Finite Group Databasebeta

← C3×#10604587 #14307414

Subgroups — C3×#10604587 #14307414

Order 1920.

568 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 148 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×#66022C211yesmaximal, characteristic
2C3×C42×AGL(1,𝔽5)C211yesmaximal, characteristic
3C3×#65682C211yesmaximal, characteristic
4C3×#64875C214yesmaximal
5#10604587C311yesmaximal, characteristic, direct product factor, Hall
6C2×C3×C4×AGL(1,𝔽5)—22no—
7C2×C3×C4×AGL(1,𝔽5)C2212yes—
8C2×C3×#700C2212yes—
9C3×C5×#125C411yescharacteristic, complemented
10C3×#4042—22no—
11C3×#4038C2212yes—
12C3×#4034C412yescomplemented
13C3×#4033—22no—
14C3×C42×D5C2211yescharacteristic
15C3×#4016C411yescharacteristic, complemented
16C3×#4004—22no—
17C3×#1994—51nomaximal, Hall
18#66022C2×C311yescharacteristic
19C42×AGL(1,𝔽5)C2×C311yescharacteristic
20#65682C2×C311yescharacteristic
21#64875C2×C314yes—
22C22×C3×AGL(1,𝔽5)—22no—
23C2×C3×C4×D5D412yes—
24C2×C3×C4×D5C2311yescharacteristic
25C2×C3×Dic10—22no—
26C2×C3×#170—22no—
27C3×C4×AGL(1,𝔽5)—28no—
28C3×C4×AGL(1,𝔽5)—22no—
29C3×C4×AGL(1,𝔽5)Q814yescomplemented
30C3×#700—24no—
31C3×C5×C4⋊C4—22no—
32C3×C5×C4⋊C4C2×C412yes—
33C3×C42×C5C2×C411yescharacteristic
34C3×#685—22no—
35C3×C4⋊Dic5C2×C412yes—
36C3×#683—22no—
37C3×#683—22no—
38C3×C4×Dic5—22no—
39C3×C4×Dic5C2×C411yescharacteristic
40C3×C4×Dic5D412yes—
41C3×#533—51no—
42C3×#423—51no—
43C3×C43—51no—
44C3×#339—54no—
45C2×C4×AGL(1,𝔽5)—22no—
46C2×C4×AGL(1,𝔽5)C22×C312yes—
47C2×#700C22×C312yes—
48C5×#125C3×C411yescharacteristic, complemented
49#4042—22no—
50#4038C22×C312yes—
51#4034C3×C412yescomplemented
52#4033—22no—
53C42×D5C22×C311yescharacteristic
54#4016C3×C411yescharacteristic, complemented
55#4004—22no—
56#1994—51noSylow
57C22×C3×D5C2×D411yescharacteristic
58C2×C3×AGL(1,𝔽5)—28no—
59C2×C3×AGL(1,𝔽5)—24no—
60C2×C3×C4×C5—22no—
61C2×C3×C4×C5C22⋊C412yes—
62C2×C3×C4×C5C22×C411yescharacteristic
63C2×C3×Dic5—22no—
64C2×C3×Dic5C22⋊C412yes—
65C2×C3×Dic5C2×D411yescharacteristic
66C3×C4×D5—24no—
67C3×C4×D5C2×Q812yes—
68C3×C4×D5#4112yes—
69C3×Dic10—44no—
70C3×#170—42no—
71C2×C3×M16—52no—
72C3×C4⋊Q8—52no—
73C3×#125—51no—
74C3×#125AGL(1,𝔽5)11yescharacteristic, complemented
75C3×C22⋊Q8—102no—
76C3×C4×Q8—102no—
77C3×#117—52no—
78C2×C3×C42—51no—
79C2×C3×C42—52no—
80C2×C3×C42—102no—
81C3×C4⋊C8—102no—
82C22×AGL(1,𝔽5)—22no—
83C2×C4×D5C3×D412yes—
84C2×C4×D5C23×C311yescharacteristic
85C2×Dic10—22no—
86C2×#170—22no—
87C4×AGL(1,𝔽5)—28no—
88C4×AGL(1,𝔽5)—22no—
89C4×AGL(1,𝔽5)C3×Q814yescomplemented
90#700—24no—
91C5×C4⋊C4—22no—
92C5×C4⋊C4C2×C3×C412yes—
93C42×C5C2×C3×C411yescharacteristic
94#685—22no—
95C4⋊Dic5C2×C3×C412yes—
96#683—22no—
97#683—22no—
98C4×Dic5—22no—
99C4×Dic5C2×C3×C411yescharacteristic
100C4×Dic5C3×D412yes—
101#533—51no—
102#423—51no—
103C43—51no—
104#339—54no—
105C22×C3×C5C2×C22⋊C411yescharacteristic
106C2×C3×D5—22no—
107C2×C3×D5C22⋊Q812yes—
108C3×AGL(1,𝔽5)—28no—
109C3×C4×C5—22no—
110C3×C4×C5—42no—
111C3×C4×C5C4×Q812yes—
112C3×Dic5—42no—
113C3×Dic5#10414yes—
114C3×Dic5#12411yescharacteristic
115C3×Dic5C4⋊Q811yescharacteristic
116C2×C3×Q8—102no—
117C22×C3×C4—51no—
118C22×C3×C4—52no—
119C22×C3×C4—102no—
120C3×M16—104no—
121C2×C3×C8—102no—
122C3×C4⋊C4—22no—
123C3×C4⋊C4—52no—
124C3×C4⋊C4—102no—
125C3×C4⋊C4—102no—
126C3×C4⋊C4C2×AGL(1,𝔽5)12yes—
127C3×C22⋊C4—102no—
128C3×C42—52no—
129C3×C42—51no—
130C3×C42—54no—
131C3×C42—102no—
132C3×C42—102no—
133C3×C42—108no—
134C3×C42C2×AGL(1,𝔽5)11yescharacteristic
135C22×D5C2×C3×D411yescharacteristic
136C2×AGL(1,𝔽5)—28no—
137C2×AGL(1,𝔽5)—24no—
138C2×C4×C5—22no—
139C2×C4×C5C3×C22⋊C412yes—
140C2×C4×C5C22×C3×C411yescharacteristic
141C2×Dic5—22no—
142C2×Dic5C3×C22⋊C412yes—
143C2×Dic5C2×C3×D411yescharacteristic
144C4×D5—24no—
145C4×D5C2×C3×Q812yes—
146C4×D5C3×#4112yes—
147Dic10—44no—
148#170—42no—
149C2×M16—52no—
150C4⋊Q8—52no—
151#125—51no—
152#125C3×AGL(1,𝔽5)11yescharacteristic, complemented
153C22⋊Q8—102no—
154C4×Q8—102no—
155#117—52no—
156C2×C42—51no—
157C2×C42—52no—
158C2×C42—102no—
159C4⋊C8—102no—
160C2×C3×C5#39111yescharacteristic
161C2×C3×C5C2×#10412yes—
162C3×D5—22no—
163C23×C3—51no—
164C3×Q8—204no—
165C2×C3×C4—22no—
166C2×C3×C4—52no—
167C2×C3×C4—51no—
168C2×C3×C4—52no—
169C2×C3×C4—52no—
170C2×C3×C4—104no—
171C2×C3×C4—104no—
172C2×C3×C4—108no—
173C2×C3×C4—102no—
174C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
175C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
176C3×C8—202no—
177C22×C5C2×C3×C22⋊C411yescharacteristic
178C2×D5—22no—
179C2×D5C3×C22⋊Q812yes—
180AGL(1,𝔽5)—28no—
181C4×C5—22no—
182C4×C5—42no—
183C4×C5C3×C4×Q812yes—
184Dic5—42no—
185Dic5C3×#10414yes—
186Dic5C3×#12411yescharacteristic
187Dic5C3×C4⋊Q811yescharacteristic
188C2×Q8—102no—
189C22×C4—51no—
190C22×C4—52no—
191C22×C4—102no—
192M16—104no—
193C2×C8—102no—
194C4⋊C4—22no—
195C4⋊C4—52no—
196C4⋊C4—102no—
197C4⋊C4—102no—
198C4⋊C4C2×C3×AGL(1,𝔽5)12yes—
199C22⋊C4—102no—
200C42—52no—
201C42—51no—
202C42—54no—
203C42—102no—
204C42—102no—
205C42—108no—
206C42C2×C3×AGL(1,𝔽5)11yescharacteristic
207C3×C5#199411yescharacteristic, complemented, Hall
208C22×C3—52no—
209C22×C3—102no—
210C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
211C3×C4—22no—
212C3×C4—42no—
213C3×C4—54no—
214C3×C4—51no—
215C3×C4—51no—
216C3×C4—108no—
217C3×C4—202no—
218C3×C4Q8×AGL(1,𝔽5)12yes—
219C2×C5C3×#39111yescharacteristic
220C2×C5C2×C3×#10412yes—
221D5—22no—
222C23—51no—
223Q8—204no—
224C2×C4—22no—
225C2×C4—51no—
226C2×C4—52no—
227C2×C4—52no—
228C2×C4—52no—
229C2×C4—104no—
230C2×C4—104no—
231C2×C4—108no—
232C2×C4—102no—
233C2×C4C3×C22⋊AGL(1,𝔽5)12yes—
234C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
235C8—202no—
236C2×C3—102no—
237C2×C3(C2×AGL(1,𝔽5))⋊Q811yescharacteristic
238C2×C3D5⋊#10412yes—
239C5C3×#199411yescharacteristic, complemented, Sylow
240C22—52no—
241C22—102no—
242C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
243C4—22no—
244C4—42no—
245C4—51no—
246C4—51no—
247C4—54no—
248C4—108no—
249C4—202no—
250C4C3×Q8×AGL(1,𝔽5)12yes—
251C3#1060458711yescharacteristic, direct product factor, Sylow
252C2—102no—
253C2C3×(C2×AGL(1,𝔽5))⋊Q811yescharacteristic
254C2C3×D5⋊#10412yes—
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C5×#125C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×C3×D5C2×D4
cosocleC2×C4×D5C23×C3
hypercenterC3×#125AGL(1,𝔽5)
hypocenterC5C3×#1994
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#125

Complements

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

NH ≅ G/Ntype
5. #10604587C3direct (N × H)
9. C3×C5×#125C4semidirect
12. C3×#4034C4semidirect
15. C3×#4016C4semidirect
29. C3×C4×AGL(1,𝔽5)Q8semidirect
48. C5×#125C3×C4semidirect
51. #4034C3×C4semidirect
54. #4016C3×C4semidirect
74. C3×#125AGL(1,𝔽5)semidirect
89. C4×AGL(1,𝔽5)C3×Q8semidirect
152. #125C3×AGL(1,𝔽5)semidirect
207. C3×C5#1994semidirect
239. C5C3×#1994semidirect
251. C3#10604587direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#660222
C3×C42×AGL(1,𝔽5)2
C3×#656822
C3×#648752
#106045873
C3×#19945