Finite Group Databasebeta

← #72254698

Subgroups — #72254698

Order 5400.

276 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 58 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
1C5×3−1+2⋊Dic5C211yesmaximal, characteristic, complemented
2C2×C5×D5×3−1+2C211yesmaximal, characteristic, complemented
3C2×C5⋊(C5⋊AGL(1,ℤ/9))C211yesmaximal, characteristic, complemented
4C3×#10597123—31nomaximal
5#14267982C311yesmaximal, characteristic, complemented
6C2×C52×3−1+2C2211yescharacteristic
7C5⋊(C5⋊AGL(1,ℤ/9))—21no—
8C5×D5×3−1+2—21no—
9#67099⋊C3—51nomaximal
10#67083⋊C3—51nomaximal
11C2×C3×C3⋊(C5⋊D5)—31no—
12C2×C32×C5×D5D311yescharacteristic
13C3×C5×Dic15—31no—
14C2×C5⋊D45C2×C311yescharacteristic, complemented
15C2×C5×C9×D5—21no—
16C2×C5×C9×D5C2×C311yescharacteristic, complemented
17C5×Dic45C2×C311yescharacteristic, complemented
18C52×3−1+2D411yescharacteristic, complemented, Hall
19#10597123—31no—
20C3×#6038—91no—
21C22×C5×3−1+2—51no—
22C2×C5⋊AGL(1,ℤ/9)—51no—
23C2×C5⋊AGL(1,ℤ/9)—51no—
24C2×C5⋊AGL(1,ℤ/9)—101no—
25C2×C5⋊AGL(1,ℤ/9)—101no—
26C2×D5×3−1+2D511yescharacteristic, complemented
273−1+2⋊Dic5D511yescharacteristic, complemented
28C5×3−1+2⋊C4—51no—
29C2×C32×C52C2×D311yescharacteristic
30C3×C3⋊(C5⋊D5)—61no—
31C32×C5×D5—21no—
32C2×C52×C9—21no—
33C2×C52×C9C22×C311yescharacteristic
34C5⋊D45—21no—
35C5×C9×D5—22no—
36C5×C9×D5—21no—
37C3×#1220—151no—
38C3×#1202—151no—
39#67099—51no—
40#67083—51no—
41C2×C3⋊(C5⋊D5)—31no—
42C2×C3×C5⋊D5—91no—
43C2×C3×C5×D5—61no—
44C2×C3×C5×D5C3×D311yescharacteristic
45C5×Dic15—31no—
46C3×C5×Dic5—91no—
47C2×C5×3−1+2—21no—
48C2×C5×3−1+2—21no—
49C2×C5×3−1+2—101no—
50C2×C5×3−1+2C2×D511yescharacteristic
51C2×C5×3−1+2C2×D511yescharacteristic
52C5⋊AGL(1,ℤ/9)—102no—
53C5⋊AGL(1,ℤ/9)—102no—
54C5⋊AGL(1,ℤ/9)—101no—
55C5⋊AGL(1,ℤ/9)—101no—
56D5×3−1+2—21no—
57C32×C52#6711yescharacteristic
58C52×C9—21no—
59C52×C9C3×D411yescharacteristic, complemented
60#611⋊C3—251noHall
61#6038—91noHall
62C22×C32×C5—51no—
63C2×C3×D15—151no—
64C2×C3×D15—151no—
65C2×C3×D15—301no—
66C2×C3×D15—301no—
67C2×C32×D5D1511yescharacteristic
68C3×Dic15—31no—
69C3×C5×Dic3—151no—
70C22×C5×C9—51no—
71C22×C5×C9—101no—
72C2×D45—51no—
73C2×D45—51no—
74C2×D45—101no—
75C2×D45—101no—
76C2×C9×D5—21no—
77C2×C9×D5C3×D511yescharacteristic, complemented
78Dic45C3×D511yescharacteristic, complemented
79C5×Dic9—51no—
80C2×C3×C52—61no—
81C2×C3×C52C2×C3×D311yescharacteristic
82C3⋊(C5⋊D5)—61no—
83C3×C5⋊D5—181no—
84C3×C5×D5—21no—
85C3×C5×D5—61no—
86C5×3−1+2—21no—
87C5×3−1+2—21no—
88C5×3−1+2D2011yescharacteristic, complemented
89C5×3−1+2C22⋊D511yescharacteristic, complemented
90#1220—151no—
91C3×C22⋊D5—451no—
92C3×D20—451no—
93#1202—151no—
94C22×3−1+2—51no—
95C2×AGL(1,ℤ/9)—251no—
963−1+2⋊C4—51no—
97C2×C5⋊D5—91no—
98C2×C5×D5AGL(1,ℤ/9)11yescharacteristic, complemented
99C5×Dic5—91no—
100C2×C32×C5—21no—
101C2×C32×C5—21no—
102C2×C32×C5—101no—
103C2×C32×C5D3×D511yescharacteristic
104C2×C32×C5C2×D1511yescharacteristic
105C3×D15—302no—
106C3×D15—302no—
107C3×D15—301no—
108C3×D15—301no—
109C32×D5—21no—
110C2×C5×C9—21no—
111C2×C5×C9—21no—
112C2×C5×C9—21no—
113C2×C5×C9—21no—
114C2×C5×C9—41no—
115C2×C5×C9—41no—
116C2×C5×C9—102no—
117C2×C5×C9—101no—
118C2×C5×C9C2×C3×D511yescharacteristic
119C2×C5×C9C2×C3×D511yescharacteristic
120D45—102no—
121D45—102no—
122D45—101no—
123D45—101no—
124C9×D5—22no—
125C9×D5—21no—
126C3×C52—61no—
127C3×C52C3×#6711yescharacteristic
128C3×#67—751no—
129#611—251no—
130C22×C3×C5—51no—
131C22×C3×C5—301no—
132C2×D15—151no—
133C2×D15—151no—
134C2×D15—301no—
135C2×D15—301no—
136C2×C3×D5—31no—
137C2×C3×D5—451no—
138C2×C3×D5—451no—
139C2×C3×D5—901no—
140C2×C3×D5—901no—
141C2×C3×D5C3×D1511yescharacteristic
142C3×C4×C5—451no—
143Dic15—31no—
144C3×Dic5—91no—
145C5×Dic3—151no—
146C2×3−1+2—101no—
147C2×3−1+2D5211yescharacteristic
148AGL(1,ℤ/9)—501no—
149C2×C52C2×AGL(1,ℤ/9)11yescharacteristic
150C5⋊D5—181no—
151C5×D5—21no—
152C32×C5—21no—
153C32×C5—21no—
154C32×C5#120211yescharacteristic
155C32×C5#122011yescharacteristic
156C5×C9—21no—
157C5×C9—21no—
158C5×C9—21no—
159C5×C9—21no—
160C5×C9—41no—
161C5×C9—41no—
162C5×C9C3×D2011yescharacteristic, complemented
163C5×C9C3×C22⋊D511yescharacteristic, complemented
164C22⋊D5—451no—
165D20—451no—
166C22×C32—51no—
167C2×C3×D3—751no—
168C3×Dic3—151no—
169C22×C9—51no—
170C22×C9—101no—
171C2×D9—251no—
172Dic9—51no—
173C2×C3×C5—21no—
174C2×C3×C5—21no—
175C2×C3×C5—31no—
176C2×C3×C5—61no—
177C2×C3×C5—61no—
178C2×C3×C5—101no—
179C2×C3×C5—121no—
180C2×C3×C5—301no—
181C2×C3×C5C3×D3×D511yescharacteristic
182C2×C3×C5C2×C3×D1511yescharacteristic
183D15—302no—
184D15—302no—
185D15—301no—
186D15—301no—
187C3×D5—21no—
188C3×D5—61no—
189C3×D5—901no—
190C3×D5—901no—
191C3×D5—902no—
192C3×D5—902no—
1933−1+2#603811yescharacteristic, complemented, Sylow
194C52#611⋊C311yescharacteristic, complemented, Sylow
195C3×D4—2251no—
196#67—751no—
197C22×C5—51no—
198C2×D5—451no—
199C2×D5—451no—
200C2×D5—901no—
201C2×D5—901no—
202C2×D5C5⋊AGL(1,ℤ/9)11yescharacteristic, complemented
203C4×C5—451no—
204Dic5—91no—
205C2×C32—101no—
206C2×C32D5×D1511yescharacteristic
207C3×D3—1501no—
208C2×C9—21no—
209C2×C9—102no—
210C2×C9—101no—
211C2×C9C3×D5211yescharacteristic
212D9—501no—
213C3×C5—21no—
214C3×C5—21no—
215C3×C5—31no—
216C3×C5—61no—
217C3×C5—61no—
218C3×C5—121no—
219C3×C5C3×#120211yescharacteristic
220C3×C5C3×#122011yescharacteristic
221C22×C3—51no—
222C22×C3—301no—
223C22×C3—2251no—
224C2×D3—751no—
225C3×C4—451no—
226Dic3—151no—
227C2×C5—21no—
228C2×C5—21no—
229C2×C5—101no—
230C2×C5D5×AGL(1,ℤ/9)11yescharacteristic
231C2×C5C2×C5⋊AGL(1,ℤ/9)11yescharacteristic
232D5—21no—
233D5—902no—
234D5—902no—
235D5—901no—
236D5—901no—
237C32#1059712311yescharacteristic
238C9—21no—
239C9C3×#603811yescharacteristic, complemented
240D4—2251noSylow
241C2×C3—61no—
242C2×C3—101no—
243C2×C3—301no—
244C2×C3—4501no—
245C2×C3C3×D5×D1511yescharacteristic
246D3—1501no—
247C5—21no—
248C5—21no—
249C5#67083⋊C311yescharacteristic, complemented
250C5#67099⋊C311yescharacteristic, complemented
251C22—51no—
252C22—2251no—
253C4—451no—
254C3—61no—
255C3C3×#1059712311yescharacteristic
256C2—101no—
257C2—4501no—
258C2D5×C5⋊AGL(1,ℤ/9)11yescharacteristic
subgroupG/H
center, hypercenterC2D5×C5⋊AGL(1,ℤ/9)
commutator, cosocleC2×C52×C9C22×C3
FrattiniC2×C3C3×D5×D15
FittingC2×C52×3−1+2C22
socleC2×C3×C52C2×C3×D3
hypocenterC52×C9C3×D4
LCSG ▷ C2×C52×C9 ▷ C52×C9
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C5×3−1+2⋊Dic5C2semidirect
2. C2×C5×D5×3−1+2C2semidirect
3. C2×C5⋊(C5⋊AGL(1,ℤ/9))C2semidirect
5. #14267982C3semidirect
14. C2×C5⋊D45C2×C3semidirect
16. C2×C5×C9×D5C2×C3semidirect
17. C5×Dic45C2×C3semidirect
18. C52×3−1+2D4semidirect
26. C2×D5×3−1+2D5semidirect
27. 3−1+2⋊Dic5D5semidirect
59. C52×C9C3×D4semidirect
77. C2×C9×D5C3×D5semidirect
78. Dic45C3×D5semidirect
88. C5×3−1+2D20semidirect
89. C5×3−1+2C22⋊D5semidirect
98. C2×C5×D5AGL(1,ℤ/9)semidirect
162. C5×C9C3×D20semidirect
163. C5×C9C3×C22⋊D5semidirect
193. 3−1+2#6038semidirect
194. C52#611⋊C3semidirect
202. C2×D5C5⋊AGL(1,ℤ/9)semidirect
239. C9C3×#6038semidirect
249. C5#67083⋊C3semidirect
250. C5#67099⋊C3semidirect

Maximal subgroups

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

subgroupindex
C5×3−1+2⋊Dic52
C2×C5×D5×3−1+22
C2×C5⋊(C5⋊AGL(1,ℤ/9))2
C3×#105971233
#142679823
#67099⋊C35
#67083⋊C35