Finite Group Databasebeta

← #14511699

Subgroups — #14511699

Order 1920.

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

normal subgroups only

This group has 210 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
1#11752105C211yesmaximal, characteristic
2C5×#5386C211yesmaximal, characteristic
3SL(2,𝔽3)×#168C211yesmaximal, characteristic
4#10601910—31nomaximal, Hall
5C4×C5×SL(2,𝔽3)C2211yescharacteristic
6C2×#6663—41nomaximal
7#91184—21no—
8C5×#859—21no—
9#73015—51nomaximal, Hall
10C5×#475—31no—
11#65381—31no—
12Q8×#168D311yescharacteristic
13#65309—31no—
14#64767—31no—
15#64706—31no—
16#64699—31no—
17C2×Dic30—41no—
18C2×C5×Dic6—41no—
19C2×C5×SL(2,𝔽3)D411yescharacteristic
20C2×C3×#168—41no—
21#6663—44no—
22#5386—51no—
23#5386D511yescharacteristic
24C8×SL(2,𝔽3)—51no—
25C5×C4⋊Q8—31no—
26C4×C5×Q8C2×D311yescharacteristic
27C2×#689—61no—
28C2×#689—61no—
29#4015—31no—
30C5×#106—61no—
31C5×C4⋊C8—31no—
32C5×#103—61no—
33#3967—61no—
34#3940—61no—
35C4⋊#168—31no—
36C4⋊#168—31no—
37C4×#168—31no—
38#1670—151noSylow
39C2×C3×C4×C5—41no—
40C2×Dic15—81no—
41Dic30—42no—
42Dic30—81no—
43C2×C5×Dic3—81no—
44C5×Dic6—42no—
45C5×Dic6—81no—
46C5×SL(2,𝔽3)Q1611yescharacteristic, complemented
47C3×#168—42no—
48C2×Dic12—201no—
49C4×SL(2,𝔽3)C2×D511yescharacteristic
50#859—21no—
51#859—101no—
52C2×C5×Q8—61no—
53C2×C5×Q8D1211yescharacteristic
54C2×Dic10—61no—
55C2×C5×C8—61no—
56C5×C4⋊C4—31no—
57C5×C4⋊C4—61no—
58C5×C4⋊C4—61no—
59C42×C5—31no—
60#689—122no—
61#689—122no—
62C4⋊Dic5—61no—
63C4⋊Dic5—61no—
64C2×#168—31no—
65C2×#168—61no—
66C2×#168S411yescharacteristic
67#494—151no—
68#475—31no—
69#475—151no—
70C8×Q8—51no—
71#335—151no—
72#332—151no—
73#332—151no—
74C22×C3×C5—41no—
75C3×C4×C5—41no—
76C3×C4×C5—41no—
77Dic15—82no—
78C5×Dic3—82no—
79C2×Dic6—41no—
80C2×Dic6—201no—
81C2×SL(2,𝔽3)C22⋊D511yescharacteristic
82C2×C3×C8—201no—
83Dic12—204no—
84C5×Q8—31no—
85C5×Q8—121no—
86C5×Q8—121no—
87C5×Q8Dic1211yescharacteristic, complemented
88C2×C4×C5—31no—
89C2×C4×C5—31no—
90C2×C4×C5—121no—
91C2×C4×C5C2×S411yescharacteristic
92C2×Dic5—121no—
93Dic10—121no—
94Dic10—121no—
95C5×C8—121no—
96#168—61no—
97#168—121no—
98#168GL(2,𝔽3)12yes—
99C2×Q16—301no—
100C2×Q16—301no—
101C4⋊Q8—31no—
102C4⋊Q8—151no—
103C4×Q8D3×D511yescharacteristic
104#106—61no—
105#106—301no—
106C4⋊C8—31no—
107C4⋊C8—151no—
108C4⋊C8—151no—
109#103—61no—
110#103—301no—
111C4×C8—151no—
112C2×C3×C5—41no—
113C2×C3×C5—41no—
114C2×C3×C5—41no—
115C2×C3×C4—41no—
116C2×Dic3—81no—
117C2×Dic3—401no—
118Dic6—42no—
119Dic6—81no—
120Dic6—202no—
121Dic6—401no—
122SL(2,𝔽3)#68911yescharacteristic, complemented
123C3×C8—202no—
124C22×C5C4⋊S411yescharacteristic
125C4×C5—31no—
126C4×C5—31no—
127C4×C5—61no—
128C4×C5—241no—
129C4×C5C2×GL(2,𝔽3)11yescharacteristic
130C4×C5#98611yescharacteristic
131Dic5—241no—
132C2×Q8—61no—
133C2×Q8—301no—
134C2×Q8#120311yescharacteristic
135Q16—602no—
136Q16—602no—
137C2×C8—51no—
138C2×C8—61no—
139C2×C8—151no—
140C2×C8—301no—
141C4⋊C4—31no—
142C4⋊C4—61no—
143C4⋊C4—61no—
144C4⋊C4—301no—
145C4⋊C4—301no—
146C42—31no—
147C3×C5—41noHall
148C22×C3—41no—
149C3×C4—41no—
150C3×C4—41no—
151Dic3—82no—
152Dic3—402no—
153C2×C5C4⋊GL(2,𝔽3)11yescharacteristic
154C2×C5#539811yescharacteristic
155C2×C5#540311yescharacteristic
156Q8—31no—
157Q8—121no—
158Q8—121no—
159Q8—601no—
160Q8—601no—
161Q8#666311yescharacteristic, complemented
162C2×C4—31no—
163C2×C4—31no—
164C2×C4—121no—
165C2×C4—601no—
166C2×C4D5×S411yescharacteristic
167C8—52no—
168C8—121no—
169C8—301no—
170C8—601no—
171C2×C3—41no—
172C2×C3—41no—
173C2×C3—41no—
174C5#7301511yescharacteristic, complemented, Sylow
175C22#9190211yescharacteristic
176C4—31no—
177C4—31no—
178C4—61no—
179C4—241no—
180C4—1201no—
181C4#9189411yescharacteristic
182C4D5×GL(2,𝔽3)11yescharacteristic
183C3—41noSylow
184C2#1175157611yescharacteristic
185C2#1175161111yescharacteristic
186C2#1175162211yescharacteristic
subgroupG/H
centerC22#91902
commutator, cosocleC4×C5×SL(2,𝔽3)C22
Frattini, hypercenterC2×C4D5×S4
FittingC4×C5×Q8C2×D3
socleC22×C5C4⋊S4
prime radicalC2×C5×SL(2,𝔽3)D4
hypocenterC5×SL(2,𝔽3)Q16
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
46. C5×SL(2,𝔽3)Q16semidirect
87. C5×Q8Dic12semidirect
122. SL(2,𝔽3)#689semidirect
161. Q8#6663semidirect
174. C5#73015semidirect

Maximal subgroups

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

subgroupindex
#117521052
C5×#53862
SL(2,𝔽3)×#1682
#106019103
C2×#66634
#730155