Finite Group Databasebeta

← C3×#10617417 #14413509

Subgroups — C3×#10617417 #14413509

Order 1920.

952 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#66266C212yesmaximal
2C3×D5×#131C211yesmaximal, characteristic
3C3×#65727C216yesmaximal, complemented
4C3×#65721C213yesmaximal, complemented
5C2×C3×#3994C213yesmaximal, complemented
6#10617417C311yesmaximal, characteristic, direct product factor, Hall
7C3×D5×#41C2211yescharacteristic
8C3×Q8×AGL(1,𝔽5)—22no—
9C3×D4×AGL(1,𝔽5)—26no—
10C3×#4130C2216yes—
11C2×C3×C4⋊AGL(1,𝔽5)C2216yes—
12C3×C5×#131C411yescharacteristic, complemented
13C3×#4094C411yescharacteristic, complemented
14C3×#4053C413yescomplemented
15C3×D5×M16C2213yes—
16C3×#4047C413yescomplemented
17C2×C3×C8×D5C2213yes—
18C3×#3994C2213yes—
19C3×#3994C2216yes—
20C3×#3994C2211yescharacteristic
21C3×#3993C2216yes—
22C3×#2913—51nomaximal, Hall
23#66266C2×C312yes—
24D5×#131C2×C311yescharacteristic
25#65727C2×C316yescomplemented
26#65721C2×C313yescomplemented
27C2×#3994C2×C313yescomplemented
28C22×C3×AGL(1,𝔽5)—26no—
29C3×C5×#41C2×C411yescharacteristic
30C3×#713Q811yescharacteristic
31C3×Q8×D5D411yescharacteristic
32C3×#711Q813yes—
33C3×D4×D5D413yes—
34C3×#709C2×C413yes—
35C2×C3×C4×D5C2313yes—
36C3×C22⋊AGL(1,𝔽5)—26no—
37C3×C4⋊AGL(1,𝔽5)—26no—
38C3×C4⋊AGL(1,𝔽5)C2312yes—
39C3×C4⋊AGL(1,𝔽5)C2316yes—
40C3×C4×AGL(1,𝔽5)—26no—
41C3×C5×M16C2×C413yescomplemented
42C2×C3×C5×C8C2×C413yescomplemented
43C3×#681C2×C413yescomplemented
44C2×C3×#168C2×C413yescomplemented
45C3×#676C2×C416yescomplemented
46C3×C8×D5C2×C413yescomplemented
47C3×C8×D5C2×C413yescomplemented
48C3×C8×D5C2313yes—
49C3×C8×D5C2311yescharacteristic
50C2×C3×#131—51no—
51C3×#520—52no—
52C3×#428—56no—
53C3×#427—53no—
54C2×C3×#107—53no—
55D5×#41C22×C311yescharacteristic
56Q8×AGL(1,𝔽5)—22no—
57D4×AGL(1,𝔽5)—26no—
58#4130C22×C316yes—
59C2×C4⋊AGL(1,𝔽5)C22×C316yes—
60C5×#131C3×C411yescharacteristic, complemented
61#4094C3×C411yescharacteristic, complemented
62#4053C3×C413yescomplemented
63D5×M16C22×C313yes—
64#4047C3×C413yescomplemented
65C2×C8×D5C22×C313yes—
66#3994C22×C313yes—
67#3994C22×C316yes—
68#3994C22×C311yescharacteristic
69#3993C22×C316yes—
70#2913—51noSylow
71C22×C3×D5C2×D413yes—
72C2×C3×AGL(1,𝔽5)—22no—
73C2×C3×AGL(1,𝔽5)—26no—
74C2×C3×AGL(1,𝔽5)—86no—
75C3×C5×Q8C4⋊C411yescharacteristic
76C3×C5×D4C4⋊C413yes—
77C2×C3×C4×C5C22×C413yes—
78C3×C22⋊D5C4⋊C416yes—
79C2×C3×Dic5C2×Q813yes—
80C3×D20C4⋊C413yes—
81C3×C4×D5C22×C413yes—
82C3×C4×D5C2×D413yes—
83C3×C4×D5C2×Q813yes—
84C3×C4×D5C2411yescharacteristic
85C3×Dic10C4⋊C413yes—
86C3×C5×C8C22×C411yescharacteristic
87C3×C5×C8C22×C413yes—
88C3×#168C22×C411yescharacteristic
89C3×#168C22×C413yes—
90C2×C3×#41—51no—
91C3×#131—51no—
92C3×#131—53no—
93C3×#131—53no—
94C3×#131AGL(1,𝔽5)11yescharacteristic, complemented
95C2×C3×M16—53no—
96C22×C3×C8—53no—
97C3×C4×Q8—102no—
98C3×C4×D4—106no—
99C3×#117—56no—
100C2×C3×C4⋊C4—56no—
101C3×#107—53no—
102C3×#107—56no—
103C3×#107—51no—
104C3×#106—56no—
105C22×AGL(1,𝔽5)—26no—
106C5×#41C2×C3×C411yescharacteristic
107#713C3×Q811yescharacteristic
108Q8×D5C3×D411yescharacteristic
109#711C3×Q813yes—
110D4×D5C3×D413yes—
111#709C2×C3×C413yes—
112C2×C4×D5C23×C313yes—
113C22⋊AGL(1,𝔽5)—26no—
114C4⋊AGL(1,𝔽5)—26no—
115C4⋊AGL(1,𝔽5)C23×C316yes—
116C4⋊AGL(1,𝔽5)C23×C312yes—
117C4×AGL(1,𝔽5)—26no—
118C5×M16C2×C3×C413yescomplemented
119C2×C5×C8C2×C3×C413yescomplemented
120#681C2×C3×C413yescomplemented
121C2×#168C2×C3×C413yescomplemented
122#676C2×C3×C416yescomplemented
123C8×D5C2×C3×C413yescomplemented
124C8×D5C2×C3×C413yescomplemented
125C8×D5C23×C313yes—
126C8×D5C23×C311yescharacteristic
127C2×#131—51no—
128#520—52no—
129#428—56no—
130#427—53no—
131C2×#107—53no—
132C22×C3×C5C2×C4⋊C413yes—
133C2×C3×D5—23no—
134C2×C3×D5—23no—
135C2×C3×D5C2×C4⋊C413yes—
136C2×C3×D5C22×D411yescharacteristic
137C3×AGL(1,𝔽5)—46no—
138C3×AGL(1,𝔽5)—42no—
139C3×C4×C5C2×C4⋊C413yes—
140C3×C4×C5C23×C411yescharacteristic
141C3×Dic5C2×C4⋊C413yes—
142C3×Dic5C22×Q811yescharacteristic
143C3×#41—51no—
144C3×#41—53no—
145C3×#41—53no—
146C3×#41C2×AGL(1,𝔽5)11yescharacteristic
147C2×C3×Q8—51no—
148C2×C3×D4—53no—
149C22×C3×C4—53no—
150C22×C3×C4—106no—
151C3×M16—53no—
152C3×M16—56no—
153C3×M16C2×AGL(1,𝔽5)13yescomplemented
154C2×C3×C8—53no—
155C2×C3×C8—53no—
156C2×C3×C8—53no—
157C2×C3×C8—53no—
158C2×C3×C8—51no—
159C2×C3×C8C2×AGL(1,𝔽5)13yescomplemented
160C3×C4⋊C4—52no—
161C3×C4⋊C4—56no—
162C3×C4⋊C4—106no—
163C3×C22⋊C4—106no—
164C3×C42—106no—
165C22×D5C2×C3×D413yes—
166C2×AGL(1,𝔽5)—22no—
167C2×AGL(1,𝔽5)—26no—
168C2×AGL(1,𝔽5)—86no—
169C5×Q8C3×C4⋊C411yescharacteristic
170C5×D4C3×C4⋊C413yes—
171C2×C4×C5C22×C3×C413yes—
172C22⋊D5C3×C4⋊C416yes—
173C2×Dic5C2×C3×Q813yes—
174D20C3×C4⋊C413yes—
175C4×D5C22×C3×C413yes—
176C4×D5C2×C3×D413yes—
177C4×D5C2×C3×Q813yes—
178C4×D5C24×C311yescharacteristic
179Dic10C3×C4⋊C413yes—
180C5×C8C22×C3×C411yescharacteristic
181C5×C8C22×C3×C413yes—
182#168C22×C3×C411yescharacteristic
183#168C22×C3×C413yes—
184C2×#41—51no—
185#131—51no—
186#131—53no—
187#131—53no—
188#131C3×AGL(1,𝔽5)11yescharacteristic, complemented
189C2×M16—53no—
190C22×C8—53no—
191C4×Q8—102no—
192C4×D4—106no—
193#117—56no—
194C2×C4⋊C4—56no—
195#107—53no—
196#107—56no—
197#107—51no—
198#106—56no—
199C2×C3×C5—23no—
200C2×C3×C5C22×C4⋊C411yescharacteristic
201C3×D5—23no—
202C3×D5#57611yescharacteristic
203C3×D5#57811yescharacteristic
204C23×C3—53no—
205C3×Q8—53no—
206C3×Q8C4⋊AGL(1,𝔽5)11yescharacteristic
207C3×D4—56no—
208C3×D4—53no—
209C3×D4C4⋊AGL(1,𝔽5)13yes—
210C2×C3×C4—53no—
211C2×C3×C4—53no—
212C2×C3×C4—53no—
213C2×C3×C4—53no—
214C2×C3×C4—51no—
215C2×C3×C4—102no—
216C2×C3×C4—106no—
217C2×C3×C4—406no—
218C2×C3×C4C22×AGL(1,𝔽5)13yes—
219C3×C8—51no—
220C3×C8—53no—
221C3×C8C22×AGL(1,𝔽5)11yescharacteristic
222C3×C8C22×AGL(1,𝔽5)13yes—
223C22×C5C2×C3×C4⋊C413yes—
224C2×D5—23no—
225C2×D5—23no—
226C2×D5C2×C3×C4⋊C413yes—
227C2×D5C22×C3×D411yescharacteristic
228AGL(1,𝔽5)—46no—
229AGL(1,𝔽5)—42no—
230C4×C5C2×C3×C4⋊C413yes—
231C4×C5C23×C3×C411yescharacteristic
232Dic5C2×C3×C4⋊C413yes—
233Dic5C22×C3×Q811yescharacteristic
234#41—51no—
235#41—53no—
236#41—53no—
237#41C2×C3×AGL(1,𝔽5)11yescharacteristic
238C2×Q8—51no—
239C2×D4—53no—
240C22×C4—53no—
241C22×C4—106no—
242M16—53no—
243M16—56no—
244M16C2×C3×AGL(1,𝔽5)13yescomplemented
245C2×C8—53no—
246C2×C8—53no—
247C2×C8—51no—
248C2×C8—53no—
249C2×C8—53no—
250C2×C8C2×C3×AGL(1,𝔽5)13yescomplemented
251C4⋊C4—52no—
252C4⋊C4—56no—
253C4⋊C4—106no—
254C22⋊C4—106no—
255C42—106no—
256C3×C5#291311yescharacteristic, complemented, Hall
257C22×C3—51no—
258C22×C3—53no—
259C22×C3—103no—
260C22×C3—103no—
261C22×C3C2×C4⋊AGL(1,𝔽5)13yes—
262C3×C4—53no—
263C3×C4—51no—
264C3×C4—202no—
265C3×C4—206no—
266C3×C4C2×C4⋊AGL(1,𝔽5)13yes—
267C3×C4C23×AGL(1,𝔽5)11yescharacteristic
268C2×C5—23no—
269C2×C5C22×C3×C4⋊C411yescharacteristic
270D5—23no—
271D5C3×#57611yescharacteristic
272D5C3×#57811yescharacteristic
273C23—53no—
274Q8—53no—
275Q8C3×C4⋊AGL(1,𝔽5)11yescharacteristic
276D4—56no—
277D4—53no—
278D4C3×C4⋊AGL(1,𝔽5)13yes—
279C2×C4—53no—
280C2×C4—53no—
281C2×C4—53no—
282C2×C4—51no—
283C2×C4—53no—
284C2×C4—102no—
285C2×C4—106no—
286C2×C4—406no—
287C2×C4C22×C3×AGL(1,𝔽5)13yes—
288C8—51no—
289C8—53no—
290C8C22×C3×AGL(1,𝔽5)11yescharacteristic
291C8C22×C3×AGL(1,𝔽5)13yes—
292C2×C3—23no—
293C2×C3—51no—
294C2×C3—51no—
295C2×C3—103no—
296C2×C3C22×C4⋊AGL(1,𝔽5)11yescharacteristic
297C5C3×#291311yescharacteristic, complemented, Sylow
298C22—51no—
299C22—53no—
300C22—103no—
301C22—103no—
302C22C2×C3×C4⋊AGL(1,𝔽5)13yes—
303C4—51no—
304C4—53no—
305C4—206no—
306C4—202no—
307C4C2×C3×C4⋊AGL(1,𝔽5)13yes—
308C4C23×C3×AGL(1,𝔽5)11yescharacteristic
309C3#1061741711yescharacteristic, direct product factor, Sylow
310C2—23no—
311C2—51no—
312C2—51no—
313C2—103no—
314C2C22×C3×C4⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC2×C3C22×C4⋊AGL(1,𝔽5)
commutatorC4×C5C23×C3×C4
FrattiniC4C23×C3×AGL(1,𝔽5)
FittingC3×C5×#131C4
socleC2×C3×C5C22×C4⋊C4
prime radicalC3×D5×#41C22
cosocleC4×D5C24×C3
hypercenterC3×#131AGL(1,𝔽5)
hypocenterC5C3×#2913
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C3×#41 ◁ C3×#131

Complements

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

NH ≅ G/Ntype
3. C3×#65727C2semidirect
4. C3×#65721C2semidirect
5. C2×C3×#3994C2semidirect
6. #10617417C3direct (N × H)
12. C3×C5×#131C4semidirect
13. C3×#4094C4semidirect
14. C3×#4053C4semidirect
16. C3×#4047C4semidirect
25. #65727C2×C3semidirect
26. #65721C2×C3semidirect
27. C2×#3994C2×C3semidirect
41. C3×C5×M16C2×C4semidirect
42. C2×C3×C5×C8C2×C4semidirect
43. C3×#681C2×C4semidirect
44. C2×C3×#168C2×C4semidirect
45. C3×#676C2×C4semidirect
46. C3×C8×D5C2×C4semidirect
47. C3×C8×D5C2×C4semidirect
60. C5×#131C3×C4semidirect
61. #4094C3×C4semidirect
62. #4053C3×C4semidirect
64. #4047C3×C4semidirect
94. C3×#131AGL(1,𝔽5)semidirect
118. C5×M16C2×C3×C4semidirect
119. C2×C5×C8C2×C3×C4semidirect
120. #681C2×C3×C4semidirect
121. C2×#168C2×C3×C4semidirect
122. #676C2×C3×C4semidirect
123. C8×D5C2×C3×C4semidirect
124. C8×D5C2×C3×C4semidirect
153. C3×M16C2×AGL(1,𝔽5)semidirect
159. C2×C3×C8C2×AGL(1,𝔽5)semidirect
188. #131C3×AGL(1,𝔽5)semidirect
244. M16C2×C3×AGL(1,𝔽5)semidirect
250. C2×C8C2×C3×AGL(1,𝔽5)semidirect
256. C3×C5#2913semidirect
297. C5C3×#2913semidirect
309. C3#10617417direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#662662
C3×D5×#1312
C3×#657272
C3×#657212
C2×C3×#39942
#106174173
C3×#29135