Finite Group Databasebeta

← C23×#6653 #14509654

Subgroups — C23×#6653 #14509654

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×C3×D20C211yesmaximal, characteristic, complemented
3C23×C5×C3⋊C8C211yesmaximal, characteristic, complemented
4C22×#6653C2128yesmaximal, direct product factor
5C23×D40—31nomaximal, Hall
6C24×D15—21no—
7C24×C3×D5—21no—
8C23×C3×C4×C5C2211yescharacteristic
9C22×D60—27no—
10C22×D60C22114yescomplemented
11C22×C3×D20—27no—
12C22×C3×D20C22114yescomplemented
13C22×C5×C3⋊C8C22114yescomplemented
14C2×#6653C221112yesdirect product factor
15C23×#213—51nomaximal, Hall
16C23×D20—31no—
17C23×D20D311yescharacteristic, complemented
18C23×C5×C8—31no—
19C22×D40—328no—
20C24×C3×C5D411yescharacteristic
21C23×D15—214no—
22C23×D15—48no—
23C23×C3×D5—214no—
24C23×C3×D5—48no—
25C22×C3×C4×C5D417yes—
26C22×C3×C4×C5C2317yes—
27C2×D60—242no—
28C2×D60C23128yescomplemented
29C2×C3×D20—242no—
30C2×C3×D20C23128yescomplemented
31C2×C5×C3⋊C8C23128yescomplemented
32#6653C23164yesdirect product factor
33C23×C3×D4—51no—
34C23×D12—51no—
35C22×#213—528no—
36C23×C3⋊C8D511yescharacteristic, complemented
37C24×D5—21no—
38C24×D5—61no—
39C23×C4×C5C2×D311yescharacteristic
40C22×D20—27no—
41C22×D20—314no—
42C22×D20—67no—
43C22×D20C2×D3114yescomplemented
44C22×C5×C8—314no—
45C2×D40—3112no—
46C23×D8—151noSylow
47C23×C3×C5D818yescomplemented
48C23×C3×C5C2×D417yes—
49C22×D15—228no—
50C22×D15—456no—
51C22×C3×D5—228no—
52C22×C3×D5—456no—
53C2×C3×C4×C5C2×D4121yes—
54C2×C3×C4×C5C2417yes—
55D60—228no—
56D60C2418yescomplemented
57C3×D20—228no—
58C3×D20C2418yescomplemented
59C5×C3⋊C8C2418yescomplemented
60C25×C3—101no—
61C24×D3—101no—
62C22×C3×D4—514no—
63C22×C3×D4—107no—
64C23×C3×C4C2×D511yescharacteristic
65C22×D12—514no—
66C22×D12—107no—
67C2×#213—5112no—
68C22×C3⋊C8C2×D5114yescomplemented
69C24×C5#6711yescharacteristic
70C23×D5—214no—
71C23×D5—48no—
72C23×D5—614no—
73C23×D5—128no—
74C22×C4×C5#6717yes—
75C22×C4×C5C22×D317yes—
76C2×D20—242no—
77C2×D20—328no—
78C2×D20—642no—
79C2×D20C22×D3128yescomplemented
80C2×C5×C8—328no—
81D40—364no—
82C23×D4—51no—
83C23×D4—151no—
84C22×D8—1528no—
85C23×C8—31no—
86C22×C3×C5C2×D8128yescomplemented
87C22×C3×C5C22×D417yes—
88C2×D15—28no—
89C2×D15—456no—
90C2×C3×D5—28no—
91C2×C3×D5—456no—
92C3×C4×C5C22×D417yes—
93C3×C4×C5C2511yescharacteristic
94C24×C3—1014no—
95C24×C3—208no—
96C24×C3D2011yescharacteristic
97C23×D3—1014no—
98C23×D3—208no—
99C2×C3×D4—528no—
100C2×C3×D4—1042no—
101C22×C3×C4D2017yes—
102C22×C3×C4C22×D517yes—
103C2×D12—528no—
104C2×D12—1042no—
105#213—564no—
106C2×C3⋊C8C22×D5128yescomplemented
107C23×C5#21318yescomplemented
108C23×C5C2×#6717yes—
109C22×D5—228no—
110C22×D5—456no—
111C22×D5—628no—
112C22×D5—1256no—
113C2×C4×C5C2×#67121yes—
114C2×C4×C5C23×D317yes—
115D20—228no—
116D20—38no—
117D20—628no—
118D20C23×D318yescomplemented
119C5×C8—38no—
120C25—101no—
121C25—301no—
122C22×D4—514no—
123C22×D4—107no—
124C22×D4—1514no—
125C22×D4—307no—
126C23×C4D3×D511yescharacteristic
127C2×D8—15112no—
128C22×C8—314no—
129C2×C3×C5C22×D8114yescomplemented
130C2×C3×C5C23×D411yescharacteristic
131D15—48no—
132C3×D5—48no—
133C23×C3—1028no—
134C23×C3—2056no—
135C23×C3D4018yescomplemented
136C23×C3C2×D2017yes—
137C22×D3—1028no—
138C22×D3—2056no—
139C3×D4—58no—
140C3×D4—1028no—
141C2×C3×C4C2×D20121yes—
142C2×C3×C4C23×D517yes—
143D12—58no—
144D12—1028no—
145C3⋊C8C23×D518yescomplemented
146C22×C5C2×#213128yescomplemented
147C22×C5C22×#6717yes—
148C2×D5—28no—
149C2×D5—456no—
150C2×D5—68no—
151C2×D5—1256no—
152C4×C5C22×#6717yes—
153C4×C5C24×D311yescharacteristic
154C24—1014no—
155C24—208no—
156C24—3014no—
157C24—608no—
158C24#120211yescharacteristic
159C2×D4—528no—
160C2×D4—1042no—
161C2×D4—1528no—
162C2×D4—3042no—
163C22×C4#120217yes—
164C22×C4C2×D3×D517yes—
165D8—1564no—
166C2×C8—328no—
167C3×C5C23×D811yescharacteristic, complemented, Hall
168C22×C3—108no—
169C22×C3—2056no—
170C22×C3C2×D40128yescomplemented
171C22×C3C22×D2017yes—
172C2×D3—108no—
173C2×D3—2056no—
174C3×C4C22×D2017yes—
175C3×C4C24×D511yescharacteristic
176C2×C5C22×#213114yescomplemented
177C2×C5C23×#6711yescharacteristic
178D5—48no—
179D5—128no—
180C23—1028no—
181C23—2056no—
182C23—3028no—
183C23—6056no—
184C23#665318yesdirect product factor
185C23C2×#120217yes—
186D4—58no—
187D4—1028no—
188D4—158no—
189D4—3028no—
190C2×C4C2×#1202121yes—
191C2×C4C22×D3×D517yes—
192C8—38no—
193C2×C3—208no—
194C2×C3C22×D40114yescomplemented
195C2×C3C23×D2011yescharacteristic
196D3—208no—
197C5C23×#21311yescharacteristic, complemented, Sylow
198C22—108no—
199C22—2056no—
200C22—308no—
201C22—6056no—
202C22C2×#6653128yesdirect product factor
203C22C22×#120217yes—
204C4C22×#120217yes—
205C4C23×D3×D511yescharacteristic
206C3C23×D4011yescharacteristic, complemented, Sylow
207C2—208no—
208C2—608no—
209C2C22×#6653114yesdirect product factor
210C2C23×#120211yescharacteristic
subgroupG/H
centerC24#1202
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×C3×D20C2semidirect
3. C23×C5×C3⋊C8C2semidirect
4. C22×#6653C2direct (N × H)
10. C22×D60C22semidirect
12. C22×C3×D20C22semidirect
13. C22×C5×C3⋊C8C22semidirect
14. C2×#6653C22direct (N × H)
17. C23×D20D3semidirect
28. C2×D60C23semidirect
30. C2×C3×D20C23semidirect
31. C2×C5×C3⋊C8C23semidirect
32. #6653C23direct (N × H)
36. C23×C3⋊C8D5semidirect
43. C22×D20C2×D3semidirect
47. C23×C3×C5D8semidirect
56. D60C24semidirect
58. C3×D20C24semidirect
59. C5×C3⋊C8C24semidirect
68. C22×C3⋊C8C2×D5semidirect
79. C2×D20C22×D3semidirect
86. C22×C3×C5C2×D8semidirect
106. C2×C3⋊C8C22×D5semidirect
107. C23×C5#213semidirect
118. D20C23×D3semidirect
129. C2×C3×C5C22×D8semidirect
135. C23×C3D40semidirect
145. C3⋊C8C23×D5semidirect
146. C22×C5C2×#213semidirect
167. C3×C5C23×D8semidirect
170. C22×C3C2×D40semidirect
176. C2×C5C22×#213semidirect
184. C23#6653direct (N × H)
194. C2×C3C22×D40semidirect
197. C5C23×#213semidirect
202. C22C2×#6653direct (N × H)
206. C3C23×D40semidirect
209. C2C22×#6653direct (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×C3×D202
C23×C5×C3⋊C82
C22×#66532
C23×D403
C23×#2135