Subgroups — C₃₁×#10827 #21467586

Order 7936.

686 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#405	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#343	C₂	1	2	yes	maximal
3	C₂×C₃₁×#341	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1736	C₂	1	2	yes	maximal
5	C₂×C₃₁×#340	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
7	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
11	C₃₁×#405	—	2	2	no	—
12	C₃₁×#405	C₄	1	4	yes	complemented
13	C₂×C₃₁×#97	—	2	1	no	—
14	C₂×C₄×C₈×C₃₁	—	2	1	no	—
15	C₃₁×#343	C₄	1	8	yes	complemented
16	C₃₁×#341	—	2	2	no	—
17	C₃₁×#340	—	2	1	no	—
18	C₃₁×#340	—	2	1	no	—
19	C₃₁×#340	—	2	2	no	—
20	C₃₁×#340	—	2	2	no	—
21	C₂³×C₄×C₃₁	—	2	2	no	—
22	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
23	C₂×M₁₆×C₃₁	—	2	1	no	—
24	C₂×M₁₆×C₃₁	—	2	2	no	—
25	C₂×M₁₆×C₃₁	—	2	2	no	—
26	C₂×M₁₆×C₃₁	—	2	2	no	—
27	C₂×M₁₆×C₃₁	—	2	2	no	—
28	C₂×M₁₆×C₃₁	—	2	2	no	—
29	C₂×M₁₆×C₃₁	—	2	2	no	—
30	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
31	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
32	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₃₁	—	2	1	no	—
34	C₂²×C₈×C₃₁	—	2	2	no	—
35	C₂²×C₈×C₃₁	—	2	2	no	—
36	C₂²×C₈×C₃₁	—	2	2	no	—
37	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₃₁	D₄	1	2	yes	—
41	C₃₁×#117	—	2	1	no	—
42	C₃₁×#117	—	2	1	no	—
43	C₃₁×#117	C₂×C₄	1	2	yes	—
44	C₃₁×#117	Q₈	1	2	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	C₂×C₄	1	4	yes	—
51	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
53	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
54	C₂×C₄²×C₃₁	—	2	1	no	—
55	C₃₁×#97	—	2	4	no	—
56	C₄×C₈×C₃₁	—	2	4	no	—
57	C₂⁴×C₃₁	#41	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	—	2	1	no	—
59	C₂²×C₄×C₃₁	—	2	1	no	—
60	C₂²×C₄×C₃₁	—	2	4	no	—
61	C₂²×C₄×C₃₁	—	4	2	no	—
62	C₂²×C₄×C₃₁	—	4	2	no	—
63	C₂²×C₄×C₃₁	—	8	2	no	—
64	C₂²×C₄×C₃₁	—	8	2	no	—
65	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
71	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
72	C₂²×C₄×C₃₁	#41	1	2	yes	—
73	C₂²×C₄×C₃₁	#41	1	2	yes	—
74	C₂²×C₄×C₃₁	#41	1	2	yes	—
75	M₁₆×C₃₁	—	2	2	no	—
76	M₁₆×C₃₁	—	2	4	no	—
77	M₁₆×C₃₁	—	2	4	no	—
78	M₁₆×C₃₁	—	2	4	no	—
79	M₁₆×C₃₁	—	4	2	no	—
80	M₁₆×C₃₁	—	4	2	no	—
81	M₁₆×C₃₁	C₄²	1	4	yes	—
82	C₂×C₈×C₃₁	—	2	2	no	—
83	C₂×C₈×C₃₁	—	2	2	no	—
84	C₂×C₈×C₃₁	—	2	2	no	—
85	C₂×C₈×C₃₁	—	2	4	no	—
86	C₂×C₈×C₃₁	—	2	2	no	—
87	C₂×C₈×C₃₁	—	2	2	no	—
88	C₂×C₈×C₃₁	—	2	4	no	—
89	C₂×C₈×C₃₁	—	2	4	no	—
90	C₂×C₈×C₃₁	—	4	2	no	—
91	C₂×C₈×C₃₁	—	4	2	no	—
92	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
93	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
94	C₄⋊C₄×C₃₁	—	2	1	no	—
95	C₄⋊C₄×C₃₁	—	2	1	no	—
96	C₄⋊C₄×C₃₁	—	2	4	no	—
97	C₄⋊C₄×C₃₁	—	2	4	no	—
98	C₄⋊C₄×C₃₁	—	4	2	no	—
99	C₄⋊C₄×C₃₁	—	4	2	no	—
100	C₄⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
101	C₄⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
102	C₂²⋊C₄×C₃₁	—	2	2	no	—
103	C₂²⋊C₄×C₃₁	—	2	2	no	—
104	C₄²×C₃₁	—	2	2	no	—
105	C₄²×C₃₁	—	2	2	no	—
106	#10827	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂³×C₃₁	—	2	2	no	—
108	C₂³×C₃₁	—	2	1	no	—
109	C₂³×C₃₁	—	2	1	no	—
110	C₂³×C₃₁	#117	1	1	yes	characteristic
111	C₂³×C₃₁	#117	1	2	yes	—
112	C₂³×C₃₁	#121	1	1	yes	characteristic
113	C₂³×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂³×C₃₁	#126	1	2	yes	—
115	C₂×C₄×C₃₁	—	2	2	no	—
116	C₂×C₄×C₃₁	—	2	2	no	—
117	C₂×C₄×C₃₁	—	4	2	no	—
118	C₂×C₄×C₃₁	—	4	1	no	—
119	C₂×C₄×C₃₁	—	4	1	no	—
120	C₂×C₄×C₃₁	—	4	4	no	—
121	C₂×C₄×C₃₁	—	8	4	no	—
122	C₂×C₄×C₃₁	—	8	2	no	—
123	C₂×C₄×C₃₁	—	8	2	no	—
124	C₂×C₄×C₃₁	—	8	4	no	—
125	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
126	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
129	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
130	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
131	C₂×C₄×C₃₁	#117	1	2	yes	—
132	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
133	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
135	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
136	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
137	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
138	C₂×C₄×C₃₁	#123	1	2	yes	—
139	C₂×C₄×C₃₁	#123	1	2	yes	—
140	C₂×C₄×C₃₁	#124	1	2	yes	—
141	C₂×C₄×C₃₁	#125	1	2	yes	—
142	C₂×C₄×C₃₁	#126	1	2	yes	—
143	C₈×C₃₁	—	2	4	no	—
144	C₈×C₃₁	—	2	2	no	—
145	C₈×C₃₁	—	2	2	no	—
146	C₈×C₃₁	—	4	4	no	—
147	C₂×#405	C₂×C₃₁	1	1	yes	characteristic
148	C₂×#343	C₂×C₃₁	1	2	yes	—
149	C₂×#341	C₂×C₃₁	1	1	yes	characteristic
150	#1736	C₂×C₃₁	1	2	yes	—
151	C₂×#340	C₂×C₃₁	1	1	yes	characteristic
152	C₂²×C₃₁	—	2	1	no	—
153	C₂²×C₃₁	—	2	1	no	—
154	C₂²×C₃₁	—	2	1	no	—
155	C₂²×C₃₁	—	2	1	no	—
156	C₂²×C₃₁	—	2	2	no	—
157	C₂²×C₃₁	—	2	2	no	—
158	C₂²×C₃₁	—	2	2	no	—
159	C₂²×C₃₁	—	2	2	no	—
160	C₂²×C₃₁	#376	1	1	yes	characteristic
161	C₂²×C₃₁	#380	1	1	yes	characteristic
162	C₂²×C₃₁	#383	1	1	yes	characteristic
163	C₂²×C₃₁	#387	1	2	yes	—
164	C₂²×C₃₁	#388	1	2	yes	—
165	C₂²×C₃₁	#471	1	2	yes	—
166	C₂²×C₃₁	#472	1	1	yes	characteristic
167	C₂²×C₃₁	#473	1	1	yes	characteristic
168	C₄×C₃₁	—	4	2	no	—
169	C₄×C₃₁	—	4	2	no	—
170	C₄×C₃₁	—	8	4	no	—
171	C₄×C₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
172	C₄×C₃₁	C₄×C₄⋊C₄	1	1	yes	characteristic
173	C₄×C₃₁	#381	1	1	yes	characteristic
174	C₄×C₃₁	#382	1	1	yes	characteristic
175	C₄×C₃₁	#387	1	2	yes	—
176	C₄×C₃₁	#388	1	2	yes	—
177	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
178	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
179	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
180	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
181	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
182	#405	—	2	2	no	—
183	#405	C₄×C₃₁	1	4	yes	complemented
184	C₂×#97	—	2	1	no	—
185	C₂×C₄×C₈	—	2	1	no	—
186	#343	C₄×C₃₁	1	8	yes	complemented
187	#341	—	2	2	no	—
188	#340	—	2	1	no	—
189	#340	—	2	1	no	—
190	#340	—	2	2	no	—
191	#340	—	2	2	no	—
192	C₂×C₃₁	—	2	1	no	—
193	C₂×C₃₁	—	2	1	no	—
194	C₂×C₃₁	—	2	2	no	—
195	C₂×C₃₁	#1434	1	1	yes	characteristic
196	C₂×C₃₁	#1815	1	1	yes	characteristic
197	C₂×C₃₁	#1816	1	1	yes	characteristic
198	C₂×C₃₁	#1930	1	2	yes	—
199	C₂×C₃₁	#1931	1	2	yes	—
200	C₂³×C₄	—	2	2	no	—
201	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
202	C₂×M₁₆	—	2	1	no	—
203	C₂×M₁₆	—	2	2	no	—
204	C₂×M₁₆	—	2	2	no	—
205	C₂×M₁₆	—	2	2	no	—
206	C₂×M₁₆	—	2	2	no	—
207	C₂×M₁₆	—	2	2	no	—
208	C₂×M₁₆	—	2	2	no	—
209	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
210	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
211	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
212	C₂²×C₈	—	2	1	no	—
213	C₂²×C₈	—	2	2	no	—
214	C₂²×C₈	—	2	2	no	—
215	C₂²×C₈	—	2	2	no	—
216	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
217	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
218	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
219	C₂²×C₈	D₄×C₃₁	1	2	yes	—
220	#117	—	2	1	no	—
221	#117	—	2	1	no	—
222	#117	C₂×C₄×C₃₁	1	2	yes	—
223	#117	Q₈×C₃₁	1	2	yes	—
224	C₂×C₄⋊C₄	—	2	2	no	—
225	C₂×C₄⋊C₄	—	2	2	no	—
226	C₂×C₄⋊C₄	—	2	2	no	—
227	C₂×C₄⋊C₄	—	2	2	no	—
228	C₂×C₄⋊C₄	—	2	2	no	—
229	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
230	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
231	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
232	C₂×C₂²⋊C₄	—	2	1	no	—
233	C₂×C₄²	—	2	1	no	—
234	#97	—	2	4	no	—
235	C₄×C₈	—	2	4	no	—
236	C₃₁	#10827	1	1	yes	characteristic, direct product factor, Sylow
237	C₂⁴	#41×C₃₁	1	1	yes	characteristic
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	—	4	2	no	—
243	C₂²×C₄	—	8	2	no	—
244	C₂²×C₄	—	8	2	no	—
245	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
246	C₂²×C₄	C₂²×C₄×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	1	yes	characteristic
250	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
251	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
252	C₂²×C₄	#41×C₃₁	1	2	yes	—
253	C₂²×C₄	#41×C₃₁	1	2	yes	—
254	C₂²×C₄	#41×C₃₁	1	2	yes	—
255	M₁₆	—	2	2	no	—
256	M₁₆	—	2	4	no	—
257	M₁₆	—	2	4	no	—
258	M₁₆	—	2	4	no	—
259	M₁₆	—	4	2	no	—
260	M₁₆	—	4	2	no	—
261	M₁₆	C₄²×C₃₁	1	4	yes	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	4	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	4	no	—
269	C₂×C₈	—	2	4	no	—
270	C₂×C₈	—	4	2	no	—
271	C₂×C₈	—	4	2	no	—
272	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
273	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	2	4	no	—
277	C₄⋊C₄	—	2	4	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	4	2	no	—
280	C₄⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
281	C₄⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
282	C₂²⋊C₄	—	2	2	no	—
283	C₂²⋊C₄	—	2	2	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	2	2	no	—
286	C₂³	—	2	2	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	C₃₁×#117	1	1	yes	characteristic
290	C₂³	C₃₁×#117	1	2	yes	—
291	C₂³	C₃₁×#121	1	1	yes	characteristic
292	C₂³	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
293	C₂³	C₃₁×#126	1	2	yes	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	8	4	no	—
301	C₂×C₄	—	8	2	no	—
302	C₂×C₄	—	8	2	no	—
303	C₂×C₄	—	8	4	no	—
304	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
305	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
306	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
307	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
308	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
309	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
310	C₂×C₄	C₃₁×#117	1	2	yes	—
311	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
312	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
313	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
314	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
315	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
316	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
317	C₂×C₄	C₃₁×#123	1	2	yes	—
318	C₂×C₄	C₃₁×#123	1	2	yes	—
319	C₂×C₄	C₃₁×#124	1	2	yes	—
320	C₂×C₄	C₃₁×#125	1	2	yes	—
321	C₂×C₄	C₃₁×#126	1	2	yes	—
322	C₈	—	2	4	no	—
323	C₈	—	2	2	no	—
324	C₈	—	2	2	no	—
325	C₈	—	4	4	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	2	2	no	—
333	C₂²	—	2	2	no	—
334	C₂²	C₃₁×#376	1	1	yes	characteristic
335	C₂²	C₃₁×#380	1	1	yes	characteristic
336	C₂²	C₃₁×#383	1	1	yes	characteristic
337	C₂²	C₃₁×#387	1	2	yes	—
338	C₂²	C₃₁×#388	1	2	yes	—
339	C₂²	C₃₁×#471	1	2	yes	—
340	C₂²	C₃₁×#472	1	1	yes	characteristic
341	C₂²	C₃₁×#473	1	1	yes	characteristic
342	C₄	—	4	2	no	—
343	C₄	—	4	2	no	—
344	C₄	—	8	4	no	—
345	C₄	C₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
346	C₄	C₄×C₄⋊C₄×C₃₁	1	1	yes	characteristic
347	C₄	C₃₁×#381	1	1	yes	characteristic
348	C₄	C₃₁×#382	1	1	yes	characteristic
349	C₄	C₃₁×#387	1	2	yes	—
350	C₄	C₃₁×#388	1	2	yes	—
351	C₂	—	2	1	no	—
352	C₂	—	2	1	no	—
353	C₂	—	2	2	no	—
354	C₂	C₃₁×#1434	1	1	yes	characteristic
355	C₂	C₃₁×#1815	1	1	yes	characteristic
356	C₂	C₃₁×#1816	1	1	yes	characteristic
357	C₂	C₃₁×#1930	1	2	yes	—
358	C₂	C₃₁×#1931	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	#121
commutator	C₂×C₄	C₂×C₄²×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
prime radical	C₂⁴×C₃₁	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₃₁×#405	C₄	semidirect
15. C₃₁×#343	C₄	semidirect
106. #10827	C₃₁	direct (N × H)
183. #405	C₄×C₃₁	semidirect
186. #343	C₄×C₃₁	semidirect
236. C₃₁	#10827	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×#405	2
C₂×C₃₁×#343	2
C₂×C₃₁×#341	2
C₃₁×#1736	2
C₂×C₃₁×#340	2
#10827	31

Subgroups — C31×#10827 #21467586

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#10827 #21467586