Finite Group Databasebeta

← #14513734

Subgroups — #14513734

Order 1920.

482 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 61 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
3#11752113C212yesmaximal, complemented
4C4⋊(C5⋊GL(2,𝔽3))C211yesmaximal, characteristic, complemented
5#11752110C211yesmaximal, characteristic, complemented
6#10614175—31nomaximal, Hall
7C22×C5×SL(2,𝔽3)C2211yescharacteristic
8C2×C5⋊GL(2,𝔽3)C2212yes—
9#91819—41nomaximal
10C4×C5×SL(2,𝔽3)C2211yescharacteristic
11C4×C5×SL(2,𝔽3)C2211yescharacteristic
12#91184C2212yes—
13#85332—51nomaximal, Hall
14C2×C4×C5×Q8D311yescharacteristic, complemented
15#65811—31no—
16#65516—32no—
17#65316—31no—
18#65316—31no—
19#65315—32no—
20#65315—32no—
21#65310—31no—
22#65310—31no—
23#65256—32no—
24#65215—31no—
25C22×C3×C4×C5—41no—
26C2×#1220—42no—
27C2×D60—41no—
28C2×C5×SL(2,𝔽3)—21no—
29C2×C5×SL(2,𝔽3)C2311yescharacteristic
30C5⋊GL(2,𝔽3)—22no—
31#6714—42no—
32C4⋊Dic15—41no—
33C2×C4×SL(2,𝔽3)D511yescharacteristic, complemented
34#5421—52no—
35#5394—52no—
36C4⋊GL(2,𝔽3)—51no—
37#5391—51no—
38C22×C5×Q8C2×D311yescharacteristic
39C2×#709—61no—
40C4×C5×Q8—31no—
41C4×C5×Q8—31no—
42C4×C5×Q8C2×D311yescharacteristic
43C4×C5×Q8C2×D311yescharacteristic
44C2×C5×C4⋊C4—31no—
45C2×C42×C5—31no—
46C2×#688—32no—
47C2×#688—32no—
48#4076—32no—
49#4076—61no—
50#4021—61no—
51#4020—31no—
52C4×D20—32no—
53C4×D20—61no—
54#4015—31no—
55#3967—32no—
56#3967—32no—
57C22⋊#168—32no—
58#3941—32no—
59#3941—32no—
60#3939—32no—
61#3939—32no—
62C4⋊#168—31no—
63C4⋊#168—31no—
64#3935—32no—
65#3112—151noSylow
66C23×C3×C5—41no—
67C22×D15—42no—
68C2×C3×C4×C5—41no—
69C2×C3×C4×C5—41no—
70C2×C3×C4×C5—82no—
71#1220—84no—
72C2×Dic15—42no—
73D60—82no—
74C5×SL(2,𝔽3)#4111yescharacteristic, complemented
75C22×SL(2,𝔽3)C2×D511yescharacteristic
76C2×GL(2,𝔽3)—52no—
77#930—201no—
78C4×SL(2,𝔽3)C2×D511yescharacteristic
79C4×SL(2,𝔽3)C2×D511yescharacteristic
80#859—52no—
81C2×C5×Q8—21no—
82C2×C5×Q8—31no—
83C2×C5×Q8—61no—
84C2×C5×Q8C22×D311yescharacteristic
85C22×C4×C5—31no—
86C22×C4×C5—31no—
87C22×C4×C5S411yescharacteristic
88C2×C22⋊D5—62no—
89#709—124no—
90C2×D20—32no—
91C2×D20—61no—
92C2×D20—61no—
93C2×C4×D5—62no—
94C2×Dic10—61no—
95C5×C4⋊C4—31no—
96C5×C4⋊C4—31no—
97C5×C4⋊C4—61no—
98C42×C5—31no—
99C42×C5—31no—
100C42×C5—31no—
101C42×C5—31no—
102#688—62no—
103#688—64no—
104#688—62no—
105#685—62no—
106#685—62no—
107C4⋊Dic5—32no—
108C4⋊Dic5—61no—
109C2×#168—32no—
110C2×#168—32no—
111#532—151no—
112C2×C4×Q8D1511yescharacteristic, complemented
113#480—152no—
114#477—151no—
115#477—151no—
116#460—151no—
117#460—151no—
118#448—152no—
119#440—152no—
120#440—152no—
121#432—151no—
122C22×C3×C5—42no—
123C22×C3×C5—41no—
124C22×C3×C5—81no—
125C22×C3×C5—81no—
126C2×D15—82no—
127C2×D15—82no—
128C2×D15—82no—
129C3×C4×C5—42no—
130C3×C4×C5—81no—
131Dic15—82no—
132C22×C3×C4—41no—
133C2×#67—202no—
134C2×D12—201no—
135C2×SL(2,𝔽3)—21no—
136C2×SL(2,𝔽3)C22×D511yescharacteristic
137GL(2,𝔽3)—102no—
138(C2×D3)⋊C4—202no—
139C4⋊Dic3—201no—
140C23×C5C2×S411yescharacteristic
141C22×D5—62no—
142C5×Q8—31no—
143C5×Q8—32no—
144C5×Q8—61no—
145C5×Q8#23511yescharacteristic
146C2×C4×C5—22no—
147C2×C4×C5—31no—
148C2×C4×C5—31no—
149C2×C4×C5—31no—
150C2×C4×C5—31no—
151C2×C4×C5—61no—
152C2×C4×C5—61no—
153C2×C4×C5—62no—
154C2×C4×C5C2×S411yescharacteristic
155C2×C4×C5C2×S411yescharacteristic
156C22⋊D5—124no—
157C2×Dic5—62no—
158D20—62no—
159D20—62no—
160D20—122no—
161D20—122no—
162C4×D5—124no—
163Dic10—122no—
164#168—62no—
165#168—62no—
166C2×#41—301no—
167C22×Q8C2×D1511yescharacteristic
168C2×SD16—152no—
169C2×SD16—152no—
170C4⋊Q8—151no—
171#127—151no—
172#124—301no—
173#121—152no—
174#121—301no—
175C4×Q8—31no—
176C4×Q8—31no—
177C4×Q8C2×D1511yescharacteristic
178C4×Q8C2×D1511yescharacteristic
179C4×D4—152no—
180C4×D4—301no—
181C2×C4⋊C4—31no—
182C2×C42—31no—
183#107—152no—
184#107—152no—
185C4⋊C8—151no—
186C4⋊C8—151no—
187#103—152no—
188#103—152no—
189#102—152no—
190#102—152no—
191C22⋊C8—152no—
192#97—152no—
193C2×C3×C5—41no—
194C2×C3×C5—41no—
195C2×C3×C5—41no—
196C2×C3×C5—82no—
197D15—162no—
198C23×C3—41no—
199C22×D3—202no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—82no—
203#67—404no—
204C2×Dic3—202no—
205D12—402no—
206SL(2,𝔽3)#70911yescharacteristic, complemented
207C22×C5—21no—
208C22×C5—21no—
209C22×C5#98612yes—
210C22×C5C22×S411yescharacteristic
211C2×D5—122no—
212C2×D5—122no—
213C2×D5—122no—
214C4×C5—21no—
215C4×C5—31no—
216C4×C5—31no—
217C4×C5—61no—
218C4×C5—61no—
219C4×C5—61no—
220C4×C5#98312yes—
221Dic5—122no—
222#41—604no—
223C2×Q8—21no—
224C2×Q8—31no—
225C2×Q8—61no—
226C2×Q8—301no—
227C2×Q8C22×D1511yescharacteristic
228C2×D4—152no—
229C2×D4—301no—
230C2×D4—302no—
231C2×D4—301no—
232C22×C4—31no—
233C22×C4—31no—
234C22×C4—302no—
235C22×C4C5⋊S411yescharacteristic
236SD16—302no—
237SD16—304no—
238SD16—302no—
239C2×C8—152no—
240C2×C8—152no—
241C4⋊C4—31no—
242C4⋊C4—31no—
243C4⋊C4—61no—
244C4⋊C4—152no—
245C4⋊C4—301no—
246C22⋊C4—302no—
247C22⋊C4—302no—
248C42—31no—
249C42—31no—
250C42—31no—
251C42—31no—
252C3×C5—41noHall
253C22×C3—42no—
254C22×C3—41no—
255C22×C3—81no—
256C22×C3—81no—
257C2×D3—402no—
258C2×D3—402no—
259C2×D3—402no—
260C3×C4—42no—
261C3×C4—81no—
262Dic3—402no—
263C2×C5—22no—
264C2×C5#591211yescharacteristic
265C2×C5C2×#98311yescharacteristic
266C2×C5C2×#98611yescharacteristic
267D5—242no—
268C23—302no—
269C23C2×C5⋊S411yescharacteristic
270Q8—31no—
271Q8—32no—
272Q8—61no—
273Q8—602no—
274Q8#681711yescharacteristic
275D4—302no—
276D4—302no—
277D4—602no—
278D4—604no—
279D4—602no—
280C2×C4—22no—
281C2×C4—31no—
282C2×C4—31no—
283C2×C4—31no—
284C2×C4—31no—
285C2×C4—61no—
286C2×C4—61no—
287C2×C4—62no—
288C2×C4—302no—
289C2×C4—604no—
290C2×C4C2×C5⋊S411yescharacteristic
291C2×C4C2×C5⋊S411yescharacteristic
292C8—302no—
293C8—302no—
294C2×C3—41no—
295C2×C3—41no—
296C2×C3—41no—
297C2×C3—82no—
298D3—802no—
299C5#8533211yescharacteristic, complemented, Sylow
300C22—21no—
301C22—21no—
302C22—602no—
303C22—602no—
304C22—602no—
305C22#9195612yes—
306C22C22×C5⋊S411yescharacteristic
307C4—21no—
308C4—31no—
309C4—31no—
310C4—61no—
311C4—61no—
312C4—61no—
313C4—602no—
314C4#9195312yes—
315C3—41noSylow
316C2—22no—
317C2—1202no—
318C2#1175689611yescharacteristic
319C2C2×#9195311yescharacteristic
320C2C2×#9195611yescharacteristic
subgroupG/H
center, FrattiniC22C22×C5⋊S4
commutator, cosocleC2×C5×SL(2,𝔽3)C23
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 ◁ C22 ◁ 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. #11752113C2semidirect
4. C4⋊(C5⋊GL(2,𝔽3))C2semidirect
5. #11752110C2semidirect
14. C2×C4×C5×Q8D3semidirect
33. C2×C4×SL(2,𝔽3)D5semidirect
74. C5×SL(2,𝔽3)#41semidirect
112. C2×C4×Q8D15semidirect
206. SL(2,𝔽3)#709semidirect
299. C5#85332semidirect

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
#117521132
C4⋊(C5⋊GL(2,𝔽3))2
#117521102
#106141753
#918194
#853325