Subgroups — C₃₁×#17689 #21474448

Order 7936.

876 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×#3504	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2970	C₂	1	2	yes	maximal
3	C₃₁×#1537	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1534	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1531	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1528	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1526	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#330	C₂	1	2	yes	maximal, complemented
9	C₃₁×#557	—	2	2	no	—
10	C₃₁×#554	—	2	2	no	—
11	C₃₁×#554	—	2	4	no	—
12	C₃₁×#544	C₄	1	4	yes	—
13	C₃₁×#544	C₄	1	2	yes	—
14	C₃₁×#533	C₂²	1	1	yes	characteristic
15	C₃₁×#533	C₂²	1	2	yes	—
16	C₂×C₃₁×#125	C₂²	1	2	yes	—
17	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
18	C₃₁×#520	C₄	1	2	yes	—
19	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
20	C₃₁×#446	—	2	2	no	—
21	C₃₁×#435	—	2	4	no	—
22	C₈×D₄×C₃₁	—	2	2	no	—
23	C₃₁×#432	C₂²	1	2	yes	—
24	C₃₁×#432	C₂²	1	4	yes	—
25	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
26	C₂×C₃₁×#97	C₂²	1	4	yes	—
27	C₃₁×#330	C₂²	1	1	yes	characteristic
28	C₃₁×#330	C₂²	1	4	yes	—
29	C₃₁×#330	C₂²	1	2	yes	—
30	C₃₁×#330	C₂²	1	2	yes	—
31	C₃₁×#330	C₂²	1	1	yes	characteristic
32	C₃₁×#330	C₂²	1	4	yes	—
33	C₃₁×#330	C₂²	1	2	yes	—
34	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
35	C₂×M₁₆×C₃₁	—	2	2	no	—
36	C₂²×C₈×C₃₁	—	2	4	no	—
37	C₃₁×C₄⋊Q₈	—	2	4	no	—
38	C₃₁×C₄⋊Q₈	—	2	2	no	—
39	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
40	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	—
41	C₃₁×#125	—	2	4	no	—
42	C₃₁×#125	—	2	2	no	—
43	C₃₁×#125	—	2	2	no	—
44	C₃₁×#125	—	2	4	no	—
45	C₃₁×#125	C₂×C₄	1	4	yes	—
46	C₃₁×#125	C₂×C₄	1	2	yes	—
47	C₃₁×#125	C₂³	1	1	yes	characteristic
48	C₃₁×#125	C₂³	1	2	yes	—
49	C₃₁×#125	C₂³	1	1	yes	characteristic
50	C₃₁×C₂²⋊Q₈	—	2	4	no	—
51	C₃₁×C₂²⋊Q₈	—	2	2	no	—
52	C₃₁×C₂²⋊Q₈	—	2	4	no	—
53	C₃₁×C₂²⋊Q₈	—	2	4	no	—
54	C₃₁×C₂²⋊Q₈	—	2	4	no	—
55	C₄×Q₈×C₃₁	—	2	2	no	—
56	C₄×Q₈×C₃₁	—	2	2	no	—
57	C₄×Q₈×C₃₁	—	2	4	no	—
58	C₄×Q₈×C₃₁	D₄	1	2	yes	—
59	C₄×D₄×C₃₁	—	2	4	no	—
60	C₄×D₄×C₃₁	—	2	2	no	—
61	C₄×D₄×C₃₁	D₄	1	2	yes	—
62	C₄×D₄×C₃₁	D₄	1	4	yes	—
63	C₃₁×#117	C₂×C₄	1	2	yes	—
64	C₃₁×#117	C₂×C₄	1	4	yes	—
65	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
66	C₃₁×#117	C₂×C₄	1	2	yes	—
67	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
68	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
69	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
70	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
71	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
72	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
73	C₃₁×C₄⋊C₈	—	2	2	no	—
74	C₃₁×C₄⋊C₈	—	2	4	no	—
75	C₃₁×C₂²⋊C₈	—	2	4	no	—
76	C₃₁×C₂²⋊C₈	—	2	2	no	—
77	C₃₁×#97	—	2	4	no	—
78	C₃₁×#97	C₂³	1	2	yes	—
79	C₃₁×#97	C₂³	1	4	yes	—
80	C₃₁×#97	C₂³	1	2	yes	—
81	C₄×C₈×C₃₁	—	2	2	no	—
82	#41×C₃₁	—	2	4	no	—
83	C₂×Q₈×C₃₁	—	2	2	no	—
84	C₂×Q₈×C₃₁	—	2	4	no	—
85	C₂×Q₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
87	C₂×D₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
88	C₂²×C₄×C₃₁	—	2	2	no	—
89	C₂²×C₄×C₃₁	—	2	4	no	—
90	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
93	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
94	M₁₆×C₃₁	—	4	4	no	—
95	C₂×C₈×C₃₁	—	2	2	no	—
96	C₂×C₈×C₃₁	—	2	2	no	—
97	C₂×C₈×C₃₁	—	2	4	no	—
98	C₂×C₈×C₃₁	—	4	8	no	—
99	C₄⋊C₄×C₃₁	—	2	4	no	—
100	C₄⋊C₄×C₃₁	—	2	4	no	—
101	C₄⋊C₄×C₃₁	—	2	4	no	—
102	C₄⋊C₄×C₃₁	—	2	4	no	—
103	C₄⋊C₄×C₃₁	—	2	2	no	—
104	C₄⋊C₄×C₃₁	—	2	2	no	—
105	C₄⋊C₄×C₃₁	—	2	4	no	—
106	C₄⋊C₄×C₃₁	—	2	2	no	—
107	C₄⋊C₄×C₃₁	—	2	4	no	—
108	C₄⋊C₄×C₃₁	—	2	2	no	—
109	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
110	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
111	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
112	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
113	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
114	C₂²⋊C₄×C₃₁	—	2	2	no	—
115	C₂²⋊C₄×C₃₁	—	2	4	no	—
116	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
117	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
118	C₄²×C₃₁	—	2	4	no	—
119	C₄²×C₃₁	—	2	2	no	—
120	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
121	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
122	C₄²×C₃₁	C₂×D₄	1	2	yes	—
123	C₄²×C₃₁	C₂×D₄	1	4	yes	—
124	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
125	#17689	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
128	Q₈×C₃₁	—	2	2	no	—
129	Q₈×C₃₁	—	8	4	no	—
130	Q₈×C₃₁	—	8	2	no	—
131	D₄×C₃₁	—	2	4	no	—
132	D₄×C₃₁	#101	1	4	yes	—
133	C₂×C₄×C₃₁	—	2	4	no	—
134	C₂×C₄×C₃₁	—	2	2	no	—
135	C₂×C₄×C₃₁	—	2	2	no	—
136	C₂×C₄×C₃₁	—	2	4	no	—
137	C₂×C₄×C₃₁	—	2	2	no	—
138	C₂×C₄×C₃₁	—	4	4	no	—
139	C₂×C₄×C₃₁	—	4	2	no	—
140	C₂×C₄×C₃₁	—	8	4	no	—
141	C₂×C₄×C₃₁	—	8	2	no	—
142	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
143	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
144	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
145	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
147	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
148	C₈×C₃₁	—	2	4	no	—
149	C₈×C₃₁	—	4	4	no	—
150	C₈×C₃₁	—	4	2	no	—
151	#3504	C₂×C₃₁	1	1	yes	characteristic
152	#2970	C₂×C₃₁	1	2	yes	—
153	#1537	C₂×C₃₁	1	1	yes	characteristic, complemented
154	#1534	C₂×C₃₁	1	2	yes	complemented
155	#1531	C₂×C₃₁	1	4	yes	complemented
156	#1528	C₂×C₃₁	1	2	yes	complemented
157	#1526	C₂×C₃₁	1	1	yes	characteristic, complemented
158	C₂×#330	C₂×C₃₁	1	2	yes	complemented
159	C₂²×C₃₁	—	2	4	no	—
160	C₂²×C₃₁	—	2	1	no	—
161	C₂²×C₃₁	—	2	2	no	—
162	C₂²×C₃₁	C₂×#101	1	4	yes	—
163	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
164	C₄×C₃₁	—	2	1	no	—
165	C₄×C₃₁	—	2	2	no	—
166	C₄×C₃₁	—	2	2	no	—
167	C₄×C₃₁	—	2	4	no	—
168	C₄×C₃₁	—	2	2	no	—
169	C₄×C₃₁	—	4	2	no	—
170	C₄×C₃₁	—	4	4	no	—
171	C₄×C₃₁	—	4	2	no	—
172	C₄×C₃₁	C₂×#101	1	2	yes	—
173	#557	—	2	2	no	—
174	#554	—	2	2	no	—
175	#554	—	2	4	no	—
176	#544	C₄×C₃₁	1	4	yes	—
177	#544	C₄×C₃₁	1	2	yes	—
178	#533	C₂²×C₃₁	1	1	yes	characteristic
179	#533	C₂²×C₃₁	1	2	yes	—
180	C₂×#125	C₂²×C₃₁	1	2	yes	—
181	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
182	#520	C₄×C₃₁	1	2	yes	—
183	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
184	#446	—	2	2	no	—
185	#435	—	2	4	no	—
186	C₈×D₄	—	2	2	no	—
187	#432	C₂²×C₃₁	1	2	yes	—
188	#432	C₂²×C₃₁	1	4	yes	—
189	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
190	C₂×#97	C₂²×C₃₁	1	4	yes	—
191	#330	C₂²×C₃₁	1	2	yes	—
192	#330	C₂²×C₃₁	1	2	yes	—
193	#330	C₂²×C₃₁	1	1	yes	characteristic
194	#330	C₂²×C₃₁	1	4	yes	—
195	#330	C₂²×C₃₁	1	2	yes	—
196	#330	C₂²×C₃₁	1	1	yes	characteristic
197	#330	C₂²×C₃₁	1	4	yes	—
198	C₂×C₃₁	—	2	4	no	—
199	C₂×C₃₁	—	4	1	no	—
200	C₂×C₃₁	C₂²×#101	1	1	yes	characteristic
201	C₂×C₃₁	#2901	1	2	yes	—
202	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
203	C₂×M₁₆	—	2	2	no	—
204	C₂²×C₈	—	2	4	no	—
205	C₄⋊Q₈	—	2	4	no	—
206	C₄⋊Q₈	—	2	2	no	—
207	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
208	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
209	#125	—	2	4	no	—
210	#125	—	2	2	no	—
211	#125	—	2	2	no	—
212	#125	—	2	4	no	—
213	#125	C₂×C₄×C₃₁	1	4	yes	—
214	#125	C₂×C₄×C₃₁	1	2	yes	—
215	#125	C₂³×C₃₁	1	1	yes	characteristic
216	#125	C₂³×C₃₁	1	2	yes	—
217	#125	C₂³×C₃₁	1	1	yes	characteristic
218	C₂²⋊Q₈	—	2	4	no	—
219	C₂²⋊Q₈	—	2	2	no	—
220	C₂²⋊Q₈	—	2	4	no	—
221	C₂²⋊Q₈	—	2	4	no	—
222	C₂²⋊Q₈	—	2	4	no	—
223	C₄×Q₈	—	2	2	no	—
224	C₄×Q₈	—	2	2	no	—
225	C₄×Q₈	—	2	4	no	—
226	C₄×Q₈	D₄×C₃₁	1	2	yes	—
227	C₄×D₄	—	2	4	no	—
228	C₄×D₄	—	2	2	no	—
229	C₄×D₄	D₄×C₃₁	1	2	yes	—
230	C₄×D₄	D₄×C₃₁	1	4	yes	—
231	#117	C₂×C₄×C₃₁	1	2	yes	—
232	#117	C₂×C₄×C₃₁	1	4	yes	—
233	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
234	#117	C₂×C₄×C₃₁	1	2	yes	—
235	C₂×C₄⋊C₄	—	2	4	no	—
236	C₂×C₄⋊C₄	—	2	2	no	—
237	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
238	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
239	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
240	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
241	C₄⋊C₈	—	2	2	no	—
242	C₄⋊C₈	—	2	4	no	—
243	C₂²⋊C₈	—	2	4	no	—
244	C₂²⋊C₈	—	2	2	no	—
245	#97	—	2	4	no	—
246	#97	C₂³×C₃₁	1	2	yes	—
247	#97	C₂³×C₃₁	1	4	yes	—
248	#97	C₂³×C₃₁	1	2	yes	—
249	C₄×C₈	—	2	2	no	—
250	C₃₁	#17689	1	1	yes	characteristic, direct product factor, Sylow
251	#41	—	2	4	no	—
252	C₂×Q₈	—	2	2	no	—
253	C₂×Q₈	—	2	4	no	—
254	C₂×Q₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
255	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
256	C₂×D₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	—	2	4	no	—
259	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
260	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
261	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
262	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
263	M₁₆	—	4	4	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	4	no	—
267	C₂×C₈	—	4	8	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	2	4	no	—
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	—	2	4	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	4	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	4	no	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
279	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
280	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
281	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
282	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
283	C₂²⋊C₄	—	2	2	no	—
284	C₂²⋊C₄	—	2	4	no	—
285	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
286	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
287	C₄²	—	2	4	no	—
288	C₄²	—	2	2	no	—
289	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
290	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
291	C₄²	C₂×D₄×C₃₁	1	2	yes	—
292	C₄²	C₂×D₄×C₃₁	1	4	yes	—
293	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
294	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
295	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
296	Q₈	—	2	2	no	—
297	Q₈	—	8	4	no	—
298	Q₈	—	8	2	no	—
299	D₄	—	2	4	no	—
300	D₄	C₃₁×#101	1	4	yes	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	8	4	no	—
309	C₂×C₄	—	8	2	no	—
310	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
311	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
312	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
313	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
314	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
315	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
316	C₈	—	2	4	no	—
317	C₈	—	4	4	no	—
318	C₈	—	4	2	no	—
319	C₂²	—	2	4	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	2	no	—
322	C₂²	C₂×C₃₁×#101	1	4	yes	—
323	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	2	no	—
326	C₄	—	2	2	no	—
327	C₄	—	2	4	no	—
328	C₄	—	2	2	no	—
329	C₄	—	4	2	no	—
330	C₄	—	4	4	no	—
331	C₄	—	4	2	no	—
332	C₄	C₂×C₃₁×#101	1	2	yes	—
333	C₂	—	2	4	no	—
334	C₂	—	4	1	no	—
335	C₂	C₂²×C₃₁×#101	1	1	yes	characteristic
336	C₂	C₃₁×#2901	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₂×#41×C₃₁	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1537	C₂	semidirect
4. C₃₁×#1534	C₂	semidirect
5. C₃₁×#1531	C₂	semidirect
6. C₃₁×#1528	C₂	semidirect
7. C₃₁×#1526	C₂	semidirect
8. C₂×C₃₁×#330	C₂	semidirect
125. #17689	C₃₁	direct (N × H)
153. #1537	C₂×C₃₁	semidirect
154. #1534	C₂×C₃₁	semidirect
155. #1531	C₂×C₃₁	semidirect
156. #1528	C₂×C₃₁	semidirect
157. #1526	C₂×C₃₁	semidirect
158. C₂×#330	C₂×C₃₁	semidirect
250. C₃₁	#17689	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3504	2
C₃₁×#2970	2
C₃₁×#1537	2
C₃₁×#1534	2
C₃₁×#1531	2
C₃₁×#1528	2
C₃₁×#1526	2
C₂×C₃₁×#330	2
#17689	31

Subgroups — C31×#17689 #21474448

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17689 #21474448