Finite Group Databasebeta

← C5×#63865 #13087189

Subgroups — C5×#63865 #13087189

Order 1440.

350 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 96 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×C5×#627C211yesmaximal, characteristic, complemented
3C3×C5×Dic3⋊C4C212yesmaximal, complemented
4C5×#3724C211yesmaximal, characteristic, complemented
5C5×#3722C212yesmaximal
6C5×#896—32nomaximal
7C22×C5×C3⋊D3C2211yescharacteristic
8C2×C32×C4×C5C2211yescharacteristic
9C2×C5×C3⋊Dic3C2211yescharacteristic
10C4×C5×C3⋊D3—22no—
11C2×C3×C5×Dic3C2212yes—
12C2×C3×C5×Dic3C2212yes—
13C5×#627—22no—
14#63865C511yesmaximal, characteristic, direct product factor, Hall
15C2×C4×C5×D3—32no—
16C2×C4×C5×D3—61no—
17C2×C5×Dic6—32no—
18C3×C5×C4⋊C4—32no—
19C5×(C2×D3)⋊C4—32no—
20C5×(C2×D3)⋊C4—32no—
21C5×Dic3⋊C4—32no—
22C5×Dic3⋊C4D312yescomplemented
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×C5×Dic3—22no—
30C3×C5×Dic3—22no—
31C5×C22⋊Q8—91noHall
32C2×C4×C3⋊D3C2×C511yescharacteristic
33C2×#627C2×C511yescharacteristic, complemented
34C3×Dic3⋊C4C2×C512yescomplemented
35#3724C2×C511yescharacteristic, complemented
36#3722C2×C512yes—
37C22×C5×D3—32no—
38C22×C5×D3—61no—
39C2×C3×C4×C5—21no—
40C2×C3×C4×C5—32no—
41C2×C3×C4×C5—32no—
42C2×C3×C4×C5C2×D312yes—
43C2×C5×Dic3—32no—
44C2×C5×Dic3—61no—
45C2×C5×Dic3C2×D312yes—
46C2×C5×Dic3C2×D312yes—
47C4×C5×D3—64no—
48C4×C5×D3—64no—
49C5×Dic6—64no—
50#896—32no—
51C2×C32×C5C2×D411yescharacteristic
52C2×C32×C5C2×Q811yescharacteristic
53C2×C32×C5#4111yescharacteristic
54C5×C3⋊D3—22no—
55C2×C5×Q8—91no—
56C22×C4×C5—91no—
57C5×C4⋊C4—32no—
58C5×C4⋊C4—91no—
59C5×C22⋊C4—92no—
60C22×C3⋊D3C22×C511yescharacteristic
61C2×C32×C4C22×C511yescharacteristic
62C2×C3⋊Dic3C22×C511yescharacteristic
63C4×C3⋊D3—22no—
64C2×C3×Dic3C22×C512yes—
65C2×C3×Dic3C22×C512yes—
66#627—22no—
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—
77C3×C4×C5—62no—
78C3×C4×C5—62no—
79C5×Dic3—22no—
80C5×Dic3—22no—
81C5×Dic3—34no—
82C5×Dic3—62no—
83C2×C4×D3—32no—
84C2×C4×D3—61no—
85C2×Dic6—32no—
86C3×C4⋊C4—32no—
87(C2×D3)⋊C4—32no—
88(C2×D3)⋊C4—32no—
89Dic3⋊C4—32no—
90Dic3⋊C4C5×D312yescomplemented
91C32×C5C22⋊Q811yescharacteristic, complemented, Hall
92C23×C5—91no—
93C5×Q8—182no—
94C2×C4×C5—32no—
95C2×C4×C5—32no—
96C2×C4×C5—91no—
97C2×C4×C5—182no—
98C2×C4×C5D3211yescharacteristic
99C22×C32C23×C511yescharacteristic
100C2×C3⋊D3—21no—
101C2×C3⋊D3—21no—
102C2×C3⋊D3C5×Q812yescomplemented
103C32×C4—21no—
104C3⋊Dic3C5×D412yes—
105C3×Dic3—22no—
106C3×Dic3—22no—
107C22⋊Q8—91noSylow
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—32no—
120C2×C3×C4—32no—
121C2×C3×C4C2×C5×D312yes—
122C2×Dic3—32no—
123C2×Dic3—61no—
124C2×Dic3C2×C5×D312yes—
125C2×Dic3C2×C5×D312yes—
126C4×D3—64no—
127C4×D3—64no—
128Dic6—64no—
129C22×C5—92no—
130C22×C5—181no—
131C22×C5—181no—
132C22×C5C2×D3211yescharacteristic
133C4×C5—21no—
134C4×C5—62no—
135C4×C5—62no—
136C4×C5—92no—
137C2×C32C2×C5×D411yescharacteristic
138C2×C32C2×C5×Q811yescharacteristic
139C2×C32C5×#4111yescharacteristic
140C3⋊D3—22no—
141C2×Q8—91no—
142C22×C4—91no—
143C4⋊C4—32no—
144C4⋊C4—91no—
145C22⋊C4—92no—
146C3×C5—21no—
147C3×C5#89612yescomplemented
148C22×C3—21no—
149C22×C3C22×C5×D312yes—
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—62no—
159C3×C4—62no—
160Dic3—22no—
161Dic3—22no—
162Dic3—34no—
163Dic3—62no—
164C2×C5—182no—
165C2×C5#379711yescharacteristic
166C2×C5#379811yescharacteristic
167C2×C5#381111yescharacteristic
168C32C5×C22⋊Q811yescharacteristic, complemented, Sylow
169C23—91no—
170Q8—182no—
171C2×C4—32no—
172C2×C4—32no—
173C2×C4—91no—
174C2×C4—182no—
175C2×C4C5×D3211yescharacteristic
176C2×C3—21no—
177C2×C3—21no—
178C2×C3—21no—
179C2×C3C5×#23512yes—
180C2×C3C5×D3×D412yes—
181C2×C3C5×D3×Q812yes—
182D3—64no—
183D3—64no—
184C5#6386511yescharacteristic, direct product factor, Sylow
185C22—92no—
186C22—181no—
187C22—181no—
188C22C2×C5×D3211yescharacteristic
189C4—21no—
190C4—62no—
191C4—62no—
192C4—92no—
193C3—21no—
194C3C5×#89612yescomplemented
195C2—182no—
196C2C5×#379711yescharacteristic
197C2C5×#379811yescharacteristic
198C2C5×#381111yescharacteristic
subgroupG/H
centerC22×C5C2×D32
commutator, cosocleC22×C32C23×C5
FrattiniC22C2×C5×D32
FittingC2×C32×C4×C5C22
socleC22×C32×C5C23
prime radicalC22×C5×C3⋊D3C22
hypercenterC2×C4×C5D32
hypocenterC32C5×C22⋊Q8
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C5 ◁ C2×C4×C5

Complements

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

NH ≅ G/Ntype
2. C2×C5×#627C2semidirect
3. C3×C5×Dic3⋊C4C2semidirect
4. C5×#3724C2semidirect
14. #63865C5direct (N × H)
22. C5×Dic3⋊C4D3semidirect
26. C2×C5×C3⋊D3Q8semidirect
33. C2×#627C2×C5semidirect
34. C3×Dic3⋊C4C2×C5semidirect
35. #3724C2×C5semidirect
90. Dic3⋊C4C5×D3semidirect
91. C32×C5C22⋊Q8semidirect
102. C2×C3⋊D3C5×Q8semidirect
147. C3×C5#896semidirect
168. C32C5×C22⋊Q8semidirect
184. C5#63865direct (N × H)
194. C3C5×#896semidirect

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×C5×#6272
C3×C5×Dic3⋊C42
C5×#37242
C5×#37222
C5×#8963
#638655