Finite Group Databasebeta

← C2×C3×#65023 #14423419

Subgroups — C2×C3×#65023 #14423419

Order 1920.

1624 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#709C211yesmaximal, characteristic, complemented
2C2×C3×#4076C211yesmaximal, characteristic, complemented
3C22×C3×D40C211yesmaximal, characteristic, complemented
4C22×C3×#677C211yesmaximal, characteristic, complemented
5C2×C3×C5×C22⋊C8C211yesmaximal, characteristic, complemented
6C2×C3×#3953C211yesmaximal, characteristic, complemented
7C2×C3×#3948C211yesmaximal, characteristic, complemented
8C3×#65023C218yesmaximal, direct product factor
9C2×#65023C311yesmaximal, characteristic, direct product factor, Hall
10C23×C3×C4×C5C2211yescharacteristic
11C22×C3×C22⋊D5—21no—
12C22×C3×C22⋊D5—21no—
13C2×C3×#709—22no—
14C2×C3×#709—24no—
15C2×C3×#709—24no—
16C2×C3×#709C2214yescomplemented
17C22×C3×D20C2211yescharacteristic
18C22×C3×D20C2211yescharacteristic
19C22×C3×C4×D5—21no—
20C22×C3×Dic10C2211yescharacteristic
21C22×C3×C5×C8C2211yescharacteristic
22C22×C3×C5×C8C2211yescharacteristic
23C3×#4076—22no—
24C3×#4076C2214yescomplemented
25C2×C3×#685—21no—
26C2×C3×C4⋊Dic5C2211yescharacteristic
27C2×C3×D40—24no—
28C2×C3×D40C2214yescomplemented
29C2×C3×#677—24no—
30C2×C3×#677C2214yescomplemented
31C3×C5×C22⋊C8C2214yescomplemented
32C3×#3953C2214yescomplemented
33C3×#3948C2214yescomplemented
34C2×C3×#449—51nomaximal, Hall
35C22×#709C2×C311yescharacteristic, complemented
36C2×#4076C2×C311yescharacteristic, complemented
37C22×D40C2×C311yescharacteristic, complemented
38C22×#677C2×C311yescharacteristic, complemented
39C2×C5×C22⋊C8C2×C311yescharacteristic, complemented
40C2×#3953C2×C311yescharacteristic, complemented
41C2×#3948C2×C311yescharacteristic, complemented
42#65023C2×C318yesdirect product factor
43C24×C3×C5D411yescharacteristic
44C23×C3×D5—21no—
45C23×C3×D5—21no—
46C22×C3×C4×C5—22no—
47C22×C3×C4×C5—22no—
48C22×C3×C4×C5D411yescharacteristic
49C22×C3×C4×C5D412yes—
50C22×C3×C4×C5C2312yes—
51C22×C3×C4×C5C2311yescharacteristic
52C2×C3×C22⋊D5—28no—
53C2×C3×C22⋊D5—24no—
54C2×C3×C22⋊D5—24no—
55C2×C3×C22⋊D5—44no—
56C22×C3×Dic5—21no—
57C22×C3×Dic5—21no—
58C3×#709—216no—
59C3×#709—28no—
60C3×#709—28no—
61C2×C3×D20—21no—
62C2×C3×D20—22no—
63C2×C3×D20—22no—
64C2×C3×D20—22no—
65C2×C3×D20—21no—
66C2×C3×D20D414yescomplemented
67C2×C3×D20C2312yes—
68C2×C3×D20C2312yes—
69C2×C3×C4×D5—28no—
70C2×C3×C4×D5—24no—
71C2×C3×Dic10—21no—
72C2×C3×Dic10—22no—
73C2×C3×Dic10D414yescomplemented
74C2×C3×Dic10C2312yes—
75C2×C3×C5×C8—22no—
76C2×C3×C5×C8—22no—
77C2×C3×C5×C8C2312yes—
78C2×C3×C5×C8C2312yes—
79C3×#685—24no—
80C3×C4⋊Dic5—21no—
81C3×C4⋊Dic5C2312yes—
82C3×D40—28no—
83C3×#677—28no—
84C22×C3×#41—51no—
85C22×C3×SD16—51no—
86C22×C3×D8—51no—
87C2×C3×#121—51no—
88C3×#449—58no—
89C2×C3×#103—51no—
90C2×C3×#102—51no—
91C2×C3×C22⋊C8D511yescharacteristic, complemented
92C23×C4×C5C22×C311yescharacteristic
93C22×C22⋊D5—21no—
94C22×C22⋊D5—21no—
95C2×#709—22no—
96C2×#709—24no—
97C2×#709—24no—
98C2×#709C22×C314yescomplemented
99C22×D20C22×C311yescharacteristic
100C22×D20C22×C311yescharacteristic
101C22×C4×D5—21no—
102C22×Dic10C22×C311yescharacteristic
103C22×C5×C8C22×C311yescharacteristic
104C22×C5×C8C22×C311yescharacteristic
105#4076—22no—
106#4076C22×C314yescomplemented
107C2×#685—21no—
108C2×C4⋊Dic5C22×C311yescharacteristic
109C2×D40—24no—
110C2×D40C22×C314yescomplemented
111C2×#677—24no—
112C2×#677C22×C314yescomplemented
113C5×C22⋊C8C22×C314yescomplemented
114#3953C22×C314yescomplemented
115#3948C22×C314yescomplemented
116C2×#449—51noSylow
117C23×C3×C5—22no—
118C23×C3×C5—22no—
119C23×C3×C5—22no—
120C23×C3×C5C2×D412yes—
121C23×C3×C5C2×D411yescharacteristic
122C22×C3×D5—24no—
123C22×C3×D5—22no—
124C22×C3×D5—22no—
125C22×C3×D5—42no—
126C22×C3×D5—42no—
127C22×C3×D5—42no—
128C22×C3×D5—42no—
129C22×C3×D5—42no—
130C2×C3×C4×C5—22no—
131C2×C3×C4×C5—24no—
132C2×C3×C4×C5—22no—
133C2×C3×C4×C5—22no—
134C2×C3×C4×C5C2×D412yes—
135C2×C3×C4×C5C2×D411yescharacteristic
136C2×C3×C4×C5C2×D412yes—
137C2×C3×C4×C5C2×D412yes—
138C2×C3×C4×C5C2411yescharacteristic
139C3×C22⋊D5—216no—
140C3×C22⋊D5—48no—
141C2×C3×Dic5—24no—
142C2×C3×Dic5—22no—
143C2×C3×Dic5—22no—
144C2×C3×Dic5—42no—
145C3×D20—24no—
146C3×D20—22no—
147C3×D20—22no—
148C3×D20—22no—
149C3×D20—42no—
150C3×D20C2×D414yescomplemented
151C3×C4×D5—216no—
152C3×Dic10—24no—
153C3×Dic10—22no—
154C3×Dic10C2×D414yescomplemented
155C3×C5×C8—22no—
156C3×C5×C8—22no—
157C2×C3×#41—54no—
158C2×C3×#41—104no—
159C2×C3×#41—104no—
160C2×C3×#41—102no—
161C22×C3×Q8—51no—
162C22×C3×D4—51no—
163C22×C3×D4—51no—
164C22×C3×D4—101no—
165C22×C3×D4—101no—
166C23×C3×C4—101no—
167C23×C3×C4C2×D511yescharacteristic
168C2×C3×SD16—54no—
169C2×C3×SD16—104no—
170C2×C3×D8—54no—
171C2×C3×D8—104no—
172C22×C3×C8C2×D511yescharacteristic
173C22×C3×C8C2×D511yescharacteristic
174C3×#121—54no—
175C3×#121—102no—
176C2×C3×C4⋊C4—51no—
177C2×C3×C22⋊C4—101no—
178C3×#103—54no—
179C3×#102—54no—
180C3×C22⋊C8C2×D514yescomplemented
181C24×C5C3×D411yescharacteristic
182C23×D5—21no—
183C23×D5—21no—
184C22×C4×C5—22no—
185C22×C4×C5—22no—
186C22×C4×C5C3×D411yescharacteristic
187C22×C4×C5C3×D412yes—
188C22×C4×C5C23×C312yes—
189C22×C4×C5C23×C311yescharacteristic
190C2×C22⋊D5—28no—
191C2×C22⋊D5—24no—
192C2×C22⋊D5—24no—
193C2×C22⋊D5—44no—
194C22×Dic5—21no—
195C22×Dic5—21no—
196#709—216no—
197#709—28no—
198#709—28no—
199C2×D20—21no—
200C2×D20—22no—
201C2×D20—22no—
202C2×D20—22no—
203C2×D20—21no—
204C2×D20C3×D414yescomplemented
205C2×D20C23×C312yes—
206C2×D20C23×C312yes—
207C2×C4×D5—28no—
208C2×C4×D5—24no—
209C2×Dic10—21no—
210C2×Dic10—22no—
211C2×Dic10C3×D414yescomplemented
212C2×Dic10C23×C312yes—
213C2×C5×C8—22no—
214C2×C5×C8—22no—
215C2×C5×C8C23×C312yes—
216C2×C5×C8C23×C312yes—
217#685—24no—
218C4⋊Dic5—21no—
219C4⋊Dic5C23×C312yes—
220D40—28no—
221#677—28no—
222C22×#41—51no—
223C22×SD16—51no—
224C22×D8—51no—
225C2×#121—51no—
226#449—58no—
227C2×#103—51no—
228C2×#102—51no—
229C2×C22⋊C8C3×D511yescharacteristic, complemented
230C22×C3×C5—22no—
231C22×C3×C5—22no—
232C22×C3×C5—22no—
233C22×C3×C5—44no—
234C22×C3×C5#12012yes—
235C22×C3×C5#13512yes—
236C22×C3×C5AGL(1,ℤ/8)12yes—
237C22×C3×C5C22×D411yescharacteristic
238C2×C3×D5—24no—
239C2×C3×D5—42no—
240C2×C3×D5—42no—
241C2×C3×D5—48no—
242C2×C3×D5—42no—
243C2×C3×D5—42no—
244C2×C3×D5—42no—
245C2×C3×D5—84no—
246C3×C4×C5—22no—
247C3×C4×C5#12012yes—
248C3×C4×C5C22×D411yescharacteristic
249C3×C4×C5C22×D411yescharacteristic
250C3×Dic5—24no—
251C3×Dic5—42no—
252C24×C3—101no—
253C24×C3—101no—
254C24×C3D2011yescharacteristic
255C3×#41—108no—
256C3×#41—108no—
257C3×#41—1016no—
258C2×C3×Q8—54no—
259C2×C3×Q8—52no—
260C2×C3×Q8—101no—
261C2×C3×Q8—102no—
262C2×C3×D4—54no—
263C2×C3×D4—52no—
264C2×C3×D4—52no—
265C2×C3×D4—108no—
266C2×C3×D4—104no—
267C2×C3×D4—104no—
268C2×C3×D4—101no—
269C2×C3×D4—101no—
270C2×C3×D4—102no—
271C2×C3×D4—102no—
272C2×C3×D4—102no—
273C2×C3×D4—204no—
274C22×C3×C4—22no—
275C22×C3×C4—22no—
276C22×C3×C4—108no—
277C22×C3×C4—101no—
278C22×C3×C4—104no—
279C22×C3×C4—101no—
280C22×C3×C4D2012yes—
281C22×C3×C4D2011yescharacteristic
282C22×C3×C4C22×D512yes—
283C22×C3×C4C22×D511yescharacteristic
284C3×SD16—108no—
285C3×D8—108no—
286C2×C3×C8—22no—
287C2×C3×C8—22no—
288C2×C3×C8C22×D512yes—
289C2×C3×C8C22×D512yes—
290C3×C4⋊C4—52no—
291C3×C4⋊C4—101no—
292C3×C22⋊C4—104no—
293C23×C5—22no—
294C23×C5—22no—
295C23×C5—22no—
296C23×C5C2×C3×D412yes—
297C23×C5C2×C3×D411yescharacteristic
298C22×D5—24no—
299C22×D5—22no—
300C22×D5—22no—
301C22×D5—42no—
302C22×D5—42no—
303C22×D5—42no—
304C22×D5—42no—
305C22×D5—42no—
306C2×C4×C5—22no—
307C2×C4×C5—24no—
308C2×C4×C5—22no—
309C2×C4×C5—22no—
310C2×C4×C5C2×C3×D412yes—
311C2×C4×C5C2×C3×D411yescharacteristic
312C2×C4×C5C2×C3×D412yes—
313C2×C4×C5C2×C3×D412yes—
314C2×C4×C5C24×C311yescharacteristic
315C22⋊D5—216no—
316C22⋊D5—48no—
317C2×Dic5—24no—
318C2×Dic5—22no—
319C2×Dic5—22no—
320C2×Dic5—42no—
321D20—24no—
322D20—22no—
323D20—22no—
324D20—22no—
325D20—42no—
326D20C2×C3×D414yescomplemented
327C4×D5—216no—
328Dic10—24no—
329Dic10—22no—
330Dic10C2×C3×D414yescomplemented
331C5×C8—22no—
332C5×C8—22no—
333C2×#41—54no—
334C2×#41—104no—
335C2×#41—104no—
336C2×#41—102no—
337C22×Q8—51no—
338C22×D4—51no—
339C22×D4—51no—
340C22×D4—101no—
341C22×D4—101no—
342C23×C4—101no—
343C23×C4C2×C3×D511yescharacteristic
344C2×SD16—54no—
345C2×SD16—104no—
346C2×D8—54no—
347C2×D8—104no—
348C22×C8C2×C3×D511yescharacteristic
349C22×C8C2×C3×D511yescharacteristic
350#121—54no—
351#121—102no—
352C2×C4⋊C4—51no—
353C2×C22⋊C4—101no—
354#103—54no—
355#102—54no—
356C22⋊C8C2×C3×D514yescomplemented
357C2×C3×C5—42no—
358C2×C3×C5#44914yescomplemented
359C2×C3×C5C2×#12011yescharacteristic
360C2×C3×C5C2×#13511yescharacteristic
361C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
362C3×D5—44no—
363C3×D5—82no—
364C23×C3—22no—
365C23×C3—22no—
366C23×C3—22no—
367C23×C3—102no—
368C23×C3—104no—
369C23×C3—102no—
370C23×C3—202no—
371C23×C3—202no—
372C23×C3—202no—
373C23×C3—202no—
374C23×C3—202no—
375C23×C3C2×D2012yes—
376C23×C3C2×D2011yescharacteristic
377C3×Q8—54no—
378C3×Q8—102no—
379C3×Q8—104no—
380C3×D4—54no—
381C3×D4—1016no—
382C3×D4—102no—
383C3×D4—102no—
384C3×D4—102no—
385C3×D4—104no—
386C3×D4—208no—
387C3×D4—202no—
388C2×C3×C4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—24no—
392C2×C3×C4—1016no—
393C2×C3×C4—102no—
394C2×C3×C4—104no—
395C2×C3×C4—102no—
396C2×C3×C4—202no—
397C2×C3×C4C2×D2012yes—
398C2×C3×C4C2×D2011yescharacteristic
399C2×C3×C4D4×D512yes—
400C2×C3×C4D4×D512yes—
401C2×C3×C4C23×D511yescharacteristic
402C3×C8—22no—
403C3×C8—22no—
404C22×C5—22no—
405C22×C5—22no—
406C22×C5—22no—
407C22×C5—44no—
408C22×C5C3×#12012yes—
409C22×C5C3×#13512yes—
410C22×C5C3×AGL(1,ℤ/8)12yes—
411C22×C5C22×C3×D411yescharacteristic
412C2×D5—24no—
413C2×D5—42no—
414C2×D5—42no—
415C2×D5—48no—
416C2×D5—42no—
417C2×D5—42no—
418C2×D5—42no—
419C2×D5—84no—
420C4×C5—22no—
421C4×C5C3×#12012yes—
422C4×C5C22×C3×D411yescharacteristic
423C4×C5C22×C3×D411yescharacteristic
424Dic5—24no—
425Dic5—42no—
426C24—101no—
427C24—101no—
428C24C3×D2011yescharacteristic
429#41—108no—
430#41—108no—
431#41—1016no—
432C2×Q8—54no—
433C2×Q8—52no—
434C2×Q8—101no—
435C2×Q8—102no—
436C2×D4—54no—
437C2×D4—52no—
438C2×D4—52no—
439C2×D4—108no—
440C2×D4—104no—
441C2×D4—104no—
442C2×D4—101no—
443C2×D4—101no—
444C2×D4—102no—
445C2×D4—102no—
446C2×D4—102no—
447C2×D4—204no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—108no—
451C22×C4—101no—
452C22×C4—104no—
453C22×C4—101no—
454C22×C4C3×D2011yescharacteristic
455C22×C4C3×D2012yes—
456C22×C4C22×C3×D512yes—
457C22×C4C22×C3×D511yescharacteristic
458SD16—108no—
459D8—108no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8C22×C3×D512yes—
463C2×C8C22×C3×D512yes—
464C4⋊C4—52no—
465C4⋊C4—101no—
466C22⋊C4—104no—
467C3×C5C2×#44911yescharacteristic, complemented, Hall
468C22×C3—22no—
469C22×C3—22no—
470C22×C3—22no—
471C22×C3—44no—
472C22×C3—104no—
473C22×C3—202no—
474C22×C3—202no—
475C22×C3—208no—
476C22×C3—202no—
477C22×C3—202no—
478C22×C3—202no—
479C22×C3—404no—
480C22×C3#402812yes—
481C22×C3#405012yes—
482C22×C3#405412yes—
483C22×C3C22×D2011yescharacteristic
484C3×C4—22no—
485C3×C4—104no—
486C3×C4—202no—
487C3×C4#402812yes—
488C3×C4C2×D4×D511yescharacteristic
489C3×C4C2×D4×D511yescharacteristic
490C2×C5—42no—
491C2×C5C3×#44914yescomplemented
492C2×C5C2×C3×#12011yescharacteristic
493C2×C5C2×C3×#13511yescharacteristic
494C2×C5C2×C3×AGL(1,ℤ/8)11yescharacteristic
495D5—44no—
496D5—82no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—102no—
501C23—104no—
502C23—102no—
503C23—202no—
504C23—202no—
505C23—202no—
506C23—202no—
507C23—202no—
508C23C2×C3×D2012yes—
509C23C2×C3×D2011yescharacteristic
510Q8—54no—
511Q8—102no—
512Q8—104no—
513D4—54no—
514D4—1016no—
515D4—102no—
516D4—102no—
517D4—102no—
518D4—104no—
519D4—208no—
520D4—202no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—1016no—
526C2×C4—102no—
527C2×C4—104no—
528C2×C4—102no—
529C2×C4—202no—
530C2×C4C2×C3×D2012yes—
531C2×C4C2×C3×D2011yescharacteristic
532C2×C4C3×D4×D512yes—
533C2×C4C3×D4×D512yes—
534C2×C4C23×C3×D511yescharacteristic
535C8—22no—
536C8—22no—
537C2×C3—42no—
538C2×C3—204no—
539C2×C3—402no—
540C2×C3#6502314yesdirect product factor
541C2×C3C2×#402811yescharacteristic
542C2×C3C2×#405011yescharacteristic
543C2×C3C2×#405411yescharacteristic
544C5C2×C3×#44911yescharacteristic, complemented, Sylow
545C22—22no—
546C22—22no—
547C22—22no—
548C22—44no—
549C22—104no—
550C22—202no—
551C22—202no—
552C22—208no—
553C22—202no—
554C22—202no—
555C22—202no—
556C22—404no—
557C22C3×#402812yes—
558C22C3×#405012yes—
559C22C3×#405412yes—
560C22C22×C3×D2011yescharacteristic
561C4—22no—
562C4—104no—
563C4—202no—
564C4C3×#402812yes—
565C4C2×C3×D4×D511yescharacteristic
566C4C2×C3×D4×D511yescharacteristic
567C3C2×#6502311yescharacteristic, direct product factor, Sylow
568C2—42no—
569C2—204no—
570C2—402no—
571C2C3×#6502314yesdirect product factor
572C2C2×C3×#402811yescharacteristic
573C2C2×C3×#405011yescharacteristic
574C2C2×C3×#405411yescharacteristic
subgroupG/H
centerC23×C3C2×D20
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC2×C3×C5×C22⋊C8C2
socleC23×C3×C5C2×D4
hypercenterC2×C3×C22⋊C8D5
hypocenterC5C2×C3×#449
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ C2×C3×C22⋊C8

Complements

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

NH ≅ G/Ntype
1. C22×C3×#709C2semidirect
2. C2×C3×#4076C2semidirect
3. C22×C3×D40C2semidirect
4. C22×C3×#677C2semidirect
5. C2×C3×C5×C22⋊C8C2semidirect
6. C2×C3×#3953C2semidirect
7. C2×C3×#3948C2semidirect
8. C3×#65023C2direct (N × H)
9. C2×#65023C3direct (N × H)
16. C2×C3×#709C22semidirect
24. C3×#4076C22semidirect
28. C2×C3×D40C22semidirect
30. C2×C3×#677C22semidirect
31. C3×C5×C22⋊C8C22semidirect
32. C3×#3953C22semidirect
33. C3×#3948C22semidirect
35. C22×#709C2×C3semidirect
36. C2×#4076C2×C3semidirect
37. C22×D40C2×C3semidirect
38. C22×#677C2×C3semidirect
39. C2×C5×C22⋊C8C2×C3semidirect
40. C2×#3953C2×C3semidirect
41. C2×#3948C2×C3semidirect
42. #65023C2×C3direct (N × H)
66. C2×C3×D20D4semidirect
73. C2×C3×Dic10D4semidirect
91. C2×C3×C22⋊C8D5semidirect
98. C2×#709C22×C3semidirect
106. #4076C22×C3semidirect
110. C2×D40C22×C3semidirect
112. C2×#677C22×C3semidirect
113. C5×C22⋊C8C22×C3semidirect
114. #3953C22×C3semidirect
115. #3948C22×C3semidirect
150. C3×D20C2×D4semidirect
154. C3×Dic10C2×D4semidirect
180. C3×C22⋊C8C2×D5semidirect
204. C2×D20C3×D4semidirect
211. C2×Dic10C3×D4semidirect
229. C2×C22⋊C8C3×D5semidirect
326. D20C2×C3×D4semidirect
330. Dic10C2×C3×D4semidirect
356. C22⋊C8C2×C3×D5semidirect
358. C2×C3×C5#449semidirect
467. C3×C5C2×#449semidirect
491. C2×C5C3×#449semidirect
540. C2×C3#65023direct (N × H)
544. C5C2×C3×#449semidirect
567. C3C2×#65023direct (N × H)
571. C2C3×#65023direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#7092
C2×C3×#40762
C22×C3×D402
C22×C3×#6772
C2×C3×C5×C22⋊C82
C2×C3×#39532
C2×C3×#39482
C3×#650232
C2×#650233
C2×C3×#4495