Finite Group Databasebeta

← #14513727

Subgroups — #14513727

Order 1920.

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

normal subgroups only

This group has 556 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
1C2×C4×C5×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2#11752140C212yesmaximal, complemented
3C4⋊(C5⋊GL(2,𝔽3))C211yesmaximal, characteristic, complemented
4C4×C5⋊GL(2,𝔽3)C212yesmaximal, complemented
5#11752105C211yesmaximal, characteristic, complemented
6#10614173—31nomaximal, Hall
7C2×#6817—41nomaximal
8C22×C5×SL(2,𝔽3)C2211yescharacteristic
9C2×C5⋊GL(2,𝔽3)C2212yes—
10C4×C5×SL(2,𝔽3)C2211yescharacteristic
11C4×C5×SL(2,𝔽3)C2211yescharacteristic
12#91184C2212yes—
13#85325—51nomaximal, Hall
14C2×C4×C5×Q8D311yescharacteristic, complemented
15#65811—31no—
16#65516—32no—
17#65316—31no—
18#65316—31no—
19C4×#688—32no—
20C4×#688—32no—
21#65309—31no—
22#65309—31no—
23#65257—32no—
24#65214—31no—
25C22×C3×C4×C5—41no—
26C2×#1220—42no—
27#6817—48no—
28C2×D60—41no—
29C2×C4×D15—42no—
30C2×Dic30—41no—
31C2×C5×SL(2,𝔽3)—21no—
32C2×C5×SL(2,𝔽3)C2311yescharacteristic
33C5⋊GL(2,𝔽3)—22no—
34C2×C4×SL(2,𝔽3)D511yescharacteristic, complemented
35#5421—52no—
36C4⋊GL(2,𝔽3)—51no—
37C4×GL(2,𝔽3)—52no—
38#5386—51no—
39C22×C5×Q8C2×D311yescharacteristic
40C2×#709—61no—
41C4×C5×Q8—31no—
42C4×C5×Q8—31no—
43C4×C5×Q8C2×D311yescharacteristic
44C4×C5×Q8C2×D311yescharacteristic
45C2×C5×C4⋊C4—31no—
46C2×C42×C5—31no—
47C2×#688—32no—
48C2×#688—32no—
49#4076—32no—
50#4076—61no—
51#4021—61no—
52#4020—31no—
53C4×D20—32no—
54C4×D20—61no—
55#4015—31no—
56#3967—32no—
57#3967—32no—
58C22⋊#168—32no—
59#3941—32no—
60#3941—32no—
61#3940—32no—
62#3940—32no—
63C4⋊#168—31no—
64C4⋊#168—31no—
65C4×#168—32no—
66#3103—151noSylow
67C23×C3×C5—41no—
68C22×D15—42no—
69C2×C3×C4×C5—44no—
70C2×C3×C4×C5—41no—
71C2×C3×C4×C5—41no—
72#1220—48no—
73C2×Dic15—42no—
74D60—44no—
75C4×D15—48no—
76Dic30—44no—
77C5×SL(2,𝔽3)#4111yescharacteristic, complemented
78C2×#235—201no—
79C22×SL(2,𝔽3)C2×D511yescharacteristic
80C2×GL(2,𝔽3)—52no—
81C4×SL(2,𝔽3)C2×D511yescharacteristic
82C4×SL(2,𝔽3)C2×D511yescharacteristic
83#859—52no—
84C2×C5×Q8—21no—
85C2×C5×Q8—31no—
86C2×C5×Q8—61no—
87C2×C5×Q8C22×D311yescharacteristic
88C22×C4×C5—31no—
89C22×C4×C5—31no—
90C22×C4×C5S411yescharacteristic
91C2×C22⋊D5—62no—
92#709—124no—
93C2×D20—32no—
94C2×D20—61no—
95C2×D20—61no—
96C2×C4×D5—62no—
97C2×Dic10—61no—
98C5×C4⋊C4—31no—
99C5×C4⋊C4—31no—
100C5×C4⋊C4—61no—
101C42×C5—31no—
102C42×C5—31no—
103C42×C5—31no—
104C42×C5—31no—
105#688—62no—
106#688—64no—
107#688—62no—
108#685—62no—
109#685—62no—
110C4⋊Dic5—32no—
111C4⋊Dic5—61no—
112C2×#168—32no—
113C2×#168—32no—
114#532—151no—
115C2×C4×Q8D1511yescharacteristic, complemented
116#481—152no—
117#475—151no—
118#475—151no—
119#460—151no—
120#460—151no—
121#448—152no—
122C4×SD16—152no—
123C4×SD16—152no—
124#431—151no—
125C22×C3×C5—42no—
126C22×C3×C5—41no—
127C22×C3×C5—81no—
128C22×C3×C5—81no—
129C2×D15—44no—
130C2×D15—82no—
131C2×D15—82no—
132C3×C4×C5—42no—
133C3×C4×C5—42no—
134Dic15—44no—
135C22×C3×C4—41no—
136C2×#67—202no—
137#235—208no—
138C2×D12—201no—
139C2×C4×D3—202no—
140C2×Dic6—201no—
141C2×SL(2,𝔽3)—21no—
142C2×SL(2,𝔽3)C22×D511yescharacteristic
143GL(2,𝔽3)—102no—
144C23×C5C2×S411yescharacteristic
145C22×D5—62no—
146C5×Q8—31no—
147C5×Q8—32no—
148C5×Q8—61no—
149C5×Q8#23511yescharacteristic, complemented
150C2×C4×C5—31no—
151C2×C4×C5—31no—
152C2×C4×C5—31no—
153C2×C4×C5—31no—
154C2×C4×C5—61no—
155C2×C4×C5—61no—
156C2×C4×C5—121no—
157C2×C4×C5GL(2,𝔽3)14yescomplemented
158C2×C4×C5C2×S411yescharacteristic
159C2×C4×C5C2×S411yescharacteristic
160C22⋊D5—124no—
161C2×Dic5—62no—
162D20—62no—
163D20—62no—
164D20—122no—
165D20—122no—
166C4×D5—124no—
167Dic10—122no—
168#168—62no—
169#168—62no—
170C2×#41—301no—
171C22×Q8C2×D1511yescharacteristic
172C2×SD16—152no—
173C2×SD16—152no—
174C4⋊Q8—151no—
175#127—151no—
176#124—301no—
177#121—152no—
178#121—301no—
179C4×Q8—31no—
180C4×Q8—31no—
181C4×Q8C2×D1511yescharacteristic
182C4×Q8C2×D1511yescharacteristic
183C4×D4—152no—
184C4×D4—301no—
185C2×C4⋊C4—31no—
186C2×C42—31no—
187#106—152no—
188#106—152no—
189C4⋊C8—151no—
190C4⋊C8—151no—
191#103—152no—
192#103—152no—
193#102—152no—
194#102—152no—
195C22⋊C8—152no—
196C4×C8—152no—
197C2×C3×C5—41no—
198C2×C3×C5—41no—
199C2×C3×C5—41no—
200C2×C3×C5—82no—
201D15—84no—
202C23×C3—41no—
203C22×D3—202no—
204C2×C3×C4—44no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207#67—208no—
208C2×Dic3—202no—
209D12—204no—
210C4×D3—208no—
211Dic6—204no—
212SL(2,𝔽3)#70911yescharacteristic, complemented
213C22×C5—21no—
214C22×C5—21no—
215C22×C5C2×GL(2,𝔽3)12yes—
216C22×C5C22×S411yescharacteristic
217C2×D5—122no—
218C2×D5—122no—
219C2×D5—122no—
220C4×C5—31no—
221C4×C5—31no—
222C4×C5—61no—
223C4×C5—61no—
224C4×C5—61no—
225C4×C5C2×GL(2,𝔽3)12yes—
226C4×C5C2×GL(2,𝔽3)12yes—
227Dic5—122no—
228#41—604no—
229C2×Q8—21no—
230C2×Q8—31no—
231C2×Q8—61no—
232C2×Q8—301no—
233C2×Q8C22×D1511yescharacteristic
234C2×D4—152no—
235C2×D4—301no—
236C2×D4—302no—
237C2×D4—301no—
238C22×C4—31no—
239C22×C4—31no—
240C22×C4—302no—
241C22×C4C5⋊S411yescharacteristic
242SD16—302no—
243SD16—304no—
244SD16—302no—
245C2×C8—152no—
246C2×C8—152no—
247C4⋊C4—31no—
248C4⋊C4—31no—
249C4⋊C4—61no—
250C4⋊C4—152no—
251C4⋊C4—301no—
252C22⋊C4—302no—
253C22⋊C4—302no—
254C42—31no—
255C42—31no—
256C42—31no—
257C42—31no—
258C3×C5—41noHall
259C22×C3—42no—
260C22×C3—41no—
261C22×C3—81no—
262C22×C3—81no—
263C2×D3—204no—
264C2×D3—402no—
265C2×D3—402no—
266C3×C4—42no—
267C3×C4—42no—
268Dic3—204no—
269C2×C5—22no—
270C2×C5#591211yescharacteristic
271C2×C5C22×GL(2,𝔽3)11yescharacteristic
272C2×C5#592311yescharacteristic
273D5—242no—
274C23—302no—
275C23C2×C5⋊S411yescharacteristic
276Q8—31no—
277Q8—32no—
278Q8—61no—
279Q8—602no—
280Q8#681711yescharacteristic, complemented
281D4—302no—
282D4—302no—
283D4—602no—
284D4—604no—
285D4—602no—
286C2×C4—31no—
287C2×C4—31no—
288C2×C4—31no—
289C2×C4—31no—
290C2×C4—61no—
291C2×C4—61no—
292C2×C4—121no—
293C2×C4—302no—
294C2×C4—604no—
295C2×C4C5⋊GL(2,𝔽3)14yescomplemented
296C2×C4C2×C5⋊S411yescharacteristic
297C2×C4C2×C5⋊S411yescharacteristic
298C8—302no—
299C8—302no—
300C2×C3—41no—
301C2×C3—41no—
302C2×C3—41no—
303C2×C3—82no—
304D3—404no—
305C5#8532511yescharacteristic, complemented, Sylow
306C22—21no—
307C22—21no—
308C22—602no—
309C22—602no—
310C22—602no—
311C22C2×C5⋊GL(2,𝔽3)12yes—
312C22C22×C5⋊S411yescharacteristic
313C4—31no—
314C4—31no—
315C4—61no—
316C4—61no—
317C4—61no—
318C4—602no—
319C4C2×C5⋊GL(2,𝔽3)12yes—
320C4C2×C5⋊GL(2,𝔽3)12yes—
321C3—41noSylow
322C2—22no—
323C2—1202no—
324C2#1175689611yescharacteristic
325C2C22×C5⋊GL(2,𝔽3)11yescharacteristic
326C2#1175690711yescharacteristic
subgroupG/H
centerC2×C4C2×C5⋊S4
commutator, cosocleC2×C5×SL(2,𝔽3)C23
FrattiniC22C22×C5⋊S4
FittingC2×C4×C5×Q8D3
socleC22×C5C22×S4
hypercenterC22×C4C5⋊S4
hypocenterC5×SL(2,𝔽3)#41
LCSG ▷ C2×C5×SL(2,𝔽3) ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C4×C5×SL(2,𝔽3)C2semidirect
2. #11752140C2semidirect
3. C4⋊(C5⋊GL(2,𝔽3))C2semidirect
4. C4×C5⋊GL(2,𝔽3)C2semidirect
5. #11752105C2semidirect
14. C2×C4×C5×Q8D3semidirect
34. C2×C4×SL(2,𝔽3)D5semidirect
77. C5×SL(2,𝔽3)#41semidirect
115. C2×C4×Q8D15semidirect
149. C5×Q8#235semidirect
157. C2×C4×C5GL(2,𝔽3)semidirect
212. SL(2,𝔽3)#709semidirect
280. Q8#6817semidirect
295. C2×C4C5⋊GL(2,𝔽3)semidirect
305. C5#85325semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C5×SL(2,𝔽3)2
#117521402
C4⋊(C5⋊GL(2,𝔽3))2
C4×C5⋊GL(2,𝔽3)2
#117521052
#106141733
C2×#68174
#853255