Subgroups — C₃₁×#9822 #21466581

Order 7936.

578 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#486	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1990	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1957	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1957	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1956	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1834	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₈²×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
10	C₃₁×#486	—	2	2	no	—
11	C₃₁×#486	C₄	1	4	yes	complemented
12	C₂×C₃₁×#103	—	2	2	no	—
13	C₂×C₃₁×#102	—	2	2	no	—
14	C₂×C₃₁×#102	—	2	2	no	—
15	C₂×C₃₁×#102	—	2	2	no	—
16	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#384	C₂²	1	1	yes	characteristic
20	C₃₁×#384	C₂²	1	1	yes	characteristic
21	C₃₁×#340	—	2	2	no	—
22	C₃₁×#340	—	2	2	no	—
23	C₈²×C₃₁	—	2	2	no	—
24	C₂²×Q₈×C₃₁	—	2	1	no	—
25	C₂²×D₄×C₃₁	—	2	1	no	—
26	C₂²×D₄×C₃₁	—	2	1	no	—
27	C₂²×D₄×C₃₁	—	2	1	no	—
28	C₂²×C₈×C₃₁	D₄	1	2	yes	—
29	C₂²×C₈×C₃₁	D₄	1	2	yes	—
30	C₂²×C₈×C₃₁	D₄	1	2	yes	—
31	C₃₁×C₄⋊Q₈	—	2	1	no	—
32	C₃₁×C₄⋊Q₈	—	2	1	no	—
33	C₃₁×C₄⋊Q₈	—	2	1	no	—
34	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
35	C₃₁×#127	—	2	1	no	—
36	C₃₁×#127	—	2	1	no	—
37	C₃₁×#127	—	2	1	no	—
38	C₃₁×#127	C₂×C₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
40	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
41	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
44	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
45	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
46	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
47	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
48	C₃₁×#103	—	4	4	no	—
49	C₃₁×#102	—	2	8	no	—
50	C₃₁×#102	—	4	4	no	—
51	C₃₁×#102	—	4	4	no	—
52	C₄×C₈×C₃₁	—	2	1	no	—
53	C₄×C₈×C₃₁	—	2	1	no	—
54	C₄×C₈×C₃₁	—	2	1	no	—
55	C₄×C₈×C₃₁	—	2	1	no	—
56	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₃₁	D₄	1	2	yes	—
58	C₂⁴×C₃₁	—	4	1	no	—
59	C₂×Q₈×C₃₁	—	2	2	no	—
60	C₂×Q₈×C₃₁	—	4	1	no	—
61	C₂×Q₈×C₃₁	—	4	2	no	—
62	C₂×Q₈×C₃₁	—	4	2	no	—
63	C₂×D₄×C₃₁	—	2	2	no	—
64	C₂×D₄×C₃₁	—	2	2	no	—
65	C₂×D₄×C₃₁	—	2	2	no	—
66	C₂×D₄×C₃₁	—	4	1	no	—
67	C₂×D₄×C₃₁	—	4	1	no	—
68	C₂×D₄×C₃₁	—	4	1	no	—
69	C₂×D₄×C₃₁	—	4	2	no	—
70	C₂×D₄×C₃₁	—	4	2	no	—
71	C₂×D₄×C₃₁	—	4	2	no	—
72	C₂×D₄×C₃₁	—	4	2	no	—
73	C₂×D₄×C₃₁	—	4	2	no	—
74	C₂×D₄×C₃₁	—	4	2	no	—
75	C₂²×C₄×C₃₁	—	4	1	no	—
76	C₂²×C₄×C₃₁	—	4	1	no	—
77	C₂²×C₄×C₃₁	—	4	1	no	—
78	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₃₁	—	2	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	—	2	2	no	—
87	C₂×C₈×C₃₁	C₂²⋊C₄	1	4	yes	—
88	C₂×C₈×C₃₁	D₈	1	4	yes	—
89	C₂×C₈×C₃₁	SD₁₆	1	4	yes	—
90	C₄⋊C₄×C₃₁	—	2	2	no	—
91	C₄⋊C₄×C₃₁	—	2	2	no	—
92	C₄⋊C₄×C₃₁	—	4	1	no	—
93	C₄⋊C₄×C₃₁	—	4	1	no	—
94	C₄⋊C₄×C₃₁	—	4	2	no	—
95	C₄⋊C₄×C₃₁	—	4	2	no	—
96	C₄⋊C₄×C₃₁	—	4	2	no	—
97	C₄⋊C₄×C₃₁	—	4	2	no	—
98	C₄⋊C₄×C₃₁	—	4	2	no	—
99	C₄⋊C₄×C₃₁	—	4	2	no	—
100	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₃₁	#41	1	1	yes	characteristic
103	C₄²×C₃₁	#41	1	1	yes	characteristic
104	#9822	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂³×C₃₁	—	4	2	no	—
106	C₂³×C₃₁	—	8	1	no	—
107	C₂³×C₃₁	—	8	1	no	—
108	C₂³×C₃₁	—	8	1	no	—
109	C₂³×C₃₁	—	8	1	no	—
110	C₂³×C₃₁	—	8	1	no	—
111	C₂³×C₃₁	—	8	1	no	—
112	C₂³×C₃₁	#127	1	1	yes	characteristic
113	Q₈×C₃₁	—	4	4	no	—
114	Q₈×C₃₁	—	8	2	no	—
115	D₄×C₃₁	—	4	4	no	—
116	D₄×C₃₁	—	4	4	no	—
117	D₄×C₃₁	—	4	4	no	—
118	D₄×C₃₁	—	8	2	no	—
119	D₄×C₃₁	—	8	2	no	—
120	D₄×C₃₁	—	8	2	no	—
121	C₂×C₄×C₃₁	—	4	2	no	—
122	C₂×C₄×C₃₁	—	8	1	no	—
123	C₂×C₄×C₃₁	—	8	1	no	—
124	C₂×C₄×C₃₁	—	8	1	no	—
125	C₂×C₄×C₃₁	—	8	1	no	—
126	C₂×C₄×C₃₁	—	8	1	no	—
127	C₂×C₄×C₃₁	—	8	1	no	—
128	C₂×C₄×C₃₁	—	8	1	no	—
129	C₂×C₄×C₃₁	—	8	1	no	—
130	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
134	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
135	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
136	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
137	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
138	C₂×C₄×C₃₁	#135	1	2	yes	—
139	C₂×C₄×C₃₁	#135	1	2	yes	—
140	C₂×C₄×C₃₁	#135	1	2	yes	—
141	C₂×C₄×C₃₁	#135	1	2	yes	—
142	C₈×C₃₁	—	2	4	no	—
143	C₈×C₃₁	—	2	4	no	—
144	C₈×C₃₁	—	2	4	no	—
145	C₂×#486	C₂×C₃₁	1	1	yes	characteristic
146	#1990	C₂×C₃₁	1	1	yes	characteristic, complemented
147	#1957	C₂×C₃₁	1	1	yes	characteristic
148	#1957	C₂×C₃₁	1	1	yes	characteristic
149	#1956	C₂×C₃₁	1	1	yes	characteristic, complemented
150	#1834	C₂×C₃₁	1	1	yes	characteristic, complemented
151	C₂×C₈²	C₂×C₃₁	1	1	yes	characteristic, complemented
152	C₂²×C₃₁	—	8	2	no	—
153	C₂²×C₃₁	—	8	2	no	—
154	C₂²×C₃₁	—	8	2	no	—
155	C₂²×C₃₁	—	16	1	no	—
156	C₂²×C₃₁	—	16	1	no	—
157	C₂²×C₃₁	—	16	1	no	—
158	C₂²×C₃₁	—	16	1	no	—
159	C₂²×C₃₁	#390	1	1	yes	characteristic
160	C₂²×C₃₁	#492	1	1	yes	characteristic
161	C₂²×C₃₁	#493	1	1	yes	characteristic
162	C₂²×C₃₁	#495	1	1	yes	characteristic
163	C₂²×C₃₁	#495	1	1	yes	characteristic
164	C₂²×C₃₁	#495	1	1	yes	characteristic
165	C₂²×C₃₁	#495	1	1	yes	characteristic
166	C₄×C₃₁	—	8	2	no	—
167	C₄×C₃₁	—	16	1	no	—
168	C₄×C₃₁	—	16	1	no	—
169	C₄×C₃₁	#416	1	2	yes	—
170	C₄×C₃₁	C₄×D₈	1	2	yes	—
171	C₄×C₃₁	C₄×SD₁₆	1	2	yes	—
172	C₄×C₃₁	#465	1	2	yes	—
173	C₄×C₃₁	#466	1	2	yes	—
174	C₄×C₃₁	#488	1	2	yes	—
175	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
176	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
177	#486	—	2	2	no	—
178	#486	C₄×C₃₁	1	4	yes	complemented
179	C₂×#103	—	2	2	no	—
180	C₂×#102	—	2	2	no	—
181	C₂×#102	—	2	2	no	—
182	C₂×#102	—	2	2	no	—
183	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
184	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
185	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
186	#384	C₂²×C₃₁	1	1	yes	characteristic
187	#384	C₂²×C₃₁	1	1	yes	characteristic
188	#340	—	2	2	no	—
189	#340	—	2	2	no	—
190	C₈²	—	2	2	no	—
191	C₂×C₃₁	—	16	2	no	—
192	C₂×C₃₁	#1706	1	1	yes	characteristic
193	C₂×C₃₁	#1708	1	1	yes	characteristic
194	C₂×C₃₁	#1709	1	1	yes	characteristic
195	C₂×C₃₁	#1711	1	1	yes	characteristic
196	C₂×C₃₁	#1958	1	1	yes	characteristic
197	C₂×C₃₁	#1968	1	1	yes	characteristic
198	C₂×C₃₁	#1970	1	1	yes	characteristic
199	C₂²×Q₈	—	2	1	no	—
200	C₂²×D₄	—	2	1	no	—
201	C₂²×D₄	—	2	1	no	—
202	C₂²×D₄	—	2	1	no	—
203	C₂²×C₈	D₄×C₃₁	1	2	yes	—
204	C₂²×C₈	D₄×C₃₁	1	2	yes	—
205	C₂²×C₈	D₄×C₃₁	1	2	yes	—
206	C₄⋊Q₈	—	2	1	no	—
207	C₄⋊Q₈	—	2	1	no	—
208	C₄⋊Q₈	—	2	1	no	—
209	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
210	#127	—	2	1	no	—
211	#127	—	2	1	no	—
212	#127	—	2	1	no	—
213	#127	C₂×C₄×C₃₁	1	2	yes	—
214	C₂×C₄⋊C₄	—	2	1	no	—
215	C₂×C₄⋊C₄	—	2	1	no	—
216	C₂×C₄⋊C₄	—	2	1	no	—
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₄⋊C₄	—	2	1	no	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
223	#103	—	4	4	no	—
224	#102	—	2	8	no	—
225	#102	—	4	4	no	—
226	#102	—	4	4	no	—
227	C₄×C₈	—	2	1	no	—
228	C₄×C₈	—	2	1	no	—
229	C₄×C₈	—	2	1	no	—
230	C₄×C₈	—	2	1	no	—
231	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
232	C₄×C₈	D₄×C₃₁	1	2	yes	—
233	C₃₁	#9822	1	1	yes	characteristic, direct product factor, Sylow
234	C₂⁴	—	4	1	no	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	—	4	1	no	—
237	C₂×Q₈	—	4	2	no	—
238	C₂×Q₈	—	4	2	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	2	2	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	4	1	no	—
243	C₂×D₄	—	4	1	no	—
244	C₂×D₄	—	4	1	no	—
245	C₂×D₄	—	4	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	2	no	—
251	C₂²×C₄	—	4	1	no	—
252	C₂²×C₄	—	4	1	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	C₂²⋊C₄×C₃₁	1	4	yes	—
264	C₂×C₈	D₈×C₃₁	1	4	yes	—
265	C₂×C₈	SD₁₆×C₃₁	1	4	yes	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
277	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
278	C₄²	#41×C₃₁	1	1	yes	characteristic
279	C₄²	#41×C₃₁	1	1	yes	characteristic
280	C₂³	—	4	2	no	—
281	C₂³	—	8	1	no	—
282	C₂³	—	8	1	no	—
283	C₂³	—	8	1	no	—
284	C₂³	—	8	1	no	—
285	C₂³	—	8	1	no	—
286	C₂³	—	8	1	no	—
287	C₂³	C₃₁×#127	1	1	yes	characteristic
288	Q₈	—	4	4	no	—
289	Q₈	—	8	2	no	—
290	D₄	—	4	4	no	—
291	D₄	—	4	4	no	—
292	D₄	—	4	4	no	—
293	D₄	—	8	2	no	—
294	D₄	—	8	2	no	—
295	D₄	—	8	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	8	1	no	—
298	C₂×C₄	—	8	1	no	—
299	C₂×C₄	—	8	1	no	—
300	C₂×C₄	—	8	1	no	—
301	C₂×C₄	—	8	1	no	—
302	C₂×C₄	—	8	1	no	—
303	C₂×C₄	—	8	1	no	—
304	C₂×C₄	—	8	1	no	—
305	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
306	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
307	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
308	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
309	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
310	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
311	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
312	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
313	C₂×C₄	C₃₁×#135	1	2	yes	—
314	C₂×C₄	C₃₁×#135	1	2	yes	—
315	C₂×C₄	C₃₁×#135	1	2	yes	—
316	C₂×C₄	C₃₁×#135	1	2	yes	—
317	C₈	—	2	4	no	—
318	C₈	—	2	4	no	—
319	C₈	—	2	4	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	16	1	no	—
324	C₂²	—	16	1	no	—
325	C₂²	—	16	1	no	—
326	C₂²	—	16	1	no	—
327	C₂²	C₃₁×#390	1	1	yes	characteristic
328	C₂²	C₃₁×#492	1	1	yes	characteristic
329	C₂²	C₃₁×#493	1	1	yes	characteristic
330	C₂²	C₃₁×#495	1	1	yes	characteristic
331	C₂²	C₃₁×#495	1	1	yes	characteristic
332	C₂²	C₃₁×#495	1	1	yes	characteristic
333	C₂²	C₃₁×#495	1	1	yes	characteristic
334	C₄	—	8	2	no	—
335	C₄	—	16	1	no	—
336	C₄	—	16	1	no	—
337	C₄	C₃₁×#416	1	2	yes	—
338	C₄	C₄×D₈×C₃₁	1	2	yes	—
339	C₄	C₄×SD₁₆×C₃₁	1	2	yes	—
340	C₄	C₃₁×#465	1	2	yes	—
341	C₄	C₃₁×#466	1	2	yes	—
342	C₄	C₃₁×#488	1	2	yes	—
343	C₂	—	16	2	no	—
344	C₂	C₃₁×#1706	1	1	yes	characteristic
345	C₂	C₃₁×#1708	1	1	yes	characteristic
346	C₂	C₃₁×#1709	1	1	yes	characteristic
347	C₂	C₃₁×#1711	1	1	yes	characteristic
348	C₂	C₃₁×#1958	1	1	yes	characteristic
349	C₂	C₃₁×#1968	1	1	yes	characteristic
350	C₂	C₃₁×#1970	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	#127
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1990	C₂	semidirect
5. C₃₁×#1956	C₂	semidirect
6. C₃₁×#1834	C₂	semidirect
7. C₂×C₈²×C₃₁	C₂	semidirect
11. C₃₁×#486	C₄	semidirect
104. #9822	C₃₁	direct (N × H)
146. #1990	C₂×C₃₁	semidirect
149. #1956	C₂×C₃₁	semidirect
150. #1834	C₂×C₃₁	semidirect
151. C₂×C₈²	C₂×C₃₁	semidirect
178. #486	C₄×C₃₁	semidirect
233. C₃₁	#9822	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#486	2
C₃₁×#1990	2
C₃₁×#1957	2
C₃₁×#1956	2
C₃₁×#1834	2
C₂×C₈²×C₃₁	2
#9822	31

Subgroups — C31×#9822 #21466581

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9822 #21466581