Subgroups — C₃₁×#24730 #21481489

Order 7936.

1018 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#461	C₂	1	2	yes	maximal
2	C₂×C₃₁×#461	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2546	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#2066	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2043	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1956	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1939	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1877	C₂	1	2	yes	maximal
11	C₃₁×#1803	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×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×C₄⋊Q₈	C₂²	1	2	yes	—
16	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
18	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
20	C₃₁×#461	—	2	8	no	—
21	C₃₁×#461	—	2	4	no	—
22	C₄×SD₁₆×C₃₁	—	2	4	no	—
23	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
26	C₂×C₃₁×#103	C₂²	1	2	yes	—
27	C₂×C₃₁×#103	C₂²	1	2	yes	—
28	C₂×C₃₁×#103	C₂²	1	2	yes	—
29	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×#103	C₂²	1	2	yes	—
31	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
33	C₃₁×#398	—	2	2	no	—
34	C₃₁×#391	C₂²	1	2	yes	—
35	C₃₁×#389	—	2	1	no	—
36	C₃₁×#389	—	2	1	no	—
37	C₃₁×#389	C₂²	1	2	yes	—
38	C₃₁×#385	—	2	2	no	—
39	C₃₁×#384	C₂²	1	1	yes	characteristic
40	C₃₁×#380	—	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	C₂³×C₄×C₃₁	—	2	2	no	—
49	C₂×SD₁₆×C₃₁	—	2	2	no	—
50	C₂×SD₁₆×C₃₁	—	2	2	no	—
51	C₂×SD₁₆×C₃₁	—	2	4	no	—
52	C₂×SD₁₆×C₃₁	—	2	4	no	—
53	C₂×SD₁₆×C₃₁	—	2	4	no	—
54	C₂×SD₁₆×C₃₁	D₄	1	4	yes	—
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×C₄⋊Q₈	—	2	4	no	—
59	C₃₁×C₄⋊Q₈	—	2	2	no	—
60	C₃₁×C₄⋊Q₈	—	2	2	no	—
61	C₃₁×C₄⋊Q₈	—	4	2	no	—
62	C₄×Q₈×C₃₁	—	2	2	no	—
63	C₄×Q₈×C₃₁	—	2	2	no	—
64	C₄×D₄×C₃₁	—	2	4	no	—
65	C₄×D₄×C₃₁	—	2	2	no	—
66	C₄×D₄×C₃₁	—	4	2	no	—
67	C₄×D₄×C₃₁	D₄	1	4	yes	—
68	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
69	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
70	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
71	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
72	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
73	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
74	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
75	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
76	C₂×C₄⋊C₄×C₃₁	—	4	4	no	—
77	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
81	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
82	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
83	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
84	C₂×C₄²×C₃₁	—	2	1	no	—
85	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
86	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
87	C₃₁×#106	—	2	2	no	—
88	C₃₁×C₄⋊C₈	—	2	4	no	—
89	C₃₁×C₄⋊C₈	—	2	2	no	—
90	C₃₁×#103	—	2	4	no	—
91	C₃₁×#103	—	2	4	no	—
92	C₃₁×#103	—	2	2	no	—
93	C₃₁×#103	—	2	4	no	—
94	C₃₁×#103	—	4	2	no	—
95	C₃₁×#102	—	2	2	no	—
96	C₄×C₈×C₃₁	—	2	2	no	—
97	C₃₁×#95	—	2	2	no	—
98	C₃₁×#95	—	2	2	no	—
99	C₃₁×#95	—	2	2	no	—
100	C₃₁×#95	D₄	1	2	yes	—
101	C₂⁴×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	1	no	—
107	C₂×Q₈×C₃₁	—	2	1	no	—
108	C₂×Q₈×C₃₁	—	2	2	no	—
109	C₂×Q₈×C₃₁	—	2	2	no	—
110	C₂×Q₈×C₃₁	—	2	2	no	—
111	C₂×Q₈×C₃₁	—	2	1	no	—
112	C₂×Q₈×C₃₁	—	2	2	no	—
113	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
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	C₂×D₄	1	2	yes	—
118	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
119	C₂×D₄×C₃₁	#41	1	2	yes	—
120	C₂²×C₄×C₃₁	—	2	1	no	—
121	C₂²×C₄×C₃₁	—	2	1	no	—
122	C₂²×C₄×C₃₁	—	2	4	no	—
123	C₂²×C₄×C₃₁	—	2	1	no	—
124	C₂²×C₄×C₃₁	—	2	2	no	—
125	C₂²×C₄×C₃₁	—	2	2	no	—
126	C₂²×C₄×C₃₁	—	4	4	no	—
127	C₂²×C₄×C₃₁	—	4	2	no	—
128	C₂²×C₄×C₃₁	—	4	2	no	—
129	C₂²×C₄×C₃₁	—	4	4	no	—
130	C₂²×C₄×C₃₁	—	8	2	no	—
131	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
132	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
134	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
136	SD₁₆×C₃₁	—	2	8	no	—
137	SD₁₆×C₃₁	—	4	8	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	2	no	—
142	C₂×C₈×C₃₁	—	2	2	no	—
143	C₂×C₈×C₃₁	—	2	2	no	—
144	C₂×C₈×C₃₁	—	2	1	no	—
145	C₂×C₈×C₃₁	—	2	1	no	—
146	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
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	—	2	1	no	—
152	C₄⋊C₄×C₃₁	—	2	2	no	—
153	C₄⋊C₄×C₃₁	—	4	4	no	—
154	C₄⋊C₄×C₃₁	—	4	4	no	—
155	C₄⋊C₄×C₃₁	—	4	2	no	—
156	C₄⋊C₄×C₃₁	—	4	2	no	—
157	C₄⋊C₄×C₃₁	—	4	2	no	—
158	C₄⋊C₄×C₃₁	—	4	4	no	—
159	C₄⋊C₄×C₃₁	—	4	2	no	—
160	C₄⋊C₄×C₃₁	—	8	1	no	—
161	C₄⋊C₄×C₃₁	SD₁₆	1	4	yes	complemented
162	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
163	C₂²⋊C₄×C₃₁	—	2	4	no	—
164	C₂²⋊C₄×C₃₁	—	4	4	no	—
165	C₄²×C₃₁	—	2	2	no	—
166	C₄²×C₃₁	—	2	2	no	—
167	C₄²×C₃₁	—	4	2	no	—
168	C₄²×C₃₁	C₂×D₄	1	2	yes	—
169	C₄²×C₃₁	#41	1	2	yes	—
170	#24730	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
171	C₂³×C₃₁	—	2	2	no	—
172	C₂³×C₃₁	—	2	2	no	—
173	C₂³×C₃₁	—	2	2	no	—
174	C₂³×C₃₁	—	4	1	no	—
175	C₂³×C₃₁	—	4	1	no	—
176	C₂³×C₃₁	—	4	1	no	—
177	C₂³×C₃₁	—	4	1	no	—
178	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
179	Q₈×C₃₁	—	2	2	no	—
180	Q₈×C₃₁	—	2	2	no	—
181	Q₈×C₃₁	—	4	1	no	—
182	Q₈×C₃₁	—	4	1	no	—
183	Q₈×C₃₁	—	4	2	no	—
184	Q₈×C₃₁	—	4	2	no	—
185	Q₈×C₃₁	—	4	2	no	—
186	Q₈×C₃₁	—	4	2	no	—
187	D₄×C₃₁	—	2	2	no	—
188	D₄×C₃₁	—	2	2	no	—
189	D₄×C₃₁	—	2	2	no	—
190	D₄×C₃₁	#121	1	4	yes	—
191	C₂×C₄×C₃₁	—	2	2	no	—
192	C₂×C₄×C₃₁	—	2	1	no	—
193	C₂×C₄×C₃₁	—	2	2	no	—
194	C₂×C₄×C₃₁	—	2	1	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	1	no	—
204	C₂×C₄×C₃₁	—	4	8	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	—	4	2	no	—
211	C₂×C₄×C₃₁	—	4	2	no	—
212	C₂×C₄×C₃₁	—	4	1	no	—
213	C₂×C₄×C₃₁	—	4	1	no	—
214	C₂×C₄×C₃₁	—	8	8	no	—
215	C₂×C₄×C₃₁	#121	1	2	yes	—
216	C₂×C₄×C₃₁	#121	1	2	yes	—
217	C₂×C₄×C₃₁	C₂×SD₁₆	1	4	yes	—
218	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
219	C₂×C₄×C₃₁	#137	1	2	yes	—
220	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
223	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
224	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
225	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
226	C₈×C₃₁	—	2	2	no	—
227	C₈×C₃₁	—	4	2	no	—
228	C₈×C₃₁	—	4	1	no	—
229	C₂×#461	C₂×C₃₁	1	2	yes	—
230	C₂×#461	C₂×C₃₁	1	1	yes	characteristic
231	C₂×C₄×SD₁₆	C₂×C₃₁	1	1	yes	characteristic
232	#2546	C₂×C₃₁	1	1	yes	characteristic, complemented
233	D₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic
234	#2066	C₂×C₃₁	1	2	yes	complemented
235	#2043	C₂×C₃₁	1	2	yes	complemented
236	#1956	C₂×C₃₁	1	1	yes	characteristic, complemented
237	#1939	C₂×C₃₁	1	1	yes	characteristic, complemented
238	#1877	C₂×C₃₁	1	2	yes	—
239	#1803	C₂×C₃₁	1	1	yes	characteristic, complemented
240	C₂²×C₃₁	—	2	4	no	—
241	C₂²×C₃₁	—	4	2	no	—
242	C₂²×C₃₁	—	4	2	no	—
243	C₂²×C₃₁	—	4	2	no	—
244	C₂²×C₃₁	—	4	2	no	—
245	C₂²×C₃₁	—	4	2	no	—
246	C₂²×C₃₁	—	4	2	no	—
247	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
248	C₂²×C₃₁	D₄²	1	1	yes	characteristic
249	C₂²×C₃₁	#548	1	1	yes	characteristic
250	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
251	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
252	C₂²×C₃₁	#577	1	1	yes	characteristic
253	C₂²×C₃₁	#578	1	1	yes	characteristic
254	C₄×C₃₁	—	2	4	no	—
255	C₄×C₃₁	—	2	2	no	—
256	C₄×C₃₁	—	4	2	no	—
257	C₄×C₃₁	—	4	2	no	—
258	C₄×C₃₁	—	4	2	no	—
259	C₄×C₃₁	—	8	2	no	—
260	C₄×C₃₁	—	8	2	no	—
261	C₄×C₃₁	—	8	1	no	—
262	C₄×C₃₁	#461	1	4	yes	complemented
263	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
264	C₄×C₃₁	#537	1	1	yes	characteristic
265	C₄×C₃₁	#547	1	1	yes	characteristic
266	C₄×C₃₁	#553	1	1	yes	characteristic
267	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
268	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
269	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
270	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
271	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
272	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
273	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
274	C₂²×C₄⋊C₄	—	2	1	no	—
275	#461	—	2	8	no	—
276	#461	—	2	4	no	—
277	C₄×SD₁₆	—	2	4	no	—
278	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
279	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
280	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
281	C₂×#103	C₂²×C₃₁	1	2	yes	—
282	C₂×#103	C₂²×C₃₁	1	2	yes	—
283	C₂×#103	C₂²×C₃₁	1	2	yes	—
284	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
285	C₂×#103	C₂²×C₃₁	1	2	yes	—
286	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
287	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
288	#398	—	2	2	no	—
289	#391	C₂²×C₃₁	1	2	yes	—
290	#389	—	2	1	no	—
291	#389	—	2	1	no	—
292	#389	C₂²×C₃₁	1	2	yes	—
293	#385	—	2	2	no	—
294	#384	C₂²×C₃₁	1	1	yes	characteristic
295	#380	—	2	1	no	—
296	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
297	#340	C₂²×C₃₁	1	2	yes	—
298	#336	C₂²×C₃₁	1	2	yes	—
299	C₂×C₃₁	—	4	4	no	—
300	C₂×C₃₁	#2416	1	1	yes	characteristic
301	C₂×C₃₁	C₂×#461	1	1	yes	characteristic
302	C₂×C₃₁	#3046	1	1	yes	characteristic
303	C₂×C₃₁	#3276	1	1	yes	characteristic
304	C₂×C₃₁	D₄×SD₁₆	1	1	yes	characteristic
305	C₂×C₃₁	#3359	1	1	yes	characteristic
306	C₂×C₃₁	#3372	1	1	yes	characteristic
307	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
308	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
309	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
310	C₂³×C₄	—	2	1	no	—
311	C₂³×C₄	—	2	2	no	—
312	C₂×SD₁₆	—	2	2	no	—
313	C₂×SD₁₆	—	2	2	no	—
314	C₂×SD₁₆	—	2	4	no	—
315	C₂×SD₁₆	—	2	4	no	—
316	C₂×SD₁₆	—	2	4	no	—
317	C₂×SD₁₆	D₄×C₃₁	1	4	yes	—
318	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
319	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
320	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
321	C₄⋊Q₈	—	2	4	no	—
322	C₄⋊Q₈	—	2	2	no	—
323	C₄⋊Q₈	—	2	2	no	—
324	C₄⋊Q₈	—	4	2	no	—
325	C₄×Q₈	—	2	2	no	—
326	C₄×Q₈	—	2	2	no	—
327	C₄×D₄	—	2	4	no	—
328	C₄×D₄	—	2	2	no	—
329	C₄×D₄	—	4	2	no	—
330	C₄×D₄	D₄×C₃₁	1	4	yes	—
331	C₂×C₄⋊C₄	—	2	1	no	—
332	C₂×C₄⋊C₄	—	2	2	no	—
333	C₂×C₄⋊C₄	—	2	1	no	—
334	C₂×C₄⋊C₄	—	2	1	no	—
335	C₂×C₄⋊C₄	—	2	2	no	—
336	C₂×C₄⋊C₄	—	2	2	no	—
337	C₂×C₄⋊C₄	—	2	1	no	—
338	C₂×C₄⋊C₄	—	2	1	no	—
339	C₂×C₄⋊C₄	—	4	4	no	—
340	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
341	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
342	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
343	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
344	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
345	C₂×C₂²⋊C₄	—	2	1	no	—
346	C₂×C₂²⋊C₄	—	2	2	no	—
347	C₂×C₄²	—	2	1	no	—
348	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
349	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
350	#106	—	2	2	no	—
351	C₄⋊C₈	—	2	4	no	—
352	C₄⋊C₈	—	2	2	no	—
353	#103	—	2	4	no	—
354	#103	—	2	4	no	—
355	#103	—	2	2	no	—
356	#103	—	2	4	no	—
357	#103	—	4	2	no	—
358	#102	—	2	2	no	—
359	C₄×C₈	—	2	2	no	—
360	#95	—	2	2	no	—
361	#95	—	2	2	no	—
362	#95	—	2	2	no	—
363	#95	D₄×C₃₁	1	2	yes	—
364	C₃₁	#24730	1	1	yes	characteristic, direct product factor, Sylow
365	C₂⁴	—	2	1	no	—
366	C₂×Q₈	—	2	1	no	—
367	C₂×Q₈	—	2	1	no	—
368	C₂×Q₈	—	2	2	no	—
369	C₂×Q₈	—	2	2	no	—
370	C₂×Q₈	—	2	1	no	—
371	C₂×Q₈	—	2	1	no	—
372	C₂×Q₈	—	2	2	no	—
373	C₂×Q₈	—	2	2	no	—
374	C₂×Q₈	—	2	2	no	—
375	C₂×Q₈	—	2	1	no	—
376	C₂×Q₈	—	2	2	no	—
377	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
378	C₂×D₄	—	2	1	no	—
379	C₂×D₄	—	2	1	no	—
380	C₂×D₄	—	2	1	no	—
381	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
382	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
383	C₂×D₄	#41×C₃₁	1	2	yes	—
384	C₂²×C₄	—	2	1	no	—
385	C₂²×C₄	—	2	1	no	—
386	C₂²×C₄	—	2	4	no	—
387	C₂²×C₄	—	2	1	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	—	4	4	no	—
391	C₂²×C₄	—	4	2	no	—
392	C₂²×C₄	—	4	2	no	—
393	C₂²×C₄	—	4	4	no	—
394	C₂²×C₄	—	8	2	no	—
395	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
396	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
397	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
398	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
399	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
400	SD₁₆	—	2	8	no	—
401	SD₁₆	—	4	8	no	—
402	C₂×C₈	—	2	1	no	—
403	C₂×C₈	—	2	1	no	—
404	C₂×C₈	—	2	1	no	—
405	C₂×C₈	—	2	2	no	—
406	C₂×C₈	—	2	2	no	—
407	C₂×C₈	—	2	2	no	—
408	C₂×C₈	—	2	1	no	—
409	C₂×C₈	—	2	1	no	—
410	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
411	C₄⋊C₄	—	2	2	no	—
412	C₄⋊C₄	—	2	1	no	—
413	C₄⋊C₄	—	2	1	no	—
414	C₄⋊C₄	—	2	2	no	—
415	C₄⋊C₄	—	2	1	no	—
416	C₄⋊C₄	—	2	2	no	—
417	C₄⋊C₄	—	4	4	no	—
418	C₄⋊C₄	—	4	4	no	—
419	C₄⋊C₄	—	4	2	no	—
420	C₄⋊C₄	—	4	2	no	—
421	C₄⋊C₄	—	4	2	no	—
422	C₄⋊C₄	—	4	4	no	—
423	C₄⋊C₄	—	4	2	no	—
424	C₄⋊C₄	—	8	1	no	—
425	C₄⋊C₄	SD₁₆×C₃₁	1	4	yes	complemented
426	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
427	C₂²⋊C₄	—	2	4	no	—
428	C₂²⋊C₄	—	4	4	no	—
429	C₄²	—	2	2	no	—
430	C₄²	—	2	2	no	—
431	C₄²	—	4	2	no	—
432	C₄²	C₂×D₄×C₃₁	1	2	yes	—
433	C₄²	#41×C₃₁	1	2	yes	—
434	C₂³	—	2	2	no	—
435	C₂³	—	2	2	no	—
436	C₂³	—	2	2	no	—
437	C₂³	—	4	1	no	—
438	C₂³	—	4	1	no	—
439	C₂³	—	4	1	no	—
440	C₂³	—	4	1	no	—
441	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
442	Q₈	—	2	2	no	—
443	Q₈	—	2	2	no	—
444	Q₈	—	4	1	no	—
445	Q₈	—	4	1	no	—
446	Q₈	—	4	2	no	—
447	Q₈	—	4	2	no	—
448	Q₈	—	4	2	no	—
449	Q₈	—	4	2	no	—
450	D₄	—	2	2	no	—
451	D₄	—	2	2	no	—
452	D₄	—	2	2	no	—
453	D₄	C₃₁×#121	1	4	yes	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	1	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	1	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	—	2	2	no	—
463	C₂×C₄	—	4	1	no	—
464	C₂×C₄	—	4	1	no	—
465	C₂×C₄	—	4	1	no	—
466	C₂×C₄	—	4	1	no	—
467	C₂×C₄	—	4	8	no	—
468	C₂×C₄	—	4	1	no	—
469	C₂×C₄	—	4	2	no	—
470	C₂×C₄	—	4	2	no	—
471	C₂×C₄	—	4	2	no	—
472	C₂×C₄	—	4	2	no	—
473	C₂×C₄	—	4	2	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	4	1	no	—
476	C₂×C₄	—	4	1	no	—
477	C₂×C₄	—	8	8	no	—
478	C₂×C₄	C₃₁×#121	1	2	yes	—
479	C₂×C₄	C₃₁×#121	1	2	yes	—
480	C₂×C₄	C₂×SD₁₆×C₃₁	1	4	yes	—
481	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
482	C₂×C₄	C₃₁×#137	1	2	yes	—
483	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
484	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
485	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
486	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
487	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
488	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
489	C₈	—	2	2	no	—
490	C₈	—	4	2	no	—
491	C₈	—	4	1	no	—
492	C₂²	—	2	4	no	—
493	C₂²	—	4	2	no	—
494	C₂²	—	4	2	no	—
495	C₂²	—	4	2	no	—
496	C₂²	—	4	2	no	—
497	C₂²	—	4	2	no	—
498	C₂²	—	4	2	no	—
499	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
500	C₂²	D₄²×C₃₁	1	1	yes	characteristic
501	C₂²	C₃₁×#548	1	1	yes	characteristic
502	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
503	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
504	C₂²	C₃₁×#577	1	1	yes	characteristic
505	C₂²	C₃₁×#578	1	1	yes	characteristic
506	C₄	—	2	4	no	—
507	C₄	—	2	2	no	—
508	C₄	—	4	2	no	—
509	C₄	—	4	2	no	—
510	C₄	—	4	2	no	—
511	C₄	—	8	2	no	—
512	C₄	—	8	2	no	—
513	C₄	—	8	1	no	—
514	C₄	C₃₁×#461	1	4	yes	complemented
515	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
516	C₄	C₃₁×#537	1	1	yes	characteristic
517	C₄	C₃₁×#547	1	1	yes	characteristic
518	C₄	C₃₁×#553	1	1	yes	characteristic
519	C₂	—	4	4	no	—
520	C₂	C₃₁×#2416	1	1	yes	characteristic
521	C₂	C₂×C₃₁×#461	1	1	yes	characteristic
522	C₂	C₃₁×#3046	1	1	yes	characteristic
523	C₂	C₃₁×#3276	1	1	yes	characteristic
524	C₂	D₄×SD₁₆×C₃₁	1	1	yes	characteristic
525	C₂	C₃₁×#3359	1	1	yes	characteristic
526	C₂	C₃₁×#3372	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
4. C₃₁×#2546	C₂	semidirect
6. C₃₁×#2066	C₂	semidirect
7. C₃₁×#2043	C₂	semidirect
8. C₃₁×#1956	C₂	semidirect
9. C₃₁×#1939	C₂	semidirect
11. C₃₁×#1803	C₂	semidirect
161. C₄⋊C₄×C₃₁	SD₁₆	semidirect
170. #24730	C₃₁	direct (N × H)
232. #2546	C₂×C₃₁	semidirect
234. #2066	C₂×C₃₁	semidirect
235. #2043	C₂×C₃₁	semidirect
236. #1956	C₂×C₃₁	semidirect
237. #1939	C₂×C₃₁	semidirect
239. #1803	C₂×C₃₁	semidirect
262. C₄×C₃₁	#461	semidirect
364. C₃₁	#24730	direct (N × H)
425. C₄⋊C₄	SD₁₆×C₃₁	semidirect
514. C₄	C₃₁×#461	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃₁×#461	2
C₂×C₄×SD₁₆×C₃₁	2
C₃₁×#2546	2
D₄×C₄⋊C₄×C₃₁	2
C₃₁×#2066	2
C₃₁×#2043	2
C₃₁×#1956	2
C₃₁×#1939	2
C₃₁×#1877	2
C₃₁×#1803	2
#24730	31

Subgroups — C31×#24730 #21481489

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24730 #21481489