Subgroups — C₃₁×#12563 #21469322

Order 7936.

570 conjugacy classes of subgroups.

The subgroup lattice has 300 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×#1383	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	2	no	—
9	C₂×C₃₁×M₃₂	—	2	2	no	—
10	C₂×C₃₁×M₃₂	C₄	1	2	yes	complemented
11	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₃₁	C₄	1	2	yes	complemented
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	2	no	—
18	C₃₁×#427	C₄	1	4	yes	—
19	C₂×C₃₁×#107	—	2	1	no	—
20	C₂×C₃₁×#106	C₂²	1	2	yes	—
21	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
22	C₂×M₁₆×C₃₁	—	2	4	no	—
23	C₂×M₁₆×C₃₁	—	2	2	no	—
24	C₂×M₁₆×C₃₁	—	4	2	no	—
25	C₂×M₁₆×C₃₁	—	4	1	no	—
26	C₂²×C₈×C₃₁	—	4	1	no	—
27	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₃₁	D₄	1	2	yes	—
33	C₂²×C₈×C₃₁	D₄	1	2	yes	—
34	C₂²×C₈×C₃₁	Q₈	1	1	yes	characteristic
35	C₂²×C₈×C₃₁	Q₈	1	1	yes	characteristic
36	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
37	C₃₁×#117	—	2	4	no	—
38	C₃₁×#117	—	4	2	no	—
39	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
40	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
41	C₂×C₄²×C₃₁	—	4	1	no	—
42	C₃₁×M₃₂	—	2	4	no	—
43	C₃₁×M₃₂	—	2	2	no	—
44	C₃₁×M₃₂	—	2	2	no	—
45	C₂×C₁₆×C₃₁	—	2	4	no	—
46	C₂×C₁₆×C₃₁	C₂×C₄	1	4	yes	—
47	C₃₁×#108	—	2	4	no	—
48	C₃₁×#108	—	2	2	no	—
49	C₃₁×#108	C₂×C₄	1	4	yes	—
50	C₃₁×#107	—	2	4	no	—
51	C₃₁×#106	—	2	2	no	—
52	C₃₁×#106	C₂×C₄	1	4	yes	—
53	C₂⁴×C₃₁	SD₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₃₁	—	4	2	no	—
55	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	D₈	1	2	yes	—
60	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	Q₁₆	1	2	yes	—
64	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
65	M₁₆×C₃₁	—	2	4	no	—
66	M₁₆×C₃₁	—	4	4	no	—
67	M₁₆×C₃₁	—	4	2	no	—
68	C₂×C₈×C₃₁	—	4	4	no	—
69	C₂×C₈×C₃₁	—	4	2	no	—
70	C₂×C₈×C₃₁	C₄²	1	2	yes	—
71	C₂×C₈×C₃₁	C₄²	1	2	yes	—
72	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
75	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
76	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
79	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
80	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
81	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
82	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
88	C₄⋊C₄×C₃₁	—	2	4	no	—
89	C₄⋊C₄×C₃₁	—	4	2	no	—
90	C₂²⋊C₄×C₃₁	—	4	4	no	—
91	C₄²×C₃₁	—	4	4	no	—
92	C₁₆×C₃₁	—	2	4	no	—
93	#12563	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₃₁	—	4	2	no	—
95	C₂³×C₃₁	#102	1	2	yes	—
96	C₂³×C₃₁	#103	1	2	yes	—
97	C₂³×C₃₁	#106	1	1	yes	characteristic
98	C₂³×C₃₁	#106	1	1	yes	characteristic
99	C₂³×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×C₄×C₃₁	—	4	4	no	—
101	C₂×C₄×C₃₁	—	8	4	no	—
102	C₂×C₄×C₃₁	#95	1	2	yes	—
103	C₂×C₄×C₃₁	#95	1	2	yes	—
104	C₂×C₄×C₃₁	#102	1	2	yes	—
105	C₂×C₄×C₃₁	#102	1	2	yes	—
106	C₂×C₄×C₃₁	#102	1	2	yes	—
107	C₂×C₄×C₃₁	#103	1	2	yes	—
108	C₂×C₄×C₃₁	#103	1	2	yes	—
109	C₂×C₄×C₃₁	#103	1	2	yes	—
110	C₂×C₄×C₃₁	#106	1	1	yes	characteristic
111	C₂×C₄×C₃₁	#106	1	1	yes	characteristic
112	C₂×C₄×C₃₁	#107	1	2	yes	—
113	C₂×C₄×C₃₁	#107	1	2	yes	—
114	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
120	C₈×C₃₁	—	4	4	no	—
121	C₈×C₃₁	#95	1	2	yes	—
122	C₈×C₃₁	#95	1	2	yes	—
123	C₈×C₃₁	C₂×C₄²	1	1	yes	characteristic
124	C₈×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₈×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₈×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂²×M₃₂	C₂×C₃₁	1	1	yes	characteristic
128	C₂²×#108	C₂×C₃₁	1	1	yes	characteristic
129	C₂×#427	C₂×C₃₁	1	1	yes	characteristic
130	#1383	C₂×C₃₁	1	4	yes	complemented
131	C₂²×C₃₁	—	2	2	no	—
132	C₂²×C₃₁	—	2	2	no	—
133	C₂²×C₃₁	—	2	2	no	—
134	C₂²×C₃₁	—	4	2	no	—
135	C₂²×C₃₁	—	4	2	no	—
136	C₂²×C₃₁	#340	1	2	yes	—
137	C₂²×C₃₁	#340	1	2	yes	—
138	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
139	C₂²×C₃₁	C₂×#103	1	1	yes	characteristic
140	C₂²×C₃₁	C₂×#106	1	1	yes	characteristic
141	C₄×C₃₁	—	8	4	no	—
142	C₄×C₃₁	#340	1	2	yes	—
143	C₄×C₃₁	#340	1	2	yes	—
144	C₄×C₃₁	C₂×#95	1	1	yes	characteristic
145	C₄×C₃₁	C₂×#102	1	1	yes	characteristic
146	C₄×C₃₁	C₂×#103	1	1	yes	characteristic
147	C₄×C₃₁	C₂×#107	1	1	yes	characteristic
148	C₂²×M₁₆	—	2	1	no	—
149	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
150	C₂×#117	—	2	1	no	—
151	C₂×M₃₂	—	2	2	no	—
152	C₂×M₃₂	—	2	2	no	—
153	C₂×M₃₂	C₄×C₃₁	1	2	yes	complemented
154	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
155	C₂²×C₁₆	C₄×C₃₁	1	2	yes	complemented
156	C₂×#108	—	2	1	no	—
157	C₂×#108	—	2	2	no	—
158	C₂×#108	C₄×C₃₁	1	4	yes	complemented
159	C₂×#108	C₂²×C₃₁	1	2	yes	—
160	#427	—	2	2	no	—
161	#427	C₄×C₃₁	1	4	yes	—
162	C₂×#107	—	2	1	no	—
163	C₂×#106	C₂²×C₃₁	1	2	yes	—
164	C₂×C₃₁	—	2	1	no	—
165	C₂×C₃₁	—	2	2	no	—
166	C₂×C₃₁	—	2	2	no	—
167	C₂×C₃₁	—	2	1	no	—
168	C₂×C₃₁	C₂×#340	1	1	yes	characteristic
169	C₂×C₃₁	#2155	1	2	yes	—
170	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
171	C₂×M₁₆	—	2	4	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	4	2	no	—
174	C₂×M₁₆	—	4	1	no	—
175	C₂²×C₈	—	4	1	no	—
176	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
177	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
178	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
179	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
180	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
181	C₂²×C₈	D₄×C₃₁	1	2	yes	—
182	C₂²×C₈	D₄×C₃₁	1	2	yes	—
183	C₂²×C₈	Q₈×C₃₁	1	1	yes	characteristic
184	C₂²×C₈	Q₈×C₃₁	1	1	yes	characteristic
185	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
186	#117	—	2	4	no	—
187	#117	—	4	2	no	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₂²⋊C₄	—	4	1	no	—
190	C₂×C₄²	—	4	1	no	—
191	M₃₂	—	2	4	no	—
192	M₃₂	—	2	2	no	—
193	M₃₂	—	2	2	no	—
194	C₂×C₁₆	—	2	4	no	—
195	C₂×C₁₆	C₂×C₄×C₃₁	1	4	yes	—
196	#108	—	2	4	no	—
197	#108	—	2	2	no	—
198	#108	C₂×C₄×C₃₁	1	4	yes	—
199	#107	—	2	4	no	—
200	#106	—	2	2	no	—
201	#106	C₂×C₄×C₃₁	1	4	yes	—
202	C₃₁	#12563	1	1	yes	characteristic, direct product factor, Sylow
203	C₂⁴	SD₁₆×C₃₁	1	1	yes	characteristic
204	C₂²×C₄	—	4	2	no	—
205	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
206	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
207	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
208	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
209	C₂²×C₄	D₈×C₃₁	1	2	yes	—
210	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
211	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
212	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
213	C₂²×C₄	Q₁₆×C₃₁	1	2	yes	—
214	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
215	M₁₆	—	2	4	no	—
216	M₁₆	—	4	4	no	—
217	M₁₆	—	4	2	no	—
218	C₂×C₈	—	4	4	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	C₄²×C₃₁	1	2	yes	—
221	C₂×C₈	C₄²×C₃₁	1	2	yes	—
222	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
223	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
224	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
225	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
226	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
227	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
228	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
229	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
230	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
231	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
232	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
233	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
234	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
235	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
236	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
237	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	4	4	no	—
241	C₄²	—	4	4	no	—
242	C₁₆	—	2	4	no	—
243	C₂³	—	4	2	no	—
244	C₂³	C₃₁×#102	1	2	yes	—
245	C₂³	C₃₁×#103	1	2	yes	—
246	C₂³	C₃₁×#106	1	1	yes	characteristic
247	C₂³	C₃₁×#106	1	1	yes	characteristic
248	C₂³	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	8	4	no	—
251	C₂×C₄	C₃₁×#95	1	2	yes	—
252	C₂×C₄	C₃₁×#95	1	2	yes	—
253	C₂×C₄	C₃₁×#102	1	2	yes	—
254	C₂×C₄	C₃₁×#102	1	2	yes	—
255	C₂×C₄	C₃₁×#102	1	2	yes	—
256	C₂×C₄	C₃₁×#103	1	2	yes	—
257	C₂×C₄	C₃₁×#103	1	2	yes	—
258	C₂×C₄	C₃₁×#103	1	2	yes	—
259	C₂×C₄	C₃₁×#106	1	1	yes	characteristic
260	C₂×C₄	C₃₁×#106	1	1	yes	characteristic
261	C₂×C₄	C₃₁×#107	1	2	yes	—
262	C₂×C₄	C₃₁×#107	1	2	yes	—
263	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
264	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
265	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
266	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
267	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
268	C₂×C₄	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
269	C₈	—	4	4	no	—
270	C₈	C₃₁×#95	1	2	yes	—
271	C₈	C₃₁×#95	1	2	yes	—
272	C₈	C₂×C₄²×C₃₁	1	1	yes	characteristic
273	C₈	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
274	C₈	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
275	C₈	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	C₃₁×#340	1	2	yes	—
282	C₂²	C₃₁×#340	1	2	yes	—
283	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
284	C₂²	C₂×C₃₁×#103	1	1	yes	characteristic
285	C₂²	C₂×C₃₁×#106	1	1	yes	characteristic
286	C₄	—	8	4	no	—
287	C₄	C₃₁×#340	1	2	yes	—
288	C₄	C₃₁×#340	1	2	yes	—
289	C₄	C₂×C₃₁×#95	1	1	yes	characteristic
290	C₄	C₂×C₃₁×#102	1	1	yes	characteristic
291	C₄	C₂×C₃₁×#103	1	1	yes	characteristic
292	C₄	C₂×C₃₁×#107	1	1	yes	characteristic
293	C₂	—	2	1	no	—
294	C₂	—	2	1	no	—
295	C₂	—	2	2	no	—
296	C₂	—	2	2	no	—
297	C₂	C₂×C₃₁×#340	1	1	yes	characteristic
298	C₂	C₃₁×#2155	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	#106
commutator	C₈	C₂×C₄²×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×#106
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×#1383	C₂	semidirect
10. C₂×C₃₁×M₃₂	C₄	semidirect
12. C₂²×C₁₆×C₃₁	C₄	semidirect
15. C₂×C₃₁×#108	C₄	semidirect
93. #12563	C₃₁	direct (N × H)
130. #1383	C₂×C₃₁	semidirect
153. C₂×M₃₂	C₄×C₃₁	semidirect
155. C₂²×C₁₆	C₄×C₃₁	semidirect
158. C₂×#108	C₄×C₃₁	semidirect
202. C₃₁	#12563	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×#1383	2
#12563	31

Subgroups — C31×#12563 #21469322

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12563 #21469322