Subgroups — C₃₁×#12561 #21469320

Order 7936.

570 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	C₂²×C₃₁×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×#108	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#427	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1382	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₃₁	—	2	1	no	—
6	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×C₃₁×#117	—	2	1	no	—
8	C₂×C₃₁×M₃₂	—	2	1	no	—
9	C₂×C₃₁×M₃₂	—	2	1	no	—
10	C₂×C₃₁×M₃₂	—	2	2	no	—
11	C₂×C₃₁×M₃₂	C₄	1	4	yes	complemented
12	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
13	C₂×C₃₁×#108	—	2	1	no	—
14	C₂×C₃₁×#108	—	2	2	no	—
15	C₂×C₃₁×#108	C₄	1	4	yes	complemented
16	C₂×C₃₁×#108	C₂²	1	2	yes	—
17	C₃₁×#427	—	2	1	no	—
18	C₃₁×#427	—	2	1	no	—
19	C₃₁×#427	C₄	1	2	yes	—
20	C₃₁×#427	C₂²	1	2	yes	—
21	C₂×C₃₁×#107	C₄	1	2	yes	—
22	C₂×C₃₁×#106	—	2	1	no	—
23	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
24	C₂×M₁₆×C₃₁	—	2	4	no	—
25	C₂×M₁₆×C₃₁	—	2	2	no	—
26	C₂×M₁₆×C₃₁	—	4	1	no	—
27	C₂×M₁₆×C₃₁	—	4	2	no	—
28	C₂²×C₈×C₃₁	—	4	1	no	—
29	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₃₁	D₄	1	2	yes	—
37	C₂²×C₈×C₃₁	Q₈	1	2	yes	—
38	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
39	C₃₁×#117	—	2	2	no	—
40	C₃₁×#117	—	2	2	no	—
41	C₃₁×#117	—	4	1	no	—
42	C₃₁×#117	—	4	1	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
44	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
45	C₂×C₄²×C₃₁	—	4	1	no	—
46	C₃₁×M₃₂	—	2	4	no	—
47	C₃₁×M₃₂	—	2	4	no	—
48	C₂×C₁₆×C₃₁	—	2	2	no	—
49	C₂×C₁₆×C₃₁	—	2	2	no	—
50	C₂×C₁₆×C₃₁	C₂×C₄	1	4	yes	complemented
51	C₃₁×#108	—	2	4	no	—
52	C₃₁×#108	—	2	2	no	—
53	C₃₁×#108	C₂×C₄	1	4	yes	complemented
54	C₃₁×#107	—	2	2	no	—
55	C₃₁×#107	C₂×C₄	1	4	yes	—
56	C₃₁×#106	—	2	2	no	—
57	C₃₁×#106	—	2	2	no	—
58	C₂⁴×C₃₁	SD₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	—	4	2	no	—
60	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
67	C₂²×C₄×C₃₁	SD₁₆	1	2	yes	—
68	C₂²×C₄×C₃₁	Q₁₆	1	2	yes	—
69	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
70	M₁₆×C₃₁	—	2	4	no	—
71	M₁₆×C₃₁	—	4	4	no	—
72	M₁₆×C₃₁	—	4	2	no	—
73	C₂×C₈×C₃₁	—	4	4	no	—
74	C₂×C₈×C₃₁	—	4	2	no	—
75	C₂×C₈×C₃₁	C₄²	1	2	yes	—
76	C₂×C₈×C₃₁	C₄²	1	2	yes	—
77	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
80	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
81	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
82	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
83	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
84	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
85	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
86	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
87	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
93	C₄⋊C₄×C₃₁	—	2	4	no	—
94	C₄⋊C₄×C₃₁	—	4	2	no	—
95	C₂²⋊C₄×C₃₁	—	4	2	no	—
96	C₂²⋊C₄×C₃₁	—	4	2	no	—
97	C₄²×C₃₁	—	4	2	no	—
98	C₄²×C₃₁	—	4	2	no	—
99	C₁₆×C₃₁	—	2	4	no	—
100	#12561	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂³×C₃₁	—	4	2	no	—
102	C₂³×C₃₁	#102	1	1	yes	characteristic
103	C₂³×C₃₁	#102	1	1	yes	characteristic
104	C₂³×C₃₁	#103	1	2	yes	—
105	C₂³×C₃₁	#106	1	2	yes	—
106	C₂³×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₃₁	—	4	4	no	—
108	C₂×C₄×C₃₁	—	8	4	no	—
109	C₂×C₄×C₃₁	#95	1	2	yes	—
110	C₂×C₄×C₃₁	#95	1	2	yes	—
111	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
112	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
113	C₂×C₄×C₃₁	#102	1	2	yes	—
114	C₂×C₄×C₃₁	#102	1	2	yes	—
115	C₂×C₄×C₃₁	#103	1	2	yes	—
116	C₂×C₄×C₃₁	#103	1	2	yes	—
117	C₂×C₄×C₃₁	#103	1	2	yes	—
118	C₂×C₄×C₃₁	#106	1	2	yes	—
119	C₂×C₄×C₃₁	#107	1	2	yes	—
120	C₂×C₄×C₃₁	#107	1	2	yes	—
121	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
127	C₈×C₃₁	—	4	4	no	—
128	C₈×C₃₁	#95	1	2	yes	—
129	C₈×C₃₁	#95	1	2	yes	—
130	C₈×C₃₁	C₂×C₄²	1	1	yes	characteristic
131	C₈×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₈×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₈×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂²×M₃₂	C₂×C₃₁	1	1	yes	characteristic
135	C₂²×#108	C₂×C₃₁	1	1	yes	characteristic
136	C₂×#427	C₂×C₃₁	1	1	yes	characteristic
137	#1382	C₂×C₃₁	1	4	yes	complemented
138	C₂²×C₃₁	—	2	2	no	—
139	C₂²×C₃₁	—	2	2	no	—
140	C₂²×C₃₁	—	2	2	no	—
141	C₂²×C₃₁	—	4	2	no	—
142	C₂²×C₃₁	—	4	2	no	—
143	C₂²×C₃₁	#340	1	2	yes	—
144	C₂²×C₃₁	#340	1	2	yes	—
145	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
146	C₂²×C₃₁	C₂×#103	1	1	yes	characteristic
147	C₂²×C₃₁	C₂×#106	1	1	yes	characteristic
148	C₄×C₃₁	—	8	4	no	—
149	C₄×C₃₁	#340	1	2	yes	—
150	C₄×C₃₁	#340	1	2	yes	—
151	C₄×C₃₁	C₂×#95	1	1	yes	characteristic
152	C₄×C₃₁	C₂×#102	1	1	yes	characteristic
153	C₄×C₃₁	C₂×#103	1	1	yes	characteristic
154	C₄×C₃₁	C₂×#107	1	1	yes	characteristic
155	C₂²×M₁₆	—	2	1	no	—
156	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
157	C₂×#117	—	2	1	no	—
158	C₂×M₃₂	—	2	1	no	—
159	C₂×M₃₂	—	2	1	no	—
160	C₂×M₃₂	—	2	2	no	—
161	C₂×M₃₂	C₄×C₃₁	1	4	yes	complemented
162	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
163	C₂×#108	—	2	1	no	—
164	C₂×#108	—	2	2	no	—
165	C₂×#108	C₄×C₃₁	1	4	yes	complemented
166	C₂×#108	C₂²×C₃₁	1	2	yes	—
167	#427	—	2	1	no	—
168	#427	—	2	1	no	—
169	#427	C₄×C₃₁	1	2	yes	—
170	#427	C₂²×C₃₁	1	2	yes	—
171	C₂×#107	C₄×C₃₁	1	2	yes	—
172	C₂×#106	—	2	1	no	—
173	C₂×C₃₁	—	2	2	no	—
174	C₂×C₃₁	—	2	2	no	—
175	C₂×C₃₁	—	2	1	no	—
176	C₂×C₃₁	—	2	1	no	—
177	C₂×C₃₁	C₂×#340	1	1	yes	characteristic
178	C₂×C₃₁	#2143	1	2	yes	—
179	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
180	C₂×M₁₆	—	2	4	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	4	1	no	—
183	C₂×M₁₆	—	4	2	no	—
184	C₂²×C₈	—	4	1	no	—
185	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
186	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
187	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
188	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
189	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
190	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
191	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
192	C₂²×C₈	D₄×C₃₁	1	2	yes	—
193	C₂²×C₈	Q₈×C₃₁	1	2	yes	—
194	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
195	#117	—	2	2	no	—
196	#117	—	2	2	no	—
197	#117	—	4	1	no	—
198	#117	—	4	1	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₂²⋊C₄	—	4	1	no	—
201	C₂×C₄²	—	4	1	no	—
202	M₃₂	—	2	4	no	—
203	M₃₂	—	2	4	no	—
204	C₂×C₁₆	—	2	2	no	—
205	C₂×C₁₆	—	2	2	no	—
206	C₂×C₁₆	C₂×C₄×C₃₁	1	4	yes	complemented
207	#108	—	2	4	no	—
208	#108	—	2	2	no	—
209	#108	C₂×C₄×C₃₁	1	4	yes	complemented
210	#107	—	2	2	no	—
211	#107	C₂×C₄×C₃₁	1	4	yes	—
212	#106	—	2	2	no	—
213	#106	—	2	2	no	—
214	C₃₁	#12561	1	1	yes	characteristic, direct product factor, Sylow
215	C₂⁴	SD₁₆×C₃₁	1	1	yes	characteristic
216	C₂²×C₄	—	4	2	no	—
217	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
218	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
219	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
220	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
221	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
222	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
223	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
224	C₂²×C₄	SD₁₆×C₃₁	1	2	yes	—
225	C₂²×C₄	Q₁₆×C₃₁	1	2	yes	—
226	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
227	M₁₆	—	2	4	no	—
228	M₁₆	—	4	4	no	—
229	M₁₆	—	4	2	no	—
230	C₂×C₈	—	4	4	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	C₄²×C₃₁	1	2	yes	—
233	C₂×C₈	C₄²×C₃₁	1	2	yes	—
234	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
235	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
236	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
237	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
238	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
239	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
240	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
241	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
242	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
243	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
244	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
245	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
246	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
247	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
248	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
249	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₂²⋊C₄	—	4	2	no	—
253	C₂²⋊C₄	—	4	2	no	—
254	C₄²	—	4	2	no	—
255	C₄²	—	4	2	no	—
256	C₁₆	—	2	4	no	—
257	C₂³	—	4	2	no	—
258	C₂³	C₃₁×#102	1	1	yes	characteristic
259	C₂³	C₃₁×#102	1	1	yes	characteristic
260	C₂³	C₃₁×#103	1	2	yes	—
261	C₂³	C₃₁×#106	1	2	yes	—
262	C₂³	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	8	4	no	—
265	C₂×C₄	C₃₁×#95	1	2	yes	—
266	C₂×C₄	C₃₁×#95	1	2	yes	—
267	C₂×C₄	C₃₁×#102	1	2	yes	—
268	C₂×C₄	C₃₁×#102	1	2	yes	—
269	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
270	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
271	C₂×C₄	C₃₁×#103	1	2	yes	—
272	C₂×C₄	C₃₁×#103	1	2	yes	—
273	C₂×C₄	C₃₁×#103	1	2	yes	—
274	C₂×C₄	C₃₁×#106	1	2	yes	—
275	C₂×C₄	C₃₁×#107	1	2	yes	—
276	C₂×C₄	C₃₁×#107	1	2	yes	—
277	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
278	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
280	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
281	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
282	C₂×C₄	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
283	C₈	—	4	4	no	—
284	C₈	C₃₁×#95	1	2	yes	—
285	C₈	C₃₁×#95	1	2	yes	—
286	C₈	C₂×C₄²×C₃₁	1	1	yes	characteristic
287	C₈	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
288	C₈	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
289	C₈	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	4	2	no	—
294	C₂²	—	4	2	no	—
295	C₂²	C₃₁×#340	1	2	yes	—
296	C₂²	C₃₁×#340	1	2	yes	—
297	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
298	C₂²	C₂×C₃₁×#103	1	1	yes	characteristic
299	C₂²	C₂×C₃₁×#106	1	1	yes	characteristic
300	C₄	—	8	4	no	—
301	C₄	C₃₁×#340	1	2	yes	—
302	C₄	C₃₁×#340	1	2	yes	—
303	C₄	C₂×C₃₁×#95	1	1	yes	characteristic
304	C₄	C₂×C₃₁×#102	1	1	yes	characteristic
305	C₄	C₂×C₃₁×#103	1	1	yes	characteristic
306	C₄	C₂×C₃₁×#107	1	1	yes	characteristic
307	C₂	—	2	1	no	—
308	C₂	—	2	1	no	—
309	C₂	—	2	2	no	—
310	C₂	—	2	2	no	—
311	C₂	C₂×C₃₁×#340	1	1	yes	characteristic
312	C₂	C₃₁×#2143	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×D₈
commutator	C₈	C₂×C₄²×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×#102
prime radical	C₂⁴×C₃₁	SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1382	C₂	semidirect
11. C₂×C₃₁×M₃₂	C₄	semidirect
15. C₂×C₃₁×#108	C₄	semidirect
50. C₂×C₁₆×C₃₁	C₂×C₄	semidirect
53. C₃₁×#108	C₂×C₄	semidirect
100. #12561	C₃₁	direct (N × H)
137. #1382	C₂×C₃₁	semidirect
161. C₂×M₃₂	C₄×C₃₁	semidirect
165. C₂×#108	C₄×C₃₁	semidirect
206. C₂×C₁₆	C₂×C₄×C₃₁	semidirect
209. #108	C₂×C₄×C₃₁	semidirect
214. C₃₁	#12561	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃₁×M₃₂	2
C₂²×C₃₁×#108	2
C₂×C₃₁×#427	2
C₃₁×#1382	2
#12561	31

Subgroups — C31×#12561 #21469320

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12561 #21469320