Finite Group Databasebeta

← C3×#65755 #11752288

Subgroups — C3×#65755 #11752288

Order 960.

380 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 112 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×#704C212yesmaximal, complemented
3C3×#4003C214yesmaximal, complemented
4#65755C311yesmaximal, characteristic, direct product factor, Hall
5C23×C3×D5C411yescharacteristic
6C22×C3×C4×C5C411yescharacteristic
7C22×C3×Dic5C2211yescharacteristic
8C2×C3×C4×D5—24no—
9C2×C3×C4×D5C412yes—
10C2×C3×C4×D5C2212yes—
11C3×#704—24no—
12C2×C3×#170C2214yescomplemented
13C3×#407—51nomaximal, Hall
14C22×C4×D5C2×C311yescharacteristic
15C2×#704C2×C312yescomplemented
16#4003C2×C314yescomplemented
17C23×C3×C5C2×C411yescharacteristic
18C22×C3×D5—22no—
19C22×C3×D5—24no—
20C22×C3×D5C2×C412yes—
21C2×C3×C4×C5—22no—
22C2×C3×C4×C5C2×C412yes—
23C2×C3×Dic5C2×C411yescharacteristic
24C2×C3×Dic5D414yes—
25C2×C3×Dic5C2311yescharacteristic
26C3×C4×D5—24no—
27C3×C4×D5—24no—
28C3×#170—24no—
29C23×C3×C4—51no—
30C2×C3×M16—52no—
31C3×C22⋊C8—54no—
32C23×D5C3×C411yescharacteristic
33C22×C4×C5C3×C411yescharacteristic
34C22×Dic5C22×C311yescharacteristic
35C2×C4×D5—24no—
36C2×C4×D5C3×C412yes—
37C2×C4×D5C22×C312yes—
38#704—24no—
39C2×#170C22×C314yescomplemented
40#407—51noSylow
41C22×C3×C5—22no—
42C22×C3×C5C22⋊C412yes—
43C22×C3×C5C22×C411yescharacteristic
44C2×C3×D5—28no—
45C2×C3×D5—22no—
46C2×C3×D5—22no—
47C2×C3×D5M1614yescomplemented
48C3×C4×C5—22no—
49C3×Dic5C22⋊C412yes—
50C3×Dic5C2×D412yes—
51C24×C3—51no—
52C22×C3×C4—51no—
53C22×C3×C4—52no—
54C22×C3×C4—52no—
55C22×C3×C4—104no—
56C22×C3×C4AGL(1,𝔽5)11yescharacteristic
57C3×M16—104no—
58C2×C3×C8—54no—
59C23×C5C2×C3×C411yescharacteristic
60C22×D5—22no—
61C22×D5—24no—
62C22×D5C2×C3×C412yes—
63C2×C4×C5—22no—
64C2×C4×C5C2×C3×C412yes—
65C2×Dic5C2×C3×C411yescharacteristic
66C2×Dic5C3×D414yes—
67C2×Dic5C23×C311yescharacteristic
68C4×D5—24no—
69C4×D5—24no—
70#170—24no—
71C23×C4—51no—
72C2×M16—52no—
73C22⋊C8—54no—
74C2×C3×C5—22no—
75C2×C3×C5C2×C22⋊C411yescharacteristic
76C2×C3×C5C2×M1612yes—
77C3×D5—24no—
78C23×C3—52no—
79C23×C3—104no—
80C23×C3—102no—
81C23×C3C2×AGL(1,𝔽5)11yescharacteristic
82C2×C3×C4—22no—
83C2×C3×C4—54no—
84C2×C3×C4—51no—
85C2×C3×C4—51no—
86C2×C3×C4—104no—
87C2×C3×C4—104no—
88C2×C3×C4C2×AGL(1,𝔽5)12yes—
89C3×C8—104no—
90C22×C5—22no—
91C22×C5C3×C22⋊C412yes—
92C22×C5C22×C3×C411yescharacteristic
93C2×D5—28no—
94C2×D5—22no—
95C2×D5—22no—
96C2×D5C3×M1614yescomplemented
97C4×C5—22no—
98Dic5C3×C22⋊C412yes—
99Dic5C2×C3×D412yes—
100C24—51no—
101C22×C4—51no—
102C22×C4—52no—
103C22×C4—52no—
104C22×C4—104no—
105C22×C4C3×AGL(1,𝔽5)11yescharacteristic
106M16—104no—
107C2×C8—54no—
108C3×C5#40711yescharacteristic, complemented, Hall
109C22×C3—22no—
110C22×C3—54no—
111C22×C3—108no—
112C22×C3—102no—
113C22×C3—102no—
114C22×C3C22⋊AGL(1,𝔽5)12yes—
115C22×C3C22×AGL(1,𝔽5)11yescharacteristic
116C3×C4—22no—
117C3×C4—52no—
118C3×C4—52no—
119C2×C5—22no—
120C2×C5C2×C3×C22⋊C411yescharacteristic
121C2×C5C2×C3×M1612yes—
122D5—24no—
123C23—52no—
124C23—104no—
125C23—102no—
126C23C2×C3×AGL(1,𝔽5)11yescharacteristic
127C2×C4—22no—
128C2×C4—54no—
129C2×C4—51no—
130C2×C4—51no—
131C2×C4—104no—
132C2×C4—104no—
133C2×C4C2×C3×AGL(1,𝔽5)12yes—
134C8—104no—
135C2×C3—22no—
136C2×C3—104no—
137C2×C3D5⋊M1612yes—
138C2×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
139C5C3×#40711yescharacteristic, complemented, Sylow
140C22—22no—
141C22—54no—
142C22—108no—
143C22—102no—
144C22—102no—
145C22C3×C22⋊AGL(1,𝔽5)12yes—
146C22C22×C3×AGL(1,𝔽5)11yescharacteristic
147C4—22no—
148C4—52no—
149C4—52no—
150C3#6575511yescharacteristic, direct product factor, Sylow
151C2—22no—
152C2—104no—
153C2C3×D5⋊M1612yes—
154C2C2×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×D5C4
cosocleC2×Dic5C23×C3
hypercenterC22×C3×C4AGL(1,𝔽5)
hypocenterC5C3×#407
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#704C2semidirect
3. C3×#4003C2semidirect
4. #65755C3direct (N × H)
12. C2×C3×#170C22semidirect
15. C2×#704C2×C3semidirect
16. #4003C2×C3semidirect
39. C2×#170C22×C3semidirect
47. C2×C3×D5M16semidirect
96. C2×D5C3×M16semidirect
108. C3×C5#407semidirect
139. C5C3×#407semidirect
150. C3#65755direct (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×#7042
C3×#40032
#657553
C3×#4075