Finite Group Databasebeta

← C23×#6654 #14509653

Subgroups — C23×#6654 #14509653

Order 1920.

3952 conjugacy classes of subgroups; only the 932 normal subgroups are shown.

normal subgroups only

This group has 3952 conjugacy classes of subgroups — too many to draw. Only its 932 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×D60C211yesmaximal, characteristic, complemented
2C23×C5×D12C211yesmaximal, characteristic, complemented
3C23×C3×#168C211yesmaximal, characteristic, complemented
4C22×#6654C2128yesmaximal, direct product factor
5C23×#686—31nomaximal, Hall
6C24×D15—21no—
7C24×C5×D3—21no—
8C23×C3×C4×C5C2211yescharacteristic
9C22×D60—27no—
10C22×D60C22114yescomplemented
11C22×C5×D12—27no—
12C22×C5×D12C22114yescomplemented
13C22×C3×#168C22114yescomplemented
14C2×#6654C221112yesdirect product factor
15C23×D24—51nomaximal, Hall
16C23×C5×D4—31no—
17C23×D20—31no—
18C22×#686—328no—
19C23×#168D311yescharacteristic, complemented
20C24×C3×C5D411yescharacteristic
21C23×D15—214no—
22C23×D15—48no—
23C23×C5×D3—214no—
24C23×C5×D3—48no—
25C22×C3×C4×C5D417yes—
26C22×C3×C4×C5C2317yes—
27C2×D60—242no—
28C2×D60C23128yescomplemented
29C2×C5×D12—242no—
30C2×C5×D12C23128yescomplemented
31C2×C3×#168C23128yescomplemented
32#6654C23164yesdirect product factor
33C23×D12—51no—
34C23×D12D511yescharacteristic, complemented
35C23×C3×C8—51no—
36C22×D24—528no—
37C25×C5—61no—
38C24×D5—61no—
39C22×C5×D4—314no—
40C22×C5×D4—67no—
41C23×C4×C5C2×D311yescharacteristic
42C22×D20—314no—
43C22×D20—67no—
44C2×#686—3112no—
45C22×#168C2×D3114yescomplemented
46C23×D8—151noSylow
47C23×C3×C5D818yescomplemented
48C23×C3×C5C2×D417yes—
49C22×D15—228no—
50C22×D15—456no—
51C22×C5×D3—228no—
52C22×C5×D3—456no—
53C2×C3×C4×C5C2×D4121yes—
54C2×C3×C4×C5C2417yes—
55D60—228no—
56D60C2418yescomplemented
57C5×D12—228no—
58C5×D12C2418yescomplemented
59C3×#168C2418yescomplemented
60C24×D3—21no—
61C24×D3—101no—
62C23×C3×C4C2×D511yescharacteristic
63C22×D12—27no—
64C22×D12—514no—
65C22×D12—107no—
66C22×D12C2×D5114yescomplemented
67C22×C3×C8—514no—
68C2×D24—5112no—
69C24×C5—614no—
70C24×C5—128no—
71C24×C5D1211yescharacteristic
72C23×D5—614no—
73C23×D5—128no—
74C2×C5×D4—328no—
75C2×C5×D4—642no—
76C22×C4×C5D1217yes—
77C22×C4×C5C22×D317yes—
78C2×D20—328no—
79C2×D20—642no—
80#686—364no—
81C2×#168C22×D3128yescomplemented
82C23×D4—31no—
83C23×D4—151no—
84C22×D8—1528no—
85C23×C8—51no—
86C22×C3×C5C2×D8128yescomplemented
87C22×C3×C5C22×D417yes—
88C2×D15—28no—
89C2×D15—456no—
90C2×C5×D3—28no—
91C2×C5×D3—456no—
92C3×C4×C5C22×D417yes—
93C3×C4×C5C2511yescharacteristic
94C24×C3C22⋊D511yescharacteristic
95C23×D3—214no—
96C23×D3—48no—
97C23×D3—1014no—
98C23×D3—208no—
99C22×C3×C4C22⋊D517yes—
100C22×C3×C4C22×D517yes—
101C2×D12—242no—
102C2×D12—528no—
103C2×D12—1042no—
104C2×D12C22×D5128yescomplemented
105C2×C3×C8—528no—
106D24—564no—
107C23×C5—628no—
108C23×C5—1256no—
109C23×C5D2418yescomplemented
110C23×C5C2×D1217yes—
111C22×D5—628no—
112C22×D5—1256no—
113C5×D4—38no—
114C5×D4—628no—
115C2×C4×C5C2×D12121yes—
116C2×C4×C5C23×D317yes—
117D20—38no—
118D20—628no—
119#168C23×D318yescomplemented
120C25—61no—
121C25—301no—
122C22×D4—314no—
123C22×D4—67no—
124C22×D4—1514no—
125C22×D4—307no—
126C23×C4D3×D511yescharacteristic
127C2×D8—15112no—
128C22×C8—514no—
129C2×C3×C5C22×D8114yescomplemented
130C2×C3×C5C23×D411yescharacteristic
131D15—48no—
132C5×D3—48no—
133C23×C3#68618yescomplemented
134C23×C3C2×C22⋊D517yes—
135C22×D3—228no—
136C22×D3—456no—
137C22×D3—1028no—
138C22×D3—2056no—
139C2×C3×C4C2×C22⋊D5121yes—
140C2×C3×C4C23×D517yes—
141D12—228no—
142D12—58no—
143D12—1028no—
144D12C23×D518yescomplemented
145C3×C8—58no—
146C22×C5—68no—
147C22×C5—1256no—
148C22×C5C2×D24128yescomplemented
149C22×C5C22×D1217yes—
150C2×D5—68no—
151C2×D5—1256no—
152C4×C5C22×D1217yes—
153C4×C5C24×D311yescharacteristic
154C24—614no—
155C24—128no—
156C24—3014no—
157C24—608no—
158C24#120311yescharacteristic
159C2×D4—328no—
160C2×D4—642no—
161C2×D4—1528no—
162C2×D4—3042no—
163C22×C4#120317yes—
164C22×C4C2×D3×D517yes—
165D8—1564no—
166C2×C8—528no—
167C3×C5C23×D811yescharacteristic, complemented, Hall
168C22×C3C2×#686128yescomplemented
169C22×C3C22×C22⋊D517yes—
170C2×D3—28no—
171C2×D3—456no—
172C2×D3—108no—
173C2×D3—2056no—
174C3×C4C22×C22⋊D517yes—
175C3×C4C24×D511yescharacteristic
176C2×C5—128no—
177C2×C5C22×D24114yescomplemented
178C2×C5C23×D1211yescharacteristic
179D5—128no—
180C23—628no—
181C23—1256no—
182C23—3028no—
183C23—6056no—
184C23#665418yesdirect product factor
185C23C2×#120317yes—
186D4—38no—
187D4—628no—
188D4—158no—
189D4—3028no—
190C2×C4C2×#1203121yes—
191C2×C4C22×D3×D517yes—
192C8—58no—
193C2×C3C22×#686114yescomplemented
194C2×C3C23×C22⋊D511yescharacteristic
195D3—48no—
196D3—208no—
197C5C23×D2411yescharacteristic, complemented, Sylow
198C22—68no—
199C22—1256no—
200C22—308no—
201C22—6056no—
202C22C2×#6654128yesdirect product factor
203C22C22×#120317yes—
204C4C22×#120317yes—
205C4C23×D3×D511yescharacteristic
206C3C23×#68611yescharacteristic, complemented, Sylow
207C2—128no—
208C2—608no—
209C2C22×#6654114yesdirect product factor
210C2C23×#120311yescharacteristic
subgroupG/H
centerC24#1203
commutator, cosocleC3×C4×C5C25
FrattiniC4C23×D3×D5
FittingC23×C3×C4×C5C22
socleC24×C3×C5D4
hypercenterC23×C4D3×D5
hypocenterC3×C5C23×D8
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C24 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. C23×D60C2semidirect
2. C23×C5×D12C2semidirect
3. C23×C3×#168C2semidirect
4. C22×#6654C2direct (N × H)
10. C22×D60C22semidirect
12. C22×C5×D12C22semidirect
13. C22×C3×#168C22semidirect
14. C2×#6654C22direct (N × H)
19. C23×#168D3semidirect
28. C2×D60C23semidirect
30. C2×C5×D12C23semidirect
31. C2×C3×#168C23semidirect
32. #6654C23direct (N × H)
34. C23×D12D5semidirect
45. C22×#168C2×D3semidirect
47. C23×C3×C5D8semidirect
56. D60C24semidirect
58. C5×D12C24semidirect
59. C3×#168C24semidirect
66. C22×D12C2×D5semidirect
81. C2×#168C22×D3semidirect
86. C22×C3×C5C2×D8semidirect
104. C2×D12C22×D5semidirect
109. C23×C5D24semidirect
119. #168C23×D3semidirect
129. C2×C3×C5C22×D8semidirect
133. C23×C3#686semidirect
144. D12C23×D5semidirect
148. C22×C5C2×D24semidirect
167. C3×C5C23×D8semidirect
168. C22×C3C2×#686semidirect
177. C2×C5C22×D24semidirect
184. C23#6654direct (N × H)
193. C2×C3C22×#686semidirect
197. C5C23×D24semidirect
202. C22C2×#6654direct (N × H)
206. C3C23×#686semidirect
209. C2C22×#6654direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D602
C23×C5×D122
C23×C3×#1682
C22×#66542
C23×#6863
C23×D245