Finite Group Databasebeta

← C3×#10604641 #14313540

Subgroups — C3×#10604641 #14313540

Order 1920.

476 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 104 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×#66058C211yesmaximal, characteristic
2C3×#65785C212yesmaximal
3C3×#64928C212yesmaximal, complemented
4C3×#64854C212yesmaximal, complemented
5#10604641C311yesmaximal, characteristic, direct product factor, Hall
6C2×C3×#713C2211yescharacteristic
7C2×C3×Q8×D5C411yescharacteristic, complemented
8C3×#4130—22no—
9C3×C5×C4⋊Q8C411yescharacteristic, complemented
10C3×#4093C2211yescharacteristic
11C3×#4090C411yescharacteristic, complemented
12C3×#4042—22no—
13C3×#4040—22no—
14C3×#4021C2211yescharacteristic
15C3×#4018C411yescharacteristic, complemented
16C3×#3999—22no—
17C3×#3999C2214yes—
18C3×#2129—51nomaximal, Hall
19#66058C2×C311yescharacteristic
20#65785C2×C312yes—
21#64928C2×C312yescomplemented
22#64854C2×C312yescomplemented
23C2×C3×C5×Q8C2×C411yescharacteristic
24C2×C3×C5×Q8C2×C411yescharacteristic
25C3×#713—42no—
26C3×Q8×D5—42no—
27C2×C3×D20D412yes—
28C2×C3×D20C2311yescharacteristic
29C2×C3×C4×D5—21no—
30C2×C3×C4×D5C2×C411yescharacteristic
31C2×C3×C4×D5D412yes—
32C2×C3×Dic10—21no—
33C2×C3×Dic10C2×C411yescharacteristic
34C3×C22⋊AGL(1,𝔽5)—24no—
35C3×C4⋊AGL(1,𝔽5)—42no—
36C3×C4×AGL(1,𝔽5)—42no—
37C3×C5×C4⋊C4—22no—
38C3×C42×C5C2×C411yescharacteristic
39C3×#685—21no—
40C3×#685—22no—
41C3×#685—22no—
42C3×#683—22no—
43C3×#683—21no—
44C3×C4×Dic5C2×C411yescharacteristic
45C3×#536—51no—
46C3×#410—52no—
47C3×#354—52no—
48C3×#354—52no—
49C2×#713C22×C311yescharacteristic
50C2×Q8×D5C3×C411yescharacteristic, complemented
51#4130—22no—
52C5×C4⋊Q8C3×C411yescharacteristic, complemented
53#4093C22×C311yescharacteristic
54#4090C3×C411yescharacteristic, complemented
55#4042—22no—
56#4040—22no—
57#4021C22×C311yescharacteristic
58#4018C3×C411yescharacteristic, complemented
59#3999—22no—
60#3999C22×C314yes—
61#2129—51noSylow
62C22×C3×D5C22⋊C411yescharacteristic
63C22×C3×D5C2×D412yes—
64C2×C3×AGL(1,𝔽5)—44no—
65C3×C5×Q8—41no—
66C3×C5×Q8—41no—
67C2×C3×C4×C5—21no—
68C2×C3×C4×C5—21no—
69C2×C3×C4×C5C22⋊C412yes—
70C2×C3×C4×C5C22×C411yescharacteristic
71C2×C3×Dic5—21no—
72C2×C3×Dic5—21no—
73C2×C3×Dic5C22⋊C411yescharacteristic
74C3×D20—42no—
75C3×D20—41no—
76C3×C4×D5—22no—
77C3×C4×D5—22no—
78C3×C4×D5—22no—
79C3×C4×D5—42no—
80C3×Dic10—42no—
81C3×Dic10—41no—
82C2×C3×#41—51no—
83C22×C3×Q8—51no—
84C3×C4⋊Q8—51no—
85C3×C4⋊Q8AGL(1,𝔽5)11yescharacteristic, complemented
86C3×#124—51no—
87C3×#124—51no—
88C3×#123—102no—
89C3×C22⋊Q8—102no—
90C3×#117—51no—
91C3×#117—102no—
92C3×#99—54no—
93C3×#99—102no—
94C2×C5×Q8C2×C3×C411yescharacteristic
95C2×C5×Q8C2×C3×C411yescharacteristic
96#713—42no—
97Q8×D5—42no—
98C2×D20C3×D412yes—
99C2×D20C23×C311yescharacteristic
100C2×C4×D5—21no—
101C2×C4×D5C2×C3×C411yescharacteristic
102C2×C4×D5C3×D412yes—
103C2×Dic10—21no—
104C2×Dic10C2×C3×C411yescharacteristic
105C22⋊AGL(1,𝔽5)—24no—
106C4⋊AGL(1,𝔽5)—42no—
107C4×AGL(1,𝔽5)—42no—
108C5×C4⋊C4—22no—
109C42×C5C2×C3×C411yescharacteristic
110#685—21no—
111#685—22no—
112#685—22no—
113#683—22no—
114#683—21no—
115C4×Dic5C2×C3×C411yescharacteristic
116#536—51no—
117#410—52no—
118#354—52no—
119#354—52no—
120C22×C3×C5C2×C22⋊C411yescharacteristic
121C2×C3×D5—21no—
122C2×C3×D5—22no—
123C2×C3×D5—41no—
124C2×C3×D5—42no—
125C3×AGL(1,𝔽5)—84no—
126C3×C4×C5—21no—
127C3×C4×C5—22no—
128C3×C4×C5—41no—
129C3×C4×C5—41no—
130C3×Dic5—41no—
131C3×Dic5—41no—
132C3×Dic5#9912yes—
133C3×#41—202no—
134C2×C3×Q8—51no—
135C2×C3×Q8—101no—
136C2×C3×Q8—202no—
137C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
138C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
139C2×C3×D4—52no—
140C2×C3×D4—51no—
141C22×C3×C4—52no—
142C22×C3×C4—51no—
143C22×C3×C4—101no—
144C3×C4⋊C4—22no—
145C3×C4⋊C4—101no—
146C3×C4⋊C4—102no—
147C3×C4⋊C4—202no—
148C3×C22⋊C4—101no—
149C3×C22⋊C4—102no—
150C3×C22⋊C4—102no—
151C3×C22⋊C4—104no—
152C3×C42—51no—
153C3×C42—202no—
154C3×C42C2×AGL(1,𝔽5)11yescharacteristic
155C22×D5C3×C22⋊C411yescharacteristic
156C22×D5C2×C3×D412yes—
157C2×AGL(1,𝔽5)—44no—
158C5×Q8—41no—
159C5×Q8—41no—
160C2×C4×C5—21no—
161C2×C4×C5—21no—
162C2×C4×C5C3×C22⋊C412yes—
163C2×C4×C5C22×C3×C411yescharacteristic
164C2×Dic5—21no—
165C2×Dic5—21no—
166C2×Dic5C3×C22⋊C411yescharacteristic
167D20—42no—
168D20—41no—
169C4×D5—22no—
170C4×D5—22no—
171C4×D5—22no—
172C4×D5—42no—
173Dic10—42no—
174Dic10—41no—
175C2×#41—51no—
176C22×Q8—51no—
177C4⋊Q8—51no—
178C4⋊Q8C3×AGL(1,𝔽5)11yescharacteristic, complemented
179#124—51no—
180#124—51no—
181#123—102no—
182C22⋊Q8—102no—
183#117—51no—
184#117—102no—
185#99—54no—
186#99—102no—
187C2×C3×C5—21no—
188C2×C3×C5C2×#9911yescharacteristic
189C3×D5—42no—
190C3×D5—41no—
191C23×C3—51no—
192C23×C3—52no—
193C3×Q8—41no—
194C3×Q8—41no—
195C3×Q8—201no—
196C3×Q8—202no—
197C3×D4—202no—
198C3×D4—201no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—51no—
202C2×C3×C4—102no—
203C2×C3×C4—102no—
204C2×C3×C4—102no—
205C2×C3×C4—101no—
206C2×C3×C4—101no—
207C2×C3×C4—202no—
208C2×C3×C4—204no—
209C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
210C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
211C22×C5C2×C3×C22⋊C411yescharacteristic
212C2×D5—21no—
213C2×D5—22no—
214C2×D5—41no—
215C2×D5—42no—
216AGL(1,𝔽5)—84no—
217C4×C5—21no—
218C4×C5—22no—
219C4×C5—41no—
220C4×C5—41no—
221Dic5—41no—
222Dic5—41no—
223Dic5C3×#9912yes—
224#41—202no—
225C2×Q8—51no—
226C2×Q8—101no—
227C2×Q8—202no—
228C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
229C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
230C2×D4—52no—
231C2×D4—51no—
232C22×C4—52no—
233C22×C4—51no—
234C22×C4—101no—
235C4⋊C4—22no—
236C4⋊C4—101no—
237C4⋊C4—102no—
238C4⋊C4—202no—
239C22⋊C4—101no—
240C22⋊C4—102no—
241C22⋊C4—102no—
242C22⋊C4—104no—
243C42—51no—
244C42—202no—
245C42C2×C3×AGL(1,𝔽5)11yescharacteristic
246C3×C5#212911yescharacteristic, complemented, Hall
247C22×C3—102no—
248C22×C3—101no—
249C22×C3—202no—
250C22×C3—201no—
251C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
252C3×C4—22no—
253C3×C4—21no—
254C3×C4—41no—
255C3×C4—41no—
256C3×C4—52no—
257C3×C4—201no—
258C3×C4—201no—
259C3×C4—404no—
260C2×C5—21no—
261C2×C5C2×C3×#9911yescharacteristic
262D5—42no—
263D5—41no—
264C23—52no—
265C23—51no—
266Q8—41no—
267Q8—41no—
268Q8—201no—
269Q8—202no—
270D4—202no—
271D4—201no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—51no—
275C2×C4—102no—
276C2×C4—102no—
277C2×C4—102no—
278C2×C4—101no—
279C2×C4—101no—
280C2×C4—202no—
281C2×C4—204no—
282C2×C4C3×C22⋊AGL(1,𝔽5)12yes—
283C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
284C2×C3—21no—
285C2×C3—202no—
286C2×C3—201no—
287C2×C3D5⋊#9911yescharacteristic
288C5C3×#212911yescharacteristic, complemented, Sylow
289C22—102no—
290C22—101no—
291C22—202no—
292C22—201no—
293C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
294C4—22no—
295C4—21no—
296C4—41no—
297C4—41no—
298C4—52no—
299C4—201no—
300C4—201no—
301C4—404no—
302C3#1060464111yescharacteristic, direct product factor, Sylow
303C2—21no—
304C2—202no—
305C2—201no—
306C2C3×D5⋊#9911yescharacteristic
subgroupG/H
centerC2×C3D5⋊#99
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C5×C4⋊Q8C4
socleC2×C3×C5C2×#99
prime radicalC2×C3×#713C22
cosocleC2×D20C23×C3
hypercenterC3×C4⋊Q8AGL(1,𝔽5)
hypocenterC5C3×#2129
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×Q8 ◁ C3×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. C3×#64928C2semidirect
4. C3×#64854C2semidirect
5. #10604641C3direct (N × H)
7. C2×C3×Q8×D5C4semidirect
9. C3×C5×C4⋊Q8C4semidirect
11. C3×#4090C4semidirect
15. C3×#4018C4semidirect
21. #64928C2×C3semidirect
22. #64854C2×C3semidirect
50. C2×Q8×D5C3×C4semidirect
52. C5×C4⋊Q8C3×C4semidirect
54. #4090C3×C4semidirect
58. #4018C3×C4semidirect
85. C3×C4⋊Q8AGL(1,𝔽5)semidirect
178. C4⋊Q8C3×AGL(1,𝔽5)semidirect
246. C3×C5#2129semidirect
288. C5C3×#2129semidirect
302. C3#10604641direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#660582
C3×#657852
C3×#649282
C3×#648542
#106046413
C3×#21295