Finite Group Databasebeta

← #13087397

Subgroups — #13087397

Order 1440.

338 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 92 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×#67149C211yesmaximal, characteristic
2#10623908C211yesmaximal, characteristic
3C3×#6712C211yesmaximal, characteristic, complemented
4C3×C4×C5×Dic3C211yesmaximal, characteristic, complemented
5#10623814C211yesmaximal, characteristic, complemented
6#10623813C211yesmaximal, characteristic
7Dic3×Dic15C211yesmaximal, characteristic, complemented
8#91761—31nomaximal
9#91403—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#67149C414yescomplemented
18#63836—51nomaximal, Hall
19C2×C4×D15—31no—
20C2×#1200—31no—
21C3×C42×C5—31no—
22#6714—31no—
23#6714—31no—
24#6714—31no—
25#6714—61no—
26#6712—31no—
27#6712D311yescharacteristic, complemented
28C4×Dic15—31no—
29C4×C5×Dic3D311yescharacteristic, complemented
30C3×#683—31no—
31#6668—31no—
32#6667—31no—
33Dic3×Dic5—31no—
34C22×C32×C5C2311yescharacteristic
35C2×C3⋊D15—21no—
36C2×C3⋊D15—21no—
37C2×C3⋊D15C2×C412yes—
38C32×C4×C5—21no—
39C3⋊Dic15—21no—
40C3×Dic15—21no—
41C3×Dic15C2×C412yes—
42C3×C5×Dic3—21no—
43C3×C5×Dic3C2×C412yes—
44#4018—91noHall
45C2×#624—51no—
46#3752—51no—
47C3×Dic3⋊C4—51no—
48C3×C4×Dic3D511yescharacteristic, complemented
49Dic3⋊Dic3—51no—
50#3722—51no—
51Dic32—51no—
52C22×D15—31no—
53C22×D15—31no—
54C22×D15—61no—
55C2×C3×C4×C5—21no—
56C2×C3×C4×C5—31no—
57C2×C3×C4×C5—31no—
58C2×C3×C4×C5C2×D311yescharacteristic
59C2×C3×C4×C5C2×D311yescharacteristic
60C2×Dic15—31no—
61C2×Dic15—31no—
62C2×Dic15—61no—
63C2×Dic15C2×D311yescharacteristic
64C2×Dic15C2×D311yescharacteristic
65C4×D15—34no—
66C2×C5×Dic3C2×D311yescharacteristic
67C2×C5×Dic3C2×D311yescharacteristic
68C2×C3×Dic5—31no—
69C2×C3×Dic5—31no—
70#1200—34no—
71#892—151no—
72#872—151no—
73C2×C32×C5C22×C411yescharacteristic
74C2×C32×C5#4111yescharacteristic
75C2×C32×C5#4111yescharacteristic
76C3⋊D15—22no—
77C2×C4×D5—91no—
78C42×C5—31no—
79#685—91no—
80#685—91no—
81#683—31no—
82#683—91no—
83C4×Dic5—91no—
84C22×C3⋊D3—51no—
85C2×C32×C4C2×D511yescharacteristic
86C2×C3⋊Dic3—51no—
87C2×C3×Dic3—51no—
88C2×C3×Dic3—51no—
89C2×C3×Dic3C2×D511yescharacteristic
90C2×C3×Dic3C2×D511yescharacteristic
91#624—54no—
92C22×C3×C5—21no—
93C22×C3×C5C22×D311yescharacteristic
94C22×C3×C5C22×D311yescharacteristic
95C2×D15—32no—
96C2×D15—32no—
97C2×D15—62no—
98C2×D15—61no—
99C2×D15—61no—
100C2×D15—61no—
101C2×D15—61no—
102C2×D15—121no—
103C2×D15—121no—
104C3×C4×C5—21no—
105C3×C4×C5—21no—
106C3×C4×C5—32no—
107C3×C4×C5—41no—
108C3×C4×C5—61no—
109Dic15—21no—
110Dic15—61no—
111Dic15—61no—
112Dic15—121no—
113Dic15C4×D312yes—
114C3×Dic5—32no—
115C3×Dic5—61no—
116C5×Dic3—21no—
117C5×Dic3C4×D312yes—
118C2×C4×D3—151no—
119C2×C4×D3—151no—
120C3×C4⋊C4—151no—
121C3×C42—31no—
122(C2×D3)⋊C4—151no—
123(C2×D3)⋊C4—151no—
124(C2×D3)⋊C4—151no—
125(C2×D3)⋊C4—151no—
126(C2×D3)⋊C4—301no—
127C4⋊Dic3—151no—
128Dic3⋊C4—51no—
129Dic3⋊C4—151no—
130C4×Dic3—151no—
131C4×Dic3—151no—
132C4×Dic3D1511yescharacteristic, complemented
133C32×C5#11711yescharacteristic, complemented, Hall
134C22×D5—91no—
135C2×C4×C5—31no—
136C2×C4×C5—31no—
137C2×C4×C5D3211yescharacteristic
138C2×Dic5—31no—
139C2×Dic5—31no—
140C2×Dic5—91no—
141C4×D5—94no—
142C22×C32C22×D511yescharacteristic
143C2×C3⋊D3—52no—
144C2×C3⋊D3—101no—
145C2×C3⋊D3—101no—
146C32×C4—21no—
147C3⋊Dic3—101no—
148C3×Dic3—21no—
149C3×Dic3—52no—
150C3×Dic3—101no—
151C3×Dic3C4×D512yes—
152#117—451noSylow
153C2×C3×C5—21no—
154C2×C3×C5—21no—
155C2×C3×C5—21no—
156C2×C3×C5C2×C4×D311yescharacteristic
157C2×C3×C5C2×C4×D311yescharacteristic
158C2×C3×C5#23511yescharacteristic
159C2×C3×C5#23511yescharacteristic
160C2×C3×C5#23711yescharacteristic
161C2×C3×C5#23911yescharacteristic
162D15—62no—
163D15—62no—
164D15—122no—
165C22×D3—151no—
166C22×D3—151no—
167C22×D3—301no—
168C2×C3×C4—21no—
169C2×C3×C4—31no—
170C2×C3×C4—31no—
171C2×C3×C4—151no—
172C2×C3×C4—151no—
173C2×C3×C4D3×D511yescharacteristic
174C2×C3×C4C2×D1511yescharacteristic
175C2×Dic3—51no—
176C2×Dic3—51no—
177C2×Dic3—151no—
178C2×Dic3—151no—
179C2×Dic3—301no—
180C2×Dic3C2×D1511yescharacteristic
181C2×Dic3C2×D1511yescharacteristic
182C4×D3—154no—
183C4×D3—154no—
184C22×C5C2×D3211yescharacteristic
185C2×D5—92no—
186C2×D5—181no—
187C2×D5—181no—
188C4×C5—21no—
189C4×C5—32no—
190C4×C5—61no—
191Dic5—32no—
192Dic5—61no—
193Dic5—181no—
194C2×C32C2×C4×D511yescharacteristic
195C2×C32#70911yescharacteristic
196C2×C32#70911yescharacteristic
197C3⋊D3—102no—
198C22×C4—451no—
199C4⋊C4—151no—
200C4⋊C4—451no—
201C22⋊C4—451no—
202C22⋊C4—451no—
203C42—31no—
204C42—451no—
205C3×C5—21no—
206C3×C5#87211yescharacteristic, complemented
207C3×C5#89211yescharacteristic, complemented
208C22×C3—21no—
209C22×C3C2×D3×D511yescharacteristic
210C22×C3C22×D1511yescharacteristic
211C2×D3—152no—
212C2×D3—152no—
213C2×D3—302no—
214C2×D3—301no—
215C2×D3—301no—
216C2×D3—301no—
217C2×D3—301no—
218C2×D3—601no—
219C2×D3—601no—
220C3×C4—21no—
221C3×C4—21no—
222C3×C4—32no—
223C3×C4—41no—
224C3×C4—61no—
225C3×C4—152no—
226C3×C4—301no—
227Dic3—21no—
228Dic3—52no—
229Dic3—101no—
230Dic3—301no—
231Dic3—301no—
232Dic3—601no—
233Dic3C4×D1512yes—
234C2×C5#379911yescharacteristic
235C2×C5C4×D3211yescharacteristic
236C2×C5#380411yescharacteristic
237D5—182no—
238C32#401811yescharacteristic, complemented, Sylow
239C23—451no—
240C2×C4—31no—
241C2×C4—31no—
242C2×C4—151no—
243C2×C4—151no—
244C2×C4—454no—
245C2×C4—451no—
246C2×C4D3×D1511yescharacteristic
247C2×C3—21no—
248C2×C3—21no—
249C2×C3—21no—
250C2×C3#676911yescharacteristic
251C2×C3C4×D3×D511yescharacteristic
252C2×C3#677911yescharacteristic
253C2×C3C2×C4×D1511yescharacteristic
254C2×C3#681711yescharacteristic
255C2×C3#681711yescharacteristic
256D3—302no—
257D3—302no—
258D3—602no—
259C5#6383611yescharacteristic, complemented, Sylow
260C22—452no—
261C22—901no—
262C22—901no—
263C22C2×D3×D1511yescharacteristic
264C4—21no—
265C4—32no—
266C4—61no—
267C4—152no—
268C4—301no—
269C4—901no—
270C3—21no—
271C3#9140311yescharacteristic, complemented
272C3#9176111yescharacteristic, complemented
273C2—902no—
274C2#1062419711yescharacteristic
275C2C4×D3×D1511yescharacteristic
276C2#1062420711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×D15
commutatorC2×C32×C5C22×C4
FittingC2×C32×C4×C5C22
socle, cosocleC22×C32×C5C23
prime radicalC22×C3⋊D15C22
hypercenterC2×C4D3×D15
hypocenterC32×C5#117
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. C3×#6712C2semidirect
4. C3×C4×C5×Dic3C2semidirect
5. #10623814C2semidirect
7. Dic3×Dic15C2semidirect
17. #67149C4semidirect
27. #6712D3semidirect
29. C4×C5×Dic3D3semidirect
48. C3×C4×Dic3D5semidirect
132. C4×Dic3D15semidirect
133. C32×C5#117semidirect
206. C3×C5#872semidirect
207. C3×C5#892semidirect
238. C32#4018semidirect
259. C5#63836semidirect
271. C3#91403semidirect
272. C3#91761semidirect

Maximal subgroups

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

subgroupindex
C2×#671492
#106239082
C3×#67122
C3×C4×C5×Dic32
#106238142
#106238132
Dic3×Dic152
#917613
#914033
#638365