Subgroups — C₃₁×#8899 #21465658

Order 7936.

388 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2991	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1848	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1574	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1489	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1476	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1299	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
9	C₈×Q₈×C₃₁	—	2	1	no	—
10	C₃₁×#436	—	2	2	no	—
11	C₃₁×#436	—	2	1	no	—
12	C₃₁×#431	—	2	1	no	—
13	C₃₁×#431	C₂²	1	1	yes	characteristic
14	C₃₁×#431	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#108	—	2	2	no	—
16	C₃₁×#423	—	2	1	no	—
17	C₃₁×#423	C₂²	1	1	yes	characteristic
18	C₃₁×#423	C₂²	1	1	yes	characteristic
19	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#335	—	2	1	no	—
22	C₃₁×#334	—	2	1	no	—
23	C₃₁×Q₈⋊C₈	—	2	1	no	—
24	C₃₁×Q₈⋊C₈	C₄	1	1	yes	characteristic
25	C₃₁×Q₈⋊C₈	C₄	1	1	yes	characteristic
26	C₃₁×D₄⋊C₈	—	2	1	no	—
27	C₃₁×D₄⋊C₈	C₄	1	1	yes	characteristic
28	C₃₁×D₄⋊C₈	C₄	1	1	yes	characteristic
29	C₂×#41×C₃₁	Q₈	1	1	yes	characteristic
30	C₂×M₁₆×C₃₁	—	2	2	no	—
31	C₂×M₁₆×C₃₁	—	2	1	no	—
32	C₂×M₁₆×C₃₁	—	2	1	no	—
33	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
34	C₂²×C₈×C₃₁	—	2	1	no	—
35	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
36	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
37	C₄×D₄×C₃₁	—	2	1	no	—
38	C₄×D₄×C₃₁	—	2	1	no	—
39	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
40	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
41	C₃₁×#117	—	2	1	no	—
42	C₃₁×#117	Q₈	1	1	yes	characteristic
43	C₂×C₄²×C₃₁	—	2	1	no	—
44	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
45	C₃₁×#108	—	4	4	no	—
46	C₃₁×C₄⋊C₈	—	2	2	no	—
47	C₃₁×C₄⋊C₈	—	2	1	no	—
48	C₃₁×C₄⋊C₈	—	2	1	no	—
49	C₃₁×C₄⋊C₈	—	2	1	no	—
50	C₃₁×C₄⋊C₈	—	2	1	no	—
51	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
52	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
54	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
55	C₃₁×C₂²⋊C₈	—	2	2	no	—
56	C₃₁×C₂²⋊C₈	—	2	1	no	—
57	C₃₁×C₂²⋊C₈	—	2	1	no	—
58	C₃₁×#97	—	2	1	no	—
59	C₄×C₈×C₃₁	—	2	2	no	—
60	C₄×C₈×C₃₁	—	2	1	no	—
61	C₄×C₈×C₃₁	—	2	1	no	—
62	C₄×C₈×C₃₁	—	2	1	no	—
63	C₄×C₈×C₃₁	—	2	1	no	—
64	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
66	#41×C₃₁	—	2	2	no	—
67	#41×C₃₁	—	4	1	no	—
68	C₂×Q₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
69	C₂×D₄×C₃₁	—	2	1	no	—
70	C₂×D₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₃₁	—	2	1	no	—
72	C₂²×C₄×C₃₁	—	2	1	no	—
73	C₂²×C₄×C₃₁	—	2	1	no	—
74	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
76	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
77	M₁₆×C₃₁	—	2	2	no	—
78	M₁₆×C₃₁	—	4	1	no	—
79	M₁₆×C₃₁	—	4	2	no	—
80	M₁₆×C₃₁	—	4	2	no	—
81	M₁₆×C₃₁	—	8	2	no	—
82	C₂×C₈×C₃₁	—	2	2	no	—
83	C₂×C₈×C₃₁	—	2	2	no	—
84	C₂×C₈×C₃₁	—	2	1	no	—
85	C₂×C₈×C₃₁	—	2	1	no	—
86	C₂×C₈×C₃₁	—	2	1	no	—
87	C₂×C₈×C₃₁	—	2	1	no	—
88	C₂×C₈×C₃₁	—	2	1	no	—
89	C₂×C₈×C₃₁	—	2	2	no	—
90	C₂×C₈×C₃₁	—	2	1	no	—
91	C₂×C₈×C₃₁	—	4	1	no	—
92	C₂×C₈×C₃₁	—	4	1	no	—
93	C₄⋊C₄×C₃₁	—	2	1	no	—
94	C₄⋊C₄×C₃₁	—	2	1	no	—
95	C₄⋊C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
96	C₄⋊C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
97	C₂²⋊C₄×C₃₁	—	2	1	no	—
98	C₂²⋊C₄×C₃₁	—	2	1	no	—
99	C₂²⋊C₄×C₃₁	—	2	1	no	—
100	C₄²×C₃₁	—	2	1	no	—
101	C₄²×C₃₁	—	2	1	no	—
102	C₄²×C₃₁	—	2	1	no	—
103	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₃₁	#41	1	1	yes	characteristic
107	#8899	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂³×C₃₁	—	2	1	no	—
109	C₂³×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
110	Q₈×C₃₁	—	2	1	no	—
111	Q₈×C₃₁	#108	1	2	yes	complemented
112	D₄×C₃₁	—	2	1	no	—
113	D₄×C₃₁	—	2	1	no	—
114	D₄×C₃₁	—	4	2	no	—
115	C₂×C₄×C₃₁	—	2	1	no	—
116	C₂×C₄×C₃₁	—	2	1	no	—
117	C₂×C₄×C₃₁	—	2	1	no	—
118	C₂×C₄×C₃₁	—	2	1	no	—
119	C₂×C₄×C₃₁	—	2	1	no	—
120	C₂×C₄×C₃₁	—	2	1	no	—
121	C₂×C₄×C₃₁	—	2	1	no	—
122	C₂×C₄×C₃₁	—	2	1	no	—
123	C₂×C₄×C₃₁	—	2	1	no	—
124	C₂×C₄×C₃₁	—	4	2	no	—
125	C₂×C₄×C₃₁	—	4	1	no	—
126	C₂×C₄×C₃₁	—	4	1	no	—
127	C₂×C₄×C₃₁	—	4	1	no	—
128	C₂×C₄×C₃₁	—	4	1	no	—
129	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
133	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
134	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
135	C₈×C₃₁	—	2	2	no	—
136	C₈×C₃₁	—	2	2	no	—
137	C₈×C₃₁	—	4	2	no	—
138	C₈×C₃₁	—	4	1	no	—
139	C₈×C₃₁	—	4	1	no	—
140	C₈×C₃₁	—	4	2	no	—
141	C₈×C₃₁	—	4	1	no	—
142	C₈×C₃₁	—	4	1	no	—
143	C₈×C₃₁	—	8	1	no	—
144	#2991	C₂×C₃₁	1	1	yes	characteristic
145	#1848	C₂×C₃₁	1	1	yes	characteristic, complemented
146	#1574	C₂×C₃₁	1	1	yes	characteristic, complemented
147	#1489	C₂×C₃₁	1	1	yes	characteristic
148	#1476	C₂×C₃₁	1	1	yes	characteristic
149	#1299	C₂×C₃₁	1	1	yes	characteristic, complemented
150	#1278	C₂×C₃₁	1	1	yes	characteristic, complemented
151	C₂²×C₃₁	—	2	1	no	—
152	C₂²×C₃₁	—	2	1	no	—
153	C₂²×C₃₁	—	2	1	no	—
154	C₂²×C₃₁	—	4	1	no	—
155	C₂²×C₃₁	—	4	1	no	—
156	C₂²×C₃₁	—	4	1	no	—
157	C₂²×C₃₁	#385	1	1	yes	characteristic
158	C₄×C₃₁	—	2	1	no	—
159	C₄×C₃₁	—	2	2	no	—
160	C₄×C₃₁	—	2	1	no	—
161	C₄×C₃₁	—	2	1	no	—
162	C₄×C₃₁	—	4	1	no	—
163	C₄×C₃₁	—	4	1	no	—
164	C₄×C₃₁	C₂×#108	1	1	yes	characteristic
165	C₄×C₃₁	#430	1	1	yes	characteristic
166	C₄×C₃₁	#443	1	1	yes	characteristic
167	C₄×C₃₁	#444	1	1	yes	characteristic
168	C₄×C₃₁	#453	1	1	yes	characteristic
169	C₄×C₃₁	#456	1	1	yes	characteristic
170	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
171	C₈×Q₈	—	2	1	no	—
172	#436	—	2	2	no	—
173	#436	—	2	1	no	—
174	#431	—	2	1	no	—
175	#431	C₂²×C₃₁	1	1	yes	characteristic
176	#431	C₂²×C₃₁	1	1	yes	characteristic
177	C₂×#108	—	2	2	no	—
178	#423	—	2	1	no	—
179	#423	C₂²×C₃₁	1	1	yes	characteristic
180	#423	C₂²×C₃₁	1	1	yes	characteristic
181	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
182	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
183	#335	—	2	1	no	—
184	#334	—	2	1	no	—
185	Q₈⋊C₈	—	2	1	no	—
186	Q₈⋊C₈	C₄×C₃₁	1	1	yes	characteristic
187	Q₈⋊C₈	C₄×C₃₁	1	1	yes	characteristic
188	D₄⋊C₈	—	2	1	no	—
189	D₄⋊C₈	C₄×C₃₁	1	1	yes	characteristic
190	D₄⋊C₈	C₄×C₃₁	1	1	yes	characteristic
191	C₂×C₃₁	—	4	1	no	—
192	C₂×C₃₁	—	8	1	no	—
193	C₂×C₃₁	#1855	1	1	yes	characteristic
194	C₂×C₃₁	#1859	1	1	yes	characteristic
195	C₂×C₃₁	#1866	1	1	yes	characteristic
196	C₂×#41	Q₈×C₃₁	1	1	yes	characteristic
197	C₂×M₁₆	—	2	2	no	—
198	C₂×M₁₆	—	2	1	no	—
199	C₂×M₁₆	—	2	1	no	—
200	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
201	C₂²×C₈	—	2	1	no	—
202	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
203	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
204	C₄×D₄	—	2	1	no	—
205	C₄×D₄	—	2	1	no	—
206	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
207	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
208	#117	—	2	1	no	—
209	#117	Q₈×C₃₁	1	1	yes	characteristic
210	C₂×C₄²	—	2	1	no	—
211	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
212	#108	—	4	4	no	—
213	C₄⋊C₈	—	2	2	no	—
214	C₄⋊C₈	—	2	1	no	—
215	C₄⋊C₈	—	2	1	no	—
216	C₄⋊C₈	—	2	1	no	—
217	C₄⋊C₈	—	2	1	no	—
218	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
219	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
220	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
221	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
222	C₂²⋊C₈	—	2	2	no	—
223	C₂²⋊C₈	—	2	1	no	—
224	C₂²⋊C₈	—	2	1	no	—
225	#97	—	2	1	no	—
226	C₄×C₈	—	2	2	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	1	yes	characteristic
232	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
233	C₃₁	#8899	1	1	yes	characteristic, direct product factor, Sylow
234	#41	—	2	2	no	—
235	#41	—	4	1	no	—
236	C₂×Q₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
237	C₂×D₄	—	2	1	no	—
238	C₂×D₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	2	1	no	—
242	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
243	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
244	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
245	M₁₆	—	2	2	no	—
246	M₁₆	—	4	1	no	—
247	M₁₆	—	4	2	no	—
248	M₁₆	—	4	2	no	—
249	M₁₆	—	8	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	—	4	1	no	—
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
264	C₄⋊C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	2	1	no	—
268	C₄²	—	2	1	no	—
269	C₄²	—	2	1	no	—
270	C₄²	—	2	1	no	—
271	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
272	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
273	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
274	C₄²	#41×C₃₁	1	1	yes	characteristic
275	C₂³	—	2	1	no	—
276	C₂³	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
277	Q₈	—	2	1	no	—
278	Q₈	C₃₁×#108	1	2	yes	complemented
279	D₄	—	2	1	no	—
280	D₄	—	2	1	no	—
281	D₄	—	4	2	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
297	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
298	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
299	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
300	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
301	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
302	C₈	—	2	2	no	—
303	C₈	—	2	2	no	—
304	C₈	—	4	2	no	—
305	C₈	—	4	1	no	—
306	C₈	—	4	1	no	—
307	C₈	—	4	2	no	—
308	C₈	—	4	1	no	—
309	C₈	—	4	1	no	—
310	C₈	—	8	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	C₃₁×#385	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	2	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	4	1	no	—
324	C₄	C₂×C₃₁×#108	1	1	yes	characteristic
325	C₄	C₃₁×#430	1	1	yes	characteristic
326	C₄	C₃₁×#443	1	1	yes	characteristic
327	C₄	C₃₁×#444	1	1	yes	characteristic
328	C₄	C₃₁×#453	1	1	yes	characteristic
329	C₄	C₃₁×#456	1	1	yes	characteristic
330	C₂	—	4	1	no	—
331	C₂	—	8	1	no	—
332	C₂	C₃₁×#1855	1	1	yes	characteristic
333	C₂	C₃₁×#1859	1	1	yes	characteristic
334	C₂	C₃₁×#1866	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#120
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	#385
prime radical	C₂×#41×C₃₁	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1848	C₂	semidirect
3. C₃₁×#1574	C₂	semidirect
6. C₃₁×#1299	C₂	semidirect
7. C₃₁×#1278	C₂	semidirect
107. #8899	C₃₁	direct (N × H)
111. Q₈×C₃₁	#108	semidirect
145. #1848	C₂×C₃₁	semidirect
146. #1574	C₂×C₃₁	semidirect
149. #1299	C₂×C₃₁	semidirect
150. #1278	C₂×C₃₁	semidirect
233. C₃₁	#8899	direct (N × H)
278. Q₈	C₃₁×#108	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#2991	2
C₃₁×#1848	2
C₃₁×#1574	2
C₃₁×#1489	2
C₃₁×#1476	2
C₃₁×#1299	2
C₃₁×#1278	2
#8899	31

Subgroups — C31×#8899 #21465658

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8899 #21465658