Subgroups — C₃₁×#8239 #21464998

Order 7936.

510 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2929	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1752	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1501	C₂	1	2	yes	maximal
4	C₃₁×#1298	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1297	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×#131	C₄	1	1	yes	characteristic, complemented
7	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₃₁×#525	C₂²	1	1	yes	characteristic
9	C₃₁×#435	—	2	2	no	—
10	C₈×D₄×C₃₁	—	2	2	no	—
11	C₃₁×#424	C₄	1	1	yes	characteristic, complemented
12	C₃₁×#408	C₄	1	1	yes	characteristic, complemented
13	C₃₁×#407	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
15	C₃₁×#405	C₄	1	1	yes	characteristic, complemented
16	C₂×C₃₁×#97	—	2	2	no	—
17	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
18	C₃₁×#336	—	2	2	no	—
19	C₃₁×#328	C₄	1	4	yes	complemented
20	C₃₁×#327	C₄	1	4	yes	complemented
21	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
23	C₃₁×#131	—	4	2	no	—
24	C₂×M₁₆×C₃₁	—	2	1	no	—
25	C₂×M₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
26	C₂×M₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₃₁	—	2	2	no	—
28	C₂²×C₈×C₃₁	—	2	1	no	—
29	C₂²×C₈×C₃₁	—	4	2	no	—
30	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₃₁	D₄	1	2	yes	—
33	C₂²×C₈×C₃₁	D₄	1	2	yes	—
34	C₃₁×#123	—	2	1	no	—
35	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
36	C₃₁×#121	C₂×C₄	1	2	yes	—
37	C₃₁×#120	—	2	1	no	—
38	C₄×D₄×C₃₁	—	2	2	no	—
39	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
40	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
41	C₂×C₄²×C₃₁	—	2	2	no	—
42	C₃₁×C₄⋊C₈	—	2	2	no	—
43	C₃₁×C₂²⋊C₈	—	2	2	no	—
44	C₃₁×C₂²⋊C₈	—	2	2	no	—
45	C₃₁×C₂²⋊C₈	—	2	2	no	—
46	C₃₁×C₂²⋊C₈	C₂×C₄	1	4	yes	—
47	C₃₁×#97	—	2	1	no	—
48	C₃₁×#97	—	4	4	no	—
49	C₄×C₈×C₃₁	—	2	1	no	—
50	C₃₁×#95	C₂×C₄	1	4	yes	—
51	C₂⁴×C₃₁	#41	1	1	yes	characteristic
52	#41×C₃₁	—	4	1	no	—
53	#41×C₃₁	—	4	1	no	—
54	C₂×Q₈×C₃₁	C₄²	1	1	yes	characteristic
55	C₂×D₄×C₃₁	—	2	1	no	—
56	C₂×D₄×C₃₁	—	2	2	no	—
57	C₂×D₄×C₃₁	C₄²	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	—	2	1	no	—
59	C₂²×C₄×C₃₁	—	2	4	no	—
60	C₂²×C₄×C₃₁	—	4	1	no	—
61	C₂²×C₄×C₃₁	—	4	1	no	—
62	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
66	M₁₆×C₃₁	—	4	2	no	—
67	M₁₆×C₃₁	—	4	1	no	—
68	M₁₆×C₃₁	—	4	1	no	—
69	C₂×C₈×C₃₁	—	2	4	no	—
70	C₂×C₈×C₃₁	—	2	2	no	—
71	C₂×C₈×C₃₁	—	2	1	no	—
72	C₂×C₈×C₃₁	—	2	1	no	—
73	C₂×C₈×C₃₁	—	2	4	no	—
74	C₂×C₈×C₃₁	—	4	1	no	—
75	C₂×C₈×C₃₁	—	4	2	no	—
76	C₂×C₈×C₃₁	—	4	2	no	—
77	C₂×C₈×C₃₁	—	4	2	no	—
78	C₂×C₈×C₃₁	—	4	2	no	—
79	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
81	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
82	C₄⋊C₄×C₃₁	—	2	2	no	—
83	C₄⋊C₄×C₃₁	C₄²	1	2	yes	—
84	C₂²⋊C₄×C₃₁	—	2	1	no	—
85	C₂²⋊C₄×C₃₁	—	2	2	no	—
86	C₂²⋊C₄×C₃₁	—	2	2	no	—
87	C₂²⋊C₄×C₃₁	—	4	4	no	—
88	C₄²×C₃₁	—	2	1	no	—
89	C₄²×C₃₁	—	4	2	no	—
90	C₄²×C₃₁	—	4	2	no	—
91	#8239	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂³×C₃₁	—	2	4	no	—
93	C₂³×C₃₁	—	2	1	no	—
94	C₂³×C₃₁	—	4	1	no	—
95	C₂³×C₃₁	#117	1	1	yes	characteristic
96	C₂³×C₃₁	C₄×D₄	1	2	yes	—
97	Q₈×C₃₁	—	4	1	no	—
98	D₄×C₃₁	—	4	1	no	—
99	D₄×C₃₁	—	4	2	no	—
100	D₄×C₃₁	—	4	4	no	—
101	C₂×C₄×C₃₁	—	2	2	no	—
102	C₂×C₄×C₃₁	—	2	1	no	—
103	C₂×C₄×C₃₁	—	2	2	no	—
104	C₂×C₄×C₃₁	—	2	1	no	—
105	C₂×C₄×C₃₁	—	2	2	no	—
106	C₂×C₄×C₃₁	—	2	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	—	4	1	no	—
111	C₂×C₄×C₃₁	—	4	4	no	—
112	C₂×C₄×C₃₁	—	4	4	no	—
113	C₂×C₄×C₃₁	—	4	4	no	—
114	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
115	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
117	C₈×C₃₁	—	2	2	no	—
118	C₈×C₃₁	—	2	1	no	—
119	C₈×C₃₁	—	4	4	no	—
120	C₈×C₃₁	—	4	1	no	—
121	C₈×C₃₁	—	4	1	no	—
122	C₈×C₃₁	#99	1	2	yes	—
123	#2929	C₂×C₃₁	1	1	yes	characteristic
124	#1752	C₂×C₃₁	1	2	yes	complemented
125	#1501	C₂×C₃₁	1	2	yes	—
126	#1298	C₂×C₃₁	1	1	yes	characteristic, complemented
127	#1297	C₂×C₃₁	1	1	yes	characteristic, complemented
128	C₂²×C₃₁	—	2	2	no	—
129	C₂²×C₃₁	—	2	2	no	—
130	C₂²×C₃₁	—	2	1	no	—
131	C₂²×C₃₁	—	2	2	no	—
132	C₂²×C₃₁	—	4	4	no	—
133	C₂²×C₃₁	—	4	1	no	—
134	C₂²×C₃₁	—	4	2	no	—
135	C₂²×C₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
136	C₂²×C₃₁	#444	1	4	yes	—
137	C₄×C₃₁	—	2	2	no	—
138	C₄×C₃₁	—	2	1	no	—
139	C₄×C₃₁	—	4	1	no	—
140	C₄×C₃₁	—	4	2	no	—
141	C₄×C₃₁	—	8	4	no	—
142	C₄×C₃₁	C₂×#99	1	1	yes	characteristic
143	C₄×C₃₁	#410	1	1	yes	characteristic
144	C₂×#131	C₄×C₃₁	1	1	yes	characteristic, complemented
145	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
146	#525	C₂²×C₃₁	1	1	yes	characteristic
147	#435	—	2	2	no	—
148	C₈×D₄	—	2	2	no	—
149	#424	C₄×C₃₁	1	1	yes	characteristic, complemented
150	#408	C₄×C₃₁	1	1	yes	characteristic, complemented
151	#407	C₂²×C₃₁	1	1	yes	characteristic
152	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
153	#405	C₄×C₃₁	1	1	yes	characteristic, complemented
154	C₂×#97	—	2	2	no	—
155	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
156	#336	—	2	2	no	—
157	#328	C₄×C₃₁	1	4	yes	complemented
158	#327	C₄×C₃₁	1	4	yes	complemented
159	C₂×C₃₁	—	2	4	no	—
160	C₂×C₃₁	—	4	1	no	—
161	C₂×C₃₁	—	8	1	no	—
162	C₂×C₃₁	C₄×#99	1	1	yes	characteristic
163	C₂×C₃₁	#1765	1	2	yes	—
164	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
165	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
166	#131	—	4	2	no	—
167	C₂×M₁₆	—	2	1	no	—
168	C₂×M₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
169	C₂×M₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	2	1	no	—
172	C₂²×C₈	—	4	2	no	—
173	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
174	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
175	C₂²×C₈	D₄×C₃₁	1	2	yes	—
176	C₂²×C₈	D₄×C₃₁	1	2	yes	—
177	#123	—	2	1	no	—
178	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
179	#121	C₂×C₄×C₃₁	1	2	yes	—
180	#120	—	2	1	no	—
181	C₄×D₄	—	2	2	no	—
182	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
183	C₂×C₂²⋊C₄	—	2	2	no	—
184	C₂×C₄²	—	2	2	no	—
185	C₄⋊C₈	—	2	2	no	—
186	C₂²⋊C₈	—	2	2	no	—
187	C₂²⋊C₈	—	2	2	no	—
188	C₂²⋊C₈	—	2	2	no	—
189	C₂²⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
190	#97	—	2	1	no	—
191	#97	—	4	4	no	—
192	C₄×C₈	—	2	1	no	—
193	#95	C₂×C₄×C₃₁	1	4	yes	—
194	C₃₁	#8239	1	1	yes	characteristic, direct product factor, Sylow
195	C₂⁴	#41×C₃₁	1	1	yes	characteristic
196	#41	—	4	1	no	—
197	#41	—	4	1	no	—
198	C₂×Q₈	C₄²×C₃₁	1	1	yes	characteristic
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	C₄²×C₃₁	1	1	yes	characteristic
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	4	1	no	—
205	C₂²×C₄	—	4	1	no	—
206	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
207	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
208	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
209	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
210	M₁₆	—	4	2	no	—
211	M₁₆	—	4	1	no	—
212	M₁₆	—	4	1	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	4	2	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
224	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
225	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	C₄²×C₃₁	1	2	yes	—
228	C₂²⋊C₄	—	2	1	no	—
229	C₂²⋊C₄	—	2	2	no	—
230	C₂²⋊C₄	—	2	2	no	—
231	C₂²⋊C₄	—	4	4	no	—
232	C₄²	—	2	1	no	—
233	C₄²	—	4	2	no	—
234	C₄²	—	4	2	no	—
235	C₂³	—	2	4	no	—
236	C₂³	—	2	1	no	—
237	C₂³	—	4	1	no	—
238	C₂³	C₃₁×#117	1	1	yes	characteristic
239	C₂³	C₄×D₄×C₃₁	1	2	yes	—
240	Q₈	—	4	1	no	—
241	D₄	—	4	1	no	—
242	D₄	—	4	2	no	—
243	D₄	—	4	4	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
258	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
259	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
260	C₈	—	2	2	no	—
261	C₈	—	2	1	no	—
262	C₈	—	4	4	no	—
263	C₈	—	4	1	no	—
264	C₈	—	4	1	no	—
265	C₈	C₃₁×#99	1	2	yes	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	4	4	no	—
271	C₂²	—	4	1	no	—
272	C₂²	—	4	2	no	—
273	C₂²	C₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
274	C₂²	C₃₁×#444	1	4	yes	—
275	C₄	—	2	2	no	—
276	C₄	—	2	1	no	—
277	C₄	—	4	1	no	—
278	C₄	—	4	2	no	—
279	C₄	—	8	4	no	—
280	C₄	C₂×C₃₁×#99	1	1	yes	characteristic
281	C₄	C₃₁×#410	1	1	yes	characteristic
282	C₂	—	2	4	no	—
283	C₂	—	4	1	no	—
284	C₂	—	8	1	no	—
285	C₂	C₄×C₃₁×#99	1	1	yes	characteristic
286	C₂	C₃₁×#1765	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
socle	C₂²×C₃₁	C₄×C₂²⋊C₄
prime radical	C₃₁×#525	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1752	C₂	semidirect
4. C₃₁×#1298	C₂	semidirect
5. C₃₁×#1297	C₂	semidirect
6. C₂×C₃₁×#131	C₄	semidirect
11. C₃₁×#424	C₄	semidirect
12. C₃₁×#408	C₄	semidirect
15. C₃₁×#405	C₄	semidirect
19. C₃₁×#328	C₄	semidirect
20. C₃₁×#327	C₄	semidirect
91. #8239	C₃₁	direct (N × H)
124. #1752	C₂×C₃₁	semidirect
126. #1298	C₂×C₃₁	semidirect
127. #1297	C₂×C₃₁	semidirect
144. C₂×#131	C₄×C₃₁	semidirect
149. #424	C₄×C₃₁	semidirect
150. #408	C₄×C₃₁	semidirect
153. #405	C₄×C₃₁	semidirect
157. #328	C₄×C₃₁	semidirect
158. #327	C₄×C₃₁	semidirect
194. C₃₁	#8239	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×#2929	2
C₃₁×#1752	2
C₃₁×#1501	2
C₃₁×#1298	2
C₃₁×#1297	2
#8239	31

Subgroups — C31×#8239 #21464998

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8239 #21464998