Finite Group Databasebeta

← C2×C3×#65336 #14423494

Subgroups — C2×C3×#65336 #14423494

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×#709C211yesmaximal, characteristic
2C2×C3×C5×C22⋊Q8C211yesmaximal, characteristic, complemented
3C22×C3×#689C211yesmaximal, characteristic, complemented
4C22×C3×#688C211yesmaximal, characteristic, complemented
5C2×C3×C22⋊#168C211yesmaximal, characteristic, complemented
6C3×#65336C218yesmaximal, direct product factor
7C2×C3×#3942C211yesmaximal, characteristic, complemented
8C2×C3×#3941C211yesmaximal, characteristic, complemented
9C2×#65336C311yesmaximal, characteristic, direct product factor, Hall
10C22×C3×C5×Q8C2211yescharacteristic
11C23×C3×C4×C5C2211yescharacteristic
12C22×C3×C22⋊D5—21no—
13C2×C3×#709—22no—
14C2×C3×#709—24no—
15C2×C3×#709—24no—
16C2×C3×#709C2214yes—
17C22×C3×D20C2211yescharacteristic
18C22×C3×C4×D5—21no—
19C22×C3×Dic10C2211yescharacteristic
20C3×C5×C22⋊Q8—22no—
21C3×C5×C22⋊Q8C2214yescomplemented
22C2×C3×C5×C4⋊C4—21no—
23C2×C3×C5×C4⋊C4C2211yescharacteristic
24C2×C3×C5×C22⋊C4—21no—
25C2×C3×#689—24no—
26C2×C3×#689C2214yescomplemented
27C2×C3×#688—24no—
28C2×C3×#688C2214yescomplemented
29C22×C3×#168C2211yescharacteristic
30C22×C3×#168C2211yescharacteristic
31C3×C22⋊#168C2214yescomplemented
32C3×#3942C2214yescomplemented
33C3×#3941C2214yescomplemented
34C2×C3×#452—51nomaximal, Hall
35C22×#709C2×C311yescharacteristic
36C2×C5×C22⋊Q8C2×C311yescharacteristic, complemented
37C22×#689C2×C311yescharacteristic, complemented
38C22×#688C2×C311yescharacteristic, complemented
39C2×C22⋊#168C2×C311yescharacteristic, complemented
40#65336C2×C318yesdirect product factor
41C2×#3942C2×C311yescharacteristic, complemented
42C2×#3941C2×C311yescharacteristic, complemented
43C24×C3×C5D411yescharacteristic
44C23×C3×D5—21no—
45C2×C3×C5×Q8—21no—
46C2×C3×C5×Q8—22no—
47C2×C3×C5×Q8—22no—
48C2×C3×C5×Q8C2312yes—
49C22×C3×C4×C5—22no—
50C22×C3×C4×C5—22no—
51C22×C3×C4×C5—21no—
52C22×C3×C4×C5—21no—
53C22×C3×C4×C5D411yescharacteristic
54C22×C3×C4×C5D412yes—
55C22×C3×C4×C5C2312yes—
56C22×C3×C4×C5C2311yescharacteristic
57C2×C3×C22⋊D5—28no—
58C2×C3×C22⋊D5—24no—
59C22×C3×Dic5—21no—
60C3×#709—216no—
61C3×#709—28no—
62C3×#709—28no—
63C2×C3×D20—21no—
64C2×C3×D20—22no—
65C2×C3×D20D414yes—
66C2×C3×D20C2312yes—
67C2×C3×C4×D5—28no—
68C2×C3×C4×D5—24no—
69C2×C3×Dic10—21no—
70C2×C3×Dic10—22no—
71C2×C3×Dic10D414yes—
72C2×C3×Dic10C2312yes—
73C3×C5×C4⋊C4—24no—
74C3×C5×C4⋊C4—21no—
75C3×C5×C4⋊C4C2312yes—
76C3×C5×C22⋊C4—24no—
77C3×#689—28no—
78C3×#688—28no—
79C2×C3×#168—22no—
80C2×C3×#168—22no—
81C2×C3×#168C2312yes—
82C2×C3×#168C2312yes—
83C22×C3×#41—51no—
84C22×C3×Q16—51no—
85C22×C3×SD16—51no—
86C2×C3×C22⋊Q8D511yescharacteristic, complemented
87C3×#452—58no—
88C2×C3×#103—51no—
89C2×C3×#102—51no—
90C2×C3×C22⋊C8—51no—
91C22×C5×Q8C22×C311yescharacteristic
92C23×C4×C5C22×C311yescharacteristic
93C22×C22⋊D5—21no—
94C2×#709—22no—
95C2×#709—24no—
96C2×#709—24no—
97C2×#709C22×C314yes—
98C22×D20C22×C311yescharacteristic
99C22×C4×D5—21no—
100C22×Dic10C22×C311yescharacteristic
101C5×C22⋊Q8—22no—
102C5×C22⋊Q8C22×C314yescomplemented
103C2×C5×C4⋊C4—21no—
104C2×C5×C4⋊C4C22×C311yescharacteristic
105C2×C5×C22⋊C4—21no—
106C2×#689—24no—
107C2×#689C22×C314yescomplemented
108C2×#688—24no—
109C2×#688C22×C314yescomplemented
110C22×#168C22×C311yescharacteristic
111C22×#168C22×C311yescharacteristic
112C22⋊#168C22×C314yescomplemented
113#3942C22×C314yescomplemented
114#3941C22×C314yescomplemented
115C2×#452—51noSylow
116C23×C3×C5—22no—
117C23×C3×C5—22no—
118C23×C3×C5—22no—
119C23×C3×C5C2×D412yes—
120C23×C3×C5C2×D411yescharacteristic
121C22×C3×D5—24no—
122C22×C3×D5—22no—
123C22×C3×D5—42no—
124C22×C3×D5—42no—
125C3×C5×Q8—22no—
126C3×C5×Q8—22no—
127C3×C5×Q8—42no—
128C2×C3×C4×C5—22no—
129C2×C3×C4×C5—24no—
130C2×C3×C4×C5—22no—
131C2×C3×C4×C5—22no—
132C2×C3×C4×C5—22no—
133C2×C3×C4×C5—22no—
134C2×C3×C4×C5—42no—
135C2×C3×C4×C5—42no—
136C2×C3×C4×C5C2×D412yes—
137C2×C3×C4×C5C2×D411yescharacteristic
138C2×C3×C4×C5C2×D412yes—
139C2×C3×C4×C5C2×D412yes—
140C2×C3×C4×C5C2411yescharacteristic
141C3×C22⋊D5—216no—
142C2×C3×Dic5—24no—
143C2×C3×Dic5—22no—
144C3×D20—24no—
145C3×D20—22no—
146C3×D20C2×D414yes—
147C3×C4×D5—216no—
148C3×Dic10—24no—
149C3×Dic10—22no—
150C3×Dic10C2×D414yes—
151C3×#168—22no—
152C3×#168—22no—
153C2×C3×#41—54no—
154C2×C3×#41—104no—
155C2×C3×#41—104no—
156C2×C3×#41—102no—
157C22×C3×Q8—51no—
158C22×C3×Q8C2×D511yescharacteristic
159C22×C3×D4—51no—
160C22×C3×D4—101no—
161C23×C3×C4—101no—
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—22no—
170C3×C22⋊Q8C2×D514yescomplemented
171C2×C3×C4⋊C4—21no—
172C2×C3×C4⋊C4C2×D511yescharacteristic
173C2×C3×C22⋊C4—21no—
174C3×#103—54no—
175C3×#102—54no—
176C3×C22⋊C8—54no—
177C24×C5C3×D411yescharacteristic
178C23×D5—21no—
179C2×C5×Q8—21no—
180C2×C5×Q8—22no—
181C2×C5×Q8—22no—
182C2×C5×Q8C23×C312yes—
183C22×C4×C5—22no—
184C22×C4×C5—22no—
185C22×C4×C5—21no—
186C22×C4×C5—21no—
187C22×C4×C5C3×D411yescharacteristic
188C22×C4×C5C3×D412yes—
189C22×C4×C5C23×C312yes—
190C22×C4×C5C23×C311yescharacteristic
191C2×C22⋊D5—28no—
192C2×C22⋊D5—24no—
193C22×Dic5—21no—
194#709—216no—
195#709—28no—
196#709—28no—
197C2×D20—22no—
198C2×D20—21no—
199C2×D20C3×D414yes—
200C2×D20C23×C312yes—
201C2×C4×D5—28no—
202C2×C4×D5—24no—
203C2×Dic10—22no—
204C2×Dic10—21no—
205C2×Dic10C3×D414yes—
206C2×Dic10C23×C312yes—
207C5×C4⋊C4—24no—
208C5×C4⋊C4—21no—
209C5×C4⋊C4C23×C312yes—
210C5×C22⋊C4—24no—
211#689—28no—
212#688—28no—
213C2×#168—22no—
214C2×#168—22no—
215C2×#168C23×C312yes—
216C2×#168C23×C312yes—
217C22×#41—51no—
218C22×Q16—51no—
219C22×SD16—51no—
220C2×C22⋊Q8C3×D511yescharacteristic, complemented
221#452—58no—
222C2×#103—51no—
223C2×#102—51no—
224C2×C22⋊C8—51no—
225C22×C3×C5—22no—
226C22×C3×C5—22no—
227C22×C3×C5—22no—
228C22×C3×C5—44no—
229C22×C3×C5#12012yes—
230C22×C3×C5#13512yes—
231C22×C3×C5#13712yes—
232C22×C3×C5C22×D411yescharacteristic
233C2×C3×D5—24no—
234C2×C3×D5—42no—
235C2×C3×D5—42no—
236C2×C3×D5—48no—
237C3×C4×C5—22no—
238C3×C4×C5—42no—
239C3×C4×C5—42no—
240C3×C4×C5#12012yes—
241C3×C4×C5C22×D411yescharacteristic
242C3×C4×C5C22×D411yescharacteristic
243C3×Dic5—24no—
244C24×C3—101no—
245C24×C3C22⋊D511yescharacteristic
246C3×#41—108no—
247C3×#41—108no—
248C3×#41—1016no—
249C2×C3×Q8—21no—
250C2×C3×Q8—22no—
251C2×C3×Q8—22no—
252C2×C3×Q8—54no—
253C2×C3×Q8—52no—
254C2×C3×Q8—102no—
255C2×C3×Q8—101no—
256C2×C3×Q8C22×D512yes—
257C2×C3×D4—54no—
258C2×C3×D4—52no—
259C2×C3×D4—108no—
260C2×C3×D4—104no—
261C2×C3×D4—102no—
262C2×C3×D4—101no—
263C22×C3×C4—22no—
264C22×C3×C4—22no—
265C22×C3×C4—21no—
266C22×C3×C4—21no—
267C22×C3×C4—108no—
268C22×C3×C4—101no—
269C22×C3×C4—104no—
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—24no—
281C3×C4⋊C4—21no—
282C3×C4⋊C4C22×D512yes—
283C3×C22⋊C4—24no—
284C23×C5—22no—
285C23×C5—22no—
286C23×C5—22no—
287C23×C5C2×C3×D412yes—
288C23×C5C2×C3×D411yescharacteristic
289C22×D5—24no—
290C22×D5—22no—
291C22×D5—42no—
292C22×D5—42no—
293C5×Q8—22no—
294C5×Q8—22no—
295C5×Q8—42no—
296C2×C4×C5—22no—
297C2×C4×C5—24no—
298C2×C4×C5—22no—
299C2×C4×C5—22no—
300C2×C4×C5—22no—
301C2×C4×C5—22no—
302C2×C4×C5—42no—
303C2×C4×C5—42no—
304C2×C4×C5C2×C3×D412yes—
305C2×C4×C5C2×C3×D412yes—
306C2×C4×C5C2×C3×D412yes—
307C2×C4×C5C2×C3×D411yescharacteristic
308C2×C4×C5C24×C311yescharacteristic
309C22⋊D5—216no—
310C2×Dic5—24no—
311C2×Dic5—22no—
312D20—24no—
313D20—22no—
314D20C2×C3×D414yes—
315C4×D5—216no—
316Dic10—24no—
317Dic10—22no—
318Dic10C2×C3×D414yes—
319#168—22no—
320#168—22no—
321C2×#41—54no—
322C2×#41—104no—
323C2×#41—104no—
324C2×#41—102no—
325C22×Q8—51no—
326C22×Q8C2×C3×D511yescharacteristic
327C22×D4—51no—
328C22×D4—101no—
329C23×C4—101no—
330C23×C4C2×C3×D511yescharacteristic
331C2×Q16—54no—
332C2×Q16—104no—
333C2×SD16—54no—
334C2×SD16—104no—
335C22×C8—51no—
336C22×C8—51no—
337C22⋊Q8—22no—
338C22⋊Q8C2×C3×D514yescomplemented
339C2×C4⋊C4—21no—
340C2×C4⋊C4C2×C3×D511yescharacteristic
341C2×C22⋊C4—21no—
342#103—54no—
343#102—54no—
344C22⋊C8—54no—
345C2×C3×C5—42no—
346C2×C3×C5#45214yescomplemented
347C2×C3×C5C2×#12011yescharacteristic
348C2×C3×C5C2×#13511yescharacteristic
349C2×C3×C5C2×#13711yescharacteristic
350C3×D5—44no—
351C23×C3—22no—
352C23×C3—22no—
353C23×C3—22no—
354C23×C3—102no—
355C23×C3—104no—
356C23×C3—202no—
357C23×C3—202no—
358C23×C3C2×C22⋊D512yes—
359C23×C3C2×C22⋊D511yescharacteristic
360C3×Q8—22no—
361C3×Q8—22no—
362C3×Q8—42no—
363C3×Q8—54no—
364C3×Q8—102no—
365C3×Q8—104no—
366C3×D4—54no—
367C3×D4—1016no—
368C3×D4—102no—
369C3×D4—104no—
370C2×C3×C4—22no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—24no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—42no—
377C2×C3×C4—42no—
378C2×C3×C4—1016no—
379C2×C3×C4—102no—
380C2×C3×C4—104no—
381C2×C3×C4D4×D512yes—
382C2×C3×C4D4×D512yes—
383C2×C3×C4C2×C22⋊D512yes—
384C2×C3×C4C2×C22⋊D511yescharacteristic
385C2×C3×C4C23×D511yescharacteristic
386C3×C8—102no—
387C3×C8—102no—
388C22×C5—22no—
389C22×C5—22no—
390C22×C5—22no—
391C22×C5—44no—
392C22×C5C3×#12012yes—
393C22×C5C3×#13512yes—
394C22×C5C3×#13712yes—
395C22×C5C22×C3×D411yescharacteristic
396C2×D5—24no—
397C2×D5—42no—
398C2×D5—42no—
399C2×D5—48no—
400C4×C5—22no—
401C4×C5—42no—
402C4×C5—42no—
403C4×C5C3×#12012yes—
404C4×C5C22×C3×D411yescharacteristic
405C4×C5C22×C3×D411yescharacteristic
406Dic5—24no—
407C24—101no—
408C24C3×C22⋊D511yescharacteristic
409#41—108no—
410#41—108no—
411#41—1016no—
412C2×Q8—21no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—54no—
416C2×Q8—52no—
417C2×Q8—102no—
418C2×Q8—101no—
419C2×Q8C22×C3×D512yes—
420C2×D4—54no—
421C2×D4—52no—
422C2×D4—108no—
423C2×D4—104no—
424C2×D4—102no—
425C2×D4—101no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—108no—
431C22×C4—101no—
432C22×C4—104no—
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—24no—
444C4⋊C4—21no—
445C4⋊C4C22×C3×D512yes—
446C22⋊C4—24no—
447C3×C5C2×#45211yescharacteristic, complemented, Hall
448C22×C3—22no—
449C22×C3—22no—
450C22×C3—22no—
451C22×C3—44no—
452C22×C3—104no—
453C22×C3—208no—
454C22×C3—202no—
455C22×C3—202no—
456C22×C3#408312yes—
457C22×C3#408812yes—
458C22×C3#409612yes—
459C22×C3C22×C22⋊D511yescharacteristic
460C3×C4—22no—
461C3×C4—42no—
462C3×C4—42no—
463C3×C4—104no—
464C3×C4#408312yes—
465C3×C4C2×D4×D511yescharacteristic
466C3×C4C2×D4×D511yescharacteristic
467C2×C5—42no—
468C2×C5C3×#45214yescomplemented
469C2×C5C2×C3×#12011yescharacteristic
470C2×C5C2×C3×#13511yescharacteristic
471C2×C5C2×C3×#13711yescharacteristic
472D5—44no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—104no—
477C23—102no—
478C23—202no—
479C23—202no—
480C23C2×C3×C22⋊D512yes—
481C23C2×C3×C22⋊D511yescharacteristic
482Q8—22no—
483Q8—22no—
484Q8—42no—
485Q8—54no—
486Q8—104no—
487Q8—102no—
488D4—54no—
489D4—1016no—
490D4—104no—
491D4—102no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—1016no—
501C2×C4—104no—
502C2×C4—102no—
503C2×C4C3×D4×D512yes—
504C2×C4C3×D4×D512yes—
505C2×C4C2×C3×C22⋊D512yes—
506C2×C4C2×C3×C22⋊D511yescharacteristic
507C2×C4C23×C3×D511yescharacteristic
508C8—102no—
509C8—102no—
510C2×C3—42no—
511C2×C3—204no—
512C2×C3#6533614yesdirect product factor
513C2×C3C2×#408311yescharacteristic
514C2×C3C2×#408811yescharacteristic
515C2×C3C2×#409611yescharacteristic
516C5C2×C3×#45211yescharacteristic, complemented, Sylow
517C22—22no—
518C22—22no—
519C22—22no—
520C22—44no—
521C22—104no—
522C22—202no—
523C22—208no—
524C22—202no—
525C22C3×#408312yes—
526C22C3×#408812yes—
527C22C3×#409612yes—
528C22C22×C3×C22⋊D511yescharacteristic
529C4—22no—
530C4—42no—
531C4—42no—
532C4—104no—
533C4C3×#408312yes—
534C4C2×C3×D4×D511yescharacteristic
535C4C2×C3×D4×D511yescharacteristic
536C3C2×#6533611yescharacteristic, direct product factor, Sylow
537C2—42no—
538C2—204no—
539C2C3×#6533614yesdirect product factor
540C2C2×C3×#408311yescharacteristic
541C2C2×C3×#408811yescharacteristic
542C2C2×C3×#409611yescharacteristic
subgroupG/H
centerC23×C3C2×C22⋊D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC2×C3×C5×C22⋊Q8C2
socleC23×C3×C5C2×D4
prime radicalC22×C3×#709C2
hypercenterC2×C3×C22⋊Q8D5
hypocenterC5C2×C3×#452
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ C2×C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×C3×C5×C22⋊Q8C2semidirect
3. C22×C3×#689C2semidirect
4. C22×C3×#688C2semidirect
5. C2×C3×C22⋊#168C2semidirect
6. C3×#65336C2direct (N × H)
7. C2×C3×#3942C2semidirect
8. C2×C3×#3941C2semidirect
9. C2×#65336C3direct (N × H)
21. C3×C5×C22⋊Q8C22semidirect
26. C2×C3×#689C22semidirect
28. C2×C3×#688C22semidirect
31. C3×C22⋊#168C22semidirect
32. C3×#3942C22semidirect
33. C3×#3941C22semidirect
36. C2×C5×C22⋊Q8C2×C3semidirect
37. C22×#689C2×C3semidirect
38. C22×#688C2×C3semidirect
39. C2×C22⋊#168C2×C3semidirect
40. #65336C2×C3direct (N × H)
41. C2×#3942C2×C3semidirect
42. C2×#3941C2×C3semidirect
86. C2×C3×C22⋊Q8D5semidirect
102. C5×C22⋊Q8C22×C3semidirect
107. C2×#689C22×C3semidirect
109. C2×#688C22×C3semidirect
112. C22⋊#168C22×C3semidirect
113. #3942C22×C3semidirect
114. #3941C22×C3semidirect
170. C3×C22⋊Q8C2×D5semidirect
220. C2×C22⋊Q8C3×D5semidirect
338. C22⋊Q8C2×C3×D5semidirect
346. C2×C3×C5#452semidirect
447. C3×C5C2×#452semidirect
468. C2×C5C3×#452semidirect
512. C2×C3#65336direct (N × H)
516. C5C2×C3×#452semidirect
536. C3C2×#65336direct (N × H)
539. C2C3×#65336direct (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×C5×C22⋊Q82
C22×C3×#6892
C22×C3×#6882
C2×C3×C22⋊#1682
C3×#653362
C2×C3×#39422
C2×C3×#39412
C2×#653363
C2×C3×#4525