Finite Group Databasebeta

← #13087332

Subgroups — #13087332

Order 1440.

274 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 74 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
1#10623908C211yesmaximal, characteristic
2C3×#6712C211yesmaximal, characteristic, complemented
3C3×C4×C5×Dic3C211yesmaximal, characteristic, complemented
4#10623815C211yesmaximal, characteristic, complemented
5#10623814C211yesmaximal, characteristic, complemented
6#10623813C211yesmaximal, characteristic
7#10623813C211yesmaximal, characteristic
8#91765—31nomaximal
9#91380—31nomaximal
10C22×C3⋊D15C2211yescharacteristic
11C2×C32×C4×C5C2211yescharacteristic
12C2×C3⋊Dic15C2211yescharacteristic
13C2×C3×Dic15C2211yescharacteristic
14C2×C3×Dic15C2211yescharacteristic
15C2×C3×C5×Dic3C2211yescharacteristic
16C2×C3×C5×Dic3C2211yescharacteristic
17#63868—51nomaximal, Hall
18C3×C42×C5—31no—
19#6714—31no—
20#6714—31no—
21#6714—31no—
22#6714—31no—
23#6714—61no—
24#6712—31no—
25#6712—31no—
26#6712D311yescharacteristic, complemented
27C4×C5×Dic3D311yescharacteristic, complemented
28C3×#683—31no—
29#6669—31no—
30#6668—31no—
31#6667—31no—
32#6667—31no—
33C22×C32×C5C2311yescharacteristic
34C2×C3⋊D15—21no—
35C2×C3⋊D15—21no—
36C2×C3⋊D15—21no—
37C32×C4×C5—21no—
38C3⋊Dic15—21no—
39C3×Dic15—21no—
40C3×Dic15—21no—
41C3×C5×Dic3—21no—
42C3×C5×Dic3—21no—
43#4022—91noHall
44#3752—51no—
45C3×Dic3⋊C4—51no—
46C3×C4×Dic3D511yescharacteristic, complemented
47#3724—51no—
48Dic3⋊Dic3—51no—
49#3722—51no—
50#3722—51no—
51C22×D15—31no—
52C22×D15—31no—
53C22×D15—61no—
54C2×C3×C4×C5—21no—
55C2×C3×C4×C5—31no—
56C2×C3×C4×C5—31no—
57C2×C3×C4×C5C2×D311yescharacteristic
58C2×C3×C4×C5C2×D311yescharacteristic
59C2×Dic15—31no—
60C2×Dic15—31no—
61C2×Dic15—61no—
62C2×Dic15C2×D311yescharacteristic
63C2×Dic15C2×D311yescharacteristic
64C2×C5×Dic3C2×D311yescharacteristic
65C2×C5×Dic3C2×D311yescharacteristic
66C2×C3×Dic5—31no—
67C2×C3×Dic5—31no—
68#898—151no—
69#876—151no—
70C2×C32×C5#4111yescharacteristic
71C2×C32×C5#4111yescharacteristic
72C2×C32×C5#4111yescharacteristic
73C3⋊D15—41no—
74C42×C5—31no—
75#685—91no—
76#685—91no—
77#685—91no—
78#683—31no—
79#683—91no—
80#683—91no—
81C22×C3⋊D3—51no—
82C2×C32×C4C2×D511yescharacteristic
83C2×C3⋊Dic3—51no—
84C2×C3×Dic3—51no—
85C2×C3×Dic3—51no—
86C2×C3×Dic3C2×D511yescharacteristic
87C2×C3×Dic3C2×D511yescharacteristic
88C22×C3×C5—21no—
89C22×C3×C5C22×D311yescharacteristic
90C22×C3×C5C22×D311yescharacteristic
91C2×D15—62no—
92C2×D15—61no—
93C2×D15—61no—
94C2×D15—61no—
95C2×D15—61no—
96C2×D15—61no—
97C2×D15—61no—
98C2×D15—121no—
99C2×D15—121no—
100C3×C4×C5—21no—
101C3×C4×C5—21no—
102C3×C4×C5—41no—
103C3×C4×C5—61no—
104C3×C4×C5—61no—
105Dic15—21no—
106Dic15—21no—
107Dic15—61no—
108Dic15—61no—
109Dic15—121no—
110C3×Dic5—61no—
111C3×Dic5—61no—
112C5×Dic3—21no—
113C5×Dic3—21no—
114C3×C4⋊C4—151no—
115C3×C42—31no—
116(C2×D3)⋊C4—151no—
117(C2×D3)⋊C4—151no—
118(C2×D3)⋊C4—151no—
119(C2×D3)⋊C4—151no—
120(C2×D3)⋊C4—151no—
121(C2×D3)⋊C4—151no—
122(C2×D3)⋊C4—301no—
123C4⋊Dic3—151no—
124Dic3⋊C4—51no—
125Dic3⋊C4—151no—
126Dic3⋊C4—151no—
127Dic3⋊C4—151no—
128C4×Dic3D1511yescharacteristic, complemented
129C32×C5#12611yescharacteristic, complemented, Hall
130C22×D5—91no—
131C2×C4×C5—31no—
132C2×C4×C5—31no—
133C2×C4×C5D3211yescharacteristic
134C2×Dic5—31no—
135C2×Dic5—31no—
136C2×Dic5—91no—
137C22×C32C22×D511yescharacteristic
138C2×C3⋊D3—101no—
139C2×C3⋊D3—101no—
140C2×C3⋊D3—101no—
141C32×C4—21no—
142C3⋊Dic3—101no—
143C3×Dic3—21no—
144C3×Dic3—21no—
145C3×Dic3—101no—
146C3×Dic3—101no—
147#126—451noSylow
148C2×C3×C5—21no—
149C2×C3×C5—21no—
150C2×C3×C5—21no—
151C2×C3×C5#23511yescharacteristic
152C2×C3×C5#23511yescharacteristic
153C2×C3×C5#23511yescharacteristic
154C2×C3×C5#23511yescharacteristic
155C2×C3×C5#23711yescharacteristic
156C2×C3×C5#23911yescharacteristic
157D15—122no—
158D15—121no—
159D15—121no—
160C22×D3—151no—
161C22×D3—151no—
162C22×D3—301no—
163C2×C3×C4—21no—
164C2×C3×C4—31no—
165C2×C3×C4—31no—
166C2×C3×C4—151no—
167C2×C3×C4—151no—
168C2×C3×C4D3×D511yescharacteristic
169C2×C3×C4C2×D1511yescharacteristic
170C2×Dic3—51no—
171C2×Dic3—51no—
172C2×Dic3—151no—
173C2×Dic3—151no—
174C2×Dic3—301no—
175C2×Dic3C2×D1511yescharacteristic
176C2×Dic3C2×D1511yescharacteristic
177C22×C5C2×D3211yescharacteristic
178C2×D5—181no—
179C2×D5—181no—
180C2×D5—181no—
181C4×C5—21no—
182C4×C5—61no—
183C4×C5—61no—
184Dic5—61no—
185Dic5—61no—
186Dic5—181no—
187C2×C32#70911yescharacteristic
188C2×C32#70911yescharacteristic
189C2×C32#70911yescharacteristic
190C3⋊D3—201no—
191C4⋊C4—151no—
192C4⋊C4—451no—
193C4⋊C4—451no—
194C22⋊C4—451no—
195C22⋊C4—451no—
196C22⋊C4—451no—
197C42—31no—
198C3×C5—21no—
199C3×C5#87611yescharacteristic, complemented
200C3×C5#89811yescharacteristic, complemented
201C22×C3—21no—
202C22×C3C2×D3×D511yescharacteristic
203C22×C3C22×D1511yescharacteristic
204C2×D3—302no—
205C2×D3—301no—
206C2×D3—301no—
207C2×D3—301no—
208C2×D3—301no—
209C2×D3—301no—
210C2×D3—301no—
211C2×D3—601no—
212C2×D3—601no—
213C3×C4—21no—
214C3×C4—21no—
215C3×C4—41no—
216C3×C4—61no—
217C3×C4—61no—
218C3×C4—301no—
219C3×C4—301no—
220Dic3—21no—
221Dic3—21no—
222Dic3—101no—
223Dic3—101no—
224Dic3—301no—
225Dic3—301no—
226Dic3—601no—
227C2×C5#379811yescharacteristic
228C2×C5#379911yescharacteristic
229C2×C5#380411yescharacteristic
230D5—361no—
231C32#402211yescharacteristic, complemented, Sylow
232C23—451no—
233C2×C4—31no—
234C2×C4—31no—
235C2×C4—151no—
236C2×C4—151no—
237C2×C4—451no—
238C2×C4D3×D1511yescharacteristic
239C2×C3—21no—
240C2×C3—21no—
241C2×C3—21no—
242C2×C3#676911yescharacteristic
243C2×C3#677111yescharacteristic
244C2×C3#677911yescharacteristic
245C2×C3#681711yescharacteristic
246C2×C3#681711yescharacteristic
247C2×C3#681711yescharacteristic
248D3—602no—
249D3—601no—
250D3—601no—
251C5#6386811yescharacteristic, complemented, Sylow
252C22—901no—
253C22—901no—
254C22—901no—
255C22C2×D3×D1511yescharacteristic
256C4—21no—
257C4—61no—
258C4—61no—
259C4—301no—
260C4—301no—
261C4—901no—
262C3—21no—
263C3#9138011yescharacteristic, complemented
264C3#9176511yescharacteristic, complemented
265C2—1801no—
266C2#1062419711yescharacteristic
267C2#1062419911yescharacteristic
268C2#1062420711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×D15
commutator, socle, cosocleC22×C32×C5C23
FittingC2×C32×C4×C5C22
prime radicalC22×C3⋊D15C22
hypercenterC2×C4D3×D15
hypocenterC32×C5#126
LCSG ▷ C22×C32×C5 ▷ C32×C5
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C3×#6712C2semidirect
3. C3×C4×C5×Dic3C2semidirect
4. #10623815C2semidirect
5. #10623814C2semidirect
26. #6712D3semidirect
27. C4×C5×Dic3D3semidirect
46. C3×C4×Dic3D5semidirect
128. C4×Dic3D15semidirect
129. C32×C5#126semidirect
199. C3×C5#876semidirect
200. C3×C5#898semidirect
231. C32#4022semidirect
251. C5#63868semidirect
263. C3#91380semidirect
264. C3#91765semidirect

Maximal subgroups

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

subgroupindex
#106239082
C3×#67122
C3×C4×C5×Dic32
#106238152
#106238142
#106238132
#917653
#913803
#638685