Finite Group Databasebeta

← C3×#65764 #11752297

Subgroups — C3×#65764 #11752297

Order 960.

436 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 120 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
1C22×C3×C4×D5C211yesmaximal, characteristic
2C2×C3×C22⋊AGL(1,𝔽5)C212yesmaximal
3C3×(C2×AGL(1,𝔽5))⋊C4C214yesmaximal, complemented
4#65764C311yesmaximal, characteristic, direct product factor, Hall
5C23×C3×D5C2211yescharacteristic
6C22×C3×AGL(1,𝔽5)C2214yes—
7C22×C3×C4×C5C411yescharacteristic, complemented
8C22×C3×Dic5C411yescharacteristic, complemented
9C2×C3×C4×D5—24no—
10C2×C3×C4×D5C412yescomplemented
11C2×C3×C4×D5C2212yes—
12C3×C22⋊AGL(1,𝔽5)—24no—
13C3×#381—51nomaximal, Hall
14C22×C4×D5C2×C311yescharacteristic
15C2×C22⋊AGL(1,𝔽5)C2×C312yes—
16(C2×AGL(1,𝔽5))⋊C4C2×C314yescomplemented
17C23×C3×C5C2×C411yescharacteristic
18C22×C3×D5—24no—
19C22×C3×D5C2×C411yescharacteristic
20C22×C3×D5D414yes—
21C22×C3×D5C2311yescharacteristic
22C2×C3×AGL(1,𝔽5)—24no—
23C2×C3×AGL(1,𝔽5)—28no—
24C2×C3×C4×C5—22no—
25C2×C3×C4×C5C2×C412yes—
26C2×C3×Dic5—22no—
27C2×C3×Dic5C2×C412yes—
28C3×C4×D5—24no—
29C3×C4×D5—24no—
30C23×C3×C4—51no—
31C2×C3×C22⋊C4—52no—
32C3×#95—54no—
33C23×D5C22×C311yescharacteristic
34C22×AGL(1,𝔽5)C22×C314yes—
35C22×C4×C5C3×C411yescharacteristic, complemented
36C22×Dic5C3×C411yescharacteristic, complemented
37C2×C4×D5—24no—
38C2×C4×D5C3×C412yescomplemented
39C2×C4×D5C22×C312yes—
40C22⋊AGL(1,𝔽5)—24no—
41#381—51noSylow
42C22×C3×C5—22no—
43C22×C3×C5C22⋊C412yes—
44C22×C3×C5C22×C411yescharacteristic
45C2×C3×D5—22no—
46C2×C3×D5—28no—
47C2×C3×D5C22⋊C412yes—
48C2×C3×D5C2×D412yes—
49C2×C3×D5#4114yes—
50C3×AGL(1,𝔽5)—44no—
51C3×C4×C5—22no—
52C3×Dic5—22no—
53C24×C3—51no—
54C22×C3×C4—51no—
55C22×C3×C4—52no—
56C22×C3×C4—52no—
57C22×C3×C4—54no—
58C22×C3×C4—104no—
59C22×C3×C4AGL(1,𝔽5)11yescharacteristic, complemented
60C3×C22⋊C4—104no—
61C23×C5C2×C3×C411yescharacteristic
62C22×D5—24no—
63C22×D5C2×C3×C411yescharacteristic
64C22×D5C3×D414yes—
65C22×D5C23×C311yescharacteristic
66C2×AGL(1,𝔽5)—24no—
67C2×AGL(1,𝔽5)—28no—
68C2×C4×C5—22no—
69C2×C4×C5C2×C3×C412yes—
70C2×Dic5—22no—
71C2×Dic5C2×C3×C412yes—
72C4×D5—24no—
73C4×D5—24no—
74C23×C4—51no—
75C2×C22⋊C4—52no—
76#95—54no—
77C2×C3×C5—22no—
78C2×C3×C5C2×C22⋊C411yescharacteristic
79C2×C3×C5#11712yes—
80C3×D5—22no—
81C3×D5#12314yes—
82C23×C3—51no—
83C23×C3—54no—
84C23×C3—51no—
85C23×C3—104no—
86C23×C3C2×AGL(1,𝔽5)11yescharacteristic
87C2×C3×C4—22no—
88C2×C3×C4—52no—
89C2×C3×C4—104no—
90C2×C3×C4—104no—
91C2×C3×C4—102no—
92C2×C3×C4—104no—
93C2×C3×C4—108no—
94C2×C3×C4C2×AGL(1,𝔽5)12yes—
95C22×C5—22no—
96C22×C5C3×C22⋊C412yes—
97C22×C5C22×C3×C411yescharacteristic
98C2×D5—22no—
99C2×D5—28no—
100C2×D5C3×C22⋊C412yes—
101C2×D5C2×C3×D412yes—
102C2×D5C3×#4114yes—
103AGL(1,𝔽5)—44no—
104C4×C5—22no—
105Dic5—22no—
106C24—51no—
107C22×C4—51no—
108C22×C4—52no—
109C22×C4—52no—
110C22×C4—54no—
111C22×C4—104no—
112C22×C4C3×AGL(1,𝔽5)11yescharacteristic, complemented
113C22⋊C4—104no—
114C3×C5#38111yescharacteristic, complemented, Hall
115C22×C3—22no—
116C22×C3—54no—
117C22×C3—52no—
118C22×C3—52no—
119C22×C3—108no—
120C22×C3—102no—
121C22×C3C22⋊AGL(1,𝔽5)12yes—
122C22×C3C22×AGL(1,𝔽5)11yescharacteristic
123C3×C4—22no—
124C3×C4—102no—
125C3×C4—204no—
126C2×C5—22no—
127C2×C5C2×C3×C22⋊C411yescharacteristic
128C2×C5C3×#11712yes—
129D5—22no—
130D5C3×#12314yes—
131C23—51no—
132C23—54no—
133C23—51no—
134C23—104no—
135C23C2×C3×AGL(1,𝔽5)11yescharacteristic
136C2×C4—22no—
137C2×C4—52no—
138C2×C4—104no—
139C2×C4—104no—
140C2×C4—102no—
141C2×C4—104no—
142C2×C4—108no—
143C2×C4C2×C3×AGL(1,𝔽5)12yes—
144C2×C3—22no—
145C2×C3—54no—
146C2×C3—102no—
147C2×C3#413012yes—
148C2×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
149C5C3×#38111yescharacteristic, complemented, Sylow
150C22—22no—
151C22—54no—
152C22—52no—
153C22—52no—
154C22—108no—
155C22—102no—
156C22C3×C22⋊AGL(1,𝔽5)12yes—
157C22C22×C3×AGL(1,𝔽5)11yescharacteristic
158C4—22no—
159C4—102no—
160C4—204no—
161C3#6576411yescharacteristic, direct product factor, Sylow
162C2—22no—
163C2—54no—
164C2—102no—
165C2C3×#413012yes—
166C2C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC22×C3C22×AGL(1,𝔽5)
commutatorC22×C5C22×C3×C4
FrattiniC22C22×C3×AGL(1,𝔽5)
FittingC22×C3×C4×C5C4
socleC22×C3×C5C22×C4
prime radicalC23×C3×D5C22
cosocleC22×D5C23×C3
hypercenterC22×C3×C4AGL(1,𝔽5)
hypocenterC5C3×#381
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
3. C3×(C2×AGL(1,𝔽5))⋊C4C2semidirect
4. #65764C3direct (N × H)
7. C22×C3×C4×C5C4semidirect
8. C22×C3×Dic5C4semidirect
10. C2×C3×C4×D5C4semidirect
16. (C2×AGL(1,𝔽5))⋊C4C2×C3semidirect
35. C22×C4×C5C3×C4semidirect
36. C22×Dic5C3×C4semidirect
38. C2×C4×D5C3×C4semidirect
59. C22×C3×C4AGL(1,𝔽5)semidirect
112. C22×C4C3×AGL(1,𝔽5)semidirect
114. C3×C5#381semidirect
149. C5C3×#381semidirect
161. C3#65764direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×D52
C2×C3×C22⋊AGL(1,𝔽5)2
C3×(C2×AGL(1,𝔽5))⋊C42
#657643
C3×#3815