Finite Group Databasebeta

← #72231449

Subgroups — #72231449

Order 2400.

1040 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 1040 conjugacy classes of subgroups — too many to draw. Only its 170 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×C3⋊(C5⋊D5)C211yesmaximal, characteristic, complemented
2C2×C52×#67C211yesmaximal, characteristic, complemented
3C2×#10597116C212yesmaximal, complemented
4C3×C22⋊#1039C211yesmaximal, characteristic, complemented
5#11925120C212yesmaximal, complemented
6#11717257—31nomaximal, Hall
7C23×C3×C52C2211yescharacteristic
8C22×C3⋊(C5⋊D5)—22no—
9C22×C3⋊(C5⋊D5)—24no—
10C22×C3⋊(C5⋊D5)C2212yes—
11C22×C52×D3C2211yescharacteristic, complemented
12C52×#67—22no—
13C2×C52×Dic3C2211yescharacteristic, complemented
14C2×C3×#1039C2212yescomplemented
15#10597116—24no—
16#91573—56nomaximal
17C23×C5⋊D5—31no—
18C2×C52×D4—31no—
19C2×C22⋊(C5⋊D5)—32no—
20C22⋊#1039D311yescharacteristic, complemented
21#87808—32no—
22C22×C3×C52—22no—
23C22×C3×C52D412yescomplemented
24C22×C3×C52C2311yescharacteristic
25C2×C3⋊(C5⋊D5)—28no—
26C2×C3⋊(C5⋊D5)—22no—
27C2×C3⋊(C5⋊D5)—22no—
28C2×C3⋊(C5⋊D5)D414yescomplemented
29C2×C52×D3—21no—
30C2×C52×D3—22no—
31C52×Dic3—21no—
32C3×#1039—22no—
33C23×D15—56no—
34C2×C5×#67D516yescomplemented
35C2×#1203—512no—
36C3×C22⋊Dic5—56no—
37#6667—512no—
38C23×C52—31no—
39C23×C52C2×D311yescharacteristic
40C22×C5⋊D5—32no—
41C22×C5⋊D5—64no—
42C22×C5⋊D5—62no—
43C52×D4—62no—
44C2×C4×C52—31no—
45C22⋊(C5⋊D5)—64no—
46C2×#1039C2×D312yescomplemented
47#4083—156no—
48C2×C3×C52—22no—
49C2×C3×C52C2×D411yescharacteristic
50C2×C3×C52C2×D412yes—
51C3⋊(C5⋊D5)—24no—
52C52×D3—41no—
53C23×C3×C5C2×D516yes—
54C22×D15—512no—
55C22×D15—1024no—
56C22×D15—1012no—
57C22×C5×D3C2×D516yescomplemented
58C5×#67—212no—
59C2×C5×Dic3C2×D516yescomplemented
60C2×C3×Dic5—512no—
61#1203—1024no—
62C22×C52—22no—
63C22×C52—61no—
64C22×C52—62no—
65C22×C52D1212yescomplemented
66C22×C52C22×D311yescharacteristic
67C2×C5⋊D5—34no—
68C2×C5⋊D5—68no—
69C2×C5⋊D5—62no—
70C2×C5⋊D5—62no—
71C4×C52—61no—
72#1039—22no—
73#882—251noHall
74C23×D5—156no—
75C2×C5×D4—36no—
76C2×C22⋊D5—1512no—
77C22⋊Dic5—56no—
78#685—1512no—
79C3×C52#12011yescharacteristic, complemented, Hall
80C22×C3×C5—212no—
81C22×C3×C5C22⋊D5112yescomplemented
82C22×C3×C5C22×D516yes—
83C2×D15—524no—
84C2×D15—1048no—
85C2×D15—1012no—
86C2×D15—1012no—
87C2×C5×D3—212no—
88C2×C5×D3—26no—
89C3×Dic5—1012no—
90C5×Dic3—26no—
91C2×C52—22no—
92C2×C52—121no—
93C2×C52C2×D1211yescharacteristic
94C2×C52D3×D412yes—
95C5⋊D5—64no—
96C23×D3—251no—
97C2×#67C5⋊D511yescharacteristic, complemented
98C2×D12—252no—
99C3×C22⋊C4—251no—
100(C2×D3)⋊C4—252no—
101C23×C5—36no—
102C23×C5D3×D516yes—
103C22×D5—1512no—
104C22×D5—3024no—
105C22×D5—3012no—
106C5×D4—612no—
107C2×C4×C5—36no—
108C22⋊D5—3024no—
109C2×Dic5—512no—
110#120—751noSylow
111C2×C3×C5—212no—
112C2×C3×C5D4×D5112yes—
113C2×C3×C5C2×C22⋊D516yes—
114D15—1024no—
115C5×D3—46no—
116C52#88211yescharacteristic, complemented, Sylow
117C23×C3C2×C5⋊D511yescharacteristic
118C22×D3—252no—
119C22×D3—504no—
120C22×D3—502no—
121C22×D3C2×C5⋊D511yescharacteristic, complemented
122C2×C3×C4—252no—
123#67—22no—
124C2×Dic3C2×C5⋊D511yescharacteristic, complemented
125D12—504no—
126C22×C5—212no—
127C22×C5—66no—
128C22×C5—612no—
129C22×C5#1203112yescomplemented
130C22×C5C2×D3×D516yes—
131C2×D5—1524no—
132C2×D5—3048no—
133C2×D5—3012no—
134C2×D5—3012no—
135C4×C5—66no—
136Dic5—1012no—
137C24—751no—
138C2×D4—31no—
139C2×D4—752no—
140C22⋊C4—251no—
141C22⋊C4—752no—
142C3×C5#408316yescomplemented
143C22×C3—22no—
144C22×C3C22⋊(C5⋊D5)12yescomplemented
145C22×C3C22×C5⋊D511yescharacteristic
146C2×D3—22no—
147C2×D3—21no—
148C2×D3—254no—
149C2×D3—508no—
150C2×D3—502no—
151C2×D3—502no—
152C3×C4—502no—
153Dic3—21no—
154C2×C5—212no—
155C2×C5—126no—
156C2×C5C2×#120316yes—
157C2×C5#6790112yes—
158D5—3024no—
159C23—31no—
160C23—752no—
161C23—1504no—
162C23—1502no—
163C23D3×C5⋊D511yescharacteristic
164D4—62no—
165D4—1504no—
166C2×C4—31no—
167C2×C4—252no—
168C2×C3—22no—
169C2×C3D4×C5⋊D512yes—
170C2×C3C2×C22⋊(C5⋊D5)11yescharacteristic
171D3—41no—
172D3—504no—
173C5#9157316yescomplemented
174C22—22no—
175C22—61no—
176C22—62no—
177C22—754no—
178C22—1508no—
179C22—1502no—
180C22—1502no—
181C22#1059711612yescomplemented
182C22C2×D3×C5⋊D511yescharacteristic
183C4—61no—
184C4—502no—
185C3#1171725711yescharacteristic, complemented, Sylow
186C2—22no—
187C2—121no—
188C2—1504no—
189C2C2×#1059711611yescharacteristic
190C2#1192561912yes—
subgroupG/H
center, FrattiniC22C2×D3×C5⋊D5
commutator, socle, cosocleC22×C3×C52C23
FittingC23×C3×C52C22
hypercenterC23D3×C5⋊D5
hypocenterC3×C52#120
LCSG ▷ C22×C3×C52 ▷ C3×C52
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C23×C3⋊(C5⋊D5)C2semidirect
2. C2×C52×#67C2semidirect
3. C2×#10597116C2semidirect
4. C3×C22⋊#1039C2semidirect
5. #11925120C2semidirect
11. C22×C52×D3C22semidirect
13. C2×C52×Dic3C22semidirect
14. C2×C3×#1039C22semidirect
20. C22⋊#1039D3semidirect
23. C22×C3×C52D4semidirect
28. C2×C3⋊(C5⋊D5)D4semidirect
34. C2×C5×#67D5semidirect
46. C2×#1039C2×D3semidirect
57. C22×C5×D3C2×D5semidirect
59. C2×C5×Dic3C2×D5semidirect
65. C22×C52D12semidirect
79. C3×C52#120semidirect
81. C22×C3×C5C22⋊D5semidirect
97. C2×#67C5⋊D5semidirect
116. C52#882semidirect
121. C22×D3C2×C5⋊D5semidirect
124. C2×Dic3C2×C5⋊D5semidirect
129. C22×C5#1203semidirect
142. C3×C5#4083semidirect
144. C22×C3C22⋊(C5⋊D5)semidirect
173. C5#91573semidirect
181. C22#10597116semidirect
185. C3#11717257semidirect

Maximal subgroups

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

subgroupindex
C23×C3⋊(C5⋊D5)2
C2×C52×#672
C2×#105971162
C3×C22⋊#10392
#119251202
#117172573
#915735