Subgroups — C₃₁×#18072 #21474831

Order 7936.

578 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3170	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2962	C₂	1	2	yes	maximal
3	C₃₁×#1593	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1587	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1579	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#334	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1481	C₂	1	2	yes	maximal
8	C₃₁×#1451	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#533	C₂²	1	2	yes	—
10	C₃₁×#500	C₄	1	1	yes	characteristic
11	C₃₁×#494	C₄	1	1	yes	characteristic
12	C₃₁×#487	C₄	1	2	yes	—
13	C₃₁×#467	C₄	1	2	yes	—
14	C₃₁×#467	C₄	1	2	yes	—
15	C₃₁×#446	—	2	4	no	—
16	C₃₁×#446	C₂²	1	4	yes	—
17	C₈×Q₈×C₃₁	C₂²	1	4	yes	—
18	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
19	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
20	C₃₁×#431	C₂²	1	1	yes	characteristic
21	C₃₁×#424	—	2	2	no	—
22	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
23	C₃₁×#405	—	2	2	no	—
24	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#334	C₂²	1	1	yes	characteristic
27	C₃₁×#334	C₂²	1	2	yes	—
28	C₃₁×#334	C₂²	1	1	yes	characteristic
29	C₃₁×Q₈⋊C₈	C₂²	1	4	yes	—
30	C₃₁×Q₈⋊C₈	C₂²	1	4	yes	—
31	C₃₁×#322	C₂²	1	2	yes	—
32	C₃₁×#322	C₂²	1	2	yes	—
33	C₂×Q₁₆×C₃₁	C₂×C₄	1	2	yes	—
34	C₂×Q₁₆×C₃₁	C₂×C₄	1	2	yes	—
35	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
36	C₂²×C₈×C₃₁	—	2	2	no	—
37	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
40	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
41	C₃₁×#125	—	2	2	no	—
42	C₃₁×C₂²⋊Q₈	—	2	2	no	—
43	C₃₁×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
44	C₄×Q₈×C₃₁	—	2	2	no	—
45	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
46	C₃₁×#117	—	2	2	no	—
47	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
48	C₃₁×#107	C₂×C₄	1	2	yes	—
49	C₃₁×#107	C₂×C₄	1	2	yes	—
50	C₃₁×C₄⋊C₈	—	2	4	no	—
51	C₃₁×C₄⋊C₈	—	2	4	no	—
52	C₃₁×C₄⋊C₈	—	2	2	no	—
53	C₃₁×C₄⋊C₈	—	2	1	no	—
54	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
55	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
56	C₃₁×#103	C₂×C₄	1	4	yes	—
57	C₃₁×#103	C₂×C₄	1	4	yes	—
58	C₃₁×C₂²⋊C₈	—	2	1	no	—
59	C₃₁×#97	—	2	4	no	—
60	C₃₁×#97	D₄	1	2	yes	—
61	C₄×C₈×C₃₁	—	2	4	no	—
62	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₃₁	C₂³	1	2	yes	—
65	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₃₁	C₂³	1	2	yes	—
68	C₂×Q₈×C₃₁	—	2	2	no	—
69	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
73	Q₁₆×C₃₁	—	2	4	no	—
74	Q₁₆×C₃₁	—	4	2	no	—
75	M₁₆×C₃₁	—	2	4	no	—
76	C₂×C₈×C₃₁	—	2	2	no	—
77	C₂×C₈×C₃₁	—	2	2	no	—
78	C₂×C₈×C₃₁	—	2	2	no	—
79	C₂×C₈×C₃₁	—	2	2	no	—
80	C₂×C₈×C₃₁	—	4	1	no	—
81	C₂×C₈×C₃₁	—	4	4	no	—
82	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
87	C₄⋊C₄×C₃₁	—	2	2	no	—
88	C₄⋊C₄×C₃₁	—	2	2	no	—
89	C₄⋊C₄×C₃₁	—	2	2	no	—
90	C₄⋊C₄×C₃₁	—	2	4	no	—
91	C₄⋊C₄×C₃₁	—	2	4	no	—
92	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
93	C₂²⋊C₄×C₃₁	—	2	4	no	—
94	C₄²×C₃₁	—	2	4	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	C₂⁴	1	1	yes	characteristic
99	#18072	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
101	Q₈×C₃₁	—	2	4	no	—
102	Q₈×C₃₁	—	2	4	no	—
103	Q₈×C₃₁	—	8	2	no	—
104	C₂×C₄×C₃₁	—	2	1	no	—
105	C₂×C₄×C₃₁	—	2	1	no	—
106	C₂×C₄×C₃₁	—	2	4	no	—
107	C₂×C₄×C₃₁	—	2	4	no	—
108	C₂×C₄×C₃₁	—	4	1	no	—
109	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₃₁	#137	1	4	yes	—
113	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
116	C₈×C₃₁	—	2	1	no	—
117	C₈×C₃₁	—	2	1	no	—
118	C₈×C₃₁	—	2	1	no	—
119	C₈×C₃₁	—	2	4	no	—
120	C₈×C₃₁	—	4	2	no	—
121	C₈×C₃₁	—	4	2	no	—
122	C₈×C₃₁	—	4	2	no	—
123	C₈×C₃₁	#131	1	2	yes	—
124	#3170	C₂×C₃₁	1	1	yes	characteristic
125	#2962	C₂×C₃₁	1	2	yes	—
126	#1593	C₂×C₃₁	1	2	yes	complemented
127	#1587	C₂×C₃₁	1	4	yes	complemented
128	#1579	C₂×C₃₁	1	2	yes	complemented
129	C₂×#334	C₂×C₃₁	1	1	yes	characteristic
130	#1481	C₂×C₃₁	1	2	yes	—
131	#1451	C₂×C₃₁	1	1	yes	characteristic
132	C₂²×C₃₁	—	2	1	no	—
133	C₂²×C₃₁	—	2	1	no	—
134	C₂²×C₃₁	#441	1	2	yes	—
135	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
136	C₄×C₃₁	—	2	1	no	—
137	C₄×C₃₁	—	2	1	no	—
138	C₄×C₃₁	—	2	1	no	—
139	C₄×C₃₁	—	4	4	no	—
140	C₄×C₃₁	—	4	4	no	—
141	C₄×C₃₁	#441	1	2	yes	—
142	C₄×C₃₁	C₂×#131	1	1	yes	characteristic
143	C₄×C₃₁	#568	1	1	yes	characteristic
144	C₄×C₃₁	C₂×#137	1	2	yes	—
145	#533	C₂²×C₃₁	1	2	yes	—
146	#500	C₄×C₃₁	1	1	yes	characteristic
147	#494	C₄×C₃₁	1	1	yes	characteristic
148	#487	C₄×C₃₁	1	2	yes	—
149	#467	C₄×C₃₁	1	2	yes	—
150	#467	C₄×C₃₁	1	2	yes	—
151	#446	—	2	4	no	—
152	#446	C₂²×C₃₁	1	4	yes	—
153	C₈×Q₈	C₂²×C₃₁	1	4	yes	—
154	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
155	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
156	#431	C₂²×C₃₁	1	1	yes	characteristic
157	#424	—	2	2	no	—
158	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
159	#405	—	2	2	no	—
160	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
161	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
162	#334	C₂²×C₃₁	1	1	yes	characteristic
163	#334	C₂²×C₃₁	1	2	yes	—
164	#334	C₂²×C₃₁	1	1	yes	characteristic
165	Q₈⋊C₈	C₂²×C₃₁	1	4	yes	—
166	Q₈⋊C₈	C₂²×C₃₁	1	4	yes	—
167	#322	C₂²×C₃₁	1	2	yes	—
168	#322	C₂²×C₃₁	1	2	yes	—
169	C₂×C₃₁	—	2	2	no	—
170	C₂×C₃₁	#2935	1	1	yes	characteristic
171	C₂×C₃₁	C₂×#441	1	1	yes	characteristic
172	C₂×C₃₁	#2957	1	1	yes	characteristic
173	C₂×Q₁₆	C₂×C₄×C₃₁	1	2	yes	—
174	C₂×Q₁₆	C₂×C₄×C₃₁	1	2	yes	—
175	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
178	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
179	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
180	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
181	#125	—	2	2	no	—
182	C₂²⋊Q₈	—	2	2	no	—
183	C₂²⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
184	C₄×Q₈	—	2	2	no	—
185	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
186	#117	—	2	2	no	—
187	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
188	#107	C₂×C₄×C₃₁	1	2	yes	—
189	#107	C₂×C₄×C₃₁	1	2	yes	—
190	C₄⋊C₈	—	2	4	no	—
191	C₄⋊C₈	—	2	4	no	—
192	C₄⋊C₈	—	2	2	no	—
193	C₄⋊C₈	—	2	1	no	—
194	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
195	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
196	#103	C₂×C₄×C₃₁	1	4	yes	—
197	#103	C₂×C₄×C₃₁	1	4	yes	—
198	C₂²⋊C₈	—	2	1	no	—
199	#97	—	2	4	no	—
200	#97	D₄×C₃₁	1	2	yes	—
201	C₄×C₈	—	2	4	no	—
202	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
203	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
204	C₄×C₈	C₂³×C₃₁	1	2	yes	—
205	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
206	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
207	C₄×C₈	C₂³×C₃₁	1	2	yes	—
208	C₃₁	#18072	1	1	yes	characteristic, direct product factor, Sylow
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
211	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
213	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
214	Q₁₆	—	2	4	no	—
215	Q₁₆	—	4	2	no	—
216	M₁₆	—	2	4	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	—	4	4	no	—
223	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
224	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
225	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
226	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
227	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
234	C₂²⋊C₄	—	2	4	no	—
235	C₄²	—	2	4	no	—
236	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
237	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
238	C₄²	#41×C₃₁	1	1	yes	characteristic
239	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
240	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
241	Q₈	—	2	4	no	—
242	Q₈	—	2	4	no	—
243	Q₈	—	8	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
250	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
251	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
252	C₂×C₄	C₃₁×#137	1	4	yes	—
253	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
254	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
255	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	2	4	no	—
260	C₈	—	4	2	no	—
261	C₈	—	4	2	no	—
262	C₈	—	4	2	no	—
263	C₈	C₃₁×#131	1	2	yes	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	C₃₁×#441	1	2	yes	—
267	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	4	4	no	—
272	C₄	—	4	4	no	—
273	C₄	C₃₁×#441	1	2	yes	—
274	C₄	C₂×C₃₁×#131	1	1	yes	characteristic
275	C₄	C₃₁×#568	1	1	yes	characteristic
276	C₄	C₂×C₃₁×#137	1	2	yes	—
277	C₂	—	2	2	no	—
278	C₂	C₃₁×#2935	1	1	yes	characteristic
279	C₂	C₂×C₃₁×#441	1	1	yes	characteristic
280	C₂	C₃₁×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₂³×C₃₁	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₄×M₁₆×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1593	C₂	semidirect
4. C₃₁×#1587	C₂	semidirect
5. C₃₁×#1579	C₂	semidirect
99. #18072	C₃₁	direct (N × H)
126. #1593	C₂×C₃₁	semidirect
127. #1587	C₂×C₃₁	semidirect
128. #1579	C₂×C₃₁	semidirect
208. C₃₁	#18072	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×#3170	2
C₃₁×#2962	2
C₃₁×#1593	2
C₃₁×#1587	2
C₃₁×#1579	2
C₂×C₃₁×#334	2
C₃₁×#1481	2
C₃₁×#1451	2
#18072	31

Subgroups — C31×#18072 #21474831

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#18072 #21474831