Finite Group Databasebeta

← C22×C4×C5×SL(2,𝔽3) #14511313

Subgroups — C22×C4×C5×SL(2,𝔽3) #14511313

Order 1920.

1038 conjugacy classes of subgroups; only the 344 normal subgroups are shown.

normal subgroups only

This group has 1038 conjugacy classes of subgroups — too many to draw. Only its 344 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
2C2×C4×C5×SL(2,𝔽3)C216yesmaximal, direct product factor
3C22×C4×C5×Q8C311yesmaximal, characteristic, complemented, Hall
4C23×C3×C4×C5—41nomaximal
5C22×C5×SL(2,𝔽3)C414yesdirect product factor
6C22×C5×SL(2,𝔽3)C2213yes—
7C4×C5×SL(2,𝔽3)C2214yesdirect product factor
8C22×C4×SL(2,𝔽3)C511yesmaximal, characteristic, direct product factor, Hall
9C23×C5×Q8C2×C311yescharacteristic
10C2×C4×C5×Q8—36no—
11C2×C4×C5×Q8C2×C316yescomplemented
12C22×C5×C4⋊C4—31no—
13C22×C42×C5—31no—
14C24×C3×C5—41no—
15C22×C3×C4×C5—48no—
16C22×C3×C4×C5—46no—
17C2×C5×SL(2,𝔽3)C2×C416yesdirect product factor
18C2×C5×SL(2,𝔽3)C2311yescharacteristic
19C23×SL(2,𝔽3)C2×C511yescharacteristic
20C2×C4×SL(2,𝔽3)C2×C516yesdirect product factor
21C22×C5×Q8—33no—
22C22×C5×Q8—34no—
23C22×C5×Q8C3×C414yescomplemented
24C22×C5×Q8C22×C313yes—
25C23×C4×C5—31no—
26C23×C4×C5—31no—
27C23×C4×C5A411yescharacteristic
28C4×C5×Q8—38no—
29C4×C5×Q8—312no—
30C4×C5×Q8C22×C314yescomplemented
31C2×C5×C4⋊C4—36no—
32C2×C5×C4⋊C4—36no—
33C2×C42×C5—36no—
34C2×C42×C5—36no—
35C22×C4×Q8C3×C511yescharacteristic, complemented, Sylow
36C23×C3×C5—44no—
37C23×C3×C5—44no—
38C23×C3×C5—44no—
39C23×C3×C5—43no—
40C2×C3×C4×C5—424no—
41C2×C3×C4×C5—44no—
42C5×SL(2,𝔽3)C22×C411yescharacteristic, direct product factor
43C23×C3×C4—41no—
44C22×SL(2,𝔽3)C4×C514yesdirect product factor
45C22×SL(2,𝔽3)C22×C513yes—
46C4×SL(2,𝔽3)C22×C514yesdirect product factor
47C24×C5C2×A411yescharacteristic
48C2×C5×Q8—32no—
49C2×C5×Q8—312no—
50C2×C5×Q8—33no—
51C2×C5×Q8—312no—
52C2×C5×Q8—36no—
53C2×C5×Q8C2×C3×C416yescomplemented
54C2×C5×Q8C23×C311yescharacteristic
55C22×C4×C5—34no—
56C22×C4×C5—34no—
57C22×C4×C5—33no—
58C22×C4×C5—33no—
59C22×C4×C5—36no—
60C22×C4×C5—64no—
61C22×C4×C5SL(2,𝔽3)18yesdirect product factor
62C22×C4×C5C2×A416yes—
63C5×C4⋊C4—34no—
64C5×C4⋊C4—312no—
65C42×C5—312no—
66C42×C5—34no—
67C23×Q8C2×C3×C511yescharacteristic
68C2×C4×Q8—36no—
69C2×C4×Q8C2×C3×C516yescomplemented
70C22×C4⋊C4—31no—
71C22×C42—31no—
72C22×C3×C5—416no—
73C22×C3×C5—46no—
74C22×C3×C5—46no—
75C22×C3×C5—46no—
76C22×C3×C5—41no—
77C3×C4×C5—48no—
78C24×C3—41no—
79C22×C3×C4—48no—
80C22×C3×C4—46no—
81C2×SL(2,𝔽3)C2×C4×C516yesdirect product factor
82C2×SL(2,𝔽3)C23×C511yescharacteristic
83C23×C5C4×A414yes—
84C23×C5C2×SL(2,𝔽3)14yes—
85C23×C5#41⋊C314yes—
86C23×C5C22×A413yes—
87C5×Q8—36no—
88C5×Q8—31no—
89C5×Q8—38no—
90C5×Q8—36no—
91C5×Q8C22×C3×C411yescharacteristic, complemented
92C2×C4×C5—36no—
93C2×C4×C5—312no—
94C2×C4×C5—36no—
95C2×C4×C5—33no—
96C2×C4×C5—31no—
97C2×C4×C5—34no—
98C2×C4×C5—612no—
99C2×C4×C5C2×SL(2,𝔽3)124yesdirect product factor
100C2×C4×C5C22×A414yes—
101C22×Q8—33no—
102C22×Q8—34no—
103C22×Q8C3×C4×C514yescomplemented
104C22×Q8C22×C3×C513yes—
105C23×C4—31no—
106C23×C4—31no—
107C23×C4C5×A411yescharacteristic
108C4×Q8—38no—
109C4×Q8—312no—
110C4×Q8C22×C3×C514yescomplemented
111C2×C4⋊C4—36no—
112C2×C4⋊C4—36no—
113C2×C42—36no—
114C2×C42—36no—
115C2×C3×C5—41no—
116C2×C3×C5—41no—
117C2×C3×C5—41no—
118C2×C3×C5—412no—
119C23×C3—44no—
120C23×C3—44no—
121C23×C3—44no—
122C23×C3—43no—
123C2×C3×C4—424no—
124C2×C3×C4—44no—
125SL(2,𝔽3)C22×C4×C511yescharacteristic, direct product factor
126C22×C5C4×SL(2,𝔽3)116yesdirect product factor
127C22×C5C2×C4×A416yes—
128C22×C5C22×SL(2,𝔽3)16yes—
129C22×C5C2×#41⋊C316yes—
130C22×C5C23×A411yescharacteristic
131C4×C5—31no—
132C4×C5—36no—
133C4×C5—31no—
134C4×C5—64no—
135C4×C5C22×SL(2,𝔽3)18yesdirect product factor
136C24C2×C5×A411yescharacteristic
137C2×Q8—32no—
138C2×Q8—312no—
139C2×Q8—33no—
140C2×Q8—312no—
141C2×Q8—36no—
142C2×Q8C2×C3×C4×C516yescomplemented
143C2×Q8C23×C3×C511yescharacteristic
144C22×C4—34no—
145C22×C4—34no—
146C22×C4—33no—
147C22×C4—33no—
148C22×C4—36no—
149C22×C4—64no—
150C22×C4C5×SL(2,𝔽3)18yesdirect product factor
151C22×C4C2×C5×A416yes—
152C4⋊C4—34no—
153C4⋊C4—312no—
154C42—34no—
155C42—312no—
156C3×C5—41noHall
157C22×C3—46no—
158C22×C3—46no—
159C22×C3—46no—
160C22×C3—416no—
161C22×C3—41no—
162C3×C4—48no—
163C2×C5C2×C4×SL(2,𝔽3)112yesdirect product factor
164C2×C5C22×C4×A411yescharacteristic
165C2×C5C23×SL(2,𝔽3)11yescharacteristic
166C2×C5C22×#41⋊C311yescharacteristic
167C23C4×C5×A414yes—
168C23C2×C5×SL(2,𝔽3)14yes—
169C23C5×#41⋊C314yes—
170C23C22×C5×A413yes—
171Q8—36no—
172Q8—31no—
173Q8—38no—
174Q8—36no—
175Q8C22×C3×C4×C511yescharacteristic, complemented
176C2×C4—36no—
177C2×C4—312no—
178C2×C4—36no—
179C2×C4—31no—
180C2×C4—33no—
181C2×C4—34no—
182C2×C4—612no—
183C2×C4C2×C5×SL(2,𝔽3)124yesdirect product factor
184C2×C4C22×C5×A414yes—
185C2×C3—41no—
186C2×C3—41no—
187C2×C3—41no—
188C2×C3—412no—
189C5C22×C4×SL(2,𝔽3)11yescharacteristic, direct product factor, Sylow
190C22C4×C5×SL(2,𝔽3)116yesdirect product factor
191C22C2×C4×C5×A416yes—
192C22C22×C5×SL(2,𝔽3)16yes—
193C22C2×C5×#41⋊C316yes—
194C22C23×C5×A411yescharacteristic
195C4—31no—
196C4—36no—
197C4—31no—
198C4—64no—
199C4C22×C5×SL(2,𝔽3)18yesdirect product factor
200C3—41noSylow
201C2C2×C4×C5×SL(2,𝔽3)112yesdirect product factor
202C2C22×C4×C5×A411yescharacteristic
203C2C23×C5×SL(2,𝔽3)11yescharacteristic
204C2C22×C5×#41⋊C311yescharacteristic
subgroupG/H
center, hypercenterC23×C4×C5A4
commutator, hypocenterQ8C22×C3×C4×C5
FrattiniC22C23×C5×A4
FittingC22×C4×C5×Q8C3
socleC24×C5C2×A4
prime radicalC23×C5×SL(2,𝔽3)C2
cosocleC2×Q8C23×C3×C5
LCSG ▷ Q8
UCSC1 ◁ C23×C4×C5

Complements

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

NH ≅ G/Ntype
2. C2×C4×C5×SL(2,𝔽3)C2direct (N × H)
3. C22×C4×C5×Q8C3semidirect
5. C22×C5×SL(2,𝔽3)C4direct (N × H)
7. C4×C5×SL(2,𝔽3)C22direct (N × H)
8. C22×C4×SL(2,𝔽3)C5direct (N × H)
11. C2×C4×C5×Q8C2×C3semidirect
17. C2×C5×SL(2,𝔽3)C2×C4direct (N × H)
20. C2×C4×SL(2,𝔽3)C2×C5direct (N × H)
23. C22×C5×Q8C3×C4semidirect
30. C4×C5×Q8C22×C3semidirect
35. C22×C4×Q8C3×C5semidirect
42. C5×SL(2,𝔽3)C22×C4direct (N × H)
44. C22×SL(2,𝔽3)C4×C5direct (N × H)
46. C4×SL(2,𝔽3)C22×C5direct (N × H)
53. C2×C5×Q8C2×C3×C4semidirect
61. C22×C4×C5SL(2,𝔽3)direct (N × H)
69. C2×C4×Q8C2×C3×C5semidirect
81. C2×SL(2,𝔽3)C2×C4×C5direct (N × H)
91. C5×Q8C22×C3×C4semidirect
99. C2×C4×C5C2×SL(2,𝔽3)direct (N × H)
103. C22×Q8C3×C4×C5semidirect
110. C4×Q8C22×C3×C5semidirect
125. SL(2,𝔽3)C22×C4×C5direct (N × H)
126. C22×C5C4×SL(2,𝔽3)direct (N × H)
135. C4×C5C22×SL(2,𝔽3)direct (N × H)
142. C2×Q8C2×C3×C4×C5semidirect
150. C22×C4C5×SL(2,𝔽3)direct (N × H)
163. C2×C5C2×C4×SL(2,𝔽3)direct (N × H)
175. Q8C22×C3×C4×C5semidirect
183. C2×C4C2×C5×SL(2,𝔽3)direct (N × H)
189. C5C22×C4×SL(2,𝔽3)direct (N × H)
190. C22C4×C5×SL(2,𝔽3)direct (N × H)
199. C4C22×C5×SL(2,𝔽3)direct (N × H)
201. C2C2×C4×C5×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
C2×C4×C5×SL(2,𝔽3)2
C22×C4×C5×Q83
C23×C3×C4×C54
C22×C4×SL(2,𝔽3)5