Finite Group Databasebeta

← #13086868

Subgroups — #13086868

Order 1440.

552 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 134 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×C4×C3⋊D15C211yesmaximal, characteristic
2C2×#67137C211yesmaximal, characteristic
3C2×#67137C211yesmaximal, characteristic
4C5×#3750C211yesmaximal, characteristic, complemented
5C32×#683C211yesmaximal, characteristic, complemented
6#10623775C211yesmaximal, characteristic, complemented
7#10623773C211yesmaximal, characteristic, complemented
8#91404—34nomaximal
9C22×C3⋊D15C2211yescharacteristic
10C2×C32×C4×C5C2211yescharacteristic
11C2×C3⋊Dic15C2211yescharacteristic
12C4×C3⋊D15—22no—
13C2×C5×C3⋊Dic3C2211yescharacteristic
14C2×C5×C3⋊Dic3C2211yescharacteristic
15C2×C32×Dic5C2211yescharacteristic
16C2×C32×Dic5C2211yescharacteristic
17#67137—22no—
18#67137C414yescomplemented
19C4⋊C4×C3⋊D3—51nomaximal, Hall
20C2×C4×D15—34no—
21C2×#1200—34no—
22C2×#1200—34no—
23C5×Dic3⋊C4—34no—
24C3×#683D314yescomplemented
25Dic3⋊Dic5—34no—
26#6668—34no—
27C22×C32×C5C2311yescharacteristic
28C2×C3⋊D15C2×C412yes—
29C2×C3⋊D15D412yes—
30C2×C3⋊D15Q812yes—
31C32×C4×C5—21no—
32C3⋊Dic15—21no—
33C5×C3⋊Dic3—21no—
34C5×C3⋊Dic3C2×C412yescomplemented
35C32×Dic5—21no—
36C32×Dic5C2×C412yescomplemented
37D5×C4⋊C4—91noHall
38C2×C4×C3⋊D3—51no—
39C2×C4×C3⋊D3—51no—
40C2×C4×C3⋊D3—51no—
41C32×C4⋊C4—51no—
42C4⋊(C3⋊Dic3)—51no—
43#3750—51no—
44#3750D511yescharacteristic, complemented
45C22×D15—34no—
46C2×C3×C4×C5C2×D314yes—
47C2×Dic15—34no—
48C4×D15—68no—
49C2×C5×Dic3—34no—
50C2×C5×Dic3—34no—
51C2×C3×Dic5C2×D314yes—
52C2×C3×Dic5C2×D314yes—
53#1200—316no—
54#1200—68no—
55D3×C4⋊C4—154no—
56C2×C32×C5C22×C411yescharacteristic
57C2×C32×C5C2×D411yescharacteristic
58C2×C32×C5C2×Q811yescharacteristic
59C3⋊D15C4⋊C414yescomplemented
60C2×C4×D5—91no—
61C2×C4×D5—91no—
62C2×C4×D5—91no—
63C5×C4⋊C4—91no—
64C4⋊Dic5—91no—
65#683—91no—
66#683C3⋊D311yescharacteristic, complemented
67C22×C3⋊D3—51no—
68C2×C32×C4—51no—
69C2×C32×C4—51no—
70C2×C32×C4C2×D511yescharacteristic
71C2×C3⋊Dic3—51no—
72C2×C3⋊Dic3C2×D511yescharacteristic
73C2×C3⋊Dic3C2×D511yescharacteristic
74C4×C3⋊D3—54no—
75C4×C3⋊D3—102no—
76C4×C3⋊D3—102no—
77C22×C3×C5C22×D314yes—
78C2×D15—38no—
79C2×D15—38no—
80C2×D15—38no—
81C3×C4×C5—24no—
82Dic15—64no—
83C3×Dic5—24no—
84C3×Dic5C4×D318yescomplemented
85C5×Dic3—38no—
86C5×Dic3—64no—
87C2×C4×D3—154no—
88C2×C4×D3—154no—
89C2×C4×D3—154no—
90C3×C4⋊C4—54no—
91C4⋊Dic3—154no—
92Dic3⋊C4—34no—
93Dic3⋊C4—154no—
94C32×C5C2×C4⋊C411yescharacteristic, complemented, Hall
95C22×D5—91no—
96C2×C4×C5—91no—
97C2×C4×C5—91no—
98C2×C4×C5C2×C3⋊D311yescharacteristic
99C2×Dic5—91no—
100C2×Dic5C2×C3⋊D311yescharacteristic
101C2×Dic5C2×C3⋊D311yescharacteristic
102C4×D5—94no—
103C4×D5—182no—
104C4×D5—182no—
105C22×C32C22×D511yescharacteristic
106C2×C3⋊D3—52no—
107C2×C3⋊D3—52no—
108C2×C3⋊D3—52no—
109C32×C4—21no—
110C32×C4—52no—
111C32×C4—101no—
112C3⋊Dic3—21no—
113C3⋊Dic3—101no—
114C3⋊Dic3C4×D512yescomplemented
115C2×C4⋊C4—451noSylow
116C2×C3×C5C2×C4×D314yes—
117C2×C3×C5D3×D414yes—
118C2×C3×C5D3×Q814yes—
119D15—316no—
120C22×D3—154no—
121C2×C3×C4—54no—
122C2×C3×C4—54no—
123C2×C3×C4D3×D514yes—
124C2×Dic3—34no—
125C2×Dic3—34no—
126C2×Dic3—154no—
127C4×D3—1516no—
128C4×D3—308no—
129C4×D3—308no—
130C22×C5C22×C3⋊D311yescharacteristic
131C2×D5—92no—
132C2×D5—92no—
133C2×D5—92no—
134C4×C5—21no—
135C4×C5—92no—
136C4×C5—181no—
137Dic5—21no—
138Dic5—181no—
139Dic5C4×C3⋊D312yescomplemented
140C2×C32C2×C4×D511yescharacteristic
141C2×C32D4×D511yescharacteristic
142C2×C32Q8×D511yescharacteristic
143C3⋊D3—54no—
144C22×C4—451no—
145C22×C4—451no—
146C22×C4—451no—
147C4⋊C4—51no—
148C4⋊C4—91no—
149C4⋊C4—451no—
150C4⋊C4—451no—
151C3×C5D3×C4⋊C414yescomplemented
152C22×C3C2×D3×D514yes—
153C2×D3—158no—
154C2×D3—158no—
155C2×D3—158no—
156C3×C4—24no—
157C3×C4—58no—
158C3×C4—104no—
159Dic3—38no—
160Dic3—64no—
161Dic3—304no—
162C2×C5C2×C4×C3⋊D311yescharacteristic
163C2×C5D4×C3⋊D311yescharacteristic
164C2×C5Q8×C3⋊D311yescharacteristic
165D5—94no—
166C32D5×C4⋊C411yescharacteristic, complemented, Sylow
167C23—451no—
168C2×C4—51no—
169C2×C4—51no—
170C2×C4—91no—
171C2×C4—91no—
172C2×C4—454no—
173C2×C4—451no—
174C2×C4—902no—
175C2×C4—902no—
176C2×C4D5×C3⋊D311yescharacteristic
177C2×C3D15⋊Q814yes—
178C2×C3C4×D3×D514yes—
179C2×C3#679014yes—
180D3—1516no—
181C5C4⋊C4×C3⋊D311yescharacteristic, complemented, Sylow
182C22—452no—
183C22—452no—
184C22—452no—
185C22C2×D5×C3⋊D311yescharacteristic
186C4—21no—
187C4—52no—
188C4—92no—
189C4—101no—
190C4—181no—
191C4—901no—
192C3#9140414yescomplemented
193C2—454no—
194C2#1062415311yescharacteristic
195C2C4×D5×C3⋊D311yescharacteristic
196C2#1062417311yescharacteristic
subgroupG/H
center, FrattiniC22C2×D5×C3⋊D3
commutatorC2×C32×C5C22×C4
FittingC2×C32×C4×C5C22
socle, cosocleC22×C32×C5C23
prime radicalC22×C3⋊D15C22
hypercenterC2×C4D5×C3⋊D3
hypocenterC32×C5C2×C4⋊C4
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. C5×#3750C2semidirect
5. C32×#683C2semidirect
6. #10623775C2semidirect
7. #10623773C2semidirect
18. #67137C4semidirect
24. C3×#683D3semidirect
34. C5×C3⋊Dic3C2×C4semidirect
36. C32×Dic5C2×C4semidirect
44. #3750D5semidirect
59. C3⋊D15C4⋊C4semidirect
66. #683C3⋊D3semidirect
84. C3×Dic5C4×D3semidirect
94. C32×C5C2×C4⋊C4semidirect
114. C3⋊Dic3C4×D5semidirect
139. Dic5C4×C3⋊D3semidirect
151. C3×C5D3×C4⋊C4semidirect
166. C32D5×C4⋊C4semidirect
181. C5C4⋊C4×C3⋊D3semidirect
192. C3#91404semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C3⋊D152
C2×#671372
C5×#37502
C32×#6832
#106237752
#106237732
#914043
C4⋊C4×C3⋊D35