Subgroups — C₃₁×#10858 #21467617

Order 7936.

830 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×#423	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#382	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#343	C₂	1	4	yes	maximal
5	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
6	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
7	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
8	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
9	C₃₁×#423	—	2	2	no	—
10	C₃₁×#423	C₄	1	4	yes	complemented
11	C₂×C₃₁×C₄⋊C₈	—	2	2	no	—
12	C₄×M₁₆×C₃₁	—	2	2	no	—
13	C₄×M₁₆×C₃₁	C₄	1	4	yes	complemented
14	C₂×C₃₁×#97	—	2	1	no	—
15	C₂×C₄×C₈×C₃₁	—	2	1	no	—
16	C₃₁×#382	—	2	1	no	—
17	C₃₁×#382	—	2	1	no	—
18	C₃₁×#382	—	2	2	no	—
19	C₂×C₃₁×#95	C₄	1	4	yes	—
20	C₃₁×#343	—	2	8	no	—
21	C₂³×C₄×C₃₁	C₂×C₄	1	4	yes	—
22	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
24	C₂×M₁₆×C₃₁	—	2	2	no	—
25	C₂×M₁₆×C₃₁	—	2	2	no	—
26	C₂×M₁₆×C₃₁	—	2	2	no	—
27	C₂×M₁₆×C₃₁	—	2	2	no	—
28	C₂×M₁₆×C₃₁	—	2	4	no	—
29	C₂×M₁₆×C₃₁	—	2	4	no	—
30	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
31	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
32	C₂×M₁₆×C₃₁	D₄	1	4	yes	—
33	C₂×M₁₆×C₃₁	Q₈	1	4	yes	—
34	C₂²×C₈×C₃₁	—	2	2	no	—
35	C₂²×C₈×C₃₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
37	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
38	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
39	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
40	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₃₁	D₄	1	2	yes	—
43	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₃₁	D₄	1	2	yes	—
46	C₂×C₄²×C₃₁	Q₈	1	2	yes	—
47	C₃₁×C₄⋊C₈	—	2	8	no	—
48	C₃₁×#97	—	2	4	no	—
49	C₄×C₈×C₃₁	—	2	4	no	—
50	C₃₁×#95	—	2	4	no	—
51	C₃₁×#95	—	2	4	no	—
52	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₃₁	—	2	4	no	—
54	C₂²×C₄×C₃₁	—	2	8	no	—
55	C₂²×C₄×C₃₁	—	4	4	no	—
56	C₂²×C₄×C₃₁	—	4	8	no	—
57	C₂²×C₄×C₃₁	C₄²	1	4	yes	—
58	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
61	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₃₁	#41	1	2	yes	—
70	C₂²×C₄×C₃₁	#41	1	2	yes	—
71	M₁₆×C₃₁	—	2	4	no	—
72	M₁₆×C₃₁	—	2	4	no	—
73	M₁₆×C₃₁	—	2	4	no	—
74	M₁₆×C₃₁	—	2	8	no	—
75	M₁₆×C₃₁	—	4	4	no	—
76	M₁₆×C₃₁	C₄⋊C₄	1	8	yes	—
77	C₂×C₈×C₃₁	—	2	4	no	—
78	C₂×C₈×C₃₁	—	2	4	no	—
79	C₂×C₈×C₃₁	—	2	4	no	—
80	C₂×C₈×C₃₁	—	2	4	no	—
81	C₂×C₈×C₃₁	—	4	2	no	—
82	C₂×C₈×C₃₁	—	4	2	no	—
83	C₄⋊C₄×C₃₁	—	4	4	no	—
84	C₄⋊C₄×C₃₁	—	4	2	no	—
85	C₄⋊C₄×C₃₁	—	4	2	no	—
86	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
87	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
88	C₄²×C₃₁	C₂²⋊C₄	1	4	yes	—
89	C₄²×C₃₁	C₄⋊C₄	1	4	yes	—
90	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
91	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
92	#10858	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂³×C₃₁	—	2	4	no	—
94	C₂³×C₃₁	C₂×C₄²	1	1	yes	characteristic
95	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂³×C₃₁	C₄×D₄	1	2	yes	—
98	C₂³×C₃₁	C₄×Q₈	1	2	yes	—
99	C₂×C₄×C₃₁	—	2	4	no	—
100	C₂×C₄×C₃₁	—	2	4	no	—
101	C₂×C₄×C₃₁	—	4	4	no	—
102	C₂×C₄×C₃₁	—	4	4	no	—
103	C₂×C₄×C₃₁	—	4	4	no	—
104	C₂×C₄×C₃₁	—	4	8	no	—
105	C₂×C₄×C₃₁	—	8	8	no	—
106	C₂×C₄×C₃₁	#95	1	4	yes	—
107	C₂×C₄×C₃₁	#95	1	4	yes	—
108	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₃₁	#117	1	2	yes	—
114	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
116	C₂×C₄×C₃₁	#121	1	2	yes	—
117	C₂×C₄×C₃₁	#121	1	2	yes	—
118	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
120	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
121	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
122	C₂×C₄×C₃₁	#125	1	2	yes	—
123	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
124	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
125	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
126	C₈×C₃₁	—	2	8	no	—
127	C₈×C₃₁	—	4	4	no	—
128	C₂×#423	C₂×C₃₁	1	1	yes	characteristic
129	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic
130	C₂×#382	C₂×C₃₁	1	1	yes	characteristic
131	C₂×#343	C₂×C₃₁	1	4	yes	—
132	C₂²×C₃₁	—	2	4	no	—
133	C₂²×C₃₁	—	2	2	no	—
134	C₂²×C₃₁	—	2	2	no	—
135	C₂²×C₃₁	—	2	4	no	—
136	C₂²×C₃₁	C₂×#95	1	1	yes	characteristic
137	C₂²×C₃₁	C₄×C₄⋊C₄	1	2	yes	—
138	C₂²×C₃₁	#389	1	2	yes	—
139	C₂²×C₃₁	#390	1	1	yes	characteristic
140	C₂²×C₃₁	#391	1	1	yes	characteristic
141	C₂²×C₃₁	#413	1	4	yes	—
142	C₄×C₃₁	—	2	4	no	—
143	C₄×C₃₁	—	2	4	no	—
144	C₄×C₃₁	—	8	4	no	—
145	C₄×C₃₁	#380	1	2	yes	—
146	C₄×C₃₁	#382	1	1	yes	characteristic
147	C₄×C₃₁	#384	1	1	yes	characteristic
148	C₄×C₃₁	#389	1	2	yes	—
149	C₄×C₃₁	#390	1	1	yes	characteristic
150	C₄×C₃₁	#391	1	1	yes	characteristic
151	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
152	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
153	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
154	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
155	#423	—	2	2	no	—
156	#423	C₄×C₃₁	1	4	yes	complemented
157	C₂×C₄⋊C₈	—	2	2	no	—
158	C₄×M₁₆	—	2	2	no	—
159	C₄×M₁₆	C₄×C₃₁	1	4	yes	complemented
160	C₂×#97	—	2	1	no	—
161	C₂×C₄×C₈	—	2	1	no	—
162	#382	—	2	1	no	—
163	#382	—	2	1	no	—
164	#382	—	2	2	no	—
165	C₂×#95	C₄×C₃₁	1	4	yes	—
166	#343	—	2	8	no	—
167	C₂×C₃₁	—	2	2	no	—
168	C₂×C₃₁	—	2	2	no	—
169	C₂×C₃₁	C₄⋊#95	1	1	yes	characteristic
170	C₂×C₃₁	#1742	1	2	yes	—
171	C₂×C₃₁	#1967	1	2	yes	—
172	C₂×C₃₁	#1998	1	2	yes	—
173	C₂³×C₄	C₂×C₄×C₃₁	1	4	yes	—
174	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
175	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	4	no	—
181	C₂×M₁₆	—	2	4	no	—
182	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
183	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
184	C₂×M₁₆	D₄×C₃₁	1	4	yes	—
185	C₂×M₁₆	Q₈×C₃₁	1	4	yes	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂×C₄⋊C₄	—	2	4	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	4	no	—
192	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
193	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
194	C₂×C₄²	D₄×C₃₁	1	2	yes	—
195	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
196	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
197	C₂×C₄²	D₄×C₃₁	1	2	yes	—
198	C₂×C₄²	Q₈×C₃₁	1	2	yes	—
199	C₄⋊C₈	—	2	8	no	—
200	#97	—	2	4	no	—
201	C₄×C₈	—	2	4	no	—
202	#95	—	2	4	no	—
203	#95	—	2	4	no	—
204	C₃₁	#10858	1	1	yes	characteristic, direct product factor, Sylow
205	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	8	no	—
208	C₂²×C₄	—	4	4	no	—
209	C₂²×C₄	—	4	8	no	—
210	C₂²×C₄	C₄²×C₃₁	1	4	yes	—
211	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
212	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
213	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
214	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
215	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
220	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
221	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
222	C₂²×C₄	#41×C₃₁	1	2	yes	—
223	C₂²×C₄	#41×C₃₁	1	2	yes	—
224	M₁₆	—	2	4	no	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	2	4	no	—
227	M₁₆	—	2	8	no	—
228	M₁₆	—	4	4	no	—
229	M₁₆	C₄⋊C₄×C₃₁	1	8	yes	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	4	2	no	—
235	C₂×C₈	—	4	2	no	—
236	C₄⋊C₄	—	4	4	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
240	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
241	C₄²	C₂²⋊C₄×C₃₁	1	4	yes	—
242	C₄²	C₄⋊C₄×C₃₁	1	4	yes	—
243	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
244	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
245	C₂³	—	2	4	no	—
246	C₂³	C₂×C₄²×C₃₁	1	1	yes	characteristic
247	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
248	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
249	C₂³	C₄×D₄×C₃₁	1	2	yes	—
250	C₂³	C₄×Q₈×C₃₁	1	2	yes	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	8	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	8	8	no	—
258	C₂×C₄	C₃₁×#95	1	4	yes	—
259	C₂×C₄	C₃₁×#95	1	4	yes	—
260	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
261	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
262	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
263	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
264	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
265	C₂×C₄	C₃₁×#117	1	2	yes	—
266	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
267	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
268	C₂×C₄	C₃₁×#121	1	2	yes	—
269	C₂×C₄	C₃₁×#121	1	2	yes	—
270	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
271	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
272	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
273	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
274	C₂×C₄	C₃₁×#125	1	2	yes	—
275	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
276	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
277	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
278	C₈	—	2	8	no	—
279	C₈	—	4	4	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	4	no	—
284	C₂²	C₂×C₃₁×#95	1	1	yes	characteristic
285	C₂²	C₄×C₄⋊C₄×C₃₁	1	2	yes	—
286	C₂²	C₃₁×#389	1	2	yes	—
287	C₂²	C₃₁×#390	1	1	yes	characteristic
288	C₂²	C₃₁×#391	1	1	yes	characteristic
289	C₂²	C₃₁×#413	1	4	yes	—
290	C₄	—	2	4	no	—
291	C₄	—	2	4	no	—
292	C₄	—	8	4	no	—
293	C₄	C₃₁×#380	1	2	yes	—
294	C₄	C₃₁×#382	1	1	yes	characteristic
295	C₄	C₃₁×#384	1	1	yes	characteristic
296	C₄	C₃₁×#389	1	2	yes	—
297	C₄	C₃₁×#390	1	1	yes	characteristic
298	C₄	C₃₁×#391	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	2	2	no	—
301	C₂	C₃₁×C₄⋊#95	1	1	yes	characteristic
302	C₂	C₃₁×#1742	1	2	yes	—
303	C₂	C₃₁×#1967	1	2	yes	—
304	C₂	C₃₁×#1998	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂×C₂²⋊C₄
commutator	C₂³	C₂×C₄²×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
prime radical	C₂⁴×C₃₁	C₂²×C₄
LCS	G ▷ C₂³ ▷ 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
10. C₃₁×#423	C₄	semidirect
13. C₄×M₁₆×C₃₁	C₄	semidirect
92. #10858	C₃₁	direct (N × H)
156. #423	C₄×C₃₁	semidirect
159. C₄×M₁₆	C₄×C₃₁	semidirect
204. C₃₁	#10858	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×#423	2
C₂×C₄×M₁₆×C₃₁	2
C₂×C₃₁×#382	2
C₂×C₃₁×#343	2
#10858	31

Subgroups — C31×#10858 #21467617

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#10858 #21467617