Finite Group Databasebeta

← #72260273

Subgroups — #72260273

Order 2400.

772 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C5×#6817C212yesmaximal, complemented
2C2×C3×C4⋊(C5⋊D5)C211yesmaximal, characteristic, complemented
3#11925664C214yesmaximal, complemented
4#11925654C214yesmaximal, complemented
5#11925652C214yesmaximal, complemented
6#11717473—31nomaximal, Hall
7C22×C3×C5⋊D5C2212yescomplemented
8C2×D15⋊D5C2214yescomplemented
9C2×C3×C4×C52C2211yescharacteristic, complemented
10C5×#1220C2214yescomplemented
11C5×D60C2212yescomplemented
12C4×C5×D15C2214yescomplemented
13C5×Dic30C2212yescomplemented
14C3×C4⋊(C5⋊D5)C2214yescomplemented
15#10597127C2218yescomplemented
16#10597126C2214yescomplemented
17#92017—52nomaximal
18C2×C4⋊(C5⋊D5)D311yescharacteristic, complemented
19C5×#709—32no—
20#87886—34no—
21D5×D20—34no—
22#87874—34no—
23C22×C3×C52C2311yescharacteristic
24C2×C5×D15C2314yes—
25C2×C3×C5⋊D5—42no—
26C2×C3×C5⋊D5C2314yes—
27D15⋊D5—44no—
28C3×C4×C52C2312yes—
29C5×Dic15C2314yes—
30#6817D512yescomplemented
31C5×#235—52no—
32C2×C3×D20—52no—
33C2×C3×D20—102no—
34D5×#67—58no—
35#6777—54no—
36D3×D20—54no—
37#6766—54no—
38#6765—54no—
39C22×C5⋊D5C2×D312yescomplemented
40C2×D52—34no—
41C2×C4×C52C2×D311yescharacteristic, complemented
42C4⋊(C5⋊D5)C2×D314yescomplemented
43C5×C22⋊D5—34no—
44C5×D20—32no—
45C4×C5×D5—34no—
46C5×Dic10—32no—
47#6038—38no—
48#6036—34no—
49#4149—152no—
50C2×C3×C52—21no—
51C2×C3×C52C2411yescharacteristic
52C5×D15—24no—
53C3×C5⋊D5—24no—
54C22×C3×D5—54no—
55C22×C3×D5—104no—
56C2×D3×D5—58no—
57C2×C3×C4×C5—22no—
58C2×C3×C4×C5C2×D512yescomplemented
59#1220C2×D514yescomplemented
60D60C2×D512yescomplemented
61C4×D15C2×D514yescomplemented
62Dic30C2×D512yescomplemented
63C5×#67—54no—
64C5×D12—52no—
65C4×C5×D3—54no—
66C5×Dic6—52no—
67C3×D20—54no—
68C3×D20—54no—
69C3×D20—108no—
70#1202—58no—
71#1201—58no—
72D5×Dic3—58no—
73C22×C52C22×D311yescharacteristic
74C2×C5⋊D5—42no—
75C2×C5⋊D5C22×D314yes—
76C2×C5×D5—34no—
77D52—124no—
78C4×C52C22×D312yes—
79C5×Dic5—34no—
80#1004—251noHall
81C5×#41—152no—
82#713—154no—
83D4×D5—154no—
84D4×D5—158no—
85#709—32no—
86#709—154no—
87C2×D20—52no—
88C2×D20—102no—
89C2×D20—154no—
90C3×C52H(5,𝔽2)11yescharacteristic, complemented, Hall
91C22×C3×C5—22no—
92C22×C3×C5C22×D512yes—
93C2×D15C22×D514yes—
94C2×C5×D3—54no—
95C2×C3×D5—58no—
96C2×C3×D5—108no—
97C2×C3×D5—204no—
98C2×C3×D5—208no—
99D3×D5—208no—
100C3×C4×C5—24no—
101C3×C4×C5C22×D512yes—
102C3×C4×C5C22×D512yes—
103Dic15C22×D514yes—
104C5×Dic3—54no—
105C2×C52—21no—
106C2×C52C23×D311yescharacteristic
107C5⋊D5—24no—
108C5×D5—64no—
109C2×C3×D4—251no—
110C2×#67—254no—
111#237—254no—
112D3×D4—254no—
113#235—52no—
114C22×D5—54no—
115C22×D5—104no—
116C22×D5—158no—
117C5×Q8—152no—
118C5×D4—154no—
119C5×D4—152no—
120C2×C4×C5—22no—
121C2×C4×C5—154no—
122C2×C4×C5D3×D512yescomplemented
123C22⋊D5—34no—
124C22⋊D5—158no—
125D20—32no—
126D20—54no—
127D20—54no—
128D20—108no—
129D20—158no—
130C4×D5—34no—
131C4×D5—158no—
132Dic10—32no—
133H(5,𝔽2)—751noSylow
134C2×C3×C5—24no—
135C2×C3×C5—22no—
136C2×C3×C5—22no—
137C2×C3×C5C23×D512yes—
138D15—24no—
139C3×D5—108no—
140C3×D5—208no—
141C5×D3—104no—
142C52#100411yescharacteristic, complemented, Sylow
143C23×C3—252no—
144C22×D3—254no—
145C3×D4—254no—
146C2×C3×C4D5211yescharacteristic, complemented
147#67—54no—
148#67—258no—
149C2×Dic3—254no—
150D12—52no—
151C4×D3—54no—
152Dic6—52no—
153C22×C5—22no—
154C22×C5—154no—
155C22×C5C2×D3×D512yes—
156C2×D5—34no—
157C2×D5—58no—
158C2×D5—108no—
159C2×D5—204no—
160C2×D5—208no—
161C2×D5—608no—
162C4×C5—24no—
163C4×C5—154no—
164C4×C5C2×D3×D512yes—
165C4×C5C2×D3×D512yes—
166Dic5—34no—
167#41—152no—
168#41—754no—
169C2×D4—251no—
170C2×D4—754no—
171C2×D4—754no—
172C3×C5—22no—
173C3×C5#414912yescomplemented
174C22×C3—254no—
175C22×C3—1002no—
176C22×C3C2×D5211yescharacteristic
177C2×D3—54no—
178C2×D3—1004no—
179C3×C4C2×D5212yes—
180Dic3—54no—
181C2×C5—24no—
182C2×C5—22no—
183C2×C5—22no—
184C2×C5—304no—
185C2×C5C22×D3×D512yes—
186D5—64no—
187D5—108no—
188D5—208no—
189C23—252no—
190C23—754no—
191Q8—152no—
192D4—154no—
193D4—152no—
194D4—254no—
195D4—758no—
196C2×C4—154no—
197C2×C4—754no—
198C2×C4D15⋊D511yescharacteristic, complemented
199C2×C3—21no—
200C2×C3—504no—
201C2×C3C22×D5211yescharacteristic
202D3—104no—
203C5—22no—
204C5#9201712yescomplemented
205C22—154no—
206C22—254no—
207C22—3004no—
208C22—1002no—
209C22C2×D15⋊D511yescharacteristic
210C4—154no—
211C4C2×D15⋊D512yes—
212C3#1171747311yescharacteristic, complemented, Sylow
213C2—21no—
214C2—304no—
215C2—504no—
216C2C22×D15⋊D511yescharacteristic
subgroupG/H
center, FrattiniC2C22×D15⋊D5
commutator, socle, cosocleC2×C3×C52C24
FittingC2×C3×C4×C52C22
hypercenterC2×C4D15⋊D5
hypocenterC3×C52H(5,𝔽2)
LCSG ▷ C2×C3×C52 ▷ C3×C52
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C5×#6817C2semidirect
2. C2×C3×C4⋊(C5⋊D5)C2semidirect
3. #11925664C2semidirect
4. #11925654C2semidirect
5. #11925652C2semidirect
7. C22×C3×C5⋊D5C22semidirect
8. C2×D15⋊D5C22semidirect
9. C2×C3×C4×C52C22semidirect
10. C5×#1220C22semidirect
11. C5×D60C22semidirect
12. C4×C5×D15C22semidirect
13. C5×Dic30C22semidirect
14. C3×C4⋊(C5⋊D5)C22semidirect
15. #10597127C22semidirect
16. #10597126C22semidirect
18. C2×C4⋊(C5⋊D5)D3semidirect
30. #6817D5semidirect
39. C22×C5⋊D5C2×D3semidirect
41. C2×C4×C52C2×D3semidirect
42. C4⋊(C5⋊D5)C2×D3semidirect
58. C2×C3×C4×C5C2×D5semidirect
59. #1220C2×D5semidirect
60. D60C2×D5semidirect
61. C4×D15C2×D5semidirect
62. Dic30C2×D5semidirect
90. C3×C52H(5,𝔽2)semidirect
122. C2×C4×C5D3×D5semidirect
142. C52#1004semidirect
146. C2×C3×C4D52semidirect
173. C3×C5#4149semidirect
198. C2×C4D15⋊D5semidirect
204. C5#92017semidirect
212. C3#11717473semidirect

Maximal subgroups

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

subgroupindex
C5×#68172
C2×C3×C4⋊(C5⋊D5)2
#119256642
#119256542
#119256522
#117174733
#920175