Subgroups — C₃₁×#8652 #21465411

Order 7936.

510 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#436	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1845	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1802	C₂	1	2	yes	maximal
4	C₂×C₃₁×#334	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1296	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆×C₃₁	—	2	1	no	—
7	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
8	C₃₁×#436	—	2	2	no	—
9	C₃₁×#436	C₄	1	4	yes	complemented
10	C₂×C₃₁×#107	—	2	1	no	—
11	C₂×C₃₁×#107	—	2	1	no	—
12	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#102	—	2	2	no	—
15	C₂×C₃₁×C₂²⋊C₈	—	2	1	no	—
16	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#384	C₂²	1	2	yes	—
18	C₃₁×#340	—	2	2	no	—
19	C₃₁×#334	—	2	2	no	—
20	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₃₁	—	2	1	no	—
22	C₂×M₁₆×C₃₁	—	2	4	no	—
23	C₂×M₁₆×C₃₁	—	4	2	no	—
24	C₂×M₁₆×C₃₁	—	4	2	no	—
25	C₂²×C₈×C₃₁	—	2	1	no	—
26	C₂²×C₈×C₃₁	—	2	2	no	—
27	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
29	C₄×D₄×C₃₁	—	2	1	no	—
30	C₄×D₄×C₃₁	—	2	1	no	—
31	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
32	C₄×D₄×C₃₁	Q₈	1	2	yes	—
33	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
34	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
38	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
39	C₃₁×#107	—	4	2	no	—
40	C₃₁×#107	—	4	2	no	—
41	C₃₁×C₄⋊C₈	—	2	1	no	—
42	C₃₁×C₄⋊C₈	—	2	2	no	—
43	C₃₁×C₄⋊C₈	—	2	2	no	—
44	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
45	C₃₁×#102	—	4	4	no	—
46	C₃₁×C₂²⋊C₈	—	2	4	no	—
47	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₃₁	D₄	1	2	yes	—
49	C₂⁴×C₃₁	—	2	1	no	—
50	C₂×D₄×C₃₁	—	2	1	no	—
51	C₂×D₄×C₃₁	—	2	2	no	—
52	C₂×D₄×C₃₁	—	2	2	no	—
53	C₂×D₄×C₃₁	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₃₁	—	2	1	no	—
55	C₂²×C₄×C₃₁	—	2	2	no	—
56	C₂²×C₄×C₃₁	—	2	2	no	—
57	C₂²×C₄×C₃₁	—	4	1	no	—
58	C₂²×C₄×C₃₁	—	4	1	no	—
59	C₂²×C₄×C₃₁	—	4	1	no	—
60	C₂²×C₄×C₃₁	—	4	1	no	—
61	C₂²×C₄×C₃₁	—	4	2	no	—
62	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
65	M₁₆×C₃₁	—	4	8	no	—
66	C₂×C₈×C₃₁	—	2	1	no	—
67	C₂×C₈×C₃₁	—	2	1	no	—
68	C₂×C₈×C₃₁	—	2	2	no	—
69	C₂×C₈×C₃₁	—	4	1	no	—
70	C₂×C₈×C₃₁	—	4	1	no	—
71	C₂×C₈×C₃₁	—	4	2	no	—
72	C₂×C₈×C₃₁	—	4	2	no	—
73	C₂×C₈×C₃₁	—	4	2	no	—
74	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
75	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
76	C₂×C₈×C₃₁	D₈	1	2	yes	—
77	C₂×C₈×C₃₁	SD₁₆	1	2	yes	—
78	C₄⋊C₄×C₃₁	—	2	1	no	—
79	C₄⋊C₄×C₃₁	—	4	2	no	—
80	C₄⋊C₄×C₃₁	—	4	2	no	—
81	C₄⋊C₄×C₃₁	—	4	2	no	—
82	C₄⋊C₄×C₃₁	—	4	2	no	—
83	C₄⋊C₄×C₃₁	—	8	2	no	—
84	C₄⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
85	C₂²⋊C₄×C₃₁	—	2	2	no	—
86	C₂²⋊C₄×C₃₁	—	2	2	no	—
87	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
90	C₄²×C₃₁	#41	1	1	yes	characteristic
91	#8652	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂³×C₃₁	—	2	2	no	—
93	C₂³×C₃₁	—	4	1	no	—
94	C₂³×C₃₁	—	4	1	no	—
95	C₂³×C₃₁	—	4	1	no	—
96	C₂³×C₃₁	—	4	1	no	—
97	C₂³×C₃₁	—	4	1	no	—
98	C₂³×C₃₁	—	4	1	no	—
99	C₂³×C₃₁	#121	1	1	yes	characteristic
100	D₄×C₃₁	—	2	4	no	—
101	D₄×C₃₁	—	4	2	no	—
102	C₂×C₄×C₃₁	—	2	1	no	—
103	C₂×C₄×C₃₁	—	2	1	no	—
104	C₂×C₄×C₃₁	—	2	2	no	—
105	C₂×C₄×C₃₁	—	4	1	no	—
106	C₂×C₄×C₃₁	—	4	1	no	—
107	C₂×C₄×C₃₁	—	4	2	no	—
108	C₂×C₄×C₃₁	—	4	2	no	—
109	C₂×C₄×C₃₁	—	4	2	no	—
110	C₂×C₄×C₃₁	—	4	2	no	—
111	C₂×C₄×C₃₁	—	8	2	no	—
112	C₂×C₄×C₃₁	—	8	2	no	—
113	C₂×C₄×C₃₁	—	8	2	no	—
114	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
117	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
118	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
122	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
124	C₂×C₄×C₃₁	#137	1	1	yes	characteristic
125	C₂×C₄×C₃₁	#137	1	2	yes	—
126	C₈×C₃₁	—	2	2	no	—
127	C₈×C₃₁	—	4	2	no	—
128	C₈×C₃₁	—	8	2	no	—
129	C₈×C₃₁	#102	1	4	yes	complemented
130	C₂×#436	C₂×C₃₁	1	1	yes	characteristic
131	#1845	C₂×C₃₁	1	1	yes	characteristic, complemented
132	#1802	C₂×C₃₁	1	2	yes	—
133	C₂×#334	C₂×C₃₁	1	1	yes	characteristic, complemented
134	#1296	C₂×C₃₁	1	2	yes	complemented
135	C₂²×C₃₁	—	4	2	no	—
136	C₂²×C₃₁	—	4	2	no	—
137	C₂²×C₃₁	—	4	2	no	—
138	C₂²×C₃₁	—	8	1	no	—
139	C₂²×C₃₁	—	8	1	no	—
140	C₂²×C₃₁	—	8	1	no	—
141	C₂²×C₃₁	—	8	1	no	—
142	C₂²×C₃₁	#380	1	1	yes	characteristic
143	C₂²×C₃₁	#459	1	1	yes	characteristic
144	C₂²×C₃₁	#461	1	1	yes	characteristic
145	C₂²×C₃₁	#469	1	1	yes	characteristic
146	C₂²×C₃₁	#470	1	1	yes	characteristic
147	C₂²×C₃₁	#472	1	1	yes	characteristic
148	C₂²×C₃₁	#473	1	1	yes	characteristic
149	C₄×C₃₁	—	2	1	no	—
150	C₄×C₃₁	—	2	1	no	—
151	C₄×C₃₁	—	4	2	no	—
152	C₄×C₃₁	—	16	2	no	—
153	C₄×C₃₁	C₂×#102	1	1	yes	characteristic
154	C₄×C₃₁	#417	1	1	yes	characteristic
155	C₄×C₃₁	#428	1	2	yes	—
156	C₄×C₃₁	#468	1	2	yes	—
157	C₄×C₃₁	#474	1	1	yes	characteristic
158	C₄×C₃₁	#476	1	1	yes	characteristic
159	C₂²×M₁₆	—	2	1	no	—
160	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
161	#436	—	2	2	no	—
162	#436	C₄×C₃₁	1	4	yes	complemented
163	C₂×#107	—	2	1	no	—
164	C₂×#107	—	2	1	no	—
165	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
166	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
167	C₂×#102	—	2	2	no	—
168	C₂×C₂²⋊C₈	—	2	1	no	—
169	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
170	#384	C₂²×C₃₁	1	2	yes	—
171	#340	—	2	2	no	—
172	#334	—	2	2	no	—
173	C₂×C₃₁	—	8	2	no	—
174	C₂×C₃₁	#1687	1	1	yes	characteristic
175	C₂×C₃₁	#1689	1	1	yes	characteristic
176	C₂×C₃₁	#1693	1	1	yes	characteristic
177	C₂×C₃₁	#1695	1	1	yes	characteristic
178	C₂×C₃₁	#1802	1	1	yes	characteristic
179	C₂×C₃₁	#1809	1	1	yes	characteristic
180	C₂×C₃₁	#1816	1	1	yes	characteristic
181	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
182	C₂³×C₄	—	2	1	no	—
183	C₂×M₁₆	—	2	4	no	—
184	C₂×M₁₆	—	4	2	no	—
185	C₂×M₁₆	—	4	2	no	—
186	C₂²×C₈	—	2	1	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
189	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
190	C₄×D₄	—	2	1	no	—
191	C₄×D₄	—	2	1	no	—
192	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
193	C₄×D₄	Q₈×C₃₁	1	2	yes	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
198	C₂×C₂²⋊C₄	—	2	1	no	—
199	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
200	#107	—	4	2	no	—
201	#107	—	4	2	no	—
202	C₄⋊C₈	—	2	1	no	—
203	C₄⋊C₈	—	2	2	no	—
204	C₄⋊C₈	—	2	2	no	—
205	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
206	#102	—	4	4	no	—
207	C₂²⋊C₈	—	2	4	no	—
208	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
209	C₄×C₈	D₄×C₃₁	1	2	yes	—
210	C₃₁	#8652	1	1	yes	characteristic, direct product factor, Sylow
211	C₂⁴	—	2	1	no	—
212	C₂×D₄	—	2	1	no	—
213	C₂×D₄	—	2	2	no	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	C₄⋊C₄×C₃₁	1	2	yes	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	4	1	no	—
220	C₂²×C₄	—	4	1	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
225	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
226	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
227	M₁₆	—	4	8	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	—	4	2	no	—
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
237	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
238	C₂×C₈	D₈×C₃₁	1	2	yes	—
239	C₂×C₈	SD₁₆×C₃₁	1	2	yes	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	8	2	no	—
246	C₄⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
247	C₂²⋊C₄	—	2	2	no	—
248	C₂²⋊C₄	—	2	2	no	—
249	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
250	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
251	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
252	C₄²	#41×C₃₁	1	1	yes	characteristic
253	C₂³	—	2	2	no	—
254	C₂³	—	4	1	no	—
255	C₂³	—	4	1	no	—
256	C₂³	—	4	1	no	—
257	C₂³	—	4	1	no	—
258	C₂³	—	4	1	no	—
259	C₂³	—	4	1	no	—
260	C₂³	C₃₁×#121	1	1	yes	characteristic
261	D₄	—	2	4	no	—
262	D₄	—	4	2	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	8	2	no	—
273	C₂×C₄	—	8	2	no	—
274	C₂×C₄	—	8	2	no	—
275	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
276	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
277	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
278	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
279	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
280	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
281	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
282	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
283	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
284	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
285	C₂×C₄	C₃₁×#137	1	1	yes	characteristic
286	C₂×C₄	C₃₁×#137	1	2	yes	—
287	C₈	—	2	2	no	—
288	C₈	—	4	2	no	—
289	C₈	—	8	2	no	—
290	C₈	C₃₁×#102	1	4	yes	complemented
291	C₂²	—	4	2	no	—
292	C₂²	—	4	2	no	—
293	C₂²	—	4	2	no	—
294	C₂²	—	8	1	no	—
295	C₂²	—	8	1	no	—
296	C₂²	—	8	1	no	—
297	C₂²	—	8	1	no	—
298	C₂²	C₃₁×#380	1	1	yes	characteristic
299	C₂²	C₃₁×#459	1	1	yes	characteristic
300	C₂²	C₃₁×#461	1	1	yes	characteristic
301	C₂²	C₃₁×#469	1	1	yes	characteristic
302	C₂²	C₃₁×#470	1	1	yes	characteristic
303	C₂²	C₃₁×#472	1	1	yes	characteristic
304	C₂²	C₃₁×#473	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	4	2	no	—
308	C₄	—	16	2	no	—
309	C₄	C₂×C₃₁×#102	1	1	yes	characteristic
310	C₄	C₃₁×#417	1	1	yes	characteristic
311	C₄	C₃₁×#428	1	2	yes	—
312	C₄	C₃₁×#468	1	2	yes	—
313	C₄	C₃₁×#474	1	1	yes	characteristic
314	C₄	C₃₁×#476	1	1	yes	characteristic
315	C₂	—	8	2	no	—
316	C₂	C₃₁×#1687	1	1	yes	characteristic
317	C₂	C₃₁×#1689	1	1	yes	characteristic
318	C₂	C₃₁×#1693	1	1	yes	characteristic
319	C₂	C₃₁×#1695	1	1	yes	characteristic
320	C₂	C₃₁×#1802	1	1	yes	characteristic
321	C₂	C₃₁×#1809	1	1	yes	characteristic
322	C₂	C₃₁×#1816	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	#121
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₂²×D₄×C₃₁	D₄
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×#1845	C₂	semidirect
4. C₂×C₃₁×#334	C₂	semidirect
5. C₃₁×#1296	C₂	semidirect
9. C₃₁×#436	C₄	semidirect
91. #8652	C₃₁	direct (N × H)
129. C₈×C₃₁	#102	semidirect
131. #1845	C₂×C₃₁	semidirect
133. C₂×#334	C₂×C₃₁	semidirect
134. #1296	C₂×C₃₁	semidirect
162. #436	C₄×C₃₁	semidirect
210. C₃₁	#8652	direct (N × H)
290. C₈	C₃₁×#102	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#436	2
C₃₁×#1845	2
C₃₁×#1802	2
C₂×C₃₁×#334	2
C₃₁×#1296	2
#8652	31

Subgroups — C31×#8652 #21465411

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8652 #21465411