Finite Group Databasebeta

← #13087040

Subgroups — #13087040

Order 1440.

468 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 106 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×#67184C211yesmaximal, characteristic
2C2×#67137C211yesmaximal, characteristic, complemented
3C5×#3757C211yesmaximal, characteristic, complemented
4C32×C22⋊Dic5C211yesmaximal, characteristic, complemented
5#10623775C211yesmaximal, characteristic, complemented
6#10623773C211yesmaximal, characteristic, complemented
7#10623772C211yesmaximal, characteristic, complemented
8#91562—34nomaximal
9C23×C32×C5C2211yescharacteristic
10C22×C3⋊D15C2211yescharacteristic
11#67184—22no—
12C2×C3⋊Dic15C2211yescharacteristic
13C2×C5×C3⋊Dic3C2211yescharacteristic
14C2×C5×C3⋊Dic3C2211yescharacteristic
15C2×C32×Dic5C2211yescharacteristic
16C2×C32×Dic5C2211yescharacteristic
17#67137—22no—
18#64092—51nomaximal, Hall
19C2×#1220—34no—
20C2×#1200—34no—
21C5×#217—34no—
22C3×C22⋊Dic5D314yescomplemented
23Dic3⋊Dic5—34no—
24#6668—34no—
25#6667—34no—
26C22×C32×C5—21no—
27C22×C32×C5—21no—
28C22×C32×C5—21no—
29C22×C32×C5C2311yescharacteristic
30C2×C3⋊D15—21no—
31C2×C3⋊D15—21no—
32C2×C3⋊D15D412yes—
33C3⋊Dic15—21no—
34C5×C3⋊Dic3—21no—
35C5×C3⋊Dic3—21no—
36C32×Dic5—21no—
37C32×Dic5—21no—
38#4029—91noHall
39C2×#638—51no—
40C2×C4×C3⋊D3—51no—
41C32×C22⋊C4—51no—
42#3757D511yescharacteristic, complemented
43#3752—51no—
44C4⋊(C3⋊Dic3)—51no—
45#3750—51no—
46C23×C3×C5C2×D314yes—
47C22×D15—34no—
48#1220—68no—
49C2×Dic15—34no—
50C2×C5×Dic3—34no—
51C2×C5×Dic3—34no—
52C2×C3×Dic5C2×D314yes—
53C2×C3×Dic5C2×D314yes—
54#1200—68no—
55#883—154no—
56C2×C32×C5—41no—
57C2×C32×C5C2×D411yescharacteristic
58C2×C32×C5#4111yescharacteristic
59C2×C32×C5#4111yescharacteristic
60C3⋊D15—22no—
61C2×C22⋊D5—91no—
62C2×C4×D5—91no—
63C5×C22⋊C4—91no—
64C22⋊Dic5C3⋊D311yescharacteristic, complemented
65#685—91no—
66C4⋊Dic5—91no—
67#683—91no—
68C23×C32C2×D511yescharacteristic
69C22×C3⋊D3—51no—
70C2×C32×C4—51no—
71C2×C32×C4—51no—
72#638—102no—
73C2×C3⋊Dic3—51no—
74C2×C3⋊Dic3C2×D511yescharacteristic
75C2×C3⋊Dic3C2×D511yescharacteristic
76C4×C3⋊D3—102no—
77C22×C3×C5—24no—
78C22×C3×C5—24no—
79C22×C3×C5—24no—
80C22×C3×C5C22×D314yes—
81C2×D15—38no—
82C2×D15—64no—
83C2×D15—64no—
84Dic15—64no—
85C3×Dic5—24no—
86C3×Dic5—24no—
87C5×Dic3—64no—
88C5×Dic3—64no—
89C2×#67—154no—
90C2×C4×D3—154no—
91C3×C22⋊C4—54no—
92#217—34no—
93(C2×D3)⋊C4—154no—
94C4⋊Dic3—154no—
95Dic3⋊C4—154no—
96C32×C5#12311yescharacteristic, complemented, Hall
97C23×C5C2×C3⋊D311yescharacteristic
98C22×D5—91no—
99C2×C4×C5—91no—
100C2×C4×C5—91no—
101C22⋊D5—182no—
102C2×Dic5—91no—
103C2×Dic5C2×C3⋊D311yescharacteristic
104C2×Dic5C2×C3⋊D311yescharacteristic
105C4×D5—182no—
106C22×C32—21no—
107C22×C32—21no—
108C22×C32—21no—
109C22×C32C22×D511yescharacteristic
110C2×C3⋊D3—52no—
111C2×C3⋊D3—101no—
112C2×C3⋊D3—101no—
113C32×C4—101no—
114C32×C4—101no—
115C3⋊Dic3—21no—
116C3⋊Dic3—21no—
117C3⋊Dic3—101no—
118#123—451noSylow
119C2×C3×C5—44no—
120C2×C3×C5#23514yes—
121C2×C3×C5D3×D414yes—
122C2×C3×C5#23714yes—
123D15—68no—
124C23×C3D3×D514yes—
125C22×D3—154no—
126C2×C3×C4—54no—
127C2×C3×C4—54no—
128#67—308no—
129C2×Dic3—34no—
130C2×Dic3—34no—
131C2×Dic3—154no—
132C4×D3—308no—
133C22×C5—21no—
134C22×C5—21no—
135C22×C5—21no—
136C22×C5C22×C3⋊D311yescharacteristic
137C2×D5—92no—
138C2×D5—181no—
139C2×D5—181no—
140C4×C5—181no—
141C4×C5—181no—
142Dic5—21no—
143Dic5—21no—
144Dic5—181no—
145C2×C32—41no—
146C2×C32#70911yescharacteristic
147C2×C32D4×D511yescharacteristic
148C2×C32#71111yescharacteristic
149C3⋊D3—102no—
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—24no—
159C22×C3—24no—
160C22×C3—24no—
161C22×C3C2×D3×D514yes—
162C2×D3—158no—
163C2×D3—304no—
164C2×D3—304no—
165C3×C4—104no—
166C3×C4—104no—
167Dic3—64no—
168Dic3—64no—
169Dic3—304no—
170C2×C5—41no—
171C2×C5#382811yescharacteristic
172C2×C5D4×C3⋊D311yescharacteristic
173C2×C5#383011yescharacteristic
174D5—182no—
175C32#402911yescharacteristic, complemented, Sylow
176C23—451no—
177C23D5×C3⋊D311yescharacteristic
178D4—902no—
179C2×C4—51no—
180C2×C4—51no—
181C2×C4—91no—
182C2×C4—91no—
183C2×C4—451no—
184C2×C4—902no—
185C2×C3—44no—
186C2×C3#677914yes—
187C2×C3#678214yes—
188C2×C3#679014yes—
189D3—308no—
190C5#6409211yescharacteristic, complemented, Sylow
191C22—21no—
192C22—21no—
193C22—21no—
194C22—452no—
195C22—901no—
196C22—901no—
197C22C2×D5×C3⋊D311yescharacteristic
198C4—101no—
199C4—101no—
200C4—181no—
201C4—181no—
202C4—901no—
203C3#9156214yescomplemented
204C2—41no—
205C2—902no—
206C2#1062416211yescharacteristic
207C2#1062416511yescharacteristic
208C2#1062417311yescharacteristic
subgroupG/H
center, FrattiniC22C2×D5×C3⋊D3
commutator, socle, cosocleC22×C32×C5C23
FittingC23×C32×C5C22
prime radicalC2×#67184C2
hypercenterC23D5×C3⋊D3
hypocenterC32×C5#123
LCSG ▷ C22×C32×C5 ▷ C32×C5
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#67137C2semidirect
3. C5×#3757C2semidirect
4. C32×C22⋊Dic5C2semidirect
5. #10623775C2semidirect
6. #10623773C2semidirect
7. #10623772C2semidirect
22. C3×C22⋊Dic5D3semidirect
42. #3757D5semidirect
64. C22⋊Dic5C3⋊D3semidirect
96. C32×C5#123semidirect
157. C3×C5#883semidirect
175. C32#4029semidirect
190. C5#64092semidirect
203. C3#91562semidirect

Maximal subgroups

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

subgroupindex
C2×#671842
C2×#671372
C5×#37572
C32×C22⋊Dic52
#106237752
#106237732
#106237722
#915623
#640925