Subgroups — C₃₁×#8710 #21465469

Order 7936.

540 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×#2285	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1797	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₁×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1279	C₂	1	4	yes	maximal, complemented
5	C₂×C₃₁×#117	—	2	1	no	—
6	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₃₁×C₂²⋊C₈	—	2	1	no	—
9	C₂×C₃₁×C₂²⋊C₈	—	2	1	no	—
10	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
11	C₃₁×#384	C₄	1	1	yes	characteristic
12	C₃₁×#384	C₄	1	1	yes	characteristic
13	C₃₁×#382	—	2	1	no	—
14	C₃₁×#380	—	2	1	no	—
15	C₃₁×#380	C₄	1	2	yes	—
16	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
17	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
18	C₃₁×#336	—	2	1	no	—
19	C₃₁×#336	—	2	1	no	—
20	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
22	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₃₁	—	2	2	no	—
24	C₂²×C₈×C₃₁	—	2	2	no	—
25	C₃₁×#117	—	4	4	no	—
26	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
27	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
28	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
29	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
30	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
32	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
33	C₂×C₄²×C₃₁	—	2	4	no	—
34	C₂×C₄²×C₃₁	—	2	2	no	—
35	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
39	C₃₁×C₄⋊C₈	—	2	4	no	—
40	C₃₁×C₂²⋊C₈	—	2	4	no	—
41	C₃₁×C₂²⋊C₈	—	4	2	no	—
42	C₄×C₈×C₃₁	—	2	4	no	—
43	C₃₁×#95	—	2	2	no	—
44	C₃₁×#95	—	2	2	no	—
45	C₂⁴×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₃₁	—	2	2	no	—
47	C₂²×C₄×C₃₁	—	2	2	no	—
48	C₂²×C₄×C₃₁	—	2	2	no	—
49	C₂²×C₄×C₃₁	—	2	2	no	—
50	C₂²×C₄×C₃₁	—	2	2	no	—
51	C₂²×C₄×C₃₁	—	2	2	no	—
52	C₂²×C₄×C₃₁	—	2	2	no	—
53	C₂²×C₄×C₃₁	—	2	2	no	—
54	C₂²×C₄×C₃₁	—	2	2	no	—
55	C₂²×C₄×C₃₁	—	4	1	no	—
56	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	SD₁₆	1	2	yes	—
60	C₂²×C₄×C₃₁	Q₁₆	1	2	yes	—
61	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₃₁	—	2	4	no	—
65	C₂×C₈×C₃₁	—	4	2	no	—
66	C₂×C₈×C₃₁	—	4	2	no	—
67	C₂×C₈×C₃₁	—	4	2	no	—
68	C₂×C₈×C₃₁	—	4	2	no	—
69	C₂×C₈×C₃₁	—	4	2	no	—
70	C₄⋊C₄×C₃₁	—	2	4	no	—
71	C₄⋊C₄×C₃₁	—	2	4	no	—
72	C₄⋊C₄×C₃₁	—	4	2	no	—
73	C₄⋊C₄×C₃₁	—	4	2	no	—
74	C₄⋊C₄×C₃₁	—	4	2	no	—
75	C₄⋊C₄×C₃₁	—	4	2	no	—
76	C₂²⋊C₄×C₃₁	—	4	4	no	—
77	C₄²×C₃₁	—	2	2	no	—
78	C₄²×C₃₁	—	2	2	no	—
79	C₄²×C₃₁	—	2	2	no	—
80	C₄²×C₃₁	—	4	4	no	—
81	C₄²×C₃₁	#41	1	4	yes	—
82	#8710	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂³×C₃₁	—	2	1	no	—
84	C₂³×C₃₁	—	2	1	no	—
85	C₂³×C₃₁	—	2	1	no	—
86	C₂³×C₃₁	—	2	1	no	—
87	C₂³×C₃₁	—	2	1	no	—
88	C₂³×C₃₁	—	2	1	no	—
89	C₂³×C₃₁	#103	1	2	yes	—
90	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₃₁	—	2	1	no	—
92	C₂×C₄×C₃₁	—	2	1	no	—
93	C₂×C₄×C₃₁	—	2	1	no	—
94	C₂×C₄×C₃₁	—	2	1	no	—
95	C₂×C₄×C₃₁	—	2	1	no	—
96	C₂×C₄×C₃₁	—	2	1	no	—
97	C₂×C₄×C₃₁	—	2	4	no	—
98	C₂×C₄×C₃₁	—	2	4	no	—
99	C₂×C₄×C₃₁	—	2	1	no	—
100	C₂×C₄×C₃₁	—	2	1	no	—
101	C₂×C₄×C₃₁	—	2	1	no	—
102	C₂×C₄×C₃₁	—	4	4	no	—
103	C₂×C₄×C₃₁	—	4	4	no	—
104	C₂×C₄×C₃₁	—	4	2	no	—
105	C₂×C₄×C₃₁	—	4	2	no	—
106	C₂×C₄×C₃₁	—	4	2	no	—
107	C₂×C₄×C₃₁	—	4	2	no	—
108	C₂×C₄×C₃₁	—	4	2	no	—
109	C₂×C₄×C₃₁	—	4	2	no	—
110	C₂×C₄×C₃₁	#103	1	2	yes	—
111	C₂×C₄×C₃₁	#104	1	2	yes	—
112	C₂×C₄×C₃₁	#117	1	2	yes	—
113	C₂×C₄×C₃₁	#123	1	2	yes	—
114	C₂×C₄×C₃₁	#123	1	2	yes	—
115	C₂×C₄×C₃₁	#131	1	2	yes	—
116	C₂×C₄×C₃₁	#131	1	2	yes	—
117	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
118	C₂×C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
119	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
120	C₈×C₃₁	—	4	4	no	—
121	C₈×C₃₁	—	8	2	no	—
122	#2285	C₂×C₃₁	1	1	yes	characteristic
123	#1797	C₂×C₃₁	1	1	yes	characteristic
124	C₄×C₂²⋊C₈	C₂×C₃₁	1	1	yes	characteristic
125	#1279	C₂×C₃₁	1	4	yes	complemented
126	C₂²×C₃₁	—	2	2	no	—
127	C₂²×C₃₁	—	2	2	no	—
128	C₂²×C₃₁	—	2	2	no	—
129	C₂²×C₃₁	—	4	1	no	—
130	C₂²×C₃₁	—	4	1	no	—
131	C₂²×C₃₁	—	4	1	no	—
132	C₂²×C₃₁	—	4	1	no	—
133	C₂²×C₃₁	#381	1	1	yes	characteristic
134	C₂²×C₃₁	#408	1	1	yes	characteristic
135	C₂²×C₃₁	#408	1	1	yes	characteristic
136	C₂²×C₃₁	C₂×#103	1	1	yes	characteristic
137	C₂²×C₃₁	#418	1	1	yes	characteristic
138	C₂²×C₃₁	C₂×#104	1	1	yes	characteristic
139	C₂²×C₃₁	#421	1	1	yes	characteristic
140	C₄×C₃₁	—	2	2	no	—
141	C₄×C₃₁	—	2	2	no	—
142	C₄×C₃₁	—	2	2	no	—
143	C₄×C₃₁	—	4	1	no	—
144	C₄×C₃₁	—	8	2	no	—
145	C₄×C₃₁	—	8	2	no	—
146	C₄×C₃₁	#433	1	4	yes	—
147	C₄×C₃₁	#481	1	2	yes	—
148	C₄×C₃₁	#484	1	2	yes	—
149	C₂×#117	—	2	1	no	—
150	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
151	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
152	C₂×C₂²⋊C₈	—	2	1	no	—
153	C₂×C₂²⋊C₈	—	2	1	no	—
154	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
155	#384	C₄×C₃₁	1	1	yes	characteristic
156	#384	C₄×C₃₁	1	1	yes	characteristic
157	#382	—	2	1	no	—
158	#380	—	2	1	no	—
159	#380	C₄×C₃₁	1	2	yes	—
160	C₄×C₄⋊C₄	—	2	1	no	—
161	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
162	#336	—	2	1	no	—
163	#336	—	2	1	no	—
164	C₂×C₃₁	—	4	2	no	—
165	C₂×C₃₁	#1481	1	1	yes	characteristic
166	C₂×C₃₁	#1484	1	1	yes	characteristic
167	C₂×C₃₁	#1493	1	1	yes	characteristic
168	C₂×C₃₁	#1496	1	1	yes	characteristic
169	C₂×C₃₁	#1818	1	1	yes	characteristic
170	C₂×C₃₁	#1824	1	1	yes	characteristic
171	C₂×C₃₁	#1826	1	1	yes	characteristic
172	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
173	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
174	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	—	2	2	no	—
177	#117	—	4	4	no	—
178	C₂×C₄⋊C₄	—	2	1	no	—
179	C₂×C₄⋊C₄	—	2	1	no	—
180	C₂×C₄⋊C₄	—	2	1	no	—
181	C₂×C₄⋊C₄	—	2	1	no	—
182	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
183	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
184	C₂×C₂²⋊C₄	—	2	2	no	—
185	C₂×C₄²	—	2	4	no	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
188	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
189	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
190	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
191	C₄⋊C₈	—	2	4	no	—
192	C₂²⋊C₈	—	2	4	no	—
193	C₂²⋊C₈	—	4	2	no	—
194	C₄×C₈	—	2	4	no	—
195	#95	—	2	2	no	—
196	#95	—	2	2	no	—
197	C₃₁	#8710	1	1	yes	characteristic, direct product factor, Sylow
198	C₂⁴	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	4	1	no	—
209	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
210	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
211	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
212	C₂²×C₄	SD₁₆×C₃₁	1	2	yes	—
213	C₂²×C₄	Q₁₆×C₃₁	1	2	yes	—
214	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	4	2	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	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	4	2	no	—
226	C₄⋊C₄	—	4	2	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄⋊C₄	—	4	2	no	—
229	C₂²⋊C₄	—	4	4	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	4	4	no	—
234	C₄²	#41×C₃₁	1	4	yes	—
235	C₂³	—	2	1	no	—
236	C₂³	—	2	1	no	—
237	C₂³	—	2	1	no	—
238	C₂³	—	2	1	no	—
239	C₂³	—	2	1	no	—
240	C₂³	—	2	1	no	—
241	C₂³	C₃₁×#103	1	2	yes	—
242	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	C₃₁×#103	1	2	yes	—
263	C₂×C₄	C₃₁×#104	1	2	yes	—
264	C₂×C₄	C₃₁×#117	1	2	yes	—
265	C₂×C₄	C₃₁×#123	1	2	yes	—
266	C₂×C₄	C₃₁×#123	1	2	yes	—
267	C₂×C₄	C₃₁×#131	1	2	yes	—
268	C₂×C₄	C₃₁×#131	1	2	yes	—
269	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
270	C₂×C₄	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
271	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
272	C₈	—	4	4	no	—
273	C₈	—	8	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	4	1	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	1	no	—
281	C₂²	C₃₁×#381	1	1	yes	characteristic
282	C₂²	C₃₁×#408	1	1	yes	characteristic
283	C₂²	C₃₁×#408	1	1	yes	characteristic
284	C₂²	C₂×C₃₁×#103	1	1	yes	characteristic
285	C₂²	C₃₁×#418	1	1	yes	characteristic
286	C₂²	C₂×C₃₁×#104	1	1	yes	characteristic
287	C₂²	C₃₁×#421	1	1	yes	characteristic
288	C₄	—	2	2	no	—
289	C₄	—	2	2	no	—
290	C₄	—	2	2	no	—
291	C₄	—	4	1	no	—
292	C₄	—	8	2	no	—
293	C₄	—	8	2	no	—
294	C₄	C₃₁×#433	1	4	yes	—
295	C₄	C₃₁×#481	1	2	yes	—
296	C₄	C₃₁×#484	1	2	yes	—
297	C₂	—	4	2	no	—
298	C₂	C₃₁×#1481	1	1	yes	characteristic
299	C₂	C₃₁×#1484	1	1	yes	characteristic
300	C₂	C₃₁×#1493	1	1	yes	characteristic
301	C₂	C₃₁×#1496	1	1	yes	characteristic
302	C₂	C₃₁×#1818	1	1	yes	characteristic
303	C₂	C₃₁×#1824	1	1	yes	characteristic
304	C₂	C₃₁×#1826	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂³×C₃₁	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₃₁	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1279	C₂	semidirect
82. #8710	C₃₁	direct (N × H)
125. #1279	C₂×C₃₁	semidirect
197. C₃₁	#8710	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×#2285	2
C₃₁×#1797	2
C₄×C₃₁×C₂²⋊C₈	2
C₃₁×#1279	2
#8710	31

Subgroups — C31×#8710 #21465469

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8710 #21465469