Finite Group Databasebeta

← C3×#10601188 #14302556

Subgroups — C3×#10601188 #14302556

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×#41×Dic5C211yesmaximal, characteristic, complemented
2C3×#66156C211yesmaximal, characteristic
3C3×C5×#421C211yesmaximal, characteristic, complemented
4C2×C3×#3957C211yesmaximal, characteristic, complemented
5C3×#65295C211yesmaximal, characteristic, complemented
6C3×#64776C211yesmaximal, characteristic, complemented
7C3×#64752C211yesmaximal, characteristic, complemented
8#10601188C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C2×C3×#709C2211yescharacteristic
11C2×C3×C5×M16C2211yescharacteristic
12C3×C5×#117C2211yescharacteristic
13C3×#4097C412yescomplemented
14C3×#4096—22no—
15C3×#4095C412yescomplemented
16C3×Q8×Dic5—21no—
17C3×#4088—21no—
18C3×D4×Dic5—21no—
19C3×D4×Dic5—22no—
20C3×#4078—21no—
21C3×#4072—21no—
22C3×#4072C2211yescharacteristic
23C2×C3×C4×Dic5—21no—
24C2×C3×C4×Dic5C2211yescharacteristic
25C2×C3×#681C2211yescharacteristic
26C3×C5×#104—22no—
27C3×#3957—21no—
28C3×#3957—21no—
29C3×#3938—22no—
30C3×#1906—51nomaximal, Hall
31#41×Dic5C2×C311yescharacteristic, complemented
32#66156C2×C311yescharacteristic
33C5×#421C2×C311yescharacteristic, complemented
34C2×#3957C2×C311yescharacteristic, complemented
35#65295C2×C311yescharacteristic, complemented
36#64776C2×C311yescharacteristic, complemented
37#64752C2×C311yescharacteristic, complemented
38C3×C5×#41—21no—
39C3×C5×#41—21no—
40C3×C5×#41C2×C412yes—
41C3×C5×#41D412yes—
42C2×C3×C5×Q8D411yescharacteristic
43C2×C3×C5×D4—21no—
44C2×C3×C5×D4D411yescharacteristic
45C22×C3×C4×C5—21no—
46C22×C3×C4×C5C2311yescharacteristic
47C2×C3×C22⋊D5—21no—
48C22×C3×Dic5—22no—
49C22×C3×Dic5—21no—
50C3×#709—21no—
51C3×#709—21no—
52C3×#709—21no—
53C3×#709—21no—
54C2×C3×D20C2×C411yescharacteristic, complemented
55C2×C3×C4×D5—21no—
56C2×C3×Dic10C2×C411yescharacteristic, complemented
57C3×C5×M16—21no—
58C3×C5×M16—21no—
59C2×C3×C5×C8—21no—
60C3×C5×C4⋊C4—21no—
61C3×C5×C22⋊C4—21no—
62C3×C42×C5—21no—
63C3×C22⋊Dic5—22no—
64C3×C22⋊Dic5D412yes—
65C3×#689—22no—
66C3×#688—22no—
67C3×#687—22no—
68C3×#686—22no—
69C3×C4⋊Dic5—22no—
70C3×C4⋊Dic5—21no—
71C3×C4×Dic5—22no—
72C3×C4×Dic5—21no—
73C3×C4×Dic5—21no—
74C3×C4×Dic5D411yescharacteristic
75C3×C4×Dic5D411yescharacteristic
76C3×#681—21no—
77C3×#681C2×C412yescomplemented
78C2×C3×#168—21no—
79C3×#575—51no—
80C3×C4×#41—51no—
81C3×#421D511yescharacteristic, complemented
82C2×C3×#104—51no—
83C3×#410—51no—
84C3×#344—51no—
85C3×#339—51no—
86C2×C5×#41C22×C311yescharacteristic
87C2×#709C22×C311yescharacteristic
88C2×C5×M16C22×C311yescharacteristic
89C5×#117C22×C311yescharacteristic
90#4097C3×C412yescomplemented
91#4096—22no—
92#4095C3×C412yescomplemented
93Q8×Dic5—21no—
94#4088—21no—
95D4×Dic5—21no—
96D4×Dic5—22no—
97#4078—21no—
98#4072—21no—
99#4072C22×C311yescharacteristic
100C2×C4×Dic5—21no—
101C2×C4×Dic5C22×C311yescharacteristic
102C2×#681C22×C311yescharacteristic
103C5×#104—22no—
104#3957—21no—
105#3957—21no—
106#3938—22no—
107#1906—51noSylow
108C23×C3×C5—21no—
109C23×C3×C5C2×D411yescharacteristic
110C22×C3×D5—21no—
111C3×C5×Q8—21no—
112C3×C5×Q8C22⋊C412yes—
113C3×C5×D4—22no—
114C3×C5×D4—22no—
115C3×C5×D4—21no—
116C3×C5×D4C22⋊C412yes—
117C2×C3×C4×C5—21no—
118C2×C3×C4×C5—22no—
119C2×C3×C4×C5—22no—
120C2×C3×C4×C5—41no—
121C2×C3×C4×C5C22×C411yescharacteristic
122C2×C3×C4×C5C2×D411yescharacteristic
123C2×C3×C4×C5C2×D411yescharacteristic
124C2×C3×C4×C5C2×D411yescharacteristic
125C2×C3×C4×C5#4111yescharacteristic
126C2×C3×C4×C5#4111yescharacteristic
127C3×C22⋊D5—41no—
128C3×C22⋊D5—41no—
129C2×C3×Dic5—21no—
130C2×C3×Dic5—21no—
131C2×C3×Dic5—22no—
132C2×C3×Dic5—21no—
133C2×C3×Dic5—42no—
134C2×C3×Dic5—44no—
135C3×D20—21no—
136C3×D20—21no—
137C3×C4×D5—41no—
138C3×C4×D5—41no—
139C3×Dic10—21no—
140C3×Dic10—21no—
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—102no—
154C3×C4×D4—101no—
155C3×#117—51no—
156C3×#117—101no—
157C3×#117C2×D511yescharacteristic
158C2×C3×C42—51no—
159C2×C3×C42—101no—
160C3×#104—22no—
161C3×#104—101no—
162C3×#104—101no—
163C3×#99—102no—
164C5×#41—21no—
165C5×#41—21no—
166C5×#41C2×C3×C412yes—
167C5×#41C3×D412yes—
168C2×C5×Q8C3×D411yescharacteristic
169C2×C5×D4—21no—
170C2×C5×D4C3×D411yescharacteristic
171C22×C4×C5—21no—
172C22×C4×C5C23×C311yescharacteristic
173C2×C22⋊D5—21no—
174C22×Dic5—22no—
175C22×Dic5—21no—
176#709—21no—
177#709—21no—
178#709—21no—
179#709—21no—
180C2×D20C2×C3×C411yescharacteristic, complemented
181C2×C4×D5—21no—
182C2×Dic10C2×C3×C411yescharacteristic, complemented
183C5×M16—21no—
184C5×M16—21no—
185C2×C5×C8—21no—
186C5×C4⋊C4—21no—
187C5×C22⋊C4—21no—
188C42×C5—21no—
189C22⋊Dic5—22no—
190C22⋊Dic5C3×D412yes—
191#689—22no—
192#688—22no—
193#687—22no—
194#686—22no—
195C4⋊Dic5—22no—
196C4⋊Dic5—21no—
197C4×Dic5—22no—
198C4×Dic5—21no—
199C4×Dic5—21no—
200C4×Dic5C3×D411yescharacteristic
201C4×Dic5C3×D411yescharacteristic
202#681—21no—
203#681C2×C3×C412yescomplemented
204C2×#168—21no—
205#575—51no—
206C4×#41—51no—
207#421C3×D511yescharacteristic, complemented
208C2×#104—51no—
209#410—51no—
210#344—51no—
211#339—51no—
212C22×C3×C5—22no—
213C22×C3×C5—41no—
214C22×C3×C5—42no—
215C22×C3×C5C4×D411yescharacteristic
216C22×C3×C5#12011yescharacteristic
217C22×C3×C5#12111yescharacteristic
218C2×C3×D5—41no—
219C2×C3×D5—81no—
220C3×C4×C5—22no—
221C3×C4×C5—81no—
222C3×C4×C5C2×C22⋊C411yescharacteristic
223C3×C4×C5C4×D411yescharacteristic
224C3×C4×C5#12111yescharacteristic
225C3×C4×C5#12311yescharacteristic
226C3×Dic5—22no—
227C3×Dic5—42no—
228C3×Dic5—41no—
229C3×Dic5—41no—
230C3×#41—21no—
231C3×#41—21no—
232C3×#41—101no—
233C3×#41—101no—
234C3×#41—101no—
235C3×#41—101no—
236C3×#41C4×D512yes—
237C3×#41C22⋊D512yes—
238C2×C3×Q8—51no—
239C2×C3×Q8D2011yescharacteristic
240C2×C3×D4—21no—
241C2×C3×D4—51no—
242C2×C3×D4—101no—
243C2×C3×D4D2011yescharacteristic
244C22×C3×C4—21no—
245C22×C3×C4—102no—
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—52no—
256C3×M16—101no—
257C2×C3×C8—21no—
258C2×C3×C8—101no—
259C3×C4⋊C4—21no—
260C3×C4⋊C4—102no—
261C3×C4⋊C4—101no—
262C3×C22⋊C4—21no—
263C3×C22⋊C4—52no—
264C3×C22⋊C4—102no—
265C3×C42—21no—
266C3×C42—51no—
267C3×C42—51no—
268C3×C42—102no—
269C3×C42—101no—
270C3×C42—101no—
271C23×C5—21no—
272C23×C5C2×C3×D411yescharacteristic
273C22×D5—21no—
274C5×Q8—21no—
275C5×Q8C3×C22⋊C412yes—
276C5×D4—22no—
277C5×D4—22no—
278C5×D4—21no—
279C5×D4C3×C22⋊C412yes—
280C2×C4×C5—21no—
281C2×C4×C5—22no—
282C2×C4×C5—22no—
283C2×C4×C5—41no—
284C2×C4×C5C22×C3×C411yescharacteristic
285C2×C4×C5C2×C3×D411yescharacteristic
286C2×C4×C5C2×C3×D411yescharacteristic
287C2×C4×C5C2×C3×D411yescharacteristic
288C2×C4×C5C3×#4111yescharacteristic
289C2×C4×C5C3×#4111yescharacteristic
290C22⋊D5—41no—
291C22⋊D5—41no—
292C2×Dic5—21no—
293C2×Dic5—21no—
294C2×Dic5—22no—
295C2×Dic5—21no—
296C2×Dic5—42no—
297C2×Dic5—44no—
298D20—21no—
299D20—21no—
300C4×D5—41no—
301C4×D5—41no—
302Dic10—21no—
303Dic10—21no—
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—102no—
317C4×D4—101no—
318#117—51no—
319#117—101no—
320#117C2×C3×D511yescharacteristic
321C2×C42—51no—
322C2×C42—101no—
323#104—22no—
324#104—101no—
325#104—101no—
326#99—102no—
327C2×C3×C5—21no—
328C2×C3×C5—21no—
329C2×C3×C5—21no—
330C2×C3×C5—42no—
331C2×C3×C5#38611yescharacteristic
332C3×D5—81no—
333C23×C3—21no—
334C23×C3—101no—
335C23×C3D4×D511yescharacteristic
336C3×Q8—21no—
337C3×Q8—101no—
338C3×Q8—101no—
339C3×Q8#68512yes—
340C3×D4—22no—
341C3×D4—22no—
342C3×D4—21no—
343C3×D4—101no—
344C3×D4—101no—
345C3×D4—201no—
346C3×D4—201no—
347C3×D4#68512yes—
348C2×C3×C4—21no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—41no—
352C2×C3×C4—101no—
353C2×C3×C4—101no—
354C2×C3×C4—101no—
355C2×C3×C4—102no—
356C2×C3×C4—202no—
357C2×C3×C4—204no—
358C2×C3×C4—201no—
359C2×C3×C4—201no—
360C2×C3×C4C2×C4×D511yescharacteristic
361C2×C3×C4C2×D2011yescharacteristic
362C2×C3×C4D4×D511yescharacteristic
363C2×C3×C4#71111yescharacteristic
364C2×C3×C4#71111yescharacteristic
365C2×C3×C4C2×C22⋊D511yescharacteristic
366C3×C8—41no—
367C3×C8—102no—
368C22×C5—22no—
369C22×C5—41no—
370C22×C5—42no—
371C22×C5C3×C4×D411yescharacteristic
372C22×C5C3×#12011yescharacteristic
373C22×C5C3×#12111yescharacteristic
374C2×D5—41no—
375C2×D5—81no—
376C4×C5—22no—
377C4×C5—81no—
378C4×C5C2×C3×C22⋊C411yescharacteristic
379C4×C5C3×C4×D411yescharacteristic
380C4×C5C3×#12111yescharacteristic
381C4×C5C3×#12311yescharacteristic
382Dic5—22no—
383Dic5—42no—
384Dic5—41no—
385Dic5—41no—
386#41—21no—
387#41—21no—
388#41—101no—
389#41—101no—
390#41—101no—
391#41—101no—
392#41C3×C4×D512yes—
393#41C3×C22⋊D512yes—
394C2×Q8—51no—
395C2×Q8C3×D2011yescharacteristic
396C2×D4—21no—
397C2×D4—51no—
398C2×D4—101no—
399C2×D4C3×D2011yescharacteristic
400C22×C4—21no—
401C22×C4—102no—
402C22×C4—101no—
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—21no—
416C4⋊C4—102no—
417C4⋊C4—101no—
418C22⋊C4—21no—
419C22⋊C4—52no—
420C22⋊C4—102no—
421C42—21no—
422C42—51no—
423C42—51no—
424C42—102no—
425C42—101no—
426C42—101no—
427C3×C5#190611yescharacteristic, complemented, Hall
428C22×C3—22no—
429C22×C3—41no—
430C22×C3—42no—
431C22×C3—201no—
432C22×C3—401no—
433C22×C3#402711yescharacteristic
434C22×C3#402811yescharacteristic
435C22×C3#408511yescharacteristic
436C3×C4—22no—
437C3×C4—81no—
438C3×C4—102no—
439C3×C4—201no—
440C3×C4—202no—
441C3×C4—201no—
442C3×C4#402711yescharacteristic
443C3×C4#403211yescharacteristic
444C3×C4C2×#68511yescharacteristic
445C3×C4#408511yescharacteristic
446C2×C5—21no—
447C2×C5—21no—
448C2×C5—21no—
449C2×C5—42no—
450C2×C5C3×#38611yescharacteristic
451D5—81no—
452C23—21no—
453C23—101no—
454C23C3×D4×D511yescharacteristic
455Q8—21no—
456Q8—101no—
457Q8—101no—
458Q8C3×#68512yes—
459D4—22no—
460D4—22no—
461D4—21no—
462D4—101no—
463D4—101no—
464D4—201no—
465D4—201no—
466D4C3×#68512yes—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—41no—
471C2×C4—101no—
472C2×C4—101no—
473C2×C4—101no—
474C2×C4—102no—
475C2×C4—202no—
476C2×C4—204no—
477C2×C4—201no—
478C2×C4—201no—
479C2×C4C2×C3×C4×D511yescharacteristic
480C2×C4C2×C3×D2011yescharacteristic
481C2×C4C3×D4×D511yescharacteristic
482C2×C4C3×#71111yescharacteristic
483C2×C4C3×#71111yescharacteristic
484C2×C4C2×C3×C22⋊D511yescharacteristic
485C8—41no—
486C8—102no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—42no—
491C2×C3—401no—
492C2×C3#6524711yescharacteristic
493C5C3×#190611yescharacteristic, complemented, Sylow
494C22—22no—
495C22—41no—
496C22—42no—
497C22—201no—
498C22—401no—
499C22C3×#402711yescharacteristic
500C22C3×#402811yescharacteristic
501C22C3×#408511yescharacteristic
502C4—22no—
503C4—81no—
504C4—102no—
505C4—201no—
506C4—202no—
507C4—201no—
508C4C3×#402711yescharacteristic
509C4C3×#403211yescharacteristic
510C4C2×C3×#68511yescharacteristic
511C4C3×#408511yescharacteristic
512C3#1060118811yescharacteristic, direct product factor, Sylow
513C2—21no—
514C2—21no—
515C2—21no—
516C2—42no—
517C2—401no—
518C2C3×#6524711yescharacteristic
subgroupG/H
centerC3×C4#4085
commutatorC2×C4×C5C22×C3×C4
FrattiniC22×C4C22×C3×D5
FittingC3×C5×#421C2
socleC2×C3×C5#386
prime radicalC3×#66156C2
cosocleC22×C4×C5C23×C3
hypercenterC3×#421D5
hypocenterC5C3×#1906
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C3×#421

Complements

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

NH ≅ G/Ntype
1. C3×#41×Dic5C2semidirect
3. C3×C5×#421C2semidirect
4. C2×C3×#3957C2semidirect
5. C3×#65295C2semidirect
6. C3×#64776C2semidirect
7. C3×#64752C2semidirect
8. #10601188C3direct (N × H)
13. C3×#4097C4semidirect
15. C3×#4095C4semidirect
31. #41×Dic5C2×C3semidirect
33. C5×#421C2×C3semidirect
34. C2×#3957C2×C3semidirect
35. #65295C2×C3semidirect
36. #64776C2×C3semidirect
37. #64752C2×C3semidirect
54. C2×C3×D20C2×C4semidirect
56. C2×C3×Dic10C2×C4semidirect
77. C3×#681C2×C4semidirect
81. C3×#421D5semidirect
90. #4097C3×C4semidirect
92. #4095C3×C4semidirect
180. C2×D20C2×C3×C4semidirect
182. C2×Dic10C2×C3×C4semidirect
203. #681C2×C3×C4semidirect
207. #421C3×D5semidirect
427. C3×C5#1906semidirect
493. C5C3×#1906semidirect
512. C3#10601188direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#41×Dic52
C3×#661562
C3×C5×#4212
C2×C3×#39572
C3×#652952
C3×#647762
C3×#647522
#106011883
C3×#19065