Subgroups — C₃₁×#22249 #21479008

Order 7936.

686 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#499	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₁×#125	C₂	1	2	yes	maximal
3	C₃₁×#1991	C₂	1	2	yes	maximal
4	C₃₁×#1921	C₂	1	2	yes	maximal
5	C₃₁×#1920	C₂	1	2	yes	maximal
6	C₃₁×#1919	C₂	1	2	yes	maximal
7	C₃₁×#1918	C₂	1	2	yes	maximal
8	C₃₁×#1835	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#1744	C₂	1	1	yes	maximal, characteristic
10	C₂×C₃₁×#125	C₂²	1	2	yes	—
11	C₃₁×#499	C₄	1	8	yes	complemented
12	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#106	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#389	C₂²	1	2	yes	—
18	C₃₁×#389	C₂²	1	2	yes	—
19	C₃₁×#387	—	2	2	no	—
20	C₃₁×#387	—	2	2	no	—
21	C₃₁×#382	C₂²	1	2	yes	—
22	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
23	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
24	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
26	C₄³×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#340	C₂²	1	2	yes	—
28	C₃₁×#340	C₂²	1	1	yes	characteristic
29	C₃₁×#340	C₂²	1	1	yes	characteristic
30	C₃₁×#340	C₂²	1	2	yes	—
31	C₃₁×#340	C₂²	1	2	yes	—
32	C₃₁×#340	C₂²	1	2	yes	—
33	C₃₁×#340	C₂²	1	2	yes	—
34	C₃₁×#336	C₂²	1	2	yes	—
35	C₃₁×#336	C₂²	1	2	yes	—
36	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
37	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
39	C₃₁×#125	—	2	4	no	—
40	C₃₁×#125	C₂×C₄	1	8	yes	—
41	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
45	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
49	C₂×C₄²×C₃₁	—	2	2	no	—
50	C₂×C₄²×C₃₁	—	2	2	no	—
51	C₂×C₄²×C₃₁	D₄	1	2	yes	—
52	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
55	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
56	C₃₁×#107	C₂×C₄	1	4	yes	—
57	C₃₁×#107	C₂×C₄	1	4	yes	—
58	C₃₁×#106	C₂×C₄	1	8	yes	—
59	C₄×C₈×C₃₁	—	2	2	no	—
60	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
61	C₃₁×#95	—	2	2	no	—
62	C₃₁×#95	—	2	2	no	—
63	C₃₁×#95	—	2	2	no	—
64	C₃₁×#95	—	2	2	no	—
65	C₃₁×#95	—	2	2	no	—
66	C₃₁×#95	—	2	2	no	—
67	C₂²×C₄×C₃₁	—	2	2	no	—
68	C₂²×C₄×C₃₁	—	2	2	no	—
69	C₂²×C₄×C₃₁	—	2	2	no	—
70	C₂²×C₄×C₃₁	—	2	2	no	—
71	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
76	C₂×C₈×C₃₁	—	2	2	no	—
77	C₂×C₈×C₃₁	—	2	2	no	—
78	C₂×C₈×C₃₁	—	2	2	no	—
79	C₂×C₈×C₃₁	—	2	2	no	—
80	C₂×C₈×C₃₁	—	2	2	no	—
81	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
83	C₄⋊C₄×C₃₁	—	2	8	no	—
84	C₄⋊C₄×C₃₁	—	2	8	no	—
85	C₄⋊C₄×C₃₁	—	2	2	no	—
86	C₄⋊C₄×C₃₁	—	2	2	no	—
87	C₄⋊C₄×C₃₁	—	2	2	no	—
88	C₄⋊C₄×C₃₁	—	2	2	no	—
89	C₄⋊C₄×C₃₁	—	2	2	no	—
90	C₄⋊C₄×C₃₁	—	2	2	no	—
91	C₄⋊C₄×C₃₁	—	4	4	no	—
92	C₄⋊C₄×C₃₁	—	4	4	no	—
93	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
94	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
95	C₄²×C₃₁	—	2	4	no	—
96	C₄²×C₃₁	—	2	2	no	—
97	C₄²×C₃₁	—	2	2	no	—
98	C₄²×C₃₁	—	4	1	no	—
99	C₄²×C₃₁	—	4	1	no	—
100	C₄²×C₃₁	—	4	4	no	—
101	C₄²×C₃₁	—	4	4	no	—
102	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
103	C₄²×C₃₁	#41	1	2	yes	—
104	#22249	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₃₁	—	2	2	no	—
107	C₂×C₄×C₃₁	—	2	2	no	—
108	C₂×C₄×C₃₁	—	2	2	no	—
109	C₂×C₄×C₃₁	—	2	1	no	—
110	C₂×C₄×C₃₁	—	2	1	no	—
111	C₂×C₄×C₃₁	—	2	4	no	—
112	C₂×C₄×C₃₁	—	2	4	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	—	4	2	no	—
117	C₂×C₄×C₃₁	—	4	2	no	—
118	C₂×C₄×C₃₁	—	4	2	no	—
119	C₂×C₄×C₃₁	—	4	2	no	—
120	C₂×C₄×C₃₁	—	4	2	no	—
121	C₂×C₄×C₃₁	—	4	2	no	—
122	C₂×C₄×C₃₁	—	4	2	no	—
123	C₂×C₄×C₃₁	—	4	2	no	—
124	C₂×C₄×C₃₁	—	4	2	no	—
125	C₂×C₄×C₃₁	—	4	2	no	—
126	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₃₁	#135	1	4	yes	—
129	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
131	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
132	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
133	C₈×C₃₁	—	2	2	no	—
134	C₈×C₃₁	—	2	2	no	—
135	C₈×C₃₁	—	4	2	no	—
136	C₂×#499	C₂×C₃₁	1	1	yes	characteristic
137	C₄×#125	C₂×C₃₁	1	2	yes	—
138	#1991	C₂×C₃₁	1	2	yes	—
139	#1921	C₂×C₃₁	1	2	yes	—
140	#1920	C₂×C₃₁	1	2	yes	—
141	#1919	C₂×C₃₁	1	2	yes	—
142	#1918	C₂×C₃₁	1	2	yes	—
143	#1835	C₂×C₃₁	1	1	yes	characteristic
144	#1744	C₂×C₃₁	1	1	yes	characteristic
145	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
146	C₂²×C₃₁	#536	1	2	yes	—
147	C₂²×C₃₁	C₂×#135	1	2	yes	—
148	C₂²×C₃₁	#575	1	1	yes	characteristic
149	C₂²×C₃₁	#575	1	1	yes	characteristic
150	C₄×C₃₁	—	2	2	no	—
151	C₄×C₃₁	—	2	2	no	—
152	C₄×C₃₁	—	2	4	no	—
153	C₄×C₃₁	—	4	4	no	—
154	C₄×C₃₁	—	4	4	no	—
155	C₄×C₃₁	—	4	2	no	—
156	C₄×C₃₁	—	8	2	no	—
157	C₄×C₃₁	—	8	2	no	—
158	C₄×C₃₁	#519	1	1	yes	characteristic
159	C₄×C₃₁	#519	1	1	yes	characteristic
160	C₄×C₃₁	#541	1	2	yes	—
161	C₂×#125	C₂²×C₃₁	1	2	yes	—
162	#499	C₄×C₃₁	1	8	yes	complemented
163	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
164	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
165	C₂×#106	C₂²×C₃₁	1	2	yes	—
166	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
167	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
168	#389	C₂²×C₃₁	1	2	yes	—
169	#389	C₂²×C₃₁	1	2	yes	—
170	#387	—	2	2	no	—
171	#387	—	2	2	no	—
172	#382	C₂²×C₃₁	1	2	yes	—
173	C₄×C₄⋊C₄	—	2	2	no	—
174	C₄×C₄⋊C₄	—	2	2	no	—
175	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
176	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
177	C₄³	C₂²×C₃₁	1	1	yes	characteristic
178	#340	C₂²×C₃₁	1	2	yes	—
179	#340	C₂²×C₃₁	1	1	yes	characteristic
180	#340	C₂²×C₃₁	1	1	yes	characteristic
181	#340	C₂²×C₃₁	1	2	yes	—
182	#340	C₂²×C₃₁	1	2	yes	—
183	#340	C₂²×C₃₁	1	2	yes	—
184	#340	C₂²×C₃₁	1	2	yes	—
185	#336	C₂²×C₃₁	1	2	yes	—
186	#336	C₂²×C₃₁	1	2	yes	—
187	C₂×C₃₁	#2327	1	1	yes	characteristic
188	C₂×C₃₁	C₄×#135	1	1	yes	characteristic
189	C₂×C₃₁	#2943	1	1	yes	characteristic
190	C₂×C₃₁	#3232	1	2	yes	—
191	C₂×C₃₁	#3233	1	2	yes	—
192	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
193	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
194	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
195	#125	—	2	4	no	—
196	#125	C₂×C₄×C₃₁	1	8	yes	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
202	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
203	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
204	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	—	2	2	no	—
207	C₂×C₄²	D₄×C₃₁	1	2	yes	—
208	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
209	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
210	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
211	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
212	#107	C₂×C₄×C₃₁	1	4	yes	—
213	#107	C₂×C₄×C₃₁	1	4	yes	—
214	#106	C₂×C₄×C₃₁	1	8	yes	—
215	C₄×C₈	—	2	2	no	—
216	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
217	#95	—	2	2	no	—
218	#95	—	2	2	no	—
219	#95	—	2	2	no	—
220	#95	—	2	2	no	—
221	#95	—	2	2	no	—
222	#95	—	2	2	no	—
223	C₃₁	#22249	1	1	yes	characteristic, direct product factor, Sylow
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
229	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
232	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
239	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
240	C₄⋊C₄	—	2	8	no	—
241	C₄⋊C₄	—	2	8	no	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	4	4	no	—
249	C₄⋊C₄	—	4	4	no	—
250	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
251	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
252	C₄²	—	2	4	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	4	1	no	—
256	C₄²	—	4	1	no	—
257	C₄²	—	4	4	no	—
258	C₄²	—	4	4	no	—
259	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
260	C₄²	#41×C₃₁	1	2	yes	—
261	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
283	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
284	C₂×C₄	C₃₁×#135	1	4	yes	—
285	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
286	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
287	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
288	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
289	C₈	—	2	2	no	—
290	C₈	—	2	2	no	—
291	C₈	—	4	2	no	—
292	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
293	C₂²	C₃₁×#536	1	2	yes	—
294	C₂²	C₂×C₃₁×#135	1	2	yes	—
295	C₂²	C₃₁×#575	1	1	yes	characteristic
296	C₂²	C₃₁×#575	1	1	yes	characteristic
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	4	no	—
300	C₄	—	4	4	no	—
301	C₄	—	4	4	no	—
302	C₄	—	4	2	no	—
303	C₄	—	8	2	no	—
304	C₄	—	8	2	no	—
305	C₄	C₃₁×#519	1	1	yes	characteristic
306	C₄	C₃₁×#519	1	1	yes	characteristic
307	C₄	C₃₁×#541	1	2	yes	—
308	C₂	C₃₁×#2327	1	1	yes	characteristic
309	C₂	C₄×C₃₁×#135	1	1	yes	characteristic
310	C₂	C₃₁×#2943	1	1	yes	characteristic
311	C₂	C₃₁×#3232	1	2	yes	—
312	C₂	C₃₁×#3233	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₃₁×#499	C₄	semidirect
104. #22249	C₃₁	direct (N × H)
162. #499	C₄×C₃₁	semidirect
223. C₃₁	#22249	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×#499	2
C₄×C₃₁×#125	2
C₃₁×#1991	2
C₃₁×#1921	2
C₃₁×#1920	2
C₃₁×#1919	2
C₃₁×#1918	2
C₃₁×#1835	2
C₃₁×#1744	2
#22249	31

Subgroups — C31×#22249 #21479008

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22249 #21479008