Finite Group Databasebeta

← C4×C5×C22⋊SL(2,𝔽3) #14511381

Subgroups — C4×C5×C22⋊SL(2,𝔽3) #14511381

Order 1920.

658 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 68 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
1C2×C5×C22⋊SL(2,𝔽3)C211yesmaximal, characteristic
2C22×C4×C5×Q8C311yesmaximal, characteristic, complemented, Hall
3C5×C22⋊SL(2,𝔽3)C411yescharacteristic, direct product factor
4C2×C4×C5×A4—41nomaximal
5C4×C5×SL(2,𝔽3)—44nomaximal
6C4×C22⋊SL(2,𝔽3)C511yesmaximal, characteristic, direct product factor, Hall
7C23×C5×Q8C2×C311yescharacteristic
8C2×C4×C5×Q8—34no—
9C2×C4×C5×Q8—34no—
10C22×C5×C4⋊C4—31no—
11C22×C42×C5—31no—
12C22×C5×A4—41no—
13C2×C5×SL(2,𝔽3)—44no—
14C4×C5×A4—42no—
15C2×C22⋊SL(2,𝔽3)C2×C511yescharacteristic
16C22×C5×Q8—34no—
17C22×C5×Q8—34no—
18C22×C5×Q8—31no—
19C22×C5×Q8C3×C411yescharacteristic, complemented
20C23×C4×C5—31no—
21C23×C4×C5—31no—
22C23×C4×C5A411yescharacteristic
23C4×C5×Q8—312no—
24C4×C5×Q8—34no—
25C4×C5×Q8—34no—
26C4×C5×Q8A414yescomplemented
27C2×C5×C4⋊C4—312no—
28C2×C42×C5—36no—
29C2×C42×C5—36no—
30C22×C4×Q8C3×C511yescharacteristic, complemented, Sylow
31C2×C5×A4—41no—
32C2×C5×A4—41no—
33C2×C5×A4—41no—
34C2×C3×C4×C5—161no—
35C5×SL(2,𝔽3)—44no—
36C22⋊SL(2,𝔽3)C4×C511yescharacteristic, direct product factor
37C2×C4×A4—41no—
38C4×SL(2,𝔽3)—44no—
39C24×C5C2×A411yescharacteristic
40C2×C5×Q8—34no—
41C2×C5×Q8—312no—
42C2×C5×Q8—312no—
43C2×C5×Q8—34no—
44C2×C5×Q8—34no—
45C2×C5×Q8C2×A414yes—
46C22×C4×C5—36no—
47C22×C4×C5—36no—
48C22×C4×C5—31no—
49C22×C4×C5—31no—
50C22×C4×C5—31no—
51C22×C4×C5—31no—
52C22×C4×C5—32no—
53C22×C4×C5—36no—
54C22×C4×C5—63no—
55C22×C4×C5—61no—
56C22×C4×C5SL(2,𝔽3)12yescomplemented
57C5×C4⋊C4—34no—
58C5×C4⋊C4—312no—
59C42×C5—312no—
60C42×C5—34no—
61C23×Q8C2×C3×C511yescharacteristic
62C2×C4×Q8—34no—
63C2×C4×Q8—34no—
64C22×C4⋊C4—31no—
65C22×C42—31no—
66C22×C3×C5—161no—
67C5×A4—41no—
68C3×C4×C5—162no—
69C22×A4—41no—
70C2×SL(2,𝔽3)—44no—
71C4×A4—42no—
72C23×C5—31no—
73C23×C5—31no—
74C23×C5—31no—
75C23×C5—31no—
76C23×C5C4×A411yescharacteristic
77C23×C5C2×SL(2,𝔽3)11yescharacteristic
78C23×C5#41⋊C311yescharacteristic
79C5×Q8—34no—
80C5×Q8—312no—
81C5×Q8—34no—
82C5×Q8C4×A414yescomplemented
83C2×C4×C5—32no—
84C2×C4×C5—36no—
85C2×C4×C5—312no—
86C2×C4×C5—36no—
87C2×C4×C5—31no—
88C2×C4×C5—34no—
89C2×C4×C5—32no—
90C2×C4×C5—32no—
91C2×C4×C5—32no—
92C2×C4×C5—34no—
93C2×C4×C5—66no—
94C2×C4×C5—66no—
95C2×C4×C5C22⋊A411yescharacteristic
96C22×Q8—34no—
97C22×Q8—34no—
98C22×Q8—31no—
99C22×Q8C3×C4×C511yescharacteristic, complemented
100C23×C4—31no—
101C23×C4—31no—
102C23×C4C5×A411yescharacteristic
103C4×Q8—312no—
104C4×Q8—34no—
105C4×Q8—34no—
106C4×Q8C5×A414yescomplemented
107C2×C4⋊C4—312no—
108C2×C42—36no—
109C2×C42—36no—
110C2×C3×C5—161no—
111C2×C3×C5—161no—
112C2×C3×C5—161no—
113C2×A4—41no—
114C2×A4—41no—
115C2×A4—41no—
116C2×C3×C4—161no—
117SL(2,𝔽3)—44no—
118C22×C5—31no—
119C22×C5—31no—
120C22×C5—31no—
121C22×C5—32no—
122C22×C5—31no—
123C22×C5—31no—
124C22×C5—31no—
125C22×C5—31no—
126C22×C5—31no—
127C22×C5—31no—
128C22×C5C4×SL(2,𝔽3)11yescharacteristic, complemented
129C22×C5C2×C22⋊A411yescharacteristic
130C4×C5—34no—
131C4×C5—34no—
132C4×C5—32no—
133C4×C5—64no—
134C4×C5C22⋊SL(2,𝔽3)12yesdirect product factor
135C24C2×C5×A411yescharacteristic
136C2×Q8—34no—
137C2×Q8—312no—
138C2×Q8—312no—
139C2×Q8—34no—
140C2×Q8—34no—
141C2×Q8C2×C5×A414yes—
142C22×C4—36no—
143C22×C4—36no—
144C22×C4—31no—
145C22×C4—31no—
146C22×C4—31no—
147C22×C4—31no—
148C22×C4—32no—
149C22×C4—36no—
150C22×C4—63no—
151C22×C4—61no—
152C22×C4C5×SL(2,𝔽3)12yescomplemented
153C4⋊C4—34no—
154C4⋊C4—312no—
155C42—312no—
156C42—34no—
157C3×C5—161noHall
158C22×C3—161no—
159A4—41no—
160C3×C4—162no—
161C2×C5—31no—
162C2×C5—31no—
163C2×C5—31no—
164C2×C5—31no—
165C2×C5C4×C22⋊A411yescharacteristic
166C2×C5C2×C22⋊SL(2,𝔽3)11yescharacteristic
167C2×C5#594811yescharacteristic
168C23—31no—
169C23—31no—
170C23—31no—
171C23—31no—
172C23C4×C5×A411yescharacteristic
173C23C2×C5×SL(2,𝔽3)11yescharacteristic
174C23C5×#41⋊C311yescharacteristic
175Q8—34no—
176Q8—34no—
177Q8—312no—
178Q8C4×C5×A414yescomplemented
179C2×C4—36no—
180C2×C4—312no—
181C2×C4—36no—
182C2×C4—32no—
183C2×C4—31no—
184C2×C4—34no—
185C2×C4—32no—
186C2×C4—32no—
187C2×C4—32no—
188C2×C4—34no—
189C2×C4—66no—
190C2×C4—66no—
191C2×C4C5×C22⋊A411yescharacteristic
192C2×C3—161no—
193C2×C3—161no—
194C2×C3—161no—
195C5C4×C22⋊SL(2,𝔽3)11yescharacteristic, direct product factor, Sylow
196C22—31no—
197C22—31no—
198C22—31no—
199C22—32no—
200C22—31no—
201C22—31no—
202C22—31no—
203C22—31no—
204C22—31no—
205C22—31no—
206C22C4×C5×SL(2,𝔽3)11yescharacteristic, complemented
207C22C2×C5×C22⋊A411yescharacteristic
208C4—34no—
209C4—34no—
210C4—32no—
211C4—64no—
212C4C5×C22⋊SL(2,𝔽3)12yesdirect product factor
213C3—161noSylow
214C2—31no—
215C2—31no—
216C2—31no—
217C2—31no—
218C2C4×C5×C22⋊A411yescharacteristic
219C2C2×C5×C22⋊SL(2,𝔽3)11yescharacteristic
220C2C5×#594811yescharacteristic
subgroupG/H
center, hypercenterC2×C4×C5C22⋊A4
commutator, hypocenterC22×Q8C3×C4×C5
FrattiniC22C2×C5×C22⋊A4
FittingC22×C4×C5×Q8C3
socleC24×C5C2×A4
prime radicalC2×C5×C22⋊SL(2,𝔽3)C2
cosocleC23×Q8C2×C3×C5
LCSG ▷ C22×Q8
UCSC1 ◁ C2×C4×C5

Complements

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

NH ≅ G/Ntype
2. C22×C4×C5×Q8C3semidirect
3. C5×C22⋊SL(2,𝔽3)C4direct (N × H)
6. C4×C22⋊SL(2,𝔽3)C5direct (N × H)
19. C22×C5×Q8C3×C4semidirect
26. C4×C5×Q8A4semidirect
30. C22×C4×Q8C3×C5semidirect
36. C22⋊SL(2,𝔽3)C4×C5direct (N × H)
56. C22×C4×C5SL(2,𝔽3)semidirect
82. C5×Q8C4×A4semidirect
99. C22×Q8C3×C4×C5semidirect
106. C4×Q8C5×A4semidirect
128. C22×C5C4×SL(2,𝔽3)semidirect
134. C4×C5C22⋊SL(2,𝔽3)direct (N × H)
152. C22×C4C5×SL(2,𝔽3)semidirect
178. Q8C4×C5×A4semidirect
195. C5C4×C22⋊SL(2,𝔽3)direct (N × H)
206. C22C4×C5×SL(2,𝔽3)semidirect
212. C4C5×C22⋊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
C2×C5×C22⋊SL(2,𝔽3)2
C22×C4×C5×Q83
C2×C4×C5×A44
C4×C5×SL(2,𝔽3)4
C4×C22⋊SL(2,𝔽3)5