Subgroups — C₂×C₃₁×#2083 #21480412

Order 7936.

1086 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#107	C₂	1	2	yes	maximal
2	C₂×C₃₁×#389	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#384	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2083	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃₁×#340	C₂	1	2	yes	maximal
6	C₂×C₃₁×#336	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
8	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
9	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
10	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
11	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
12	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#107	—	2	8	no	—
16	C₂×C₃₁×#107	C₂²	1	8	yes	—
17	C₃₁×#389	—	2	2	no	—
18	C₃₁×#389	C₂²	1	4	yes	—
19	C₃₁×#384	—	2	2	no	—
20	C₃₁×#384	C₂²	1	4	yes	—
21	C₂×C₃₁×#95	—	2	1	no	—
22	C₃₁×#340	C₂²	1	8	yes	—
23	C₃₁×#336	C₂²	1	4	yes	—
24	C₂³×C₄×C₃₁	—	2	1	no	—
25	C₂³×C₄×C₃₁	—	2	2	no	—
26	C₂³×C₄×C₃₁	—	2	1	no	—
27	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₃₁	—	2	4	no	—
31	C₂²×C₈×C₃₁	—	2	4	no	—
32	C₂²×C₈×C₃₁	Q₈	1	8	yes	—
33	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
34	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
35	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
37	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
38	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
39	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
40	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
41	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
44	C₂×C₄⋊C₄×C₃₁	—	4	4	no	—
45	C₂×C₄⋊C₄×C₃₁	—	4	4	no	—
46	C₂×C₄⋊C₄×C₃₁	—	4	4	no	—
47	C₂×C₄⋊C₄×C₃₁	D₄	1	4	yes	—
48	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
49	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
51	C₂×C₄²×C₃₁	—	2	4	no	—
52	C₂×C₄²×C₃₁	D₄	1	2	yes	—
53	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
54	C₃₁×#107	—	2	16	no	—
55	C₃₁×#95	—	2	4	no	—
56	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	—	2	2	no	—
58	C₂²×C₄×C₃₁	—	2	2	no	—
59	C₂²×C₄×C₃₁	—	2	2	no	—
60	C₂²×C₄×C₃₁	—	2	2	no	—
61	C₂²×C₄×C₃₁	—	2	4	no	—
62	C₂²×C₄×C₃₁	—	2	2	no	—
63	C₂²×C₄×C₃₁	—	2	2	no	—
64	C₂²×C₄×C₃₁	—	2	4	no	—
65	C₂²×C₄×C₃₁	—	4	2	no	—
66	C₂²×C₄×C₃₁	—	4	2	no	—
67	C₂²×C₄×C₃₁	—	4	4	no	—
68	C₂²×C₄×C₃₁	—	4	2	no	—
69	C₂²×C₄×C₃₁	—	4	4	no	—
70	C₂²×C₄×C₃₁	—	4	2	no	—
71	C₂²×C₄×C₃₁	—	4	4	no	—
72	C₂²×C₄×C₃₁	—	4	2	no	—
73	C₂²×C₄×C₃₁	D₈	1	4	yes	—
74	C₂²×C₄×C₃₁	Q₁₆	1	4	yes	—
75	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
80	C₂²×C₄×C₃₁	#41	1	2	yes	—
81	C₂²×C₄×C₃₁	#41	1	4	yes	—
82	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
83	C₂×C₈×C₃₁	—	2	16	no	—
84	C₂×C₈×C₃₁	—	2	4	no	—
85	C₂×C₈×C₃₁	—	2	4	no	—
86	C₂×C₈×C₃₁	C₂×Q₈	1	8	yes	—
87	C₄⋊C₄×C₃₁	—	2	4	no	—
88	C₄⋊C₄×C₃₁	—	2	2	no	—
89	C₄⋊C₄×C₃₁	—	2	4	no	—
90	C₄⋊C₄×C₃₁	—	2	4	no	—
91	C₄⋊C₄×C₃₁	—	2	4	no	—
92	C₄⋊C₄×C₃₁	—	4	8	no	—
93	C₄⋊C₄×C₃₁	—	4	4	no	—
94	C₄⋊C₄×C₃₁	—	4	2	no	—
95	C₄⋊C₄×C₃₁	—	4	8	no	—
96	C₄⋊C₄×C₃₁	—	4	8	no	—
97	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
98	C₄²×C₃₁	—	2	4	no	—
99	C₄²×C₃₁	—	2	4	no	—
100	C₂×#2083	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂³×C₃₁	#121	1	2	yes	—
102	C₂³×C₃₁	#123	1	2	yes	—
103	C₂³×C₃₁	C₄⋊Q₈	1	2	yes	—
104	C₂³×C₃₁	C₂×D₈	1	2	yes	—
105	C₂³×C₃₁	C₂×Q₁₆	1	2	yes	—
106	C₂³×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
107	C₂³×C₃₁	#137	1	2	yes	—
108	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₃₁	—	2	2	no	—
110	C₂×C₄×C₃₁	—	2	2	no	—
111	C₂×C₄×C₃₁	—	2	2	no	—
112	C₂×C₄×C₃₁	—	2	2	no	—
113	C₂×C₄×C₃₁	—	2	8	no	—
114	C₂×C₄×C₃₁	—	2	8	no	—
115	C₂×C₄×C₃₁	—	2	4	no	—
116	C₂×C₄×C₃₁	—	4	2	no	—
117	C₂×C₄×C₃₁	—	4	2	no	—
118	C₂×C₄×C₃₁	—	4	8	no	—
119	C₂×C₄×C₃₁	—	4	4	no	—
120	C₂×C₄×C₃₁	—	4	2	no	—
121	C₂×C₄×C₃₁	—	4	2	no	—
122	C₂×C₄×C₃₁	—	4	4	no	—
123	C₂×C₄×C₃₁	—	4	4	no	—
124	C₂×C₄×C₃₁	—	4	2	no	—
125	C₂×C₄×C₃₁	—	8	8	no	—
126	C₂×C₄×C₃₁	—	8	4	no	—
127	C₂×C₄×C₃₁	#121	1	2	yes	—
128	C₂×C₄×C₃₁	C₂²⋊Q₈	1	8	yes	—
129	C₂×C₄×C₃₁	#123	1	2	yes	—
130	C₂×C₄×C₃₁	#125	1	8	yes	—
131	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
132	C₂×C₄×C₃₁	C₂×D₈	1	4	yes	—
133	C₂×C₄×C₃₁	C₂×Q₁₆	1	4	yes	—
134	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
136	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
137	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
138	C₈×C₃₁	—	2	8	no	—
139	C₂²×#107	C₂×C₃₁	1	2	yes	—
140	C₂×#389	C₂×C₃₁	1	1	yes	characteristic
141	C₂×#384	C₂×C₃₁	1	1	yes	characteristic
142	#2083	C₂×C₃₁	1	8	yes	direct product factor
143	C₂×#340	C₂×C₃₁	1	2	yes	—
144	C₂×#336	C₂×C₃₁	1	1	yes	characteristic
145	C₂²×C₃₁	#398	1	4	yes	—
146	C₂²×C₃₁	#459	1	4	yes	—
147	C₂²×C₃₁	#462	1	4	yes	—
148	C₂²×C₃₁	#480	1	4	yes	—
149	C₂²×C₃₁	#484	1	4	yes	—
150	C₂²×C₃₁	#500	1	4	yes	—
151	C₂²×C₃₁	#501	1	4	yes	—
152	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
153	C₂²×C₃₁	C₂×#123	1	1	yes	characteristic
154	C₂²×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
155	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
156	C₂²×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
157	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
158	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
159	C₄×C₃₁	—	2	4	no	—
160	C₄×C₃₁	—	2	4	no	—
161	C₄×C₃₁	—	4	4	no	—
162	C₄×C₃₁	—	8	2	no	—
163	C₄×C₃₁	—	8	4	no	—
164	C₄×C₃₁	#398	1	4	yes	—
165	C₄×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
166	C₄×C₃₁	C₂×#125	1	2	yes	—
167	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
168	C₂²×C₄⋊C₄	—	2	1	no	—
169	C₂²×C₄⋊C₄	—	2	1	no	—
170	C₂²×C₄⋊C₄	—	2	1	no	—
171	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
172	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
173	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
174	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
175	C₂×#107	—	2	8	no	—
176	C₂×#107	C₂²×C₃₁	1	8	yes	—
177	#389	—	2	2	no	—
178	#389	C₂²×C₃₁	1	4	yes	—
179	#384	—	2	2	no	—
180	#384	C₂²×C₃₁	1	4	yes	—
181	C₂×#95	—	2	1	no	—
182	#340	C₂²×C₃₁	1	8	yes	—
183	#336	C₂²×C₃₁	1	4	yes	—
184	C₂×C₃₁	#2083	1	8	yes	direct product factor
185	C₂×C₃₁	C₂×#398	1	1	yes	characteristic
186	C₂×C₃₁	C₂×#459	1	1	yes	characteristic
187	C₂×C₃₁	C₂×#462	1	1	yes	characteristic
188	C₂×C₃₁	C₂×#480	1	1	yes	characteristic
189	C₂×C₃₁	C₂×#484	1	1	yes	characteristic
190	C₂×C₃₁	C₂×#500	1	1	yes	characteristic
191	C₂×C₃₁	C₂×#501	1	1	yes	characteristic
192	C₂³×C₄	—	2	1	no	—
193	C₂³×C₄	—	2	2	no	—
194	C₂³×C₄	—	2	1	no	—
195	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
196	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
197	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
198	C₂²×C₈	—	2	4	no	—
199	C₂²×C₈	—	2	4	no	—
200	C₂²×C₈	Q₈×C₃₁	1	8	yes	—
201	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
202	C₂×C₄⋊C₄	—	2	1	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	4	no	—
205	C₂×C₄⋊C₄	—	2	4	no	—
206	C₂×C₄⋊C₄	—	2	4	no	—
207	C₂×C₄⋊C₄	—	2	4	no	—
208	C₂×C₄⋊C₄	—	2	4	no	—
209	C₂×C₄⋊C₄	—	2	4	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	—	2	1	no	—
212	C₂×C₄⋊C₄	—	4	4	no	—
213	C₂×C₄⋊C₄	—	4	4	no	—
214	C₂×C₄⋊C₄	—	4	4	no	—
215	C₂×C₄⋊C₄	D₄×C₃₁	1	4	yes	—
216	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
217	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
218	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
219	C₂×C₄²	—	2	4	no	—
220	C₂×C₄²	D₄×C₃₁	1	2	yes	—
221	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
222	#107	—	2	16	no	—
223	#95	—	2	4	no	—
224	C₃₁	C₂×#2083	1	1	yes	characteristic, direct product factor, Sylow
225	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	—	4	2	no	—
236	C₂²×C₄	—	4	4	no	—
237	C₂²×C₄	—	4	2	no	—
238	C₂²×C₄	—	4	4	no	—
239	C₂²×C₄	—	4	2	no	—
240	C₂²×C₄	—	4	4	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	D₈×C₃₁	1	4	yes	—
243	C₂²×C₄	Q₁₆×C₃₁	1	4	yes	—
244	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
245	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
246	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
247	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
248	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
249	C₂²×C₄	#41×C₃₁	1	2	yes	—
250	C₂²×C₄	#41×C₃₁	1	4	yes	—
251	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
252	C₂×C₈	—	2	16	no	—
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	—	2	4	no	—
255	C₂×C₈	C₂×Q₈×C₃₁	1	8	yes	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	4	8	no	—
262	C₄⋊C₄	—	4	4	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	8	no	—
265	C₄⋊C₄	—	4	8	no	—
266	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
267	C₄²	—	2	4	no	—
268	C₄²	—	2	4	no	—
269	C₂³	C₃₁×#121	1	2	yes	—
270	C₂³	C₃₁×#123	1	2	yes	—
271	C₂³	C₃₁×C₄⋊Q₈	1	2	yes	—
272	C₂³	C₂×D₈×C₃₁	1	2	yes	—
273	C₂³	C₂×Q₁₆×C₃₁	1	2	yes	—
274	C₂³	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
275	C₂³	C₃₁×#137	1	2	yes	—
276	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	—	2	8	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	4	8	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	8	8	no	—
294	C₂×C₄	—	8	4	no	—
295	C₂×C₄	C₃₁×#121	1	2	yes	—
296	C₂×C₄	C₃₁×C₂²⋊Q₈	1	8	yes	—
297	C₂×C₄	C₃₁×#123	1	2	yes	—
298	C₂×C₄	C₃₁×#125	1	8	yes	—
299	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
300	C₂×C₄	C₂×D₈×C₃₁	1	4	yes	—
301	C₂×C₄	C₂×Q₁₆×C₃₁	1	4	yes	—
302	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
303	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
304	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
305	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
306	C₈	—	2	8	no	—
307	C₂²	C₃₁×#398	1	4	yes	—
308	C₂²	C₃₁×#459	1	4	yes	—
309	C₂²	C₃₁×#462	1	4	yes	—
310	C₂²	C₃₁×#480	1	4	yes	—
311	C₂²	C₃₁×#484	1	4	yes	—
312	C₂²	C₃₁×#500	1	4	yes	—
313	C₂²	C₃₁×#501	1	4	yes	—
314	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
315	C₂²	C₂×C₃₁×#123	1	1	yes	characteristic
316	C₂²	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
317	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
318	C₂²	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
319	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
320	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
321	C₄	—	2	4	no	—
322	C₄	—	2	4	no	—
323	C₄	—	4	4	no	—
324	C₄	—	8	2	no	—
325	C₄	—	8	4	no	—
326	C₄	C₃₁×#398	1	4	yes	—
327	C₄	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
328	C₄	C₂×C₃₁×#125	1	2	yes	—
329	C₂	C₃₁×#2083	1	8	yes	direct product factor
330	C₂	C₂×C₃₁×#398	1	1	yes	characteristic
331	C₂	C₂×C₃₁×#459	1	1	yes	characteristic
332	C₂	C₂×C₃₁×#462	1	1	yes	characteristic
333	C₂	C₂×C₃₁×#480	1	1	yes	characteristic
334	C₂	C₂×C₃₁×#484	1	1	yes	characteristic
335	C₂	C₂×C₃₁×#500	1	1	yes	characteristic
336	C₂	C₂×C₃₁×#501	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₃₁×#2083	C₂	direct (N × H)
100. C₂×#2083	C₃₁	direct (N × H)
142. #2083	C₂×C₃₁	direct (N × H)
184. C₂×C₃₁	#2083	direct (N × H)
224. C₃₁	C₂×#2083	direct (N × H)
329. C₂	C₃₁×#2083	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×#107	2
C₂×C₃₁×#389	2
C₂×C₃₁×#384	2
C₃₁×#2083	2
C₂×C₃₁×#340	2
C₂×C₃₁×#336	2
C₂×#2083	31

Subgroups — C2×C31×#2083 #21480412

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2083 #21480412