Subgroups — C₃₁×#20779 #21477538

Order 7936.

802 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	C₂×C₃₁×#376	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1840	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1832	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1817	C₂	1	2	yes	maximal
5	C₃₁×#1797	C₂	1	2	yes	maximal
6	C₃₁×#1768	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1749	C₂	1	1	yes	maximal, characteristic
8	C₄×C₃₁×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#1746	C₂	1	2	yes	maximal, complemented
10	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
14	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
15	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃₁×#97	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
18	C₃₁×#376	C₄	1	2	yes	—
19	C₃₁×#376	C₄	1	2	yes	—
20	C₃₁×#376	C₂²	1	1	yes	characteristic
21	C₃₁×#376	C₂²	1	1	yes	characteristic
22	C₃₁×#376	C₂²	1	1	yes	characteristic
23	C₃₁×#376	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#95	C₄	1	2	yes	—
25	C₂×C₃₁×#95	C₄	1	2	yes	—
26	C₃₁×#336	C₂²	1	4	yes	—
27	C₃₁×#336	C₂²	1	4	yes	—
28	C₃₁×#336	C₂²	1	4	yes	—
29	C₃₁×#336	C₂²	1	2	yes	—
30	C₃₁×#336	C₂²	1	2	yes	—
31	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
33	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
34	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
37	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
38	C₂×C₄²×C₃₁	—	2	4	no	—
39	C₂×C₄²×C₃₁	—	2	4	no	—
40	C₂×C₄²×C₃₁	—	2	4	no	—
41	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
51	C₃₁×C₄⋊C₈	—	2	8	no	—
52	C₃₁×C₂²⋊C₈	—	2	4	no	—
53	C₃₁×C₂²⋊C₈	—	2	4	no	—
54	C₃₁×#97	—	2	4	no	—
55	C₄×C₈×C₃₁	—	2	4	no	—
56	C₃₁×#95	C₂×C₄	1	2	yes	—
57	C₃₁×#95	C₂×C₄	1	2	yes	—
58	C₃₁×#95	C₂×C₄	1	2	yes	—
59	C₃₁×#95	C₂×C₄	1	2	yes	—
60	C₃₁×#95	C₂×C₄	1	2	yes	—
61	C₃₁×#95	C₂×C₄	1	2	yes	—
62	C₃₁×#95	C₂×C₄	1	2	yes	—
63	C₃₁×#95	C₂×C₄	1	2	yes	—
64	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	—	2	2	no	—
66	C₂²×C₄×C₃₁	—	2	2	no	—
67	C₂²×C₄×C₃₁	—	2	2	no	—
68	C₂²×C₄×C₃₁	—	2	2	no	—
69	C₂²×C₄×C₃₁	—	2	4	no	—
70	C₂²×C₄×C₃₁	—	2	4	no	—
71	C₂²×C₄×C₃₁	—	2	4	no	—
72	C₂²×C₄×C₃₁	—	2	4	no	—
73	C₂²×C₄×C₃₁	—	2	4	no	—
74	C₂²×C₄×C₃₁	—	4	2	no	—
75	C₂²×C₄×C₃₁	—	4	2	no	—
76	C₂²×C₄×C₃₁	—	4	2	no	—
77	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₃₁	#41	1	4	yes	—
87	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
88	C₂×C₈×C₃₁	—	2	4	no	—
89	C₂×C₈×C₃₁	—	2	4	no	—
90	C₂×C₈×C₃₁	—	2	4	no	—
91	C₂×C₈×C₃₁	—	2	4	no	—
92	C₂×C₈×C₃₁	—	2	4	no	—
93	C₂×C₈×C₃₁	—	2	4	no	—
94	C₄²×C₃₁	—	2	4	no	—
95	C₄²×C₃₁	—	2	4	no	—
96	C₄²×C₃₁	—	2	2	no	—
97	C₄²×C₃₁	—	2	2	no	—
98	C₄²×C₃₁	—	2	2	no	—
99	C₄²×C₃₁	—	2	2	no	—
100	C₄²×C₃₁	—	2	2	no	—
101	C₄²×C₃₁	—	2	2	no	—
102	C₄²×C₃₁	#41	1	4	yes	—
103	C₄²×C₃₁	#41	1	4	yes	—
104	#20779	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂³×C₃₁	—	2	1	no	—
106	C₂³×C₃₁	—	2	1	no	—
107	C₂³×C₃₁	—	2	2	no	—
108	C₂³×C₃₁	#117	1	2	yes	—
109	C₂³×C₃₁	#131	1	4	yes	—
110	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₃₁	—	2	1	no	—
112	C₂×C₄×C₃₁	—	2	1	no	—
113	C₂×C₄×C₃₁	—	2	2	no	—
114	C₂×C₄×C₃₁	—	2	2	no	—
115	C₂×C₄×C₃₁	—	2	2	no	—
116	C₂×C₄×C₃₁	—	2	2	no	—
117	C₂×C₄×C₃₁	—	2	8	no	—
118	C₂×C₄×C₃₁	—	2	1	no	—
119	C₂×C₄×C₃₁	—	2	1	no	—
120	C₂×C₄×C₃₁	—	2	1	no	—
121	C₂×C₄×C₃₁	—	2	1	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	2	no	—
129	C₂×C₄×C₃₁	—	2	2	no	—
130	C₂×C₄×C₃₁	—	2	2	no	—
131	C₂×C₄×C₃₁	—	2	2	no	—
132	C₂×C₄×C₃₁	—	4	8	no	—
133	C₂×C₄×C₃₁	—	4	8	no	—
134	C₂×C₄×C₃₁	—	4	4	no	—
135	C₂×C₄×C₃₁	—	4	4	no	—
136	C₂×C₄×C₃₁	#117	1	2	yes	—
137	C₂×C₄×C₃₁	#117	1	4	yes	—
138	C₂×C₄×C₃₁	#117	1	4	yes	—
139	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
140	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
141	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
142	C₈×C₃₁	—	4	4	no	—
143	C₈×C₃₁	—	4	4	no	—
144	C₂×#376	C₂×C₃₁	1	1	yes	characteristic
145	#1840	C₂×C₃₁	1	2	yes	complemented
146	#1832	C₂×C₃₁	1	2	yes	complemented
147	#1817	C₂×C₃₁	1	2	yes	—
148	#1797	C₂×C₃₁	1	2	yes	—
149	#1768	C₂×C₃₁	1	2	yes	complemented
150	#1749	C₂×C₃₁	1	1	yes	characteristic
151	C₄×C₂²⋊C₈	C₂×C₃₁	1	1	yes	characteristic
152	#1746	C₂×C₃₁	1	2	yes	complemented
153	C₂²×C₃₁	—	2	2	no	—
154	C₂²×C₃₁	—	2	4	no	—
155	C₂²×C₃₁	—	2	4	no	—
156	C₂²×C₃₁	—	2	2	no	—
157	C₂²×C₃₁	#433	1	4	yes	—
158	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
159	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
160	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
161	C₂²×C₃₁	C₂×#131	1	2	yes	—
162	C₂²×C₃₁	#568	1	2	yes	—
163	C₄×C₃₁	—	2	4	no	—
164	C₄×C₃₁	—	2	4	no	—
165	C₄×C₃₁	—	2	2	no	—
166	C₄×C₃₁	—	4	1	no	—
167	C₄×C₃₁	—	4	1	no	—
168	C₄×C₃₁	—	4	1	no	—
169	C₄×C₃₁	—	4	1	no	—
170	C₄×C₃₁	—	4	2	no	—
171	C₄×C₃₁	—	4	2	no	—
172	C₄×C₃₁	#529	1	4	yes	—
173	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
174	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
175	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
176	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
177	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
178	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
179	C₂×#97	C₂²×C₃₁	1	2	yes	—
180	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
181	#376	C₄×C₃₁	1	2	yes	—
182	#376	C₄×C₃₁	1	2	yes	—
183	#376	C₂²×C₃₁	1	1	yes	characteristic
184	#376	C₂²×C₃₁	1	1	yes	characteristic
185	#376	C₂²×C₃₁	1	1	yes	characteristic
186	#376	C₂²×C₃₁	1	1	yes	characteristic
187	C₂×#95	C₄×C₃₁	1	2	yes	—
188	C₂×#95	C₄×C₃₁	1	2	yes	—
189	#336	C₂²×C₃₁	1	4	yes	—
190	#336	C₂²×C₃₁	1	4	yes	—
191	#336	C₂²×C₃₁	1	4	yes	—
192	#336	C₂²×C₃₁	1	2	yes	—
193	#336	C₂²×C₃₁	1	2	yes	—
194	C₂×C₃₁	—	2	4	no	—
195	C₂×C₃₁	#2283	1	1	yes	characteristic
196	C₂×C₃₁	C₂×#433	1	1	yes	characteristic
197	C₂×C₃₁	#2921	1	1	yes	characteristic
198	C₂×C₃₁	#2924	1	2	yes	—
199	C₂×C₃₁	#2924	1	2	yes	—
200	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
201	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
202	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
203	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
204	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
205	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
206	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
207	C₂×C₄²	—	2	4	no	—
208	C₂×C₄²	—	2	4	no	—
209	C₂×C₄²	—	2	4	no	—
210	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
211	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
212	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
213	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
214	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
215	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
216	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
217	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
218	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
219	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
220	C₄⋊C₈	—	2	8	no	—
221	C₂²⋊C₈	—	2	4	no	—
222	C₂²⋊C₈	—	2	4	no	—
223	#97	—	2	4	no	—
224	C₄×C₈	—	2	4	no	—
225	#95	C₂×C₄×C₃₁	1	2	yes	—
226	#95	C₂×C₄×C₃₁	1	2	yes	—
227	#95	C₂×C₄×C₃₁	1	2	yes	—
228	#95	C₂×C₄×C₃₁	1	2	yes	—
229	#95	C₂×C₄×C₃₁	1	2	yes	—
230	#95	C₂×C₄×C₃₁	1	2	yes	—
231	#95	C₂×C₄×C₃₁	1	2	yes	—
232	#95	C₂×C₄×C₃₁	1	2	yes	—
233	C₃₁	#20779	1	1	yes	characteristic, direct product factor, Sylow
234	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	4	no	—
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	2	4	no	—
242	C₂²×C₄	—	2	4	no	—
243	C₂²×C₄	—	2	4	no	—
244	C₂²×C₄	—	4	2	no	—
245	C₂²×C₄	—	4	2	no	—
246	C₂²×C₄	—	4	2	no	—
247	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
248	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
249	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
250	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
251	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
252	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
253	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
254	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
255	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
256	C₂²×C₄	#41×C₃₁	1	4	yes	—
257	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	2	4	no	—
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	—	2	4	no	—
264	C₄²	—	2	4	no	—
265	C₄²	—	2	4	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	#41×C₃₁	1	4	yes	—
273	C₄²	#41×C₃₁	1	4	yes	—
274	C₂³	—	2	1	no	—
275	C₂³	—	2	1	no	—
276	C₂³	—	2	2	no	—
277	C₂³	C₃₁×#117	1	2	yes	—
278	C₂³	C₃₁×#131	1	4	yes	—
279	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	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	8	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	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	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	4	8	no	—
302	C₂×C₄	—	4	8	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	—	4	4	no	—
305	C₂×C₄	C₃₁×#117	1	2	yes	—
306	C₂×C₄	C₃₁×#117	1	4	yes	—
307	C₂×C₄	C₃₁×#117	1	4	yes	—
308	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
309	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
310	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
311	C₈	—	4	4	no	—
312	C₈	—	4	4	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	2	4	no	—
315	C₂²	—	2	4	no	—
316	C₂²	—	2	2	no	—
317	C₂²	C₃₁×#433	1	4	yes	—
318	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
319	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
320	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
321	C₂²	C₂×C₃₁×#131	1	2	yes	—
322	C₂²	C₃₁×#568	1	2	yes	—
323	C₄	—	2	4	no	—
324	C₄	—	2	4	no	—
325	C₄	—	2	2	no	—
326	C₄	—	4	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	2	no	—
331	C₄	—	4	2	no	—
332	C₄	C₃₁×#529	1	4	yes	—
333	C₂	—	2	4	no	—
334	C₂	C₃₁×#2283	1	1	yes	characteristic
335	C₂	C₂×C₃₁×#433	1	1	yes	characteristic
336	C₂	C₃₁×#2921	1	1	yes	characteristic
337	C₂	C₃₁×#2924	1	2	yes	—
338	C₂	C₃₁×#2924	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator	C₂³	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂³×C₄
prime radical	C₂⁴×C₃₁	C₂²×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
2. C₃₁×#1840	C₂	semidirect
3. C₃₁×#1832	C₂	semidirect
6. C₃₁×#1768	C₂	semidirect
9. C₃₁×#1746	C₂	semidirect
104. #20779	C₃₁	direct (N × H)
145. #1840	C₂×C₃₁	semidirect
146. #1832	C₂×C₃₁	semidirect
149. #1768	C₂×C₃₁	semidirect
152. #1746	C₂×C₃₁	semidirect
233. C₃₁	#20779	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×#376	2
C₃₁×#1840	2
C₃₁×#1832	2
C₃₁×#1817	2
C₃₁×#1797	2
C₃₁×#1768	2
C₃₁×#1749	2
C₄×C₃₁×C₂²⋊C₈	2
C₃₁×#1746	2
#20779	31

Subgroups — C31×#20779 #21477538

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20779 #21477538