Finite Group Databasebeta

← #10617539⋊C3 #14512578

Subgroups — #10617539⋊C3 #14512578

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×#4150⋊C3C211yesmaximal, characteristic
2#66262⋊C3C212yesmaximal, complemented
3#10617539C311yesmaximal, characteristic, complemented, Hall
4C2×C5×#41⋊C3C411yescharacteristic, complemented
5#4150⋊C3—22no—
6C2×D5×SL(2,𝔽3)C411yescharacteristic, complemented
7C2×D5×SL(2,𝔽3)C2211yescharacteristic
8AGL(1,𝔽5)×SL(2,𝔽3)—22no—
9C3×#3999—41nomaximal
10#2884⋊C3—51nomaximal, Hall
11C2×#4150C2×C311yescharacteristic
12#66262C2×C312yescomplemented
13#65771—32no—
14#65745—31no—
15#65745—31no—
16C2×C3×D20—41no—
17C5×#41⋊C3—21no—
18C2×C5×SL(2,𝔽3)C2×C411yescharacteristic
19C3×C22⋊AGL(1,𝔽5)—42no—
20D5×SL(2,𝔽3)—21no—
21D5×SL(2,𝔽3)D412yes—
22C2×2−1+4⋊C3—51no—
23#519⋊C3—52no—
24C2×C5×#41C3×C411yescharacteristic, complemented
25#4150—22no—
26#4150—62no—
27C2×Q8×D5C3×C411yescharacteristic, complemented
28C2×Q8×D5C22×C311yescharacteristic
29C2×#711—31no—
30C2×#711—31no—
31C2×#709—31no—
32C22×Dic10—31no—
33Q8×AGL(1,𝔽5)—22no—
34Q8×AGL(1,𝔽5)—62no—
35#4130—32no—
36#4130—32no—
37#4011—32no—
38#4011—32no—
39#3999—31no—
40#3999A411yescharacteristic
41#2884—51noSylow
42C22×C3×D5—41no—
43C22×C3×D5—41no—
44C2×C3×AGL(1,𝔽5)—82no—
45C2×C3×C4×C5—41no—
46C3×D20—82no—
47C5×SL(2,𝔽3)C22⋊C411yescharacteristic
482−1+4⋊C3—102no—
49C2×#41⋊C3AGL(1,𝔽5)11yescharacteristic, complemented
50C22×SL(2,𝔽3)—51no—
51C22×SL(2,𝔽3)—51no—
52C4×SL(2,𝔽3)—102no—
53C3×#99—201no—
54C5×#41—21no—
55C5×#41—61no—
56C2×C5×Q8C2×C3×C411yescharacteristic
57C2×C5×D4—31no—
58C22×C4×C5—31no—
59C2×C22⋊D5—31no—
60C2×C22⋊D5—31no—
61C22×Dic5—32no—
62Q8×D5—21no—
63Q8×D5—32no—
64Q8×D5—61no—
65Q8×D5C3×D412yes—
66#711—62no—
67#711—62no—
68#711—62no—
69#711—62no—
70#709—62no—
71#709—62no—
72C2×D20C2×A411yescharacteristic
73C2×C4×D5—31no—
74C2×C4×D5—31no—
75C2×Dic10—31no—
76C2×Dic10—31no—
77C2×Dic10—31no—
78C2×Dic10—62no—
79C2×Dic10—62no—
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—
92C2×C3×D5—42no—
93C2×C3×D5—81no—
94C2×C3×D5—161no—
95C2×C3×D5—161no—
96C3×AGL(1,𝔽5)—162no—
97C3×C4×C5—81no—
98C2×C3×D4—201no—
99#41⋊C3—21no—
100C2×SL(2,𝔽3)—52no—
101C2×SL(2,𝔽3)—101no—
102C2×SL(2,𝔽3)C2×AGL(1,𝔽5)11yescharacteristic
103C3×C22⋊C4—202no—
104C23×C5—31no—
105C22×D5C4×A411yescharacteristic
106C22×D5C22×A411yescharacteristic
107C2×AGL(1,𝔽5)—22no—
108C2×AGL(1,𝔽5)—62no—
109C5×Q8—31no—
110C5×Q8C3×C22⋊C411yescharacteristic
111C5×D4—61no—
112C5×D4—61no—
113C2×C4×C5—31no—
114C2×C4×C5—61no—
115C2×C4×C5—61no—
116C2×C4×C5C4×A411yescharacteristic
117C22⋊D5—62no—
118C22⋊D5—62no—
119C2×Dic5—31no—
120C2×Dic5—31no—
121C2×Dic5—64no—
122D20—22no—
123C4×D5—32no—
124C4×D5—32no—
125C4×D5—61no—
126C4×D5—61no—
127Dic10—32no—
128Dic10—32no—
129Dic10—62no—
130Dic10—61no—
131Dic10—61no—
1322−1+4—102no—
1332−1+4—302no—
134C2×#41—151no—
135C2×#41—151no—
136C2×#41—151no—
137C2×#41C3×AGL(1,𝔽5)11yescharacteristic, complemented
138C22×Q8—51no—
139C22×Q8—51no—
140C22×Q8—151no—
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—
151C3×D5—82no—
152C3×D5—161no—
153C23×C3—201no—
154C23×C3—201no—
155C3×D4—402no—
156C2×C3×C4—41no—
157C2×C3×C4—402no—
158SL(2,𝔽3)C22⋊AGL(1,𝔽5)11yescharacteristic
159C22×C5—61no—
160C22×C5—121no—
161C22×C5C2×C4×A411yescharacteristic
162C2×D5—21no—
163C2×D5—41no—
164C2×D5—41no—
165C2×D5D4×A412yes—
166AGL(1,𝔽5)—42no—
167AGL(1,𝔽5)—122no—
168C4×C5—21no—
169C4×C5—31no—
170C4×C5—31no—
171Dic5—32no—
172Dic5—61no—
173#41—21no—
174#41—61no—
175#41—302no—
176#41—302no—
177#41—302no—
178#41—302no—
179#41—302no—
180#41—302no—
181C2×Q8—52no—
182C2×Q8—101no—
183C2×Q8—151no—
184C2×Q8—151no—
185C2×Q8—151no—
186C2×Q8—152no—
187C2×Q8—302no—
188C2×Q8—302no—
189C2×Q8—301no—
190C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
191C2×D4—31no—
192C2×D4—51no—
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—202no—
208C22×C3—401no—
209C22×C3—801no—
210C22×C3—801no—
211C3×C4—81no—
212C3×C4—802no—
213C2×C5—21no—
214C2×C5—121no—
215C2×C5A4×C22⋊C411yescharacteristic
216D5—22no—
217D5—41no—
218C23—31no—
219C23—51no—
220C23—51no—
221Q8—31no—
222Q8—152no—
223Q8—152no—
224Q8—302no—
225Q8—301no—
226Q8—301no—
227Q8C3×C22⋊AGL(1,𝔽5)11yescharacteristic
228D4—61no—
229D4—61no—
230D4—102no—
231D4—302no—
232D4—302no—
233C2×C4—31no—
234C2×C4—61no—
235C2×C4—61no—
236C2×C4—102no—
237C2×C4—152no—
238C2×C4—152no—
239C2×C4—151no—
240C2×C4—151no—
241C2×C4—301no—
242C2×C4—301no—
243C2×C4—304no—
244C2×C4—302no—
245C2×C4A4×AGL(1,𝔽5)11yescharacteristic
246C2×C3—41no—
247C2×C3—81no—
248C2×C3—402no—
249C2×C3—801no—
250C5#2884⋊C311yescharacteristic, complemented, Sylow
251C22—52no—
252C22—61no—
253C22—101no—
254C22—121no—
255C22—201no—
256C22—201no—
257C22C2×A4×AGL(1,𝔽5)11yescharacteristic
258C4—21no—
259C4—31no—
260C4—31no—
261C4—152no—
262C4—202no—
263C4—301no—
264C4—602no—
265C3—41noSylow
266C2—21no—
267C2—102no—
268C2—121no—
269C2—201no—
270C2A4×C22⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC2A4×C22⋊AGL(1,𝔽5)
commutatorC2×C5×Q8C2×C3×C4
FrattiniC22C2×A4×AGL(1,𝔽5)
FittingC2×C5×#41C3×C4
socleC2×C5A4×C22⋊C4
prime radicalC2×#4150⋊C3C2
cosocleC2×Q8×D5C22×C3
hypercenterC2×C4A4×AGL(1,𝔽5)
hypocenterC5×Q8C3×C22⋊C4
LCSG ▷ C2×C5×Q8 ▷ C5×Q8
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #66262⋊C3C2semidirect
3. #10617539C3semidirect
4. C2×C5×#41⋊C3C4semidirect
6. C2×D5×SL(2,𝔽3)C4semidirect
12. #66262C2×C3semidirect
24. C2×C5×#41C3×C4semidirect
27. C2×Q8×D5C3×C4semidirect
49. C2×#41⋊C3AGL(1,𝔽5)semidirect
137. C2×#41C3×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×#4150⋊C32
#66262⋊C32
#106175393
C3×#39994
#2884⋊C35