Subgroups — C₃₁×#21020 #21477779

Order 7936.

942 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×#424	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2296	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1916	C₂	1	8	yes	maximal, complemented
5	C₃₁×#1729	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1729	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1729	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#1728	C₂	1	1	yes	maximal, characteristic
9	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#424	—	2	4	no	—
17	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
18	C₂×C₄×C₈×C₃₁	—	2	4	no	—
19	C₂×C₄×C₈×C₃₁	C₂²	1	4	yes	—
20	C₃₁×#389	C₄	1	4	yes	—
21	C₃₁×#385	C₄	1	4	yes	—
22	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
23	C₃₁×#336	C₂²	1	4	yes	—
24	C₃₁×#336	C₂²	1	4	yes	—
25	C₃₁×#336	C₂²	1	4	yes	—
26	C₃₁×#336	C₂²	1	4	yes	—
27	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
28	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
29	C₂×M₁₆×C₃₁	—	2	2	no	—
30	C₂×M₁₆×C₃₁	—	2	2	no	—
31	C₂×M₁₆×C₃₁	—	2	2	no	—
32	C₂×M₁₆×C₃₁	—	2	2	no	—
33	C₂×M₁₆×C₃₁	—	2	2	no	—
34	C₂×M₁₆×C₃₁	—	2	2	no	—
35	C₂²×C₈×C₃₁	—	2	2	no	—
36	C₂²×C₈×C₃₁	—	2	2	no	—
37	C₂²×C₈×C₃₁	—	2	2	no	—
38	C₂²×C₈×C₃₁	—	2	2	no	—
39	C₂²×C₈×C₃₁	D₄	1	4	yes	—
40	C₂²×C₈×C₃₁	Q₈	1	4	yes	—
41	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
42	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
43	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
44	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
45	C₃₁×#117	—	2	2	no	—
46	C₃₁×#117	—	2	2	no	—
47	C₃₁×#117	—	2	2	no	—
48	C₃₁×#117	—	2	2	no	—
49	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	8	yes	—
52	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
53	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₃₁	—	2	2	no	—
55	C₂×C₄²×C₃₁	—	2	2	no	—
56	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
57	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
58	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
59	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
60	C₃₁×C₄⋊C₈	—	2	8	no	—
61	C₄×C₈×C₃₁	—	2	8	no	—
62	C₃₁×#95	C₂×C₄	1	8	yes	—
63	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	—	2	4	no	—
65	C₂²×C₄×C₃₁	—	2	8	no	—
66	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
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	C₂²×C₄×C₃₁	#41	1	2	yes	—
76	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
77	M₁₆×C₃₁	—	4	4	no	—
78	M₁₆×C₃₁	—	4	4	no	—
79	C₂×C₈×C₃₁	—	2	8	no	—
80	C₂×C₈×C₃₁	—	2	8	no	—
81	C₂×C₈×C₃₁	—	2	2	no	—
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	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	2	no	—
91	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
92	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
93	C₄⋊C₄×C₃₁	—	2	4	no	—
94	C₄⋊C₄×C₃₁	—	2	4	no	—
95	C₄⋊C₄×C₃₁	—	2	16	no	—
96	C₂²⋊C₄×C₃₁	—	2	4	no	—
97	C₂²⋊C₄×C₃₁	—	2	4	no	—
98	C₄²×C₃₁	—	2	4	no	—
99	C₄²×C₃₁	—	2	2	no	—
100	C₄²×C₃₁	—	2	2	no	—
101	C₄²×C₃₁	—	2	2	no	—
102	C₄²×C₃₁	—	2	2	no	—
103	C₄²×C₃₁	—	2	2	no	—
104	C₄²×C₃₁	—	2	2	no	—
105	#21020	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂³×C₃₁	—	2	4	no	—
107	C₂³×C₃₁	#117	1	2	yes	—
108	C₂³×C₃₁	C₄×D₄	1	2	yes	—
109	C₂³×C₃₁	C₄×Q₈	1	2	yes	—
110	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₃₁	—	2	16	no	—
112	C₂×C₄×C₃₁	—	2	4	no	—
113	C₂×C₄×C₃₁	—	2	4	no	—
114	C₂×C₄×C₃₁	—	2	4	no	—
115	C₂×C₄×C₃₁	—	2	4	no	—
116	C₂×C₄×C₃₁	—	2	8	no	—
117	C₂×C₄×C₃₁	—	2	8	no	—
118	C₂×C₄×C₃₁	#117	1	2	yes	—
119	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
121	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
122	C₂×C₄×C₃₁	#123	1	4	yes	—
123	C₂×C₄×C₃₁	#125	1	4	yes	—
124	C₂×C₄×C₃₁	#126	1	4	yes	—
125	C₂×C₄×C₃₁	#131	1	8	yes	—
126	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
128	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
129	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
130	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
131	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
132	C₈×C₃₁	—	2	4	no	—
133	C₈×C₃₁	—	2	4	no	—
134	C₈×C₃₁	—	4	2	no	—
135	C₈×C₃₁	—	4	2	no	—
136	C₂×#424	C₂×C₃₁	1	1	yes	characteristic
137	C₂²×C₄×C₈	C₂×C₃₁	1	1	yes	characteristic
138	#2296	C₂×C₃₁	1	1	yes	characteristic
139	#1916	C₂×C₃₁	1	8	yes	complemented
140	#1729	C₂×C₃₁	1	1	yes	characteristic
141	#1729	C₂×C₃₁	1	1	yes	characteristic
142	#1729	C₂×C₃₁	1	1	yes	characteristic
143	#1728	C₂×C₃₁	1	1	yes	characteristic
144	C₂²×C₃₁	—	2	4	no	—
145	C₂²×C₃₁	—	2	4	no	—
146	C₂²×C₃₁	—	2	4	no	—
147	C₂²×C₃₁	#387	1	4	yes	—
148	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
149	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
150	C₂²×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
151	C₂²×C₃₁	C₂×#131	1	4	yes	—
152	C₄×C₃₁	—	2	8	no	—
153	C₄×C₃₁	—	4	4	no	—
154	C₄×C₃₁	—	4	4	no	—
155	C₄×C₃₁	#387	1	4	yes	—
156	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
157	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
158	C₄×C₃₁	C₂×#125	1	1	yes	characteristic
159	C₄×C₃₁	C₂×#126	1	1	yes	characteristic
160	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
161	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
162	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
163	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
164	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
165	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
166	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
167	#424	—	2	4	no	—
168	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
169	C₂×C₄×C₈	—	2	4	no	—
170	C₂×C₄×C₈	C₂²×C₃₁	1	4	yes	—
171	#389	C₄×C₃₁	1	4	yes	—
172	#385	C₄×C₃₁	1	4	yes	—
173	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
174	#336	C₂²×C₃₁	1	4	yes	—
175	#336	C₂²×C₃₁	1	4	yes	—
176	#336	C₂²×C₃₁	1	4	yes	—
177	#336	C₂²×C₃₁	1	4	yes	—
178	C₂×C₃₁	—	2	4	no	—
179	C₂×C₃₁	C₂×#387	1	1	yes	characteristic
180	C₂×C₃₁	#2922	1	2	yes	—
181	C₂×C₃₁	#2929	1	2	yes	—
182	C₂×C₃₁	#2960	1	2	yes	—
183	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
184	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂²×C₈	—	2	2	no	—
192	C₂²×C₈	—	2	2	no	—
193	C₂²×C₈	—	2	2	no	—
194	C₂²×C₈	—	2	2	no	—
195	C₂²×C₈	D₄×C₃₁	1	4	yes	—
196	C₂²×C₈	Q₈×C₃₁	1	4	yes	—
197	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
198	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
199	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
200	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
201	#117	—	2	2	no	—
202	#117	—	2	2	no	—
203	#117	—	2	2	no	—
204	#117	—	2	2	no	—
205	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
206	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
207	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	8	yes	—
208	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
209	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
210	C₂×C₄²	—	2	2	no	—
211	C₂×C₄²	—	2	2	no	—
212	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
213	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
214	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
215	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
216	C₄⋊C₈	—	2	8	no	—
217	C₄×C₈	—	2	8	no	—
218	#95	C₂×C₄×C₃₁	1	8	yes	—
219	C₃₁	#21020	1	1	yes	characteristic, direct product factor, Sylow
220	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	2	8	no	—
223	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
224	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
225	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
226	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
228	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
229	C₂²×C₄	#41×C₃₁	1	2	yes	—
230	C₂²×C₄	#41×C₃₁	1	2	yes	—
231	C₂²×C₄	#41×C₃₁	1	2	yes	—
232	C₂²×C₄	#41×C₃₁	1	2	yes	—
233	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
234	M₁₆	—	4	4	no	—
235	M₁₆	—	4	4	no	—
236	C₂×C₈	—	2	8	no	—
237	C₂×C₈	—	2	8	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
249	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	2	16	no	—
253	C₂²⋊C₄	—	2	4	no	—
254	C₂²⋊C₄	—	2	4	no	—
255	C₄²	—	2	4	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	2	no	—
262	C₂³	—	2	4	no	—
263	C₂³	C₃₁×#117	1	2	yes	—
264	C₂³	C₄×D₄×C₃₁	1	2	yes	—
265	C₂³	C₄×Q₈×C₃₁	1	2	yes	—
266	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
267	C₂×C₄	—	2	16	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	8	no	—
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	C₃₁×#117	1	2	yes	—
275	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
276	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
277	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
278	C₂×C₄	C₃₁×#123	1	4	yes	—
279	C₂×C₄	C₃₁×#125	1	4	yes	—
280	C₂×C₄	C₃₁×#126	1	4	yes	—
281	C₂×C₄	C₃₁×#131	1	8	yes	—
282	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
283	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
284	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
285	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
286	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
287	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
288	C₈	—	2	4	no	—
289	C₈	—	2	4	no	—
290	C₈	—	4	2	no	—
291	C₈	—	4	2	no	—
292	C₂²	—	2	4	no	—
293	C₂²	—	2	4	no	—
294	C₂²	—	2	4	no	—
295	C₂²	C₃₁×#387	1	4	yes	—
296	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
297	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
298	C₂²	C₂×C₄×Q₈×C₃₁	1	1	yes	characteristic
299	C₂²	C₂×C₃₁×#131	1	4	yes	—
300	C₄	—	2	8	no	—
301	C₄	—	4	4	no	—
302	C₄	—	4	4	no	—
303	C₄	C₃₁×#387	1	4	yes	—
304	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
305	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
306	C₄	C₂×C₃₁×#125	1	1	yes	characteristic
307	C₄	C₂×C₃₁×#126	1	1	yes	characteristic
308	C₂	—	2	4	no	—
309	C₂	C₂×C₃₁×#387	1	1	yes	characteristic
310	C₂	C₃₁×#2922	1	2	yes	—
311	C₂	C₃₁×#2929	1	2	yes	—
312	C₂	C₃₁×#2960	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
4. C₃₁×#1916	C₂	semidirect
105. #21020	C₃₁	direct (N × H)
139. #1916	C₂×C₃₁	semidirect
219. C₃₁	#21020	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×#424	2
C₂²×C₄×C₈×C₃₁	2
C₃₁×#2296	2
C₃₁×#1916	2
C₃₁×#1729	2
C₃₁×#1728	2
#21020	31

Subgroups — C31×#21020 #21477779

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21020 #21477779