Subgroups — C₃₁×#8247 #21465006

Order 7936.

444 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2966	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1726	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1524	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1500	C₂	1	1	yes	maximal, characteristic
5	C₈×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1302	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1300	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#529	C₂²	1	1	yes	characteristic
9	C₃₁×#446	C₄	1	1	yes	characteristic, complemented
10	C₈×Q₈×C₃₁	C₄	1	1	yes	characteristic, complemented
11	C₃₁×#436	C₄	1	1	yes	characteristic, complemented
12	C₃₁×#435	—	2	1	no	—
13	C₈×D₄×C₃₁	—	2	1	no	—
14	C₈×D₄×C₃₁	C₄	1	1	yes	characteristic, complemented
15	C₃₁×#433	—	2	1	no	—
16	C₃₁×#432	C₂²	1	1	yes	characteristic
17	C₃₁×#431	C₂²	1	1	yes	characteristic
18	C₃₁×#405	—	2	1	no	—
19	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#97	—	2	1	no	—
21	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₃₁	—	2	1	no	—
23	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#376	C₂²	1	1	yes	characteristic
25	C₃₁×#330	C₄	1	2	yes	complemented
26	C₃₁×#329	C₄	1	2	yes	complemented
27	C₃₁×#328	C₄	1	2	yes	—
28	C₃₁×#327	C₄	1	2	yes	—
29	C₃₁×#322	—	2	1	no	—
30	C₈²×C₃₁	—	2	1	no	—
31	C₂×M₁₆×C₃₁	—	2	1	no	—
32	C₂×M₁₆×C₃₁	—	2	1	no	—
33	C₂²×C₈×C₃₁	—	2	1	no	—
34	C₂²×C₈×C₃₁	—	2	1	no	—
35	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
37	C₃₁×#126	—	2	1	no	—
38	C₃₁×#125	C₂×C₄	1	1	yes	characteristic
39	C₃₁×#124	C₂×C₄	1	1	yes	characteristic
40	C₃₁×#123	—	2	1	no	—
41	C₃₁×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
42	C₃₁×#121	C₂×C₄	1	1	yes	characteristic
43	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₄×D₄×C₃₁	—	2	1	no	—
45	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
46	C₃₁×#117	—	2	1	no	—
47	C₂×C₄²×C₃₁	—	2	1	no	—
48	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
49	C₃₁×C₄⋊C₈	—	2	1	no	—
50	C₃₁×C₄⋊C₈	—	2	1	no	—
51	C₃₁×C₄⋊C₈	—	2	1	no	—
52	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
54	C₃₁×C₂²⋊C₈	—	2	1	no	—
55	C₃₁×C₂²⋊C₈	—	2	1	no	—
56	C₃₁×C₂²⋊C₈	—	2	1	no	—
57	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
58	C₃₁×#97	—	2	4	no	—
59	C₃₁×#97	—	2	1	no	—
60	C₃₁×#97	—	2	1	no	—
61	C₃₁×#97	—	2	1	no	—
62	C₃₁×#97	C₂×C₄	1	2	yes	—
63	C₃₁×#97	C₂×C₄	1	2	yes	—
64	C₄×C₈×C₃₁	—	2	4	no	—
65	C₄×C₈×C₃₁	—	2	1	no	—
66	C₄×C₈×C₃₁	—	2	1	no	—
67	C₄×C₈×C₃₁	—	2	1	no	—
68	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
69	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
71	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
72	C₃₁×#95	—	2	1	no	—
73	C₃₁×#95	C₂×C₄	1	2	yes	—
74	C₂×Q₈×C₃₁	C₄²	1	1	yes	characteristic
75	C₂×D₄×C₃₁	—	2	1	no	—
76	C₂×D₄×C₃₁	C₄²	1	1	yes	characteristic
77	C₂²×C₄×C₃₁	—	2	2	no	—
78	C₂²×C₄×C₃₁	—	2	1	no	—
79	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
82	M₁₆×C₃₁	—	2	4	no	—
83	M₁₆×C₃₁	—	4	2	no	—
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	2	no	—
90	C₂×C₈×C₃₁	—	2	1	no	—
91	C₂×C₈×C₃₁	—	2	1	no	—
92	C₂×C₈×C₃₁	—	2	2	no	—
93	C₂×C₈×C₃₁	—	4	2	no	—
94	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
98	C₄⋊C₄×C₃₁	—	2	1	no	—
99	C₄⋊C₄×C₃₁	—	2	1	no	—
100	C₄⋊C₄×C₃₁	—	2	1	no	—
101	C₄⋊C₄×C₃₁	—	2	1	no	—
102	C₄⋊C₄×C₃₁	C₄²	1	1	yes	characteristic
103	C₄⋊C₄×C₃₁	C₄²	1	1	yes	characteristic
104	C₂²⋊C₄×C₃₁	—	2	1	no	—
105	C₂²⋊C₄×C₃₁	—	2	1	no	—
106	C₂²⋊C₄×C₃₁	—	2	1	no	—
107	C₂²⋊C₄×C₃₁	—	2	1	no	—
108	C₂²⋊C₄×C₃₁	—	2	1	no	—
109	C₄²×C₃₁	—	2	2	no	—
110	C₄²×C₃₁	—	2	2	no	—
111	C₄²×C₃₁	—	2	1	no	—
112	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₃₁	#41	1	1	yes	characteristic
116	#8247	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂³×C₃₁	—	2	1	no	—
118	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
119	Q₈×C₃₁	—	4	1	no	—
120	D₄×C₃₁	—	4	1	no	—
121	D₄×C₃₁	—	4	2	no	—
122	C₂×C₄×C₃₁	—	2	2	no	—
123	C₂×C₄×C₃₁	—	2	1	no	—
124	C₂×C₄×C₃₁	—	2	1	no	—
125	C₂×C₄×C₃₁	—	2	4	no	—
126	C₂×C₄×C₃₁	—	2	1	no	—
127	C₂×C₄×C₃₁	—	2	1	no	—
128	C₂×C₄×C₃₁	—	2	1	no	—
129	C₂×C₄×C₃₁	—	4	1	no	—
130	C₂×C₄×C₃₁	—	4	2	no	—
131	C₂×C₄×C₃₁	—	4	2	no	—
132	C₂×C₄×C₃₁	—	4	1	no	—
133	C₂×C₄×C₃₁	—	4	1	no	—
134	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
135	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
137	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
140	C₈×C₃₁	—	2	4	no	—
141	C₈×C₃₁	—	2	4	no	—
142	C₈×C₃₁	—	2	1	no	—
143	C₈×C₃₁	—	2	1	no	—
144	C₈×C₃₁	—	4	2	no	—
145	C₈×C₃₁	—	4	1	no	—
146	C₈×C₃₁	—	4	1	no	—
147	C₈×C₃₁	#104	1	2	yes	—
148	C₈×C₃₁	#104	1	2	yes	—
149	#2966	C₂×C₃₁	1	1	yes	characteristic
150	#1726	C₂×C₃₁	1	1	yes	characteristic, complemented
151	#1524	C₂×C₃₁	1	1	yes	characteristic
152	#1500	C₂×C₃₁	1	1	yes	characteristic
153	C₈×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
154	#1302	C₂×C₃₁	1	1	yes	characteristic, complemented
155	#1300	C₂×C₃₁	1	1	yes	characteristic, complemented
156	C₂²×C₃₁	—	2	1	no	—
157	C₂²×C₃₁	—	2	1	no	—
158	C₂²×C₃₁	—	4	1	no	—
159	C₂²×C₃₁	—	4	1	no	—
160	C₂²×C₃₁	—	4	1	no	—
161	C₂²×C₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
162	C₂²×C₃₁	#443	1	2	yes	—
163	C₄×C₃₁	—	2	1	no	—
164	C₄×C₃₁	—	2	1	no	—
165	C₄×C₃₁	—	2	1	no	—
166	C₄×C₃₁	—	2	1	no	—
167	C₄×C₃₁	—	2	1	no	—
168	C₄×C₃₁	—	4	4	no	—
169	C₄×C₃₁	—	4	1	no	—
170	C₄×C₃₁	—	4	1	no	—
171	C₄×C₃₁	—	4	1	no	—
172	C₄×C₃₁	C₂×#104	1	1	yes	characteristic
173	C₄×C₃₁	C₂×#104	1	1	yes	characteristic
174	#529	C₂²×C₃₁	1	1	yes	characteristic
175	#446	C₄×C₃₁	1	1	yes	characteristic, complemented
176	C₈×Q₈	C₄×C₃₁	1	1	yes	characteristic, complemented
177	#436	C₄×C₃₁	1	1	yes	characteristic, complemented
178	#435	—	2	1	no	—
179	C₈×D₄	—	2	1	no	—
180	C₈×D₄	C₄×C₃₁	1	1	yes	characteristic, complemented
181	#433	—	2	1	no	—
182	#432	C₂²×C₃₁	1	1	yes	characteristic
183	#431	C₂²×C₃₁	1	1	yes	characteristic
184	#405	—	2	1	no	—
185	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
186	C₂×#97	—	2	1	no	—
187	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
188	C₂×C₄×C₈	—	2	1	no	—
189	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
190	#376	C₂²×C₃₁	1	1	yes	characteristic
191	#330	C₄×C₃₁	1	2	yes	complemented
192	#329	C₄×C₃₁	1	2	yes	complemented
193	#328	C₄×C₃₁	1	2	yes	—
194	#327	C₄×C₃₁	1	2	yes	—
195	#322	—	2	1	no	—
196	C₈²	—	2	1	no	—
197	C₂×C₃₁	—	2	2	no	—
198	C₂×C₃₁	—	8	1	no	—
199	C₂×C₃₁	#1757	1	1	yes	characteristic
200	C₂×C₃₁	C₄×#104	1	1	yes	characteristic
201	C₂×C₃₁	#1764	1	1	yes	characteristic
202	C₂×M₁₆	—	2	1	no	—
203	C₂×M₁₆	—	2	1	no	—
204	C₂²×C₈	—	2	1	no	—
205	C₂²×C₈	—	2	1	no	—
206	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
207	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
208	#126	—	2	1	no	—
209	#125	C₂×C₄×C₃₁	1	1	yes	characteristic
210	#124	C₂×C₄×C₃₁	1	1	yes	characteristic
211	#123	—	2	1	no	—
212	C₂²⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
213	#121	C₂×C₄×C₃₁	1	1	yes	characteristic
214	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
215	C₄×D₄	—	2	1	no	—
216	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
217	#117	—	2	1	no	—
218	C₂×C₄²	—	2	1	no	—
219	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
220	C₄⋊C₈	—	2	1	no	—
221	C₄⋊C₈	—	2	1	no	—
222	C₄⋊C₈	—	2	1	no	—
223	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
224	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
225	C₂²⋊C₈	—	2	1	no	—
226	C₂²⋊C₈	—	2	1	no	—
227	C₂²⋊C₈	—	2	1	no	—
228	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
229	#97	—	2	4	no	—
230	#97	—	2	1	no	—
231	#97	—	2	1	no	—
232	#97	—	2	1	no	—
233	#97	C₂×C₄×C₃₁	1	2	yes	—
234	#97	C₂×C₄×C₃₁	1	2	yes	—
235	C₄×C₈	—	2	4	no	—
236	C₄×C₈	—	2	1	no	—
237	C₄×C₈	—	2	1	no	—
238	C₄×C₈	—	2	1	no	—
239	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
240	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
241	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
242	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
243	#95	—	2	1	no	—
244	#95	C₂×C₄×C₃₁	1	2	yes	—
245	C₃₁	#8247	1	1	yes	characteristic, direct product factor, Sylow
246	C₂×Q₈	C₄²×C₃₁	1	1	yes	characteristic
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	C₄²×C₃₁	1	1	yes	characteristic
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
253	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
254	M₁₆	—	2	4	no	—
255	M₁₆	—	4	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	4	2	no	—
266	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
267	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
268	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
269	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
270	C₄⋊C₄	—	2	1	no	—
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	C₄²×C₃₁	1	1	yes	characteristic
275	C₄⋊C₄	C₄²×C₃₁	1	1	yes	characteristic
276	C₂²⋊C₄	—	2	1	no	—
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	2	1	no	—
280	C₂²⋊C₄	—	2	1	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	2	1	no	—
284	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
285	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
286	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
287	C₄²	#41×C₃₁	1	1	yes	characteristic
288	C₂³	—	2	1	no	—
289	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
290	Q₈	—	4	1	no	—
291	D₄	—	4	1	no	—
292	D₄	—	4	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
306	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
307	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
308	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
309	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
310	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
311	C₈	—	2	4	no	—
312	C₈	—	2	4	no	—
313	C₈	—	2	1	no	—
314	C₈	—	2	1	no	—
315	C₈	—	4	2	no	—
316	C₈	—	4	1	no	—
317	C₈	—	4	1	no	—
318	C₈	C₃₁×#104	1	2	yes	—
319	C₈	C₃₁×#104	1	2	yes	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	C₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
326	C₂²	C₃₁×#443	1	2	yes	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	4	4	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	4	1	no	—
336	C₄	C₂×C₃₁×#104	1	1	yes	characteristic
337	C₄	C₂×C₃₁×#104	1	1	yes	characteristic
338	C₂	—	2	2	no	—
339	C₂	—	8	1	no	—
340	C₂	C₃₁×#1757	1	1	yes	characteristic
341	C₂	C₄×C₃₁×#104	1	1	yes	characteristic
342	C₂	C₃₁×#1764	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₃₁	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	C₄×C₂²⋊C₄
prime radical	C₃₁×#121	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1726	C₂	semidirect
5. C₈×M₁₆×C₃₁	C₂	semidirect
6. C₃₁×#1302	C₂	semidirect
7. C₃₁×#1300	C₂	semidirect
9. C₃₁×#446	C₄	semidirect
10. C₈×Q₈×C₃₁	C₄	semidirect
11. C₃₁×#436	C₄	semidirect
14. C₈×D₄×C₃₁	C₄	semidirect
25. C₃₁×#330	C₄	semidirect
26. C₃₁×#329	C₄	semidirect
116. #8247	C₃₁	direct (N × H)
150. #1726	C₂×C₃₁	semidirect
153. C₈×M₁₆	C₂×C₃₁	semidirect
154. #1302	C₂×C₃₁	semidirect
155. #1300	C₂×C₃₁	semidirect
175. #446	C₄×C₃₁	semidirect
176. C₈×Q₈	C₄×C₃₁	semidirect
177. #436	C₄×C₃₁	semidirect
180. C₈×D₄	C₄×C₃₁	semidirect
191. #330	C₄×C₃₁	semidirect
192. #329	C₄×C₃₁	semidirect
245. C₃₁	#8247	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×#2966	2
C₃₁×#1726	2
C₃₁×#1524	2
C₃₁×#1500	2
C₈×M₁₆×C₃₁	2
C₃₁×#1302	2
C₃₁×#1300	2
#8247	31

Subgroups — C31×#8247 #21465006

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8247 #21465006