Finite Group Databasebeta

← #13087772

Subgroups — #13087772

Order 1440.

366 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 75 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×C5×#638C211yesmaximal, characteristic
2C2×#67155C211yesmaximal, characteristic, complemented
3C3×#6719C212yesmaximal, complemented
4#10623831C212yesmaximal, complemented
5#10623830C211yesmaximal, characteristic, complemented
6#91555—32nomaximal
7C23×C32×C5C2211yescharacteristic
8C22×C5×C3⋊D3C2211yescharacteristic
9C5×#638—22no—
10C2×C5×C3⋊Dic3C2211yescharacteristic
11C2×C3×Dic15C2212yes—
12C2×C3×Dic15C2212yes—
13#67155—22no—
14#63975—51nomaximal, Hall
15C2×C5×#67—32no—
16C2×C5×#67—61no—
17C2×D3×Dic5—32no—
18#6719D312yescomplemented
19C3×C22⋊Dic5—32no—
20#6669—32no—
21#6668—32no—
22(C2×D3)⋊Dic5—32no—
23C22×C32×C5—21no—
24C22×C32×C5—22no—
25C22×C32×C5C2311yescharacteristic
26C2×C5×C3⋊D3—22no—
27C2×C5×C3⋊D3D412yes—
28C5×C3⋊Dic3—21no—
29C3×Dic15—22no—
30C3×Dic15—22no—
31#4081—91noHall
32C2×#638D511yescharacteristic
33C2×#624—51no—
34C3×#217—52no—
35Dic3⋊Dic3—52no—
36#3722—51no—
37C23×C3×C5—21no—
38C23×C3×C5C2×D312yes—
39C22×C5×D3—32no—
40C22×C5×D3—61no—
41C2×Dic15C2×D312yes—
42C2×Dic15C2×D312yes—
43C5×#67—64no—
44C5×#67—64no—
45C2×C5×Dic3—32no—
46C2×C5×Dic3—61no—
47C2×C3×Dic5—32no—
48C2×C3×Dic5—32no—
49D3×Dic5—64no—
50#883—152no—
51C2×C32×C5—41no—
52C2×C32×C5C2×D411yescharacteristic
53C2×C32×C5#4112yes—
54C5×C3⋊D3—22no—
55C2×C5×D4—91no—
56C22×Dic5—91no—
57C22⋊Dic5—32no—
58C22⋊Dic5—91no—
59#683—92no—
60C23×C32C2×D511yescharacteristic
61C22×C3⋊D3C2×D511yescharacteristic
62#638—22no—
63C2×C3⋊Dic3C2×D511yescharacteristic
64C2×C3×Dic3—52no—
65C2×C3×Dic3—52no—
66#624—102no—
67C22×C3×C5—22no—
68C22×C3×C5—22no—
69C22×C3×C5—22no—
70C22×C3×C5—22no—
71C22×C3×C5—21no—
72C22×C3×C5—42no—
73C22×C3×C5C22×D312yes—
74C2×C5×D3—34no—
75C2×C5×D3—62no—
76C2×C5×D3—62no—
77C2×C5×D3—62no—
78C2×C5×D3—122no—
79Dic15—22no—
80Dic15—22no—
81C3×Dic5—62no—
82C3×Dic5—62no—
83C5×Dic3—62no—
84C5×Dic3—62no—
85C2×#67—32no—
86C2×#67—61no—
87C2×C4×D3—152no—
88C3×C22⋊C4—152no—
89#217—52no—
90(C2×D3)⋊C4—152no—
91C4⋊Dic3—152no—
92Dic3⋊C4—152no—
93C32×C5#12311yescharacteristic, complemented, Hall
94C23×C5—91no—
95C23×C5D3211yescharacteristic
96C5×D4—182no—
97C2×C4×C5—91no—
98C2×Dic5—32no—
99C2×Dic5—32no—
100C2×Dic5—182no—
101C22×C32—21no—
102C22×C32—22no—
103C22×C32C22×D511yescharacteristic
104C2×C3⋊D3—22no—
105C2×C3⋊D3C22⋊D512yes—
106C3⋊Dic3—21no—
107C3×Dic3—102no—
108C3×Dic3—102no—
109#123—451noSylow
110C2×C3×C5—22no—
111C2×C3×C5—21no—
112C2×C3×C5—42no—
113C2×C3×C5—42no—
114C2×C3×C5#23512yes—
115C2×C3×C5D3×D412yes—
116C2×C3×C5#23712yes—
117C5×D3—64no—
118C5×D3—122no—
119C23×C3—21no—
120C23×C3D3×D512yes—
121C22×D3—32no—
122C22×D3—61no—
123C2×C3×C4—152no—
124C2×C3×C4—152no—
125#67—64no—
126#67—64no—
127C2×Dic3—32no—
128C2×Dic3—52no—
129C2×Dic3—52no—
130C2×Dic3—61no—
131C4×D3—304no—
132C22×C5—22no—
133C22×C5—21no—
134C22×C5—92no—
135C22×C5—182no—
136C22×C5C2×D3211yescharacteristic
137C4×C5—181no—
138Dic5—62no—
139Dic5—62no—
140C2×C32—41no—
141C2×C32#71112yes—
142C2×C32C2×C22⋊D511yescharacteristic
143C3⋊D3—22no—
144C2×D4—91no—
145C22×C4—451no—
146C4⋊C4—452no—
147C22⋊C4—152no—
148C22⋊C4—451no—
149C3×C5—21no—
150C3×C5#88312yescomplemented
151C22×C3—22no—
152C22×C3—22no—
153C22×C3—22no—
154C22×C3—22no—
155C22×C3—21no—
156C22×C3—42no—
157C22×C3C2×D3×D512yes—
158C2×D3—34no—
159C2×D3—62no—
160C2×D3—62no—
161C2×D3—62no—
162C2×D3—122no—
163C3×C4—302no—
164C3×C4—302no—
165Dic3—62no—
166Dic3—62no—
167Dic3—102no—
168Dic3—102no—
169C2×C5—41no—
170C2×C5—182no—
171C2×C5#380412yes—
172C2×C5#381111yescharacteristic
173C32#408111yescharacteristic, complemented, Sylow
174C23—91no—
175C23(C3⋊D3)⋊D511yescharacteristic
176D4—182no—
177C2×C4—91no—
178C2×C4—152no—
179C2×C4—152no—
180C2×C4—902no—
181C2×C3—22no—
182C2×C3—21no—
183C2×C3—42no—
184C2×C3—42no—
185C2×C3#678012yes—
186C2×C3#678212yes—
187C2×C3D3×C22⋊D512yes—
188D3—64no—
189D3—122no—
190C5#6397511yescharacteristic, complemented, Sylow
191C22—22no—
192C22—21no—
193C22—92no—
194C22—182no—
195C22C2×(C3⋊D3)⋊D511yescharacteristic
196C4—181no—
197C4—302no—
198C4—302no—
199C3—21no—
200C3#9155512yescomplemented
201C2—41no—
202C2—182no—
203C2#1062423012yes—
204C2#1062423611yescharacteristic
subgroupG/H
center, FrattiniC22C2×(C3⋊D3)⋊D5
commutator, socle, cosocleC22×C32×C5C23
FittingC23×C32×C5C22
prime radicalC2×C5×#638C2
hypercenterC23(C3⋊D3)⋊D5
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×#67155C2semidirect
3. C3×#6719C2semidirect
4. #10623831C2semidirect
5. #10623830C2semidirect
18. #6719D3semidirect
93. C32×C5#123semidirect
150. C3×C5#883semidirect
173. C32#4081semidirect
190. C5#63975semidirect
200. C3#91555semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#6382
C2×#671552
C3×#67192
#106238312
#106238302
#915553
#639755