Finite Group Databasebeta

← C3×#10604385 #14298637

Subgroups — C3×#10604385 #14298637

Order 1920.

592 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 124 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×#65855C211yesmaximal, characteristic
2C2×C3×#3999C212yesmaximal
3C3×#64866C214yesmaximal, complemented
4#10604385C311yesmaximal, characteristic, direct product factor, Hall
5C22×C3×D20C2211yescharacteristic
6C2×C3×C22⋊AGL(1,𝔽5)C2214yes—
7C3×C5×#117C411yescharacteristic, complemented
8C3×#4072C411yescharacteristic, complemented
9C3×#4039—22no—
10C3×D5×C22⋊C4C412yescomplemented
11C3×D5×C22⋊C4C2212yes—
12C3×C4×D20—22no—
13C3×#3999—22no—
14C3×#3999—22no—
15C3×#1828—51nomaximal, Hall
16#65855C2×C311yescharacteristic
17C2×#3999C2×C312yes—
18#64866C2×C314yescomplemented
19C23×C3×D5C2×C411yescharacteristic
20C23×C3×D5C2311yescharacteristic
21C22×C3×AGL(1,𝔽5)—24no—
22C22×C3×C4×C5C2×C411yescharacteristic
23C2×C3×D20—24no—
24C2×C3×D20D412yes—
25C2×C3×D20D412yes—
26C2×C3×C4×D5—22no—
27C2×C3×C4×D5—22no—
28C3×C22⋊AGL(1,𝔽5)—24no—
29C3×C22⋊AGL(1,𝔽5)—24no—
30C3×C5×C4⋊C4—21no—
31C3×C5×C22⋊C4C2×C412yes—
32C3×C42×C5—21no—
33C3×C22⋊Dic5C2×C412yes—
34C3×#685—22no—
35C3×#685—22no—
36C3×C4⋊Dic5—21no—
37C3×C4×Dic5—21no—
38C3×#518—51no—
39C2×C3×#99—52no—
40C3×#342—54no—
41C22×D20C22×C311yescharacteristic
42C2×C22⋊AGL(1,𝔽5)C22×C314yes—
43C5×#117C3×C411yescharacteristic, complemented
44#4072C3×C411yescharacteristic, complemented
45#4039—22no—
46D5×C22⋊C4C3×C412yescomplemented
47D5×C22⋊C4C22×C312yes—
48C4×D20—22no—
49#3999—22no—
50#3999—22no—
51#1828—51noSylow
52C23×C3×C5C22×C411yescharacteristic
53C22×C3×D5—22no—
54C22×C3×D5—81no—
55C22×C3×D5—81no—
56C22×C3×D5C22⋊C412yes—
57C22×C3×D5C2×D412yes—
58C22×C3×D5#4114yes—
59C2×C3×AGL(1,𝔽5)—44no—
60C2×C3×AGL(1,𝔽5)—84no—
61C2×C3×C4×C5—22no—
62C2×C3×C4×C5—22no—
63C2×C3×C4×C5C22⋊C411yescharacteristic
64C2×C3×C4×C5C22⋊C411yescharacteristic
65C2×C3×Dic5—22no—
66C3×D20—24no—
67C3×D20—24no—
68C3×C4×D5—82no—
69C3×C4×D5—82no—
70C22×C3×D4—51no—
71C3×C4×D4—102no—
72C3×C4×D4—102no—
73C3×#117—51no—
74C3×#117AGL(1,𝔽5)11yescharacteristic, complemented
75C2×C3×C22⋊C4—52no—
76C2×C3×C22⋊C4—52no—
77C2×C3×C22⋊C4—54no—
78C3×#99—102no—
79C3×#99—102no—
80C23×D5C2×C3×C411yescharacteristic
81C23×D5C23×C311yescharacteristic
82C22×AGL(1,𝔽5)—24no—
83C22×C4×C5C2×C3×C411yescharacteristic
84C2×D20—24no—
85C2×D20C3×D412yes—
86C2×D20C3×D412yes—
87C2×C4×D5—22no—
88C2×C4×D5—22no—
89C22⋊AGL(1,𝔽5)—24no—
90C22⋊AGL(1,𝔽5)—24no—
91C5×C4⋊C4—21no—
92C5×C22⋊C4C2×C3×C412yes—
93C42×C5—21no—
94C22⋊Dic5C2×C3×C412yes—
95#685—22no—
96#685—22no—
97C4⋊Dic5—21no—
98C4×Dic5—21no—
99#518—51no—
100C2×#99—52no—
101#342—54no—
102C22×C3×C5—41no—
103C22×C3×C5C2×C22⋊C411yescharacteristic
104C22×C3×C5#11712yes—
105C2×C3×D5—22no—
106C2×C3×D5—42no—
107C2×C3×D5—44no—
108C2×C3×D5—42no—
109C2×C3×D5—82no—
110C2×C3×D5#12314yes—
111C3×AGL(1,𝔽5)—84no—
112C3×C4×C5—21no—
113C3×C4×C5—21no—
114C3×C4×C5—42no—
115C3×Dic5—42no—
116C24×C3—51no—
117C24×C3—51no—
118C2×C3×D4—52no—
119C2×C3×D4—52no—
120C2×C3×D4—104no—
121C22×C3×C4—102no—
122C22×C3×C4—102no—
123C22×C3×C4—104no—
124C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
125C3×C4⋊C4—21no—
126C3×C4⋊C4—101no—
127C3×C22⋊C4—52no—
128C3×C22⋊C4—102no—
129C3×C22⋊C4—102no—
130C3×C22⋊C4—104no—
131C3×C22⋊C4—104no—
132C3×C22⋊C4C2×AGL(1,𝔽5)12yes—
133C3×C42—21no—
134C3×C42—101no—
135C23×C5C22×C3×C411yescharacteristic
136C22×D5—22no—
137C22×D5—81no—
138C22×D5—81no—
139C22×D5C3×C22⋊C412yes—
140C22×D5C2×C3×D412yes—
141C22×D5C3×#4114yes—
142C2×AGL(1,𝔽5)—44no—
143C2×AGL(1,𝔽5)—84no—
144C2×C4×C5—22no—
145C2×C4×C5—22no—
146C2×C4×C5C3×C22⋊C411yescharacteristic
147C2×C4×C5C3×C22⋊C411yescharacteristic
148C2×Dic5—22no—
149D20—24no—
150D20—24no—
151C4×D5—82no—
152C4×D5—82no—
153C22×D4—51no—
154C4×D4—102no—
155C4×D4—102no—
156#117—51no—
157#117C3×AGL(1,𝔽5)11yescharacteristic, complemented
158C2×C22⋊C4—52no—
159C2×C22⋊C4—52no—
160C2×C22⋊C4—54no—
161#99—102no—
162#99—102no—
163C2×C3×C5—21no—
164C2×C3×C5—22no—
165C2×C3×C5#38111yescharacteristic
166C3×D5—24no—
167C3×D5—42no—
168C23×C3—54no—
169C23×C3—52no—
170C23×C3—52no—
171C23×C3—102no—
172C23×C3—401no—
173C23×C3—401no—
174C23×C3C22×AGL(1,𝔽5)11yescharacteristic
175C3×D4—104no—
176C3×D4—104no—
177C2×C3×C4—22no—
178C2×C3×C4—22no—
179C2×C3×C4—102no—
180C2×C3×C4—204no—
181C2×C3×C4—402no—
182C2×C3×C4—402no—
183C2×C3×C4—404no—
184C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
185C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
186C22×C5—41no—
187C22×C5C2×C3×C22⋊C411yescharacteristic
188C22×C5C3×#11712yes—
189C2×D5—22no—
190C2×D5—42no—
191C2×D5—44no—
192C2×D5—42no—
193C2×D5—82no—
194C2×D5C3×#12314yes—
195AGL(1,𝔽5)—84no—
196C4×C5—21no—
197C4×C5—21no—
198C4×C5—42no—
199Dic5—42no—
200C24—51no—
201C24—51no—
202C2×D4—52no—
203C2×D4—52no—
204C2×D4—104no—
205C22×C4—102no—
206C22×C4—102no—
207C22×C4—104no—
208C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
209C4⋊C4—21no—
210C4⋊C4—101no—
211C22⋊C4—52no—
212C22⋊C4—102no—
213C22⋊C4—102no—
214C22⋊C4—104no—
215C22⋊C4—104no—
216C22⋊C4C2×C3×AGL(1,𝔽5)12yes—
217C42—21no—
218C42—101no—
219C3×C5#182811yescharacteristic, complemented, Hall
220C22×C3—41no—
221C22×C3—54no—
222C22×C3—102no—
223C22×C3—202no—
224C22×C3—204no—
225C22×C3—202no—
226C22×C3—402no—
227C22×C3#413012yes—
228C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
229C3×C4—21no—
230C3×C4—21no—
231C3×C4—42no—
232C3×C4—202no—
233C3×C4—404no—
234C2×C5—21no—
235C2×C5—22no—
236C2×C5C3×#38111yescharacteristic
237D5—24no—
238D5—42no—
239C23—54no—
240C23—52no—
241C23—52no—
242C23—102no—
243C23—401no—
244C23—401no—
245C23C22×C3×AGL(1,𝔽5)11yescharacteristic
246D4—104no—
247D4—104no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—102no—
251C2×C4—204no—
252C2×C4—402no—
253C2×C4—402no—
254C2×C4—404no—
255C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
256C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
257C2×C3—21no—
258C2×C3—22no—
259C2×C3—104no—
260C2×C3—202no—
261C2×C3#6576411yescharacteristic
262C5C3×#182811yescharacteristic, complemented, Sylow
263C22—41no—
264C22—54no—
265C22—102no—
266C22—202no—
267C22—204no—
268C22—202no—
269C22—402no—
270C22C3×#413012yes—
271C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
272C4—21no—
273C4—21no—
274C4—42no—
275C4—202no—
276C4—404no—
277C3#1060438511yescharacteristic, direct product factor, Sylow
278C2—21no—
279C2—22no—
280C2—104no—
281C2—202no—
282C2C3×#6576411yescharacteristic
subgroupG/H
centerC2×C3#65764
commutatorC23×C5C22×C3×C4
FrattiniC23C22×C3×AGL(1,𝔽5)
FittingC3×C5×#117C4
socleC2×C3×C5#381
prime radicalC22×C3×D20C22
cosocleC23×D5C23×C3
hypercenterC3×#117AGL(1,𝔽5)
hypocenterC5C3×#1828
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C3×#117

Complements

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

NH ≅ G/Ntype
3. C3×#64866C2semidirect
4. #10604385C3direct (N × H)
7. C3×C5×#117C4semidirect
8. C3×#4072C4semidirect
10. C3×D5×C22⋊C4C4semidirect
18. #64866C2×C3semidirect
43. C5×#117C3×C4semidirect
44. #4072C3×C4semidirect
46. D5×C22⋊C4C3×C4semidirect
74. C3×#117AGL(1,𝔽5)semidirect
157. #117C3×AGL(1,𝔽5)semidirect
219. C3×C5#1828semidirect
262. C5C3×#1828semidirect
277. C3#10604385direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#658552
C2×C3×#39992
C3×#648662
#106043853
C3×#18285