Subgroups — C₃₁×#19646 #21476405

Order 7936.

732 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	C₃₁×#3210	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3182	C₂	1	2	yes	maximal, complemented
3	C₃₁×#3169	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#3160	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1720	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1714	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1713	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1710	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1706	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1706	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#533	C₂²	1	2	yes	—
13	C₃₁×#532	C₂²	1	2	yes	—
14	C₃₁×#500	C₂²	1	1	yes	characteristic
15	C₃₁×#500	C₂²	1	1	yes	characteristic
16	C₃₁×#500	C₂²	1	1	yes	characteristic
17	C₃₁×#500	C₂²	1	2	yes	—
18	C₃₁×#499	—	2	2	no	—
19	C₃₁×#498	C₂²	1	2	yes	—
20	C₃₁×#494	C₂²	1	2	yes	—
21	C₃₁×#493	C₂²	1	1	yes	characteristic
22	C₃₁×#492	C₂²	1	2	yes	—
23	C₃₁×#492	C₂²	1	2	yes	—
24	C₃₁×#487	C₂²	1	2	yes	—
25	C₃₁×#487	C₂²	1	2	yes	—
26	C₃₁×#486	C₂²	1	2	yes	—
27	C₃₁×#486	C₂²	1	2	yes	—
28	C₃₁×#486	C₂²	1	2	yes	—
29	C₃₁×#468	—	2	4	no	—
30	C₃₁×#466	—	2	2	no	—
31	C₃₁×#464	—	2	4	no	—
32	C₃₁×#452	—	2	4	no	—
33	C₃₁×#440	—	2	4	no	—
34	C₄×D₈×C₃₁	—	2	2	no	—
35	C₃₁×#431	C₂²	1	1	yes	characteristic
36	C₃₁×#427	—	2	2	no	—
37	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
39	C₃₁×#322	C₂²	1	1	yes	characteristic
40	C₃₁×#322	C₂²	1	1	yes	characteristic
41	C₈²×C₃₁	C₂²	1	1	yes	characteristic
42	C₈²×C₃₁	C₂²	1	1	yes	characteristic
43	C₂×#41×C₃₁	—	2	2	no	—
44	C₂×Q₁₆×C₃₁	—	2	4	no	—
45	C₂×SD₁₆×C₃₁	—	2	4	no	—
46	C₂×SD₁₆×C₃₁	—	2	4	no	—
47	C₂×D₈×C₃₁	—	2	2	no	—
48	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
49	C₂²×C₈×C₃₁	D₄	1	2	yes	—
50	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
51	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
52	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
53	C₃₁×#127	C₂³	1	2	yes	—
54	C₃₁×#125	—	2	2	no	—
55	C₃₁×#124	—	2	2	no	—
56	C₃₁×C₂²⋊Q₈	—	2	2	no	—
57	C₃₁×C₂²⋊Q₈	—	2	2	no	—
58	C₃₁×#121	—	2	2	no	—
59	C₃₁×#121	—	2	2	no	—
60	C₄×Q₈×C₃₁	—	2	2	no	—
61	C₄×Q₈×C₃₁	—	2	2	no	—
62	C₄×D₄×C₃₁	—	2	2	no	—
63	C₄×D₄×C₃₁	—	2	2	no	—
64	C₃₁×#117	—	2	2	no	—
65	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
66	C₃₁×#107	—	2	2	no	—
67	C₃₁×#107	—	2	2	no	—
68	C₃₁×#107	—	2	2	no	—
69	C₃₁×#107	—	2	4	no	—
70	C₃₁×#106	—	2	4	no	—
71	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
72	C₃₁×#103	—	2	4	no	—
73	C₃₁×#103	—	2	4	no	—
74	C₃₁×#102	—	2	4	no	—
75	C₃₁×#102	—	2	4	no	—
76	C₃₁×#102	—	2	4	no	—
77	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
78	C₃₁×#97	D₄	1	2	yes	—
79	C₄×C₈×C₃₁	D₄	1	2	yes	—
80	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
82	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
83	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
86	#41×C₃₁	—	8	4	no	—
87	C₂×Q₈×C₃₁	—	2	2	no	—
88	C₂×Q₈×C₃₁	—	2	2	no	—
89	C₂×Q₈×C₃₁	—	2	2	no	—
90	C₂×D₄×C₃₁	—	2	2	no	—
91	C₂×D₄×C₃₁	—	2	2	no	—
92	C₂×D₄×C₃₁	—	2	2	no	—
93	C₂×D₄×C₃₁	—	4	2	no	—
94	C₂²×C₄×C₃₁	—	4	2	no	—
95	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
98	Q₁₆×C₃₁	—	8	4	no	—
99	SD₁₆×C₃₁	—	4	8	no	—
100	SD₁₆×C₃₁	—	8	4	no	—
101	D₈×C₃₁	—	4	4	no	—
102	M₁₆×C₃₁	—	2	4	no	—
103	C₂×C₈×C₃₁	—	2	4	no	—
104	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
110	C₄⋊C₄×C₃₁	—	2	2	no	—
111	C₄⋊C₄×C₃₁	—	2	2	no	—
112	C₄⋊C₄×C₃₁	—	2	2	no	—
113	C₄⋊C₄×C₃₁	—	2	2	no	—
114	C₄⋊C₄×C₃₁	—	2	2	no	—
115	C₄⋊C₄×C₃₁	—	2	2	no	—
116	C₄⋊C₄×C₃₁	—	4	2	no	—
117	C₄⋊C₄×C₃₁	—	4	2	no	—
118	C₂²⋊C₄×C₃₁	—	4	2	no	—
119	C₂²⋊C₄×C₃₁	—	4	2	no	—
120	C₂²⋊C₄×C₃₁	—	4	2	no	—
121	C₄²×C₃₁	—	4	2	no	—
122	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
126	#19646	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂³×C₃₁	—	4	2	no	—
128	C₂³×C₃₁	#127	1	1	yes	characteristic
129	Q₈×C₃₁	—	4	4	no	—
130	Q₈×C₃₁	—	4	4	no	—
131	Q₈×C₃₁	—	4	4	no	—
132	D₄×C₃₁	—	4	4	no	—
133	D₄×C₃₁	—	4	4	no	—
134	D₄×C₃₁	—	4	4	no	—
135	D₄×C₃₁	—	8	4	no	—
136	C₂×C₄×C₃₁	—	2	2	no	—
137	C₂×C₄×C₃₁	—	2	2	no	—
138	C₂×C₄×C₃₁	—	2	2	no	—
139	C₂×C₄×C₃₁	—	4	2	no	—
140	C₂×C₄×C₃₁	—	4	2	no	—
141	C₂×C₄×C₃₁	—	4	2	no	—
142	C₂×C₄×C₃₁	—	8	4	no	—
143	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
144	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
145	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
146	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
149	C₈×C₃₁	—	2	2	no	—
150	C₈×C₃₁	—	2	2	no	—
151	C₈×C₃₁	—	2	2	no	—
152	C₈×C₃₁	—	2	2	no	—
153	C₈×C₃₁	#135	1	4	yes	—
154	C₈×C₃₁	#137	1	4	yes	—
155	#3210	C₂×C₃₁	1	2	yes	complemented
156	#3182	C₂×C₃₁	1	2	yes	complemented
157	#3169	C₂×C₃₁	1	1	yes	characteristic
158	#3160	C₂×C₃₁	1	1	yes	characteristic, complemented
159	#1720	C₂×C₃₁	1	2	yes	complemented
160	#1714	C₂×C₃₁	1	2	yes	complemented
161	#1713	C₂×C₃₁	1	1	yes	characteristic, complemented
162	#1710	C₂×C₃₁	1	1	yes	characteristic, complemented
163	#1706	C₂×C₃₁	1	1	yes	characteristic, complemented
164	#1706	C₂×C₃₁	1	1	yes	characteristic, complemented
165	C₈×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
166	C₂²×C₃₁	—	2	1	no	—
167	C₂²×C₃₁	—	2	1	no	—
168	C₂²×C₃₁	—	2	1	no	—
169	C₂²×C₃₁	—	8	2	no	—
170	C₂²×C₃₁	—	8	2	no	—
171	C₂²×C₃₁	—	8	2	no	—
172	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
173	C₄×C₃₁	—	2	1	no	—
174	C₄×C₃₁	—	2	1	no	—
175	C₄×C₃₁	—	2	1	no	—
176	C₄×C₃₁	—	8	2	no	—
177	C₄×C₃₁	—	8	2	no	—
178	C₄×C₃₁	—	8	2	no	—
179	C₄×C₃₁	C₂×#135	1	2	yes	—
180	C₄×C₃₁	C₂×#137	1	2	yes	—
181	C₄×C₃₁	#576	1	2	yes	—
182	#533	C₂²×C₃₁	1	2	yes	—
183	#532	C₂²×C₃₁	1	2	yes	—
184	#500	C₂²×C₃₁	1	1	yes	characteristic
185	#500	C₂²×C₃₁	1	1	yes	characteristic
186	#500	C₂²×C₃₁	1	1	yes	characteristic
187	#500	C₂²×C₃₁	1	2	yes	—
188	#499	—	2	2	no	—
189	#498	C₂²×C₃₁	1	2	yes	—
190	#494	C₂²×C₃₁	1	2	yes	—
191	#493	C₂²×C₃₁	1	1	yes	characteristic
192	#492	C₂²×C₃₁	1	2	yes	—
193	#492	C₂²×C₃₁	1	2	yes	—
194	#487	C₂²×C₃₁	1	2	yes	—
195	#487	C₂²×C₃₁	1	2	yes	—
196	#486	C₂²×C₃₁	1	2	yes	—
197	#486	C₂²×C₃₁	1	2	yes	—
198	#486	C₂²×C₃₁	1	2	yes	—
199	#468	—	2	4	no	—
200	#466	—	2	2	no	—
201	#464	—	2	4	no	—
202	#452	—	2	4	no	—
203	#440	—	2	4	no	—
204	C₄×D₈	—	2	2	no	—
205	#431	C₂²×C₃₁	1	1	yes	characteristic
206	#427	—	2	2	no	—
207	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
208	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
209	#322	C₂²×C₃₁	1	1	yes	characteristic
210	#322	C₂²×C₃₁	1	1	yes	characteristic
211	C₈²	C₂²×C₃₁	1	1	yes	characteristic
212	C₈²	C₂²×C₃₁	1	1	yes	characteristic
213	C₂×C₃₁	—	4	1	no	—
214	C₂×C₃₁	—	16	2	no	—
215	C₂×C₃₁	#3147	1	1	yes	characteristic
216	C₂×C₃₁	#3152	1	1	yes	characteristic
217	C₂×C₃₁	#3155	1	1	yes	characteristic
218	C₂×#41	—	2	2	no	—
219	C₂×Q₁₆	—	2	4	no	—
220	C₂×SD₁₆	—	2	4	no	—
221	C₂×SD₁₆	—	2	4	no	—
222	C₂×D₈	—	2	2	no	—
223	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
224	C₂²×C₈	D₄×C₃₁	1	2	yes	—
225	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
226	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
227	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
228	#127	C₂³×C₃₁	1	2	yes	—
229	#125	—	2	2	no	—
230	#124	—	2	2	no	—
231	C₂²⋊Q₈	—	2	2	no	—
232	C₂²⋊Q₈	—	2	2	no	—
233	#121	—	2	2	no	—
234	#121	—	2	2	no	—
235	C₄×Q₈	—	2	2	no	—
236	C₄×Q₈	—	2	2	no	—
237	C₄×D₄	—	2	2	no	—
238	C₄×D₄	—	2	2	no	—
239	#117	—	2	2	no	—
240	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
241	#107	—	2	2	no	—
242	#107	—	2	2	no	—
243	#107	—	2	2	no	—
244	#107	—	2	4	no	—
245	#106	—	2	4	no	—
246	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
247	#103	—	2	4	no	—
248	#103	—	2	4	no	—
249	#102	—	2	4	no	—
250	#102	—	2	4	no	—
251	#102	—	2	4	no	—
252	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
253	#97	D₄×C₃₁	1	2	yes	—
254	C₄×C₈	D₄×C₃₁	1	2	yes	—
255	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
256	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
257	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
258	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
259	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
260	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
261	C₃₁	#19646	1	1	yes	characteristic, direct product factor, Sylow
262	#41	—	8	4	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	2	2	no	—
265	C₂×Q₈	—	2	2	no	—
266	C₂×D₄	—	2	2	no	—
267	C₂×D₄	—	2	2	no	—
268	C₂×D₄	—	2	2	no	—
269	C₂×D₄	—	4	2	no	—
270	C₂²×C₄	—	4	2	no	—
271	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
272	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
273	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
274	Q₁₆	—	8	4	no	—
275	SD₁₆	—	4	8	no	—
276	SD₁₆	—	8	4	no	—
277	D₈	—	4	4	no	—
278	M₁₆	—	2	4	no	—
279	C₂×C₈	—	2	4	no	—
280	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
281	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
282	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
283	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
284	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
285	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	4	2	no	—
293	C₄⋊C₄	—	4	2	no	—
294	C₂²⋊C₄	—	4	2	no	—
295	C₂²⋊C₄	—	4	2	no	—
296	C₂²⋊C₄	—	4	2	no	—
297	C₄²	—	4	2	no	—
298	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
299	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
300	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
301	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
302	C₂³	—	4	2	no	—
303	C₂³	C₃₁×#127	1	1	yes	characteristic
304	Q₈	—	4	4	no	—
305	Q₈	—	4	4	no	—
306	Q₈	—	4	4	no	—
307	D₄	—	4	4	no	—
308	D₄	—	4	4	no	—
309	D₄	—	4	4	no	—
310	D₄	—	8	4	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	8	4	no	—
318	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
319	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
320	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
321	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
322	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
323	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
324	C₈	—	2	2	no	—
325	C₈	—	2	2	no	—
326	C₈	—	2	2	no	—
327	C₈	—	2	2	no	—
328	C₈	C₃₁×#135	1	4	yes	—
329	C₈	C₃₁×#137	1	4	yes	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	8	2	no	—
334	C₂²	—	8	2	no	—
335	C₂²	—	8	2	no	—
336	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	8	2	no	—
341	C₄	—	8	2	no	—
342	C₄	—	8	2	no	—
343	C₄	C₂×C₃₁×#135	1	2	yes	—
344	C₄	C₂×C₃₁×#137	1	2	yes	—
345	C₄	C₃₁×#576	1	2	yes	—
346	C₂	—	4	1	no	—
347	C₂	—	16	2	no	—
348	C₂	C₃₁×#3147	1	1	yes	characteristic
349	C₂	C₃₁×#3152	1	1	yes	characteristic
350	C₂	C₃₁×#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3169	C₂
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
1. C₃₁×#3210	C₂	semidirect
2. C₃₁×#3182	C₂	semidirect
4. C₃₁×#3160	C₂	semidirect
5. C₃₁×#1720	C₂	semidirect
6. C₃₁×#1714	C₂	semidirect
7. C₃₁×#1713	C₂	semidirect
8. C₃₁×#1710	C₂	semidirect
9. C₃₁×#1706	C₂	semidirect
10. C₃₁×#1706	C₂	semidirect
11. C₈×M₁₆×C₃₁	C₂	semidirect
126. #19646	C₃₁	direct (N × H)
155. #3210	C₂×C₃₁	semidirect
156. #3182	C₂×C₃₁	semidirect
158. #3160	C₂×C₃₁	semidirect
159. #1720	C₂×C₃₁	semidirect
160. #1714	C₂×C₃₁	semidirect
161. #1713	C₂×C₃₁	semidirect
162. #1710	C₂×C₃₁	semidirect
163. #1706	C₂×C₃₁	semidirect
164. #1706	C₂×C₃₁	semidirect
165. C₈×M₁₆	C₂×C₃₁	semidirect
261. C₃₁	#19646	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×#3210	2
C₃₁×#3182	2
C₃₁×#3169	2
C₃₁×#3160	2
C₃₁×#1720	2
C₃₁×#1714	2
C₃₁×#1713	2
C₃₁×#1710	2
C₃₁×#1706	2
C₈×M₁₆×C₃₁	2
#19646	31

Subgroups — C31×#19646 #21476405

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19646 #21476405