Finite Group Databasebeta

← #10615313⋊C3 #14512261

Subgroups — #10615313⋊C3 #14512261

Order 1920.

250 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 45 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
1C5×C4⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
2C4⋊(C5⋊(#41⋊C3))C211yesmaximal, characteristic, complemented
3SL(2,𝔽3)×#168C211yesmaximal, characteristic, complemented
4#10615313C311yesmaximal, characteristic, complemented, Hall
5C2×C5×#41⋊C3—21no—
6C2×C5⋊(#41⋊C3)—21no—
7C4×C5×SL(2,𝔽3)C2211yescharacteristic
8C3×#3939—41nomaximal
9#3391⋊C3—51nomaximal, Hall
10C5×#550C2×C311yescharacteristic
11#65911C2×C311yescharacteristic
12#65362—31no—
13#65361—31no—
14Q8×#168C2×C311yescharacteristic
15#65304—31no—
16C4×#686—31no—
17C5×#41⋊C3—41no—
18C2×C5×SL(2,𝔽3)D411yescharacteristic
19C5⋊(#41⋊C3)—41no—
20C3×C5×C4⋊C4—41no—
21C3×C4⋊Dic5—41no—
22C2×C3×#168—41no—
23C4⋊(#41⋊C3)—51no—
24C4⋊(#41⋊C3)D511yescharacteristic, complemented
25C8×SL(2,𝔽3)—51no—
26C2×C5×#41—21no—
27C2×#713—21no—
28C5×#127—31no—
29C5×#121—61no—
30C4×C5×Q8C22×C311yescharacteristic
31C4×C5×D4—31no—
32C2×#686—32no—
33C2×#686—31no—
34#4041—61no—
35#4020—31no—
36C4×D20—31no—
37#3964—31no—
38#3941—31no—
39#3939A411yescharacteristic
40C4⋊#168—31no—
41C4×#168—31no—
42#3391—51noSylow
43C2×C3×C4×C5—41no—
44C2×C3×C4×C5—81no—
45C2×C3×Dic5—81no—
46C5×SL(2,𝔽3)D811yescharacteristic, complemented
47C3×#168—42no—
48C2×#41⋊C3—21no—
49C2×#41⋊C3—101no—
50C4×SL(2,𝔽3)C2×D511yescharacteristic
51C3×#107—201no—
52C5×#41—41no—
53C5×#41—121no—
54C2×C5×Q8C3×D411yescharacteristic
55C2×C5×D4—31no—
56C2×C5×D4—61no—
57C2×C5×D4—61no—
58C22×C4×C5—61no—
59#713—41no—
60#713—121no—
61C2×D20—31no—
62C2×D20—61no—
63C2×D20—61no—
64C2×C4×D5—61no—
65C5×C4⋊C4—31no—
66C5×C4⋊C4C2×A411yescharacteristic
67C5×C22⋊C4—61no—
68C42×C5—31no—
69#686—62no—
70#686—122no—
71#685—61no—
72C4⋊Dic5C2×A411yescharacteristic
73C2×#168—31no—
74C2×#168C2×A411yescharacteristic
75#550—51no—
76#550C3×D511yescharacteristic
77#493—151no—
78#459—151no—
79#459—151no—
80C8×Q8—51no—
81C4×D8—151no—
82C22×C3×C5—41no—
83C3×C4×C5—41no—
84C3×C4×C5—41no—
85C3×C4×C5—161no—
86C3×Dic5—161no—
87#41⋊C3—41no—
88#41⋊C3—201no—
89C2×SL(2,𝔽3)C22⋊D511yescharacteristic
90C2×C3×C8—201no—
91C3×C4⋊C4—41no—
92C3×C4⋊C4—201no—
93C23×C5—61no—
94C22×D5—61no—
95C5×Q8—31no—
96C5×Q8C3×D811yescharacteristic
97C5×D4—61no—
98C5×D4—61no—
99C5×D4—121no—
100C5×D4—121no—
101C5×D4—241no—
102C2×C4×C5—21no—
103C2×C4×C5—31no—
104C2×C4×C5—31no—
105C2×C4×C5—121no—
106C2×C4×C5—121no—
107C2×C4×C5C22×A411yescharacteristic
108C2×Dic5—21no—
109D20—61no—
110D20—61no—
111D20—121no—
112D20—121no—
113D20—241no—
114C4×D5—121no—
115C4×D5—121no—
116#168—61no—
117#168#41⋊C312yescomplemented
118C2×#41—21no—
119C2×#41—101no—
120C2×D8—151no—
121C2×D8—152no—
122#127—31no—
123#127—151no—
124#121—61no—
125#121—301no—
126C4×Q8C2×C3×D511yescharacteristic
127C4×D4—31no—
128C4×D4—151no—
129#107—51no—
130C4⋊C8—151no—
131#102—151no—
132#102—151no—
133C4×C8—151no—
134C2×C3×C5—41no—
135C2×C3×C5—41no—
136C2×C3×C5—41no—
137C2×C3×C4—41no—
138C2×C3×C4—81no—
139C2×C3×C4—401no—
140SL(2,𝔽3)#68611yescharacteristic, complemented
141C3×C8—202no—
142C22×C5—121no—
143C22×C5—121no—
144C22×C5—121no—
145C22×C5D4×A411yescharacteristic
146C2×D5—121no—
147C2×D5—121no—
148C2×D5—121no—
149C4×C5—31no—
150C4×C5—31no—
151C4×C5—41no—
152C4×C5—61no—
153C4×C5C2×#41⋊C311yescharacteristic
154C4×C5#99411yescharacteristic
155Dic5—41no—
156#41—41no—
157#41—121no—
158#41—201no—
159#41—601no—
160C2×Q8C3×C22⋊D511yescharacteristic
161C2×D4—31no—
162C2×D4—61no—
163C2×D4—61no—
164C2×D4—151no—
165C2×D4—301no—
166C2×D4—301no—
167C22×C4—61no—
168C22×C4—301no—
169D8—302no—
170D8—602no—
171C2×C8—51no—
172C2×C8—151no—
173C4⋊C4—31no—
174C4⋊C4—51no—
175C4⋊C4D5×A411yescharacteristic
176C22⋊C4—61no—
177C22⋊C4—301no—
178C42—31no—
179C3×C5—41noHall
180C22×C3—41no—
181C3×C4—41no—
182C3×C4—41no—
183C3×C4—161no—
184C3×C4—801no—
185C2×C5—241no—
186C2×C5C4⋊(#41⋊C3)11yescharacteristic
187C2×C5A4×D811yescharacteristic
188C2×C5#576⋊C311yescharacteristic
189D5—241no—
190C23—61no—
191C23—301no—
192Q8—31no—
193Q8C3×#68611yescharacteristic
194D4—61no—
195D4—61no—
196D4—121no—
197D4—121no—
198D4—241no—
199D4—301no—
200D4—301no—
201D4—601no—
202D4—601no—
203D4—1201no—
204C2×C4—21no—
205C2×C4—31no—
206C2×C4—31no—
207C2×C4—101no—
208C2×C4—121no—
209C2×C4—121no—
210C2×C4—601no—
211C2×C4—601no—
212C2×C4C2×D5×A411yescharacteristic
213C8—52no—
214C8—301no—
215C2×C3—41no—
216C2×C3—41no—
217C2×C3—41no—
218C5#3391⋊C311yescharacteristic, complemented, Sylow
219C22—121no—
220C22—121no—
221C22—121no—
222C22—601no—
223C22—601no—
224C22—601no—
225C22A4×C22⋊D511yescharacteristic
226C4—31no—
227C4—31no—
228C4—41no—
229C4—61no—
230C4—201no—
231C4C5⋊#99411yescharacteristic
232C4D5×#41⋊C311yescharacteristic
233C3—41noSylow
234C2—241no—
235C2—1201no—
236C2A4×#68611yescharacteristic
237C2#66168⋊C311yescharacteristic
238C2#66169⋊C311yescharacteristic
subgroupG/H
centerC22A4×C22⋊D5
commutator, cosocleC4×C5×Q8C22×C3
FrattiniC2×C4C2×D5×A4
FittingC5×#550C2×C3
socleC22×C5D4×A4
hypercenterC4⋊C4D5×A4
hypocenterC5×Q8C3×D8
LCSG ▷ C4×C5×Q8 ▷ C2×C5×Q8 ▷ C5×Q8
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C5×C4⋊(#41⋊C3)C2semidirect
2. C4⋊(C5⋊(#41⋊C3))C2semidirect
3. SL(2,𝔽3)×#168C2semidirect
4. #10615313C3semidirect
24. C4⋊(#41⋊C3)D5semidirect
46. C5×SL(2,𝔽3)D8semidirect
117. #168#41⋊C3semidirect
140. SL(2,𝔽3)#686semidirect
218. C5#3391⋊C3semidirect

Maximal subgroups

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

subgroupindex
C5×C4⋊(#41⋊C3)2
C4⋊(C5⋊(#41⋊C3))2
SL(2,𝔽3)×#1682
#106153133
C3×#39394
#3391⋊C35