Finite Group Databasebeta

← C2×C5×#543⋊C3 #14511387

Subgroups — C2×C5×#543⋊C3 #14511387

Order 1920.

612 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 66 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
1C5×#543⋊C3C211yesmaximal, characteristic, direct product factor
2C2×C5×#543C311yesmaximal, characteristic, complemented, Hall
3C23×C5×A4—41nomaximal
4C22×C5×SL(2,𝔽3)—44nomaximal
5C2×#543⋊C3C511yesmaximal, characteristic, direct product factor, Hall
6C5×#543—34no—
7C5×#543—31no—
8C5×#543C2×C311yescharacteristic, complemented
9C2×C5×C22⋊Q8—34no—
10C22×C5×C22⋊C4—31no—
11C22×C5×A4—44no—
12C22×C5×A4—42no—
13C22×C5×A4—41no—
14C2×C5×SL(2,𝔽3)—48no—
15C2×C5×SL(2,𝔽3)—44no—
16#543⋊C3C2×C511yescharacteristic, direct product factor
17C25×C5A411yescharacteristic
18C22×C5×Q8A414yescomplemented
19C23×C4×C5—32no—
20C5×C22⋊Q8—312no—
21C5×C22⋊Q8—34no—
22C5×C22⋊Q8—38no—
23C5×C22⋊Q8—34no—
24C5×C22⋊Q8—34no—
25C2×C5×C4⋊C4—34no—
26C2×C5×C22⋊C4—34no—
27C2×C5×C22⋊C4—34no—
28C2×C5×C22⋊C4—32no—
29C2×C5×C22⋊C4—31no—
30C2×C5×C22⋊C4—31no—
31C2×#543C3×C511yescharacteristic, complemented, Sylow
32C23×C3×C5—161no—
33C2×C5×A4—42no—
34C2×C5×A4—44no—
35C2×C5×A4—41no—
36C5×SL(2,𝔽3)—44no—
37C23×A4—41no—
38C22×SL(2,𝔽3)—44no—
39C24×C5—31no—
40C24×C5—31no—
41C24×C5—32no—
42C24×C5—44no—
43C24×C5SL(2,𝔽3)12yescomplemented
44C24×C5C2×A411yescharacteristic
45C2×C5×Q8—34no—
46C2×C5×Q8—48no—
47C2×C5×Q8C2×A414yescomplemented
48C22×C4×C5—34no—
49C22×C4×C5—32no—
50C22×C4×C5—34no—
51C22×C4×C5—32no—
52C22×C4×C5—124no—
53C5×C4⋊C4—312no—
54C5×C4⋊C4—34no—
55C5×C22⋊C4—34no—
56C5×C22⋊C4—38no—
57C5×C22⋊C4—34no—
58#543—34no—
59#543—31no—
60#543C2×C3×C511yescharacteristic, complemented
61C2×C22⋊Q8—34no—
62C22×C22⋊C4—31no—
63C22×C3×C5—162no—
64C22×C3×C5—164no—
65C22×C3×C5—161no—
66C5×A4—41no—
67C22×A4—44no—
68C22×A4—42no—
69C22×A4—41no—
70C2×SL(2,𝔽3)—48no—
71C2×SL(2,𝔽3)—44no—
72C23×C5—32no—
73C23×C5—32no—
74C23×C5—32no—
75C23×C5—31no—
76C23×C5—31no—
77C23×C5—32no—
78C23×C5—31no—
79C23×C5—44no—
80C23×C5—42no—
81C23×C5—64no—
82C23×C5—122no—
83C23×C5—124no—
84C23×C5C2×SL(2,𝔽3)11yescharacteristic, complemented
85C23×C5C22⋊A411yescharacteristic
86C5×Q8—44no—
87C5×Q8—124no—
88C2×C4×C5—34no—
89C2×C4×C5—34no—
90C2×C4×C5—68no—
91C2×C4×C5—122no—
92C2×C4×C5—124no—
93C2×C4×C5—122no—
94C25C5×A411yescharacteristic
95C22×Q8C5×A414yescomplemented
96C23×C4—32no—
97C22⋊Q8—312no—
98C22⋊Q8—38no—
99C22⋊Q8—34no—
100C22⋊Q8—34no—
101C22⋊Q8—34no—
102C2×C4⋊C4—34no—
103C2×C22⋊C4—34no—
104C2×C22⋊C4—34no—
105C2×C22⋊C4—32no—
106C2×C22⋊C4—31no—
107C2×C22⋊C4—31no—
108C2×C3×C5—161no—
109C2×C3×C5—162no—
110C2×C3×C5—164no—
111C23×C3—161no—
112C2×A4—42no—
113C2×A4—44no—
114C2×A4—41no—
115SL(2,𝔽3)—44no—
116C22×C5—31no—
117C22×C5—32no—
118C22×C5—31no—
119C22×C5—41no—
120C22×C5—62no—
121C22×C5—62no—
122C22×C5—64no—
123C22×C5—64no—
124C22×C5—122no—
125C22×C5—121no—
126C22×C5—122no—
127C22×C5C22⋊SL(2,𝔽3)12yes—
128C22×C5#99714yes—
129C22×C5C2×C22⋊A411yescharacteristic
130C4×C5—64no—
131C4×C5—64no—
132C24—31no—
133C24—31no—
134C24—32no—
135C24—44no—
136C24C5×SL(2,𝔽3)12yescomplemented
137C24C2×C5×A411yescharacteristic
138C2×Q8—34no—
139C2×Q8—48no—
140C2×Q8C2×C5×A414yescomplemented
141C22×C4—34no—
142C22×C4—32no—
143C22×C4—34no—
144C22×C4—32no—
145C22×C4—124no—
146C4⋊C4—34no—
147C4⋊C4—312no—
148C22⋊C4—34no—
149C22⋊C4—38no—
150C22⋊C4—34no—
151C3×C5—161noHall
152C22×C3—162no—
153C22×C3—164no—
154C22×C3—161no—
155A4—41no—
156C2×C5—61no—
157C2×C5—61no—
158C2×C5—62no—
159C2×C5#543⋊C314yesdirect product factor
160C2×C5C2×C22⋊SL(2,𝔽3)11yescharacteristic
161C2×C5C2×#99712yes—
162C23—32no—
163C23—32no—
164C23—32no—
165C23—31no—
166C23—31no—
167C23—32no—
168C23—31no—
169C23—44no—
170C23—42no—
171C23—64no—
172C23—122no—
173C23—124no—
174C23C2×C5×SL(2,𝔽3)11yescharacteristic, complemented
175C23C5×C22⋊A411yescharacteristic
176Q8—44no—
177Q8—124no—
178C2×C4—34no—
179C2×C4—34no—
180C2×C4—68no—
181C2×C4—122no—
182C2×C4—124no—
183C2×C4—122no—
184C2×C3—161no—
185C2×C3—162no—
186C2×C3—164no—
187C5C2×#543⋊C311yescharacteristic, direct product factor, Sylow
188C22—31no—
189C22—32no—
190C22—31no—
191C22—41no—
192C22—62no—
193C22—62no—
194C22—64no—
195C22—64no—
196C22—122no—
197C22—121no—
198C22—122no—
199C22C5×C22⋊SL(2,𝔽3)12yes—
200C22C5×#99714yes—
201C22C2×C5×C22⋊A411yescharacteristic
202C4—64no—
203C4—64no—
204C3—161noSylow
205C2—61no—
206C2—61no—
207C2—62no—
208C2C5×#543⋊C314yesdirect product factor
209C2C2×C5×C22⋊SL(2,𝔽3)11yescharacteristic
210C2C2×C5×#99712yes—
subgroupG/H
center, socle, hypercenterC23×C5C22⋊A4
commutator, cosocle, hypocenter#543C2×C3×C5
FrattiniC22C2×C5×C22⋊A4
FittingC2×C5×#543C3
LCSG ▷ #543
UCSC1 ◁ C23×C5

Complements

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

NH ≅ G/Ntype
1. C5×#543⋊C3C2direct (N × H)
2. C2×C5×#543C3semidirect
5. C2×#543⋊C3C5direct (N × H)
8. C5×#543C2×C3semidirect
16. #543⋊C3C2×C5direct (N × H)
18. C22×C5×Q8A4semidirect
31. C2×#543C3×C5semidirect
43. C24×C5SL(2,𝔽3)semidirect
47. C2×C5×Q8C2×A4semidirect
60. #543C2×C3×C5semidirect
84. C23×C5C2×SL(2,𝔽3)semidirect
95. C22×Q8C5×A4semidirect
136. C24C5×SL(2,𝔽3)semidirect
140. C2×Q8C2×C5×A4semidirect
159. C2×C5#543⋊C3direct (N × H)
174. C23C2×C5×SL(2,𝔽3)semidirect
187. C5C2×#543⋊C3direct (N × H)
208. C2C5×#543⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#543⋊C32
C2×C5×#5433
C23×C5×A44
C22×C5×SL(2,𝔽3)4
C2×#543⋊C35