Subgroups — C₃₁×#29185 #21485944

Order 7936.

590 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2647	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2330	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2088	C₂	1	2	yes	maximal
4	C₃₁×#2088	C₂	1	2	yes	maximal
5	C₃₁×#1992	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1991	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1920	C₂	1	2	yes	maximal
8	C₃₁×#1919	C₂	1	2	yes	maximal
9	C₃₁×#1844	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#1836	C₂	1	1	yes	maximal, characteristic
11	C₃₁×#1768	C₂	1	1	yes	maximal, characteristic
12	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#106	C₂²	1	2	yes	—
14	C₂×C₃₁×#106	C₂²	1	2	yes	—
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#400	—	2	2	no	—
20	C₃₁×#398	—	2	2	no	—
21	C₃₁×#389	—	2	2	no	—
22	C₃₁×#389	C₂²	1	2	yes	—
23	C₃₁×#389	C₂²	1	2	yes	—
24	C₃₁×#389	C₂²	1	2	yes	—
25	C₃₁×#387	—	2	2	no	—
26	C₃₁×#384	C₂²	1	1	yes	characteristic
27	C₃₁×#384	C₂²	1	1	yes	characteristic
28	C₃₁×#382	C₂²	1	1	yes	characteristic
29	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
30	C₃₁×#376	C₂²	1	1	yes	characteristic
31	C₃₁×#340	C₂²	1	2	yes	—
32	C₃₁×#340	C₂²	1	2	yes	—
33	C₃₁×#340	C₂²	1	2	yes	—
34	C₃₁×#340	C₂²	1	2	yes	—
35	C₃₁×#340	C₂²	1	2	yes	—
36	C₃₁×#340	C₂²	1	2	yes	—
37	C₃₁×#336	C₂²	1	1	yes	characteristic
38	C₃₁×#336	C₂²	1	1	yes	characteristic
39	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
43	C₃₁×#125	—	2	2	no	—
44	C₃₁×#125	Q₈	1	4	yes	—
45	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
46	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
47	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
48	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
49	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
50	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
51	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
55	C₂×C₄²×C₃₁	—	2	2	no	—
56	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
59	C₃₁×#106	—	2	4	no	—
60	C₃₁×#106	—	2	4	no	—
61	C₃₁×C₄⋊C₈	—	2	2	no	—
62	C₃₁×C₄⋊C₈	—	2	2	no	—
63	C₃₁×#97	—	2	2	no	—
64	C₄×C₈×C₃₁	—	2	2	no	—
65	C₃₁×#95	—	2	2	no	—
66	C₃₁×#95	—	2	2	no	—
67	C₃₁×#95	—	2	2	no	—
68	C₃₁×#95	—	2	2	no	—
69	C₃₁×#95	—	2	2	no	—
70	C₃₁×#95	D₄	1	2	yes	—
71	C₃₁×#95	D₄	1	2	yes	—
72	C₂²×C₄×C₃₁	—	2	2	no	—
73	C₂²×C₄×C₃₁	—	2	2	no	—
74	C₂²×C₄×C₃₁	—	2	2	no	—
75	C₂²×C₄×C₃₁	—	2	2	no	—
76	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
81	C₂×C₈×C₃₁	—	2	1	no	—
82	C₂×C₈×C₃₁	—	2	1	no	—
83	C₂×C₈×C₃₁	—	2	1	no	—
84	C₂×C₈×C₃₁	—	2	1	no	—
85	C₂×C₈×C₃₁	—	2	1	no	—
86	C₂×C₈×C₃₁	—	2	1	no	—
87	C₂×C₈×C₃₁	—	2	1	no	—
88	C₂×C₈×C₃₁	—	2	1	no	—
89	C₂×C₈×C₃₁	—	2	1	no	—
90	C₂×C₈×C₃₁	—	2	1	no	—
91	C₂×C₈×C₃₁	#41	1	2	yes	—
92	C₂×C₈×C₃₁	#41	1	2	yes	—
93	C₄⋊C₄×C₃₁	—	2	8	no	—
94	C₄⋊C₄×C₃₁	—	2	2	no	—
95	C₄⋊C₄×C₃₁	—	2	2	no	—
96	C₄⋊C₄×C₃₁	—	2	2	no	—
97	C₄⋊C₄×C₃₁	—	2	2	no	—
98	C₄⋊C₄×C₃₁	—	2	2	no	—
99	C₄⋊C₄×C₃₁	—	2	2	no	—
100	C₄⋊C₄×C₃₁	—	2	2	no	—
101	C₄⋊C₄×C₃₁	—	8	2	no	—
102	C₄⋊C₄×C₃₁	—	8	2	no	—
103	C₄⋊C₄×C₃₁	—	8	2	no	—
104	C₄⋊C₄×C₃₁	—	8	2	no	—
105	C₄⋊C₄×C₃₁	—	8	2	no	—
106	C₄⋊C₄×C₃₁	C₂×Q₈	1	4	yes	—
107	C₄²×C₃₁	—	2	1	no	—
108	C₄²×C₃₁	—	2	1	no	—
109	C₄²×C₃₁	—	2	1	no	—
110	C₄²×C₃₁	—	2	1	no	—
111	C₄²×C₃₁	—	8	2	no	—
112	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
113	C₄²×C₃₁	#41	1	2	yes	—
114	#29185	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₃₁	—	2	2	no	—
117	C₂×C₄×C₃₁	—	2	2	no	—
118	C₂×C₄×C₃₁	—	2	4	no	—
119	C₂×C₄×C₃₁	—	2	1	no	—
120	C₂×C₄×C₃₁	—	2	1	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	—	4	2	no	—
127	C₂×C₄×C₃₁	—	4	2	no	—
128	C₂×C₄×C₃₁	—	4	2	no	—
129	C₂×C₄×C₃₁	—	4	2	no	—
130	C₂×C₄×C₃₁	—	4	2	no	—
131	C₂×C₄×C₃₁	—	4	2	no	—
132	C₂×C₄×C₃₁	—	4	2	no	—
133	C₂×C₄×C₃₁	—	4	2	no	—
134	C₂×C₄×C₃₁	—	4	2	no	—
135	C₂×C₄×C₃₁	—	4	2	no	—
136	C₂×C₄×C₃₁	—	4	2	no	—
137	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
138	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
139	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
143	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
144	C₈×C₃₁	—	2	2	no	—
145	C₈×C₃₁	—	2	2	no	—
146	C₈×C₃₁	—	4	1	no	—
147	C₈×C₃₁	—	4	1	no	—
148	#2647	C₂×C₃₁	1	1	yes	characteristic
149	#2330	C₂×C₃₁	1	1	yes	characteristic
150	#2088	C₂×C₃₁	1	2	yes	—
151	#2088	C₂×C₃₁	1	2	yes	—
152	#1992	C₂×C₃₁	1	1	yes	characteristic
153	#1991	C₂×C₃₁	1	1	yes	characteristic
154	#1920	C₂×C₃₁	1	2	yes	—
155	#1919	C₂×C₃₁	1	2	yes	—
156	#1844	C₂×C₃₁	1	1	yes	characteristic
157	#1836	C₂×C₃₁	1	1	yes	characteristic
158	#1768	C₂×C₃₁	1	1	yes	characteristic
159	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
160	C₂²×C₃₁	#532	1	1	yes	characteristic
161	C₂²×C₃₁	#536	1	1	yes	characteristic
162	C₂²×C₃₁	#577	1	2	yes	—
163	C₂²×C₃₁	#577	1	2	yes	—
164	C₄×C₃₁	—	2	4	no	—
165	C₄×C₃₁	—	4	4	no	—
166	C₄×C₃₁	—	4	1	no	—
167	C₄×C₃₁	—	4	1	no	—
168	C₄×C₃₁	—	4	2	no	—
169	C₄×C₃₁	—	8	2	no	—
170	C₄×C₃₁	—	8	2	no	—
171	C₄×C₃₁	—	8	2	no	—
172	C₄×C₃₁	#552	1	1	yes	characteristic
173	C₄×C₃₁	#552	1	1	yes	characteristic
174	C₄×C₃₁	#557	1	1	yes	characteristic
175	C₄×C₃₁	#557	1	1	yes	characteristic
176	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
177	C₂×#106	C₂²×C₃₁	1	2	yes	—
178	C₂×#106	C₂²×C₃₁	1	2	yes	—
179	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
180	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
181	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
182	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
183	#400	—	2	2	no	—
184	#398	—	2	2	no	—
185	#389	—	2	2	no	—
186	#389	C₂²×C₃₁	1	2	yes	—
187	#389	C₂²×C₃₁	1	2	yes	—
188	#389	C₂²×C₃₁	1	2	yes	—
189	#387	—	2	2	no	—
190	#384	C₂²×C₃₁	1	1	yes	characteristic
191	#384	C₂²×C₃₁	1	1	yes	characteristic
192	#382	C₂²×C₃₁	1	1	yes	characteristic
193	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
194	#376	C₂²×C₃₁	1	1	yes	characteristic
195	#340	C₂²×C₃₁	1	2	yes	—
196	#340	C₂²×C₃₁	1	2	yes	—
197	#340	C₂²×C₃₁	1	2	yes	—
198	#340	C₂²×C₃₁	1	2	yes	—
199	#340	C₂²×C₃₁	1	2	yes	—
200	#340	C₂²×C₃₁	1	2	yes	—
201	#336	C₂²×C₃₁	1	1	yes	characteristic
202	#336	C₂²×C₃₁	1	1	yes	characteristic
203	C₂×C₃₁	#2565	1	1	yes	characteristic
204	C₂×C₃₁	#3084	1	2	yes	—
205	C₂×C₃₁	#3174	1	1	yes	characteristic
206	C₂×C₃₁	#3174	1	1	yes	characteristic
207	C₂×C₃₁	#3244	1	2	yes	—
208	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
209	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
210	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
211	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
212	#125	—	2	2	no	—
213	#125	Q₈×C₃₁	1	4	yes	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	—	2	2	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₄⋊C₄	—	2	2	no	—
220	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
221	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
222	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
223	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
224	C₂×C₄²	—	2	2	no	—
225	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
226	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
227	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
228	#106	—	2	4	no	—
229	#106	—	2	4	no	—
230	C₄⋊C₈	—	2	2	no	—
231	C₄⋊C₈	—	2	2	no	—
232	#97	—	2	2	no	—
233	C₄×C₈	—	2	2	no	—
234	#95	—	2	2	no	—
235	#95	—	2	2	no	—
236	#95	—	2	2	no	—
237	#95	—	2	2	no	—
238	#95	—	2	2	no	—
239	#95	D₄×C₃₁	1	2	yes	—
240	#95	D₄×C₃₁	1	2	yes	—
241	C₃₁	#29185	1	1	yes	characteristic, direct product factor, Sylow
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
247	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
249	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
250	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	#41×C₃₁	1	2	yes	—
262	C₂×C₈	#41×C₃₁	1	2	yes	—
263	C₄⋊C₄	—	2	8	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	8	2	no	—
272	C₄⋊C₄	—	8	2	no	—
273	C₄⋊C₄	—	8	2	no	—
274	C₄⋊C₄	—	8	2	no	—
275	C₄⋊C₄	—	8	2	no	—
276	C₄⋊C₄	C₂×Q₈×C₃₁	1	4	yes	—
277	C₄²	—	2	1	no	—
278	C₄²	—	2	1	no	—
279	C₄²	—	2	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	8	2	no	—
282	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
283	C₄²	#41×C₃₁	1	2	yes	—
284	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	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	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
307	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
308	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
309	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
310	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
311	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
312	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
313	C₈	—	2	2	no	—
314	C₈	—	2	2	no	—
315	C₈	—	4	1	no	—
316	C₈	—	4	1	no	—
317	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
318	C₂²	C₃₁×#532	1	1	yes	characteristic
319	C₂²	C₃₁×#536	1	1	yes	characteristic
320	C₂²	C₃₁×#577	1	2	yes	—
321	C₂²	C₃₁×#577	1	2	yes	—
322	C₄	—	2	4	no	—
323	C₄	—	4	4	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	2	no	—
327	C₄	—	8	2	no	—
328	C₄	—	8	2	no	—
329	C₄	—	8	2	no	—
330	C₄	C₃₁×#552	1	1	yes	characteristic
331	C₄	C₃₁×#552	1	1	yes	characteristic
332	C₄	C₃₁×#557	1	1	yes	characteristic
333	C₄	C₃₁×#557	1	1	yes	characteristic
334	C₂	C₃₁×#2565	1	1	yes	characteristic
335	C₂	C₃₁×#3084	1	2	yes	—
336	C₂	C₃₁×#3174	1	1	yes	characteristic
337	C₂	C₃₁×#3174	1	1	yes	characteristic
338	C₂	C₃₁×#3244	1	2	yes	—

	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 ◁ C₃₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
114. #29185	C₃₁	direct (N × H)
241. C₃₁	#29185	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#2647	2
C₃₁×#2330	2
C₃₁×#2088	2
C₃₁×#1992	2
C₃₁×#1991	2
C₃₁×#1920	2
C₃₁×#1919	2
C₃₁×#1844	2
C₃₁×#1836	2
C₃₁×#1768	2
#29185	31

Subgroups — C31×#29185 #21485944

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#29185 #21485944