Finite Group Databasebeta

← C3×C4×S5 #13088300

Subgroups — C3×C4×S5 #13088300

Order 1440.

207 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 207 conjugacy classes of subgroups — too many to draw. Only its 22 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×S5C211yesmaximal, characteristic
2C3×C4×A5C211yesmaximal, characteristic, complemented
3C3×A5⋊C4C211yesmaximal, characteristic, complemented
4C4×S5C311yesmaximal, characteristic, direct product factor
5C2×C3×A5C2211yescharacteristic
6C3×S5C412yesdirect product factor
7C3×C4×S4—51nomaximal, Hall
8C2×S5C2×C311yescharacteristic
9C3×C4×AGL(1,𝔽5)—61nomaximal
10C4×A5C2×C311yescharacteristic, complemented
11A5⋊C4C2×C311yescharacteristic, complemented
12C3×A5C2×C411yescharacteristic, complemented
13C2×C3×S4—51no—
14C2×C3×C4×D3—101nomaximal
15C3×C4×A4—51no—
16C3×SL(2,ℤ/4)—51no—
17C2×C3×AGL(1,𝔽5)—61no—
18C2×C3×AGL(1,𝔽5)—61no—
19C2×A5C22×C311yescharacteristic
20S5C3×C412yesdirect product factor
21C3×C4×D5—61no—
22C4×S4—51no—
23C3×C4×D4—151no—
24C4×AGL(1,𝔽5)—61no—
25C22×C3×D3—101no—
26C2×C3×A4—51no—
27C3×S4—52no—
28C2×C32×C4—101no—
29C2×C3×Dic3—101no—
30C3×C4×D3—101no—
31C3×C4×D3—101no—
32C3×C4×D3—101no—
33C3×C4×D3—101no—
34C2×C3×D5—61no—
35C3×AGL(1,𝔽5)—62no—
36C3×AGL(1,𝔽5)—62no—
37A5C2×C3×C411yescharacteristic, complemented
38C3×C4×C5—61no—
39C3×Dic5—61no—
40C2×S4—51no—
41C2×C3×D4—151no—
42C22×C3×C4—51no—
43C22×C3×C4—151no—
44C2×C4×D3—101no—
45C4×A4—51no—
46C4×A4—101no—
47SL(2,ℤ/4)—51no—
48C3×C4⋊C4—151no—
49C3×C22⋊C4—151no—
50C3×C22⋊C4—151no—
51C3×C42—151no—
52C2×AGL(1,𝔽5)—61no—
53C2×AGL(1,𝔽5)—61no—
54C4×D5—61no—
55C22×C32—101no—
56C2×C3×D3—101no—
57C2×C3×D3—102no—
58C2×C3×D3—101no—
59C2×C3×D3—102no—
60C3×A4—51no—
61C32×C4—101no—
62C32×C4—101no—
63C3×Dic3—101no—
64C3×Dic3—101no—
65C4×D4—151noSylow
66C2×C3×C5—61no—
67C3×D5—61no—
68C3×D5—61no—
69C23×C3—51no—
70C23×C3—151no—
71C22×D3—101no—
72C2×A4—51no—
73C2×A4—101no—
74S4—52no—
75C3×D4—152no—
76C3×D4—152no—
77C2×C3×C4—101no—
78C2×C3×C4—101no—
79C2×C3×C4—151no—
80C2×C3×C4—151no—
81C2×C3×C4—151no—
82C2×C3×C4—151no—
83C2×C3×C4—151no—
84C2×C3×C4—201no—
85C2×C3×C4—301no—
86C2×Dic3—101no—
87C4×D3—101no—
88C4×D3—101no—
89C4×D3—101no—
90C4×D3—101no—
91C2×D5—61no—
92AGL(1,𝔽5)—62no—
93AGL(1,𝔽5)—62no—
94C4×C5—61no—
95Dic5—61no—
96C2×C32—102no—
97C2×C32—101no—
98C3×D3—102no—
99C3×D3—101no—
100C3×D3—101no—
101C2×D4—151no—
102C22×C4—51no—
103C22×C4—151no—
104C4⋊C4—151no—
105C22⋊C4—151no—
106C22⋊C4—151no—
107C42—151no—
108C3×C5—61no—
109C22×C3—51no—
110C22×C3—101no—
111C22×C3—101no—
112C22×C3—152no—
113C22×C3—151no—
114C22×C3—151no—
115C22×C3—201no—
116C22×C3—301no—
117C2×D3—101no—
118C2×D3—101no—
119C2×D3—102no—
120C2×D3—102no—
121A4—51no—
122A4—101no—
123C3×C4—101no—
124C3×C4—101no—
125C3×C4—101no—
126C3×C4—152no—
127C3×C4—151no—
128C3×C4—201no—
129C3×C4—201no—
130C3×C4—301no—
131C3×C4S511yescharacteristic, direct product factor
132Dic3—101no—
133Dic3—101no—
134C2×C5—61no—
135D5—61no—
136D5—61no—
137C32—101noSylow
138C23—51no—
139C23—151no—
140D4—152no—
141D4—152no—
142C2×C4—101no—
143C2×C4—151no—
144C2×C4—151no—
145C2×C4—151no—
146C2×C4—151no—
147C2×C4—151no—
148C2×C4—301no—
149C2×C3—102no—
150C2×C3—102no—
151C2×C3—101no—
152C2×C3—151no—
153C2×C3—151no—
154C2×C3—202no—
155C2×C3—201no—
156C2×C3C2×S511yescharacteristic
157D3—102no—
158D3—101no—
159D3—101no—
160C5—61noSylow
161C22—51no—
162C22—101no—
163C22—152no—
164C22—151no—
165C22—151no—
166C22—301no—
167C4—101no—
168C4—152no—
169C4—151no—
170C4—301no—
171C4C3×S511yescharacteristic, direct product factor
172C3—101no—
173C3—201no—
174C3C4×S511yescharacteristic, direct product factor
175C2—102no—
176C2—151no—
177C2—151no—
178C2C2×C3×S511yescharacteristic
subgroupG/H
center, Fitting, hypercenterC3×C4S5
commutator, hypocenterA5C2×C3×C4
FrattiniC2C2×C3×S5
socleC2×C3×A5C22
prime radicalC2×C3×S5C2
cosocleC2×A5C22×C3
LCSG ▷ A5
UCSC1 ◁ C3×C4

Complements

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

NH ≅ G/Ntype
2. C3×C4×A5C2semidirect
3. C3×A5⋊C4C2semidirect
4. C4×S5C3direct (N × H)
6. C3×S5C4direct (N × H)
10. C4×A5C2×C3semidirect
11. A5⋊C4C2×C3semidirect
12. C3×A5C2×C4semidirect
20. S5C3×C4direct (N × H)
37. A5C2×C3×C4semidirect
131. C3×C4S5direct (N × H)
171. C4C3×S5direct (N × H)
174. C3C4×S5direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×S52
C3×C4×A52
C3×A5⋊C42
C4×S53
C3×C4×S45
C3×C4×AGL(1,𝔽5)6
C2×C3×C4×D310