Subgroups — C₃₁×#20545 #21477304

Order 7936.

934 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#431	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2283	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1770	C₂	1	2	yes	maximal, complemented
5	C₈×C₄⋊C₄×C₃₁	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1768	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1752	C₂	1	2	yes	maximal
8	C₈×C₂²⋊C₄×C₃₁	C₂	1	2	yes	maximal
9	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
10	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#431	C₄	1	8	yes	complemented
12	C₂×C₃₁×C₄⋊C₈	C₄	1	8	yes	complemented
13	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#97	C₂²	1	4	yes	—
20	C₂×C₄×C₈×C₃₁	C₂²	1	4	yes	—
21	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
22	C₃₁×#383	C₄	1	2	yes	—
23	C₃₁×#382	C₄	1	2	yes	—
24	C₃₁×#381	C₄	1	2	yes	—
25	C₃₁×#380	C₄	1	2	yes	—
26	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
27	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
28	C₃₁×#376	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	1	yes	characteristic
32	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₃₁	—	4	4	no	—
35	C₂²×C₈×C₃₁	C₂×C₄	1	8	yes	—
36	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
37	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
38	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
39	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
40	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
41	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
47	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
49	C₃₁×C₄⋊C₈	C₂×C₄	1	8	yes	—
50	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
51	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
52	C₃₁×C₄⋊C₈	C₂×C₄	1	8	yes	—
53	C₃₁×C₂²⋊C₈	C₂×C₄	1	4	yes	—
54	C₃₁×C₂²⋊C₈	C₂×C₄	1	4	yes	—
55	C₃₁×#97	—	4	4	no	—
56	C₄×C₈×C₃₁	—	4	4	no	—
57	C₄×C₈×C₃₁	C₂×C₄	1	8	yes	—
58	C₃₁×#95	C₂×C₄	1	4	yes	—
59	C₃₁×#95	C₂×C₄	1	4	yes	—
60	C₃₁×#95	C₂×C₄	1	4	yes	—
61	C₃₁×#95	C₂×C₄	1	4	yes	—
62	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	—	2	4	no	—
64	C₂²×C₄×C₃₁	—	2	4	no	—
65	C₂²×C₄×C₃₁	—	2	2	no	—
66	C₂²×C₄×C₃₁	—	2	2	no	—
67	C₂²×C₄×C₃₁	C₄²	1	4	yes	—
68	C₂²×C₄×C₃₁	C₄²	1	4	yes	—
69	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
70	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
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	2	yes	—
76	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
78	C₂×C₈×C₃₁	—	2	2	no	—
79	C₂×C₈×C₃₁	—	2	4	no	—
80	C₂×C₈×C₃₁	—	2	2	no	—
81	C₂×C₈×C₃₁	—	2	4	no	—
82	C₂×C₈×C₃₁	—	2	2	no	—
83	C₂×C₈×C₃₁	—	2	2	no	—
84	C₂×C₈×C₃₁	—	4	8	no	—
85	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
88	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
89	C₄⋊C₄×C₃₁	—	2	8	no	—
90	C₂²⋊C₄×C₃₁	—	2	8	no	—
91	C₄²×C₃₁	—	2	4	no	—
92	C₄²×C₃₁	—	2	4	no	—
93	C₄²×C₃₁	C₄²	1	4	yes	—
94	C₄²×C₃₁	C₄²	1	4	yes	—
95	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
96	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
97	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
98	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
99	#20545	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂³×C₃₁	—	2	2	no	—
101	C₂³×C₃₁	—	2	2	no	—
102	C₂³×C₃₁	C₂×C₄²	1	2	yes	—
103	C₂³×C₃₁	#131	1	4	yes	—
104	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₃₁	—	2	2	no	—
106	C₂×C₄×C₃₁	—	2	2	no	—
107	C₂×C₄×C₃₁	—	2	2	no	—
108	C₂×C₄×C₃₁	—	2	2	no	—
109	C₂×C₄×C₃₁	—	2	8	no	—
110	C₂×C₄×C₃₁	—	2	2	no	—
111	C₂×C₄×C₃₁	—	2	2	no	—
112	C₂×C₄×C₃₁	—	2	4	no	—
113	C₂×C₄×C₃₁	—	2	8	no	—
114	C₂×C₄×C₃₁	—	2	8	no	—
115	C₂×C₄×C₃₁	—	2	8	no	—
116	C₂×C₄×C₃₁	—	2	8	no	—
117	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
118	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
119	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
120	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
121	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
122	C₂×C₄×C₃₁	#131	1	4	yes	—
123	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
126	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
127	C₈×C₃₁	—	2	4	no	—
128	C₈×C₃₁	—	2	4	no	—
129	C₈×C₃₁	—	2	4	no	—
130	C₈×C₃₁	—	2	4	no	—
131	C₂×#431	C₂×C₃₁	1	1	yes	characteristic
132	C₂²×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic
133	#2283	C₂×C₃₁	1	1	yes	characteristic
134	#1770	C₂×C₃₁	1	2	yes	complemented
135	C₈×C₄⋊C₄	C₂×C₃₁	1	2	yes	complemented
136	#1768	C₂×C₃₁	1	4	yes	complemented
137	#1752	C₂×C₃₁	1	2	yes	—
138	C₈×C₂²⋊C₄	C₂×C₃₁	1	2	yes	—
139	C₂²×C₃₁	—	2	2	no	—
140	C₂²×C₃₁	—	2	8	no	—
141	C₂²×C₃₁	—	2	2	no	—
142	C₂²×C₃₁	#405	1	4	yes	—
143	C₂²×C₃₁	C₂²×C₄²	1	1	yes	characteristic
144	C₂²×C₃₁	#520	1	2	yes	—
145	C₂²×C₃₁	C₂×#131	1	2	yes	—
146	C₂²×C₃₁	C₂×#131	1	2	yes	—
147	C₄×C₃₁	—	2	2	no	—
148	C₄×C₃₁	—	2	2	no	—
149	C₄×C₃₁	—	2	2	no	—
150	C₄×C₃₁	—	2	2	no	—
151	C₄×C₃₁	—	4	8	no	—
152	C₄×C₃₁	#405	1	4	yes	—
153	C₄×C₃₁	#518	1	1	yes	characteristic
154	C₄×C₃₁	#519	1	1	yes	characteristic
155	C₄×C₃₁	#520	1	2	yes	—
156	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
157	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
158	#431	C₄×C₃₁	1	8	yes	complemented
159	C₂×C₄⋊C₈	C₄×C₃₁	1	8	yes	complemented
160	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
161	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
162	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
163	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
164	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
165	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
166	C₂×#97	C₂²×C₃₁	1	4	yes	—
167	C₂×C₄×C₈	C₂²×C₃₁	1	4	yes	—
168	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
169	#383	C₄×C₃₁	1	2	yes	—
170	#382	C₄×C₃₁	1	2	yes	—
171	#381	C₄×C₃₁	1	2	yes	—
172	#380	C₄×C₃₁	1	2	yes	—
173	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
174	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
175	#376	C₂²×C₃₁	1	2	yes	—
176	#336	C₂²×C₃₁	1	4	yes	—
177	#336	C₂²×C₃₁	1	4	yes	—
178	C₂×C₃₁	—	2	4	no	—
179	C₂×C₃₁	#2274	1	1	yes	characteristic
180	C₂×C₃₁	C₂×#405	1	1	yes	characteristic
181	C₂×C₃₁	C₂×#405	1	1	yes	characteristic
182	C₂×C₃₁	#2994	1	4	yes	—
183	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
184	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
185	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
186	C₂²×C₈	—	4	4	no	—
187	C₂²×C₈	C₂×C₄×C₃₁	1	8	yes	—
188	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
189	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
190	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
191	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
192	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
193	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
194	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
195	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
196	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
197	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
198	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
199	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
200	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
201	C₄⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
202	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
203	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
204	C₄⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
205	C₂²⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
206	C₂²⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
207	#97	—	4	4	no	—
208	C₄×C₈	—	4	4	no	—
209	C₄×C₈	C₂×C₄×C₃₁	1	8	yes	—
210	#95	C₂×C₄×C₃₁	1	4	yes	—
211	#95	C₂×C₄×C₃₁	1	4	yes	—
212	#95	C₂×C₄×C₃₁	1	4	yes	—
213	#95	C₂×C₄×C₃₁	1	4	yes	—
214	C₃₁	#20545	1	1	yes	characteristic, direct product factor, Sylow
215	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	C₄²×C₃₁	1	4	yes	—
221	C₂²×C₄	C₄²×C₃₁	1	4	yes	—
222	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
223	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
224	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
225	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
226	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
227	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
228	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
229	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
230	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	4	8	no	—
238	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
239	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
240	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
241	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
242	C₄⋊C₄	—	2	8	no	—
243	C₂²⋊C₄	—	2	8	no	—
244	C₄²	—	2	4	no	—
245	C₄²	—	2	4	no	—
246	C₄²	C₄²×C₃₁	1	4	yes	—
247	C₄²	C₄²×C₃₁	1	4	yes	—
248	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
249	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
250	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
251	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
252	C₂³	—	2	2	no	—
253	C₂³	—	2	2	no	—
254	C₂³	C₂×C₄²×C₃₁	1	2	yes	—
255	C₂³	C₃₁×#131	1	4	yes	—
256	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	8	no	—
267	C₂×C₄	—	2	8	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
270	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
271	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
272	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
273	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
274	C₂×C₄	C₃₁×#131	1	4	yes	—
275	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
276	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
277	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
278	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
279	C₈	—	2	4	no	—
280	C₈	—	2	4	no	—
281	C₈	—	2	4	no	—
282	C₈	—	2	4	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	8	no	—
285	C₂²	—	2	2	no	—
286	C₂²	C₃₁×#405	1	4	yes	—
287	C₂²	C₂²×C₄²×C₃₁	1	1	yes	characteristic
288	C₂²	C₃₁×#520	1	2	yes	—
289	C₂²	C₂×C₃₁×#131	1	2	yes	—
290	C₂²	C₂×C₃₁×#131	1	2	yes	—
291	C₄	—	2	2	no	—
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	2	no	—
295	C₄	—	4	8	no	—
296	C₄	C₃₁×#405	1	4	yes	—
297	C₄	C₃₁×#518	1	1	yes	characteristic
298	C₄	C₃₁×#519	1	1	yes	characteristic
299	C₄	C₃₁×#520	1	2	yes	—
300	C₂	—	2	4	no	—
301	C₂	C₃₁×#2274	1	1	yes	characteristic
302	C₂	C₂×C₃₁×#405	1	1	yes	characteristic
303	C₂	C₂×C₃₁×#405	1	1	yes	characteristic
304	C₂	C₃₁×#2994	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1770	C₂	semidirect
5. C₈×C₄⋊C₄×C₃₁	C₂	semidirect
6. C₃₁×#1768	C₂	semidirect
11. C₃₁×#431	C₄	semidirect
12. C₂×C₃₁×C₄⋊C₈	C₄	semidirect
99. #20545	C₃₁	direct (N × H)
134. #1770	C₂×C₃₁	semidirect
135. C₈×C₄⋊C₄	C₂×C₃₁	semidirect
136. #1768	C₂×C₃₁	semidirect
158. #431	C₄×C₃₁	semidirect
159. C₂×C₄⋊C₈	C₄×C₃₁	semidirect
214. C₃₁	#20545	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×#431	2
C₂²×C₃₁×C₄⋊C₈	2
C₃₁×#2283	2
C₃₁×#1770	2
C₈×C₄⋊C₄×C₃₁	2
C₃₁×#1768	2
C₃₁×#1752	2
C₈×C₂²⋊C₄×C₃₁	2
#20545	31

Subgroups — C31×#20545 #21477304

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20545 #21477304