Finite Group Databasebeta

← #14512490

Subgroups — #14512490

Order 1920.

380 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 38 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×#4147⋊C3C211yesmaximal, characteristic
2#11756720C212yesmaximal, complemented
3#10604974—31nomaximal, Hall
4C22×C5×SL(2,𝔽3)C411yescharacteristic, complemented
5#4147⋊C3—22no—
6C2×D5×SL(2,𝔽3)C411yescharacteristic, complemented
7C2×C5⋊(#41⋊C3)C2211yescharacteristic
8#91885—22no—
9#91242—41nomaximal
10#73154—51nomaximal, Hall
11C2×#4147D311yescharacteristic
12#65784—32no—
13#65782—32no—
14#65761—31no—
15#65745—31no—
16C2×C3×C22⋊D5—41no—
17C2×C5×SL(2,𝔽3)—21no—
18C2×C5×SL(2,𝔽3)C2×C411yescharacteristic
19#6762—42no—
20D5×SL(2,𝔽3)—21no—
21C5⋊(#41⋊C3)D412yes—
22C2×2−1+4⋊C3—51no—
23#5423—52no—
24C22×C5×Q8Dic311yescharacteristic
25#4147—22no—
26#4147—62no—
27C2×#713—31no—
28C2×#713C2×D311yescharacteristic
29C2×Q8×D5—31no—
30C2×Q8×D5Dic311yescharacteristic
31C2×#709—31no—
32C2×#709—31no—
33#4130—32no—
34D5⋊M16—32no—
35#4013—61no—
36#4011—61no—
37#4010—62no—
38#4010—62no—
39#4009—64no—
40#4005—61no—
41#3999—61no—
42#1796—151noSylow
43C23×C3×C5—41no—
44C22×C3×D5—41no—
45C3×C22⋊D5—82no—
46C2×C3×Dic5—41no—
47C5×SL(2,𝔽3)C22⋊C411yescharacteristic
48C3⋊#170—82no—
492−1+4⋊C3—102no—
50C2×#41⋊C3—51no—
51C22×SL(2,𝔽3)—51no—
52C22×SL(2,𝔽3)AGL(1,𝔽5)11yescharacteristic, complemented
53GU(2,𝔽3)—102no—
54#833—201no—
55C2×C5×Q8—21no—
56C2×C5×Q8—31no—
57C2×C5×Q8—61no—
58C2×C5×Q8C2×Dic311yescharacteristic
59C22×C4×C5—31no—
60C2×C22⋊D5—31no—
61C2×C22⋊D5S411yescharacteristic
62#713—32no—
63#713—62no—
64#713—62no—
65#713#6712yes—
66Q8×D5—21no—
67Q8×D5—34no—
68Q8×D5—61no—
69Q8×D5—62no—
70#709—62no—
71#709—62no—
72#709—62no—
73#709—62no—
74C2×D20—31no—
75C2×D20—31no—
76C2×C4×D5—31no—
77C2×C4×D5—31no—
78C2×C4×D5—61no—
79C2×Dic10—31no—
80C2×Dic10—31no—
81C22⋊AGL(1,𝔽5)—62no—
82#704—62no—
83C4⋊AGL(1,𝔽5)—62no—
84C4×AGL(1,𝔽5)—62no—
85#700—62no—
86D5⋊C8—62no—
87C2×2−1+4—51no—
88#421—152no—
89#419—152no—
90#413—151no—
91#410—151no—
92C22×C3×C5—41no—
93C22×C3×C5—81no—
94C22×C3×C5—161no—
95C2×C3×D5—81no—
96C2×C3×D5—161no—
97C3×Dic5—42no—
98C2×C3×D4—201no—
99#41⋊C3—52no—
100C2×SL(2,𝔽3)—21no—
101C2×SL(2,𝔽3)—101no—
102C2×SL(2,𝔽3)C2×AGL(1,𝔽5)11yescharacteristic
103#208—202no—
104C23×C5SL(2,ℤ/4)11yescharacteristic
105C22×D5—31no—
106C22×D5SL(2,ℤ/4)11yescharacteristic
107C2×AGL(1,𝔽5)—122no—
108C5×Q8—31no—
109C5×Q8—32no—
110C5×Q8—61no—
111C5×Q8#21711yescharacteristic
112C2×C4×C5—31no—
113C2×C4×C5—31no—
114C2×C4×C5—61no—
115C2×C4×C5—61no—
116C22⋊D5—22no—
117C22⋊D5—62no—
118C2×Dic5—31no—
119C2×Dic5C2×S411yescharacteristic
120D20—32no—
121D20—61no—
122D20—62no—
123C4×D5—32no—
124C4×D5—32no—
125C4×D5—61no—
126C4×D5—61no—
127C4×D5—64no—
128Dic10—32no—
129Dic10—62no—
130Dic10—61no—
131#170—122no—
1322−1+4—102no—
1332−1+4—302no—
134C2×#41—51no—
135C2×#41—151no—
136C2×#41—151no—
137C2×#41—151no—
138C22×Q8—51no—
139C22×Q8—151no—
140C22×Q8ΓL(1,𝔽16)11yescharacteristic
141C2×M16—152no—
142#117—152no—
143#104—302no—
144#104—302no—
145#103—304no—
146#101—301no—
147#100—301no—
148#99—301no—
149#99—301no—
150C2×C3×C5—41no—
151C2×C3×C5—81no—
152C2×C3×C5—161no—
153C3×D5—161no—
154C23×C3—41no—
155C23×C3—201no—
156C3×D4—402no—
157C2×C3×C4—201no—
158SL(2,𝔽3)C22⋊AGL(1,𝔽5)11yescharacteristic
159C3⋊C8—402no—
160C22×C5—21no—
161C22×C5—41no—
162C22×C5C2×SL(2,ℤ/4)11yescharacteristic
163C2×D5—21no—
164C2×D5—32no—
165C2×D5—41no—
166C2×D5—121no—
167AGL(1,𝔽5)—242no—
168C4×C5—31no—
169C4×C5—31no—
170C4×C5—61no—
171Dic5—61no—
172Dic5GL(2,ℤ/4)12yes—
173#41—52no—
174#41—152no—
175#41—302no—
176#41—302no—
177#41—302no—
178#41—302no—
179#41—302no—
180#41—302no—
181C2×Q8—21no—
182C2×Q8—31no—
183C2×Q8—61no—
184C2×Q8—101no—
185C2×Q8—151no—
186C2×Q8—151no—
187C2×Q8—154no—
188C2×Q8—302no—
189C2×Q8—301no—
190C2×Q8C2×ΓL(1,𝔽16)11yescharacteristic
191C2×D4—51no—
192C2×D4—151no—
193C2×D4—151no—
194C2×D4—151no—
195C22×C4—31no—
196C22×C4—151no—
197C22×C4—151no—
198C22×C4—301no—
199M16—302no—
200M16—302no—
201C2×C8—302no—
202C4⋊C4—302no—
203C22⋊C4—302no—
204C42—302no—
205C3×C5—41noHall
206C22×C3—41no—
207C22×C3—81no—
208C22×C3—161no—
209C22×C3—401no—
210C22×C3—801no—
211C3×C4—202no—
212C2×C5—21no—
213C2×C5—41no—
214C2×C5#543211yescharacteristic
215D5—41no—
216D5—62no—
217C23—51no—
218C23—151no—
219C23A4⋊AGL(1,𝔽5)11yescharacteristic
220Q8—31no—
221Q8—32no—
222Q8—61no—
223Q8—152no—
224Q8—301no—
225Q8—302no—
226Q8#676311yescharacteristic
227D4—102no—
228D4—152no—
229D4—301no—
230D4—302no—
231D4—302no—
232C2×C4—31no—
233C2×C4—31no—
234C2×C4—51no—
235C2×C4—61no—
236C2×C4—61no—
237C2×C4—152no—
238C2×C4—152no—
239C2×C4—151no—
240C2×C4—301no—
241C2×C4—301no—
242C2×C4—304no—
243C2×C4—602no—
244C8—602no—
245C2×C3—41no—
246C2×C3—81no—
247C2×C3—161no—
248C2×C3—801no—
249C5#7315411yescharacteristic, complemented, Sylow
250C22—21no—
251C22—41no—
252C22—101no—
253C22—152no—
254C22—201no—
255C22—601no—
256C22C2×A4⋊AGL(1,𝔽5)11yescharacteristic
257C4—31no—
258C4—31no—
259C4—52no—
260C4—61no—
261C4—301no—
262C4—1202no—
263C3—41noSylow
264C2—21no—
265C2—41no—
266C2—201no—
267C2—302no—
268C2#1175672911yescharacteristic
subgroupG/H
centerC2#11756729
commutatorC2×C5×SL(2,𝔽3)C2×C4
FrattiniC22C2×A4⋊AGL(1,𝔽5)
FittingC22×C5×Q8Dic3
socleC2×C5#5432
prime radicalC2×#4147⋊C3C2
cosocleC2×C5⋊(#41⋊C3)C22
hypercenterC23A4⋊AGL(1,𝔽5)
hypocenterC5×SL(2,𝔽3)C22⋊C4
LCSG ▷ C2×C5×SL(2,𝔽3) ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. #11756720C2semidirect
4. C22×C5×SL(2,𝔽3)C4semidirect
6. C2×D5×SL(2,𝔽3)C4semidirect
52. C22×SL(2,𝔽3)AGL(1,𝔽5)semidirect
249. C5#73154semidirect

Maximal subgroups

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

subgroupindex
C2×#4147⋊C32
#117567202
#106049743
#912424
#731545