Subgroups — C₃₁×#10818 #21467577

Order 7936.

734 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#341	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1735	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#339	C₂	1	2	yes	maximal
5	C₃₁×#1729	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
7	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#117	C₂²	1	2	yes	—
10	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₃₁	—	2	4	no	—
12	C₂×C₄×C₈×C₃₁	C₄	1	4	yes	complemented
13	C₃₁×#341	C₄	1	4	yes	complemented
14	C₃₁×#340	C₄	1	4	yes	complemented
15	C₃₁×#339	—	2	4	no	—
16	C₃₁×#336	—	2	4	no	—
17	C₂³×C₄×C₃₁	—	2	1	no	—
18	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
19	C₂×M₁₆×C₃₁	—	2	2	no	—
20	C₂×M₁₆×C₃₁	—	2	2	no	—
21	C₂×M₁₆×C₃₁	—	2	2	no	—
22	C₂×M₁₆×C₃₁	—	2	2	no	—
23	C₂×M₁₆×C₃₁	—	2	2	no	—
24	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₃₁	—	2	2	no	—
26	C₂²×C₈×C₃₁	—	2	2	no	—
27	C₂²×C₈×C₃₁	—	2	2	no	—
28	C₂²×C₈×C₃₁	—	2	2	no	—
29	C₂²×C₈×C₃₁	—	2	2	no	—
30	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₃₁	D₄	1	2	yes	—
35	C₂²×C₈×C₃₁	Q₈	1	2	yes	—
36	C₃₁×#117	—	2	2	no	—
37	C₃₁×#117	—	2	2	no	—
38	C₃₁×#117	—	2	2	no	—
39	C₃₁×#117	—	2	2	no	—
40	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
41	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
42	C₂×C₄²×C₃₁	—	2	2	no	—
43	C₂×C₄²×C₃₁	—	2	1	no	—
44	C₂×C₄²×C₃₁	—	2	2	no	—
45	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₃₁	D₄	1	4	yes	—
47	C₄×C₈×C₃₁	—	2	8	no	—
48	C₂⁴×C₃₁	#41	1	1	yes	characteristic
49	C₂²×C₄×C₃₁	—	2	8	no	—
50	C₂²×C₄×C₃₁	—	2	2	no	—
51	C₂²×C₄×C₃₁	—	2	4	no	—
52	C₂²×C₄×C₃₁	—	2	4	no	—
53	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	#41	1	2	yes	—
60	C₂²×C₄×C₃₁	#41	1	2	yes	—
61	C₂²×C₄×C₃₁	#41	1	2	yes	—
62	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
63	M₁₆×C₃₁	—	2	4	no	—
64	M₁₆×C₃₁	—	2	4	no	—
65	M₁₆×C₃₁	—	4	2	no	—
66	M₁₆×C₃₁	—	4	2	no	—
67	C₂×C₈×C₃₁	—	2	2	no	—
68	C₂×C₈×C₃₁	—	2	2	no	—
69	C₂×C₈×C₃₁	—	2	2	no	—
70	C₂×C₈×C₃₁	—	2	2	no	—
71	C₂×C₈×C₃₁	—	2	4	no	—
72	C₂×C₈×C₃₁	—	2	4	no	—
73	C₂×C₈×C₃₁	—	2	4	no	—
74	C₂×C₈×C₃₁	—	2	4	no	—
75	C₂×C₈×C₃₁	—	2	4	no	—
76	C₂×C₈×C₃₁	—	4	2	no	—
77	C₂×C₈×C₃₁	—	4	2	no	—
78	C₂×C₈×C₃₁	C₄²	1	4	yes	—
79	C₂×C₈×C₃₁	C₄⋊C₄	1	4	yes	—
80	C₄⋊C₄×C₃₁	—	2	4	no	—
81	C₄⋊C₄×C₃₁	—	2	4	no	—
82	C₂²⋊C₄×C₃₁	—	4	2	no	—
83	C₂²⋊C₄×C₃₁	—	4	2	no	—
84	C₄²×C₃₁	—	2	2	no	—
85	C₄²×C₃₁	—	2	4	no	—
86	C₄²×C₃₁	—	4	2	no	—
87	C₄²×C₃₁	—	4	2	no	—
88	C₄²×C₃₁	C₂²⋊C₄	1	4	yes	—
89	#10818	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₃₁	—	2	4	no	—
91	C₂³×C₃₁	#117	1	1	yes	characteristic
92	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
93	C₂³×C₃₁	C₄×Q₈	1	1	yes	characteristic
94	C₂³×C₃₁	#123	1	2	yes	—
95	C₂³×C₃₁	#126	1	2	yes	—
96	C₂×C₄×C₃₁	—	2	16	no	—
97	C₂×C₄×C₃₁	—	2	8	no	—
98	C₂×C₄×C₃₁	—	2	4	no	—
99	C₂×C₄×C₃₁	—	4	4	no	—
100	C₂×C₄×C₃₁	—	4	8	no	—
101	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
105	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
106	C₂×C₄×C₃₁	#117	1	2	yes	—
107	C₂×C₄×C₃₁	#117	1	2	yes	—
108	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
109	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₃₁	C₄×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₃₁	#121	1	2	yes	—
114	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₃₁	#123	1	2	yes	—
116	C₂×C₄×C₃₁	#124	1	2	yes	—
117	C₂×C₄×C₃₁	#125	1	2	yes	—
118	C₂×C₄×C₃₁	#126	1	2	yes	—
119	C₈×C₃₁	—	2	4	no	—
120	C₈×C₃₁	—	2	4	no	—
121	C₈×C₃₁	—	4	4	no	—
122	C₂²×C₄×C₈	C₂×C₃₁	1	1	yes	characteristic
123	C₂×#341	C₂×C₃₁	1	1	yes	characteristic
124	#1735	C₂×C₃₁	1	1	yes	characteristic
125	C₂×#339	C₂×C₃₁	1	2	yes	—
126	#1729	C₂×C₃₁	1	2	yes	—
127	C₂²×C₃₁	—	2	4	no	—
128	C₂²×C₃₁	—	2	4	no	—
129	C₂²×C₃₁	—	2	2	no	—
130	C₂²×C₃₁	—	2	2	no	—
131	C₂²×C₃₁	C₄×C₄⋊C₄	1	1	yes	characteristic
132	C₂²×C₃₁	#381	1	1	yes	characteristic
133	C₂²×C₃₁	#383	1	1	yes	characteristic
134	C₂²×C₃₁	#387	1	2	yes	—
135	C₂²×C₃₁	#388	1	2	yes	—
136	C₂²×C₃₁	#443	1	4	yes	—
137	C₄×C₃₁	—	2	8	no	—
138	C₄×C₃₁	—	8	4	no	—
139	C₄×C₃₁	#376	1	1	yes	characteristic
140	C₄×C₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
141	C₄×C₃₁	#380	1	1	yes	characteristic
142	C₄×C₃₁	#382	1	1	yes	characteristic
143	C₄×C₃₁	#387	1	2	yes	—
144	C₄×C₃₁	#388	1	2	yes	—
145	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
146	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
147	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
148	C₂×#117	C₂²×C₃₁	1	2	yes	—
149	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
150	C₂×C₄×C₈	—	2	4	no	—
151	C₂×C₄×C₈	C₄×C₃₁	1	4	yes	complemented
152	#341	C₄×C₃₁	1	4	yes	complemented
153	#340	C₄×C₃₁	1	4	yes	complemented
154	#339	—	2	4	no	—
155	#336	—	2	4	no	—
156	C₂×C₃₁	—	2	1	no	—
157	C₂×C₃₁	—	2	1	no	—
158	C₂×C₃₁	—	2	2	no	—
159	C₂×C₃₁	#1434	1	1	yes	characteristic
160	C₂×C₃₁	#1772	1	2	yes	—
161	C₂×C₃₁	#1937	1	4	yes	—
162	C₂³×C₄	—	2	1	no	—
163	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	2	2	no	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
176	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
177	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
178	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
179	C₂²×C₈	D₄×C₃₁	1	2	yes	—
180	C₂²×C₈	Q₈×C₃₁	1	2	yes	—
181	#117	—	2	2	no	—
182	#117	—	2	2	no	—
183	#117	—	2	2	no	—
184	#117	—	2	2	no	—
185	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
186	C₂×C₂²⋊C₄	—	2	2	no	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	—	2	1	no	—
189	C₂×C₄²	—	2	2	no	—
190	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
191	C₂×C₄²	D₄×C₃₁	1	4	yes	—
192	C₄×C₈	—	2	8	no	—
193	C₃₁	#10818	1	1	yes	characteristic, direct product factor, Sylow
194	C₂⁴	#41×C₃₁	1	1	yes	characteristic
195	C₂²×C₄	—	2	8	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
200	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
201	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
203	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
204	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
205	C₂²×C₄	#41×C₃₁	1	2	yes	—
206	C₂²×C₄	#41×C₃₁	1	2	yes	—
207	C₂²×C₄	#41×C₃₁	1	2	yes	—
208	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
209	M₁₆	—	2	4	no	—
210	M₁₆	—	2	4	no	—
211	M₁₆	—	4	2	no	—
212	M₁₆	—	4	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	C₄²×C₃₁	1	4	yes	—
225	C₂×C₈	C₄⋊C₄×C₃₁	1	4	yes	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₂²⋊C₄	—	4	2	no	—
229	C₂²⋊C₄	—	4	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	4	no	—
232	C₄²	—	4	2	no	—
233	C₄²	—	4	2	no	—
234	C₄²	C₂²⋊C₄×C₃₁	1	4	yes	—
235	C₂³	—	2	4	no	—
236	C₂³	C₃₁×#117	1	1	yes	characteristic
237	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
238	C₂³	C₄×Q₈×C₃₁	1	1	yes	characteristic
239	C₂³	C₃₁×#123	1	2	yes	—
240	C₂³	C₃₁×#126	1	2	yes	—
241	C₂×C₄	—	2	16	no	—
242	C₂×C₄	—	2	8	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	4	8	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
247	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
249	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
250	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
251	C₂×C₄	C₃₁×#117	1	2	yes	—
252	C₂×C₄	C₃₁×#117	1	2	yes	—
253	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
254	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
255	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
256	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
257	C₂×C₄	C₄×Q₈×C₃₁	1	1	yes	characteristic
258	C₂×C₄	C₃₁×#121	1	2	yes	—
259	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
260	C₂×C₄	C₃₁×#123	1	2	yes	—
261	C₂×C₄	C₃₁×#124	1	2	yes	—
262	C₂×C₄	C₃₁×#125	1	2	yes	—
263	C₂×C₄	C₃₁×#126	1	2	yes	—
264	C₈	—	2	4	no	—
265	C₈	—	2	4	no	—
266	C₈	—	4	4	no	—
267	C₂²	—	2	4	no	—
268	C₂²	—	2	4	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	C₄×C₄⋊C₄×C₃₁	1	1	yes	characteristic
272	C₂²	C₃₁×#381	1	1	yes	characteristic
273	C₂²	C₃₁×#383	1	1	yes	characteristic
274	C₂²	C₃₁×#387	1	2	yes	—
275	C₂²	C₃₁×#388	1	2	yes	—
276	C₂²	C₃₁×#443	1	4	yes	—
277	C₄	—	2	8	no	—
278	C₄	—	8	4	no	—
279	C₄	C₃₁×#376	1	1	yes	characteristic
280	C₄	C₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
281	C₄	C₃₁×#380	1	1	yes	characteristic
282	C₄	C₃₁×#382	1	1	yes	characteristic
283	C₄	C₃₁×#387	1	2	yes	—
284	C₄	C₃₁×#388	1	2	yes	—
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	—	2	2	no	—
288	C₂	C₃₁×#1434	1	1	yes	characteristic
289	C₂	C₃₁×#1772	1	2	yes	—
290	C₂	C₃₁×#1937	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
socle	C₂³×C₃₁	C₄×D₄
prime radical	C₂⁴×C₃₁	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₂×C₄×C₈×C₃₁	C₄	semidirect
13. C₃₁×#341	C₄	semidirect
14. C₃₁×#340	C₄	semidirect
89. #10818	C₃₁	direct (N × H)
151. C₂×C₄×C₈	C₄×C₃₁	semidirect
152. #341	C₄×C₃₁	semidirect
153. #340	C₄×C₃₁	semidirect
193. C₃₁	#10818	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₈×C₃₁	2
C₂×C₃₁×#341	2
C₃₁×#1735	2
C₂×C₃₁×#339	2
C₃₁×#1729	2
#10818	31

Subgroups — C31×#10818 #21467577

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#10818 #21467577