Finite Group Databasebeta

← #14512740

Subgroups — #14512740

Order 1920.

480 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 53 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
1C4×C5⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
2#11752110C211yesmaximal, characteristic, complemented
3C5×C4⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
4Dic5×GL(2,𝔽3)C212yesmaximal, complemented
5#11751599C212yesmaximal, complemented
6#10612092—31nomaximal, Hall
7C2×C5⋊(#41⋊C3)C2211yescharacteristic
8C2×C5×GL(2,𝔽3)C2212yes—
9C4×C5×SL(2,𝔽3)C2211yescharacteristic
10Dic5×D12—41nomaximal
11Dic5×SL(2,𝔽3)C2211yescharacteristic
12#91184C2212yes—
13#85345—51nomaximal, Hall
14#66050—31no—
15C4×#713D311yescharacteristic, complemented
16C5×#460—31no—
17#65461—32no—
18SD16×Dic5—32no—
19#65361—31no—
20#65310—31no—
21#65131—31no—
22#65122—31no—
23#65068—32no—
24#65045—32no—
25C2×C5×D12—41no—
26C2×C5×SL(2,𝔽3)C2311yescharacteristic
27C2×D3×Dic5—42no—
28C5⋊(#41⋊C3)—21no—
29C5×GL(2,𝔽3)—22no—
30C4⋊Dic15—41no—
31C3×C4×Dic5—41no—
32(C2×D3)⋊Dic5—42no—
33C4×#41⋊C3—51no—
34#5427—52no—
35C4⋊GL(2,𝔽3)D511yescharacteristic, complemented
36C4×GL(2,𝔽3)—52no—
37#5391—51no—
38C2×#713C2×D311yescharacteristic
39C2×#711—61no—
40C2×C5×SD16—32no—
41C5×#127—31no—
42C4×C5×Q8C2×D311yescharacteristic
43Q8×Dic5C2×D311yescharacteristic
44#4084—32no—
45#4084—61no—
46#4082—61no—
47D4×Dic5—32no—
48D4×Dic5—61no—
49C2×#686—32no—
50#4039—31no—
51#4038—31no—
52C4×D20—31no—
53C42×D5—31no—
54#4015—31no—
55C5×C4⋊C8—31no—
56C5×#102—32no—
57#3967—32no—
58#3964—32no—
59#3953—32no—
60#3952—32no—
61#3949—32no—
62C8×Dic5—32no—
63#3939—32no—
64C4⋊#168—31no—
65#3106—151noSylow
66C22×C5×D3—42no—
67C2×C3×C4×C5—41no—
68C2×Dic15—42no—
69C5×D12—44no—
70C2×C3×Dic5—41no—
71C2×C3×Dic5—41no—
72C5×SL(2,𝔽3)#4111yescharacteristic, complemented
73D3×Dic5—84no—
74C2×#41⋊C3—51no—
75C2×GL(2,𝔽3)C2×D512yes—
76C4×D12—201no—
77C4×SL(2,𝔽3)—51no—
78C4×SL(2,𝔽3)C2×D511yescharacteristic
79#859—52no—
80C2×C5×Q8C22×D311yescharacteristic
81C2×C5×D4—32no—
82C2×C5×D4—61no—
83C2×C5×D4—61no—
84C2×C22⋊D5—62no—
85C22×Dic5—62no—
86#713—21no—
87#713—61no—
88#711—124no—
89C2×D20—31no—
90C2×C4×D5—31no—
91C2×C4×D5—31no—
92C2×C4×D5—31no—
93C2×Dic10—61no—
94C5×SD16—62no—
95C5×SD16—62no—
96C2×C5×C8—32no—
97C5×C4⋊C4—31no—
98C42×C5—31no—
99C22⋊Dic5—62no—
100C22⋊Dic5—62no—
101#686—64no—
102#685—61no—
103C4⋊Dic5—31no—
104C4⋊Dic5—32no—
105C4⋊Dic5—61no—
106C4×Dic5—31no—
107C4×Dic5—31no—
108C4×Dic5S411yescharacteristic
109C2×#168—32no—
110#532—151no—
111C4×#41—51no—
112#482—152no—
113#477—151no—
114#476—151no—
115#460—31no—
116#459—151no—
117#449—152no—
118C4×SD16—152no—
119C4×D8—152no—
120#431—151no—
121C22×C3×C5—41no—
122C2×C5×D3—44no—
123C2×C5×D3—82no—
124C2×C5×D3—82no—
125C3×C4×C5—42no—
126Dic15—82no—
127C3×Dic5—42no—
128C3×Dic5—81no—
129C2×D12—41no—
130C2×C4×D3—202no—
131#41⋊C3—101no—
132C2×SL(2,𝔽3)C22×D511yescharacteristic
133GL(2,𝔽3)—22no—
134C3×C42—201no—
135(C2×D3)⋊C4—202no—
136C4⋊Dic3—201no—
137C23×C5—62no—
138C22×D5—31no—
139C5×Q8—31no—
140C5×Q8#23511yescharacteristic
141C5×D4—62no—
142C5×D4—62no—
143C5×D4—122no—
144C5×D4—122no—
145C2×C4×C5—31no—
146C2×C4×C5—31no—
147C2×C4×C5C2×S411yescharacteristic
148C22⋊D5—124no—
149C2×Dic5—31no—
150C2×Dic5—62no—
151C2×Dic5—124no—
152C2×Dic5C2×S411yescharacteristic
153C2×Dic5C2×S411yescharacteristic
154D20—32no—
155D20—61no—
156C4×D5—61no—
157C4×D5—61no—
158C4×D5—62no—
159C4×D5—62no—
160Dic10—122no—
161C5×C8—62no—
162#168—62no—
163C2×#41—51no—
164C2×#41—301no—
165C2×SD16—32no—
166C2×D8—152no—
167C4⋊Q8—151no—
168#127—31no—
169#124—301no—
170#121—152no—
171#121—301no—
172C4×Q8—51no—
173C4×Q8D3×D511yescharacteristic
174C4×D4—151no—
175C4×D4—151no—
176C4×D4—152no—
177C4×D4—301no—
178#117—151no—
179C2×C42—151no—
180#107—152no—
181#106—152no—
182C4⋊C8—31no—
183C4⋊C8—151no—
184#103—152no—
185#102—32no—
186#102—152no—
187#102—152no—
188C22⋊C8—152no—
189C4×C8—152no—
190C2×C3×C5—41no—
191C2×C3×C5—41no—
192C2×C3×C5—41no—
193C5×D3—84no—
194C22×D3—42no—
195C2×C3×C4—41no—
196C2×C3×C4—201no—
197C2×C3×C4—201no—
198C2×Dic3—202no—
199D12—44no—
200C4×D3—404no—
201SL(2,𝔽3)#71111yescharacteristic, complemented
202C22×C5—122no—
203C22×C5—122no—
204C22×C5—122no—
205C22×C5C22×S411yescharacteristic
206C2×D5—61no—
207C2×D5—61no—
208C2×D5—61no—
209C4×C5—31no—
210C4×C5—31no—
211C4×C5—61no—
212C4×C5#98512yes—
213Dic5—21no—
214Dic5—61no—
215Dic5—122no—
216Dic5#98612yes—
217#41—101no—
218#41—301no—
219#41—604no—
220C2×Q8—301no—
221C2×Q8C2×D3×D511yescharacteristic
222C2×D4—32no—
223C2×D4—61no—
224C2×D4—61no—
225C2×D4—151no—
226C2×D4—302no—
227C22×C4—151no—
228C22×C4—151no—
229C22×C4—151no—
230C22×C4—302no—
231SD16—62no—
232SD16—62no—
233D8—304no—
234C2×C8—32no—
235C2×C8—152no—
236C4⋊C4—31no—
237C4⋊C4—151no—
238C4⋊C4—152no—
239C4⋊C4—301no—
240C22⋊C4—301no—
241C22⋊C4—302no—
242C22⋊C4—302no—
243C42—31no—
244C42—51no—
245C42—151no—
246C42—151no—
247C3×C5—41noHall
248C22×C3—41no—
249C2×D3—44no—
250C2×D3—82no—
251C2×D3—82no—
252C3×C4—42no—
253C3×C4—202no—
254C3×C4—401no—
255Dic3—402no—
256C2×C5—242no—
257C2×C5#591211yescharacteristic
258C2×C5C2×#98511yescharacteristic
259C2×C5C2×#98611yescharacteristic
260D5—121no—
261C23—62no—
262C23—151no—
263Q8—31no—
264Q8—602no—
265Q8#677211yescharacteristic
266D4—62no—
267D4—62no—
268D4—122no—
269D4—122no—
270D4—152no—
271D4—301no—
272D4—604no—
273C2×C4—31no—
274C2×C4—31no—
275C2×C4—51no—
276C2×C4—51no—
277C2×C4—151no—
278C2×C4—301no—
279C2×C4—301no—
280C2×C4—302no—
281C2×C4—302no—
282C2×C4—302no—
283C2×C4—604no—
284C2×C4D5×S411yescharacteristic
285C8—62no—
286C8—302no—
287C2×C3—41no—
288C2×C3—41no—
289C2×C3—41no—
290D3—84no—
291C5#8534511yescharacteristic, complemented, Sylow
292C22—122no—
293C22—122no—
294C22—122no—
295C22—301no—
296C22—301no—
297C22—301no—
298C22C2×D5×S411yescharacteristic
299C4—31no—
300C4—31no—
301C4—52no—
302C4—61no—
303C4—101no—
304C4—301no—
305C4—602no—
306C4#9189212yes—
307C3—41noSylow
308C2—242no—
309C2—601no—
310C2#1175675311yescharacteristic
311C2C2×#9189211yescharacteristic
312C2D5×#98611yescharacteristic
subgroupG/H
center, FrattiniC22C2×D5×S4
commutator, cosocleC2×C5×SL(2,𝔽3)C23
FittingC4×C5×Q8C2×D3
socleC22×C5C22×S4
hypercenterC2×C4D5×S4
hypocenterC5×SL(2,𝔽3)#41
LCSG ▷ C2×C5×SL(2,𝔽3) ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C4×C5⋊(#41⋊C3)C2semidirect
2. #11752110C2semidirect
3. C5×C4⋊GL(2,𝔽3)C2semidirect
4. Dic5×GL(2,𝔽3)C2semidirect
5. #11751599C2semidirect
15. C4×#713D3semidirect
35. C4⋊GL(2,𝔽3)D5semidirect
72. C5×SL(2,𝔽3)#41semidirect
201. SL(2,𝔽3)#711semidirect
291. C5#85345semidirect

Maximal subgroups

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

subgroupindex
C4×C5⋊(#41⋊C3)2
#117521102
C5×C4⋊GL(2,𝔽3)2
Dic5×GL(2,𝔽3)2
#117515992
#106120923
Dic5×D124
#853455