Finite Group Databasebeta

← C3×#10604961 #14284340

Subgroups — C3×#10604961 #14284340

Order 1920.

496 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 116 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
1C2×C3×#4093C211yesmaximal, characteristic
2C3×#65757C212yesmaximal, complemented
3C3×#64927C214yesmaximal, complemented
4#10604961C311yesmaximal, characteristic, direct product factor, Hall
5C22×C3×C5×Q8C411yescharacteristic
6C22×C3×D20C411yescharacteristic
7C3×#4093—22no—
8C3×#4093C412yes—
9C3×#4093C2212yes—
10C2×C3×#685—22no—
11C2×C3×C4×Dic5C2211yescharacteristic
12C3×C22⋊#170—22no—
13C3×#4004—22no—
14C3×#4002C2214yescomplemented
15C3×#1533—51nomaximal, Hall
16C2×#4093C2×C311yescharacteristic
17#65757C2×C312yescomplemented
18#64927C2×C314yescomplemented
19C23×C3×D5—21no—
20C2×C3×C5×Q8—21no—
21C2×C3×C5×Q8—42no—
22C2×C3×C5×Q8C2×C412yes—
23C22×C3×C4×C5—21no—
24C22×C3×C4×C5C2×C411yescharacteristic
25C22×C3×Dic5D412yes—
26C2×C3×D20—21no—
27C2×C3×D20—42no—
28C2×C3×D20C2×C412yes—
29C2×C3×#170—24no—
30C3×#685—24no—
31C3×#685—24no—
32C3×C4×Dic5C2×C411yescharacteristic
33C3×C4×Dic5D412yes—
34C3×C4×Dic5C2311yescharacteristic
35C2×C3×#124—51no—
36C3×#432—52no—
37C3×#329—54no—
38C22×C5×Q8C3×C411yescharacteristic
39C22×D20C3×C411yescharacteristic
40#4093—22no—
41#4093C3×C412yes—
42#4093C22×C312yes—
43C2×#685—22no—
44C2×C4×Dic5C22×C311yescharacteristic
45C22⋊#170—22no—
46#4004—22no—
47#4002C22×C314yescomplemented
48#1533—51noSylow
49C23×C3×C5C22⋊C411yescharacteristic
50C22×C3×D5—22no—
51C22×C3×D5—44no—
52C22×C3×D5—42no—
53C3×C5×Q8—42no—
54C3×C5×Q8—42no—
55C2×C3×C4×C5—22no—
56C2×C3×C4×C5—22no—
57C2×C3×C4×C5—42no—
58C2×C3×C4×C5C22⋊C411yescharacteristic
59C2×C3×C4×C5C22×C411yescharacteristic
60C2×C3×Dic5—42no—
61C2×C3×Dic5C22⋊C412yes—
62C2×C3×Dic5C2×D412yes—
63C3×D20—42no—
64C3×D20—42no—
65C3×#170—44no—
66C22×C3×Q8AGL(1,𝔽5)11yescharacteristic
67C22×C3×D4—51no—
68C3×#124—52no—
69C3×#124—52no—
70C3×#124—102no—
71C2×C3×C22⋊C4—102no—
72C2×C3×C42—51no—
73C3×C4⋊C8—102no—
74C3×C22⋊C8—102no—
75C3×#97—54no—
76C23×D5—21no—
77C2×C5×Q8—21no—
78C2×C5×Q8—42no—
79C2×C5×Q8C2×C3×C412yes—
80C22×C4×C5—21no—
81C22×C4×C5C2×C3×C411yescharacteristic
82C22×Dic5C3×D412yes—
83C2×D20—21no—
84C2×D20—42no—
85C2×D20C2×C3×C412yes—
86C2×#170—24no—
87#685—24no—
88#685—24no—
89C4×Dic5C2×C3×C411yescharacteristic
90C4×Dic5C3×D412yes—
91C4×Dic5C23×C311yescharacteristic
92C2×#124—51no—
93#432—52no—
94#329—54no—
95C22×C3×C5—22no—
96C22×C3×C5#10012yes—
97C22×C3×C5C2×C22⋊C411yescharacteristic
98C2×C3×D5—42no—
99C2×C3×D5—44no—
100C2×C3×D5—84no—
101C3×C4×C5—21no—
102C3×C4×C5—21no—
103C3×C4×C5—42no—
104C3×Dic5—22no—
105C3×Dic5—22no—
106C24×C3—101no—
107C2×C3×Q8—21no—
108C2×C3×Q8—42no—
109C2×C3×Q8C2×AGL(1,𝔽5)12yes—
110C2×C3×D4—52no—
111C2×C3×D4—101no—
112C2×C3×D4—202no—
113C22×C3×C4—21no—
114C22×C3×C4—52no—
115C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
116C2×C3×C8—104no—
117C3×C22⋊C4—104no—
118C3×C22⋊C4—104no—
119C3×C42—52no—
120C3×C42—51no—
121C3×C42—51no—
122C23×C5C3×C22⋊C411yescharacteristic
123C22×D5—22no—
124C22×D5—44no—
125C22×D5—42no—
126C5×Q8—42no—
127C5×Q8—42no—
128C2×C4×C5—22no—
129C2×C4×C5—22no—
130C2×C4×C5—42no—
131C2×C4×C5C3×C22⋊C411yescharacteristic
132C2×C4×C5C22×C3×C411yescharacteristic
133C2×Dic5—42no—
134C2×Dic5C3×C22⋊C412yes—
135C2×Dic5C2×C3×D412yes—
136D20—42no—
137D20—42no—
138#170—44no—
139C22×Q8C3×AGL(1,𝔽5)11yescharacteristic
140C22×D4—51no—
141#124—52no—
142#124—52no—
143#124—102no—
144C2×C22⋊C4—102no—
145C2×C42—51no—
146C4⋊C8—102no—
147C22⋊C8—102no—
148#97—54no—
149C2×C3×C5—22no—
150C2×C3×C5C2×#10011yescharacteristic
151C2×C3×C5#42112yes—
152C3×D5—82no—
153C23×C3—102no—
154C23×C3—204no—
155C23×C3—202no—
156C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
157C3×Q8—42no—
158C3×Q8—42no—
159C3×D4—202no—
160C3×D4—202no—
161C2×C3×C4—22no—
162C2×C3×C4—22no—
163C2×C3×C4—42no—
164C2×C3×C4—52no—
165C2×C3×C4—52no—
166C2×C3×C4—202no—
167C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
168C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
169C3×C8—204no—
170C22×C5—22no—
171C22×C5C3×#10012yes—
172C22×C5C2×C3×C22⋊C411yescharacteristic
173C2×D5—42no—
174C2×D5—44no—
175C2×D5—84no—
176C4×C5—21no—
177C4×C5—21no—
178C4×C5—42no—
179Dic5—22no—
180Dic5—22no—
181C24—101no—
182C2×Q8—21no—
183C2×Q8—42no—
184C2×Q8C2×C3×AGL(1,𝔽5)12yes—
185C2×D4—52no—
186C2×D4—101no—
187C2×D4—202no—
188C22×C4—21no—
189C22×C4—52no—
190C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
191C2×C8—104no—
192C22⋊C4—104no—
193C22⋊C4—104no—
194C42—52no—
195C42—51no—
196C42—51no—
197C3×C5#153311yescharacteristic, complemented, Hall
198C22×C3—22no—
199C22×C3—202no—
200C22×C3—204no—
201C22×C3—404no—
202C22×C3#401312yes—
203C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
204C3×C4—21no—
205C3×C4—21no—
206C3×C4—42no—
207C3×C4—102no—
208C3×C4—102no—
209C2×C5—22no—
210C2×C5C2×C3×#10011yescharacteristic
211C2×C5C3×#42112yes—
212D5—82no—
213C23—102no—
214C23—204no—
215C23—202no—
216C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
217Q8—42no—
218Q8—42no—
219D4—202no—
220D4—202no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—42no—
224C2×C4—52no—
225C2×C4—52no—
226C2×C4—202no—
227C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
228C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
229C8—204no—
230C2×C3—22no—
231C2×C3—402no—
232C2×C3#6578412yes—
233C2×C3C2×#401311yescharacteristic
234C5C3×#153311yescharacteristic, complemented, Sylow
235C22—22no—
236C22—204no—
237C22—202no—
238C22—404no—
239C22C3×#401312yes—
240C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
241C4—21no—
242C4—21no—
243C4—42no—
244C4—102no—
245C4—102no—
246C3#1060496111yescharacteristic, direct product factor, Sylow
247C2—22no—
248C2—402no—
249C2C3×#6578412yes—
250C2C2×C3×#401311yescharacteristic
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC22×C3×C5×Q8C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×C3×D20C4
cosocleC4×Dic5C23×C3
hypercenterC22×C3×Q8AGL(1,𝔽5)
hypocenterC5C3×#1533
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C22×C3×Q8

Complements

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

NH ≅ G/Ntype
2. C3×#65757C2semidirect
3. C3×#64927C2semidirect
4. #10604961C3direct (N × H)
14. C3×#4002C22semidirect
17. #65757C2×C3semidirect
18. #64927C2×C3semidirect
47. #4002C22×C3semidirect
197. C3×C5#1533semidirect
234. C5C3×#1533semidirect
246. C3#10604961direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#40932
C3×#657572
C3×#649272
#106049613
C3×#15335