Finite Group Databasebeta

← #13088435

Subgroups — #13088435

Order 1440.

270 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 64 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
1#10624052C211yesmaximal, characteristic
2#10623983C211yesmaximal, characteristic, complemented
3#10623976C212yesmaximal, complemented
4#10623967C211yesmaximal, characteristic, complemented
5C3×#6738C212yesmaximal, complemented
6#91914—32nomaximal
7#67157C411yescharacteristic
8#67151C412yes—
9C5×#627C411yescharacteristic
10#67137C2211yescharacteristic
11C3×#1200C2212yes—
12C3⋊(C3⋊#170)C2211yescharacteristic
13C3×C3⋊#170C2212yes—
14C32×#170C2211yescharacteristic
15#63803—51nomaximal, Hall
16#6773—32no—
17#6758—32no—
18C3×D5⋊C8—32no—
19#6740—32no—
20#6740—32no—
21#6738—32no—
22#6738—61no—
23#6738D312yescomplemented
24C2×C3⋊D15C2×C411yescharacteristic
25C2×C3×D15C2×C412yes—
26C5×C3⋊Dic3C2×C411yescharacteristic
27C3×C5×Dic3C2×C412yes—
28C32×Dic5C2311yescharacteristic
29#4133—91noHall
30#3798—51no—
31C8×C3⋊D3—51no—
32C3×D3×C8—52no—
33#3712—51no—
34#3711—52no—
35D60—32no—
36C5×Dic6—32no—
37C3×C4×D5—32no—
38#1202—32no—
39#1202—32no—
40#1200—32no—
41#1200—61no—
42#1200C2×D312yes—
43C3⋊#170—32no—
44C3⋊#170—61no—
45C3⋊#170C2×D312yes—
46C3×#170—21no—
47C3×#170—32no—
48C3×#170C2×D312yes—
49#901—152no—
50C2×C32×C5C22×C411yescharacteristic
51C3⋊D15—21no—
52C3×D15—22no—
53#713—91no—
54#700—91no—
55#700—92no—
56D5⋊C8—32no—
57D5⋊C8—91no—
58C4×C3⋊D3—51no—
59C3×C4×D3—52no—
60#627AGL(1,𝔽5)11yescharacteristic
61#626—52no—
62#625—51no—
63C32×C8—51no—
64C3⋊(C3⋊C8)—51no—
65C3×C3⋊C8—52no—
66C2×D15—32no—
67C2×D15—61no—
68C2×D15C4×D312yes—
69C2×C3×D5—32no—
70C3×C4×C5—32no—
71C3×Dic5—21no—
72C3×Dic5C22×D312yes—
73C5×Dic3—32no—
74C5×Dic3—61no—
75C5×Dic3C4×D312yes—
76#235—152no—
77C2×C3×C8—152no—
78#208—152no—
79ΓL(1,𝔽25)—152no—
80ΓL(1,𝔽25)—152no—
81D3×C8—52no—
82D3×C8—152no—
83D3×C8—301no—
84C32×C5#13111yescharacteristic, complemented, Hall
85C5×Q8—91no—
86D20—92no—
87D20—91no—
88C4×D5—32no—
89C4×D5—91no—
90#170—32no—
91#170—91no—
92#170D3211yescharacteristic
93C2×C3⋊D3—51no—
94C2×C3×D3—52no—
95C32×C4—51no—
96C3⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
97C3×Dic3C2×AGL(1,𝔽5)12yes—
98#131—451noSylow
99C2×C3×C5—21no—
100C2×C3×C5C2×C4×D312yes—
101D15—22no—
102D15—62no—
103D15—62no—
104C3×D5—62no—
105C2×C3×C4—152no—
106#67—152no—
107#67—152no—
108D12—152no—
109C4×D3—52no—
110C4×D3—152no—
111C4×D3—301no—
112Dic6—32no—
113C3×C8—52no—
114C3×C8—101no—
115C3×C8—152no—
116C3⋊C8—52no—
117C3⋊C8—152no—
118C3⋊C8—301no—
119C2×D5—32no—
120C2×D5—91no—
121C4×C5—32no—
122C4×C5—91no—
123Dic5C2×D3211yescharacteristic
124C2×C32C22×AGL(1,𝔽5)11yescharacteristic
125C3⋊D3—101no—
126C3×D3—102no—
127#41—451no—
128M16—451no—
129M16—452no—
130C2×C8—152no—
131C2×C8—451no—
132C3×C5—21no—
133C3×C5#90112yescomplemented
134C22×C3—152no—
135C2×D3—52no—
136C2×D3—152no—
137C2×D3—301no—
138C3×C4—32no—
139C3×C4—52no—
140C3×C4—101no—
141Dic3—32no—
142Dic3—61no—
143Dic3D3×AGL(1,𝔽5)12yes—
144C2×C5C4×D3211yescharacteristic
145D5—62no—
146D5—181no—
147C32#413311yescharacteristic, complemented, Sylow
148Q8—91no—
149D4—452no—
150D4—451no—
151C2×C4—152no—
152C2×C4—451no—
153C8—51no—
154C8—152no—
155C8—451no—
156C2×C3—21no—
157C2×C3—302no—
158C2×C3C2×D3×AGL(1,𝔽5)12yes—
159D3—102no—
160D3—302no—
161D3—302no—
162C5#6380311yescharacteristic, complemented, Sylow
163C22—152no—
164C22—451no—
165C4—32no—
166C4—51no—
167C4—91no—
168C3—21no—
169C3#9191412yescomplemented
170C2—302no—
171C2—901no—
172C2D32×AGL(1,𝔽5)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D32×AGL(1,𝔽5)
commutator, Fitting, socleC2×C32×C5C22×C4
prime radical#10624052C2
cosocleC32×Dic5C23
hypocenterC32×C5#131
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
2. #10623983C2semidirect
3. #10623976C2semidirect
4. #10623967C2semidirect
5. C3×#6738C2semidirect
23. #6738D3semidirect
84. C32×C5#131semidirect
133. C3×C5#901semidirect
147. C32#4133semidirect
162. C5#63803semidirect
169. C3#91914semidirect

Maximal subgroups

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

subgroupindex
#106240522
#106239832
#106239762
#106239672
C3×#67382
#919143
#638035