Finite Group Databasebeta

← #14511707

Subgroups — #14511707

Order 1920.

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

normal subgroups only

This group has 262 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
1SL(2,𝔽3)×Dic10C211yesmaximal, characteristic, complemented
2C4⋊(C5⋊GL(2,𝔽3))C211yesmaximal, characteristic
3C5×SL(2,𝔽3)⋊C8C211yesmaximal, characteristic, complemented
4#10599057—31nomaximal, Hall
5C2×C5⋊GL(2,𝔽3)—21no—
6C4×C5×SL(2,𝔽3)C2211yescharacteristic
7C2×#6658—41nomaximal
8Dic5×SL(2,𝔽3)—21no—
9#73100—51nomaximal, Hall
10Q8×Dic10D311yescharacteristic, complemented
11#65316—31no—
12#65137—31no—
13#64982—31no—
14C5×Q8⋊C8—31no—
15#64701—31no—
16#64676—31no—
17C2×D60—41no—
18C2×C3×Dic10—41no—
19C2×C5×SL(2,𝔽3)D411yescharacteristic
20C5⋊GL(2,𝔽3)—41no—
21C2×C5×C3⋊C8—41no—
22#6658—44no—
23Q8×SL(2,𝔽3)—51no—
24C4⋊GL(2,𝔽3)—51no—
25SL(2,𝔽3)⋊C8D511yescharacteristic, complemented
26C4×C5×Q8C2×D311yescharacteristic
27Q8×Dic5—21no—
28C2×#688—61no—
29C2×#677—62no—
30C2×#677—61no—
31#4034—61no—
32#4020—31no—
33#4015—31no—
34C4×Dic10—31no—
35C5×C4⋊C8—31no—
36C4×C5×C8—31no—
37#3953—61no—
38#3941—61no—
39C4⋊#168—31no—
40#1623—151noSylow
41C22×D15—81no—
42C2×C3×C4×C5—41no—
43D60—42no—
44D60—81no—
45C2×C3×Dic5—81no—
46C3×Dic10—42no—
47C3×Dic10—81no—
48C5×SL(2,𝔽3)SD1611yescharacteristic, complemented
49C5×C3⋊C8—42no—
50C2×GL(2,𝔽3)—101no—
51C2×#215—201no—
52C4×SL(2,𝔽3)—101no—
53C4×SL(2,𝔽3)C2×D511yescharacteristic
54C2×C5×Q8#6711yescharacteristic
55C2×D20—61no—
56C2×D20—61no—
57C2×D20—61no—
58C2×Dic10—32no—
59C2×Dic10S411yescharacteristic
60C2×C5×C8—61no—
61C2×C5×C8—61no—
62C5×C4⋊C4—31no—
63C42×C5—31no—
64#688—121no—
65#688—121no—
66C4⋊Dic5—31no—
67C4⋊Dic5—61no—
68#683—61no—
69C4×Dic5—61no—
70C2×#168—61no—
71#677—124no—
72#677—122no—
73Q82—51no—
74#492—151no—
75#460—151no—
76#460—151no—
77#333—151no—
78Q8⋊C8—31no—
79Q8⋊C8—151no—
80C22×C3×C5—41no—
81C2×D15—82no—
82C2×D15—161no—
83C2×D15—161no—
84C3×C4×C5—41no—
85C3×C4×C5—41no—
86C3×Dic5—82no—
87C2×C3×Q8—201no—
88C2×D12—201no—
89C2×SL(2,𝔽3)D2011yescharacteristic
90GL(2,𝔽3)—201no—
91#215—204no—
92C2×C3⋊C8—41no—
93C22×D5—121no—
94C5×Q8—31no—
95C5×Q8#21511yescharacteristic, complemented
96C2×C4×C5—31no—
97C2×C4×C5—31no—
98C2×C4×C5C2×S411yescharacteristic
99C2×Dic5—21no—
100C2×Dic5—61no—
101D20—121no—
102D20—121no—
103D20—122no—
104D20—121no—
105D20—121no—
106Dic10—21no—
107Dic10—122no—
108Dic10GL(2,𝔽3)12yescomplemented
109C5×C8—62no—
110C5×C8—121no—
111#168—121no—
112C2×SD16—301no—
113C2×SD16—302no—
114C2×SD16—301no—
115C4⋊Q8—151no—
116C4⋊Q8—301no—
117#127—151no—
118C4×Q8—101no—
119C4×Q8—151no—
120C4×Q8D3×D511yescharacteristic
121C4⋊C8—31no—
122C4⋊C8—151no—
123#102—301no—
124#102—301no—
125C4×C8—31no—
126C2×C3×C5—41no—
127C2×C3×C5—41no—
128C2×C3×C5—41no—
129D15—162no—
130C22×D3—401no—
131C3×Q8—202no—
132C3×Q8—401no—
133C2×C3×C4—41no—
134C2×C3×C4—401no—
135D12—202no—
136D12—401no—
137SL(2,𝔽3)#67711yescharacteristic, complemented
138C3⋊C8—42no—
139C22×C5GL(2,ℤ/4)11yescharacteristic
140C2×D5—241no—
141C2×D5—241no—
142C2×D5—241no—
143C4×C5—31no—
144C4×C5—31no—
145C4×C5—61no—
146C4×C5C2×GL(2,𝔽3)11yescharacteristic
147C4×C5#98311yescharacteristic
148Dic5—22no—
149Dic5—121no—
150C2×Q8—51no—
151C2×Q8—152no—
152C2×Q8#120211yescharacteristic
153C2×D4—301no—
154C2×D4—301no—
155C2×D4—301no—
156SD16—604no—
157SD16—602no—
158SD16—601no—
159SD16—601no—
160C2×C8—61no—
161C2×C8—61no—
162C2×C8—301no—
163C4⋊C4—31no—
164C4⋊C4—151no—
165C4⋊C4—301no—
166C4⋊C4—301no—
167C42—31no—
168C42—301no—
169C3×C5—41noHall
170C22×C3—41no—
171C2×D3—402no—
172C2×D3—801no—
173C2×D3—801no—
174C3×C4—41no—
175C3×C4—41no—
176C3×C4—402no—
177C2×C5#541711yescharacteristic
178C2×C5#542111yescharacteristic
179C2×C5#542911yescharacteristic
180D5—481no—
181C23—601no—
182Q8—31no—
183Q8—52no—
184Q8—101no—
185Q8—602no—
186Q8#665811yescharacteristic, complemented
187D4—601no—
188D4—601no—
189D4—602no—
190D4—601no—
191D4—601no—
192C2×C4—31no—
193C2×C4—31no—
194C2×C4—101no—
195C2×C4—301no—
196C2×C4D5×S411yescharacteristic
197C8—62no—
198C8—121no—
199C8—601no—
200C2×C3—41no—
201C2×C3—41no—
202C2×C3—41no—
203D3—802no—
204C5#7310011yescharacteristic, complemented, Sylow
205C22—1201no—
206C22—1201no—
207C22—1201no—
208C22#9190511yescharacteristic
209C4—31no—
210C4—31no—
211C4—61no—
212C4—102no—
213C4—601no—
214C4#9189711yescharacteristic
215C4D5×GL(2,𝔽3)11yescharacteristic
216C3—41noSylow
217C2—2401no—
218C2#1175158411yescharacteristic
219C2#1175161411yescharacteristic
220C2#1175163011yescharacteristic
subgroupG/H
centerC22#91905
commutator, cosocleC4×C5×SL(2,𝔽3)C22
Frattini, hypercenterC2×C4D5×S4
FittingC4×C5×Q8C2×D3
socleC22×C5GL(2,ℤ/4)
prime radicalC4⋊(C5⋊GL(2,𝔽3))C2
hypocenterC5×SL(2,𝔽3)SD16
LCSG ▷ C4×C5×SL(2,𝔽3) ▷ C2×C5×SL(2,𝔽3) ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. SL(2,𝔽3)×Dic10C2semidirect
3. C5×SL(2,𝔽3)⋊C8C2semidirect
10. Q8×Dic10D3semidirect
25. SL(2,𝔽3)⋊C8D5semidirect
48. C5×SL(2,𝔽3)SD16semidirect
95. C5×Q8#215semidirect
108. Dic10GL(2,𝔽3)semidirect
137. SL(2,𝔽3)#677semidirect
186. Q8#6658semidirect
204. C5#73100semidirect

Maximal subgroups

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

subgroupindex
SL(2,𝔽3)×Dic102
C4⋊(C5⋊GL(2,𝔽3))2
C5×SL(2,𝔽3)⋊C82
#105990573
C2×#66584
#731005