Finite Group Databasebeta

← C3×#10604918 #14313505

Subgroups — C3×#10604918 #14313505

Order 1920.

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

normal subgroups only

This group has 540 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×#66139C211yesmaximal, characteristic
2C3×#65745C212yesmaximal, complemented
3C3×#64922C214yesmaximal, complemented
4#10604918C311yesmaximal, characteristic, direct product factor, Hall
5C22×C3×C5×D4C411yescharacteristic, complemented
6C2×C3×#709C2211yescharacteristic
7C3×#4130—22no—
8C3×#4099—22no—
9C3×#4086C2212yes—
10C3×#4084—22no—
11C3×#4082C412yescomplemented
12C3×#4072C411yescharacteristic, complemented
13C3×#4011—22no—
14C3×#3999C2214yescomplemented
15C3×#2128—51nomaximal, Hall
16#66139C2×C311yescharacteristic
17#65745C2×C312yescomplemented
18#64922C2×C314yescomplemented
19C24×C3×C5—21no—
20C2×C3×C5×D4—21no—
21C2×C3×C5×D4—42no—
22C2×C3×C5×D4C2×C412yes—
23C22×C3×C4×C5C2×C411yescharacteristic
24C2×C3×C22⋊D5—24no—
25C2×C3×C22⋊D5D412yes—
26C3×#709—42no—
27C2×C3×D20C2311yescharacteristic
28C2×C3×C4×D5D412yes—
29C2×C3×Dic10C2×C411yescharacteristic
30C3×C22⋊AGL(1,𝔽5)—24no—
31C3×C4⋊AGL(1,𝔽5)—42no—
32C3×C4×AGL(1,𝔽5)—42no—
33C3×C22⋊Dic5—21no—
34C3×C22⋊Dic5—24no—
35C3×C4⋊Dic5—21no—
36C3×C4×Dic5C2×C412yes—
37C3×#535—51no—
38C3×#410—52no—
39C3×#353—54no—
40C22×C5×D4C3×C411yescharacteristic, complemented
41C2×#709C22×C311yescharacteristic
42#4130—22no—
43#4099—22no—
44#4086C22×C312yes—
45#4084—22no—
46#4082C3×C412yescomplemented
47#4072C3×C411yescharacteristic, complemented
48#4011—22no—
49#3999C22×C314yescomplemented
50#2128—51noSylow
51C23×C3×C5—22no—
52C23×C3×C5—42no—
53C23×C3×C5—82no—
54C23×C3×C5C22⋊C411yescharacteristic
55C22×C3×D5C2×D412yes—
56C2×C3×AGL(1,𝔽5)—44no—
57C3×C5×D4—42no—
58C3×C5×D4—42no—
59C2×C3×C4×C5—22no—
60C2×C3×C4×C5C22⋊C411yescharacteristic
61C2×C3×C4×C5C22×C411yescharacteristic
62C3×C22⋊D5—42no—
63C3×C22⋊D5—84no—
64C2×C3×Dic5—22no—
65C2×C3×Dic5C22⋊C412yes—
66C3×D20—41no—
67C3×C4×D5—22no—
68C3×C4×D5—22no—
69C3×Dic10—41no—
70C2×C3×#41—51no—
71C22×C3×D4AGL(1,𝔽5)11yescharacteristic, complemented
72C3×#127—52no—
73C3×#124—52no—
74C3×#121—102no—
75C3×#120—102no—
76C3×#117—51no—
77C3×#117—102no—
78C3×#99—54no—
79C3×#99—102no—
80C24×C5—21no—
81C2×C5×D4—21no—
82C2×C5×D4—42no—
83C2×C5×D4C2×C3×C412yes—
84C22×C4×C5C2×C3×C411yescharacteristic
85C2×C22⋊D5—24no—
86C2×C22⋊D5C3×D412yes—
87#709—42no—
88C2×D20C23×C311yescharacteristic
89C2×C4×D5C3×D412yes—
90C2×Dic10C2×C3×C411yescharacteristic
91C22⋊AGL(1,𝔽5)—24no—
92C4⋊AGL(1,𝔽5)—42no—
93C4×AGL(1,𝔽5)—42no—
94C22⋊Dic5—21no—
95C22⋊Dic5—24no—
96C4⋊Dic5—21no—
97C4×Dic5C2×C3×C412yes—
98#535—51no—
99#410—52no—
100#353—54no—
101C22×C3×C5—21no—
102C22×C3×C5—41no—
103C22×C3×C5—44no—
104C22×C3×C5—42no—
105C22×C3×C5—84no—
106C22×C3×C5C2×C22⋊C411yescharacteristic
107C2×C3×D5—22no—
108C2×C3×D5—42no—
109C3×AGL(1,𝔽5)—84no—
110C3×C4×C5—21no—
111C3×C4×C5#9912yes—
112C3×Dic5—22no—
113C3×Dic5—42no—
114C24×C3—21no—
115C3×#41—202no—
116C2×C3×Q8—51no—
117C2×C3×D4—21no—
118C2×C3×D4—42no—
119C2×C3×D4—52no—
120C2×C3×D4—51no—
121C2×C3×D4—104no—
122C2×C3×D4C2×AGL(1,𝔽5)12yes—
123C22×C3×C4—52no—
124C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
125C3×C4⋊C4—101no—
126C3×C4⋊C4—202no—
127C3×C22⋊C4—104no—
128C3×C22⋊C4—101no—
129C3×C22⋊C4—104no—
130C3×C42—52no—
131C3×C42—202no—
132C23×C5—22no—
133C23×C5—42no—
134C23×C5—82no—
135C23×C5C3×C22⋊C411yescharacteristic
136C22×D5C2×C3×D412yes—
137C2×AGL(1,𝔽5)—44no—
138C5×D4—42no—
139C5×D4—42no—
140C2×C4×C5—22no—
141C2×C4×C5C3×C22⋊C411yescharacteristic
142C2×C4×C5C22×C3×C411yescharacteristic
143C22⋊D5—42no—
144C22⋊D5—84no—
145C2×Dic5—22no—
146C2×Dic5C3×C22⋊C412yes—
147D20—41no—
148C4×D5—22no—
149C4×D5—22no—
150Dic10—41no—
151C2×#41—51no—
152C22×D4C3×AGL(1,𝔽5)11yescharacteristic, complemented
153#127—52no—
154#124—52no—
155#121—102no—
156#120—102no—
157#117—51no—
158#117—102no—
159#99—54no—
160#99—102no—
161C2×C3×C5—21no—
162C2×C3×C5—41no—
163C2×C3×C5—82no—
164C2×C3×C5C2×#9911yescharacteristic
165C3×D5—42no—
166C23×C3—22no—
167C23×C3—42no—
168C23×C3—52no—
169C23×C3—82no—
170C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
171C3×Q8—201no—
172C3×D4—42no—
173C3×D4—42no—
174C3×D4—201no—
175C3×D4—202no—
176C3×D4—404no—
177C2×C3×C4—22no—
178C2×C3×C4—52no—
179C2×C3×C4—102no—
180C2×C3×C4—102no—
181C2×C3×C4—102no—
182C2×C3×C4—204no—
183C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
184C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
185C22×C5—21no—
186C22×C5—41no—
187C22×C5—44no—
188C22×C5—42no—
189C22×C5—84no—
190C22×C5C2×C3×C22⋊C411yescharacteristic
191C2×D5—22no—
192C2×D5—42no—
193AGL(1,𝔽5)—84no—
194C4×C5—21no—
195C4×C5C3×#9912yes—
196Dic5—22no—
197Dic5—42no—
198C24—21no—
199#41—202no—
200C2×Q8—51no—
201C2×D4—21no—
202C2×D4—42no—
203C2×D4—52no—
204C2×D4—51no—
205C2×D4—104no—
206C2×D4C2×C3×AGL(1,𝔽5)12yes—
207C22×C4—52no—
208C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
209C4⋊C4—101no—
210C4⋊C4—202no—
211C22⋊C4—104no—
212C22⋊C4—101no—
213C22⋊C4—104no—
214C42—52no—
215C42—202no—
216C3×C5#212811yescharacteristic, complemented, Hall
217C22×C3—21no—
218C22×C3—41no—
219C22×C3—44no—
220C22×C3—42no—
221C22×C3—84no—
222C22×C3—102no—
223C22×C3—202no—
224C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
225C3×C4—21no—
226C3×C4—102no—
227C3×C4—202no—
228C3×C4—404no—
229C3×C4#401112yes—
230C2×C5—21no—
231C2×C5—41no—
232C2×C5—82no—
233C2×C5C2×C3×#9911yescharacteristic
234D5—42no—
235C23—22no—
236C23—42no—
237C23—52no—
238C23—82no—
239C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
240Q8—201no—
241D4—42no—
242D4—42no—
243D4—201no—
244D4—202no—
245D4—404no—
246C2×C4—22no—
247C2×C4—52no—
248C2×C4—102no—
249C2×C4—102no—
250C2×C4—102no—
251C2×C4—204no—
252C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
253C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
254C2×C3—21no—
255C2×C3—41no—
256C2×C3—82no—
257C2×C3—202no—
258C2×C3C2×#401111yescharacteristic
259C5C3×#212811yescharacteristic, complemented, Sylow
260C22—21no—
261C22—41no—
262C22—44no—
263C22—42no—
264C22—84no—
265C22—102no—
266C22—202no—
267C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
268C4—21no—
269C4—102no—
270C4—202no—
271C4—404no—
272C4C3×#401112yes—
273C3#1060491811yescharacteristic, direct product factor, Sylow
274C2—21no—
275C2—41no—
276C2—82no—
277C2—202no—
278C2C2×C3×#401111yescharacteristic
subgroupG/H
centerC2×C3C2×#4011
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC22×C3×C5×D4C4
socleC2×C3×C5C2×#99
prime radicalC3×#66139C2
cosocleC2×D20C23×C3
hypercenterC22×C3×D4AGL(1,𝔽5)
hypocenterC5C3×#2128
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C22×C3×C4 ◁ C22×C3×D4

Complements

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

NH ≅ G/Ntype
2. C3×#65745C2semidirect
3. C3×#64922C2semidirect
4. #10604918C3direct (N × H)
5. C22×C3×C5×D4C4semidirect
11. C3×#4082C4semidirect
12. C3×#4072C4semidirect
14. C3×#3999C22semidirect
17. #65745C2×C3semidirect
18. #64922C2×C3semidirect
40. C22×C5×D4C3×C4semidirect
46. #4082C3×C4semidirect
47. #4072C3×C4semidirect
49. #3999C22×C3semidirect
71. C22×C3×D4AGL(1,𝔽5)semidirect
152. C22×D4C3×AGL(1,𝔽5)semidirect
216. C3×C5#2128semidirect
259. C5C3×#2128semidirect
273. C3#10604918direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#661392
C3×#657452
C3×#649222
#106049183
C3×#21285