Finite Group Databasebeta

← C3×#10599832 #14302565

Subgroups — C3×#10599832 #14302565

Order 1920.

664 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#66082C211yesmaximal, characteristic
2C3×#65856C211yesmaximal, characteristic, complemented
3C3×#65428C211yesmaximal, characteristic, complemented
4C2×C3×#3957C211yesmaximal, characteristic, complemented
5C3×#65295C211yesmaximal, characteristic, complemented
6C3×C5×#344C211yesmaximal, characteristic, complemented
7C3×#64775C211yesmaximal, characteristic, complemented
8#10599832C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×#709C2211yescharacteristic
10C2×C3×#709C2211yescharacteristic
11C2×C3×C5×M16C2211yescharacteristic
12C2×C3×C5×M16C2211yescharacteristic
13C3×C5×#117C2211yescharacteristic
14C3×#4072C2211yescharacteristic
15C2×C3×C4×Dic5C2211yescharacteristic
16C3×#4055—21no—
17C3×#4055C412yescomplemented
18C3×#4054—21no—
19C3×#4054C412yescomplemented
20C3×#4050—22no—
21C3×#4039—21no—
22C3×#4033—21no—
23C3×#4027—22no—
24C3×#4018—21no—
25C3×C42×D5—21no—
26C3×#3957—22no—
27C3×#3957—21no—
28C3×#3957—21no—
29C3×#3938—22no—
30C3×#1906—51nomaximal, Hall
31#66082C2×C311yescharacteristic
32#65856C2×C311yescharacteristic, complemented
33#65428C2×C311yescharacteristic, complemented
34C2×#3957C2×C311yescharacteristic, complemented
35#65295C2×C311yescharacteristic, complemented
36C5×#344C2×C311yescharacteristic, complemented
37#64775C2×C311yescharacteristic, complemented
38C22×C3×C4×C5C2311yescharacteristic
39C2×C3×C22⋊D5—21no—
40C2×C3×C22⋊D5—21no—
41C22×C3×Dic5—21no—
42C3×#709—21no—
43C3×#709—21no—
44C3×#709—21no—
45C3×#709—21no—
46C3×#709—21no—
47C3×#709—21no—
48C3×#709C2×C412yes—
49C3×#709D412yes—
50C2×C3×D20C2×C411yescharacteristic, complemented
51C2×C3×D20D411yescharacteristic
52C2×C3×C4×D5—21no—
53C2×C3×C4×D5—22no—
54C2×C3×C4×D5—21no—
55C2×C3×Dic10C2×C411yescharacteristic, complemented
56C2×C3×Dic10D411yescharacteristic
57C3×C5×M16—21no—
58C3×C5×M16—21no—
59C3×C5×M16—21no—
60C3×C5×M16C2×C412yescomplemented
61C2×C3×C5×C8—21no—
62C2×C3×C5×C8—21no—
63C3×C5×C4⋊C4—21no—
64C3×C5×C22⋊C4D412yes—
65C3×C42×C5—21no—
66C3×C22⋊Dic5—21no—
67C3×#685—22no—
68C3×C4⋊Dic5—21no—
69C3×#683—22no—
70C3×C4×Dic5—22no—
71C3×C4×Dic5—21no—
72C3×C4×Dic5—21no—
73C3×C4×Dic5D411yescharacteristic
74C3×C4×Dic5D411yescharacteristic
75C3×Dic20—22no—
76C3×D40—22no—
77C3×#677—22no—
78C3×#677—22no—
79C3×#575—51no—
80C3×C4×#41—51no—
81C3×#421—51no—
82C2×C3×#104—51no—
83C3×#410—51no—
84C3×#344D511yescharacteristic, complemented
85C3×#339—51no—
86C2×#709C22×C311yescharacteristic
87C2×#709C22×C311yescharacteristic
88C2×C5×M16C22×C311yescharacteristic
89C2×C5×M16C22×C311yescharacteristic
90C5×#117C22×C311yescharacteristic
91#4072C22×C311yescharacteristic
92C2×C4×Dic5C22×C311yescharacteristic
93#4055—21no—
94#4055C3×C412yescomplemented
95#4054—21no—
96#4054C3×C412yescomplemented
97#4050—22no—
98#4039—21no—
99#4033—21no—
100#4027—22no—
101#4018—21no—
102C42×D5—21no—
103#3957—22no—
104#3957—21no—
105#3957—21no—
106#3938—22no—
107#1906—51noSylow
108C23×C3×C5C2×D411yescharacteristic
109C22×C3×D5—21no—
110C22×C3×D5—21no—
111C2×C3×C4×C5—22no—
112C2×C3×C4×C5C22×C411yescharacteristic
113C2×C3×C4×C5C2×D411yescharacteristic
114C2×C3×C4×C5C2×D411yescharacteristic
115C2×C3×C4×C5C2×D411yescharacteristic
116C2×C3×C4×C5#4111yescharacteristic
117C2×C3×C4×C5#4111yescharacteristic
118C3×C22⋊D5—22no—
119C3×C22⋊D5—22no—
120C3×C22⋊D5—41no—
121C3×C22⋊D5—41no—
122C2×C3×Dic5—21no—
123C2×C3×Dic5—21no—
124C2×C3×Dic5—21no—
125C2×C3×Dic5—21no—
126C2×C3×Dic5—42no—
127C2×C3×Dic5—41no—
128C3×D20—21no—
129C3×D20—21no—
130C3×D20—21no—
131C3×D20C22⋊C412yes—
132C3×C4×D5—22no—
133C3×C4×D5—22no—
134C3×C4×D5—44no—
135C3×C4×D5—41no—
136C3×C4×D5—41no—
137C3×Dic10—21no—
138C3×Dic10—21no—
139C3×Dic10—21no—
140C3×Dic10C22⋊C412yes—
141C3×C5×C8—22no—
142C3×C5×C8—41no—
143C2×C3×#41—51no—
144C2×C3×#41—51no—
145C3×#137—52no—
146C3×#137—101no—
147C3×AGL(1,ℤ/8)—52no—
148C3×AGL(1,ℤ/8)—101no—
149C3×#135—102no—
150C2×C3×M16C2×D511yescharacteristic
151C2×C3×M16C2×D511yescharacteristic
152C3×C4×Q8—101no—
153C3×C4×D4—101no—
154C3×C4×D4—102no—
155C3×#117—51no—
156C3×#117—101no—
157C3×#117C2×D511yescharacteristic
158C2×C3×C42—51no—
159C2×C3×C42—101no—
160C3×#104—101no—
161C3×#104—101no—
162C3×#104—102no—
163C3×#99—102no—
164C22×C4×C5C23×C311yescharacteristic
165C2×C22⋊D5—21no—
166C2×C22⋊D5—21no—
167C22×Dic5—21no—
168#709—21no—
169#709—21no—
170#709—21no—
171#709—21no—
172#709—21no—
173#709—21no—
174#709C2×C3×C412yes—
175#709C3×D412yes—
176C2×D20C2×C3×C411yescharacteristic, complemented
177C2×D20C3×D411yescharacteristic
178C2×C4×D5—21no—
179C2×C4×D5—22no—
180C2×C4×D5—21no—
181C2×Dic10C2×C3×C411yescharacteristic, complemented
182C2×Dic10C3×D411yescharacteristic
183C5×M16—21no—
184C5×M16—21no—
185C5×M16—21no—
186C5×M16C2×C3×C412yescomplemented
187C2×C5×C8—21no—
188C2×C5×C8—21no—
189C5×C4⋊C4—21no—
190C5×C22⋊C4C3×D412yes—
191C42×C5—21no—
192C22⋊Dic5—21no—
193#685—22no—
194C4⋊Dic5—21no—
195#683—22no—
196C4×Dic5—22no—
197C4×Dic5—21no—
198C4×Dic5—21no—
199C4×Dic5C3×D411yescharacteristic
200C4×Dic5C3×D411yescharacteristic
201Dic20—22no—
202D40—22no—
203#677—22no—
204#677—22no—
205#575—51no—
206C4×#41—51no—
207#421—51no—
208C2×#104—51no—
209#410—51no—
210#344C3×D511yescharacteristic, complemented
211#339—51no—
212C22×C3×C5—41no—
213C22×C3×C5C4×D411yescharacteristic
214C22×C3×C5#12011yescharacteristic
215C22×C3×C5#12111yescharacteristic
216C2×C3×D5—22no—
217C2×C3×D5—42no—
218C2×C3×D5—41no—
219C2×C3×D5—81no—
220C3×C4×C5—42no—
221C3×C4×C5C2×C22⋊C411yescharacteristic
222C3×C4×C5C4×D411yescharacteristic
223C3×C4×C5#12111yescharacteristic
224C3×C4×C5#12311yescharacteristic
225C3×Dic5—22no—
226C3×Dic5—22no—
227C3×Dic5—41no—
228C3×Dic5—41no—
229C3×Dic5—81no—
230C3×#41—52no—
231C3×#41—52no—
232C3×#41—101no—
233C3×#41—101no—
234C3×#41—101no—
235C3×#41—101no—
236C3×#41—101no—
237C3×#41—101no—
238C2×C3×Q8—51no—
239C2×C3×Q8—51no—
240C2×C3×D4—51no—
241C2×C3×D4—51no—
242C2×C3×D4—101no—
243C2×C3×D4—101no—
244C22×C3×C4—102no—
245C22×C3×C4—101no—
246C22×C3×C4—101no—
247C22×C3×C4—101no—
248C22×C3×C4C22×D511yescharacteristic
249C3×Q16—102no—
250C3×SD16—102no—
251C3×SD16—102no—
252C3×D8—102no—
253C3×M16—21no—
254C3×M16—21no—
255C3×M16—21no—
256C3×M16C4×D512yescomplemented
257C2×C3×C8—21no—
258C2×C3×C8—21no—
259C3×C4⋊C4—21no—
260C3×C4⋊C4—102no—
261C3×C4⋊C4—101no—
262C3×C22⋊C4—102no—
263C3×C22⋊C4—101no—
264C3×C22⋊C4D2012yes—
265C3×C42—21no—
266C3×C42—51no—
267C3×C42—51no—
268C3×C42—102no—
269C3×C42—101no—
270C3×C42—101no—
271C23×C5C2×C3×D411yescharacteristic
272C22×D5—21no—
273C22×D5—21no—
274C2×C4×C5—22no—
275C2×C4×C5C22×C3×C411yescharacteristic
276C2×C4×C5C2×C3×D411yescharacteristic
277C2×C4×C5C2×C3×D411yescharacteristic
278C2×C4×C5C2×C3×D411yescharacteristic
279C2×C4×C5C3×#4111yescharacteristic
280C2×C4×C5C3×#4111yescharacteristic
281C22⋊D5—22no—
282C22⋊D5—22no—
283C22⋊D5—41no—
284C22⋊D5—41no—
285C2×Dic5—21no—
286C2×Dic5—21no—
287C2×Dic5—21no—
288C2×Dic5—21no—
289C2×Dic5—42no—
290C2×Dic5—41no—
291D20—21no—
292D20—21no—
293D20—21no—
294D20C3×C22⋊C412yes—
295C4×D5—22no—
296C4×D5—22no—
297C4×D5—44no—
298C4×D5—41no—
299C4×D5—41no—
300Dic10—21no—
301Dic10—21no—
302Dic10—21no—
303Dic10C3×C22⋊C412yes—
304C5×C8—22no—
305C5×C8—41no—
306C2×#41—51no—
307C2×#41—51no—
308#137—52no—
309#137—101no—
310AGL(1,ℤ/8)—52no—
311AGL(1,ℤ/8)—101no—
312#135—102no—
313C2×M16C2×C3×D511yescharacteristic
314C2×M16C2×C3×D511yescharacteristic
315C4×Q8—101no—
316C4×D4—101no—
317C4×D4—102no—
318#117—51no—
319#117—101no—
320#117C2×C3×D511yescharacteristic
321C2×C42—51no—
322C2×C42—101no—
323#104—101no—
324#104—101no—
325#104—102no—
326#99—102no—
327C2×C3×C5—21no—
328C2×C3×C5—21no—
329C2×C3×C5—21no—
330C2×C3×C5#38611yescharacteristic
331C3×D5—42no—
332C3×D5—81no—
333C23×C3—101no—
334C23×C3—101no—
335C23×C3C2×D2011yescharacteristic
336C3×Q8—52no—
337C3×Q8—101no—
338C3×Q8—101no—
339C3×Q8—101no—
340C3×D4—52no—
341C3×D4—102no—
342C3×D4—102no—
343C3×D4—101no—
344C3×D4—101no—
345C3×D4—101no—
346C3×D4—201no—
347C3×D4—201no—
348C2×C3×C4—22no—
349C2×C3×C4—101no—
350C2×C3×C4—101no—
351C2×C3×C4—102no—
352C2×C3×C4—102no—
353C2×C3×C4—101no—
354C2×C3×C4—101no—
355C2×C3×C4—204no—
356C2×C3×C4—202no—
357C2×C3×C4—201no—
358C2×C3×C4—201no—
359C2×C3×C4—201no—
360C2×C3×C4C2×C4×D511yescharacteristic
361C2×C3×C4#70911yescharacteristic
362C2×C3×C4D4×D511yescharacteristic
363C2×C3×C4D4×D511yescharacteristic
364C2×C3×C4D4×D511yescharacteristic
365C2×C3×C4#71311yescharacteristic
366C3×C8—22no—
367C3×C8—41no—
368C22×C5—41no—
369C22×C5C3×C4×D411yescharacteristic
370C22×C5C3×#12011yescharacteristic
371C22×C5C3×#12111yescharacteristic
372C2×D5—22no—
373C2×D5—42no—
374C2×D5—41no—
375C2×D5—81no—
376C4×C5—42no—
377C4×C5C2×C3×C22⋊C411yescharacteristic
378C4×C5C3×C4×D411yescharacteristic
379C4×C5C3×#12111yescharacteristic
380C4×C5C3×#12311yescharacteristic
381Dic5—22no—
382Dic5—22no—
383Dic5—41no—
384Dic5—41no—
385Dic5—81no—
386#41—52no—
387#41—52no—
388#41—101no—
389#41—101no—
390#41—101no—
391#41—101no—
392#41—101no—
393#41—101no—
394C2×Q8—51no—
395C2×Q8—51no—
396C2×D4—51no—
397C2×D4—51no—
398C2×D4—101no—
399C2×D4—101no—
400C22×C4—102no—
401C22×C4—101no—
402C22×C4—101no—
403C22×C4—101no—
404C22×C4C22×C3×D511yescharacteristic
405Q16—102no—
406SD16—102no—
407SD16—102no—
408D8—102no—
409M16—21no—
410M16—21no—
411M16—21no—
412M16C3×C4×D512yescomplemented
413C2×C8—21no—
414C2×C8—21no—
415C4⋊C4—21no—
416C4⋊C4—102no—
417C4⋊C4—101no—
418C22⋊C4—102no—
419C22⋊C4—101no—
420C22⋊C4C3×D2012yes—
421C42—21no—
422C42—51no—
423C42—51no—
424C42—102no—
425C42—101no—
426C42—101no—
427C3×C5#190611yescharacteristic, complemented, Hall
428C22×C3—41no—
429C22×C3—102no—
430C22×C3—202no—
431C22×C3—201no—
432C22×C3—401no—
433C22×C3C4×D2011yescharacteristic
434C22×C3#402811yescharacteristic
435C22×C3#404111yescharacteristic
436C3×C4—42no—
437C3×C4—102no—
438C3×C4—102no—
439C3×C4—201no—
440C3×C4—201no—
441C3×C4—401no—
442C3×C4D5×C22⋊C411yescharacteristic
443C3×C4#403011yescharacteristic
444C3×C4#403911yescharacteristic
445C3×C4#404011yescharacteristic
446C2×C5—21no—
447C2×C5—21no—
448C2×C5—21no—
449C2×C5C3×#38611yescharacteristic
450D5—42no—
451D5—81no—
452C23—101no—
453C23—101no—
454C23C2×C3×D2011yescharacteristic
455Q8—52no—
456Q8—101no—
457Q8—101no—
458Q8—101no—
459D4—52no—
460D4—102no—
461D4—102no—
462D4—101no—
463D4—101no—
464D4—101no—
465D4—201no—
466D4—201no—
467C2×C4—22no—
468C2×C4—101no—
469C2×C4—101no—
470C2×C4—102no—
471C2×C4—102no—
472C2×C4—101no—
473C2×C4—101no—
474C2×C4—204no—
475C2×C4—202no—
476C2×C4—201no—
477C2×C4—201no—
478C2×C4—201no—
479C2×C4C2×C3×C4×D511yescharacteristic
480C2×C4C3×#70911yescharacteristic
481C2×C4C3×D4×D511yescharacteristic
482C2×C4C3×D4×D511yescharacteristic
483C2×C4C3×D4×D511yescharacteristic
484C2×C4C3×#71311yescharacteristic
485C8—22no—
486C8—41no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—202no—
491C2×C3—401no—
492C2×C3#6495411yescharacteristic
493C5C3×#190611yescharacteristic, complemented, Sylow
494C22—41no—
495C22—102no—
496C22—202no—
497C22—201no—
498C22—401no—
499C22C3×C4×D2011yescharacteristic
500C22C3×#402811yescharacteristic
501C22C3×#404111yescharacteristic
502C4—42no—
503C4—102no—
504C4—102no—
505C4—201no—
506C4—201no—
507C4—401no—
508C4C3×D5×C22⋊C411yescharacteristic
509C4C3×#403011yescharacteristic
510C4C3×#403911yescharacteristic
511C4C3×#404011yescharacteristic
512C3#1059983211yescharacteristic, direct product factor, Sylow
513C2—21no—
514C2—21no—
515C2—21no—
516C2—202no—
517C2—401no—
518C2C3×#6495411yescharacteristic
subgroupG/H
centerC3×C4#4030
commutatorC2×C4×C5C22×C3×C4
FrattiniC22×C4C22×C3×D5
FittingC3×C5×#344C2
socleC2×C3×C5#386
prime radicalC3×#66082C2
cosocleC22×C4×C5C23×C3
hypercenterC3×#344D5
hypocenterC5C3×#1906
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C3×#344

Complements

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

NH ≅ G/Ntype
2. C3×#65856C2semidirect
3. C3×#65428C2semidirect
4. C2×C3×#3957C2semidirect
5. C3×#65295C2semidirect
6. C3×C5×#344C2semidirect
7. C3×#64775C2semidirect
8. #10599832C3direct (N × H)
17. C3×#4055C4semidirect
19. C3×#4054C4semidirect
32. #65856C2×C3semidirect
33. #65428C2×C3semidirect
34. C2×#3957C2×C3semidirect
35. #65295C2×C3semidirect
36. C5×#344C2×C3semidirect
37. #64775C2×C3semidirect
50. C2×C3×D20C2×C4semidirect
55. C2×C3×Dic10C2×C4semidirect
60. C3×C5×M16C2×C4semidirect
84. C3×#344D5semidirect
94. #4055C3×C4semidirect
96. #4054C3×C4semidirect
176. C2×D20C2×C3×C4semidirect
181. C2×Dic10C2×C3×C4semidirect
186. C5×M16C2×C3×C4semidirect
210. #344C3×D5semidirect
256. C3×M16C4×D5semidirect
412. M16C3×C4×D5semidirect
427. C3×C5#1906semidirect
493. C5C3×#1906semidirect
512. C3#10599832direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#660822
C3×#658562
C3×#654282
C2×C3×#39572
C3×#652952
C3×C5×#3442
C3×#647752
#105998323
C3×#19065