Subgroups — C₃₁×#9589 #21466348

Order 7936.

326 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	C₃₁×#1986	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1950	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1542	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1493	C₂	1	1	yes	maximal, characteristic
5	C₈×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1303	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1279	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#533	C₂²	1	1	yes	characteristic
9	C₃₁×#431	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×#108	—	2	1	no	—
11	C₃₁×#423	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#341	—	2	1	no	—
17	C₃₁×#339	—	2	1	no	—
18	C₃₁×#339	—	2	1	no	—
19	C₃₁×#333	C₄	1	1	yes	characteristic, complemented
20	C₃₁×#333	C₄	1	1	yes	characteristic, complemented
21	C₃₁×#332	C₄	1	1	yes	characteristic, complemented
22	C₃₁×#332	C₄	1	1	yes	characteristic, complemented
23	C₃₁×Q₈⋊C₈	—	2	1	no	—
24	C₃₁×Q₈⋊C₈	—	2	1	no	—
25	C₃₁×#322	—	2	1	no	—
26	C₈²×C₃₁	—	2	1	no	—
27	C₂×M₁₆×C₃₁	—	2	1	no	—
28	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
29	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₃₁	—	2	1	no	—
31	C₂²×C₈×C₃₁	—	2	1	no	—
32	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
33	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
34	C₃₁×#125	—	2	1	no	—
35	C₃₁×C₂²⋊Q₈	—	2	1	no	—
36	C₃₁×C₂²⋊Q₈	—	2	1	no	—
37	C₄×Q₈×C₃₁	—	2	1	no	—
38	C₄×Q₈×C₃₁	—	2	1	no	—
39	C₃₁×#117	—	2	1	no	—
40	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
41	C₂×C₄²×C₃₁	—	2	1	no	—
42	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
43	C₃₁×#108	—	4	2	no	—
44	C₃₁×C₄⋊C₈	—	2	1	no	—
45	C₃₁×C₄⋊C₈	—	2	1	no	—
46	C₃₁×C₄⋊C₈	—	2	1	no	—
47	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
48	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
49	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
50	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
51	C₃₁×C₂²⋊C₈	—	2	1	no	—
52	C₃₁×#97	—	2	1	no	—
53	C₄×C₈×C₃₁	—	2	1	no	—
54	C₄×C₈×C₃₁	—	2	1	no	—
55	C₄×C₈×C₃₁	—	2	1	no	—
56	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
57	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
58	C₃₁×#95	—	2	1	no	—
59	C₂×Q₈×C₃₁	—	2	1	no	—
60	C₂×Q₈×C₃₁	—	2	1	no	—
61	C₂²×C₄×C₃₁	—	4	1	no	—
62	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
65	M₁₆×C₃₁	—	2	1	no	—
66	M₁₆×C₃₁	—	2	1	no	—
67	M₁₆×C₃₁	—	4	1	no	—
68	M₁₆×C₃₁	—	4	1	no	—
69	M₁₆×C₃₁	SD₁₆	1	2	yes	—
70	M₁₆×C₃₁	Q₁₆	1	2	yes	—
71	C₂×C₈×C₃₁	—	2	1	no	—
72	C₂×C₈×C₃₁	—	2	1	no	—
73	C₂×C₈×C₃₁	—	2	1	no	—
74	C₂×C₈×C₃₁	—	2	1	no	—
75	C₂×C₈×C₃₁	—	2	1	no	—
76	C₂×C₈×C₃₁	—	2	1	no	—
77	C₂×C₈×C₃₁	—	2	1	no	—
78	C₂×C₈×C₃₁	—	2	1	no	—
79	C₂×C₈×C₃₁	—	4	2	no	—
80	C₂×C₈×C₃₁	—	4	1	no	—
81	C₂×C₈×C₃₁	—	4	1	no	—
82	C₂×C₈×C₃₁	—	4	1	no	—
83	C₄⋊C₄×C₃₁	—	2	2	no	—
84	C₄⋊C₄×C₃₁	—	2	1	no	—
85	C₄⋊C₄×C₃₁	—	2	1	no	—
86	C₄⋊C₄×C₃₁	—	2	1	no	—
87	C₄⋊C₄×C₃₁	—	2	1	no	—
88	C₄⋊C₄×C₃₁	—	4	1	no	—
89	C₄⋊C₄×C₃₁	—	4	1	no	—
90	C₄⋊C₄×C₃₁	—	4	1	no	—
91	C₂²⋊C₄×C₃₁	—	4	1	no	—
92	C₄²×C₃₁	—	4	1	no	—
93	C₄²×C₃₁	—	4	1	no	—
94	C₄²×C₃₁	—	4	1	no	—
95	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₃₁	#41	1	1	yes	characteristic
98	C₄²×C₃₁	#41	1	1	yes	characteristic
99	#9589	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂³×C₃₁	#124	1	1	yes	characteristic
101	Q₈×C₃₁	—	4	1	no	—
102	Q₈×C₃₁	—	4	1	no	—
103	Q₈×C₃₁	—	8	1	no	—
104	C₂×C₄×C₃₁	—	2	1	no	—
105	C₂×C₄×C₃₁	—	2	1	no	—
106	C₂×C₄×C₃₁	—	4	1	no	—
107	C₂×C₄×C₃₁	—	4	2	no	—
108	C₂×C₄×C₃₁	—	4	1	no	—
109	C₂×C₄×C₃₁	—	4	1	no	—
110	C₂×C₄×C₃₁	—	8	1	no	—
111	C₂×C₄×C₃₁	—	8	1	no	—
112	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
113	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
116	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
117	C₂×C₄×C₃₁	#126	1	1	yes	characteristic
118	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
120	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
121	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
122	C₈×C₃₁	—	2	2	no	—
123	C₈×C₃₁	—	2	2	no	—
124	C₈×C₃₁	—	2	2	no	—
125	C₈×C₃₁	—	4	1	no	—
126	C₈×C₃₁	—	4	1	no	—
127	C₈×C₃₁	—	4	1	no	—
128	C₈×C₃₁	—	4	1	no	—
129	C₈×C₃₁	—	4	1	no	—
130	C₈×C₃₁	—	8	1	no	—
131	#1986	C₂×C₃₁	1	1	yes	characteristic
132	#1950	C₂×C₃₁	1	1	yes	characteristic
133	#1542	C₂×C₃₁	1	1	yes	characteristic
134	#1493	C₂×C₃₁	1	1	yes	characteristic
135	C₈×M₁₆	C₂×C₃₁	1	1	yes	characteristic
136	#1303	C₂×C₃₁	1	1	yes	characteristic
137	#1279	C₂×C₃₁	1	1	yes	characteristic
138	C₂²×C₃₁	—	2	1	no	—
139	C₂²×C₃₁	—	2	1	no	—
140	C₂²×C₃₁	#388	1	1	yes	characteristic
141	C₂²×C₃₁	#487	1	1	yes	characteristic
142	C₂²×C₃₁	#488	1	1	yes	characteristic
143	C₄×C₃₁	—	2	1	no	—
144	C₄×C₃₁	—	2	1	no	—
145	C₄×C₃₁	—	2	1	no	—
146	C₄×C₃₁	—	8	2	no	—
147	C₄×C₃₁	—	8	1	no	—
148	C₄×C₃₁	—	8	1	no	—
149	C₄×C₃₁	C₄×SD₁₆	1	1	yes	characteristic
150	C₄×C₃₁	C₄×Q₁₆	1	1	yes	characteristic
151	C₄×C₃₁	#443	1	1	yes	characteristic
152	C₄×C₃₁	#444	1	1	yes	characteristic
153	C₄×C₃₁	#465	1	1	yes	characteristic
154	C₄×C₃₁	#467	1	1	yes	characteristic
155	#533	C₂²×C₃₁	1	1	yes	characteristic
156	#431	C₂²×C₃₁	1	1	yes	characteristic
157	C₂×#108	—	2	1	no	—
158	#423	C₂²×C₃₁	1	1	yes	characteristic
159	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
160	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
161	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
162	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
163	#341	—	2	1	no	—
164	#339	—	2	1	no	—
165	#339	—	2	1	no	—
166	#333	C₄×C₃₁	1	1	yes	characteristic, complemented
167	#333	C₄×C₃₁	1	1	yes	characteristic, complemented
168	#332	C₄×C₃₁	1	1	yes	characteristic, complemented
169	#332	C₄×C₃₁	1	1	yes	characteristic, complemented
170	Q₈⋊C₈	—	2	1	no	—
171	Q₈⋊C₈	—	2	1	no	—
172	#322	—	2	1	no	—
173	C₈²	—	2	1	no	—
174	C₂×C₃₁	—	2	1	no	—
175	C₂×C₃₁	—	2	1	no	—
176	C₂×C₃₁	#1929	1	1	yes	characteristic
177	C₂×C₃₁	#1932	1	1	yes	characteristic
178	C₂×C₃₁	#1933	1	1	yes	characteristic
179	C₂×M₁₆	—	2	1	no	—
180	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
181	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
182	C₂²×C₈	—	2	1	no	—
183	C₂²×C₈	—	2	1	no	—
184	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
185	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
186	#125	—	2	1	no	—
187	C₂²⋊Q₈	—	2	1	no	—
188	C₂²⋊Q₈	—	2	1	no	—
189	C₄×Q₈	—	2	1	no	—
190	C₄×Q₈	—	2	1	no	—
191	#117	—	2	1	no	—
192	C₂×C₄⋊C₄	—	2	1	no	—
193	C₂×C₄²	—	2	1	no	—
194	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
195	#108	—	4	2	no	—
196	C₄⋊C₈	—	2	1	no	—
197	C₄⋊C₈	—	2	1	no	—
198	C₄⋊C₈	—	2	1	no	—
199	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
200	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
201	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
202	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
203	C₂²⋊C₈	—	2	1	no	—
204	#97	—	2	1	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	2	1	no	—
208	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
209	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
210	#95	—	2	1	no	—
211	C₃₁	#9589	1	1	yes	characteristic, direct product factor, Sylow
212	C₂×Q₈	—	2	1	no	—
213	C₂×Q₈	—	2	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
216	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
217	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
218	M₁₆	—	2	1	no	—
219	M₁₆	—	2	1	no	—
220	M₁₆	—	4	1	no	—
221	M₁₆	—	4	1	no	—
222	M₁₆	SD₁₆×C₃₁	1	2	yes	—
223	M₁₆	Q₁₆×C₃₁	1	2	yes	—
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	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	4	1	no	—
234	C₂×C₈	—	4	1	no	—
235	C₂×C₈	—	4	1	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	—	4	1	no	—
242	C₄⋊C₄	—	4	1	no	—
243	C₄⋊C₄	—	4	1	no	—
244	C₂²⋊C₄	—	4	1	no	—
245	C₄²	—	4	1	no	—
246	C₄²	—	4	1	no	—
247	C₄²	—	4	1	no	—
248	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
249	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
250	C₄²	#41×C₃₁	1	1	yes	characteristic
251	C₄²	#41×C₃₁	1	1	yes	characteristic
252	C₂³	C₃₁×#124	1	1	yes	characteristic
253	Q₈	—	4	1	no	—
254	Q₈	—	4	1	no	—
255	Q₈	—	8	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	8	1	no	—
263	C₂×C₄	—	8	1	no	—
264	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
265	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
266	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
267	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
268	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
269	C₂×C₄	C₃₁×#126	1	1	yes	characteristic
270	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
271	C₂×C₄	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
272	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
273	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
274	C₈	—	2	2	no	—
275	C₈	—	2	2	no	—
276	C₈	—	2	2	no	—
277	C₈	—	4	1	no	—
278	C₈	—	4	1	no	—
279	C₈	—	4	1	no	—
280	C₈	—	4	1	no	—
281	C₈	—	4	1	no	—
282	C₈	—	8	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	C₃₁×#388	1	1	yes	characteristic
286	C₂²	C₃₁×#487	1	1	yes	characteristic
287	C₂²	C₃₁×#488	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	8	2	no	—
292	C₄	—	8	1	no	—
293	C₄	—	8	1	no	—
294	C₄	C₄×SD₁₆×C₃₁	1	1	yes	characteristic
295	C₄	C₄×Q₁₆×C₃₁	1	1	yes	characteristic
296	C₄	C₃₁×#443	1	1	yes	characteristic
297	C₄	C₃₁×#444	1	1	yes	characteristic
298	C₄	C₃₁×#465	1	1	yes	characteristic
299	C₄	C₃₁×#467	1	1	yes	characteristic
300	C₂	—	2	1	no	—
301	C₂	—	2	1	no	—
302	C₂	C₃₁×#1929	1	1	yes	characteristic
303	C₂	C₃₁×#1932	1	1	yes	characteristic
304	C₂	C₃₁×#1933	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#121
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	#388
prime radical	C₂³×C₃₁	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
19. C₃₁×#333	C₄	semidirect
20. C₃₁×#333	C₄	semidirect
21. C₃₁×#332	C₄	semidirect
22. C₃₁×#332	C₄	semidirect
99. #9589	C₃₁	direct (N × H)
166. #333	C₄×C₃₁	semidirect
167. #333	C₄×C₃₁	semidirect
168. #332	C₄×C₃₁	semidirect
169. #332	C₄×C₃₁	semidirect
211. C₃₁	#9589	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×#1986	2
C₃₁×#1950	2
C₃₁×#1542	2
C₃₁×#1493	2
C₈×M₁₆×C₃₁	2
C₃₁×#1303	2
C₃₁×#1279	2
#9589	31

Subgroups — C31×#9589 #21466348

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9589 #21466348