Subgroups — C₃₁×#33216 #21489975

Order 7936.

794 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	C₂×C₃₁×#499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#405	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2629	C₂	1	2	yes	maximal
4	C₃₁×#1992	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1943	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1942	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1845	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1809	C₂	1	2	yes	maximal
9	C₃₁×#1808	C₂	1	2	yes	maximal
10	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#125	C₂²	1	2	yes	—
13	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
14	C₃₁×#499	—	2	4	no	—
15	C₂×C₃₁×#107	C₂²	1	2	yes	—
16	C₂×C₃₁×#107	C₂²	1	2	yes	—
17	C₂×C₃₁×#107	C₂²	1	2	yes	—
18	C₂×C₃₁×#106	C₂²	1	2	yes	—
19	C₃₁×#405	—	2	4	no	—
20	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#398	—	2	4	no	—
25	C₃₁×#396	—	2	4	no	—
26	C₃₁×#389	C₂²	1	4	yes	—
27	C₃₁×#384	C₂²	1	2	yes	—
28	C₃₁×#380	C₂²	1	4	yes	—
29	C₃₁×#340	C₂²	1	4	yes	—
30	C₃₁×#340	C₂²	1	4	yes	—
31	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
32	C₂×M₁₆×C₃₁	—	2	2	no	—
33	C₂×M₁₆×C₃₁	—	2	2	no	—
34	C₂×M₁₆×C₃₁	Q₈	1	4	yes	—
35	C₂²×C₈×C₃₁	—	2	2	no	—
36	C₂²×C₈×C₃₁	—	2	2	no	—
37	C₂²×C₈×C₃₁	Q₈	1	4	yes	—
38	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
42	C₃₁×#125	—	2	4	no	—
43	C₃₁×#125	—	2	4	no	—
44	C₃₁×#117	—	2	2	no	—
45	C₃₁×#117	—	2	2	no	—
46	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
47	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
48	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
49	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
51	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
52	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
53	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
54	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
56	C₃₁×#107	—	2	4	no	—
57	C₃₁×#107	—	2	4	no	—
58	C₃₁×#107	—	2	4	no	—
59	C₃₁×#106	—	2	4	no	—
60	C₃₁×#97	—	2	2	no	—
61	C₃₁×#97	—	2	2	no	—
62	C₄×C₈×C₃₁	—	2	2	no	—
63	C₄×C₈×C₃₁	D₄	1	4	yes	—
64	C₃₁×#95	—	2	4	no	—
65	C₃₁×#95	—	2	4	no	—
66	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₃₁	—	2	1	no	—
68	C₂²×C₄×C₃₁	—	2	1	no	—
69	C₂²×C₄×C₃₁	—	2	1	no	—
70	C₂²×C₄×C₃₁	—	2	1	no	—
71	C₂²×C₄×C₃₁	—	2	4	no	—
72	C₂²×C₄×C₃₁	—	2	4	no	—
73	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
79	M₁₆×C₃₁	—	2	8	no	—
80	C₂×C₈×C₃₁	—	2	1	no	—
81	C₂×C₈×C₃₁	—	2	1	no	—
82	C₂×C₈×C₃₁	—	2	1	no	—
83	C₂×C₈×C₃₁	—	2	1	no	—
84	C₂×C₈×C₃₁	—	2	8	no	—
85	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
88	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
89	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
90	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
91	C₂×C₈×C₃₁	#41	1	2	yes	—
92	C₂×C₈×C₃₁	#41	1	2	yes	—
93	C₄⋊C₄×C₃₁	—	2	4	no	—
94	C₄⋊C₄×C₃₁	—	2	4	no	—
95	C₄⋊C₄×C₃₁	—	2	4	no	—
96	C₄⋊C₄×C₃₁	—	2	4	no	—
97	C₄⋊C₄×C₃₁	—	2	4	no	—
98	C₄⋊C₄×C₃₁	—	4	8	no	—
99	C₄⋊C₄×C₃₁	—	8	4	no	—
100	C₄⋊C₄×C₃₁	—	8	4	no	—
101	C₂²⋊C₄×C₃₁	—	2	4	no	—
102	C₂²⋊C₄×C₃₁	—	8	4	no	—
103	C₄²×C₃₁	—	2	1	no	—
104	C₄²×C₃₁	—	2	1	no	—
105	C₄²×C₃₁	C₂×D₄	1	2	yes	—
106	C₄²×C₃₁	#41	1	2	yes	—
107	#33216	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂³×C₃₁	—	2	1	no	—
109	C₂³×C₃₁	—	2	1	no	—
110	C₂³×C₃₁	—	2	2	no	—
111	C₂³×C₃₁	—	2	2	no	—
112	C₂³×C₃₁	C₄⋊Q₈	1	2	yes	—
113	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₃₁	—	2	1	no	—
115	C₂×C₄×C₃₁	—	2	1	no	—
116	C₂×C₄×C₃₁	—	2	2	no	—
117	C₂×C₄×C₃₁	—	2	2	no	—
118	C₂×C₄×C₃₁	—	2	2	no	—
119	C₂×C₄×C₃₁	—	2	2	no	—
120	C₂×C₄×C₃₁	—	2	2	no	—
121	C₂×C₄×C₃₁	—	2	2	no	—
122	C₂×C₄×C₃₁	—	2	2	no	—
123	C₂×C₄×C₃₁	—	2	2	no	—
124	C₂×C₄×C₃₁	—	2	2	no	—
125	C₂×C₄×C₃₁	—	4	4	no	—
126	C₂×C₄×C₃₁	—	4	4	no	—
127	C₂×C₄×C₃₁	—	4	4	no	—
128	C₂×C₄×C₃₁	—	4	4	no	—
129	C₂×C₄×C₃₁	—	4	4	no	—
130	C₂×C₄×C₃₁	—	4	4	no	—
131	C₂×C₄×C₃₁	#121	1	4	yes	—
132	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
133	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
135	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
136	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
139	C₈×C₃₁	—	2	2	no	—
140	C₈×C₃₁	—	2	2	no	—
141	C₈×C₃₁	C₂²⋊Q₈	1	4	yes	—
142	C₈×C₃₁	C₂²⋊Q₈	1	4	yes	—
143	C₂×#499	C₂×C₃₁	1	1	yes	characteristic, complemented
144	C₂×#405	C₂×C₃₁	1	1	yes	characteristic
145	#2629	C₂×C₃₁	1	2	yes	—
146	#1992	C₂×C₃₁	1	2	yes	complemented
147	#1943	C₂×C₃₁	1	2	yes	complemented
148	#1942	C₂×C₃₁	1	2	yes	complemented
149	#1845	C₂×C₃₁	1	1	yes	characteristic, complemented
150	#1809	C₂×C₃₁	1	2	yes	—
151	#1808	C₂×C₃₁	1	2	yes	—
152	C₂²×C₃₁	—	2	2	no	—
153	C₂²×C₃₁	—	2	4	no	—
154	C₂²×C₃₁	—	4	1	no	—
155	C₂²×C₃₁	—	4	1	no	—
156	C₂²×C₃₁	—	4	2	no	—
157	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
158	C₂²×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
159	C₂²×C₃₁	#532	1	1	yes	characteristic
160	C₂²×C₃₁	#576	1	2	yes	—
161	C₂²×C₃₁	#578	1	2	yes	—
162	C₄×C₃₁	—	2	2	no	—
163	C₄×C₃₁	—	2	4	no	—
164	C₄×C₃₁	—	4	2	no	—
165	C₄×C₃₁	—	8	4	no	—
166	C₄×C₃₁	—	8	4	no	—
167	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
168	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
169	C₄×C₃₁	#533	1	1	yes	characteristic
170	C₄×C₃₁	#533	1	1	yes	characteristic
171	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
172	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
173	C₂×#125	C₂²×C₃₁	1	2	yes	—
174	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
175	#499	—	2	4	no	—
176	C₂×#107	C₂²×C₃₁	1	2	yes	—
177	C₂×#107	C₂²×C₃₁	1	2	yes	—
178	C₂×#107	C₂²×C₃₁	1	2	yes	—
179	C₂×#106	C₂²×C₃₁	1	2	yes	—
180	#405	—	2	4	no	—
181	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
182	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
183	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
184	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
185	#398	—	2	4	no	—
186	#396	—	2	4	no	—
187	#389	C₂²×C₃₁	1	4	yes	—
188	#384	C₂²×C₃₁	1	2	yes	—
189	#380	C₂²×C₃₁	1	4	yes	—
190	#340	C₂²×C₃₁	1	4	yes	—
191	#340	C₂²×C₃₁	1	4	yes	—
192	C₂×C₃₁	—	4	2	no	—
193	C₂×C₃₁	#2857	1	1	yes	characteristic
194	C₂×C₃₁	#3061	1	2	yes	—
195	C₂×C₃₁	#3164	1	2	yes	—
196	C₂×C₃₁	#3176	1	1	yes	characteristic
197	C₂×C₃₁	#3177	1	1	yes	characteristic
198	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
199	C₂×M₁₆	—	2	2	no	—
200	C₂×M₁₆	—	2	2	no	—
201	C₂×M₁₆	Q₈×C₃₁	1	4	yes	—
202	C₂²×C₈	—	2	2	no	—
203	C₂²×C₈	—	2	2	no	—
204	C₂²×C₈	Q₈×C₃₁	1	4	yes	—
205	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
206	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
207	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
208	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
209	#125	—	2	4	no	—
210	#125	—	2	4	no	—
211	#117	—	2	2	no	—
212	#117	—	2	2	no	—
213	C₂×C₄⋊C₄	—	2	4	no	—
214	C₂×C₄⋊C₄	—	2	4	no	—
215	C₂×C₄⋊C₄	—	2	4	no	—
216	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
217	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
218	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
219	C₂×C₂²⋊C₄	—	2	4	no	—
220	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
221	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
222	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
223	#107	—	2	4	no	—
224	#107	—	2	4	no	—
225	#107	—	2	4	no	—
226	#106	—	2	4	no	—
227	#97	—	2	2	no	—
228	#97	—	2	2	no	—
229	C₄×C₈	—	2	2	no	—
230	C₄×C₈	D₄×C₃₁	1	4	yes	—
231	#95	—	2	4	no	—
232	#95	—	2	4	no	—
233	C₃₁	#33216	1	1	yes	characteristic, direct product factor, Sylow
234	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	2	4	no	—
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
242	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
243	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
244	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
245	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
246	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
247	M₁₆	—	2	8	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	8	no	—
253	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
254	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
255	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
256	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
257	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
258	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
259	C₂×C₈	#41×C₃₁	1	2	yes	—
260	C₂×C₈	#41×C₃₁	1	2	yes	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	2	4	no	—
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	4	8	no	—
267	C₄⋊C₄	—	8	4	no	—
268	C₄⋊C₄	—	8	4	no	—
269	C₂²⋊C₄	—	2	4	no	—
270	C₂²⋊C₄	—	8	4	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	2	1	no	—
273	C₄²	C₂×D₄×C₃₁	1	2	yes	—
274	C₄²	#41×C₃₁	1	2	yes	—
275	C₂³	—	2	1	no	—
276	C₂³	—	2	1	no	—
277	C₂³	—	2	2	no	—
278	C₂³	—	2	2	no	—
279	C₂³	C₃₁×C₄⋊Q₈	1	2	yes	—
280	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	C₃₁×#121	1	4	yes	—
299	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
300	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
301	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
302	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
303	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
304	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
305	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
306	C₈	—	2	2	no	—
307	C₈	—	2	2	no	—
308	C₈	C₃₁×C₂²⋊Q₈	1	4	yes	—
309	C₈	C₃₁×C₂²⋊Q₈	1	4	yes	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	4	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	4	2	no	—
315	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
316	C₂²	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
317	C₂²	C₃₁×#532	1	1	yes	characteristic
318	C₂²	C₃₁×#576	1	2	yes	—
319	C₂²	C₃₁×#578	1	2	yes	—
320	C₄	—	2	2	no	—
321	C₄	—	2	4	no	—
322	C₄	—	4	2	no	—
323	C₄	—	8	4	no	—
324	C₄	—	8	4	no	—
325	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
326	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
327	C₄	C₃₁×#533	1	1	yes	characteristic
328	C₄	C₃₁×#533	1	1	yes	characteristic
329	C₂	—	4	2	no	—
330	C₂	C₃₁×#2857	1	1	yes	characteristic
331	C₂	C₃₁×#3061	1	2	yes	—
332	C₂	C₃₁×#3164	1	2	yes	—
333	C₂	C₃₁×#3176	1	1	yes	characteristic
334	C₂	C₃₁×#3177	1	1	yes	characteristic

	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 ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#499	C₂	semidirect
4. C₃₁×#1992	C₂	semidirect
5. C₃₁×#1943	C₂	semidirect
6. C₃₁×#1942	C₂	semidirect
7. C₃₁×#1845	C₂	semidirect
107. #33216	C₃₁	direct (N × H)
143. C₂×#499	C₂×C₃₁	semidirect
146. #1992	C₂×C₃₁	semidirect
147. #1943	C₂×C₃₁	semidirect
148. #1942	C₂×C₃₁	semidirect
149. #1845	C₂×C₃₁	semidirect
233. C₃₁	#33216	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×#499	2
C₂×C₃₁×#405	2
C₃₁×#2629	2
C₃₁×#1992	2
C₃₁×#1943	2
C₃₁×#1942	2
C₃₁×#1845	2
C₃₁×#1809	2
C₃₁×#1808	2
#33216	31

Subgroups — C31×#33216 #21489975

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33216 #21489975