Subgroups — C₃₁×#21732 #21478491

Order 7936.

702 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3160	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2960	C₂	1	2	yes	maximal
3	C₃₁×#1994	C₂	1	2	yes	maximal
4	C₃₁×#1945	C₂	1	4	yes	maximal
5	C₃₁×#1848	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1772	C₂	1	4	yes	maximal
7	C₄²×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#533	C₂²	1	2	yes	—
9	C₃₁×#500	C₄	1	2	yes	complemented
10	C₃₁×#499	C₄	1	4	yes	complemented
11	C₃₁×#498	C₄	1	2	yes	complemented
12	C₃₁×#446	—	2	4	no	—
13	C₈×Q₈×C₃₁	—	2	4	no	—
14	C₂×C₃₁×#108	C₂²	1	4	yes	—
15	C₃₁×#427	C₂²	1	4	yes	—
16	C₃₁×#424	C₂²	1	4	yes	—
17	C₃₁×#423	C₂²	1	2	yes	—
18	C₃₁×#405	C₂²	1	4	yes	—
19	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₄³×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#339	C₂²	1	8	yes	—
25	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
26	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
28	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
29	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	—
30	C₃₁×#125	C₂×C₄	1	4	yes	—
31	C₃₁×C₂²⋊Q₈	—	2	4	no	—
32	C₄×Q₈×C₃₁	—	2	4	no	—
33	C₃₁×#117	C₂³	1	4	yes	—
34	C₂×C₄²×C₃₁	—	2	2	no	—
35	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
36	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
37	C₃₁×#108	—	2	8	no	—
38	C₃₁×#107	C₂×C₄	1	8	yes	—
39	C₃₁×#106	C₂×C₄	1	8	yes	—
40	C₃₁×C₄⋊C₈	—	2	8	no	—
41	C₃₁×C₄⋊C₈	—	2	4	no	—
42	C₃₁×#97	—	2	4	no	—
43	C₄×C₈×C₃₁	—	2	4	no	—
44	C₄×C₈×C₃₁	—	2	4	no	—
45	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₃₁	D₄	1	2	yes	—
49	C₄×C₈×C₃₁	D₄	1	2	yes	—
50	C₄×C₈×C₃₁	Q₈	1	8	yes	complemented
51	C₂×Q₈×C₃₁	—	2	4	no	—
52	C₂²×C₄×C₃₁	—	2	2	no	—
53	C₂²×C₄×C₃₁	#41	1	2	yes	—
54	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
55	M₁₆×C₃₁	—	2	8	no	—
56	C₂×C₈×C₃₁	—	2	2	no	—
57	C₂×C₈×C₃₁	—	2	4	no	—
58	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
59	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
60	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
64	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
65	C₂×C₈×C₃₁	#41	1	4	yes	—
66	C₄⋊C₄×C₃₁	—	2	8	no	—
67	C₄⋊C₄×C₃₁	—	2	4	no	—
68	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
69	C₂²⋊C₄×C₃₁	—	2	4	no	—
70	C₄²×C₃₁	—	2	8	no	—
71	C₄²×C₃₁	—	2	4	no	—
72	C₄²×C₃₁	—	2	2	no	—
73	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₃₁	C₂×D₄	1	2	yes	—
76	C₄²×C₃₁	C₂×Q₈	1	4	yes	—
77	#21732	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
79	Q₈×C₃₁	—	4	8	no	—
80	C₂×C₄×C₃₁	—	2	8	no	—
81	C₂×C₄×C₃₁	—	2	4	no	—
82	C₂×C₄×C₃₁	—	2	4	no	—
83	C₂×C₄×C₃₁	—	2	8	no	—
84	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
86	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
87	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
88	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
89	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
90	C₈×C₃₁	—	2	2	no	—
91	C₈×C₃₁	—	4	4	no	—
92	C₈×C₃₁	C₄×Q₈	1	4	yes	—
93	C₈×C₃₁	C₄×Q₈	1	4	yes	—
94	C₈×C₃₁	C₄⋊Q₈	1	2	yes	—
95	C₈×C₃₁	C₄⋊Q₈	1	2	yes	—
96	#3160	C₂×C₃₁	1	1	yes	characteristic
97	#2960	C₂×C₃₁	1	2	yes	—
98	#1994	C₂×C₃₁	1	2	yes	—
99	#1945	C₂×C₃₁	1	4	yes	—
100	#1848	C₂×C₃₁	1	1	yes	characteristic
101	#1772	C₂×C₃₁	1	4	yes	—
102	C₄²×C₈	C₂×C₃₁	1	1	yes	characteristic
103	C₂²×C₃₁	—	2	2	no	—
104	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
105	C₂²×C₃₁	#533	1	2	yes	—
106	C₄×C₃₁	—	2	8	no	—
107	C₄×C₃₁	—	2	2	no	—
108	C₄×C₃₁	—	4	8	no	—
109	C₄×C₃₁	#443	1	4	yes	—
110	C₄×C₃₁	C₂×C₄×Q₈	1	2	yes	—
111	C₄×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
112	C₄×C₃₁	#532	1	1	yes	characteristic
113	#533	C₂²×C₃₁	1	2	yes	—
114	#500	C₄×C₃₁	1	2	yes	complemented
115	#499	C₄×C₃₁	1	4	yes	complemented
116	#498	C₄×C₃₁	1	2	yes	complemented
117	#446	—	2	4	no	—
118	C₈×Q₈	—	2	4	no	—
119	C₂×#108	C₂²×C₃₁	1	4	yes	—
120	#427	C₂²×C₃₁	1	4	yes	—
121	#424	C₂²×C₃₁	1	4	yes	—
122	#423	C₂²×C₃₁	1	2	yes	—
123	#405	C₂²×C₃₁	1	4	yes	—
124	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
125	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
126	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
127	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
128	C₄³	C₂²×C₃₁	1	1	yes	characteristic
129	#339	C₂²×C₃₁	1	8	yes	—
130	C₂×C₃₁	—	2	2	no	—
131	C₂×C₃₁	C₄×C₄⋊Q₈	1	1	yes	characteristic
132	C₂×C₃₁	C₂×#443	1	2	yes	—
133	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
134	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
135	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
136	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
137	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
138	#125	C₂×C₄×C₃₁	1	4	yes	—
139	C₂²⋊Q₈	—	2	4	no	—
140	C₄×Q₈	—	2	4	no	—
141	#117	C₂³×C₃₁	1	4	yes	—
142	C₂×C₄²	—	2	2	no	—
143	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
144	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
145	#108	—	2	8	no	—
146	#107	C₂×C₄×C₃₁	1	8	yes	—
147	#106	C₂×C₄×C₃₁	1	8	yes	—
148	C₄⋊C₈	—	2	8	no	—
149	C₄⋊C₈	—	2	4	no	—
150	#97	—	2	4	no	—
151	C₄×C₈	—	2	4	no	—
152	C₄×C₈	—	2	4	no	—
153	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
154	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
155	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
156	C₄×C₈	D₄×C₃₁	1	2	yes	—
157	C₄×C₈	D₄×C₃₁	1	2	yes	—
158	C₄×C₈	Q₈×C₃₁	1	8	yes	complemented
159	C₃₁	#21732	1	1	yes	characteristic, direct product factor, Sylow
160	C₂×Q₈	—	2	4	no	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	#41×C₃₁	1	2	yes	—
163	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
164	M₁₆	—	2	8	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	4	no	—
167	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
168	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
169	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
170	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
171	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
172	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
173	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
174	C₂×C₈	#41×C₃₁	1	4	yes	—
175	C₄⋊C₄	—	2	8	no	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
178	C₂²⋊C₄	—	2	4	no	—
179	C₄²	—	2	8	no	—
180	C₄²	—	2	4	no	—
181	C₄²	—	2	2	no	—
182	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
183	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
184	C₄²	C₂×D₄×C₃₁	1	2	yes	—
185	C₄²	C₂×Q₈×C₃₁	1	4	yes	—
186	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
187	Q₈	—	4	8	no	—
188	C₂×C₄	—	2	8	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	8	no	—
192	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
193	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
194	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
195	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
196	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
197	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
198	C₈	—	2	2	no	—
199	C₈	—	4	4	no	—
200	C₈	C₄×Q₈×C₃₁	1	4	yes	—
201	C₈	C₄×Q₈×C₃₁	1	4	yes	—
202	C₈	C₃₁×C₄⋊Q₈	1	2	yes	—
203	C₈	C₃₁×C₄⋊Q₈	1	2	yes	—
204	C₂²	—	2	2	no	—
205	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
206	C₂²	C₃₁×#533	1	2	yes	—
207	C₄	—	2	8	no	—
208	C₄	—	2	2	no	—
209	C₄	—	4	8	no	—
210	C₄	C₃₁×#443	1	4	yes	—
211	C₄	C₂×C₄×Q₈×C₃₁	1	2	yes	—
212	C₄	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
213	C₄	C₃₁×#532	1	1	yes	characteristic
214	C₂	—	2	2	no	—
215	C₂	C₄×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
216	C₂	C₂×C₃₁×#443	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
9. C₃₁×#500	C₄	semidirect
10. C₃₁×#499	C₄	semidirect
11. C₃₁×#498	C₄	semidirect
50. C₄×C₈×C₃₁	Q₈	semidirect
77. #21732	C₃₁	direct (N × H)
114. #500	C₄×C₃₁	semidirect
115. #499	C₄×C₃₁	semidirect
116. #498	C₄×C₃₁	semidirect
158. C₄×C₈	Q₈×C₃₁	semidirect
159. C₃₁	#21732	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×#3160	2
C₃₁×#2960	2
C₃₁×#1994	2
C₃₁×#1945	2
C₃₁×#1848	2
C₃₁×#1772	2
C₄²×C₈×C₃₁	2
#21732	31

Subgroups — C31×#21732 #21478491

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21732 #21478491