Subgroups — C₃₁×#9785 #21466544

Order 7936.

594 conjugacy classes of subgroups.

The subgroup lattice has 356 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×#1807	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1752	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	C₄	1	1	yes	characteristic, complemented
14	C₂×C₃₁×#106	—	2	1	no	—
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×C₃₁×#97	—	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×#106	—	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	C₃₁×#97	—	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	4	no	—
100	C₂²⋊C₄×C₃₁	—	4	2	no	—
101	C₂²⋊C₄×C₃₁	—	4	2	no	—
102	C₂²⋊C₄×C₃₁	—	4	4	no	—
103	C₄²×C₃₁	—	4	1	no	—
104	C₄²×C₃₁	—	4	1	no	—
105	#9785	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂³×C₃₁	—	4	1	no	—
107	C₂³×C₃₁	—	4	1	no	—
108	C₂³×C₃₁	—	4	2	no	—
109	C₂³×C₃₁	—	4	4	no	—
110	C₂³×C₃₁	—	8	2	no	—
111	C₂³×C₃₁	—	8	2	no	—
112	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
113	C₂³×C₃₁	#121	1	1	yes	characteristic
114	C₂³×C₃₁	#127	1	1	yes	characteristic
115	C₂³×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
116	D₄×C₃₁	—	4	2	no	—
117	D₄×C₃₁	—	4	2	no	—
118	D₄×C₃₁	—	8	2	no	—
119	D₄×C₃₁	—	8	4	no	—
120	D₄×C₃₁	—	8	2	no	—
121	D₄×C₃₁	—	8	2	no	—
122	C₂×C₄×C₃₁	—	2	1	no	—
123	C₂×C₄×C₃₁	—	2	1	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	—	4	1	no	—
129	C₂×C₄×C₃₁	—	4	1	no	—
130	C₂×C₄×C₃₁	—	4	1	no	—
131	C₂×C₄×C₃₁	—	4	2	no	—
132	C₂×C₄×C₃₁	—	4	2	no	—
133	C₂×C₄×C₃₁	—	4	2	no	—
134	C₂×C₄×C₃₁	—	4	4	no	—
135	C₂×C₄×C₃₁	—	4	2	no	—
136	C₂×C₄×C₃₁	—	8	2	no	—
137	C₂×C₄×C₃₁	—	8	2	no	—
138	C₂×C₄×C₃₁	—	8	2	no	—
139	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
142	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
143	C₈×C₃₁	—	2	1	no	—
144	C₈×C₃₁	—	2	1	no	—
145	C₈×C₃₁	—	2	1	no	—
146	C₈×C₃₁	—	4	2	no	—
147	C₈×C₃₁	#99	1	2	yes	complemented
148	C₂×#466	C₂×C₃₁	1	1	yes	characteristic
149	#1964	C₂×C₃₁	1	2	yes	complemented
150	#1807	C₂×C₃₁	1	1	yes	characteristic, complemented
151	#1752	C₂×C₃₁	1	1	yes	characteristic, complemented
152	#1519	C₂×C₃₁	1	2	yes	complemented
153	C₂²×C₃₁	—	2	1	no	—
154	C₂²×C₃₁	—	2	1	no	—
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	—	4	4	no	—
160	C₂²×C₃₁	—	8	2	no	—
161	C₂²×C₃₁	—	8	2	no	—
162	C₂²×C₃₁	—	8	2	no	—
163	C₂²×C₃₁	—	16	4	no	—
164	C₂²×C₃₁	#390	1	1	yes	characteristic
165	C₂²×C₃₁	#442	1	2	yes	—
166	C₂²×C₃₁	#469	1	2	yes	—
167	C₂²×C₃₁	#496	1	2	yes	—
168	C₄×C₃₁	—	2	1	no	—
169	C₄×C₃₁	—	2	1	no	—
170	C₄×C₃₁	—	2	1	no	—
171	C₄×C₃₁	—	8	2	no	—
172	C₄×C₃₁	—	8	2	no	—
173	C₄×C₃₁	—	16	2	no	—
174	C₄×C₃₁	C₂×#99	1	1	yes	characteristic
175	C₄×C₃₁	#410	1	1	yes	characteristic
176	C₂²×D₈	C₄×C₃₁	1	1	yes	characteristic, complemented
177	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
178	C₂×#121	C₂²×C₃₁	1	2	yes	—
179	#466	—	2	1	no	—
180	#466	—	2	1	no	—
181	#466	—	2	1	no	—
182	#466	—	2	1	no	—
183	C₂×#107	C₄×C₃₁	1	1	yes	characteristic, complemented
184	C₂×#106	—	2	1	no	—
185	C₂×#102	C₄×C₃₁	1	2	yes	complemented
186	C₂×#99	—	2	4	no	—
187	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
188	C₂×#97	—	2	1	no	—
189	#380	C₂²×C₃₁	1	2	yes	—
190	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
191	#340	—	2	1	no	—
192	#336	—	2	1	no	—
193	#327	—	2	4	no	—
194	C₂×C₃₁	—	2	2	no	—
195	C₂×C₃₁	—	2	2	no	—
196	C₂×C₃₁	—	2	2	no	—
197	C₂×C₃₁	—	16	2	no	—
198	C₂×C₃₁	#1964	1	1	yes	characteristic
199	C₂×C₃₁	#1972	1	1	yes	characteristic
200	C₂×C₃₁	#1976	1	1	yes	characteristic
201	C₂²×D₄	—	2	2	no	—
202	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
203	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
204	C₂×D₈	—	2	1	no	—
205	C₂×D₈	—	2	1	no	—
206	C₂×D₈	—	4	2	no	—
207	C₂²×C₈	—	2	1	no	—
208	C₂²×C₈	—	4	2	no	—
209	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
210	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
211	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
212	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
213	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
214	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
215	#121	—	2	2	no	—
216	#121	—	2	2	no	—
217	#121	—	2	2	no	—
218	#121	—	2	2	no	—
219	C₂×C₄⋊C₄	—	2	2	no	—
220	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
221	C₂×C₂²⋊C₄	—	2	2	no	—
222	C₂×C₂²⋊C₄	—	2	2	no	—
223	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
224	C₂×C₄²	—	2	1	no	—
225	#107	—	2	1	no	—
226	#107	—	2	1	no	—
227	#106	—	4	2	no	—
228	#102	—	2	2	no	—
229	#102	—	2	2	no	—
230	#99	—	4	8	no	—
231	C₂²⋊C₈	—	2	1	no	—
232	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
233	#97	—	4	2	no	—
234	#95	—	2	1	no	—
235	#95	—	2	2	no	—
236	C₃₁	#9785	1	1	yes	characteristic, direct product factor, Sylow
237	C₂⁴	—	2	2	no	—
238	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	2	2	no	—
241	C₂×D₄	—	4	2	no	—
242	C₂×D₄	—	4	2	no	—
243	C₂×D₄	—	4	2	no	—
244	C₂×D₄	—	4	2	no	—
245	C₂×D₄	—	4	4	no	—
246	C₂×D₄	—	4	4	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	—	4	2	no	—
252	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
254	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
256	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
257	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
258	D₈	—	4	2	no	—
259	D₈	—	4	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	4	1	no	—
265	C₂×C₈	—	4	1	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	4	1	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
270	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
271	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
272	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₂²⋊C₄	—	2	4	no	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₂²⋊C₄	—	4	2	no	—
280	C₂²⋊C₄	—	4	4	no	—
281	C₄²	—	4	1	no	—
282	C₄²	—	4	1	no	—
283	C₂³	—	4	1	no	—
284	C₂³	—	4	1	no	—
285	C₂³	—	4	2	no	—
286	C₂³	—	4	4	no	—
287	C₂³	—	8	2	no	—
288	C₂³	—	8	2	no	—
289	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
290	C₂³	C₃₁×#121	1	1	yes	characteristic
291	C₂³	C₃₁×#127	1	1	yes	characteristic
292	C₂³	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
293	D₄	—	4	2	no	—
294	D₄	—	4	2	no	—
295	D₄	—	8	2	no	—
296	D₄	—	8	4	no	—
297	D₄	—	8	2	no	—
298	D₄	—	8	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	4	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	8	2	no	—
314	C₂×C₄	—	8	2	no	—
315	C₂×C₄	—	8	2	no	—
316	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
317	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
318	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
319	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
320	C₈	—	2	1	no	—
321	C₈	—	2	1	no	—
322	C₈	—	2	1	no	—
323	C₈	—	4	2	no	—
324	C₈	C₃₁×#99	1	2	yes	complemented
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	4	4	no	—
332	C₂²	—	8	2	no	—
333	C₂²	—	8	2	no	—
334	C₂²	—	8	2	no	—
335	C₂²	—	16	4	no	—
336	C₂²	C₃₁×#390	1	1	yes	characteristic
337	C₂²	C₃₁×#442	1	2	yes	—
338	C₂²	C₃₁×#469	1	2	yes	—
339	C₂²	C₃₁×#496	1	2	yes	—
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	8	2	no	—
344	C₄	—	8	2	no	—
345	C₄	—	16	2	no	—
346	C₄	C₂×C₃₁×#99	1	1	yes	characteristic
347	C₄	C₃₁×#410	1	1	yes	characteristic
348	C₂	—	2	2	no	—
349	C₂	—	2	2	no	—
350	C₂	—	2	2	no	—
351	C₂	—	16	2	no	—
352	C₂	C₃₁×#1964	1	1	yes	characteristic
353	C₂	C₃₁×#1972	1	1	yes	characteristic
354	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×#1807	C₂	semidirect
4. C₃₁×#1752	C₂	semidirect
5. C₃₁×#1519	C₂	semidirect
6. C₂²×D₈×C₃₁	C₄	semidirect
13. C₂×C₃₁×#107	C₄	semidirect
15. C₂×C₃₁×#102	C₄	semidirect
105. #9785	C₃₁	direct (N × H)
147. C₈×C₃₁	#99	semidirect
149. #1964	C₂×C₃₁	semidirect
150. #1807	C₂×C₃₁	semidirect
151. #1752	C₂×C₃₁	semidirect
152. #1519	C₂×C₃₁	semidirect
176. C₂²×D₈	C₄×C₃₁	semidirect
183. C₂×#107	C₄×C₃₁	semidirect
185. C₂×#102	C₄×C₃₁	semidirect
236. C₃₁	#9785	direct (N × H)
324. 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×#1807	2
C₃₁×#1752	2
C₃₁×#1519	2
#9785	31

Subgroups — C31×#9785 #21466544

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9785 #21466544