Finite Group Databasebeta

← #13086927

Subgroups — #13086927

Order 1440.

468 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 108 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×#67138C211yesmaximal, characteristic
2C2×C3⋊D3×Dic5C211yesmaximal, characteristic, complemented
3C4⋊(C3⋊Dic15)C211yesmaximal, characteristic, complemented
4C5×#3752C211yesmaximal, characteristic, complemented
5C32×#685C211yesmaximal, characteristic, complemented
6#10623775C211yesmaximal, characteristic, complemented
7#10623770C211yesmaximal, characteristic, complemented
8#91457—34nomaximal
9C22×C5×C3⋊D3C2211yescharacteristic
10C22×C32×D5C2211yescharacteristic
11C2×C32×C4×C5C2211yescharacteristic
12C2×C3⋊Dic15C2211yescharacteristic
13C2×C3⋊Dic15C2211yescharacteristic
14C2×C5×C3⋊Dic3C2211yescharacteristic
15C2×C32×Dic5C2211yescharacteristic
16#67138—22no—
17C3⋊D3×Dic5—22no—
18#64092—51nomaximal, Hall
19C2×#1201—34no—
20C2×D3×Dic5—34no—
21C4⋊Dic15—34no—
22C5×(C2×D3)⋊C4—34no—
23C3×#685D314yescomplemented
24Dic3⋊Dic5—34no—
25#6665—34no—
26C22×C32×C5C2311yescharacteristic
27C2×C5×C3⋊D3—21no—
28C2×C5×C3⋊D3—21no—
29C2×C5×C3⋊D3D412yes—
30C2×C32×D5—21no—
31C2×C32×D5—21no—
32C2×C32×D5—21no—
33C32×C4×C5—21no—
34C3⋊Dic15—21no—
35C3⋊Dic15—21no—
36C5×C3⋊Dic3—21no—
37C32×Dic5—21no—
38#4032—91noHall
39C2×#638—51no—
40C2×C4×C3⋊D3—51no—
41C32×C22⋊C4—51no—
42#3757—51no—
43#3752D511yescharacteristic, complemented
44C4⋊(C3⋊Dic3)—51no—
45#3750—51no—
46C22×C5×D3—34no—
47C22×C3×D5C2×D314yes—
48C2×C3×C4×C5C2×D314yes—
49C2×Dic15—34no—
50C2×Dic15—34no—
51C2×C5×Dic3—34no—
52C2×C3×Dic5C2×D314yes—
53#1201—68no—
54D3×Dic5—68no—
55#883—154no—
56C2×C32×C5C2×D411yescharacteristic
57C2×C32×C5#4111yescharacteristic
58C2×C32×C5#4111yescharacteristic
59C5×C3⋊D3—22no—
60C32×D5—41no—
61C2×C22⋊D5—91no—
62C22×Dic5—91no—
63C5×C22⋊C4—91no—
64#685—91no—
65#685C3⋊D311yescharacteristic, complemented
66C4⋊Dic5—91no—
67C4⋊Dic5—91no—
68C23×C32—51no—
69C22×C3⋊D3C2×D511yescharacteristic
70C2×C32×C4—51no—
71C2×C32×C4C2×D511yescharacteristic
72#638—102no—
73C2×C3⋊Dic3—51no—
74C2×C3⋊Dic3—51no—
75C2×C3⋊Dic3C2×D511yescharacteristic
76C4×C3⋊D3—102no—
77C22×C3×C5C22×D314yes—
78C2×C5×D3—38no—
79C2×C5×D3—64no—
80C2×C5×D3—64no—
81C2×C3×D5—24no—
82C2×C3×D5—24no—
83C2×C3×D5—24no—
84C3×C4×C5—24no—
85Dic15—64no—
86Dic15—64no—
87C3×Dic5—24no—
88C5×Dic3—64no—
89C2×#67—154no—
90C2×C4×D3—154no—
91C3×C22⋊C4—54no—
92#217—154no—
93(C2×D3)⋊C4—34no—
94C4⋊Dic3—154no—
95Dic3⋊C4—154no—
96C32×C5#12311yescharacteristic, complemented, Hall
97C23×C5—91no—
98C22×D5C2×C3⋊D311yescharacteristic
99C2×C4×C5—91no—
100C2×C4×C5C2×C3⋊D311yescharacteristic
101C22⋊D5—182no—
102C2×Dic5—91no—
103C2×Dic5—91no—
104C2×Dic5—182no—
105C2×Dic5C2×C3⋊D311yescharacteristic
106C22×C32—101no—
107C22×C32—101no—
108C22×C32—101no—
109C22×C32C22×D511yescharacteristic
110C2×C3⋊D3—21no—
111C2×C3⋊D3—21no—
112C2×C3⋊D3D2012yes—
113C32×C4—21no—
114C32×C4—101no—
115C3⋊Dic3—21no—
116C3⋊Dic3—101no—
117C3⋊Dic3—101no—
118#123—451noSylow
119C2×C3×C5#23514yes—
120C2×C3×C5D3×D414yes—
121C2×C3×C5#23714yes—
122C3×D5—44no—
123C5×D3—68no—
124C23×C3—54no—
125C22×D3—34no—
126C2×C3×C4—54no—
127C2×C3×C4D3×D514yes—
128#67—308no—
129C2×Dic3—34no—
130C2×Dic3—154no—
131C2×Dic3—154no—
132C4×D3—308no—
133C22×C5—92no—
134C22×C5—181no—
135C22×C5—181no—
136C22×C5C22×C3⋊D311yescharacteristic
137C2×D5—21no—
138C2×D5—21no—
139C2×D5—21no—
140C4×C5—21no—
141C4×C5—181no—
142Dic5—21no—
143Dic5—181no—
144Dic5—181no—
145C2×C32—201no—
146C2×C32C2×D2011yescharacteristic
147C2×C32#71111yescharacteristic
148C2×C32#71111yescharacteristic
149C3⋊D3—22no—
150C2×D4—451no—
151C22×C4—451no—
152C4⋊C4—451no—
153C4⋊C4—451no—
154C22⋊C4—51no—
155C22⋊C4—91no—
156C22⋊C4—451no—
157C3×C5#88314yescomplemented
158C22×C3—104no—
159C22×C3—104no—
160C22×C3—104no—
161C22×C3C2×D3×D514yes—
162C2×D3—38no—
163C2×D3—64no—
164C2×D3—64no—
165C3×C4—24no—
166C3×C4—104no—
167Dic3—64no—
168Dic3—304no—
169Dic3—304no—
170C2×C5—182no—
171C2×C5#382811yescharacteristic
172C2×C5D4×C3⋊D311yescharacteristic
173C2×C5#383011yescharacteristic
174D5—41no—
175C32#403211yescharacteristic, complemented, Sylow
176C23—51no—
177C23—91no—
178D4—902no—
179C2×C4—51no—
180C2×C4—91no—
181C2×C4—451no—
182C2×C4—451no—
183C2×C4—902no—
184C2×C4D5×C3⋊D311yescharacteristic
185C2×C3—204no—
186C2×C3#677214yes—
187C2×C3D3×D2014yes—
188C2×C3#678014yes—
189D3—68no—
190C5#6409211yescharacteristic, complemented, Sylow
191C22—92no—
192C22—101no—
193C22—101no—
194C22—101no—
195C22—181no—
196C22—181no—
197C22C2×D5×C3⋊D311yescharacteristic
198C4—21no—
199C4—101no—
200C4—181no—
201C4—901no—
202C4—901no—
203C3#9145714yescomplemented
204C2—182no—
205C2—201no—
206C2#1062415511yescharacteristic
207C2C3⋊D3×D2011yescharacteristic
208C2#1062416311yescharacteristic
subgroupG/H
center, FrattiniC22C2×D5×C3⋊D3
commutator, socle, cosocleC22×C32×C5C23
FittingC2×C32×C4×C5C22
prime radicalC2×#67138C2
hypercenterC2×C4D5×C3⋊D3
hypocenterC32×C5#123
LCSG ▷ C22×C32×C5 ▷ C32×C5
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊D3×Dic5C2semidirect
3. C4⋊(C3⋊Dic15)C2semidirect
4. C5×#3752C2semidirect
5. C32×#685C2semidirect
6. #10623775C2semidirect
7. #10623770C2semidirect
23. C3×#685D3semidirect
43. #3752D5semidirect
65. #685C3⋊D3semidirect
96. C32×C5#123semidirect
157. C3×C5#883semidirect
175. C32#4032semidirect
190. C5#64092semidirect
203. C3#91457semidirect

Maximal subgroups

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

subgroupindex
C2×#671382
C2×C3⋊D3×Dic52
C4⋊(C3⋊Dic15)2
C5×#37522
C32×#6852
#106237752
#106237702
#914573
#640925