Finite Group Databasebeta

← #14512822

Subgroups — #14512822

Order 1920.

452 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 59 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)×D20C211yesmaximal, characteristic
2C4×C5⋊#226C211yesmaximal, characteristic, complemented
3C5×#5391C211yesmaximal, characteristic, complemented
4D5×#859C212yesmaximal, complemented
5#11751607C212yesmaximal, complemented
6#10612850—31nomaximal, Hall
7C2×D5×SL(2,𝔽3)C2212yes—
8C2×C5⋊#226C2211yescharacteristic
9C4×C5×SL(2,𝔽3)C2211yescharacteristic
10#91434—41nomaximal
11#91184C2211yescharacteristic
12C5×#859C2212yes—
13#85393—51nomaximal, Hall
14#66064—31no—
15Q8×D20D311yescharacteristic
16C5×#477—31no—
17#65382—31no—
18C4×#689—31no—
19#65176—32no—
20#65173—32no—
21#65138—31no—
22#65123—31no—
23#65097—32no—
24D5×#103—32no—
25C2×C3×D20—41no—
26C2×C5×SL(2,𝔽3)C2311yescharacteristic
27C2×D5×Dic3—42no—
28D5×SL(2,𝔽3)—22no—
29C5⋊#226—21no—
30C4×Dic15—41no—
31C5×C4⋊Dic3—41no—
32#6665—42no—
33D4×SL(2,𝔽3)—51no—
34#5419—52no—
35C2×#859—52no—
36#5391D511yescharacteristic, complemented
37C4×#226—51no—
38C2×Q8×D5C2×D312yes—
39C5×C4⋊Q8—31no—
40C4×C5×Q8C2×D311yescharacteristic
41#4092—62no—
42C2×#689—31no—
43C2×Dic20—32no—
44C2×C8×D5—32no—
45#4043—32no—
46#4043—32no—
47#4039—61no—
48D5×C4⋊C4—32no—
49#4036—61no—
50C4×D20—31no—
51#4015—31no—
52C4×Dic10—31no—
53C5×#107—32no—
54C5×C4⋊C8—31no—
55C5×#103—32no—
56#3967—31no—
57#3952—32no—
58#3948—32no—
59#3942—31no—
60#3942—32no—
61#3942—32no—
62#3939—31no—
63C4×#168—31no—
64#3299—151noSylow
65C22×C3×D5—42no—
66C2×C3×C4×C5—41no—
67C2×Dic15—41no—
68C2×Dic15—41no—
69C2×C5×Dic3—42no—
70C3×D20—44no—
71C5×SL(2,𝔽3)#4111yescharacteristic
72D5×Dic3—84no—
73C22×SL(2,𝔽3)—52no—
74C2×#226—51no—
75D4×Dic3—201no—
76C4×SL(2,𝔽3)C2×D511yescharacteristic
77#859—51no—
78#859C2×D512yes—
79C2×C5×Q8—61no—
80C2×C5×Q8C22×D311yescharacteristic
81Q8×D5—22no—
82Q8×D5—62no—
83C2×D20S411yescharacteristic
84C2×C4×D5—32no—
85C2×C4×D5—62no—
86C2×Dic10—32no—
87C2×Dic10—31no—
88C2×C5×C8—32no—
89C5×C4⋊C4—31no—
90C5×C4⋊C4—32no—
91C5×C4⋊C4—61no—
92C42×C5—31no—
93#689—61no—
94#689—61no—
95#685—32no—
96#685—62no—
97C4⋊Dic5—31no—
98C4⋊Dic5—31no—
99C4⋊Dic5—61no—
100C4⋊Dic5—62no—
101#683—62no—
102#683—61no—
103C4×Dic5—61no—
104C2×#168—31no—
105C2×#168—31no—
106Dic20—122no—
107C8×D5—122no—
108#554—151no—
109D4×Q8—51no—
110#487—151no—
111#484—152no—
112#477—31no—
113#467—152no—
114#462—151no—
115#451—152no—
116C4×Q16—151no—
117C8×D4—151no—
118C2×#103—152no—
119C22×C3×C5—41no—
120C2×C3×D5—44no—
121C2×C3×D5—82no—
122C2×C3×D5—82no—
123C3×C4×C5—42no—
124Dic15—42no—
125Dic15—81no—
126C5×Dic3—82no—
127C2×C3×D4—201no—
128C22×Dic3—202no—
129C2×SL(2,𝔽3)—102no—
130C2×SL(2,𝔽3)C22×D511yescharacteristic
131#226—101no—
132#217—202no—
133C4⋊Dic3—41no—
134C4×Dic3—201no—
135C22×D5C2×S412yes—
136C5×Q8—31no—
137C5×Q8—241no—
138C5×Q8#23711yescharacteristic
139C2×C4×C5—31no—
140C2×C4×C5—31no—
141C2×C4×C5—62no—
142C2×C4×C5C2×S411yescharacteristic
143C2×Dic5—32no—
144C2×Dic5—61no—
145C2×Dic5—61no—
146D20#22614yescomplemented
147C4×D5—62no—
148C4×D5—62no—
149C4×D5—242no—
150Dic10—62no—
151Dic10—62no—
152Dic10—61no—
153Dic10—61no—
154C5×C8—62no—
155#168—61no—
156#168—61no—
157C22×Q8—52no—
158C2×Q16—152no—
159C2×Q16—151no—
160C22×C8—152no—
161C4⋊Q8—31no—
162C4⋊Q8—151no—
163#125—301no—
164C22⋊Q8—152no—
165C22⋊Q8—152no—
166C22⋊Q8—302no—
167C4×Q8—151no—
168C4×Q8D3×D511yescharacteristic
169C4×D4—151no—
170C4×D4—301no—
171C2×C4⋊C4—152no—
172#107—32no—
173#107—151no—
174C4⋊C8—31no—
175#103—32no—
176#103—152no—
177#103—151no—
178#103—152no—
179#103—152no—
180#103—151no—
181C22⋊C8—152no—
182C4×C8—151no—
183C2×C3×C5—41no—
184C2×C3×C5—41no—
185C2×C3×C5—41no—
186C3×D5—84no—
187C23×C3—202no—
188C3×D4—204no—
189C2×C3×C4—41no—
190C2×Dic3—42no—
191C2×Dic3—201no—
192C2×Dic3—201no—
193C2×Dic3—404no—
194SL(2,𝔽3)#71311yescharacteristic
195C22×C5C22×S411yescharacteristic
196C2×D5—22no—
197C2×D5—22no—
198C2×D5C2×#22614yes—
199C4×C5—31no—
200C4×C5—31no—
201C4×C5—61no—
202C4×C5—122no—
203C4×C5C2×#22612yes—
204Dic5—62no—
205Dic5—121no—
206Dic5—121no—
207C2×Q8—61no—
208C2×Q8—102no—
209C2×Q8—152no—
210C2×Q8—151no—
211C2×Q8—302no—
212C2×Q8C2×D3×D511yescharacteristic
213C2×D4—51no—
214C22×C4—152no—
215C22×C4—302no—
216Q16—301no—
217Q16—301no—
218Q16—602no—
219C2×C8—32no—
220C2×C8—151no—
221C2×C8—151no—
222C2×C8—602no—
223C4⋊C4—31no—
224C4⋊C4—32no—
225C4⋊C4—61no—
226C4⋊C4—151no—
227C4⋊C4—151no—
228C4⋊C4—301no—
229C4⋊C4—301no—
230C4⋊C4—302no—
231C4⋊C4—302no—
232C22⋊C4—152no—
233C22⋊C4—302no—
234C42—31no—
235C42—301no—
236C3×C5—41noHall
237C22×C3—41no—
238C22×C3—204no—
239C22×C3—402no—
240C22×C3—402no—
241C3×C4—42no—
242Dic3—82no—
243Dic3—202no—
244Dic3—401no—
245C2×C5#591411yescharacteristic
246C2×C5C22×#22611yescharacteristic
247C2×C5#592611yescharacteristic
248D5—24no—
249C23—52no—
250Q8—31no—
251Q8—241no—
252Q8—302no—
253Q8—302no—
254Q8—301no—
255Q8—301no—
256Q8#676611yescharacteristic
257D4—54no—
258C2×C4—31no—
259C2×C4—31no—
260C2×C4—62no—
261C2×C4—152no—
262C2×C4—302no—
263C2×C4—302no—
264C2×C4—301no—
265C2×C4—301no—
266C2×C4—1202no—
267C2×C4D5×S411yescharacteristic
268C8—62no—
269C8—301no—
270C8—301no—
271C2×C3—41no—
272C2×C3—41no—
273C2×C3—41no—
274C2×C3—404no—
275C5#8539311yescharacteristic, complemented, Sylow
276C22—54no—
277C22—102no—
278C22—102no—
279C22C2×D5×S411yescharacteristic
280C4—31no—
281C4—31no—
282C4—61no—
283C4—122no—
284C4—302no—
285C4—601no—
286C4—601no—
287C4D5×#22612yes—
288C3—41noSylow
289C2—104no—
290C2#1175675111yescharacteristic
291C2C2×D5×#22611yescharacteristic
292C2#1175678611yescharacteristic
subgroupG/H
center, FrattiniC22C2×D5×S4
commutator, cosocleC2×C5×SL(2,𝔽3)C23
FittingC4×C5×Q8C2×D3
socleC22×C5C22×S4
prime radicalSL(2,𝔽3)×D20C2
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
2. C4×C5⋊#226C2semidirect
3. C5×#5391C2semidirect
4. D5×#859C2semidirect
5. #11751607C2semidirect
36. #5391D5semidirect
146. D20#226semidirect
275. C5#85393semidirect

Maximal subgroups

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

subgroupindex
SL(2,𝔽3)×D202
C4×C5⋊#2262
C5×#53912
D5×#8592
#117516072
#106128503
#914344
#853935