Finite Group Databasebeta

← #14512666

Subgroups — #14512666

Order 1920.

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

normal subgroups only

This group has 412 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
1SL(2,𝔽3)×Dic10C211yesmaximal, characteristic, complemented
2C4⋊(C5⋊GL(2,𝔽3))C211yesmaximal, characteristic
3C4×C5×#226C211yesmaximal, characteristic, complemented
4#11751604C212yesmaximal, complemented
5#11751603C212yesmaximal, complemented
6#10611434—31nomaximal, Hall
7C2×C5⋊GL(2,𝔽3)C2212yes—
8C2×C5×#226C2211yescharacteristic
9C4×C5×SL(2,𝔽3)C2211yescharacteristic
10#91374—41nomaximal
11Dic5×SL(2,𝔽3)C2212yes—
12C5×#859C2211yescharacteristic
13#85433—51nomaximal, Hall
14#65919—31no—
15Q8×Dic10D311yescharacteristic, complemented
16C4×C5×Q16—31no—
17#65474—32no—
18#65316—31no—
19#65310—31no—
20#65179—32no—
21#65106—32no—
22#65084—32no—
23#64983—31no—
24#64968—31no—
25C2×D60—41no—
26C2×C3×Dic10—41no—
27C2×C5×SL(2,𝔽3)C2311yescharacteristic
28C5⋊GL(2,𝔽3)—22no—
29C5×#226—21no—
30C4×C5×Dic3—41no—
31#6667—42no—
32#6667—42no—
33Q8×SL(2,𝔽3)—51no—
34#5395—52no—
35#5394—52no—
36C4⋊GL(2,𝔽3)—51no—
37C4×#226D511yescharacteristic, complemented
38C2×C5×Q16—31no—
39C4×C5×Q8—31no—
40C4×C5×Q8C2×D311yescharacteristic
41#4093—32no—
42Q8×Dic5C2×D312yes—
43C2×#688—32no—
44#4040—62no—
45#4039—32no—
46#4034—32no—
47#4021—61no—
48#4020—31no—
49C4×D20—61no—
50#4015—31no—
51C4×Dic10—31no—
52C5×#107—31no—
53C5×#103—31no—
54C5×#103—31no—
55C4×C5×C8—31no—
56#3967—32no—
57#3953—32no—
58#3953—32no—
59#3947—32no—
60#3946—32no—
61#3941—32no—
62#3939—32no—
63C4⋊#168—31no—
64#3375—151noSylow
65C22×D15—42no—
66C2×C3×C4×C5—41no—
67D60—82no—
68C2×C5×Dic3—41no—
69C2×C5×Dic3—41no—
70C2×C3×Dic5—42no—
71C3×Dic10—82no—
72C5×SL(2,𝔽3)#4111yescharacteristic
73C2×GL(2,𝔽3)—52no—
74C2×#226C2×D511yescharacteristic
75#947—201no—
76C4×SL(2,𝔽3)—52no—
77C4×SL(2,𝔽3)C2×D511yescharacteristic
78#859C2×D511yescharacteristic
79C2×C5×Q8—31no—
80C2×C5×Q8C22×D311yescharacteristic
81C2×D20—32no—
82C2×D20—61no—
83C2×D20—61no—
84C2×C4×D5—62no—
85C2×Dic10—61no—
86C2×Dic10S411yescharacteristic
87C5×Q16—61no—
88C5×Q16—61no—
89C2×C5×C8—31no—
90C2×C5×C8—31no—
91C5×C4⋊C4—31no—
92C5×C4⋊C4—31no—
93C5×C4⋊C4—61no—
94C42×C5—31no—
95C42×C5—61no—
96#688—62no—
97#688—62no—
98#685—62no—
99#685—62no—
100#685—62no—
101C4⋊Dic5—31no—
102C4⋊Dic5—32no—
103#683—32no—
104C4×Dic5—32no—
105C2×#168—32no—
106#559—151no—
107Q82—51no—
108#489—152no—
109#486—151no—
110#477—151no—
111#474—152no—
112#464—152no—
113#460—151no—
114#446—151no—
115#440—152no—
116C4×Q16—31no—
117C22×C3×C5—41no—
118C2×D15—82no—
119C2×D15—82no—
120C2×D15—82no—
121C3×C4×C5—42no—
122C3×Dic5—82no—
123C5×Dic3—81no—
124C5×Dic3—81no—
125C2×C3×Q8—201no—
126C2×D12—201no—
127C2×SL(2,𝔽3)C22×D511yescharacteristic
128GL(2,𝔽3)—102no—
129#226—21no—
130(C2×D3)⋊C4—202no—
131(C2×D3)⋊C4—202no—
132C4×Dic3—41no—
133C22×D5—62no—
134C5×Q8—31no—
135C5×Q8—61no—
136C5×Q8—61no—
137C5×Q8#23911yescharacteristic
138C2×C4×C5—31no—
139C2×C4×C5—31no—
140C2×C4×C5—61no—
141C2×C4×C5—61no—
142C2×C4×C5C2×S411yescharacteristic
143C2×Dic5—32no—
144C2×Dic5C2×S412yes—
145D20—62no—
146D20—62no—
147D20—122no—
148D20—241no—
149C4×D5—242no—
150Dic10—22no—
151Dic10—241no—
152C5×C8—61no—
153C5×C8—61no—
154#168—62no—
155C2×Q16—31no—
156C2×SD16—152no—
157C4⋊Q8—151no—
158C4⋊Q8—152no—
159#127—151no—
160#124—152no—
161#124—301no—
162#123—302no—
163C4×Q8—31no—
164C4×Q8—52no—
165C4×Q8—151no—
166C4×Q8D3×D511yescharacteristic
167C4×D4—152no—
168C4×D4—301no—
169#107—31no—
170#107—152no—
171C4⋊C8—151no—
172C4⋊C8—152no—
173#103—31no—
174#103—31no—
175#103—152no—
176#102—152no—
177#102—152no—
178#102—152no—
179#97—152no—
180C4×C8—31no—
181C2×C3×C5—41no—
182C2×C3×C5—41no—
183C2×C3×C5—41no—
184D15—162no—
185C22×D3—202no—
186C3×Q8—402no—
187C2×C3×C4—41no—
188C2×C3×C4—202no—
189C2×Dic3—41no—
190C2×Dic3—41no—
191D12—402no—
192SL(2,𝔽3)#70911yescharacteristic
193C22×C5C22×S411yescharacteristic
194C2×D5—122no—
195C2×D5—122no—
196C2×D5—122no—
197C4×C5—31no—
198C4×C5—31no—
199C4×C5—61no—
200C4×C5—121no—
201C4×C5—121no—
202C4×C5#98312yes—
203Dic5—22no—
204Dic5—62no—
205C2×Q8—31no—
206C2×Q8—51no—
207C2×Q8—301no—
208C2×Q8C2×D3×D511yescharacteristic
209C2×D4—152no—
210C2×D4—301no—
211C2×D4—301no—
212C22×C4—302no—
213Q16—61no—
214Q16—61no—
215SD16—302no—
216SD16—302no—
217C2×C8—31no—
218C2×C8—31no—
219C2×C8—152no—
220C4⋊C4—31no—
221C4⋊C4—31no—
222C4⋊C4—61no—
223C4⋊C4—152no—
224C4⋊C4—152no—
225C4⋊C4—151no—
226C22⋊C4—302no—
227C22⋊C4—302no—
228C22⋊C4—302no—
229C42—31no—
230C42—61no—
231C42—152no—
232C3×C5—41noHall
233C22×C3—41no—
234C2×D3—402no—
235C2×D3—402no—
236C2×D3—402no—
237C3×C4—42no—
238C3×C4—402no—
239Dic3—81no—
240Dic3—81no—
241C2×C5C2×#98311yescharacteristic
242C2×C5#591911yescharacteristic
243C2×C5#592511yescharacteristic
244D5—242no—
245C23—302no—
246Q8—31no—
247Q8—61no—
248Q8—61no—
249Q8—102no—
250Q8—1201no—
251Q8#676911yescharacteristic
252D4—302no—
253D4—302no—
254D4—602no—
255D4—1201no—
256C2×C4—31no—
257C2×C4—31no—
258C2×C4—52no—
259C2×C4—61no—
260C2×C4—61no—
261C2×C4—152no—
262C2×C4—1202no—
263C2×C4D5×S411yescharacteristic
264C8—61no—
265C8—61no—
266C8—302no—
267C2×C3—41no—
268C2×C3—41no—
269C2×C3—41no—
270D3—802no—
271C5#8543311yescharacteristic, complemented, Sylow
272C22—602no—
273C22—602no—
274C22—602no—
275C22C2×D5×S411yescharacteristic
276C4—31no—
277C4—31no—
278C4—61no—
279C4—102no—
280C4—121no—
281C4—121no—
282C4—302no—
283C4#9189712yes—
284C3—41noSylow
285C2—1202no—
286C2#1175674711yescharacteristic
287C2C2×#9189711yescharacteristic
288C2#1175677511yescharacteristic
subgroupG/H
center, FrattiniC22C2×D5×S4
commutator, cosocleC2×C5×SL(2,𝔽3)C23
FittingC4×C5×Q8C2×D3
socleC22×C5C22×S4
prime radicalC4⋊(C5⋊GL(2,𝔽3))C2
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. SL(2,𝔽3)×Dic10C2semidirect
3. C4×C5×#226C2semidirect
4. #11751604C2semidirect
5. #11751603C2semidirect
15. Q8×Dic10D3semidirect
37. C4×#226D5semidirect
271. C5#85433semidirect

Maximal subgroups

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

subgroupindex
SL(2,𝔽3)×Dic102
C4⋊(C5⋊GL(2,𝔽3))2
C4×C5×#2262
#117516042
#117516032
#106114343
#913744
#854335