Subgroups — C₃₁×#9287 #21466046

Order 7936.

478 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3227	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2932	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1993	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1548	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1517	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1286	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1275	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#131	—	2	1	no	—
9	C₃₁×#532	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₃₁×#498	C₄	1	2	yes	complemented
12	C₃₁×#486	C₄	1	2	yes	complemented
13	C₃₁×#481	—	2	2	no	—
14	C₃₁×#476	—	2	2	no	—
15	C₃₁×#435	—	2	2	no	—
16	C₃₁×#435	—	2	2	no	—
17	C₃₁×#431	C₂²	1	1	yes	characteristic
18	C₃₁×#431	C₂²	1	1	yes	characteristic
19	C₃₁×#423	C₂²	1	1	yes	characteristic
20	C₃₁×#408	—	2	1	no	—
21	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
22	C₃₁×#382	C₂²	1	1	yes	characteristic
23	C₃₁×#340	—	2	1	no	—
24	C₃₁×#340	—	2	1	no	—
25	C₃₁×#333	—	2	2	no	—
26	C₃₁×#327	—	2	2	no	—
27	C₂×#41×C₃₁	—	2	1	no	—
28	C₂²×Q₈×C₃₁	—	2	1	no	—
29	C₃₁×#131	—	4	2	no	—
30	C₃₁×#131	—	4	2	no	—
31	C₂×M₁₆×C₃₁	—	2	2	no	—
32	C₂×M₁₆×C₃₁	—	2	1	no	—
33	C₂²×C₈×C₃₁	—	2	2	no	—
34	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
36	C₃₁×C₄⋊Q₈	—	2	1	no	—
37	C₃₁×C₄⋊Q₈	—	2	1	no	—
38	C₃₁×C₄⋊Q₈	—	2	1	no	—
39	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
40	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
41	C₃₁×#127	C₂×C₄	1	1	yes	characteristic
42	C₃₁×#124	—	2	1	no	—
43	C₃₁×#121	—	2	1	no	—
44	C₃₁×#121	D₄	1	2	yes	—
45	C₄×D₄×C₃₁	—	2	1	no	—
46	C₄×D₄×C₃₁	D₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
48	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
49	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
50	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
51	C₃₁×#106	—	2	4	no	—
52	C₃₁×C₄⋊C₈	—	2	1	no	—
53	C₃₁×C₄⋊C₈	—	2	1	no	—
54	C₃₁×C₄⋊C₈	—	2	1	no	—
55	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
56	C₃₁×#102	—	2	4	no	—
57	C₃₁×C₂²⋊C₈	—	2	4	no	—
58	C₃₁×C₂²⋊C₈	—	2	1	no	—
59	C₃₁×C₂²⋊C₈	—	2	1	no	—
60	C₃₁×#97	—	2	1	no	—
61	C₃₁×#97	—	2	1	no	—
62	C₄×C₈×C₃₁	—	2	1	no	—
63	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
64	C₃₁×#95	—	2	1	no	—
65	C₃₁×#95	—	2	1	no	—
66	#41×C₃₁	—	4	1	no	—
67	#41×C₃₁	—	4	1	no	—
68	#41×C₃₁	—	4	1	no	—
69	#41×C₃₁	—	4	1	no	—
70	C₂×Q₈×C₃₁	—	2	1	no	—
71	C₂×Q₈×C₃₁	—	2	2	no	—
72	C₂×Q₈×C₃₁	—	4	1	no	—
73	C₂×Q₈×C₃₁	—	8	2	no	—
74	C₂×D₄×C₃₁	—	2	2	no	—
75	C₂×D₄×C₃₁	—	2	1	no	—
76	C₂×D₄×C₃₁	—	2	1	no	—
77	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
78	C₂²×C₄×C₃₁	—	2	2	no	—
79	C₂²×C₄×C₃₁	—	4	1	no	—
80	C₂²×C₄×C₃₁	—	4	1	no	—
81	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
84	M₁₆×C₃₁	—	4	4	no	—
85	M₁₆×C₃₁	—	4	1	no	—
86	M₁₆×C₃₁	—	4	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	—	2	2	no	—
92	C₂×C₈×C₃₁	—	2	1	no	—
93	C₂×C₈×C₃₁	—	2	1	no	—
94	C₂×C₈×C₃₁	—	4	4	no	—
95	C₂×C₈×C₃₁	—	4	1	no	—
96	C₂×C₈×C₃₁	SD₁₆	1	2	yes	—
97	C₄⋊C₄×C₃₁	—	2	1	no	—
98	C₄⋊C₄×C₃₁	—	2	2	no	—
99	C₄⋊C₄×C₃₁	—	2	2	no	—
100	C₄⋊C₄×C₃₁	—	4	1	no	—
101	C₄⋊C₄×C₃₁	—	4	1	no	—
102	C₄⋊C₄×C₃₁	—	4	1	no	—
103	C₄⋊C₄×C₃₁	—	4	1	no	—
104	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
105	C₂²⋊C₄×C₃₁	—	2	2	no	—
106	C₂²⋊C₄×C₃₁	—	2	2	no	—
107	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₃₁	#41	1	1	yes	characteristic
111	#9287	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂³×C₃₁	—	2	2	no	—
113	C₂³×C₃₁	#121	1	1	yes	characteristic
114	Q₈×C₃₁	—	4	1	no	—
115	Q₈×C₃₁	—	4	1	no	—
116	Q₈×C₃₁	—	8	2	no	—
117	Q₈×C₃₁	—	8	2	no	—
118	D₄×C₃₁	—	4	2	no	—
119	D₄×C₃₁	—	4	2	no	—
120	D₄×C₃₁	—	4	2	no	—
121	D₄×C₃₁	—	4	1	no	—
122	D₄×C₃₁	—	4	1	no	—
123	D₄×C₃₁	—	8	1	no	—
124	C₂×C₄×C₃₁	—	2	2	no	—
125	C₂×C₄×C₃₁	—	4	1	no	—
126	C₂×C₄×C₃₁	—	4	1	no	—
127	C₂×C₄×C₃₁	—	4	2	no	—
128	C₂×C₄×C₃₁	—	4	2	no	—
129	C₂×C₄×C₃₁	—	4	2	no	—
130	C₂×C₄×C₃₁	—	8	2	no	—
131	C₂×C₄×C₃₁	—	8	1	no	—
132	C₂×C₄×C₃₁	—	8	1	no	—
133	C₂×C₄×C₃₁	—	8	1	no	—
134	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
138	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
139	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
140	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
141	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
142	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₄×C₃₁	#137	1	1	yes	characteristic
144	C₈×C₃₁	—	2	2	no	—
145	C₈×C₃₁	—	4	2	no	—
146	C₈×C₃₁	—	4	2	no	—
147	C₈×C₃₁	—	4	1	no	—
148	C₈×C₃₁	—	8	1	no	—
149	#3227	C₂×C₃₁	1	1	yes	characteristic
150	#2932	C₂×C₃₁	1	1	yes	characteristic
151	#1993	C₂×C₃₁	1	1	yes	characteristic, complemented
152	#1548	C₂×C₃₁	1	1	yes	characteristic, complemented
153	#1517	C₂×C₃₁	1	1	yes	characteristic
154	#1286	C₂×C₃₁	1	1	yes	characteristic, complemented
155	#1275	C₂×C₃₁	1	1	yes	characteristic, complemented
156	C₂²×C₃₁	—	2	1	no	—
157	C₂²×C₃₁	—	2	1	no	—
158	C₂²×C₃₁	—	4	2	no	—
159	C₂²×C₃₁	—	4	2	no	—
160	C₂²×C₃₁	—	4	2	no	—
161	C₂²×C₃₁	#386	1	1	yes	characteristic
162	C₂²×C₃₁	#461	1	1	yes	characteristic
163	C₂²×C₃₁	#463	1	1	yes	characteristic
164	C₄×C₃₁	—	2	1	no	—
165	C₄×C₃₁	—	2	1	no	—
166	C₄×C₃₁	—	2	1	no	—
167	C₄×C₃₁	—	2	1	no	—
168	C₄×C₃₁	—	4	2	no	—
169	C₄×C₃₁	—	8	2	no	—
170	C₄×C₃₁	—	16	1	no	—
171	C₄×C₃₁	C₄×SD₁₆	1	1	yes	characteristic
172	C₄×C₃₁	#440	1	1	yes	characteristic
173	C₄×C₃₁	#450	1	1	yes	characteristic
174	C₄×C₃₁	#452	1	1	yes	characteristic
175	C₂×#131	—	2	1	no	—
176	#532	C₂²×C₃₁	1	1	yes	characteristic
177	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
178	#498	C₄×C₃₁	1	2	yes	complemented
179	#486	C₄×C₃₁	1	2	yes	complemented
180	#481	—	2	2	no	—
181	#476	—	2	2	no	—
182	#435	—	2	2	no	—
183	#435	—	2	2	no	—
184	#431	C₂²×C₃₁	1	1	yes	characteristic
185	#431	C₂²×C₃₁	1	1	yes	characteristic
186	#423	C₂²×C₃₁	1	1	yes	characteristic
187	#408	—	2	1	no	—
188	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
189	#382	C₂²×C₃₁	1	1	yes	characteristic
190	#340	—	2	1	no	—
191	#340	—	2	1	no	—
192	#333	—	2	2	no	—
193	#327	—	2	2	no	—
194	C₂×C₃₁	—	2	1	no	—
195	C₂×C₃₁	—	2	1	no	—
196	C₂×C₃₁	—	8	2	no	—
197	C₂×C₃₁	#1877	1	1	yes	characteristic
198	C₂×C₃₁	#1903	1	1	yes	characteristic
199	C₂×C₃₁	#1909	1	1	yes	characteristic
200	C₂×#41	—	2	1	no	—
201	C₂²×Q₈	—	2	1	no	—
202	#131	—	4	2	no	—
203	#131	—	4	2	no	—
204	C₂×M₁₆	—	2	2	no	—
205	C₂×M₁₆	—	2	1	no	—
206	C₂²×C₈	—	2	2	no	—
207	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
208	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
209	C₄⋊Q₈	—	2	1	no	—
210	C₄⋊Q₈	—	2	1	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
213	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
214	#127	C₂×C₄×C₃₁	1	1	yes	characteristic
215	#124	—	2	1	no	—
216	#121	—	2	1	no	—
217	#121	D₄×C₃₁	1	2	yes	—
218	C₄×D₄	—	2	1	no	—
219	C₄×D₄	D₄×C₃₁	1	2	yes	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
224	#106	—	2	4	no	—
225	C₄⋊C₈	—	2	1	no	—
226	C₄⋊C₈	—	2	1	no	—
227	C₄⋊C₈	—	2	1	no	—
228	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
229	#102	—	2	4	no	—
230	C₂²⋊C₈	—	2	4	no	—
231	C₂²⋊C₈	—	2	1	no	—
232	C₂²⋊C₈	—	2	1	no	—
233	#97	—	2	1	no	—
234	#97	—	2	1	no	—
235	C₄×C₈	—	2	1	no	—
236	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
237	#95	—	2	1	no	—
238	#95	—	2	1	no	—
239	C₃₁	#9287	1	1	yes	characteristic, direct product factor, Sylow
240	#41	—	4	1	no	—
241	#41	—	4	1	no	—
242	#41	—	4	1	no	—
243	#41	—	4	1	no	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	—	8	2	no	—
248	C₂×D₄	—	2	2	no	—
249	C₂×D₄	—	2	1	no	—
250	C₂×D₄	—	2	1	no	—
251	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
257	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
258	M₁₆	—	4	4	no	—
259	M₁₆	—	4	1	no	—
260	M₁₆	—	4	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	2	1	no	—
268	C₂×C₈	—	4	4	no	—
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	SD₁₆×C₃₁	1	2	yes	—
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
279	C₂²⋊C₄	—	2	2	no	—
280	C₂²⋊C₄	—	2	2	no	—
281	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
282	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
283	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
284	C₄²	#41×C₃₁	1	1	yes	characteristic
285	C₂³	—	2	2	no	—
286	C₂³	C₃₁×#121	1	1	yes	characteristic
287	Q₈	—	4	1	no	—
288	Q₈	—	4	1	no	—
289	Q₈	—	8	2	no	—
290	Q₈	—	8	2	no	—
291	D₄	—	4	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	4	2	no	—
294	D₄	—	4	1	no	—
295	D₄	—	4	1	no	—
296	D₄	—	8	1	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	8	2	no	—
304	C₂×C₄	—	8	1	no	—
305	C₂×C₄	—	8	1	no	—
306	C₂×C₄	—	8	1	no	—
307	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
308	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
309	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
310	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
311	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
312	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
313	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
314	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
315	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
316	C₂×C₄	C₃₁×#137	1	1	yes	characteristic
317	C₈	—	2	2	no	—
318	C₈	—	4	2	no	—
319	C₈	—	4	2	no	—
320	C₈	—	4	1	no	—
321	C₈	—	8	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	4	2	no	—
325	C₂²	—	4	2	no	—
326	C₂²	—	4	2	no	—
327	C₂²	C₃₁×#386	1	1	yes	characteristic
328	C₂²	C₃₁×#461	1	1	yes	characteristic
329	C₂²	C₃₁×#463	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	4	2	no	—
335	C₄	—	8	2	no	—
336	C₄	—	16	1	no	—
337	C₄	C₄×SD₁₆×C₃₁	1	1	yes	characteristic
338	C₄	C₃₁×#440	1	1	yes	characteristic
339	C₄	C₃₁×#450	1	1	yes	characteristic
340	C₄	C₃₁×#452	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	8	2	no	—
344	C₂	C₃₁×#1877	1	1	yes	characteristic
345	C₂	C₃₁×#1903	1	1	yes	characteristic
346	C₂	C₃₁×#1909	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	#386
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₃₁×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1993	C₂	semidirect
4. C₃₁×#1548	C₂	semidirect
6. C₃₁×#1286	C₂	semidirect
7. C₃₁×#1275	C₂	semidirect
11. C₃₁×#498	C₄	semidirect
12. C₃₁×#486	C₄	semidirect
111. #9287	C₃₁	direct (N × H)
151. #1993	C₂×C₃₁	semidirect
152. #1548	C₂×C₃₁	semidirect
154. #1286	C₂×C₃₁	semidirect
155. #1275	C₂×C₃₁	semidirect
178. #498	C₄×C₃₁	semidirect
179. #486	C₄×C₃₁	semidirect
239. C₃₁	#9287	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3227	2
C₃₁×#2932	2
C₃₁×#1993	2
C₃₁×#1548	2
C₃₁×#1517	2
C₃₁×#1286	2
C₃₁×#1275	2
#9287	31

Subgroups — C31×#9287 #21466046

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9287 #21466046