Subgroups — C₃₁×#33212 #21489971

Order 7936.

914 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2857	C₂	1	2	yes	maximal
4	C₃₁×#1990	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1944	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1844	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1809	C₂	1	4	yes	maximal
8	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
9	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₃₁×C₂²⋊Q₈	—	2	2	no	—
11	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
12	C₃₁×#500	—	2	4	no	—
13	C₂×C₃₁×#107	C₂²	1	4	yes	—
14	C₂×C₃₁×#106	C₂²	1	4	yes	—
15	C₄×M₁₆×C₃₁	—	2	4	no	—
16	C₂×C₃₁×#97	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#391	—	2	2	no	—
20	C₃₁×#389	—	2	2	no	—
21	C₃₁×#389	C₂²	1	4	yes	—
22	C₃₁×#384	C₂²	1	2	yes	—
23	C₃₁×#380	—	2	2	no	—
24	C₃₁×#380	C₂²	1	4	yes	—
25	C₃₁×#340	C₂²	1	8	yes	—
26	C₂²×Q₈×C₃₁	—	2	2	no	—
27	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
30	C₂×M₁₆×C₃₁	—	2	4	no	—
31	C₂×M₁₆×C₃₁	—	2	4	no	—
32	C₂×M₁₆×C₃₁	Q₈	1	8	yes	complemented
33	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
34	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
35	C₃₁×C₄⋊Q₈	—	2	4	no	—
36	C₃₁×C₄⋊Q₈	—	2	4	no	—
37	C₃₁×C₂²⋊Q₈	—	4	8	no	—
38	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
39	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
40	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
41	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
42	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
43	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
44	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
45	C₂×C₄²×C₃₁	—	2	2	no	—
46	C₂×C₄²×C₃₁	—	2	2	no	—
47	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
51	C₃₁×#107	—	2	8	no	—
52	C₃₁×#106	—	2	8	no	—
53	C₃₁×#97	—	2	4	no	—
54	C₄×C₈×C₃₁	—	2	2	no	—
55	C₄×C₈×C₃₁	D₄	1	4	yes	—
56	C₃₁×#95	—	2	4	no	—
57	C₃₁×#95	—	2	4	no	—
58	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
59	C₂×Q₈×C₃₁	—	4	4	no	—
60	C₂×Q₈×C₃₁	—	4	4	no	—
61	C₂×Q₈×C₃₁	—	4	4	no	—
62	C₂²×C₄×C₃₁	—	2	1	no	—
63	C₂²×C₄×C₃₁	—	2	1	no	—
64	C₂²×C₄×C₃₁	—	2	2	no	—
65	C₂²×C₄×C₃₁	—	2	2	no	—
66	C₂²×C₄×C₃₁	—	2	2	no	—
67	C₂²×C₄×C₃₁	—	2	4	no	—
68	C₂²×C₄×C₃₁	—	2	4	no	—
69	C₂²×C₄×C₃₁	—	2	4	no	—
70	C₂²×C₄×C₃₁	—	2	4	no	—
71	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
77	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
78	M₁₆×C₃₁	—	2	16	no	—
79	C₂×C₈×C₃₁	—	2	2	no	—
80	C₂×C₈×C₃₁	—	2	2	no	—
81	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
82	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
83	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
84	C₂×C₈×C₃₁	#41	1	4	yes	—
85	C₄⋊C₄×C₃₁	—	2	4	no	—
86	C₄⋊C₄×C₃₁	—	2	8	no	—
87	C₄⋊C₄×C₃₁	—	2	4	no	—
88	C₄⋊C₄×C₃₁	—	4	8	no	—
89	C₄⋊C₄×C₃₁	—	4	4	no	—
90	C₄⋊C₄×C₃₁	—	4	4	no	—
91	C₄⋊C₄×C₃₁	—	4	4	no	—
92	C₂²⋊C₄×C₃₁	—	4	8	no	—
93	C₄²×C₃₁	—	2	2	no	—
94	C₄²×C₃₁	—	2	2	no	—
95	C₄²×C₃₁	—	2	1	no	—
96	C₄²×C₃₁	—	2	1	no	—
97	C₄²×C₃₁	C₂×D₄	1	2	yes	—
98	C₄²×C₃₁	#41	1	2	yes	—
99	#33212	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂³×C₃₁	—	2	1	no	—
101	C₂³×C₃₁	—	2	1	no	—
102	C₂³×C₃₁	—	2	2	no	—
103	C₂³×C₃₁	—	2	2	no	—
104	C₂³×C₃₁	C₄⋊Q₈	1	2	yes	—
105	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
106	Q₈×C₃₁	—	8	8	no	—
107	C₂×C₄×C₃₁	—	2	1	no	—
108	C₂×C₄×C₃₁	—	2	1	no	—
109	C₂×C₄×C₃₁	—	2	4	no	—
110	C₂×C₄×C₃₁	—	2	4	no	—
111	C₂×C₄×C₃₁	—	2	1	no	—
112	C₂×C₄×C₃₁	—	2	1	no	—
113	C₂×C₄×C₃₁	—	2	2	no	—
114	C₂×C₄×C₃₁	—	2	2	no	—
115	C₂×C₄×C₃₁	—	4	4	no	—
116	C₂×C₄×C₃₁	—	4	4	no	—
117	C₂×C₄×C₃₁	—	4	4	no	—
118	C₂×C₄×C₃₁	—	4	4	no	—
119	C₂×C₄×C₃₁	—	4	4	no	—
120	C₂×C₄×C₃₁	—	4	4	no	—
121	C₂×C₄×C₃₁	—	4	4	no	—
122	C₂×C₄×C₃₁	—	4	4	no	—
123	C₂×C₄×C₃₁	#121	1	2	yes	—
124	C₂×C₄×C₃₁	#121	1	2	yes	—
125	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
126	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
127	C₂×C₄×C₃₁	#137	1	4	yes	—
128	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
130	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
131	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
132	C₈×C₃₁	—	2	4	no	—
133	C₈×C₃₁	C₂²⋊Q₈	1	8	yes	complemented
134	C₂×#500	C₂×C₃₁	1	1	yes	characteristic, complemented
135	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic
136	#2857	C₂×C₃₁	1	2	yes	—
137	#1990	C₂×C₃₁	1	2	yes	complemented
138	#1944	C₂×C₃₁	1	4	yes	complemented
139	#1844	C₂×C₃₁	1	1	yes	characteristic, complemented
140	#1809	C₂×C₃₁	1	4	yes	—
141	C₂²×C₃₁	—	2	2	no	—
142	C₂²×C₃₁	—	2	4	no	—
143	C₂²×C₃₁	—	4	2	no	—
144	C₂²×C₃₁	—	4	1	no	—
145	C₂²×C₃₁	—	4	1	no	—
146	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
147	C₂²×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
148	C₂²×C₃₁	#532	1	1	yes	characteristic
149	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
150	C₂²×C₃₁	C₂×#137	1	2	yes	—
151	C₄×C₃₁	—	2	2	no	—
152	C₄×C₃₁	—	4	1	no	—
153	C₄×C₃₁	—	4	1	no	—
154	C₄×C₃₁	—	8	4	no	—
155	C₄×C₃₁	—	8	4	no	—
156	C₄×C₃₁	#469	1	4	yes	—
157	C₄×C₃₁	#470	1	4	yes	—
158	C₄×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
159	C₄×C₃₁	#533	1	2	yes	—
160	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
161	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
162	C₂×C₂²⋊Q₈	—	2	2	no	—
163	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
164	#500	—	2	4	no	—
165	C₂×#107	C₂²×C₃₁	1	4	yes	—
166	C₂×#106	C₂²×C₃₁	1	4	yes	—
167	C₄×M₁₆	—	2	4	no	—
168	C₂×#97	C₂²×C₃₁	1	2	yes	—
169	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
170	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
171	#391	—	2	2	no	—
172	#389	—	2	2	no	—
173	#389	C₂²×C₃₁	1	4	yes	—
174	#384	C₂²×C₃₁	1	2	yes	—
175	#380	—	2	2	no	—
176	#380	C₂²×C₃₁	1	4	yes	—
177	#340	C₂²×C₃₁	1	8	yes	—
178	C₂×C₃₁	—	4	2	no	—
179	C₂×C₃₁	#2857	1	1	yes	characteristic
180	C₂×C₃₁	C₂×#469	1	1	yes	characteristic
181	C₂×C₃₁	C₂×#470	1	1	yes	characteristic
182	C₂×C₃₁	#3165	1	2	yes	—
183	C₂×C₃₁	#3182	1	2	yes	—
184	C₂²×Q₈	—	2	2	no	—
185	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
186	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
187	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
188	C₂×M₁₆	—	2	4	no	—
189	C₂×M₁₆	—	2	4	no	—
190	C₂×M₁₆	Q₈×C₃₁	1	8	yes	complemented
191	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
192	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
193	C₄⋊Q₈	—	2	4	no	—
194	C₄⋊Q₈	—	2	4	no	—
195	C₂²⋊Q₈	—	4	8	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	4	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
201	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
202	C₂×C₂²⋊C₄	—	2	4	no	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	—	2	2	no	—
205	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
206	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
207	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
208	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
209	#107	—	2	8	no	—
210	#106	—	2	8	no	—
211	#97	—	2	4	no	—
212	C₄×C₈	—	2	2	no	—
213	C₄×C₈	D₄×C₃₁	1	4	yes	—
214	#95	—	2	4	no	—
215	#95	—	2	4	no	—
216	C₃₁	#33212	1	1	yes	characteristic, direct product factor, Sylow
217	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
218	C₂×Q₈	—	4	4	no	—
219	C₂×Q₈	—	4	4	no	—
220	C₂×Q₈	—	4	4	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
235	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
236	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
237	M₁₆	—	2	16	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
241	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
242	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
243	C₂×C₈	#41×C₃₁	1	4	yes	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	2	8	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	4	8	no	—
248	C₄⋊C₄	—	4	4	no	—
249	C₄⋊C₄	—	4	4	no	—
250	C₄⋊C₄	—	4	4	no	—
251	C₂²⋊C₄	—	4	8	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	C₂×D₄×C₃₁	1	2	yes	—
257	C₄²	#41×C₃₁	1	2	yes	—
258	C₂³	—	2	1	no	—
259	C₂³	—	2	1	no	—
260	C₂³	—	2	2	no	—
261	C₂³	—	2	2	no	—
262	C₂³	C₃₁×C₄⋊Q₈	1	2	yes	—
263	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
264	Q₈	—	8	8	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	C₃₁×#121	1	2	yes	—
282	C₂×C₄	C₃₁×#121	1	2	yes	—
283	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
284	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
285	C₂×C₄	C₃₁×#137	1	4	yes	—
286	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
287	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
288	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
289	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
290	C₈	—	2	4	no	—
291	C₈	C₃₁×C₂²⋊Q₈	1	8	yes	complemented
292	C₂²	—	2	2	no	—
293	C₂²	—	2	4	no	—
294	C₂²	—	4	2	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
298	C₂²	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
299	C₂²	C₃₁×#532	1	1	yes	characteristic
300	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
301	C₂²	C₂×C₃₁×#137	1	2	yes	—
302	C₄	—	2	2	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	8	4	no	—
306	C₄	—	8	4	no	—
307	C₄	C₃₁×#469	1	4	yes	—
308	C₄	C₃₁×#470	1	4	yes	—
309	C₄	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
310	C₄	C₃₁×#533	1	2	yes	—
311	C₂	—	4	2	no	—
312	C₂	C₃₁×#2857	1	1	yes	characteristic
313	C₂	C₂×C₃₁×#469	1	1	yes	characteristic
314	C₂	C₂×C₃₁×#470	1	1	yes	characteristic
315	C₂	C₃₁×#3165	1	2	yes	—
316	C₂	C₃₁×#3182	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#500	C₂	semidirect
4. C₃₁×#1990	C₂	semidirect
5. C₃₁×#1944	C₂	semidirect
6. C₃₁×#1844	C₂	semidirect
32. C₂×M₁₆×C₃₁	Q₈	semidirect
99. #33212	C₃₁	direct (N × H)
133. C₈×C₃₁	C₂²⋊Q₈	semidirect
134. C₂×#500	C₂×C₃₁	semidirect
137. #1990	C₂×C₃₁	semidirect
138. #1944	C₂×C₃₁	semidirect
139. #1844	C₂×C₃₁	semidirect
190. C₂×M₁₆	Q₈×C₃₁	semidirect
216. C₃₁	#33212	direct (N × H)
291. C₈	C₃₁×C₂²⋊Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#500	2
C₂×C₄×M₁₆×C₃₁	2
C₃₁×#2857	2
C₃₁×#1990	2
C₃₁×#1944	2
C₃₁×#1844	2
C₃₁×#1809	2
#33212	31

Subgroups — C31×#33212 #21489971

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33212 #21489971