Finite Group Databasebeta

← C3×#10604305 #14295376

Subgroups — C3×#10604305 #14295376

Order 1920.

824 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4039C211yesmaximal, characteristic
2C3×C4×C22⋊AGL(1,𝔽5)C212yesmaximal, complemented
3C2×C3×#3999C212yesmaximal
4C3×#64917C212yesmaximal, complemented
5#10604305C311yesmaximal, characteristic, direct product factor, Hall
6C22×C3×D20C2211yescharacteristic
7C22×C3×C4×D5C411yescharacteristic, complemented
8C22×C3×C4×D5C2211yescharacteristic
9C2×C3×C22⋊AGL(1,𝔽5)C2214yes—
10C2×C3×C4×AGL(1,𝔽5)—22no—
11C2×C3×C5×C4⋊C4C411yescharacteristic, complemented
12C2×C3×#685C411yescharacteristic, complemented
13C2×C3×#685C2211yescharacteristic
14C2×C3×C4×Dic5C411yescharacteristic, complemented
15C3×#4039—22no—
16C3×#4039—22no—
17C3×(C2×AGL(1,𝔽5))⋊C4—22no—
18C3×#3999C418yescomplemented
19C3×C4×#99—51nomaximal, Hall
20C2×#4039C2×C311yescharacteristic
21C4×C22⋊AGL(1,𝔽5)C2×C312yescomplemented
22C2×#3999C2×C312yes—
23#64917C2×C312yescomplemented
24C23×C3×D5C2×C411yescharacteristic
25C23×C3×D5C2311yescharacteristic
26C22×C3×AGL(1,𝔽5)—24no—
27C22×C3×C4×C5C2×C411yescharacteristic
28C22×C3×C4×C5C2×C411yescharacteristic
29C22×C3×C4×C5C2×C411yescharacteristic
30C22×C3×Dic5C2×C411yescharacteristic
31C22×C3×Dic5C2×C411yescharacteristic
32C2×C3×D20—22no—
33C2×C3×D20—22no—
34C2×C3×D20C2×C414yes—
35C2×C3×C4×D5—21no—
36C2×C3×C4×D5—21no—
37C2×C3×C4×D5—42no—
38C2×C3×C4×D5—42no—
39C2×C3×C4×D5D412yes—
40C2×C3×C4×D5D412yes—
41C3×C22⋊AGL(1,𝔽5)—24no—
42C3×C22⋊AGL(1,𝔽5)C2×C418yescomplemented
43C3×C4×AGL(1,𝔽5)—44no—
44C3×C5×C4⋊C4—22no—
45C3×#685—22no—
46C3×#685—22no—
47C3×C4×Dic5—21no—
48C3×C4×Dic5—21no—
49C2×C3×C4×D4—51no—
50C2×C3×#99—52no—
51C3×C4×C22⋊C4—52no—
52C3×#342—52no—
53C22×D20C22×C311yescharacteristic
54C22×C4×D5C3×C411yescharacteristic, complemented
55C22×C4×D5C22×C311yescharacteristic
56C2×C22⋊AGL(1,𝔽5)C22×C314yes—
57C2×C4×AGL(1,𝔽5)—22no—
58C2×C5×C4⋊C4C3×C411yescharacteristic, complemented
59C2×#685C3×C411yescharacteristic, complemented
60C2×#685C22×C311yescharacteristic
61C2×C4×Dic5C3×C411yescharacteristic, complemented
62#4039—22no—
63#4039—22no—
64(C2×AGL(1,𝔽5))⋊C4—22no—
65#3999C3×C418yescomplemented
66C4×#99—51noSylow
67C23×C3×C5C22×C411yescharacteristic
68C22×C3×D5—21no—
69C22×C3×D5—21no—
70C22×C3×D5—41no—
71C22×C3×D5—41no—
72C22×C3×D5—41no—
73C22×C3×D5—41no—
74C22×C3×D5C4212yescomplemented
75C22×C3×D5C22×C412yes—
76C22×C3×D5C2×D412yes—
77C22×C3×D5#4112yes—
78C2×C3×AGL(1,𝔽5)—28no—
79C2×C3×AGL(1,𝔽5)—44no—
80C2×C3×AGL(1,𝔽5)—44no—
81C2×C3×C4×C5—21no—
82C2×C3×C4×C5—21no—
83C2×C3×C4×C5—21no—
84C2×C3×C4×C5—41no—
85C2×C3×C4×C5—41no—
86C2×C3×C4×C5C4212yescomplemented
87C2×C3×C4×C5C22⋊C412yes—
88C2×C3×Dic5—22no—
89C2×C3×Dic5—21no—
90C2×C3×Dic5—41no—
91C2×C3×Dic5C22⋊C411yescharacteristic
92C2×C3×Dic5C22⋊C411yescharacteristic
93C3×D20—28no—
94C3×C4×D5—24no—
95C3×C4×D5—42no—
96C3×C4×D5—42no—
97C3×C4×D5—42no—
98C22×C3×D4—51no—
99C23×C3×C4—51no—
100C23×C3×C4—51no—
101C3×C4×D4—102no—
102C3×C4×D4—102no—
103C2×C3×C4⋊C4AGL(1,𝔽5)11yescharacteristic, complemented
104C2×C3×C22⋊C4—54no—
105C2×C3×C22⋊C4—51no—
106C2×C3×C22⋊C4—51no—
107C2×C3×C42—51no—
108C2×C3×C42—102no—
109C3×#99—58no—
110C3×#95—102no—
111C23×D5C2×C3×C411yescharacteristic
112C23×D5C23×C311yescharacteristic
113C22×AGL(1,𝔽5)—24no—
114C22×C4×C5C2×C3×C411yescharacteristic
115C22×C4×C5C2×C3×C411yescharacteristic
116C22×C4×C5C2×C3×C411yescharacteristic
117C22×Dic5C2×C3×C411yescharacteristic
118C22×Dic5C2×C3×C411yescharacteristic
119C2×D20—22no—
120C2×D20—22no—
121C2×D20C2×C3×C414yes—
122C2×C4×D5—21no—
123C2×C4×D5—21no—
124C2×C4×D5—42no—
125C2×C4×D5—42no—
126C2×C4×D5C3×D412yes—
127C2×C4×D5C3×D412yes—
128C22⋊AGL(1,𝔽5)—24no—
129C22⋊AGL(1,𝔽5)C2×C3×C418yescomplemented
130C4×AGL(1,𝔽5)—44no—
131C5×C4⋊C4—22no—
132#685—22no—
133#685—22no—
134C4×Dic5—21no—
135C4×Dic5—21no—
136C2×C4×D4—51no—
137C2×#99—52no—
138C4×C22⋊C4—52no—
139#342—52no—
140C22×C3×C5—21no—
141C22×C3×C5—21no—
142C22×C3×C5C2×C4211yescharacteristic
143C22×C3×C5C2×C22⋊C411yescharacteristic
144C22×C3×C5#11711yescharacteristic
145C2×C3×D5—22no—
146C2×C3×D5—22no—
147C2×C3×D5—22no—
148C2×C3×D5—42no—
149C2×C3×D5—42no—
150C2×C3×D5—42no—
151C2×C3×D5—41no—
152C2×C3×D5—41no—
153C2×C3×D5—42no—
154C2×C3×D5C4×D414yes—
155C3×AGL(1,𝔽5)—48no—
156C3×C4×C5—22no—
157C3×C4×C5—22no—
158C3×C4×C5—41no—
159C3×Dic5—21no—
160C3×Dic5—41no—
161C3×Dic5#9912yescomplemented
162C24×C3—51no—
163C24×C3—51no—
164C2×C3×D4—54no—
165C2×C3×D4—102no—
166C2×C3×D4—102no—
167C22×C3×C4—51no—
168C22×C3×C4—52no—
169C22×C3×C4—52no—
170C22×C3×C4—51no—
171C22×C3×C4—101no—
172C22×C3×C4—101no—
173C22×C3×C4—104no—
174C22×C3×C4—202no—
175C22×C3×C4—202no—
176C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
177C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
178C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
179C3×C4⋊C4—22no—
180C3×C22⋊C4—58no—
181C3×C22⋊C4—102no—
182C3×C22⋊C4—104no—
183C3×C22⋊C4—102no—
184C3×C42—101no—
185C3×C42—101no—
186C3×C42—204no—
187C23×C5C22×C3×C411yescharacteristic
188C22×D5—21no—
189C22×D5—21no—
190C22×D5—41no—
191C22×D5—41no—
192C22×D5—41no—
193C22×D5—41no—
194C22×D5C3×C4212yescomplemented
195C22×D5C22×C3×C412yes—
196C22×D5C2×C3×D412yes—
197C22×D5C3×#4112yes—
198C2×AGL(1,𝔽5)—28no—
199C2×AGL(1,𝔽5)—44no—
200C2×AGL(1,𝔽5)—44no—
201C2×C4×C5—21no—
202C2×C4×C5—21no—
203C2×C4×C5—21no—
204C2×C4×C5—41no—
205C2×C4×C5—41no—
206C2×C4×C5C3×C4212yescomplemented
207C2×C4×C5C3×C22⋊C412yes—
208C2×Dic5—22no—
209C2×Dic5—21no—
210C2×Dic5—41no—
211C2×Dic5C3×C22⋊C411yescharacteristic
212C2×Dic5C3×C22⋊C411yescharacteristic
213D20—28no—
214C4×D5—24no—
215C4×D5—42no—
216C4×D5—42no—
217C4×D5—42no—
218C22×D4—51no—
219C23×C4—51no—
220C23×C4—51no—
221C4×D4—102no—
222C4×D4—102no—
223C2×C4⋊C4C3×AGL(1,𝔽5)11yescharacteristic, complemented
224C2×C22⋊C4—54no—
225C2×C22⋊C4—51no—
226C2×C22⋊C4—51no—
227C2×C42—51no—
228C2×C42—102no—
229#99—58no—
230#95—102no—
231C2×C3×C5—21no—
232C2×C3×C5—21no—
233C2×C3×C5C4×C22⋊C411yescharacteristic
234C2×C3×C5C2×#9911yescharacteristic
235C2×C3×C5#41011yescharacteristic
236C3×D5—24no—
237C3×D5—42no—
238C23×C3—52no—
239C23×C3—52no—
240C23×C3—52no—
241C23×C3—52no—
242C23×C3—101no—
243C23×C3—101no—
244C23×C3—201no—
245C23×C3—201no—
246C23×C3—201no—
247C23×C3—201no—
248C23×C3C22×AGL(1,𝔽5)11yescharacteristic
249C3×D4—108no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—51no—
256C2×C3×C4—51no—
257C2×C3×C4—104no—
258C2×C3×C4—102no—
259C2×C3×C4—108no—
260C2×C3×C4—101no—
261C2×C3×C4—202no—
262C2×C3×C4—202no—
263C2×C3×C4—202no—
264C2×C3×C4—204no—
265C2×C3×C4—204no—
266C2×C3×C4—201no—
267C2×C3×C4C4×AGL(1,𝔽5)12yescomplemented
268C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
269C22×C5—21no—
270C22×C5—21no—
271C22×C5C2×C3×C4211yescharacteristic
272C22×C5C2×C3×C22⋊C411yescharacteristic
273C22×C5C3×#11711yescharacteristic
274C2×D5—22no—
275C2×D5—22no—
276C2×D5—22no—
277C2×D5—42no—
278C2×D5—42no—
279C2×D5—42no—
280C2×D5—41no—
281C2×D5—41no—
282C2×D5—42no—
283C2×D5C3×C4×D414yes—
284AGL(1,𝔽5)—48no—
285C4×C5—22no—
286C4×C5—22no—
287C4×C5—41no—
288Dic5—21no—
289Dic5—41no—
290Dic5C3×#9912yescomplemented
291C24—51no—
292C24—51no—
293C2×D4—54no—
294C2×D4—102no—
295C2×D4—102no—
296C22×C4—51no—
297C22×C4—52no—
298C22×C4—52no—
299C22×C4—51no—
300C22×C4—101no—
301C22×C4—101no—
302C22×C4—104no—
303C22×C4—202no—
304C22×C4—202no—
305C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
306C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
307C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
308C4⋊C4—22no—
309C22⋊C4—58no—
310C22⋊C4—102no—
311C22⋊C4—104no—
312C22⋊C4—102no—
313C42—101no—
314C42—101no—
315C42—204no—
316C3×C5C4×#9911yescharacteristic, complemented, Hall
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—54no—
320C22×C3—102no—
321C22×C3—102no—
322C22×C3—102no—
323C22×C3—202no—
324C22×C3—202no—
325C22×C3—202no—
326C22×C3—202no—
327C22×C3—201no—
328C22×C3—201no—
329C22×C3C2×C4×AGL(1,𝔽5)11yescharacteristic
330C22×C3#413011yescharacteristic
331C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
332C3×C4—22no—
333C3×C4—22no—
334C3×C4—41no—
335C3×C4—52no—
336C3×C4—101no—
337C3×C4—208no—
338C3×C4—201no—
339C2×C5—21no—
340C2×C5—21no—
341C2×C5C3×C4×C22⋊C411yescharacteristic
342C2×C5C2×C3×#9911yescharacteristic
343C2×C5C3×#41011yescharacteristic
344D5—24no—
345D5—42no—
346C23—52no—
347C23—52no—
348C23—52no—
349C23—52no—
350C23—101no—
351C23—101no—
352C23—201no—
353C23—201no—
354C23—201no—
355C23—201no—
356C23C22×C3×AGL(1,𝔽5)11yescharacteristic
357D4—108no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—51no—
364C2×C4—51no—
365C2×C4—104no—
366C2×C4—102no—
367C2×C4—108no—
368C2×C4—101no—
369C2×C4—202no—
370C2×C4—202no—
371C2×C4—202no—
372C2×C4—204no—
373C2×C4—204no—
374C2×C4—201no—
375C2×C4C3×C4×AGL(1,𝔽5)12yescomplemented
376C2×C4C3×C22⋊AGL(1,𝔽5)12yes—
377C2×C3—21no—
378C2×C3—21no—
379C2×C3—104no—
380C2×C3—202no—
381C2×C3C4×C22⋊AGL(1,𝔽5)11yescharacteristic
382C2×C3D5⋊#9911yescharacteristic
383C2×C3#6578511yescharacteristic
384C5C3×C4×#9911yescharacteristic, complemented, Sylow
385C22—21no—
386C22—21no—
387C22—54no—
388C22—102no—
389C22—102no—
390C22—102no—
391C22—202no—
392C22—202no—
393C22—202no—
394C22—202no—
395C22—201no—
396C22—201no—
397C22C2×C3×C4×AGL(1,𝔽5)11yescharacteristic
398C22C3×#413011yescharacteristic
399C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
400C4—22no—
401C4—22no—
402C4—41no—
403C4—52no—
404C4—101no—
405C4—208no—
406C4—201no—
407C3#1060430511yescharacteristic, direct product factor, Sylow
408C2—21no—
409C2—21no—
410C2—104no—
411C2—202no—
412C2C3×C4×C22⋊AGL(1,𝔽5)11yescharacteristic
413C2C3×D5⋊#9911yescharacteristic
414C2C3×#6578511yescharacteristic
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC22×C5C2×C3×C42
FrattiniC23C22×C3×AGL(1,𝔽5)
FittingC2×C3×C5×C4⋊C4C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×C3×D20C22
cosocleC23×D5C23×C3
hypercenterC2×C3×C4⋊C4AGL(1,𝔽5)
hypocenterC5C3×C4×#99
LCSG ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×C4×C22⋊AGL(1,𝔽5)C2semidirect
4. C3×#64917C2semidirect
5. #10604305C3direct (N × H)
7. C22×C3×C4×D5C4semidirect
11. C2×C3×C5×C4⋊C4C4semidirect
12. C2×C3×#685C4semidirect
14. C2×C3×C4×Dic5C4semidirect
18. C3×#3999C4semidirect
21. C4×C22⋊AGL(1,𝔽5)C2×C3semidirect
23. #64917C2×C3semidirect
42. C3×C22⋊AGL(1,𝔽5)C2×C4semidirect
54. C22×C4×D5C3×C4semidirect
58. C2×C5×C4⋊C4C3×C4semidirect
59. C2×#685C3×C4semidirect
61. C2×C4×Dic5C3×C4semidirect
65. #3999C3×C4semidirect
74. C22×C3×D5C42semidirect
86. C2×C3×C4×C5C42semidirect
103. C2×C3×C4⋊C4AGL(1,𝔽5)semidirect
129. C22⋊AGL(1,𝔽5)C2×C3×C4semidirect
161. C3×Dic5#99semidirect
194. C22×D5C3×C42semidirect
206. C2×C4×C5C3×C42semidirect
223. C2×C4⋊C4C3×AGL(1,𝔽5)semidirect
267. C2×C3×C4C4×AGL(1,𝔽5)semidirect
290. Dic5C3×#99semidirect
316. C3×C5C4×#99semidirect
375. C2×C4C3×C4×AGL(1,𝔽5)semidirect
384. C5C3×C4×#99semidirect
407. C3#10604305direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#40392
C3×C4×C22⋊AGL(1,𝔽5)2
C2×C3×#39992
C3×#649172
#106043053
C3×C4×#995