Subgroups — C₃₁×#24732 #21481491

Order 7936.

914 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#462	C₂	1	2	yes	maximal
2	C₂×C₃₁×#462	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2546	C₂	1	1	yes	maximal, characteristic
5	Q₈×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#2068	C₂	1	2	yes	maximal
7	C₃₁×#2043	C₂	1	2	yes	maximal
8	C₃₁×#1956	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#1939	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#1878	C₂	1	2	yes	maximal
11	C₃₁×#1803	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
13	C₂²×Q₁₆×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×Q₈×C₃₁	C₂²	1	2	yes	—
18	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#462	—	2	8	no	—
20	C₃₁×#462	—	2	4	no	—
21	C₄×Q₁₆×C₃₁	—	2	4	no	—
22	C₂×C₃₁×#107	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	2	yes	—
28	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×#103	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
32	C₃₁×#398	—	2	2	no	—
33	C₃₁×#391	C₂²	1	2	yes	—
34	C₃₁×#389	—	2	1	no	—
35	C₃₁×#389	—	2	1	no	—
36	C₃₁×#389	—	2	2	no	—
37	C₃₁×#389	C₂²	1	2	yes	—
38	C₃₁×#384	—	2	1	no	—
39	C₃₁×#384	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	2	yes	—
45	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₂×Q₁₆×C₃₁	—	2	2	no	—
48	C₂×Q₁₆×C₃₁	—	2	2	no	—
49	C₂×Q₁₆×C₃₁	—	2	4	no	—
50	C₂×Q₁₆×C₃₁	—	2	4	no	—
51	C₂×Q₁₆×C₃₁	—	2	4	no	—
52	C₂×Q₁₆×C₃₁	D₄	1	4	yes	—
53	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
55	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
56	C₃₁×C₄⋊Q₈	—	2	4	no	—
57	C₃₁×C₄⋊Q₈	—	2	2	no	—
58	C₃₁×C₄⋊Q₈	—	2	2	no	—
59	C₃₁×C₄⋊Q₈	—	4	2	no	—
60	C₄×Q₈×C₃₁	—	2	2	no	—
61	C₄×Q₈×C₃₁	—	2	4	no	—
62	C₄×Q₈×C₃₁	—	2	2	no	—
63	C₄×Q₈×C₃₁	—	2	2	no	—
64	C₄×Q₈×C₃₁	—	4	2	no	—
65	C₄×Q₈×C₃₁	D₄	1	4	yes	—
66	C₂×C₄⋊C₄×C₃₁	—	2	1	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	2	no	—
71	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
72	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
73	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
74	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
75	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
76	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
79	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
81	C₂×C₄²×C₃₁	—	2	1	no	—
82	C₂×C₄²×C₃₁	—	2	2	no	—
83	C₂×C₄²×C₃₁	—	2	1	no	—
84	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
85	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
86	C₃₁×#107	—	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	—	2	2	no	—
93	C₃₁×#103	—	2	4	no	—
94	C₃₁×#103	—	4	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	C₂×Q₈×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	1	no	—
105	C₂×Q₈×C₃₁	—	2	2	no	—
106	C₂×Q₈×C₃₁	—	2	2	no	—
107	C₂×Q₈×C₃₁	—	2	1	no	—
108	C₂×Q₈×C₃₁	—	2	1	no	—
109	C₂×Q₈×C₃₁	—	2	2	no	—
110	C₂×Q₈×C₃₁	—	2	2	no	—
111	C₂×Q₈×C₃₁	—	2	2	no	—
112	C₂×Q₈×C₃₁	—	2	1	no	—
113	C₂×Q₈×C₃₁	—	2	2	no	—
114	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
115	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
116	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
117	C₂×Q₈×C₃₁	#41	1	2	yes	—
118	C₂²×C₄×C₃₁	—	2	1	no	—
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	C₂×D₄	1	2	yes	—
125	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
127	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
129	Q₁₆×C₃₁	—	2	8	no	—
130	Q₁₆×C₃₁	—	4	8	no	—
131	C₂×C₈×C₃₁	—	2	1	no	—
132	C₂×C₈×C₃₁	—	2	1	no	—
133	C₂×C₈×C₃₁	—	2	1	no	—
134	C₂×C₈×C₃₁	—	2	2	no	—
135	C₂×C₈×C₃₁	—	2	2	no	—
136	C₂×C₈×C₃₁	—	2	2	no	—
137	C₂×C₈×C₃₁	—	2	1	no	—
138	C₂×C₈×C₃₁	—	2	1	no	—
139	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
140	C₄⋊C₄×C₃₁	—	2	4	no	—
141	C₄⋊C₄×C₃₁	—	2	2	no	—
142	C₄⋊C₄×C₃₁	—	2	1	no	—
143	C₄⋊C₄×C₃₁	—	2	2	no	—
144	C₄⋊C₄×C₃₁	—	2	1	no	—
145	C₄⋊C₄×C₃₁	—	2	1	no	—
146	C₄⋊C₄×C₃₁	—	2	2	no	—
147	C₄⋊C₄×C₃₁	—	4	4	no	—
148	C₄⋊C₄×C₃₁	—	4	4	no	—
149	C₄⋊C₄×C₃₁	—	4	4	no	—
150	C₄⋊C₄×C₃₁	—	4	2	no	—
151	C₄⋊C₄×C₃₁	—	4	2	no	—
152	C₄⋊C₄×C₃₁	—	4	2	no	—
153	C₄⋊C₄×C₃₁	—	4	2	no	—
154	C₄⋊C₄×C₃₁	—	8	1	no	—
155	C₄⋊C₄×C₃₁	—	8	2	no	—
156	C₄⋊C₄×C₃₁	Q₁₆	1	4	yes	complemented
157	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
158	C₄²×C₃₁	—	2	4	no	—
159	C₄²×C₃₁	—	2	2	no	—
160	C₄²×C₃₁	—	2	2	no	—
161	C₄²×C₃₁	—	4	4	no	—
162	C₄²×C₃₁	—	4	2	no	—
163	C₄²×C₃₁	C₂×D₄	1	2	yes	—
164	C₄²×C₃₁	#41	1	2	yes	—
165	#24732	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
166	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
167	Q₈×C₃₁	—	2	2	no	—
168	Q₈×C₃₁	—	2	2	no	—
169	Q₈×C₃₁	—	2	2	no	—
170	Q₈×C₃₁	—	2	2	no	—
171	Q₈×C₃₁	—	2	2	no	—
172	Q₈×C₃₁	—	4	1	no	—
173	Q₈×C₃₁	—	4	1	no	—
174	Q₈×C₃₁	—	4	2	no	—
175	Q₈×C₃₁	—	4	2	no	—
176	Q₈×C₃₁	—	4	2	no	—
177	Q₈×C₃₁	—	4	2	no	—
178	Q₈×C₃₁	#121	1	4	yes	—
179	C₂×C₄×C₃₁	—	2	2	no	—
180	C₂×C₄×C₃₁	—	2	2	no	—
181	C₂×C₄×C₃₁	—	2	2	no	—
182	C₂×C₄×C₃₁	—	2	2	no	—
183	C₂×C₄×C₃₁	—	2	1	no	—
184	C₂×C₄×C₃₁	—	2	2	no	—
185	C₂×C₄×C₃₁	—	2	1	no	—
186	C₂×C₄×C₃₁	—	2	2	no	—
187	C₂×C₄×C₃₁	—	2	2	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	—	4	1	no	—
192	C₂×C₄×C₃₁	—	4	1	no	—
193	C₂×C₄×C₃₁	—	4	1	no	—
194	C₂×C₄×C₃₁	—	4	1	no	—
195	C₂×C₄×C₃₁	—	4	1	no	—
196	C₂×C₄×C₃₁	—	4	2	no	—
197	C₂×C₄×C₃₁	—	4	2	no	—
198	C₂×C₄×C₃₁	—	4	2	no	—
199	C₂×C₄×C₃₁	—	4	2	no	—
200	C₂×C₄×C₃₁	—	4	2	no	—
201	C₂×C₄×C₃₁	—	4	2	no	—
202	C₂×C₄×C₃₁	—	4	1	no	—
203	C₂×C₄×C₃₁	—	4	1	no	—
204	C₂×C₄×C₃₁	#121	1	2	yes	—
205	C₂×C₄×C₃₁	#121	1	2	yes	—
206	C₂×C₄×C₃₁	C₂×Q₁₆	1	4	yes	—
207	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
208	C₂×C₄×C₃₁	#137	1	2	yes	—
209	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
210	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
211	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
212	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
213	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
214	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
215	C₈×C₃₁	—	2	2	no	—
216	C₈×C₃₁	—	4	2	no	—
217	C₈×C₃₁	—	4	1	no	—
218	C₂×#462	C₂×C₃₁	1	2	yes	—
219	C₂×#462	C₂×C₃₁	1	1	yes	characteristic
220	C₂×C₄×Q₁₆	C₂×C₃₁	1	1	yes	characteristic
221	#2546	C₂×C₃₁	1	1	yes	characteristic
222	Q₈×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic
223	#2068	C₂×C₃₁	1	2	yes	—
224	#2043	C₂×C₃₁	1	2	yes	—
225	#1956	C₂×C₃₁	1	1	yes	characteristic
226	#1939	C₂×C₃₁	1	1	yes	characteristic
227	#1878	C₂×C₃₁	1	2	yes	—
228	#1803	C₂×C₃₁	1	1	yes	characteristic
229	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
230	C₂²×C₃₁	D₄²	1	1	yes	characteristic
231	C₂²×C₃₁	#548	1	1	yes	characteristic
232	C₂²×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
233	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
234	C₂²×C₃₁	#577	1	1	yes	characteristic
235	C₂²×C₃₁	#578	1	1	yes	characteristic
236	C₄×C₃₁	—	2	4	no	—
237	C₄×C₃₁	—	2	4	no	—
238	C₄×C₃₁	—	2	2	no	—
239	C₄×C₃₁	—	4	2	no	—
240	C₄×C₃₁	—	4	2	no	—
241	C₄×C₃₁	—	4	2	no	—
242	C₄×C₃₁	—	8	2	no	—
243	C₄×C₃₁	—	8	2	no	—
244	C₄×C₃₁	—	8	1	no	—
245	C₄×C₃₁	#462	1	4	yes	complemented
246	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
247	C₄×C₃₁	#537	1	1	yes	characteristic
248	C₄×C₃₁	#547	1	1	yes	characteristic
249	C₄×C₃₁	#553	1	1	yes	characteristic
250	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
251	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
252	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
253	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
254	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
255	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
256	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
257	#462	—	2	8	no	—
258	#462	—	2	4	no	—
259	C₄×Q₁₆	—	2	4	no	—
260	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
261	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
262	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
263	C₂×#103	C₂²×C₃₁	1	2	yes	—
264	C₂×#103	C₂²×C₃₁	1	2	yes	—
265	C₂×#103	C₂²×C₃₁	1	2	yes	—
266	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
267	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
268	C₂×#103	C₂²×C₃₁	1	2	yes	—
269	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
270	#398	—	2	2	no	—
271	#391	C₂²×C₃₁	1	2	yes	—
272	#389	—	2	1	no	—
273	#389	—	2	1	no	—
274	#389	—	2	2	no	—
275	#389	C₂²×C₃₁	1	2	yes	—
276	#384	—	2	1	no	—
277	#384	C₂²×C₃₁	1	1	yes	characteristic
278	C₄×C₄⋊C₄	—	2	1	no	—
279	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
280	#340	C₂²×C₃₁	1	2	yes	—
281	#336	C₂²×C₃₁	1	2	yes	—
282	C₂×C₃₁	#2416	1	1	yes	characteristic
283	C₂×C₃₁	C₂×#462	1	1	yes	characteristic
284	C₂×C₃₁	#3046	1	1	yes	characteristic
285	C₂×C₃₁	#3278	1	1	yes	characteristic
286	C₂×C₃₁	D₄×Q₁₆	1	1	yes	characteristic
287	C₂×C₃₁	#3363	1	1	yes	characteristic
288	C₂×C₃₁	#3374	1	1	yes	characteristic
289	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
290	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
291	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
292	C₂×Q₁₆	—	2	2	no	—
293	C₂×Q₁₆	—	2	2	no	—
294	C₂×Q₁₆	—	2	4	no	—
295	C₂×Q₁₆	—	2	4	no	—
296	C₂×Q₁₆	—	2	4	no	—
297	C₂×Q₁₆	D₄×C₃₁	1	4	yes	—
298	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
299	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
300	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
301	C₄⋊Q₈	—	2	4	no	—
302	C₄⋊Q₈	—	2	2	no	—
303	C₄⋊Q₈	—	2	2	no	—
304	C₄⋊Q₈	—	4	2	no	—
305	C₄×Q₈	—	2	2	no	—
306	C₄×Q₈	—	2	4	no	—
307	C₄×Q₈	—	2	2	no	—
308	C₄×Q₈	—	2	2	no	—
309	C₄×Q₈	—	4	2	no	—
310	C₄×Q₈	D₄×C₃₁	1	4	yes	—
311	C₂×C₄⋊C₄	—	2	1	no	—
312	C₂×C₄⋊C₄	—	2	2	no	—
313	C₂×C₄⋊C₄	—	2	1	no	—
314	C₂×C₄⋊C₄	—	2	1	no	—
315	C₂×C₄⋊C₄	—	2	2	no	—
316	C₂×C₄⋊C₄	—	2	2	no	—
317	C₂×C₄⋊C₄	—	2	1	no	—
318	C₂×C₄⋊C₄	—	2	1	no	—
319	C₂×C₄⋊C₄	—	2	2	no	—
320	C₂×C₄⋊C₄	—	2	1	no	—
321	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
322	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
323	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
324	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
325	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
326	C₂×C₄²	—	2	1	no	—
327	C₂×C₄²	—	2	2	no	—
328	C₂×C₄²	—	2	1	no	—
329	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
330	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
331	#107	—	2	2	no	—
332	C₄⋊C₈	—	2	4	no	—
333	C₄⋊C₈	—	2	2	no	—
334	#103	—	2	2	no	—
335	#103	—	2	4	no	—
336	#103	—	2	4	no	—
337	#103	—	2	2	no	—
338	#103	—	2	4	no	—
339	#103	—	4	2	no	—
340	C₄×C₈	—	2	2	no	—
341	#95	—	2	2	no	—
342	#95	—	2	2	no	—
343	#95	—	2	2	no	—
344	#95	D₄×C₃₁	1	2	yes	—
345	C₃₁	#24732	1	1	yes	characteristic, direct product factor, Sylow
346	C₂×Q₈	—	2	1	no	—
347	C₂×Q₈	—	2	1	no	—
348	C₂×Q₈	—	2	1	no	—
349	C₂×Q₈	—	2	1	no	—
350	C₂×Q₈	—	2	1	no	—
351	C₂×Q₈	—	2	2	no	—
352	C₂×Q₈	—	2	2	no	—
353	C₂×Q₈	—	2	1	no	—
354	C₂×Q₈	—	2	1	no	—
355	C₂×Q₈	—	2	2	no	—
356	C₂×Q₈	—	2	2	no	—
357	C₂×Q₈	—	2	2	no	—
358	C₂×Q₈	—	2	1	no	—
359	C₂×Q₈	—	2	2	no	—
360	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
361	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
362	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
363	C₂×Q₈	#41×C₃₁	1	2	yes	—
364	C₂²×C₄	—	2	1	no	—
365	C₂²×C₄	—	2	1	no	—
366	C₂²×C₄	—	2	1	no	—
367	C₂²×C₄	—	2	1	no	—
368	C₂²×C₄	—	2	2	no	—
369	C₂²×C₄	—	2	2	no	—
370	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
371	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
372	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
373	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
374	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
375	Q₁₆	—	2	8	no	—
376	Q₁₆	—	4	8	no	—
377	C₂×C₈	—	2	1	no	—
378	C₂×C₈	—	2	1	no	—
379	C₂×C₈	—	2	1	no	—
380	C₂×C₈	—	2	2	no	—
381	C₂×C₈	—	2	2	no	—
382	C₂×C₈	—	2	2	no	—
383	C₂×C₈	—	2	1	no	—
384	C₂×C₈	—	2	1	no	—
385	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
386	C₄⋊C₄	—	2	4	no	—
387	C₄⋊C₄	—	2	2	no	—
388	C₄⋊C₄	—	2	1	no	—
389	C₄⋊C₄	—	2	2	no	—
390	C₄⋊C₄	—	2	1	no	—
391	C₄⋊C₄	—	2	1	no	—
392	C₄⋊C₄	—	2	2	no	—
393	C₄⋊C₄	—	4	4	no	—
394	C₄⋊C₄	—	4	4	no	—
395	C₄⋊C₄	—	4	4	no	—
396	C₄⋊C₄	—	4	2	no	—
397	C₄⋊C₄	—	4	2	no	—
398	C₄⋊C₄	—	4	2	no	—
399	C₄⋊C₄	—	4	2	no	—
400	C₄⋊C₄	—	8	1	no	—
401	C₄⋊C₄	—	8	2	no	—
402	C₄⋊C₄	Q₁₆×C₃₁	1	4	yes	complemented
403	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
404	C₄²	—	2	4	no	—
405	C₄²	—	2	2	no	—
406	C₄²	—	2	2	no	—
407	C₄²	—	4	4	no	—
408	C₄²	—	4	2	no	—
409	C₄²	C₂×D₄×C₃₁	1	2	yes	—
410	C₄²	#41×C₃₁	1	2	yes	—
411	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
412	Q₈	—	2	2	no	—
413	Q₈	—	2	2	no	—
414	Q₈	—	2	2	no	—
415	Q₈	—	2	2	no	—
416	Q₈	—	2	2	no	—
417	Q₈	—	4	1	no	—
418	Q₈	—	4	1	no	—
419	Q₈	—	4	2	no	—
420	Q₈	—	4	2	no	—
421	Q₈	—	4	2	no	—
422	Q₈	—	4	2	no	—
423	Q₈	C₃₁×#121	1	4	yes	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	4	1	no	—
437	C₂×C₄	—	4	1	no	—
438	C₂×C₄	—	4	1	no	—
439	C₂×C₄	—	4	1	no	—
440	C₂×C₄	—	4	1	no	—
441	C₂×C₄	—	4	2	no	—
442	C₂×C₄	—	4	2	no	—
443	C₂×C₄	—	4	2	no	—
444	C₂×C₄	—	4	2	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	4	2	no	—
447	C₂×C₄	—	4	1	no	—
448	C₂×C₄	—	4	1	no	—
449	C₂×C₄	C₃₁×#121	1	2	yes	—
450	C₂×C₄	C₃₁×#121	1	2	yes	—
451	C₂×C₄	C₂×Q₁₆×C₃₁	1	4	yes	—
452	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
453	C₂×C₄	C₃₁×#137	1	2	yes	—
454	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
455	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
456	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
457	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
458	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
459	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
460	C₈	—	2	2	no	—
461	C₈	—	4	2	no	—
462	C₈	—	4	1	no	—
463	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
464	C₂²	D₄²×C₃₁	1	1	yes	characteristic
465	C₂²	C₃₁×#548	1	1	yes	characteristic
466	C₂²	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
467	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
468	C₂²	C₃₁×#577	1	1	yes	characteristic
469	C₂²	C₃₁×#578	1	1	yes	characteristic
470	C₄	—	2	4	no	—
471	C₄	—	2	4	no	—
472	C₄	—	2	2	no	—
473	C₄	—	4	2	no	—
474	C₄	—	4	2	no	—
475	C₄	—	4	2	no	—
476	C₄	—	8	2	no	—
477	C₄	—	8	2	no	—
478	C₄	—	8	1	no	—
479	C₄	C₃₁×#462	1	4	yes	complemented
480	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
481	C₄	C₃₁×#537	1	1	yes	characteristic
482	C₄	C₃₁×#547	1	1	yes	characteristic
483	C₄	C₃₁×#553	1	1	yes	characteristic
484	C₂	C₃₁×#2416	1	1	yes	characteristic
485	C₂	C₂×C₃₁×#462	1	1	yes	characteristic
486	C₂	C₃₁×#3046	1	1	yes	characteristic
487	C₂	C₃₁×#3278	1	1	yes	characteristic
488	C₂	D₄×Q₁₆×C₃₁	1	1	yes	characteristic
489	C₂	C₃₁×#3363	1	1	yes	characteristic
490	C₂	C₃₁×#3374	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
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
156. C₄⋊C₄×C₃₁	Q₁₆	semidirect
165. #24732	C₃₁	direct (N × H)
245. C₄×C₃₁	#462	semidirect
345. C₃₁	#24732	direct (N × H)
402. C₄⋊C₄	Q₁₆×C₃₁	semidirect
479. C₄	C₃₁×#462	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#462	2
C₂×C₄×Q₁₆×C₃₁	2
C₃₁×#2546	2
Q₈×C₄⋊C₄×C₃₁	2
C₃₁×#2068	2
C₃₁×#2043	2
C₃₁×#1956	2
C₃₁×#1939	2
C₃₁×#1878	2
C₃₁×#1803	2
#24732	31

Subgroups — C31×#24732 #21481491

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24732 #21481491