Finite Group Databasebeta

← #14511697

Subgroups — #14511697

Order 1920.

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

normal subgroups only

This group has 236 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
1C4⋊(C5⋊GL(2,𝔽3))C211yesmaximal, characteristic
2C5×#5386C211yesmaximal, characteristic, complemented
3SL(2,𝔽3)×#168C211yesmaximal, characteristic, complemented
4#10601909—31nomaximal, Hall
5C2×C5⋊GL(2,𝔽3)—21no—
6C4×C5×SL(2,𝔽3)C2211yescharacteristic
7C2×#6660—41nomaximal
8C5×#859—21no—
9#73021—51nomaximal, Hall
10C5×#475—31no—
11#65374—31no—
12#65316—31no—
13Q8×#168D311yescharacteristic, complemented
14#64767—31no—
15#64705—31no—
16#64700—31no—
17C2×D60—41no—
18C2×C5×Dic6—41no—
19C2×C5×SL(2,𝔽3)D411yescharacteristic
20C5⋊GL(2,𝔽3)—41no—
21C2×C3×#168—41no—
22#6660—44no—
23C4⋊GL(2,𝔽3)—51no—
24#5386D511yescharacteristic, complemented
25C8×SL(2,𝔽3)—51no—
26C5×C4⋊Q8—31no—
27C4×C5×Q8C2×D311yescharacteristic
28C2×#688—61no—
29C2×#688—61no—
30C2×#688—61no—
31#4020—31no—
32C5×#106—61no—
33C5×C4⋊C8—31no—
34C5×#103—61no—
35#3941—61no—
36C4⋊#168—31no—
37C4⋊#168—31no—
38C4×#168—31no—
39#1666—151noSylow
40C22×D15—81no—
41C2×C3×C4×C5—41no—
42D60—42no—
43D60—81no—
44C2×C5×Dic3—81no—
45C5×Dic6—42no—
46C5×Dic6—81no—
47C5×SL(2,𝔽3)SD1611yescharacteristic, complemented
48C3×#168—42no—
49C2×GL(2,𝔽3)—101no—
50C2×#204—201no—
51C4×SL(2,𝔽3)C2×D511yescharacteristic
52#859—21no—
53C2×C5×Q8—61no—
54C2×C5×Q8D1211yescharacteristic
55C2×D20—61no—
56C2×D20—61no—
57C2×D20—61no—
58C2×C5×C8—61no—
59C5×C4⋊C4—31no—
60C5×C4⋊C4—61no—
61C5×C4⋊C4—61no—
62C42×C5—31no—
63#688—122no—
64#688—121no—
65#688—121no—
66#688—122no—
67C2×#168—31no—
68C2×#168—61no—
69C2×#168S411yescharacteristic
70#492—151no—
71#475—31no—
72#460—151no—
73C8×Q8—51no—
74#334—151no—
75#332—151no—
76#331—151no—
77C22×C3×C5—41no—
78C2×D15—82no—
79C2×D15—161no—
80C2×D15—161no—
81C3×C4×C5—41no—
82C3×C4×C5—41no—
83C5×Dic3—82no—
84C2×D12—201no—
85C2×Dic6—41no—
86C2×SL(2,𝔽3)C22⋊D511yescharacteristic
87GL(2,𝔽3)—201no—
88C2×C3×C8—201no—
89#204—204no—
90C22×D5—121no—
91C5×Q8—31no—
92C5×Q8—121no—
93C5×Q8—121no—
94C5×Q8#20411yescharacteristic, complemented
95C2×C4×C5—31no—
96C2×C4×C5—31no—
97C2×C4×C5—121no—
98C2×C4×C5C2×S411yescharacteristic
99D20—121no—
100D20—121no—
101D20—121no—
102D20—121no—
103D20—241no—
104C5×C8—121no—
105#168—61no—
106#168—121no—
107#168GL(2,𝔽3)12yescomplemented
108C2×SD16—301no—
109C2×SD16—301no—
110C2×SD16—301no—
111C4⋊Q8—31no—
112#127—151no—
113C4×Q8D3×D511yescharacteristic
114#106—61no—
115C4⋊C8—31no—
116C4⋊C8—151no—
117C4⋊C8—151no—
118#103—61no—
119#102—301no—
120C4×C8—151no—
121C2×C3×C5—41no—
122C2×C3×C5—41no—
123C2×C3×C5—41no—
124D15—162no—
125C22×D3—401no—
126C2×C3×C4—41no—
127C2×Dic3—81no—
128D12—202no—
129D12—401no—
130Dic6—42no—
131Dic6—81no—
132SL(2,𝔽3)#68811yescharacteristic, complemented
133C3×C8—202no—
134C22×C5C4⋊S411yescharacteristic
135C2×D5—241no—
136C2×D5—241no—
137C2×D5—241no—
138C4×C5—31no—
139C4×C5—31no—
140C4×C5—61no—
141C4×C5—241no—
142C4×C5C2×GL(2,𝔽3)11yescharacteristic
143C4×C5#98611yescharacteristic
144C2×Q8—61no—
145C2×Q8#120311yescharacteristic
146C2×D4—301no—
147C2×D4—301no—
148C2×D4—301no—
149SD16—602no—
150SD16—602no—
151SD16—601no—
152SD16—601no—
153C2×C8—51no—
154C2×C8—61no—
155C2×C8—151no—
156C2×C8—301no—
157C4⋊C4—31no—
158C4⋊C4—61no—
159C4⋊C4—61no—
160C42—31no—
161C3×C5—41noHall
162C22×C3—41no—
163C2×D3—402no—
164C2×D3—801no—
165C2×D3—801no—
166C3×C4—41no—
167C3×C4—41no—
168Dic3—82no—
169C2×C5C4⋊GL(2,𝔽3)11yescharacteristic
170C2×C5#540111yescharacteristic
171C2×C5#540611yescharacteristic
172D5—481no—
173C23—601no—
174Q8—31no—
175Q8—121no—
176Q8—121no—
177Q8#666011yescharacteristic, complemented
178D4—601no—
179D4—601no—
180D4—601no—
181D4—601no—
182D4—1201no—
183C2×C4—31no—
184C2×C4—31no—
185C2×C4—121no—
186C2×C4D5×S411yescharacteristic
187C8—52no—
188C8—121no—
189C8—301no—
190C8—601no—
191C2×C3—41no—
192C2×C3—41no—
193C2×C3—41no—
194D3—802no—
195C5#7302111yescharacteristic, complemented, Sylow
196C22—1201no—
197C22—1201no—
198C22—1201no—
199C22#9190211yescharacteristic
200C4—31no—
201C4—31no—
202C4—61no—
203C4—241no—
204C4#9189411yescharacteristic
205C4D5×GL(2,𝔽3)11yescharacteristic
206C3—41noSylow
207C2—2401no—
208C2#1175157411yescharacteristic
209C2#1175161111yescharacteristic
210C2#1175162011yescharacteristic
subgroupG/H
centerC22#91902
commutator, cosocleC4×C5×SL(2,𝔽3)C22
Frattini, hypercenterC2×C4D5×S4
FittingC4×C5×Q8C2×D3
socleC22×C5C4⋊S4
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
2. C5×#5386C2semidirect
3. SL(2,𝔽3)×#168C2semidirect
13. Q8×#168D3semidirect
24. #5386D5semidirect
47. C5×SL(2,𝔽3)SD16semidirect
94. C5×Q8#204semidirect
107. #168GL(2,𝔽3)semidirect
132. SL(2,𝔽3)#688semidirect
177. Q8#6660semidirect
195. C5#73021semidirect

Maximal subgroups

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

subgroupindex
C4⋊(C5⋊GL(2,𝔽3))2
C5×#53862
SL(2,𝔽3)×#1682
#106019093
C2×#66604
#730215