Finite Group Databasebeta

← #14512714

Subgroups — #14512714

Order 1920.

542 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 59 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
1SL(2,𝔽3)×Dic10C211yesmaximal, characteristic, complemented
2C4⋊(C5⋊GL(2,𝔽3))C211yesmaximal, characteristic, complemented
3C4×C5×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#11751611C212yesmaximal, complemented
5#11751610C212yesmaximal, complemented
6#10611353—31nomaximal, Hall
7C2×#6769—41nomaximal
8C2×C5⋊GL(2,𝔽3)C2212yes—
9C2×C5×GL(2,𝔽3)C2211yescharacteristic
10C4×C5×SL(2,𝔽3)C2211yescharacteristic
11Dic5×SL(2,𝔽3)C2212yes—
12C5×#859C2211yescharacteristic
13#85430—51nomaximal, Hall
14Q8×Dic10D311yescharacteristic, complemented
15#65887—31no—
16C4×C5×SD16—31no—
17#65452—32no—
18#65316—31no—
19#65297—31no—
20#65159—32no—
21#65077—32no—
22#65049—32no—
23#64983—31no—
24C8×Dic10—31no—
25C2×D60—41no—
26C2×C4×C5×D3—41no—
27C2×C3×Dic10—41no—
28C2×C5×SL(2,𝔽3)C2311yescharacteristic
29C2×#1203—42no—
30C2×#1200—42no—
31#6769—48no—
32C5⋊GL(2,𝔽3)—22no—
33C5×GL(2,𝔽3)—21no—
34Q8×SL(2,𝔽3)—51no—
35C4⋊GL(2,𝔽3)—52no—
36C4⋊GL(2,𝔽3)—51no—
37C4×GL(2,𝔽3)—52no—
38C4×GL(2,𝔽3)D511yescharacteristic, complemented
39C2×#709—61no—
40C2×C5×SD16—31no—
41C4×C5×Q8C2×D311yescharacteristic
42C4×C5×D4—31no—
43Q8×Dic5C2×D312yes—
44C2×#688—32no—
45#4086—32no—
46#4076—61no—
47#4039—32no—
48#4034—32no—
49#4030—62no—
50#4020—31no—
51#4015—31no—
52C4×Dic10—31no—
53C5×#106—31no—
54C5×#103—31no—
55C5×#102—31no—
56C4×C5×C8—31no—
57#3964—32no—
58#3953—32no—
59#3953—32no—
60#3946—32no—
61C8×Dic5—32no—
62#3941—32no—
63#3940—32no—
64C4⋊#168—31no—
65#3362—151noSylow
66C22×D15—42no—
67C22×C5×D3—41no—
68C2×C3×C4×C5—41no—
69D60—44no—
70C2×C5×Dic3—41no—
71C4×C5×D3—44no—
72C2×C3×Dic5—42no—
73C3×Dic10—44no—
74C5×SL(2,𝔽3)#4111yescharacteristic, complemented
75#1203—48no—
76#1200—48no—
77C2×#239—201no—
78C2×GL(2,𝔽3)—52no—
79C2×GL(2,𝔽3)C2×D511yescharacteristic
80C4×SL(2,𝔽3)—52no—
81C4×SL(2,𝔽3)C2×D511yescharacteristic
82#859C2×D511yescharacteristic
83C2×C5×Q8C22×D311yescharacteristic
84C2×C5×D4—31no—
85C22×C4×C5—61no—
86C2×C22⋊D5—62no—
87C2×C22⋊D5—62no—
88#709—124no—
89C2×D20—32no—
90C2×D20—61no—
91C2×D20—61no—
92C2×C4×D5—62no—
93C2×Dic10—61no—
94C2×Dic10S411yescharacteristic
95C5×SD16—61no—
96C5×SD16—61no—
97C2×C5×C8—31no—
98C2×C5×C8—31no—
99C5×C4⋊C4—31no—
100C5×C4⋊C4—31no—
101C5×C22⋊C4—61no—
102C42×C5—31no—
103#688—62no—
104#688—62no—
105#685—62no—
106C4⋊Dic5—31no—
107C4⋊Dic5—32no—
108#683—32no—
109C4×Dic5—32no—
110C2×#168—32no—
111Q82—51no—
112#550—151no—
113#486—151no—
114#486—152no—
115#476—152no—
116#476—151no—
117#460—152no—
118#460—151no—
119C8×Q8—151no—
120C4×SD16—31no—
121C4×SD16—152no—
122C22×C3×C5—41no—
123C2×D15—44no—
124C2×D15—82no—
125C2×D15—82no—
126C2×C5×D3—42no—
127C2×C5×D3—81no—
128C2×C5×D3—81no—
129C3×C4×C5—42no—
130C3×Dic5—44no—
131C5×Dic3—42no—
132C2×C3×Q8—201no—
133#239—208no—
134C2×D12—201no—
135C2×D12—202no—
136C2×C4×D3—41no—
137C2×C4×D3—202no—
138C2×SL(2,𝔽3)C22×D511yescharacteristic
139GL(2,𝔽3)—21no—
140GL(2,𝔽3)—102no—
141C23×C5—61no—
142C22×D5—62no—
143C5×Q8—31no—
144C5×Q8#23911yescharacteristic, complemented
145C5×D4—61no—
146C5×D4—61no—
147C2×C4×C5—31no—
148C2×C4×C5—31no—
149C2×C4×C5—61no—
150C2×C4×C5—122no—
151C2×C4×C5C2×S411yescharacteristic
152C22⋊D5—124no—
153C22⋊D5—242no—
154C2×Dic5—32no—
155C2×Dic5C2×S412yes—
156D20—62no—
157D20—62no—
158D20—122no—
159D20—241no—
160C4×D5—124no—
161Dic10—241no—
162Dic10GL(2,𝔽3)14yescomplemented
163C5×C8—61no—
164C5×C8—61no—
165#168—62no—
166C2×#41—301no—
167C2×SD16—31no—
168C2×SD16—152no—
169C4⋊Q8—151no—
170C4⋊Q8—152no—
171#127—151no—
172#127—152no—
173#121—302no—
174#121—301no—
175C4×Q8—52no—
176C4×Q8—151no—
177C4×Q8D3×D511yescharacteristic
178C4×D4—31no—
179C4×D4—152no—
180#106—31no—
181#106—152no—
182C4⋊C8—151no—
183C4⋊C8—152no—
184#103—31no—
185#102—31no—
186#102—152no—
187#102—152no—
188#102—152no—
189#102—152no—
190C4×C8—31no—
191C4×C8—152no—
192C2×C3×C5—41no—
193C2×C3×C5—41no—
194C2×C3×C5—41no—
195D15—84no—
196C5×D3—82no—
197C22×D3—41no—
198C22×D3—202no—
199C3×Q8—204no—
200C2×C3×C4—41no—
201C2×C3×C4—202no—
202C2×Dic3—41no—
203D12—204no—
204D12—208no—
205C4×D3—44no—
206C4×D3—208no—
207SL(2,𝔽3)#70911yescharacteristic, complemented
208C22×C5—121no—
209C22×C5—121no—
210C22×C5—121no—
211C22×C5C22×S411yescharacteristic
212C2×D5—122no—
213C2×D5—122no—
214C2×D5—122no—
215C4×C5—31no—
216C4×C5—31no—
217C4×C5—61no—
218C4×C5—121no—
219C4×C5C2×GL(2,𝔽3)12yes—
220Dic5—62no—
221Dic5C2×GL(2,𝔽3)14yes—
222#41—604no—
223C2×Q8—51no—
224C2×Q8—301no—
225C2×Q8C2×D3×D511yescharacteristic
226C2×D4—31no—
227C2×D4—152no—
228C2×D4—301no—
229C2×D4—302no—
230C2×D4—302no—
231C2×D4—301no—
232C22×C4—61no—
233C22×C4—302no—
234SD16—61no—
235SD16—61no—
236SD16—302no—
237SD16—302no—
238C2×C8—31no—
239C2×C8—31no—
240C2×C8—152no—
241C4⋊C4—31no—
242C4⋊C4—31no—
243C4⋊C4—152no—
244C4⋊C4—152no—
245C4⋊C4—151no—
246C22⋊C4—61no—
247C22⋊C4—302no—
248C42—31no—
249C42—152no—
250C3×C5—41noHall
251C22×C3—41no—
252C2×D3—42no—
253C2×D3—81no—
254C2×D3—81no—
255C2×D3—204no—
256C2×D3—402no—
257C2×D3—402no—
258C3×C4—42no—
259C3×C4—204no—
260Dic3—42no—
261C2×C5—241no—
262C2×C5C22×GL(2,𝔽3)11yescharacteristic
263C2×C5#591911yescharacteristic
264C2×C5#592511yescharacteristic
265D5—242no—
266C23—61no—
267C23—302no—
268Q8—31no—
269Q8—54no—
270Q8—1201no—
271Q8#676911yescharacteristic, complemented
272D4—61no—
273D4—61no—
274D4—302no—
275D4—302no—
276D4—604no—
277D4—602no—
278D4—1201no—
279D4—1202no—
280C2×C4—31no—
281C2×C4—31no—
282C2×C4—52no—
283C2×C4—61no—
284C2×C4—122no—
285C2×C4—152no—
286C2×C4—604no—
287C2×C4D5×S411yescharacteristic
288C8—61no—
289C8—61no—
290C8—302no—
291C2×C3—41no—
292C2×C3—41no—
293C2×C3—41no—
294D3—82no—
295D3—404no—
296C5#8543011yescharacteristic, complemented, Sylow
297C22—121no—
298C22—121no—
299C22—121no—
300C22—602no—
301C22—602no—
302C22—602no—
303C22C2×D5×S411yescharacteristic
304C4—31no—
305C4—31no—
306C4—54no—
307C4—61no—
308C4—121no—
309C4—302no—
310C4D5×GL(2,𝔽3)12yes—
311C3—41noSylow
312C2—241no—
313C2—1202no—
314C2#1175674711yescharacteristic
315C2C2×D5×GL(2,𝔽3)11yescharacteristic
316C2#1175677511yescharacteristic
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. SL(2,𝔽3)×Dic10C2semidirect
2. C4⋊(C5⋊GL(2,𝔽3))C2semidirect
3. C4×C5×GL(2,𝔽3)C2semidirect
4. #11751611C2semidirect
5. #11751610C2semidirect
14. Q8×Dic10D3semidirect
38. C4×GL(2,𝔽3)D5semidirect
74. C5×SL(2,𝔽3)#41semidirect
144. C5×Q8#239semidirect
162. Dic10GL(2,𝔽3)semidirect
207. SL(2,𝔽3)#709semidirect
271. Q8#6769semidirect
296. C5#85430semidirect

Maximal subgroups

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

subgroupindex
SL(2,𝔽3)×Dic102
C4⋊(C5⋊GL(2,𝔽3))2
C4×C5×GL(2,𝔽3)2
#117516112
#117516102
#106113533
C2×#67694
#854305