Subgroups — C₃₁×#20800 #21477559

Order 7936.

1190 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#428	C₂	1	4	yes	maximal
2	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2285	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#384	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1847	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1845	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1844	C₂	1	2	yes	maximal, complemented
8	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
9	C₂×C₃₁×#117	C₂²	1	2	yes	—
10	C₂²×C₄⋊C₄×C₃₁	—	2	2	no	—
11	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
12	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#428	—	2	16	no	—
14	C₂×C₃₁×#107	C₂²	1	8	yes	—
15	C₂×C₃₁×#106	C₂²	1	8	yes	—
16	C₄×M₁₆×C₃₁	C₄	1	8	yes	complemented
17	C₂×C₃₁×#97	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
19	C₃₁×#384	—	2	2	no	—
20	C₃₁×#384	C₂²	1	1	yes	characteristic
21	C₃₁×#384	C₂²	1	2	yes	—
22	C₃₁×#384	C₂²	1	1	yes	characteristic
23	C₃₁×#384	C₂²	1	2	yes	—
24	C₃₁×#382	C₂²	1	2	yes	—
25	C₃₁×#380	—	2	2	no	—
26	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
27	C₂³×C₄×C₃₁	—	2	2	no	—
28	C₂³×C₄×C₃₁	D₄	1	2	yes	—
29	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
30	C₂×M₁₆×C₃₁	C₂×C₄	1	8	yes	—
31	C₂×M₁₆×C₃₁	D₄	1	8	yes	—
32	C₂×M₁₆×C₃₁	Q₈	1	8	yes	—
33	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
34	C₃₁×#117	—	2	8	no	—
35	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
36	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
37	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
38	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
39	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
40	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
41	C₂×C₄⋊C₄×C₃₁	—	4	8	no	—
42	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
44	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
45	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
46	C₂×C₄²×C₃₁	—	2	2	no	—
47	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
48	C₂×C₄²×C₃₁	D₄	1	2	yes	—
49	C₂×C₄²×C₃₁	Q₈	1	4	yes	—
50	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
52	C₃₁×#107	—	2	16	no	—
53	C₃₁×#106	—	2	16	no	—
54	C₃₁×#97	C₂×C₄	1	8	yes	complemented
55	C₄×C₈×C₃₁	C₂×C₄	1	8	yes	complemented
56	C₃₁×#95	—	2	4	no	—
57	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	—	2	4	no	—
59	C₂²×C₄×C₃₁	—	2	4	no	—
60	C₂²×C₄×C₃₁	—	2	2	no	—
61	C₂²×C₄×C₃₁	—	2	2	no	—
62	C₂²×C₄×C₃₁	—	2	4	no	—
63	C₂²×C₄×C₃₁	—	4	8	no	—
64	C₂²×C₄×C₃₁	—	4	4	no	—
65	C₂²×C₄×C₃₁	C₄⋊C₄	1	8	yes	—
66	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
70	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
72	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
73	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
74	M₁₆×C₃₁	C₄⋊C₄	1	32	yes	complemented
75	C₂×C₈×C₃₁	C₂²×C₄	1	8	yes	—
76	C₂×C₈×C₃₁	C₂×D₄	1	8	yes	—
77	C₂×C₈×C₃₁	C₂×Q₈	1	8	yes	—
78	C₄⋊C₄×C₃₁	—	2	4	no	—
79	C₄⋊C₄×C₃₁	—	2	8	no	—
80	C₄⋊C₄×C₃₁	—	2	4	no	—
81	C₄⋊C₄×C₃₁	—	4	8	no	—
82	C₄⋊C₄×C₃₁	—	4	4	no	—
83	C₄⋊C₄×C₃₁	—	4	4	no	—
84	C₄⋊C₄×C₃₁	—	4	8	no	—
85	C₄⋊C₄×C₃₁	—	4	4	no	—
86	C₂²⋊C₄×C₃₁	—	4	4	no	—
87	C₄²×C₃₁	—	4	4	no	—
88	C₄²×C₃₁	C₄⋊C₄	1	8	yes	—
89	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
90	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
91	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
92	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
93	#20800	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₃₁	—	2	2	no	—
95	C₂³×C₃₁	—	2	2	no	—
96	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
97	C₂³×C₃₁	#127	1	2	yes	—
98	C₂³×C₃₁	C₄⋊Q₈	1	2	yes	—
99	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄×C₃₁	—	2	2	no	—
101	C₂×C₄×C₃₁	—	2	2	no	—
102	C₂×C₄×C₃₁	—	2	16	no	—
103	C₂×C₄×C₃₁	—	4	2	no	—
104	C₂×C₄×C₃₁	—	4	2	no	—
105	C₂×C₄×C₃₁	—	4	4	no	—
106	C₂×C₄×C₃₁	—	4	8	no	—
107	C₂×C₄×C₃₁	—	4	4	no	—
108	C₂×C₄×C₃₁	—	4	4	no	—
109	C₂×C₄×C₃₁	—	8	8	no	—
110	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
111	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	8	yes	—
112	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
114	C₂×C₄×C₃₁	#127	1	2	yes	—
115	C₂×C₄×C₃₁	C₄⋊Q₈	1	8	yes	—
116	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
117	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
118	C₂×C₄×C₃₁	#137	1	4	yes	—
119	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
121	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
122	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
123	C₈×C₃₁	C₂×C₄⋊C₄	1	16	yes	complemented
124	C₂×#428	C₂×C₃₁	1	4	yes	—
125	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic
126	#2285	C₂×C₃₁	1	1	yes	characteristic
127	C₂×#384	C₂×C₃₁	1	1	yes	characteristic
128	#1847	C₂×C₃₁	1	4	yes	complemented
129	#1845	C₂×C₃₁	1	2	yes	complemented
130	#1844	C₂×C₃₁	1	2	yes	complemented
131	C₂²×C₃₁	—	2	2	no	—
132	C₂²×C₃₁	—	2	2	no	—
133	C₂²×C₃₁	—	2	4	no	—
134	C₂²×C₃₁	—	2	4	no	—
135	C₂²×C₃₁	#384	1	4	yes	—
136	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
137	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
138	C₂²×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
139	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
140	C₂²×C₃₁	C₂×#137	1	2	yes	—
141	C₄×C₃₁	—	2	4	no	—
142	C₄×C₃₁	—	8	2	no	—
143	C₄×C₃₁	—	8	4	no	—
144	C₄×C₃₁	—	8	2	no	—
145	C₄×C₃₁	#384	1	4	yes	—
146	C₄×C₃₁	#428	1	8	yes	complemented
147	C₄×C₃₁	C₂²×C₄⋊C₄	1	2	yes	—
148	C₄×C₃₁	C₂×C₄⋊Q₈	1	2	yes	—
149	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
150	C₂×#117	C₂²×C₃₁	1	2	yes	—
151	C₂²×C₄⋊C₄	—	2	2	no	—
152	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
153	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
154	#428	—	2	16	no	—
155	C₂×#107	C₂²×C₃₁	1	8	yes	—
156	C₂×#106	C₂²×C₃₁	1	8	yes	—
157	C₄×M₁₆	C₄×C₃₁	1	8	yes	complemented
158	C₂×#97	C₂²×C₃₁	1	2	yes	—
159	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
160	#384	—	2	2	no	—
161	#384	C₂²×C₃₁	1	1	yes	characteristic
162	#384	C₂²×C₃₁	1	2	yes	—
163	#384	C₂²×C₃₁	1	1	yes	characteristic
164	#384	C₂²×C₃₁	1	2	yes	—
165	#382	C₂²×C₃₁	1	2	yes	—
166	#380	—	2	2	no	—
167	C₄×C₄⋊C₄	—	2	2	no	—
168	C₂×C₃₁	—	2	4	no	—
169	C₂×C₃₁	C₂×#384	1	1	yes	characteristic
170	C₂×C₃₁	C₂×#428	1	2	yes	—
171	C₂×C₃₁	#3151	1	1	yes	characteristic
172	C₂×C₃₁	#3152	1	1	yes	characteristic
173	C₂×C₃₁	#3165	1	2	yes	—
174	C₂³×C₄	—	2	2	no	—
175	C₂³×C₄	D₄×C₃₁	1	2	yes	—
176	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
177	C₂×M₁₆	C₂×C₄×C₃₁	1	8	yes	—
178	C₂×M₁₆	D₄×C₃₁	1	8	yes	—
179	C₂×M₁₆	Q₈×C₃₁	1	8	yes	—
180	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
181	#117	—	2	8	no	—
182	C₂×C₄⋊C₄	—	2	8	no	—
183	C₂×C₄⋊C₄	—	2	4	no	—
184	C₂×C₄⋊C₄	—	2	4	no	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	—	4	8	no	—
189	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
190	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
191	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
192	C₂×C₂²⋊C₄	—	2	2	no	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
195	C₂×C₄²	D₄×C₃₁	1	2	yes	—
196	C₂×C₄²	Q₈×C₃₁	1	4	yes	—
197	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
198	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
199	#107	—	2	16	no	—
200	#106	—	2	16	no	—
201	#97	C₂×C₄×C₃₁	1	8	yes	complemented
202	C₄×C₈	C₂×C₄×C₃₁	1	8	yes	complemented
203	#95	—	2	4	no	—
204	C₃₁	#20800	1	1	yes	characteristic, direct product factor, Sylow
205	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	4	8	no	—
212	C₂²×C₄	—	4	4	no	—
213	C₂²×C₄	C₄⋊C₄×C₃₁	1	8	yes	—
214	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
215	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
216	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
218	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
219	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
220	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
221	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
222	M₁₆	C₄⋊C₄×C₃₁	1	32	yes	complemented
223	C₂×C₈	C₂²×C₄×C₃₁	1	8	yes	—
224	C₂×C₈	C₂×D₄×C₃₁	1	8	yes	—
225	C₂×C₈	C₂×Q₈×C₃₁	1	8	yes	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	2	8	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	4	8	no	—
230	C₄⋊C₄	—	4	4	no	—
231	C₄⋊C₄	—	4	4	no	—
232	C₄⋊C₄	—	4	8	no	—
233	C₄⋊C₄	—	4	4	no	—
234	C₂²⋊C₄	—	4	4	no	—
235	C₄²	—	4	4	no	—
236	C₄²	C₄⋊C₄×C₃₁	1	8	yes	—
237	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
238	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
239	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
240	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
244	C₂³	C₃₁×#127	1	2	yes	—
245	C₂³	C₃₁×C₄⋊Q₈	1	2	yes	—
246	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	16	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	8	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	8	8	no	—
257	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
258	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
259	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	8	yes	—
260	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
261	C₂×C₄	C₃₁×#127	1	2	yes	—
262	C₂×C₄	C₃₁×C₄⋊Q₈	1	8	yes	—
263	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
264	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
265	C₂×C₄	C₃₁×#137	1	4	yes	—
266	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
267	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
268	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
269	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
270	C₈	C₂×C₄⋊C₄×C₃₁	1	16	yes	complemented
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	2	4	no	—
275	C₂²	C₃₁×#384	1	4	yes	—
276	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
277	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
278	C₂²	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
279	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
280	C₂²	C₂×C₃₁×#137	1	2	yes	—
281	C₄	—	2	4	no	—
282	C₄	—	8	4	no	—
283	C₄	—	8	2	no	—
284	C₄	—	8	2	no	—
285	C₄	C₃₁×#384	1	4	yes	—
286	C₄	C₃₁×#428	1	8	yes	complemented
287	C₄	C₂²×C₄⋊C₄×C₃₁	1	2	yes	—
288	C₄	C₂×C₃₁×C₄⋊Q₈	1	2	yes	—
289	C₂	—	2	4	no	—
290	C₂	C₂×C₃₁×#384	1	1	yes	characteristic
291	C₂	C₂×C₃₁×#428	1	2	yes	—
292	C₂	C₃₁×#3151	1	1	yes	characteristic
293	C₂	C₃₁×#3152	1	1	yes	characteristic
294	C₂	C₃₁×#3165	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	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₃₁×#1847	C₂	semidirect
6. C₃₁×#1845	C₂	semidirect
7. C₃₁×#1844	C₂	semidirect
16. C₄×M₁₆×C₃₁	C₄	semidirect
54. C₃₁×#97	C₂×C₄	semidirect
55. C₄×C₈×C₃₁	C₂×C₄	semidirect
74. M₁₆×C₃₁	C₄⋊C₄	semidirect
93. #20800	C₃₁	direct (N × H)
123. C₈×C₃₁	C₂×C₄⋊C₄	semidirect
128. #1847	C₂×C₃₁	semidirect
129. #1845	C₂×C₃₁	semidirect
130. #1844	C₂×C₃₁	semidirect
146. C₄×C₃₁	#428	semidirect
157. C₄×M₁₆	C₄×C₃₁	semidirect
201. #97	C₂×C₄×C₃₁	semidirect
202. C₄×C₈	C₂×C₄×C₃₁	semidirect
204. C₃₁	#20800	direct (N × H)
222. M₁₆	C₄⋊C₄×C₃₁	semidirect
270. C₈	C₂×C₄⋊C₄×C₃₁	semidirect
286. C₄	C₃₁×#428	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#428	2
C₂×C₄×M₁₆×C₃₁	2
C₃₁×#2285	2
C₂×C₃₁×#384	2
C₃₁×#1847	2
C₃₁×#1845	2
C₃₁×#1844	2
#20800	31

Subgroups — C31×#20800 #21477559

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20800 #21477559