Finite Group Databasebeta

← #14512788

Subgroups — #14512788

Order 1920.

462 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 51 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
1#65904⋊C3C211yesmaximal, characteristic, complemented
2C4⋊(C5⋊GL(2,𝔽3))C211yesmaximal, characteristic, complemented
3C4×C5×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#11751611C212yesmaximal, complemented
5#11751603C212yesmaximal, complemented
6#10611356—31nomaximal, Hall
7C2×C5⋊GL(2,𝔽3)C2212yes—
8C2×C5×GL(2,𝔽3)C2211yescharacteristic
9C4×C5×SL(2,𝔽3)C2211yescharacteristic
10#91460—41nomaximal
11Dic5×SL(2,𝔽3)C2212yes—
12C5×#859C2211yescharacteristic
13#85422—51nomaximal, Hall
14#65904D311yescharacteristic, complemented
15#65887—31no—
16C4×C5×SD16—31no—
17#65452—32no—
18#65316—31no—
19#65296—31no—
20#65160—32no—
21#65106—32no—
22#65053—32no—
23#64983—31no—
24#64968—31no—
25C2×D60—41no—
26C2×C4×C5×D3—41no—
27C2×C5×SL(2,𝔽3)C2311yescharacteristic
28C2×#1203—42no—
29C5⋊GL(2,𝔽3)—22no—
30C5×GL(2,𝔽3)—21no—
31C3×C4⋊Dic5—41no—
32#6667—42no—
33#557⋊C3—51no—
34#5394—52no—
35C4⋊GL(2,𝔽3)—52no—
36C4⋊GL(2,𝔽3)—51no—
37C4×GL(2,𝔽3)D511yescharacteristic, complemented
38C2×#709—61no—
39C2×C5×SD16—31no—
40C4×C5×Q8C2×D311yescharacteristic
41C4×C5×D4—31no—
42Q8×Dic5C2×D312yes—
43C2×#688—32no—
44#4086—32no—
45#4076—61no—
46#4039—32no—
47#4036—32no—
48#4030—62no—
49#4020—31no—
50#4015—31no—
51C4×Dic10—31no—
52C5×#106—31no—
53C5×#103—31no—
54C5×#102—31no—
55C4×C5×C8—31no—
56#3964—32no—
57#3953—32no—
58#3953—32no—
59#3947—32no—
60#3946—32no—
61#3941—32no—
62#3939—32no—
63C4⋊#168—31no—
64#3368—151noSylow
65C22×D15—42no—
66C22×C5×D3—41no—
67C2×C3×C4×C5—41no—
68D60—82no—
69C2×C5×Dic3—41no—
70C4×C5×D3—82no—
71C2×C3×Dic5—42no—
72C5×SL(2,𝔽3)#4111yescharacteristic, complemented
73#1203—84no—
74C2×GL(2,𝔽3)—52no—
75C2×GL(2,𝔽3)C2×D511yescharacteristic
76#895—201no—
77C4×SL(2,𝔽3)—52no—
78C4×SL(2,𝔽3)C2×D511yescharacteristic
79#859C2×D511yescharacteristic
80C2×C5×Q8C22×D311yescharacteristic
81C2×C5×D4—31no—
82C22×C4×C5—61no—
83C2×C22⋊D5—62no—
84C2×C22⋊D5—62no—
85#709—124no—
86C2×D20—32no—
87C2×D20—61no—
88C2×D20—61no—
89C2×C4×D5—62no—
90C2×Dic10—31no—
91C5×SD16—61no—
92C5×SD16—61no—
93C2×C5×C8—31no—
94C2×C5×C8—31no—
95C5×C4⋊C4—31no—
96C5×C4⋊C4—31no—
97C5×C22⋊C4—61no—
98C42×C5—31no—
99#688—62no—
100#688—62no—
101#685—62no—
102C4⋊Dic5—32no—
103C4⋊Dic5—61no—
104C4⋊Dic5S411yescharacteristic
105#683—32no—
106C4×Dic5—32no—
107C2×#168—32no—
108#557—51no—
109#550—151no—
110#490—152no—
111#486—151no—
112#478—152no—
113#474—151no—
114#460—152no—
115#460—151no—
116#446—151no—
117#440—152no—
118C4×SD16—31no—
119C22×C3×C5—41no—
120C2×D15—82no—
121C2×D15—82no—
122C2×D15—82no—
123C2×C5×D3—42no—
124C2×C5×D3—81no—
125C2×C5×D3—81no—
126C3×C4×C5—42no—
127C3×Dic5—82no—
128C5×Dic3—81no—
129C2×D12—201no—
130C2×D12—202no—
131C2×C4×D3—41no—
132C2×SL(2,𝔽3)C22×D511yescharacteristic
133GL(2,𝔽3)—21no—
134GL(2,𝔽3)—102no—
135C3×C4⋊C4—201no—
136(C2×D3)⋊C4—202no—
137C23×C5—61no—
138C22×D5—62no—
139C5×Q8—31no—
140C5×Q8#23911yescharacteristic
141C5×D4—61no—
142C5×D4—61no—
143C2×C4×C5—31no—
144C2×C4×C5—31no—
145C2×C4×C5—61no—
146C2×C4×C5—122no—
147C2×C4×C5C2×S411yescharacteristic
148C22⋊D5—124no—
149C22⋊D5—242no—
150C2×Dic5—32no—
151C2×Dic5C2×S412yes—
152D20—62no—
153D20—62no—
154D20—122no—
155D20—241no—
156C4×D5—124no—
157Dic10—62no—
158C5×C8—61no—
159C5×C8—61no—
160#168—62no—
161C2×#41—301no—
162C2×SD16—31no—
163C2×SD16—152no—
164C4⋊Q8—151no—
165#127—151no—
166#127—152no—
167#125—152no—
168#121—302no—
169#121—301no—
170C4×Q8—52no—
171C4×Q8—151no—
172C4×Q8D3×D511yescharacteristic
173C4×D4—31no—
174C4×D4—152no—
175#107—152no—
176#106—31no—
177C4⋊C8—151no—
178C4⋊C8—152no—
179#103—31no—
180#102—31no—
181#102—152no—
182#102—152no—
183#102—152no—
184#102—152no—
185#97—152no—
186C4×C8—31no—
187C2×C3×C5—41no—
188C2×C3×C5—41no—
189C2×C3×C5—41no—
190D15—162no—
191C5×D3—82no—
192C22×D3—41no—
193C22×D3—202no—
194C2×C3×C4—41no—
195C2×C3×C4—202no—
196C2×Dic3—41no—
197D12—402no—
198D12—404no—
199C4×D3—82no—
200SL(2,𝔽3)#70911yescharacteristic, complemented
201C22×C5—121no—
202C22×C5—121no—
203C22×C5—121no—
204C22×C5C22×S411yescharacteristic
205C2×D5—122no—
206C2×D5—122no—
207C2×D5—122no—
208C4×C5—31no—
209C4×C5—31no—
210C4×C5—61no—
211C4×C5—121no—
212C4×C5#98612yes—
213Dic5—22no—
214Dic5—62no—
215#41—604no—
216C2×Q8—151no—
217C2×Q8C2×D3×D511yescharacteristic
218C2×D4—31no—
219C2×D4—152no—
220C2×D4—301no—
221C2×D4—302no—
222C2×D4—302no—
223C2×D4—301no—
224C22×C4—61no—
225C22×C4—302no—
226SD16—61no—
227SD16—61no—
228SD16—302no—
229SD16—302no—
230C2×C8—31no—
231C2×C8—31no—
232C2×C8—152no—
233C4⋊C4—31no—
234C4⋊C4—31no—
235C4⋊C4—51no—
236C4⋊C4—152no—
237C4⋊C4—152no—
238C4⋊C4—301no—
239C22⋊C4—61no—
240C22⋊C4—302no—
241C42—31no—
242C42—152no—
243C3×C5—41noHall
244C22×C3—41no—
245C2×D3—42no—
246C2×D3—81no—
247C2×D3—81no—
248C2×D3—402no—
249C2×D3—402no—
250C2×D3—402no—
251C3×C4—42no—
252C3×C4—402no—
253Dic3—81no—
254C2×C5—241no—
255C2×C5#591911yescharacteristic
256C2×C5C2×#98611yescharacteristic
257C2×C5#592611yescharacteristic
258D5—242no—
259C23—61no—
260C23—302no—
261Q8—31no—
262Q8—302no—
263Q8#676911yescharacteristic
264D4—61no—
265D4—61no—
266D4—302no—
267D4—302no—
268D4—604no—
269D4—602no—
270D4—1201no—
271D4—1202no—
272C2×C4—31no—
273C2×C4—31no—
274C2×C4—52no—
275C2×C4—61no—
276C2×C4—122no—
277C2×C4—152no—
278C2×C4—604no—
279C2×C4D5×S411yescharacteristic
280C8—61no—
281C8—61no—
282C8—302no—
283C2×C3—41no—
284C2×C3—41no—
285C2×C3—41no—
286D3—82no—
287D3—802no—
288C5#8542211yescharacteristic, complemented, Sylow
289C22—121no—
290C22—121no—
291C22—121no—
292C22—602no—
293C22—602no—
294C22—602no—
295C22C2×D5×S411yescharacteristic
296C4—31no—
297C4—31no—
298C4—61no—
299C4—102no—
300C4—121no—
301C4—302no—
302C4#9189412yes—
303C3—41noSylow
304C2—241no—
305C2—1202no—
306C2#1175674711yescharacteristic
307C2C2×#9189411yescharacteristic
308C2#1175678511yescharacteristic
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. #65904⋊C3C2semidirect
2. C4⋊(C5⋊GL(2,𝔽3))C2semidirect
3. C4×C5×GL(2,𝔽3)C2semidirect
4. #11751611C2semidirect
5. #11751603C2semidirect
14. #65904D3semidirect
37. C4×GL(2,𝔽3)D5semidirect
72. C5×SL(2,𝔽3)#41semidirect
200. SL(2,𝔽3)#709semidirect
288. C5#85422semidirect

Maximal subgroups

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

subgroupindex
#65904⋊C32
C4⋊(C5⋊GL(2,𝔽3))2
C4×C5×GL(2,𝔽3)2
#117516112
#117516032
#106113563
#914604
#854225