Subgroups — C₃₁×#33065 #21489824

Order 7936.

746 conjugacy classes of subgroups.

The subgroup lattice has 354 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	C₃₁×#3080	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3039	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2875	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1998	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1966	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1930	C₂	1	4	yes	maximal, complemented
8	C₃₁×#1908	C₂	1	4	yes	maximal
9	C₃₁×#1838	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#137	C₂²	1	2	yes	—
11	C₂×C₃₁×#131	C₂²	1	2	yes	—
12	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
14	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#491	—	2	1	no	—
16	C₃₁×#491	—	2	1	no	—
17	C₃₁×#491	—	2	2	no	—
18	C₃₁×#479	—	2	2	no	—
19	C₃₁×#478	—	2	2	no	—
20	C₃₁×#462	—	2	2	no	—
21	C₃₁×#461	—	2	2	no	—
22	C₃₁×#428	C₂²	1	2	yes	—
23	C₃₁×#423	C₂²	1	1	yes	characteristic
24	C₃₁×#423	C₂²	1	1	yes	characteristic
25	C₃₁×#419	C₂²	1	2	yes	—
26	C₃₁×#417	C₂²	1	2	yes	—
27	C₂×C₃₁×#103	C₂²	1	4	yes	—
28	C₂×C₃₁×#101	C₂²	1	4	yes	—
29	C₃₁×#405	—	2	2	no	—
30	C₃₁×#405	C₂²	1	2	yes	—
31	C₄×M₁₆×C₃₁	—	2	1	no	—
32	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
33	C₂×C₃₁×#97	—	2	1	no	—
34	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
35	C₃₁×#343	C₂²	1	4	yes	—
36	C₃₁×#341	C₂²	1	4	yes	—
37	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
38	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
39	C₃₁×#137	—	2	8	no	—
40	C₂×Q₁₆×C₃₁	—	2	2	no	—
41	C₂×SD₁₆×C₃₁	—	2	2	no	—
42	C₃₁×#131	—	2	4	no	—
43	C₃₁×#131	—	2	4	no	—
44	C₂×M₁₆×C₃₁	—	2	2	no	—
45	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
46	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
47	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
48	C₂²×C₈×C₃₁	—	2	2	no	—
49	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
50	C₃₁×C₄⋊Q₈	—	2	2	no	—
51	C₃₁×C₄⋊Q₈	—	2	1	no	—
52	C₃₁×C₄⋊Q₈	—	2	1	no	—
53	C₃₁×#125	—	2	2	no	—
54	C₃₁×#125	—	2	1	no	—
55	C₃₁×#125	—	2	1	no	—
56	C₄×Q₈×C₃₁	D₄	1	2	yes	—
57	C₄×D₄×C₃₁	—	2	2	no	—
58	C₄×D₄×C₃₁	D₄	1	2	yes	—
59	C₃₁×#117	—	2	1	no	—
60	C₃₁×#117	C₂³	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
62	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
64	C₂×C₄²×C₃₁	—	2	1	no	—
65	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
66	C₃₁×#107	—	2	2	no	—
67	C₃₁×#106	—	2	2	no	—
68	C₃₁×C₄⋊C₈	—	2	2	no	—
69	C₃₁×C₄⋊C₈	—	2	2	no	—
70	C₃₁×#103	—	2	2	no	—
71	C₃₁×#103	—	2	4	no	—
72	C₃₁×#103	—	2	4	no	—
73	C₃₁×#103	—	2	4	no	—
74	C₃₁×#102	—	2	2	no	—
75	C₃₁×#101	—	2	8	no	—
76	C₃₁×#97	—	2	2	no	—
77	C₃₁×#97	—	2	1	no	—
78	C₃₁×#97	D₄	1	1	yes	characteristic
79	C₃₁×#97	D₄	1	1	yes	characteristic
80	C₃₁×#97	D₄	1	2	yes	—
81	C₄×C₈×C₃₁	—	2	2	no	—
82	C₄×C₈×C₃₁	—	2	1	no	—
83	#41×C₃₁	—	2	2	no	—
84	#41×C₃₁	#41	1	4	yes	—
85	C₂×Q₈×C₃₁	—	2	2	no	—
86	C₂×Q₈×C₃₁	—	2	2	no	—
87	C₂×Q₈×C₃₁	—	2	4	no	—
88	C₂×Q₈×C₃₁	—	2	4	no	—
89	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
90	C₂×D₄×C₃₁	—	2	1	no	—
91	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₃₁	—	2	1	no	—
93	C₂²×C₄×C₃₁	—	2	2	no	—
94	C₂²×C₄×C₃₁	—	2	2	no	—
95	C₂²×C₄×C₃₁	—	2	2	no	—
96	C₂²×C₄×C₃₁	#41	1	2	yes	—
97	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
98	Q₁₆×C₃₁	—	4	4	no	—
99	SD₁₆×C₃₁	—	4	4	no	—
100	M₁₆×C₃₁	—	2	8	no	—
101	M₁₆×C₃₁	—	2	2	no	—
102	M₁₆×C₃₁	—	2	4	no	—
103	M₁₆×C₃₁	—	2	4	no	—
104	M₁₆×C₃₁	#41	1	4	yes	—
105	C₂×C₈×C₃₁	—	2	8	no	—
106	C₂×C₈×C₃₁	—	2	2	no	—
107	C₂×C₈×C₃₁	—	2	2	no	—
108	C₂×C₈×C₃₁	—	2	2	no	—
109	C₂×C₈×C₃₁	—	2	2	no	—
110	C₂×C₈×C₃₁	—	2	2	no	—
111	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
112	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
113	C₄⋊C₄×C₃₁	—	2	2	no	—
114	C₄⋊C₄×C₃₁	—	2	2	no	—
115	C₄⋊C₄×C₃₁	—	2	2	no	—
116	C₄⋊C₄×C₃₁	—	4	2	no	—
117	C₄⋊C₄×C₃₁	—	4	2	no	—
118	C₄⋊C₄×C₃₁	—	4	2	no	—
119	C₄⋊C₄×C₃₁	—	4	2	no	—
120	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
121	C₂²⋊C₄×C₃₁	—	2	1	no	—
122	C₂²⋊C₄×C₃₁	—	2	2	no	—
123	C₄²×C₃₁	—	2	2	no	—
124	C₄²×C₃₁	—	2	1	no	—
125	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₃₁	C₂×D₄	1	2	yes	—
128	#33065	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂³×C₃₁	—	2	1	no	—
130	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
131	Q₈×C₃₁	—	2	1	no	—
132	Q₈×C₃₁	—	4	4	no	—
133	Q₈×C₃₁	—	4	2	no	—
134	Q₈×C₃₁	—	4	2	no	—
135	Q₈×C₃₁	#124	1	2	yes	—
136	D₄×C₃₁	—	2	4	no	—
137	D₄×C₃₁	—	2	1	no	—
138	D₄×C₃₁	#124	1	2	yes	—
139	C₂×C₄×C₃₁	—	2	1	no	—
140	C₂×C₄×C₃₁	—	2	4	no	—
141	C₂×C₄×C₃₁	—	2	2	no	—
142	C₂×C₄×C₃₁	—	4	2	no	—
143	C₂×C₄×C₃₁	—	4	2	no	—
144	C₂×C₄×C₃₁	—	4	2	no	—
145	C₂×C₄×C₃₁	—	4	4	no	—
146	C₂×C₄×C₃₁	—	8	2	no	—
147	C₂×C₄×C₃₁	—	8	2	no	—
148	C₂×C₄×C₃₁	#121	1	2	yes	—
149	C₂×C₄×C₃₁	#121	1	2	yes	—
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	2	yes	—
153	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
154	C₈×C₃₁	—	2	2	no	—
155	C₈×C₃₁	—	2	4	no	—
156	C₈×C₃₁	—	2	2	no	—
157	C₈×C₃₁	—	4	2	no	—
158	C₈×C₃₁	—	4	2	no	—
159	C₂×#491	C₂×C₃₁	1	1	yes	characteristic, complemented
160	#3080	C₂×C₃₁	1	1	yes	characteristic
161	#3039	C₂×C₃₁	1	1	yes	characteristic
162	#2875	C₂×C₃₁	1	1	yes	characteristic
163	#1998	C₂×C₃₁	1	1	yes	characteristic, complemented
164	#1966	C₂×C₃₁	1	1	yes	characteristic, complemented
165	#1930	C₂×C₃₁	1	4	yes	complemented
166	#1908	C₂×C₃₁	1	4	yes	—
167	#1838	C₂×C₃₁	1	1	yes	characteristic, complemented
168	C₂²×C₃₁	—	2	2	no	—
169	C₂²×C₃₁	—	2	2	no	—
170	C₂²×C₃₁	—	4	2	no	—
171	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
172	C₂²×C₃₁	#529	1	2	yes	—
173	C₄×C₃₁	—	2	2	no	—
174	C₄×C₃₁	—	2	2	no	—
175	C₄×C₃₁	—	2	2	no	—
176	C₄×C₃₁	—	4	2	no	—
177	C₄×C₃₁	—	8	2	no	—
178	C₄×C₃₁	—	8	2	no	—
179	C₄×C₃₁	#525	1	2	yes	—
180	C₄×C₃₁	C₂×#124	1	1	yes	characteristic
181	C₄×C₃₁	#532	1	1	yes	characteristic
182	C₂×#137	C₂²×C₃₁	1	2	yes	—
183	C₂×#131	C₂²×C₃₁	1	2	yes	—
184	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
185	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
186	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
187	#491	—	2	1	no	—
188	#491	—	2	1	no	—
189	#491	—	2	2	no	—
190	#479	—	2	2	no	—
191	#478	—	2	2	no	—
192	#462	—	2	2	no	—
193	#461	—	2	2	no	—
194	#428	C₂²×C₃₁	1	2	yes	—
195	#423	C₂²×C₃₁	1	1	yes	characteristic
196	#423	C₂²×C₃₁	1	1	yes	characteristic
197	#419	C₂²×C₃₁	1	2	yes	—
198	#417	C₂²×C₃₁	1	2	yes	—
199	C₂×#103	C₂²×C₃₁	1	4	yes	—
200	C₂×#101	C₂²×C₃₁	1	4	yes	—
201	#405	—	2	2	no	—
202	#405	C₂²×C₃₁	1	2	yes	—
203	C₄×M₁₆	—	2	1	no	—
204	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
205	C₂×#97	—	2	1	no	—
206	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
207	#343	C₂²×C₃₁	1	4	yes	—
208	#341	C₂²×C₃₁	1	4	yes	—
209	C₂×C₃₁	—	2	2	no	—
210	C₂×C₃₁	—	4	2	no	—
211	C₂×C₃₁	#2850	1	1	yes	characteristic
212	C₂×C₃₁	#3069	1	2	yes	—
213	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
214	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
215	#137	—	2	8	no	—
216	C₂×Q₁₆	—	2	2	no	—
217	C₂×SD₁₆	—	2	2	no	—
218	#131	—	2	4	no	—
219	#131	—	2	4	no	—
220	C₂×M₁₆	—	2	2	no	—
221	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
222	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
223	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
224	C₂²×C₈	—	2	2	no	—
225	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
226	C₄⋊Q₈	—	2	2	no	—
227	C₄⋊Q₈	—	2	1	no	—
228	C₄⋊Q₈	—	2	1	no	—
229	#125	—	2	2	no	—
230	#125	—	2	1	no	—
231	#125	—	2	1	no	—
232	C₄×Q₈	D₄×C₃₁	1	2	yes	—
233	C₄×D₄	—	2	2	no	—
234	C₄×D₄	D₄×C₃₁	1	2	yes	—
235	#117	—	2	1	no	—
236	#117	C₂³×C₃₁	1	1	yes	characteristic
237	C₂×C₄⋊C₄	—	2	2	no	—
238	C₂×C₄⋊C₄	—	2	2	no	—
239	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
240	C₂×C₄²	—	2	1	no	—
241	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
242	#107	—	2	2	no	—
243	#106	—	2	2	no	—
244	C₄⋊C₈	—	2	2	no	—
245	C₄⋊C₈	—	2	2	no	—
246	#103	—	2	2	no	—
247	#103	—	2	4	no	—
248	#103	—	2	4	no	—
249	#103	—	2	4	no	—
250	#102	—	2	2	no	—
251	#101	—	2	8	no	—
252	#97	—	2	2	no	—
253	#97	—	2	1	no	—
254	#97	D₄×C₃₁	1	1	yes	characteristic
255	#97	D₄×C₃₁	1	1	yes	characteristic
256	#97	D₄×C₃₁	1	2	yes	—
257	C₄×C₈	—	2	2	no	—
258	C₄×C₈	—	2	1	no	—
259	C₃₁	#33065	1	1	yes	characteristic, direct product factor, Sylow
260	#41	—	2	2	no	—
261	#41	#41×C₃₁	1	4	yes	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	2	4	no	—
265	C₂×Q₈	—	2	4	no	—
266	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
267	C₂×D₄	—	2	1	no	—
268	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
269	C₂²×C₄	—	2	1	no	—
270	C₂²×C₄	—	2	2	no	—
271	C₂²×C₄	—	2	2	no	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	#41×C₃₁	1	2	yes	—
274	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
275	Q₁₆	—	4	4	no	—
276	SD₁₆	—	4	4	no	—
277	M₁₆	—	2	8	no	—
278	M₁₆	—	2	2	no	—
279	M₁₆	—	2	4	no	—
280	M₁₆	—	2	4	no	—
281	M₁₆	#41×C₃₁	1	4	yes	—
282	C₂×C₈	—	2	8	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
289	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	4	2	no	—
294	C₄⋊C₄	—	4	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	1	no	—
299	C₂²⋊C₄	—	2	2	no	—
300	C₄²	—	2	2	no	—
301	C₄²	—	2	1	no	—
302	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
303	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
304	C₄²	C₂×D₄×C₃₁	1	2	yes	—
305	C₂³	—	2	1	no	—
306	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
307	Q₈	—	2	1	no	—
308	Q₈	—	4	4	no	—
309	Q₈	—	4	2	no	—
310	Q₈	—	4	2	no	—
311	Q₈	C₃₁×#124	1	2	yes	—
312	D₄	—	2	4	no	—
313	D₄	—	2	1	no	—
314	D₄	C₃₁×#124	1	2	yes	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	4	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	4	no	—
322	C₂×C₄	—	8	2	no	—
323	C₂×C₄	—	8	2	no	—
324	C₂×C₄	C₃₁×#121	1	2	yes	—
325	C₂×C₄	C₃₁×#121	1	2	yes	—
326	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
327	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
328	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
329	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
330	C₈	—	2	2	no	—
331	C₈	—	2	4	no	—
332	C₈	—	2	2	no	—
333	C₈	—	4	2	no	—
334	C₈	—	4	2	no	—
335	C₂²	—	2	2	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	4	2	no	—
338	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
339	C₂²	C₃₁×#529	1	2	yes	—
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	2	no	—
343	C₄	—	4	2	no	—
344	C₄	—	8	2	no	—
345	C₄	—	8	2	no	—
346	C₄	C₃₁×#525	1	2	yes	—
347	C₄	C₂×C₃₁×#124	1	1	yes	characteristic
348	C₄	C₃₁×#532	1	1	yes	characteristic
349	C₂	—	2	2	no	—
350	C₂	—	4	2	no	—
351	C₂	C₃₁×#2850	1	1	yes	characteristic
352	C₂	C₃₁×#3069	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×#41×C₃₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ 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. C₂×C₃₁×#491	C₂	semidirect
5. C₃₁×#1998	C₂	semidirect
6. C₃₁×#1966	C₂	semidirect
7. C₃₁×#1930	C₂	semidirect
9. C₃₁×#1838	C₂	semidirect
128. #33065	C₃₁	direct (N × H)
159. C₂×#491	C₂×C₃₁	semidirect
163. #1998	C₂×C₃₁	semidirect
164. #1966	C₂×C₃₁	semidirect
165. #1930	C₂×C₃₁	semidirect
167. #1838	C₂×C₃₁	semidirect
259. C₃₁	#33065	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
C₃₁×#3080	2
C₃₁×#3039	2
C₃₁×#2875	2
C₃₁×#1998	2
C₃₁×#1966	2
C₃₁×#1930	2
C₃₁×#1908	2
C₃₁×#1838	2
#33065	31

Subgroups — C31×#33065 #21489824

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33065 #21489824