Subgroups — C₃₁×#19687 #21476446

Order 7936.

764 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×#3228	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3182	C₂	1	2	yes	maximal, complemented
3	C₃₁×#3170	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3147	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1723	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1722	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1717	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1714	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1451	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#135	—	2	2	no	—
11	C₃₁×#533	C₂²	1	2	yes	—
12	C₃₁×#532	C₂²	1	2	yes	—
13	C₃₁×#500	C₂²	1	1	yes	characteristic
14	C₃₁×#500	C₂²	1	2	yes	—
15	C₃₁×#500	C₂²	1	2	yes	—
16	C₃₁×#498	C₂²	1	2	yes	—
17	C₃₁×#495	—	2	2	no	—
18	C₃₁×#494	C₂²	1	1	yes	characteristic
19	C₃₁×#494	C₂²	1	1	yes	characteristic
20	C₃₁×#493	C₂²	1	1	yes	characteristic
21	C₃₁×#492	C₂²	1	2	yes	—
22	C₃₁×#492	C₂²	1	2	yes	—
23	C₃₁×#487	C₂²	1	2	yes	—
24	C₃₁×#487	C₂²	1	2	yes	—
25	C₃₁×#486	C₂²	1	2	yes	—
26	C₃₁×#486	C₂²	1	2	yes	—
27	C₃₁×#486	C₂²	1	2	yes	—
28	C₃₁×#482	—	2	4	no	—
29	C₃₁×#478	—	2	4	no	—
30	C₃₁×#468	—	2	4	no	—
31	C₃₁×#467	—	2	2	no	—
32	C₃₁×#440	—	2	4	no	—
33	C₄×Q₁₆×C₃₁	—	2	2	no	—
34	C₃₁×#431	C₂²	1	1	yes	characteristic
35	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#322	C₂²	1	2	yes	—
38	C₃₁×#322	C₂²	1	2	yes	—
39	C₂×#41×C₃₁	—	2	2	no	—
40	C₃₁×#135	—	4	8	no	—
41	C₂×Q₁₆×C₃₁	—	2	2	no	—
42	C₂×Q₁₆×C₃₁	—	2	2	no	—
43	C₂×SD₁₆×C₃₁	—	2	4	no	—
44	C₂×SD₁₆×C₃₁	—	2	4	no	—
45	C₂×D₈×C₃₁	—	2	2	no	—
46	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
47	C₂²×C₈×C₃₁	D₄	1	2	yes	—
48	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
49	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
50	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
51	C₃₁×#127	C₂³	1	2	yes	—
52	C₃₁×#125	—	2	2	no	—
53	C₃₁×#124	—	2	2	no	—
54	C₃₁×C₂²⋊Q₈	—	2	2	no	—
55	C₃₁×C₂²⋊Q₈	—	2	2	no	—
56	C₃₁×#121	—	2	2	no	—
57	C₃₁×#121	—	2	2	no	—
58	C₄×Q₈×C₃₁	—	2	2	no	—
59	C₄×Q₈×C₃₁	—	2	2	no	—
60	C₄×D₄×C₃₁	—	2	2	no	—
61	C₄×D₄×C₃₁	—	2	2	no	—
62	C₃₁×#117	—	2	2	no	—
63	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
64	C₃₁×#107	—	2	2	no	—
65	C₃₁×#107	—	2	4	no	—
66	C₃₁×#107	—	2	4	no	—
67	C₃₁×#106	—	2	4	no	—
68	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
69	C₃₁×#103	—	2	4	no	—
70	C₃₁×#103	—	2	4	no	—
71	C₃₁×#102	—	2	4	no	—
72	C₃₁×#102	—	2	4	no	—
73	C₃₁×#102	—	2	4	no	—
74	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
75	C₃₁×#97	D₄	1	2	yes	—
76	C₄×C₈×C₃₁	D₄	1	2	yes	—
77	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₃₁	C₂³	1	2	yes	—
80	C₄×C₈×C₃₁	C₂³	1	2	yes	—
81	#41×C₃₁	—	4	8	no	—
82	C₂×Q₈×C₃₁	—	2	2	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	2	no	—
87	C₂×D₄×C₃₁	—	2	2	no	—
88	C₂×D₄×C₃₁	—	4	2	no	—
89	C₂²×C₄×C₃₁	—	4	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	C₂×D₄	1	1	yes	characteristic
93	Q₁₆×C₃₁	—	4	4	no	—
94	Q₁₆×C₃₁	—	4	4	no	—
95	SD₁₆×C₃₁	—	4	8	no	—
96	SD₁₆×C₃₁	—	4	8	no	—
97	D₈×C₃₁	—	4	4	no	—
98	M₁₆×C₃₁	—	4	2	no	—
99	C₂×C₈×C₃₁	—	2	4	no	—
100	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
102	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
103	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
104	C₄⋊C₄×C₃₁	—	2	2	no	—
105	C₄⋊C₄×C₃₁	—	2	2	no	—
106	C₄⋊C₄×C₃₁	—	2	2	no	—
107	C₄⋊C₄×C₃₁	—	2	2	no	—
108	C₄⋊C₄×C₃₁	—	2	2	no	—
109	C₄⋊C₄×C₃₁	—	2	2	no	—
110	C₄⋊C₄×C₃₁	—	4	2	no	—
111	C₄⋊C₄×C₃₁	—	4	2	no	—
112	C₂²⋊C₄×C₃₁	—	4	2	no	—
113	C₂²⋊C₄×C₃₁	—	4	2	no	—
114	C₂²⋊C₄×C₃₁	—	4	2	no	—
115	C₄²×C₃₁	—	4	2	no	—
116	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
120	#19687	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂³×C₃₁	—	4	2	no	—
122	C₂³×C₃₁	#127	1	1	yes	characteristic
123	Q₈×C₃₁	—	4	4	no	—
124	Q₈×C₃₁	—	4	4	no	—
125	Q₈×C₃₁	—	4	4	no	—
126	D₄×C₃₁	—	4	4	no	—
127	D₄×C₃₁	—	4	4	no	—
128	D₄×C₃₁	—	4	4	no	—
129	D₄×C₃₁	—	8	4	no	—
130	C₂×C₄×C₃₁	—	2	2	no	—
131	C₂×C₄×C₃₁	—	2	2	no	—
132	C₂×C₄×C₃₁	—	2	2	no	—
133	C₂×C₄×C₃₁	—	4	2	no	—
134	C₂×C₄×C₃₁	—	4	2	no	—
135	C₂×C₄×C₃₁	—	4	2	no	—
136	C₂×C₄×C₃₁	—	8	4	no	—
137	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
138	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
139	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
140	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
143	C₈×C₃₁	—	2	2	no	—
144	C₈×C₃₁	—	2	4	no	—
145	C₈×C₃₁	—	2	4	no	—
146	C₈×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
147	#3228	C₂×C₃₁	1	2	yes	complemented
148	#3182	C₂×C₃₁	1	2	yes	complemented
149	#3170	C₂×C₃₁	1	1	yes	characteristic, complemented
150	#3147	C₂×C₃₁	1	1	yes	characteristic
151	#1723	C₂×C₃₁	1	2	yes	complemented
152	#1722	C₂×C₃₁	1	2	yes	complemented
153	#1717	C₂×C₃₁	1	2	yes	complemented
154	#1714	C₂×C₃₁	1	2	yes	complemented
155	#1451	C₂×C₃₁	1	1	yes	characteristic, complemented
156	C₂²×C₃₁	—	2	1	no	—
157	C₂²×C₃₁	—	2	1	no	—
158	C₂²×C₃₁	—	2	1	no	—
159	C₂²×C₃₁	—	8	2	no	—
160	C₂²×C₃₁	—	8	2	no	—
161	C₂²×C₃₁	—	8	2	no	—
162	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
163	C₄×C₃₁	—	2	1	no	—
164	C₄×C₃₁	—	2	1	no	—
165	C₄×C₃₁	—	2	1	no	—
166	C₄×C₃₁	—	8	2	no	—
167	C₄×C₃₁	—	8	2	no	—
168	C₄×C₃₁	—	8	2	no	—
169	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
170	C₄×C₃₁	#575	1	2	yes	—
171	C₄×C₃₁	#578	1	2	yes	—
172	C₂×#135	—	2	2	no	—
173	#533	C₂²×C₃₁	1	2	yes	—
174	#532	C₂²×C₃₁	1	2	yes	—
175	#500	C₂²×C₃₁	1	1	yes	characteristic
176	#500	C₂²×C₃₁	1	2	yes	—
177	#500	C₂²×C₃₁	1	2	yes	—
178	#498	C₂²×C₃₁	1	2	yes	—
179	#495	—	2	2	no	—
180	#494	C₂²×C₃₁	1	1	yes	characteristic
181	#494	C₂²×C₃₁	1	1	yes	characteristic
182	#493	C₂²×C₃₁	1	1	yes	characteristic
183	#492	C₂²×C₃₁	1	2	yes	—
184	#492	C₂²×C₃₁	1	2	yes	—
185	#487	C₂²×C₃₁	1	2	yes	—
186	#487	C₂²×C₃₁	1	2	yes	—
187	#486	C₂²×C₃₁	1	2	yes	—
188	#486	C₂²×C₃₁	1	2	yes	—
189	#486	C₂²×C₃₁	1	2	yes	—
190	#482	—	2	4	no	—
191	#478	—	2	4	no	—
192	#468	—	2	4	no	—
193	#467	—	2	2	no	—
194	#440	—	2	4	no	—
195	C₄×Q₁₆	—	2	2	no	—
196	#431	C₂²×C₃₁	1	1	yes	characteristic
197	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
198	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
199	#322	C₂²×C₃₁	1	2	yes	—
200	#322	C₂²×C₃₁	1	2	yes	—
201	C₂×C₃₁	—	4	1	no	—
202	C₂×C₃₁	—	16	2	no	—
203	C₂×C₃₁	#3150	1	1	yes	characteristic
204	C₂×C₃₁	#3151	1	1	yes	characteristic
205	C₂×C₃₁	#3156	1	1	yes	characteristic
206	C₂×#41	—	2	2	no	—
207	#135	—	4	8	no	—
208	C₂×Q₁₆	—	2	2	no	—
209	C₂×Q₁₆	—	2	2	no	—
210	C₂×SD₁₆	—	2	4	no	—
211	C₂×SD₁₆	—	2	4	no	—
212	C₂×D₈	—	2	2	no	—
213	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
214	C₂²×C₈	D₄×C₃₁	1	2	yes	—
215	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
216	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
217	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
218	#127	C₂³×C₃₁	1	2	yes	—
219	#125	—	2	2	no	—
220	#124	—	2	2	no	—
221	C₂²⋊Q₈	—	2	2	no	—
222	C₂²⋊Q₈	—	2	2	no	—
223	#121	—	2	2	no	—
224	#121	—	2	2	no	—
225	C₄×Q₈	—	2	2	no	—
226	C₄×Q₈	—	2	2	no	—
227	C₄×D₄	—	2	2	no	—
228	C₄×D₄	—	2	2	no	—
229	#117	—	2	2	no	—
230	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
231	#107	—	2	2	no	—
232	#107	—	2	4	no	—
233	#107	—	2	4	no	—
234	#106	—	2	4	no	—
235	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
236	#103	—	2	4	no	—
237	#103	—	2	4	no	—
238	#102	—	2	4	no	—
239	#102	—	2	4	no	—
240	#102	—	2	4	no	—
241	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
242	#97	D₄×C₃₁	1	2	yes	—
243	C₄×C₈	D₄×C₃₁	1	2	yes	—
244	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
245	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
246	C₄×C₈	C₂³×C₃₁	1	2	yes	—
247	C₄×C₈	C₂³×C₃₁	1	2	yes	—
248	C₃₁	#19687	1	1	yes	characteristic, direct product factor, Sylow
249	#41	—	4	8	no	—
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	—	2	2	no	—
252	C₂×Q₈	—	2	2	no	—
253	C₂×D₄	—	2	2	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	4	2	no	—
257	C₂²×C₄	—	4	2	no	—
258	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
259	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
260	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
261	Q₁₆	—	4	4	no	—
262	Q₁₆	—	4	4	no	—
263	SD₁₆	—	4	8	no	—
264	SD₁₆	—	4	8	no	—
265	D₈	—	4	4	no	—
266	M₁₆	—	4	2	no	—
267	C₂×C₈	—	2	4	no	—
268	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
269	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
270	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
271	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	4	2	no	—
280	C₂²⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₄²	—	4	2	no	—
284	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
285	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
286	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
287	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
288	C₂³	—	4	2	no	—
289	C₂³	C₃₁×#127	1	1	yes	characteristic
290	Q₈	—	4	4	no	—
291	Q₈	—	4	4	no	—
292	Q₈	—	4	4	no	—
293	D₄	—	4	4	no	—
294	D₄	—	4	4	no	—
295	D₄	—	4	4	no	—
296	D₄	—	8	4	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	8	4	no	—
304	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
305	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
306	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
307	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
308	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
309	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
310	C₈	—	2	2	no	—
311	C₈	—	2	4	no	—
312	C₈	—	2	4	no	—
313	C₈	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	8	2	no	—
318	C₂²	—	8	2	no	—
319	C₂²	—	8	2	no	—
320	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	8	2	no	—
325	C₄	—	8	2	no	—
326	C₄	—	8	2	no	—
327	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
328	C₄	C₃₁×#575	1	2	yes	—
329	C₄	C₃₁×#578	1	2	yes	—
330	C₂	—	4	1	no	—
331	C₂	—	16	2	no	—
332	C₂	C₃₁×#3150	1	1	yes	characteristic
333	C₂	C₃₁×#3151	1	1	yes	characteristic
334	C₂	C₃₁×#3156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3147	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×#3228	C₂	semidirect
2. C₃₁×#3182	C₂	semidirect
3. C₃₁×#3170	C₂	semidirect
5. C₃₁×#1723	C₂	semidirect
6. C₃₁×#1722	C₂	semidirect
7. C₃₁×#1717	C₂	semidirect
8. C₃₁×#1714	C₂	semidirect
9. C₃₁×#1451	C₂	semidirect
120. #19687	C₃₁	direct (N × H)
147. #3228	C₂×C₃₁	semidirect
148. #3182	C₂×C₃₁	semidirect
149. #3170	C₂×C₃₁	semidirect
151. #1723	C₂×C₃₁	semidirect
152. #1722	C₂×C₃₁	semidirect
153. #1717	C₂×C₃₁	semidirect
154. #1714	C₂×C₃₁	semidirect
155. #1451	C₂×C₃₁	semidirect
248. C₃₁	#19687	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×#3228	2
C₃₁×#3182	2
C₃₁×#3170	2
C₃₁×#3147	2
C₃₁×#1723	2
C₃₁×#1722	2
C₃₁×#1717	2
C₃₁×#1714	2
C₃₁×#1451	2
#19687	31

Subgroups — C31×#19687 #21476446

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19687 #21476446