Finite Group Databasebeta

← C2×C3×#65785 #14411928

Subgroups — C2×C3×#65785 #14411928

Order 1920.

1576 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#713C211yesmaximal, characteristic
2C2×C3×#4130C212yesmaximal, complemented
3C3×#65785C218yesmaximal, direct product factor
4C2×C3×#3999C214yesmaximal, complemented
5C2×#65785C311yesmaximal, characteristic, direct product factor, Hall
6C22×C3×C5×Q8C411yescharacteristic, complemented
7C2×C3×#713—22no—
8C2×C3×#713—22no—
9C2×C3×#713—24no—
10C2×C3×#713C414yescomplemented
11C2×C3×#713C2214yes—
12C22×C3×D20C411yescharacteristic, complemented
13C22×C3×D20C2212yes—
14C22×C3×C4×D5C412yescomplemented
15C22×C3×C4×D5C2211yescharacteristic
16C2×C3×C22⋊AGL(1,𝔽5)C2214yes—
17C3×#4130—24no—
18C3×#4130C2218yescomplemented
19C2×C3×C4⋊AGL(1,𝔽5)—22no—
20C2×C3×C4×AGL(1,𝔽5)—22no—
21C3×#3999C22116yescomplemented
22C2×C3×#410—51nomaximal, Hall
23C22×#713C2×C311yescharacteristic
24C2×#4130C2×C312yescomplemented
25#65785C2×C318yesdirect product factor
26C2×#3999C2×C314yescomplemented
27C23×C3×D5C2×C412yes—
28C23×C3×D5C2311yescharacteristic
29C22×C3×AGL(1,𝔽5)—24no—
30C2×C3×C5×Q8—22no—
31C2×C3×C5×Q8—22no—
32C2×C3×C5×Q8C2×C414yescomplemented
33C22×C3×C4×C5C2×C412yes—
34C22×C3×C4×C5C2×C411yescharacteristic
35C22×C3×Dic5C2×C411yescharacteristic
36C3×#713—28no—
37C3×#713—28no—
38C3×#713—28no—
39C3×#713—28no—
40C2×C3×D20—24no—
41C2×C3×D20—24no—
42C2×C3×D20—22no—
43C2×C3×D20—22no—
44C2×C3×D20C2×C414yes—
45C2×C3×D20C2×C414yescomplemented
46C2×C3×D20C2314yes—
47C2×C3×C4×D5—24no—
48C2×C3×C4×D5—22no—
49C2×C3×C4×D5—22no—
50C2×C3×C4×D5C2×C412yes—
51C2×C3×C4×D5C2×C418yescomplemented
52C2×C3×C4×D5D412yes—
53C2×C3×C4×D5D412yes—
54C2×C3×C4×D5D414yes—
55C2×C3×C4×D5C2312yes—
56C3×C22⋊AGL(1,𝔽5)—24no—
57C3×C22⋊AGL(1,𝔽5)C2318yes—
58C3×C4⋊AGL(1,𝔽5)—28no—
59C3×C4×AGL(1,𝔽5)—28no—
60C22×C3×#41—51no—
61C2×C3×#117—52no—
62C3×#410—58no—
63C2×C3×#99—54no—
64C22×C5×Q8C3×C411yescharacteristic, complemented
65C2×#713—22no—
66C2×#713—22no—
67C2×#713—24no—
68C2×#713C3×C414yescomplemented
69C2×#713C22×C314yes—
70C22×D20C3×C411yescharacteristic, complemented
71C22×D20C22×C312yes—
72C22×C4×D5C3×C412yescomplemented
73C22×C4×D5C22×C311yescharacteristic
74C2×C22⋊AGL(1,𝔽5)C22×C314yes—
75#4130—24no—
76#4130C22×C318yescomplemented
77C2×C4⋊AGL(1,𝔽5)—22no—
78C2×C4×AGL(1,𝔽5)—22no—
79#3999C22×C3116yescomplemented
80C2×#410—51noSylow
81C23×C3×C5C22×C411yescharacteristic
82C22×C3×D5—22no—
83C22×C3×D5—22no—
84C22×C3×D5—42no—
85C22×C3×D5—44no—
86C22×C3×D5C22×C414yes—
87C22×C3×D5C22×C411yescharacteristic
88C22×C3×D5C2×D412yes—
89C22×C3×D5C2×D412yes—
90C22×C3×D5C2411yescharacteristic
91C2×C3×AGL(1,𝔽5)—28no—
92C2×C3×AGL(1,𝔽5)—48no—
93C3×C5×Q8—28no—
94C2×C3×C4×C5—22no—
95C2×C3×C4×C5—22no—
96C2×C3×C4×C5C22⋊C412yes—
97C2×C3×C4×C5C22⋊C412yes—
98C2×C3×C4×C5C22×C414yes—
99C2×C3×C4×C5C22×C412yes—
100C2×C3×Dic5C22⋊C412yes—
101C2×C3×Dic5C22⋊C411yescharacteristic
102C2×C3×Dic5C22⋊C411yescharacteristic
103C2×C3×Dic5C22×C412yes—
104C3×D20—28no—
105C3×D20—28no—
106C3×D20—28no—
107C3×C4×D5—28no—
108C3×C4×D5—28no—
109C3×C4×D5C22⋊C414yes—
110C3×C4×D5C22⋊C414yes—
111C3×C4×D5C2×D414yes—
112C3×C4×D5C2×D414yes—
113C2×C3×#41—54no—
114C2×C3×#41—54no—
115C2×C3×#41—102no—
116C2×C3×#41—102no—
117C2×C3×#41—104no—
118C22×C3×Q8AGL(1,𝔽5)11yescharacteristic, complemented
119C22×C3×D4—52no—
120C22×C3×D4—51no—
121C23×C3×C4—51no—
122C23×C3×C4—52no—
123C3×#117—58no—
124C3×#117—104no—
125C2×C3×C4⋊C4—102no—
126C2×C3×C22⋊C4—54no—
127C2×C3×C42—102no—
128C3×#99—516no—
129C23×D5C2×C3×C412yes—
130C23×D5C23×C311yescharacteristic
131C22×AGL(1,𝔽5)—24no—
132C2×C5×Q8—22no—
133C2×C5×Q8—22no—
134C2×C5×Q8C2×C3×C414yescomplemented
135C22×C4×C5C2×C3×C412yes—
136C22×C4×C5C2×C3×C411yescharacteristic
137C22×Dic5C2×C3×C411yescharacteristic
138#713—28no—
139#713—28no—
140#713—28no—
141#713—28no—
142C2×D20—24no—
143C2×D20—24no—
144C2×D20—22no—
145C2×D20—22no—
146C2×D20C2×C3×C414yes—
147C2×D20C2×C3×C414yescomplemented
148C2×D20C23×C314yes—
149C2×C4×D5—24no—
150C2×C4×D5—22no—
151C2×C4×D5—22no—
152C2×C4×D5C2×C3×C412yes—
153C2×C4×D5C2×C3×C418yescomplemented
154C2×C4×D5C3×D412yes—
155C2×C4×D5C3×D412yes—
156C2×C4×D5C3×D414yes—
157C2×C4×D5C23×C312yes—
158C22⋊AGL(1,𝔽5)—24no—
159C22⋊AGL(1,𝔽5)C23×C318yes—
160C4⋊AGL(1,𝔽5)—28no—
161C4×AGL(1,𝔽5)—28no—
162C22×#41—51no—
163C2×#117—52no—
164#410—58no—
165C2×#99—54no—
166C22×C3×C5—22no—
167C22×C3×C5C2×C22⋊C411yescharacteristic
168C22×C3×C5C2×C22⋊C411yescharacteristic
169C22×C3×C5C23×C411yescharacteristic
170C2×C3×D5—24no—
171C2×C3×D5—24no—
172C2×C3×D5—41no—
173C2×C3×D5—42no—
174C2×C3×D5—41no—
175C2×C3×D5—44no—
176C2×C3×D5—44no—
177C2×C3×D5—44no—
178C2×C3×D5C2×C22⋊C412yes—
179C2×C3×D5C22×D412yes—
180C3×AGL(1,𝔽5)—48no—
181C3×C4×C5—24no—
182C3×C4×C5C2×C22⋊C414yes—
183C3×Dic5C2×C22⋊C412yes—
184C3×Dic5C2×C22⋊C412yes—
185C24×C3—51no—
186C24×C3—52no—
187C3×#41—108no—
188C3×#41—108no—
189C3×#41—108no—
190C3×#41—108no—
191C2×C3×Q8—22no—
192C2×C3×Q8—22no—
193C2×C3×Q8C2×AGL(1,𝔽5)14yescomplemented
194C2×C3×D4—54no—
195C2×C3×D4—54no—
196C2×C3×D4—54no—
197C2×C3×D4—104no—
198C2×C3×D4—104no—
199C2×C3×D4—102no—
200C2×C3×D4—102no—
201C22×C3×C4—51no—
202C22×C3×C4—52no—
203C22×C3×C4—52no—
204C22×C3×C4—54no—
205C22×C3×C4—52no—
206C22×C3×C4—52no—
207C22×C3×C4—58no—
208C22×C3×C4—104no—
209C22×C3×C4—102no—
210C22×C3×C4—102no—
211C22×C3×C4—104no—
212C22×C3×C4C2×AGL(1,𝔽5)12yes—
213C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
214C3×C4⋊C4—108no—
215C3×C22⋊C4—58no—
216C3×C22⋊C4—104no—
217C3×C42—108no—
218C23×C5C22×C3×C411yescharacteristic
219C22×D5—22no—
220C22×D5—22no—
221C22×D5—42no—
222C22×D5—44no—
223C22×D5C22×C3×C414yes—
224C22×D5C22×C3×C411yescharacteristic
225C22×D5C2×C3×D412yes—
226C22×D5C2×C3×D412yes—
227C22×D5C24×C311yescharacteristic
228C2×AGL(1,𝔽5)—28no—
229C2×AGL(1,𝔽5)—48no—
230C5×Q8—28no—
231C2×C4×C5—22no—
232C2×C4×C5—22no—
233C2×C4×C5C3×C22⋊C412yes—
234C2×C4×C5C3×C22⋊C412yes—
235C2×C4×C5C22×C3×C414yes—
236C2×C4×C5C22×C3×C412yes—
237C2×Dic5C3×C22⋊C412yes—
238C2×Dic5C3×C22⋊C411yescharacteristic
239C2×Dic5C3×C22⋊C411yescharacteristic
240C2×Dic5C22×C3×C412yes—
241D20—28no—
242D20—28no—
243D20—28no—
244C4×D5—28no—
245C4×D5—28no—
246C4×D5C3×C22⋊C414yes—
247C4×D5C3×C22⋊C414yes—
248C4×D5C2×C3×D414yes—
249C4×D5C2×C3×D414yes—
250C2×#41—54no—
251C2×#41—54no—
252C2×#41—102no—
253C2×#41—102no—
254C2×#41—104no—
255C22×Q8C3×AGL(1,𝔽5)11yescharacteristic, complemented
256C22×D4—51no—
257C22×D4—52no—
258C23×C4—51no—
259C23×C4—52no—
260#117—58no—
261#117—104no—
262C2×C4⋊C4—102no—
263C2×C22⋊C4—54no—
264C2×C42—102no—
265#99—516no—
266C2×C3×C5—21no—
267C2×C3×C5—21no—
268C2×C3×C5#41012yescomplemented
269C2×C3×C5C22×C22⋊C411yescharacteristic
270C3×D5—22no—
271C3×D5—22no—
272C3×D5—44no—
273C23×C3—52no—
274C23×C3—52no—
275C23×C3—51no—
276C23×C3—51no—
277C23×C3—54no—
278C23×C3—102no—
279C23×C3—102no—
280C23×C3—202no—
281C23×C3—204no—
282C23×C3C22×AGL(1,𝔽5)11yescharacteristic
283C3×Q8—28no—
284C3×D4—108no—
285C3×D4—108no—
286C3×D4—108no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—52no—
290C2×C3×C4—51no—
291C2×C3×C4—51no—
292C2×C3×C4—52no—
293C2×C3×C4—54no—
294C2×C3×C4—54no—
295C2×C3×C4—54no—
296C2×C3×C4—54no—
297C2×C3×C4—108no—
298C2×C3×C4—108no—
299C2×C3×C4—108no—
300C2×C3×C4—208no—
301C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
302C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
303C2×C3×C4C22×AGL(1,𝔽5)14yes—
304C2×C3×C4C22×AGL(1,𝔽5)12yes—
305C22×C5—22no—
306C22×C5C2×C3×C22⋊C411yescharacteristic
307C22×C5C2×C3×C22⋊C411yescharacteristic
308C22×C5C23×C3×C411yescharacteristic
309C2×D5—24no—
310C2×D5—24no—
311C2×D5—41no—
312C2×D5—42no—
313C2×D5—41no—
314C2×D5—44no—
315C2×D5—44no—
316C2×D5—44no—
317C2×D5C2×C3×C22⋊C412yes—
318C2×D5C22×C3×D412yes—
319AGL(1,𝔽5)—48no—
320C4×C5—24no—
321C4×C5C2×C3×C22⋊C414yes—
322Dic5C2×C3×C22⋊C412yes—
323Dic5C2×C3×C22⋊C412yes—
324C24—51no—
325C24—52no—
326#41—108no—
327#41—108no—
328#41—108no—
329#41—108no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8C2×C3×AGL(1,𝔽5)14yescomplemented
333C2×D4—54no—
334C2×D4—54no—
335C2×D4—54no—
336C2×D4—104no—
337C2×D4—104no—
338C2×D4—102no—
339C2×D4—102no—
340C22×C4—51no—
341C22×C4—52no—
342C22×C4—52no—
343C22×C4—54no—
344C22×C4—52no—
345C22×C4—52no—
346C22×C4—58no—
347C22×C4—104no—
348C22×C4—102no—
349C22×C4—102no—
350C22×C4—104no—
351C22×C4C2×C3×AGL(1,𝔽5)12yes—
352C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
353C4⋊C4—108no—
354C22⋊C4—58no—
355C22⋊C4—104no—
356C42—108no—
357C3×C5C2×#41011yescharacteristic, complemented, Hall
358C22×C3—22no—
359C22×C3—52no—
360C22×C3—52no—
361C22×C3—104no—
362C22×C3—104no—
363C22×C3—202no—
364C22×C3—201no—
365C22×C3—201no—
366C22×C3—204no—
367C22×C3—204no—
368C22×C3—204no—
369C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
370C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
371C22×C3C23×AGL(1,𝔽5)11yescharacteristic
372C3×C4—24no—
373C3×C4—52no—
374C3×C4—52no—
375C3×C4—208no—
376C3×C4C2×C22⋊AGL(1,𝔽5)14yes—
377C2×C5—21no—
378C2×C5—21no—
379C2×C5C3×#41012yescomplemented
380C2×C5C22×C3×C22⋊C411yescharacteristic
381D5—22no—
382D5—22no—
383D5—44no—
384C23—52no—
385C23—52no—
386C23—51no—
387C23—51no—
388C23—54no—
389C23—102no—
390C23—102no—
391C23—202no—
392C23—204no—
393C23C22×C3×AGL(1,𝔽5)11yescharacteristic
394Q8—28no—
395D4—108no—
396D4—108no—
397D4—108no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—52no—
401C2×C4—51no—
402C2×C4—51no—
403C2×C4—52no—
404C2×C4—54no—
405C2×C4—54no—
406C2×C4—54no—
407C2×C4—54no—
408C2×C4—108no—
409C2×C4—108no—
410C2×C4—108no—
411C2×C4—208no—
412C2×C4C3×C22⋊AGL(1,𝔽5)12yes—
413C2×C4C3×C22⋊AGL(1,𝔽5)12yes—
414C2×C4C22×C3×AGL(1,𝔽5)14yes—
415C2×C4C22×C3×AGL(1,𝔽5)12yes—
416C2×C3—21no—
417C2×C3—21no—
418C2×C3—102no—
419C2×C3—102no—
420C2×C3—204no—
421C2×C3#6578512yesdirect product factor
422C2×C3C22×C22⋊AGL(1,𝔽5)11yescharacteristic
423C5C2×C3×#41011yescharacteristic, complemented, Sylow
424C22—22no—
425C22—52no—
426C22—52no—
427C22—104no—
428C22—104no—
429C22—202no—
430C22—201no—
431C22—201no—
432C22—204no—
433C22—204no—
434C22—204no—
435C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
436C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
437C22C23×C3×AGL(1,𝔽5)11yescharacteristic
438C4—24no—
439C4—52no—
440C4—52no—
441C4—208no—
442C4C2×C3×C22⋊AGL(1,𝔽5)14yes—
443C3C2×#6578511yescharacteristic, direct product factor, Sylow
444C2—21no—
445C2—21no—
446C2—102no—
447C2—102no—
448C2—204no—
449C2C3×#6578512yesdirect product factor
450C2C22×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC22×C5C23×C3×C4
FrattiniC22C23×C3×AGL(1,𝔽5)
FittingC22×C3×C5×Q8C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×C3×#713C2
cosocleC22×D5C24×C3
hypercenterC22×C3×Q8AGL(1,𝔽5)
hypocenterC5C2×C3×#410
LCSG ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C22×C3×Q8

Complements

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

NH ≅ G/Ntype
2. C2×C3×#4130C2semidirect
3. C3×#65785C2direct (N × H)
4. C2×C3×#3999C2semidirect
5. C2×#65785C3direct (N × H)
6. C22×C3×C5×Q8C4semidirect
10. C2×C3×#713C4semidirect
12. C22×C3×D20C4semidirect
14. C22×C3×C4×D5C4semidirect
18. C3×#4130C22semidirect
21. C3×#3999C22semidirect
24. C2×#4130C2×C3semidirect
25. #65785C2×C3direct (N × H)
26. C2×#3999C2×C3semidirect
32. C2×C3×C5×Q8C2×C4semidirect
45. C2×C3×D20C2×C4semidirect
51. C2×C3×C4×D5C2×C4semidirect
64. C22×C5×Q8C3×C4semidirect
68. C2×#713C3×C4semidirect
70. C22×D20C3×C4semidirect
72. C22×C4×D5C3×C4semidirect
76. #4130C22×C3semidirect
79. #3999C22×C3semidirect
118. C22×C3×Q8AGL(1,𝔽5)semidirect
134. C2×C5×Q8C2×C3×C4semidirect
147. C2×D20C2×C3×C4semidirect
153. C2×C4×D5C2×C3×C4semidirect
193. C2×C3×Q8C2×AGL(1,𝔽5)semidirect
255. C22×Q8C3×AGL(1,𝔽5)semidirect
268. C2×C3×C5#410semidirect
332. C2×Q8C2×C3×AGL(1,𝔽5)semidirect
357. C3×C5C2×#410semidirect
379. C2×C5C3×#410semidirect
421. C2×C3#65785direct (N × H)
423. C5C2×C3×#410semidirect
443. C3C2×#65785direct (N × H)
449. C2C3×#65785direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#7132
C2×C3×#41302
C3×#657852
C2×C3×#39992
C2×#657853
C2×C3×#4105