Finite Group Databasebeta

← C3×C4⋊(C5⋊S4) #13089058

Subgroups — C3×C4⋊(C5⋊S4) #13089058

Order 1440.

236 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 42 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×C3×C5⋊S4C212yesmaximal, complemented
2C3×C4×C5×A4C211yesmaximal, characteristic, complemented
3C4⋊(C5⋊S4)C311yesmaximal, characteristic, direct product factor
4C3×#4076—31nomaximal
5C2×C3×C5×A4C2211yescharacteristic
6C3×C5⋊S4—22no—
7C3×D60—41nomaximal
8C3×C4⋊S4—51nomaximal, Hall
9C2×C5⋊S4C2×C312yescomplemented
10C22×C3×C4×C5D311yescharacteristic, complemented
11C2×C3×C22⋊D5—32no—
12C2×C3×D20—31no—
13C4×C5×A4—21no—
14C4×C5×A4C2×C311yescharacteristic, complemented
15C3×#685—32no—
16C3×C4⋊Dic5—31no—
17C2×C3×D15—42no—
18C3×C5×A4D411yescharacteristic, complemented
19C32×C4×C5—41no—
20#4076—31noHall
21C2×C3×S4—52no—
22C3×C4×A4D511yescharacteristic, complemented
23C23×C3×C5C2×D311yescharacteristic
24C22×C3×D5—32no—
25C2×C5×A4—21no—
26C2×C5×A4C22×C311yescharacteristic
27C5⋊S4—22no—
28C2×C3×C4×C5—31no—
29C2×C3×C4×C5—31no—
30D60—41no—
31C3×C22⋊D5—62no—
32C3×C22⋊D5—62no—
33C2×C3×Dic5—32no—
34C3×D20—61no—
35C3×D20—61no—
36C4⋊S4—51no—
37C3×#121—151no—
38C2×C32×C5—41no—
39C3×D15—82no—
40C22×C4×C5C3×D311yescharacteristic, complemented
41C2×C22⋊D5—32no—
42C2×D20—31no—
43#685—32no—
44C4⋊Dic5—31no—
45C2×C3×A4C2×D511yescharacteristic
46C3×S4—102no—
47C3×D12—201no—
48C22×C3×C5—31no—
49C22×C3×C5—31no—
50C22×C3×C5D1211yescharacteristic, complemented
51C2×D15—42no—
52C2×C3×D5—62no—
53C2×C3×D5—62no—
54C2×C3×D5—62no—
55C5×A4—21no—
56C5×A4C3×D411yescharacteristic, complemented
57C3×C4×C5—31no—
58C3×C4×C5—41no—
59C3×C4×C5—81no—
60C3×C4×C5S411yescharacteristic, complemented
61C3×Dic5—62no—
62C2×S4—52no—
63C2×C3×D4—151no—
64C2×C3×D4—152no—
65C22×C3×C4D1511yescharacteristic, complemented
66C4×A4—21no—
67C4×A4C3×D511yescharacteristic, complemented
68C3×C4⋊C4—151no—
69C3×C22⋊C4—152no—
70C32×C5—41noHall
71C23×C5C2×C3×D311yescharacteristic
72C22×D5—32no—
73C2×C4×C5—31no—
74C2×C4×C5—31no—
75C22⋊D5—62no—
76C22⋊D5—62no—
77C2×Dic5—32no—
78D20—61no—
79D20—61no—
80C2×C3×D3—202no—
81C3×A4D2011yescharacteristic, complemented
82C32×C4—41no—
83#121—151noSylow
84C2×C3×C5—31no—
85C2×C3×C5—31no—
86C2×C3×C5—41no—
87C2×C3×C5—81no—
88C2×C3×C5C2×S411yescharacteristic
89D15—82no—
90C3×D5—122no—
91C23×C3—152no—
92C23×C3C2×D1511yescharacteristic
93C2×A4—21no—
94C2×A4C2×C3×D511yescharacteristic
95S4—102no—
96C3×D4—301no—
97C3×D4—302no—
98C3×D4—302no—
99C3×D4—301no—
100C2×C3×C4—31no—
101C2×C3×C4—31no—
102C2×C3×C4—152no—
103D12—201no—
104C22×C5—31no—
105C22×C5—31no—
106C22×C5C3×D1211yescharacteristic, complemented
107C2×D5—62no—
108C2×D5—62no—
109C2×D5—62no—
110C4×C5—31no—
111C4×C5C3×S411yescharacteristic, complemented
112Dic5—62no—
113C2×C32—41no—
114C3×D3—402no—
115C2×D4—151no—
116C2×D4—152no—
117C22×C4C3×D1511yescharacteristic, complemented
118C4⋊C4—151no—
119C22⋊C4—152no—
120C3×C5—41no—
121C3×C5—81no—
122C3×C5C4⋊S411yescharacteristic, complemented
123C22×C3—31no—
124C22×C3—31no—
125C22×C3—302no—
126C22×C3—302no—
127C22×C3—302no—
128C22×C3D6011yescharacteristic, complemented
129C2×D3—202no—
130A4—21no—
131A4C3×D2011yescharacteristic, complemented
132C3×C4—31no—
133C3×C4—41no—
134C3×C4—81no—
135C3×C4—302no—
136C3×C4C5⋊S411yescharacteristic, complemented
137C2×C5—31no—
138C2×C5—31no—
139C2×C5C2×C3×S411yescharacteristic
140D5—122no—
141C32—41noSylow
142C23—152no—
143C23C2×C3×D1511yescharacteristic
144D4—301no—
145D4—302no—
146D4—302no—
147D4—301no—
148C2×C4—31no—
149C2×C4—31no—
150C2×C4—152no—
151C2×C3—31no—
152C2×C3—31no—
153C2×C3—41no—
154C2×C3—81no—
155C2×C3—602no—
156C2×C3C2×C5⋊S411yescharacteristic
157D3—402no—
158C5C3×C4⋊S411yescharacteristic, complemented, Sylow
159C22—31no—
160C22—31no—
161C22—302no—
162C22—302no—
163C22—302no—
164C22C3×D6011yescharacteristic, complemented
165C4—31no—
166C4—302no—
167C4C3×C5⋊S411yescharacteristic, complemented
168C3—41no—
169C3—81no—
170C3C4⋊(C5⋊S4)11yescharacteristic, direct product factor
171C2—31no—
172C2—31no—
173C2—602no—
174C2C2×C3×C5⋊S411yescharacteristic
subgroupG/H
centerC2×C3C2×C5⋊S4
commutator, cosocleC2×C5×A4C22×C3
FrattiniC2C2×C3×C5⋊S4
FittingC22×C3×C4×C5D3
socleC23×C3×C5C2×D3
hypercenterC3×C4C5⋊S4
hypocenterC5×A4C3×D4
LCSG ▷ C2×C5×A4 ▷ C5×A4
UCSC1 ◁ C2×C3 ◁ C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5⋊S4C2semidirect
2. C3×C4×C5×A4C2semidirect
3. C4⋊(C5⋊S4)C3direct (N × H)
9. C2×C5⋊S4C2×C3semidirect
10. C22×C3×C4×C5D3semidirect
14. C4×C5×A4C2×C3semidirect
18. C3×C5×A4D4semidirect
22. C3×C4×A4D5semidirect
40. C22×C4×C5C3×D3semidirect
50. C22×C3×C5D12semidirect
56. C5×A4C3×D4semidirect
60. C3×C4×C5S4semidirect
65. C22×C3×C4D15semidirect
67. C4×A4C3×D5semidirect
81. C3×A4D20semidirect
106. C22×C5C3×D12semidirect
111. C4×C5C3×S4semidirect
117. C22×C4C3×D15semidirect
122. C3×C5C4⋊S4semidirect
128. C22×C3D60semidirect
131. A4C3×D20semidirect
136. C3×C4C5⋊S4semidirect
158. C5C3×C4⋊S4semidirect
164. C22C3×D60semidirect
167. C4C3×C5⋊S4semidirect
170. C3C4⋊(C5⋊S4)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C5⋊S42
C3×C4×C5×A42
C4⋊(C5⋊S4)3
C3×#40763
C3×D604
C3×C4⋊S45