Finite Group Databasebeta

← C2×C3×#65528 #14423500

Subgroups — C2×C3×#65528 #14423500

Order 1920.

1520 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C5×#41C211yesmaximal, characteristic
2C22×C3×#689C211yesmaximal, characteristic, complemented
3C22×C3×#687C211yesmaximal, characteristic, complemented
4C2×C3×#4070C211yesmaximal, characteristic, complemented
5C3×#65528C218yesmaximal, direct product factor
6C2×C3×#3967C211yesmaximal, characteristic, complemented
7C2×C3×#3964C211yesmaximal, characteristic, complemented
8C2×C3×C22⋊#168C211yesmaximal, characteristic, complemented
9C2×#65528C311yesmaximal, characteristic, direct product factor, Hall
10C2×C3×C5×#41—24no—
11C2×C3×C5×#41—24no—
12C2×C3×C5×#41—22no—
13C2×C3×C5×#41C2214yes—
14C22×C3×C5×Q8C2211yescharacteristic
15C22×C3×C5×D4—21no—
16C22×C3×C5×D4C2211yescharacteristic
17C23×C3×C4×C5—21no—
18C23×C3×C4×C5C2211yescharacteristic
19C22×C3×Dic10C2211yescharacteristic
20C2×C3×C22⋊Dic5—21no—
21C2×C3×#689—24no—
22C2×C3×#689C2214yescomplemented
23C2×C3×#687—24no—
24C2×C3×#687C2214yescomplemented
25C2×C3×C4⋊Dic5C2211yescharacteristic
26C3×#4070—22no—
27C3×#4070C2214yescomplemented
28C2×C3×#683—21no—
29C22×C3×#168C2211yescharacteristic
30C22×C3×#168C2211yescharacteristic
31C3×#3967C2214yescomplemented
32C3×#3964C2214yescomplemented
33C3×C22⋊#168C2214yescomplemented
34C2×C3×#452—51nomaximal, Hall
35C22×C5×#41C2×C311yescharacteristic
36C22×#689C2×C311yescharacteristic, complemented
37C22×#687C2×C311yescharacteristic, complemented
38C2×#4070C2×C311yescharacteristic, complemented
39#65528C2×C318yesdirect product factor
40C2×#3967C2×C311yescharacteristic, complemented
41C2×#3964C2×C311yescharacteristic, complemented
42C2×C22⋊#168C2×C311yescharacteristic, complemented
43C24×C3×C5—21no—
44C24×C3×C5D411yescharacteristic
45C3×C5×#41—216no—
46C3×C5×#41—28no—
47C3×C5×#41—28no—
48C2×C3×C5×Q8—22no—
49C2×C3×C5×Q8—21no—
50C2×C3×C5×Q8D414yes—
51C2×C3×C5×Q8C2312yes—
52C2×C3×C5×D4—28no—
53C2×C3×C5×D4—24no—
54C2×C3×C5×D4—21no—
55C2×C3×C5×D4—22no—
56C2×C3×C5×D4D414yes—
57C2×C3×C5×D4C2312yes—
58C22×C3×C4×C5—28no—
59C22×C3×C4×C5—21no—
60C22×C3×C4×C5—24no—
61C22×C3×C4×C5—22no—
62C22×C3×C4×C5—22no—
63C22×C3×C4×C5D411yescharacteristic
64C22×C3×C4×C5D412yes—
65C22×C3×C4×C5C2312yes—
66C22×C3×C4×C5C2311yescharacteristic
67C22×C3×Dic5—21no—
68C22×C3×Dic5—21no—
69C2×C3×Dic10—22no—
70C2×C3×Dic10—22no—
71C2×C3×Dic10—21no—
72C2×C3×Dic10C2312yes—
73C3×C22⋊Dic5—24no—
74C3×#689—28no—
75C3×#687—28no—
76C3×C4⋊Dic5—21no—
77C3×C4⋊Dic5C2312yes—
78C3×#683—24no—
79C2×C3×#168—22no—
80C2×C3×#168—22no—
81C2×C3×#168C2312yes—
82C2×C3×#168C2312yes—
83C22×C3×#41D511yescharacteristic
84C22×C3×Q16—51no—
85C22×C3×SD16—51no—
86C2×C3×C22⋊Q8—51no—
87C3×#452—58no—
88C2×C3×#103—51no—
89C2×C3×#102—51no—
90C2×C3×C22⋊C8—51no—
91C2×C5×#41—24no—
92C2×C5×#41—24no—
93C2×C5×#41—22no—
94C2×C5×#41C22×C314yes—
95C22×C5×Q8C22×C311yescharacteristic
96C22×C5×D4—21no—
97C22×C5×D4C22×C311yescharacteristic
98C23×C4×C5—21no—
99C23×C4×C5C22×C311yescharacteristic
100C22×Dic10C22×C311yescharacteristic
101C2×C22⋊Dic5—21no—
102C2×#689—24no—
103C2×#689C22×C314yescomplemented
104C2×#687—24no—
105C2×#687C22×C314yescomplemented
106C2×C4⋊Dic5C22×C311yescharacteristic
107#4070—22no—
108#4070C22×C314yescomplemented
109C2×#683—21no—
110C22×#168C22×C311yescharacteristic
111C22×#168C22×C311yescharacteristic
112#3967C22×C314yescomplemented
113#3964C22×C314yescomplemented
114C22⋊#168C22×C314yescomplemented
115C2×#452—51noSylow
116C23×C3×C5—22no—
117C23×C3×C5—24no—
118C23×C3×C5—22no—
119C23×C3×C5—22no—
120C23×C3×C5—22no—
121C23×C3×C5—42no—
122C23×C3×C5—42no—
123C23×C3×C5C2×D412yes—
124C23×C3×C5C2×D411yescharacteristic
125C3×C5×Q8—24no—
126C3×C5×Q8—22no—
127C3×C5×Q8C2×D414yes—
128C3×C5×D4—216no—
129C3×C5×D4—24no—
130C3×C5×D4—22no—
131C3×C5×D4C2×D414yes—
132C2×C3×C4×C5—216no—
133C2×C3×C4×C5—22no—
134C2×C3×C4×C5—24no—
135C2×C3×C4×C5—22no—
136C2×C3×C4×C5—24no—
137C2×C3×C4×C5—22no—
138C2×C3×C4×C5—22no—
139C2×C3×C4×C5C2×D412yes—
140C2×C3×C4×C5C2×D411yescharacteristic
141C2×C3×C4×C5C2×D412yes—
142C2×C3×C4×C5C2×D412yes—
143C2×C3×C4×C5C2411yescharacteristic
144C2×C3×Dic5—22no—
145C2×C3×Dic5—22no—
146C2×C3×Dic5—42no—
147C2×C3×Dic5—42no—
148C3×Dic10—22no—
149C3×Dic10—22no—
150C3×Dic10—42no—
151C3×#168—22no—
152C3×#168—22no—
153C2×C3×#41—24no—
154C2×C3×#41—24no—
155C2×C3×#41—22no—
156C2×C3×#41C2×D514yes—
157C22×C3×Q8—51no—
158C22×C3×Q8C2×D511yescharacteristic
159C22×C3×D4—21no—
160C22×C3×D4C2×D511yescharacteristic
161C23×C3×C4—21no—
162C23×C3×C4C2×D511yescharacteristic
163C2×C3×Q16—54no—
164C2×C3×Q16—104no—
165C2×C3×SD16—54no—
166C2×C3×SD16—104no—
167C22×C3×C8—51no—
168C22×C3×C8—51no—
169C3×C22⋊Q8—54no—
170C3×C22⋊Q8—102no—
171C2×C3×C4⋊C4—51no—
172C2×C3×C4⋊C4—101no—
173C2×C3×C22⋊C4—101no—
174C3×#103—54no—
175C3×#102—54no—
176C3×C22⋊C8—54no—
177C24×C5—21no—
178C24×C5C3×D411yescharacteristic
179C5×#41—216no—
180C5×#41—28no—
181C5×#41—28no—
182C2×C5×Q8—22no—
183C2×C5×Q8—21no—
184C2×C5×Q8C3×D414yes—
185C2×C5×Q8C23×C312yes—
186C2×C5×D4—28no—
187C2×C5×D4—24no—
188C2×C5×D4—21no—
189C2×C5×D4—22no—
190C2×C5×D4C3×D414yes—
191C2×C5×D4C23×C312yes—
192C22×C4×C5—28no—
193C22×C4×C5—21no—
194C22×C4×C5—24no—
195C22×C4×C5—22no—
196C22×C4×C5—22no—
197C22×C4×C5C3×D411yescharacteristic
198C22×C4×C5C3×D412yes—
199C22×C4×C5C23×C312yes—
200C22×C4×C5C23×C311yescharacteristic
201C22×Dic5—21no—
202C22×Dic5—21no—
203C2×Dic10—22no—
204C2×Dic10—22no—
205C2×Dic10—21no—
206C2×Dic10C23×C312yes—
207C22⋊Dic5—24no—
208#689—28no—
209#687—28no—
210C4⋊Dic5—21no—
211C4⋊Dic5C23×C312yes—
212#683—24no—
213C2×#168—22no—
214C2×#168—22no—
215C2×#168C23×C312yes—
216C2×#168C23×C312yes—
217C22×#41C3×D511yescharacteristic
218C22×Q16—51no—
219C22×SD16—51no—
220C2×C22⋊Q8—51no—
221#452—58no—
222C2×#103—51no—
223C2×#102—51no—
224C2×C22⋊C8—51no—
225C22×C3×C5—22no—
226C22×C3×C5—24no—
227C22×C3×C5—22no—
228C22×C3×C5—22no—
229C22×C3×C5—42no—
230C22×C3×C5—42no—
231C22×C3×C5—44no—
232C22×C3×C5—48no—
233C22×C3×C5#12012yes—
234C22×C3×C5#13512yes—
235C22×C3×C5#13712yes—
236C22×C3×C5C22×D411yescharacteristic
237C3×C4×C5—22no—
238C3×C4×C5—24no—
239C3×C4×C5#12012yes—
240C3×C4×C5C22×D411yescharacteristic
241C3×C4×C5C22×D411yescharacteristic
242C3×Dic5—42no—
243C3×Dic5—42no—
244C24×C3—21no—
245C24×C3C22⋊D511yescharacteristic
246C3×#41—216no—
247C3×#41—28no—
248C3×#41—28no—
249C2×C3×Q8—22no—
250C2×C3×Q8—21no—
251C2×C3×Q8—52no—
252C2×C3×Q8—101no—
253C2×C3×Q8—102no—
254C2×C3×Q8—102no—
255C2×C3×Q8C22⋊D514yes—
256C2×C3×Q8C22×D512yes—
257C2×C3×D4—28no—
258C2×C3×D4—24no—
259C2×C3×D4—21no—
260C2×C3×D4—22no—
261C2×C3×D4C22⋊D514yes—
262C2×C3×D4C22×D512yes—
263C22×C3×C4—28no—
264C22×C3×C4—21no—
265C22×C3×C4—24no—
266C22×C3×C4—22no—
267C22×C3×C4—22no—
268C22×C3×C4—101no—
269C22×C3×C4—101no—
270C22×C3×C4C22⋊D512yes—
271C22×C3×C4C22⋊D511yescharacteristic
272C22×C3×C4C22×D512yes—
273C22×C3×C4C22×D511yescharacteristic
274C3×Q16—108no—
275C3×SD16—108no—
276C2×C3×C8—52no—
277C2×C3×C8—52no—
278C2×C3×C8—102no—
279C2×C3×C8—102no—
280C3×C4⋊C4—52no—
281C3×C4⋊C4—104no—
282C3×C4⋊C4—101no—
283C3×C22⋊C4—104no—
284C23×C5—22no—
285C23×C5—24no—
286C23×C5—22no—
287C23×C5—22no—
288C23×C5—22no—
289C23×C5—42no—
290C23×C5—42no—
291C23×C5C2×C3×D412yes—
292C23×C5C2×C3×D411yescharacteristic
293C5×Q8—24no—
294C5×Q8—22no—
295C5×Q8C2×C3×D414yes—
296C5×D4—216no—
297C5×D4—24no—
298C5×D4—22no—
299C5×D4C2×C3×D414yes—
300C2×C4×C5—216no—
301C2×C4×C5—22no—
302C2×C4×C5—24no—
303C2×C4×C5—22no—
304C2×C4×C5—24no—
305C2×C4×C5—22no—
306C2×C4×C5—22no—
307C2×C4×C5C2×C3×D412yes—
308C2×C4×C5C2×C3×D411yescharacteristic
309C2×C4×C5C2×C3×D412yes—
310C2×C4×C5C2×C3×D412yes—
311C2×C4×C5C24×C311yescharacteristic
312C2×Dic5—22no—
313C2×Dic5—22no—
314C2×Dic5—42no—
315C2×Dic5—42no—
316Dic10—22no—
317Dic10—22no—
318Dic10—42no—
319#168—22no—
320#168—22no—
321C2×#41—24no—
322C2×#41—24no—
323C2×#41—22no—
324C2×#41C2×C3×D514yes—
325C22×Q8—51no—
326C22×Q8C2×C3×D511yescharacteristic
327C22×D4—21no—
328C22×D4C2×C3×D511yescharacteristic
329C23×C4—21no—
330C23×C4C2×C3×D511yescharacteristic
331C2×Q16—54no—
332C2×Q16—104no—
333C2×SD16—54no—
334C2×SD16—104no—
335C22×C8—51no—
336C22×C8—51no—
337C22⋊Q8—54no—
338C22⋊Q8—102no—
339C2×C4⋊C4—51no—
340C2×C4⋊C4—101no—
341C2×C22⋊C4—101no—
342#103—54no—
343#102—54no—
344C22⋊C8—54no—
345C2×C3×C5—42no—
346C2×C3×C5—44no—
347C2×C3×C5#45214yescomplemented
348C2×C3×C5C2×#12011yescharacteristic
349C2×C3×C5C2×#13511yescharacteristic
350C2×C3×C5C2×#13711yescharacteristic
351C23×C3—22no—
352C23×C3—22no—
353C23×C3—22no—
354C23×C3—24no—
355C23×C3—22no—
356C23×C3—42no—
357C23×C3—42no—
358C23×C3C2×C22⋊D512yes—
359C23×C3C2×C22⋊D511yescharacteristic
360C3×Q8—24no—
361C3×Q8—22no—
362C3×Q8—102no—
363C3×Q8—102no—
364C3×Q8—202no—
365C3×Q8C2×C22⋊D514yes—
366C3×D4—216no—
367C3×D4—24no—
368C3×D4—22no—
369C3×D4C2×C22⋊D514yes—
370C2×C3×C4—216no—
371C2×C3×C4—22no—
372C2×C3×C4—24no—
373C2×C3×C4—22no—
374C2×C3×C4—24no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—102no—
378C2×C3×C4—102no—
379C2×C3×C4—202no—
380C2×C3×C4—202no—
381C2×C3×C4C2×C22⋊D512yes—
382C2×C3×C4C2×C22⋊D511yescharacteristic
383C2×C3×C4C2×C22⋊D512yes—
384C2×C3×C4C2×C22⋊D512yes—
385C2×C3×C4C23×D511yescharacteristic
386C3×C8—102no—
387C3×C8—102no—
388C22×C5—22no—
389C22×C5—22no—
390C22×C5—22no—
391C22×C5—24no—
392C22×C5—42no—
393C22×C5—42no—
394C22×C5—44no—
395C22×C5—48no—
396C22×C5C3×#12012yes—
397C22×C5C3×#13512yes—
398C22×C5C3×#13712yes—
399C22×C5C22×C3×D411yescharacteristic
400C4×C5—22no—
401C4×C5—24no—
402C4×C5C3×#12012yes—
403C4×C5C22×C3×D411yescharacteristic
404C4×C5C22×C3×D411yescharacteristic
405Dic5—42no—
406Dic5—42no—
407C24—21no—
408C24C3×C22⋊D511yescharacteristic
409#41—216no—
410#41—28no—
411#41—28no—
412C2×Q8—22no—
413C2×Q8—21no—
414C2×Q8—52no—
415C2×Q8—101no—
416C2×Q8—102no—
417C2×Q8—102no—
418C2×Q8C3×C22⋊D514yes—
419C2×Q8C22×C3×D512yes—
420C2×D4—28no—
421C2×D4—24no—
422C2×D4—21no—
423C2×D4—22no—
424C2×D4C3×C22⋊D514yes—
425C2×D4C22×C3×D512yes—
426C22×C4—28no—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—101no—
432C22×C4—101no—
433C22×C4C3×C22⋊D511yescharacteristic
434C22×C4C3×C22⋊D512yes—
435C22×C4C22×C3×D512yes—
436C22×C4C22×C3×D511yescharacteristic
437Q16—108no—
438SD16—108no—
439C2×C8—52no—
440C2×C8—52no—
441C2×C8—102no—
442C2×C8—102no—
443C4⋊C4—52no—
444C4⋊C4—101no—
445C4⋊C4—104no—
446C22⋊C4—104no—
447C3×C5C2×#45211yescharacteristic, complemented, Hall
448C22×C3—22no—
449C22×C3—22no—
450C22×C3—22no—
451C22×C3—24no—
452C22×C3—42no—
453C22×C3—42no—
454C22×C3—44no—
455C22×C3—48no—
456C22×C3#409612yes—
457C22×C3#409712yes—
458C22×C3#409912yes—
459C22×C3C22×C22⋊D511yescharacteristic
460C3×C4—22no—
461C3×C4—24no—
462C3×C4—202no—
463C3×C4—202no—
464C3×C4#409912yes—
465C3×C4C22×C22⋊D511yescharacteristic
466C3×C4C22×C22⋊D511yescharacteristic
467C2×C5—42no—
468C2×C5—44no—
469C2×C5C3×#45214yescomplemented
470C2×C5C2×C3×#12011yescharacteristic
471C2×C5C2×C3×#13511yescharacteristic
472C2×C5C2×C3×#13711yescharacteristic
473C23—22no—
474C23—22no—
475C23—22no—
476C23—24no—
477C23—22no—
478C23—42no—
479C23—42no—
480C23C2×C3×C22⋊D512yes—
481C23C2×C3×C22⋊D511yescharacteristic
482Q8—24no—
483Q8—22no—
484Q8—102no—
485Q8—102no—
486Q8—202no—
487Q8C2×C3×C22⋊D514yes—
488D4—216no—
489D4—24no—
490D4—22no—
491D4C2×C3×C22⋊D514yes—
492C2×C4—216no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—102no—
500C2×C4—102no—
501C2×C4—202no—
502C2×C4—202no—
503C2×C4C2×C3×C22⋊D512yes—
504C2×C4C2×C3×C22⋊D511yescharacteristic
505C2×C4C2×C3×C22⋊D512yes—
506C2×C4C2×C3×C22⋊D512yes—
507C2×C4C23×C3×D511yescharacteristic
508C8—102no—
509C8—102no—
510C2×C3—42no—
511C2×C3—44no—
512C2×C3#6552814yesdirect product factor
513C2×C3C2×#409611yescharacteristic
514C2×C3C2×#409711yescharacteristic
515C2×C3C2×#409911yescharacteristic
516C5C2×C3×#45211yescharacteristic, complemented, Sylow
517C22—22no—
518C22—22no—
519C22—22no—
520C22—24no—
521C22—42no—
522C22—42no—
523C22—44no—
524C22—48no—
525C22C3×#409612yes—
526C22C3×#409712yes—
527C22C3×#409912yes—
528C22C22×C3×C22⋊D511yescharacteristic
529C4—22no—
530C4—24no—
531C4—202no—
532C4—202no—
533C4C3×#409912yes—
534C4C22×C3×C22⋊D511yescharacteristic
535C4C22×C3×C22⋊D511yescharacteristic
536C3C2×#6552811yescharacteristic, direct product factor, Sylow
537C2—42no—
538C2—44no—
539C2C3×#6552814yesdirect product factor
540C2C2×C3×#409611yescharacteristic
541C2C2×C3×#409711yescharacteristic
542C2C2×C3×#409911yescharacteristic
subgroupG/H
centerC23×C3C2×C22⋊D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
Fitting, prime radicalC22×C3×C5×#41C2
socleC23×C3×C5C2×D4
hypercenterC22×C3×#41D5
hypocenterC5C2×C3×#452
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ C22×C3×#41

Complements

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

NH ≅ G/Ntype
2. C22×C3×#689C2semidirect
3. C22×C3×#687C2semidirect
4. C2×C3×#4070C2semidirect
5. C3×#65528C2direct (N × H)
6. C2×C3×#3967C2semidirect
7. C2×C3×#3964C2semidirect
8. C2×C3×C22⋊#168C2semidirect
9. C2×#65528C3direct (N × H)
22. C2×C3×#689C22semidirect
24. C2×C3×#687C22semidirect
27. C3×#4070C22semidirect
31. C3×#3967C22semidirect
32. C3×#3964C22semidirect
33. C3×C22⋊#168C22semidirect
36. C22×#689C2×C3semidirect
37. C22×#687C2×C3semidirect
38. C2×#4070C2×C3semidirect
39. #65528C2×C3direct (N × H)
40. C2×#3967C2×C3semidirect
41. C2×#3964C2×C3semidirect
42. C2×C22⋊#168C2×C3semidirect
103. C2×#689C22×C3semidirect
105. C2×#687C22×C3semidirect
108. #4070C22×C3semidirect
112. #3967C22×C3semidirect
113. #3964C22×C3semidirect
114. C22⋊#168C22×C3semidirect
347. C2×C3×C5#452semidirect
447. C3×C5C2×#452semidirect
469. C2×C5C3×#452semidirect
512. C2×C3#65528direct (N × H)
516. C5C2×C3×#452semidirect
536. C3C2×#65528direct (N × H)
539. C2C3×#65528direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C5×#412
C22×C3×#6892
C22×C3×#6872
C2×C3×#40702
C3×#655282
C2×C3×#39672
C2×C3×#39642
C2×C3×C22⋊#1682
C2×#655283
C2×C3×#4525