Finite Group Databasebeta

← #14513756

Subgroups — #14513756

Order 1920.

606 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 73 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
1C22×C5⋊GL(2,𝔽3)C211yesmaximal, characteristic
2C5×#519⋊C3C211yesmaximal, characteristic, complemented
3#11752139C211yesmaximal, characteristic, complemented
4#11752113C212yesmaximal, complemented
5C4×C5⋊GL(2,𝔽3)C212yesmaximal, complemented
6#10614279—31nomaximal, Hall
7C22×C5×SL(2,𝔽3)C2211yescharacteristic
8C2×C5⋊GL(2,𝔽3)C414yescomplemented
9C2×C5⋊GL(2,𝔽3)C2211yescharacteristic
10C2×C5⋊GL(2,𝔽3)C2211yescharacteristic
11C22⋊C4×D15—41nomaximal
12C4×C5×SL(2,𝔽3)C2212yescomplemented
13#91184C2212yescomplemented
14#85353—51nomaximal, Hall
15C5×#519D311yescharacteristic, complemented
16C22×#688—31no—
17#65855—31no—
18#65514—31no—
19#65315—32no—
20#65315—32no—
21C4×#688—32no—
22C4×#688—32no—
23#65290—31no—
24#65287—31no—
25#65286—31no—
26C23×D15—41no—
27C2×C4×D15—42no—
28C2×C5×SL(2,𝔽3)C2×C412yescomplemented
29C2×C5×SL(2,𝔽3)C2311yescharacteristic
30C5⋊GL(2,𝔽3)C2×C412yescomplemented
31C5⋊GL(2,𝔽3)C2×C412yescomplemented
32C3×C5×C22⋊C4—41no—
33#6719—41no—
34#6714—42no—
35C22×GL(2,𝔽3)—51no—
36#519⋊C3D511yescharacteristic, complemented
37#5420—51no—
38#5394—52no—
39C4×GL(2,𝔽3)—52no—
40C22×C5×Q8C2×D311yescharacteristic
41C22×D20—31no—
42C4×C5×Q8—32no—
43C4×C5×Q8C2×D312yescomplemented
44C5×#117—31no—
45C5×#117—31no—
46C2×#688—31no—
47C2×#688—31no—
48C2×#688—34no—
49C2×#688—34no—
50C2×#688—31no—
51C2×#688—31no—
52#4072—31no—
53C2×#681—31no—
54C22×#168—31no—
55#4039—62no—
56D5×C22⋊C4—61no—
57D5×C22⋊C4—61no—
58C4×D20—32no—
59C4×D20—32no—
60#3967—32no—
61#3967—32no—
62#3941—32no—
63#3941—32no—
64#3940—32no—
65#3939—32no—
66#3935—32no—
67C4×#168—32no—
68#2949—151noSylow
69C23×C3×C5—41no—
70C22×D15—41no—
71C22×D15—44no—
72C22×D15—41no—
73C22×D15—84no—
74C2×C3×C4×C5—42no—
75C2×Dic15—42no—
76C4×D15—84no—
77C5×SL(2,𝔽3)C22×C411yescharacteristic
78C22×SL(2,𝔽3)C2×D511yescharacteristic
79C2×GL(2,𝔽3)—54no—
80C2×GL(2,𝔽3)—51no—
81C2×GL(2,𝔽3)—51no—
82D3×C22⋊C4—201no—
83C4×SL(2,𝔽3)C2×D512yescomplemented
84#859—52no—
85C23×D5—61no—
86C2×C5×Q8—31no—
87C2×C5×Q8—32no—
88C2×C5×Q8C4×D312yescomplemented
89C2×C5×Q8C22×D311yescharacteristic
90C22×C4×C5—31no—
91C2×D20—31no—
92C2×D20—31no—
93C2×D20—34no—
94C2×D20—62no—
95C2×D20—61no—
96C2×C4×D5—62no—
97C2×C4×D5—62no—
98C5×C4⋊C4—32no—
99C5×C4⋊C4—32no—
100C5×C22⋊C4—31no—
101C5×C22⋊C4S411yescharacteristic
102C42×C5—32no—
103C42×C5—61no—
104C22⋊Dic5—61no—
105#688—34no—
106#688—34no—
107#688—32no—
108#688—32no—
109#688—32no—
110#688—32no—
111#685—62no—
112#685—62no—
113C4⋊Dic5—32no—
114C4×Dic5—61no—
115#681—62no—
116C2×#168—31no—
117C2×#168—31no—
118C2×#168—34no—
119C2×#168—32no—
120C22×SD16—151no—
121#519D1511yescharacteristic, complemented
122#518—151no—
123#440—152no—
124#440—152no—
125C4×SD16—152no—
126C4×SD16—152no—
127#427—151no—
128#419—151no—
129#418—151no—
130#405—151no—
131C22×C3×C5—42no—
132C22×C3×C5—41no—
133C22×C3×C5—82no—
134C2×D15—42no—
135C2×D15—42no—
136C2×D15—44no—
137C2×D15—82no—
138C2×D15—88no—
139C3×C4×C5—82no—
140Dic15—82no—
141C23×D3—201no—
142C2×C4×D3—202no—
143C2×SL(2,𝔽3)C4×D512yescomplemented
144C2×SL(2,𝔽3)C22×D511yescharacteristic
145GL(2,𝔽3)—52no—
146GL(2,𝔽3)—52no—
147C3×C22⋊C4—41no—
148#217—201no—
149(C2×D3)⋊C4—202no—
150C23×C5C2×S411yescharacteristic
151C22×D5—61no—
152C22×D5—61no—
153C22×D5—64no—
154C22×D5—242no—
155C5×Q8—32no—
156C5×Q8—31no—
157C5×Q8—61no—
158C5×Q8C2×C4×D311yescharacteristic
159C2×C4×C5—32no—
160C2×C4×C5—32no—
161C2×C4×C5—31no—
162C2×C4×C5—31no—
163C2×C4×C5—32no—
164C2×C4×C5C2×S412yes—
165C2×Dic5—62no—
166D20—32no—
167D20—32no—
168D20—61no—
169D20—61no—
170D20—64no—
171C4×D5—242no—
172C4×D5—242no—
173#168—32no—
174#168—32no—
175#168—62no—
176C22×Q8C2×D1511yescharacteristic
177C22×D4—151no—
178C2×SD16—151no—
179C2×SD16—151no—
180C2×SD16—154no—
181C2×SD16—154no—
182C2×SD16—151no—
183C2×SD16—151no—
184C2×M16—151no—
185C22×C8—151no—
186C4×Q8—32no—
187C4×Q8C2×D1512yescomplemented
188C4×D4—152no—
189C4×D4—152no—
190C4×D4—302no—
191#117—31no—
192#117—31no—
193#117—151no—
194C2×C22⋊C4—301no—
195C2×C22⋊C4—301no—
196#107—152no—
197#106—152no—
198#103—152no—
199#103—152no—
200#102—152no—
201#102—152no—
202#97—152no—
203C4×C8—152no—
204C2×C3×C5—41no—
205C2×C3×C5—42no—
206C2×C3×C5—82no—
207D15—44no—
208D15—82no—
209C23×C3—41no—
210C22×D3—201no—
211C22×D3—201no—
212C22×D3—204no—
213C22×D3—404no—
214C2×C3×C4—42no—
215C2×Dic3—202no—
216C4×D3—404no—
217SL(2,𝔽3)C2×C4×D511yescharacteristic
218C22×C5—22no—
219C22×C5C4×S412yes—
220C22×C5C22×S411yescharacteristic
221C2×D5—62no—
222C2×D5—62no—
223C2×D5—122no—
224C2×D5—122no—
225C2×D5—244no—
226C4×C5—22no—
227C4×C5—31no—
228C4×C5—31no—
229C4×C5—32no—
230C4×C5—62no—
231Dic5—122no—
232C24—301no—
233C2×Q8—31no—
234C2×Q8—32no—
235C2×Q8C4×D1512yescomplemented
236C2×Q8C22×D1511yescharacteristic
237C2×D4—151no—
238C2×D4—151no—
239C2×D4—154no—
240C2×D4—301no—
241C2×D4—302no—
242C22×C4—31no—
243C22×C4—302no—
244C22×C4—302no—
245SD16—154no—
246SD16—154no—
247SD16—152no—
248SD16—152no—
249SD16—152no—
250SD16—152no—
251M16—302no—
252C2×C8—151no—
253C2×C8—151no—
254C2×C8—154no—
255C2×C8—152no—
256C4⋊C4—32no—
257C4⋊C4—32no—
258C4⋊C4—152no—
259C22⋊C4—31no—
260C22⋊C4—302no—
261C22⋊C4—301no—
262C22⋊C4—302no—
263C22⋊C4C5⋊S411yescharacteristic
264C42—32no—
265C42—61no—
266C42—301no—
267C3×C5—41noHall
268C22×C3—42no—
269C22×C3—41no—
270C22×C3—82no—
271C2×D3—202no—
272C2×D3—202no—
273C2×D3—204no—
274C2×D3—402no—
275C2×D3—408no—
276C3×C4—82no—
277Dic3—402no—
278C2×C5—22no—
279C2×C5C2×C4×S411yescharacteristic
280C2×C5#592612yes—
281D5—122no—
282D5—122no—
283C23—301no—
284C23—301no—
285C23—304no—
286C23—1202no—
287C23C2×C5⋊S411yescharacteristic
288Q8—32no—
289Q8—31no—
290Q8—61no—
291Q8C2×C4×D1511yescharacteristic
292D4—152no—
293D4—152no—
294D4—301no—
295D4—301no—
296D4—304no—
297C2×C4—32no—
298C2×C4—32no—
299C2×C4—31no—
300C2×C4—31no—
301C2×C4—32no—
302C2×C4—302no—
303C2×C4—1202no—
304C2×C4—1202no—
305C2×C4C2×C5⋊S412yes—
306C8—152no—
307C8—152no—
308C8—302no—
309C2×C3—41no—
310C2×C3—42no—
311C2×C3—82no—
312D3—204no—
313D3—402no—
314C5#8535311yescharacteristic, complemented, Sylow
315C22—22no—
316C22—302no—
317C22—302no—
318C22—602no—
319C22—602no—
320C22—1204no—
321C22C4×C5⋊S412yes—
322C22C22×C5⋊S411yescharacteristic
323C4—22no—
324C4—31no—
325C4—31no—
326C4—32no—
327C4—62no—
328C4—602no—
329C3—41noSylow
330C2—22no—
331C2—602no—
332C2—602no—
333C2C2×C4×C5⋊S411yescharacteristic
334C2#1175691012yes—
subgroupG/H
center, FrattiniC22C22×C5⋊S4
commutator, hypocenterC5×SL(2,𝔽3)C22×C4
FittingC5×#519D3
socleC22×C5C22×S4
prime radicalC22×C5⋊GL(2,𝔽3)C2
cosocleC2×C5×SL(2,𝔽3)C23
hypercenterC22⋊C4C5⋊S4
LCSG ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C5×#519⋊C3C2semidirect
3. #11752139C2semidirect
4. #11752113C2semidirect
5. C4×C5⋊GL(2,𝔽3)C2semidirect
8. C2×C5⋊GL(2,𝔽3)C4semidirect
12. C4×C5×SL(2,𝔽3)C22semidirect
13. #91184C22semidirect
15. C5×#519D3semidirect
28. C2×C5×SL(2,𝔽3)C2×C4semidirect
30. C5⋊GL(2,𝔽3)C2×C4semidirect
31. C5⋊GL(2,𝔽3)C2×C4semidirect
36. #519⋊C3D5semidirect
43. C4×C5×Q8C2×D3semidirect
83. C4×SL(2,𝔽3)C2×D5semidirect
88. C2×C5×Q8C4×D3semidirect
121. #519D15semidirect
143. C2×SL(2,𝔽3)C4×D5semidirect
187. C4×Q8C2×D15semidirect
235. C2×Q8C4×D15semidirect
314. C5#85353semidirect

Maximal subgroups

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

subgroupindex
C22×C5⋊GL(2,𝔽3)2
C5×#519⋊C32
#117521392
#117521132
C4×C5⋊GL(2,𝔽3)2
#106142793
C22⋊C4×D154
#853535