Finite Group Databasebeta

← C3×#10601184 #14302571

Subgroups — C3×#10601184 #14302571

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×#66156C211yesmaximal, characteristic
2C3×C4×#709C211yesmaximal, characteristic, complemented
3C2×C3×C5×#104C211yesmaximal, characteristic, complemented
4C3×#65529C211yesmaximal, characteristic, complemented
5C3×#65428C211yesmaximal, characteristic, complemented
6C3×#64776C211yesmaximal, characteristic, complemented
7C3×#64743C211yesmaximal, characteristic, complemented
8#10601184C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C2×C3×#709C2211yescharacteristic
11C2×C3×C5×M16C2211yescharacteristic
12C2×C3×C42×C5C2211yescharacteristic
13C3×#4097—21no—
14C3×#4096—22no—
15C3×#4095—21no—
16C3×#4088C412yescomplemented
17C3×#4078C412yescomplemented
18C3×C4×C22⋊D5—22no—
19C3×#4072C2211yescharacteristic
20C3×#4072C2211yescharacteristic
21C2×C3×#681C2211yescharacteristic
22C3×C4×D20—21no—
23C3×#4018—21no—
24C3×C42×D5—21no—
25C3×C4×Dic10—21no—
26C3×C5×#104—21no—
27C3×C5×#104—21no—
28C3×#3966—22no—
29C3×#3957—22no—
30C3×#1906—51nomaximal, Hall
31#66156C2×C311yescharacteristic
32C4×#709C2×C311yescharacteristic, complemented
33C2×C5×#104C2×C311yescharacteristic, complemented
34#65529C2×C311yescharacteristic, complemented
35#65428C2×C311yescharacteristic, complemented
36#64776C2×C311yescharacteristic, complemented
37#64743C2×C311yescharacteristic, complemented
38C3×C5×#41—21no—
39C3×C5×#41—21no—
40C3×C5×#41—21no—
41C3×C5×#41—21no—
42C2×C3×C5×Q8C2×C411yescharacteristic, complemented
43C2×C3×C5×D4—21no—
44C2×C3×C5×D4C2×C411yescharacteristic, complemented
45C22×C3×C4×C5—21no—
46C22×C3×C4×C5—21no—
47C22×C3×C4×C5C2311yescharacteristic
48C2×C3×C22⋊D5—21no—
49C3×#709—21no—
50C3×#709—21no—
51C3×#709C2×C412yes—
52C3×#709D412yes—
53C2×C3×D20D411yescharacteristic
54C2×C3×C4×D5—21no—
55C2×C3×C4×D5—22no—
56C2×C3×Dic10D411yescharacteristic
57C3×C5×M16—21no—
58C3×C5×M16—21no—
59C2×C3×C5×C8—21no—
60C3×C42×C5—21no—
61C3×C42×C5D411yescharacteristic
62C3×C42×C5D411yescharacteristic
63C3×C22⋊Dic5—21no—
64C3×C22⋊Dic5D412yes—
65C3×#689—22no—
66C3×#688—22no—
67C3×#687—22no—
68C3×#686—22no—
69C3×#685—22no—
70C3×C4⋊Dic5—21no—
71C3×C4⋊Dic5—21no—
72C3×#683—22no—
73C3×C4×Dic5—22no—
74C3×C4×Dic5—21no—
75C3×C4×Dic5—21no—
76C3×#681—21no—
77C3×#681C2×C412yescomplemented
78C2×C3×#168—21no—
79C3×#575—51no—
80C3×C4×#41—51no—
81C3×#421—51no—
82C2×C3×#104D511yescharacteristic, complemented
83C3×#410—51no—
84C3×#344—51no—
85C3×#339—51no—
86C2×C5×#41C22×C311yescharacteristic
87C2×#709C22×C311yescharacteristic
88C2×C5×M16C22×C311yescharacteristic
89C2×C42×C5C22×C311yescharacteristic
90#4097—21no—
91#4096—22no—
92#4095—21no—
93#4088C3×C412yescomplemented
94#4078C3×C412yescomplemented
95C4×C22⋊D5—22no—
96#4072C22×C311yescharacteristic
97#4072C22×C311yescharacteristic
98C2×#681C22×C311yescharacteristic
99C4×D20—21no—
100#4018—21no—
101C42×D5—21no—
102C4×Dic10—21no—
103C5×#104—21no—
104C5×#104—21no—
105#3966—22no—
106#3957—22no—
107#1906—51noSylow
108C23×C3×C5—21no—
109C23×C3×C5C2×D411yescharacteristic
110C22×C3×D5—21no—
111C3×C5×Q8—21no—
112C3×C5×Q8—21no—
113C3×C5×D4—21no—
114C3×C5×D4—21no—
115C3×C5×D4—41no—
116C3×C5×D4—41no—
117C2×C3×C4×C5—21no—
118C2×C3×C4×C5—21no—
119C2×C3×C4×C5—21no—
120C2×C3×C4×C5—42no—
121C2×C3×C4×C5—41no—
122C2×C3×C4×C5—41no—
123C2×C3×C4×C5C22×C411yescharacteristic
124C2×C3×C4×C5C2×D411yescharacteristic
125C2×C3×C4×C5C2×D411yescharacteristic
126C2×C3×C4×C5C2×D411yescharacteristic
127C2×C3×C4×C5#4111yescharacteristic
128C2×C3×C4×C5#4111yescharacteristic
129C3×C22⋊D5—22no—
130C3×C22⋊D5—22no—
131C2×C3×Dic5—21no—
132C2×C3×Dic5—22no—
133C2×C3×Dic5—41no—
134C3×D20—21no—
135C3×D20C22⋊C412yes—
136C3×C4×D5—22no—
137C3×C4×D5—22no—
138C3×C4×D5—44no—
139C3×Dic10—21no—
140C3×Dic10C22⋊C412yes—
141C3×C5×C8—41no—
142C3×#168—22no—
143C2×C3×#41—51no—
144C2×C3×#41C2×D511yescharacteristic
145C3×#137—52no—
146C3×#137—101no—
147C3×AGL(1,ℤ/8)—52no—
148C3×AGL(1,ℤ/8)—101no—
149C3×#135—102no—
150C2×C3×M16—51no—
151C2×C3×M16C2×D511yescharacteristic
152C3×C4×Q8—101no—
153C3×C4×D4—101no—
154C3×C4×D4—102no—
155C3×#117—51no—
156C3×#117—51no—
157C3×#117—101no—
158C2×C3×C42—101no—
159C2×C3×C42C2×D511yescharacteristic
160C3×#104—21no—
161C3×#104—21no—
162C3×#104—102no—
163C3×#99—102no—
164C5×#41—21no—
165C5×#41—21no—
166C5×#41—21no—
167C5×#41—21no—
168C2×C5×Q8C2×C3×C411yescharacteristic, complemented
169C2×C5×D4—21no—
170C2×C5×D4C2×C3×C411yescharacteristic, complemented
171C22×C4×C5—21no—
172C22×C4×C5—21no—
173C22×C4×C5C23×C311yescharacteristic
174C2×C22⋊D5—21no—
175#709—21no—
176#709—21no—
177#709C2×C3×C412yes—
178#709C3×D412yes—
179C2×D20C3×D411yescharacteristic
180C2×C4×D5—21no—
181C2×C4×D5—22no—
182C2×Dic10C3×D411yescharacteristic
183C5×M16—21no—
184C5×M16—21no—
185C2×C5×C8—21no—
186C42×C5—21no—
187C42×C5C3×D411yescharacteristic
188C42×C5C3×D411yescharacteristic
189C22⋊Dic5—21no—
190C22⋊Dic5C3×D412yes—
191#689—22no—
192#688—22no—
193#687—22no—
194#686—22no—
195#685—22no—
196C4⋊Dic5—21no—
197C4⋊Dic5—21no—
198#683—22no—
199C4×Dic5—22no—
200C4×Dic5—21no—
201C4×Dic5—21no—
202#681—21no—
203#681C2×C3×C412yescomplemented
204C2×#168—21no—
205#575—51no—
206C4×#41—51no—
207#421—51no—
208C2×#104C3×D511yescharacteristic, complemented
209#410—51no—
210#344—51no—
211#339—51no—
212C22×C3×C5—41no—
213C22×C3×C5—41no—
214C22×C3×C5—81no—
215C22×C3×C5C4×D411yescharacteristic
216C22×C3×C5#12011yescharacteristic
217C22×C3×C5#12111yescharacteristic
218C2×C3×D5—22no—
219C2×C3×D5—42no—
220C3×C4×C5—22no—
221C3×C4×C5—41no—
222C3×C4×C5—41no—
223C3×C4×C5C2×C22⋊C411yescharacteristic
224C3×C4×C5C4×D411yescharacteristic
225C3×C4×C5#12111yescharacteristic
226C3×C4×C5#12311yescharacteristic
227C3×Dic5—22no—
228C3×Dic5—42no—
229C3×Dic5—81no—
230C3×#41—21no—
231C3×#41—21no—
232C3×#41—21no—
233C3×#41—21no—
234C3×#41—52no—
235C3×#41—52no—
236C3×#41—101no—
237C3×#41—101no—
238C2×C3×Q8—51no—
239C2×C3×Q8C4×D511yescharacteristic, complemented
240C2×C3×D4—21no—
241C2×C3×D4—51no—
242C2×C3×D4—101no—
243C2×C3×D4C4×D511yescharacteristic, complemented
244C22×C3×C4—21no—
245C22×C3×C4—21no—
246C22×C3×C4—102no—
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—52no—
256C3×M16—101no—
257C2×C3×C8—21no—
258C2×C3×C8—101no—
259C3×C4⋊C4—102no—
260C3×C4⋊C4—101no—
261C3×C4⋊C4—101no—
262C3×C22⋊C4—52no—
263C3×C22⋊C4—102no—
264C3×C22⋊C4—101no—
265C3×C42—21no—
266C3×C42—102no—
267C3×C42—101no—
268C3×C42—101no—
269C3×C42C22⋊D511yescharacteristic
270C3×C42C22⋊D511yescharacteristic
271C23×C5—21no—
272C23×C5C2×C3×D411yescharacteristic
273C22×D5—21no—
274C5×Q8—21no—
275C5×Q8—21no—
276C5×D4—21no—
277C5×D4—21no—
278C5×D4—41no—
279C5×D4—41no—
280C2×C4×C5—21no—
281C2×C4×C5—21no—
282C2×C4×C5—21no—
283C2×C4×C5—42no—
284C2×C4×C5—41no—
285C2×C4×C5—41no—
286C2×C4×C5C22×C3×C411yescharacteristic
287C2×C4×C5C2×C3×D411yescharacteristic
288C2×C4×C5C2×C3×D411yescharacteristic
289C2×C4×C5C2×C3×D411yescharacteristic
290C2×C4×C5C3×#4111yescharacteristic
291C2×C4×C5C3×#4111yescharacteristic
292C22⋊D5—22no—
293C22⋊D5—22no—
294C2×Dic5—21no—
295C2×Dic5—22no—
296C2×Dic5—41no—
297D20—21no—
298D20C3×C22⋊C412yes—
299C4×D5—22no—
300C4×D5—22no—
301C4×D5—44no—
302Dic10—21no—
303Dic10C3×C22⋊C412yes—
304C5×C8—41no—
305#168—22no—
306C2×#41—51no—
307C2×#41C2×C3×D511yescharacteristic
308#137—52no—
309#137—101no—
310AGL(1,ℤ/8)—52no—
311AGL(1,ℤ/8)—101no—
312#135—102no—
313C2×M16—51no—
314C2×M16C2×C3×D511yescharacteristic
315C4×Q8—101no—
316C4×D4—101no—
317C4×D4—102no—
318#117—51no—
319#117—51no—
320#117—101no—
321C2×C42—101no—
322C2×C42C2×C3×D511yescharacteristic
323#104—21no—
324#104—21no—
325#104—102no—
326#99—102no—
327C2×C3×C5—21no—
328C2×C3×C5—21no—
329C2×C3×C5—21no—
330C2×C3×C5—81no—
331C2×C3×C5#38611yescharacteristic
332C3×D5—42no—
333C23×C3—21no—
334C23×C3—101no—
335C23×C3D4×D511yescharacteristic
336C3×Q8—21no—
337C3×Q8—21no—
338C3×Q8—52no—
339C3×Q8—101no—
340C3×D4—21no—
341C3×D4—21no—
342C3×D4—41no—
343C3×D4—41no—
344C3×D4—52no—
345C3×D4—102no—
346C3×D4—102no—
347C3×D4—101no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—42no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—101no—
355C2×C3×C4—102no—
356C2×C3×C4—102no—
357C2×C3×C4—102no—
358C2×C3×C4—204no—
359C2×C3×C4—201no—
360C2×C3×C4C2×C4×D511yescharacteristic
361C2×C3×C4#70911yescharacteristic
362C2×C3×C4D4×D511yescharacteristic
363C2×C3×C4D4×D511yescharacteristic
364C2×C3×C4#71111yescharacteristic
365C2×C3×C4C2×C22⋊D511yescharacteristic
366C3×C8—41no—
367C3×C8—102no—
368C22×C5—41no—
369C22×C5—41no—
370C22×C5—81no—
371C22×C5C3×C4×D411yescharacteristic
372C22×C5C3×#12011yescharacteristic
373C22×C5C3×#12111yescharacteristic
374C2×D5—22no—
375C2×D5—42no—
376C4×C5—22no—
377C4×C5—41no—
378C4×C5—41no—
379C4×C5C2×C3×C22⋊C411yescharacteristic
380C4×C5C3×C4×D411yescharacteristic
381C4×C5C3×#12111yescharacteristic
382C4×C5C3×#12311yescharacteristic
383Dic5—22no—
384Dic5—42no—
385Dic5—81no—
386#41—21no—
387#41—21no—
388#41—21no—
389#41—21no—
390#41—52no—
391#41—52no—
392#41—101no—
393#41—101no—
394C2×Q8—51no—
395C2×Q8C3×C4×D511yescharacteristic, complemented
396C2×D4—21no—
397C2×D4—51no—
398C2×D4—101no—
399C2×D4C3×C4×D511yescharacteristic, complemented
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—102no—
403C22×C4—101no—
404C22×C4C22×C3×D511yescharacteristic
405Q16—102no—
406SD16—102no—
407SD16—102no—
408D8—102no—
409M16—21no—
410M16—21no—
411M16—52no—
412M16—101no—
413C2×C8—21no—
414C2×C8—101no—
415C4⋊C4—102no—
416C4⋊C4—101no—
417C4⋊C4—101no—
418C22⋊C4—52no—
419C22⋊C4—102no—
420C22⋊C4—101no—
421C42—21no—
422C42—102no—
423C42—101no—
424C42—101no—
425C42C3×C22⋊D511yescharacteristic
426C42C3×C22⋊D511yescharacteristic
427C3×C5#190611yescharacteristic, complemented, Hall
428C22×C3—41no—
429C22×C3—41no—
430C22×C3—81no—
431C22×C3—102no—
432C22×C3—202no—
433C22×C3#402711yescharacteristic
434C22×C3#403011yescharacteristic
435C22×C3#408311yescharacteristic
436C3×C4—22no—
437C3×C4—41no—
438C3×C4—41no—
439C3×C4—102no—
440C3×C4—202no—
441C3×C4—401no—
442C3×C4D5×C22⋊C411yescharacteristic
443C3×C4#402911yescharacteristic
444C3×C4C4×C22⋊D511yescharacteristic
445C3×C4#408411yescharacteristic
446C2×C5—21no—
447C2×C5—21no—
448C2×C5—21no—
449C2×C5—81no—
450C2×C5C3×#38611yescharacteristic
451D5—42no—
452C23—21no—
453C23—101no—
454C23C3×D4×D511yescharacteristic
455Q8—21no—
456Q8—21no—
457Q8—52no—
458Q8—101no—
459D4—21no—
460D4—21no—
461D4—41no—
462D4—41no—
463D4—52no—
464D4—102no—
465D4—102no—
466D4—101no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—101no—
474C2×C4—102no—
475C2×C4—102no—
476C2×C4—102no—
477C2×C4—204no—
478C2×C4—201no—
479C2×C4C2×C3×C4×D511yescharacteristic
480C2×C4C3×#70911yescharacteristic
481C2×C4C3×D4×D511yescharacteristic
482C2×C4C3×D4×D511yescharacteristic
483C2×C4C3×#71111yescharacteristic
484C2×C4C2×C3×C22⋊D511yescharacteristic
485C8—41no—
486C8—102no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—81no—
491C2×C3—202no—
492C2×C3#6524511yescharacteristic
493C5C3×#190611yescharacteristic, complemented, Sylow
494C22—41no—
495C22—41no—
496C22—81no—
497C22—102no—
498C22—202no—
499C22C3×#402711yescharacteristic
500C22C3×#403011yescharacteristic
501C22C3×#408311yescharacteristic
502C4—22no—
503C4—41no—
504C4—41no—
505C4—102no—
506C4—202no—
507C4—401no—
508C4C3×D5×C22⋊C411yescharacteristic
509C4C3×#402911yescharacteristic
510C4C3×C4×C22⋊D511yescharacteristic
511C4C3×#408411yescharacteristic
512C3#1060118411yescharacteristic, direct product factor, Sylow
513C2—21no—
514C2—21no—
515C2—21no—
516C2—81no—
517C2—202no—
518C2C3×#6524511yescharacteristic
subgroupG/H
centerC3×C4#4084
commutatorC2×C4×C5C22×C3×C4
FrattiniC22×C4C22×C3×D5
FittingC2×C3×C5×#104C2
socleC2×C3×C5#386
prime radicalC3×#66156C2
cosocleC22×C4×C5C23×C3
hypercenterC2×C3×#104D5
hypocenterC5C3×#1906
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C2×C3×#104

Complements

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

NH ≅ G/Ntype
2. C3×C4×#709C2semidirect
3. C2×C3×C5×#104C2semidirect
4. C3×#65529C2semidirect
5. C3×#65428C2semidirect
6. C3×#64776C2semidirect
7. C3×#64743C2semidirect
8. #10601184C3direct (N × H)
16. C3×#4088C4semidirect
17. C3×#4078C4semidirect
32. C4×#709C2×C3semidirect
33. C2×C5×#104C2×C3semidirect
34. #65529C2×C3semidirect
35. #65428C2×C3semidirect
36. #64776C2×C3semidirect
37. #64743C2×C3semidirect
42. C2×C3×C5×Q8C2×C4semidirect
44. C2×C3×C5×D4C2×C4semidirect
77. C3×#681C2×C4semidirect
82. C2×C3×#104D5semidirect
93. #4088C3×C4semidirect
94. #4078C3×C4semidirect
168. C2×C5×Q8C2×C3×C4semidirect
170. C2×C5×D4C2×C3×C4semidirect
203. #681C2×C3×C4semidirect
208. C2×#104C3×D5semidirect
239. C2×C3×Q8C4×D5semidirect
243. C2×C3×D4C4×D5semidirect
395. C2×Q8C3×C4×D5semidirect
399. C2×D4C3×C4×D5semidirect
427. C3×C5#1906semidirect
493. C5C3×#1906semidirect
512. C3#10601184direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#661562
C3×C4×#7092
C2×C3×C5×#1042
C3×#655292
C3×#654282
C3×#647762
C3×#647432
#106011843
C3×#19065