Finite Group Databasebeta

← C3×#91561 #13086704

Subgroups — C3×#91561 #13086704

Order 1440.

562 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 136 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×D15C211yesmaximal, characteristic
2C2×C3×#1200C212yesmaximal, complemented
3C3×C5×#217C211yesmaximal, characteristic, complemented
4C32×C22⋊Dic5C211yesmaximal, characteristic, complemented
5C3×#6667C212yesmaximal, complemented
6C3×D5×C22⋊C4—31nomaximal
7#91561C311yesmaximal, characteristic, direct product factor
8C23×C32×C5C2211yescharacteristic
9C22×C3×D15—24no—
10C22×C3×D15C414yescomplemented
11C22×C3×D15C2211yescharacteristic
12C22×C3×D15C2211yescharacteristic
13C2×C3×C5×Dic3C2212yescomplemented
14C2×C32×Dic5C2212yescomplemented
15C3×#1200—24no—
16C3×D3×C22⋊C4—51nomaximal, Hall
17C23×D15C2×C311yescharacteristic
18C23×C3×D5—31no—
19C2×C3×C4×D5—32no—
20C2×#1200C2×C312yescomplemented
21C3×C5×C22⋊C4—31no—
22C5×#217C2×C311yescharacteristic, complemented
23C3×C22⋊Dic5—21no—
24C3×C22⋊Dic5D311yescharacteristic, complemented
25C3×C22⋊Dic5C2×C311yescharacteristic, complemented
26C3×#685—32no—
27#6667C2×C312yescomplemented
28C22×C32×C5—22no—
29C22×C32×C5C2×C412yescomplemented
30C22×C32×C5C2311yescharacteristic
31C2×C3×D15—22no—
32C2×C3×D15—28no—
33C2×C3×D15C2×C412yes—
34C2×C3×D15C2×C412yescomplemented
35C2×C3×D15D414yes—
36C3×C5×Dic3—22no—
37C32×Dic5—22no—
38D5×C22⋊C4—31noHall
39C23×C3×D3—51no—
40C2×C3×C4×D3—52no—
41C32×C22⋊C4—51no—
42C3×#217D511yescharacteristic, complemented
43C3×(C2×D3)⋊C4—52no—
44C23×C3×C5—21no—
45C23×C3×C5C2×D311yescharacteristic
46C23×C3×C5C22×C311yescharacteristic
47C22×D15—24no—
48C22×D15C3×C414yescomplemented
49C22×D15C22×C311yescharacteristic
50C22×D15C22×C311yescharacteristic
51C22×C3×D5—31no—
52C22×C3×D5—34no—
53C22×C3×D5—31no—
54C22×C3×D5—64no—
55C2×C3×C4×C5—32no—
56C2×C5×Dic3C22×C312yescomplemented
57C2×C3×Dic5—22no—
58C2×C3×Dic5C2×D312yescomplemented
59C2×C3×Dic5C22×C312yescomplemented
60C3×C4×D5—64no—
61#1200—24no—
62C2×C3×C22⋊C4—151no—
63D3×C22⋊C4—51no—
64C2×C32×C5—22no—
65C2×C32×C5C22×C411yescharacteristic
66C2×C32×C5C2×D412yes—
67C3×D15—22no—
68C3×D15C22⋊C414yescomplemented
69C23×D5—31no—
70C2×C4×D5—32no—
71C5×C22⋊C4—31no—
72C22⋊Dic5C3×D311yescharacteristic, complemented
73#685—32no—
74C23×C32C2×D511yescharacteristic
75C22×C3×D3—51no—
76C22×C3×D3—54no—
77C22×C3×D3—51no—
78C22×C3×D3—104no—
79C2×C32×C4—52no—
80C2×C3×Dic3C2×D512yescomplemented
81C3×C4×D3—104no—
82C22×C3×C5—22no—
83C22×C3×C5—22no—
84C22×C3×C5—22no—
85C22×C3×C5—21no—
86C22×C3×C5—42no—
87C22×C3×C5C4×D312yescomplemented
88C22×C3×C5C2×C3×C412yescomplemented
89C22×C3×C5C22×D311yescharacteristic
90C22×C3×C5C23×C311yescharacteristic
91C2×D15—22no—
92C2×D15—28no—
93C2×D15C2×C3×C412yes—
94C2×D15C2×C3×C412yescomplemented
95C2×D15C3×D414yes—
96C2×C3×D5—32no—
97C2×C3×D5—34no—
98C2×C3×D5—32no—
99C2×C3×D5—68no—
100C2×C3×D5—62no—
101C3×C4×C5—62no—
102C3×Dic5—22no—
103C3×Dic5—22no—
104C3×Dic5—42no—
105C5×Dic3—22no—
106C24×C3—151no—
107C23×D3—51no—
108C22×C3×C4—152no—
109C2×C4×D3—52no—
110C3×C22⋊C4—31no—
111C3×C22⋊C4—51no—
112C3×C22⋊C4—51no—
113C3×C22⋊C4—101no—
114C3×C22⋊C4—152no—
115#217C3×D511yescharacteristic, complemented
116(C2×D3)⋊C4—52no—
117C32×C5C2×C22⋊C411yescharacteristic, complemented, Hall
118C23×C5C2×C3×D311yescharacteristic
119C22×D5—31no—
120C22×D5—34no—
121C22×D5—31no—
122C22×D5—64no—
123C2×C4×C5—32no—
124C2×Dic5C2×C3×D312yescomplemented
125C4×D5—64no—
126C22×C32—22no—
127C22×C32C4×D512yescomplemented
128C22×C32C22×D511yescharacteristic
129C2×C3×D3—52no—
130C2×C3×D3—54no—
131C2×C3×D3—52no—
132C2×C3×D3—108no—
133C2×C3×D3—102no—
134C32×C4—102no—
135C3×Dic3—22no—
136C2×C22⋊C4—151noSylow
137C2×C3×C5—22no—
138C2×C3×C5—22no—
139C2×C3×C5—21no—
140C2×C3×C5—22no—
141C2×C3×C5—42no—
142C2×C3×C5C2×C4×D311yescharacteristic
143C2×C3×C5D3×D412yes—
144C2×C3×C5C22×C3×C411yescharacteristic
145C2×C3×C5C2×C3×D412yes—
146D15—22no—
147D15C3×C22⋊C414yescomplemented
148C3×D5—34no—
149C3×D5—62no—
150C23×C3—21no—
151C23×C3—151no—
152C23×C3—154no—
153C23×C3—151no—
154C23×C3—304no—
155C23×C3D3×D511yescharacteristic
156C23×C3C2×C3×D511yescharacteristic
157C22×D3—51no—
158C22×D3—54no—
159C22×D3—51no—
160C22×D3—104no—
161C2×C3×C4—32no—
162C2×C3×C4—52no—
163C2×C3×C4—52no—
164C2×C3×C4—102no—
165C2×C3×C4—304no—
166C2×Dic3C2×C3×D512yescomplemented
167C4×D3—104no—
168C22×C5—22no—
169C22×C5C3×C4×D312yescomplemented
170C22×C5C22×C3×D311yescharacteristic
171C2×D5—32no—
172C2×D5—34no—
173C2×D5—32no—
174C2×D5—68no—
175C2×D5—62no—
176C4×C5—62no—
177Dic5—22no—
178C2×C32—22no—
179C2×C32C2×C4×D511yescharacteristic
180C2×C32D4×D512yes—
181C3×D3—54no—
182C3×D3—102no—
183C24—151no—
184C22×C4—152no—
185C22⋊C4—31no—
186C22⋊C4—51no—
187C22⋊C4—152no—
188C3×C5—21no—
189C3×C5D3×C22⋊C411yescharacteristic, complemented
190C3×C5C2×C3×C22⋊C411yescharacteristic, complemented
191C22×C3—22no—
192C22×C3—22no—
193C22×C3—22no—
194C22×C3—21no—
195C22×C3—42no—
196C22×C3—152no—
197C22×C3—154no—
198C22×C3—152no—
199C22×C3—308no—
200C22×C3—302no—
201C22×C3#120012yescomplemented
202C22×C3C3×C4×D512yescomplemented
203C22×C3C2×D3×D511yescharacteristic
204C22×C3C22×C3×D511yescharacteristic
205C2×D3—52no—
206C2×D3—54no—
207C2×D3—52no—
208C2×D3—108no—
209C2×D3—102no—
210C3×C4—62no—
211C3×C4—102no—
212C3×C4—102no—
213C3×C4—202no—
214Dic3—22no—
215C2×C5—22no—
216C2×C5C2×C3×C4×D311yescharacteristic
217C2×C5C3×D3×D412yes—
218D5—34no—
219D5—62no—
220C32D5×C22⋊C411yescharacteristic, complemented, Sylow
221C23—151no—
222C23—154no—
223C23—151no—
224C23—304no—
225C23C3×D3×D511yescharacteristic
226C2×C4—32no—
227C2×C4—52no—
228C2×C4—304no—
229C2×C3—22no—
230C2×C3—22no—
231C2×C3—21no—
232C2×C3—22no—
233C2×C3—42no—
234C2×C3—154no—
235C2×C3—302no—
236C2×C3C2×#120011yescharacteristic
237C2×C3#679012yes—
238C2×C3C2×C3×C4×D511yescharacteristic
239C2×C3C3×D4×D512yes—
240D3—54no—
241D3—102no—
242C5C3×D3×C22⋊C411yescharacteristic, complemented, Sylow
243C22—22no—
244C22—152no—
245C22—154no—
246C22—152no—
247C22—308no—
248C22—302no—
249C22C3×#120012yescomplemented
250C22C2×C3×D3×D511yescharacteristic
251C4—62no—
252C4—102no—
253C3—21no—
254C3#9156111yescharacteristic, direct product factor
255C3C3×D5×C22⋊C411yescharacteristic, complemented
256C2—22no—
257C2—154no—
258C2—302no—
259C2C2×C3×#120011yescharacteristic
260C2C3×#679012yes—
subgroupG/H
centerC22×C3C2×D3×D5
commutatorC2×C3×C5C22×C3×C4
FrattiniC22C2×C3×D3×D5
FittingC23×C32×C5C22
socleC22×C32×C5C23
prime radicalC23×C3×D15C2
cosocleC22×C3×C5C23×C3
hypercenterC23×C3D3×D5
hypocenterC3×C5C2×C3×C22⋊C4
LCSG ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. C2×C3×#1200C2semidirect
3. C3×C5×#217C2semidirect
4. C32×C22⋊Dic5C2semidirect
5. C3×#6667C2semidirect
7. #91561C3direct (N × H)
10. C22×C3×D15C4semidirect
13. C2×C3×C5×Dic3C22semidirect
14. C2×C32×Dic5C22semidirect
20. C2×#1200C2×C3semidirect
22. C5×#217C2×C3semidirect
24. C3×C22⋊Dic5D3semidirect
25. C3×C22⋊Dic5C2×C3semidirect
27. #6667C2×C3semidirect
29. C22×C32×C5C2×C4semidirect
34. C2×C3×D15C2×C4semidirect
42. C3×#217D5semidirect
48. C22×D15C3×C4semidirect
56. C2×C5×Dic3C22×C3semidirect
58. C2×C3×Dic5C2×D3semidirect
59. C2×C3×Dic5C22×C3semidirect
68. C3×D15C22⋊C4semidirect
72. C22⋊Dic5C3×D3semidirect
80. C2×C3×Dic3C2×D5semidirect
87. C22×C3×C5C4×D3semidirect
88. C22×C3×C5C2×C3×C4semidirect
94. C2×D15C2×C3×C4semidirect
115. #217C3×D5semidirect
117. C32×C5C2×C22⋊C4semidirect
124. C2×Dic5C2×C3×D3semidirect
127. C22×C32C4×D5semidirect
147. D15C3×C22⋊C4semidirect
166. C2×Dic3C2×C3×D5semidirect
169. C22×C5C3×C4×D3semidirect
189. C3×C5D3×C22⋊C4semidirect
190. C3×C5C2×C3×C22⋊C4semidirect
201. C22×C3#1200semidirect
202. C22×C3C3×C4×D5semidirect
220. C32D5×C22⋊C4semidirect
242. C5C3×D3×C22⋊C4semidirect
249. C22C3×#1200semidirect
254. C3#91561direct (N × H)
255. C3C3×D5×C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C23×C3×D152
C2×C3×#12002
C3×C5×#2172
C32×C22⋊Dic52
C3×#66672
C3×D5×C22⋊C43
#915613
C3×D3×C22⋊C45