Finite Group Databasebeta

← C3×#10615142 #14424535

Subgroups — C3×#10615142 #14424535

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
2C3×C5×#536C211yesmaximal, characteristic, complemented
3C22×C3×#689C211yesmaximal, characteristic, complemented
4C2×C3×#4088C211yesmaximal, characteristic, complemented
5C3×#65523C211yesmaximal, characteristic, complemented
6C3×#65339C214yesmaximal, complemented
7C3×#65336C214yesmaximal, complemented
8C3×#65293C211yesmaximal, characteristic, complemented
9C3×#65291C211yesmaximal, characteristic, complemented
10#10615142C311yesmaximal, characteristic, direct product factor, Hall
11C2×C3×C5×#41C2211yescharacteristic
12C22×C3×C5×Q8C2211yescharacteristic
13C3×#4150—24no—
14C3×#4150—24no—
15C2×C3×Q8×D5—21no—
16C2×C3×#711—22no—
17C2×C3×#711—21no—
18C2×C3×#709C2212yes—
19C2×C3×#709C2211yescharacteristic
20C22×C3×Dic10C2212yes—
21C22×C3×Dic10C2211yescharacteristic
22C3×C5×C4⋊Q8—21no—
23C3×C5×#124—21no—
24C3×C5×#123—22no—
25C3×C5×C22⋊Q8C2214yes—
26C3×C5×#117C2211yescharacteristic
27C2×C3×#689—24no—
28C2×C3×#689C2212yes—
29C2×C3×#689C2214yes—
30C3×#4088—24no—
31C2×C3×#688C2212yes—
32C2×C3×#681C2211yescharacteristic
33C22×C3×#168C2211yescharacteristic
34C3×C22⋊#168C2214yes—
35C3×#3942C2212yes—
36C3×#3942C2214yes—
37C3×#3941C2212yes—
38C3×#3011—51nomaximal, Hall
39C2×#4150C2×C311yescharacteristic
40C5×#536C2×C311yescharacteristic, complemented
41C22×#689C2×C311yescharacteristic, complemented
42C2×#4088C2×C311yescharacteristic, complemented
43#65523C2×C311yescharacteristic, complemented
44#65339C2×C314yescomplemented
45#65336C2×C314yescomplemented
46#65293C2×C311yescharacteristic, complemented
47#65291C2×C311yescharacteristic, complemented
48C3×C5×#41—22no—
49C3×C5×#41—22no—
50C2×C3×C5×Q8—21no—
51C2×C3×C5×Q8—22no—
52C2×C3×C5×Q8—22no—
53C2×C3×C5×Q8D411yescharacteristic
54C2×C3×C5×Q8C2312yes—
55C2×C3×C5×D4D412yes—
56C2×C3×C5×D4D411yescharacteristic
57C22×C3×C4×C5—21no—
58C22×C3×C4×C5C2312yes—
59C22×C3×C4×C5C2311yescharacteristic
60C2×C3×C22⋊D5—22no—
61C2×C3×C22⋊D5—21no—
62C22×C3×Dic5—22no—
63C22×C3×Dic5—21no—
64C3×Q8×D5—28no—
65C3×#711—216no—
66C3×#711—28no—
67C3×#709—22no—
68C3×#709—24no—
69C3×#709—24no—
70C3×#709D414yes—
71C2×C3×D20C2311yescharacteristic
72C2×C3×C4×D5—22no—
73C2×C3×C4×D5—21no—
74C2×C3×Dic10—22no—
75C2×C3×Dic10—21no—
76C2×C3×Dic10—22no—
77C2×C3×Dic10—24no—
78C2×C3×Dic10—24no—
79C2×C3×Dic10D414yes—
80C2×C3×Dic10C2311yescharacteristic
81C2×C3×Dic10C2312yes—
82C3×C5×C4⋊C4—24no—
83C3×C5×C4⋊C4C2312yes—
84C3×C5×C22⋊C4—21no—
85C3×C5×C22⋊C4—24no—
86C3×C42×C5—21no—
87C3×#689—24no—
88C3×#689—28no—
89C3×#688—24no—
90C3×#681—22no—
91C2×C3×#168—22no—
92C2×C3×#168C2312yes—
93C2×C3×#168C2312yes—
94C2×C3×2−1+4—51no—
95C2×C3×#137—51no—
96C22×C3×Q16—51no—
97C3×#536D511yescharacteristic, complemented
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×D5—21no—
108C2×#711—22no—
109C2×#711—21no—
110C2×#709C22×C312yes—
111C2×#709C22×C311yescharacteristic
112C22×Dic10C22×C312yes—
113C22×Dic10C22×C311yescharacteristic
114C5×C4⋊Q8—21no—
115C5×#124—21no—
116C5×#123—22no—
117C5×C22⋊Q8C22×C314yes—
118C5×#117C22×C311yescharacteristic
119C2×#689—24no—
120C2×#689C22×C314yes—
121C2×#689C22×C312yes—
122#4088—24no—
123C2×#688C22×C312yes—
124C2×#681C22×C311yescharacteristic
125C22×#168C22×C311yescharacteristic
126C22⋊#168C22×C314yes—
127#3942C22×C312yes—
128#3942C22×C314yes—
129#3941C22×C312yes—
130#3011—51noSylow
131C23×C3×C5C2×D412yes—
132C23×C3×C5C2×D411yescharacteristic
133C22×C3×D5—21no—
134C3×C5×Q8—22no—
135C3×C5×Q8—22no—
136C3×C5×Q8—22no—
137C3×C5×Q8—42no—
138C3×C5×D4—24no—
139C3×C5×D4—22no—
140C2×C3×C4×C5—22no—
141C2×C3×C4×C5—22no—
142C2×C3×C4×C5—22no—
143C2×C3×C4×C5—22no—
144C2×C3×C4×C5—42no—
145C2×C3×C4×C5C2×D412yes—
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—21no—
155C2×C3×Dic5—22no—
156C3×D20—21no—
157C3×D20C2×D412yes—
158C3×C4×D5—28no—
159C3×C4×D5—24no—
160C3×Dic10—28no—
161C3×Dic10—24no—
162C3×Dic10—22no—
163C3×Dic10—21no—
164C3×Dic10C2×D412yes—
165C3×Dic10C2×D414yes—
166C3×#168—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—102no—
173C2×C3×#41—101no—
174C2×C3×#41C2×D511yescharacteristic
175C22×C3×Q8—51no—
176C22×C3×Q8—52no—
177C22×C3×Q8—101no—
178C22×C3×Q8C2×D511yescharacteristic
179C3×#137—104no—
180C2×C3×Q16—54no—
181C2×C3×Q16—52no—
182C2×C3×Q16—104no—
183C2×C3×SD16—52no—
184C2×C3×M16—51no—
185C22×C3×C8—51no—
186C3×C4⋊Q8—21no—
187C3×#124—21no—
188C3×#123—22no—
189C3×C22⋊Q8C2×D514yes—
190C3×#117C2×D511yescharacteristic
191C3×#103—52no—
192C3×#103—54no—
193C3×#102—52no—
194C3×C22⋊C8—54no—
195C5×#41—22no—
196C5×#41—22no—
197C2×C5×Q8—21no—
198C2×C5×Q8—22no—
199C2×C5×Q8—22no—
200C2×C5×Q8C3×D411yescharacteristic
201C2×C5×Q8C23×C312yes—
202C2×C5×D4C3×D412yes—
203C2×C5×D4C3×D411yescharacteristic
204C22×C4×C5—21no—
205C22×C4×C5C23×C312yes—
206C22×C4×C5C23×C311yescharacteristic
207C2×C22⋊D5—22no—
208C2×C22⋊D5—21no—
209C22×Dic5—22no—
210C22×Dic5—21no—
211Q8×D5—28no—
212#711—216no—
213#711—28no—
214#709—22no—
215#709—24no—
216#709—24no—
217#709C3×D414yes—
218C2×D20C23×C311yescharacteristic
219C2×C4×D5—22no—
220C2×C4×D5—21no—
221C2×Dic10—22no—
222C2×Dic10—21no—
223C2×Dic10—22no—
224C2×Dic10—24no—
225C2×Dic10—24no—
226C2×Dic10C3×D414yes—
227C2×Dic10C23×C311yescharacteristic
228C2×Dic10C23×C312yes—
229C5×C4⋊C4—24no—
230C5×C4⋊C4C23×C312yes—
231C5×C22⋊C4—21no—
232C5×C22⋊C4—24no—
233C42×C5—21no—
234#689—24no—
235#689—28no—
236#688—24no—
237#681—22no—
238C2×#168—22no—
239C2×#168C23×C312yes—
240C2×#168C23×C312yes—
241C2×2−1+4—51no—
242C2×#137—51no—
243C22×Q16—51no—
244#536C3×D511yescharacteristic, complemented
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—24no—
253C22×C3×C5#12012yes—
254C22×C3×C5C22×D411yescharacteristic
255C2×C3×D5—22no—
256C2×C3×D5—42no—
257C3×C4×C5—22no—
258C3×C4×C5—42no—
259C3×C4×C5—42no—
260C3×C4×C5#12012yes—
261C3×C4×C5C22×D411yescharacteristic
262C3×C4×C5C22×D411yescharacteristic
263C3×Dic5—22no—
264C3×Dic5—24no—
265C3×#41—22no—
266C3×#41—22no—
267C3×#41—54no—
268C3×#41—104no—
269C3×#41—104no—
270C3×#41—1016no—
271C3×#41—108no—
272C3×#41—102no—
273C2×C3×Q8—21no—
274C2×C3×Q8—22no—
275C2×C3×Q8—22no—
276C2×C3×Q8—54no—
277C2×C3×Q8—52no—
278C2×C3×Q8—51no—
279C2×C3×Q8—104no—
280C2×C3×Q8—104no—
281C2×C3×Q8—108no—
282C2×C3×Q8—102no—
283C2×C3×Q8—102no—
284C2×C3×Q8—101no—
285C2×C3×Q8C22⋊D511yescharacteristic
286C2×C3×Q8C22×D512yes—
287C2×C3×D4—51no—
288C2×C3×D4—102no—
289C2×C3×D4—101no—
290C2×C3×D4C22⋊D512yes—
291C2×C3×D4C22⋊D511yescharacteristic
292C22×C3×C4—21no—
293C22×C3×C4—102no—
294C22×C3×C4—101no—
295C22×C3×C4—102no—
296C22×C3×C4—101no—
297C22×C3×C4C22×D512yes—
298C22×C3×C4C22×D511yescharacteristic
299C3×Q16—108no—
300C3×Q16—104no—
301C3×SD16—104no—
302C3×M16—102no—
303C2×C3×C8—52no—
304C2×C3×C8—52no—
305C2×C3×C8—102no—
306C3×C4⋊C4—24no—
307C3×C4⋊C4C22×D512yes—
308C3×C22⋊C4—21no—
309C3×C22⋊C4—24no—
310C3×C42—21no—
311C23×C5C2×C3×D412yes—
312C23×C5C2×C3×D411yescharacteristic
313C22×D5—21no—
314C5×Q8—22no—
315C5×Q8—22no—
316C5×Q8—22no—
317C5×Q8—42no—
318C5×D4—24no—
319C5×D4—22no—
320C2×C4×C5—22no—
321C2×C4×C5—22no—
322C2×C4×C5—22no—
323C2×C4×C5—22no—
324C2×C4×C5—42no—
325C2×C4×C5C2×C3×D412yes—
326C2×C4×C5C2×C3×D412yes—
327C2×C4×C5C2×C3×D412yes—
328C2×C4×C5C2×C3×D411yescharacteristic
329C2×C4×C5C24×C311yescharacteristic
330C22⋊D5—28no—
331C22⋊D5—24no—
332C2×Dic5—28no—
333C2×Dic5—24no—
334C2×Dic5—21no—
335C2×Dic5—22no—
336D20—21no—
337D20C2×C3×D412yes—
338C4×D5—28no—
339C4×D5—24no—
340Dic10—28no—
341Dic10—24no—
342Dic10—22no—
343Dic10—21no—
344Dic10C2×C3×D412yes—
345Dic10C2×C3×D414yes—
346#168—22no—
347#168—22no—
3482−1+4—104no—
3492−1+4—104no—
350C2×#41—51no—
351C2×#41—52no—
352C2×#41—102no—
353C2×#41—101no—
354C2×#41C2×C3×D511yescharacteristic
355C22×Q8—51no—
356C22×Q8—52no—
357C22×Q8—101no—
358C22×Q8C2×C3×D511yescharacteristic
359#137—104no—
360C2×Q16—54no—
361C2×Q16—52no—
362C2×Q16—104no—
363C2×SD16—52no—
364C2×M16—51no—
365C22×C8—51no—
366C4⋊Q8—21no—
367#124—21no—
368#123—22no—
369C22⋊Q8C2×C3×D514yes—
370#117C2×C3×D511yescharacteristic
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—101no—
381C23×C3C2×C22⋊D512yes—
382C23×C3C2×C22⋊D511yescharacteristic
383C3×Q8—22no—
384C3×Q8—22no—
385C3×Q8—22no—
386C3×Q8—42no—
387C3×Q8—54no—
388C3×Q8—52no—
389C3×Q8—108no—
390C3×Q8—102no—
391C3×Q8—101no—
392C3×Q8—104no—
393C3×D4—24no—
394C3×D4—22no—
395C3×D4—52no—
396C3×D4—108no—
397C3×D4—104no—
398C3×D4—101no—
399C2×C3×C4—22no—
400C2×C3×C4—22no—
401C2×C3×C4—22no—
402C2×C3×C4—22no—
403C2×C3×C4—42no—
404C2×C3×C4—108no—
405C2×C3×C4—108no—
406C2×C3×C4—102no—
407C2×C3×C4—101no—
408C2×C3×C4—104no—
409C2×C3×C4—104no—
410C2×C3×C4D4×D512yes—
411C2×C3×C4D4×D512yes—
412C2×C3×C4C2×C22⋊D512yes—
413C2×C3×C4C2×C22⋊D511yescharacteristic
414C2×C3×C4C23×D511yescharacteristic
415C3×C8—102no—
416C3×C8—102no—
417C22×C5—22no—
418C22×C5—24no—
419C22×C5—22no—
420C22×C5C3×#12012yes—
421C22×C5C22×C3×D411yescharacteristic
422C2×D5—22no—
423C2×D5—42no—
424C4×C5—22no—
425C4×C5—42no—
426C4×C5—42no—
427C4×C5C3×#12012yes—
428C4×C5C22×C3×D411yescharacteristic
429C4×C5C22×C3×D411yescharacteristic
430Dic5—22no—
431Dic5—24no—
432#41—22no—
433#41—22no—
434#41—54no—
435#41—104no—
436#41—104no—
437#41—1016no—
438#41—108no—
439#41—102no—
440C2×Q8—21no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—54no—
444C2×Q8—52no—
445C2×Q8—51no—
446C2×Q8—104no—
447C2×Q8—104no—
448C2×Q8—108no—
449C2×Q8—102no—
450C2×Q8—102no—
451C2×Q8—101no—
452C2×Q8C3×C22⋊D511yescharacteristic
453C2×Q8C22×C3×D512yes—
454C2×D4—51no—
455C2×D4—102no—
456C2×D4—101no—
457C2×D4C3×C22⋊D512yes—
458C2×D4C3×C22⋊D511yescharacteristic
459C22×C4—21no—
460C22×C4—102no—
461C22×C4—101no—
462C22×C4—102no—
463C22×C4—101no—
464C22×C4C22×C3×D512yes—
465C22×C4C22×C3×D511yescharacteristic
466Q16—108no—
467Q16—104no—
468SD16—104no—
469M16—102no—
470C2×C8—52no—
471C2×C8—52no—
472C2×C8—102no—
473C4⋊C4—24no—
474C4⋊C4C22×C3×D512yes—
475C22⋊C4—21no—
476C22⋊C4—24no—
477C42—21no—
478C3×C5#301111yescharacteristic, complemented, Hall
479C22×C3—22no—
480C22×C3—22no—
481C22×C3—24no—
482C22×C3—102no—
483C22×C3—202no—
484C22×C3#408312yes—
485C22×C3C22×C22⋊D511yescharacteristic
486C3×C4—22no—
487C3×C4—42no—
488C3×C4—42no—
489C3×C4—104no—
490C3×C4—102no—
491C3×C4#408312yes—
492C3×C4C2×D4×D511yescharacteristic
493C3×C4C2×D4×D511yescharacteristic
494C2×C5—22no—
495C2×C5—42no—
496C2×C5C2×C3×#12011yescharacteristic
497C2×C5C3×#57812yes—
498D5—42no—
499C23—101no—
500C23C2×C3×C22⋊D512yes—
501C23C2×C3×C22⋊D511yescharacteristic
502Q8—22no—
503Q8—22no—
504Q8—22no—
505Q8—42no—
506Q8—54no—
507Q8—52no—
508Q8—108no—
509Q8—104no—
510Q8—102no—
511Q8—101no—
512D4—24no—
513D4—22no—
514D4—52no—
515D4—108no—
516D4—104no—
517D4—101no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—42no—
523C2×C4—108no—
524C2×C4—108no—
525C2×C4—104no—
526C2×C4—102no—
527C2×C4—101no—
528C2×C4—104no—
529C2×C4C3×D4×D512yes—
530C2×C4C3×D4×D512yes—
531C2×C4C2×C3×C22⋊D512yes—
532C2×C4C2×C3×C22⋊D511yescharacteristic
533C2×C4C23×C3×D511yescharacteristic
534C8—102no—
535C8—102no—
536C2×C3—22no—
537C2×C3—42no—
538C2×C3—202no—
539C2×C3C2×#408311yescharacteristic
540C2×C3#6617212yes—
541C5C3×#301111yescharacteristic, complemented, Sylow
542C22—22no—
543C22—22no—
544C22—24no—
545C22—102no—
546C22—202no—
547C22C3×#408312yes—
548C22C22×C3×C22⋊D511yescharacteristic
549C4—22no—
550C4—42no—
551C4—42no—
552C4—104no—
553C4—102no—
554C4C3×#408312yes—
555C4C2×C3×D4×D511yescharacteristic
556C4C2×C3×D4×D511yescharacteristic
557C3#1061514211yescharacteristic, direct product factor, Sylow
558C2—22no—
559C2—42no—
560C2—202no—
561C2C2×C3×#408311yescharacteristic
562C2C3×#6617212yes—
subgroupG/H
centerC22×C3C22×C22⋊D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC3×C5×#536C2
socleC22×C3×C5C22×D4
prime radicalC2×C3×#4150C2
hypercenterC3×#536D5
hypocenterC5C3×#3011
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C2×C3×#41 ◁ C3×#536

Complements

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

NH ≅ G/Ntype
2. C3×C5×#536C2semidirect
3. C22×C3×#689C2semidirect
4. C2×C3×#4088C2semidirect
5. C3×#65523C2semidirect
6. C3×#65339C2semidirect
7. C3×#65336C2semidirect
8. C3×#65293C2semidirect
9. C3×#65291C2semidirect
10. #10615142C3direct (N × H)
40. C5×#536C2×C3semidirect
41. C22×#689C2×C3semidirect
42. C2×#4088C2×C3semidirect
43. #65523C2×C3semidirect
44. #65339C2×C3semidirect
45. #65336C2×C3semidirect
46. #65293C2×C3semidirect
47. #65291C2×C3semidirect
97. C3×#536D5semidirect
244. #536C3×D5semidirect
478. C3×C5#3011semidirect
541. C5C3×#3011semidirect
557. C3#10615142direct (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
C3×C5×#5362
C22×C3×#6892
C2×C3×#40882
C3×#655232
C3×#653392
C3×#653362
C3×#652932
C3×#652912
#106151423
C3×#30115