Finite Group Databasebeta

← C23×#6652 #14509652

Subgroups — C23×#6652 #14509652

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×C5×D12C211yesmaximal, characteristic, complemented
2C23×C3×D20C211yesmaximal, characteristic, complemented
3C23×C3⋊#168C211yesmaximal, characteristic, complemented
4C22×#6652C2128yesmaximal, direct product factor
5C23×#686—31nomaximal, Hall
6C24×C5×D3—21no—
7C24×C3×D5—21no—
8C23×C3×C4×C5C2211yescharacteristic
9C22×C5×D12—27no—
10C22×C5×D12C22114yescomplemented
11C22×C3×D20—27no—
12C22×C3×D20C22114yescomplemented
13C22×C3⋊#168C22114yescomplemented
14C2×#6652C221112yesdirect product factor
15C23×#213—51nomaximal, Hall
16C23×C5×D4—31no—
17C23×D20D311yescharacteristic, complemented
18C22×#686—328no—
19C23×#168—31no—
20C24×C3×C5D411yescharacteristic
21C23×C5×D3—214no—
22C23×C5×D3—48no—
23C23×C3×D5—214no—
24C23×C3×D5—48no—
25C22×C3×C4×C5D417yes—
26C22×C3×C4×C5C2317yes—
27C2×C5×D12—242no—
28C2×C5×D12C23128yescomplemented
29C2×C3×D20—242no—
30C2×C3×D20C23128yescomplemented
31C2×C3⋊#168C23128yescomplemented
32#6652C23164yesdirect product factor
33C23×C3×D4—51no—
34C23×D12D511yescharacteristic, complemented
35C22×#213—528no—
36C23×C3⋊C8—51no—
37C25×C5—61no—
38C24×D5—21no—
39C22×C5×D4—314no—
40C22×C5×D4—67no—
41C23×C4×C5C2×D311yescharacteristic
42C22×D20—27no—
43C22×D20C2×D3114yescomplemented
44C2×#686—3112no—
45C22×#168—314no—
46C23×D8—151noSylow
47C23×C3×C5D818yescomplemented
48C23×C3×C5C2×D417yes—
49C22×C5×D3—228no—
50C22×C5×D3—456no—
51C22×C3×D5—228no—
52C22×C3×D5—456no—
53C2×C3×C4×C5C2×D4121yes—
54C2×C3×C4×C5C2417yes—
55C5×D12—228no—
56C5×D12C2418yescomplemented
57C3×D20—228no—
58C3×D20C2418yescomplemented
59C3⋊#168C2418yescomplemented
60C25×C3—101no—
61C24×D3—21no—
62C22×C3×D4—514no—
63C22×C3×D4—107no—
64C23×C3×C4C2×D511yescharacteristic
65C22×D12—27no—
66C22×D12C2×D5114yescomplemented
67C2×#213—5112no—
68C22×C3⋊C8—514no—
69C24×C5—614no—
70C24×C5—128no—
71C24×C5#6711yescharacteristic
72C23×D5—214no—
73C23×D5—48no—
74C2×C5×D4—328no—
75C2×C5×D4—642no—
76C22×C4×C5#6717yes—
77C22×C4×C5C22×D317yes—
78C2×D20—242no—
79C2×D20C22×D3128yescomplemented
80#686—364no—
81C2×#168—328no—
82C23×D4—31no—
83C23×D4—51no—
84C22×D8—1528no—
85C23×C8—151no—
86C22×C3×C5C2×D8128yescomplemented
87C22×C3×C5C22×D417yes—
88C2×C5×D3—28no—
89C2×C5×D3—456no—
90C2×C3×D5—28no—
91C2×C3×D5—456no—
92C3×C4×C5C22×D417yes—
93C3×C4×C5C2511yescharacteristic
94C24×C3—1014no—
95C24×C3—208no—
96C24×C3C22⋊D511yescharacteristic
97C23×D3—214no—
98C23×D3—48no—
99C2×C3×D4—528no—
100C2×C3×D4—1042no—
101C22×C3×C4C22⋊D517yes—
102C22×C3×C4C22×D517yes—
103C2×D12—242no—
104C2×D12C22×D5128yescomplemented
105#213—564no—
106C2×C3⋊C8—528no—
107C23×C5—628no—
108C23×C5—1256no—
109C23×C5#21318yescomplemented
110C23×C5C2×#6717yes—
111C22×D5—228no—
112C22×D5—456no—
113C5×D4—38no—
114C5×D4—628no—
115C2×C4×C5C2×#67121yes—
116C2×C4×C5C23×D317yes—
117D20—228no—
118D20C23×D318yescomplemented
119#168—38no—
120C25—61no—
121C25—101no—
122C22×D4—314no—
123C22×D4—514no—
124C22×D4—67no—
125C22×D4—107no—
126C23×C4D3×D511yescharacteristic
127C2×D8—15112no—
128C22×C8—1514no—
129C2×C3×C5C22×D8114yescomplemented
130C2×C3×C5C23×D411yescharacteristic
131C3×D5—48no—
132C5×D3—48no—
133C23×C3—1028no—
134C23×C3—2056no—
135C23×C3#68618yescomplemented
136C23×C3C2×C22⋊D517yes—
137C22×D3—228no—
138C22×D3—456no—
139C3×D4—58no—
140C3×D4—1028no—
141C2×C3×C4C2×C22⋊D5121yes—
142C2×C3×C4C23×D517yes—
143D12—228no—
144D12C23×D518yescomplemented
145C3⋊C8—58no—
146C22×C5—68no—
147C22×C5—1256no—
148C22×C5C2×#213128yescomplemented
149C22×C5C22×#6717yes—
150C2×D5—28no—
151C2×D5—456no—
152C4×C5C22×#6717yes—
153C4×C5C24×D311yescharacteristic
154C24—614no—
155C24—1014no—
156C24—128no—
157C24—208no—
158C24#120111yescharacteristic
159C2×D4—328no—
160C2×D4—528no—
161C2×D4—642no—
162C2×D4—1042no—
163C22×C4#120117yes—
164C22×C4C2×D3×D517yes—
165D8—1564no—
166C2×C8—1528no—
167C3×C5C23×D811yescharacteristic, complemented, Hall
168C22×C3—108no—
169C22×C3—2056no—
170C22×C3C2×#686128yescomplemented
171C22×C3C22×C22⋊D517yes—
172C2×D3—28no—
173C2×D3—456no—
174C3×C4C22×C22⋊D517yes—
175C3×C4C24×D511yescharacteristic
176C2×C5—128no—
177C2×C5C22×#213114yescomplemented
178C2×C5C23×#6711yescharacteristic
179D5—48no—
180C23—628no—
181C23—1028no—
182C23—1256no—
183C23—2056no—
184C23#665218yesdirect product factor
185C23C2×#120117yes—
186D4—38no—
187D4—58no—
188D4—628no—
189D4—1028no—
190C2×C4C2×#1201121yes—
191C2×C4C22×D3×D517yes—
192C8—158no—
193C2×C3—208no—
194C2×C3C22×#686114yescomplemented
195C2×C3C23×C22⋊D511yescharacteristic
196D3—48no—
197C5C23×#21311yescharacteristic, complemented, Sylow
198C22—68no—
199C22—108no—
200C22—1256no—
201C22—2056no—
202C22C2×#6652128yesdirect product factor
203C22C22×#120117yes—
204C4C22×#120117yes—
205C4C23×D3×D511yescharacteristic
206C3C23×#68611yescharacteristic, complemented, Sylow
207C2—128no—
208C2—208no—
209C2C22×#6652114yesdirect product factor
210C2C23×#120111yescharacteristic
subgroupG/H
centerC24#1201
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×C5×D12C2semidirect
2. C23×C3×D20C2semidirect
3. C23×C3⋊#168C2semidirect
4. C22×#6652C2direct (N × H)
10. C22×C5×D12C22semidirect
12. C22×C3×D20C22semidirect
13. C22×C3⋊#168C22semidirect
14. C2×#6652C22direct (N × H)
17. C23×D20D3semidirect
28. C2×C5×D12C23semidirect
30. C2×C3×D20C23semidirect
31. C2×C3⋊#168C23semidirect
32. #6652C23direct (N × H)
34. C23×D12D5semidirect
43. C22×D20C2×D3semidirect
47. C23×C3×C5D8semidirect
56. C5×D12C24semidirect
58. C3×D20C24semidirect
59. C3⋊#168C24semidirect
66. C22×D12C2×D5semidirect
79. C2×D20C22×D3semidirect
86. C22×C3×C5C2×D8semidirect
104. C2×D12C22×D5semidirect
109. C23×C5#213semidirect
118. D20C23×D3semidirect
129. C2×C3×C5C22×D8semidirect
135. C23×C3#686semidirect
144. D12C23×D5semidirect
148. C22×C5C2×#213semidirect
167. C3×C5C23×D8semidirect
170. C22×C3C2×#686semidirect
177. C2×C5C22×#213semidirect
184. C23#6652direct (N × H)
194. C2×C3C22×#686semidirect
197. C5C23×#213semidirect
202. C22C2×#6652direct (N × H)
206. C3C23×#686semidirect
209. C2C22×#6652direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C5×D122
C23×C3×D202
C23×C3⋊#1682
C22×#66522
C23×#6863
C23×#2135