Finite Group Databasebeta

← C2×C3×#6717 #13088040

Subgroups — C2×C3×#6717 #13088040

Order 1440.

310 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 110 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×C32×C5×Q8C211yesmaximal, characteristic, complemented
2C2×C3×D60C211yesmaximal, characteristic
3C3×#6717C214yesmaximal, direct product factor
4C2×C3×C3⋊#168C211yesmaximal, characteristic, complemented
5C2×#6717C311yesmaximal, characteristic, direct product factor
6C2×C3×#688—31nomaximal
7C22×C3×D15—21no—
8C32×C5×Q8—21no—
9C32×C5×Q8C2212yescomplemented
10C2×C32×C4×C5—21no—
11C2×C32×C4×C5C2211yescharacteristic
12C3×D60—21no—
13C3×D60C2212yes—
14C3×C3⋊#168C2212yescomplemented
15C2×C3×#215—51nomaximal, Hall
16C2×C3×C5×Q8—21no—
17C2×C3×C5×Q8D311yescharacteristic, complemented
18C2×C3×C5×Q8C2×C311yescharacteristic, complemented
19C2×D60C2×C311yescharacteristic
20C2×C3×D20—31no—
21#6717C2×C314yesdirect product factor
22C2×C3⋊#168C2×C311yescharacteristic, complemented
23C3×#688—34no—
24C2×C3×#168—31no—
25C22×C32×C5D411yescharacteristic
26C2×C3×D15—22no—
27C2×C3×D15—42no—
28C32×C4×C5—22no—
29C32×C4×C5D411yescharacteristic
30C32×C4×C5C2311yescharacteristic
31C2×#688—31noHall
32C2×C32×Q8D511yescharacteristic, complemented
33C2×C3×D12—51no—
34C3×#215—54no—
35C2×C3×C3⋊C8—51no—
36C22×D15—21no—
37C22×C3×D5—61no—
38C3×C5×Q8—22no—
39C3×C5×Q8—22no—
40C3×C5×Q8—21no—
41C3×C5×Q8—21no—
42C3×C5×Q8C2×D312yescomplemented
43C3×C5×Q8C22×C312yescomplemented
44C2×C3×C4×C5—22no—
45C2×C3×C4×C5—21no—
46C2×C3×C4×C5—21no—
47C2×C3×C4×C5—21no—
48C2×C3×C4×C5C2×D311yescharacteristic
49C2×C3×C4×C5C22×C311yescharacteristic
50D60—21no—
51D60C22×C312yes—
52C3×D20—32no—
53C3×D20—61no—
54C3⋊#168C22×C312yescomplemented
55C3×#168—32no—
56C2×C3×SD16—151no—
57C2×#215—51no—
58C2×C32×C5SD1612yescomplemented
59C2×C32×C5C2×D411yescharacteristic
60C3×D15—42no—
61C2×C5×Q8C3×D311yescharacteristic, complemented
62C2×D20—31no—
63#688—34no—
64C2×#168—31no—
65C22×C3×D3—101no—
66C32×Q8—21no—
67C32×Q8C2×D512yescomplemented
68C2×C32×C4—21no—
69C2×C32×C4C2×D511yescharacteristic
70C3×D12—52no—
71C3×D12—101no—
72C3×C3⋊C8—52no—
73C22×C3×C5—21no—
74C22×C3×C5#6711yescharacteristic
75C22×C3×C5C3×D411yescharacteristic
76C2×D15—22no—
77C2×D15—42no—
78C2×C3×D5—62no—
79C2×C3×D5—122no—
80C3×C4×C5—24no—
81C3×C4×C5—22no—
82C3×C4×C5—22no—
83C3×C4×C5—21no—
84C3×C4×C5—21no—
85C3×C4×C5#6711yescharacteristic
86C3×C4×C5C3×D411yescharacteristic
87C3×C4×C5C22×D311yescharacteristic
88C3×C4×C5C23×C311yescharacteristic
89C2×C3×Q8—21no—
90C2×C3×Q8C3×D511yescharacteristic, complemented
91C2×C3×Q8D1511yescharacteristic, complemented
92C2×C3×D4—151no—
93C2×D12—51no—
94C3×SD16—154no—
95C2×C3×C8—151no—
96#215—54no—
97C2×C3⋊C8—51no—
98C32×C5C2×SD1611yescharacteristic, complemented, Hall
99C22×D5—61no—
100C5×Q8—21no—
101C5×Q8C2×C3×D312yescomplemented
102C2×C4×C5—21no—
103C2×C4×C5C2×C3×D311yescharacteristic
104D20—32no—
105D20—61no—
106#168—32no—
107C22×C32C22⋊D511yescharacteristic
108C2×C3×D3—102no—
109C2×C3×D3—202no—
110C32×C4—22no—
111C32×C4C22⋊D511yescharacteristic
112C32×C4C22×D511yescharacteristic
113C2×SD16—151noSylow
114C2×C3×C5—22no—
115C2×C3×C5—21no—
116C2×C3×C5#21512yescomplemented
117C2×C3×C5C3×SD1612yescomplemented
118C2×C3×C5C2×#6711yescharacteristic
119C2×C3×C5C2×C3×D411yescharacteristic
120D15—42no—
121C3×D5—122no—
122C23×C3—301no—
123C22×D3—101no—
124C3×Q8—22no—
125C3×Q8—21no—
126C3×Q8—21no—
127C3×Q8—22no—
128C3×Q8C2×C3×D512yescomplemented
129C3×Q8C2×D1512yescomplemented
130C3×D4—152no—
131C3×D4—301no—
132C2×C3×C4—22no—
133C2×C3×C4—21no—
134C2×C3×C4—21no—
135C2×C3×C4—21no—
136C2×C3×C4C2×C3×D511yescharacteristic
137C2×C3×C4C2×D1511yescharacteristic
138D12—52no—
139D12—101no—
140C3×C8—152no—
141C3⋊C8—52no—
142C22×C5C3×#6711yescharacteristic
143C2×D5—62no—
144C2×D5—122no—
145C4×C5—22no—
146C4×C5C3×#6711yescharacteristic
147C4×C5C22×C3×D311yescharacteristic
148C2×C32#68812yescomplemented
149C2×C32C2×C22⋊D511yescharacteristic
150C3×D3—202no—
151C2×Q8C3×D1511yescharacteristic, complemented
152C2×D4—151no—
153SD16—154no—
154C2×C8—151no—
155C3×C5—21no—
156C3×C5C2×#21511yescharacteristic, complemented
157C3×C5C2×C3×SD1611yescharacteristic, complemented
158C22×C3—21no—
159C22×C3—302no—
160C22×C3—602no—
161C22×C3C3×C22⋊D511yescharacteristic
162C22×C3#122011yescharacteristic
163C2×D3—102no—
164C2×D3—202no—
165C3×C4—24no—
166C3×C4—22no—
167C3×C4—22no—
168C3×C4—21no—
169C3×C4—21no—
170C3×C4C3×C22⋊D511yescharacteristic
171C3×C4#122011yescharacteristic
172C3×C4C22×C3×D511yescharacteristic
173C3×C4C22×D1511yescharacteristic
174C2×C5C3×#21512yescomplemented
175C2×C5C2×C3×#6711yescharacteristic
176D5—122no—
177C32C2×#68811yescharacteristic, complemented, Sylow
178C23—301no—
179Q8—21no—
180Q8C2×C3×D1512yescomplemented
181D4—152no—
182D4—301no—
183C2×C4—21no—
184C2×C4C2×C3×D1511yescharacteristic
185C8—152no—
186C2×C3—21no—
187C2×C3—22no—
188C2×C3—602no—
189C2×C3C3×#68812yescomplemented
190C2×C3#671712yesdirect product factor
191C2×C3C2×C3×C22⋊D511yescharacteristic
192C2×C3C2×#122011yescharacteristic
193D3—202no—
194C5C2×C3×#21511yescharacteristic, complemented, Sylow
195C22—302no—
196C22—602no—
197C22C3×#122011yescharacteristic
198C4—22no—
199C4C3×#122011yescharacteristic
200C4C22×C3×D1511yescharacteristic
201C3—21no—
202C3C2×C3×#68811yescharacteristic, complemented
203C3C2×#671711yescharacteristic, direct product factor
204C2—602no—
205C2C3×#671712yesdirect product factor
206C2C2×C3×#122011yescharacteristic
subgroupG/H
centerC22×C3#1220
commutator, cosocleC3×C4×C5C23×C3
FrattiniC4C22×C3×D15
FittingC2×C32×C5×Q8C2
socleC22×C32×C5D4
prime radicalC2×C3×D60C2
hypercenterC2×C3×Q8D15
hypocenterC3×C5C2×C3×SD16
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C32×C5×Q8C2semidirect
3. C3×#6717C2direct (N × H)
4. C2×C3×C3⋊#168C2semidirect
5. C2×#6717C3direct (N × H)
9. C32×C5×Q8C22semidirect
14. C3×C3⋊#168C22semidirect
17. C2×C3×C5×Q8D3semidirect
18. C2×C3×C5×Q8C2×C3semidirect
21. #6717C2×C3direct (N × H)
22. C2×C3⋊#168C2×C3semidirect
32. C2×C32×Q8D5semidirect
42. C3×C5×Q8C2×D3semidirect
43. C3×C5×Q8C22×C3semidirect
54. C3⋊#168C22×C3semidirect
58. C2×C32×C5SD16semidirect
61. C2×C5×Q8C3×D3semidirect
67. C32×Q8C2×D5semidirect
90. C2×C3×Q8C3×D5semidirect
91. C2×C3×Q8D15semidirect
98. C32×C5C2×SD16semidirect
101. C5×Q8C2×C3×D3semidirect
116. C2×C3×C5#215semidirect
117. C2×C3×C5C3×SD16semidirect
128. C3×Q8C2×C3×D5semidirect
129. C3×Q8C2×D15semidirect
148. C2×C32#688semidirect
151. C2×Q8C3×D15semidirect
156. C3×C5C2×#215semidirect
157. C3×C5C2×C3×SD16semidirect
174. C2×C5C3×#215semidirect
177. C32C2×#688semidirect
180. Q8C2×C3×D15semidirect
189. C2×C3C3×#688semidirect
190. C2×C3#6717direct (N × H)
194. C5C2×C3×#215semidirect
202. C3C2×C3×#688semidirect
203. C3C2×#6717direct (N × H)
205. C2C3×#6717direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×C5×Q82
C2×C3×D602
C3×#67172
C2×C3×C3⋊#1682
C2×#67173
C2×C3×#6883
C2×C3×#2155