Finite Group Databasebeta

← #14512014

Subgroups — #14512014

Order 1920.

262 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 41 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×Q8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C4⋊(C5⋊GL(2,𝔽3))C211yesmaximal, characteristic
3SL(2,𝔽3)⋊#168C211yesmaximal, characteristic, complemented
4#10602732—31nomaximal, Hall
5C2×C5⋊GL(2,𝔽3)—21no—
6C2×#6717—41nomaximal
7C4×C5×SL(2,𝔽3)—21no—
8C4×C5×SL(2,𝔽3)C2211yescharacteristic
9#73100—51nomaximal, Hall
10C5×Q82D311yescharacteristic, complemented
11#65374—31no—
12#65316—31no—
13#65316—31no—
14#64766—31no—
15Q8⋊#168—31no—
16Q8⋊#168—31no—
17C2×C3×C5×Q8—41no—
18C2×D60—41no—
19C2×C5×SL(2,𝔽3)D411yescharacteristic
20C5⋊GL(2,𝔽3)—41no—
21#6717—44no—
22C2×C3⋊#168—41no—
23Q8×SL(2,𝔽3)D511yescharacteristic, complemented
24C4⋊GL(2,𝔽3)—51no—
25SL(2,𝔽3)⋊C8—51no—
26C5×C4⋊Q8—31no—
27C5×C4⋊Q8—61no—
28C4×C5×Q8—21no—
29C4×C5×Q8—31no—
30C4×C5×Q8C2×D311yescharacteristic
31C2×#688—62no—
32C2×#688—61no—
33C2×#688—61no—
34#4020—31no—
35#3941—61no—
36#3941—61no—
37C4⋊#168—31no—
38C4⋊#168—31no—
39C4×#168—31no—
40#1623—151noSylow
41C22×D15—81no—
42C3×C5×Q8—42no—
43C3×C5×Q8—81no—
44C2×C3×C4×C5—41no—
45C2×C3×C4×C5—81no—
46D60—42no—
47D60—81no—
48C5×SL(2,𝔽3)SD1611yescharacteristic, complemented
49C3⋊#168—42no—
50C2×GL(2,𝔽3)—101no—
51C2×#215—201no—
52C4×SL(2,𝔽3)—21no—
53C4×SL(2,𝔽3)C2×D511yescharacteristic
54C2×C5×Q8—32no—
55C2×C5×Q8#6711yescharacteristic
56C2×C5×Q8S411yescharacteristic
57C2×D20—61no—
58C2×D20—61no—
59C2×D20—61no—
60C5×C4⋊C4—31no—
61C5×C4⋊C4—31no—
62C5×C4⋊C4—61no—
63C5×C4⋊C4—61no—
64C42×C5—31no—
65C42×C5—61no—
66#688—124no—
67#688—121no—
68#688—121no—
69#688—122no—
70C2×#168—61no—
71C2×#168—61no—
72C2×#168—61no—
73Q82D1511yescharacteristic, complemented
74#492—151no—
75#460—151no—
76#460—151no—
77#333—151no—
78Q8⋊C8—151no—
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×C4×C5—82no—
87C2×C3×Q8—41no—
88C2×D12—201no—
89C2×SL(2,𝔽3)C22⋊D511yescharacteristic
90GL(2,𝔽3)—201no—
91#215—204no—
92C2×C3⋊C8—201no—
93C22×D5—121no—
94C5×Q8—21no—
95C5×Q8—31no—
96C5×Q8—122no—
97C5×Q8#21511yescharacteristic, complemented
98C5×Q8GL(2,𝔽3)12yescomplemented
99C2×C4×C5—21no—
100C2×C4×C5—31no—
101C2×C4×C5—31no—
102C2×C4×C5—61no—
103C2×C4×C5C2×S411yescharacteristic
104D20—121no—
105D20—121no—
106D20—122no—
107D20—121no—
108D20—121no—
109#168—62no—
110#168—121no—
111#168—121no—
112C2×SD16—302no—
113C2×SD16—301no—
114C2×SD16—301no—
115C4⋊Q8—31no—
116C4⋊Q8—61no—
117#127—151no—
118C4×Q8—21no—
119C4×Q8—31no—
120C4×Q8C2×D1511yescharacteristic
121C4⋊C8—151no—
122C4⋊C8—151no—
123#102—301no—
124#102—301no—
125C4×C8—151no—
126C2×C3×C5—41no—
127C2×C3×C5—41no—
128C2×C3×C5—41no—
129D15—162no—
130C22×D3—401no—
131C3×Q8—42no—
132C3×Q8—81no—
133C2×C3×C4—41no—
134C2×C3×C4—81no—
135D12—202no—
136D12—401no—
137SL(2,𝔽3)#68811yescharacteristic, complemented
138C3⋊C8—202no—
139C22×C5GL(2,ℤ/4)11yescharacteristic
140C2×D5—241no—
141C2×D5—241no—
142C2×D5—241no—
143C4×C5—22no—
144C4×C5—31no—
145C4×C5—31no—
146C4×C5—61no—
147C4×C5—121no—
148C4×C5C2×GL(2,𝔽3)11yescharacteristic
149C4×C5#98311yescharacteristic
150C2×Q8—32no—
151C2×Q8#122011yescharacteristic
152C2×Q8C5⋊S411yescharacteristic
153C2×D4—301no—
154C2×D4—301no—
155C2×D4—301no—
156SD16—604no—
157SD16—601no—
158SD16—601no—
159SD16—602no—
160C2×C8—301no—
161C2×C8—301no—
162C2×C8—301no—
163C4⋊C4—31no—
164C4⋊C4—31no—
165C4⋊C4—61no—
166C4⋊C4—61no—
167C42—31no—
168C42—61no—
169C3×C5—41noHall
170C22×C3—41no—
171C2×D3—402no—
172C2×D3—801no—
173C2×D3—801no—
174C3×C4—41no—
175C3×C4—41no—
176C3×C4—82no—
177C2×C5#541711yescharacteristic
178C2×C5#542111yescharacteristic
179C2×C5#542911yescharacteristic
180D5—481no—
181C23—601no—
182Q8—21no—
183Q8—31no—
184Q8—122no—
185Q8#671711yescharacteristic, complemented
186Q8C5⋊GL(2,𝔽3)12yescomplemented
187D4—601no—
188D4—601no—
189D4—602no—
190D4—601no—
191D4—601no—
192C2×C4—21no—
193C2×C4—31no—
194C2×C4—31no—
195C2×C4—61no—
196C2×C4C2×C5⋊S411yescharacteristic
197C8—302no—
198C8—601no—
199C8—601no—
200C2×C3—41no—
201C2×C3—41no—
202C2×C3—41no—
203D3—802no—
204C5#7310011yescharacteristic, complemented, Sylow
205C22—1201no—
206C22—1201no—
207C22—1201no—
208C22#9195811yescharacteristic
209C4—22no—
210C4—31no—
211C4—31no—
212C4—61no—
213C4—121no—
214C4C2×C5⋊GL(2,𝔽3)11yescharacteristic
215C4#9195311yescharacteristic
216C3—41noSylow
217C2—2401no—
218C2#1175213611yescharacteristic
219C2#1175214011yescharacteristic
220C2#1175214811yescharacteristic
subgroupG/H
centerC22#91958
commutator, cosocleC4×C5×SL(2,𝔽3)C22
FrattiniC2×C4C2×C5⋊S4
FittingC5×Q82D3
socleC22×C5GL(2,ℤ/4)
prime radicalC4⋊(C5⋊GL(2,𝔽3))C2
hypercenterC2×Q8C5⋊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 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C5×Q8×SL(2,𝔽3)C2semidirect
3. SL(2,𝔽3)⋊#168C2semidirect
10. C5×Q82D3semidirect
23. Q8×SL(2,𝔽3)D5semidirect
48. C5×SL(2,𝔽3)SD16semidirect
73. Q82D15semidirect
97. C5×Q8#215semidirect
98. C5×Q8GL(2,𝔽3)semidirect
137. SL(2,𝔽3)#688semidirect
185. Q8#6717semidirect
186. Q8C5⋊GL(2,𝔽3)semidirect
204. C5#73100semidirect

Maximal subgroups

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

subgroupindex
C5×Q8×SL(2,𝔽3)2
C4⋊(C5⋊GL(2,𝔽3))2
SL(2,𝔽3)⋊#1682
#106027323
C2×#67174
#731005