Subgroups — C₃₁×#10711 #21467470

Order 7936.

842 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#405	C₂	1	2	yes	maximal
2	C₂×C₃₁×#405	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#344	C₂	1	2	yes	maximal
4	C₃₁×#1742	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1740	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
7	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#117	C₂²	1	2	yes	—
10	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
11	C₃₁×#405	—	2	4	no	—
12	C₃₁×#405	—	2	2	no	—
13	C₃₁×#405	—	2	2	no	—
14	C₃₁×#405	C₄	1	8	yes	complemented
15	C₂×C₃₁×#97	—	2	2	no	—
16	C₂×C₃₁×#97	C₄	1	2	yes	complemented
17	C₂×C₄×C₈×C₃₁	—	2	2	no	—
18	C₂×C₄×C₈×C₃₁	C₄	1	2	yes	complemented
19	C₃₁×#344	C₄	1	8	yes	complemented
20	C₃₁×#343	C₄	1	4	yes	complemented
21	C₃₁×#342	C₄	1	4	yes	—
22	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
23	C₂×M₁₆×C₃₁	—	2	2	no	—
24	C₂×M₁₆×C₃₁	—	2	2	no	—
25	C₂×M₁₆×C₃₁	—	2	1	no	—
26	C₂×M₁₆×C₃₁	—	2	1	no	—
27	C₂×M₁₆×C₃₁	—	2	2	no	—
28	C₂×M₁₆×C₃₁	—	2	2	no	—
29	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
30	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
31	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
32	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₃₁	—	2	2	no	—
34	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
37	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₃₁	D₄	1	2	yes	—
39	C₂²×C₈×C₃₁	D₄	1	2	yes	—
40	C₂²×C₈×C₃₁	D₄	1	2	yes	—
41	C₂²×C₈×C₃₁	Q₈	1	2	yes	—
42	C₃₁×#117	—	2	2	no	—
43	C₃₁×#117	—	2	2	no	—
44	C₃₁×#117	—	2	2	no	—
45	C₃₁×#117	—	2	2	no	—
46	C₃₁×#117	C₂×C₄	1	4	yes	—
47	C₃₁×#117	C₂×C₄	1	4	yes	—
48	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
49	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
50	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
51	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
52	C₂×C₄²×C₃₁	—	2	2	no	—
53	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
54	C₃₁×#97	—	2	8	no	—
55	C₃₁×#97	—	2	4	no	—
56	C₄×C₈×C₃₁	—	2	8	no	—
57	C₄×C₈×C₃₁	—	2	2	no	—
58	C₄×C₈×C₃₁	—	2	2	no	—
59	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₃₁	—	2	4	no	—
61	C₂²×C₄×C₃₁	C₄²	1	4	yes	—
62	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
70	C₂²×C₄×C₃₁	#41	1	2	yes	—
71	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
72	M₁₆×C₃₁	—	2	4	no	—
73	M₁₆×C₃₁	—	2	4	no	—
74	M₁₆×C₃₁	—	2	4	no	—
75	M₁₆×C₃₁	—	2	4	no	—
76	M₁₆×C₃₁	—	2	4	no	—
77	M₁₆×C₃₁	C₄²	1	8	yes	—
78	C₂×C₈×C₃₁	—	2	4	no	—
79	C₂×C₈×C₃₁	—	2	2	no	—
80	C₂×C₈×C₃₁	—	2	2	no	—
81	C₂×C₈×C₃₁	—	2	4	no	—
82	C₂×C₈×C₃₁	—	2	4	no	—
83	C₂×C₈×C₃₁	C₄²	1	4	yes	—
84	C₂×C₈×C₃₁	C₄²	1	2	yes	—
85	C₂×C₈×C₃₁	C₄²	1	2	yes	—
86	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₈×C₃₁	C₂²⋊C₄	1	4	yes	—
88	C₂×C₈×C₃₁	C₂²⋊C₄	1	4	yes	—
89	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
90	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
92	C₂×C₈×C₃₁	C₄⋊C₄	1	4	yes	—
93	C₂×C₈×C₃₁	C₄⋊C₄	1	4	yes	—
94	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
95	C₄⋊C₄×C₃₁	—	2	4	no	—
96	C₄⋊C₄×C₃₁	—	2	4	no	—
97	C₄⋊C₄×C₃₁	—	2	4	no	—
98	C₂²⋊C₄×C₃₁	—	2	4	no	—
99	C₂²⋊C₄×C₃₁	—	2	4	no	—
100	C₂²⋊C₄×C₃₁	C₄²	1	8	yes	—
101	C₄²×C₃₁	—	2	4	no	—
102	C₄²×C₃₁	—	2	2	no	—
103	C₄²×C₃₁	—	2	2	no	—
104	C₄²×C₃₁	—	2	4	no	—
105	#10711	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂³×C₃₁	—	4	2	no	—
107	C₂³×C₃₁	C₂×C₄²	1	1	yes	characteristic
108	C₂³×C₃₁	C₂×C₄²	1	2	yes	—
109	C₂³×C₃₁	C₄×D₄	1	2	yes	—
110	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
111	C₂³×C₃₁	C₄×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₃₁	—	2	4	no	—
113	C₂×C₄×C₃₁	—	2	8	no	—
114	C₂×C₄×C₃₁	—	4	8	no	—
115	C₂×C₄×C₃₁	—	4	4	no	—
116	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
118	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
119	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
120	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
124	C₂×C₄×C₃₁	#117	1	2	yes	—
125	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
126	C₂×C₄×C₃₁	#117	1	2	yes	—
127	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
130	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
131	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
132	C₂×C₄×C₃₁	C₄×Q₈	1	1	yes	characteristic
133	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
134	C₈×C₃₁	—	2	8	no	—
135	C₈×C₃₁	—	2	4	no	—
136	C₈×C₃₁	#95	1	4	yes	—
137	C₈×C₃₁	#95	1	2	yes	—
138	C₈×C₃₁	#95	1	2	yes	—
139	C₂×#405	C₂×C₃₁	1	2	yes	—
140	C₂×#405	C₂×C₃₁	1	1	yes	characteristic
141	C₂×#344	C₂×C₃₁	1	2	yes	—
142	#1742	C₂×C₃₁	1	1	yes	characteristic
143	#1740	C₂×C₃₁	1	1	yes	characteristic
144	C₂²×C₃₁	—	2	2	no	—
145	C₂²×C₃₁	—	2	4	no	—
146	C₂²×C₃₁	—	4	2	no	—
147	C₂²×C₃₁	—	4	1	no	—
148	C₂²×C₃₁	—	4	1	no	—
149	C₂²×C₃₁	C₄³	1	1	yes	characteristic
150	C₂²×C₃₁	C₄×C₂²⋊C₄	1	2	yes	—
151	C₂²×C₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
152	C₂²×C₃₁	C₄×C₄⋊C₄	1	1	yes	characteristic
153	C₂²×C₃₁	C₄×C₄⋊C₄	1	2	yes	—
154	C₄×C₃₁	—	4	8	no	—
155	C₄×C₃₁	—	4	4	no	—
156	C₄×C₃₁	C₂×#95	1	1	yes	characteristic
157	C₄×C₃₁	#376	1	1	yes	characteristic
158	C₄×C₃₁	C₄×C₂²⋊C₄	1	2	yes	—
159	C₄×C₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
160	C₄×C₃₁	C₄×C₄⋊C₄	1	1	yes	characteristic
161	C₄×C₃₁	C₄×C₄⋊C₄	1	2	yes	—
162	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
163	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
164	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
165	C₂×#117	C₂²×C₃₁	1	2	yes	—
166	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
167	#405	—	2	4	no	—
168	#405	—	2	2	no	—
169	#405	—	2	2	no	—
170	#405	C₄×C₃₁	1	8	yes	complemented
171	C₂×#97	—	2	2	no	—
172	C₂×#97	C₄×C₃₁	1	2	yes	complemented
173	C₂×C₄×C₈	—	2	2	no	—
174	C₂×C₄×C₈	C₄×C₃₁	1	2	yes	complemented
175	#344	C₄×C₃₁	1	8	yes	complemented
176	#343	C₄×C₃₁	1	4	yes	complemented
177	#342	C₄×C₃₁	1	4	yes	—
178	C₂×C₃₁	—	2	2	no	—
179	C₂×C₃₁	—	2	1	no	—
180	C₂×C₃₁	—	2	1	no	—
181	C₂×C₃₁	—	2	2	no	—
182	C₂×C₃₁	C₄×#95	1	1	yes	characteristic
183	C₂×C₃₁	#1757	1	2	yes	—
184	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	1	no	—
188	C₂×M₁₆	—	2	1	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
192	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
193	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
194	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
195	C₂²×C₈	—	2	2	no	—
196	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
197	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
198	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
199	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
200	C₂²×C₈	D₄×C₃₁	1	2	yes	—
201	C₂²×C₈	D₄×C₃₁	1	2	yes	—
202	C₂²×C₈	D₄×C₃₁	1	2	yes	—
203	C₂²×C₈	Q₈×C₃₁	1	2	yes	—
204	#117	—	2	2	no	—
205	#117	—	2	2	no	—
206	#117	—	2	2	no	—
207	#117	—	2	2	no	—
208	#117	C₂×C₄×C₃₁	1	4	yes	—
209	#117	C₂×C₄×C₃₁	1	4	yes	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
212	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
213	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
214	C₂×C₄²	—	2	2	no	—
215	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
216	#97	—	2	8	no	—
217	#97	—	2	4	no	—
218	C₄×C₈	—	2	8	no	—
219	C₄×C₈	—	2	2	no	—
220	C₄×C₈	—	2	2	no	—
221	C₃₁	#10711	1	1	yes	characteristic, direct product factor, Sylow
222	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	C₄²×C₃₁	1	4	yes	—
225	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
226	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
227	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
228	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
229	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
230	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
231	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
232	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
233	C₂²×C₄	#41×C₃₁	1	2	yes	—
234	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
235	M₁₆	—	2	4	no	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	2	4	no	—
240	M₁₆	C₄²×C₃₁	1	8	yes	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	C₄²×C₃₁	1	4	yes	—
247	C₂×C₈	C₄²×C₃₁	1	2	yes	—
248	C₂×C₈	C₄²×C₃₁	1	2	yes	—
249	C₂×C₈	C₂²⋊C₄×C₃₁	1	4	yes	—
250	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
251	C₂×C₈	C₂²⋊C₄×C₃₁	1	4	yes	—
252	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
253	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
254	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
255	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
256	C₂×C₈	C₄⋊C₄×C₃₁	1	4	yes	—
257	C₂×C₈	C₄⋊C₄×C₃₁	1	4	yes	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₂²⋊C₄	—	2	4	no	—
262	C₂²⋊C₄	—	2	4	no	—
263	C₂²⋊C₄	C₄²×C₃₁	1	8	yes	—
264	C₄²	—	2	4	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	2	4	no	—
268	C₂³	—	4	2	no	—
269	C₂³	C₂×C₄²×C₃₁	1	1	yes	characteristic
270	C₂³	C₂×C₄²×C₃₁	1	2	yes	—
271	C₂³	C₄×D₄×C₃₁	1	2	yes	—
272	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
273	C₂³	C₄×Q₈×C₃₁	1	1	yes	characteristic
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	4	8	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
280	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
281	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
282	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
283	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
284	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
285	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
286	C₂×C₄	C₃₁×#117	1	2	yes	—
287	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
288	C₂×C₄	C₃₁×#117	1	2	yes	—
289	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
290	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
291	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
292	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
293	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
294	C₂×C₄	C₄×Q₈×C₃₁	1	1	yes	characteristic
295	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
296	C₈	—	2	8	no	—
297	C₈	—	2	4	no	—
298	C₈	C₃₁×#95	1	2	yes	—
299	C₈	C₃₁×#95	1	4	yes	—
300	C₈	C₃₁×#95	1	2	yes	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	4	no	—
303	C₂²	—	4	2	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	C₄³×C₃₁	1	1	yes	characteristic
307	C₂²	C₄×C₂²⋊C₄×C₃₁	1	2	yes	—
308	C₂²	C₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
309	C₂²	C₄×C₄⋊C₄×C₃₁	1	1	yes	characteristic
310	C₂²	C₄×C₄⋊C₄×C₃₁	1	2	yes	—
311	C₄	—	4	8	no	—
312	C₄	—	4	4	no	—
313	C₄	C₂×C₃₁×#95	1	1	yes	characteristic
314	C₄	C₃₁×#376	1	1	yes	characteristic
315	C₄	C₄×C₂²⋊C₄×C₃₁	1	2	yes	—
316	C₄	C₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
317	C₄	C₄×C₄⋊C₄×C₃₁	1	1	yes	characteristic
318	C₄	C₄×C₄⋊C₄×C₃₁	1	2	yes	—
319	C₂	—	2	2	no	—
320	C₂	—	2	1	no	—
321	C₂	—	2	1	no	—
322	C₂	—	2	2	no	—
323	C₂	C₄×C₃₁×#95	1	1	yes	characteristic
324	C₂	C₃₁×#1757	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₃₁	C₂²⋊C₄
commutator	C₂²	C₄³×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
socle	C₂²×C₃₁	C₄×C₂²⋊C₄
prime radical	C₂⁴×C₃₁	C₂²×C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
14. C₃₁×#405	C₄	semidirect
16. C₂×C₃₁×#97	C₄	semidirect
18. C₂×C₄×C₈×C₃₁	C₄	semidirect
19. C₃₁×#344	C₄	semidirect
20. C₃₁×#343	C₄	semidirect
105. #10711	C₃₁	direct (N × H)
170. #405	C₄×C₃₁	semidirect
172. C₂×#97	C₄×C₃₁	semidirect
174. C₂×C₄×C₈	C₄×C₃₁	semidirect
175. #344	C₄×C₃₁	semidirect
176. #343	C₄×C₃₁	semidirect
221. C₃₁	#10711	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×#405	2
C₂×C₃₁×#344	2
C₃₁×#1742	2
C₃₁×#1740	2
#10711	31

Subgroups — C31×#10711 #21467470

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#10711 #21467470