Subgroups — C₃₁×#18065 #21474824

Order 7936.

652 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2990	C₂	1	2	yes	maximal, complemented
2	C₄×C₃₁×#135	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1593	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1582	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1580	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1566	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1489	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1451	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#135	C₄	1	1	yes	characteristic
10	C₄×#41×C₃₁	C₂²	1	2	yes	—
11	C₃₁×#446	C₂²	1	4	yes	—
12	C₄×Q₁₆×C₃₁	C₄	1	1	yes	characteristic
13	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₄×SD₁₆×C₃₁	C₄	1	2	yes	—
15	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
16	C₄×D₈×C₃₁	C₄	1	1	yes	characteristic
17	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#435	—	2	4	no	—
19	C₈×D₄×C₃₁	C₂²	1	4	yes	—
20	C₃₁×#432	—	2	2	no	—
21	C₃₁×#431	C₂²	1	1	yes	characteristic
22	C₃₁×#427	C₄	1	1	yes	characteristic
23	C₃₁×#423	C₂²	1	2	yes	—
24	C₂×C₃₁×C₄⋊C₈	—	2	2	no	—
25	C₃₁×#416	C₄	1	2	yes	—
26	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#334	C₂²	1	2	yes	—
29	C₃₁×#334	C₂²	1	2	yes	—
30	C₃₁×Q₈⋊C₈	C₂²	1	4	yes	—
31	C₃₁×D₄⋊C₈	C₂²	1	4	yes	—
32	C₃₁×#322	C₂²	1	2	yes	—
33	C₃₁×#322	C₂²	1	2	yes	—
34	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
35	C₃₁×#135	—	2	4	no	—
36	C₂×Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
37	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	—
38	C₂×D₈×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₂×M₁₆×C₃₁	—	2	1	no	—
40	C₂×M₁₆×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	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
45	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
46	C₄×D₄×C₃₁	—	2	2	no	—
47	C₄×D₄×C₃₁	—	2	2	no	—
48	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
49	C₄×D₄×C₃₁	C₂³	1	2	yes	—
50	C₃₁×#117	—	2	2	no	—
51	C₃₁×#117	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₃₁	—	2	2	no	—
53	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
54	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
55	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
56	C₃₁×#106	C₂×C₄	1	2	yes	—
57	C₃₁×C₄⋊C₈	—	2	4	no	—
58	C₃₁×C₄⋊C₈	—	2	2	no	—
59	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
60	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
61	C₃₁×#103	C₂×C₄	1	2	yes	—
62	C₃₁×#103	C₂×C₄	1	2	yes	—
63	C₃₁×#102	C₂×C₄	1	2	yes	—
64	C₃₁×#102	C₂×C₄	1	2	yes	—
65	C₃₁×C₂²⋊C₈	—	2	4	no	—
66	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
67	C₃₁×#97	—	2	4	no	—
68	C₃₁×#97	—	2	1	no	—
69	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₃₁	C₂³	1	2	yes	—
72	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₃₁	C₂³	1	2	yes	—
75	#41×C₃₁	—	2	4	no	—
76	#41×C₃₁	—	4	2	no	—
77	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
78	C₂×D₄×C₃₁	—	2	2	no	—
79	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₃₁	—	2	2	no	—
81	C₂²×C₄×C₃₁	—	2	2	no	—
82	C₂²×C₄×C₃₁	—	2	2	no	—
83	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
86	Q₁₆×C₃₁	—	2	2	no	—
87	SD₁₆×C₃₁	—	2	4	no	—
88	D₈×C₃₁	M₁₆	1	4	yes	complemented
89	M₁₆×C₃₁	—	4	2	no	—
90	M₁₆×C₃₁	—	8	2	no	—
91	C₂×C₈×C₃₁	—	2	2	no	—
92	C₂×C₈×C₃₁	—	2	4	no	—
93	C₂×C₈×C₃₁	—	2	2	no	—
94	C₂×C₈×C₃₁	—	4	1	no	—
95	C₂×C₈×C₃₁	—	8	4	no	—
96	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
101	C₄⋊C₄×C₃₁	—	2	2	no	—
102	C₄⋊C₄×C₃₁	—	2	2	no	—
103	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
104	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
105	C₂²⋊C₄×C₃₁	—	2	2	no	—
106	C₂²⋊C₄×C₃₁	—	2	2	no	—
107	C₂²⋊C₄×C₃₁	—	2	2	no	—
108	C₄²×C₃₁	—	2	2	no	—
109	C₄²×C₃₁	—	2	2	no	—
110	C₄²×C₃₁	—	2	2	no	—
111	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₃₁	#41	1	1	yes	characteristic
114	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
115	#18065	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂³×C₃₁	—	2	2	no	—
117	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
118	Q₈×C₃₁	—	2	2	no	—
119	Q₈×C₃₁	—	2	2	no	—
120	D₄×C₃₁	—	2	2	no	—
121	D₄×C₃₁	—	4	4	no	—
122	D₄×C₃₁	C₂×M₁₆	1	4	yes	—
123	C₂×C₄×C₃₁	—	2	1	no	—
124	C₂×C₄×C₃₁	—	2	1	no	—
125	C₂×C₄×C₃₁	—	2	2	no	—
126	C₂×C₄×C₃₁	—	2	2	no	—
127	C₂×C₄×C₃₁	—	2	2	no	—
128	C₂×C₄×C₃₁	—	2	2	no	—
129	C₂×C₄×C₃₁	—	2	2	no	—
130	C₂×C₄×C₃₁	—	4	4	no	—
131	C₂×C₄×C₃₁	—	4	2	no	—
132	C₂×C₄×C₃₁	—	4	2	no	—
133	C₂×C₄×C₃₁	—	4	4	no	—
134	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
140	C₈×C₃₁	—	2	1	no	—
141	C₈×C₃₁	—	2	1	no	—
142	C₈×C₃₁	—	2	1	no	—
143	C₈×C₃₁	—	2	4	no	—
144	C₈×C₃₁	—	4	2	no	—
145	C₈×C₃₁	—	4	4	no	—
146	C₈×C₃₁	C₂×M₁₆	1	2	yes	—
147	#2990	C₂×C₃₁	1	2	yes	complemented
148	C₄×#135	C₂×C₃₁	1	1	yes	characteristic
149	#1593	C₂×C₃₁	1	2	yes	complemented
150	#1582	C₂×C₃₁	1	4	yes	complemented
151	#1580	C₂×C₃₁	1	2	yes	complemented
152	#1566	C₂×C₃₁	1	1	yes	characteristic, complemented
153	#1489	C₂×C₃₁	1	2	yes	complemented
154	#1451	C₂×C₃₁	1	1	yes	characteristic, complemented
155	C₂²×C₃₁	—	2	1	no	—
156	C₂²×C₃₁	—	2	4	no	—
157	C₂²×C₃₁	—	2	1	no	—
158	C₂²×C₃₁	—	2	1	no	—
159	C₂²×C₃₁	—	4	2	no	—
160	C₂²×C₃₁	—	4	2	no	—
161	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
162	C₄×C₃₁	—	2	1	no	—
163	C₄×C₃₁	—	2	1	no	—
164	C₄×C₃₁	—	2	1	no	—
165	C₄×C₃₁	—	4	2	no	—
166	C₄×C₃₁	—	4	2	no	—
167	C₄×C₃₁	—	4	2	no	—
168	C₄×C₃₁	#444	1	2	yes	—
169	C₄×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
170	C₄×C₃₁	#568	1	1	yes	characteristic
171	C₄×C₃₁	#577	1	2	yes	—
172	C₂×#135	C₄×C₃₁	1	1	yes	characteristic
173	C₄×#41	C₂²×C₃₁	1	2	yes	—
174	#446	C₂²×C₃₁	1	4	yes	—
175	C₄×Q₁₆	C₄×C₃₁	1	1	yes	characteristic
176	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
177	C₄×SD₁₆	C₄×C₃₁	1	2	yes	—
178	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
179	C₄×D₈	C₄×C₃₁	1	1	yes	characteristic
180	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
181	#435	—	2	4	no	—
182	C₈×D₄	C₂²×C₃₁	1	4	yes	—
183	#432	—	2	2	no	—
184	#431	C₂²×C₃₁	1	1	yes	characteristic
185	#427	C₄×C₃₁	1	1	yes	characteristic
186	#423	C₂²×C₃₁	1	2	yes	—
187	C₂×C₄⋊C₈	—	2	2	no	—
188	#416	C₄×C₃₁	1	2	yes	—
189	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
190	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
191	#334	C₂²×C₃₁	1	2	yes	—
192	#334	C₂²×C₃₁	1	2	yes	—
193	Q₈⋊C₈	C₂²×C₃₁	1	4	yes	—
194	D₄⋊C₈	C₂²×C₃₁	1	4	yes	—
195	#322	C₂²×C₃₁	1	2	yes	—
196	#322	C₂²×C₃₁	1	2	yes	—
197	C₂×C₃₁	—	4	1	no	—
198	C₂×C₃₁	—	4	4	no	—
199	C₂×C₃₁	D₄×M₁₆	1	1	yes	characteristic
200	C₂×C₃₁	#2952	1	1	yes	characteristic
201	C₂×C₃₁	C₂×#444	1	1	yes	characteristic
202	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
203	#135	—	2	4	no	—
204	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
205	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	—
206	C₂×D₈	C₂×C₄×C₃₁	1	1	yes	characteristic
207	C₂×M₁₆	—	2	1	no	—
208	C₂×M₁₆	—	2	2	no	—
209	C₂²×C₈	—	2	4	no	—
210	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
211	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
212	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
213	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
214	C₄×D₄	—	2	2	no	—
215	C₄×D₄	—	2	2	no	—
216	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
217	C₄×D₄	C₂³×C₃₁	1	2	yes	—
218	#117	—	2	2	no	—
219	#117	C₂×C₄×C₃₁	1	2	yes	—
220	C₂×C₄²	—	2	2	no	—
221	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
222	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
223	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
224	#106	C₂×C₄×C₃₁	1	2	yes	—
225	C₄⋊C₈	—	2	4	no	—
226	C₄⋊C₈	—	2	2	no	—
227	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
228	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
229	#103	C₂×C₄×C₃₁	1	2	yes	—
230	#103	C₂×C₄×C₃₁	1	2	yes	—
231	#102	C₂×C₄×C₃₁	1	2	yes	—
232	#102	C₂×C₄×C₃₁	1	2	yes	—
233	C₂²⋊C₈	—	2	4	no	—
234	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
235	#97	—	2	4	no	—
236	#97	—	2	1	no	—
237	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
238	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
239	C₄×C₈	C₂³×C₃₁	1	2	yes	—
240	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
241	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
242	C₄×C₈	C₂³×C₃₁	1	2	yes	—
243	C₃₁	#18065	1	1	yes	characteristic, direct product factor, Sylow
244	#41	—	2	4	no	—
245	#41	—	4	2	no	—
246	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
247	C₂×D₄	—	2	2	no	—
248	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
254	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
255	Q₁₆	—	2	2	no	—
256	SD₁₆	—	2	4	no	—
257	D₈	M₁₆×C₃₁	1	4	yes	complemented
258	M₁₆	—	4	2	no	—
259	M₁₆	—	8	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	—	8	4	no	—
265	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
266	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
267	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
268	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
269	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
273	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
274	C₂²⋊C₄	—	2	2	no	—
275	C₂²⋊C₄	—	2	2	no	—
276	C₂²⋊C₄	—	2	2	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
281	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
282	C₄²	#41×C₃₁	1	1	yes	characteristic
283	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
284	C₂³	—	2	2	no	—
285	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
286	Q₈	—	2	2	no	—
287	Q₈	—	2	2	no	—
288	D₄	—	2	2	no	—
289	D₄	—	4	4	no	—
290	D₄	C₂×M₁₆×C₃₁	1	4	yes	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
303	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
304	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
305	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
307	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	4	no	—
312	C₈	—	4	2	no	—
313	C₈	—	4	4	no	—
314	C₈	C₂×M₁₆×C₃₁	1	2	yes	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	4	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	4	2	no	—
326	C₄	—	4	2	no	—
327	C₄	—	4	2	no	—
328	C₄	C₃₁×#444	1	2	yes	—
329	C₄	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
330	C₄	C₃₁×#568	1	1	yes	characteristic
331	C₄	C₃₁×#577	1	2	yes	—
332	C₂	—	4	1	no	—
333	C₂	—	4	4	no	—
334	C₂	D₄×M₁₆×C₃₁	1	1	yes	characteristic
335	C₂	C₃₁×#2952	1	1	yes	characteristic
336	C₂	C₂×C₃₁×#444	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	C₂×C₃₁×#135	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₃₁×#431 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#2990	C₂	semidirect
3. C₃₁×#1593	C₂	semidirect
4. C₃₁×#1582	C₂	semidirect
5. C₃₁×#1580	C₂	semidirect
6. C₃₁×#1566	C₂	semidirect
7. C₃₁×#1489	C₂	semidirect
8. C₃₁×#1451	C₂	semidirect
88. D₈×C₃₁	M₁₆	semidirect
115. #18065	C₃₁	direct (N × H)
147. #2990	C₂×C₃₁	semidirect
149. #1593	C₂×C₃₁	semidirect
150. #1582	C₂×C₃₁	semidirect
151. #1580	C₂×C₃₁	semidirect
152. #1566	C₂×C₃₁	semidirect
153. #1489	C₂×C₃₁	semidirect
154. #1451	C₂×C₃₁	semidirect
243. C₃₁	#18065	direct (N × H)
257. D₈	M₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#2990	2
C₄×C₃₁×#135	2
C₃₁×#1593	2
C₃₁×#1582	2
C₃₁×#1580	2
C₃₁×#1566	2
C₃₁×#1489	2
C₃₁×#1451	2
#18065	31

Subgroups — C31×#18065 #21474824

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#18065 #21474824