Finite Group Databasebeta

← C2×C3×#65070 #14423416

Subgroups — C2×C3×#65070 #14423416

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×#711C211yesmaximal, characteristic, complemented
2C22×C3×#686C211yesmaximal, characteristic, complemented
3C22×C3×#677C211yesmaximal, characteristic, complemented
4C2×C3×#4041C211yesmaximal, characteristic, complemented
5C2×C3×C5×#102C211yesmaximal, characteristic, complemented
6C2×C3×#3952C211yesmaximal, characteristic, complemented
7C2×C3×#3942C211yesmaximal, characteristic, complemented
8C3×#65070C218yesmaximal, direct product factor
9C2×#65070C311yesmaximal, characteristic, direct product factor, Hall
10C22×C3×C5×D4C2211yescharacteristic
11C22×C3×C22⋊D5—21no—
12C23×C3×Dic5—21no—
13C2×C3×#711—24no—
14C2×C3×#711—22no—
15C2×C3×#711—24no—
16C2×C3×#711C2214yescomplemented
17C22×C3×D20—21no—
18C22×C3×D20C2211yescharacteristic
19C22×C3×C4×D5C2211yescharacteristic
20C22×C3×Dic10C2211yescharacteristic
21C22×C3×C5×C8C2211yescharacteristic
22C2×C3×C5×C4⋊C4C2211yescharacteristic
23C2×C3×#686—24no—
24C2×C3×#686C2214yescomplemented
25C2×C3×#685—21no—
26C22×C3×#168C2211yescharacteristic
27C2×C3×#677—24no—
28C2×C3×#677C2214yescomplemented
29C3×#4041—22no—
30C3×#4041C2214yescomplemented
31C3×C5×#102C2214yescomplemented
32C3×#3952C2214yescomplemented
33C3×#3942C2214yescomplemented
34C2×C3×#449—51nomaximal, Hall
35C22×#711C2×C311yescharacteristic, complemented
36C22×#686C2×C311yescharacteristic, complemented
37C22×#677C2×C311yescharacteristic, complemented
38C2×#4041C2×C311yescharacteristic, complemented
39C2×C5×#102C2×C311yescharacteristic, complemented
40C2×#3952C2×C311yescharacteristic, complemented
41C2×#3942C2×C311yescharacteristic, complemented
42#65070C2×C318yesdirect product factor
43C24×C3×C5—21no—
44C23×C3×D5—21no—
45C23×C3×D5D411yescharacteristic
46C2×C3×C5×D4—21no—
47C2×C3×C5×D4—22no—
48C2×C3×C5×D4D414yescomplemented
49C2×C3×C5×D4C2312yes—
50C22×C3×C4×C5—21no—
51C22×C3×C4×C5C2311yescharacteristic
52C2×C3×C22⋊D5—28no—
53C2×C3×C22⋊D5—24no—
54C22×C3×Dic5—28no—
55C22×C3×Dic5—21no—
56C22×C3×Dic5—24no—
57C22×C3×Dic5D411yescharacteristic
58C3×#711—216no—
59C3×#711—28no—
60C3×#711—28no—
61C2×C3×D20—22no—
62C2×C3×D20—22no—
63C2×C3×D20—24no—
64C2×C3×D20—21no—
65C2×C3×D20—44no—
66C2×C3×D20C2312yes—
67C2×C3×C4×D5—22no—
68C2×C3×C4×D5—22no—
69C2×C3×C4×D5D412yes—
70C2×C3×C4×D5C2312yes—
71C2×C3×Dic10—21no—
72C2×C3×Dic10—22no—
73C2×C3×Dic10D414yescomplemented
74C2×C3×Dic10C2312yes—
75C2×C3×C5×C8—22no—
76C2×C3×C5×C8C2312yes—
77C3×C5×C4⋊C4—21no—
78C3×C5×C4⋊C4C2312yes—
79C3×#686—28no—
80C3×#685—24no—
81C2×C3×#168—22no—
82C2×C3×#168C2312yes—
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×#102D511yescharacteristic, complemented
91C2×C3×C22⋊C8—51no—
92C22×C5×D4C22×C311yescharacteristic
93C22×C22⋊D5—21no—
94C23×Dic5—21no—
95C2×#711—24no—
96C2×#711—22no—
97C2×#711—24no—
98C2×#711C22×C314yescomplemented
99C22×D20—21no—
100C22×D20C22×C311yescharacteristic
101C22×C4×D5C22×C311yescharacteristic
102C22×Dic10C22×C311yescharacteristic
103C22×C5×C8C22×C311yescharacteristic
104C2×C5×C4⋊C4C22×C311yescharacteristic
105C2×#686—24no—
106C2×#686C22×C314yescomplemented
107C2×#685—21no—
108C22×#168C22×C311yescharacteristic
109C2×#677—24no—
110C2×#677C22×C314yescomplemented
111#4041—22no—
112#4041C22×C314yescomplemented
113C5×#102C22×C314yescomplemented
114#3952C22×C314yescomplemented
115#3942C22×C314yescomplemented
116C2×#449—51noSylow
117C23×C3×C5—24no—
118C23×C3×C5—22no—
119C23×C3×C5—42no—
120C23×C3×C5—42no—
121C23×C3×C5C2×D411yescharacteristic
122C22×C3×D5—22no—
123C22×C3×D5—22no—
124C22×C3×D5—22no—
125C22×C3×D5—22no—
126C22×C3×D5—42no—
127C22×C3×D5—42no—
128C22×C3×D5—42no—
129C22×C3×D5C2×D412yes—
130C3×C5×D4—24no—
131C3×C5×D4—22no—
132C3×C5×D4C2×D414yescomplemented
133C2×C3×C4×C5—22no—
134C2×C3×C4×C5—42no—
135C2×C3×C4×C5C2×D411yescharacteristic
136C2×C3×C4×C5C2×D412yes—
137C2×C3×C4×C5C2×D412yes—
138C2×C3×C4×C5C2411yescharacteristic
139C3×C22⋊D5—216no—
140C2×C3×Dic5—216no—
141C2×C3×Dic5—24no—
142C2×C3×Dic5—22no—
143C2×C3×Dic5—22no—
144C2×C3×Dic5C2×D412yes—
145C3×D20—22no—
146C3×D20—22no—
147C3×D20—48no—
148C3×D20—42no—
149C3×C4×D5—24no—
150C3×C4×D5—22no—
151C3×C4×D5—22no—
152C3×Dic10—24no—
153C3×Dic10—22no—
154C3×Dic10C2×D414yescomplemented
155C3×C5×C8—22no—
156C3×#168—22no—
157C2×C3×#41—54no—
158C2×C3×#41—102no—
159C2×C3×#41—104no—
160C2×C3×#41—104no—
161C22×C3×Q8—51no—
162C22×C3×D4—51no—
163C22×C3×D4—101no—
164C22×C3×D4—101no—
165C22×C3×D4C2×D511yescharacteristic
166C23×C3×C4—51no—
167C23×C3×C4—101no—
168C2×C3×SD16—54no—
169C2×C3×SD16—104no—
170C2×C3×D8—54no—
171C2×C3×D8—104no—
172C22×C3×C8—51no—
173C22×C3×C8C2×D511yescharacteristic
174C3×#121—54no—
175C3×#121—102no—
176C2×C3×C4⋊C4C2×D511yescharacteristic
177C2×C3×C22⋊C4—101no—
178C3×#103—54no—
179C3×#102C2×D514yescomplemented
180C3×C22⋊C8—54no—
181C24×C5—21no—
182C23×D5—21no—
183C23×D5C3×D411yescharacteristic
184C2×C5×D4—21no—
185C2×C5×D4—22no—
186C2×C5×D4C3×D414yescomplemented
187C2×C5×D4C23×C312yes—
188C22×C4×C5—21no—
189C22×C4×C5C23×C311yescharacteristic
190C2×C22⋊D5—28no—
191C2×C22⋊D5—24no—
192C22×Dic5—28no—
193C22×Dic5—21no—
194C22×Dic5—24no—
195C22×Dic5C3×D411yescharacteristic
196#711—216no—
197#711—28no—
198#711—28no—
199C2×D20—22no—
200C2×D20—22no—
201C2×D20—24no—
202C2×D20—21no—
203C2×D20—44no—
204C2×D20C23×C312yes—
205C2×C4×D5—22no—
206C2×C4×D5—22no—
207C2×C4×D5C3×D412yes—
208C2×C4×D5C23×C312yes—
209C2×Dic10—22no—
210C2×Dic10—21no—
211C2×Dic10C3×D414yescomplemented
212C2×Dic10C23×C312yes—
213C2×C5×C8—22no—
214C2×C5×C8C23×C312yes—
215C5×C4⋊C4—21no—
216C5×C4⋊C4C23×C312yes—
217#686—28no—
218#685—24no—
219C2×#168—22no—
220C2×#168C23×C312yes—
221#677—28no—
222C22×#41—51no—
223C22×SD16—51no—
224C22×D8—51no—
225C2×#121—51no—
226#449—58no—
227C2×#103—51no—
228C2×#102C3×D511yescharacteristic, complemented
229C2×C22⋊C8—51no—
230C22×C3×C5—24no—
231C22×C3×C5—42no—
232C22×C3×C5—42no—
233C22×C3×C5—48no—
234C22×C3×C5#12012yes—
235C22×C3×C5#13512yes—
236C22×C3×C5AGL(1,ℤ/8)12yes—
237C22×C3×C5C22×D411yescharacteristic
238C2×C3×D5—22no—
239C2×C3×D5—22no—
240C2×C3×D5—22no—
241C2×C3×D5—44no—
242C2×C3×D5—42no—
243C2×C3×D5—42no—
244C2×C3×D5—42no—
245C2×C3×D5—84no—
246C3×C4×C5—42no—
247C3×C4×C5#12012yes—
248C3×C4×C5C22×D411yescharacteristic
249C3×C4×C5C22×D411yescharacteristic
250C3×Dic5—24no—
251C3×Dic5—22no—
252C24×C3—21no—
253C24×C3—51no—
254C24×C3—101no—
255C3×#41—108no—
256C3×#41—1016no—
257C3×#41—108no—
258C2×C3×Q8—54no—
259C2×C3×Q8—52no—
260C2×C3×Q8—102no—
261C2×C3×Q8—101no—
262C2×C3×D4—22no—
263C2×C3×D4—21no—
264C2×C3×D4—52no—
265C2×C3×D4—104no—
266C2×C3×D4—104no—
267C2×C3×D4—108no—
268C2×C3×D4—102no—
269C2×C3×D4—102no—
270C2×C3×D4—101no—
271C2×C3×D4—204no—
272C2×C3×D4D2014yescomplemented
273C2×C3×D4C22×D512yes—
274C22×C3×C4—21no—
275C22×C3×C4—51no—
276C22×C3×C4—52no—
277C22×C3×C4—52no—
278C22×C3×C4—104no—
279C22×C3×C4—108no—
280C22×C3×C4—101no—
281C22×C3×C4—102no—
282C22×C3×C4—102no—
283C22×C3×C4C22×D511yescharacteristic
284C3×SD16—108no—
285C3×D8—108no—
286C2×C3×C8—22no—
287C2×C3×C8—52no—
288C2×C3×C8—102no—
289C2×C3×C8C22×D512yes—
290C3×C4⋊C4—21no—
291C3×C4⋊C4C22×D512yes—
292C3×C22⋊C4—104no—
293C23×C5—24no—
294C23×C5—22no—
295C23×C5—42no—
296C23×C5—42no—
297C23×C5C2×C3×D411yescharacteristic
298C22×D5—22no—
299C22×D5—22no—
300C22×D5—22no—
301C22×D5—22no—
302C22×D5—42no—
303C22×D5—42no—
304C22×D5—42no—
305C22×D5C2×C3×D412yes—
306C5×D4—24no—
307C5×D4—22no—
308C5×D4C2×C3×D414yescomplemented
309C2×C4×C5—22no—
310C2×C4×C5—42no—
311C2×C4×C5C2×C3×D412yes—
312C2×C4×C5C2×C3×D411yescharacteristic
313C2×C4×C5C2×C3×D412yes—
314C2×C4×C5C24×C311yescharacteristic
315C22⋊D5—216no—
316C2×Dic5—216no—
317C2×Dic5—24no—
318C2×Dic5—22no—
319C2×Dic5—22no—
320C2×Dic5C2×C3×D412yes—
321D20—22no—
322D20—22no—
323D20—48no—
324D20—42no—
325C4×D5—24no—
326C4×D5—22no—
327C4×D5—22no—
328Dic10—24no—
329Dic10—22no—
330Dic10C2×C3×D414yescomplemented
331C5×C8—22no—
332#168—22no—
333C2×#41—54no—
334C2×#41—102no—
335C2×#41—104no—
336C2×#41—104no—
337C22×Q8—51no—
338C22×D4—51no—
339C22×D4—101no—
340C22×D4—101no—
341C22×D4C2×C3×D511yescharacteristic
342C23×C4—51no—
343C23×C4—101no—
344C2×SD16—54no—
345C2×SD16—104no—
346C2×D8—54no—
347C2×D8—104no—
348C22×C8—51no—
349C22×C8C2×C3×D511yescharacteristic
350#121—54no—
351#121—102no—
352C2×C4⋊C4C2×C3×D511yescharacteristic
353C2×C22⋊C4—101no—
354#103—54no—
355#102C2×C3×D514yescomplemented
356C22⋊C8—54no—
357C2×C3×C5—44no—
358C2×C3×C5#44914yescomplemented
359C2×C3×C5C2×#12011yescharacteristic
360C2×C3×C5C2×#13511yescharacteristic
361C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
362C3×D5—42no—
363C3×D5—82no—
364C23×C3—22no—
365C23×C3—24no—
366C23×C3—42no—
367C23×C3—42no—
368C23×C3—52no—
369C23×C3—102no—
370C23×C3—102no—
371C23×C3—102no—
372C23×C3—102no—
373C23×C3—202no—
374C23×C3—202no—
375C23×C3—202no—
376C23×C3D4×D511yescharacteristic
377C3×Q8—54no—
378C3×Q8—102no—
379C3×Q8—104no—
380C3×D4—22no—
381C3×D4—24no—
382C3×D4—1016no—
383C3×D4—102no—
384C3×D4—102no—
385C3×D4—208no—
386C3×D4—202no—
387C3×D4C2×D2014yescomplemented
388C2×C3×C4—22no—
389C2×C3×C4—42no—
390C2×C3×C4—52no—
391C2×C3×C4—1016no—
392C2×C3×C4—102no—
393C2×C3×C4—102no—
394C2×C3×C4—104no—
395C2×C3×C4—102no—
396C2×C3×C4—102no—
397C2×C3×C4—104no—
398C2×C3×C4C2×D2012yes—
399C2×C3×C4D4×D512yes—
400C2×C3×C4D4×D511yescharacteristic
401C2×C3×C4C23×D511yescharacteristic
402C3×C8—22no—
403C3×C8—102no—
404C22×C5—24no—
405C22×C5—42no—
406C22×C5—42no—
407C22×C5—48no—
408C22×C5C3×#12012yes—
409C22×C5C3×#13512yes—
410C22×C5C3×AGL(1,ℤ/8)12yes—
411C22×C5C22×C3×D411yescharacteristic
412C2×D5—22no—
413C2×D5—22no—
414C2×D5—22no—
415C2×D5—44no—
416C2×D5—42no—
417C2×D5—42no—
418C2×D5—42no—
419C2×D5—84no—
420C4×C5—42no—
421C4×C5C3×#12012yes—
422C4×C5C22×C3×D411yescharacteristic
423C4×C5C22×C3×D411yescharacteristic
424Dic5—24no—
425Dic5—22no—
426C24—21no—
427C24—51no—
428C24—101no—
429#41—108no—
430#41—1016no—
431#41—108no—
432C2×Q8—54no—
433C2×Q8—52no—
434C2×Q8—102no—
435C2×Q8—101no—
436C2×D4—22no—
437C2×D4—21no—
438C2×D4—52no—
439C2×D4—104no—
440C2×D4—104no—
441C2×D4—108no—
442C2×D4—102no—
443C2×D4—102no—
444C2×D4—101no—
445C2×D4—204no—
446C2×D4C3×D2014yescomplemented
447C2×D4C22×C3×D512yes—
448C22×C4—21no—
449C22×C4—51no—
450C22×C4—52no—
451C22×C4—52no—
452C22×C4—104no—
453C22×C4—108no—
454C22×C4—101no—
455C22×C4—102no—
456C22×C4—102no—
457C22×C4C22×C3×D511yescharacteristic
458SD16—108no—
459D8—108no—
460C2×C8—22no—
461C2×C8—52no—
462C2×C8—102no—
463C2×C8C22×C3×D512yes—
464C4⋊C4—21no—
465C4⋊C4C22×C3×D512yes—
466C22⋊C4—104no—
467C3×C5C2×#44911yescharacteristic, complemented, Hall
468C22×C3—24no—
469C22×C3—42no—
470C22×C3—48no—
471C22×C3—42no—
472C22×C3—102no—
473C22×C3—102no—
474C22×C3—102no—
475C22×C3—204no—
476C22×C3—202no—
477C22×C3—202no—
478C22×C3—202no—
479C22×C3—404no—
480C22×C3#402812yes—
481C22×C3#405712yes—
482C22×C3#406212yes—
483C22×C3C2×D4×D511yescharacteristic
484C3×C4—42no—
485C3×C4—102no—
486C3×C4—104no—
487C3×C4#402812yes—
488C3×C4C22×D2011yescharacteristic
489C3×C4C2×D4×D511yescharacteristic
490C2×C5—44no—
491C2×C5C3×#44914yescomplemented
492C2×C5C2×C3×#12011yescharacteristic
493C2×C5C2×C3×#13511yescharacteristic
494C2×C5C2×C3×AGL(1,ℤ/8)11yescharacteristic
495D5—42no—
496D5—82no—
497C23—22no—
498C23—24no—
499C23—42no—
500C23—42no—
501C23—52no—
502C23—102no—
503C23—102no—
504C23—102no—
505C23—102no—
506C23—202no—
507C23—202no—
508C23—202no—
509C23C3×D4×D511yescharacteristic
510Q8—54no—
511Q8—102no—
512Q8—104no—
513D4—22no—
514D4—24no—
515D4—1016no—
516D4—102no—
517D4—102no—
518D4—208no—
519D4—202no—
520D4C2×C3×D2014yescomplemented
521C2×C4—22no—
522C2×C4—42no—
523C2×C4—52no—
524C2×C4—1016no—
525C2×C4—102no—
526C2×C4—102no—
527C2×C4—104no—
528C2×C4—102no—
529C2×C4—102no—
530C2×C4—104no—
531C2×C4C2×C3×D2012yes—
532C2×C4C3×D4×D512yes—
533C2×C4C3×D4×D511yescharacteristic
534C2×C4C23×C3×D511yescharacteristic
535C8—22no—
536C8—102no—
537C2×C3—44no—
538C2×C3—202no—
539C2×C3—402no—
540C2×C3#6507014yesdirect product factor
541C2×C3C2×#402811yescharacteristic
542C2×C3C2×#405711yescharacteristic
543C2×C3C2×#406211yescharacteristic
544C5C2×C3×#44911yescharacteristic, complemented, Sylow
545C22—24no—
546C22—48no—
547C22—42no—
548C22—42no—
549C22—102no—
550C22—102no—
551C22—102no—
552C22—204no—
553C22—202no—
554C22—202no—
555C22—202no—
556C22—404no—
557C22C3×#402812yes—
558C22C3×#405712yes—
559C22C3×#406212yes—
560C22C2×C3×D4×D511yescharacteristic
561C4—42no—
562C4—102no—
563C4—104no—
564C4C3×#402812yes—
565C4C22×C3×D2011yescharacteristic
566C4C2×C3×D4×D511yescharacteristic
567C3C2×#6507011yescharacteristic, direct product factor, Sylow
568C2—44no—
569C2—202no—
570C2—402no—
571C2C3×#6507014yesdirect product factor
572C2C2×C3×#402811yescharacteristic
573C2C2×C3×#405711yescharacteristic
574C2C2×C3×#406211yescharacteristic
subgroupG/H
centerC23×C3D4×D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC2×C3×C5×#102C2
socleC23×C3×C5C2×D4
hypercenterC2×C3×#102D5
hypocenterC5C2×C3×#449
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C2×C3×#102

Complements

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

NH ≅ G/Ntype
1. C22×C3×#711C2semidirect
2. C22×C3×#686C2semidirect
3. C22×C3×#677C2semidirect
4. C2×C3×#4041C2semidirect
5. C2×C3×C5×#102C2semidirect
6. C2×C3×#3952C2semidirect
7. C2×C3×#3942C2semidirect
8. C3×#65070C2direct (N × H)
9. C2×#65070C3direct (N × H)
16. C2×C3×#711C22semidirect
24. C2×C3×#686C22semidirect
28. C2×C3×#677C22semidirect
30. C3×#4041C22semidirect
31. C3×C5×#102C22semidirect
32. C3×#3952C22semidirect
33. C3×#3942C22semidirect
35. C22×#711C2×C3semidirect
36. C22×#686C2×C3semidirect
37. C22×#677C2×C3semidirect
38. C2×#4041C2×C3semidirect
39. C2×C5×#102C2×C3semidirect
40. C2×#3952C2×C3semidirect
41. C2×#3942C2×C3semidirect
42. #65070C2×C3direct (N × H)
48. C2×C3×C5×D4D4semidirect
73. C2×C3×Dic10D4semidirect
90. C2×C3×#102D5semidirect
98. C2×#711C22×C3semidirect
106. C2×#686C22×C3semidirect
110. C2×#677C22×C3semidirect
112. #4041C22×C3semidirect
113. C5×#102C22×C3semidirect
114. #3952C22×C3semidirect
115. #3942C22×C3semidirect
132. C3×C5×D4C2×D4semidirect
154. C3×Dic10C2×D4semidirect
179. C3×#102C2×D5semidirect
186. C2×C5×D4C3×D4semidirect
211. C2×Dic10C3×D4semidirect
228. C2×#102C3×D5semidirect
272. C2×C3×D4D20semidirect
308. C5×D4C2×C3×D4semidirect
330. Dic10C2×C3×D4semidirect
355. #102C2×C3×D5semidirect
358. C2×C3×C5#449semidirect
387. C3×D4C2×D20semidirect
446. C2×D4C3×D20semidirect
467. C3×C5C2×#449semidirect
491. C2×C5C3×#449semidirect
520. D4C2×C3×D20semidirect
540. C2×C3#65070direct (N × H)
544. C5C2×C3×#449semidirect
567. C3C2×#65070direct (N × H)
571. C2C3×#65070direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#7112
C22×C3×#6862
C22×C3×#6772
C2×C3×#40412
C2×C3×C5×#1022
C2×C3×#39522
C2×C3×#39422
C3×#650702
C2×#650703
C2×C3×#4495