Finite Group Databasebeta

← C3×#10610659 #14424211

Subgroups — C3×#10610659 #14424211

Order 1920.

1568 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4144C211yesmaximal, characteristic, complemented
2C22×C3×#686C211yesmaximal, characteristic, complemented
3C2×C3×#4054C211yesmaximal, characteristic, complemented
4C3×#65859C211yesmaximal, characteristic, complemented
5C3×C5×#418C211yesmaximal, characteristic, complemented
6C3×#65418C211yesmaximal, characteristic, complemented
7C3×#65291C211yesmaximal, characteristic, complemented
8C3×#65070C214yesmaximal, complemented
9C3×#65062C214yesmaximal, complemented
10#10610659C311yesmaximal, 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×#711C2212yescomplemented
17C2×C3×D4×D5—21no—
18C2×C3×D4×D5C2212yescomplemented
19C2×C3×#709C2211yescharacteristic
20C2×C3×#709C2211yescharacteristic
21C22×C3×D20C2211yescharacteristic, complemented
22C2×C3×C5×M16C2211yescharacteristic, complemented
23C3×C5×#117C2211yescharacteristic, complemented
24C2×C3×#686—24no—
25C2×C3×#686C2214yescomplemented
26C22×C3×#168C2211yescharacteristic, complemented
27C3×#4054—24no—
28C2×C3×D40C2212yescomplemented
29C2×C3×#677C2212yescomplemented
30C3×#4041C2214yescomplemented
31C3×#4028—22no—
32C3×#4021—21no—
33C3×#4020—21no—
34C3×C5×#102C2214yescomplemented
35C3×#3952C2214yescomplemented
36C3×#3942C2212yescomplemented
37C3×#3941C2212yescomplemented
38C3×#3008—51nomaximal, Hall
39C2×#4144C2×C311yescharacteristic, complemented
40C22×#686C2×C311yescharacteristic, complemented
41C2×#4054C2×C311yescharacteristic, complemented
42#65859C2×C311yescharacteristic, complemented
43C5×#418C2×C311yescharacteristic, complemented
44#65418C2×C311yescharacteristic, complemented
45#65291C2×C311yescharacteristic, complemented
46#65070C2×C314yescomplemented
47#65062C2×C314yescomplemented
48C24×C3×C5—21no—
49C23×C3×D5—22no—
50C23×C3×D5—21no—
51C2×C3×C5×D4—21no—
52C2×C3×C5×D4—22no—
53C2×C3×C5×D4D414yescomplemented
54C2×C3×C5×D4C2312yes—
55C22×C3×C4×C5C2311yescharacteristic
56C2×C3×C22⋊D5—216no—
57C2×C3×C22⋊D5—21no—
58C2×C3×C22⋊D5—22no—
59C2×C3×C22⋊D5—22no—
60C2×C3×C22⋊D5D412yes—
61C22×C3×Dic5—22no—
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×#709D414yescomplemented
72C2×C3×D20—22no—
73C2×C3×D20—22no—
74C2×C3×D20—21no—
75C2×C3×D20—24no—
76C2×C3×D20D411yescharacteristic
77C2×C3×D20C2312yes—
78C2×C3×D20C2311yescharacteristic
79C2×C3×C4×D5—21no—
80C2×C3×C4×D5C2312yes—
81C2×C3×Dic10D411yescharacteristic
82C2×C3×Dic10C2311yescharacteristic
83C3×C5×M16—22no—
84C2×C3×C5×C8C2312yes—
85C3×C5×C4⋊C4C2312yes—
86C3×C5×C22⋊C4—21no—
87C3×C42×C5—21no—
88C3×#686—28no—
89C3×#685—24no—
90C2×C3×#168—22no—
91C2×C3×#168C2312yes—
92C3×D40—24no—
93C3×#677—24no—
94C2×C3×H(5,𝔽2)—51no—
95C2×C3×AGL(1,ℤ/8)—51no—
96C22×C3×D8—51no—
97C3×#535—51no—
98C3×#449—54no—
99C3×#447—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×C312yescomplemented
109C2×D4×D5—21no—
110C2×D4×D5C22×C312yescomplemented
111C2×#709C22×C311yescharacteristic
112C2×#709C22×C311yescharacteristic
113C22×D20C22×C311yescharacteristic, complemented
114C2×C5×M16C22×C311yescharacteristic, complemented
115C5×#117C22×C311yescharacteristic, complemented
116C2×#686—24no—
117C2×#686C22×C314yescomplemented
118C22×#168C22×C311yescharacteristic, complemented
119#4054—24no—
120C2×D40C22×C312yescomplemented
121C2×#677C22×C312yescomplemented
122#4041C22×C314yescomplemented
123#4028—22no—
124#4021—21no—
125#4020—21no—
126C5×#102C22×C314yescomplemented
127#3952C22×C314yescomplemented
128#3942C22×C312yescomplemented
129#3941C22×C312yescomplemented
130#3008—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—22no—
138C22×C3×D5—42no—
139C22×C3×D5—84no—
140C22×C3×D5—82no—
141C22×C3×D5C2×D412yes—
142C3×C5×D4—24no—
143C3×C5×D4—22no—
144C3×C5×D4C2×D414yes—
145C2×C3×C4×C5—22no—
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—28no—
152C3×C22⋊D5—24no—
153C3×C22⋊D5—24no—
154C2×C3×Dic5—28no—
155C2×C3×Dic5—21no—
156C2×C3×Dic5C2×D412yes—
157C3×D20—22no—
158C3×D20—21no—
159C3×D20—22no—
160C3×D20—22no—
161C3×D20—42no—
162C3×D20—84no—
163C3×D20C2×D412yes—
164C3×C4×D5—24no—
165C3×C4×D5—22no—
166C3×C4×D5—22no—
167C3×Dic10—22no—
168C3×Dic10—21no—
169C3×Dic10C2×D412yes—
170C3×C5×C8—22no—
171C3×#168—22no—
172C3×H(5,𝔽2)—104no—
173C3×H(5,𝔽2)—104no—
174C2×C3×#41—51no—
175C2×C3×#41—52no—
176C2×C3×#41—51no—
177C2×C3×#41—101no—
178C22×C3×D4—52no—
179C22×C3×D4—51no—
180C22×C3×D4—101no—
181C22×C3×D4—102no—
182C22×C3×D4C2×D511yescharacteristic
183C3×AGL(1,ℤ/8)—104no—
184C2×C3×SD16—52no—
185C2×C3×D8—54no—
186C2×C3×D8—52no—
187C2×C3×D8—104no—
188C2×C3×M16C2×D511yescharacteristic, complemented
189C22×C3×C8—51no—
190C3×#127—101no—
191C3×#124—101no—
192C3×#121—54no—
193C3×#120—102no—
194C3×#117C2×D511yescharacteristic, complemented
195C3×#103—52no—
196C3×#102—52no—
197C3×#102C2×D514yescomplemented
198C3×C22⋊C8—54no—
199C24×C5—21no—
200C23×D5—22no—
201C23×D5—21no—
202C2×C5×D4—21no—
203C2×C5×D4—22no—
204C2×C5×D4C3×D414yescomplemented
205C2×C5×D4C23×C312yes—
206C22×C4×C5C23×C311yescharacteristic
207C2×C22⋊D5—216no—
208C2×C22⋊D5—21no—
209C2×C22⋊D5—22no—
210C2×C22⋊D5—22no—
211C2×C22⋊D5C3×D412yes—
212C22×Dic5—22no—
213#711—28no—
214#711—24no—
215#711—24no—
216D4×D5—28no—
217D4×D5—24no—
218D4×D5—24no—
219#709—22no—
220#709—22no—
221#709—22no—
222#709C3×D414yescomplemented
223C2×D20—22no—
224C2×D20—22no—
225C2×D20—21no—
226C2×D20—24no—
227C2×D20C3×D411yescharacteristic
228C2×D20C23×C312yes—
229C2×D20C23×C311yescharacteristic
230C2×C4×D5—21no—
231C2×C4×D5C23×C312yes—
232C2×Dic10C3×D411yescharacteristic
233C2×Dic10C23×C311yescharacteristic
234C5×M16—22no—
235C2×C5×C8C23×C312yes—
236C5×C4⋊C4C23×C312yes—
237C5×C22⋊C4—21no—
238C42×C5—21no—
239#686—28no—
240#685—24no—
241C2×#168—22no—
242C2×#168C23×C312yes—
243D40—24no—
244#677—24no—
245C2×H(5,𝔽2)—51no—
246C2×AGL(1,ℤ/8)—51no—
247C22×D8—51no—
248#535—51no—
249#449—54no—
250#447—54no—
251#418C3×D511yescharacteristic, complemented
252#416—51no—
253#408—51no—
254C22×C3×C5—22no—
255C22×C3×C5—24no—
256C22×C3×C5—44no—
257C22×C3×C5—84no—
258C22×C3×C5#12012yes—
259C22×C3×C5C22×D411yescharacteristic
260C2×C3×D5—22no—
261C2×C3×D5—22no—
262C2×C3×D5—24no—
263C2×C3×D5—42no—
264C2×C3×D5—42no—
265C2×C3×D5—44no—
266C2×C3×D5—88no—
267C2×C3×D5—84no—
268C3×C4×C5—42no—
269C3×C4×C5#12012yes—
270C3×C4×C5C22×D411yescharacteristic
271C3×C4×C5C22×D411yescharacteristic
272C3×Dic5—22no—
273C3×Dic5—22no—
274C24×C3—21no—
275C24×C3—102no—
276C24×C3—101no—
277C3×#41—54no—
278C3×#41—104no—
279C3×#41—108no—
280C3×#41—104no—
281C3×#41—102no—
282C3×#41—102no—
283C3×#41—102no—
284C2×C3×Q8—51no—
285C2×C3×Q8—51no—
286C2×C3×D4—21no—
287C2×C3×D4—22no—
288C2×C3×D4—52no—
289C2×C3×D4—51no—
290C2×C3×D4—51no—
291C2×C3×D4—52no—
292C2×C3×D4—104no—
293C2×C3×D4—102no—
294C2×C3×D4—104no—
295C2×C3×D4—108no—
296C2×C3×D4—102no—
297C2×C3×D4—1016no—
298C2×C3×D4—101no—
299C2×C3×D4—104no—
300C2×C3×D4—102no—
301C2×C3×D4—102no—
302C2×C3×D4—101no—
303C2×C3×D4D2014yescomplemented
304C2×C3×D4C22×D512yes—
305C22×C3×C4—52no—
306C22×C3×C4—102no—
307C22×C3×C4—101no—
308C22×C3×C4C22×D511yescharacteristic
309C3×SD16—104no—
310C3×D8—104no—
311C3×D8—108no—
312C3×M16—22no—
313C2×C3×C8—52no—
314C2×C3×C8—102no—
315C2×C3×C8C22×D512yes—
316C3×C4⋊C4C22×D512yes—
317C3×C22⋊C4—21no—
318C3×C22⋊C4—104no—
319C3×C42—21no—
320C23×C5—22no—
321C23×C5—24no—
322C23×C5—82no—
323C23×C5C2×C3×D411yescharacteristic
324C22×D5—21no—
325C22×D5—28no—
326C22×D5—22no—
327C22×D5—42no—
328C22×D5—84no—
329C22×D5—82no—
330C22×D5C2×C3×D412yes—
331C5×D4—24no—
332C5×D4—22no—
333C5×D4C2×C3×D414yes—
334C2×C4×C5—22no—
335C2×C4×C5C2×C3×D412yes—
336C2×C4×C5C2×C3×D411yescharacteristic
337C2×C4×C5C2×C3×D412yes—
338C2×C4×C5C24×C311yescharacteristic
339C22⋊D5—28no—
340C22⋊D5—28no—
341C22⋊D5—24no—
342C22⋊D5—24no—
343C2×Dic5—28no—
344C2×Dic5—21no—
345C2×Dic5C2×C3×D412yes—
346D20—22no—
347D20—21no—
348D20—22no—
349D20—22no—
350D20—42no—
351D20—84no—
352D20C2×C3×D412yes—
353C4×D5—24no—
354C4×D5—22no—
355C4×D5—22no—
356Dic10—22no—
357Dic10—21no—
358Dic10C2×C3×D412yes—
359C5×C8—22no—
360#168—22no—
361H(5,𝔽2)—104no—
362H(5,𝔽2)—104no—
363C2×#41—51no—
364C2×#41—52no—
365C2×#41—51no—
366C2×#41—101no—
367C22×D4—52no—
368C22×D4—51no—
369C22×D4—101no—
370C22×D4—102no—
371C22×D4C2×C3×D511yescharacteristic
372AGL(1,ℤ/8)—104no—
373C2×SD16—52no—
374C2×D8—54no—
375C2×D8—52no—
376C2×D8—104no—
377C2×M16C2×C3×D511yescharacteristic, complemented
378C22×C8—51no—
379#127—101no—
380#124—101no—
381#121—54no—
382#120—102no—
383#117C2×C3×D511yescharacteristic, complemented
384#103—52no—
385#102—52no—
386#102C2×C3×D514yescomplemented
387C22⋊C8—54no—
388C2×C3×C5—22no—
389C2×C3×C5—44no—
390C2×C3×C5C2×#12011yescharacteristic
391C2×C3×C5#57612yes—
392C3×D5—42no—
393C3×D5—42no—
394C3×D5—82no—
395C23×C3—22no—
396C23×C3—24no—
397C23×C3—52no—
398C23×C3—82no—
399C23×C3—108no—
400C23×C3—101no—
401C23×C3—102no—
402C23×C3—202no—
403C23×C3—402no—
404C23×C3—404no—
405C23×C3D4×D511yescharacteristic
406C3×Q8—52no—
407C3×Q8—101no—
408C3×Q8—102no—
409C3×D4—22no—
410C3×D4—24no—
411C3×D4—52no—
412C3×D4—108no—
413C3×D4—108no—
414C3×D4—104no—
415C3×D4—104no—
416C3×D4—101no—
417C3×D4—102no—
418C3×D4—102no—
419C3×D4—102no—
420C3×D4—202no—
421C3×D4—404no—
422C3×D4C2×D2014yes—
423C2×C3×C4—22no—
424C2×C3×C4—52no—
425C2×C3×C4—108no—
426C2×C3×C4—101no—
427C2×C3×C4—102no—
428C2×C3×C4—102no—
429C2×C3×C4—104no—
430C2×C3×C4C2×D2012yes—
431C2×C3×C4D4×D512yes—
432C2×C3×C4D4×D511yescharacteristic
433C2×C3×C4C23×D511yescharacteristic
434C3×C8—22no—
435C3×C8—102no—
436C22×C5—22no—
437C22×C5—24no—
438C22×C5—44no—
439C22×C5—84no—
440C22×C5C3×#12012yes—
441C22×C5C22×C3×D411yescharacteristic
442C2×D5—24no—
443C2×D5—22no—
444C2×D5—22no—
445C2×D5—42no—
446C2×D5—44no—
447C2×D5—42no—
448C2×D5—88no—
449C2×D5—84no—
450C4×C5—42no—
451C4×C5C3×#12012yes—
452C4×C5C22×C3×D411yescharacteristic
453C4×C5C22×C3×D411yescharacteristic
454Dic5—22no—
455Dic5—22no—
456C24—21no—
457C24—102no—
458C24—101no—
459#41—54no—
460#41—104no—
461#41—108no—
462#41—104no—
463#41—102no—
464#41—102no—
465#41—102no—
466C2×Q8—51no—
467C2×Q8—51no—
468C2×D4—21no—
469C2×D4—22no—
470C2×D4—52no—
471C2×D4—51no—
472C2×D4—51no—
473C2×D4—52no—
474C2×D4—104no—
475C2×D4—102no—
476C2×D4—104no—
477C2×D4—108no—
478C2×D4—102no—
479C2×D4—1016no—
480C2×D4—101no—
481C2×D4—104no—
482C2×D4—102no—
483C2×D4—102no—
484C2×D4—101no—
485C2×D4C3×D2014yescomplemented
486C2×D4C22×C3×D512yes—
487C22×C4—52no—
488C22×C4—102no—
489C22×C4—101no—
490C22×C4C22×C3×D511yescharacteristic
491SD16—104no—
492D8—104no—
493D8—108no—
494M16—22no—
495C2×C8—52no—
496C2×C8—102no—
497C2×C8C22×C3×D512yes—
498C4⋊C4C22×C3×D512yes—
499C22⋊C4—21no—
500C22⋊C4—104no—
501C42—21no—
502C3×C5#300811yescharacteristic, complemented, Hall
503C22×C3—22no—
504C22×C3—24no—
505C22×C3—44no—
506C22×C3—84no—
507C22×C3—102no—
508C22×C3—102no—
509C22×C3—104no—
510C22×C3—202no—
511C22×C3—202no—
512C22×C3—204no—
513C22×C3—404no—
514C22×C3—408no—
515C22×C3#402812yes—
516C22×C3C2×D4×D511yescharacteristic
517C3×C4—42no—
518C3×C4—102no—
519C3×C4—102no—
520C3×C4#402812yes—
521C3×C4C22×D2011yescharacteristic
522C3×C4C2×D4×D511yescharacteristic
523C2×C5—22no—
524C2×C5—44no—
525C2×C5C2×C3×#12011yescharacteristic
526C2×C5C3×#57612yes—
527D5—42no—
528D5—42no—
529D5—82no—
530C23—22no—
531C23—24no—
532C23—52no—
533C23—82no—
534C23—108no—
535C23—101no—
536C23—102no—
537C23—202no—
538C23—402no—
539C23—404no—
540C23C3×D4×D511yescharacteristic
541Q8—52no—
542Q8—101no—
543Q8—102no—
544D4—22no—
545D4—24no—
546D4—52no—
547D4—108no—
548D4—108no—
549D4—104no—
550D4—104no—
551D4—101no—
552D4—102no—
553D4—102no—
554D4—102no—
555D4—202no—
556D4—404no—
557D4C2×C3×D2014yes—
558C2×C4—22no—
559C2×C4—52no—
560C2×C4—108no—
561C2×C4—101no—
562C2×C4—102no—
563C2×C4—102no—
564C2×C4—104no—
565C2×C4C2×C3×D2012yes—
566C2×C4C3×D4×D512yes—
567C2×C4C3×D4×D511yescharacteristic
568C2×C4C23×C3×D511yescharacteristic
569C8—22no—
570C8—102no—
571C2×C3—22no—
572C2×C3—44no—
573C2×C3—202no—
574C2×C3—202no—
575C2×C3—402no—
576C2×C3C2×#402811yescharacteristic
577C2×C3#6610812yes—
578C5C3×#300811yescharacteristic, complemented, Sylow
579C22—22no—
580C22—24no—
581C22—44no—
582C22—84no—
583C22—104no—
584C22—102no—
585C22—102no—
586C22—202no—
587C22—204no—
588C22—202no—
589C22—404no—
590C22—408no—
591C22C3×#402812yes—
592C22C2×C3×D4×D511yescharacteristic
593C4—42no—
594C4—102no—
595C4—102no—
596C4C3×#402812yes—
597C4C22×C3×D2011yescharacteristic
598C4C2×C3×D4×D511yescharacteristic
599C3#1061065911yescharacteristic, direct product factor, Sylow
600C2—22no—
601C2—44no—
602C2—202no—
603C2—202no—
604C2—402no—
605C2C2×C3×#402811yescharacteristic
606C2C3×#6610812yes—
subgroupG/H
centerC22×C3C2×D4×D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC3×C5×#418C2
socleC22×C3×C5C22×D4
hypercenterC3×#418D5
hypocenterC5C3×#3008
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
1. C2×C3×#4144C2semidirect
2. C22×C3×#686C2semidirect
3. C2×C3×#4054C2semidirect
4. C3×#65859C2semidirect
5. C3×C5×#418C2semidirect
6. C3×#65418C2semidirect
7. C3×#65291C2semidirect
8. C3×#65070C2semidirect
9. C3×#65062C2semidirect
10. #10610659C3direct (N × H)
16. C2×C3×#711C22semidirect
18. C2×C3×D4×D5C22semidirect
21. C22×C3×D20C22semidirect
22. C2×C3×C5×M16C22semidirect
23. C3×C5×#117C22semidirect
25. C2×C3×#686C22semidirect
26. C22×C3×#168C22semidirect
28. C2×C3×D40C22semidirect
29. C2×C3×#677C22semidirect
30. C3×#4041C22semidirect
34. C3×C5×#102C22semidirect
35. C3×#3952C22semidirect
36. C3×#3942C22semidirect
37. C3×#3941C22semidirect
39. C2×#4144C2×C3semidirect
40. C22×#686C2×C3semidirect
41. C2×#4054C2×C3semidirect
42. #65859C2×C3semidirect
43. C5×#418C2×C3semidirect
44. #65418C2×C3semidirect
45. #65291C2×C3semidirect
46. #65070C2×C3semidirect
47. #65062C2×C3semidirect
53. C2×C3×C5×D4D4semidirect
71. C3×#709D4semidirect
100. C3×#418D5semidirect
108. C2×#711C22×C3semidirect
110. C2×D4×D5C22×C3semidirect
113. C22×D20C22×C3semidirect
114. C2×C5×M16C22×C3semidirect
115. C5×#117C22×C3semidirect
117. C2×#686C22×C3semidirect
118. C22×#168C22×C3semidirect
120. C2×D40C22×C3semidirect
121. C2×#677C22×C3semidirect
122. #4041C22×C3semidirect
126. C5×#102C22×C3semidirect
127. #3952C22×C3semidirect
128. #3942C22×C3semidirect
129. #3941C22×C3semidirect
188. C2×C3×M16C2×D5semidirect
194. C3×#117C2×D5semidirect
197. C3×#102C2×D5semidirect
204. C2×C5×D4C3×D4semidirect
222. #709C3×D4semidirect
251. #418C3×D5semidirect
303. C2×C3×D4D20semidirect
377. C2×M16C2×C3×D5semidirect
383. #117C2×C3×D5semidirect
386. #102C2×C3×D5semidirect
485. C2×D4C3×D20semidirect
502. C3×C5#3008semidirect
578. C5C3×#3008semidirect
599. C3#10610659direct (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×#6862
C2×C3×#40542
C3×#658592
C3×C5×#4182
C3×#654182
C3×#652912
C3×#650702
C3×#650622
#106106593
C3×#30085