Subgroups — C₃₁×#9783 #21466542

Order 7936.

606 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#466	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1964	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1808	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1519	C₂	1	2	yes	maximal, complemented
6	C₂²×D₈×C₃₁	C₄	1	1	yes	characteristic, complemented
7	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₃₁×#121	C₂²	1	2	yes	—
9	C₃₁×#466	—	2	1	no	—
10	C₃₁×#466	—	2	1	no	—
11	C₃₁×#466	—	2	1	no	—
12	C₃₁×#466	—	2	1	no	—
13	C₂×C₃₁×#107	—	2	1	no	—
14	C₂×C₃₁×#107	C₄	1	1	yes	characteristic, complemented
15	C₂×C₃₁×#102	C₄	1	2	yes	complemented
16	C₂×C₃₁×#99	—	2	4	no	—
17	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₃₁	—	2	1	no	—
19	C₃₁×#380	C₂²	1	2	yes	—
20	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#340	—	2	1	no	—
22	C₃₁×#336	—	2	1	no	—
23	C₃₁×#327	—	2	4	no	—
24	C₂²×D₄×C₃₁	—	2	2	no	—
25	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
27	C₂×D₈×C₃₁	—	2	1	no	—
28	C₂×D₈×C₃₁	—	2	1	no	—
29	C₂×D₈×C₃₁	—	4	2	no	—
30	C₂²×C₈×C₃₁	—	2	1	no	—
31	C₂²×C₈×C₃₁	—	4	2	no	—
32	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₃₁	C₂×C₄	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	1	yes	characteristic
37	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
38	C₃₁×#121	—	2	2	no	—
39	C₃₁×#121	—	2	2	no	—
40	C₃₁×#121	—	2	2	no	—
41	C₃₁×#121	—	2	2	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
44	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
45	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
46	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
47	C₂×C₄²×C₃₁	—	2	1	no	—
48	C₃₁×#107	—	2	1	no	—
49	C₃₁×#107	—	2	1	no	—
50	C₃₁×#107	—	4	2	no	—
51	C₃₁×#102	—	2	2	no	—
52	C₃₁×#102	—	2	2	no	—
53	C₃₁×#99	—	4	8	no	—
54	C₃₁×C₂²⋊C₈	—	2	1	no	—
55	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
56	C₄×C₈×C₃₁	—	4	2	no	—
57	C₃₁×#95	—	2	1	no	—
58	C₃₁×#95	—	2	2	no	—
59	C₂⁴×C₃₁	—	2	2	no	—
60	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
61	C₂×D₄×C₃₁	—	2	2	no	—
62	C₂×D₄×C₃₁	—	2	2	no	—
63	C₂×D₄×C₃₁	—	4	2	no	—
64	C₂×D₄×C₃₁	—	4	2	no	—
65	C₂×D₄×C₃₁	—	4	2	no	—
66	C₂×D₄×C₃₁	—	4	2	no	—
67	C₂×D₄×C₃₁	—	4	4	no	—
68	C₂×D₄×C₃₁	—	4	4	no	—
69	C₂²×C₄×C₃₁	—	2	2	no	—
70	C₂²×C₄×C₃₁	—	2	2	no	—
71	C₂²×C₄×C₃₁	—	4	1	no	—
72	C₂²×C₄×C₃₁	—	4	1	no	—
73	C₂²×C₄×C₃₁	—	4	2	no	—
74	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
79	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
80	D₈×C₃₁	—	4	2	no	—
81	D₈×C₃₁	—	4	2	no	—
82	C₂×C₈×C₃₁	—	2	2	no	—
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	—	4	1	no	—
87	C₂×C₈×C₃₁	—	4	1	no	—
88	C₂×C₈×C₃₁	—	4	1	no	—
89	C₂×C₈×C₃₁	—	4	1	no	—
90	C₂×C₈×C₃₁	—	4	1	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	2	no	—
96	C₄⋊C₄×C₃₁	—	2	2	no	—
97	C₄⋊C₄×C₃₁	—	4	2	no	—
98	C₄⋊C₄×C₃₁	—	4	2	no	—
99	C₂²⋊C₄×C₃₁	—	2	2	no	—
100	C₂²⋊C₄×C₃₁	—	4	2	no	—
101	C₂²⋊C₄×C₃₁	—	4	2	no	—
102	C₂²⋊C₄×C₃₁	—	4	4	no	—
103	C₂²⋊C₄×C₃₁	D₈	1	4	yes	complemented
104	C₄²×C₃₁	—	4	1	no	—
105	C₄²×C₃₁	—	4	1	no	—
106	#9783	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂³×C₃₁	—	4	1	no	—
108	C₂³×C₃₁	—	4	1	no	—
109	C₂³×C₃₁	—	4	2	no	—
110	C₂³×C₃₁	—	4	4	no	—
111	C₂³×C₃₁	—	8	2	no	—
112	C₂³×C₃₁	—	8	2	no	—
113	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
114	C₂³×C₃₁	#121	1	1	yes	characteristic
115	C₂³×C₃₁	#127	1	1	yes	characteristic
116	C₂³×C₃₁	C₂×D₈	1	2	yes	—
117	C₂³×C₃₁	#135	1	2	yes	—
118	D₄×C₃₁	—	4	2	no	—
119	D₄×C₃₁	—	4	2	no	—
120	D₄×C₃₁	—	8	2	no	—
121	D₄×C₃₁	—	8	4	no	—
122	D₄×C₃₁	—	8	2	no	—
123	D₄×C₃₁	—	8	2	no	—
124	C₂×C₄×C₃₁	—	2	1	no	—
125	C₂×C₄×C₃₁	—	2	1	no	—
126	C₂×C₄×C₃₁	—	2	1	no	—
127	C₂×C₄×C₃₁	—	2	1	no	—
128	C₂×C₄×C₃₁	—	2	1	no	—
129	C₂×C₄×C₃₁	—	2	1	no	—
130	C₂×C₄×C₃₁	—	2	4	no	—
131	C₂×C₄×C₃₁	—	4	1	no	—
132	C₂×C₄×C₃₁	—	4	1	no	—
133	C₂×C₄×C₃₁	—	4	1	no	—
134	C₂×C₄×C₃₁	—	4	2	no	—
135	C₂×C₄×C₃₁	—	4	2	no	—
136	C₂×C₄×C₃₁	—	4	2	no	—
137	C₂×C₄×C₃₁	—	4	4	no	—
138	C₂×C₄×C₃₁	—	8	2	no	—
139	C₂×C₄×C₃₁	—	8	2	no	—
140	C₂×C₄×C₃₁	—	8	2	no	—
141	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
144	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
145	C₈×C₃₁	—	2	1	no	—
146	C₈×C₃₁	—	2	1	no	—
147	C₈×C₃₁	—	2	1	no	—
148	C₈×C₃₁	—	4	2	no	—
149	C₈×C₃₁	#99	1	2	yes	complemented
150	C₂×#466	C₂×C₃₁	1	1	yes	characteristic
151	#1964	C₂×C₃₁	1	2	yes	complemented
152	#1808	C₂×C₃₁	1	1	yes	characteristic, complemented
153	C₈×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
154	#1519	C₂×C₃₁	1	2	yes	complemented
155	C₂²×C₃₁	—	2	1	no	—
156	C₂²×C₃₁	—	2	1	no	—
157	C₂²×C₃₁	—	2	1	no	—
158	C₂²×C₃₁	—	2	1	no	—
159	C₂²×C₃₁	—	2	1	no	—
160	C₂²×C₃₁	—	2	1	no	—
161	C₂²×C₃₁	—	4	2	no	—
162	C₂²×C₃₁	—	4	2	no	—
163	C₂²×C₃₁	—	8	2	no	—
164	C₂²×C₃₁	—	8	2	no	—
165	C₂²×C₃₁	—	8	2	no	—
166	C₂²×C₃₁	—	16	4	no	—
167	C₂²×C₃₁	#390	1	1	yes	characteristic
168	C₂²×C₃₁	C₄×D₈	1	2	yes	—
169	C₂²×C₃₁	#466	1	2	yes	—
170	C₂²×C₃₁	#493	1	1	yes	characteristic
171	C₂²×C₃₁	#495	1	1	yes	characteristic
172	C₄×C₃₁	—	2	1	no	—
173	C₄×C₃₁	—	2	1	no	—
174	C₄×C₃₁	—	2	1	no	—
175	C₄×C₃₁	—	4	4	no	—
176	C₄×C₃₁	—	8	2	no	—
177	C₄×C₃₁	—	16	2	no	—
178	C₄×C₃₁	C₂×#99	1	1	yes	characteristic
179	C₄×C₃₁	#410	1	1	yes	characteristic
180	C₂²×D₈	C₄×C₃₁	1	1	yes	characteristic, complemented
181	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
182	C₂×#121	C₂²×C₃₁	1	2	yes	—
183	#466	—	2	1	no	—
184	#466	—	2	1	no	—
185	#466	—	2	1	no	—
186	#466	—	2	1	no	—
187	C₂×#107	—	2	1	no	—
188	C₂×#107	C₄×C₃₁	1	1	yes	characteristic, complemented
189	C₂×#102	C₄×C₃₁	1	2	yes	complemented
190	C₂×#99	—	2	4	no	—
191	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
192	C₂×C₄×C₈	—	2	1	no	—
193	#380	C₂²×C₃₁	1	2	yes	—
194	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
195	#340	—	2	1	no	—
196	#336	—	2	1	no	—
197	#327	—	2	4	no	—
198	C₂×C₃₁	—	2	2	no	—
199	C₂×C₃₁	—	2	1	no	—
200	C₂×C₃₁	—	2	2	no	—
201	C₂×C₃₁	—	2	1	no	—
202	C₂×C₃₁	—	16	2	no	—
203	C₂×C₃₁	#1964	1	1	yes	characteristic
204	C₂×C₃₁	#1970	1	1	yes	characteristic
205	C₂×C₃₁	#1976	1	1	yes	characteristic
206	C₂²×D₄	—	2	2	no	—
207	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
208	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
209	C₂×D₈	—	2	1	no	—
210	C₂×D₈	—	2	1	no	—
211	C₂×D₈	—	4	2	no	—
212	C₂²×C₈	—	2	1	no	—
213	C₂²×C₈	—	4	2	no	—
214	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
215	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
216	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
217	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
218	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
219	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
220	#121	—	2	2	no	—
221	#121	—	2	2	no	—
222	#121	—	2	2	no	—
223	#121	—	2	2	no	—
224	C₂×C₄⋊C₄	—	2	2	no	—
225	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
226	C₂×C₂²⋊C₄	—	2	2	no	—
227	C₂×C₂²⋊C₄	—	2	2	no	—
228	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
229	C₂×C₄²	—	2	1	no	—
230	#107	—	2	1	no	—
231	#107	—	2	1	no	—
232	#107	—	4	2	no	—
233	#102	—	2	2	no	—
234	#102	—	2	2	no	—
235	#99	—	4	8	no	—
236	C₂²⋊C₈	—	2	1	no	—
237	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
238	C₄×C₈	—	4	2	no	—
239	#95	—	2	1	no	—
240	#95	—	2	2	no	—
241	C₃₁	#9783	1	1	yes	characteristic, direct product factor, Sylow
242	C₂⁴	—	2	2	no	—
243	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	4	2	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂×D₄	—	4	2	no	—
249	C₂×D₄	—	4	2	no	—
250	C₂×D₄	—	4	4	no	—
251	C₂×D₄	—	4	4	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	—	4	2	no	—
257	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
258	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
259	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
260	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
261	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
262	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
263	D₈	—	4	2	no	—
264	D₈	—	4	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	—	4	1	no	—
271	C₂×C₈	—	4	1	no	—
272	C₂×C₈	—	4	1	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
275	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
276	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
277	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	4	2	no	—
281	C₄⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	2	2	no	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₂²⋊C₄	—	4	2	no	—
285	C₂²⋊C₄	—	4	4	no	—
286	C₂²⋊C₄	D₈×C₃₁	1	4	yes	complemented
287	C₄²	—	4	1	no	—
288	C₄²	—	4	1	no	—
289	C₂³	—	4	1	no	—
290	C₂³	—	4	1	no	—
291	C₂³	—	4	2	no	—
292	C₂³	—	4	4	no	—
293	C₂³	—	8	2	no	—
294	C₂³	—	8	2	no	—
295	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
296	C₂³	C₃₁×#121	1	1	yes	characteristic
297	C₂³	C₃₁×#127	1	1	yes	characteristic
298	C₂³	C₂×D₈×C₃₁	1	2	yes	—
299	C₂³	C₃₁×#135	1	2	yes	—
300	D₄	—	4	2	no	—
301	D₄	—	4	2	no	—
302	D₄	—	8	2	no	—
303	D₄	—	8	4	no	—
304	D₄	—	8	2	no	—
305	D₄	—	8	2	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	4	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	4	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	8	2	no	—
321	C₂×C₄	—	8	2	no	—
322	C₂×C₄	—	8	2	no	—
323	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
324	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
325	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
326	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
327	C₈	—	2	1	no	—
328	C₈	—	2	1	no	—
329	C₈	—	2	1	no	—
330	C₈	—	4	2	no	—
331	C₈	C₃₁×#99	1	2	yes	complemented
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	4	2	no	—
339	C₂²	—	4	2	no	—
340	C₂²	—	8	2	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	—	16	4	no	—
344	C₂²	C₃₁×#390	1	1	yes	characteristic
345	C₂²	C₄×D₈×C₃₁	1	2	yes	—
346	C₂²	C₃₁×#466	1	2	yes	—
347	C₂²	C₃₁×#493	1	1	yes	characteristic
348	C₂²	C₃₁×#495	1	1	yes	characteristic
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	4	4	no	—
353	C₄	—	8	2	no	—
354	C₄	—	16	2	no	—
355	C₄	C₂×C₃₁×#99	1	1	yes	characteristic
356	C₄	C₃₁×#410	1	1	yes	characteristic
357	C₂	—	2	2	no	—
358	C₂	—	2	1	no	—
359	C₂	—	2	2	no	—
360	C₂	—	2	1	no	—
361	C₂	—	16	2	no	—
362	C₂	C₃₁×#1964	1	1	yes	characteristic
363	C₂	C₃₁×#1970	1	1	yes	characteristic
364	C₂	C₃₁×#1976	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	#390
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
prime radical	C₂×C₃₁×#466	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1964	C₂	semidirect
3. C₃₁×#1808	C₂	semidirect
4. C₈×C₂²⋊C₄×C₃₁	C₂	semidirect
5. C₃₁×#1519	C₂	semidirect
6. C₂²×D₈×C₃₁	C₄	semidirect
14. C₂×C₃₁×#107	C₄	semidirect
15. C₂×C₃₁×#102	C₄	semidirect
103. C₂²⋊C₄×C₃₁	D₈	semidirect
106. #9783	C₃₁	direct (N × H)
149. C₈×C₃₁	#99	semidirect
151. #1964	C₂×C₃₁	semidirect
152. #1808	C₂×C₃₁	semidirect
153. C₈×C₂²⋊C₄	C₂×C₃₁	semidirect
154. #1519	C₂×C₃₁	semidirect
180. C₂²×D₈	C₄×C₃₁	semidirect
188. C₂×#107	C₄×C₃₁	semidirect
189. C₂×#102	C₄×C₃₁	semidirect
241. C₃₁	#9783	direct (N × H)
286. C₂²⋊C₄	D₈×C₃₁	semidirect
331. C₈	C₃₁×#99	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#466	2
C₃₁×#1964	2
C₃₁×#1808	2
C₈×C₂²⋊C₄×C₃₁	2
C₃₁×#1519	2
#9783	31

Subgroups — C31×#9783 #21466542

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9783 #21466542