Finite Group Databasebeta

← C3×#10604354 #14313507

Subgroups — C3×#10604354 #14313507

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×#65859C211yesmaximal, characteristic
2C3×#65745C212yesmaximal, complemented
3C3×#64853C214yesmaximal, complemented
4#10604354C311yesmaximal, characteristic, direct product factor, Hall
5C2×C3×#709C2211yescharacteristic
6C22×C3×D20C411yescharacteristic, complemented
7C3×#4130—22no—
8C3×C5×#117C411yescharacteristic, complemented
9C3×#4041—22no—
10C3×#4028—22no—
11C3×#4021C412yescomplemented
12C3×#4020C2212yes—
13C3×#4011—22no—
14C3×#3999C2214yescomplemented
15C3×#2128—51nomaximal, Hall
16#65859C2×C311yescharacteristic
17#65745C2×C312yescomplemented
18#64853C2×C314yescomplemented
19C23×C3×D5—21no—
20C22×C3×C4×C5C2×C411yescharacteristic
21C2×C3×C22⋊D5D412yes—
22C3×#709—42no—
23C2×C3×D20—21no—
24C2×C3×D20—24no—
25C2×C3×D20—42no—
26C2×C3×D20C2×C412yes—
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×C5×C4⋊C4—21no—
34C3×C5×C22⋊C4—21no—
35C3×C42×C5C2×C412yes—
36C3×#685—24no—
37C3×#535—51no—
38C3×#410—52no—
39C3×#353—54no—
40C2×#709C22×C311yescharacteristic
41C22×D20C3×C411yescharacteristic, complemented
42#4130—22no—
43C5×#117C3×C411yescharacteristic, complemented
44#4041—22no—
45#4028—22no—
46#4021C3×C412yescomplemented
47#4020C22×C312yes—
48#4011—22no—
49#3999C22×C314yescomplemented
50#2128—51noSylow
51C23×C3×C5C22⋊C411yescharacteristic
52C22×C3×D5—22no—
53C22×C3×D5—42no—
54C22×C3×D5—82no—
55C22×C3×D5C2×D412yes—
56C2×C3×AGL(1,𝔽5)—44no—
57C2×C3×C4×C5—22no—
58C2×C3×C4×C5—22no—
59C2×C3×C4×C5C22⋊C411yescharacteristic
60C2×C3×C4×C5C22×C411yescharacteristic
61C3×C22⋊D5—42no—
62C2×C3×Dic5C22⋊C412yes—
63C3×D20—41no—
64C3×D20—42no—
65C3×D20—42no—
66C3×D20—84no—
67C3×C4×D5—22no—
68C3×C4×D5—22no—
69C3×Dic10—41no—
70C2×C3×#41—51no—
71C22×C3×D4—51no—
72C3×#127—52no—
73C3×#124—52no—
74C3×#121—102no—
75C3×#120—102no—
76C3×#117—102no—
77C3×#117AGL(1,𝔽5)11yescharacteristic, complemented
78C3×#99—54no—
79C3×#99—102no—
80C23×D5—21no—
81C22×C4×C5C2×C3×C411yescharacteristic
82C2×C22⋊D5C3×D412yes—
83#709—42no—
84C2×D20—21no—
85C2×D20—24no—
86C2×D20—42no—
87C2×D20C2×C3×C412yes—
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—
94C5×C4⋊C4—21no—
95C5×C22⋊C4—21no—
96C42×C5C2×C3×C412yes—
97#685—24no—
98#535—51no—
99#410—52no—
100#353—54no—
101C22×C3×C5—21no—
102C22×C3×C5—41no—
103C22×C3×C5C2×C22⋊C411yescharacteristic
104C2×C3×D5—22no—
105C2×C3×D5—42no—
106C2×C3×D5—44no—
107C2×C3×D5—42no—
108C2×C3×D5—84no—
109C3×AGL(1,𝔽5)—84no—
110C3×C4×C5—21no—
111C3×C4×C5—42no—
112C3×C4×C5#9912yes—
113C3×Dic5—22no—
114C24×C3—101no—
115C3×#41—202no—
116C2×C3×Q8—51no—
117C2×C3×D4—52no—
118C2×C3×D4—51no—
119C2×C3×D4—52no—
120C2×C3×D4—104no—
121C2×C3×D4—101no—
122C2×C3×D4—202no—
123C22×C3×C4—52no—
124C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
125C3×C4⋊C4—21no—
126C3×C4⋊C4—202no—
127C3×C22⋊C4—21no—
128C3×C22⋊C4—104no—
129C3×C22⋊C4—104no—
130C3×C42—202no—
131C3×C42C2×AGL(1,𝔽5)12yes—
132C23×C5C3×C22⋊C411yescharacteristic
133C22×D5—22no—
134C22×D5—42no—
135C22×D5—82no—
136C22×D5C2×C3×D412yes—
137C2×AGL(1,𝔽5)—44no—
138C2×C4×C5—22no—
139C2×C4×C5—22no—
140C2×C4×C5C3×C22⋊C411yescharacteristic
141C2×C4×C5C22×C3×C411yescharacteristic
142C22⋊D5—42no—
143C2×Dic5C3×C22⋊C412yes—
144D20—41no—
145D20—42no—
146D20—42no—
147D20—84no—
148C4×D5—22no—
149C4×D5—22no—
150Dic10—41no—
151C2×#41—51no—
152C22×D4—51no—
153#127—52no—
154#124—52no—
155#121—102no—
156#120—102no—
157#117—102no—
158#117C3×AGL(1,𝔽5)11yescharacteristic, complemented
159#99—54no—
160#99—102no—
161C2×C3×C5—21no—
162C2×C3×C5—41no—
163C2×C3×C5C2×#9911yescharacteristic
164C3×D5—42no—
165C3×D5—82no—
166C23×C3—52no—
167C23×C3—102no—
168C23×C3—202no—
169C23×C3—402no—
170C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
171C3×Q8—201no—
172C3×D4—201no—
173C3×D4—202no—
174C3×D4—202no—
175C3×D4—202no—
176C3×D4—404no—
177C2×C3×C4—22no—
178C2×C3×C4—22no—
179C2×C3×C4—52no—
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×C5C2×C3×C22⋊C411yescharacteristic
188C2×D5—22no—
189C2×D5—42no—
190C2×D5—44no—
191C2×D5—42no—
192C2×D5—84no—
193AGL(1,𝔽5)—84no—
194C4×C5—21no—
195C4×C5—42no—
196C4×C5C3×#9912yes—
197Dic5—22no—
198C24—101no—
199#41—202no—
200C2×Q8—51no—
201C2×D4—52no—
202C2×D4—51no—
203C2×D4—52no—
204C2×D4—104no—
205C2×D4—101no—
206C2×D4—202no—
207C22×C4—52no—
208C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
209C4⋊C4—21no—
210C4⋊C4—202no—
211C22⋊C4—21no—
212C22⋊C4—104no—
213C22⋊C4—104no—
214C42—202no—
215C42C2×C3×AGL(1,𝔽5)12yes—
216C3×C5#212811yescharacteristic, complemented, Hall
217C22×C3—21no—
218C22×C3—41no—
219C22×C3—102no—
220C22×C3—202no—
221C22×C3—204no—
222C22×C3—202no—
223C22×C3—404no—
224C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
225C3×C4—21no—
226C3×C4—42no—
227C3×C4—102no—
228C3×C4—404no—
229C3×C4#399912yes—
230C2×C5—21no—
231C2×C5—41no—
232C2×C5C2×C3×#9911yescharacteristic
233D5—42no—
234D5—82no—
235C23—52no—
236C23—102no—
237C23—202no—
238C23—402no—
239C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
240Q8—201no—
241D4—201no—
242D4—202no—
243D4—202no—
244D4—202no—
245D4—404no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—52no—
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—202no—
257C2×C3—402no—
258C2×C3C2×#399911yescharacteristic
259C5C3×#212811yescharacteristic, complemented, Sylow
260C22—21no—
261C22—41no—
262C22—102no—
263C22—202no—
264C22—204no—
265C22—202no—
266C22—404no—
267C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
268C4—21no—
269C4—42no—
270C4—102no—
271C4—404no—
272C4C3×#399912yes—
273C3#1060435411yescharacteristic, direct product factor, Sylow
274C2—21no—
275C2—41no—
276C2—202no—
277C2—402no—
278C2C2×C3×#399911yescharacteristic
subgroupG/H
centerC2×C3C2×#3999
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C5×#117C4
socleC2×C3×C5C2×#99
prime radicalC3×#65859C2
cosocleC2×D20C23×C3
hypercenterC3×#117AGL(1,𝔽5)
hypocenterC5C3×#2128
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C22×C3×C4 ◁ C3×#117

Complements

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

NH ≅ G/Ntype
2. C3×#65745C2semidirect
3. C3×#64853C2semidirect
4. #10604354C3direct (N × H)
6. C22×C3×D20C4semidirect
8. C3×C5×#117C4semidirect
11. C3×#4021C4semidirect
14. C3×#3999C22semidirect
17. #65745C2×C3semidirect
18. #64853C2×C3semidirect
41. C22×D20C3×C4semidirect
43. C5×#117C3×C4semidirect
46. #4021C3×C4semidirect
49. #3999C22×C3semidirect
77. C3×#117AGL(1,𝔽5)semidirect
158. #117C3×AGL(1,𝔽5)semidirect
216. C3×C5#2128semidirect
259. C5C3×#2128semidirect
273. C3#10604354direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#658592
C3×#657452
C3×#648532
#106043543
C3×#21285