Finite Group Databasebeta

← C3×#10617208 #14319775

Subgroups — C3×#10617208 #14319775

Order 1920.

1256 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×Q8×AGL(1,𝔽5)C212yesmaximal
2C3×C4×Q8×D5C211yesmaximal, characteristic
3C3×C4⋊C4×AGL(1,𝔽5)C216yesmaximal
4C3×C4×C4⋊AGL(1,𝔽5)C213yesmaximal
5C3×#65686C213yesmaximal
6#10617208C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×Q8×D5C2211yescharacteristic
8C3×Q8×AGL(1,𝔽5)C4116yescomplemented
9C2×C3×C4⋊AGL(1,𝔽5)C2216yes—
10C2×C3×C4×AGL(1,𝔽5)C2216yes—
11C2×C3×C4×AGL(1,𝔽5)C2216yes—
12C3×C4×C5×Q8C411yescharacteristic, complemented
13C3×Q8×Dic5C411yescharacteristic, complemented
14C3×D5×C4⋊C4C2213yes—
15C3×#4033C413yescomplemented
16C3×C42×D5C2213yes—
17C3×C4×Dic10C413yescomplemented
18C3×(C2×AGL(1,𝔽5))⋊C4C2211yescharacteristic
19C3×(C2×AGL(1,𝔽5))⋊C4C2213yes—
20C3×(C2×AGL(1,𝔽5))⋊C4C2216yes—
21C3×#2276—51nomaximal, Hall
22C2×Q8×AGL(1,𝔽5)C2×C312yes—
23C4×Q8×D5C2×C311yescharacteristic
24C4⋊C4×AGL(1,𝔽5)C2×C316yes—
25C4×C4⋊AGL(1,𝔽5)C2×C313yes—
26#65686C2×C313yes—
27C22×C3×AGL(1,𝔽5)C2316yes—
28C22×C3×AGL(1,𝔽5)C2312yes—
29C2×C3×C5×Q8C2×C411yescharacteristic
30C3×Q8×D5C2×C418yes—
31C2×C3×C4×D5C2313yes—
32C2×C3×C4×D5C2311yescharacteristic
33C2×C3×C4×D5C2313yes—
34C2×C3×Dic10C2×C413yes—
35C3×C4⋊AGL(1,𝔽5)C2×C4124yes—
36C3×C4×AGL(1,𝔽5)—46no—
37C3×C4×AGL(1,𝔽5)C2×C4124yes—
38C3×C5×C4⋊C4C2×C413yes—
39C3×C42×C5C2×C413yes—
40C3×C4⋊Dic5C2×C413yes—
41C3×#683C2×C416yes—
42C3×C4×Dic5C2×C413yes—
43C3×C4×Dic5C2×C413yes—
44C3×C4×Dic5C2×C413yes—
45C2×C3×C4×Q8—52no—
46C2×C3×C4×Q8—51no—
47C3×C4×C4⋊C4—53no—
48C3×C4×C4⋊C4—56no—
49C3×#376—53no—
50C2×Q8×D5C22×C311yescharacteristic
51Q8×AGL(1,𝔽5)C3×C4116yescomplemented
52C2×C4⋊AGL(1,𝔽5)C22×C316yes—
53C2×C4×AGL(1,𝔽5)C22×C316yes—
54C2×C4×AGL(1,𝔽5)C22×C316yes—
55C4×C5×Q8C3×C411yescharacteristic, complemented
56Q8×Dic5C3×C411yescharacteristic, complemented
57D5×C4⋊C4C22×C313yes—
58#4033C3×C413yescomplemented
59C42×D5C22×C313yes—
60C4×Dic10C3×C413yescomplemented
61(C2×AGL(1,𝔽5))⋊C4C22×C311yescharacteristic
62(C2×AGL(1,𝔽5))⋊C4C22×C313yes—
63(C2×AGL(1,𝔽5))⋊C4C22×C316yes—
64#2276—51noSylow
65C22×C3×D5C2411yescharacteristic
66C2×C3×AGL(1,𝔽5)—26no—
67C2×C3×AGL(1,𝔽5)—22no—
68C2×C3×AGL(1,𝔽5)—22no—
69C2×C3×AGL(1,𝔽5)—26no—
70C2×C3×AGL(1,𝔽5)C22×C414yes—
71C2×C3×AGL(1,𝔽5)C22×C4112yes—
72C3×C5×Q8C4214yescomplemented
73C2×C3×C4×C5C22×C411yescharacteristic
74C2×C3×C4×C5C22×C413yes—
75C2×C3×C4×C5C22×C413yes—
76C2×C3×Dic5C22×C411yescharacteristic
77C2×C3×Dic5C22×C413yes—
78C2×C3×Dic5C22×C413yes—
79C3×C4×D5—23no—
80C3×C4×D5—22no—
81C3×C4×D5—23no—
82C3×C4×D5C22×C4112yes—
83C3×Dic10C42112yescomplemented
84C22×C3×Q8—51no—
85C3×C4×Q8—516no—
86C3×C4×Q8—51no—
87C3×C4×Q8—53no—
88C3×C4×Q8—53no—
89C3×C4×Q8AGL(1,𝔽5)11yescharacteristic, complemented
90C2×C3×C4⋊C4—56no—
91C2×C3×C4⋊C4—53no—
92C2×C3×C42—56no—
93C2×C3×C42—53no—
94C2×C3×C42—56no—
95C3×#95—51no—
96C3×#95—56no—
97C3×#95—53no—
98C22×AGL(1,𝔽5)C23×C316yes—
99C22×AGL(1,𝔽5)C23×C312yes—
100C2×C5×Q8C2×C3×C411yescharacteristic
101Q8×D5C2×C3×C418yes—
102C2×C4×D5C23×C313yes—
103C2×C4×D5C23×C311yescharacteristic
104C2×C4×D5C23×C313yes—
105C2×Dic10C2×C3×C413yes—
106C4⋊AGL(1,𝔽5)C2×C3×C4124yes—
107C4×AGL(1,𝔽5)—46no—
108C4×AGL(1,𝔽5)C2×C3×C4124yes—
109C5×C4⋊C4C2×C3×C413yes—
110C42×C5C2×C3×C413yes—
111C4⋊Dic5C2×C3×C413yes—
112#683C2×C3×C416yes—
113C4×Dic5C2×C3×C413yes—
114C4×Dic5C2×C3×C413yes—
115C4×Dic5C2×C3×C413yes—
116C2×C4×Q8—52no—
117C2×C4×Q8—51no—
118C4×C4⋊C4—53no—
119C4×C4⋊C4—56no—
120#376—53no—
121C22×C3×C5C23×C411yescharacteristic
122C2×C3×D5C23×C412yes—
123C2×C3×D5H(5,𝔽2)11yescharacteristic
124C2×C3×D52−1+411yescharacteristic
125C2×C3×D52−1+412yes—
126C3×AGL(1,𝔽5)—24no—
127C3×AGL(1,𝔽5)—212no—
128C3×C4×C5—21no—
129C3×C4×C5—23no—
130C3×C4×C5C2×C4216yes—
131C3×Dic5—21no—
132C3×Dic5—23no—
133C3×Dic5C2×C4216yes—
134C2×C3×Q8—58no—
135C2×C3×Q8—53no—
136C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
137C22×C3×C4—53no—
138C22×C3×C4—52no—
139C22×C3×C4—56no—
140C22×C3×C4—51no—
141C22×C3×C4—53no—
142C3×C4⋊C4—524no—
143C3×C4⋊C4—56no—
144C3×C4⋊C4—53no—
145C3×C4⋊C4C2×AGL(1,𝔽5)13yes—
146C3×C42—524no—
147C3×C42—53no—
148C3×C42—53no—
149C3×C42—53no—
150C3×C42—206no—
151C3×C42C2×AGL(1,𝔽5)13yes—
152C22×D5C24×C311yescharacteristic
153C2×AGL(1,𝔽5)—26no—
154C2×AGL(1,𝔽5)—22no—
155C2×AGL(1,𝔽5)—22no—
156C2×AGL(1,𝔽5)—26no—
157C2×AGL(1,𝔽5)C22×C3×C414yes—
158C2×AGL(1,𝔽5)C22×C3×C4112yes—
159C5×Q8C3×C4214yescomplemented
160C2×C4×C5C22×C3×C411yescharacteristic
161C2×C4×C5C22×C3×C413yes—
162C2×C4×C5C22×C3×C413yes—
163C2×Dic5C22×C3×C411yescharacteristic
164C2×Dic5C22×C3×C413yes—
165C2×Dic5C22×C3×C413yes—
166C4×D5—23no—
167C4×D5—22no—
168C4×D5—23no—
169C4×D5C22×C3×C4112yes—
170Dic10C3×C42112yescomplemented
171C22×Q8—51no—
172C4×Q8—516no—
173C4×Q8—51no—
174C4×Q8—53no—
175C4×Q8—53no—
176C4×Q8C3×AGL(1,𝔽5)11yescharacteristic, complemented
177C2×C4⋊C4—56no—
178C2×C4⋊C4—53no—
179C2×C42—56no—
180C2×C42—53no—
181C2×C42—56no—
182#95—51no—
183#95—56no—
184#95—53no—
185C2×C3×C5C22×C4211yescharacteristic
186C2×C3×C5#51911yescharacteristic
187C2×C3×C5#52011yescharacteristic
188C3×D5#51912yes—
189C3×D5#52012yes—
190C23×C3—51no—
191C3×Q8—512no—
192C3×Q8C4×AGL(1,𝔽5)14yescomplemented
193C2×C3×C4—53no—
194C2×C3×C4—512no—
195C2×C3×C4—54no—
196C2×C3×C4—512no—
197C2×C3×C4—51no—
198C2×C3×C4—53no—
199C2×C3×C4—103no—
200C2×C3×C4—102no—
201C2×C3×C4—103no—
202C2×C3×C4—106no—
203C2×C3×C4—102no—
204C2×C3×C4—106no—
205C2×C3×C4—102no—
206C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
207C2×C3×C4C22×AGL(1,𝔽5)13yes—
208C2×C3×C4C22×AGL(1,𝔽5)13yes—
209C22×C5C23×C3×C411yescharacteristic
210C2×D5C23×C3×C412yes—
211C2×D5C3×H(5,𝔽2)11yescharacteristic
212C2×D5C3×2−1+411yescharacteristic
213C2×D5C3×2−1+412yes—
214AGL(1,𝔽5)—24no—
215AGL(1,𝔽5)—212no—
216C4×C5—21no—
217C4×C5—23no—
218C4×C5C2×C3×C4216yes—
219Dic5—21no—
220Dic5—23no—
221Dic5C2×C3×C4216yes—
222C2×Q8—58no—
223C2×Q8—53no—
224C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
225C22×C4—53no—
226C22×C4—52no—
227C22×C4—56no—
228C22×C4—51no—
229C22×C4—53no—
230C4⋊C4—524no—
231C4⋊C4—56no—
232C4⋊C4—53no—
233C4⋊C4C2×C3×AGL(1,𝔽5)13yes—
234C42—524no—
235C42—53no—
236C42—53no—
237C42—53no—
238C42—206no—
239C42C2×C3×AGL(1,𝔽5)13yes—
240C3×C5#227611yescharacteristic, complemented, Hall
241C22×C3—52no—
242C22×C3—52no—
243C22×C3—51no—
244C22×C3—51no—
245C22×C3C23×AGL(1,𝔽5)11yescharacteristic
246C3×C4—21no—
247C3×C4—23no—
248C3×C4—56no—
249C3×C4—1012no—
250C3×C4—104no—
251C3×C4—101no—
252C3×C4—103no—
253C3×C4C2×C4×AGL(1,𝔽5)16yes—
254C2×C5C22×C3×C4211yescharacteristic
255C2×C5C3×#51911yescharacteristic
256C2×C5C3×#52011yescharacteristic
257D5C3×#51912yes—
258D5C3×#52012yes—
259C23—51no—
260Q8—512no—
261Q8C3×C4×AGL(1,𝔽5)14yescomplemented
262C2×C4—53no—
263C2×C4—512no—
264C2×C4—54no—
265C2×C4—512no—
266C2×C4—51no—
267C2×C4—53no—
268C2×C4—103no—
269C2×C4—102no—
270C2×C4—103no—
271C2×C4—106no—
272C2×C4—102no—
273C2×C4—106no—
274C2×C4—102no—
275C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
276C2×C4C22×C3×AGL(1,𝔽5)13yes—
277C2×C4C22×C3×AGL(1,𝔽5)13yes—
278C2×C3—52no—
279C2×C3—52no—
280C2×C3C22×C4×AGL(1,𝔽5)11yescharacteristic
281C2×C3#6626211yescharacteristic
282C2×C3#6626611yescharacteristic
283C5C3×#227611yescharacteristic, complemented, Sylow
284C22—52no—
285C22—52no—
286C22—51no—
287C22—51no—
288C22C23×C3×AGL(1,𝔽5)11yescharacteristic
289C4—21no—
290C4—23no—
291C4—56no—
292C4—1012no—
293C4—104no—
294C4—101no—
295C4—103no—
296C4C2×C3×C4×AGL(1,𝔽5)16yes—
297C3#1061720811yescharacteristic, direct product factor, Sylow
298C2—52no—
299C2—52no—
300C2C22×C3×C4×AGL(1,𝔽5)11yescharacteristic
301C2C3×#6626211yescharacteristic
302C2C3×#6626611yescharacteristic
subgroupG/H
centerC22×C3C23×AGL(1,𝔽5)
commutatorC2×C5C22×C3×C42
FrattiniC22C23×C3×AGL(1,𝔽5)
FittingC3×C4×C5×Q8C4
socleC22×C3×C5C23×C4
prime radicalC22×C3×D5C24
cosocleC22×D5C24×C3
hypercenterC3×C4×Q8AGL(1,𝔽5)
hypocenterC5C3×#2276
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×C4×Q8

Complements

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

NH ≅ G/Ntype
6. #10617208C3direct (N × H)
8. C3×Q8×AGL(1,𝔽5)C4semidirect
12. C3×C4×C5×Q8C4semidirect
13. C3×Q8×Dic5C4semidirect
15. C3×#4033C4semidirect
17. C3×C4×Dic10C4semidirect
51. Q8×AGL(1,𝔽5)C3×C4semidirect
55. C4×C5×Q8C3×C4semidirect
56. Q8×Dic5C3×C4semidirect
58. #4033C3×C4semidirect
60. C4×Dic10C3×C4semidirect
72. C3×C5×Q8C42semidirect
83. C3×Dic10C42semidirect
89. C3×C4×Q8AGL(1,𝔽5)semidirect
159. C5×Q8C3×C42semidirect
170. Dic10C3×C42semidirect
176. C4×Q8C3×AGL(1,𝔽5)semidirect
192. C3×Q8C4×AGL(1,𝔽5)semidirect
240. C3×C5#2276semidirect
261. Q8C3×C4×AGL(1,𝔽5)semidirect
283. C5C3×#2276semidirect
297. C3#10617208direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×Q8×AGL(1,𝔽5)2
C3×C4×Q8×D52
C3×C4⋊C4×AGL(1,𝔽5)2
C3×C4×C4⋊AGL(1,𝔽5)2
C3×#656862
#106172083
C3×#22765