Subgroups — C₃₁×#20451 #21477210

Order 7936.

1142 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#427	C₂	1	2	yes	maximal
2	C₂²×C₃₁×#97	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₁×#117	C₂	1	2	yes	maximal
4	C₃₁×#1776	C₂	1	8	yes	maximal, complemented
5	C₃₁×#1735	C₂	1	2	yes	maximal
6	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₃₁×#117	C₂²	1	4	yes	—
9	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
10	C₃₁×#427	C₄	1	16	yes	complemented
11	C₂×C₃₁×#107	C₂²	1	4	yes	—
12	C₂×C₃₁×#106	C₂²	1	4	yes	—
13	C₂×C₃₁×#97	C₄	1	8	yes	complemented
14	C₂×C₃₁×#97	C₂²	1	4	yes	—
15	C₄×C₄⋊C₄×C₃₁	C₂²	1	8	yes	—
16	C₄×C₂²⋊C₄×C₃₁	—	2	4	no	—
17	C₃₁×#376	—	2	2	no	—
18	C₄³×C₃₁	—	2	2	no	—
19	C₃₁×#340	C₂²	1	8	yes	—
20	C₂³×C₄×C₃₁	D₄	1	2	yes	—
21	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
22	C₂²×C₈×C₃₁	—	2	4	no	—
23	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₃₁	C₂×C₄	1	8	yes	—
26	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
27	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
28	C₃₁×#117	—	2	8	no	—
29	C₃₁×#117	C₂×C₄	1	16	yes	—
30	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
31	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
32	C₂×C₄²×C₃₁	—	2	4	no	—
33	C₂×C₄²×C₃₁	—	2	8	no	—
34	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₃₁	D₄	1	2	yes	—
36	C₂×C₄²×C₃₁	Q₈	1	4	yes	—
37	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
38	C₃₁×#107	C₂×C₄	1	16	yes	complemented
39	C₃₁×#106	C₂×C₄	1	16	yes	complemented
40	C₃₁×#97	C₂×C₄	1	16	yes	—
41	C₃₁×#95	—	2	8	no	—
42	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₃₁	—	2	8	no	—
44	C₂²×C₄×C₃₁	—	4	4	no	—
45	C₂²×C₄×C₃₁	C₄⋊C₄	1	8	yes	—
46	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
48	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
49	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
51	C₂²×C₄×C₃₁	#41	1	4	yes	—
52	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
53	C₂×C₈×C₃₁	—	2	8	no	—
54	C₂×C₈×C₃₁	—	2	4	no	—
55	C₂×C₈×C₃₁	C₄²	1	16	yes	complemented
56	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
57	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
58	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
59	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
60	C₄⋊C₄×C₃₁	—	2	8	no	—
61	C₄⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
62	C₂²⋊C₄×C₃₁	—	2	16	no	—
63	C₄²×C₃₁	—	2	16	no	—
64	C₄²×C₃₁	—	4	16	no	—
65	C₄²×C₃₁	C₄⋊C₄	1	8	yes	—
66	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
67	C₄²×C₃₁	C₂×Q₈	1	4	yes	—
68	#20451	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂³×C₃₁	—	2	4	no	—
70	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
71	C₂³×C₃₁	C₄×D₄	1	4	yes	—
72	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
73	C₂×C₄×C₃₁	—	2	8	no	—
74	C₂×C₄×C₃₁	—	2	16	no	—
75	C₂×C₄×C₃₁	—	4	8	no	—
76	C₂×C₄×C₃₁	—	4	16	no	—
77	C₂×C₄×C₃₁	C₂×C₄²	1	4	yes	—
78	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	8	yes	—
79	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
80	C₂×C₄×C₃₁	#117	1	4	yes	—
81	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
82	C₂×C₄×C₃₁	C₄×Q₈	1	8	yes	—
83	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
84	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
85	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
86	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
87	C₈×C₃₁	—	2	4	no	—
88	C₈×C₃₁	C₂×C₄²	1	8	yes	—
89	C₂×#427	C₂×C₃₁	1	2	yes	—
90	C₂²×#97	C₂×C₃₁	1	1	yes	characteristic
91	C₄×#117	C₂×C₃₁	1	2	yes	—
92	#1776	C₂×C₃₁	1	8	yes	complemented
93	#1735	C₂×C₃₁	1	2	yes	—
94	C₂²×C₃₁	—	2	4	no	—
95	C₂²×C₃₁	—	2	8	no	—
96	C₂²×C₃₁	C₄×C₄⋊C₄	1	4	yes	—
97	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
98	C₂²×C₃₁	C₂×C₄×D₄	1	2	yes	—
99	C₂²×C₃₁	#575	1	4	yes	—
100	C₄×C₃₁	—	2	8	no	—
101	C₄×C₃₁	—	4	16	no	—
102	C₄×C₃₁	C₄×C₄⋊C₄	1	4	yes	—
103	C₄×C₃₁	C₂²×C₄²	1	1	yes	characteristic
104	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
105	C₄×C₃₁	C₂×C₄×Q₈	1	2	yes	—
106	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
107	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
108	C₂×#117	C₂²×C₃₁	1	4	yes	—
109	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
110	#427	C₄×C₃₁	1	16	yes	complemented
111	C₂×#107	C₂²×C₃₁	1	4	yes	—
112	C₂×#106	C₂²×C₃₁	1	4	yes	—
113	C₂×#97	C₄×C₃₁	1	8	yes	complemented
114	C₂×#97	C₂²×C₃₁	1	4	yes	—
115	C₄×C₄⋊C₄	C₂²×C₃₁	1	8	yes	—
116	C₄×C₂²⋊C₄	—	2	4	no	—
117	#376	—	2	2	no	—
118	C₄³	—	2	2	no	—
119	#340	C₂²×C₃₁	1	8	yes	—
120	C₂×C₃₁	—	2	4	no	—
121	C₂×C₃₁	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₃₁	#2911	1	2	yes	—
123	C₂×C₃₁	#2943	1	4	yes	—
124	C₂³×C₄	D₄×C₃₁	1	2	yes	—
125	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
126	C₂²×C₈	—	2	4	no	—
127	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
128	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
129	C₂²×C₈	C₂×C₄×C₃₁	1	8	yes	—
130	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
131	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
132	#117	—	2	8	no	—
133	#117	C₂×C₄×C₃₁	1	16	yes	—
134	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
135	C₂×C₂²⋊C₄	—	2	4	no	—
136	C₂×C₄²	—	2	4	no	—
137	C₂×C₄²	—	2	8	no	—
138	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
139	C₂×C₄²	D₄×C₃₁	1	2	yes	—
140	C₂×C₄²	Q₈×C₃₁	1	4	yes	—
141	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
142	#107	C₂×C₄×C₃₁	1	16	yes	complemented
143	#106	C₂×C₄×C₃₁	1	16	yes	complemented
144	#97	C₂×C₄×C₃₁	1	16	yes	—
145	#95	—	2	8	no	—
146	C₃₁	#20451	1	1	yes	characteristic, direct product factor, Sylow
147	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
148	C₂²×C₄	—	2	8	no	—
149	C₂²×C₄	—	4	4	no	—
150	C₂²×C₄	C₄⋊C₄×C₃₁	1	8	yes	—
151	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
152	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
153	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
154	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
155	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
156	C₂²×C₄	#41×C₃₁	1	4	yes	—
157	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
158	C₂×C₈	—	2	8	no	—
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	C₄²×C₃₁	1	16	yes	complemented
161	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
162	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
163	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
164	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
165	C₄⋊C₄	—	2	8	no	—
166	C₄⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
167	C₂²⋊C₄	—	2	16	no	—
168	C₄²	—	2	16	no	—
169	C₄²	—	4	16	no	—
170	C₄²	C₄⋊C₄×C₃₁	1	8	yes	—
171	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
172	C₄²	C₂×Q₈×C₃₁	1	4	yes	—
173	C₂³	—	2	4	no	—
174	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
175	C₂³	C₄×D₄×C₃₁	1	4	yes	—
176	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	2	16	no	—
179	C₂×C₄	—	4	8	no	—
180	C₂×C₄	—	4	16	no	—
181	C₂×C₄	C₂×C₄²×C₃₁	1	4	yes	—
182	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	8	yes	—
183	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
184	C₂×C₄	C₃₁×#117	1	4	yes	—
185	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
186	C₂×C₄	C₄×Q₈×C₃₁	1	8	yes	—
187	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
188	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
189	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
190	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
191	C₈	—	2	4	no	—
192	C₈	C₂×C₄²×C₃₁	1	8	yes	—
193	C₂²	—	2	4	no	—
194	C₂²	—	2	8	no	—
195	C₂²	C₄×C₄⋊C₄×C₃₁	1	4	yes	—
196	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
197	C₂²	C₂×C₄×D₄×C₃₁	1	2	yes	—
198	C₂²	C₃₁×#575	1	4	yes	—
199	C₄	—	2	8	no	—
200	C₄	—	4	16	no	—
201	C₄	C₄×C₄⋊C₄×C₃₁	1	4	yes	—
202	C₄	C₂²×C₄²×C₃₁	1	1	yes	characteristic
203	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
204	C₄	C₂×C₄×Q₈×C₃₁	1	2	yes	—
205	C₂	—	2	4	no	—
206	C₂	C₂×C₄×C₄⋊C₄×C₃₁	1	1	yes	characteristic
207	C₂	C₃₁×#2911	1	2	yes	—
208	C₂	C₃₁×#2943	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₄	C₂²×C₄²×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂²×D₄
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ 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×#1776	C₂	semidirect
10. C₃₁×#427	C₄	semidirect
13. C₂×C₃₁×#97	C₄	semidirect
38. C₃₁×#107	C₂×C₄	semidirect
39. C₃₁×#106	C₂×C₄	semidirect
55. C₂×C₈×C₃₁	C₄²	semidirect
68. #20451	C₃₁	direct (N × H)
92. #1776	C₂×C₃₁	semidirect
110. #427	C₄×C₃₁	semidirect
113. C₂×#97	C₄×C₃₁	semidirect
142. #107	C₂×C₄×C₃₁	semidirect
143. #106	C₂×C₄×C₃₁	semidirect
146. C₃₁	#20451	direct (N × H)
160. C₂×C₈	C₄²×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#427	2
C₂²×C₃₁×#97	2
C₄×C₃₁×#117	2
C₃₁×#1776	2
C₃₁×#1735	2
#20451	31

Subgroups — C31×#20451 #21477210

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20451 #21477210