Finite Group Databasebeta

← C2×C3×#65108 #14425939

Subgroups — C2×C3×#65108 #14425939

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
1C2×C3×#4093C211yesmaximal, characteristic, complemented
2C22×C3×#688C211yesmaximal, characteristic, complemented
3C22×C3×D40C211yesmaximal, characteristic
4C2×C3×#4039C211yesmaximal, characteristic, complemented
5C2×C3×C5×#103C211yesmaximal, characteristic, complemented
6C2×C3×#3946C211yesmaximal, characteristic, complemented
7C2×C3×#3941C211yesmaximal, characteristic, complemented
8C3×#65108C218yesmaximal, direct product factor
9C2×#65108C311yesmaximal, characteristic, direct product factor, Hall
10C22×C3×C5×Q8C2211yescharacteristic
11C22×C3×D20C2211yescharacteristic
12C22×C3×D20C2211yescharacteristic
13C22×C3×C4×D5—21no—
14C22×C3×C5×C8C2211yescharacteristic
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C3×#4093—22no—
17C3×#4093C2214yescomplemented
18C2×C3×#688—24no—
19C2×C3×#688C2214yescomplemented
20C2×C3×#685—21no—
21C2×C3×#685—21no—
22C2×C3×#685—21no—
23C2×C3×C4×Dic5C2211yescharacteristic
24C22×C3×#168C2211yescharacteristic
25C2×C3×D40—24no—
26C2×C3×D40C2214yes—
27C3×#4039—22no—
28C3×#4039C2214yescomplemented
29C3×C5×#103C2214yescomplemented
30C3×#3946C2214yescomplemented
31C3×#3941C2214yescomplemented
32C2×C3×#463—51nomaximal, Hall
33C2×#4093C2×C311yescharacteristic, complemented
34C22×#688C2×C311yescharacteristic, complemented
35C22×D40C2×C311yescharacteristic
36C2×#4039C2×C311yescharacteristic, complemented
37C2×C5×#103C2×C311yescharacteristic, complemented
38C2×#3946C2×C311yescharacteristic, complemented
39C2×#3941C2×C311yescharacteristic, complemented
40#65108C2×C318yesdirect product factor
41C23×C3×D5—21no—
42C23×C3×D5—21no—
43C2×C3×C5×Q8—21no—
44C2×C3×C5×Q8—22no—
45C2×C3×C5×Q8—22no—
46C2×C3×C5×Q8C2312yes—
47C22×C3×C4×C5—21no—
48C22×C3×C4×C5—21no—
49C22×C3×C4×C5C2311yescharacteristic
50C22×C3×Dic5D411yescharacteristic
51C22×C3×Dic5D411yescharacteristic
52C2×C3×D20—21no—
53C2×C3×D20—22no—
54C2×C3×D20—22no—
55C2×C3×D20—22no—
56C2×C3×D20—21no—
57C2×C3×D20D414yes—
58C2×C3×D20C2312yes—
59C2×C3×D20C2312yes—
60C2×C3×C4×D5—24no—
61C2×C3×C4×D5—44no—
62C2×C3×C5×C8—22no—
63C2×C3×C5×C8C2312yes—
64C3×C5×C4⋊C4—21no—
65C3×C5×C4⋊C4C2312yes—
66C3×#688—28no—
67C3×#685—24no—
68C3×#685—24no—
69C3×#685—24no—
70C3×C4×Dic5D412yes—
71C3×C4×Dic5C2312yes—
72C2×C3×#168—22no—
73C2×C3×#168C2312yes—
74C3×D40—28no—
75C22×C3×SD16—51no—
76C22×C3×D8—51no—
77C2×C3×#124—51no—
78C2×C3×C4×D4—51no—
79C3×#463—58no—
80C2×C3×C4⋊C8—51no—
81C2×C3×#103D511yescharacteristic, complemented
82C2×C3×#102—51no—
83C22×C5×Q8C22×C311yescharacteristic
84C22×D20C22×C311yescharacteristic
85C22×D20C22×C311yescharacteristic
86C22×C4×D5—21no—
87C22×C5×C8C22×C311yescharacteristic
88C2×C5×C4⋊C4C22×C311yescharacteristic
89#4093—22no—
90#4093C22×C314yescomplemented
91C2×#688—24no—
92C2×#688C22×C314yescomplemented
93C2×#685—21no—
94C2×#685—21no—
95C2×#685—21no—
96C2×C4×Dic5C22×C311yescharacteristic
97C22×#168C22×C311yescharacteristic
98C2×D40—24no—
99C2×D40C22×C314yes—
100#4039—22no—
101#4039C22×C314yescomplemented
102C5×#103C22×C314yescomplemented
103#3946C22×C314yescomplemented
104#3941C22×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—
115C3×C5×Q8—22no—
116C3×C5×Q8—22no—
117C3×C5×Q8—42no—
118C2×C3×C4×C5—22no—
119C2×C3×C4×C5—22no—
120C2×C3×C4×C5—42no—
121C2×C3×C4×C5—42no—
122C2×C3×C4×C5C2×D411yescharacteristic
123C2×C3×C4×C5C2×D412yes—
124C2×C3×C4×C5#4112yes—
125C2×C3×C4×C5C2411yescharacteristic
126C2×C3×Dic5—22no—
127C2×C3×Dic5—22no—
128C2×C3×Dic5C2×D412yes—
129C2×C3×Dic5C2×D412yes—
130C3×D20—24no—
131C3×D20—22no—
132C3×D20—22no—
133C3×D20—22no—
134C3×D20—42no—
135C3×D20C2×D414yes—
136C3×C4×D5—48no—
137C3×C5×C8—22no—
138C3×#168—22no—
139C22×C3×Q8C2×D511yescharacteristic
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×C8—51no—
148C22×C3×C8C2×D511yescharacteristic
149C3×#124—54no—
150C3×#124—102no—
151C3×C4×D4—54no—
152C3×C4×D4—102no—
153C2×C3×C4⋊C4C2×D511yescharacteristic
154C2×C3×C22⋊C4—101no—
155C2×C3×C22⋊C4—101no—
156C2×C3×C22⋊C4—101no—
157C2×C3×C42—51no—
158C3×C4⋊C8—54no—
159C3×#103C2×D514yescomplemented
160C3×#102—54no—
161C23×D5—21no—
162C23×D5—21no—
163C2×C5×Q8—21no—
164C2×C5×Q8—22no—
165C2×C5×Q8—22no—
166C2×C5×Q8C23×C312yes—
167C22×C4×C5—21no—
168C22×C4×C5—21no—
169C22×C4×C5C23×C311yescharacteristic
170C22×Dic5C3×D411yescharacteristic
171C22×Dic5C3×D411yescharacteristic
172C2×D20—21no—
173C2×D20—22no—
174C2×D20—22no—
175C2×D20—22no—
176C2×D20—21no—
177C2×D20C3×D414yes—
178C2×D20C23×C312yes—
179C2×D20C23×C312yes—
180C2×C4×D5—24no—
181C2×C4×D5—44no—
182C2×C5×C8—22no—
183C2×C5×C8C23×C312yes—
184C5×C4⋊C4—21no—
185C5×C4⋊C4C23×C312yes—
186#688—28no—
187#685—24no—
188#685—24no—
189#685—24no—
190C4×Dic5C3×D412yes—
191C4×Dic5C23×C312yes—
192C2×#168—22no—
193C2×#168C23×C312yes—
194D40—28no—
195C22×SD16—51no—
196C22×D8—51no—
197C2×#124—51no—
198C2×C4×D4—51no—
199#463—58no—
200C2×C4⋊C8—51no—
201C2×#103C3×D511yescharacteristic, complemented
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—42no—
216C3×C4×C5—42no—
217C3×C4×C5#12112yes—
218C3×C4×C5C22×D411yescharacteristic
219C3×C4×C5C2×#4111yescharacteristic
220C3×Dic5—22no—
221C3×Dic5—22no—
222C24×C3—101no—
223C24×C3—101no—
224C2×C3×Q8—21no—
225C2×C3×Q8—22no—
226C2×C3×Q8—22no—
227C2×C3×Q8C22×D512yes—
228C2×C3×D4—54no—
229C2×C3×D4—52no—
230C2×C3×D4—52no—
231C2×C3×D4—102no—
232C2×C3×D4—102no—
233C2×C3×D4—101no—
234C2×C3×D4—102no—
235C2×C3×D4—101no—
236C22×C3×C4—21no—
237C22×C3×C4—21no—
238C22×C3×C4—51no—
239C22×C3×C4—51no—
240C22×C3×C4—104no—
241C22×C3×C4—204no—
242C22×C3×C4C22×D511yescharacteristic
243C3×SD16—108no—
244C3×D8—108no—
245C2×C3×C8—22no—
246C2×C3×C8—52no—
247C2×C3×C8—102no—
248C2×C3×C8C22×D512yes—
249C3×C4⋊C4—21no—
250C3×C4⋊C4C22×D512yes—
251C3×C22⋊C4—104no—
252C3×C22⋊C4—104no—
253C3×C22⋊C4—104no—
254C3×C42—52no—
255C3×C42—52no—
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—
265C5×Q8—22no—
266C5×Q8—22no—
267C5×Q8—42no—
268C2×C4×C5—22no—
269C2×C4×C5—22no—
270C2×C4×C5—42no—
271C2×C4×C5—42no—
272C2×C4×C5C2×C3×D411yescharacteristic
273C2×C4×C5C2×C3×D412yes—
274C2×C4×C5C3×#4112yes—
275C2×C4×C5C24×C311yescharacteristic
276C2×Dic5—22no—
277C2×Dic5—22no—
278C2×Dic5C2×C3×D412yes—
279C2×Dic5C2×C3×D412yes—
280D20—24no—
281D20—22no—
282D20—22no—
283D20—22no—
284D20—42no—
285D20C2×C3×D414yes—
286C4×D5—48no—
287C5×C8—22no—
288#168—22no—
289C22×Q8C2×C3×D511yescharacteristic
290C22×D4—51no—
291C22×D4—51no—
292C23×C4—101no—
293C2×SD16—54no—
294C2×SD16—104no—
295C2×D8—54no—
296C2×D8—104no—
297C22×C8—51no—
298C22×C8C2×C3×D511yescharacteristic
299#124—54no—
300#124—102no—
301C4×D4—54no—
302C4×D4—102no—
303C2×C4⋊C4C2×C3×D511yescharacteristic
304C2×C22⋊C4—101no—
305C2×C22⋊C4—101no—
306C2×C22⋊C4—101no—
307C2×C42—51no—
308C4⋊C8—54no—
309#103C2×C3×D514yescomplemented
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×C3D4×D511yescharacteristic
326C3×Q8—22no—
327C3×Q8—22no—
328C3×Q8—42no—
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—42no—
338C2×C3×C4—42no—
339C2×C3×C4—52no—
340C2×C3×C4—52no—
341C2×C3×C4—102no—
342C2×C3×C4—102no—
343C2×C3×C4—208no—
344C2×C3×C4#70912yes—
345C2×C3×C4D4×D512yes—
346C2×C3×C4D4×D511yescharacteristic
347C2×C3×C4C23×D511yescharacteristic
348C3×C8—22no—
349C3×C8—102no—
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—42no—
363C4×C5—42no—
364C4×C5C3×#12112yes—
365C4×C5C22×C3×D411yescharacteristic
366C4×C5C2×C3×#4111yescharacteristic
367Dic5—22no—
368Dic5—22no—
369C24—101no—
370C24—101no—
371C2×Q8—21no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8C22×C3×D512yes—
375C2×D4—54no—
376C2×D4—52no—
377C2×D4—52no—
378C2×D4—102no—
379C2×D4—102no—
380C2×D4—101no—
381C2×D4—102no—
382C2×D4—101no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—51no—
386C22×C4—51no—
387C22×C4—104no—
388C22×C4—204no—
389C22×C4C22×C3×D511yescharacteristic
390SD16—108no—
391D8—108no—
392C2×C8—22no—
393C2×C8—52no—
394C2×C8—102no—
395C2×C8C22×C3×D512yes—
396C4⋊C4—21no—
397C4⋊C4C22×C3×D512yes—
398C22⋊C4—104no—
399C22⋊C4—104no—
400C22⋊C4—104no—
401C42—52no—
402C42—52no—
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#403012yes—
413C22×C3#406012yes—
414C22×C3#406512yes—
415C22×C3C2×D4×D511yescharacteristic
416C3×C4—42no—
417C3×C4—42no—
418C3×C4—102no—
419C3×C4—102no—
420C3×C4#403012yes—
421C3×C4C2×#70911yescharacteristic
422C3×C4C2×D4×D511yescharacteristic
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—
437C23C3×D4×D511yescharacteristic
438Q8—22no—
439Q8—22no—
440Q8—42no—
441D4—54no—
442D4—102no—
443D4—102no—
444D4—102no—
445D4—104no—
446D4—202no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—52no—
452C2×C4—52no—
453C2×C4—102no—
454C2×C4—102no—
455C2×C4—208no—
456C2×C4C3×#70912yes—
457C2×C4C3×D4×D512yes—
458C2×C4C3×D4×D511yescharacteristic
459C2×C4C23×C3×D511yescharacteristic
460C8—22no—
461C8—102no—
462C2×C3—204no—
463C2×C3—402no—
464C2×C3#6510814yesdirect product factor
465C2×C3C2×#403011yescharacteristic
466C2×C3C2×#406011yescharacteristic
467C2×C3C2×#406511yescharacteristic
468C5C2×C3×#46311yescharacteristic, complemented, Sylow
469C22—104no—
470C22—202no—
471C22—208no—
472C22—202no—
473C22—202no—
474C22—202no—
475C22—202no—
476C22—404no—
477C22C3×#403012yes—
478C22C3×#406012yes—
479C22C3×#406512yes—
480C22C2×C3×D4×D511yescharacteristic
481C4—42no—
482C4—42no—
483C4—102no—
484C4—102no—
485C4C3×#403012yes—
486C4C2×C3×#70911yescharacteristic
487C4C2×C3×D4×D511yescharacteristic
488C3C2×#6510811yescharacteristic, direct product factor, Sylow
489C2—204no—
490C2—402no—
491C2C3×#6510814yesdirect product factor
492C2C2×C3×#403011yescharacteristic
493C2C2×C3×#406011yescharacteristic
494C2C2×C3×#406511yescharacteristic
subgroupG/H
centerC23×C3D4×D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC2×C3×C5×#103C2
socleC23×C3×C5C2×D4
prime radicalC22×C3×D40C2
hypercenterC2×C3×#103D5
hypocenterC5C2×C3×#463
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C2×C3×#103

Complements

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

NH ≅ G/Ntype
1. C2×C3×#4093C2semidirect
2. C22×C3×#688C2semidirect
4. C2×C3×#4039C2semidirect
5. C2×C3×C5×#103C2semidirect
6. C2×C3×#3946C2semidirect
7. C2×C3×#3941C2semidirect
8. C3×#65108C2direct (N × H)
9. C2×#65108C3direct (N × H)
17. C3×#4093C22semidirect
19. C2×C3×#688C22semidirect
28. C3×#4039C22semidirect
29. C3×C5×#103C22semidirect
30. C3×#3946C22semidirect
31. C3×#3941C22semidirect
33. C2×#4093C2×C3semidirect
34. C22×#688C2×C3semidirect
36. C2×#4039C2×C3semidirect
37. C2×C5×#103C2×C3semidirect
38. C2×#3946C2×C3semidirect
39. C2×#3941C2×C3semidirect
40. #65108C2×C3direct (N × H)
81. C2×C3×#103D5semidirect
90. #4093C22×C3semidirect
92. C2×#688C22×C3semidirect
101. #4039C22×C3semidirect
102. C5×#103C22×C3semidirect
103. #3946C22×C3semidirect
104. #3941C22×C3semidirect
159. C3×#103C2×D5semidirect
201. C2×#103C3×D5semidirect
309. #103C2×C3×D5semidirect
311. C2×C3×C5#463semidirect
403. C3×C5C2×#463semidirect
423. C2×C5C3×#463semidirect
464. C2×C3#65108direct (N × H)
468. C5C2×C3×#463semidirect
488. C3C2×#65108direct (N × H)
491. C2C3×#65108direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#40932
C22×C3×#6882
C22×C3×D402
C2×C3×#40392
C2×C3×C5×#1032
C2×C3×#39462
C2×C3×#39412
C3×#651082
C2×#651083
C2×C3×#4635