Finite Group Databasebeta

← C3×#91782 #13087991

Subgroups — C3×#91782 #13087991

Order 1440.

402 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 130 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×D60C211yesmaximal, characteristic
2C2×C3×C4×D15C212yesmaximal, complemented
3C32×C5×C4⋊C4C211yesmaximal, characteristic, complemented
4C3×#6714C212yesmaximal, complemented
5C3×C4×Dic15C211yesmaximal, characteristic, complemented
6#91782C311yesmaximal, characteristic, direct product factor
7C3×#4039—31nomaximal
8C22×C3×D15C2212yes—
9C2×C32×C4×C5C2212yes—
10C2×C32×C4×C5C2211yescharacteristic
11C2×C3×Dic15C2211yescharacteristic
12C2×C3×Dic15C2211yescharacteristic
13C3×D60C414yescomplemented
14C3×C4×D15—24no—
15C3×#893—51nomaximal, Hall
16C2×D60C2×C311yescharacteristic
17C2×C4×D15C2×C312yescomplemented
18C2×C3×D20—31no—
19C2×C3×C4×D5—32no—
20C3×C5×C4⋊C4—21no—
21C3×C5×C4⋊C4D311yescharacteristic, complemented
22C3×C5×C4⋊C4C2×C311yescharacteristic, complemented
23#6714C2×C312yescomplemented
24C4×Dic15C2×C311yescharacteristic, complemented
25C3×#685—32no—
26C3×C4×Dic5—31no—
27C22×C32×C5C2311yescharacteristic
28C2×C3×D15—22no—
29C2×C3×D15—22no—
30C2×C3×D15C2×C414yescomplemented
31C32×C4×C5—22no—
32C32×C4×C5C2×C412yescomplemented
33C3×Dic15—21no—
34C3×Dic15D412yescomplemented
35#4039—31noHall
36C2×C3×D12—51no—
37C2×C3×C4×D3—52no—
38C32×C4⋊C4D511yescharacteristic, complemented
39C3×(C2×D3)⋊C4—52no—
40C3×C4×Dic3—51no—
41C22×D15C22×C312yes—
42C22×C3×D5—32no—
43C2×C3×C4×C5—21no—
44C2×C3×C4×C5—22no—
45C2×C3×C4×C5C2×D312yes—
46C2×C3×C4×C5C2×D311yescharacteristic
47C2×C3×C4×C5C22×C312yes—
48C2×C3×C4×C5C22×C311yescharacteristic
49C2×Dic15C22×C311yescharacteristic
50C2×Dic15C22×C311yescharacteristic
51D60C3×C414yescomplemented
52C4×D15—24no—
53C2×C3×Dic5—31no—
54C2×C3×Dic5—31no—
55C3×D20—34no—
56C3×C4×D5—64no—
57C3×C4×D4—151no—
58#893—51no—
59C2×C32×C5C22×C411yescharacteristic
60C2×C32×C5C2×D411yescharacteristic
61C2×C32×C5#4111yescharacteristic
62C3×D15—24no—
63C2×D20—31no—
64C2×C4×D5—32no—
65C5×C4⋊C4C3×D311yescharacteristic, complemented
66#685—32no—
67C4×Dic5—31no—
68C22×C3×D3—52no—
69C2×C32×C4C2×D512yes—
70C2×C32×C4C2×D511yescharacteristic
71C2×C3×Dic3—51no—
72C2×C3×Dic3—51no—
73C3×D12—54no—
74C3×C4×D3—104no—
75C22×C3×C5—21no—
76C22×C3×C5C22×D311yescharacteristic
77C22×C3×C5C23×C311yescharacteristic
78C2×D15—22no—
79C2×D15—22no—
80C2×D15C2×C3×C414yescomplemented
81C2×C3×D5—34no—
82C2×C3×D5—62no—
83C2×C3×D5—62no—
84C3×C4×C5—22no—
85C3×C4×C5—22no—
86C3×C4×C5—22no—
87C3×C4×C5—42no—
88C3×C4×C5C4×D312yescomplemented
89C3×C4×C5C2×C3×C412yescomplemented
90Dic15—21no—
91Dic15C3×D412yescomplemented
92C3×Dic5—32no—
93C3×Dic5—61no—
94C2×C3×D4—151no—
95C22×C3×C4—152no—
96C2×D12—51no—
97C2×C4×D3—52no—
98C3×C4⋊C4—21no—
99C3×C4⋊C4C3×D511yescharacteristic, complemented
100C3×C4⋊C4D1511yescharacteristic, complemented
101C3×C22⋊C4—152no—
102C3×C42—151no—
103(C2×D3)⋊C4—52no—
104C4×Dic3—51no—
105C32×C5C4×D411yescharacteristic, complemented, Hall
106C22×D5—32no—
107C2×C4×C5C2×C3×D312yes—
108C2×C4×C5C2×C3×D311yescharacteristic
109C2×Dic5—31no—
110C2×Dic5—31no—
111D20—34no—
112C4×D5—64no—
113C22×C32C22×D511yescharacteristic
114C2×C3×D3—54no—
115C2×C3×D3—102no—
116C2×C3×D3—102no—
117C32×C4—22no—
118C32×C4C4×D512yescomplemented
119C3×Dic3—52no—
120C3×Dic3—101no—
121C4×D4—151noSylow
122C2×C3×C5—21no—
123C2×C3×C5—21no—
124C2×C3×C5—21no—
125C2×C3×C5C2×C4×D311yescharacteristic
126C2×C3×C5D3×D411yescharacteristic
127C2×C3×C5#23911yescharacteristic
128C2×C3×C5C22×C3×C411yescharacteristic
129C2×C3×C5C2×C3×D411yescharacteristic
130C2×C3×C5C3×#4111yescharacteristic
131D15—24no—
132C3×D5—64no—
133C23×C3—152no—
134C22×D3—52no—
135C3×D4—154no—
136C2×C3×C4—21no—
137C2×C3×C4—22no—
138C2×C3×C4—151no—
139C2×C3×C4—151no—
140C2×C3×C4—304no—
141C2×C3×C4C2×C3×D512yes—
142C2×C3×C4C2×C3×D511yescharacteristic
143C2×C3×C4C2×D1512yes—
144C2×C3×C4C2×D1511yescharacteristic
145C2×Dic3—51no—
146C2×Dic3—51no—
147D12—54no—
148C4×D3—104no—
149C22×C5C22×C3×D311yescharacteristic
150C2×D5—34no—
151C2×D5—62no—
152C2×D5—62no—
153C4×C5—22no—
154C4×C5C3×C4×D312yescomplemented
155Dic5—32no—
156Dic5—61no—
157C2×C32C2×C4×D511yescharacteristic
158C2×C32D4×D511yescharacteristic
159C2×C32#71311yescharacteristic
160C3×D3—104no—
161C2×D4—151no—
162C22×C4—152no—
163C4⋊C4C3×D1511yescharacteristic, complemented
164C22⋊C4—152no—
165C42—151no—
166C3×C5—21no—
167C3×C5#89311yescharacteristic, complemented
168C3×C5C3×C4×D411yescharacteristic, complemented
169C22×C3—21no—
170C22×C3—154no—
171C22×C3—302no—
172C22×C3—302no—
173C22×C3C22×C3×D511yescharacteristic
174C22×C3C22×D1511yescharacteristic
175C2×D3—54no—
176C2×D3—102no—
177C2×D3—102no—
178C3×C4—22no—
179C3×C4—22no—
180C3×C4—22no—
181C3×C4—42no—
182C3×C4—152no—
183C3×C4—301no—
184C3×C4C3×C4×D512yescomplemented
185C3×C4C4×D1512yescomplemented
186Dic3—52no—
187Dic3—101no—
188C2×C5C2×C3×C4×D311yescharacteristic
189C2×C5C3×D3×D411yescharacteristic
190C2×C5C3×#23911yescharacteristic
191D5—64no—
192C32#403911yescharacteristic, complemented, Sylow
193C23—152no—
194D4—154no—
195C2×C4—151no—
196C2×C4—151no—
197C2×C4—304no—
198C2×C4C2×C3×D1512yes—
199C2×C4C2×C3×D1511yescharacteristic
200C2×C3—21no—
201C2×C3—21no—
202C2×C3—21no—
203C2×C3—304no—
204C2×C3C2×C3×C4×D511yescharacteristic
205C2×C3C3×D4×D511yescharacteristic
206C2×C3C3×#71311yescharacteristic
207C2×C3C2×C4×D1511yescharacteristic
208C2×C3D4×D1511yescharacteristic
209C2×C3#682111yescharacteristic
210D3—104no—
211C5C3×#89311yescharacteristic, complemented, Sylow
212C22—154no—
213C22—302no—
214C22—302no—
215C22C22×C3×D1511yescharacteristic
216C4—22no—
217C4—152no—
218C4—301no—
219C4C3×C4×D1512yescomplemented
220C3—21no—
221C3C3×#403911yescharacteristic, complemented
222C3#9178211yescharacteristic, direct product factor
223C2—304no—
224C2C2×C3×C4×D1511yescharacteristic
225C2C3×D4×D1511yescharacteristic
226C2C3×#682111yescharacteristic
subgroupG/H
centerC22×C3C22×D15
commutatorC2×C3×C5C22×C3×C4
FrattiniC22C22×C3×D15
FittingC32×C5×C4⋊C4C2
socleC22×C32×C5C23
prime radicalC2×C3×D60C2
cosocleC22×C3×C5C23×C3
hypercenterC3×C4⋊C4D15
hypocenterC3×C5C3×C4×D4
LCSG ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22×C3 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4×D15C2semidirect
3. C32×C5×C4⋊C4C2semidirect
4. C3×#6714C2semidirect
5. C3×C4×Dic15C2semidirect
6. #91782C3direct (N × H)
13. C3×D60C4semidirect
17. C2×C4×D15C2×C3semidirect
21. C3×C5×C4⋊C4D3semidirect
22. C3×C5×C4⋊C4C2×C3semidirect
23. #6714C2×C3semidirect
24. C4×Dic15C2×C3semidirect
30. C2×C3×D15C2×C4semidirect
32. C32×C4×C5C2×C4semidirect
34. C3×Dic15D4semidirect
38. C32×C4⋊C4D5semidirect
51. D60C3×C4semidirect
65. C5×C4⋊C4C3×D3semidirect
80. C2×D15C2×C3×C4semidirect
88. C3×C4×C5C4×D3semidirect
89. C3×C4×C5C2×C3×C4semidirect
91. Dic15C3×D4semidirect
99. C3×C4⋊C4C3×D5semidirect
100. C3×C4⋊C4D15semidirect
105. C32×C5C4×D4semidirect
118. C32×C4C4×D5semidirect
154. C4×C5C3×C4×D3semidirect
163. C4⋊C4C3×D15semidirect
167. C3×C5#893semidirect
168. C3×C5C3×C4×D4semidirect
184. C3×C4C3×C4×D5semidirect
185. C3×C4C4×D15semidirect
192. C32#4039semidirect
211. C5C3×#893semidirect
219. C4C3×C4×D15semidirect
221. C3C3×#4039semidirect
222. C3#91782direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D602
C2×C3×C4×D152
C32×C5×C4⋊C42
C3×#67142
C3×C4×Dic152
#917823
C3×#40393
C3×#8935