Finite Group Databasebeta

← #10617514⋊C3 #14512567

Subgroups — #10617514⋊C3 #14512567

Order 1920.

382 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 44 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#66262⋊C3C212yesmaximal, complemented
3#10617514C311yesmaximal, characteristic, complemented, Hall
4C22×C5×SL(2,𝔽3)C411yescharacteristic, complemented
5#4147⋊C3—22no—
6C2×D5×SL(2,𝔽3)C2211yescharacteristic
7C2×C5⋊(#41⋊C3)C411yescharacteristic, complemented
8AGL(1,𝔽5)×SL(2,𝔽3)—22no—
9C3×#4011—41nomaximal
10#2884⋊C3—51nomaximal, Hall
11C2×#4147C2×C311yescharacteristic
12#66262C2×C312yescomplemented
13#65785—32no—
14#65745—31no—
15#65745—31no—
16C2×C3×C22⋊D5—41no—
17C2×C5×SL(2,𝔽3)—21no—
18C2×C5×SL(2,𝔽3)C2×C411yescharacteristic
19C3×C22⋊AGL(1,𝔽5)—42no—
20D5×SL(2,𝔽3)D412yes—
21C5⋊(#41⋊C3)—21no—
22C2×2−1+4⋊C3—51no—
23#519⋊C3—52no—
24C22×C5×Q8C3×C411yescharacteristic, complemented
25#4147—22no—
26#4147—62no—
27C2×#713—31no—
28C2×#713C3×C411yescharacteristic, complemented
29C2×Q8×D5—31no—
30C2×Q8×D5C22×C311yescharacteristic
31C2×#709—31no—
32C2×#709—31no—
33Q8×AGL(1,𝔽5)—22no—
34Q8×AGL(1,𝔽5)—62no—
35#4130—32no—
36#4130—32no—
37#4011—31no—
38#4011A411yescharacteristic
39#3999—32no—
40#3999—32no—
41#2884—51noSylow
42C23×C3×C5—41no—
43C22×C3×D5—41no—
44C2×C3×AGL(1,𝔽5)—82no—
45C3×C22⋊D5—82no—
46C2×C3×Dic5—41no—
47C5×SL(2,𝔽3)C22⋊C411yescharacteristic
482−1+4⋊C3—102no—
49C2×#41⋊C3—51no—
50C22×SL(2,𝔽3)—51no—
51C22×SL(2,𝔽3)AGL(1,𝔽5)11yescharacteristic, complemented
52C4×SL(2,𝔽3)—102no—
53C3×#99—201no—
54C2×C5×Q8—21no—
55C2×C5×Q8—31no—
56C2×C5×Q8—61no—
57C2×C5×Q8C2×C3×C411yescharacteristic
58C22×C4×C5—31no—
59C2×C22⋊D5—31no—
60C2×C22⋊D5C2×A411yescharacteristic
61#713—21no—
62#713—62no—
63#713—62no—
64#713—61no—
65Q8×D5—32no—
66Q8×D5—62no—
67Q8×D5—62no—
68Q8×D5C3×D412yes—
69#709—62no—
70#709—62no—
71#709—62no—
72#709—62no—
73C2×D20—31no—
74C2×D20—31no—
75C2×C4×D5—31no—
76C2×C4×D5—32no—
77C2×C4×D5—31no—
78C2×Dic10—31no—
79C2×Dic10—31no—
80C22⋊AGL(1,𝔽5)—32no—
81C22⋊AGL(1,𝔽5)C2×A412yes—
82C4⋊AGL(1,𝔽5)—62no—
83C4⋊AGL(1,𝔽5)—62no—
84C4×AGL(1,𝔽5)—62no—
85C4×AGL(1,𝔽5)—62no—
86C2×2−1+4—51no—
87#519—52no—
88#410—151no—
89#410—151no—
90#410—152no—
91C22×C3×C5—41no—
92C22×C3×C5—81no—
93C22×C3×C5—161no—
94C2×C3×D5—42no—
95C2×C3×D5—161no—
96C3×AGL(1,𝔽5)—162no—
97C3×Dic5—81no—
98C2×C3×D4—201no—
99#41⋊C3—101no—
100C2×SL(2,𝔽3)—21no—
101C2×SL(2,𝔽3)—52no—
102C2×SL(2,𝔽3)C2×AGL(1,𝔽5)11yescharacteristic
103C3×C22⋊C4—202no—
104C23×C5C4×A411yescharacteristic
105C22×D5—31no—
106C22×D5C22×A411yescharacteristic
107C2×AGL(1,𝔽5)—22no—
108C2×AGL(1,𝔽5)—62no—
109C5×Q8—31no—
110C5×Q8—61no—
111C5×Q8—61no—
112C5×Q8C3×C22⋊C411yescharacteristic
113C2×C4×C5—31no—
114C2×C4×C5—31no—
115C2×C4×C5—61no—
116C2×C4×C5—61no—
117C22⋊D5—22no—
118C22⋊D5—62no—
119C2×Dic5—31no—
120C2×Dic5C4×A411yescharacteristic
121D20—61no—
122D20—61no—
123D20—62no—
124C4×D5—32no—
125C4×D5—32no—
126C4×D5—61no—
127C4×D5—61no—
128C4×D5—64no—
129Dic10—32no—
130Dic10—32no—
131Dic10—62no—
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×Q8C3×AGL(1,𝔽5)11yescharacteristic, complemented
141C4×Q8—102no—
142C4×Q8—302no—
143#117—152no—
144#117—152no—
145#99—51no—
146#99—152no—
147#99—151no—
148#99—152no—
149C2×C3×C5—41no—
150C2×C3×C5—81no—
151C2×C3×C5—161no—
152C3×D5—82no—
153C23×C3—41no—
154C23×C3—201no—
155C3×D4—402no—
156C2×C3×C4—201no—
157C2×C3×C4—402no—
158SL(2,𝔽3)C22⋊AGL(1,𝔽5)11yescharacteristic
159C22×C5—21no—
160C22×C5—41no—
161C22×C5C2×C4×A411yescharacteristic
162C2×D5—41no—
163C2×D5—61no—
164C2×D5—121no—
165C2×D5D4×A412yes—
166AGL(1,𝔽5)—42no—
167AGL(1,𝔽5)—122no—
168C4×C5—31no—
169C4×C5—31no—
170C4×C5—61no—
171Dic5—21no—
172Dic5—32no—
173#41—101no—
174#41—302no—
175#41—302no—
176#41—302no—
177#41—302no—
178#41—302no—
179#41—302no—
180#41—301no—
181C2×Q8—21no—
182C2×Q8—31no—
183C2×Q8—52no—
184C2×Q8—61no—
185C2×Q8—151no—
186C2×Q8—151no—
187C2×Q8—152no—
188C2×Q8—302no—
189C2×Q8—302no—
190C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
191C2×D4—51no—
192C2×D4—151no—
193C2×D4—151no—
194C2×D4—151no—
195C22×C4—31no—
196C22×C4—151no—
197C22×C4—151no—
198C22×C4—152no—
199C4⋊C4—302no—
200C4⋊C4—302no—
201C22⋊C4—52no—
202C22⋊C4—152no—
203C42—302no—
204C42—302no—
205C3×C5—41noHall
206C22×C3—41no—
207C22×C3—81no—
208C22×C3—161no—
209C22×C3—202no—
210C22×C3—801no—
211C3×C4—401no—
212C3×C4—802no—
213C2×C5—21no—
214C2×C5—41no—
215C2×C5A4×C22⋊C411yescharacteristic
216D5—22no—
217D5—121no—
218C23—51no—
219C23—151no—
220C23A4×AGL(1,𝔽5)11yescharacteristic
221Q8—31no—
222Q8—61no—
223Q8—61no—
224Q8—152no—
225Q8—152no—
226Q8—302no—
227Q8C3×C22⋊AGL(1,𝔽5)11yescharacteristic
228D4—102no—
229D4—301no—
230D4—301no—
231D4—302no—
232D4—302no—
233C2×C4—31no—
234C2×C4—31no—
235C2×C4—51no—
236C2×C4—61no—
237C2×C4—61no—
238C2×C4—102no—
239C2×C4—152no—
240C2×C4—152no—
241C2×C4—151no—
242C2×C4—301no—
243C2×C4—301no—
244C2×C4—304no—
245C2×C4—302no—
246C2×C3—41no—
247C2×C3—81no—
248C2×C3—161no—
249C2×C3—402no—
250C5#2884⋊C311yescharacteristic, complemented, Sylow
251C22—21no—
252C22—41no—
253C22—52no—
254C22—201no—
255C22—301no—
256C22—601no—
257C22C2×A4×AGL(1,𝔽5)11yescharacteristic
258C4—31no—
259C4—31no—
260C4—61no—
261C4—101no—
262C4—152no—
263C4—202no—
264C4—602no—
265C3—41noSylow
266C2—21no—
267C2—41no—
268C2—102no—
269C2—601no—
270C2A4×C22⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC2A4×C22⋊AGL(1,𝔽5)
commutatorC2×C5×Q8C2×C3×C4
FrattiniC22C2×A4×AGL(1,𝔽5)
FittingC22×C5×Q8C3×C4
socleC2×C5A4×C22⋊C4
prime radicalC2×#4147⋊C3C2
cosocleC2×Q8×D5C22×C3
hypercenterC23A4×AGL(1,𝔽5)
hypocenterC5×Q8C3×C22⋊C4
LCSG ▷ C2×C5×Q8 ▷ C5×Q8
UCSC1 ◁ C2 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. #66262⋊C3C2semidirect
3. #10617514C3semidirect
4. C22×C5×SL(2,𝔽3)C4semidirect
7. C2×C5⋊(#41⋊C3)C4semidirect
12. #66262C2×C3semidirect
24. C22×C5×Q8C3×C4semidirect
28. C2×#713C3×C4semidirect
51. C22×SL(2,𝔽3)AGL(1,𝔽5)semidirect
140. C22×Q8C3×AGL(1,𝔽5)semidirect
250. C5#2884⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#4147⋊C32
#66262⋊C32
#106175143
C3×#40114
#2884⋊C35