Finite Group Databasebeta

← #13088916

Subgroups — #13088916

Order 1440.

694 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 144 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#10624227C211yesmaximal, characteristic, complemented
2#10624193C212yesmaximal, complemented
3C4×C5×D32C211yesmaximal, characteristic, complemented
4C3⋊D3×Dic10C211yesmaximal, characteristic, complemented
5C3×#6769C212yesmaximal, complemented
6#10624053C214yesmaximal, complemented
7#10624034C212yesmaximal, complemented
8#10624022C212yesmaximal, complemented
9D3×#709—32nomaximal
10C2×(C3⋊D3)⋊D5C2212yes—
11C2×C5×D32C2211yescharacteristic
12#67180C2211yescharacteristic, complemented
13C4×C5×C3⋊D3C2211yescharacteristic
14C3×D60C2212yescomplemented
15C3×C4×C5×D3C2212yescomplemented
16C32×Dic10C2211yescharacteristic, complemented
17#67157C2212yescomplemented
18#67150C2214yescomplemented
19Dic3×D15C2214yescomplemented
20C5×#624C2211yescharacteristic
21C5×D3×Dic3C2212yescomplemented
22#67141C2212yescomplemented
23C3⋊D3×Dic5C2212yes—
24C3×#1203C2214yescomplemented
25C3×#1200C2214yescomplemented
26#64319—51nomaximal, Hall
27#6817—32no—
28C2×C4×C5×D3—32no—
29C3×#709—32no—
30D3×C22⋊D5—34no—
31#6779—34no—
32D3×D20—32no—
33C4×D3×D5—34no—
34#6771—34no—
35#6769D312yescomplemented
36D3×Dic10—32no—
37D3×Dic10—61no—
38#6765—32no—
39C2×C5×C3⋊D3C2311yescharacteristic
40C2×C3×D15C2314yes—
41C2×C3×C5×D3C2312yes—
42(C3⋊D3)⋊D5—24no—
43C5×D32—22no—
44C32×C4×C5C2311yescharacteristic
45C3⋊Dic15C2312yes—
46C5×C3⋊Dic3C2311yescharacteristic
47C3×C5×Dic3C2312yes—
48C32×Dic5C2312yes—
49C2×#709—91noHall
50Q8×C3⋊D3—51no—
51C3×#239—52no—
52#3802—52no—
53#3802—51no—
54C4×D32—52no—
55C4×D32D511yescharacteristic, complemented
56#3795—52no—
57#3795—54no—
58C22×C5×D3—32no—
59C2×D3×D5—34no—
60C2×C3×C4×C5—32no—
61#1220—34no—
62D60C2×D312yescomplemented
63C4×D15—34no—
64Dic30—32no—
65Dic30—61no—
66C2×C5×Dic3—32no—
67C4×C5×D3—32no—
68C4×C5×D3—32no—
69C4×C5×D3—32no—
70C4×C5×D3—61no—
71C4×C5×D3C2×D312yescomplemented
72C3×C22⋊D5—34no—
73C3×D20—32no—
74C3×C4×D5—34no—
75C3×Dic10—21no—
76C3×Dic10C2×D312yescomplemented
77#1204—34no—
78#1204—62no—
79#1203C2×D314yescomplemented
80#1202—34no—
81#1201—34no—
82#1200C2×D314yescomplemented
83D3×Dic5—34no—
84D3×Dic5—62no—
85D5×Dic3—34no—
86D3×#41—152no—
87C2×C32×C5C2411yescharacteristic
88C5×C3⋊D3#4112yescomplemented
89C3×D15—24no—
90C3×C5×D3—22no—
91C22×C4×C5—91no—
92C2×C22⋊D5—92no—
93#709—32no—
94#709—94no—
95#709—92no—
96C2×D20—91no—
97C2×C4×D5—92no—
98C2×Dic10—91no—
99C2×D32—52no—
100C2×D32C2×D511yescharacteristic
101C32×Q8—51no—
102C4×C3⋊D3—52no—
103C4×C3⋊D3C2×D511yescharacteristic
104#634—51no—
105#634—52no—
106C3×D12—54no—
107C3×D12—52no—
108C3×C4×D3—54no—
109C3×C4×D3C2×D512yescomplemented
110#625—54no—
111#625—52no—
112#624C2×D511yescharacteristic
113D3×Dic3—54no—
114D3×Dic3C2×D512yescomplemented
115C22×C3×C5—32no—
116C2×D15C22×D314yes—
117C2×C5×D3—32no—
118C2×C5×D3—61no—
119C2×C5×D3—64no—
120C2×C5×D3C22×D312yes—
121C2×C3×D5—34no—
122D3×D5—68no—
123C3×C4×C5—21no—
124C3×C4×C5—32no—
125C3×C4×C5C22×D312yes—
126Dic15—34no—
127Dic15—62no—
128C3×Dic5—22no—
129C3×Dic5C22×D314yes—
130C5×Dic3—32no—
131C5×Dic3—61no—
132C5×Dic3C22×D312yes—
133C3×#41—152no—
134#239—52no—
135D3×Q8—152no—
136D3×Q8—301no—
137#237—154no—
138#237—152no—
139D3×D4—152no—
140D3×D4—154no—
141#235—154no—
142#235—152no—
143C2×C4×D3—32no—
144C2×C4×D3—154no—
145C32×C5C2×#4111yescharacteristic, complemented, Hall
146C23×C5—91no—
147C22×D5—92no—
148C2×C4×C5—32no—
149C2×C4×C5—91no—
150C2×C4×C5—92no—
151C2×C4×C5—91no—
152C22⋊D5—34no—
153C22⋊D5—94no—
154C2×Dic5—92no—
155D20—32no—
156D20—92no—
157C4×D5—34no—
158C4×D5—94no—
159Dic10—91no—
160Dic10—92no—
161Dic10D3211yescharacteristic, complemented
162C2×C3⋊D3C22×D511yescharacteristic
163C2×C3×D3—54no—
164C2×C3×D3C22×D512yes—
165D32—22no—
166D32—104no—
167C32×C4—52no—
168C32×C4C22×D511yescharacteristic
169C3⋊Dic3—52no—
170C3⋊Dic3C22×D511yescharacteristic
171C3×Dic3C22×D512yes—
172C2×#41—451noSylow
173C2×C3×C5—21no—
174C2×C3×C5—62no—
175C2×C3×C5C23×D312yes—
176D15—24no—
177C3×D5—64no—
178C5×D3—22no—
179C5×D3—34no—
180C5×D3—62no—
181C22×D3—32no—
182C22×D3—154no—
183C3×Q8—52no—
184C3×Q8—101no—
185C3×D4—154no—
186C3×D4—152no—
187C2×C3×C4—32no—
188C2×C3×C4—154no—
189#67—154no—
190#67—154no—
191#67—154no—
192C2×Dic3—32no—
193C2×Dic3—154no—
194D12—54no—
195D12—52no—
196C4×D3—32no—
197C4×D3—32no—
198C4×D3—32no—
199C4×D3—54no—
200C4×D3—61no—
201C4×D3—154no—
202C4×D3—154no—
203C4×D3—302no—
204C4×D3D3×D512yescomplemented
205Dic6—154no—
206Dic6—152no—
207Dic6—302no—
208Dic6—301no—
209C22×C5—32no—
210C22×C5—91no—
211C22×C5—182no—
212C2×D5—34no—
213C2×D5—184no—
214C4×C5—32no—
215C4×C5—91no—
216C4×C5C2×D3211yescharacteristic
217Dic5—92no—
218Dic5C2×D3212yes—
219C2×C32C23×D511yescharacteristic
220C3⋊D3#70912yescomplemented
221C3×D3—22no—
222C3×D3—104no—
223#41—152no—
224#41—452no—
225#41—454no—
226C2×Q8—451no—
227C2×D4—451no—
228C2×D4—452no—
229C22×C4—91no—
230C22×C4—452no—
231C3×C5—21no—
232C3×C5D3×#4112yescomplemented
233C22×C3—32no—
234C22×C3—154no—
235C2×D3—32no—
236C2×D3—54no—
237C2×D3—61no—
238C2×D3—64no—
239C2×D3—308no—
240C2×D3C2×D3×D512yes—
241C3×C4—21no—
242C3×C4—32no—
243C3×C4—54no—
244C3×C4—102no—
245C3×C4C2×D3×D512yes—
246Dic3—32no—
247Dic3—61no—
248Dic3—154no—
249Dic3—302no—
250Dic3C2×D3×D512yes—
251C2×C5—62no—
252C2×C5—92no—
253C2×C5C22×D3211yescharacteristic
254D5—64no—
255C32C2×#70911yescharacteristic, complemented, Sylow
256C23—91no—
257C23—452no—
258Q8—51no—
259Q8—452no—
260Q8—451no—
261D4—154no—
262D4—152no—
263D4—452no—
264D4—454no—
265C2×C4—32no—
266C2×C4—91no—
267C2×C4—92no—
268C2×C4—91no—
269C2×C4—154no—
270C2×C4—454no—
271C2×C4—452no—
272C2×C3—21no—
273C2×C3—62no—
274C2×C3—304no—
275C2×C3C22×D3×D512yes—
276D3—22no—
277D3—34no—
278D3—62no—
279D3—104no—
280C5#6431911yescharacteristic, complemented, Sylow
281C22—32no—
282C22—91no—
283C22—154no—
284C22—182no—
285C22—904no—
286C4—32no—
287C4—52no—
288C4—91no—
289C4—452no—
290C4D32×D511yescharacteristic
291C3—21no—
292C3D3×#70912yescomplemented
293C2—62no—
294C2—92no—
295C2—304no—
296C2C2×D32×D511yescharacteristic
subgroupG/H
center, FrattiniC2C2×D32×D5
commutator, socle, cosocleC2×C32×C5C24
FittingC32×C4×C5C23
hypercenterC4D32×D5
hypocenterC32×C5C2×#41
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. #10624227C2semidirect
2. #10624193C2semidirect
3. C4×C5×D32C2semidirect
4. C3⋊D3×Dic10C2semidirect
5. C3×#6769C2semidirect
6. #10624053C2semidirect
7. #10624034C2semidirect
8. #10624022C2semidirect
12. #67180C22semidirect
14. C3×D60C22semidirect
15. C3×C4×C5×D3C22semidirect
16. C32×Dic10C22semidirect
17. #67157C22semidirect
18. #67150C22semidirect
19. Dic3×D15C22semidirect
21. C5×D3×Dic3C22semidirect
22. #67141C22semidirect
24. C3×#1203C22semidirect
25. C3×#1200C22semidirect
35. #6769D3semidirect
55. C4×D32D5semidirect
62. D60C2×D3semidirect
71. C4×C5×D3C2×D3semidirect
76. C3×Dic10C2×D3semidirect
79. #1203C2×D3semidirect
82. #1200C2×D3semidirect
88. C5×C3⋊D3#41semidirect
109. C3×C4×D3C2×D5semidirect
114. D3×Dic3C2×D5semidirect
145. C32×C5C2×#41semidirect
161. Dic10D32semidirect
204. C4×D3D3×D5semidirect
220. C3⋊D3#709semidirect
232. C3×C5D3×#41semidirect
255. C32C2×#709semidirect
280. C5#64319semidirect
292. C3D3×#709semidirect

Maximal subgroups

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

subgroupindex
#106242272
#106241932
C4×C5×D322
C3⋊D3×Dic102
C3×#67692
#106240532
#106240342
#106240222
D3×#7093
#643195