Finite Group Databasebeta

← #13089277

Subgroups — #13089277

Order 1440.

876 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 876 conjugacy classes of subgroups — too many to draw. Only its 200 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#10624285C211yesmaximal, characteristic, complemented
2C5×#3830C211yesmaximal, characteristic, complemented
3C32×#711C211yesmaximal, characteristic, complemented
4C2×#67141C212yesmaximal, complemented
5#10624165C212yesmaximal, complemented
6#10624163C212yesmaximal, complemented
7#10624162C212yesmaximal, complemented
8#10624154C211yesmaximal, characteristic, complemented
9#10624153C211yesmaximal, characteristic, complemented
10C3⋊D3×Dic10C211yesmaximal, characteristic, complemented
11D5×#634C211yesmaximal, characteristic, complemented
12#92020—34nomaximal
13C32×C5×D4C2211yescharacteristic
14#67184C2212yes—
15C2×C3⋊Dic15C2212yes—
16C4×C3⋊D15C2211yescharacteristic
17#67180C2211yescharacteristic
18C5×#638C2212yes—
19C2×C5×C3⋊Dic3C2212yes—
20C4×C5×C3⋊D3C2211yescharacteristic
21C5×#634C2211yescharacteristic
22C32×C22⋊D5C2212yes—
23C2×C32×Dic5C2212yes—
24C32×C4×D5C2211yescharacteristic
25C32×Dic10C2211yescharacteristic
26#67141C2212yes—
27#67141C2212yes—
28#67141C2212yes—
29#67141C2211yescharacteristic
30#67140C2211yescharacteristic
31#67139C2211yescharacteristic
32#67138C2211yescharacteristic
33#67137C2212yes—
34C3⋊D3×Dic5C2212yes—
35D5×C3⋊Dic3C2212yes—
36#64367—51nomaximal, Hall
37#6819—34no—
38C5×#237—34no—
39C3×#711D314yescomplemented
40C2×#1204—38no—
41#6782—38no—
42#6780—38no—
43#6779—38no—
44#6771—34no—
45D15⋊Q8—34no—
46D3×Dic10—34no—
47D5×Dic6—34no—
48C22×C32×C5C2312yes—
49C2×C3⋊D15C2311yescharacteristic
50C2×C5×C3⋊D3C2311yescharacteristic
51C2×C32×D5C2311yescharacteristic
52C32×C4×C5C2311yescharacteristic
53C3⋊Dic15C2312yes—
54C3⋊Dic15C2311yescharacteristic
55C5×C3⋊Dic3C2312yes—
56C5×C3⋊Dic3C2311yescharacteristic
57C32×Dic5C2312yes—
58C32×Dic5C2311yescharacteristic
59#4150—91noHall
60C32×#41—51no—
61Q8×C3⋊D3—51no—
62Q8×C3⋊D3—51no—
63#3830—51no—
64#3830—52no—
65#3830—52no—
66#3830D511yescharacteristic, complemented
67#3828—52no—
68#3828—51no—
69C2×#634—52no—
70C2×#634—51no—
71C3×C5×D4C2×D314yes—
72#1220—38no—
73C2×Dic15—38no—
74C4×D15—34no—
75Dic30—34no—
76C5×#67—38no—
77C2×C5×Dic3—38no—
78C4×C5×D3—34no—
79C5×Dic6—34no—
80C3×C22⋊D5C2×D318yes—
81C2×C3×Dic5C2×D318yes—
82C3×C4×D5C2×D314yes—
83C3×Dic10C2×D314yes—
84#1204—38no—
85#1204—38no—
86#1204—38no—
87#1204—34no—
88#1203—34no—
89#1202—34no—
90#1201—34no—
91#1200—38no—
92D3×Dic5—38no—
93D5×Dic3—38no—
94#1010—154no—
95C2×C32×C5—22no—
96C2×C32×C5C2411yescharacteristic
97C3⋊D15—21no—
98C5×C3⋊D3—21no—
99C32×D5—21no—
100C5×#41—91no—
101Q8×D5—91no—
102Q8×D5—91no—
103#711—91no—
104#711—92no—
105#711—92no—
106#711C3⋊D311yescharacteristic, complemented
107#709—92no—
108#709—91no—
109C2×Dic10—92no—
110C2×Dic10—91no—
111C32×Q8—51no—
112C32×D4—52no—
113C32×D4C2×D511yescharacteristic
114C2×C32×C4—52no—
115C2×C32×C4—51no—
116#638—52no—
117#638—51no—
118#638—51no—
119#638C2×D512yes—
120C2×C3⋊Dic3—52no—
121C2×C3⋊Dic3—52no—
122C2×C3⋊Dic3C2×D512yes—
123C4⋊(C3⋊D3)—51no—
124C4×C3⋊D3—52no—
125C4×C3⋊D3—51no—
126C4×C3⋊D3—52no—
127C4×C3⋊D3C2×D511yescharacteristic
128#634—51no—
129#634—51no—
130#634—52no—
131#634—52no—
132#634—52no—
133#634C2×D511yescharacteristic
134C22×C3×C5C22×D318yes—
135C2×D15—34no—
136C2×C5×D3—34no—
137C2×C3×D5C22×D314yes—
138C3×C4×C5C22×D314yes—
139Dic15—38no—
140Dic15—34no—
141C3×Dic5C22×D318yes—
142C3×Dic5C22×D314yes—
143C5×Dic3—38no—
144C5×Dic3—34no—
145C3×#41—54no—
146D3×Q8—154no—
147D3×Q8—154no—
148#237—34no—
149#237—158no—
150#237—158no—
151#237—154no—
152#235—154no—
153#235—158no—
154C2×Dic6—154no—
155C2×Dic6—158no—
156C32×C52−1+411yescharacteristic, complemented, Hall
157C5×Q8—91no—
158C5×D4—92no—
159C5×D4C2×C3⋊D311yescharacteristic
160C2×C4×C5—92no—
161C2×C4×C5—91no—
162C22⋊D5—92no—
163C22⋊D5—91no—
164C22⋊D5—91no—
165C22⋊D5C2×C3⋊D312yes—
166C2×Dic5—92no—
167C2×Dic5—92no—
168C2×Dic5C2×C3⋊D312yes—
169D20—91no—
170C4×D5—92no—
171C4×D5—91no—
172C4×D5—92no—
173C4×D5C2×C3⋊D311yescharacteristic
174Dic10—91no—
175Dic10—91no—
176Dic10—92no—
177Dic10—92no—
178Dic10—92no—
179Dic10C2×C3⋊D311yescharacteristic
180C22×C32—51no—
181C22×C32C22×D512yes—
182C2×C3⋊D3—51no—
183C2×C3⋊D3C22×D511yescharacteristic
184C32×C4—51no—
185C32×C4—52no—
186C32×C4C22×D511yescharacteristic
187C3⋊Dic3—52no—
188C3⋊Dic3—51no—
189C3⋊Dic3C22×D511yescharacteristic
190C3⋊Dic3C22×D512yes—
1912−1+4—451noSylow
192C2×C3×C5—28no—
193C2×C3×C5C23×D314yes—
194D15—64no—
195C3×D5—24no—
196C5×D3—64no—
197C3×Q8—54no—
198C3×D4—58no—
199C3×D4D3×D514yes—
200C2×C3×C4—58no—
201C2×C3×C4—54no—
202#67—38no—
203#67—154no—
204#67—154no—
205#67—158no—
206C2×Dic3—38no—
207C2×Dic3—158no—
208C2×Dic3—158no—
209D12—154no—
210C4×D3—34no—
211C4×D3—158no—
212C4×D3—158no—
213C4×D3—154no—
214Dic6—34no—
215Dic6—158no—
216Dic6—154no—
217Dic6—158no—
218Dic6—158no—
219Dic6—154no—
220C22×C5—91no—
221C22×C5C22×C3⋊D312yes—
222C2×D5—91no—
223C2×D5C22×C3⋊D311yescharacteristic
224C4×C5—91no—
225C4×C5—92no—
226C4×C5C22×C3⋊D311yescharacteristic
227Dic5—92no—
228Dic5—91no—
229Dic5C22×C3⋊D311yescharacteristic
230Dic5C22×C3⋊D312yes—
231C2×C32—22no—
232C2×C32—101no—
233C2×C32C23×D511yescharacteristic
234C3⋊D3—21no—
235C3⋊D3—101no—
236#41—51no—
237#41—91no—
238#41—451no—
239#41—452no—
240#41—452no—
241#41—452no—
242#41—451no—
243C2×Q8—452no—
244C2×Q8—451no—
245C2×Q8—451no—
246C2×Q8—451no—
247C3×C5#101014yescomplemented
248C22×C3—54no—
249C22×C3C2×D3×D518yes—
250C2×D3—34no—
251C2×D3—154no—
252C3×C4—54no—
253C3×C4—58no—
254C3×C4C2×D3×D514yes—
255Dic3—38no—
256Dic3—34no—
257Dic3—158no—
258Dic3—154no—
259C2×C5—22no—
260C2×C5—181no—
261C2×C5C23×C3⋊D311yescharacteristic
262D5—21no—
263D5—181no—
264C32#415011yescharacteristic, complemented, Sylow
265Q8—51no—
266Q8—91no—
267Q8—451no—
268Q8—452no—
269Q8—452no—
270Q8—452no—
271Q8—451no—
272D4—52no—
273D4—92no—
274D4—451no—
275D4—451no—
276D4—451no—
277D4—452no—
278D4D5×C3⋊D311yescharacteristic
279C2×C4—52no—
280C2×C4—51no—
281C2×C4—92no—
282C2×C4—91no—
283C2×C4—452no—
284C2×C4—452no—
285C2×C4—452no—
286C2×C4—452no—
287C2×C4—451no—
288C2×C3—28no—
289C2×C3—104no—
290C2×C3C22×D3×D514yes—
291D3—64no—
292D3—304no—
293C5#6436711yescharacteristic, complemented, Sylow
294C22—51no—
295C22—91no—
296C22—451no—
297C22C2×D5×C3⋊D312yes—
298C4—51no—
299C4—52no—
300C4—91no—
301C4—92no—
302C4—452no—
303C4—451no—
304C4C2×D5×C3⋊D311yescharacteristic
305C3#9202014yescomplemented
306C2—22no—
307C2—101no—
308C2—181no—
309C2—901no—
310C2C22×D5×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC2C22×D5×C3⋊D3
commutator, socle, cosocleC2×C32×C5C24
FittingC32×C5×D4C22
hypercenterD4D5×C3⋊D3
hypocenterC32×C52−1+4
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ D4

Complements

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

NH ≅ G/Ntype
1. #10624285C2semidirect
2. C5×#3830C2semidirect
3. C32×#711C2semidirect
4. C2×#67141C2semidirect
5. #10624165C2semidirect
6. #10624163C2semidirect
7. #10624162C2semidirect
8. #10624154C2semidirect
9. #10624153C2semidirect
10. C3⋊D3×Dic10C2semidirect
11. D5×#634C2semidirect
39. C3×#711D3semidirect
66. #3830D5semidirect
106. #711C3⋊D3semidirect
156. C32×C52−1+4semidirect
247. C3×C5#1010semidirect
264. C32#4150semidirect
293. C5#64367semidirect
305. C3#92020semidirect

Maximal subgroups

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

subgroupindex
#106242852
C5×#38302
C32×#7112
C2×#671412
#106241652
#106241632
#106241622
#106241542
#106241532
C3⋊D3×Dic102
D5×#6342
#920203
#643675