Finite Group Databasebeta

← C3×#10604355 #14313538

Subgroups — C3×#10604355 #14313538

Order 1920.

476 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 108 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×#65860C211yesmaximal, characteristic
2C3×#65745C212yesmaximal
3C3×#64854C214yesmaximal, complemented
4#10604355C311yesmaximal, characteristic, direct product factor, Hall
5C2×C3×#709C2211yescharacteristic
6C22×C3×Dic10C411yescharacteristic, complemented
7C3×#4130—22no—
8C3×C5×#117C411yescharacteristic, complemented
9C3×#4043—22no—
10C3×#4032—22no—
11C3×#4021C2212yes—
12C3×#4015C412yescomplemented
13C3×#4011—22no—
14C3×#3999C2214yes—
15C3×#2129—51nomaximal, Hall
16#65860C2×C311yescharacteristic
17#65745C2×C312yes—
18#64854C2×C314yescomplemented
19C22×C3×C4×C5C2×C411yescharacteristic
20C2×C3×C22⋊D5D412yes—
21C22×C3×Dic5—21no—
22C3×#709—42no—
23C2×C3×D20C2311yescharacteristic
24C2×C3×C4×D5D412yes—
25C2×C3×Dic10—21no—
26C2×C3×Dic10—42no—
27C2×C3×Dic10C2×C412yes—
28C2×C3×Dic10C2×C411yescharacteristic
29C3×C22⋊AGL(1,𝔽5)—24no—
30C3×C4⋊AGL(1,𝔽5)—42no—
31C3×C4×AGL(1,𝔽5)—42no—
32C3×C5×C4⋊C4—21no—
33C3×C5×C22⋊C4—21no—
34C3×C42×C5C2×C412yes—
35C3×#685—24no—
36C3×C4⋊Dic5—24no—
37C3×#536—51no—
38C3×#410—52no—
39C3×#354—54no—
40C2×#709C22×C311yescharacteristic
41C22×Dic10C3×C411yescharacteristic, complemented
42#4130—22no—
43C5×#117C3×C411yescharacteristic, complemented
44#4043—22no—
45#4032—22no—
46#4021C22×C312yes—
47#4015C3×C412yescomplemented
48#4011—22no—
49#3999C22×C314yes—
50#2129—51noSylow
51C23×C3×C5C22⋊C411yescharacteristic
52C22×C3×D5C2×D412yes—
53C2×C3×AGL(1,𝔽5)—44no—
54C2×C3×C4×C5—22no—
55C2×C3×C4×C5—22no—
56C2×C3×C4×C5C22⋊C411yescharacteristic
57C2×C3×C4×C5C22×C411yescharacteristic
58C3×C22⋊D5—42no—
59C2×C3×Dic5—22no—
60C2×C3×Dic5—42no—
61C2×C3×Dic5C22⋊C412yes—
62C3×D20—41no—
63C3×C4×D5—22no—
64C3×C4×D5—22no—
65C3×Dic10—41no—
66C3×Dic10—42no—
67C3×Dic10—42no—
68C2×C3×#41—51no—
69C22×C3×Q8—51no—
70C3×C4⋊Q8—52no—
71C3×#124—52no—
72C3×#123—102no—
73C3×C22⋊Q8—102no—
74C3×#117—102no—
75C3×#117AGL(1,𝔽5)11yescharacteristic, complemented
76C3×#99—54no—
77C3×#99—102no—
78C22×C4×C5C2×C3×C411yescharacteristic
79C2×C22⋊D5C3×D412yes—
80C22×Dic5—21no—
81#709—42no—
82C2×D20C23×C311yescharacteristic
83C2×C4×D5C3×D412yes—
84C2×Dic10—21no—
85C2×Dic10—42no—
86C2×Dic10C2×C3×C412yes—
87C2×Dic10C2×C3×C411yescharacteristic
88C22⋊AGL(1,𝔽5)—24no—
89C4⋊AGL(1,𝔽5)—42no—
90C4×AGL(1,𝔽5)—42no—
91C5×C4⋊C4—21no—
92C5×C22⋊C4—21no—
93C42×C5C2×C3×C412yes—
94#685—24no—
95C4⋊Dic5—24no—
96#536—51no—
97#410—52no—
98#354—54no—
99C22×C3×C5—21no—
100C22×C3×C5—41no—
101C22×C3×C5C2×C22⋊C411yescharacteristic
102C2×C3×D5—22no—
103C2×C3×D5—42no—
104C3×AGL(1,𝔽5)—84no—
105C3×C4×C5—21no—
106C3×C4×C5—42no—
107C3×C4×C5#9912yes—
108C3×Dic5—22no—
109C3×Dic5—42no—
110C3×#41—202no—
111C2×C3×Q8—51no—
112C2×C3×Q8—52no—
113C2×C3×Q8—101no—
114C2×C3×Q8—202no—
115C2×C3×D4—52no—
116C2×C3×D4—51no—
117C22×C3×C4—52no—
118C22×C3×C4—101no—
119C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
120C3×C4⋊C4—21no—
121C3×C4⋊C4—104no—
122C3×C4⋊C4—202no—
123C3×C22⋊C4—21no—
124C3×C22⋊C4—104no—
125C3×C22⋊C4—104no—
126C3×C42—202no—
127C3×C42C2×AGL(1,𝔽5)12yes—
128C23×C5C3×C22⋊C411yescharacteristic
129C22×D5C2×C3×D412yes—
130C2×AGL(1,𝔽5)—44no—
131C2×C4×C5—22no—
132C2×C4×C5—22no—
133C2×C4×C5C3×C22⋊C411yescharacteristic
134C2×C4×C5C22×C3×C411yescharacteristic
135C22⋊D5—42no—
136C2×Dic5—22no—
137C2×Dic5—42no—
138C2×Dic5C3×C22⋊C412yes—
139D20—41no—
140C4×D5—22no—
141C4×D5—22no—
142Dic10—41no—
143Dic10—42no—
144Dic10—42no—
145C2×#41—51no—
146C22×Q8—51no—
147C4⋊Q8—52no—
148#124—52no—
149#123—102no—
150C22⋊Q8—102no—
151#117—102no—
152#117C3×AGL(1,𝔽5)11yescharacteristic, complemented
153#99—54no—
154#99—102no—
155C2×C3×C5—21no—
156C2×C3×C5—41no—
157C2×C3×C5C2×#9911yescharacteristic
158C3×D5—42no—
159C23×C3—52no—
160C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
161C3×Q8—201no—
162C3×Q8—202no—
163C3×Q8—202no—
164C3×D4—201no—
165C3×D4—202no—
166C2×C3×C4—22no—
167C2×C3×C4—22no—
168C2×C3×C4—52no—
169C2×C3×C4—102no—
170C2×C3×C4—102no—
171C2×C3×C4—102no—
172C2×C3×C4—202no—
173C2×C3×C4—204no—
174C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
175C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
176C22×C5—21no—
177C22×C5—41no—
178C22×C5C2×C3×C22⋊C411yescharacteristic
179C2×D5—22no—
180C2×D5—42no—
181AGL(1,𝔽5)—84no—
182C4×C5—21no—
183C4×C5—42no—
184C4×C5C3×#9912yes—
185Dic5—22no—
186Dic5—42no—
187#41—202no—
188C2×Q8—51no—
189C2×Q8—52no—
190C2×Q8—101no—
191C2×Q8—202no—
192C2×D4—52no—
193C2×D4—51no—
194C22×C4—52no—
195C22×C4—101no—
196C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
197C4⋊C4—21no—
198C4⋊C4—104no—
199C4⋊C4—202no—
200C22⋊C4—21no—
201C22⋊C4—104no—
202C22⋊C4—104no—
203C42—202no—
204C42C2×C3×AGL(1,𝔽5)12yes—
205C3×C5#212911yescharacteristic, complemented, Hall
206C22×C3—21no—
207C22×C3—41no—
208C22×C3—102no—
209C22×C3—202no—
210C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
211C3×C4—21no—
212C3×C4—42no—
213C3×C4—102no—
214C3×C4—202no—
215C3×C4—404no—
216C3×C4#399912yes—
217C2×C5—21no—
218C2×C5—41no—
219C2×C5C2×C3×#9911yescharacteristic
220D5—42no—
221C23—52no—
222C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
223Q8—201no—
224Q8—202no—
225Q8—202no—
226D4—201no—
227D4—202no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—52no—
231C2×C4—102no—
232C2×C4—102no—
233C2×C4—102no—
234C2×C4—202no—
235C2×C4—204no—
236C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
237C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
238C2×C3—21no—
239C2×C3—41no—
240C2×C3—202no—
241C2×C3C2×#399911yescharacteristic
242C5C3×#212911yescharacteristic, complemented, Sylow
243C22—21no—
244C22—41no—
245C22—102no—
246C22—202no—
247C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
248C4—21no—
249C4—42no—
250C4—102no—
251C4—202no—
252C4—404no—
253C4C3×#399912yes—
254C3#1060435511yescharacteristic, direct product factor, Sylow
255C2—21no—
256C2—41no—
257C2—202no—
258C2C2×C3×#399911yescharacteristic
subgroupG/H
centerC2×C3C2×#3999
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C5×#117C4
socleC2×C3×C5C2×#99
prime radicalC2×C3×#709C22
cosocleC2×D20C23×C3
hypercenterC3×#117AGL(1,𝔽5)
hypocenterC5C3×#2129
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C22×C3×C4 ◁ C3×#117

Complements

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

NH ≅ G/Ntype
3. C3×#64854C2semidirect
4. #10604355C3direct (N × H)
6. C22×C3×Dic10C4semidirect
8. C3×C5×#117C4semidirect
12. C3×#4015C4semidirect
18. #64854C2×C3semidirect
41. C22×Dic10C3×C4semidirect
43. C5×#117C3×C4semidirect
47. #4015C3×C4semidirect
75. C3×#117AGL(1,𝔽5)semidirect
152. #117C3×AGL(1,𝔽5)semidirect
205. C3×C5#2129semidirect
242. C5C3×#2129semidirect
254. C3#10604355direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#658602
C3×#657452
C3×#648542
#106043553
C3×#21295