Finite Group Databasebeta

← #14512010

Subgroups — #14512010

Order 1920.

226 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 37 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×#557⋊C3C211yesmaximal, characteristic, complemented
2C4⋊(C5⋊GL(2,𝔽3))C211yesmaximal, characteristic
3SL(2,𝔽3)⋊#168C211yesmaximal, characteristic, complemented
4#10602728—31nomaximal, Hall
5C2×C5⋊GL(2,𝔽3)—21no—
6C4×C5×SL(2,𝔽3)—21no—
7C4×C5×SL(2,𝔽3)C2211yescharacteristic
8#91093—41nomaximal
9#73096—51nomaximal, Hall
10C5×#557D311yescharacteristic, complemented
11#65377—31no—
12#65316—31no—
13#65316—31no—
14#64767—31no—
15Q8⋊#168—31no—
16Q8⋊#168—31no—
17C2×D60—41no—
18C2×C5×SL(2,𝔽3)D411yescharacteristic
19C5⋊GL(2,𝔽3)—41no—
20C3×C5×C4⋊C4—41no—
21C2×C3⋊#168—41no—
22#557⋊C3D511yescharacteristic, complemented
23C4⋊GL(2,𝔽3)—51no—
24SL(2,𝔽3)⋊C8—51no—
25C5×C4⋊Q8—31no—
26C5×#125—61no—
27C4×C5×Q8—21no—
28C4×C5×Q8—31no—
29C4×C5×Q8C2×D311yescharacteristic
30C2×#688—61no—
31C2×#688—61no—
32#4020—31no—
33#3941—61no—
34#3941—62no—
35#3941—61no—
36C4⋊#168—31no—
37C4⋊#168—31no—
38C4×#168—31no—
39#1626—151noSylow
40C22×D15—81no—
41C2×C3×C4×C5—41no—
42C2×C3×C4×C5—81no—
43D60—42no—
44D60—81no—
45C5×SL(2,𝔽3)SD1611yescharacteristic, complemented
46C3⋊#168—81no—
47C2×GL(2,𝔽3)—101no—
48C4×SL(2,𝔽3)—21no—
49C4×SL(2,𝔽3)C2×D511yescharacteristic
50#809—201no—
51C2×C5×Q8—31no—
52C2×C5×Q8#6711yescharacteristic
53C2×D20—61no—
54C2×D20—61no—
55C2×D20—61no—
56C5×C4⋊C4—31no—
57C5×C4⋊C4—32no—
58C5×C4⋊C4—61no—
59C5×C4⋊C4—61no—
60C5×C4⋊C4S411yescharacteristic
61C42×C5—31no—
62C42×C5—61no—
63#688—121no—
64#688—122no—
65#688—121no—
66C2×#168—61no—
67C2×#168—61no—
68C2×#168—61no—
69#557D1511yescharacteristic, complemented
70#486—151no—
71#460—151no—
72#460—151no—
73#332—151no—
74Q8⋊C8—151no—
75Q8⋊C8—151no—
76C22×C3×C5—41no—
77C2×D15—82no—
78C2×D15—161no—
79C2×D15—161no—
80C3×C4×C5—41no—
81C3×C4×C5—41no—
82C3×C4×C5—161no—
83C2×D12—201no—
84C2×SL(2,𝔽3)C22⋊D511yescharacteristic
85GL(2,𝔽3)—201no—
86C3×C4⋊C4—41no—
87C2×C3⋊C8—201no—
88C22×D5—121no—
89C5×Q8—31no—
90C5×Q8—61no—
91C5×Q8—61no—
92C5×Q8#21511yescharacteristic
93C2×C4×C5—21no—
94C2×C4×C5—31no—
95C2×C4×C5—31no—
96C2×C4×C5—61no—
97C2×C4×C5C2×S411yescharacteristic
98D20—121no—
99D20—121no—
100D20—122no—
101D20—121no—
102D20—121no—
103#168—121no—
104#168—121no—
105#168—121no—
106C2×SD16—301no—
107C2×SD16—301no—
108C4⋊Q8—31no—
109#127—151no—
110#125—61no—
111C4×Q8—21no—
112C4×Q8—31no—
113C4×Q8C2×D1511yescharacteristic
114C4⋊C8—151no—
115C4⋊C8—151no—
116#102—301no—
117#102—302no—
118#102—301no—
119C4×C8—151no—
120C2×C3×C5—41no—
121C2×C3×C5—41no—
122C2×C3×C5—41no—
123D15—162no—
124C22×D3—401no—
125C2×C3×C4—41no—
126C2×C3×C4—81no—
127D12—202no—
128D12—401no—
129SL(2,𝔽3)#68811yescharacteristic, complemented
130C3⋊C8—401no—
131C22×C5GL(2,ℤ/4)11yescharacteristic
132C2×D5—241no—
133C2×D5—241no—
134C2×D5—241no—
135C4×C5—31no—
136C4×C5—31no—
137C4×C5—41no—
138C4×C5—61no—
139C4×C5—121no—
140C4×C5#98511yescharacteristic
141C4×C5#98611yescharacteristic
142C2×Q8—31no—
143C2×Q8#122011yescharacteristic
144C2×D4—301no—
145C2×D4—301no—
146C2×D4—301no—
147SD16—601no—
148SD16—602no—
149SD16—601no—
150C2×C8—301no—
151C2×C8—301no—
152C2×C8—301no—
153C4⋊C4—31no—
154C4⋊C4—32no—
155C4⋊C4—61no—
156C4⋊C4—61no—
157C4⋊C4C5⋊S411yescharacteristic
158C42—31no—
159C42—61no—
160C3×C5—41noHall
161C22×C3—41no—
162C2×D3—402no—
163C2×D3—801no—
164C2×D3—801no—
165C3×C4—41no—
166C3×C4—41no—
167C3×C4—161no—
168C2×C5#541711yescharacteristic
169C2×C5#542711yescharacteristic
170C2×C5#543111yescharacteristic
171D5—481no—
172C23—601no—
173Q8—31no—
174Q8—61no—
175Q8—61no—
176Q8#671711yescharacteristic
177D4—601no—
178D4—601no—
179D4—602no—
180D4—601no—
181D4—601no—
182C2×C4—21no—
183C2×C4—31no—
184C2×C4—31no—
185C2×C4—61no—
186C2×C4C2×C5⋊S411yescharacteristic
187C8—601no—
188C8—601no—
189C8—601no—
190C2×C3—41no—
191C2×C3—41no—
192C2×C3—41no—
193D3—802no—
194C5#7309611yescharacteristic, complemented, Sylow
195C22—1201no—
196C22—1201no—
197C22—1201no—
198C22#9195811yescharacteristic
199C4—31no—
200C4—31no—
201C4—41no—
202C4—61no—
203C4—121no—
204C4#9195511yescharacteristic
205C4#9195611yescharacteristic
206C3—41noSylow
207C2—2401no—
208C2#1175213611yescharacteristic
209C2#1175214611yescharacteristic
210C2#1175215011yescharacteristic
subgroupG/H
centerC22#91958
commutator, cosocleC4×C5×SL(2,𝔽3)C22
FrattiniC2×C4C2×C5⋊S4
FittingC5×#557D3
socleC22×C5GL(2,ℤ/4)
prime radicalC4⋊(C5⋊GL(2,𝔽3))C2
hypercenterC4⋊C4C5⋊S4
hypocenterC5×SL(2,𝔽3)SD16
LCSG ▷ C4×C5×SL(2,𝔽3) ▷ C2×C5×SL(2,𝔽3) ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C5×#557⋊C3C2semidirect
3. SL(2,𝔽3)⋊#168C2semidirect
10. C5×#557D3semidirect
22. #557⋊C3D5semidirect
45. C5×SL(2,𝔽3)SD16semidirect
69. #557D15semidirect
129. SL(2,𝔽3)#688semidirect
194. C5#73096semidirect

Maximal subgroups

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

subgroupindex
C5×#557⋊C32
C4⋊(C5⋊GL(2,𝔽3))2
SL(2,𝔽3)⋊#1682
#106027283
#910934
#730965