Finite Group Databasebeta

← #13089409

Subgroups — #13089409

Order 1440.

710 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 147 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×C4×C5×C3⋊D3C211yesmaximal, characteristic, complemented
2C3×#6817C212yesmaximal, complemented
3#10624236C212yesmaximal, complemented
4#10624229C212yesmaximal, complemented
5#10624227C211yesmaximal, characteristic, complemented
6C4×(C3⋊D3)⋊D5C212yesmaximal, complemented
7#10624223C212yesmaximal, complemented
8#10624222C211yesmaximal, characteristic, complemented
9#10624220C212yesmaximal, complemented
10D3×#709—32nomaximal
11C22×C5×C3⋊D3C2211yescharacteristic
12C2×(C3⋊D3)⋊D5C2212yes—
13C2×C32×C4×C5C2211yescharacteristic, complemented
14C2×C5×C3⋊Dic3C2211yescharacteristic, complemented
15C4×C5×C3⋊D3C2211yescharacteristic
16C4×C5×C3⋊D3C2211yescharacteristic
17C4×C5×C3⋊D3C2211yescharacteristic, complemented
18C4×C5×C3⋊D3C2211yescharacteristic, complemented
19C3×#1220C2214yescomplemented
20C3×D60C2212yescomplemented
21C3×C4×D15C2214yescomplemented
22C3×Dic30C2212yescomplemented
23#67158C2212yescomplemented
24#67157C2212yescomplemented
25#67156C2214yescomplemented
26#67155C2212yes—
27#67154C2214yescomplemented
28#64306—51nomaximal, Hall
29#6817D312yescomplemented
30C2×C4×C5×D3—32no—
31C2×C4×C5×D3—61no—
32C3×#709—32no—
33D3×C22⋊D5—34no—
34#6779—34no—
35D3×D20—32no—
36C4×D3×D5—34no—
37#6771—34no—
38#6769—32no—
39D3×Dic10—32no—
40#6765—32no—
41C22×C32×C5C2311yescharacteristic
42C2×C5×C3⋊D3—22no—
43C2×C5×C3⋊D3C2311yescharacteristic
44C2×C5×C3⋊D3C2311yescharacteristic
45C2×C3×D15C2314yes—
46(C3⋊D3)⋊D5—24no—
47C32×C4×C5C2311yescharacteristic
48C32×C4×C5C2311yescharacteristic
49C5×C3⋊Dic3C2311yescharacteristic
50C5×C3⋊Dic3C2311yescharacteristic
51C3×Dic15C2314yes—
52C2×#709—91noHall
53C2×C4×C3⋊D3D511yescharacteristic, complemented
54C3×#235—52no—
55#3811—52no—
56#3805—52no—
57#3802—51no—
58C4×D32—52no—
59#3798—52no—
60#3797—51no—
61#3796—52no—
62C22×C5×D3—32no—
63C22×C5×D3—61no—
64C2×D3×D5—34no—
65C2×C3×C4×C5—21no—
66C2×C3×C4×C5C2×D312yescomplemented
67#1220C2×D314yescomplemented
68D60C2×D312yescomplemented
69C4×D15C2×D314yescomplemented
70Dic30C2×D312yescomplemented
71C2×C5×Dic3—32no—
72C2×C5×Dic3—61no—
73C4×C5×D3—32no—
74C4×C5×D3—32no—
75C4×C5×D3—32no—
76C4×C5×D3—32no—
77C4×C5×D3—61no—
78C4×C5×D3—61no—
79C4×C5×D3—61no—
80C4×C5×D3—61no—
81C3×C22⋊D5—34no—
82C3×D20—32no—
83C3×C4×D5—34no—
84C3×Dic10—32no—
85#1204—34no—
86#1203—34no—
87#1202—34no—
88#1201—34no—
89#1200—34no—
90D3×Dic5—34no—
91D5×Dic3—34no—
92D3×#41—152no—
93C2×C32×C5—21no—
94C2×C32×C5C2411yescharacteristic
95C5×C3⋊D3—21no—
96C5×C3⋊D3#4112yescomplemented
97C3×D15—24no—
98C22×C4×C5—91no—
99C2×C22⋊D5—92no—
100#709—32no—
101#709—92no—
102#709—92no—
103#709—92no—
104C2×D20—91no—
105C2×C4×D5—92no—
106C2×Dic10—91no—
107C22×C3⋊D3C2×D511yescharacteristic
108C2×D32—52no—
109C2×C32×C4C2×D511yescharacteristic, complemented
110C2×C3⋊Dic3C2×D511yescharacteristic, complemented
111C4×C3⋊D3C2×D511yescharacteristic
112C4×C3⋊D3C2×D511yescharacteristic
113C4×C3⋊D3C2×D511yescharacteristic, complemented
114C4×C3⋊D3C2×D511yescharacteristic, complemented
115C3×#67—54no—
116C3×D12—52no—
117C3×C4×D3—54no—
118C3×Dic6—52no—
119#627—52no—
120#626—54no—
121#625—52no—
122#624—52no—
123D3×Dic3—54no—
124C22×C3×C5—21no—
125C22×C3×C5C22×D312yes—
126C2×D15C22×D314yes—
127C2×C5×D3—32no—
128C2×C5×D3—32no—
129C2×C5×D3—61no—
130C2×C5×D3—61no—
131C2×C5×D3—64no—
132C2×C5×D3—64no—
133C2×C3×D5—34no—
134D3×D5—68no—
135C3×C4×C5—21no—
136C3×C4×C5—21no—
137C3×C4×C5C22×D312yes—
138C3×C4×C5C22×D312yes—
139Dic15C22×D314yes—
140C3×Dic5—34no—
141C5×Dic3—32no—
142C5×Dic3—32no—
143C5×Dic3—61no—
144C5×Dic3—61no—
145C3×#41—152no—
146#239—152no—
147D3×Q8—152no—
148#237—152no—
149#237—154no—
150D3×D4—152no—
151D3×D4—154no—
152#235—52no—
153#235—154no—
154C2×C4×D3—32no—
155C2×C4×D3—61no—
156C2×C4×D3—154no—
157C32×C5C2×#4111yescharacteristic, complemented, Hall
158C23×C5—91no—
159C22×D5—92no—
160C2×C4×C5—91no—
161C2×C4×C5—91no—
162C2×C4×C5—91no—
163C2×C4×C5—91no—
164C2×C4×C5—91no—
165C2×C4×C5D3211yescharacteristic, complemented
166C22⋊D5—34no—
167C22⋊D5—94no—
168C2×Dic5—92no—
169D20—32no—
170D20—92no—
171C4×D5—34no—
172C4×D5—94no—
173Dic10—32no—
174Dic10—92no—
175C22×C32C22×D511yescharacteristic
176C2×C3⋊D3—22no—
177C2×C3⋊D3C22×D511yescharacteristic
178C2×C3⋊D3C22×D511yescharacteristic
179C2×C3×D3—54no—
180D32—104no—
181C32×C4C22×D511yescharacteristic
182C32×C4C22×D511yescharacteristic
183C3⋊Dic3C22×D511yescharacteristic
184C3⋊Dic3C22×D511yescharacteristic
185C3×Dic3—54no—
186C2×#41—451noSylow
187C2×C3×C5—22no—
188C2×C3×C5—22no—
189C2×C3×C5—21no—
190C2×C3×C5C23×D312yes—
191D15—24no—
192C3×D5—64no—
193C5×D3—34no—
194C5×D3—62no—
195C5×D3—62no—
196C5×D3—62no—
197C22×D3—32no—
198C22×D3—61no—
199C22×D3—154no—
200C3×Q8—152no—
201C3×D4—154no—
202C3×D4—152no—
203C2×C3×C4—21no—
204C2×C3×C4—154no—
205C2×C3×C4D3×D512yescomplemented
206#67—54no—
207#67—154no—
208#67—154no—
209C2×Dic3—32no—
210C2×Dic3—61no—
211C2×Dic3—154no—
212D12—52no—
213D12—154no—
214C4×D3—32no—
215C4×D3—32no—
216C4×D3—32no—
217C4×D3—32no—
218C4×D3—54no—
219C4×D3—61no—
220C4×D3—61no—
221C4×D3—61no—
222C4×D3—61no—
223C4×D3—154no—
224C4×D3—154no—
225Dic6—52no—
226Dic6—154no—
227C22×C5—91no—
228C22×C5—91no—
229C22×C5—182no—
230C22×C5C2×D3211yescharacteristic
231C2×D5—34no—
232C2×D5—184no—
233C4×C5—91no—
234C4×C5—91no—
235C4×C5C2×D3211yescharacteristic
236C4×C5C2×D3211yescharacteristic
237Dic5—34no—
238C2×C32—21no—
239C2×C32C23×D511yescharacteristic
240C3⋊D3—21no—
241C3⋊D3#70912yescomplemented
242C3×D3—104no—
243#41—152no—
244#41—452no—
245#41—452no—
246#41—452no—
247C2×Q8—451no—
248C2×D4—451no—
249C2×D4—452no—
250C22×C4—91no—
251C22×C4—452no—
252C3×C5—21no—
253C3×C5D3×#4112yescomplemented
254C22×C3—21no—
255C22×C3—154no—
256C22×C3C2×D3×D512yes—
257C2×D3—32no—
258C2×D3—32no—
259C2×D3—54no—
260C2×D3—61no—
261C2×D3—61no—
262C2×D3—64no—
263C2×D3—64no—
264C2×D3—308no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4—154no—
268C3×C4C2×D3×D512yes—
269C3×C4C2×D3×D512yes—
270Dic3—32no—
271Dic3—32no—
272Dic3—54no—
273Dic3—61no—
274Dic3—61no—
275C2×C5—21no—
276C2×C5—92no—
277C2×C5—181no—
278C2×C5C22×D3211yescharacteristic
279D5—64no—
280C32C2×#70911yescharacteristic, complemented, Sylow
281C23—91no—
282C23—452no—
283Q8—152no—
284Q8—452no—
285D4—154no—
286D4—152no—
287D4—452no—
288D4—454no—
289C2×C4—91no—
290C2×C4—91no—
291C2×C4—91no—
292C2×C4—91no—
293C2×C4—91no—
294C2×C4—154no—
295C2×C4—454no—
296C2×C4—452no—
297C2×C4(C3⋊D3)⋊D511yescharacteristic, complemented
298C2×C3—22no—
299C2×C3—22no—
300C2×C3—21no—
301C2×C3—304no—
302C2×C3C22×D3×D512yes—
303D3—34no—
304D3—62no—
305D3—62no—
306D3—62no—
307D3—104no—
308C5#6430611yescharacteristic, complemented, Sylow
309C22—91no—
310C22—91no—
311C22—154no—
312C22—182no—
313C22—904no—
314C22C2×(C3⋊D3)⋊D511yescharacteristic
315C4—91no—
316C4—91no—
317C4—154no—
318C4C2×(C3⋊D3)⋊D511yescharacteristic
319C4C2×(C3⋊D3)⋊D511yescharacteristic
320C3—21no—
321C3D3×#70912yescomplemented
322C2—21no—
323C2—92no—
324C2—181no—
325C2—304no—
326C2C22×(C3⋊D3)⋊D511yescharacteristic
subgroupG/H
centerC4C2×(C3⋊D3)⋊D5
commutator, socle, cosocleC2×C32×C5C24
FrattiniC2C22×(C3⋊D3)⋊D5
FittingC2×C32×C4×C5C22
hypercenterC2×C4(C3⋊D3)⋊D5
hypocenterC32×C5C2×#41
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C4×C5×C3⋊D3C2semidirect
2. C3×#6817C2semidirect
3. #10624236C2semidirect
4. #10624229C2semidirect
5. #10624227C2semidirect
6. C4×(C3⋊D3)⋊D5C2semidirect
7. #10624223C2semidirect
8. #10624222C2semidirect
9. #10624220C2semidirect
13. C2×C32×C4×C5C22semidirect
14. C2×C5×C3⋊Dic3C22semidirect
17. C4×C5×C3⋊D3C22semidirect
18. C4×C5×C3⋊D3C22semidirect
19. C3×#1220C22semidirect
20. C3×D60C22semidirect
21. C3×C4×D15C22semidirect
22. C3×Dic30C22semidirect
23. #67158C22semidirect
24. #67157C22semidirect
25. #67156C22semidirect
27. #67154C22semidirect
29. #6817D3semidirect
53. C2×C4×C3⋊D3D5semidirect
66. C2×C3×C4×C5C2×D3semidirect
67. #1220C2×D3semidirect
68. D60C2×D3semidirect
69. C4×D15C2×D3semidirect
70. Dic30C2×D3semidirect
96. C5×C3⋊D3#41semidirect
109. C2×C32×C4C2×D5semidirect
110. C2×C3⋊Dic3C2×D5semidirect
113. C4×C3⋊D3C2×D5semidirect
114. C4×C3⋊D3C2×D5semidirect
157. C32×C5C2×#41semidirect
165. C2×C4×C5D32semidirect
205. C2×C3×C4D3×D5semidirect
241. C3⋊D3#709semidirect
253. C3×C5D3×#41semidirect
280. C32C2×#709semidirect
297. C2×C4(C3⋊D3)⋊D5semidirect
308. C5#64306semidirect
321. C3D3×#709semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C5×C3⋊D32
C3×#68172
#106242362
#106242292
#106242272
C4×(C3⋊D3)⋊D52
#106242232
#106242222
#106242202
D3×#7093
#643065