Subgroups — C₃₁×#21440 #21478199

Order 7936.

1708 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#435	C₂	1	3	yes	maximal, complemented
3	C₄×C₃₁×#120	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1869	C₂	1	3	yes	maximal, complemented
5	C₃₁×#1850	C₂	1	3	yes	maximal, complemented
6	C₃₁×#1779	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1752	C₂	1	3	yes	maximal, complemented
8	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
9	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
10	C₂³×C₈×C₃₁	—	2	12	no	—
11	C₂³×C₈×C₃₁	C₂²	1	3	yes	—
12	C₂³×C₈×C₃₁	C₂²	1	3	yes	—
13	C₂×C₃₁×#120	C₄	1	1	yes	characteristic
14	C₂×C₄×D₄×C₃₁	C₂²	1	3	yes	—
15	C₃₁×#435	—	2	12	no	—
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	3	yes	complemented
17	C₂×C₃₁×C₂²⋊C₈	C₂²	1	3	yes	—
18	C₂×C₃₁×C₂²⋊C₈	C₂²	1	3	yes	—
19	C₂×C₃₁×C₂²⋊C₈	C₂²	1	3	yes	complemented
20	C₂×C₃₁×#97	C₂²	1	3	yes	complemented
21	C₃₁×#386	C₄	1	3	yes	—
22	C₃₁×#385	C₄	1	3	yes	—
23	C₃₁×#379	C₄	1	1	yes	characteristic
24	C₄×C₂²⋊C₄×C₃₁	C₂²	1	3	yes	—
25	C₃₁×#336	C₂²	1	6	yes	complemented
26	C₂⁵×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×D₄×C₃₁	C₂×C₄	1	3	yes	—
28	C₂³×C₄×C₃₁	—	2	6	no	—
29	C₂³×C₄×C₃₁	—	2	6	no	—
30	C₂³×C₄×C₃₁	C₂×C₄	1	3	yes	—
31	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
32	C₂³×C₄×C₃₁	C₂³	1	3	yes	—
33	C₂×M₁₆×C₃₁	—	2	6	no	—
34	C₂²×C₈×C₃₁	—	2	32	no	—
35	C₂²×C₈×C₃₁	—	2	6	no	—
36	C₂²×C₈×C₃₁	—	2	12	no	—
37	C₂²×C₈×C₃₁	—	2	12	no	—
38	C₂²×C₈×C₃₁	D₄	1	12	yes	complemented
39	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
40	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
42	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
43	C₃₁×#120	—	2	4	no	—
44	C₄×D₄×C₃₁	—	2	6	no	—
45	C₄×D₄×C₃₁	—	2	6	no	—
46	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	3	yes	—
47	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	3	yes	—
48	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	3	yes	—
49	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	3	yes	—
50	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	3	yes	—
51	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
52	C₃₁×C₄⋊C₈	—	2	6	no	—
53	C₃₁×C₂²⋊C₈	—	2	6	no	—
54	C₃₁×C₂²⋊C₈	—	2	6	no	—
55	C₃₁×C₂²⋊C₈	—	2	6	no	—
56	C₃₁×#97	—	2	6	no	—
57	C₃₁×#95	C₂×C₄	1	6	yes	—
58	C₂⁴×C₃₁	—	2	6	no	—
59	C₂⁴×C₃₁	—	2	6	no	—
60	C₂⁴×C₃₁	M₁₆	1	4	yes	complemented
61	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
62	C₂⁴×C₃₁	C₂²×C₄	1	3	yes	—
63	C₂×D₄×C₃₁	—	2	6	no	—
64	C₂×D₄×C₃₁	—	2	6	no	—
65	C₂×D₄×C₃₁	—	2	6	no	—
66	C₂²×C₄×C₃₁	—	2	8	no	—
67	C₂²×C₄×C₃₁	—	2	24	no	—
68	C₂²×C₄×C₃₁	—	2	6	no	—
69	C₂²×C₄×C₃₁	—	2	12	no	—
70	C₂²×C₄×C₃₁	—	2	6	no	—
71	C₂²×C₄×C₃₁	—	2	6	no	—
72	C₂²×C₄×C₃₁	—	2	3	no	—
73	C₂²×C₄×C₃₁	—	2	3	no	—
74	C₂²×C₄×C₃₁	C₂²×C₄	1	3	yes	—
75	C₂²×C₄×C₃₁	C₂²×C₄	1	3	yes	—
76	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₃₁	C₂²×C₄	1	3	yes	—
78	C₂²×C₄×C₃₁	C₂×D₄	1	6	yes	—
79	C₂²×C₄×C₃₁	#41	1	6	yes	—
80	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
81	M₁₆×C₃₁	—	4	4	no	—
82	C₂×C₈×C₃₁	—	2	24	no	—
83	C₂×C₈×C₃₁	—	2	6	no	—
84	C₂×C₈×C₃₁	—	2	24	no	—
85	C₂×C₈×C₃₁	—	2	3	no	—
86	C₂×C₈×C₃₁	—	2	3	no	—
87	C₂×C₈×C₃₁	—	2	6	no	—
88	C₂×C₈×C₃₁	C₂×D₄	1	6	yes	—
89	C₂×C₈×C₃₁	C₂×D₄	1	6	yes	—
90	C₄⋊C₄×C₃₁	—	2	6	no	—
91	C₂²⋊C₄×C₃₁	—	2	6	no	—
92	C₂²⋊C₄×C₃₁	—	2	6	no	—
93	C₂²⋊C₄×C₃₁	—	2	6	no	—
94	C₂²⋊C₄×C₃₁	—	2	6	no	—
95	C₄²×C₃₁	—	2	3	no	—
96	C₄²×C₃₁	—	2	3	no	—
97	#21440	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂³×C₃₁	—	2	8	no	—
99	C₂³×C₃₁	—	2	24	no	—
100	C₂³×C₃₁	—	2	6	no	—
101	C₂³×C₃₁	—	2	6	no	—
102	C₂³×C₃₁	—	2	24	no	—
103	C₂³×C₃₁	—	2	3	no	—
104	C₂³×C₃₁	—	2	1	no	—
105	C₂³×C₃₁	—	2	3	no	—
106	C₂³×C₃₁	C₄×D₄	1	6	yes	—
107	C₂³×C₃₁	C₂×M₁₆	1	6	yes	—
108	C₂³×C₃₁	#131	1	6	yes	—
109	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
110	D₄×C₃₁	—	4	12	no	—
111	C₂×C₄×C₃₁	—	2	24	no	—
112	C₂×C₄×C₃₁	—	2	6	no	—
113	C₂×C₄×C₃₁	—	2	6	no	—
114	C₂×C₄×C₃₁	—	2	6	no	—
115	C₂×C₄×C₃₁	—	2	12	no	—
116	C₂×C₄×C₃₁	—	2	3	no	—
117	C₂×C₄×C₃₁	—	2	3	no	—
118	C₂×C₄×C₃₁	—	2	3	no	—
119	C₂×C₄×C₃₁	—	2	3	no	—
120	C₂×C₄×C₃₁	—	2	6	no	—
121	C₂×C₄×C₃₁	—	2	3	no	—
122	C₂×C₄×C₃₁	—	4	1	no	—
123	C₂×C₄×C₃₁	—	4	3	no	—
124	C₂×C₄×C₃₁	C₄×D₄	1	6	yes	—
125	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
126	C₂×C₄×C₃₁	C₂×#41	1	3	yes	—
127	C₈×C₃₁	—	2	6	no	—
128	C₈×C₃₁	—	4	1	no	—
129	C₈×C₃₁	—	4	3	no	—
130	C₈×C₃₁	#120	1	4	yes	complemented
131	C₂⁴×C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
132	C₂×#435	C₂×C₃₁	1	3	yes	complemented
133	C₄×#120	C₂×C₃₁	1	1	yes	characteristic
134	#1869	C₂×C₃₁	1	3	yes	complemented
135	#1850	C₂×C₃₁	1	3	yes	complemented
136	#1779	C₂×C₃₁	1	1	yes	characteristic, complemented
137	#1752	C₂×C₃₁	1	3	yes	complemented
138	C₂²×C₃₁	—	2	32	no	—
139	C₂²×C₃₁	—	2	12	no	—
140	C₂²×C₃₁	—	2	6	no	—
141	C₂²×C₃₁	—	2	12	no	—
142	C₂²×C₃₁	—	2	6	no	—
143	C₂²×C₃₁	—	4	3	no	—
144	C₂²×C₃₁	—	4	3	no	—
145	C₂²×C₃₁	—	4	1	no	—
146	C₂²×C₃₁	#435	1	12	yes	complemented
147	C₂²×C₃₁	C₂×C₄×D₄	1	3	yes	—
148	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
149	C₂²×C₃₁	C₂×#131	1	3	yes	—
150	C₄×C₃₁	—	2	6	no	—
151	C₄×C₃₁	—	4	3	no	—
152	C₄×C₃₁	—	4	3	no	—
153	C₄×C₃₁	—	4	1	no	—
154	C₄×C₃₁	C₂×#120	1	1	yes	characteristic
155	C₄×C₃₁	#525	1	3	yes	—
156	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
157	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
158	C₂³×C₈	—	2	12	no	—
159	C₂³×C₈	C₂²×C₃₁	1	3	yes	—
160	C₂³×C₈	C₂²×C₃₁	1	3	yes	—
161	C₂×#120	C₄×C₃₁	1	1	yes	characteristic
162	C₂×C₄×D₄	C₂²×C₃₁	1	3	yes	—
163	#435	—	2	12	no	—
164	C₂×C₄⋊C₈	C₂²×C₃₁	1	3	yes	complemented
165	C₂×C₂²⋊C₈	C₂²×C₃₁	1	3	yes	—
166	C₂×C₂²⋊C₈	C₂²×C₃₁	1	3	yes	—
167	C₂×C₂²⋊C₈	C₂²×C₃₁	1	3	yes	complemented
168	C₂×#97	C₂²×C₃₁	1	3	yes	complemented
169	#386	C₄×C₃₁	1	3	yes	—
170	#385	C₄×C₃₁	1	3	yes	—
171	#379	C₄×C₃₁	1	1	yes	characteristic
172	C₄×C₂²⋊C₄	C₂²×C₃₁	1	3	yes	—
173	#336	C₂²×C₃₁	1	6	yes	complemented
174	C₂×C₃₁	—	2	12	no	—
175	C₂×C₃₁	—	8	1	no	—
176	C₂×C₃₁	C₄×#120	1	1	yes	characteristic
177	C₂×C₃₁	C₂×#435	1	3	yes	—
178	C₂×C₃₁	#2929	1	3	yes	—
179	C₂⁵	C₂×C₄×C₃₁	1	1	yes	characteristic
180	C₂²×D₄	C₂×C₄×C₃₁	1	3	yes	—
181	C₂³×C₄	—	2	6	no	—
182	C₂³×C₄	—	2	6	no	—
183	C₂³×C₄	C₂×C₄×C₃₁	1	3	yes	—
184	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
185	C₂³×C₄	C₂³×C₃₁	1	3	yes	—
186	C₂×M₁₆	—	2	6	no	—
187	C₂²×C₈	—	2	32	no	—
188	C₂²×C₈	—	2	6	no	—
189	C₂²×C₈	—	2	12	no	—
190	C₂²×C₈	—	2	12	no	—
191	C₂²×C₈	D₄×C₃₁	1	12	yes	complemented
192	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
193	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
194	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
195	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
196	#120	—	2	4	no	—
197	C₄×D₄	—	2	6	no	—
198	C₄×D₄	—	2	6	no	—
199	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	3	yes	—
200	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	3	yes	—
201	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	3	yes	—
202	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	3	yes	—
203	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	3	yes	—
204	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
205	C₄⋊C₈	—	2	6	no	—
206	C₂²⋊C₈	—	2	6	no	—
207	C₂²⋊C₈	—	2	6	no	—
208	C₂²⋊C₈	—	2	6	no	—
209	#97	—	2	6	no	—
210	#95	C₂×C₄×C₃₁	1	6	yes	—
211	C₃₁	#21440	1	1	yes	characteristic, direct product factor, Sylow
212	C₂⁴	—	2	6	no	—
213	C₂⁴	—	2	6	no	—
214	C₂⁴	M₁₆×C₃₁	1	4	yes	complemented
215	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
216	C₂⁴	C₂²×C₄×C₃₁	1	3	yes	—
217	C₂×D₄	—	2	6	no	—
218	C₂×D₄	—	2	6	no	—
219	C₂×D₄	—	2	6	no	—
220	C₂²×C₄	—	2	8	no	—
221	C₂²×C₄	—	2	24	no	—
222	C₂²×C₄	—	2	6	no	—
223	C₂²×C₄	—	2	12	no	—
224	C₂²×C₄	—	2	6	no	—
225	C₂²×C₄	—	2	6	no	—
226	C₂²×C₄	—	2	3	no	—
227	C₂²×C₄	—	2	3	no	—
228	C₂²×C₄	C₂²×C₄×C₃₁	1	3	yes	—
229	C₂²×C₄	C₂²×C₄×C₃₁	1	3	yes	—
230	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
231	C₂²×C₄	C₂²×C₄×C₃₁	1	3	yes	—
232	C₂²×C₄	C₂×D₄×C₃₁	1	6	yes	—
233	C₂²×C₄	#41×C₃₁	1	6	yes	—
234	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
235	M₁₆	—	4	4	no	—
236	C₂×C₈	—	2	24	no	—
237	C₂×C₈	—	2	6	no	—
238	C₂×C₈	—	2	24	no	—
239	C₂×C₈	—	2	3	no	—
240	C₂×C₈	—	2	3	no	—
241	C₂×C₈	—	2	6	no	—
242	C₂×C₈	C₂×D₄×C₃₁	1	6	yes	—
243	C₂×C₈	C₂×D₄×C₃₁	1	6	yes	—
244	C₄⋊C₄	—	2	6	no	—
245	C₂²⋊C₄	—	2	6	no	—
246	C₂²⋊C₄	—	2	6	no	—
247	C₂²⋊C₄	—	2	6	no	—
248	C₂²⋊C₄	—	2	6	no	—
249	C₄²	—	2	3	no	—
250	C₄²	—	2	3	no	—
251	C₂³	—	2	8	no	—
252	C₂³	—	2	24	no	—
253	C₂³	—	2	24	no	—
254	C₂³	—	2	6	no	—
255	C₂³	—	2	6	no	—
256	C₂³	—	2	3	no	—
257	C₂³	—	2	1	no	—
258	C₂³	—	2	3	no	—
259	C₂³	C₄×D₄×C₃₁	1	6	yes	—
260	C₂³	C₂×M₁₆×C₃₁	1	6	yes	—
261	C₂³	C₃₁×#131	1	6	yes	—
262	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
263	D₄	—	4	12	no	—
264	C₂×C₄	—	2	24	no	—
265	C₂×C₄	—	2	6	no	—
266	C₂×C₄	—	2	6	no	—
267	C₂×C₄	—	2	6	no	—
268	C₂×C₄	—	2	12	no	—
269	C₂×C₄	—	2	3	no	—
270	C₂×C₄	—	2	3	no	—
271	C₂×C₄	—	2	3	no	—
272	C₂×C₄	—	2	3	no	—
273	C₂×C₄	—	2	6	no	—
274	C₂×C₄	—	2	3	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	3	no	—
277	C₂×C₄	C₄×D₄×C₃₁	1	6	yes	—
278	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
279	C₂×C₄	C₂×#41×C₃₁	1	3	yes	—
280	C₈	—	2	6	no	—
281	C₈	—	4	1	no	—
282	C₈	—	4	3	no	—
283	C₈	C₃₁×#120	1	4	yes	complemented
284	C₂²	—	2	32	no	—
285	C₂²	—	2	12	no	—
286	C₂²	—	2	6	no	—
287	C₂²	—	2	12	no	—
288	C₂²	—	2	6	no	—
289	C₂²	—	4	3	no	—
290	C₂²	—	4	3	no	—
291	C₂²	—	4	1	no	—
292	C₂²	C₃₁×#435	1	12	yes	complemented
293	C₂²	C₂×C₄×D₄×C₃₁	1	3	yes	—
294	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
295	C₂²	C₂×C₃₁×#131	1	3	yes	—
296	C₄	—	2	6	no	—
297	C₄	—	4	3	no	—
298	C₄	—	4	3	no	—
299	C₄	—	4	1	no	—
300	C₄	C₂×C₃₁×#120	1	1	yes	characteristic
301	C₄	C₃₁×#525	1	3	yes	—
302	C₂	—	2	12	no	—
303	C₂	—	8	1	no	—
304	C₂	C₄×C₃₁×#120	1	1	yes	characteristic
305	C₂	C₂×C₃₁×#435	1	3	yes	—
306	C₂	C₃₁×#2929	1	3	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	C₂×C₃₁×#120	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
1. C₂⁴×C₈×C₃₁	C₂	semidirect
2. C₂×C₃₁×#435	C₂	semidirect
4. C₃₁×#1869	C₂	semidirect
5. C₃₁×#1850	C₂	semidirect
6. C₃₁×#1779	C₂	semidirect
7. C₃₁×#1752	C₂	semidirect
9. C₂²×M₁₆×C₃₁	C₂²	semidirect
16. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
19. C₂×C₃₁×C₂²⋊C₈	C₂²	semidirect
20. C₂×C₃₁×#97	C₂²	semidirect
25. C₃₁×#336	C₂²	semidirect
38. C₂²×C₈×C₃₁	D₄	semidirect
60. C₂⁴×C₃₁	M₁₆	semidirect
97. #21440	C₃₁	direct (N × H)
130. C₈×C₃₁	#120	semidirect
131. C₂⁴×C₈	C₂×C₃₁	semidirect
132. C₂×#435	C₂×C₃₁	semidirect
134. #1869	C₂×C₃₁	semidirect
135. #1850	C₂×C₃₁	semidirect
136. #1779	C₂×C₃₁	semidirect
137. #1752	C₂×C₃₁	semidirect
146. C₂²×C₃₁	#435	semidirect
157. C₂²×M₁₆	C₂²×C₃₁	semidirect
164. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
167. C₂×C₂²⋊C₈	C₂²×C₃₁	semidirect
168. C₂×#97	C₂²×C₃₁	semidirect
173. #336	C₂²×C₃₁	semidirect
191. C₂²×C₈	D₄×C₃₁	semidirect
211. C₃₁	#21440	direct (N × H)
214. C₂⁴	M₁₆×C₃₁	semidirect
283. C₈	C₃₁×#120	semidirect
292. C₂²	C₃₁×#435	semidirect

Maximal subgroups

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

subgroup	index
C₂⁴×C₈×C₃₁	2
C₂×C₃₁×#435	2
C₄×C₃₁×#120	2
C₃₁×#1869	2
C₃₁×#1850	2
C₃₁×#1779	2
C₃₁×#1752	2
#21440	31

Subgroups — C31×#21440 #21478199

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21440 #21478199