Finite Group Databasebeta

← #13087733

Subgroups — #13087733

Order 1440.

280 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 59 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
1C5×#3830C211yesmaximal, characteristic
2#10624222C211yesmaximal, characteristic, complemented
3C3×#6716C212yesmaximal, complemented
4#10623826C212yesmaximal, complemented
5#10623820C211yesmaximal, characteristic, complemented
6#91486—32nomaximal
7C32×C5×D4C2211yescharacteristic
8C5×#638—21no—
9C2×C5×C3⋊Dic3—21no—
10C4×C5×C3⋊D3C2211yescharacteristic
11C5×#634C2211yescharacteristic
12C3×Dic30C2212yes—
13#67158—21no—
14#67155—21no—
15C3×C3⋊#168C2212yes—
16#63931—51nomaximal, Hall
17C5×#237—32no—
18C5×#237—61no—
19D3×Dic10—32no—
20#6716D312yescomplemented
21C3×#687—32no—
22#6663—32no—
23#6661—32no—
24#6650—32no—
25C22×C32×C5—21no—
26C2×C5×C3⋊D3D411yescharacteristic
27C32×C4×C5C2311yescharacteristic
28C5×C3⋊Dic3—21no—
29C5×C3⋊Dic3D411yescharacteristic
30C3×Dic15—22no—
31#4097—91noHall
32#3830D511yescharacteristic
33#3797—51no—
34C3×#214—52no—
35#3719—52no—
36#3712—51no—
37C3×C5×D4—21no—
38C3×C5×D4C2×D312yes—
39Dic30C2×D312yes—
40C5×#67—62no—
41C5×#67—62no—
42C2×C5×Dic3—62no—
43C2×C5×Dic3—62no—
44C4×C5×D3—32no—
45C4×C5×D3—61no—
46C5×Dic6—32no—
47C5×Dic6—61no—
48C3×Dic10—32no—
49#1204—62no—
50D3×Dic5—62no—
51C3⋊#168C2×D312yes—
52C3×#168—32no—
53#915—152no—
54C2×C32×C5—41no—
55C2×C32×C5C2×D411yescharacteristic
56C5×C3⋊D3—21no—
57C5×#41—91no—
58C2×Dic10—91no—
59#689—92no—
60#687—32no—
61#681—91no—
62C32×D4C2×D511yescharacteristic
63#638—21no—
64C2×C3⋊Dic3—21no—
65C4×C3⋊D3C2×D511yescharacteristic
66#634C2×D511yescharacteristic
67C3×Dic6—52no—
68#627—101no—
69#624—101no—
70C3×C3⋊C8—52no—
71C22×C3×C5—22no—
72C22×C3×C5—22no—
73C2×C5×D3—32no—
74C2×C5×D3—61no—
75C3×C4×C5—21no—
76C3×C4×C5C22×D312yes—
77Dic15—22no—
78C3×Dic5—62no—
79C5×Dic3—32no—
80C5×Dic3—61no—
81C5×Dic3—62no—
82C5×Dic3—62no—
83D3×Q8—152no—
84#237—32no—
85#237—61no—
86C3×SD16—152no—
87#216—152no—
88#214—52no—
89Dic12—152no—
90ΓL(1,𝔽25)—152no—
91C32×C5#13711yescharacteristic, complemented, Hall
92C5×Q8—91no—
93C5×D4—181no—
94C5×D4D3211yescharacteristic
95C2×C4×C5—91no—
96C2×C4×C5—181no—
97C2×Dic5—181no—
98Dic10—32no—
99Dic10—181no—
100#168—32no—
101C22×C32—21no—
102C2×C3⋊D3C22⋊D511yescharacteristic
103C32×C4C22×D511yescharacteristic
104C3⋊Dic3—21no—
105C3⋊Dic3C22⋊D511yescharacteristic
106C3×Dic3—102no—
107#137—451noSylow
108C2×C3×C5—21no—
109C2×C3×C5—42no—
110C2×C3×C5—42no—
111C2×C3×C5D3×D412yes—
112C5×D3—62no—
113C5×D3—121no—
114C3×Q8—152no—
115C3×D4—21no—
116C3×D4D3×D512yes—
117#67—62no—
118#67—62no—
119C2×Dic3—62no—
120C2×Dic3—62no—
121C4×D3—32no—
122C4×D3—61no—
123C4×D3—302no—
124Dic6—32no—
125Dic6—52no—
126Dic6—61no—
127Dic6—302no—
128C3×C8—152no—
129C3⋊C8—52no—
130C22×C5—21no—
131C22×C5—91no—
132C4×C5—91no—
133C4×C5—181no—
134C4×C5C2×D3211yescharacteristic
135Dic5—62no—
136C2×C32—41no—
137C2×C32C2×C22⋊D511yescharacteristic
138C3⋊D3—21no—
139#41—91no—
140C2×Q8—451no—
141Q16—452no—
142SD16—152no—
143M16—451no—
144C3×C5—21no—
145C3×C5#91512yescomplemented
146C22×C3—22no—
147C22×C3—22no—
148C2×D3—32no—
149C2×D3—61no—
150C3×C4—21no—
151C3×C4—302no—
152C3×C4C2×D3×D512yes—
153Dic3—32no—
154Dic3—61no—
155Dic3—62no—
156Dic3—62no—
157Dic3—102no—
158C2×C5—41no—
159C2×C5—181no—
160C2×C5#381111yescharacteristic
161C32#409711yescharacteristic, complemented, Sylow
162Q8—91no—
163Q8—152no—
164Q8—901no—
165D4—181no—
166D4(C3⋊D3)⋊D511yescharacteristic
167C2×C4—91no—
168C2×C4—181no—
169C2×C4—901no—
170C8—152no—
171C2×C3—21no—
172C2×C3—42no—
173C2×C3—42no—
174C2×C3D3×C22⋊D512yes—
175D3—62no—
176D3—121no—
177C5#6393111yescharacteristic, complemented, Sylow
178C22—21no—
179C22—91no—
180C4—91no—
181C4—181no—
182C4—302no—
183C4C2×(C3⋊D3)⋊D511yescharacteristic
184C3—21no—
185C3#9148612yescomplemented
186C2—41no—
187C2—181no—
188C2#1062423611yescharacteristic
subgroupG/H
centerC2#10624236
commutator, cosocleC32×C4×C5C23
FrattiniC4C2×(C3⋊D3)⋊D5
FittingC32×C5×D4C22
socleC2×C32×C5C2×D4
prime radicalC5×#3830C2
hypercenterD4(C3⋊D3)⋊D5
hypocenterC32×C5#137
LCSG ▷ C32×C4×C5 ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C4 ◁ D4

Complements

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

NH ≅ G/Ntype
2. #10624222C2semidirect
3. C3×#6716C2semidirect
4. #10623826C2semidirect
5. #10623820C2semidirect
20. #6716D3semidirect
91. C32×C5#137semidirect
145. C3×C5#915semidirect
161. C32#4097semidirect
177. C5#63931semidirect
185. C3#91486semidirect

Maximal subgroups

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

subgroupindex
C5×#38302
#106242222
C3×#67162
#106238262
#106238202
#914863
#639315