Finite Group Databasebeta

← C3×#10604664 #14306043

Subgroups — C3×#10604664 #14306043

Order 1920.

640 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 140 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×#66059C211yesmaximal, characteristic
2C2×C3×#4010C214yesmaximal, complemented
3C3×#65686C211yesmaximal, characteristic, complemented
4C3×#65682C211yesmaximal, characteristic, complemented
5#10604664C311yesmaximal, characteristic, direct product factor, Hall
6C2×C3×#713C2212yes—
7C2×C3×C4×AGL(1,𝔽5)C2212yes—
8C2×C3×#700C2212yes—
9C3×C5×C4⋊Q8C411yescharacteristic, complemented
10C3×#4093C412yescomplemented
11C3×#4041—22no—
12C3×#4039—22no—
13C3×#4020C411yescharacteristic, complemented
14C3×C42×D5C2211yescharacteristic
15C3×#4010—28no—
16C3×(C2×AGL(1,𝔽5))⋊C4—22no—
17C3×#4004—22no—
18C3×#1963—51nomaximal, Hall
19#66059C2×C311yescharacteristic
20C2×#4010C2×C314yescomplemented
21#65686C2×C311yescharacteristic, complemented
22#65682C2×C311yescharacteristic, complemented
23C22×C3×AGL(1,𝔽5)—22no—
24C2×C3×C5×Q8C2×C412yes—
25C3×#713—24no—
26C3×#713—24no—
27C2×C3×D20—22no—
28C2×C3×D20—22no—
29C2×C3×D20C2×C412yes—
30C2×C3×C4×D5—22no—
31C2×C3×C4×D5D412yes—
32C2×C3×C4×D5C2311yescharacteristic
33C2×C3×#170—22no—
34C3×C4×AGL(1,𝔽5)—22no—
35C3×C4×AGL(1,𝔽5)D414yescomplemented
36C3×#700—24no—
37C3×C5×C4⋊C4—22no—
38C3×C42×C5C2×C411yescharacteristic
39C3×#685—24no—
40C3×C4×Dic5C2×C411yescharacteristic
41C3×C4×Dic5D412yes—
42C3×#532—51no—
43C3×#423—51no—
44C2×C3×#104—54no—
45C3×#376—51no—
46C2×#713C22×C312yes—
47C2×C4×AGL(1,𝔽5)C22×C312yes—
48C2×#700C22×C312yes—
49C5×C4⋊Q8C3×C411yescharacteristic, complemented
50#4093C3×C412yescomplemented
51#4041—22no—
52#4039—22no—
53#4020C3×C411yescharacteristic, complemented
54C42×D5C22×C311yescharacteristic
55#4010—28no—
56(C2×AGL(1,𝔽5))⋊C4—22no—
57#4004—22no—
58#1963—51noSylow
59C22×C3×D5—22no—
60C22×C3×D5C2×D411yescharacteristic
61C2×C3×AGL(1,𝔽5)—24no—
62C2×C3×AGL(1,𝔽5)—44no—
63C3×C5×Q8—24no—
64C2×C3×C4×C5—22no—
65C2×C3×C4×C5C22⋊C412yes—
66C2×C3×C4×C5C22×C411yescharacteristic
67C2×C3×Dic5C22⋊C412yes—
68C2×C3×Dic5C2×D411yescharacteristic
69C3×D20—24no—
70C3×D20—44no—
71C3×D20—44no—
72C3×C4×D5—42no—
73C3×C4×D5—42no—
74C3×C4×D5—42no—
75C3×C4×D5C2×D412yes—
76C3×C4×D5#4112yes—
77C3×#170—42no—
78C2×C3×#41—52no—
79C2×C3×M16—52no—
80C3×C4⋊Q8AGL(1,𝔽5)11yescharacteristic, complemented
81C3×#127—51no—
82C3×#124—52no—
83C3×#121—102no—
84C3×C4×D4—102no—
85C2×C3×C42—51no—
86C2×C3×C42—52no—
87C3×C4⋊C8—102no—
88C3×#104—108no—
89C3×#95—102no—
90C22×AGL(1,𝔽5)—22no—
91C2×C5×Q8C2×C3×C412yes—
92#713—24no—
93#713—24no—
94C2×D20—22no—
95C2×D20—22no—
96C2×D20C2×C3×C412yes—
97C2×C4×D5—22no—
98C2×C4×D5C3×D412yes—
99C2×C4×D5C23×C311yescharacteristic
100C2×#170—22no—
101C4×AGL(1,𝔽5)—22no—
102C4×AGL(1,𝔽5)C3×D414yescomplemented
103#700—24no—
104C5×C4⋊C4—22no—
105C42×C5C2×C3×C411yescharacteristic
106#685—24no—
107C4×Dic5C2×C3×C411yescharacteristic
108C4×Dic5C3×D412yes—
109#532—51no—
110#423—51no—
111C2×#104—54no—
112#376—51no—
113C22×C3×C5C2×C22⋊C411yescharacteristic
114C2×C3×D5—22no—
115C2×C3×D5—42no—
116C2×C3×D5—44no—
117C2×C3×D5#12112yes—
118C3×AGL(1,𝔽5)—44no—
119C3×C4×C5—22no—
120C3×C4×C5—42no—
121C3×C4×C5C4×D412yes—
122C3×Dic5—22no—
123C3×Dic5#12411yescharacteristic
124C3×Dic5#12711yescharacteristic
125C3×#41—104no—
126C3×#41—104no—
127C2×C3×Q8C2×AGL(1,𝔽5)12yes—
128C2×C3×D4—52no—
129C2×C3×D4—102no—
130C2×C3×D4—102no—
131C22×C3×C4—51no—
132C22×C3×C4—52no—
133C22×C3×C4—102no—
134C22×C3×C4—102no—
135C3×M16—104no—
136C2×C3×C8—102no—
137C3×C4⋊C4—22no—
138C3×C22⋊C4—104no—
139C3×C42—52no—
140C3×C42—51no—
141C3×C42—54no—
142C3×C42—102no—
143C3×C42C2×AGL(1,𝔽5)11yescharacteristic
144C22×D5—22no—
145C22×D5C2×C3×D411yescharacteristic
146C2×AGL(1,𝔽5)—24no—
147C2×AGL(1,𝔽5)—44no—
148C5×Q8—24no—
149C2×C4×C5—22no—
150C2×C4×C5C3×C22⋊C412yes—
151C2×C4×C5C22×C3×C411yescharacteristic
152C2×Dic5C3×C22⋊C412yes—
153C2×Dic5C2×C3×D411yescharacteristic
154D20—24no—
155D20—44no—
156D20—44no—
157C4×D5—42no—
158C4×D5—42no—
159C4×D5—42no—
160C4×D5C2×C3×D412yes—
161C4×D5C3×#4112yes—
162#170—42no—
163C2×#41—52no—
164C2×M16—52no—
165C4⋊Q8C3×AGL(1,𝔽5)11yescharacteristic, complemented
166#127—51no—
167#124—52no—
168#121—102no—
169C4×D4—102no—
170C2×C42—51no—
171C2×C42—52no—
172C4⋊C8—102no—
173#104—108no—
174#95—102no—
175C2×C3×C5#39011yescharacteristic
176C2×C3×C5#42112yes—
177C3×D5—22no—
178C3×D5—82no—
179C23×C3—51no—
180C23×C3—102no—
181C3×Q8—24no—
182C3×D4—104no—
183C3×D4—204no—
184C3×D4—204no—
185C2×C3×C4—22no—
186C2×C3×C4—51no—
187C2×C3×C4—52no—
188C2×C3×C4—52no—
189C2×C3×C4—52no—
190C2×C3×C4—104no—
191C2×C3×C4—202no—
192C2×C3×C4—202no—
193C2×C3×C4—202no—
194C2×C3×C4—204no—
195C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
196C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
197C3×C8—202no—
198C22×C5C2×C3×C22⋊C411yescharacteristic
199C2×D5—22no—
200C2×D5—42no—
201C2×D5—44no—
202C2×D5C3×#12112yes—
203AGL(1,𝔽5)—44no—
204C4×C5—22no—
205C4×C5—42no—
206C4×C5C3×C4×D412yes—
207Dic5—22no—
208Dic5C3×#12411yescharacteristic
209Dic5C3×#12711yescharacteristic
210#41—104no—
211#41—104no—
212C2×Q8C2×C3×AGL(1,𝔽5)12yes—
213C2×D4—52no—
214C2×D4—102no—
215C2×D4—102no—
216C22×C4—51no—
217C22×C4—52no—
218C22×C4—102no—
219C22×C4—102no—
220M16—104no—
221C2×C8—102no—
222C4⋊C4—22no—
223C22⋊C4—104no—
224C42—52no—
225C42—51no—
226C42—54no—
227C42—102no—
228C42C2×C3×AGL(1,𝔽5)11yescharacteristic
229C3×C5#196311yescharacteristic, complemented, Hall
230C22×C3—52no—
231C22×C3—102no—
232C22×C3—202no—
233C22×C3—204no—
234C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
235C3×C4—22no—
236C3×C4—42no—
237C3×C4—51no—
238C3×C4—51no—
239C3×C4—102no—
240C3×C4—204no—
241C3×C4D4×AGL(1,𝔽5)12yes—
242C2×C5C3×#39011yescharacteristic
243C2×C5C3×#42112yes—
244D5—22no—
245D5—82no—
246C23—51no—
247C23—102no—
248Q8—24no—
249D4—104no—
250D4—204no—
251D4—204no—
252C2×C4—22no—
253C2×C4—51no—
254C2×C4—52no—
255C2×C4—52no—
256C2×C4—52no—
257C2×C4—104no—
258C2×C4—202no—
259C2×C4—202no—
260C2×C4—202no—
261C2×C4—204no—
262C2×C4C3×C22⋊AGL(1,𝔽5)12yes—
263C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
264C8—202no—
265C2×C3—102no—
266C2×C3—402no—
267C2×C3#6578011yescharacteristic
268C2×C3#6578412yes—
269C5C3×#196311yescharacteristic, complemented, Sylow
270C22—52no—
271C22—102no—
272C22—202no—
273C22—204no—
274C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
275C4—22no—
276C4—42no—
277C4—51no—
278C4—51no—
279C4—102no—
280C4—204no—
281C4C3×D4×AGL(1,𝔽5)12yes—
282C3#1060466411yescharacteristic, direct product factor, Sylow
283C2—102no—
284C2—402no—
285C2C3×#6578011yescharacteristic
286C2C3×#6578412yes—
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C5×C4⋊Q8C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC3×#66059C2
cosocleC2×C4×D5C23×C3
hypercenterC3×C4⋊Q8AGL(1,𝔽5)
hypocenterC5C3×#1963
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×C3×#4010C2semidirect
3. C3×#65686C2semidirect
4. C3×#65682C2semidirect
5. #10604664C3direct (N × H)
9. C3×C5×C4⋊Q8C4semidirect
10. C3×#4093C4semidirect
13. C3×#4020C4semidirect
20. C2×#4010C2×C3semidirect
21. #65686C2×C3semidirect
22. #65682C2×C3semidirect
35. C3×C4×AGL(1,𝔽5)D4semidirect
49. C5×C4⋊Q8C3×C4semidirect
50. #4093C3×C4semidirect
53. #4020C3×C4semidirect
80. C3×C4⋊Q8AGL(1,𝔽5)semidirect
102. C4×AGL(1,𝔽5)C3×D4semidirect
165. C4⋊Q8C3×AGL(1,𝔽5)semidirect
229. C3×C5#1963semidirect
269. C5C3×#1963semidirect
282. C3#10604664direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#660592
C2×C3×#40102
C3×#656862
C3×#656822
#106046643
C3×#19635