2	1	no	—
382	C₂²×C₄	—	2	2	no	—
383	C₂²×C₄	—	2	2	no	—
384	C₂²×C₄	—	4	2	no	—
385	C₂²×C₄	—	4	2	no	—
386	C₂²×C₄	—	4	2	no	—
387	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
388	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
389	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
390	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
391	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
392	Q₁₆	—	4	8	no	—
393	SD₁₆	—	2	8	no	—
394	C₂×C₈	—	2	1	no	—
395	C₂×C₈	—	2	1	no	—
396	C₂×C₈	—	2	1	no	—
397	C₂×C₈	—	2	2	no	—
398	C₂×C₈	—	2	2	no	—
399	C₂×C₈	—	2	2	no	—
400	C₂×C₈	—	2	1	no	—
401	C₂×C₈	—	2	1	no	—
402	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
403	C₄⋊C₄	—	2	4	no	—
404	C₄⋊C₄	—	2	2	no	—
405	C₄⋊C₄	—	2	2	no	—
406	C₄⋊C₄	—	2	1	no	—
407	C₄⋊C₄	—	2	2	no	—
408	C₄⋊C₄	—	2	1	no	—
409	C₄⋊C₄	—	2	1	no	—
410	C₄⋊C₄	—	2	2	no	—
411	C₄⋊C₄	—	4	4	no	—
412	C₄⋊C₄	—	4	2	no	—
413	C₄⋊C₄	—	4	2	no	—
414	C₄⋊C₄	—	4	2	no	—
415	C₄⋊C₄	—	8	1	no	—
416	C₄⋊C₄	—	8	2	no	—
417	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
418	C₂²⋊C₄	—	4	4	no	—
419	C₂²⋊C₄	—	4	4	no	—
420	C₂²⋊C₄	—	4	2	no	—
421	C₄²	—	2	4	no	—
422	C₄²	—	2	2	no	—
423	C₄²	—	2	2	no	—
424	C₄²	—	4	4	no	—
425	C₄²	C₂×D₄×C₃₁	1	2	yes	—
426	C₄²	#41×C₃₁	1	2	yes	—
427	C₂³	—	2	2	no	—
428	C₂³	—	4	1	no	—
429	C₂³	—	4	1	no	—
430	C₂³	—	4	1	no	—
431	C₂³	—	4	1	no	—
432	C₂³	—	4	1	no	—
433	C₂³	—	4	1	no	—
434	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
435	Q₈	—	2	2	no	—
436	Q₈	—	2	2	no	—
437	Q₈	—	2	2	no	—
438	Q₈	—	4	2	no	—
439	Q₈	—	4	2	no	—
440	Q₈	—	4	2	no	—
441	Q₈	—	4	2	no	—
442	Q₈	C₃₁×#121	1	4	yes	—
443	D₄	—	2	2	no	—
444	D₄	—	2	2	no	—
445	D₄	—	4	1	no	—
446	D₄	—	4	1	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	1	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	—	4	1	no	—
460	C₂×C₄	—	4	1	no	—
461	C₂×C₄	—	4	1	no	—
462	C₂×C₄	—	4	2	no	—
463	C₂×C₄	—	4	2	no	—
464	C₂×C₄	—	4	2	no	—
465	C₂×C₄	—	4	2	no	—
466	C₂×C₄	—	4	2	no	—
467	C₂×C₄	—	4	1	no	—
468	C₂×C₄	—	4	1	no	—
469	C₂×C₄	—	4	2	no	—
470	C₂×C₄	—	8	4	no	—
471	C₂×C₄	C₃₁×#121	1	2	yes	—
472	C₂×C₄	C₃₁×#121	1	2	yes	—
473	C₂×C₄	C₃₁×#135	1	2	yes	—
474	C₂×C₄	C₃₁×#137	1	2	yes	—
475	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
476	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
477	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
478	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
479	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
480	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
481	C₈	—	2	2	no	—
482	C₈	—	4	2	no	—
483	C₈	—	4	1	no	—
484	C₂²	—	4	2	no	—
485	C₂²	—	4	2	no	—
486	C₂²	—	4	2	no	—
487	C₂²	—	8	1	no	—
488	C₂²	—	8	1	no	—
489	C₂²	—	8	1	no	—
490	C₂²	—	8	1	no	—
491	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
492	C₂²	D₄²×C₃₁	1	1	yes	characteristic
493	C₂²	C₃₁×#548	1	1	yes	characteristic
494	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
495	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
496	C₂²	C₃₁×#577	1	1	yes	characteristic
497	C₂²	C₃₁×#578	1	1	yes	characteristic
498	C₄	—	2	4	no	—
499	C₄	—	2	2	no	—
500	C₄	—	4	2	no	—
501	C₄	—	4	2	no	—
502	C₄	—	4	2	no	—
503	C₄	—	8	2	no	—
504	C₄	—	8	2	no	—
505	C₄	—	8	1	no	—
506	C₄	C₃₁×#464	1	4	yes	complemented
507	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
508	C₄	C₃₁×#537	1	1	yes	characteristic
509	C₄	C₃₁×#547	1	1	yes	characteristic
510	C₄	C₃₁×#553	1	1	yes	characteristic
511	C₂	—	8	2	no	—
512	C₂	C₃₁×#2416	1	1	yes	characteristic
513	C₂	C₂×C₃₁×#464	1	1	yes	characteristic
514	C₂	C₃₁×#3046	1	1	yes	characteristic
515	C₂	C₃₁×#3276	1	1	yes	characteristic
516	C₂	C₃₁×#3284	1	1	yes	characteristic
517	C₂	C₃₁×#3364	1	1	yes	characteristic
518	C₂	C₃₁×#3369	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂³
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
2. C₂×C₃₁×#462	C₂	semidirect
5. Q₈×C₄⋊C₄×C₃₁	C₂	semidirect
6. C₃₁×#2066	C₂	semidirect
8. C₃₁×#1956	C₂	semidirect
9. C₃₁×#1939	C₂	semidirect
10. C₃₁×#1878	C₂	semidirect
167. #24729	C₃₁	direct (N × H)
226. C₂×#462	C₂×C₃₁	semidirect
229. Q₈×C₄⋊C₄	C₂×C₃₁	semidirect
230. #2066	C₂×C₃₁	semidirect
232. #1956	C₂×C₃₁	semidirect
233. #1939	C₂×C₃₁	semidirect
234. #1878	C₂×C₃₁	semidirect
258. C₄×C₃₁	#464	semidirect
359. C₃₁	#24729	direct (N × H)
506. C₄	C₃₁×#464	semidirect

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×#462	2
C₂×C₄×SD₁₆×C₃₁	2
C₃₁×#2547	2
Q₈×C₄⋊C₄×C₃₁	2
C₃₁×#2066	2
C₃₁×#2042	2
C₃₁×#1956	2
C₃₁×#1939	2
C₃₁×#1878	2
C₃₁×#1802	2
#24729	31

Subgroups — C31×#24729 #21481488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24729 #21481488