Subgroups — C₃₁×#8245 #21465004

Order 7936.

444 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2931	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1543	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1525	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1451	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1299	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1299	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#131	C₄	1	2	yes	—
9	C₃₁×#532	C₂²	1	1	yes	characteristic
10	C₃₁×#436	—	2	1	no	—
11	C₃₁×#436	—	2	1	no	—
12	C₈×D₄×C₃₁	—	2	1	no	—
13	C₈×D₄×C₃₁	—	2	1	no	—
14	C₃₁×#431	C₂²	1	1	yes	characteristic
15	C₃₁×#431	C₂²	1	1	yes	characteristic
16	C₃₁×#423	C₂²	1	1	yes	characteristic
17	C₃₁×#408	C₄	1	2	yes	—
18	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
19	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#333	C₄	1	2	yes	—
22	C₃₁×#331	C₄	1	2	yes	—
23	C₃₁×#329	C₄	1	4	yes	—
24	C₃₁×#322	—	2	1	no	—
25	C₃₁×#322	—	2	1	no	—
26	C₃₁×#322	—	2	1	no	—
27	C₈²×C₃₁	—	2	1	no	—
28	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
29	C₃₁×#131	—	4	4	no	—
30	C₂×M₁₆×C₃₁	—	2	1	no	—
31	C₂×M₁₆×C₃₁	—	2	1	no	—
32	C₂×M₁₆×C₃₁	—	2	2	no	—
33	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₃₁	—	2	2	no	—
35	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
37	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
38	C₃₁×#127	C₂×C₄	1	1	yes	characteristic
39	C₃₁×#124	C₂×C₄	1	2	yes	—
40	C₃₁×#121	—	2	1	no	—
41	C₃₁×#121	—	2	1	no	—
42	C₄×D₄×C₃₁	—	2	1	no	—
43	C₄×D₄×C₃₁	—	2	1	no	—
44	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
45	C₃₁×C₄⋊C₈	—	2	1	no	—
46	C₃₁×C₄⋊C₈	—	2	1	no	—
47	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
48	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
49	C₃₁×C₂²⋊C₈	—	2	1	no	—
50	C₃₁×C₂²⋊C₈	—	2	1	no	—
51	C₃₁×C₂²⋊C₈	—	2	2	no	—
52	C₃₁×C₂²⋊C₈	—	2	2	no	—
53	C₃₁×#97	C₂×C₄	1	2	yes	—
54	C₃₁×#97	C₂×C₄	1	2	yes	—
55	C₄×C₈×C₃₁	—	2	1	no	—
56	C₄×C₈×C₃₁	—	2	1	no	—
57	C₄×C₈×C₃₁	—	2	1	no	—
58	C₄×C₈×C₃₁	—	2	1	no	—
59	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
60	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
61	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
62	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
63	#41×C₃₁	—	4	2	no	—
64	#41×C₃₁	—	4	2	no	—
65	C₂×Q₈×C₃₁	C₄²	1	2	yes	—
66	C₂×D₄×C₃₁	—	2	2	no	—
67	C₂×D₄×C₃₁	—	2	1	no	—
68	C₂×D₄×C₃₁	—	2	1	no	—
69	C₂×D₄×C₃₁	C₄²	1	2	yes	—
70	C₂²×C₄×C₃₁	—	2	2	no	—
71	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
73	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
74	M₁₆×C₃₁	—	2	4	no	—
75	M₁₆×C₃₁	—	4	2	no	—
76	M₁₆×C₃₁	—	4	4	no	—
77	M₁₆×C₃₁	—	4	2	no	—
78	C₂×C₈×C₃₁	—	2	2	no	—
79	C₂×C₈×C₃₁	—	2	2	no	—
80	C₂×C₈×C₃₁	—	2	2	no	—
81	C₂×C₈×C₃₁	—	2	2	no	—
82	C₂×C₈×C₃₁	—	2	2	no	—
83	C₂×C₈×C₃₁	—	2	2	no	—
84	C₂×C₈×C₃₁	—	4	1	no	—
85	C₂×C₈×C₃₁	—	4	4	no	—
86	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
90	C₄⋊C₄×C₃₁	—	2	1	no	—
91	C₄⋊C₄×C₃₁	—	2	1	no	—
92	C₂²⋊C₄×C₃₁	—	2	2	no	—
93	C₂²⋊C₄×C₃₁	—	2	2	no	—
94	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
98	#8245	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂³×C₃₁	—	2	2	no	—
100	C₂³×C₃₁	#117	1	1	yes	characteristic
101	Q₈×C₃₁	—	4	2	no	—
102	D₄×C₃₁	—	4	2	no	—
103	D₄×C₃₁	—	4	4	no	—
104	D₄×C₃₁	—	4	2	no	—
105	D₄×C₃₁	—	4	2	no	—
106	C₂×C₄×C₃₁	—	2	1	no	—
107	C₂×C₄×C₃₁	—	2	1	no	—
108	C₂×C₄×C₃₁	—	2	2	no	—
109	C₂×C₄×C₃₁	—	2	2	no	—
110	C₂×C₄×C₃₁	—	2	2	no	—
111	C₂×C₄×C₃₁	—	4	2	no	—
112	C₂×C₄×C₃₁	—	4	2	no	—
113	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
119	C₈×C₃₁	—	2	4	no	—
120	C₈×C₃₁	—	2	1	no	—
121	C₈×C₃₁	—	2	1	no	—
122	C₈×C₃₁	—	2	1	no	—
123	C₈×C₃₁	—	4	2	no	—
124	C₈×C₃₁	—	4	2	no	—
125	C₈×C₃₁	—	4	2	no	—
126	C₈×C₃₁	—	4	2	no	—
127	C₈×C₃₁	#100	1	2	yes	—
128	#2931	C₂×C₃₁	1	1	yes	characteristic
129	#1543	C₂×C₃₁	1	1	yes	characteristic
130	#1525	C₂×C₃₁	1	1	yes	characteristic
131	#1451	C₂×C₃₁	1	1	yes	characteristic, complemented
132	C₈×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
133	#1299	C₂×C₃₁	1	1	yes	characteristic, complemented
134	#1299	C₂×C₃₁	1	1	yes	characteristic, complemented
135	C₂²×C₃₁	—	2	1	no	—
136	C₂²×C₃₁	—	2	1	no	—
137	C₂²×C₃₁	—	2	1	no	—
138	C₂²×C₃₁	—	4	2	no	—
139	C₂²×C₃₁	—	4	2	no	—
140	C₂²×C₃₁	—	4	2	no	—
141	C₂²×C₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
142	C₄×C₃₁	—	2	1	no	—
143	C₄×C₃₁	—	2	1	no	—
144	C₄×C₃₁	—	2	1	no	—
145	C₄×C₃₁	—	4	2	no	—
146	C₄×C₃₁	C₂×#100	1	1	yes	characteristic
147	C₄×C₃₁	#413	1	1	yes	characteristic
148	C₄×C₃₁	#443	1	2	yes	—
149	C₄×C₃₁	#444	1	2	yes	—
150	C₂×#131	C₄×C₃₁	1	2	yes	—
151	#532	C₂²×C₃₁	1	1	yes	characteristic
152	#436	—	2	1	no	—
153	#436	—	2	1	no	—
154	C₈×D₄	—	2	1	no	—
155	C₈×D₄	—	2	1	no	—
156	#431	C₂²×C₃₁	1	1	yes	characteristic
157	#431	C₂²×C₃₁	1	1	yes	characteristic
158	#423	C₂²×C₃₁	1	1	yes	characteristic
159	#408	C₄×C₃₁	1	2	yes	—
160	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
161	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
162	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
163	#333	C₄×C₃₁	1	2	yes	—
164	#331	C₄×C₃₁	1	2	yes	—
165	#329	C₄×C₃₁	1	4	yes	—
166	#322	—	2	1	no	—
167	#322	—	2	1	no	—
168	#322	—	2	1	no	—
169	C₈²	—	2	1	no	—
170	C₂×C₃₁	—	4	1	no	—
171	C₂×C₃₁	—	8	2	no	—
172	C₂×C₃₁	C₄×#100	1	1	yes	characteristic
173	C₂×C₃₁	#1764	1	1	yes	characteristic
174	C₂×C₃₁	#1765	1	1	yes	characteristic
175	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
176	#131	—	4	4	no	—
177	C₂×M₁₆	—	2	1	no	—
178	C₂×M₁₆	—	2	1	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
183	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
184	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
185	#127	C₂×C₄×C₃₁	1	1	yes	characteristic
186	#124	C₂×C₄×C₃₁	1	2	yes	—
187	#121	—	2	1	no	—
188	#121	—	2	1	no	—
189	C₄×D₄	—	2	1	no	—
190	C₄×D₄	—	2	1	no	—
191	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
192	C₄⋊C₈	—	2	1	no	—
193	C₄⋊C₈	—	2	1	no	—
194	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
195	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
196	C₂²⋊C₈	—	2	1	no	—
197	C₂²⋊C₈	—	2	1	no	—
198	C₂²⋊C₈	—	2	2	no	—
199	C₂²⋊C₈	—	2	2	no	—
200	#97	C₂×C₄×C₃₁	1	2	yes	—
201	#97	C₂×C₄×C₃₁	1	2	yes	—
202	C₄×C₈	—	2	1	no	—
203	C₄×C₈	—	2	1	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
207	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
208	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
209	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
210	C₃₁	#8245	1	1	yes	characteristic, direct product factor, Sylow
211	#41	—	4	2	no	—
212	#41	—	4	2	no	—
213	C₂×Q₈	C₄²×C₃₁	1	2	yes	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	2	1	no	—
216	C₂×D₄	—	2	1	no	—
217	C₂×D₄	C₄²×C₃₁	1	2	yes	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
220	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
221	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
222	M₁₆	—	2	4	no	—
223	M₁₆	—	4	2	no	—
224	M₁₆	—	4	4	no	—
225	M₁₆	—	4	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	4	4	no	—
234	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
235	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
236	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
237	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	—	2	1	no	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₂²⋊C₄	—	2	2	no	—
242	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
243	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
244	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
245	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
246	C₂³	—	2	2	no	—
247	C₂³	C₃₁×#117	1	1	yes	characteristic
248	Q₈	—	4	2	no	—
249	D₄	—	4	2	no	—
250	D₄	—	4	4	no	—
251	D₄	—	4	2	no	—
252	D₄	—	4	2	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
261	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
262	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
263	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
264	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
265	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
266	C₈	—	2	4	no	—
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	2	1	no	—
270	C₈	—	4	2	no	—
271	C₈	—	4	2	no	—
272	C₈	—	4	2	no	—
273	C₈	—	4	2	no	—
274	C₈	C₃₁×#100	1	2	yes	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	C₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	4	2	no	—
286	C₄	C₂×C₃₁×#100	1	1	yes	characteristic
287	C₄	C₃₁×#413	1	1	yes	characteristic
288	C₄	C₃₁×#443	1	2	yes	—
289	C₄	C₃₁×#444	1	2	yes	—
290	C₂	—	4	1	no	—
291	C₂	—	8	2	no	—
292	C₂	C₄×C₃₁×#100	1	1	yes	characteristic
293	C₂	C₃₁×#1764	1	1	yes	characteristic
294	C₂	C₃₁×#1765	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	C₄×C₂²⋊C₄
prime radical	C₃₁×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1451	C₂	semidirect
5. C₈×M₁₆×C₃₁	C₂	semidirect
6. C₃₁×#1299	C₂	semidirect
7. C₃₁×#1299	C₂	semidirect
98. #8245	C₃₁	direct (N × H)
131. #1451	C₂×C₃₁	semidirect
132. C₈×M₁₆	C₂×C₃₁	semidirect
133. #1299	C₂×C₃₁	semidirect
134. #1299	C₂×C₃₁	semidirect
210. C₃₁	#8245	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×#2931	2
C₃₁×#1543	2
C₃₁×#1525	2
C₃₁×#1451	2
C₈×M₁₆×C₃₁	2
C₃₁×#1299	2
#8245	31

Subgroups — C31×#8245 #21465004

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8245 #21465004