Finite Group Databasebeta

← C3×AGL(1,𝔽5)×#131 #14411459

Subgroups — C3×AGL(1,𝔽5)×#131 #14411459

Order 1920.

1144 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#41×AGL(1,𝔽5)C211yesmaximal, characteristic
2C3×D5⋊#131C211yesmaximal, characteristic
3C3×D5×#131C211yesmaximal, characteristic
4C3×#65728C213yesmaximal, complemented
5C3×M16×AGL(1,𝔽5)C213yesmaximal, complemented
6C3×#65718C213yesmaximal, complemented
7C2×C3×C8×AGL(1,𝔽5)C213yesmaximal, complemented
8AGL(1,𝔽5)×#131C311yesmaximal, characteristic, direct product factor, Hall
9C3×D5×#41C2211yescharacteristic
10C3×Q8×AGL(1,𝔽5)C412yes—
11C3×#4133C412yescomplemented
12C3×D4×AGL(1,𝔽5)C416yes—
13C3×#4131C416yescomplemented
14C3×#4130C2213yes—
15C2×C3×C4×AGL(1,𝔽5)C2213yes—
16C3×D5⋊M16C2213yes—
17C2×C3×D5⋊C8C2213yes—
18C3×C5×#131C411yescharacteristic, complemented
19C3×#4094C411yescharacteristic, complemented
20C3×#4053C413yescomplemented
21C3×D5×M16C2213yes—
22C3×#4047C413yescomplemented
23C2×C3×C8×D5C2213yes—
24C3×#3992C2216yes—
25C3×C8×AGL(1,𝔽5)C2211yescharacteristic
26C3×C8×AGL(1,𝔽5)C2213yes—
27C3×C8×AGL(1,𝔽5)C2213yes—
28C3×C8×AGL(1,𝔽5)C2213yes—
29C3×C4×#131—51nomaximal, Hall
30#41×AGL(1,𝔽5)C2×C311yescharacteristic
31D5⋊#131C2×C311yescharacteristic
32D5×#131C2×C311yescharacteristic
33#65728C2×C313yescomplemented
34M16×AGL(1,𝔽5)C2×C313yescomplemented
35#65718C2×C313yescomplemented
36C2×C8×AGL(1,𝔽5)C2×C313yescomplemented
37C22×C3×AGL(1,𝔽5)C2×C416yes—
38C3×C5×#41C2×C411yescharacteristic
39C3×#713C2×C411yescharacteristic
40C3×Q8×D5C2×C411yescharacteristic
41C3×#711C2×C413yes—
42C3×D4×D5C2×C413yes—
43C3×#709C2×C413yes—
44C2×C3×C4×D5C2313yes—
45C3×C22⋊AGL(1,𝔽5)C2×C416yes—
46C3×#704C2×C416yescomplemented
47C2×C3×#170C2×C416yescomplemented
48C3×C4⋊AGL(1,𝔽5)C2×C416yes—
49C3×C4×AGL(1,𝔽5)C2×C416yes—
50C3×C4×AGL(1,𝔽5)C2311yescharacteristic
51C3×C4×AGL(1,𝔽5)C2313yes—
52C3×#700C2×C416yescomplemented
53C3×D5⋊C8C2×C416yescomplemented
54C3×D5⋊C8C2311yescharacteristic
55C3×D5⋊C8C2313yes—
56C3×C5×M16C2×C413yescomplemented
57C2×C3×C5×C8C2×C413yescomplemented
58C3×#681C2×C413yescomplemented
59C2×C3×#168C2×C413yescomplemented
60C3×#676C2×C416yescomplemented
61C3×C8×D5C2×C413yescomplemented
62C3×C8×D5C2×C413yescomplemented
63C3×C8×D5C2311yescharacteristic
64C3×C8×D5C2313yes—
65C2×C3×#131—51no—
66C2×C3×#131—51no—
67C3×C4×#41—51no—
68C3×#405—53no—
69C3×#405—53no—
70C3×C4×M16—53no—
71C2×C3×C4×C8—53no—
72D5×#41C22×C311yescharacteristic
73Q8×AGL(1,𝔽5)C3×C412yes—
74#4133C3×C412yescomplemented
75D4×AGL(1,𝔽5)C3×C416yes—
76#4131C3×C416yescomplemented
77#4130C22×C313yes—
78C2×C4×AGL(1,𝔽5)C22×C313yes—
79D5⋊M16C22×C313yes—
80C2×D5⋊C8C22×C313yes—
81C5×#131C3×C411yescharacteristic, complemented
82#4094C3×C411yescharacteristic, complemented
83#4053C3×C413yescomplemented
84D5×M16C22×C313yes—
85#4047C3×C413yescomplemented
86C2×C8×D5C22×C313yes—
87#3992C22×C316yes—
88C8×AGL(1,𝔽5)C22×C311yescharacteristic
89C8×AGL(1,𝔽5)C22×C313yes—
90C8×AGL(1,𝔽5)C22×C313yes—
91C8×AGL(1,𝔽5)C22×C313yes—
92C4×#131—51noSylow
93C22×C3×D5C22×C413yes—
94C2×C3×AGL(1,𝔽5)—212no—
95C2×C3×AGL(1,𝔽5)C22×C416yes—
96C2×C3×AGL(1,𝔽5)C22×C412yes—
97C3×C5×Q8C4211yescharacteristic
98C3×C5×D4C4213yes—
99C2×C3×C4×C5C22×C413yes—
100C3×C22⋊D5C4216yes—
101C2×C3×Dic5C22×C413yes—
102C3×D20C4213yes—
103C3×C4×D5C22×C413yes—
104C3×C4×D5C22×C413yes—
105C3×C4×D5C22×C413yes—
106C3×C4×D5C2411yescharacteristic
107C3×Dic10C4213yes—
108C3×#170C22×C412yes—
109C3×#170C22×C416yes—
110C3×C5×C8C22×C413yes—
111C3×C5×C8C22×C411yescharacteristic
112C3×#168C22×C411yescharacteristic
113C3×#168C22×C413yes—
114C2×C3×#41—51no—
115C3×#131—56no—
116C3×#131—52no—
117C3×#131—51no—
118C3×#131—53no—
119C3×#131—53no—
120C3×#131AGL(1,𝔽5)11yescharacteristic, direct product factor
121C2×C3×M16—53no—
122C2×C3×M16—53no—
123C22×C3×C8—53no—
124C22×C3×C8—53no—
125C3×C4×Q8—52no—
126C3×C4×D4—56no—
127C3×#117—53no—
128C2×C3×C42—53no—
129C3×#97—56no—
130C3×C4×C8—51no—
131C3×C4×C8—53no—
132C3×C4×C8—53no—
133C3×C4×C8—53no—
134C22×AGL(1,𝔽5)C2×C3×C416yes—
135C5×#41C2×C3×C411yescharacteristic
136#713C2×C3×C411yescharacteristic
137Q8×D5C2×C3×C411yescharacteristic
138#711C2×C3×C413yes—
139D4×D5C2×C3×C413yes—
140#709C2×C3×C413yes—
141C2×C4×D5C23×C313yes—
142C22⋊AGL(1,𝔽5)C2×C3×C416yes—
143#704C2×C3×C416yescomplemented
144C2×#170C2×C3×C416yescomplemented
145C4⋊AGL(1,𝔽5)C2×C3×C416yes—
146C4×AGL(1,𝔽5)C2×C3×C416yes—
147C4×AGL(1,𝔽5)C23×C311yescharacteristic
148C4×AGL(1,𝔽5)C23×C313yes—
149#700C2×C3×C416yescomplemented
150D5⋊C8C2×C3×C416yescomplemented
151D5⋊C8C23×C311yescharacteristic
152D5⋊C8C23×C313yes—
153C5×M16C2×C3×C413yescomplemented
154C2×C5×C8C2×C3×C413yescomplemented
155#681C2×C3×C413yescomplemented
156C2×#168C2×C3×C413yescomplemented
157#676C2×C3×C416yescomplemented
158C8×D5C2×C3×C413yescomplemented
159C8×D5C2×C3×C413yescomplemented
160C8×D5C23×C311yescharacteristic
161C8×D5C23×C313yes—
162C2×#131—51no—
163C2×#131—51no—
164C4×#41—51no—
165#405—53no—
166#405—53no—
167C4×M16—53no—
168C2×C4×C8—53no—
169C22×C3×C5C2×C4213yes—
170C2×C3×D5—26no—
171C2×C3×D5C2×C4213yes—
172C2×C3×D5C23×C411yescharacteristic
173C3×AGL(1,𝔽5)—26no—
174C3×AGL(1,𝔽5)#13114yesdirect product factor
175C3×C4×C5C2×C4213yes—
176C3×C4×C5C23×C411yescharacteristic
177C3×Dic5C2×C4213yes—
178C3×Dic5C23×C411yescharacteristic
179C3×#41—51no—
180C3×#41—53no—
181C3×#41—53no—
182C3×#41C2×AGL(1,𝔽5)11yescharacteristic
183C2×C3×Q8—51no—
184C2×C3×D4—53no—
185C22×C3×C4—53no—
186C22×C3×C4—56no—
187C3×M16—56no—
188C3×M16—56no—
189C3×M16—53no—
190C3×M16—56no—
191C3×M16C2×AGL(1,𝔽5)13yescomplemented
192C2×C3×C8—51no—
193C2×C3×C8—53no—
194C2×C3×C8—56no—
195C2×C3×C8—56no—
196C2×C3×C8—53no—
197C2×C3×C8—53no—
198C2×C3×C8—53no—
199C2×C3×C8—53no—
200C2×C3×C8—51no—
201C2×C3×C8C2×AGL(1,𝔽5)13yescomplemented
202C3×C4⋊C4—56no—
203C3×C22⋊C4—56no—
204C3×C42—51no—
205C3×C42—53no—
206C3×C42—56no—
207C22×D5C22×C3×C413yes—
208C2×AGL(1,𝔽5)—212no—
209C2×AGL(1,𝔽5)C22×C3×C416yes—
210C2×AGL(1,𝔽5)C22×C3×C412yes—
211C5×Q8C3×C4211yescharacteristic
212C5×D4C3×C4213yes—
213C2×C4×C5C22×C3×C413yes—
214C22⋊D5C3×C4216yes—
215C2×Dic5C22×C3×C413yes—
216D20C3×C4213yes—
217C4×D5C22×C3×C413yes—
218C4×D5C22×C3×C413yes—
219C4×D5C22×C3×C413yes—
220C4×D5C24×C311yescharacteristic
221Dic10C3×C4213yes—
222#170C22×C3×C412yes—
223#170C22×C3×C416yes—
224C5×C8C22×C3×C411yescharacteristic
225C5×C8C22×C3×C413yes—
226#168C22×C3×C411yescharacteristic
227#168C22×C3×C413yes—
228C2×#41—51no—
229#131—56no—
230#131—52no—
231#131—51no—
232#131—53no—
233#131—53no—
234#131C3×AGL(1,𝔽5)11yescharacteristic, direct product factor
235C2×M16—53no—
236C2×M16—53no—
237C22×C8—53no—
238C22×C8—53no—
239C4×Q8—52no—
240C4×D4—56no—
241#117—53no—
242C2×C42—53no—
243#97—56no—
244C4×C8—51no—
245C4×C8—53no—
246C4×C8—53no—
247C4×C8—53no—
248C2×C3×C5—23no—
249C2×C3×C5C22×C4211yescharacteristic
250C3×D5—23no—
251C3×D5C2×#13112yes—
252C23×C3—53no—
253C3×Q8—53no—
254C3×Q8C4×AGL(1,𝔽5)11yescharacteristic
255C3×D4—56no—
256C3×D4—53no—
257C3×D4C4×AGL(1,𝔽5)13yes—
258C2×C3×C4—52no—
259C2×C3×C4—53no—
260C2×C3×C4—53no—
261C2×C3×C4—53no—
262C2×C3×C4—53no—
263C2×C3×C4—51no—
264C2×C3×C4—56no—
265C2×C3×C4—1012no—
266C2×C3×C4C22×AGL(1,𝔽5)13yes—
267C3×C8—52no—
268C3×C8—56no—
269C3×C8—51no—
270C3×C8—53no—
271C3×C8C22×AGL(1,𝔽5)13yes—
272C3×C8C22×AGL(1,𝔽5)11yescharacteristic
273C22×C5C2×C3×C4213yes—
274C2×D5—26no—
275C2×D5C2×C3×C4213yes—
276C2×D5C23×C3×C411yescharacteristic
277AGL(1,𝔽5)—26no—
278AGL(1,𝔽5)C3×#13114yesdirect product factor
279C4×C5C2×C3×C4213yes—
280C4×C5C23×C3×C411yescharacteristic
281Dic5C2×C3×C4213yes—
282Dic5C23×C3×C411yescharacteristic
283#41—51no—
284#41—53no—
285#41—53no—
286#41C2×C3×AGL(1,𝔽5)11yescharacteristic
287C2×Q8—51no—
288C2×D4—53no—
289C22×C4—53no—
290C22×C4—56no—
291M16—56no—
292M16—56no—
293M16—53no—
294M16—56no—
295M16C2×C3×AGL(1,𝔽5)13yescomplemented
296C2×C8—51no—
297C2×C8—53no—
298C2×C8—56no—
299C2×C8—56no—
300C2×C8—53no—
301C2×C8—51no—
302C2×C8—53no—
303C2×C8—53no—
304C2×C8—53no—
305C2×C8C2×C3×AGL(1,𝔽5)13yescomplemented
306C4⋊C4—56no—
307C22⋊C4—56no—
308C42—51no—
309C42—53no—
310C42—56no—
311C3×C5C4×#13111yescharacteristic, complemented, Hall
312C22×C3—51no—
313C22×C3—53no—
314C22×C3—106no—
315C22×C3C2×C4×AGL(1,𝔽5)13yes—
316C3×C4—54no—
317C3×C4—53no—
318C3×C4—51no—
319C3×C4—106no—
320C3×C4C2×C4×AGL(1,𝔽5)13yes—
321C3×C4C23×AGL(1,𝔽5)11yescharacteristic
322C2×C5—23no—
323C2×C5C22×C3×C4211yescharacteristic
324D5—23no—
325D5C2×C3×#13112yes—
326C23—53no—
327Q8—53no—
328Q8C3×C4×AGL(1,𝔽5)11yescharacteristic
329D4—56no—
330D4—53no—
331D4C3×C4×AGL(1,𝔽5)13yes—
332C2×C4—52no—
333C2×C4—53no—
334C2×C4—53no—
335C2×C4—53no—
336C2×C4—51no—
337C2×C4—53no—
338C2×C4—56no—
339C2×C4—1012no—
340C2×C4C22×C3×AGL(1,𝔽5)13yes—
341C8—52no—
342C8—56no—
343C8—51no—
344C8—53no—
345C8C22×C3×AGL(1,𝔽5)11yescharacteristic
346C8C22×C3×AGL(1,𝔽5)13yes—
347C2×C3—23no—
348C2×C3—52no—
349C2×C3—103no—
350C2×C3C22×C4×AGL(1,𝔽5)11yescharacteristic
351C5C3×C4×#13111yescharacteristic, complemented, Sylow
352C22—51no—
353C22—53no—
354C22—106no—
355C22C2×C3×C4×AGL(1,𝔽5)13yes—
356C4—54no—
357C4—51no—
358C4—53no—
359C4—106no—
360C4C2×C3×C4×AGL(1,𝔽5)13yes—
361C4C23×C3×AGL(1,𝔽5)11yescharacteristic
362C3AGL(1,𝔽5)×#13111yescharacteristic, direct product factor, Sylow
363C2—23no—
364C2—52no—
365C2—103no—
366C2C22×C3×C4×AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC3×C8C22×AGL(1,𝔽5)
commutatorC2×C5C22×C3×C42
FrattiniC4C23×C3×AGL(1,𝔽5)
FittingC3×C5×#131C4
socleC2×C3×C5C22×C42
prime radicalC3×D5×#41C22
cosocleC4×D5C24×C3
hypercenterC3×#131AGL(1,𝔽5)
hypocenterC5C3×C4×#131
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C3×C8 ◁ C3×#131

Complements

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

NH ≅ G/Ntype
4. C3×#65728C2semidirect
5. C3×M16×AGL(1,𝔽5)C2semidirect
6. C3×#65718C2semidirect
7. C2×C3×C8×AGL(1,𝔽5)C2semidirect
8. AGL(1,𝔽5)×#131C3direct (N × H)
11. C3×#4133C4semidirect
13. C3×#4131C4semidirect
18. C3×C5×#131C4semidirect
19. C3×#4094C4semidirect
20. C3×#4053C4semidirect
22. C3×#4047C4semidirect
33. #65728C2×C3semidirect
34. M16×AGL(1,𝔽5)C2×C3semidirect
35. #65718C2×C3semidirect
36. C2×C8×AGL(1,𝔽5)C2×C3semidirect
46. C3×#704C2×C4semidirect
47. C2×C3×#170C2×C4semidirect
52. C3×#700C2×C4semidirect
53. C3×D5⋊C8C2×C4semidirect
56. C3×C5×M16C2×C4semidirect
57. C2×C3×C5×C8C2×C4semidirect
58. C3×#681C2×C4semidirect
59. C2×C3×#168C2×C4semidirect
60. C3×#676C2×C4semidirect
61. C3×C8×D5C2×C4semidirect
62. C3×C8×D5C2×C4semidirect
74. #4133C3×C4semidirect
76. #4131C3×C4semidirect
81. C5×#131C3×C4semidirect
82. #4094C3×C4semidirect
83. #4053C3×C4semidirect
85. #4047C3×C4semidirect
120. C3×#131AGL(1,𝔽5)direct (N × H)
143. #704C2×C3×C4semidirect
144. C2×#170C2×C3×C4semidirect
149. #700C2×C3×C4semidirect
150. D5⋊C8C2×C3×C4semidirect
153. C5×M16C2×C3×C4semidirect
154. C2×C5×C8C2×C3×C4semidirect
155. #681C2×C3×C4semidirect
156. C2×#168C2×C3×C4semidirect
157. #676C2×C3×C4semidirect
158. C8×D5C2×C3×C4semidirect
159. C8×D5C2×C3×C4semidirect
174. C3×AGL(1,𝔽5)#131direct (N × H)
191. C3×M16C2×AGL(1,𝔽5)semidirect
201. C2×C3×C8C2×AGL(1,𝔽5)semidirect
234. #131C3×AGL(1,𝔽5)direct (N × H)
278. AGL(1,𝔽5)C3×#131direct (N × H)
295. M16C2×C3×AGL(1,𝔽5)semidirect
305. C2×C8C2×C3×AGL(1,𝔽5)semidirect
311. C3×C5C4×#131semidirect
351. C5C3×C4×#131semidirect
362. C3AGL(1,𝔽5)×#131direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#41×AGL(1,𝔽5)2
C3×D5⋊#1312
C3×D5×#1312
C3×#657282
C3×M16×AGL(1,𝔽5)2
C3×#657182
C2×C3×C8×AGL(1,𝔽5)2
AGL(1,𝔽5)×#1313
C3×C4×#1315