Finite Group Databasebeta

← C3×C4×C5⋊S4 #13089057

Subgroups — C3×C4×C5⋊S4 #13089057

Order 1440.

236 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 46 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⋊S4C211yesmaximal, characteristic
2C3×C4×C5×A4C211yesmaximal, characteristic, complemented
3C3×A4⋊Dic5C211yesmaximal, characteristic, complemented
4C4×C5⋊S4C311yesmaximal, characteristic, direct product factor
5C3×C4×C22⋊D5—31nomaximal
6C2×C3×C5×A4C2211yescharacteristic
7C3×C5⋊S4C412yesdirect product factor
8C3×C4×D15—41nomaximal
9C3×C4×S4—51nomaximal, Hall
10C2×C5⋊S4C2×C311yescharacteristic
11C22×C3×C4×C5D311yescharacteristic, complemented
12C2×C3×C22⋊D5—31no—
13C2×C3×C4×D5—31no—
14C4×C5×A4—21no—
15C4×C5×A4C2×C311yescharacteristic, complemented
16A4⋊Dic5C2×C311yescharacteristic, complemented
17C3×C22⋊Dic5—31no—
18C3×#685—31no—
19C3×#683—31no—
20C3×C4×Dic5—31no—
21C2×C3×D15—41no—
22C3×C5×A4C2×C411yescharacteristic, complemented
23C32×C4×C5—41no—
24C3×Dic15—41no—
25C4×C22⋊D5—31noHall
26C2×C3×S4—51no—
27C3×C4×A4D511yescharacteristic, complemented
28C3×SL(2,ℤ/4)—51no—
29C23×C3×C5C2×D311yescharacteristic
30C22×C3×D5—31no—
31C2×C5×A4—21no—
32C2×C5×A4C22×C311yescharacteristic
33C5⋊S4C3×C412yesdirect product factor
34C2×C3×C4×C5—31no—
35C2×C3×C4×C5—31no—
36C4×D15—41no—
37C3×C22⋊D5—32no—
38C3×C22⋊D5—32no—
39C2×C3×Dic5—31no—
40C2×C3×Dic5—31no—
41C2×C3×Dic5—31no—
42C3×C4×D5—61no—
43C3×C4×D5—61no—
44C4×S4—51no—
45C3×C4×D4—151no—
46C2×C32×C5—41no—
47C3×D15—42no—
48C22×C4×C5C3×D311yescharacteristic, complemented
49C2×C22⋊D5—31no—
50C2×C4×D5—31no—
51C22⋊Dic5—31no—
52#685—31no—
53#683—31no—
54C4×Dic5—31no—
55C2×C3×A4C2×D511yescharacteristic
56C3×S4—52no—
57C3×C4×D3—201no—
58C22×C3×C5—31no—
59C22×C3×C5—31no—
60C22×C3×C5C4×D311yescharacteristic, complemented
61C2×D15—41no—
62C2×C3×D5—32no—
63C2×C3×D5—61no—
64C2×C3×D5—61no—
65C5×A4—21no—
66C5×A4C2×C3×C411yescharacteristic, complemented
67C3×C4×C5—31no—
68C3×C4×C5—41no—
69C3×C4×C5—81no—
70C3×C4×C5S411yescharacteristic, complemented
71Dic15—41no—
72C3×Dic5—32no—
73C3×Dic5—61no—
74C3×Dic5—61no—
75C2×S4—51no—
76C2×C3×D4—151no—
77C22×C3×C4—151no—
78C22×C3×C4D1511yescharacteristic, complemented
79C4×A4—21no—
80C4×A4C3×D511yescharacteristic, complemented
81SL(2,ℤ/4)—51no—
82C3×C4⋊C4—151no—
83C3×C22⋊C4—151no—
84C3×C22⋊C4—151no—
85C3×C42—151no—
86C32×C5—41noHall
87C23×C5C2×C3×D311yescharacteristic
88C22×D5—31no—
89C2×C4×C5—31no—
90C2×C4×C5—31no—
91C22⋊D5—32no—
92C22⋊D5—32no—
93C2×Dic5—31no—
94C2×Dic5—31no—
95C2×Dic5—31no—
96C4×D5—61no—
97C4×D5—61no—
98C2×C3×D3—201no—
99C3×A4C4×D511yescharacteristic, complemented
100C32×C4—41no—
101C3×Dic3—201no—
102C4×D4—151noSylow
103C2×C3×C5—31no—
104C2×C3×C5—31no—
105C2×C3×C5—41no—
106C2×C3×C5—81no—
107C2×C3×C5C2×S411yescharacteristic
108D15—42no—
109C3×D5—62no—
110C23×C3—151no—
111C23×C3C2×D1511yescharacteristic
112C2×A4—21no—
113C2×A4C2×C3×D511yescharacteristic
114S4—52no—
115C3×D4—152no—
116C3×D4—152no—
117C2×C3×C4—31no—
118C2×C3×C4—31no—
119C2×C3×C4—151no—
120C2×C3×C4—151no—
121C2×C3×C4—151no—
122C2×C3×C4—301no—
123C2×C3×C4—301no—
124C4×D3—201no—
125C22×C5—31no—
126C22×C5—31no—
127C22×C5C3×C4×D311yescharacteristic, complemented
128C2×D5—32no—
129C2×D5—61no—
130C2×D5—61no—
131C4×C5—31no—
132C4×C5C3×S411yescharacteristic, complemented
133Dic5—32no—
134Dic5—61no—
135Dic5—61no—
136C2×C32—41no—
137C3×D3—202no—
138C2×D4—151no—
139C22×C4—151no—
140C22×C4C3×D1511yescharacteristic, complemented
141C4⋊C4—151no—
142C22⋊C4—151no—
143C22⋊C4—151no—
144C42—151no—
145C3×C5—41no—
146C3×C5—81no—
147C3×C5C4×S411yescharacteristic, complemented
148C22×C3—31no—
149C22×C3—31no—
150C22×C3—152no—
151C22×C3—301no—
152C22×C3—301no—
153C22×C3C4×D1511yescharacteristic, complemented
154C2×D3—201no—
155A4—21no—
156A4C3×C4×D511yescharacteristic, complemented
157C3×C4—31no—
158C3×C4—41no—
159C3×C4—81no—
160C3×C4—152no—
161C3×C4—301no—
162C3×C4—301no—
163C3×C4C5⋊S411yescharacteristic, direct product factor
164Dic3—201no—
165C2×C5—31no—
166C2×C5—31no—
167C2×C5C2×C3×S411yescharacteristic
168D5—62no—
169C32—41noSylow
170C23—151no—
171C23C2×C3×D1511yescharacteristic
172D4—152no—
173D4—152no—
174C2×C4—31no—
175C2×C4—31no—
176C2×C4—151no—
177C2×C4—151no—
178C2×C4—151no—
179C2×C4—301no—
180C2×C4—301no—
181C2×C3—31no—
182C2×C3—31no—
183C2×C3—41no—
184C2×C3—81no—
185C2×C3—302no—
186C2×C3C2×C5⋊S411yescharacteristic
187D3—202no—
188C5C3×C4×S411yescharacteristic, complemented, Sylow
189C22—31no—
190C22—31no—
191C22—152no—
192C22—301no—
193C22—301no—
194C22C3×C4×D1511yescharacteristic, complemented
195C4—31no—
196C4—152no—
197C4—301no—
198C4—301no—
199C4C3×C5⋊S411yescharacteristic, direct product factor
200C3—41no—
201C3—81no—
202C3C4×C5⋊S411yescharacteristic, direct product factor
203C2—31no—
204C2—31no—
205C2—302no—
206C2C2×C3×C5⋊S411yescharacteristic
subgroupG/H
center, hypercenterC3×C4C5⋊S4
commutator, hypocenterC5×A4C2×C3×C4
FrattiniC2C2×C3×C5⋊S4
FittingC22×C3×C4×C5D3
socleC23×C3×C5C2×D3
prime radicalC2×C3×C5⋊S4C2
cosocleC2×C5×A4C22×C3
LCSG ▷ C5×A4
UCSC1 ◁ C3×C4

Complements

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

NH ≅ G/Ntype
2. C3×C4×C5×A4C2semidirect
3. C3×A4⋊Dic5C2semidirect
4. C4×C5⋊S4C3direct (N × H)
7. C3×C5⋊S4C4direct (N × H)
11. C22×C3×C4×C5D3semidirect
15. C4×C5×A4C2×C3semidirect
16. A4⋊Dic5C2×C3semidirect
22. C3×C5×A4C2×C4semidirect
27. C3×C4×A4D5semidirect
33. C5⋊S4C3×C4direct (N × H)
48. C22×C4×C5C3×D3semidirect
60. C22×C3×C5C4×D3semidirect
66. C5×A4C2×C3×C4semidirect
70. C3×C4×C5S4semidirect
78. C22×C3×C4D15semidirect
80. C4×A4C3×D5semidirect
99. C3×A4C4×D5semidirect
127. C22×C5C3×C4×D3semidirect
132. C4×C5C3×S4semidirect
140. C22×C4C3×D15semidirect
147. C3×C5C4×S4semidirect
153. C22×C3C4×D15semidirect
156. A4C3×C4×D5semidirect
163. C3×C4C5⋊S4direct (N × H)
188. C5C3×C4×S4semidirect
194. C22C3×C4×D15semidirect
199. C4C3×C5⋊S4direct (N × H)
202. C3C4×C5⋊S4direct (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
C3×A4⋊Dic52
C4×C5⋊S43
C3×C4×C22⋊D53
C3×C4×D154
C3×C4×S45