Subgroups — C₃₁×#21656 #21478415

Order 7936.

970 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	C₂×C₃₁×#490	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₁×#127	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₁×#125	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1992	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1961	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1934	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1836	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1762	C₂	1	4	yes	maximal, complemented
9	C₄×C₃₁×#97	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₄×C₃₁	—	2	2	no	—
13	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
14	C₃₁×#490	C₄	1	8	yes	complemented
15	C₂×C₃₁×#102	C₂²	1	4	yes	—
16	C₂×C₃₁×#102	C₂²	1	4	yes	—
17	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#97	C₂²	1	2	yes	—
19	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
21	C₃₁×#390	—	2	2	no	—
22	C₃₁×#390	C₂²	1	2	yes	—
23	C₃₁×#389	C₂²	1	2	yes	—
24	C₃₁×#387	—	2	2	no	—
25	C₃₁×#384	C₂²	1	1	yes	characteristic
26	C₃₁×#382	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
28	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
29	C₄³×C₃₁	C₂²	1	1	yes	characteristic
30	C₃₁×#340	C₂²	1	4	yes	—
31	C₃₁×#340	C₂²	1	4	yes	—
32	C₂²×D₄×C₃₁	—	2	2	no	—
33	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
34	C₂³×C₄×C₃₁	—	2	2	no	—
35	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
36	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
37	C₃₁×#127	—	2	2	no	—
38	C₃₁×#127	C₂×C₄	1	4	yes	—
39	C₃₁×#125	—	2	2	no	—
40	C₃₁×#125	C₂×C₄	1	4	yes	complemented
41	C₄×D₄×C₃₁	—	2	8	no	—
42	C₄×D₄×C₃₁	—	4	8	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
45	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
46	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
49	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
50	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
51	C₂×C₄²×C₃₁	—	2	2	no	—
52	C₂×C₄²×C₃₁	D₄	1	2	yes	—
53	C₂×C₄²×C₃₁	D₄	1	2	yes	—
54	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
57	C₃₁×#102	—	2	8	no	—
58	C₃₁×#102	C₂×C₄	1	16	yes	complemented
59	C₃₁×#97	—	2	2	no	—
60	C₃₁×#97	—	2	4	no	—
61	C₃₁×#97	C₂×C₄	1	4	yes	complemented
62	C₄×C₈×C₃₁	—	2	4	no	—
63	C₃₁×#95	—	2	2	no	—
64	C₃₁×#95	—	2	2	no	—
65	C₃₁×#95	—	2	2	no	—
66	C₂⁴×C₃₁	—	2	2	no	—
67	C₂×D₄×C₃₁	—	2	8	no	—
68	C₂×D₄×C₃₁	—	2	4	no	—
69	C₂×D₄×C₃₁	—	2	4	no	—
70	C₂×D₄×C₃₁	—	2	2	no	—
71	C₂×D₄×C₃₁	—	4	4	no	—
72	C₂×D₄×C₃₁	—	4	4	no	—
73	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₃₁	—	2	2	no	—
75	C₂²×C₄×C₃₁	—	2	2	no	—
76	C₂²×C₄×C₃₁	—	2	2	no	—
77	C₂²×C₄×C₃₁	—	4	8	no	—
78	C₂²×C₄×C₃₁	—	4	4	no	—
79	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
84	C₂×C₈×C₃₁	—	2	2	no	—
85	C₂×C₈×C₃₁	—	2	2	no	—
86	C₂×C₈×C₃₁	—	2	2	no	—
87	C₂×C₈×C₃₁	—	2	2	no	—
88	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
89	C₂×C₈×C₃₁	#41	1	4	yes	—
90	C₄⋊C₄×C₃₁	—	2	8	no	—
91	C₄⋊C₄×C₃₁	—	2	2	no	—
92	C₄⋊C₄×C₃₁	—	2	4	no	—
93	C₄⋊C₄×C₃₁	—	2	4	no	—
94	C₄⋊C₄×C₃₁	—	4	4	no	—
95	C₄⋊C₄×C₃₁	—	4	4	no	—
96	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
97	C₂²⋊C₄×C₃₁	—	4	4	no	—
98	C₂²⋊C₄×C₃₁	—	4	8	no	—
99	C₄²×C₃₁	—	2	4	no	—
100	C₄²×C₃₁	—	2	2	no	—
101	C₄²×C₃₁	—	4	2	no	—
102	C₄²×C₃₁	—	4	4	no	—
103	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
104	C₄²×C₃₁	#41	1	2	yes	—
105	C₄²×C₃₁	#41	1	4	yes	—
106	#21656	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂³×C₃₁	—	2	4	no	—
108	C₂³×C₃₁	—	4	2	no	—
109	C₂³×C₃₁	—	4	2	no	—
110	C₂³×C₃₁	—	4	4	no	—
111	C₂³×C₃₁	—	4	4	no	—
112	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
113	D₄×C₃₁	—	2	8	no	—
114	D₄×C₃₁	—	4	16	no	—
115	D₄×C₃₁	—	4	4	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	4	no	—
123	C₂×C₄×C₃₁	—	4	2	no	—
124	C₂×C₄×C₃₁	—	4	2	no	—
125	C₂×C₄×C₃₁	—	4	2	no	—
126	C₂×C₄×C₃₁	—	4	2	no	—
127	C₂×C₄×C₃₁	—	4	4	no	—
128	C₂×C₄×C₃₁	—	4	2	no	—
129	C₂×C₄×C₃₁	—	4	2	no	—
130	C₂×C₄×C₃₁	—	8	8	no	—
131	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
132	C₂×C₄×C₃₁	#124	1	2	yes	—
133	C₂×C₄×C₃₁	#124	1	2	yes	—
134	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
135	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
137	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
138	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
139	C₈×C₃₁	—	2	4	no	—
140	C₈×C₃₁	—	2	4	no	—
141	C₂×#490	C₂×C₃₁	1	1	yes	characteristic
142	C₄×#127	C₂×C₃₁	1	1	yes	characteristic
143	C₄×#125	C₂×C₃₁	1	1	yes	characteristic, complemented
144	#1992	C₂×C₃₁	1	1	yes	characteristic, complemented
145	#1961	C₂×C₃₁	1	1	yes	characteristic
146	#1934	C₂×C₃₁	1	4	yes	complemented
147	#1836	C₂×C₃₁	1	1	yes	characteristic, complemented
148	#1762	C₂×C₃₁	1	4	yes	complemented
149	C₄×#97	C₂×C₃₁	1	1	yes	characteristic, complemented
150	C₂²×C₃₁	—	4	8	no	—
151	C₂²×C₃₁	—	4	4	no	—
152	C₂²×C₃₁	—	8	2	no	—
153	C₂²×C₃₁	—	8	2	no	—
154	C₂²×C₃₁	—	8	4	no	—
155	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
156	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
157	C₂²×C₃₁	#532	1	1	yes	characteristic
158	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
159	C₂²×C₃₁	#575	1	2	yes	—
160	C₄×C₃₁	—	2	2	no	—
161	C₄×C₃₁	—	2	2	no	—
162	C₄×C₃₁	—	2	4	no	—
163	C₄×C₃₁	—	2	4	no	—
164	C₄×C₃₁	—	4	4	no	—
165	C₄×C₃₁	—	8	2	no	—
166	C₄×C₃₁	—	8	2	no	—
167	C₄×C₃₁	C₄×#41	1	2	yes	—
168	C₄×C₃₁	#529	1	2	yes	—
169	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
170	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
171	C₂×C₄×D₄	—	2	2	no	—
172	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
173	#490	C₄×C₃₁	1	8	yes	complemented
174	C₂×#102	C₂²×C₃₁	1	4	yes	—
175	C₂×#102	C₂²×C₃₁	1	4	yes	—
176	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
177	C₂×#97	C₂²×C₃₁	1	2	yes	—
178	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
179	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
180	#390	—	2	2	no	—
181	#390	C₂²×C₃₁	1	2	yes	—
182	#389	C₂²×C₃₁	1	2	yes	—
183	#387	—	2	2	no	—
184	#384	C₂²×C₃₁	1	1	yes	characteristic
185	#382	C₂²×C₃₁	1	1	yes	characteristic
186	C₄×C₄⋊C₄	—	2	2	no	—
187	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
188	C₄³	C₂²×C₃₁	1	1	yes	characteristic
189	#340	C₂²×C₃₁	1	4	yes	—
190	#340	C₂²×C₃₁	1	4	yes	—
191	C₂×C₃₁	—	8	4	no	—
192	C₂×C₃₁	C₄×#124	1	1	yes	characteristic
193	C₂×C₃₁	C₄×AGL(1,ℤ/8)	1	2	yes	—
194	C₂×C₃₁	#3138	1	1	yes	characteristic
195	C₂×C₃₁	#3142	1	1	yes	characteristic
196	C₂×C₃₁	#3180	1	2	yes	—
197	C₂²×D₄	—	2	2	no	—
198	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
199	C₂³×C₄	—	2	2	no	—
200	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
201	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
202	#127	—	2	2	no	—
203	#127	C₂×C₄×C₃₁	1	4	yes	—
204	#125	—	2	2	no	—
205	#125	C₂×C₄×C₃₁	1	4	yes	complemented
206	C₄×D₄	—	2	8	no	—
207	C₄×D₄	—	4	8	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
212	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
213	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
214	C₂×C₂²⋊C₄	—	2	4	no	—
215	C₂×C₂²⋊C₄	—	2	2	no	—
216	C₂×C₄²	—	2	2	no	—
217	C₂×C₄²	D₄×C₃₁	1	2	yes	—
218	C₂×C₄²	D₄×C₃₁	1	2	yes	—
219	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
220	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
221	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
222	#102	—	2	8	no	—
223	#102	C₂×C₄×C₃₁	1	16	yes	complemented
224	#97	—	2	2	no	—
225	#97	—	2	4	no	—
226	#97	C₂×C₄×C₃₁	1	4	yes	complemented
227	C₄×C₈	—	2	4	no	—
228	#95	—	2	2	no	—
229	#95	—	2	2	no	—
230	#95	—	2	2	no	—
231	C₃₁	#21656	1	1	yes	characteristic, direct product factor, Sylow
232	C₂⁴	—	2	2	no	—
233	C₂×D₄	—	2	8	no	—
234	C₂×D₄	—	2	4	no	—
235	C₂×D₄	—	2	4	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	4	4	no	—
238	C₂×D₄	—	4	4	no	—
239	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	4	8	no	—
244	C₂²×C₄	—	4	4	no	—
245	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
246	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
247	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
249	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
255	C₂×C₈	#41×C₃₁	1	4	yes	—
256	C₄⋊C₄	—	2	8	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	4	4	no	—
261	C₄⋊C₄	—	4	4	no	—
262	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
263	C₂²⋊C₄	—	4	4	no	—
264	C₂²⋊C₄	—	4	8	no	—
265	C₄²	—	2	4	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	4	2	no	—
268	C₄²	—	4	4	no	—
269	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
270	C₄²	#41×C₃₁	1	2	yes	—
271	C₄²	#41×C₃₁	1	4	yes	—
272	C₂³	—	2	4	no	—
273	C₂³	—	4	2	no	—
274	C₂³	—	4	2	no	—
275	C₂³	—	4	4	no	—
276	C₂³	—	4	4	no	—
277	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
278	D₄	—	2	8	no	—
279	D₄	—	4	16	no	—
280	D₄	—	4	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	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	4	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	8	8	no	—
296	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
297	C₂×C₄	C₃₁×#124	1	2	yes	—
298	C₂×C₄	C₃₁×#124	1	2	yes	—
299	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
300	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
301	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
302	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
303	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
304	C₈	—	2	4	no	—
305	C₈	—	2	4	no	—
306	C₂²	—	4	8	no	—
307	C₂²	—	4	4	no	—
308	C₂²	—	8	2	no	—
309	C₂²	—	8	2	no	—
310	C₂²	—	8	4	no	—
311	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
312	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
313	C₂²	C₃₁×#532	1	1	yes	characteristic
314	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
315	C₂²	C₃₁×#575	1	2	yes	—
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	2	4	no	—
319	C₄	—	2	4	no	—
320	C₄	—	4	4	no	—
321	C₄	—	8	2	no	—
322	C₄	—	8	2	no	—
323	C₄	C₄×#41×C₃₁	1	2	yes	—
324	C₄	C₃₁×#529	1	2	yes	—
325	C₂	—	8	4	no	—
326	C₂	C₄×C₃₁×#124	1	1	yes	characteristic
327	C₂	C₄×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
328	C₂	C₃₁×#3138	1	1	yes	characteristic
329	C₂	C₃₁×#3142	1	1	yes	characteristic
330	C₂	C₃₁×#3180	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×C₃₁×#125	C₂	semidirect
4. C₃₁×#1992	C₂	semidirect
6. C₃₁×#1934	C₂	semidirect
7. C₃₁×#1836	C₂	semidirect
8. C₃₁×#1762	C₂	semidirect
9. C₄×C₃₁×#97	C₂	semidirect
14. C₃₁×#490	C₄	semidirect
40. C₃₁×#125	C₂×C₄	semidirect
58. C₃₁×#102	C₂×C₄	semidirect
61. C₃₁×#97	C₂×C₄	semidirect
106. #21656	C₃₁	direct (N × H)
143. C₄×#125	C₂×C₃₁	semidirect
144. #1992	C₂×C₃₁	semidirect
146. #1934	C₂×C₃₁	semidirect
147. #1836	C₂×C₃₁	semidirect
148. #1762	C₂×C₃₁	semidirect
149. C₄×#97	C₂×C₃₁	semidirect
173. #490	C₄×C₃₁	semidirect
205. #125	C₂×C₄×C₃₁	semidirect
223. #102	C₂×C₄×C₃₁	semidirect
226. #97	C₂×C₄×C₃₁	semidirect
231. C₃₁	#21656	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×#490	2
C₄×C₃₁×#127	2
C₄×C₃₁×#125	2
C₃₁×#1992	2
C₃₁×#1961	2
C₃₁×#1934	2
C₃₁×#1836	2
C₃₁×#1762	2
C₄×C₃₁×#97	2
#21656	31

Subgroups — C31×#21656 #21478415

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21656 #21478415