Finite Group Databasebeta

← C2×C3×#65133 #14425933

Subgroups — C2×C3×#65133 #14425933

Order 1920.

1128 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×D40C211yesmaximal, characteristic
2C22×C3×#677C211yesmaximal, characteristic, complemented
3C2×C3×#4021C211yesmaximal, characteristic, complemented
4C2×C3×C4×D20C211yesmaximal, characteristic, complemented
5C2×C3×C5×C4⋊C8C211yesmaximal, characteristic, complemented
6C2×C3×#3953C211yesmaximal, characteristic, complemented
7C2×C3×#3948C211yesmaximal, characteristic, complemented
8C3×#65133C218yesmaximal, direct product factor
9C2×#65133C311yesmaximal, characteristic, direct product factor, Hall
10C22×C3×D20C2211yescharacteristic
11C22×C3×D20C2211yescharacteristic
12C22×C3×C4×D5—21no—
13C22×C3×Dic10C2211yescharacteristic
14C22×C3×C5×C8C2211yescharacteristic
15C22×C3×C5×C8C2211yescharacteristic
16C2×C3×C42×C5C2211yescharacteristic
17C2×C3×#685—21no—
18C2×C3×#685—21no—
19C2×C3×#685—21no—
20C2×C3×C4⋊Dic5C2211yescharacteristic
21C2×C3×D40—24no—
22C2×C3×D40C2214yes—
23C2×C3×#677—24no—
24C2×C3×#677C2214yescomplemented
25C3×#4021—22no—
26C3×#4021C2214yescomplemented
27C3×C4×D20—22no—
28C3×C4×D20C2214yescomplemented
29C3×C5×C4⋊C8C2214yescomplemented
30C3×#3953C2214yescomplemented
31C3×#3948C2214yescomplemented
32C2×C3×#463—51nomaximal, Hall
33C22×D40C2×C311yescharacteristic
34C22×#677C2×C311yescharacteristic, complemented
35C2×#4021C2×C311yescharacteristic, complemented
36C2×C4×D20C2×C311yescharacteristic, complemented
37C2×C5×C4⋊C8C2×C311yescharacteristic, complemented
38C2×#3953C2×C311yescharacteristic, complemented
39C2×#3948C2×C311yescharacteristic, complemented
40#65133C2×C318yesdirect product factor
41C23×C3×D5—21no—
42C23×C3×D5—21no—
43C22×C3×C4×C5D411yescharacteristic
44C22×C3×C4×C5D411yescharacteristic
45C22×C3×C4×C5C2311yescharacteristic
46C22×C3×Dic5—21no—
47C22×C3×Dic5—21no—
48C2×C3×D20—21no—
49C2×C3×D20—22no—
50C2×C3×D20—22no—
51C2×C3×D20—22no—
52C2×C3×D20—21no—
53C2×C3×D20D414yes—
54C2×C3×D20C2312yes—
55C2×C3×D20C2312yes—
56C2×C3×C4×D5—24no—
57C2×C3×C4×D5—44no—
58C2×C3×Dic10—22no—
59C2×C3×Dic10—22no—
60C2×C3×Dic10—21no—
61C2×C3×Dic10C2312yes—
62C2×C3×C5×C8—22no—
63C2×C3×C5×C8—22no—
64C2×C3×C5×C8C2312yes—
65C2×C3×C5×C8C2312yes—
66C3×C42×C5D412yes—
67C3×C42×C5C2312yes—
68C3×#685—24no—
69C3×#685—24no—
70C3×#685—24no—
71C3×C4⋊Dic5—21no—
72C3×C4⋊Dic5C2312yes—
73C3×D40—28no—
74C3×#677—28no—
75C22×C3×SD16—51no—
76C22×C3×D8—51no—
77C2×C3×#124—51no—
78C2×C3×C4×D4—51no—
79C3×#463—58no—
80C2×C3×C4⋊C8D511yescharacteristic, complemented
81C2×C3×#103—51no—
82C2×C3×#102—51no—
83C22×D20C22×C311yescharacteristic
84C22×D20C22×C311yescharacteristic
85C22×C4×D5—21no—
86C22×Dic10C22×C311yescharacteristic
87C22×C5×C8C22×C311yescharacteristic
88C22×C5×C8C22×C311yescharacteristic
89C2×C42×C5C22×C311yescharacteristic
90C2×#685—21no—
91C2×#685—21no—
92C2×#685—21no—
93C2×C4⋊Dic5C22×C311yescharacteristic
94C2×D40—24no—
95C2×D40C22×C314yes—
96C2×#677—24no—
97C2×#677C22×C314yescomplemented
98#4021—22no—
99#4021C22×C314yescomplemented
100C4×D20—22no—
101C4×D20C22×C314yescomplemented
102C5×C4⋊C8C22×C314yescomplemented
103#3953C22×C314yescomplemented
104#3948C22×C314yescomplemented
105C2×#463—51noSylow
106C23×C3×C5C2×D411yescharacteristic
107C22×C3×D5—24no—
108C22×C3×D5—22no—
109C22×C3×D5—22no—
110C22×C3×D5—42no—
111C22×C3×D5—42no—
112C22×C3×D5—42no—
113C22×C3×D5—42no—
114C22×C3×D5—42no—
115C2×C3×C4×C5—22no—
116C2×C3×C4×C5—22no—
117C2×C3×C4×C5C2×D412yes—
118C2×C3×C4×C5C2×D412yes—
119C2×C3×C4×C5C2×D411yescharacteristic
120C2×C3×C4×C5C2×D412yes—
121C2×C3×C4×C5#4112yes—
122C2×C3×C4×C5C2411yescharacteristic
123C2×C3×Dic5—22no—
124C2×C3×Dic5—22no—
125C2×C3×Dic5—42no—
126C2×C3×Dic5—42no—
127C3×D20—24no—
128C3×D20—22no—
129C3×D20—22no—
130C3×D20—22no—
131C3×D20—42no—
132C3×D20C2×D414yes—
133C3×C4×D5—48no—
134C3×Dic10—22no—
135C3×Dic10—22no—
136C3×Dic10—42no—
137C3×C5×C8—22no—
138C3×C5×C8—22no—
139C22×C3×Q8—51no—
140C22×C3×D4—51no—
141C22×C3×D4—51no—
142C23×C3×C4—101no—
143C2×C3×SD16—54no—
144C2×C3×SD16—104no—
145C2×C3×D8—54no—
146C2×C3×D8—104no—
147C22×C3×C8C2×D511yescharacteristic
148C22×C3×C8C2×D511yescharacteristic
149C3×#124—54no—
150C3×#124—102no—
151C3×C4×D4—54no—
152C3×C4×D4—102no—
153C2×C3×C4⋊C4—51no—
154C2×C3×C22⋊C4—101no—
155C2×C3×C22⋊C4—101no—
156C2×C3×C22⋊C4—101no—
157C2×C3×C42C2×D511yescharacteristic
158C3×C4⋊C8C2×D514yescomplemented
159C3×#103—54no—
160C3×#102—54no—
161C23×D5—21no—
162C23×D5—21no—
163C22×C4×C5C3×D411yescharacteristic
164C22×C4×C5C3×D411yescharacteristic
165C22×C4×C5C23×C311yescharacteristic
166C22×Dic5—21no—
167C22×Dic5—21no—
168C2×D20—21no—
169C2×D20—22no—
170C2×D20—22no—
171C2×D20—22no—
172C2×D20—21no—
173C2×D20C3×D414yes—
174C2×D20C23×C312yes—
175C2×D20C23×C312yes—
176C2×C4×D5—24no—
177C2×C4×D5—44no—
178C2×Dic10—22no—
179C2×Dic10—22no—
180C2×Dic10—21no—
181C2×Dic10C23×C312yes—
182C2×C5×C8—22no—
183C2×C5×C8—22no—
184C2×C5×C8C23×C312yes—
185C2×C5×C8C23×C312yes—
186C42×C5C3×D412yes—
187C42×C5C23×C312yes—
188#685—24no—
189#685—24no—
190#685—24no—
191C4⋊Dic5—21no—
192C4⋊Dic5C23×C312yes—
193D40—28no—
194#677—28no—
195C22×SD16—51no—
196C22×D8—51no—
197C2×#124—51no—
198C2×C4×D4—51no—
199#463—58no—
200C2×C4⋊C8C3×D511yescharacteristic, complemented
201C2×#103—51no—
202C2×#102—51no—
203C22×C3×C5#12112yes—
204C22×C3×C5#13512yes—
205C22×C3×C5AGL(1,ℤ/8)12yes—
206C22×C3×C5C22×D411yescharacteristic
207C2×C3×D5—24no—
208C2×C3×D5—42no—
209C2×C3×D5—42no—
210C2×C3×D5—48no—
211C2×C3×D5—42no—
212C2×C3×D5—42no—
213C2×C3×D5—42no—
214C2×C3×D5—84no—
215C3×C4×C5—22no—
216C3×C4×C5—22no—
217C3×C4×C5#12112yes—
218C3×C4×C5C22×D411yescharacteristic
219C3×C4×C5C2×#4111yescharacteristic
220C3×Dic5—42no—
221C3×Dic5—42no—
222C24×C3—101no—
223C24×C3—101no—
224C2×C3×Q8—52no—
225C2×C3×Q8—101no—
226C2×C3×Q8—102no—
227C2×C3×Q8—102no—
228C2×C3×D4—54no—
229C2×C3×D4—52no—
230C2×C3×D4—52no—
231C2×C3×D4—101no—
232C2×C3×D4—101no—
233C2×C3×D4—102no—
234C2×C3×D4—102no—
235C2×C3×D4—102no—
236C22×C3×C4—101no—
237C22×C3×C4—101no—
238C22×C3×C4—104no—
239C22×C3×C4—204no—
240C22×C3×C4D2011yescharacteristic
241C22×C3×C4D2011yescharacteristic
242C22×C3×C4C22×D511yescharacteristic
243C3×SD16—108no—
244C3×D8—108no—
245C2×C3×C8—22no—
246C2×C3×C8—22no—
247C2×C3×C8C22×D512yes—
248C2×C3×C8C22×D512yes—
249C3×C4⋊C4—52no—
250C3×C4⋊C4—101no—
251C3×C22⋊C4—104no—
252C3×C22⋊C4—104no—
253C3×C22⋊C4—104no—
254C3×C42D2012yes—
255C3×C42C22×D512yes—
256C23×C5C2×C3×D411yescharacteristic
257C22×D5—24no—
258C22×D5—22no—
259C22×D5—22no—
260C22×D5—42no—
261C22×D5—42no—
262C22×D5—42no—
263C22×D5—42no—
264C22×D5—42no—
265C2×C4×C5—22no—
266C2×C4×C5—22no—
267C2×C4×C5C2×C3×D412yes—
268C2×C4×C5C2×C3×D412yes—
269C2×C4×C5C2×C3×D411yescharacteristic
270C2×C4×C5C2×C3×D412yes—
271C2×C4×C5C3×#4112yes—
272C2×C4×C5C24×C311yescharacteristic
273C2×Dic5—22no—
274C2×Dic5—22no—
275C2×Dic5—42no—
276C2×Dic5—42no—
277D20—24no—
278D20—22no—
279D20—22no—
280D20—22no—
281D20—42no—
282D20C2×C3×D414yes—
283C4×D5—48no—
284Dic10—22no—
285Dic10—22no—
286Dic10—42no—
287C5×C8—22no—
288C5×C8—22no—
289C22×Q8—51no—
290C22×D4—51no—
291C22×D4—51no—
292C23×C4—101no—
293C2×SD16—54no—
294C2×SD16—104no—
295C2×D8—54no—
296C2×D8—104no—
297C22×C8C2×C3×D511yescharacteristic
298C22×C8C2×C3×D511yescharacteristic
299#124—54no—
300#124—102no—
301C4×D4—54no—
302C4×D4—102no—
303C2×C4⋊C4—51no—
304C2×C22⋊C4—101no—
305C2×C22⋊C4—101no—
306C2×C22⋊C4—101no—
307C2×C42C2×C3×D511yescharacteristic
308C4⋊C8C2×C3×D514yescomplemented
309#103—54no—
310#102—54no—
311C2×C3×C5#46314yescomplemented
312C2×C3×C5C2×#12111yescharacteristic
313C2×C3×C5C2×#13511yescharacteristic
314C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
315C3×D5—44no—
316C3×D5—82no—
317C23×C3—102no—
318C23×C3—104no—
319C23×C3—102no—
320C23×C3—202no—
321C23×C3—202no—
322C23×C3—202no—
323C23×C3—202no—
324C23×C3—202no—
325C23×C3C2×D2011yescharacteristic
326C3×Q8—102no—
327C3×Q8—102no—
328C3×Q8—202no—
329C3×D4—54no—
330C3×D4—102no—
331C3×D4—102no—
332C3×D4—102no—
333C3×D4—104no—
334C3×D4—202no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—102no—
338C2×C3×C4—102no—
339C2×C3×C4—208no—
340C2×C3×C4—202no—
341C2×C3×C4—202no—
342C2×C3×C4C2×D2012yes—
343C2×C3×C4C2×D2012yes—
344C2×C3×C4C2×D2011yescharacteristic
345C2×C3×C4D4×D512yes—
346C2×C3×C4#71312yes—
347C2×C3×C4C23×D511yescharacteristic
348C3×C8—22no—
349C3×C8—22no—
350C22×C5C3×#12112yes—
351C22×C5C3×#13512yes—
352C22×C5C3×AGL(1,ℤ/8)12yes—
353C22×C5C22×C3×D411yescharacteristic
354C2×D5—24no—
355C2×D5—42no—
356C2×D5—42no—
357C2×D5—48no—
358C2×D5—42no—
359C2×D5—42no—
360C2×D5—42no—
361C2×D5—84no—
362C4×C5—22no—
363C4×C5—22no—
364C4×C5C3×#12112yes—
365C4×C5C22×C3×D411yescharacteristic
366C4×C5C2×C3×#4111yescharacteristic
367Dic5—42no—
368Dic5—42no—
369C24—101no—
370C24—101no—
371C2×Q8—52no—
372C2×Q8—101no—
373C2×Q8—102no—
374C2×Q8—102no—
375C2×D4—54no—
376C2×D4—52no—
377C2×D4—52no—
378C2×D4—101no—
379C2×D4—101no—
380C2×D4—102no—
381C2×D4—102no—
382C2×D4—102no—
383C22×C4—101no—
384C22×C4—101no—
385C22×C4—104no—
386C22×C4—204no—
387C22×C4C3×D2011yescharacteristic
388C22×C4C3×D2011yescharacteristic
389C22×C4C22×C3×D511yescharacteristic
390SD16—108no—
391D8—108no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8C22×C3×D512yes—
395C2×C8C22×C3×D512yes—
396C4⋊C4—52no—
397C4⋊C4—101no—
398C22⋊C4—104no—
399C22⋊C4—104no—
400C22⋊C4—104no—
401C42C3×D2012yes—
402C42C22×C3×D512yes—
403C3×C5C2×#46311yescharacteristic, complemented, Hall
404C22×C3—104no—
405C22×C3—202no—
406C22×C3—202no—
407C22×C3—208no—
408C22×C3—202no—
409C22×C3—202no—
410C22×C3—202no—
411C22×C3—404no—
412C22×C3#404112yes—
413C22×C3#405012yes—
414C22×C3#405412yes—
415C22×C3C22×D2011yescharacteristic
416C3×C4—22no—
417C3×C4—22no—
418C3×C4—202no—
419C3×C4—202no—
420C3×C4#404112yes—
421C3×C4C2×D4×D511yescharacteristic
422C3×C4C2×#71311yescharacteristic
423C2×C5C3×#46314yescomplemented
424C2×C5C2×C3×#12111yescharacteristic
425C2×C5C2×C3×#13511yescharacteristic
426C2×C5C2×C3×AGL(1,ℤ/8)11yescharacteristic
427D5—44no—
428D5—82no—
429C23—102no—
430C23—104no—
431C23—102no—
432C23—202no—
433C23—202no—
434C23—202no—
435C23—202no—
436C23—202no—
437C23C2×C3×D2011yescharacteristic
438Q8—102no—
439Q8—102no—
440Q8—202no—
441D4—54no—
442D4—102no—
443D4—102no—
444D4—102no—
445D4—104no—
446D4—202no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—102no—
450C2×C4—102no—
451C2×C4—208no—
452C2×C4—202no—
453C2×C4—202no—
454C2×C4C2×C3×D2012yes—
455C2×C4C2×C3×D2012yes—
456C2×C4C2×C3×D2011yescharacteristic
457C2×C4C3×D4×D512yes—
458C2×C4C3×#71312yes—
459C2×C4C23×C3×D511yescharacteristic
460C8—22no—
461C8—22no—
462C2×C3—204no—
463C2×C3—402no—
464C2×C3#6513314yesdirect product factor
465C2×C3C2×#404111yescharacteristic
466C2×C3C2×#405011yescharacteristic
467C2×C3C2×#405411yescharacteristic
468C5C2×C3×#46311yescharacteristic, complemented, Sylow
469C22—104no—
470C22—208no—
471C22—202no—
472C22—202no—
473C22—202no—
474C22—202no—
475C22—202no—
476C22—404no—
477C22C3×#404112yes—
478C22C3×#405012yes—
479C22C3×#405412yes—
480C22C22×C3×D2011yescharacteristic
481C4—22no—
482C4—22no—
483C4—202no—
484C4—202no—
485C4C3×#404112yes—
486C4C2×C3×D4×D511yescharacteristic
487C4C2×C3×#71311yescharacteristic
488C3C2×#6513311yescharacteristic, direct product factor, Sylow
489C2—204no—
490C2—402no—
491C2C3×#6513314yesdirect product factor
492C2C2×C3×#404111yescharacteristic
493C2C2×C3×#405011yescharacteristic
494C2C2×C3×#405411yescharacteristic
subgroupG/H
centerC23×C3C2×D20
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC2×C3×C5×C4⋊C8C2
socleC23×C3×C5C2×D4
prime radicalC22×C3×D40C2
hypercenterC2×C3×C4⋊C8D5
hypocenterC5C2×C3×#463
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ C2×C3×C4⋊C8

Complements

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

NH ≅ G/Ntype
2. C22×C3×#677C2semidirect
3. C2×C3×#4021C2semidirect
4. C2×C3×C4×D20C2semidirect
5. C2×C3×C5×C4⋊C8C2semidirect
6. C2×C3×#3953C2semidirect
7. C2×C3×#3948C2semidirect
8. C3×#65133C2direct (N × H)
9. C2×#65133C3direct (N × H)
24. C2×C3×#677C22semidirect
26. C3×#4021C22semidirect
28. C3×C4×D20C22semidirect
29. C3×C5×C4⋊C8C22semidirect
30. C3×#3953C22semidirect
31. C3×#3948C22semidirect
34. C22×#677C2×C3semidirect
35. C2×#4021C2×C3semidirect
36. C2×C4×D20C2×C3semidirect
37. C2×C5×C4⋊C8C2×C3semidirect
38. C2×#3953C2×C3semidirect
39. C2×#3948C2×C3semidirect
40. #65133C2×C3direct (N × H)
80. C2×C3×C4⋊C8D5semidirect
97. C2×#677C22×C3semidirect
99. #4021C22×C3semidirect
101. C4×D20C22×C3semidirect
102. C5×C4⋊C8C22×C3semidirect
103. #3953C22×C3semidirect
104. #3948C22×C3semidirect
158. C3×C4⋊C8C2×D5semidirect
200. C2×C4⋊C8C3×D5semidirect
308. C4⋊C8C2×C3×D5semidirect
311. C2×C3×C5#463semidirect
403. C3×C5C2×#463semidirect
423. C2×C5C3×#463semidirect
464. C2×C3#65133direct (N × H)
468. C5C2×C3×#463semidirect
488. C3C2×#65133direct (N × H)
491. C2C3×#65133direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×D402
C22×C3×#6772
C2×C3×#40212
C2×C3×C4×D202
C2×C3×C5×C4⋊C82
C2×C3×#39532
C2×C3×#39482
C3×#651332
C2×#651333
C2×C3×#4635