Finite Group Databasebeta

← C3×#10602720 #14289053

Subgroups — C3×#10602720 #14289053

Order 1920.

480 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 114 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
1C3×C5×#555C211yesmaximal, characteristic, complemented
2C3×#65374C211yesmaximal, characteristic, complemented
3C3×#65304C212yesmaximal, complemented
4C3×#64766C211yesmaximal, characteristic, complemented
5C3×D4⋊#168C212yesmaximal, complemented
6#10602720C311yesmaximal, characteristic, direct product factor, Hall
7C3×C5×C4⋊Q8C2211yescharacteristic
8C3×C5×#124—22no—
9C3×C5×#121—22no—
10C3×C4×C5×D4C2212yes—
11C3×C5×#117—22no—
12C2×C3×#688—22no—
13C2×C3×#686—22no—
14C3×#4020C2211yescharacteristic
15C3×#3941—22no—
16C3×C4⋊#168C2212yes—
17C3×C4×#168C2211yescharacteristic
18C3×#1624—51nomaximal, Hall
19C5×#555C2×C311yescharacteristic, complemented
20#65374C2×C311yescharacteristic, complemented
21#65304C2×C312yescomplemented
22#64766C2×C311yescharacteristic, complemented
23D4⋊#168C2×C312yescomplemented
24C2×C3×C5×Q8D412yes—
25C2×C3×C5×D4—22no—
26C2×C3×C5×D4D412yes—
27C22×C3×C4×C5—22no—
28C2×C3×D20—22no—
29C2×C3×D20—21no—
30C3×C5×C4⋊C4—22no—
31C3×C5×C4⋊C4D412yes—
32C3×C5×C22⋊C4—24no—
33C3×C5×C22⋊C4—22no—
34C3×C42×C5—22no—
35C3×C42×C5C2311yescharacteristic
36C3×#688—44no—
37C3×#686—44no—
38C2×C3×#168—21no—
39C2×C3×#168—22no—
40C3×#555D511yescharacteristic, complemented
41C3×#492—51no—
42C3×#459—52no—
43C3×#333—51no—
44C3×D4⋊C8—52no—
45C5×C4⋊Q8C22×C311yescharacteristic
46C5×#124—22no—
47C5×#121—22no—
48C4×C5×D4C22×C312yes—
49C5×#117—22no—
50C2×#688—22no—
51C2×#686—22no—
52#4020C22×C311yescharacteristic
53#3941—22no—
54C4⋊#168C22×C312yes—
55C4×#168C22×C311yescharacteristic
56#1624—51noSylow
57C23×C3×C5—22no—
58C22×C3×D5—41no—
59C3×C5×Q8—42no—
60C3×C5×D4—22no—
61C3×C5×D4—22no—
62C3×C5×D4—82no—
63C2×C3×C4×C5—22no—
64C2×C3×C4×C5—22no—
65C2×C3×C4×C5—42no—
66C2×C3×C4×C5—42no—
67C2×C3×C4×C5C2×D411yescharacteristic
68C2×C3×C4×C5C2×D412yes—
69C3×D20—42no—
70C3×D20—42no—
71C3×D20—42no—
72C3×#168—22no—
73C3×#168—42no—
74C2×C3×SD16—102no—
75C2×C3×D8—102no—
76C3×C4⋊Q8C2×D511yescharacteristic
77C3×#127—51no—
78C3×#124—22no—
79C3×#121—22no—
80C3×C4×D4C2×D512yes—
81C3×#117—22no—
82C3×C4⋊C8—52no—
83C3×#102—102no—
84C3×C4×C8—51no—
85C2×C5×Q8C3×D412yes—
86C2×C5×D4—22no—
87C2×C5×D4C3×D412yes—
88C22×C4×C5—22no—
89C2×D20—22no—
90C2×D20—21no—
91C5×C4⋊C4—22no—
92C5×C4⋊C4C3×D412yes—
93C5×C22⋊C4—24no—
94C5×C22⋊C4—22no—
95C42×C5—22no—
96C42×C5C23×C311yescharacteristic
97#688—44no—
98#686—44no—
99C2×#168—21no—
100C2×#168—22no—
101#555C3×D511yescharacteristic, complemented
102#492—51no—
103#459—52no—
104#333—51no—
105D4⋊C8—52no—
106C22×C3×C5—42no—
107C22×C3×C5—42no—
108C22×C3×C5—42no—
109C22×C3×C5#12011yescharacteristic
110C2×C3×D5—82no—
111C2×C3×D5—81no—
112C3×C4×C5—21no—
113C3×C4×C5—42no—
114C3×C4×C5—42no—
115C3×C4×C5#13512yes—
116C3×C4×C5AGL(1,ℤ/8)12yes—
117C2×C3×Q8C22⋊D512yes—
118C2×C3×D4—22no—
119C2×C3×D4—102no—
120C2×C3×D4—101no—
121C2×C3×D4C22⋊D512yes—
122C22×C3×C4—22no—
123C3×SD16—204no—
124C3×D8—204no—
125C2×C3×C8—101no—
126C2×C3×C8—102no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4C22⋊D512yes—
129C3×C22⋊C4—24no—
130C3×C22⋊C4—22no—
131C3×C42—22no—
132C3×C42C22×D511yescharacteristic
133C23×C5—22no—
134C22×D5—41no—
135C5×Q8—42no—
136C5×D4—22no—
137C5×D4—22no—
138C5×D4—82no—
139C2×C4×C5—22no—
140C2×C4×C5—22no—
141C2×C4×C5—42no—
142C2×C4×C5—42no—
143C2×C4×C5C2×C3×D411yescharacteristic
144C2×C4×C5C2×C3×D412yes—
145D20—42no—
146D20—42no—
147D20—42no—
148#168—22no—
149#168—42no—
150C2×SD16—102no—
151C2×D8—102no—
152C4⋊Q8C2×C3×D511yescharacteristic
153#127—51no—
154#124—22no—
155#121—22no—
156C4×D4C2×C3×D512yes—
157#117—22no—
158C4⋊C8—52no—
159#102—102no—
160C4×C8—51no—
161C2×C3×C5—82no—
162C2×C3×C5#44912yes—
163C2×C3×C5#45311yescharacteristic
164C3×D5—161no—
165C23×C3—22no—
166C23×C3—201no—
167C3×Q8—42no—
168C3×D4—22no—
169C3×D4—22no—
170C3×D4—82no—
171C3×D4—202no—
172C3×D4—202no—
173C3×D4—202no—
174C2×C3×C4—22no—
175C2×C3×C4—22no—
176C2×C3×C4—42no—
177C2×C3×C4—42no—
178C2×C3×C4C2×C22⋊D511yescharacteristic
179C2×C3×C4C2×C22⋊D512yes—
180C3×C8—102no—
181C3×C8—202no—
182C22×C5—42no—
183C22×C5—42no—
184C22×C5—42no—
185C22×C5C3×#12011yescharacteristic
186C2×D5—82no—
187C2×D5—81no—
188C4×C5—21no—
189C4×C5—42no—
190C4×C5—42no—
191C4×C5C3×#13512yes—
192C4×C5C3×AGL(1,ℤ/8)12yes—
193C2×Q8C3×C22⋊D512yes—
194C2×D4—22no—
195C2×D4—102no—
196C2×D4—101no—
197C2×D4C3×C22⋊D512yes—
198C22×C4—22no—
199SD16—204no—
200D8—204no—
201C2×C8—101no—
202C2×C8—102no—
203C4⋊C4—22no—
204C4⋊C4C3×C22⋊D512yes—
205C22⋊C4—24no—
206C22⋊C4—22no—
207C42—22no—
208C42C22×C3×D511yescharacteristic
209C3×C5#162411yescharacteristic, complemented, Hall
210C22×C3—42no—
211C22×C3—42no—
212C22×C3—42no—
213C22×C3—402no—
214C22×C3—401no—
215C22×C3#409911yescharacteristic
216C3×C4—21no—
217C3×C4—42no—
218C3×C4—42no—
219C3×C4#409512yes—
220C3×C4#409612yes—
221C2×C5—82no—
222C2×C5C3×#44912yes—
223C2×C5C3×#45311yescharacteristic
224D5—161no—
225C23—22no—
226C23—201no—
227Q8—42no—
228D4—22no—
229D4—22no—
230D4—82no—
231D4—202no—
232D4—202no—
233D4—202no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4C2×C3×C22⋊D511yescharacteristic
239C2×C4C2×C3×C22⋊D512yes—
240C8—102no—
241C8—202no—
242C2×C3—82no—
243C2×C3—801no—
244C2×C3#6552712yes—
245C2×C3#6553011yescharacteristic
246C5C3×#162411yescharacteristic, complemented, Sylow
247C22—42no—
248C22—42no—
249C22—42no—
250C22—402no—
251C22—401no—
252C22C3×#409911yescharacteristic
253C4—21no—
254C4—42no—
255C4—42no—
256C4C3×#409512yes—
257C4C3×#409612yes—
258C3#1060272011yescharacteristic, direct product factor, Sylow
259C2—82no—
260C2—801no—
261C2C3×#6552712yes—
262C2C3×#6553011yescharacteristic
subgroupG/H
centerC22×C3#4099
commutator, cosocleC42×C5C23×C3
FrattiniC42C22×C3×D5
FittingC3×C5×#555C2
socleC22×C3×C5#120
hypercenterC3×#555D5
hypocenterC5C3×#1624
LCSG ▷ C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#555

Complements

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

NH ≅ G/Ntype
1. C3×C5×#555C2semidirect
2. C3×#65374C2semidirect
3. C3×#65304C2semidirect
4. C3×#64766C2semidirect
5. C3×D4⋊#168C2semidirect
6. #10602720C3direct (N × H)
19. C5×#555C2×C3semidirect
20. #65374C2×C3semidirect
21. #65304C2×C3semidirect
22. #64766C2×C3semidirect
23. D4⋊#168C2×C3semidirect
40. C3×#555D5semidirect
101. #555C3×D5semidirect
209. C3×C5#1624semidirect
246. C5C3×#1624semidirect
258. C3#10602720direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5552
C3×#653742
C3×#653042
C3×#647662
C3×D4⋊#1682
#106027203
C3×#16245