Finite Group Databasebeta

← C3×#10610660 #14424322

Subgroups — C3×#10610660 #14424322

Order 1920.

1504 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4144C211yesmaximal, characteristic
2C22×C3×#687C211yesmaximal, characteristic, complemented
3C2×C3×#4055C211yesmaximal, characteristic, complemented
4C3×#65860C211yesmaximal, characteristic, complemented
5C3×C5×#418C211yesmaximal, characteristic, complemented
6C3×#65418C211yesmaximal, characteristic, complemented
7C3×#65291C211yesmaximal, characteristic, complemented
8C3×#65072C214yesmaximal, complemented
9C3×#65065C214yesmaximal, complemented
10#10610660C311yesmaximal, characteristic, direct product factor, Hall
11C22×C3×C5×D4C2211yescharacteristic
12C22×C3×C22⋊D5—22no—
13C3×#4144—24no—
14C3×#4144—24no—
15C2×C3×#711—21no—
16C2×C3×#711C2212yes—
17C2×C3×D4×D5—21no—
18C2×C3×D4×D5C2212yes—
19C2×C3×#709C2211yescharacteristic
20C2×C3×#709C2211yescharacteristic
21C22×C3×Dic10C2211yescharacteristic, complemented
22C2×C3×C5×M16C2211yescharacteristic, complemented
23C3×C5×#117C2211yescharacteristic, complemented
24C2×C3×#687—24no—
25C2×C3×#687C2214yescomplemented
26C22×C3×#168C2211yescharacteristic, complemented
27C3×#4055—24no—
28C2×C3×Dic20C2212yescomplemented
29C2×C3×#677C2212yescomplemented
30C3×#4043C2214yescomplemented
31C3×#4032—22no—
32C3×#4021—21no—
33C3×#4015—21no—
34C3×C5×#102C2214yescomplemented
35C3×#3952C2214yescomplemented
36C3×#3942C2212yescomplemented
37C3×#3941C2212yescomplemented
38C3×#3009—51nomaximal, Hall
39C2×#4144C2×C311yescharacteristic
40C22×#687C2×C311yescharacteristic, complemented
41C2×#4055C2×C311yescharacteristic, complemented
42#65860C2×C311yescharacteristic, complemented
43C5×#418C2×C311yescharacteristic, complemented
44#65418C2×C311yescharacteristic, complemented
45#65291C2×C311yescharacteristic, complemented
46#65072C2×C314yescomplemented
47#65065C2×C314yescomplemented
48C24×C3×C5—21no—
49C23×C3×D5—22no—
50C2×C3×C5×D4—21no—
51C2×C3×C5×D4—22no—
52C2×C3×C5×D4D414yes—
53C2×C3×C5×D4C2312yes—
54C22×C3×C4×C5C2311yescharacteristic
55C2×C3×C22⋊D5—216no—
56C2×C3×C22⋊D5—21no—
57C2×C3×C22⋊D5—22no—
58C2×C3×C22⋊D5—22no—
59C2×C3×C22⋊D5D412yes—
60C22×C3×Dic5—22no—
61C22×C3×Dic5—21no—
62C3×#711—28no—
63C3×#711—24no—
64C3×#711—24no—
65C3×D4×D5—28no—
66C3×D4×D5—24no—
67C3×D4×D5—24no—
68C3×#709—22no—
69C3×#709—22no—
70C3×#709—22no—
71C3×#709D414yes—
72C2×C3×D20D411yescharacteristic
73C2×C3×D20C2311yescharacteristic
74C2×C3×C4×D5—21no—
75C2×C3×C4×D5C2312yes—
76C2×C3×Dic10—22no—
77C2×C3×Dic10—22no—
78C2×C3×Dic10—21no—
79C2×C3×Dic10D411yescharacteristic
80C2×C3×Dic10C2312yes—
81C2×C3×Dic10C2311yescharacteristic
82C3×C5×M16—22no—
83C2×C3×C5×C8C2312yes—
84C3×C5×C4⋊C4C2312yes—
85C3×C5×C22⋊C4—21no—
86C3×C42×C5—21no—
87C3×#687—28no—
88C3×#685—24no—
89C3×C4⋊Dic5—24no—
90C2×C3×#168—22no—
91C2×C3×#168C2312yes—
92C3×Dic20—24no—
93C3×#677—24no—
94C2×C3×H(5,𝔽2)—51no—
95C2×C3×#137—51no—
96C22×C3×SD16—51no—
97C3×#536—51no—
98C3×#452—54no—
99C3×#450—54no—
100C3×#418D511yescharacteristic, complemented
101C3×#416—51no—
102C3×#408—51no—
103C22×C5×D4C22×C311yescharacteristic
104C22×C22⋊D5—22no—
105#4144—24no—
106#4144—24no—
107C2×#711—21no—
108C2×#711C22×C312yes—
109C2×D4×D5—21no—
110C2×D4×D5C22×C312yes—
111C2×#709C22×C311yescharacteristic
112C2×#709C22×C311yescharacteristic
113C22×Dic10C22×C311yescharacteristic, complemented
114C2×C5×M16C22×C311yescharacteristic, complemented
115C5×#117C22×C311yescharacteristic, complemented
116C2×#687—24no—
117C2×#687C22×C314yescomplemented
118C22×#168C22×C311yescharacteristic, complemented
119#4055—24no—
120C2×Dic20C22×C312yescomplemented
121C2×#677C22×C312yescomplemented
122#4043C22×C314yescomplemented
123#4032—22no—
124#4021—21no—
125#4015—21no—
126C5×#102C22×C314yescomplemented
127#3952C22×C314yescomplemented
128#3942C22×C312yescomplemented
129#3941C22×C312yescomplemented
130#3009—51noSylow
131C23×C3×C5—22no—
132C23×C3×C5—24no—
133C23×C3×C5—82no—
134C23×C3×C5C2×D411yescharacteristic
135C22×C3×D5—21no—
136C22×C3×D5—28no—
137C22×C3×D5—84no—
138C22×C3×D5C2×D412yes—
139C3×C5×D4—24no—
140C3×C5×D4—22no—
141C3×C5×D4C2×D414yes—
142C2×C3×C4×C5—22no—
143C2×C3×C4×C5C2×D411yescharacteristic
144C2×C3×C4×C5C2×D412yes—
145C2×C3×C4×C5C2×D412yes—
146C2×C3×C4×C5C2411yescharacteristic
147C3×C22⋊D5—28no—
148C3×C22⋊D5—28no—
149C3×C22⋊D5—24no—
150C3×C22⋊D5—24no—
151C2×C3×Dic5—28no—
152C2×C3×Dic5—21no—
153C2×C3×Dic5—22no—
154C2×C3×Dic5—42no—
155C2×C3×Dic5C2×D412yes—
156C3×D20—22no—
157C3×D20—21no—
158C3×D20C2×D412yes—
159C3×C4×D5—24no—
160C3×C4×D5—22no—
161C3×C4×D5—22no—
162C3×Dic10—22no—
163C3×Dic10—21no—
164C3×Dic10—22no—
165C3×Dic10—22no—
166C3×Dic10—42no—
167C3×Dic10C2×D412yes—
168C3×C5×C8—22no—
169C3×#168—22no—
170C3×H(5,𝔽2)—104no—
171C3×H(5,𝔽2)—104no—
172C2×C3×#41—51no—
173C2×C3×#41—52no—
174C2×C3×#41—51no—
175C2×C3×#41—101no—
176C22×C3×Q8—51no—
177C22×C3×D4—52no—
178C22×C3×D4—101no—
179C22×C3×D4—102no—
180C22×C3×D4C2×D511yescharacteristic
181C3×#137—104no—
182C2×C3×Q16—52no—
183C2×C3×SD16—54no—
184C2×C3×SD16—52no—
185C2×C3×SD16—104no—
186C2×C3×M16C2×D511yescharacteristic, complemented
187C22×C3×C8—51no—
188C3×C4⋊Q8—101no—
189C3×#124—101no—
190C3×#123—102no—
191C3×C22⋊Q8—54no—
192C3×#117C2×D511yescharacteristic, complemented
193C3×#103—52no—
194C3×#102—52no—
195C3×#102C2×D514yescomplemented
196C3×C22⋊C8—54no—
197C24×C5—21no—
198C23×D5—22no—
199C2×C5×D4—21no—
200C2×C5×D4—22no—
201C2×C5×D4C3×D414yes—
202C2×C5×D4C23×C312yes—
203C22×C4×C5C23×C311yescharacteristic
204C2×C22⋊D5—216no—
205C2×C22⋊D5—21no—
206C2×C22⋊D5—22no—
207C2×C22⋊D5—22no—
208C2×C22⋊D5C3×D412yes—
209C22×Dic5—22no—
210C22×Dic5—21no—
211#711—28no—
212#711—24no—
213#711—24no—
214D4×D5—28no—
215D4×D5—24no—
216D4×D5—24no—
217#709—22no—
218#709—22no—
219#709—22no—
220#709C3×D414yes—
221C2×D20C3×D411yescharacteristic
222C2×D20C23×C311yescharacteristic
223C2×C4×D5—21no—
224C2×C4×D5C23×C312yes—
225C2×Dic10—22no—
226C2×Dic10—22no—
227C2×Dic10—21no—
228C2×Dic10C3×D411yescharacteristic
229C2×Dic10C23×C312yes—
230C2×Dic10C23×C311yescharacteristic
231C5×M16—22no—
232C2×C5×C8C23×C312yes—
233C5×C4⋊C4C23×C312yes—
234C5×C22⋊C4—21no—
235C42×C5—21no—
236#687—28no—
237#685—24no—
238C4⋊Dic5—24no—
239C2×#168—22no—
240C2×#168C23×C312yes—
241Dic20—24no—
242#677—24no—
243C2×H(5,𝔽2)—51no—
244C2×#137—51no—
245C22×SD16—51no—
246#536—51no—
247#452—54no—
248#450—54no—
249#418C3×D511yescharacteristic, complemented
250#416—51no—
251#408—51no—
252C22×C3×C5—22no—
253C22×C3×C5—24no—
254C22×C3×C5—44no—
255C22×C3×C5—84no—
256C22×C3×C5#12012yes—
257C22×C3×C5C22×D411yescharacteristic
258C2×C3×D5—22no—
259C2×C3×D5—22no—
260C2×C3×D5—24no—
261C2×C3×D5—42no—
262C2×C3×D5—88no—
263C3×C4×C5—42no—
264C3×C4×C5#12012yes—
265C3×C4×C5C22×D411yescharacteristic
266C3×C4×C5C22×D411yescharacteristic
267C3×Dic5—22no—
268C3×Dic5—22no—
269C3×Dic5—42no—
270C24×C3—21no—
271C24×C3—102no—
272C3×#41—54no—
273C3×#41—104no—
274C3×#41—108no—
275C3×#41—104no—
276C3×#41—102no—
277C3×#41—102no—
278C3×#41—102no—
279C2×C3×Q8—51no—
280C2×C3×Q8—51no—
281C2×C3×Q8—52no—
282C2×C3×Q8—101no—
283C2×C3×Q8—102no—
284C2×C3×Q8—102no—
285C2×C3×D4—21no—
286C2×C3×D4—22no—
287C2×C3×D4—52no—
288C2×C3×D4—51no—
289C2×C3×D4—51no—
290C2×C3×D4—102no—
291C2×C3×D4—104no—
292C2×C3×D4—108no—
293C2×C3×D4—102no—
294C2×C3×D4—1016no—
295C2×C3×D4—101no—
296C2×C3×D4—104no—
297C2×C3×D4D2014yes—
298C2×C3×D4C22×D512yes—
299C22×C3×C4—52no—
300C22×C3×C4—102no—
301C22×C3×C4—101no—
302C22×C3×C4—101no—
303C22×C3×C4C22×D511yescharacteristic
304C3×Q16—104no—
305C3×SD16—104no—
306C3×SD16—108no—
307C3×M16—22no—
308C2×C3×C8—52no—
309C2×C3×C8—102no—
310C2×C3×C8C22×D512yes—
311C3×C4⋊C4—104no—
312C3×C4⋊C4C22×D512yes—
313C3×C22⋊C4—21no—
314C3×C22⋊C4—104no—
315C3×C42—21no—
316C23×C5—22no—
317C23×C5—24no—
318C23×C5—82no—
319C23×C5C2×C3×D411yescharacteristic
320C22×D5—21no—
321C22×D5—28no—
322C22×D5—84no—
323C22×D5C2×C3×D412yes—
324C5×D4—24no—
325C5×D4—22no—
326C5×D4C2×C3×D414yes—
327C2×C4×C5—22no—
328C2×C4×C5C2×C3×D412yes—
329C2×C4×C5C2×C3×D411yescharacteristic
330C2×C4×C5C2×C3×D412yes—
331C2×C4×C5C24×C311yescharacteristic
332C22⋊D5—28no—
333C22⋊D5—28no—
334C22⋊D5—24no—
335C22⋊D5—24no—
336C2×Dic5—28no—
337C2×Dic5—21no—
338C2×Dic5—22no—
339C2×Dic5—42no—
340C2×Dic5C2×C3×D412yes—
341D20—22no—
342D20—21no—
343D20C2×C3×D412yes—
344C4×D5—24no—
345C4×D5—22no—
346C4×D5—22no—
347Dic10—22no—
348Dic10—21no—
349Dic10—22no—
350Dic10—22no—
351Dic10—42no—
352Dic10C2×C3×D412yes—
353C5×C8—22no—
354#168—22no—
355H(5,𝔽2)—104no—
356H(5,𝔽2)—104no—
357C2×#41—51no—
358C2×#41—52no—
359C2×#41—51no—
360C2×#41—101no—
361C22×Q8—51no—
362C22×D4—52no—
363C22×D4—101no—
364C22×D4—102no—
365C22×D4C2×C3×D511yescharacteristic
366#137—104no—
367C2×Q16—52no—
368C2×SD16—54no—
369C2×SD16—52no—
370C2×SD16—104no—
371C2×M16C2×C3×D511yescharacteristic, complemented
372C22×C8—51no—
373C4⋊Q8—101no—
374#124—101no—
375#123—102no—
376C22⋊Q8—54no—
377#117C2×C3×D511yescharacteristic, complemented
378#103—52no—
379#102—52no—
380#102C2×C3×D514yescomplemented
381C22⋊C8—54no—
382C2×C3×C5—22no—
383C2×C3×C5—44no—
384C2×C3×C5C2×#12011yescharacteristic
385C2×C3×C5#57712yes—
386C3×D5—42no—
387C3×D5—42no—
388C23×C3—22no—
389C23×C3—24no—
390C23×C3—52no—
391C23×C3—82no—
392C23×C3—108no—
393C23×C3—101no—
394C23×C3—404no—
395C23×C3D4×D511yescharacteristic
396C3×Q8—52no—
397C3×Q8—101no—
398C3×Q8—102no—
399C3×Q8—102no—
400C3×Q8—102no—
401C3×Q8—202no—
402C3×D4—22no—
403C3×D4—24no—
404C3×D4—52no—
405C3×D4—108no—
406C3×D4—108no—
407C3×D4—104no—
408C3×D4—104no—
409C3×D4—101no—
410C3×D4—102no—
411C3×D4C2×D2014yes—
412C2×C3×C4—22no—
413C2×C3×C4—52no—
414C2×C3×C4—108no—
415C2×C3×C4—101no—
416C2×C3×C4—102no—
417C2×C3×C4—102no—
418C2×C3×C4—104no—
419C2×C3×C4—102no—
420C2×C3×C4—202no—
421C2×C3×C4C2×D2012yes—
422C2×C3×C4D4×D512yes—
423C2×C3×C4D4×D511yescharacteristic
424C2×C3×C4C23×D511yescharacteristic
425C3×C8—22no—
426C3×C8—102no—
427C22×C5—22no—
428C22×C5—24no—
429C22×C5—44no—
430C22×C5—84no—
431C22×C5C3×#12012yes—
432C22×C5C22×C3×D411yescharacteristic
433C2×D5—24no—
434C2×D5—22no—
435C2×D5—22no—
436C2×D5—42no—
437C2×D5—88no—
438C4×C5—42no—
439C4×C5C3×#12012yes—
440C4×C5C22×C3×D411yescharacteristic
441C4×C5C22×C3×D411yescharacteristic
442Dic5—22no—
443Dic5—22no—
444Dic5—42no—
445C24—21no—
446C24—102no—
447#41—54no—
448#41—104no—
449#41—108no—
450#41—104no—
451#41—102no—
452#41—102no—
453#41—102no—
454C2×Q8—51no—
455C2×Q8—51no—
456C2×Q8—52no—
457C2×Q8—101no—
458C2×Q8—102no—
459C2×Q8—102no—
460C2×D4—21no—
461C2×D4—22no—
462C2×D4—52no—
463C2×D4—51no—
464C2×D4—51no—
465C2×D4—102no—
466C2×D4—104no—
467C2×D4—108no—
468C2×D4—102no—
469C2×D4—1016no—
470C2×D4—101no—
471C2×D4—104no—
472C2×D4C3×D2014yes—
473C2×D4C22×C3×D512yes—
474C22×C4—52no—
475C22×C4—102no—
476C22×C4—101no—
477C22×C4—101no—
478C22×C4C22×C3×D511yescharacteristic
479Q16—104no—
480SD16—104no—
481SD16—108no—
482M16—22no—
483C2×C8—52no—
484C2×C8—102no—
485C2×C8C22×C3×D512yes—
486C4⋊C4—104no—
487C4⋊C4C22×C3×D512yes—
488C22⋊C4—21no—
489C22⋊C4—104no—
490C42—21no—
491C3×C5#300911yescharacteristic, complemented, Hall
492C22×C3—22no—
493C22×C3—24no—
494C22×C3—44no—
495C22×C3—84no—
496C22×C3—102no—
497C22×C3—102no—
498C22×C3—104no—
499C22×C3—202no—
500C22×C3—408no—
501C22×C3#402812yes—
502C22×C3C2×D4×D511yescharacteristic
503C3×C4—42no—
504C3×C4—102no—
505C3×C4—102no—
506C3×C4—202no—
507C3×C4#402812yes—
508C3×C4C22×D2011yescharacteristic
509C3×C4C2×D4×D511yescharacteristic
510C2×C5—22no—
511C2×C5—44no—
512C2×C5C2×C3×#12011yescharacteristic
513C2×C5C3×#57712yes—
514D5—42no—
515D5—42no—
516C23—22no—
517C23—24no—
518C23—52no—
519C23—82no—
520C23—108no—
521C23—101no—
522C23—404no—
523C23C3×D4×D511yescharacteristic
524Q8—52no—
525Q8—101no—
526Q8—102no—
527Q8—102no—
528Q8—102no—
529Q8—202no—
530D4—22no—
531D4—24no—
532D4—52no—
533D4—108no—
534D4—108no—
535D4—104no—
536D4—104no—
537D4—101no—
538D4—102no—
539D4C2×C3×D2014yes—
540C2×C4—22no—
541C2×C4—52no—
542C2×C4—108no—
543C2×C4—101no—
544C2×C4—102no—
545C2×C4—102no—
546C2×C4—104no—
547C2×C4—102no—
548C2×C4—202no—
549C2×C4C2×C3×D2012yes—
550C2×C4C3×D4×D512yes—
551C2×C4C3×D4×D511yescharacteristic
552C2×C4C23×C3×D511yescharacteristic
553C8—22no—
554C8—102no—
555C2×C3—22no—
556C2×C3—44no—
557C2×C3—202no—
558C2×C3—202no—
559C2×C3C2×#402811yescharacteristic
560C2×C3#6610912yes—
561C5C3×#300911yescharacteristic, complemented, Sylow
562C22—22no—
563C22—24no—
564C22—44no—
565C22—84no—
566C22—104no—
567C22—102no—
568C22—102no—
569C22—202no—
570C22—408no—
571C22C3×#402812yes—
572C22C2×C3×D4×D511yescharacteristic
573C4—42no—
574C4—102no—
575C4—102no—
576C4—202no—
577C4C3×#402812yes—
578C4C22×C3×D2011yescharacteristic
579C4C2×C3×D4×D511yescharacteristic
580C3#1061066011yescharacteristic, direct product factor, Sylow
581C2—22no—
582C2—44no—
583C2—202no—
584C2—202no—
585C2C2×C3×#402811yescharacteristic
586C2C3×#6610912yes—
subgroupG/H
centerC22×C3C2×D4×D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC3×C5×#418C2
socleC22×C3×C5C22×D4
prime radicalC2×C3×#4144C2
hypercenterC3×#418D5
hypocenterC5C3×#3009
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#418

Complements

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

NH ≅ G/Ntype
2. C22×C3×#687C2semidirect
3. C2×C3×#4055C2semidirect
4. C3×#65860C2semidirect
5. C3×C5×#418C2semidirect
6. C3×#65418C2semidirect
7. C3×#65291C2semidirect
8. C3×#65072C2semidirect
9. C3×#65065C2semidirect
10. #10610660C3direct (N × H)
21. C22×C3×Dic10C22semidirect
22. C2×C3×C5×M16C22semidirect
23. C3×C5×#117C22semidirect
25. C2×C3×#687C22semidirect
26. C22×C3×#168C22semidirect
28. C2×C3×Dic20C22semidirect
29. C2×C3×#677C22semidirect
30. C3×#4043C22semidirect
34. C3×C5×#102C22semidirect
35. C3×#3952C22semidirect
36. C3×#3942C22semidirect
37. C3×#3941C22semidirect
40. C22×#687C2×C3semidirect
41. C2×#4055C2×C3semidirect
42. #65860C2×C3semidirect
43. C5×#418C2×C3semidirect
44. #65418C2×C3semidirect
45. #65291C2×C3semidirect
46. #65072C2×C3semidirect
47. #65065C2×C3semidirect
100. C3×#418D5semidirect
113. C22×Dic10C22×C3semidirect
114. C2×C5×M16C22×C3semidirect
115. C5×#117C22×C3semidirect
117. C2×#687C22×C3semidirect
118. C22×#168C22×C3semidirect
120. C2×Dic20C22×C3semidirect
121. C2×#677C22×C3semidirect
122. #4043C22×C3semidirect
126. C5×#102C22×C3semidirect
127. #3952C22×C3semidirect
128. #3942C22×C3semidirect
129. #3941C22×C3semidirect
186. C2×C3×M16C2×D5semidirect
192. C3×#117C2×D5semidirect
195. C3×#102C2×D5semidirect
249. #418C3×D5semidirect
371. C2×M16C2×C3×D5semidirect
377. #117C2×C3×D5semidirect
380. #102C2×C3×D5semidirect
491. C3×C5#3009semidirect
561. C5C3×#3009semidirect
580. C3#10610660direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#41442
C22×C3×#6872
C2×C3×#40552
C3×#658602
C3×C5×#4182
C3×#654182
C3×#652912
C3×#650722
C3×#650652
#106106603
C3×#30095