Subgroups — C₃₁×#19333 #21476092

Order 7936.

636 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3254	C₂	1	2	yes	maximal
2	C₃₁×#3170	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2931	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1699	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1673	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1669	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1567	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1561	C₂	1	2	yes	maximal, complemented
9	C₂×C₃₁×#131	—	2	1	no	—
10	C₃₁×#533	C₂²	1	2	yes	—
11	C₃₁×#532	C₂²	1	1	yes	characteristic
12	C₃₁×#500	C₂²	1	1	yes	characteristic
13	C₃₁×#494	C₂²	1	1	yes	characteristic
14	C₃₁×#487	C₂²	1	2	yes	—
15	C₃₁×#477	—	2	2	no	—
16	C₃₁×#474	C₂²	1	4	yes	—
17	C₃₁×#468	—	2	4	no	—
18	C₃₁×#467	—	2	1	no	—
19	C₃₁×#467	—	2	1	no	—
20	C₃₁×#461	C₂²	1	4	yes	—
21	C₃₁×#460	—	2	2	no	—
22	C₄×Q₁₆×C₃₁	—	2	1	no	—
23	C₄×Q₁₆×C₃₁	—	2	1	no	—
24	C₃₁×#436	—	2	1	no	—
25	C₃₁×#436	C₂²	1	2	yes	—
26	C₈×D₄×C₃₁	—	2	1	no	—
27	C₈×D₄×C₃₁	C₂²	1	2	yes	—
28	C₃₁×#423	C₂²	1	1	yes	characteristic
29	C₃₁×#423	C₂²	1	2	yes	—
30	C₃₁×#408	—	2	1	no	—
31	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
32	C₃₁×#335	C₂²	1	2	yes	—
33	C₃₁×#335	C₂²	1	2	yes	—
34	C₃₁×#332	C₂²	1	4	yes	—
35	C₃₁×#332	C₂²	1	4	yes	—
36	C₂×#41×C₃₁	—	2	1	no	—
37	C₂×Q₁₆×C₃₁	—	2	1	no	—
38	C₂×Q₁₆×C₃₁	—	2	1	no	—
39	C₂×SD₁₆×C₃₁	—	2	4	no	—
40	C₃₁×#131	—	4	4	no	—
41	C₂×M₁₆×C₃₁	—	2	1	no	—
42	C₂×M₁₆×C₃₁	—	2	2	no	—
43	C₂×M₁₆×C₃₁	—	2	2	no	—
44	C₂²×C₈×C₃₁	—	2	2	no	—
45	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
47	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
48	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
49	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
50	C₃₁×#127	D₄	1	1	yes	characteristic
51	C₃₁×#125	—	2	2	no	—
52	C₃₁×#124	—	2	1	no	—
53	C₃₁×C₂²⋊Q₈	—	2	2	no	—
54	C₃₁×C₂²⋊Q₈	—	2	2	no	—
55	C₃₁×#121	—	2	1	no	—
56	C₃₁×#121	D₄	1	2	yes	—
57	C₄×Q₈×C₃₁	—	2	2	no	—
58	C₄×Q₈×C₃₁	—	2	2	no	—
59	C₄×D₄×C₃₁	—	2	1	no	—
60	C₄×D₄×C₃₁	C₂³	1	2	yes	—
61	C₃₁×#117	—	2	2	no	—
62	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
63	C₃₁×#107	—	2	1	no	—
64	C₃₁×#107	—	2	1	no	—
65	C₃₁×#107	—	2	4	no	—
66	C₃₁×C₄⋊C₈	—	2	1	no	—
67	C₃₁×C₄⋊C₈	—	2	2	no	—
68	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
69	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
70	C₃₁×#103	—	2	2	no	—
71	C₃₁×#103	—	2	2	no	—
72	C₃₁×#103	—	2	4	no	—
73	C₃₁×#102	—	2	4	no	—
74	C₃₁×C₂²⋊C₈	—	2	2	no	—
75	C₃₁×C₂²⋊C₈	—	2	2	no	—
76	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
77	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
78	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
80	#41×C₃₁	—	4	2	no	—
81	#41×C₃₁	—	4	2	no	—
82	C₂×Q₈×C₃₁	—	2	1	no	—
83	C₂×Q₈×C₃₁	—	2	2	no	—
84	C₂×Q₈×C₃₁	—	2	2	no	—
85	C₂×D₄×C₃₁	—	2	2	no	—
86	C₂×D₄×C₃₁	—	2	1	no	—
87	C₂×D₄×C₃₁	—	2	1	no	—
88	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₃₁	—	2	2	no	—
90	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
93	Q₁₆×C₃₁	—	4	2	no	—
94	Q₁₆×C₃₁	—	4	2	no	—
95	SD₁₆×C₃₁	—	8	4	no	—
96	M₁₆×C₃₁	—	4	4	no	—
97	M₁₆×C₃₁	—	4	2	no	—
98	M₁₆×C₃₁	—	8	2	no	—
99	C₂×C₈×C₃₁	—	2	2	no	—
100	C₂×C₈×C₃₁	—	2	2	no	—
101	C₂×C₈×C₃₁	—	2	4	no	—
102	C₂×C₈×C₃₁	—	4	1	no	—
103	C₂×C₈×C₃₁	—	4	4	no	—
104	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
108	C₄⋊C₄×C₃₁	—	2	1	no	—
109	C₄⋊C₄×C₃₁	—	2	2	no	—
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	—	4	2	no	—
114	C₄⋊C₄×C₃₁	—	4	2	no	—
115	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
116	C₂²⋊C₄×C₃₁	—	2	2	no	—
117	C₂²⋊C₄×C₃₁	—	2	2	no	—
118	C₂²⋊C₄×C₃₁	—	4	2	no	—
119	C₄²×C₃₁	—	4	2	no	—
120	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₃₁	#41	1	1	yes	characteristic
123	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
124	#19333	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂³×C₃₁	—	2	2	no	—
126	C₂³×C₃₁	#121	1	1	yes	characteristic
127	Q₈×C₃₁	—	4	2	no	—
128	Q₈×C₃₁	—	4	2	no	—
129	Q₈×C₃₁	—	4	2	no	—
130	Q₈×C₃₁	—	4	2	no	—
131	Q₈×C₃₁	—	4	2	no	—
132	D₄×C₃₁	—	2	2	no	—
133	D₄×C₃₁	—	2	2	no	—
134	D₄×C₃₁	—	4	2	no	—
135	D₄×C₃₁	—	4	4	no	—
136	D₄×C₃₁	—	4	2	no	—
137	C₂×C₄×C₃₁	—	2	1	no	—
138	C₂×C₄×C₃₁	—	2	1	no	—
139	C₂×C₄×C₃₁	—	2	1	no	—
140	C₂×C₄×C₃₁	—	2	1	no	—
141	C₂×C₄×C₃₁	—	2	2	no	—
142	C₂×C₄×C₃₁	—	2	2	no	—
143	C₂×C₄×C₃₁	—	4	2	no	—
144	C₂×C₄×C₃₁	—	4	2	no	—
145	C₂×C₄×C₃₁	—	4	2	no	—
146	C₂×C₄×C₃₁	—	4	2	no	—
147	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
148	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
149	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
150	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
153	C₈×C₃₁	—	2	1	no	—
154	C₈×C₃₁	—	2	1	no	—
155	C₈×C₃₁	—	2	1	no	—
156	C₈×C₃₁	—	4	2	no	—
157	C₈×C₃₁	—	4	4	no	—
158	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
159	#3254	C₂×C₃₁	1	2	yes	—
160	#3170	C₂×C₃₁	1	1	yes	characteristic, complemented
161	#2931	C₂×C₃₁	1	1	yes	characteristic
162	#1699	C₂×C₃₁	1	4	yes	complemented
163	#1673	C₂×C₃₁	1	2	yes	complemented
164	#1669	C₂×C₃₁	1	2	yes	complemented
165	#1567	C₂×C₃₁	1	1	yes	characteristic, complemented
166	#1561	C₂×C₃₁	1	2	yes	complemented
167	C₂²×C₃₁	—	2	1	no	—
168	C₂²×C₃₁	—	2	1	no	—
169	C₂²×C₃₁	—	2	1	no	—
170	C₂²×C₃₁	—	4	2	no	—
171	C₂²×C₃₁	—	4	2	no	—
172	C₂²×C₃₁	—	4	2	no	—
173	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
174	C₄×C₃₁	—	2	1	no	—
175	C₄×C₃₁	—	2	1	no	—
176	C₄×C₃₁	—	2	1	no	—
177	C₄×C₃₁	—	4	2	no	—
178	C₄×C₃₁	—	8	2	no	—
179	C₄×C₃₁	—	8	2	no	—
180	C₄×C₃₁	#473	1	2	yes	—
181	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
182	C₄×C₃₁	#575	1	1	yes	characteristic
183	C₄×C₃₁	#576	1	1	yes	characteristic
184	C₄×C₃₁	#578	1	1	yes	characteristic
185	C₂×#131	—	2	1	no	—
186	#533	C₂²×C₃₁	1	2	yes	—
187	#532	C₂²×C₃₁	1	1	yes	characteristic
188	#500	C₂²×C₃₁	1	1	yes	characteristic
189	#494	C₂²×C₃₁	1	1	yes	characteristic
190	#487	C₂²×C₃₁	1	2	yes	—
191	#477	—	2	2	no	—
192	#474	C₂²×C₃₁	1	4	yes	—
193	#468	—	2	4	no	—
194	#467	—	2	1	no	—
195	#467	—	2	1	no	—
196	#461	C₂²×C₃₁	1	4	yes	—
197	#460	—	2	2	no	—
198	C₄×Q₁₆	—	2	1	no	—
199	C₄×Q₁₆	—	2	1	no	—
200	#436	—	2	1	no	—
201	#436	C₂²×C₃₁	1	2	yes	—
202	C₈×D₄	—	2	1	no	—
203	C₈×D₄	C₂²×C₃₁	1	2	yes	—
204	#423	C₂²×C₃₁	1	1	yes	characteristic
205	#423	C₂²×C₃₁	1	2	yes	—
206	#408	—	2	1	no	—
207	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
208	#335	C₂²×C₃₁	1	2	yes	—
209	#335	C₂²×C₃₁	1	2	yes	—
210	#332	C₂²×C₃₁	1	4	yes	—
211	#332	C₂²×C₃₁	1	4	yes	—
212	C₂×C₃₁	—	4	1	no	—
213	C₂×C₃₁	—	8	2	no	—
214	C₂×C₃₁	#3036	1	1	yes	characteristic
215	C₂×C₃₁	#3061	1	1	yes	characteristic
216	C₂×C₃₁	C₂×#473	1	1	yes	characteristic
217	C₂×#41	—	2	1	no	—
218	C₂×Q₁₆	—	2	1	no	—
219	C₂×Q₁₆	—	2	1	no	—
220	C₂×SD₁₆	—	2	4	no	—
221	#131	—	4	4	no	—
222	C₂×M₁₆	—	2	1	no	—
223	C₂×M₁₆	—	2	2	no	—
224	C₂×M₁₆	—	2	2	no	—
225	C₂²×C₈	—	2	2	no	—
226	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
227	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
228	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
229	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
230	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
231	#127	D₄×C₃₁	1	1	yes	characteristic
232	#125	—	2	2	no	—
233	#124	—	2	1	no	—
234	C₂²⋊Q₈	—	2	2	no	—
235	C₂²⋊Q₈	—	2	2	no	—
236	#121	—	2	1	no	—
237	#121	D₄×C₃₁	1	2	yes	—
238	C₄×Q₈	—	2	2	no	—
239	C₄×Q₈	—	2	2	no	—
240	C₄×D₄	—	2	1	no	—
241	C₄×D₄	C₂³×C₃₁	1	2	yes	—
242	#117	—	2	2	no	—
243	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
244	#107	—	2	1	no	—
245	#107	—	2	1	no	—
246	#107	—	2	4	no	—
247	C₄⋊C₈	—	2	1	no	—
248	C₄⋊C₈	—	2	2	no	—
249	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
250	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
251	#103	—	2	2	no	—
252	#103	—	2	2	no	—
253	#103	—	2	4	no	—
254	#102	—	2	4	no	—
255	C₂²⋊C₈	—	2	2	no	—
256	C₂²⋊C₈	—	2	2	no	—
257	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
258	C₄×C₈	D₄×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	#19333	1	1	yes	characteristic, direct product factor, Sylow
262	#41	—	4	2	no	—
263	#41	—	4	2	no	—
264	C₂×Q₈	—	2	1	no	—
265	C₂×Q₈	—	2	2	no	—
266	C₂×Q₈	—	2	2	no	—
267	C₂×D₄	—	2	2	no	—
268	C₂×D₄	—	2	1	no	—
269	C₂×D₄	—	2	1	no	—
270	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
271	C₂²×C₄	—	2	2	no	—
272	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
273	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
274	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
275	Q₁₆	—	4	2	no	—
276	Q₁₆	—	4	2	no	—
277	SD₁₆	—	8	4	no	—
278	M₁₆	—	4	4	no	—
279	M₁₆	—	4	2	no	—
280	M₁₆	—	8	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	4	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	4	4	no	—
286	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
287	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
288	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
289	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	4	2	no	—
296	C₄⋊C₄	—	4	2	no	—
297	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
298	C₂²⋊C₄	—	2	2	no	—
299	C₂²⋊C₄	—	2	2	no	—
300	C₂²⋊C₄	—	4	2	no	—
301	C₄²	—	4	2	no	—
302	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
303	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
304	C₄²	#41×C₃₁	1	1	yes	characteristic
305	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
306	C₂³	—	2	2	no	—
307	C₂³	C₃₁×#121	1	1	yes	characteristic
308	Q₈	—	4	2	no	—
309	Q₈	—	4	2	no	—
310	Q₈	—	4	2	no	—
311	Q₈	—	4	2	no	—
312	Q₈	—	4	2	no	—
313	D₄	—	2	2	no	—
314	D₄	—	2	2	no	—
315	D₄	—	4	2	no	—
316	D₄	—	4	4	no	—
317	D₄	—	4	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
329	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
330	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
331	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
332	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
333	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	2	1	no	—
337	C₈	—	4	2	no	—
338	C₈	—	4	4	no	—
339	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	4	2	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	4	2	no	—
346	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	4	2	no	—
351	C₄	—	8	2	no	—
352	C₄	—	8	2	no	—
353	C₄	C₃₁×#473	1	2	yes	—
354	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
355	C₄	C₃₁×#575	1	1	yes	characteristic
356	C₄	C₃₁×#576	1	1	yes	characteristic
357	C₄	C₃₁×#578	1	1	yes	characteristic
358	C₂	—	4	1	no	—
359	C₂	—	8	2	no	—
360	C₂	C₃₁×#3036	1	1	yes	characteristic
361	C₂	C₃₁×#3061	1	1	yes	characteristic
362	C₂	C₂×C₃₁×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×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
2. C₃₁×#3170	C₂	semidirect
4. C₃₁×#1699	C₂	semidirect
5. C₃₁×#1673	C₂	semidirect
6. C₃₁×#1669	C₂	semidirect
7. C₃₁×#1567	C₂	semidirect
8. C₃₁×#1561	C₂	semidirect
124. #19333	C₃₁	direct (N × H)
160. #3170	C₂×C₃₁	semidirect
162. #1699	C₂×C₃₁	semidirect
163. #1673	C₂×C₃₁	semidirect
164. #1669	C₂×C₃₁	semidirect
165. #1567	C₂×C₃₁	semidirect
166. #1561	C₂×C₃₁	semidirect
261. C₃₁	#19333	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×#3254	2
C₃₁×#3170	2
C₃₁×#2931	2
C₃₁×#1699	2
C₃₁×#1673	2
C₃₁×#1669	2
C₃₁×#1567	2
C₃₁×#1561	2
#19333	31

Subgroups — C31×#19333 #21476092

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19333 #21476092