Subgroups — C₃₁×#8595 #21465354

Order 7936.

598 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1797	C₂	1	2	yes	maximal
3	C₂×C₃₁×#334	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1277	C₂	1	2	yes	maximal, complemented
5	C₂³×C₈×C₃₁	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₃₁	C₄	1	1	yes	characteristic
7	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
9	C₂×C₃₁×C₂²⋊C₈	—	2	2	no	—
10	C₂×C₄×C₈×C₃₁	—	2	4	no	—
11	C₂×C₄×C₈×C₃₁	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₃₁	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#336	—	2	2	no	—
16	C₃₁×#334	—	2	4	no	—
17	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
19	C₂³×C₄×C₃₁	Q₈	1	1	yes	characteristic
20	C₂²×C₈×C₃₁	—	2	2	no	—
21	C₂²×C₈×C₃₁	—	2	2	no	—
22	C₂²×C₈×C₃₁	—	2	2	no	—
23	C₂²×C₈×C₃₁	—	2	2	no	—
24	C₂²×C₈×C₃₁	—	2	2	no	—
25	C₂²×C₈×C₃₁	—	2	4	no	—
26	C₂²×C₈×C₃₁	C₈	1	4	yes	complemented
27	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₃₁	—	2	2	no	—
32	C₂×C₄²×C₃₁	—	2	2	no	—
33	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₃₁	Q₈	1	1	yes	characteristic
36	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
37	C₃₁×C₄⋊C₈	—	2	4	no	—
38	C₃₁×C₄⋊C₈	—	2	4	no	—
39	C₃₁×C₂²⋊C₈	—	4	4	no	—
40	C₄×C₈×C₃₁	—	2	2	no	—
41	C₄×C₈×C₃₁	—	2	2	no	—
42	C₄×C₈×C₃₁	—	2	2	no	—
43	C₄×C₈×C₃₁	D₄	1	4	yes	—
44	C₂⁴×C₃₁	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₃₁	—	2	2	no	—
46	C₂²×C₄×C₃₁	—	2	2	no	—
47	C₂²×C₄×C₃₁	—	2	1	no	—
48	C₂²×C₄×C₃₁	—	2	1	no	—
49	C₂²×C₄×C₃₁	—	2	2	no	—
50	C₂²×C₄×C₃₁	—	2	2	no	—
51	C₂²×C₄×C₃₁	—	2	1	no	—
52	C₂²×C₄×C₃₁	—	2	1	no	—
53	C₂²×C₄×C₃₁	—	2	2	no	—
54	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	C₂×C₈	1	2	yes	—
58	C₂²×C₄×C₃₁	M₁₆	1	2	yes	—
59	C₂²×C₄×C₃₁	SD₁₆	1	4	yes	—
60	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₈×C₃₁	—	2	8	no	—
64	C₂×C₈×C₃₁	—	2	1	no	—
65	C₂×C₈×C₃₁	—	2	1	no	—
66	C₂×C₈×C₃₁	—	2	1	no	—
67	C₂×C₈×C₃₁	—	2	1	no	—
68	C₂×C₈×C₃₁	—	2	1	no	—
69	C₂×C₈×C₃₁	—	2	1	no	—
70	C₂×C₈×C₃₁	—	2	8	no	—
71	C₂×C₈×C₃₁	—	4	4	no	—
72	C₂×C₈×C₃₁	—	4	4	no	—
73	C₂×C₈×C₃₁	—	4	4	no	—
74	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
75	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
76	C₂×C₈×C₃₁	C₂×C₈	1	2	yes	—
77	C₂×C₈×C₃₁	C₂×C₈	1	2	yes	—
78	C₂×C₈×C₃₁	M₁₆	1	2	yes	—
79	C₂×C₈×C₃₁	M₁₆	1	2	yes	—
80	C₄²×C₃₁	—	2	1	no	—
81	C₄²×C₃₁	—	2	1	no	—
82	C₄²×C₃₁	—	2	2	no	—
83	C₄²×C₃₁	—	2	2	no	—
84	C₄²×C₃₁	C₂×D₄	1	2	yes	—
85	C₄²×C₃₁	#41	1	2	yes	—
86	#8595	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂³×C₃₁	—	2	1	no	—
88	C₂³×C₃₁	—	2	1	no	—
89	C₂³×C₃₁	—	2	1	no	—
90	C₂³×C₃₁	—	2	1	no	—
91	C₂³×C₃₁	C₄⋊C₈	1	2	yes	—
92	C₂³×C₃₁	#106	1	2	yes	—
93	C₂³×C₃₁	#108	1	2	yes	—
94	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₃₁	—	2	8	no	—
96	C₂×C₄×C₃₁	—	2	1	no	—
97	C₂×C₄×C₃₁	—	2	1	no	—
98	C₂×C₄×C₃₁	—	2	1	no	—
99	C₂×C₄×C₃₁	—	2	1	no	—
100	C₂×C₄×C₃₁	—	2	1	no	—
101	C₂×C₄×C₃₁	—	2	1	no	—
102	C₂×C₄×C₃₁	—	2	1	no	—
103	C₂×C₄×C₃₁	—	2	1	no	—
104	C₂×C₄×C₃₁	—	2	2	no	—
105	C₂×C₄×C₃₁	—	2	2	no	—
106	C₂×C₄×C₃₁	—	2	2	no	—
107	C₂×C₄×C₃₁	—	2	2	no	—
108	C₂×C₄×C₃₁	—	2	8	no	—
109	C₂×C₄×C₃₁	C₄⋊C₈	1	2	yes	—
110	C₂×C₄×C₃₁	#106	1	2	yes	—
111	C₂×C₄×C₃₁	#108	1	2	yes	—
112	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
114	C₂×C₄×C₃₁	#121	1	2	yes	—
115	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
118	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
119	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
120	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
121	C₂×C₄×C₃₁	#135	1	2	yes	—
122	C₈×C₃₁	—	2	2	no	—
123	C₈×C₃₁	—	2	2	no	—
124	C₈×C₃₁	—	2	2	no	—
125	C₈×C₃₁	—	8	4	no	—
126	C₈×C₃₁	C₂²⋊C₈	1	4	yes	complemented
127	C₂²×C₄×C₈	C₂×C₃₁	1	1	yes	characteristic
128	#1797	C₂×C₃₁	1	2	yes	—
129	C₂×#334	C₂×C₃₁	1	2	yes	complemented
130	#1277	C₂×C₃₁	1	2	yes	complemented
131	C₂²×C₃₁	—	2	2	no	—
132	C₂²×C₃₁	—	2	2	no	—
133	C₂²×C₃₁	—	2	2	no	—
134	C₂²×C₃₁	—	2	2	no	—
135	C₂²×C₃₁	—	2	2	no	—
136	C₂²×C₃₁	—	2	2	no	—
137	C₂²×C₃₁	#334	1	4	yes	complemented
138	C₂²×C₃₁	#380	1	1	yes	characteristic
139	C₂²×C₃₁	C₂×C₄⋊C₈	1	1	yes	characteristic
140	C₂²×C₃₁	#423	1	1	yes	characteristic
141	C₂²×C₃₁	C₂×#106	1	1	yes	characteristic
142	C₂²×C₃₁	#427	1	1	yes	characteristic
143	C₂²×C₃₁	C₂×#108	1	1	yes	characteristic
144	C₂²×C₃₁	C₂×#108	1	1	yes	characteristic
145	C₄×C₃₁	—	2	2	no	—
146	C₄×C₃₁	—	2	2	no	—
147	C₄×C₃₁	—	2	2	no	—
148	C₄×C₃₁	—	2	2	no	—
149	C₄×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
150	C₄×C₃₁	#407	1	1	yes	characteristic
151	C₄×C₃₁	#431	1	1	yes	characteristic
152	C₄×C₃₁	#432	1	1	yes	characteristic
153	C₄×C₃₁	#465	1	4	yes	—
154	C₂³×C₈	C₄×C₃₁	1	1	yes	characteristic
155	C₂³×C₈	C₄×C₃₁	1	1	yes	characteristic
156	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
157	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
158	C₂×C₂²⋊C₈	—	2	2	no	—
159	C₂×C₄×C₈	—	2	4	no	—
160	C₂×C₄×C₈	C₄×C₃₁	1	1	yes	characteristic
161	C₂×C₄×C₈	C₄×C₃₁	1	1	yes	characteristic
162	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
163	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
164	#336	—	2	2	no	—
165	#334	—	2	4	no	—
166	C₂×C₃₁	—	2	4	no	—
167	C₂×C₃₁	C₂×#334	1	1	yes	characteristic
168	C₂×C₃₁	#1564	1	1	yes	characteristic
169	C₂×C₃₁	#1566	1	1	yes	characteristic
170	C₂×C₃₁	#1568	1	1	yes	characteristic
171	C₂×C₃₁	#1797	1	1	yes	characteristic
172	C₂×C₃₁	#1807	1	1	yes	characteristic
173	C₂×C₃₁	#1810	1	1	yes	characteristic
174	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
175	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
176	C₂³×C₄	Q₈×C₃₁	1	1	yes	characteristic
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	2	4	no	—
183	C₂²×C₈	C₈×C₃₁	1	4	yes	complemented
184	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
185	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
186	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
187	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	—	2	2	no	—
190	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
191	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
192	C₂×C₄²	Q₈×C₃₁	1	1	yes	characteristic
193	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
194	C₄⋊C₈	—	2	4	no	—
195	C₄⋊C₈	—	2	4	no	—
196	C₂²⋊C₈	—	4	4	no	—
197	C₄×C₈	—	2	2	no	—
198	C₄×C₈	—	2	2	no	—
199	C₄×C₈	—	2	2	no	—
200	C₄×C₈	D₄×C₃₁	1	4	yes	—
201	C₃₁	#8595	1	1	yes	characteristic, direct product factor, Sylow
202	C₂⁴	C₄⋊C₄×C₃₁	1	1	yes	characteristic
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
213	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
214	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₈×C₃₁	1	2	yes	—
216	C₂²×C₄	M₁₆×C₃₁	1	2	yes	—
217	C₂²×C₄	SD₁₆×C₃₁	1	4	yes	—
218	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
220	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
221	C₂×C₈	—	2	8	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	8	no	—
229	C₂×C₈	—	4	4	no	—
230	C₂×C₈	—	4	4	no	—
231	C₂×C₈	—	4	4	no	—
232	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
233	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
234	C₂×C₈	C₂×C₈×C₃₁	1	2	yes	—
235	C₂×C₈	C₂×C₈×C₃₁	1	2	yes	—
236	C₂×C₈	M₁₆×C₃₁	1	2	yes	—
237	C₂×C₈	M₁₆×C₃₁	1	2	yes	—
238	C₄²	—	2	1	no	—
239	C₄²	—	2	1	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	C₂×D₄×C₃₁	1	2	yes	—
243	C₄²	#41×C₃₁	1	2	yes	—
244	C₂³	—	2	1	no	—
245	C₂³	—	2	1	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	C₃₁×C₄⋊C₈	1	2	yes	—
249	C₂³	C₃₁×#106	1	2	yes	—
250	C₂³	C₃₁×#108	1	2	yes	—
251	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
252	C₂×C₄	—	2	8	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	8	no	—
266	C₂×C₄	C₃₁×C₄⋊C₈	1	2	yes	—
267	C₂×C₄	C₃₁×#106	1	2	yes	—
268	C₂×C₄	C₃₁×#108	1	2	yes	—
269	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
270	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
271	C₂×C₄	C₃₁×#121	1	2	yes	—
272	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
273	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
274	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
275	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
276	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
277	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
278	C₂×C₄	C₃₁×#135	1	2	yes	—
279	C₈	—	2	2	no	—
280	C₈	—	2	2	no	—
281	C₈	—	2	2	no	—
282	C₈	—	8	4	no	—
283	C₈	C₃₁×C₂²⋊C₈	1	4	yes	complemented
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	C₃₁×#334	1	4	yes	complemented
291	C₂²	C₃₁×#380	1	1	yes	characteristic
292	C₂²	C₂×C₃₁×C₄⋊C₈	1	1	yes	characteristic
293	C₂²	C₃₁×#423	1	1	yes	characteristic
294	C₂²	C₂×C₃₁×#106	1	1	yes	characteristic
295	C₂²	C₃₁×#427	1	1	yes	characteristic
296	C₂²	C₂×C₃₁×#108	1	1	yes	characteristic
297	C₂²	C₂×C₃₁×#108	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	2	2	no	—
302	C₄	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
303	C₄	C₃₁×#407	1	1	yes	characteristic
304	C₄	C₃₁×#431	1	1	yes	characteristic
305	C₄	C₃₁×#432	1	1	yes	characteristic
306	C₄	C₃₁×#465	1	4	yes	—
307	C₂	—	2	4	no	—
308	C₂	C₂×C₃₁×#334	1	1	yes	characteristic
309	C₂	C₃₁×#1564	1	1	yes	characteristic
310	C₂	C₃₁×#1566	1	1	yes	characteristic
311	C₂	C₃₁×#1568	1	1	yes	characteristic
312	C₂	C₃₁×#1797	1	1	yes	characteristic
313	C₂	C₃₁×#1807	1	1	yes	characteristic
314	C₂	C₃₁×#1810	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂³×C₃₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₃₁	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₃₁×#334	C₂	semidirect
4. C₃₁×#1277	C₂	semidirect
26. C₂²×C₈×C₃₁	C₈	semidirect
86. #8595	C₃₁	direct (N × H)
126. C₈×C₃₁	C₂²⋊C₈	semidirect
129. C₂×#334	C₂×C₃₁	semidirect
130. #1277	C₂×C₃₁	semidirect
137. C₂²×C₃₁	#334	semidirect
183. C₂²×C₈	C₈×C₃₁	semidirect
201. C₃₁	#8595	direct (N × H)
283. C₈	C₃₁×C₂²⋊C₈	semidirect
290. C₂²	C₃₁×#334	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₈×C₃₁	2
C₃₁×#1797	2
C₂×C₃₁×#334	2
C₃₁×#1277	2
#8595	31

Subgroups — C31×#8595 #21465354

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8595 #21465354