Finite Group Databasebeta

← #14513283

Subgroups — #14513283

Order 1920.

606 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 75 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
2#66146⋊C3C211yesmaximal, characteristic, complemented
3#11752139C211yesmaximal, characteristic, complemented
4Dic5×GL(2,𝔽3)C212yesmaximal, complemented
5#11751600C212yesmaximal, complemented
6#10615574—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
11D3×C22⋊Dic5—41nomaximal
12Dic5×SL(2,𝔽3)C2212yescomplemented
13#91184C2212yescomplemented
14#85353—51nomaximal, Hall
15C22×C5×SD16—31no—
16#66146D311yescharacteristic, complemented
17#66132—31no—
18#65514—31no—
19#65504—31no—
20#65453—32no—
21#65453—32no—
22SD16×Dic5—32no—
23SD16×Dic5—32no—
24#65395—31no—
25#65390—31no—
26C23×C5×D3—41no—
27C2×C5×SL(2,𝔽3)C2×C412yescomplemented
28C2×C5×SL(2,𝔽3)C2311yescharacteristic
29C2×D3×Dic5—42no—
30C5×GL(2,𝔽3)C2×C412yescomplemented
31C5×GL(2,𝔽3)C2×C412yescomplemented
32#6719—41no—
33C3×C22⋊Dic5—41no—
34(C2×D3)⋊Dic5—42no—
35C22×GL(2,𝔽3)D511yescharacteristic
36#519⋊C3—51no—
37#5420—51no—
38#5394—52no—
39C4×GL(2,𝔽3)—52no—
40C22×C5×Q8C2×D311yescharacteristic
41C22×C5×D4—31no—
42C2×C5×SD16—31no—
43C2×C5×SD16—31no—
44C2×C5×SD16—34no—
45C2×C5×SD16—34no—
46C2×C5×SD16—31no—
47C2×C5×SD16—31no—
48C22×C5×C8—31no—
49C2×C22⋊Dic5—61no—
50C2×C22⋊Dic5—61no—
51Q8×Dic5—32no—
52Q8×Dic5C2×D312yescomplemented
53D4×Dic5—32no—
54D4×Dic5—32no—
55D4×Dic5—62no—
56#4072—31no—
57#4072—31no—
58#4072—31no—
59C2×#681—31no—
60#3967—32no—
61#3967—32no—
62#3964—32no—
63#3964—32no—
64#3950—32no—
65#3949—32no—
66#3947—32no—
67C8×Dic5—32no—
68#2949—151noSylow
69C23×C3×C5—41no—
70C22×C5×D3—41no—
71C22×C5×D3—41no—
72C22×C5×D3—44no—
73C22×C5×D3—84no—
74C2×Dic15—42no—
75C2×C3×Dic5—42no—
76C5×SL(2,𝔽3)C22×C411yescharacteristic
77D3×Dic5—84no—
78C22×SL(2,𝔽3)C2×D511yescharacteristic
79C2×GL(2,𝔽3)Dic514yescomplemented
80C2×GL(2,𝔽3)C2×D511yescharacteristic
81C2×GL(2,𝔽3)C2×D511yescharacteristic
82D3×C22⋊C4—201no—
83C4×SL(2,𝔽3)—52no—
84#859—52no—
85C24×C5—61no—
86C2×C5×Q8—32no—
87C2×C5×Q8—31no—
88C2×C5×Q8C4×D312yescomplemented
89C2×C5×Q8C22×D311yescharacteristic
90C2×C5×D4—34no—
91C2×C5×D4—31no—
92C2×C5×D4—31no—
93C2×C5×D4—62no—
94C2×C5×D4—61no—
95C22×C4×C5—31no—
96C22×Dic5—62no—
97C22×Dic5—62no—
98C5×SD16—34no—
99C5×SD16—34no—
100C5×SD16—32no—
101C5×SD16—32no—
102C5×SD16—32no—
103C5×SD16—32no—
104C2×C5×C8—34no—
105C2×C5×C8—31no—
106C2×C5×C8—31no—
107C22⋊Dic5—31no—
108C22⋊Dic5—62no—
109C22⋊Dic5—61no—
110C22⋊Dic5—62no—
111C22⋊Dic5S411yescharacteristic
112C4⋊Dic5—32no—
113C4⋊Dic5—32no—
114C4⋊Dic5—32no—
115C4×Dic5—32no—
116C4×Dic5—61no—
117C4×Dic5—61no—
118#681—62no—
119C2×#168—32no—
120C22×SD16—31no—
121#519—51no—
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×C5×D3—42no—
135C2×C5×D3—42no—
136C2×C5×D3—44no—
137C2×C5×D3—82no—
138C2×C5×D3—88no—
139Dic15—82no—
140C3×Dic5—82no—
141C23×D3—41no—
142C2×C4×D3—202no—
143C2×SL(2,𝔽3)C2×Dic512yescomplemented
144C2×SL(2,𝔽3)C22×D511yescharacteristic
145GL(2,𝔽3)C2×Dic512yescomplemented
146GL(2,𝔽3)C2×Dic512yescomplemented
147C3×C22⋊C4—201no—
148#217—201no—
149(C2×D3)⋊C4—202no—
150C23×C5—61no—
151C23×C5—61no—
152C23×C5—64no—
153C23×C5—242no—
154C23×C5C2×S411yescharacteristic
155C5×Q8—32no—
156C5×Q8—31no—
157C5×Q8—61no—
158C5×Q8C2×C4×D311yescharacteristic
159C5×D4—32no—
160C5×D4—32no—
161C5×D4—61no—
162C5×D4—61no—
163C5×D4—64no—
164C2×C4×C5—32no—
165C2×C4×C5—32no—
166C2×C4×C5—31no—
167C2×C4×C5—31no—
168C2×Dic5—32no—
169C2×Dic5—62no—
170C2×Dic5—242no—
171C2×Dic5—242no—
172C2×Dic5C2×S412yes—
173C5×C8—32no—
174C5×C8—32no—
175#168—62no—
176C22×Q8D3×D511yescharacteristic
177C22×D4—31no—
178C2×SD16—34no—
179C2×SD16—31no—
180C2×SD16—31no—
181C2×SD16—34no—
182C2×SD16—31no—
183C2×SD16—31no—
184C2×M16—151no—
185C22×C8—31no—
186C4×Q8—52no—
187C4×Q8—152no—
188C4×D4—152no—
189C4×D4—152no—
190C4×D4—302no—
191#117—151no—
192#117—151no—
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—
207C5×D3—44no—
208C5×D3—82no—
209C23×C3—41no—
210C22×D3—41no—
211C22×D3—41no—
212C22×D3—44no—
213C22×D3—84no—
214C2×C3×C4—202no—
215C2×Dic3—202no—
216C4×D3—404no—
217SL(2,𝔽3)C22×Dic511yescharacteristic
218C22×C5—22no—
219C22×C5—62no—
220C22×C5—62no—
221C22×C5—122no—
222C22×C5—122no—
223C22×C5—244no—
224C22×C5C4×S412yes—
225C22×C5C22×S411yescharacteristic
226C4×C5—31no—
227C4×C5—31no—
228C4×C5—32no—
229Dic5—22no—
230Dic5—62no—
231Dic5—122no—
232C24—61no—
233C2×Q8—31no—
234C2×Q8—32no—
235C2×Q8D3×Dic512yescomplemented
236C2×Q8C2×D3×D511yescharacteristic
237C2×D4—31no—
238C2×D4—31no—
239C2×D4—34no—
240C2×D4—61no—
241C2×D4—62no—
242C22×C4—31no—
243C22×C4—302no—
244C22×C4—302no—
245SD16—34no—
246SD16—34no—
247SD16—32no—
248SD16—32no—
249SD16—32no—
250SD16—32no—
251M16—302no—
252C2×C8—31no—
253C2×C8—31no—
254C2×C8—34no—
255C2×C8—152no—
256C4⋊C4—152no—
257C4⋊C4—152no—
258C4⋊C4—152no—
259C22⋊C4—51no—
260C22⋊C4—151no—
261C22⋊C4—302no—
262C22⋊C4—302no—
263C22⋊C4—301no—
264C42—152no—
265C42—301no—
266C42—301no—
267C3×C5—41noHall
268C22×C3—41no—
269C22×C3—42no—
270C22×C3—82no—
271C2×D3—42no—
272C2×D3—42no—
273C2×D3—44no—
274C2×D3—82no—
275C2×D3—88no—
276C3×C4—402no—
277Dic3—402no—
278C2×C5—22no—
279C2×C5—122no—
280C2×C5—122no—
281C2×C5C2×C4×S411yescharacteristic
282C2×C5#592612yes—
283C23—61no—
284C23—61no—
285C23—64no—
286C23—242no—
287C23D5×S411yescharacteristic
288Q8—32no—
289Q8—31no—
290Q8—61no—
291Q8C2×D3×Dic511yescharacteristic
292D4—32no—
293D4—32no—
294D4—61no—
295D4—61no—
296D4—64no—
297C2×C4—31no—
298C2×C4—31no—
299C2×C4—32no—
300C2×C4—32no—
301C2×C4—52no—
302C2×C4—152no—
303C2×C4—302no—
304C2×C4—1202no—
305C2×C4—1202no—
306C8—32no—
307C8—32no—
308C8—302no—
309C2×C3—42no—
310C2×C3—41no—
311C2×C3—82no—
312D3—44no—
313D3—82no—
314C5#8535311yescharacteristic, complemented, Sylow
315C22—22no—
316C22—62no—
317C22—62no—
318C22—122no—
319C22—122no—
320C22—244no—
321C22Dic5×S412yes—
322C22C2×D5×S411yescharacteristic
323C4—31no—
324C4—31no—
325C4—32no—
326C4—102no—
327C4—302no—
328C4—602no—
329C3—41noSylow
330C2—22no—
331C2—122no—
332C2—122no—
333C2#1175677112yes—
334C2C2×Dic5×S411yescharacteristic
subgroupG/H
center, FrattiniC22C2×D5×S4
commutator, hypocenterC5×SL(2,𝔽3)C22×C4
FittingC22×C5×Q8C2×D3
socleC22×C5C22×S4
prime radicalC22×C5×GL(2,𝔽3)C2
cosocleC2×C5×SL(2,𝔽3)C23
hypercenterC23D5×S4
LCSG ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. #66146⋊C3C2semidirect
3. #11752139C2semidirect
4. Dic5×GL(2,𝔽3)C2semidirect
5. #11751600C2semidirect
8. C2×C5×GL(2,𝔽3)C4semidirect
12. Dic5×SL(2,𝔽3)C22semidirect
13. #91184C22semidirect
16. #66146D3semidirect
27. C2×C5×SL(2,𝔽3)C2×C4semidirect
30. C5×GL(2,𝔽3)C2×C4semidirect
31. C5×GL(2,𝔽3)C2×C4semidirect
52. Q8×Dic5C2×D3semidirect
79. C2×GL(2,𝔽3)Dic5semidirect
88. C2×C5×Q8C4×D3semidirect
143. C2×SL(2,𝔽3)C2×Dic5semidirect
145. GL(2,𝔽3)C2×Dic5semidirect
146. GL(2,𝔽3)C2×Dic5semidirect
235. C2×Q8D3×Dic5semidirect
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
#66146⋊C32
#117521392
Dic5×GL(2,𝔽3)2
#117516002
#106155743
D3×C22⋊Dic54
#853535