Finite Group Databasebeta

← C23×D5×SL(2,𝔽3) #14514651

Subgroups — C23×D5×SL(2,𝔽3) #14514651

Order 1920.

3558 conjugacy classes of subgroups; only the 607 normal subgroups are shown.

normal subgroups only

This group has 3558 conjugacy classes of subgroups — too many to draw. Only its 607 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
1C23×C5×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C22×D5×SL(2,𝔽3)C2114yesmaximal, direct product factor
3C23×Q8×D5C311yesmaximal, characteristic, complemented, Hall
4C24×C3×D5—41nomaximal
5C22×C5×SL(2,𝔽3)C2217yescomplemented
6C2×D5×SL(2,𝔽3)C22128yesdirect product factor
7C24×SL(2,𝔽3)—51nomaximal, Hall
8C23×C5×Q8C2×C311yescharacteristic, complemented
9C22×Q8×D5—314no—
10C22×Q8×D5C2×C3114yescomplemented
11C23×C4×D5—31no—
12C23×Dic10—31no—
13C24×C3×C5—41no—
14C23×C3×D5—416no—
15C23×C3×D5—414no—
16C2×C5×SL(2,𝔽3)C2317yescomplemented
17D5×SL(2,𝔽3)C2318yesdirect product factor
18C23×SL(2,𝔽3)—514no—
19C23×SL(2,𝔽3)D511yescharacteristic, direct product factor
20C24×D5A411yescharacteristic
21C22×C5×Q8—37no—
22C22×C5×Q8C22×C317yescomplemented
23C23×C4×C5—31no—
24C23×Dic5—31no—
25C2×Q8×D5—356no—
26C2×Q8×D5—384no—
27C2×Q8×D5C22×C3128yescomplemented
28C22×C4×D5—314no—
29C22×C4×D5—314no—
30C22×Dic10—314no—
31C22×Dic10—314no—
32C24×Q8—51noSylow
33C23×C3×C5—48no—
34C23×C3×C5—47no—
35C22×C3×D5—4112no—
36C22×C3×D5—428no—
37C5×SL(2,𝔽3)C2411yescharacteristic, complemented
38C25×C3—201no—
39C22×SL(2,𝔽3)—528no—
40C22×SL(2,𝔽3)C2×D517yesdirect product factor
41C24×C5C2×A411yescharacteristic
42C23×D5SL(2,𝔽3)116yesdirect product factor
43C23×D5C2×A4114yes—
44C2×C5×Q8—314no—
45C2×C5×Q8—321no—
46C2×C5×Q8C23×C317yescomplemented
47C22×C4×C5—37no—
48C22×C4×C5—37no—
49C22×Dic5—314no—
50Q8×D5—3112no—
51Q8×D5—356no—
52Q8×D5C23×C318yescomplemented
53C2×C4×D5—384no—
54C2×C4×D5—328no—
55C2×Dic10—384no—
56C2×Dic10—328no—
57C23×Q8—514no—
58C23×Q8—151no—
59C23×Q8—1514no—
60C23×Q8C3×D511yescharacteristic, complemented
61C24×C4—151no—
62C22×C3×C5—428no—
63C22×C3×C5—47no—
64C2×C3×D5—4112no—
65C2×C3×D5—48no—
66C24×C3—41no—
67C24×C3—2014no—
68C24×C3—2016no—
69C2×SL(2,𝔽3)—58no—
70C2×SL(2,𝔽3)C22×D517yesdirect product factor
71C23×C5C2×SL(2,𝔽3)18yescomplemented
72C23×C5C22×A417yes—
73C22×D5C2×SL(2,𝔽3)1112yesdirect product factor
74C22×D5C22×A4128yes—
75C5×Q8—37no—
76C5×Q8—314no—
77C5×Q8C24×C311yescharacteristic, complemented
78C2×C4×C5—321no—
79C2×C4×C5—37no—
80C2×Dic5—328no—
81C4×D5—356no—
82C4×D5—38no—
83Dic10—356no—
84Dic10—38no—
85C25—51no—
86C22×Q8—37no—
87C22×Q8—528no—
88C22×Q8—1514no—
89C22×Q8—1514no—
90C22×Q8—1556no—
91C22×Q8—1584no—
92C22×Q8C2×C3×D517yescomplemented
93C23×C4—31no—
94C23×C4—1514no—
95C23×C4—1514no—
96C23×C4—151no—
97C2×C3×C5—414no—
98C2×C3×C5—41no—
99C3×D5—416no—
100C23×C3—48no—
101C23×C3—47no—
102C23×C3—2028no—
103C23×C3—20112no—
104SL(2,𝔽3)C23×D511yescharacteristic, direct product factor
105C22×C5C22×SL(2,𝔽3)128yescomplemented
106C22×C5C23×A417yes—
107C2×D5C22×SL(2,𝔽3)1112yesdirect product factor
108C2×D5C23×A418yes—
109C4×C5—37no—
110C4×C5—31no—
111Dic5—38no—
112C24—516no—
113C24—514no—
114C24D5×A411yescharacteristic
115C2×Q8—314no—
116C2×Q8—321no—
117C2×Q8—58no—
118C2×Q8—1584no—
119C2×Q8—1528no—
120C2×Q8—15112no—
121C2×Q8—1556no—
122C2×Q8C22×C3×D517yescomplemented
123C22×C4—37no—
124C22×C4—37no—
125C22×C4—1528no—
126C22×C4—1584no—
127C22×C4—1514no—
128C3×C5—41noHall
129C22×C3—428no—
130C22×C3—47no—
131C22×C3—20112no—
132C22×C3—208no—
133C2×C5C23×SL(2,𝔽3)114yescomplemented
134C2×C5C24×A411yescharacteristic
135D5C23×SL(2,𝔽3)116yesdirect product factor
136C23—5112no—
137C23—528no—
138C23D5×SL(2,𝔽3)18yesdirect product factor
139C23C2×D5×A417yes—
140Q8—37no—
141Q8—314no—
142Q8—158no—
143Q8—1556no—
144Q8C23×C3×D511yescharacteristic, complemented
145C2×C4—37no—
146C2×C4—321no—
147C2×C4—158no—
148C2×C4—1556no—
149C2×C4—1528no—
150C2×C3—41no—
151C2×C3—414no—
152C2×C3—2016no—
153C5C24×SL(2,𝔽3)11yescharacteristic, complemented, Sylow
154C22—58no—
155C22—5112no—
156C22C2×D5×SL(2,𝔽3)128yesdirect product factor
157C22C22×D5×A417yes—
158C4—31no—
159C4—37no—
160C4—158no—
161C3—41noSylow
162C2—516no—
163C2C22×D5×SL(2,𝔽3)114yesdirect product factor
164C2C23×D5×A411yescharacteristic
subgroupG/H
center, hypercenterC24D5×A4
commutator, cosocle, hypocenterC5×Q8C24×C3
FrattiniC2C23×D5×A4
FittingC23×C5×Q8C2×C3
socleC24×C5C2×A4
LCSG ▷ C5×Q8
UCSC1 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C23×C5×SL(2,𝔽3)C2semidirect
2. C22×D5×SL(2,𝔽3)C2direct (N × H)
3. C23×Q8×D5C3semidirect
5. C22×C5×SL(2,𝔽3)C22semidirect
6. C2×D5×SL(2,𝔽3)C22direct (N × H)
8. C23×C5×Q8C2×C3semidirect
10. C22×Q8×D5C2×C3semidirect
16. C2×C5×SL(2,𝔽3)C23semidirect
17. D5×SL(2,𝔽3)C23direct (N × H)
19. C23×SL(2,𝔽3)D5direct (N × H)
22. C22×C5×Q8C22×C3semidirect
27. C2×Q8×D5C22×C3semidirect
37. C5×SL(2,𝔽3)C24semidirect
40. C22×SL(2,𝔽3)C2×D5direct (N × H)
42. C23×D5SL(2,𝔽3)direct (N × H)
46. C2×C5×Q8C23×C3semidirect
52. Q8×D5C23×C3semidirect
60. C23×Q8C3×D5semidirect
70. C2×SL(2,𝔽3)C22×D5direct (N × H)
71. C23×C5C2×SL(2,𝔽3)semidirect
73. C22×D5C2×SL(2,𝔽3)direct (N × H)
77. C5×Q8C24×C3semidirect
92. C22×Q8C2×C3×D5semidirect
104. SL(2,𝔽3)C23×D5direct (N × H)
105. C22×C5C22×SL(2,𝔽3)semidirect
107. C2×D5C22×SL(2,𝔽3)direct (N × H)
122. C2×Q8C22×C3×D5semidirect
133. C2×C5C23×SL(2,𝔽3)semidirect
135. D5C23×SL(2,𝔽3)direct (N × H)
138. C23D5×SL(2,𝔽3)direct (N × H)
144. Q8C23×C3×D5semidirect
153. C5C24×SL(2,𝔽3)semidirect
156. C22C2×D5×SL(2,𝔽3)direct (N × H)
163. C2C22×D5×SL(2,𝔽3)direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C5×SL(2,𝔽3)2
C22×D5×SL(2,𝔽3)2
C23×Q8×D53
C24×C3×D54
C24×SL(2,𝔽3)5