Finite Group Databasebeta

← C3×#10615135 #14424313

Subgroups — C3×#10615135 #14424313

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×#4149C211yesmaximal, characteristic
2C3×C5×#536C211yesmaximal, characteristic, complemented
3C2×C3×#4088C211yesmaximal, characteristic, complemented
4C22×C3×#688C211yesmaximal, characteristic, complemented
5C3×#65523C211yesmaximal, characteristic, complemented
6C3×#65336C214yesmaximal, complemented
7C3×#65335C214yesmaximal, complemented
8C3×#65291C211yesmaximal, characteristic, complemented
9C3×#65290C211yesmaximal, characteristic, complemented
10#10615135C311yesmaximal, characteristic, direct product factor, Hall
11C2×C3×C5×#41C2211yescharacteristic
12C22×C3×C5×Q8C2211yescharacteristic, complemented
13C3×#4149—24no—
14C3×#4149—24no—
15C2×C3×#713—21no—
16C2×C3×D4×D5—22no—
17C2×C3×D4×D5—21no—
18C2×C3×#709C2212yes—
19C2×C3×#709C2211yescharacteristic
20C22×C3×D20C2212yes—
21C22×C3×D20C2211yescharacteristic
22C3×C5×C4⋊Q8—21no—
23C3×C5×#124—21no—
24C3×C5×#123—22no—
25C3×C5×C22⋊Q8C2214yescomplemented
26C3×C5×#117C2211yescharacteristic, complemented
27C2×C3×#689C2212yescomplemented
28C3×#4088—24no—
29C2×C3×#688—24no—
30C2×C3×#688C2214yescomplemented
31C2×C3×#688C2212yescomplemented
32C2×C3×#681C2211yescharacteristic, complemented
33C22×C3×#168C2211yescharacteristic, complemented
34C3×C22⋊#168C2214yescomplemented
35C3×#3942C2212yescomplemented
36C3×#3941C2214yescomplemented
37C3×#3941C2212yescomplemented
38C3×#3009—51nomaximal, Hall
39C2×#4149C2×C311yescharacteristic
40C5×#536C2×C311yescharacteristic, complemented
41C2×#4088C2×C311yescharacteristic, complemented
42C22×#688C2×C311yescharacteristic, complemented
43#65523C2×C311yescharacteristic, complemented
44#65336C2×C314yescomplemented
45#65335C2×C314yescomplemented
46#65291C2×C311yescharacteristic, complemented
47#65290C2×C311yescharacteristic, complemented
48C23×C3×D5—22no—
49C23×C3×D5—21no—
50C3×C5×#41—22no—
51C3×C5×#41—22no—
52C2×C3×C5×Q8—21no—
53C2×C3×C5×Q8—22no—
54C2×C3×C5×Q8—22no—
55C2×C3×C5×Q8D411yescharacteristic
56C2×C3×C5×Q8C2312yes—
57C2×C3×C5×D4D412yes—
58C2×C3×C5×D4D411yescharacteristic
59C22×C3×C4×C5—21no—
60C22×C3×C4×C5C2312yes—
61C22×C3×C4×C5C2311yescharacteristic
62C2×C3×C22⋊D5—22no—
63C2×C3×C22⋊D5—21no—
64C3×#713—28no—
65C3×D4×D5—216no—
66C3×D4×D5—28no—
67C3×#709—22no—
68C3×#709—24no—
69C3×#709—24no—
70C3×#709D414yes—
71C2×C3×D20—22no—
72C2×C3×D20—21no—
73C2×C3×D20—22no—
74C2×C3×D20—24no—
75C2×C3×D20—24no—
76C2×C3×D20D414yes—
77C2×C3×D20C2311yescharacteristic
78C2×C3×D20C2312yes—
79C2×C3×C4×D5—22no—
80C2×C3×C4×D5—21no—
81C2×C3×Dic10C2311yescharacteristic
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×#688—28no—
89C3×#688—24no—
90C3×#681—22no—
91C2×C3×#168—22no—
92C2×C3×#168C2312yes—
93C2×C3×#168C2312yes—
94C2×C3×H(5,𝔽2)—51no—
95C2×C3×#137—51no—
96C22×C3×SD16—51no—
97C3×#536D511yescharacteristic, complemented
98C3×#452—54no—
99C3×#450—54no—
100C3×#418—51no—
101C3×#416—51no—
102C3×#408—51no—
103C2×C5×#41C22×C311yescharacteristic
104C22×C5×Q8C22×C311yescharacteristic, complemented
105#4149—24no—
106#4149—24no—
107C2×#713—21no—
108C2×D4×D5—22no—
109C2×D4×D5—21no—
110C2×#709C22×C312yes—
111C2×#709C22×C311yescharacteristic
112C22×D20C22×C312yes—
113C22×D20C22×C311yescharacteristic
114C5×C4⋊Q8—21no—
115C5×#124—21no—
116C5×#123—22no—
117C5×C22⋊Q8C22×C314yescomplemented
118C5×#117C22×C311yescharacteristic, complemented
119C2×#689C22×C312yescomplemented
120#4088—24no—
121C2×#688—24no—
122C2×#688C22×C312yescomplemented
123C2×#688C22×C314yescomplemented
124C2×#681C22×C311yescharacteristic, complemented
125C22×#168C22×C311yescharacteristic, complemented
126C22⋊#168C22×C314yescomplemented
127#3942C22×C312yescomplemented
128#3941C22×C314yescomplemented
129#3941C22×C312yescomplemented
130#3009—51noSylow
131C23×C3×C5C2×D412yes—
132C23×C3×C5C2×D411yescharacteristic
133C22×C3×D5—22no—
134C22×C3×D5—21no—
135C22×C3×D5—28no—
136C22×C3×D5—24no—
137C22×C3×D5—82no—
138C22×C3×D5—84no—
139C3×C5×Q8—22no—
140C3×C5×Q8—22no—
141C3×C5×Q8—22no—
142C3×C5×Q8—42no—
143C3×C5×D4—24no—
144C3×C5×D4—22no—
145C2×C3×C4×C5—22no—
146C2×C3×C4×C5—22no—
147C2×C3×C4×C5—22no—
148C2×C3×C4×C5—22no—
149C2×C3×C4×C5—42no—
150C2×C3×C4×C5C2×D412yes—
151C2×C3×C4×C5C2×D411yescharacteristic
152C2×C3×C4×C5C2×D412yes—
153C2×C3×C4×C5C2×D412yes—
154C2×C3×C4×C5C2411yescharacteristic
155C3×C22⋊D5—28no—
156C3×C22⋊D5—24no—
157C2×C3×Dic5—21no—
158C3×D20—28no—
159C3×D20—24no—
160C3×D20—21no—
161C3×D20—22no—
162C3×D20C2×D412yes—
163C3×D20C2×D414yes—
164C3×C4×D5—28no—
165C3×C4×D5—24no—
166C3×Dic10—21no—
167C3×Dic10C2×D412yes—
168C3×#168—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—101no—
175C2×C3×#41C2×D511yescharacteristic
176C22×C3×Q8C2×D511yescharacteristic, complemented
177C22×C3×D4—51no—
178C22×C3×D4—52no—
179C22×C3×D4—102no—
180C22×C3×D4—101no—
181C3×#137—104no—
182C2×C3×Q16—52no—
183C2×C3×SD16—54no—
184C2×C3×SD16—52no—
185C2×C3×SD16—104no—
186C2×C3×M16—51no—
187C22×C3×C8—51no—
188C3×C4⋊Q8—21no—
189C3×#124—21no—
190C3×#123—22no—
191C3×C22⋊Q8C2×D514yescomplemented
192C3×#117C2×D511yescharacteristic, complemented
193C3×#103—52no—
194C3×#102—52no—
195C3×#102—54no—
196C3×C22⋊C8—54no—
197C23×D5—22no—
198C23×D5—21no—
199C5×#41—22no—
200C5×#41—22no—
201C2×C5×Q8—21no—
202C2×C5×Q8—22no—
203C2×C5×Q8—22no—
204C2×C5×Q8C3×D411yescharacteristic
205C2×C5×Q8C23×C312yes—
206C2×C5×D4C3×D412yes—
207C2×C5×D4C3×D411yescharacteristic
208C22×C4×C5—21no—
209C22×C4×C5C23×C312yes—
210C22×C4×C5C23×C311yescharacteristic
211C2×C22⋊D5—22no—
212C2×C22⋊D5—21no—
213#713—28no—
214D4×D5—216no—
215D4×D5—28no—
216#709—22no—
217#709—24no—
218#709—24no—
219#709C3×D414yes—
220C2×D20—22no—
221C2×D20—21no—
222C2×D20—22no—
223C2×D20—24no—
224C2×D20—24no—
225C2×D20C3×D414yes—
226C2×D20C23×C311yescharacteristic
227C2×D20C23×C312yes—
228C2×C4×D5—22no—
229C2×C4×D5—21no—
230C2×Dic10C23×C311yescharacteristic
231C5×C4⋊C4—24no—
232C5×C4⋊C4C23×C312yes—
233C5×C22⋊C4—21no—
234C5×C22⋊C4—24no—
235C42×C5—21no—
236#689—24no—
237#688—28no—
238#688—24no—
239#681—22no—
240C2×#168—22no—
241C2×#168C23×C312yes—
242C2×#168C23×C312yes—
243C2×H(5,𝔽2)—51no—
244C2×#137—51no—
245C22×SD16—51no—
246#536C3×D511yescharacteristic, complemented
247#452—54no—
248#450—54no—
249#418—51no—
250#416—51no—
251#408—51no—
252C22×C3×C5—22no—
253C22×C3×C5—22no—
254C22×C3×C5—24no—
255C22×C3×C5#12012yes—
256C22×C3×C5C22×D411yescharacteristic
257C2×C3×D5—24no—
258C2×C3×D5—22no—
259C2×C3×D5—44no—
260C2×C3×D5—42no—
261C2×C3×D5—84no—
262C2×C3×D5—88no—
263C3×C4×C5—22no—
264C3×C4×C5—42no—
265C3×C4×C5—42no—
266C3×C4×C5#12012yes—
267C3×C4×C5C22×D411yescharacteristic
268C3×C4×C5C22×D411yescharacteristic
269C3×Dic5—22no—
270C24×C3—102no—
271C24×C3—101no—
272C3×#41—22no—
273C3×#41—22no—
274C3×#41—54no—
275C3×#41—104no—
276C3×#41—104no—
277C3×#41—108no—
278C3×#41—102no—
279C2×C3×Q8—21no—
280C2×C3×Q8—22no—
281C2×C3×Q8—22no—
282C2×C3×Q8—51no—
283C2×C3×Q8C22⋊D511yescharacteristic
284C2×C3×Q8C22×D512yes—
285C2×C3×D4—54no—
286C2×C3×D4—52no—
287C2×C3×D4—51no—
288C2×C3×D4—104no—
289C2×C3×D4—104no—
290C2×C3×D4—1016no—
291C2×C3×D4—108no—
292C2×C3×D4—102no—
293C2×C3×D4—101no—
294C2×C3×D4—102no—
295C2×C3×D4—102no—
296C2×C3×D4—101no—
297C2×C3×D4C22⋊D512yes—
298C2×C3×D4C22⋊D511yescharacteristic
299C22×C3×C4—21no—
300C22×C3×C4—102no—
301C22×C3×C4—101no—
302C22×C3×C4C22×D512yes—
303C22×C3×C4C22×D511yescharacteristic
304C3×Q16—104no—
305C3×SD16—108no—
306C3×SD16—104no—
307C3×M16—102no—
308C2×C3×C8—52no—
309C2×C3×C8—52no—
310C2×C3×C8—102no—
311C3×C4⋊C4—24no—
312C3×C4⋊C4C22×D512yes—
313C3×C22⋊C4—21no—
314C3×C22⋊C4—24no—
315C3×C42—21no—
316C23×C5C2×C3×D412yes—
317C23×C5C2×C3×D411yescharacteristic
318C22×D5—22no—
319C22×D5—21no—
320C22×D5—28no—
321C22×D5—24no—
322C22×D5—82no—
323C22×D5—84no—
324C5×Q8—22no—
325C5×Q8—22no—
326C5×Q8—22no—
327C5×Q8—42no—
328C5×D4—24no—
329C5×D4—22no—
330C2×C4×C5—22no—
331C2×C4×C5—22no—
332C2×C4×C5—22no—
333C2×C4×C5—22no—
334C2×C4×C5—42no—
335C2×C4×C5C2×C3×D412yes—
336C2×C4×C5C2×C3×D412yes—
337C2×C4×C5C2×C3×D412yes—
338C2×C4×C5C2×C3×D411yescharacteristic
339C2×C4×C5C24×C311yescharacteristic
340C22⋊D5—28no—
341C22⋊D5—24no—
342C2×Dic5—21no—
343D20—28no—
344D20—24no—
345D20—21no—
346D20—22no—
347D20C2×C3×D412yes—
348D20C2×C3×D414yes—
349C4×D5—28no—
350C4×D5—24no—
351Dic10—21no—
352Dic10C2×C3×D412yes—
353#168—22no—
354#168—22no—
355H(5,𝔽2)—104no—
356H(5,𝔽2)—104no—
357C2×#41—51no—
358C2×#41—52no—
359C2×#41—101no—
360C2×#41C2×C3×D511yescharacteristic
361C22×Q8C2×C3×D511yescharacteristic, complemented
362C22×D4—51no—
363C22×D4—52no—
364C22×D4—102no—
365C22×D4—101no—
366#137—104no—
367C2×Q16—52no—
368C2×SD16—54no—
369C2×SD16—52no—
370C2×SD16—104no—
371C2×M16—51no—
372C22×C8—51no—
373C4⋊Q8—21no—
374#124—21no—
375#123—22no—
376C22⋊Q8C2×C3×D514yescomplemented
377#117C2×C3×D511yescharacteristic, complemented
378#103—52no—
379#102—52no—
380#102—54no—
381C22⋊C8—54no—
382C2×C3×C5—22no—
383C2×C3×C5—42no—
384C2×C3×C5C2×#12011yescharacteristic
385C2×C3×C5#57712yes—
386C3×D5—44no—
387C3×D5—42no—
388C23×C3—108no—
389C23×C3—102no—
390C23×C3—101no—
391C23×C3—104no—
392C23×C3—402no—
393C23×C3—404no—
394C23×C3C2×C22⋊D512yes—
395C23×C3C2×C22⋊D511yescharacteristic
396C3×Q8—22no—
397C3×Q8—22no—
398C3×Q8—22no—
399C3×Q8—42no—
400C3×Q8—52no—
401C3×Q8—101no—
402C3×D4—24no—
403C3×D4—22no—
404C3×D4—54no—
405C3×D4—52no—
406C3×D4—108no—
407C3×D4—108no—
408C3×D4—104no—
409C3×D4—101no—
410C3×D4—102no—
411C3×D4—104no—
412C2×C3×C4—22no—
413C2×C3×C4—22no—
414C2×C3×C4—22no—
415C2×C3×C4—22no—
416C2×C3×C4—42no—
417C2×C3×C4—108no—
418C2×C3×C4—101no—
419C2×C3×C4—104no—
420C2×C3×C4D4×D512yes—
421C2×C3×C4D4×D512yes—
422C2×C3×C4C2×C22⋊D512yes—
423C2×C3×C4C2×C22⋊D511yescharacteristic
424C2×C3×C4C23×D511yescharacteristic
425C3×C8—102no—
426C3×C8—102no—
427C22×C5—22no—
428C22×C5—24no—
429C22×C5—22no—
430C22×C5C3×#12012yes—
431C22×C5C22×C3×D411yescharacteristic
432C2×D5—24no—
433C2×D5—22no—
434C2×D5—44no—
435C2×D5—42no—
436C2×D5—84no—
437C2×D5—88no—
438C4×C5—22no—
439C4×C5—42no—
440C4×C5—42no—
441C4×C5C3×#12012yes—
442C4×C5C22×C3×D411yescharacteristic
443C4×C5C22×C3×D411yescharacteristic
444Dic5—22no—
445C24—102no—
446C24—101no—
447#41—22no—
448#41—22no—
449#41—54no—
450#41—104no—
451#41—104no—
452#41—108no—
453#41—102no—
454C2×Q8—21no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—51no—
458C2×Q8C3×C22⋊D511yescharacteristic
459C2×Q8C22×C3×D512yes—
460C2×D4—54no—
461C2×D4—52no—
462C2×D4—51no—
463C2×D4—104no—
464C2×D4—104no—
465C2×D4—1016no—
466C2×D4—108no—
467C2×D4—102no—
468C2×D4—101no—
469C2×D4—102no—
470C2×D4—102no—
471C2×D4—101no—
472C2×D4C3×C22⋊D512yes—
473C2×D4C3×C22⋊D511yescharacteristic
474C22×C4—21no—
475C22×C4—102no—
476C22×C4—101no—
477C22×C4C22×C3×D512yes—
478C22×C4C22×C3×D511yescharacteristic
479Q16—104no—
480SD16—108no—
481SD16—104no—
482M16—102no—
483C2×C8—52no—
484C2×C8—52no—
485C2×C8—102no—
486C4⋊C4—24no—
487C4⋊C4C22×C3×D512yes—
488C22⋊C4—21no—
489C22⋊C4—24no—
490C42—21no—
491C3×C5#300911yescharacteristic, complemented, Hall
492C22×C3—22no—
493C22×C3—22no—
494C22×C3—24no—
495C22×C3—104no—
496C22×C3—102no—
497C22×C3—204no—
498C22×C3—202no—
499C22×C3—404no—
500C22×C3—408no—
501C22×C3#408312yes—
502C22×C3C22×C22⋊D511yescharacteristic
503C3×C4—22no—
504C3×C4—42no—
505C3×C4—42no—
506C3×C4—102no—
507C3×C4#408312yes—
508C3×C4C2×D4×D511yescharacteristic
509C3×C4C2×D4×D511yescharacteristic
510C2×C5—22no—
511C2×C5—42no—
512C2×C5C2×C3×#12011yescharacteristic
513C2×C5C3×#57712yes—
514D5—44no—
515D5—42no—
516C23—108no—
517C23—102no—
518C23—101no—
519C23—104no—
520C23—402no—
521C23—404no—
522C23C2×C3×C22⋊D512yes—
523C23C2×C3×C22⋊D511yescharacteristic
524Q8—22no—
525Q8—22no—
526Q8—22no—
527Q8—42no—
528Q8—52no—
529Q8—101no—
530D4—24no—
531D4—22no—
532D4—54no—
533D4—52no—
534D4—108no—
535D4—108no—
536D4—104no—
537D4—104no—
538D4—101no—
539D4—102no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—42no—
545C2×C4—108no—
546C2×C4—101no—
547C2×C4—104no—
548C2×C4C3×D4×D512yes—
549C2×C4C3×D4×D512yes—
550C2×C4C2×C3×C22⋊D512yes—
551C2×C4C2×C3×C22⋊D511yescharacteristic
552C2×C4C23×C3×D511yescharacteristic
553C8—102no—
554C8—102no—
555C2×C3—22no—
556C2×C3—42no—
557C2×C3—204no—
558C2×C3—202no—
559C2×C3C2×#408311yescharacteristic
560C2×C3#6617112yes—
561C5C3×#300911yescharacteristic, complemented, Sylow
562C22—22no—
563C22—22no—
564C22—24no—
565C22—104no—
566C22—102no—
567C22—204no—
568C22—202no—
569C22—404no—
570C22—408no—
571C22C3×#408312yes—
572C22C22×C3×C22⋊D511yescharacteristic
573C4—22no—
574C4—42no—
575C4—42no—
576C4—102no—
577C4C3×#408312yes—
578C4C2×C3×D4×D511yescharacteristic
579C4C2×C3×D4×D511yescharacteristic
580C3#1061513511yescharacteristic, direct product factor, Sylow
581C2—22no—
582C2—42no—
583C2—204no—
584C2—202no—
585C2C2×C3×#408311yescharacteristic
586C2C3×#6617112yes—
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×#4149C2
hypercenterC3×#536D5
hypocenterC5C3×#3009
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. C2×C3×#4088C2semidirect
4. C22×C3×#688C2semidirect
5. C3×#65523C2semidirect
6. C3×#65336C2semidirect
7. C3×#65335C2semidirect
8. C3×#65291C2semidirect
9. C3×#65290C2semidirect
10. #10615135C3direct (N × H)
12. C22×C3×C5×Q8C22semidirect
25. C3×C5×C22⋊Q8C22semidirect
26. C3×C5×#117C22semidirect
27. C2×C3×#689C22semidirect
30. C2×C3×#688C22semidirect
31. C2×C3×#688C22semidirect
32. C2×C3×#681C22semidirect
33. C22×C3×#168C22semidirect
34. C3×C22⋊#168C22semidirect
35. C3×#3942C22semidirect
36. C3×#3941C22semidirect
37. C3×#3941C22semidirect
40. C5×#536C2×C3semidirect
41. C2×#4088C2×C3semidirect
42. C22×#688C2×C3semidirect
43. #65523C2×C3semidirect
44. #65336C2×C3semidirect
45. #65335C2×C3semidirect
46. #65291C2×C3semidirect
47. #65290C2×C3semidirect
97. C3×#536D5semidirect
104. C22×C5×Q8C22×C3semidirect
117. C5×C22⋊Q8C22×C3semidirect
118. C5×#117C22×C3semidirect
119. C2×#689C22×C3semidirect
122. C2×#688C22×C3semidirect
123. C2×#688C22×C3semidirect
124. C2×#681C22×C3semidirect
125. C22×#168C22×C3semidirect
126. C22⋊#168C22×C3semidirect
127. #3942C22×C3semidirect
128. #3941C22×C3semidirect
129. #3941C22×C3semidirect
176. C22×C3×Q8C2×D5semidirect
191. C3×C22⋊Q8C2×D5semidirect
192. C3×#117C2×D5semidirect
246. #536C3×D5semidirect
361. C22×Q8C2×C3×D5semidirect
376. C22⋊Q8C2×C3×D5semidirect
377. #117C2×C3×D5semidirect
491. C3×C5#3009semidirect
561. C5C3×#3009semidirect
580. C3#10615135direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#41492
C3×C5×#5362
C2×C3×#40882
C22×C3×#6882
C3×#655232
C3×#653362
C3×#653352
C3×#652912
C3×#652902
#106151353
C3×#30095