Subgroups — C₃₁×#8830 #21465589

Order 7936.

556 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1797	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₁×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#334	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1277	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈×C₃₁	—	2	1	no	—
7	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
9	C₂×C₃₁×C₄⋊C₈	—	2	4	no	—
10	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₃₁×C₂²⋊C₈	—	2	1	no	—
14	C₂×C₃₁×C₂²⋊C₈	C₄	1	1	yes	characteristic
15	C₂×C₃₁×C₂²⋊C₈	C₄	1	1	yes	characteristic
16	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
17	C₃₁×#336	—	2	1	no	—
18	C₃₁×#336	C₄	1	2	yes	—
19	C₃₁×#334	—	2	4	no	—
20	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
22	C₂³×C₄×C₃₁	Q₈	1	1	yes	characteristic
23	C₂²×C₈×C₃₁	—	2	4	no	—
24	C₂²×C₈×C₃₁	—	2	2	no	—
25	C₂²×C₈×C₃₁	—	2	1	no	—
26	C₂²×C₈×C₃₁	—	2	1	no	—
27	C₂²×C₈×C₃₁	—	4	2	no	—
28	C₂²×C₈×C₃₁	—	4	2	no	—
29	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₃₁	—	2	2	no	—
32	C₂×C₄²×C₃₁	—	2	2	no	—
33	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₃₁	Q₈	1	1	yes	characteristic
36	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
37	C₃₁×C₄⋊C₈	—	2	4	no	—
38	C₃₁×C₄⋊C₈	—	2	2	no	—
39	C₃₁×C₄⋊C₈	—	2	2	no	—
40	C₃₁×C₄⋊C₈	—	2	2	no	—
41	C₃₁×C₄⋊C₈	D₄	1	4	yes	—
42	C₃₁×C₂²⋊C₈	—	2	2	no	—
43	C₃₁×C₂²⋊C₈	—	4	2	no	—
44	C₃₁×C₂²⋊C₈	C₈	1	4	yes	complemented
45	C₄×C₈×C₃₁	—	2	2	no	—
46	C₄×C₈×C₃₁	—	2	2	no	—
47	C₂⁴×C₃₁	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₃₁	—	2	2	no	—
49	C₂²×C₄×C₃₁	—	2	1	no	—
50	C₂²×C₄×C₃₁	—	2	1	no	—
51	C₂²×C₄×C₃₁	—	2	2	no	—
52	C₂²×C₄×C₃₁	—	2	1	no	—
53	C₂²×C₄×C₃₁	—	2	1	no	—
54	C₂²×C₄×C₃₁	—	2	2	no	—
55	C₂²×C₄×C₃₁	—	2	2	no	—
56	C₂²×C₄×C₃₁	—	4	1	no	—
57	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₃₁	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₃₁	M₁₆	1	2	yes	—
62	C₂²×C₄×C₃₁	SD₁₆	1	2	yes	—
63	C₂²×C₄×C₃₁	SD₁₆	1	2	yes	—
64	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₈×C₃₁	—	2	2	no	—
68	C₂×C₈×C₃₁	—	2	2	no	—
69	C₂×C₈×C₃₁	—	2	2	no	—
70	C₂×C₈×C₃₁	—	2	4	no	—
71	C₂×C₈×C₃₁	—	2	2	no	—
72	C₂×C₈×C₃₁	—	2	2	no	—
73	C₂×C₈×C₃₁	—	4	8	no	—
74	C₂×C₈×C₃₁	—	4	2	no	—
75	C₂×C₈×C₃₁	—	4	2	no	—
76	C₂×C₈×C₃₁	—	4	1	no	—
77	C₂×C₈×C₃₁	—	4	1	no	—
78	C₂×C₈×C₃₁	—	4	1	no	—
79	C₂×C₈×C₃₁	—	4	1	no	—
80	C₂×C₈×C₃₁	—	4	1	no	—
81	C₂×C₈×C₃₁	—	4	1	no	—
82	C₂×C₈×C₃₁	C₂×C₈	1	4	yes	complemented
83	C₄²×C₃₁	—	2	2	no	—
84	C₄²×C₃₁	—	2	1	no	—
85	C₄²×C₃₁	—	2	2	no	—
86	C₄²×C₃₁	—	2	1	no	—
87	C₄²×C₃₁	C₂×D₄	1	2	yes	—
88	C₄²×C₃₁	#41	1	2	yes	—
89	#8830	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₃₁	—	2	1	no	—
91	C₂³×C₃₁	—	2	1	no	—
92	C₂³×C₃₁	—	2	1	no	—
93	C₂³×C₃₁	—	2	1	no	—
94	C₂³×C₃₁	C₄⋊C₈	1	2	yes	—
95	C₂³×C₃₁	#106	1	2	yes	—
96	C₂³×C₃₁	#108	1	2	yes	—
97	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₃₁	—	2	1	no	—
99	C₂×C₄×C₃₁	—	2	1	no	—
100	C₂×C₄×C₃₁	—	2	1	no	—
101	C₂×C₄×C₃₁	—	2	1	no	—
102	C₂×C₄×C₃₁	—	2	1	no	—
103	C₂×C₄×C₃₁	—	2	1	no	—
104	C₂×C₄×C₃₁	—	2	4	no	—
105	C₂×C₄×C₃₁	—	2	4	no	—
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	2	no	—
110	C₂×C₄×C₃₁	—	2	1	no	—
111	C₂×C₄×C₃₁	—	2	1	no	—
112	C₂×C₄×C₃₁	—	2	1	no	—
113	C₂×C₄×C₃₁	—	4	4	no	—
114	C₂×C₄×C₃₁	C₄⋊C₈	1	2	yes	—
115	C₂×C₄×C₃₁	#106	1	2	yes	—
116	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
118	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
120	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
122	C₂×C₄×C₃₁	#131	1	2	yes	—
123	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₂×C₄×C₃₁	#137	1	1	yes	characteristic
127	C₈×C₃₁	—	2	4	no	—
128	C₈×C₃₁	—	4	2	no	—
129	C₈×C₃₁	—	4	4	no	—
130	C₈×C₃₁	—	8	1	no	—
131	C₈×C₃₁	—	8	1	no	—
132	C₂²×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic
133	#1797	C₂×C₃₁	1	1	yes	characteristic
134	C₄×C₂²⋊C₈	C₂×C₃₁	1	1	yes	characteristic
135	C₂×#334	C₂×C₃₁	1	2	yes	complemented
136	#1277	C₂×C₃₁	1	2	yes	complemented
137	C₂²×C₃₁	—	2	2	no	—
138	C₂²×C₃₁	—	2	2	no	—
139	C₂²×C₃₁	—	2	2	no	—
140	C₂²×C₃₁	—	2	2	no	—
141	C₂²×C₃₁	—	2	2	no	—
142	C₂²×C₃₁	—	2	2	no	—
143	C₂²×C₃₁	#334	1	4	yes	complemented
144	C₂²×C₃₁	#385	1	1	yes	characteristic
145	C₂²×C₃₁	C₂×C₄⋊C₈	1	1	yes	characteristic
146	C₂²×C₃₁	#424	1	1	yes	characteristic
147	C₂²×C₃₁	C₂×#106	1	1	yes	characteristic
148	C₂²×C₃₁	#428	1	1	yes	characteristic
149	C₂²×C₃₁	C₂×#108	1	1	yes	characteristic
150	C₂²×C₃₁	#430	1	1	yes	characteristic
151	C₄×C₃₁	—	2	2	no	—
152	C₄×C₃₁	—	2	2	no	—
153	C₄×C₃₁	—	2	2	no	—
154	C₄×C₃₁	—	4	1	no	—
155	C₄×C₃₁	C₈×D₄	1	2	yes	—
156	C₄×C₃₁	#435	1	2	yes	—
157	C₄×C₃₁	#448	1	1	yes	characteristic
158	C₄×C₃₁	#450	1	1	yes	characteristic
159	C₄×C₃₁	#481	1	1	yes	characteristic
160	C₄×C₃₁	#483	1	1	yes	characteristic
161	C₂³×C₈	—	2	1	no	—
162	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
163	C₂×C₄⋊C₈	—	2	1	no	—
164	C₂×C₄⋊C₈	—	2	4	no	—
165	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
166	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
167	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
168	C₂×C₂²⋊C₈	—	2	1	no	—
169	C₂×C₂²⋊C₈	C₄×C₃₁	1	1	yes	characteristic
170	C₂×C₂²⋊C₈	C₄×C₃₁	1	1	yes	characteristic
171	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
172	#336	—	2	1	no	—
173	#336	C₄×C₃₁	1	2	yes	—
174	#334	—	2	4	no	—
175	C₂×C₃₁	—	2	4	no	—
176	C₂×C₃₁	C₂×#334	1	1	yes	characteristic
177	C₂×C₃₁	#1565	1	1	yes	characteristic
178	C₂×C₃₁	#1570	1	1	yes	characteristic
179	C₂×C₃₁	#1573	1	1	yes	characteristic
180	C₂×C₃₁	#1850	1	1	yes	characteristic
181	C₂×C₃₁	#1853	1	1	yes	characteristic
182	C₂×C₃₁	#1855	1	1	yes	characteristic
183	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
184	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
185	C₂³×C₄	Q₈×C₃₁	1	1	yes	characteristic
186	C₂²×C₈	—	2	4	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	—	4	2	no	—
191	C₂²×C₈	—	4	2	no	—
192	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
193	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
197	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
198	C₂×C₄²	Q₈×C₃₁	1	1	yes	characteristic
199	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
200	C₄⋊C₈	—	2	4	no	—
201	C₄⋊C₈	—	2	2	no	—
202	C₄⋊C₈	—	2	2	no	—
203	C₄⋊C₈	—	2	2	no	—
204	C₄⋊C₈	D₄×C₃₁	1	4	yes	—
205	C₂²⋊C₈	—	2	2	no	—
206	C₂²⋊C₈	—	4	2	no	—
207	C₂²⋊C₈	C₈×C₃₁	1	4	yes	complemented
208	C₄×C₈	—	2	2	no	—
209	C₄×C₈	—	2	2	no	—
210	C₃₁	#8830	1	1	yes	characteristic, direct product factor, Sylow
211	C₂⁴	C₄⋊C₄×C₃₁	1	1	yes	characteristic
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	4	1	no	—
221	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
222	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
223	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₈×C₃₁	1	2	yes	—
225	C₂²×C₄	M₁₆×C₃₁	1	2	yes	—
226	C₂²×C₄	SD₁₆×C₃₁	1	2	yes	—
227	C₂²×C₄	SD₁₆×C₃₁	1	2	yes	—
228	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
230	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	4	8	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	1	no	—
241	C₂×C₈	—	4	1	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	—	4	1	no	—
244	C₂×C₈	—	4	1	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	C₂×C₈×C₃₁	1	4	yes	complemented
247	C₄²	—	2	2	no	—
248	C₄²	—	2	1	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	1	no	—
251	C₄²	C₂×D₄×C₃₁	1	2	yes	—
252	C₄²	#41×C₃₁	1	2	yes	—
253	C₂³	—	2	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	—	2	1	no	—
256	C₂³	—	2	1	no	—
257	C₂³	C₃₁×C₄⋊C₈	1	2	yes	—
258	C₂³	C₃₁×#106	1	2	yes	—
259	C₂³	C₃₁×#108	1	2	yes	—
260	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	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	—	2	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	C₃₁×C₄⋊C₈	1	2	yes	—
278	C₂×C₄	C₃₁×#106	1	2	yes	—
279	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
280	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
281	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
282	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
283	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
284	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
285	C₂×C₄	C₃₁×#131	1	2	yes	—
286	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
287	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
288	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
289	C₂×C₄	C₃₁×#137	1	1	yes	characteristic
290	C₈	—	2	4	no	—
291	C₈	—	4	2	no	—
292	C₈	—	4	4	no	—
293	C₈	—	8	1	no	—
294	C₈	—	8	1	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	C₃₁×#334	1	4	yes	complemented
302	C₂²	C₃₁×#385	1	1	yes	characteristic
303	C₂²	C₂×C₃₁×C₄⋊C₈	1	1	yes	characteristic
304	C₂²	C₃₁×#424	1	1	yes	characteristic
305	C₂²	C₂×C₃₁×#106	1	1	yes	characteristic
306	C₂²	C₃₁×#428	1	1	yes	characteristic
307	C₂²	C₂×C₃₁×#108	1	1	yes	characteristic
308	C₂²	C₃₁×#430	1	1	yes	characteristic
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	—	4	1	no	—
313	C₄	C₈×D₄×C₃₁	1	2	yes	—
314	C₄	C₃₁×#435	1	2	yes	—
315	C₄	C₃₁×#448	1	1	yes	characteristic
316	C₄	C₃₁×#450	1	1	yes	characteristic
317	C₄	C₃₁×#481	1	1	yes	characteristic
318	C₄	C₃₁×#483	1	1	yes	characteristic
319	C₂	—	2	4	no	—
320	C₂	C₂×C₃₁×#334	1	1	yes	characteristic
321	C₂	C₃₁×#1565	1	1	yes	characteristic
322	C₂	C₃₁×#1570	1	1	yes	characteristic
323	C₂	C₃₁×#1573	1	1	yes	characteristic
324	C₂	C₃₁×#1850	1	1	yes	characteristic
325	C₂	C₃₁×#1853	1	1	yes	characteristic
326	C₂	C₃₁×#1855	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂³×C₃₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₃₁	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₃₁×#334	C₂	semidirect
5. C₃₁×#1277	C₂	semidirect
44. C₃₁×C₂²⋊C₈	C₈	semidirect
82. C₂×C₈×C₃₁	C₂×C₈	semidirect
89. #8830	C₃₁	direct (N × H)
135. C₂×#334	C₂×C₃₁	semidirect
136. #1277	C₂×C₃₁	semidirect
143. C₂²×C₃₁	#334	semidirect
207. C₂²⋊C₈	C₈×C₃₁	semidirect
210. C₃₁	#8830	direct (N × H)
246. C₂×C₈	C₂×C₈×C₃₁	semidirect
301. C₂²	C₃₁×#334	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×C₄⋊C₈	2
C₃₁×#1797	2
C₄×C₃₁×C₂²⋊C₈	2
C₂×C₃₁×#334	2
C₃₁×#1277	2
#8830	31

Subgroups — C31×#8830 #21465589

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8830 #21465589