Finite Group Databasebeta

← #13087719

Subgroups — #13087719

Order 1440.

350 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 79 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×C5×C3⋊D3C211yesmaximal, characteristic
2C2×#67158C211yesmaximal, characteristic, complemented
3C3×#6712C212yesmaximal, complemented
4#10623832C211yesmaximal, characteristic, complemented
5#10623830C212yesmaximal
6#91352—32nomaximal
7C22×C5×C3⋊D3C2211yescharacteristic
8C2×C32×C4×C5C2211yescharacteristic
9C2×C5×C3⋊Dic3C2211yescharacteristic
10C4×C5×C3⋊D3—22no—
11C2×C3×Dic15C2212yes—
12C2×C3×Dic15C2212yes—
13#67158—22no—
14#63865—51nomaximal, Hall
15C2×C4×C5×D3—32no—
16C2×C4×C5×D3—61no—
17C2×#1204—32no—
18#6712D312yescomplemented
19C3×#683—32no—
20Dic3⋊Dic5—32no—
21(C2×D3)⋊Dic5—32no—
22(C2×D3)⋊Dic5—32no—
23C22×C32×C5C2311yescharacteristic
24C2×C5×C3⋊D3—21no—
25C2×C5×C3⋊D3—21no—
26C2×C5×C3⋊D3Q812yescomplemented
27C32×C4×C5—21no—
28C5×C3⋊Dic3D412yes—
29C3×Dic15—22no—
30C3×Dic15—22no—
31#4070—91noHall
32C2×C4×C3⋊D3D511yescharacteristic
33C2×#627—51no—
34C3×Dic3⋊C4—52no—
35#3724—51no—
36#3722—52no—
37C22×C5×D3—32no—
38C22×C5×D3—61no—
39C2×C3×C4×C5—21no—
40C2×C3×C4×C5C2×D312yes—
41C2×Dic15C2×D312yes—
42C2×Dic15C2×D312yes—
43C2×C5×Dic3—32no—
44C2×C5×Dic3—61no—
45C4×C5×D3—64no—
46C4×C5×D3—64no—
47C2×C3×Dic5—32no—
48C2×C3×Dic5—32no—
49#1204—64no—
50#896—152no—
51C2×C32×C5C2×D411yescharacteristic
52C2×C32×C5C2×Q811yescharacteristic
53C2×C32×C5#4111yescharacteristic
54C5×C3⋊D3—22no—
55C22×C4×C5—91no—
56C2×Dic10—91no—
57C22⋊Dic5—92no—
58C4⋊Dic5—91no—
59#683—32no—
60C22×C3⋊D3C2×D511yescharacteristic
61C2×C32×C4C2×D511yescharacteristic
62C2×C3⋊Dic3C2×D511yescharacteristic
63C4×C3⋊D3—22no—
64C2×C3×Dic3—52no—
65C2×C3×Dic3—52no—
66#627—102no—
67C22×C3×C5—21no—
68C22×C3×C5C22×D312yes—
69C2×C5×D3—34no—
70C2×C5×D3—62no—
71C2×C5×D3—62no—
72C2×C5×D3—62no—
73C2×C5×D3—62no—
74C2×C5×D3—62no—
75C3×C4×C5—22no—
76C3×C4×C5—22no—
77Dic15—22no—
78Dic15—22no—
79C3×Dic5—62no—
80C3×Dic5—62no—
81C5×Dic3—34no—
82C5×Dic3—62no—
83C2×C4×D3—32no—
84C2×C4×D3—61no—
85C2×Dic6—152no—
86C3×C4⋊C4—152no—
87(C2×D3)⋊C4—152no—
88(C2×D3)⋊C4—152no—
89Dic3⋊C4—52no—
90Dic3⋊C4—152no—
91C32×C5C22⋊Q811yescharacteristic, complemented, Hall
92C23×C5—91no—
93C2×C4×C5—91no—
94C2×C4×C5—182no—
95C2×C4×C5D3211yescharacteristic
96C2×Dic5—32no—
97C2×Dic5—32no—
98Dic10—182no—
99C22×C32C22×D511yescharacteristic
100C2×C3⋊D3—21no—
101C2×C3⋊D3—21no—
102C2×C3⋊D3Dic1012yescomplemented
103C32×C4—21no—
104C3⋊Dic3C22⋊D512yes—
105C3×Dic3—102no—
106C3×Dic3—102no—
107C22⋊Q8—451noSylow
108C2×C3×C5—21no—
109C2×C3×C5—21no—
110C2×C3×C5—21no—
111C2×C3×C5#23512yes—
112C2×C3×C5D3×D412yes—
113C2×C3×C5D3×Q812yes—
114C5×D3—64no—
115C5×D3—64no—
116C22×D3—32no—
117C22×D3—61no—
118C2×C3×C4—21no—
119C2×C3×C4—152no—
120C2×C3×C4—152no—
121C2×C3×C4D3×D512yes—
122C2×Dic3—32no—
123C2×Dic3—52no—
124C2×Dic3—52no—
125C2×Dic3—61no—
126C4×D3—64no—
127C4×D3—64no—
128Dic6—304no—
129C22×C5—92no—
130C22×C5—181no—
131C22×C5—181no—
132C22×C5C2×D3211yescharacteristic
133C4×C5—21no—
134C4×C5—92no—
135Dic5—62no—
136Dic5—62no—
137C2×C32C2×Dic1011yescharacteristic
138C2×C32#70911yescharacteristic
139C2×C32C2×C22⋊D511yescharacteristic
140C3⋊D3—22no—
141C2×Q8—451no—
142C22×C4—91no—
143C4⋊C4—152no—
144C4⋊C4—451no—
145C22⋊C4—452no—
146C3×C5—21no—
147C3×C5#89612yescomplemented
148C22×C3—21no—
149C22×C3C2×D3×D512yes—
150C2×D3—34no—
151C2×D3—62no—
152C2×D3—62no—
153C2×D3—62no—
154C2×D3—62no—
155C2×D3—62no—
156C3×C4—22no—
157C3×C4—22no—
158C3×C4—302no—
159C3×C4—302no—
160Dic3—34no—
161Dic3—62no—
162Dic3—102no—
163Dic3—102no—
164C2×C5—182no—
165C2×C5#379711yescharacteristic
166C2×C5#379811yescharacteristic
167C2×C5#381111yescharacteristic
168C32#407011yescharacteristic, complemented, Sylow
169C23—91no—
170Q8—902no—
171C2×C4—91no—
172C2×C4—152no—
173C2×C4—152no—
174C2×C4—182no—
175C2×C4(C3⋊D3)⋊D511yescharacteristic
176C2×C3—21no—
177C2×C3—21no—
178C2×C3—21no—
179C2×C3D3×Dic1012yes—
180C2×C3#677112yes—
181C2×C3D3×C22⋊D512yes—
182D3—64no—
183D3—64no—
184C5#6386511yescharacteristic, complemented, Sylow
185C22—92no—
186C22—181no—
187C22—181no—
188C22C2×(C3⋊D3)⋊D511yescharacteristic
189C4—21no—
190C4—92no—
191C4—302no—
192C4—302no—
193C3—21no—
194C3#9135212yescomplemented
195C2—182no—
196C2#1062422211yescharacteristic
197C2#1062422311yescharacteristic
198C2#1062423611yescharacteristic
subgroupG/H
center, FrattiniC22C2×(C3⋊D3)⋊D5
commutator, socle, cosocleC22×C32×C5C23
FittingC2×C32×C4×C5C22
prime radicalC22×C5×C3⋊D3C22
hypercenterC2×C4(C3⋊D3)⋊D5
hypocenterC32×C5C22⋊Q8
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×#67158C2semidirect
3. C3×#6712C2semidirect
4. #10623832C2semidirect
18. #6712D3semidirect
26. C2×C5×C3⋊D3Q8semidirect
91. C32×C5C22⋊Q8semidirect
102. C2×C3⋊D3Dic10semidirect
147. C3×C5#896semidirect
168. C32#4070semidirect
184. C5#63865semidirect
194. C3#91352semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C5×C3⋊D32
C2×#671582
C3×#67122
#106238322
#106238302
#913523
#638655