Finite Group Databasebeta

← C3×#10615714 #14424538

Subgroups — C3×#10615714 #14424538

Order 1920.

1400 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4150C211yesmaximal, characteristic
2C2×C3×C5×#137C211yesmaximal, characteristic, complemented
3C3×#66150C211yesmaximal, characteristic, complemented
4C22×C3×#689C211yesmaximal, characteristic, complemented
5C3×#65522C211yesmaximal, characteristic, complemented
6C3×#65475C214yesmaximal, complemented
7C3×#65462C214yesmaximal, complemented
8C3×#65418C211yesmaximal, characteristic, complemented
9C3×#65409C211yesmaximal, characteristic, complemented
10#10615714C311yesmaximal, characteristic, direct product factor, Hall
11C2×C3×C5×#41C2211yescharacteristic
12C22×C3×C5×Q8C2211yescharacteristic
13C3×#4150—24no—
14C3×#4150—24no—
15C2×C3×Q8×D5C2212yes—
16C2×C3×#711—22no—
17C2×C3×#711C2212yes—
18C2×C3×#709—21no—
19C2×C3×#709C2211yescharacteristic
20C22×C3×Dic10—21no—
21C22×C3×Dic10C2211yescharacteristic
22C3×C5×#137—24no—
23C2×C3×C5×Q16C2212yes—
24C2×C3×C5×SD16C2212yes—
25C2×C3×C5×M16C2211yescharacteristic
26C3×#4093—21no—
27C3×#4092C2214yes—
28C3×#4090—21no—
29C2×C3×#689—24no—
30C2×C3×#689C2214yes—
31C3×#4075—22no—
32C3×#4072C2211yescharacteristic
33C22×C3×#168C2211yescharacteristic
34C3×#3967C2212yes—
35C3×#3964C2212yes—
36C3×#3952C2214yes—
37C3×#3948C2214yes—
38C3×#3011—51nomaximal, Hall
39C2×#4150C2×C311yescharacteristic
40C2×C5×#137C2×C311yescharacteristic, complemented
41#66150C2×C311yescharacteristic, complemented
42C22×#689C2×C311yescharacteristic, complemented
43#65522C2×C311yescharacteristic, complemented
44#65475C2×C314yescomplemented
45#65462C2×C314yescomplemented
46#65418C2×C311yescharacteristic, complemented
47#65409C2×C311yescharacteristic, complemented
48C3×C5×#41—22no—
49C3×C5×#41D414yes—
50C2×C3×C5×Q8—21no—
51C2×C3×C5×Q8—22no—
52C2×C3×C5×Q8—22no—
53C2×C3×C5×Q8C2312yes—
54C2×C3×C5×Q8C2311yescharacteristic
55C2×C3×C5×D4—21no—
56C2×C3×C5×D4C2311yescharacteristic
57C22×C3×C4×C5—21no—
58C22×C3×C4×C5—21no—
59C22×C3×C4×C5C2311yescharacteristic
60C2×C3×C22⋊D5—22no—
61C2×C3×C22⋊D5D412yes—
62C22×C3×Dic5—21no—
63C22×C3×Dic5—22no—
64C3×Q8×D5—24no—
65C3×Q8×D5—24no—
66C3×#711—216no—
67C3×#711—24no—
68C3×#711—24no—
69C3×#709—28no—
70C3×#709—22no—
71C3×#709—22no—
72C2×C3×D20D411yescharacteristic
73C2×C3×C4×D5—22no—
74C2×C3×C4×D5C2312yes—
75C2×C3×Dic10—28no—
76C2×C3×Dic10—21no—
77C2×C3×Dic10—22no—
78C2×C3×Dic10—22no—
79C2×C3×Dic10D411yescharacteristic
80C2×C3×Dic10D414yes—
81C2×C3×Dic10C2312yes—
82C3×C5×Q16—24no—
83C3×C5×SD16—24no—
84C3×C5×M16—22no—
85C2×C3×C5×C8C2312yes—
86C3×C22⋊Dic5—21no—
87C3×#689—28no—
88C3×#685—24no—
89C3×C4⋊Dic5C2312yes—
90C3×#683—24no—
91C3×C4×Dic5—21no—
92C2×C3×#168—22no—
93C2×C3×#168C2312yes—
94C2×C3×2−1+4—51no—
95C2×C3×#137D511yescharacteristic, complemented
96C22×C3×Q16—51no—
97C3×#536—51no—
98C3×#452—54no—
99C3×#451—54no—
100C3×#419—51no—
101C3×#416—51no—
102C3×#408—51no—
103C2×C5×#41C22×C311yescharacteristic
104C22×C5×Q8C22×C311yescharacteristic
105#4150—24no—
106#4150—24no—
107C2×Q8×D5C22×C312yes—
108C2×#711—22no—
109C2×#711C22×C312yes—
110C2×#709—21no—
111C2×#709C22×C311yescharacteristic
112C22×Dic10—21no—
113C22×Dic10C22×C311yescharacteristic
114C5×#137—24no—
115C2×C5×Q16C22×C312yes—
116C2×C5×SD16C22×C312yes—
117C2×C5×M16C22×C311yescharacteristic
118#4093—21no—
119#4092C22×C314yes—
120#4090—21no—
121C2×#689—24no—
122C2×#689C22×C314yes—
123#4075—22no—
124#4072C22×C311yescharacteristic
125C22×#168C22×C311yescharacteristic
126#3967C22×C312yes—
127#3964C22×C312yes—
128#3952C22×C314yes—
129#3948C22×C314yes—
130#3011—51noSylow
131C23×C3×C5—21no—
132C23×C3×C5C2×D411yescharacteristic
133C22×C3×D5C2×D412yes—
134C3×C5×Q8—21no—
135C3×C5×Q8—22no—
136C3×C5×Q8—22no—
137C3×C5×Q8—42no—
138C3×C5×Q8C2×D412yes—
139C3×C5×D4—24no—
140C3×C5×D4—21no—
141C3×C5×D4C2×D412yes—
142C2×C3×C4×C5—21no—
143C2×C3×C4×C5—24no—
144C2×C3×C4×C5—22no—
145C2×C3×C4×C5—42no—
146C2×C3×C4×C5C2×D411yescharacteristic
147C2×C3×C4×C5C2×D412yes—
148C2×C3×C4×C5C2×D412yes—
149C2×C3×C4×C5C2411yescharacteristic
150C3×C22⋊D5—28no—
151C3×C22⋊D5—24no—
152C2×C3×Dic5—28no—
153C2×C3×Dic5—24no—
154C2×C3×Dic5—22no—
155C2×C3×Dic5—22no—
156C2×C3×Dic5C2×D412yes—
157C3×D20—22no—
158C3×C4×D5—28no—
159C3×C4×D5—22no—
160C3×C4×D5—22no—
161C3×Dic10—28no—
162C3×Dic10—22no—
163C3×Dic10—24no—
164C3×Dic10—22no—
165C3×Dic10C2×D414yes—
166C3×C5×C8—22no—
167C3×#168—22no—
168C3×2−1+4—104no—
169C3×2−1+4—104no—
170C2×C3×#41—51no—
171C2×C3×#41—52no—
172C2×C3×#41—101no—
173C2×C3×#41—102no—
174C2×C3×#41C2×D511yescharacteristic
175C22×C3×Q8—52no—
176C22×C3×Q8—51no—
177C22×C3×Q8—101no—
178C22×C3×Q8C2×D511yescharacteristic
179C3×#137—24no—
180C2×C3×Q16—54no—
181C2×C3×Q16—104no—
182C2×C3×Q16C2×D512yes—
183C2×C3×SD16C2×D512yes—
184C2×C3×M16C2×D511yescharacteristic
185C22×C3×C8—51no—
186C3×C4⋊Q8—101no—
187C3×#124—101no—
188C3×#123—102no—
189C3×C22⋊Q8—54no—
190C3×#117—51no—
191C3×#103—52no—
192C3×#103—54no—
193C3×#102—52no—
194C3×C22⋊C8—54no—
195C5×#41—22no—
196C5×#41C3×D414yes—
197C2×C5×Q8—21no—
198C2×C5×Q8—22no—
199C2×C5×Q8—22no—
200C2×C5×Q8C23×C312yes—
201C2×C5×Q8C23×C311yescharacteristic
202C2×C5×D4—21no—
203C2×C5×D4C23×C311yescharacteristic
204C22×C4×C5—21no—
205C22×C4×C5—21no—
206C22×C4×C5C23×C311yescharacteristic
207C2×C22⋊D5—22no—
208C2×C22⋊D5C3×D412yes—
209C22×Dic5—21no—
210C22×Dic5—22no—
211Q8×D5—24no—
212Q8×D5—24no—
213#711—216no—
214#711—24no—
215#711—24no—
216#709—28no—
217#709—22no—
218#709—22no—
219C2×D20C3×D411yescharacteristic
220C2×C4×D5—22no—
221C2×C4×D5C23×C312yes—
222C2×Dic10—28no—
223C2×Dic10—21no—
224C2×Dic10—22no—
225C2×Dic10—22no—
226C2×Dic10C3×D411yescharacteristic
227C2×Dic10C3×D414yes—
228C2×Dic10C23×C312yes—
229C5×Q16—24no—
230C5×SD16—24no—
231C5×M16—22no—
232C2×C5×C8C23×C312yes—
233C22⋊Dic5—21no—
234#689—28no—
235#685—24no—
236C4⋊Dic5C23×C312yes—
237#683—24no—
238C4×Dic5—21no—
239C2×#168—22no—
240C2×#168C23×C312yes—
241C2×2−1+4—51no—
242C2×#137C3×D511yescharacteristic, complemented
243C22×Q16—51no—
244#536—51no—
245#452—54no—
246#451—54no—
247#419—51no—
248#416—51no—
249#408—51no—
250C22×C3×C5—22no—
251C22×C3×C5—22no—
252C22×C3×C5—42no—
253C22×C3×C5#12012yes—
254C22×C3×C5C22×D411yescharacteristic
255C2×C3×D5—22no—
256C2×C3×D5—24no—
257C3×C4×C5—22no—
258C3×C4×C5—42no—
259C3×C4×C5#12012yes—
260C3×C4×C5C22×D411yescharacteristic
261C3×C4×C5C22×D411yescharacteristic
262C3×Dic5—22no—
263C3×Dic5—24no—
264C3×Dic5—42no—
265C3×#41—22no—
266C3×#41—108no—
267C3×#41—104no—
268C3×#41—1016no—
269C3×#41—104no—
270C3×#41—102no—
271C3×#41—102no—
272C3×#41C22⋊D514yes—
273C2×C3×Q8—21no—
274C2×C3×Q8—22no—
275C2×C3×Q8—22no—
276C2×C3×Q8—51no—
277C2×C3×Q8—54no—
278C2×C3×Q8—52no—
279C2×C3×Q8—102no—
280C2×C3×Q8—108no—
281C2×C3×Q8—104no—
282C2×C3×Q8—104no—
283C2×C3×Q8—101no—
284C2×C3×Q8—102no—
285C2×C3×Q8C22×D512yes—
286C2×C3×Q8C22×D511yescharacteristic
287C2×C3×D4—21no—
288C2×C3×D4—52no—
289C2×C3×D4—51no—
290C2×C3×D4—102no—
291C2×C3×D4C22×D511yescharacteristic
292C22×C3×C4—21no—
293C22×C3×C4—21no—
294C22×C3×C4—52no—
295C22×C3×C4—102no—
296C22×C3×C4—102no—
297C22×C3×C4—101no—
298C22×C3×C4C22×D511yescharacteristic
299C3×Q16—24no—
300C3×Q16—108no—
301C3×SD16—24no—
302C3×M16—22no—
303C2×C3×C8—52no—
304C2×C3×C8—102no—
305C2×C3×C8C22×D512yes—
306C3×C4⋊C4—52no—
307C3×C4⋊C4—104no—
308C3×C22⋊C4—104no—
309C3×C22⋊C4—101no—
310C3×C42—101no—
311C23×C5—21no—
312C23×C5C2×C3×D411yescharacteristic
313C22×D5C2×C3×D412yes—
314C5×Q8—21no—
315C5×Q8—22no—
316C5×Q8—22no—
317C5×Q8—42no—
318C5×Q8C2×C3×D412yes—
319C5×D4—24no—
320C5×D4—21no—
321C5×D4C2×C3×D412yes—
322C2×C4×C5—21no—
323C2×C4×C5—24no—
324C2×C4×C5—22no—
325C2×C4×C5—42no—
326C2×C4×C5C2×C3×D411yescharacteristic
327C2×C4×C5C2×C3×D412yes—
328C2×C4×C5C2×C3×D412yes—
329C2×C4×C5C24×C311yescharacteristic
330C22⋊D5—28no—
331C22⋊D5—24no—
332C2×Dic5—28no—
333C2×Dic5—24no—
334C2×Dic5—22no—
335C2×Dic5—22no—
336C2×Dic5C2×C3×D412yes—
337D20—22no—
338C4×D5—28no—
339C4×D5—22no—
340C4×D5—22no—
341Dic10—28no—
342Dic10—22no—
343Dic10—24no—
344Dic10—22no—
345Dic10C2×C3×D414yes—
346C5×C8—22no—
347#168—22no—
3482−1+4—104no—
3492−1+4—104no—
350C2×#41—51no—
351C2×#41—52no—
352C2×#41—101no—
353C2×#41—102no—
354C2×#41C2×C3×D511yescharacteristic
355C22×Q8—52no—
356C22×Q8—51no—
357C22×Q8—101no—
358C22×Q8C2×C3×D511yescharacteristic
359#137—24no—
360C2×Q16—54no—
361C2×Q16—104no—
362C2×Q16C2×C3×D512yes—
363C2×SD16C2×C3×D512yes—
364C2×M16C2×C3×D511yescharacteristic
365C22×C8—51no—
366C4⋊Q8—101no—
367#124—101no—
368#123—102no—
369C22⋊Q8—54no—
370#117—51no—
371#103—52no—
372#103—54no—
373#102—52no—
374C22⋊C8—54no—
375C2×C3×C5—22no—
376C2×C3×C5—42no—
377C2×C3×C5C2×#12011yescharacteristic
378C2×C3×C5#57812yes—
379C3×D5—42no—
380C23×C3—21no—
381C23×C3—52no—
382C23×C3D4×D511yescharacteristic
383C3×Q8—21no—
384C3×Q8—22no—
385C3×Q8—22no—
386C3×Q8—42no—
387C3×Q8—54no—
388C3×Q8—108no—
389C3×Q8—102no—
390C3×Q8—102no—
391C3×Q8—104no—
392C3×Q8C2×C22⋊D512yes—
393C3×D4—24no—
394C3×D4—21no—
395C3×D4—108no—
396C3×D4—104no—
397C3×D4—102no—
398C3×D4C2×C22⋊D512yes—
399C2×C3×C4—21no—
400C2×C3×C4—24no—
401C2×C3×C4—22no—
402C2×C3×C4—42no—
403C2×C3×C4—52no—
404C2×C3×C4—108no—
405C2×C3×C4—108no—
406C2×C3×C4—102no—
407C2×C3×C4—104no—
408C2×C3×C4—102no—
409C2×C3×C4—102no—
410C2×C3×C4—102no—
411C2×C3×C4D4×D512yes—
412C2×C3×C4D4×D511yescharacteristic
413C2×C3×C4C2×C22⋊D512yes—
414C2×C3×C4C23×D511yescharacteristic
415C3×C8—22no—
416C3×C8—102no—
417C22×C5—22no—
418C22×C5—22no—
419C22×C5—42no—
420C22×C5C3×#12012yes—
421C22×C5C22×C3×D411yescharacteristic
422C2×D5—24no—
423C2×D5—22no—
424C4×C5—22no—
425C4×C5—42no—
426C4×C5C3×#12012yes—
427C4×C5C22×C3×D411yescharacteristic
428C4×C5C22×C3×D411yescharacteristic
429Dic5—22no—
430Dic5—24no—
431Dic5—42no—
432#41—22no—
433#41—108no—
434#41—104no—
435#41—1016no—
436#41—104no—
437#41—102no—
438#41—102no—
439#41C3×C22⋊D514yes—
440C2×Q8—21no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—51no—
444C2×Q8—54no—
445C2×Q8—52no—
446C2×Q8—102no—
447C2×Q8—108no—
448C2×Q8—104no—
449C2×Q8—104no—
450C2×Q8—101no—
451C2×Q8—102no—
452C2×Q8C22×C3×D512yes—
453C2×Q8C22×C3×D511yescharacteristic
454C2×D4—21no—
455C2×D4—52no—
456C2×D4—51no—
457C2×D4—102no—
458C2×D4C22×C3×D511yescharacteristic
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—52no—
462C22×C4—102no—
463C22×C4—102no—
464C22×C4—101no—
465C22×C4C22×C3×D511yescharacteristic
466Q16—24no—
467Q16—108no—
468SD16—24no—
469M16—22no—
470C2×C8—52no—
471C2×C8—102no—
472C2×C8C22×C3×D512yes—
473C4⋊C4—52no—
474C4⋊C4—104no—
475C22⋊C4—104no—
476C22⋊C4—101no—
477C42—101no—
478C3×C5#301111yescharacteristic, complemented, Hall
479C22×C3—22no—
480C22×C3—22no—
481C22×C3—42no—
482C22×C3—104no—
483C22×C3—102no—
484C22×C3#408312yes—
485C22×C3C2×D4×D511yescharacteristic
486C3×C4—22no—
487C3×C4—42no—
488C3×C4—102no—
489C3×C4—104no—
490C3×C4—202no—
491C3×C4#408312yes—
492C3×C4C2×D4×D511yescharacteristic
493C3×C4C22×C22⋊D511yescharacteristic
494C2×C5—22no—
495C2×C5—42no—
496C2×C5C2×C3×#12011yescharacteristic
497C2×C5C3×#57812yes—
498D5—42no—
499C23—21no—
500C23—52no—
501C23C3×D4×D511yescharacteristic
502Q8—21no—
503Q8—22no—
504Q8—22no—
505Q8—42no—
506Q8—54no—
507Q8—108no—
508Q8—102no—
509Q8—104no—
510Q8—102no—
511Q8C2×C3×C22⋊D512yes—
512D4—24no—
513D4—21no—
514D4—108no—
515D4—104no—
516D4—102no—
517D4C2×C3×C22⋊D512yes—
518C2×C4—21no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—42no—
522C2×C4—52no—
523C2×C4—108no—
524C2×C4—108no—
525C2×C4—104no—
526C2×C4—102no—
527C2×C4—102no—
528C2×C4—102no—
529C2×C4—102no—
530C2×C4C3×D4×D511yescharacteristic
531C2×C4C3×D4×D512yes—
532C2×C4C2×C3×C22⋊D512yes—
533C2×C4C23×C3×D511yescharacteristic
534C8—22no—
535C8—102no—
536C2×C3—22no—
537C2×C3—42no—
538C2×C3—202no—
539C2×C3#6611312yes—
540C2×C3C2×#408311yescharacteristic
541C5C3×#301111yescharacteristic, complemented, Sylow
542C22—22no—
543C22—22no—
544C22—42no—
545C22—104no—
546C22—102no—
547C22C3×#408312yes—
548C22C2×C3×D4×D511yescharacteristic
549C4—22no—
550C4—42no—
551C4—102no—
552C4—104no—
553C4—202no—
554C4C3×#408312yes—
555C4C2×C3×D4×D511yescharacteristic
556C4C22×C3×C22⋊D511yescharacteristic
557C3#1061571411yescharacteristic, direct product factor, Sylow
558C2—22no—
559C2—42no—
560C2—202no—
561C2C3×#6611312yes—
562C2C2×C3×#408311yescharacteristic
subgroupG/H
centerC22×C3C2×D4×D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC2×C3×C5×#137C2
socleC22×C3×C5C22×D4
prime radicalC2×C3×#4150C2
hypercenterC2×C3×#137D5
hypocenterC5C3×#3011
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#137

Complements

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

NH ≅ G/Ntype
2. C2×C3×C5×#137C2semidirect
3. C3×#66150C2semidirect
4. C22×C3×#689C2semidirect
5. C3×#65522C2semidirect
6. C3×#65475C2semidirect
7. C3×#65462C2semidirect
8. C3×#65418C2semidirect
9. C3×#65409C2semidirect
10. #10615714C3direct (N × H)
40. C2×C5×#137C2×C3semidirect
41. #66150C2×C3semidirect
42. C22×#689C2×C3semidirect
43. #65522C2×C3semidirect
44. #65475C2×C3semidirect
45. #65462C2×C3semidirect
46. #65418C2×C3semidirect
47. #65409C2×C3semidirect
95. C2×C3×#137D5semidirect
242. C2×#137C3×D5semidirect
478. C3×C5#3011semidirect
541. C5C3×#3011semidirect
557. C3#10615714direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#41502
C2×C3×C5×#1372
C3×#661502
C22×C3×#6892
C3×#655222
C3×#654752
C3×#654622
C3×#654182
C3×#654092
#106157143
C3×#30115