Subgroups — C₃₁×#20780 #21477539

Order 7936.

758 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1840	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1831	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1817	C₂	1	2	yes	maximal
5	C₃₁×#1798	C₂	1	2	yes	maximal
6	C₃₁×#1768	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1750	C₂	1	2	yes	maximal
8	C₃₁×#1745	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1744	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
14	C₂×C₃₁×C₂²⋊C₈	C₂²	1	4	yes	—
15	C₂×C₃₁×#97	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
17	C₄×C₄⋊C₄×C₃₁	C₄	1	2	yes	—
18	C₄×C₄⋊C₄×C₃₁	C₄	1	2	yes	—
19	C₄×C₂²⋊C₄×C₃₁	C₄	1	2	yes	—
20	C₄×C₂²⋊C₄×C₃₁	C₄	1	2	yes	—
21	C₃₁×#376	C₂²	1	1	yes	characteristic
22	C₃₁×#376	C₂²	1	1	yes	characteristic
23	C₄³×C₃₁	C₂²	1	1	yes	characteristic
24	C₄³×C₃₁	C₂²	1	1	yes	characteristic
25	C₃₁×#336	C₂²	1	2	yes	—
26	C₃₁×#336	C₂²	1	2	yes	—
27	C₃₁×#336	C₂²	1	2	yes	—
28	C₃₁×#336	C₂²	1	2	yes	—
29	C₃₁×#336	C₂²	1	4	yes	—
30	C₃₁×#336	C₂²	1	4	yes	—
31	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
32	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
34	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
35	C₃₁×#117	—	2	4	no	—
36	C₃₁×#117	—	2	4	no	—
37	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
38	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
40	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
41	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
42	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₃₁	—	2	4	no	—
44	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
55	C₃₁×C₄⋊C₈	—	2	8	no	—
56	C₃₁×C₂²⋊C₈	—	2	8	no	—
57	C₃₁×#97	—	2	4	no	—
58	C₄×C₈×C₃₁	—	2	4	no	—
59	C₃₁×#95	C₂×C₄	1	2	yes	—
60	C₃₁×#95	C₂×C₄	1	2	yes	—
61	C₃₁×#95	C₂×C₄	1	2	yes	—
62	C₃₁×#95	C₂×C₄	1	2	yes	—
63	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	—	2	2	no	—
65	C₂²×C₄×C₃₁	—	2	2	no	—
66	C₂²×C₄×C₃₁	—	2	4	no	—
67	C₂²×C₄×C₃₁	—	2	4	no	—
68	C₂²×C₄×C₃₁	—	4	2	no	—
69	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₃₁	#41	1	4	yes	—
79	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
80	C₂×C₈×C₃₁	—	2	4	no	—
81	C₂×C₈×C₃₁	—	2	4	no	—
82	C₂×C₈×C₃₁	—	2	4	no	—
83	C₂×C₈×C₃₁	—	2	4	no	—
84	C₂×C₈×C₃₁	—	2	4	no	—
85	C₂×C₈×C₃₁	—	2	4	no	—
86	C₄⋊C₄×C₃₁	—	2	2	no	—
87	C₄⋊C₄×C₃₁	—	2	2	no	—
88	C₄⋊C₄×C₃₁	—	2	4	no	—
89	C₂²⋊C₄×C₃₁	—	2	2	no	—
90	C₂²⋊C₄×C₃₁	—	2	2	no	—
91	C₂²⋊C₄×C₃₁	—	2	4	no	—
92	C₄²×C₃₁	—	2	4	no	—
93	C₄²×C₃₁	—	2	4	no	—
94	C₄²×C₃₁	—	2	4	no	—
95	C₄²×C₃₁	—	2	2	no	—
96	C₄²×C₃₁	—	2	2	no	—
97	C₄²×C₃₁	—	2	2	no	—
98	C₄²×C₃₁	—	2	2	no	—
99	C₄²×C₃₁	—	4	2	no	—
100	C₄²×C₃₁	—	4	2	no	—
101	C₄²×C₃₁	#41	1	4	yes	—
102	C₄²×C₃₁	#41	1	4	yes	—
103	#20780	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂³×C₃₁	—	2	1	no	—
105	C₂³×C₃₁	—	2	1	no	—
106	C₂³×C₃₁	—	2	2	no	—
107	C₂³×C₃₁	—	2	2	no	—
108	C₂³×C₃₁	#117	1	2	yes	—
109	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₃₁	—	2	2	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	—	2	2	no	—
116	C₂×C₄×C₃₁	—	2	2	no	—
117	C₂×C₄×C₃₁	—	2	2	no	—
118	C₂×C₄×C₃₁	—	2	8	no	—
119	C₂×C₄×C₃₁	—	2	2	no	—
120	C₂×C₄×C₃₁	—	2	1	no	—
121	C₂×C₄×C₃₁	—	2	1	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	—	2	2	no	—
126	C₂×C₄×C₃₁	—	2	2	no	—
127	C₂×C₄×C₃₁	—	2	2	no	—
128	C₂×C₄×C₃₁	—	2	2	no	—
129	C₂×C₄×C₃₁	—	4	8	no	—
130	C₂×C₄×C₃₁	#117	1	2	yes	—
131	C₂×C₄×C₃₁	#117	1	2	yes	—
132	C₂×C₄×C₃₁	#117	1	2	yes	—
133	C₂×C₄×C₃₁	#117	1	4	yes	—
134	C₂×C₄×C₃₁	#131	1	4	yes	—
135	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
136	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
137	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
138	C₈×C₃₁	—	4	4	no	—
139	C₈×C₃₁	—	4	4	no	—
140	C₄×#117	C₂×C₃₁	1	1	yes	characteristic
141	#1840	C₂×C₃₁	1	2	yes	complemented
142	#1831	C₂×C₃₁	1	2	yes	complemented
143	#1817	C₂×C₃₁	1	2	yes	—
144	#1798	C₂×C₃₁	1	2	yes	—
145	#1768	C₂×C₃₁	1	2	yes	complemented
146	#1750	C₂×C₃₁	1	2	yes	—
147	#1745	C₂×C₃₁	1	1	yes	characteristic, complemented
148	#1744	C₂×C₃₁	1	1	yes	characteristic, complemented
149	C₂²×C₃₁	—	2	4	no	—
150	C₂²×C₃₁	—	2	2	no	—
151	C₂²×C₃₁	—	4	1	no	—
152	C₂²×C₃₁	—	4	1	no	—
153	C₂²×C₃₁	—	4	2	no	—
154	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
155	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
156	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
157	C₂²×C₃₁	C₂×#131	1	2	yes	—
158	C₂²×C₃₁	#568	1	2	yes	—
159	C₄×C₃₁	—	2	2	no	—
160	C₄×C₃₁	—	2	2	no	—
161	C₄×C₃₁	—	2	4	no	—
162	C₄×C₃₁	—	2	4	no	—
163	C₄×C₃₁	—	2	2	no	—
164	C₄×C₃₁	—	4	1	no	—
165	C₄×C₃₁	—	4	1	no	—
166	C₄×C₃₁	—	4	2	no	—
167	C₄×C₃₁	—	4	2	no	—
168	C₄×C₃₁	#529	1	4	yes	—
169	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
170	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
171	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
172	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
173	C₂×C₂²⋊C₈	C₂²×C₃₁	1	4	yes	—
174	C₂×#97	C₂²×C₃₁	1	2	yes	—
175	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
176	C₄×C₄⋊C₄	C₄×C₃₁	1	2	yes	—
177	C₄×C₄⋊C₄	C₄×C₃₁	1	2	yes	—
178	C₄×C₂²⋊C₄	C₄×C₃₁	1	2	yes	—
179	C₄×C₂²⋊C₄	C₄×C₃₁	1	2	yes	—
180	#376	C₂²×C₃₁	1	1	yes	characteristic
181	#376	C₂²×C₃₁	1	1	yes	characteristic
182	C₄³	C₂²×C₃₁	1	1	yes	characteristic
183	C₄³	C₂²×C₃₁	1	1	yes	characteristic
184	#336	C₂²×C₃₁	1	2	yes	—
185	#336	C₂²×C₃₁	1	2	yes	—
186	#336	C₂²×C₃₁	1	2	yes	—
187	#336	C₂²×C₃₁	1	2	yes	—
188	#336	C₂²×C₃₁	1	4	yes	—
189	#336	C₂²×C₃₁	1	4	yes	—
190	C₂×C₃₁	—	4	2	no	—
191	C₂×C₃₁	#2283	1	1	yes	characteristic
192	C₂×C₃₁	#2922	1	1	yes	characteristic
193	C₂×C₃₁	#2923	1	1	yes	characteristic
194	C₂×C₃₁	#2924	1	2	yes	—
195	C₂×C₃₁	#2925	1	2	yes	—
196	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
197	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
198	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
199	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
200	#117	—	2	4	no	—
201	#117	—	2	4	no	—
202	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
203	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
204	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
205	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
206	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
207	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
208	C₂×C₄²	—	2	4	no	—
209	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
210	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
211	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
212	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
213	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
214	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
215	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
216	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
217	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
218	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
219	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
220	C₄⋊C₈	—	2	8	no	—
221	C₂²⋊C₈	—	2	8	no	—
222	#97	—	2	4	no	—
223	C₄×C₈	—	2	4	no	—
224	#95	C₂×C₄×C₃₁	1	2	yes	—
225	#95	C₂×C₄×C₃₁	1	2	yes	—
226	#95	C₂×C₄×C₃₁	1	2	yes	—
227	#95	C₂×C₄×C₃₁	1	2	yes	—
228	C₃₁	#20780	1	1	yes	characteristic, direct product factor, Sylow
229	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
236	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
237	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
238	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
239	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
240	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
241	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
242	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
243	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
244	C₂²×C₄	#41×C₃₁	1	4	yes	—
245	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	—	2	4	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₂²⋊C₄	—	2	2	no	—
256	C₂²⋊C₄	—	2	2	no	—
257	C₂²⋊C₄	—	2	4	no	—
258	C₄²	—	2	4	no	—
259	C₄²	—	2	4	no	—
260	C₄²	—	2	4	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	2	2	no	—
265	C₄²	—	4	2	no	—
266	C₄²	—	4	2	no	—
267	C₄²	#41×C₃₁	1	4	yes	—
268	C₄²	#41×C₃₁	1	4	yes	—
269	C₂³	—	2	1	no	—
270	C₂³	—	2	1	no	—
271	C₂³	—	2	2	no	—
272	C₂³	—	2	2	no	—
273	C₂³	C₃₁×#117	1	2	yes	—
274	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	8	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	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	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	8	no	—
295	C₂×C₄	C₃₁×#117	1	2	yes	—
296	C₂×C₄	C₃₁×#117	1	2	yes	—
297	C₂×C₄	C₃₁×#117	1	2	yes	—
298	C₂×C₄	C₃₁×#117	1	4	yes	—
299	C₂×C₄	C₃₁×#131	1	4	yes	—
300	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
301	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
302	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
303	C₈	—	4	4	no	—
304	C₈	—	4	4	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	1	no	—
309	C₂²	—	4	2	no	—
310	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
311	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
312	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
313	C₂²	C₂×C₃₁×#131	1	2	yes	—
314	C₂²	C₃₁×#568	1	2	yes	—
315	C₄	—	2	2	no	—
316	C₄	—	2	2	no	—
317	C₄	—	2	4	no	—
318	C₄	—	2	4	no	—
319	C₄	—	2	2	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	2	no	—
323	C₄	—	4	2	no	—
324	C₄	C₃₁×#529	1	4	yes	—
325	C₂	—	4	2	no	—
326	C₂	C₃₁×#2283	1	1	yes	characteristic
327	C₂	C₃₁×#2922	1	1	yes	characteristic
328	C₂	C₃₁×#2923	1	1	yes	characteristic
329	C₂	C₃₁×#2924	1	2	yes	—
330	C₂	C₃₁×#2925	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1840	C₂	semidirect
3. C₃₁×#1831	C₂	semidirect
6. C₃₁×#1768	C₂	semidirect
8. C₃₁×#1745	C₂	semidirect
9. C₃₁×#1744	C₂	semidirect
103. #20780	C₃₁	direct (N × H)
141. #1840	C₂×C₃₁	semidirect
142. #1831	C₂×C₃₁	semidirect
145. #1768	C₂×C₃₁	semidirect
147. #1745	C₂×C₃₁	semidirect
148. #1744	C₂×C₃₁	semidirect
228. C₃₁	#20780	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₃₁×#117	2
C₃₁×#1840	2
C₃₁×#1831	2
C₃₁×#1817	2
C₃₁×#1798	2
C₃₁×#1768	2
C₃₁×#1750	2
C₃₁×#1745	2
C₃₁×#1744	2
#20780	31

Subgroups — C31×#20780 #21477539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20780 #21477539