Subgroups — C₃₁×#22240 #21478999

Order 7936.

750 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#446	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1984	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1984	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1917	C₂	1	4	yes	maximal
7	C₃₁×#1916	C₂	1	4	yes	maximal
8	C₃₁×#1831	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#1744	C₂	1	1	yes	maximal, characteristic
10	C₂×C₃₁×C₄⋊Q₈	C₄	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#446	—	2	4	no	—
14	C₈×Q₈×C₃₁	—	2	4	no	—
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
19	C₂×C₃₁×#97	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₃₁×#391	C₄	1	1	yes	characteristic
24	C₃₁×#391	C₄	1	1	yes	characteristic
25	C₃₁×#389	C₄	1	4	yes	—
26	C₃₁×#384	C₄	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
28	C₄³×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#336	C₂²	1	4	yes	—
30	C₃₁×#336	C₂²	1	2	yes	—
31	C₃₁×#336	C₂²	1	4	yes	—
32	C₃₁×#336	C₂²	1	4	yes	—
33	C₃₁×#336	C₂²	1	2	yes	—
34	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
35	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
37	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
38	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
39	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
41	C₃₁×C₄⋊Q₈	—	2	4	no	—
42	C₄×Q₈×C₃₁	—	2	2	no	—
43	C₄×Q₈×C₃₁	—	2	2	no	—
44	C₄×Q₈×C₃₁	—	2	2	no	—
45	C₄×Q₈×C₃₁	—	2	2	no	—
46	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
49	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
51	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
52	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
56	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
57	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
58	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
59	C₃₁×C₄⋊C₈	—	2	4	no	—
60	C₃₁×C₄⋊C₈	—	2	4	no	—
61	C₃₁×C₄⋊C₈	—	2	2	no	—
62	C₃₁×C₄⋊C₈	—	2	2	no	—
63	C₃₁×#97	—	2	4	no	—
64	C₄×C₈×C₃₁	—	2	4	no	—
65	C₄×C₈×C₃₁	—	2	2	no	—
66	C₄×C₈×C₃₁	D₄	1	4	yes	—
67	C₃₁×#95	C₂×C₄	1	4	yes	—
68	C₃₁×#95	C₂×C₄	1	4	yes	—
69	C₂×Q₈×C₃₁	—	2	2	no	—
70	C₂×Q₈×C₃₁	—	2	2	no	—
71	C₂×Q₈×C₃₁	—	2	2	no	—
72	C₂×Q₈×C₃₁	—	2	2	no	—
73	C₂×Q₈×C₃₁	—	2	2	no	—
74	C₂×Q₈×C₃₁	M₁₆	1	4	yes	—
75	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
84	C₂×C₈×C₃₁	—	2	2	no	—
85	C₂×C₈×C₃₁	—	2	2	no	—
86	C₂×C₈×C₃₁	—	2	2	no	—
87	C₂×C₈×C₃₁	—	2	2	no	—
88	C₂×C₈×C₃₁	—	2	2	no	—
89	C₂×C₈×C₃₁	—	2	4	no	—
90	C₂×C₈×C₃₁	—	2	4	no	—
91	C₂×C₈×C₃₁	—	2	2	no	—
92	C₂×C₈×C₃₁	—	2	2	no	—
93	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
95	C₄⋊C₄×C₃₁	—	2	4	no	—
96	C₄⋊C₄×C₃₁	—	2	2	no	—
97	C₄⋊C₄×C₃₁	—	2	2	no	—
98	C₄⋊C₄×C₃₁	—	2	4	no	—
99	C₄⋊C₄×C₃₁	—	2	8	no	—
100	C₄⋊C₄×C₃₁	—	2	8	no	—
101	C₄²×C₃₁	—	2	2	no	—
102	C₄²×C₃₁	—	2	4	no	—
103	C₄²×C₃₁	—	2	2	no	—
104	C₄²×C₃₁	—	2	2	no	—
105	C₄²×C₃₁	—	2	2	no	—
106	C₄²×C₃₁	—	2	2	no	—
107	C₄²×C₃₁	—	2	2	no	—
108	C₄²×C₃₁	—	2	2	no	—
109	C₄²×C₃₁	—	2	2	no	—
110	C₄²×C₃₁	C₂×D₄	1	2	yes	—
111	C₄²×C₃₁	#41	1	2	yes	—
112	#22240	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
114	Q₈×C₃₁	—	2	8	no	—
115	Q₈×C₃₁	—	4	4	no	—
116	C₂×C₄×C₃₁	—	2	2	no	—
117	C₂×C₄×C₃₁	—	2	2	no	—
118	C₂×C₄×C₃₁	—	2	2	no	—
119	C₂×C₄×C₃₁	—	2	2	no	—
120	C₂×C₄×C₃₁	—	2	2	no	—
121	C₂×C₄×C₃₁	—	2	2	no	—
122	C₂×C₄×C₃₁	—	2	2	no	—
123	C₂×C₄×C₃₁	—	2	2	no	—
124	C₂×C₄×C₃₁	—	2	2	no	—
125	C₂×C₄×C₃₁	—	2	2	no	—
126	C₂×C₄×C₃₁	—	2	2	no	—
127	C₂×C₄×C₃₁	—	2	2	no	—
128	C₂×C₄×C₃₁	—	2	4	no	—
129	C₂×C₄×C₃₁	—	2	4	no	—
130	C₂×C₄×C₃₁	—	2	2	no	—
131	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
132	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
133	C₂×C₄×C₃₁	C₂×M₁₆	1	4	yes	—
134	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
135	C₂×C₄×C₃₁	#131	1	4	yes	—
136	C₂×C₄×C₃₁	#131	1	2	yes	—
137	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
139	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
140	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
141	C₈×C₃₁	—	2	2	no	—
142	C₈×C₃₁	—	2	2	no	—
143	C₈×C₃₁	—	4	2	no	—
144	C₈×C₃₁	—	4	2	no	—
145	C₈×C₃₁	—	4	2	no	—
146	C₂×#446	C₂×C₃₁	1	1	yes	characteristic
147	C₂×C₈×Q₈	C₂×C₃₁	1	1	yes	characteristic
148	C₄×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic
149	#1984	C₂×C₃₁	1	1	yes	characteristic
150	#1984	C₂×C₃₁	1	1	yes	characteristic
151	#1917	C₂×C₃₁	1	4	yes	—
152	#1916	C₂×C₃₁	1	4	yes	—
153	#1831	C₂×C₃₁	1	1	yes	characteristic
154	#1744	C₂×C₃₁	1	1	yes	characteristic
155	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
156	C₂²×C₃₁	#519	1	2	yes	—
157	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
158	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
159	C₂²×C₃₁	C₂×#131	1	2	yes	—
160	C₄×C₃₁	—	2	4	no	—
161	C₄×C₃₁	—	2	2	no	—
162	C₄×C₃₁	—	2	4	no	—
163	C₄×C₃₁	—	4	2	no	—
164	C₄×C₃₁	—	4	2	no	—
165	C₄×C₃₁	—	4	2	no	—
166	C₄×C₃₁	—	4	2	no	—
167	C₄×C₃₁	#435	1	4	yes	—
168	C₄×C₃₁	#536	1	1	yes	characteristic
169	C₄×C₃₁	#536	1	1	yes	characteristic
170	C₄×C₃₁	#541	1	2	yes	—
171	C₂×C₄⋊Q₈	C₄×C₃₁	1	1	yes	characteristic
172	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
173	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
174	#446	—	2	4	no	—
175	C₈×Q₈	—	2	4	no	—
176	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
177	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
178	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
179	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
180	C₂×#97	C₂²×C₃₁	1	2	yes	—
181	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
182	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
183	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
184	#391	C₄×C₃₁	1	1	yes	characteristic
185	#391	C₄×C₃₁	1	1	yes	characteristic
186	#389	C₄×C₃₁	1	4	yes	—
187	#384	C₄×C₃₁	1	1	yes	characteristic
188	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
189	C₄³	C₂²×C₃₁	1	1	yes	characteristic
190	#336	C₂²×C₃₁	1	4	yes	—
191	#336	C₂²×C₃₁	1	2	yes	—
192	#336	C₂²×C₃₁	1	4	yes	—
193	#336	C₂²×C₃₁	1	4	yes	—
194	#336	C₂²×C₃₁	1	2	yes	—
195	C₂×C₃₁	#2327	1	1	yes	characteristic
196	C₂×C₃₁	C₂×#435	1	1	yes	characteristic
197	C₂×C₃₁	#2931	1	1	yes	characteristic
198	C₂×C₃₁	#2984	1	2	yes	—
199	C₂×C₃₁	#2986	1	2	yes	—
200	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
201	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
202	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
203	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
204	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
205	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
206	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
207	C₄⋊Q₈	—	2	4	no	—
208	C₄×Q₈	—	2	2	no	—
209	C₄×Q₈	—	2	2	no	—
210	C₄×Q₈	—	2	2	no	—
211	C₄×Q₈	—	2	2	no	—
212	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
213	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
214	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
215	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
216	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
217	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
218	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
219	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
220	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
221	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
222	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
223	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
224	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
225	C₄⋊C₈	—	2	4	no	—
226	C₄⋊C₈	—	2	4	no	—
227	C₄⋊C₈	—	2	2	no	—
228	C₄⋊C₈	—	2	2	no	—
229	#97	—	2	4	no	—
230	C₄×C₈	—	2	4	no	—
231	C₄×C₈	—	2	2	no	—
232	C₄×C₈	D₄×C₃₁	1	4	yes	—
233	#95	C₂×C₄×C₃₁	1	4	yes	—
234	#95	C₂×C₄×C₃₁	1	4	yes	—
235	C₃₁	#22240	1	1	yes	characteristic, direct product factor, Sylow
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	M₁₆×C₃₁	1	4	yes	—
242	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
243	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
244	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
245	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
246	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
247	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
248	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
249	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
250	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
261	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
262	C₄⋊C₄	—	2	4	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	2	8	no	—
267	C₄⋊C₄	—	2	8	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	4	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	C₂×D₄×C₃₁	1	2	yes	—
278	C₄²	#41×C₃₁	1	2	yes	—
279	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
280	Q₈	—	2	8	no	—
281	Q₈	—	4	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
298	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
299	C₂×C₄	C₂×M₁₆×C₃₁	1	4	yes	—
300	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
301	C₂×C₄	C₃₁×#131	1	4	yes	—
302	C₂×C₄	C₃₁×#131	1	2	yes	—
303	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
304	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
305	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
306	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
307	C₈	—	2	2	no	—
308	C₈	—	2	2	no	—
309	C₈	—	4	2	no	—
310	C₈	—	4	2	no	—
311	C₈	—	4	2	no	—
312	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
313	C₂²	C₃₁×#519	1	2	yes	—
314	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
315	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
316	C₂²	C₂×C₃₁×#131	1	2	yes	—
317	C₄	—	2	4	no	—
318	C₄	—	2	2	no	—
319	C₄	—	2	4	no	—
320	C₄	—	4	2	no	—
321	C₄	—	4	2	no	—
322	C₄	—	4	2	no	—
323	C₄	—	4	2	no	—
324	C₄	C₃₁×#435	1	4	yes	—
325	C₄	C₃₁×#536	1	1	yes	characteristic
326	C₄	C₃₁×#536	1	1	yes	characteristic
327	C₄	C₃₁×#541	1	2	yes	—
328	C₂	C₃₁×#2327	1	1	yes	characteristic
329	C₂	C₂×C₃₁×#435	1	1	yes	characteristic
330	C₂	C₃₁×#2931	1	1	yes	characteristic
331	C₂	C₃₁×#2984	1	2	yes	—
332	C₂	C₃₁×#2986	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
112. #22240	C₃₁	direct (N × H)
235. C₃₁	#22240	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×#446	2
C₂×C₈×Q₈×C₃₁	2
C₄×C₃₁×C₄⋊Q₈	2
C₃₁×#1984	2
C₃₁×#1917	2
C₃₁×#1916	2
C₃₁×#1831	2
C₃₁×#1744	2
#22240	31

Subgroups — C31×#22240 #21478999

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22240 #21478999