Finite Group Databasebeta

← C5×#91821 #72230986

Subgroups — C5×#91821 #72230986

Order 2400.

380 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 94 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
1C2×C3×C52×D4C211yesmaximal, characteristic, complemented
2C5×#6817C211yesmaximal, characteristic, complemented
3C5×#6716C212yesmaximal, complemented
4C5×#6715C212yesmaximal, complemented
5C5×#6710C211yesmaximal, characteristic, complemented
6C5×#4078—31nomaximal, Hall
7C23×C3×C52—21no—
8C3×C52×D4—21no—
9C3×C52×D4C2212yes—
10C2×C3×C4×C52C2211yescharacteristic
11C5×#1220—21no—
12C5×D60C2211yescharacteristic, complemented
13C4×C5×D15—21no—
14C5×Dic30C2211yescharacteristic, complemented
15C5×C3⋊#168C2212yescomplemented
16#91821C511yesmaximal, characteristic, direct product factor
17C5×#932—51nomaximal
18C2×C52×D4D311yescharacteristic, complemented
19C5×#709—31no—
20C5×#687—32no—
21C5×#686—32no—
22C5×#681—31no—
23C22×C3×C52—22no—
24C22×C3×C52—42no—
25C22×C3×C52D411yescharacteristic
26C2×C5×D15—21no—
27C3×C4×C52D411yescharacteristic
28C3×C4×C52C2311yescharacteristic
29C5×Dic15—21no—
30C2×C3×C5×D4—22no—
31C2×C3×C5×D4D511yescharacteristic, complemented
32C2×C3×C5×D4C2×C511yescharacteristic, complemented
33#6817C2×C511yescharacteristic, complemented
34C5×#235—51no—
35#6716C2×C512yescomplemented
36#6715C2×C512yescomplemented
37#6710C2×C511yescharacteristic, complemented
38C5×#214—52no—
39C5×#213—52no—
40C5×#208—51no—
41C23×C52—21no—
42C52×D4—21no—
43C52×D4C2×D312yes—
44C2×C4×C52C2×D311yescharacteristic
45C5×C22⋊D5—61no—
46C5×D20—31no—
47C4×C5×D5—61no—
48C5×Dic10—31no—
49C5×#168—32no—
50C5×AGL(1,ℤ/8)—151no—
51#4078—31no—
52C2×C3×C52—21no—
53C2×C3×C52—42no—
54C2×C3×C52C2×D411yescharacteristic
55C5×D15—41no—
56C23×C3×C5—22no—
57C23×C3×C5—22no—
58C23×C3×C5—21no—
59C23×C3×C5—21no—
60C3×C5×D4—22no—
61C3×C5×D4—22no—
62C3×C5×D4—24no—
63C3×C5×D4—21no—
64C3×C5×D4—21no—
65C3×C5×D4C2×D512yes—
66C3×C5×D4C22×C512yes—
67C2×C3×C4×C5—22no—
68C2×C3×C4×C5C2×D511yescharacteristic
69C2×C3×C4×C5C22×C511yescharacteristic
70#1220—21no—
71D60C22×C511yescharacteristic, complemented
72C4×D15—21no—
73Dic30C22×C511yescharacteristic, complemented
74C5×#67—101no—
75C5×D12—51no—
76C4×C5×D3—101no—
77C5×Dic6—51no—
78C3⋊#168C22×C512yescomplemented
79C5×C3⋊C8—52no—
80C22×C52—22no—
81C22×C52—42no—
82C22×C52#6711yescharacteristic
83C2×C5×D5—61no—
84C4×C52#6711yescharacteristic
85C4×C52C22×D311yescharacteristic
86C5×Dic5—61no—
87#932—51noHall
88C5×#41—151no—
89C2×C5×D4—22no—
90C2×C5×D4C5×D311yescharacteristic, complemented
91C2×C5×D4D1511yescharacteristic, complemented
92#709—31no—
93C5×SD16—152no—
94C5×D8—152no—
95C5×M16—151no—
96#687—32no—
97#686—32no—
98#681—31no—
99C3×C52AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
100C22×C3×C5—24no—
101C22×C3×C5—24no—
102C22×C3×C5—22no—
103C22×C3×C5—22no—
104C22×C3×C5—22no—
105C22×C3×C5—44no—
106C22×C3×C5—44no—
107C22×C3×C5—42no—
108C22×C3×C5—42no—
109C22×C3×C5C22⋊D511yescharacteristic
110C22×C3×C5C5×D411yescharacteristic
111C2×D15—21no—
112C2×C5×D3—101no—
113C3×C4×C5—22no—
114C3×C4×C5—22no—
115C3×C4×C5C22⋊D511yescharacteristic
116C3×C4×C5C5×D411yescharacteristic
117C3×C4×C5C22×D511yescharacteristic
118C3×C4×C5C23×C511yescharacteristic
119Dic15—21no—
120C5×Dic3—101no—
121C2×C52—21no—
122C2×C52—42no—
123C2×C52C2×#6711yescharacteristic
124C5×D5—121no—
125C2×C3×D4C5×D511yescharacteristic, complemented
126#235—51no—
127#214—52no—
128#213—52no—
129#208—51no—
130C23×C5—22no—
131C23×C5—22no—
132C23×C5—21no—
133C23×C5—21no—
134C5×Q8—151no—
135C5×D4—22no—
136C5×D4—22no—
137C5×D4—21no—
138C5×D4—21no—
139C5×D4—24no—
140C5×D4—151no—
141C5×D4—301no—
142C5×D4C2×C5×D312yes—
143C5×D4C2×D1512yes—
144C2×C4×C5—22no—
145C2×C4×C5—301no—
146C2×C4×C5C2×C5×D311yescharacteristic
147C2×C4×C5C2×D1511yescharacteristic
148C22⋊D5—61no—
149D20—31no—
150C4×D5—61no—
151Dic10—31no—
152C5×C8—152no—
153#168—32no—
154AGL(1,ℤ/8)—151noSylow
155C2×C3×C5—22no—
156C2×C3×C5—22no—
157C2×C3×C5—21no—
158C2×C3×C5—21no—
159C2×C3×C5—22no—
160C2×C3×C5—44no—
161C2×C3×C5—44no—
162C2×C3×C5—42no—
163C2×C3×C5—42no—
164C2×C3×C5C2×C22⋊D511yescharacteristic
165C2×C3×C5C2×C5×D411yescharacteristic
166D15—41no—
167C5×D3—201no—
168C52#93211yescharacteristic, complemented, Sylow
169C23×C3—21no—
170C3×D4—21no—
171C3×D4C2×C5×D512yes—
172C2×C3×C4C2×C5×D511yescharacteristic
173#67—101no—
174D12—51no—
175C4×D3—101no—
176Dic6—51no—
177C3⋊C8—52no—
178C22×C5—24no—
179C22×C5—24no—
180C22×C5—22no—
181C22×C5—22no—
182C22×C5—22no—
183C22×C5—44no—
184C22×C5—44no—
185C22×C5—42no—
186C22×C5—42no—
187C22×C5—301no—
188C22×C5C5×#6711yescharacteristic
189C22×C5#122011yescharacteristic
190C2×D5—61no—
191C4×C5—22no—
192C4×C5—22no—
193C4×C5—301no—
194C4×C5C5×#6711yescharacteristic
195C4×C5#122011yescharacteristic
196C4×C5C22×C5×D311yescharacteristic
197C4×C5C22×D1511yescharacteristic
198Dic5—61no—
199#41—151no—
200C2×D4C5×D1511yescharacteristic, complemented
201SD16—152no—
202D8—152no—
203M16—151no—
204C3×C5—22no—
205C3×C5#407811yescharacteristic, complemented
206C3×C5C5×AGL(1,ℤ/8)11yescharacteristic, complemented
207C22×C3—22no—
208C22×C3—42no—
209C22×C3C5×C22⋊D511yescharacteristic
210C2×D3—101no—
211C3×C4C5×C22⋊D511yescharacteristic
212C3×C4C22×C5×D511yescharacteristic
213Dic3—101no—
214C2×C5—22no—
215C2×C5—22no—
216C2×C5—21no—
217C2×C5—21no—
218C2×C5—22no—
219C2×C5—44no—
220C2×C5—44no—
221C2×C5—42no—
222C2×C5—42no—
223C2×C5—601no—
224C2×C5C2×C5×#6711yescharacteristic
225C2×C5C2×#122011yescharacteristic
226D5—121no—
227C23—21no—
228Q8—151no—
229D4—21no—
230D4—151no—
231D4—301no—
232D4C2×C5×D1512yes—
233C2×C4—301no—
234C2×C4C2×C5×D1511yescharacteristic
235C8—152no—
236C2×C3—21no—
237C2×C3—42no—
238C2×C3C2×C5×C22⋊D511yescharacteristic
239D3—201no—
240C5—22no—
241C5C5×#93211yescharacteristic, complemented
242C5#9182111yescharacteristic, direct product factor
243C22—22no—
244C22—42no—
245C22—301no—
246C22C5×#122011yescharacteristic
247C4—301no—
248C4C5×#122011yescharacteristic
249C4C22×C5×D1511yescharacteristic
250C3C5×#407811yescharacteristic, complemented, Sylow
251C2—21no—
252C2—42no—
253C2—601no—
254C2C2×C5×#122011yescharacteristic
subgroupG/H
centerC2×C5C2×#1220
commutator, cosocleC3×C4×C5C23×C5
FrattiniC4C22×C5×D15
FittingC2×C3×C52×D4C2
socleC2×C3×C52C2×D4
hypercenterC2×C5×D4D15
hypocenterC3×C5C5×AGL(1,ℤ/8)
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C2×C5×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C52×D4C2semidirect
2. C5×#6817C2semidirect
3. C5×#6716C2semidirect
4. C5×#6715C2semidirect
5. C5×#6710C2semidirect
12. C5×D60C22semidirect
14. C5×Dic30C22semidirect
15. C5×C3⋊#168C22semidirect
16. #91821C5direct (N × H)
18. C2×C52×D4D3semidirect
31. C2×C3×C5×D4D5semidirect
32. C2×C3×C5×D4C2×C5semidirect
33. #6817C2×C5semidirect
35. #6716C2×C5semidirect
36. #6715C2×C5semidirect
37. #6710C2×C5semidirect
71. D60C22×C5semidirect
73. Dic30C22×C5semidirect
78. C3⋊#168C22×C5semidirect
90. C2×C5×D4C5×D3semidirect
91. C2×C5×D4D15semidirect
99. C3×C52AGL(1,ℤ/8)semidirect
125. C2×C3×D4C5×D5semidirect
168. C52#932semidirect
200. C2×D4C5×D15semidirect
205. C3×C5#4078semidirect
206. C3×C5C5×AGL(1,ℤ/8)semidirect
241. C5C5×#932semidirect
242. C5#91821direct (N × H)
250. C3C5×#4078semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C52×D42
C5×#68172
C5×#67162
C5×#67152
C5×#67102
C5×#40783
#918215
C5×#9325