Subgroups — C₃₁×#8166 #21464925

Order 7936.

494 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#436	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₁×#100	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1460	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1451	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1283	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1283	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×M₁₆×C₃₁	C₄	1	2	yes	—
9	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
10	C₃₁×#436	—	2	1	no	—
11	C₃₁×#436	—	2	1	no	—
12	C₃₁×#436	—	2	1	no	—
13	C₃₁×#436	—	2	1	no	—
14	C₃₁×#431	C₂²	1	1	yes	characteristic
15	C₃₁×#431	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#100	C₄	1	2	yes	—
18	C₂×C₃₁×C₂²⋊C₈	C₄	1	2	yes	—
19	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
20	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₃₁×#343	C₄	1	2	yes	—
23	C₃₁×#323	C₄	1	4	yes	—
24	C₃₁×#322	—	2	1	no	—
25	C₃₁×#322	—	2	1	no	—
26	C₃₁×#322	—	2	1	no	—
27	C₈²×C₃₁	—	2	1	no	—
28	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
30	C₂×M₁₆×C₃₁	—	2	2	no	—
31	C₂×M₁₆×C₃₁	—	2	2	no	—
32	C₂×M₁₆×C₃₁	—	4	4	no	—
33	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
34	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
38	C₄×D₄×C₃₁	—	2	1	no	—
39	C₄×D₄×C₃₁	—	2	1	no	—
40	C₄×D₄×C₃₁	—	2	1	no	—
41	C₄×D₄×C₃₁	—	2	1	no	—
42	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
45	C₃₁×C₄⋊C₈	—	2	1	no	—
46	C₃₁×C₄⋊C₈	—	2	1	no	—
47	C₃₁×C₄⋊C₈	—	2	1	no	—
48	C₃₁×C₄⋊C₈	—	2	1	no	—
49	C₃₁×#100	—	2	4	no	—
50	C₃₁×C₂²⋊C₈	—	2	2	no	—
51	C₃₁×C₂²⋊C₈	—	2	2	no	—
52	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
53	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
54	C₃₁×#97	—	2	1	no	—
55	C₃₁×#97	—	2	1	no	—
56	C₄×C₈×C₃₁	—	2	1	no	—
57	C₄×C₈×C₃₁	—	2	1	no	—
58	C₄×C₈×C₃₁	—	2	1	no	—
59	C₄×C₈×C₃₁	—	2	1	no	—
60	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
61	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
62	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
63	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
64	C₂⁴×C₃₁	C₄²	1	2	yes	—
65	C₂×D₄×C₃₁	—	2	1	no	—
66	C₂×D₄×C₃₁	—	2	1	no	—
67	C₂×D₄×C₃₁	—	2	2	no	—
68	C₂×D₄×C₃₁	—	2	2	no	—
69	C₂²×C₄×C₃₁	—	2	2	no	—
70	C₂²×C₄×C₃₁	—	2	2	no	—
71	C₂²×C₄×C₃₁	—	4	2	no	—
72	C₂²×C₄×C₃₁	—	4	2	no	—
73	C₂²×C₄×C₃₁	C₄²	1	2	yes	—
74	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
77	M₁₆×C₃₁	—	2	2	no	—
78	M₁₆×C₃₁	—	2	2	no	—
79	M₁₆×C₃₁	—	2	2	no	—
80	M₁₆×C₃₁	—	4	4	no	—
81	M₁₆×C₃₁	—	4	4	no	—
82	M₁₆×C₃₁	M₁₆	1	4	yes	complemented
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	—	4	1	no	—
90	C₂×C₈×C₃₁	—	4	2	no	—
91	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
95	C₄⋊C₄×C₃₁	—	2	1	no	—
96	C₄⋊C₄×C₃₁	—	2	1	no	—
97	C₂²⋊C₄×C₃₁	—	2	2	no	—
98	C₂²⋊C₄×C₃₁	—	2	2	no	—
99	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₃₁	#41	1	1	yes	characteristic
102	C₄²×C₃₁	#41	1	1	yes	characteristic
103	#8166	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂³×C₃₁	—	2	2	no	—
105	C₂³×C₃₁	—	2	2	no	—
106	C₂³×C₃₁	—	2	2	no	—
107	C₂³×C₃₁	—	4	2	no	—
108	C₂³×C₃₁	—	4	2	no	—
109	C₂³×C₃₁	C₂×C₄²	1	1	yes	characteristic
110	D₄×C₃₁	—	4	2	no	—
111	D₄×C₃₁	—	4	2	no	—
112	C₂×C₄×C₃₁	—	2	1	no	—
113	C₂×C₄×C₃₁	—	2	1	no	—
114	C₂×C₄×C₃₁	—	2	2	no	—
115	C₂×C₄×C₃₁	—	4	1	no	—
116	C₂×C₄×C₃₁	—	4	4	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	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
122	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
127	C₂×C₄×C₃₁	#131	1	2	yes	—
128	C₈×C₃₁	—	2	1	no	—
129	C₈×C₃₁	—	2	1	no	—
130	C₈×C₃₁	—	2	1	no	—
131	C₈×C₃₁	—	2	4	no	—
132	C₈×C₃₁	—	4	2	no	—
133	C₈×C₃₁	—	4	2	no	—
134	C₈×C₃₁	—	4	2	no	—
135	C₈×C₃₁	—	4	2	no	—
136	C₈×C₃₁	#100	1	2	yes	complemented
137	C₂×#436	C₂×C₃₁	1	1	yes	characteristic
138	C₄×#100	C₂×C₃₁	1	1	yes	characteristic
139	#1460	C₂×C₃₁	1	1	yes	characteristic
140	#1451	C₂×C₃₁	1	1	yes	characteristic, complemented
141	C₈×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
142	#1283	C₂×C₃₁	1	1	yes	characteristic, complemented
143	#1283	C₂×C₃₁	1	1	yes	characteristic, complemented
144	C₂²×C₃₁	—	2	1	no	—
145	C₂²×C₃₁	—	2	1	no	—
146	C₂²×C₃₁	—	4	2	no	—
147	C₂²×C₃₁	—	4	2	no	—
148	C₂²×C₃₁	—	4	2	no	—
149	C₂²×C₃₁	—	8	2	no	—
150	C₂²×C₃₁	—	8	2	no	—
151	C₂²×C₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
152	C₂²×C₃₁	C₄×M₁₆	1	1	yes	characteristic
153	C₂²×C₃₁	#405	1	1	yes	characteristic
154	C₄×C₃₁	—	2	1	no	—
155	C₄×C₃₁	—	2	1	no	—
156	C₄×C₃₁	—	2	1	no	—
157	C₄×C₃₁	—	4	2	no	—
158	C₄×C₃₁	C₂×#100	1	1	yes	characteristic
159	C₄×C₃₁	#413	1	1	yes	characteristic
160	C₄×C₃₁	#435	1	2	yes	—
161	C₄×C₃₁	#436	1	2	yes	—
162	C₂²×M₁₆	C₄×C₃₁	1	2	yes	—
163	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
164	#436	—	2	1	no	—
165	#436	—	2	1	no	—
166	#436	—	2	1	no	—
167	#436	—	2	1	no	—
168	#431	C₂²×C₃₁	1	1	yes	characteristic
169	#431	C₂²×C₃₁	1	1	yes	characteristic
170	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
171	C₂×#100	C₄×C₃₁	1	2	yes	—
172	C₂×C₂²⋊C₈	C₄×C₃₁	1	2	yes	—
173	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
174	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
175	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
176	#343	C₄×C₃₁	1	2	yes	—
177	#323	C₄×C₃₁	1	4	yes	—
178	#322	—	2	1	no	—
179	#322	—	2	1	no	—
180	#322	—	2	1	no	—
181	C₈²	—	2	1	no	—
182	C₂×C₃₁	—	2	1	no	—
183	C₂×C₃₁	—	2	1	no	—
184	C₂×C₃₁	—	8	2	no	—
185	C₂×C₃₁	#1753	1	1	yes	characteristic
186	C₂×C₃₁	C₄×#100	1	1	yes	characteristic
187	C₂×C₃₁	#1757	1	1	yes	characteristic
188	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
189	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	4	4	no	—
193	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
194	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
195	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
196	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
197	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
198	C₄×D₄	—	2	1	no	—
199	C₄×D₄	—	2	1	no	—
200	C₄×D₄	—	2	1	no	—
201	C₄×D₄	—	2	1	no	—
202	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
203	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
204	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
205	C₄⋊C₈	—	2	1	no	—
206	C₄⋊C₈	—	2	1	no	—
207	C₄⋊C₈	—	2	1	no	—
208	C₄⋊C₈	—	2	1	no	—
209	#100	—	2	4	no	—
210	C₂²⋊C₈	—	2	2	no	—
211	C₂²⋊C₈	—	2	2	no	—
212	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
213	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
214	#97	—	2	1	no	—
215	#97	—	2	1	no	—
216	C₄×C₈	—	2	1	no	—
217	C₄×C₈	—	2	1	no	—
218	C₄×C₈	—	2	1	no	—
219	C₄×C₈	—	2	1	no	—
220	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
221	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
222	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
223	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
224	C₃₁	#8166	1	1	yes	characteristic, direct product factor, Sylow
225	C₂⁴	C₄²×C₃₁	1	2	yes	—
226	C₂×D₄	—	2	1	no	—
227	C₂×D₄	—	2	1	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	4	2	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	C₄²×C₃₁	1	2	yes	—
235	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
236	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
237	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
238	M₁₆	—	2	2	no	—
239	M₁₆	—	2	2	no	—
240	M₁₆	—	2	2	no	—
241	M₁₆	—	4	4	no	—
242	M₁₆	—	4	4	no	—
243	M₁₆	M₁₆×C₃₁	1	4	yes	complemented
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	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	4	1	no	—
251	C₂×C₈	—	4	2	no	—
252	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
253	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
254	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
255	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	2	1	no	—
258	C₂²⋊C₄	—	2	2	no	—
259	C₂²⋊C₄	—	2	2	no	—
260	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
261	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
262	C₄²	#41×C₃₁	1	1	yes	characteristic
263	C₄²	#41×C₃₁	1	1	yes	characteristic
264	C₂³	—	2	2	no	—
265	C₂³	—	2	2	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	4	2	no	—
268	C₂³	—	4	2	no	—
269	C₂³	C₂×C₄²×C₃₁	1	1	yes	characteristic
270	D₄	—	4	2	no	—
271	D₄	—	4	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
281	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
282	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
283	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
284	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
285	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
286	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
287	C₂×C₄	C₃₁×#131	1	2	yes	—
288	C₈	—	2	1	no	—
289	C₈	—	2	1	no	—
290	C₈	—	2	1	no	—
291	C₈	—	2	4	no	—
292	C₈	—	4	2	no	—
293	C₈	—	4	2	no	—
294	C₈	—	4	2	no	—
295	C₈	—	4	2	no	—
296	C₈	C₃₁×#100	1	2	yes	complemented
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	4	2	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	8	2	no	—
303	C₂²	—	8	2	no	—
304	C₂²	C₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
305	C₂²	C₄×M₁₆×C₃₁	1	1	yes	characteristic
306	C₂²	C₃₁×#405	1	1	yes	characteristic
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	4	2	no	—
311	C₄	C₂×C₃₁×#100	1	1	yes	characteristic
312	C₄	C₃₁×#413	1	1	yes	characteristic
313	C₄	C₃₁×#435	1	2	yes	—
314	C₄	C₃₁×#436	1	2	yes	—
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	8	2	no	—
318	C₂	C₃₁×#1753	1	1	yes	characteristic
319	C₂	C₄×C₃₁×#100	1	1	yes	characteristic
320	C₂	C₃₁×#1757	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×C₂²⋊C₄
commutator	C₂³	C₂×C₄²×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	C₄×C₂²⋊C₄
prime radical	C₂²×D₄×C₃₁	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1451	C₂	semidirect
5. C₈×M₁₆×C₃₁	C₂	semidirect
6. C₃₁×#1283	C₂	semidirect
7. C₃₁×#1283	C₂	semidirect
82. M₁₆×C₃₁	M₁₆	semidirect
103. #8166	C₃₁	direct (N × H)
136. C₈×C₃₁	#100	semidirect
140. #1451	C₂×C₃₁	semidirect
141. C₈×M₁₆	C₂×C₃₁	semidirect
142. #1283	C₂×C₃₁	semidirect
143. #1283	C₂×C₃₁	semidirect
224. C₃₁	#8166	direct (N × H)
243. M₁₆	M₁₆×C₃₁	semidirect
296. C₈	C₃₁×#100	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#436	2
C₄×C₃₁×#100	2
C₃₁×#1460	2
C₃₁×#1451	2
C₈×M₁₆×C₃₁	2
C₃₁×#1283	2
#8166	31

Subgroups — C31×#8166 #21464925

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8166 #21464925