Subgroups — C₃₁×#33059 #21489818

Order 7936.

870 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	C₂×C₃₁×#491	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2857	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2851	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1992	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1960	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1933	C₂	1	4	yes	maximal, complemented
8	C₃₁×#1894	C₂	1	4	yes	maximal
9	C₃₁×#1836	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
11	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
14	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#491	—	2	4	no	—
16	C₂×C₃₁×#103	C₂²	1	4	yes	—
17	C₂×C₃₁×#103	C₂²	1	4	yes	—
18	C₄×M₁₆×C₃₁	—	2	4	no	—
19	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
22	C₃₁×#391	C₂²	1	2	yes	—
23	C₃₁×#389	—	2	2	no	—
24	C₃₁×#389	C₂²	1	2	yes	—
25	C₃₁×#387	—	2	2	no	—
26	C₃₁×#385	—	2	2	no	—
27	C₃₁×#384	C₂²	1	1	yes	characteristic
28	C₃₁×#382	C₂²	1	1	yes	characteristic
29	C₃₁×#380	—	2	2	no	—
30	C₃₁×#380	C₂²	1	2	yes	—
31	C₃₁×#340	C₂²	1	4	yes	—
32	C₃₁×#340	C₂²	1	4	yes	—
33	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
34	C₂³×C₄×C₃₁	D₄	1	2	yes	—
35	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₃₁	—	2	4	no	—
37	C₂×M₁₆×C₃₁	—	2	4	no	—
38	C₂×M₁₆×C₃₁	—	2	4	no	—
39	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
40	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
41	C₃₁×C₄⋊Q₈	—	2	2	no	—
42	C₃₁×C₄⋊Q₈	—	2	2	no	—
43	C₃₁×#125	—	2	2	no	—
44	C₃₁×#125	—	2	2	no	—
45	C₃₁×C₂²⋊Q₈	—	2	8	no	—
46	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
47	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
48	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
49	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
50	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
51	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
54	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
55	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
56	C₂×C₄²×C₃₁	—	2	2	no	—
57	C₂×C₄²×C₃₁	—	2	2	no	—
58	C₂×C₄²×C₃₁	D₄	1	2	yes	—
59	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
61	C₃₁×#103	—	2	8	no	—
62	C₃₁×#103	—	2	8	no	—
63	C₃₁×#97	—	2	2	no	—
64	C₃₁×#97	D₄	1	4	yes	—
65	C₄×C₈×C₃₁	—	2	4	no	—
66	C₃₁×#95	—	2	4	no	—
67	C₃₁×#95	—	2	2	no	—
68	C₃₁×#95	—	2	2	no	—
69	C₃₁×#95	—	2	2	no	—
70	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
71	C₂×Q₈×C₃₁	—	2	2	no	—
72	C₂×Q₈×C₃₁	—	2	4	no	—
73	C₂×Q₈×C₃₁	—	2	4	no	—
74	C₂×Q₈×C₃₁	—	2	2	no	—
75	C₂²×C₄×C₃₁	—	2	1	no	—
76	C₂²×C₄×C₃₁	—	2	1	no	—
77	C₂²×C₄×C₃₁	—	2	2	no	—
78	C₂²×C₄×C₃₁	—	2	4	no	—
79	C₂²×C₄×C₃₁	—	2	4	no	—
80	C₂²×C₄×C₃₁	—	2	2	no	—
81	C₂²×C₄×C₃₁	—	2	2	no	—
82	C₂²×C₄×C₃₁	—	2	2	no	—
83	C₂²×C₄×C₃₁	—	2	2	no	—
84	C₂²×C₄×C₃₁	—	4	2	no	—
85	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₃₁	#41	1	4	yes	—
89	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
90	M₁₆×C₃₁	—	2	16	no	—
91	C₂×C₈×C₃₁	—	2	2	no	—
92	C₂×C₈×C₃₁	—	2	2	no	—
93	C₂×C₈×C₃₁	—	2	2	no	—
94	C₂×C₈×C₃₁	—	2	2	no	—
95	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
96	C₂×C₈×C₃₁	#41	1	4	yes	—
97	C₄⋊C₄×C₃₁	—	2	2	no	—
98	C₄⋊C₄×C₃₁	—	2	4	no	—
99	C₄⋊C₄×C₃₁	—	2	4	no	—
100	C₄⋊C₄×C₃₁	—	2	2	no	—
101	C₄⋊C₄×C₃₁	—	4	4	no	—
102	C₄⋊C₄×C₃₁	—	4	4	no	—
103	C₄⋊C₄×C₃₁	—	4	4	no	—
104	C₄⋊C₄×C₃₁	—	4	4	no	—
105	C₄⋊C₄×C₃₁	—	4	4	no	—
106	C₂²⋊C₄×C₃₁	—	4	4	no	—
107	C₂²⋊C₄×C₃₁	—	4	4	no	—
108	C₄²×C₃₁	—	2	4	no	—
109	C₄²×C₃₁	—	2	1	no	—
110	C₄²×C₃₁	—	2	1	no	—
111	C₄²×C₃₁	C₂×D₄	1	2	yes	—
112	C₄²×C₃₁	#41	1	2	yes	—
113	#33059	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂³×C₃₁	—	2	1	no	—
115	C₂³×C₃₁	—	2	1	no	—
116	C₂³×C₃₁	—	2	2	no	—
117	C₂³×C₃₁	—	2	2	no	—
118	C₂³×C₃₁	#124	1	2	yes	—
119	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
120	Q₈×C₃₁	—	4	4	no	—
121	Q₈×C₃₁	—	4	4	no	—
122	C₂×C₄×C₃₁	—	2	1	no	—
123	C₂×C₄×C₃₁	—	2	2	no	—
124	C₂×C₄×C₃₁	—	2	1	no	—
125	C₂×C₄×C₃₁	—	2	2	no	—
126	C₂×C₄×C₃₁	—	2	2	no	—
127	C₂×C₄×C₃₁	—	2	2	no	—
128	C₂×C₄×C₃₁	—	2	4	no	—
129	C₂×C₄×C₃₁	—	2	4	no	—
130	C₂×C₄×C₃₁	—	4	8	no	—
131	C₂×C₄×C₃₁	—	4	2	no	—
132	C₂×C₄×C₃₁	—	4	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	—	4	2	no	—
137	C₂×C₄×C₃₁	—	4	2	no	—
138	C₂×C₄×C₃₁	—	4	2	no	—
139	C₂×C₄×C₃₁	—	4	2	no	—
140	C₂×C₄×C₃₁	—	4	2	no	—
141	C₂×C₄×C₃₁	—	4	4	no	—
142	C₂×C₄×C₃₁	#121	1	4	yes	—
143	C₂×C₄×C₃₁	#124	1	2	yes	—
144	C₂×C₄×C₃₁	#137	1	4	yes	—
145	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
147	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
148	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
149	C₈×C₃₁	—	2	4	no	—
150	C₈×C₃₁	—	2	4	no	—
151	C₂×#491	C₂×C₃₁	1	1	yes	characteristic, complemented
152	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic
153	#2857	C₂×C₃₁	1	1	yes	characteristic
154	#2851	C₂×C₃₁	1	1	yes	characteristic
155	#1992	C₂×C₃₁	1	1	yes	characteristic, complemented
156	#1960	C₂×C₃₁	1	1	yes	characteristic, complemented
157	#1933	C₂×C₃₁	1	4	yes	complemented
158	#1894	C₂×C₃₁	1	4	yes	—
159	#1836	C₂×C₃₁	1	1	yes	characteristic, complemented
160	C₂²×C₃₁	—	2	2	no	—
161	C₂²×C₃₁	—	2	2	no	—
162	C₂²×C₃₁	—	2	2	no	—
163	C₂²×C₃₁	—	4	2	no	—
164	C₂²×C₃₁	—	4	2	no	—
165	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
166	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
167	C₂²×C₃₁	#532	1	1	yes	characteristic
168	C₂²×C₃₁	C₂×#137	1	2	yes	—
169	C₂²×C₃₁	#575	1	2	yes	—
170	C₄×C₃₁	—	2	2	no	—
171	C₄×C₃₁	—	2	4	no	—
172	C₄×C₃₁	—	2	4	no	—
173	C₄×C₃₁	—	8	2	no	—
174	C₄×C₃₁	—	8	2	no	—
175	C₄×C₃₁	—	8	2	no	—
176	C₄×C₃₁	—	8	2	no	—
177	C₄×C₃₁	#525	1	2	yes	—
178	C₄×C₃₁	#529	1	2	yes	—
179	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
180	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
181	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
182	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
183	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
184	#491	—	2	4	no	—
185	C₂×#103	C₂²×C₃₁	1	4	yes	—
186	C₂×#103	C₂²×C₃₁	1	4	yes	—
187	C₄×M₁₆	—	2	4	no	—
188	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
189	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
190	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
191	#391	C₂²×C₃₁	1	2	yes	—
192	#389	—	2	2	no	—
193	#389	C₂²×C₃₁	1	2	yes	—
194	#387	—	2	2	no	—
195	#385	—	2	2	no	—
196	#384	C₂²×C₃₁	1	1	yes	characteristic
197	#382	C₂²×C₃₁	1	1	yes	characteristic
198	#380	—	2	2	no	—
199	#380	C₂²×C₃₁	1	2	yes	—
200	#340	C₂²×C₃₁	1	4	yes	—
201	#340	C₂²×C₃₁	1	4	yes	—
202	C₂×C₃₁	—	4	2	no	—
203	C₂×C₃₁	#2850	1	1	yes	characteristic
204	C₂×C₃₁	#3056	1	2	yes	—
205	C₂×C₃₁	#3139	1	1	yes	characteristic
206	C₂×C₃₁	#3142	1	1	yes	characteristic
207	C₂×C₃₁	#3172	1	2	yes	—
208	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
209	C₂³×C₄	D₄×C₃₁	1	2	yes	—
210	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
211	C₂×M₁₆	—	2	4	no	—
212	C₂×M₁₆	—	2	4	no	—
213	C₂×M₁₆	—	2	4	no	—
214	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
215	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
216	C₄⋊Q₈	—	2	2	no	—
217	C₄⋊Q₈	—	2	2	no	—
218	#125	—	2	2	no	—
219	#125	—	2	2	no	—
220	C₂²⋊Q₈	—	2	8	no	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	C₂×C₄⋊C₄	—	2	2	no	—
224	C₂×C₄⋊C₄	—	2	2	no	—
225	C₂×C₄⋊C₄	—	2	2	no	—
226	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
227	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
228	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
229	C₂×C₂²⋊C₄	—	2	2	no	—
230	C₂×C₂²⋊C₄	—	2	2	no	—
231	C₂×C₄²	—	2	2	no	—
232	C₂×C₄²	—	2	2	no	—
233	C₂×C₄²	D₄×C₃₁	1	2	yes	—
234	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
235	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
236	#103	—	2	8	no	—
237	#103	—	2	8	no	—
238	#97	—	2	2	no	—
239	#97	D₄×C₃₁	1	4	yes	—
240	C₄×C₈	—	2	4	no	—
241	#95	—	2	4	no	—
242	#95	—	2	2	no	—
243	#95	—	2	2	no	—
244	#95	—	2	2	no	—
245	C₃₁	#33059	1	1	yes	characteristic, direct product factor, Sylow
246	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	2	4	no	—
249	C₂×Q₈	—	2	4	no	—
250	C₂×Q₈	—	2	2	no	—
251	C₂²×C₄	—	2	1	no	—
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	2	4	no	—
255	C₂²×C₄	—	2	4	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	4	2	no	—
261	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
262	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
263	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
264	C₂²×C₄	#41×C₃₁	1	4	yes	—
265	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
266	M₁₆	—	2	16	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
272	C₂×C₈	#41×C₃₁	1	4	yes	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	4	no	—
275	C₄⋊C₄	—	2	4	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	4	4	no	—
278	C₄⋊C₄	—	4	4	no	—
279	C₄⋊C₄	—	4	4	no	—
280	C₄⋊C₄	—	4	4	no	—
281	C₄⋊C₄	—	4	4	no	—
282	C₂²⋊C₄	—	4	4	no	—
283	C₂²⋊C₄	—	4	4	no	—
284	C₄²	—	2	4	no	—
285	C₄²	—	2	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	C₂×D₄×C₃₁	1	2	yes	—
288	C₄²	#41×C₃₁	1	2	yes	—
289	C₂³	—	2	1	no	—
290	C₂³	—	2	1	no	—
291	C₂³	—	2	2	no	—
292	C₂³	—	2	2	no	—
293	C₂³	C₃₁×#124	1	2	yes	—
294	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
295	Q₈	—	4	4	no	—
296	Q₈	—	4	4	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	—	4	8	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	4	no	—
317	C₂×C₄	C₃₁×#121	1	4	yes	—
318	C₂×C₄	C₃₁×#124	1	2	yes	—
319	C₂×C₄	C₃₁×#137	1	4	yes	—
320	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
321	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
322	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
323	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
324	C₈	—	2	4	no	—
325	C₈	—	2	4	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	4	2	no	—
330	C₂²	—	4	2	no	—
331	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
332	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
333	C₂²	C₃₁×#532	1	1	yes	characteristic
334	C₂²	C₂×C₃₁×#137	1	2	yes	—
335	C₂²	C₃₁×#575	1	2	yes	—
336	C₄	—	2	2	no	—
337	C₄	—	2	4	no	—
338	C₄	—	2	4	no	—
339	C₄	—	8	2	no	—
340	C₄	—	8	2	no	—
341	C₄	—	8	2	no	—
342	C₄	—	8	2	no	—
343	C₄	C₃₁×#525	1	2	yes	—
344	C₄	C₃₁×#529	1	2	yes	—
345	C₂	—	4	2	no	—
346	C₂	C₃₁×#2850	1	1	yes	characteristic
347	C₂	C₃₁×#3056	1	2	yes	—
348	C₂	C₃₁×#3139	1	1	yes	characteristic
349	C₂	C₃₁×#3142	1	1	yes	characteristic
350	C₂	C₃₁×#3172	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×#491	C₂	semidirect
5. C₃₁×#1992	C₂	semidirect
6. C₃₁×#1960	C₂	semidirect
7. C₃₁×#1933	C₂	semidirect
9. C₃₁×#1836	C₂	semidirect
113. #33059	C₃₁	direct (N × H)
151. C₂×#491	C₂×C₃₁	semidirect
155. #1992	C₂×C₃₁	semidirect
156. #1960	C₂×C₃₁	semidirect
157. #1933	C₂×C₃₁	semidirect
159. #1836	C₂×C₃₁	semidirect
245. C₃₁	#33059	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#491	2
C₂×C₄×M₁₆×C₃₁	2
C₃₁×#2857	2
C₃₁×#2851	2
C₃₁×#1992	2
C₃₁×#1960	2
C₃₁×#1933	2
C₃₁×#1894	2
C₃₁×#1836	2
#33059	31

Subgroups — C31×#33059 #21489818

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33059 #21489818