Subgroups — C₃₁×#18066 #21474825

Order 7936.

722 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3169	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2935	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1589	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1581	C₂	1	4	yes	maximal, complemented
5	C₈×D₈×C₃₁	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#335	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1480	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#131	—	2	2	no	—
10	C₃₁×#532	C₂²	1	2	yes	—
11	C₃₁×#500	C₄	1	1	yes	characteristic
12	C₃₁×#493	C₄	1	1	yes	characteristic
13	C₃₁×#486	C₄	1	2	yes	—
14	C₃₁×#466	C₄	1	2	yes	—
15	C₃₁×#466	C₄	1	2	yes	—
16	C₄×D₈×C₃₁	C₂²	1	2	yes	—
17	C₄×D₈×C₃₁	C₂²	1	2	yes	—
18	C₃₁×#436	C₂²	1	4	yes	—
19	C₃₁×#435	—	2	4	no	—
20	C₈×D₄×C₃₁	C₂²	1	4	yes	—
21	C₃₁×#431	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
23	C₃₁×#408	—	2	2	no	—
24	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#335	C₂²	1	1	yes	characteristic
27	C₃₁×#335	C₂²	1	2	yes	—
28	C₃₁×#335	C₂²	1	1	yes	characteristic
29	C₃₁×D₄⋊C₈	C₂²	1	4	yes	—
30	C₃₁×D₄⋊C₈	C₂²	1	4	yes	—
31	C₃₁×#322	C₂²	1	2	yes	—
32	C₈²×C₃₁	C₂²	1	2	yes	—
33	C₂×#41×C₃₁	—	2	2	no	—
34	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
35	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
36	C₃₁×#131	—	4	8	no	—
37	C₂×M₁₆×C₃₁	—	2	4	no	—
38	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
39	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₃₁	—	2	2	no	—
41	C₂²×C₈×C₃₁	—	2	4	no	—
42	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
45	C₃₁×#127	C₂×C₄	1	2	yes	—
46	C₃₁×#124	—	2	2	no	—
47	C₃₁×#121	—	2	2	no	—
48	C₃₁×#121	C₂×C₄	1	4	yes	—
49	C₄×D₄×C₃₁	—	2	2	no	—
50	C₄×D₄×C₃₁	C₂³	1	4	yes	—
51	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
52	C₃₁×#107	C₂×C₄	1	2	yes	—
53	C₃₁×#107	C₂×C₄	1	2	yes	—
54	C₃₁×C₄⋊C₈	—	2	2	no	—
55	C₃₁×C₄⋊C₈	—	2	1	no	—
56	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
57	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
58	C₃₁×#102	C₂×C₄	1	4	yes	—
59	C₃₁×#102	C₂×C₄	1	4	yes	—
60	C₃₁×C₂²⋊C₈	—	2	1	no	—
61	C₃₁×C₂²⋊C₈	—	2	4	no	—
62	C₃₁×C₂²⋊C₈	—	2	4	no	—
63	C₃₁×#97	D₄	1	2	yes	—
64	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₃₁	C₂³	1	2	yes	—
67	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₃₁	C₂³	1	2	yes	—
70	#41×C₃₁	—	4	4	no	—
71	#41×C₃₁	—	4	4	no	—
72	C₂×Q₈×C₃₁	—	2	2	no	—
73	C₂×D₄×C₃₁	—	2	4	no	—
74	C₂×D₄×C₃₁	—	2	2	no	—
75	C₂×D₄×C₃₁	—	2	2	no	—
76	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
77	C₂²×C₄×C₃₁	—	2	4	no	—
78	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
81	D₈×C₃₁	—	2	4	no	—
82	D₈×C₃₁	—	4	2	no	—
83	M₁₆×C₃₁	—	2	2	no	—
84	M₁₆×C₃₁	—	4	8	no	—
85	M₁₆×C₃₁	D₈	1	4	yes	complemented
86	C₂×C₈×C₃₁	—	2	2	no	—
87	C₂×C₈×C₃₁	—	2	2	no	—
88	C₂×C₈×C₃₁	—	2	2	no	—
89	C₂×C₈×C₃₁	—	2	2	no	—
90	C₂×C₈×C₃₁	—	4	1	no	—
91	C₂×C₈×C₃₁	—	4	4	no	—
92	C₂×C₈×C₃₁	—	4	8	no	—
93	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
99	C₄⋊C₄×C₃₁	—	2	2	no	—
100	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
101	C₂²⋊C₄×C₃₁	—	2	4	no	—
102	C₂²⋊C₄×C₃₁	—	2	4	no	—
103	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₃₁	#41	1	1	yes	characteristic
106	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
107	#18066	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂³×C₃₁	—	2	4	no	—
109	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
110	Q₈×C₃₁	—	4	4	no	—
111	D₄×C₃₁	—	2	4	no	—
112	D₄×C₃₁	—	2	4	no	—
113	D₄×C₃₁	—	4	8	no	—
114	D₄×C₃₁	—	4	4	no	—
115	D₄×C₃₁	—	8	2	no	—
116	C₂×C₄×C₃₁	—	2	1	no	—
117	C₂×C₄×C₃₁	—	2	1	no	—
118	C₂×C₄×C₃₁	—	2	4	no	—
119	C₂×C₄×C₃₁	—	4	1	no	—
120	C₂×C₄×C₃₁	—	4	4	no	—
121	C₂×C₄×C₃₁	—	4	4	no	—
122	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
126	C₂×C₄×C₃₁	#135	1	2	yes	—
127	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
130	C₈×C₃₁	—	2	1	no	—
131	C₈×C₃₁	—	2	1	no	—
132	C₈×C₃₁	—	2	1	no	—
133	C₈×C₃₁	—	2	2	no	—
134	C₈×C₃₁	—	4	2	no	—
135	C₈×C₃₁	—	4	2	no	—
136	C₈×C₃₁	—	4	2	no	—
137	C₈×C₃₁	#131	1	2	yes	complemented
138	C₈×C₃₁	C₂×D₈	1	4	yes	—
139	#3169	C₂×C₃₁	1	1	yes	characteristic
140	#2935	C₂×C₃₁	1	2	yes	complemented
141	#1589	C₂×C₃₁	1	2	yes	complemented
142	#1581	C₂×C₃₁	1	4	yes	complemented
143	C₈×D₈	C₂×C₃₁	1	2	yes	complemented
144	C₂×#335	C₂×C₃₁	1	1	yes	characteristic, complemented
145	#1480	C₂×C₃₁	1	2	yes	complemented
146	C₈×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
147	C₂²×C₃₁	—	2	1	no	—
148	C₂²×C₃₁	—	2	1	no	—
149	C₂²×C₃₁	—	4	4	no	—
150	C₂²×C₃₁	—	4	4	no	—
151	C₂²×C₃₁	—	4	4	no	—
152	C₂²×C₃₁	C₄×D₈	1	2	yes	—
153	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
154	C₄×C₃₁	—	2	1	no	—
155	C₄×C₃₁	—	2	1	no	—
156	C₄×C₃₁	—	2	1	no	—
157	C₄×C₃₁	—	4	4	no	—
158	C₄×C₃₁	C₄×D₈	1	2	yes	—
159	C₄×C₃₁	C₂×#131	1	1	yes	characteristic
160	C₄×C₃₁	#568	1	1	yes	characteristic
161	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
162	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
163	C₂×#131	—	2	2	no	—
164	#532	C₂²×C₃₁	1	2	yes	—
165	#500	C₄×C₃₁	1	1	yes	characteristic
166	#493	C₄×C₃₁	1	1	yes	characteristic
167	#486	C₄×C₃₁	1	2	yes	—
168	#466	C₄×C₃₁	1	2	yes	—
169	#466	C₄×C₃₁	1	2	yes	—
170	C₄×D₈	C₂²×C₃₁	1	2	yes	—
171	C₄×D₈	C₂²×C₃₁	1	2	yes	—
172	#436	C₂²×C₃₁	1	4	yes	—
173	#435	—	2	4	no	—
174	C₈×D₄	C₂²×C₃₁	1	4	yes	—
175	#431	C₂²×C₃₁	1	1	yes	characteristic
176	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
177	#408	—	2	2	no	—
178	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
179	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
180	#335	C₂²×C₃₁	1	1	yes	characteristic
181	#335	C₂²×C₃₁	1	2	yes	—
182	#335	C₂²×C₃₁	1	1	yes	characteristic
183	D₄⋊C₈	C₂²×C₃₁	1	4	yes	—
184	D₄⋊C₈	C₂²×C₃₁	1	4	yes	—
185	#322	C₂²×C₃₁	1	2	yes	—
186	C₈²	C₂²×C₃₁	1	2	yes	—
187	C₂×C₃₁	—	2	2	no	—
188	C₂×C₃₁	—	8	4	no	—
189	C₂×C₃₁	#2935	1	1	yes	characteristic
190	C₂×C₃₁	C₂×C₄×D₈	1	1	yes	characteristic
191	C₂×C₃₁	#2957	1	1	yes	characteristic
192	C₂×#41	—	2	2	no	—
193	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
194	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
195	#131	—	4	8	no	—
196	C₂×M₁₆	—	2	4	no	—
197	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
198	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
199	C₂²×C₈	—	2	2	no	—
200	C₂²×C₈	—	2	4	no	—
201	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
202	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
203	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
204	#127	C₂×C₄×C₃₁	1	2	yes	—
205	#124	—	2	2	no	—
206	#121	—	2	2	no	—
207	#121	C₂×C₄×C₃₁	1	4	yes	—
208	C₄×D₄	—	2	2	no	—
209	C₄×D₄	C₂³×C₃₁	1	4	yes	—
210	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
211	#107	C₂×C₄×C₃₁	1	2	yes	—
212	#107	C₂×C₄×C₃₁	1	2	yes	—
213	C₄⋊C₈	—	2	2	no	—
214	C₄⋊C₈	—	2	1	no	—
215	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
216	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
217	#102	C₂×C₄×C₃₁	1	4	yes	—
218	#102	C₂×C₄×C₃₁	1	4	yes	—
219	C₂²⋊C₈	—	2	1	no	—
220	C₂²⋊C₈	—	2	4	no	—
221	C₂²⋊C₈	—	2	4	no	—
222	#97	D₄×C₃₁	1	2	yes	—
223	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
224	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
225	C₄×C₈	C₂³×C₃₁	1	2	yes	—
226	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
227	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
228	C₄×C₈	C₂³×C₃₁	1	2	yes	—
229	C₃₁	#18066	1	1	yes	characteristic, direct product factor, Sylow
230	#41	—	4	4	no	—
231	#41	—	4	4	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×D₄	—	2	4	no	—
234	C₂×D₄	—	2	2	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
240	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
241	D₈	—	2	4	no	—
242	D₈	—	4	2	no	—
243	M₁₆	—	2	2	no	—
244	M₁₆	—	4	8	no	—
245	M₁₆	D₈×C₃₁	1	4	yes	complemented
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	4	1	no	—
251	C₂×C₈	—	4	4	no	—
252	C₂×C₈	—	4	8	no	—
253	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
254	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
255	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
256	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
257	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
258	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
261	C₂²⋊C₄	—	2	4	no	—
262	C₂²⋊C₄	—	2	4	no	—
263	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
264	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
265	C₄²	#41×C₃₁	1	1	yes	characteristic
266	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
267	C₂³	—	2	4	no	—
268	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
269	Q₈	—	4	4	no	—
270	D₄	—	2	4	no	—
271	D₄	—	2	4	no	—
272	D₄	—	4	8	no	—
273	D₄	—	4	4	no	—
274	D₄	—	8	2	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
282	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
283	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
284	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
285	C₂×C₄	C₃₁×#135	1	2	yes	—
286	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
287	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
288	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
289	C₈	—	2	1	no	—
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	2	2	no	—
293	C₈	—	4	2	no	—
294	C₈	—	4	2	no	—
295	C₈	—	4	2	no	—
296	C₈	C₃₁×#131	1	2	yes	complemented
297	C₈	C₂×D₈×C₃₁	1	4	yes	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	4	4	no	—
301	C₂²	—	4	4	no	—
302	C₂²	—	4	4	no	—
303	C₂²	C₄×D₈×C₃₁	1	2	yes	—
304	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	4	4	no	—
309	C₄	C₄×D₈×C₃₁	1	2	yes	—
310	C₄	C₂×C₃₁×#131	1	1	yes	characteristic
311	C₄	C₃₁×#568	1	1	yes	characteristic
312	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
313	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
314	C₂	—	2	2	no	—
315	C₂	—	8	4	no	—
316	C₂	C₃₁×#2935	1	1	yes	characteristic
317	C₂	C₂×C₄×D₈×C₃₁	1	1	yes	characteristic
318	C₂	C₃₁×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₃₁×#3169	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₄×M₁₆×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2935	C₂	semidirect
3. C₃₁×#1589	C₂	semidirect
4. C₃₁×#1581	C₂	semidirect
5. C₈×D₈×C₃₁	C₂	semidirect
6. C₂×C₃₁×#335	C₂	semidirect
7. C₃₁×#1480	C₂	semidirect
8. C₈×M₁₆×C₃₁	C₂	semidirect
85. M₁₆×C₃₁	D₈	semidirect
107. #18066	C₃₁	direct (N × H)
137. C₈×C₃₁	#131	semidirect
140. #2935	C₂×C₃₁	semidirect
141. #1589	C₂×C₃₁	semidirect
142. #1581	C₂×C₃₁	semidirect
143. C₈×D₈	C₂×C₃₁	semidirect
144. C₂×#335	C₂×C₃₁	semidirect
145. #1480	C₂×C₃₁	semidirect
146. C₈×M₁₆	C₂×C₃₁	semidirect
229. C₃₁	#18066	direct (N × H)
245. M₁₆	D₈×C₃₁	semidirect
296. C₈	C₃₁×#131	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3169	2
C₃₁×#2935	2
C₃₁×#1589	2
C₃₁×#1581	2
C₈×D₈×C₃₁	2
C₂×C₃₁×#335	2
C₃₁×#1480	2
C₈×M₁₆×C₃₁	2
#18066	31

Subgroups — C31×#18066 #21474825

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#18066 #21474825