Finite Group Databasebeta

← C2×C3×#65476 #14423497

Subgroups — C2×C3×#65476 #14423497

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×#713C211yesmaximal, characteristic
2C22×C3×C5×Q16C211yesmaximal, characteristic, complemented
3C2×C3×#4092C211yesmaximal, characteristic, complemented
4C22×C3×#688C211yesmaximal, characteristic, complemented
5C2×C3×#3967C211yesmaximal, characteristic, complemented
6C3×#65476C218yesmaximal, direct product factor
7C2×C3×#3953C211yesmaximal, characteristic, complemented
8C2×C3×#3952C211yesmaximal, characteristic, complemented
9C2×#65476C311yesmaximal, characteristic, direct product factor, Hall
10C22×C3×C5×Q8C2211yescharacteristic
11C22×C3×C5×Q8C2211yescharacteristic
12C2×C3×#713—24no—
13C2×C3×#713—24no—
14C2×C3×#713—22no—
15C2×C3×#713C2214yes—
16C22×C3×D20—21no—
17C22×C3×D20C2211yescharacteristic
18C22×C3×C4×D5—21no—
19C22×C3×C4×D5C2211yescharacteristic
20C2×C3×C5×Q16—24no—
21C2×C3×C5×Q16C2214yescomplemented
22C22×C3×C5×C8C2211yescharacteristic
23C3×#4092—22no—
24C3×#4092C2214yescomplemented
25C2×C3×#688—24no—
26C2×C3×#688C2214yescomplemented
27C2×C3×#685—21no—
28C2×C3×C4⋊Dic5C2211yescharacteristic
29C2×C3×#683—21no—
30C22×C3×#168C2211yescharacteristic
31C3×#3967C2214yescomplemented
32C3×#3953C2214yescomplemented
33C3×#3952C2214yescomplemented
34C2×C3×#452—51nomaximal, Hall
35C22×#713C2×C311yescharacteristic
36C22×C5×Q16C2×C311yescharacteristic, complemented
37C2×#4092C2×C311yescharacteristic, complemented
38C22×#688C2×C311yescharacteristic, complemented
39C2×#3967C2×C311yescharacteristic, complemented
40#65476C2×C318yesdirect product factor
41C2×#3953C2×C311yescharacteristic, complemented
42C2×#3952C2×C311yescharacteristic, complemented
43C23×C3×D5—21no—
44C23×C3×D5D411yescharacteristic
45C2×C3×C5×Q8—22no—
46C2×C3×C5×Q8—21no—
47C2×C3×C5×Q8—21no—
48C2×C3×C5×Q8—22no—
49C2×C3×C5×Q8—22no—
50C2×C3×C5×Q8D414yes—
51C2×C3×C5×Q8C2312yes—
52C2×C3×C5×Q8C2312yes—
53C22×C3×C4×C5—21no—
54C22×C3×C4×C5—21no—
55C22×C3×C4×C5C2311yescharacteristic
56C22×C3×Dic5—21no—
57C22×C3×Dic5D411yescharacteristic
58C3×#713—216no—
59C3×#713—28no—
60C3×#713—28no—
61C2×C3×D20—28no—
62C2×C3×D20—24no—
63C2×C3×D20—21no—
64C2×C3×D20—22no—
65C2×C3×D20D414yes—
66C2×C3×D20C2312yes—
67C2×C3×C4×D5—28no—
68C2×C3×C4×D5—22no—
69C2×C3×C4×D5—22no—
70C2×C3×C4×D5—24no—
71C2×C3×C4×D5D412yes—
72C2×C3×C4×D5C2312yes—
73C3×C5×Q16—28no—
74C2×C3×C5×C8—22no—
75C2×C3×C5×C8C2312yes—
76C3×#688—28no—
77C3×#685—24no—
78C3×C4⋊Dic5—21no—
79C3×C4⋊Dic5C2312yes—
80C3×#683—24no—
81C2×C3×#168—22no—
82C2×C3×#168C2312yes—
83C22×C3×#41—51no—
84C22×C3×Q16D511yescharacteristic, complemented
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—
91C22×C5×Q8C22×C311yescharacteristic
92C22×C5×Q8C22×C311yescharacteristic
93C2×#713—24no—
94C2×#713—24no—
95C2×#713—22no—
96C2×#713C22×C314yes—
97C22×D20—21no—
98C22×D20C22×C311yescharacteristic
99C22×C4×D5—21no—
100C22×C4×D5C22×C311yescharacteristic
101C2×C5×Q16—24no—
102C2×C5×Q16C22×C314yescomplemented
103C22×C5×C8C22×C311yescharacteristic
104#4092—22no—
105#4092C22×C314yescomplemented
106C2×#688—24no—
107C2×#688C22×C314yescomplemented
108C2×#685—21no—
109C2×C4⋊Dic5C22×C311yescharacteristic
110C2×#683—21no—
111C22×#168C22×C311yescharacteristic
112#3967C22×C314yescomplemented
113#3953C22×C314yescomplemented
114#3952C22×C314yescomplemented
115C2×#452—51noSylow
116C23×C3×C5C2×D411yescharacteristic
117C22×C3×D5—22no—
118C22×C3×D5—24no—
119C22×C3×D5—22no—
120C22×C3×D5—22no—
121C22×C3×D5—22no—
122C22×C3×D5—42no—
123C22×C3×D5—42no—
124C22×C3×D5C2×D412yes—
125C3×C5×Q8—24no—
126C3×C5×Q8—22no—
127C3×C5×Q8—22no—
128C3×C5×Q8—22no—
129C3×C5×Q8—42no—
130C3×C5×Q8C2×D414yes—
131C2×C3×C4×C5—24no—
132C2×C3×C4×C5—22no—
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
139C2×C3×Dic5—22no—
140C2×C3×Dic5—22no—
141C2×C3×Dic5—42no—
142C2×C3×Dic5C2×D412yes—
143C3×D20—216no—
144C3×D20—24no—
145C3×D20—22no—
146C3×D20C2×D414yes—
147C3×C4×D5—216no—
148C3×C4×D5—24no—
149C3×C4×D5—22no—
150C3×C4×D5—22no—
151C3×C5×C8—22no—
152C3×#168—22no—
153C2×C3×#41—54no—
154C2×C3×#41—102no—
155C2×C3×#41—104no—
156C2×C3×#41—104no—
157C22×C3×Q8C2×D511yescharacteristic
158C22×C3×Q8C2×D511yescharacteristic
159C22×C3×D4—51no—
160C22×C3×D4—101no—
161C23×C3×C4—51no—
162C23×C3×C4—101no—
163C2×C3×Q16—24no—
164C2×C3×Q16C2×D514yescomplemented
165C2×C3×SD16—54no—
166C2×C3×SD16—104no—
167C22×C3×C8—51no—
168C22×C3×C8C2×D511yescharacteristic
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—
177C23×D5—21no—
178C23×D5C3×D411yescharacteristic
179C2×C5×Q8—22no—
180C2×C5×Q8—21no—
181C2×C5×Q8—21no—
182C2×C5×Q8—22no—
183C2×C5×Q8—22no—
184C2×C5×Q8C3×D414yes—
185C2×C5×Q8C23×C312yes—
186C2×C5×Q8C23×C312yes—
187C22×C4×C5—21no—
188C22×C4×C5—21no—
189C22×C4×C5C23×C311yescharacteristic
190C22×Dic5—21no—
191C22×Dic5C3×D411yescharacteristic
192#713—216no—
193#713—28no—
194#713—28no—
195C2×D20—28no—
196C2×D20—24no—
197C2×D20—21no—
198C2×D20—22no—
199C2×D20C3×D414yes—
200C2×D20C23×C312yes—
201C2×C4×D5—28no—
202C2×C4×D5—22no—
203C2×C4×D5—22no—
204C2×C4×D5—24no—
205C2×C4×D5C3×D412yes—
206C2×C4×D5C23×C312yes—
207C5×Q16—28no—
208C2×C5×C8—22no—
209C2×C5×C8C23×C312yes—
210#688—28no—
211#685—24no—
212C4⋊Dic5—21no—
213C4⋊Dic5C23×C312yes—
214#683—24no—
215C2×#168—22no—
216C2×#168C23×C312yes—
217C22×#41—51no—
218C22×Q16C3×D511yescharacteristic, complemented
219C22×SD16—51no—
220C2×C22⋊Q8—51no—
221#452—58no—
222C2×#103—51no—
223C2×#102—51no—
224C2×C22⋊C8—51no—
225C22×C3×C5#12012yes—
226C22×C3×C5#13512yes—
227C22×C3×C5#13712yes—
228C22×C3×C5C22×D411yescharacteristic
229C2×C3×D5—22no—
230C2×C3×D5—24no—
231C2×C3×D5—22no—
232C2×C3×D5—22no—
233C2×C3×D5—42no—
234C2×C3×D5—42no—
235C2×C3×D5—44no—
236C2×C3×D5—48no—
237C3×C4×C5—24no—
238C3×C4×C5—42no—
239C3×C4×C5#12012yes—
240C3×C4×C5C22×D411yescharacteristic
241C3×C4×C5C22×D411yescharacteristic
242C3×Dic5—22no—
243C3×Dic5—42no—
244C24×C3—51no—
245C24×C3—101no—
246C3×#41—108no—
247C3×#41—1016no—
248C3×#41—108no—
249C2×C3×Q8—21no—
250C2×C3×Q8—22no—
251C2×C3×Q8—21no—
252C2×C3×Q8—22no—
253C2×C3×Q8—22no—
254C2×C3×Q8C22⋊D514yes—
255C2×C3×Q8C22×D512yes—
256C2×C3×Q8C22×D512yes—
257C2×C3×D4—54no—
258C2×C3×D4—52no—
259C2×C3×D4—104no—
260C2×C3×D4—108no—
261C2×C3×D4—101no—
262C2×C3×D4—102no—
263C22×C3×C4—21no—
264C22×C3×C4—21no—
265C22×C3×C4—51no—
266C22×C3×C4—52no—
267C22×C3×C4—52no—
268C22×C3×C4—104no—
269C22×C3×C4—108no—
270C22×C3×C4—102no—
271C22×C3×C4—102no—
272C22×C3×C4—101no—
273C22×C3×C4C22×D511yescharacteristic
274C3×Q16—28no—
275C3×SD16—108no—
276C2×C3×C8—22no—
277C2×C3×C8—52no—
278C2×C3×C8—102no—
279C2×C3×C8C22×D512yes—
280C3×C4⋊C4—52no—
281C3×C4⋊C4—104no—
282C3×C4⋊C4—101no—
283C3×C22⋊C4—104no—
284C23×C5C2×C3×D411yescharacteristic
285C22×D5—22no—
286C22×D5—24no—
287C22×D5—22no—
288C22×D5—22no—
289C22×D5—22no—
290C22×D5—42no—
291C22×D5—42no—
292C22×D5C2×C3×D412yes—
293C5×Q8—24no—
294C5×Q8—22no—
295C5×Q8—22no—
296C5×Q8—22no—
297C5×Q8—42no—
298C5×Q8C2×C3×D414yes—
299C2×C4×C5—24no—
300C2×C4×C5—22no—
301C2×C4×C5—22no—
302C2×C4×C5—42no—
303C2×C4×C5C2×C3×D411yescharacteristic
304C2×C4×C5C2×C3×D412yes—
305C2×C4×C5C2×C3×D412yes—
306C2×C4×C5C24×C311yescharacteristic
307C2×Dic5—22no—
308C2×Dic5—22no—
309C2×Dic5—42no—
310C2×Dic5C2×C3×D412yes—
311D20—216no—
312D20—24no—
313D20—22no—
314D20C2×C3×D414yes—
315C4×D5—216no—
316C4×D5—24no—
317C4×D5—22no—
318C4×D5—22no—
319C5×C8—22no—
320#168—22no—
321C2×#41—54no—
322C2×#41—102no—
323C2×#41—104no—
324C2×#41—104no—
325C22×Q8C2×C3×D511yescharacteristic
326C22×Q8C2×C3×D511yescharacteristic
327C22×D4—51no—
328C22×D4—101no—
329C23×C4—51no—
330C23×C4—101no—
331C2×Q16—24no—
332C2×Q16C2×C3×D514yescomplemented
333C2×SD16—54no—
334C2×SD16—104no—
335C22×C8—51no—
336C22×C8C2×C3×D511yescharacteristic
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#45214yescomplemented
346C2×C3×C5C2×#12011yescharacteristic
347C2×C3×C5C2×#13511yescharacteristic
348C2×C3×C5C2×#13711yescharacteristic
349C3×D5—42no—
350C3×D5—44no—
351C23×C3—52no—
352C23×C3—102no—
353C23×C3—102no—
354C23×C3—102no—
355C23×C3—104no—
356C23×C3—102no—
357C23×C3—202no—
358C23×C3—202no—
359C23×C3D4×D511yescharacteristic
360C3×Q8—22no—
361C3×Q8—24no—
362C3×Q8—22no—
363C3×Q8—22no—
364C3×Q8—42no—
365C3×Q8C2×C22⋊D514yes—
366C3×D4—54no—
367C3×D4—1016no—
368C3×D4—102no—
369C3×D4—104no—
370C2×C3×C4—22no—
371C2×C3×C4—24no—
372C2×C3×C4—22no—
373C2×C3×C4—42no—
374C2×C3×C4—52no—
375C2×C3×C4—1016no—
376C2×C3×C4—102no—
377C2×C3×C4—102no—
378C2×C3×C4—102no—
379C2×C3×C4—104no—
380C2×C3×C4—102no—
381C2×C3×C4—202no—
382C2×C3×C4D4×D512yes—
383C2×C3×C4D4×D511yescharacteristic
384C2×C3×C4C2×C22⋊D512yes—
385C2×C3×C4C23×D511yescharacteristic
386C3×C8—22no—
387C3×C8—102no—
388C22×C5C3×#12012yes—
389C22×C5C3×#13512yes—
390C22×C5C3×#13712yes—
391C22×C5C22×C3×D411yescharacteristic
392C2×D5—22no—
393C2×D5—22no—
394C2×D5—22no—
395C2×D5—24no—
396C2×D5—42no—
397C2×D5—42no—
398C2×D5—44no—
399C2×D5—48no—
400C4×C5—24no—
401C4×C5—42no—
402C4×C5C3×#12012yes—
403C4×C5C22×C3×D411yescharacteristic
404C4×C5C22×C3×D411yescharacteristic
405Dic5—22no—
406Dic5—42no—
407C24—51no—
408C24—101no—
409#41—108no—
410#41—1016no—
411#41—108no—
412C2×Q8—22no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8C3×C22⋊D514yes—
418C2×Q8C22×C3×D512yes—
419C2×Q8C22×C3×D512yes—
420C2×D4—54no—
421C2×D4—52no—
422C2×D4—104no—
423C2×D4—108no—
424C2×D4—101no—
425C2×D4—102no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—51no—
429C22×C4—52no—
430C22×C4—52no—
431C22×C4—104no—
432C22×C4—108no—
433C22×C4—102no—
434C22×C4—102no—
435C22×C4—101no—
436C22×C4C22×C3×D511yescharacteristic
437Q16—28no—
438SD16—108no—
439C2×C8—22no—
440C2×C8—52no—
441C2×C8—102no—
442C2×C8C22×C3×D512yes—
443C4⋊C4—52no—
444C4⋊C4—104no—
445C4⋊C4—101no—
446C22⋊C4—104no—
447C3×C5C2×#45211yescharacteristic, complemented, Hall
448C22×C3—102no—
449C22×C3—102no—
450C22×C3—104no—
451C22×C3—102no—
452C22×C3—202no—
453C22×C3—202no—
454C22×C3—204no—
455C22×C3—208no—
456C22×C3#406412yes—
457C22×C3#406512yes—
458C22×C3#408312yes—
459C22×C3C2×D4×D511yescharacteristic
460C3×C4—24no—
461C3×C4—42no—
462C3×C4—102no—
463C3×C4—202no—
464C3×C4#408312yes—
465C3×C4C2×D4×D511yescharacteristic
466C3×C4C22×C22⋊D511yescharacteristic
467C2×C5C3×#45214yescomplemented
468C2×C5C2×C3×#12011yescharacteristic
469C2×C5C2×C3×#13511yescharacteristic
470C2×C5C2×C3×#13711yescharacteristic
471D5—42no—
472D5—44no—
473C23—52no—
474C23—104no—
475C23—102no—
476C23—102no—
477C23—102no—
478C23—102no—
479C23—202no—
480C23—202no—
481C23C3×D4×D511yescharacteristic
482Q8—24no—
483Q8—22no—
484Q8—22no—
485Q8—22no—
486Q8—42no—
487Q8C2×C3×C22⋊D514yes—
488D4—54no—
489D4—1016no—
490D4—104no—
491D4—102no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—42no—
496C2×C4—52no—
497C2×C4—1016no—
498C2×C4—102no—
499C2×C4—104no—
500C2×C4—102no—
501C2×C4—102no—
502C2×C4—102no—
503C2×C4—202no—
504C2×C4C3×D4×D511yescharacteristic
505C2×C4C3×D4×D512yes—
506C2×C4C2×C3×C22⋊D512yes—
507C2×C4C23×C3×D511yescharacteristic
508C8—22no—
509C8—102no—
510C2×C3—202no—
511C2×C3—204no—
512C2×C3#6547614yesdirect product factor
513C2×C3C2×#406411yescharacteristic
514C2×C3C2×#406511yescharacteristic
515C2×C3C2×#408311yescharacteristic
516C5C2×C3×#45211yescharacteristic, complemented, Sylow
517C22—102no—
518C22—104no—
519C22—102no—
520C22—102no—
521C22—202no—
522C22—204no—
523C22—208no—
524C22—202no—
525C22C3×#406412yes—
526C22C3×#406512yes—
527C22C3×#408312yes—
528C22C2×C3×D4×D511yescharacteristic
529C4—24no—
530C4—42no—
531C4—102no—
532C4—202no—
533C4C3×#408312yes—
534C4C2×C3×D4×D511yescharacteristic
535C4C22×C3×C22⋊D511yescharacteristic
536C3C2×#6547611yescharacteristic, direct product factor, Sylow
537C2—202no—
538C2—204no—
539C2C3×#6547614yesdirect product factor
540C2C2×C3×#406411yescharacteristic
541C2C2×C3×#406511yescharacteristic
542C2C2×C3×#408311yescharacteristic
subgroupG/H
centerC23×C3D4×D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC22×C3×C5×Q16C2
socleC23×C3×C5C2×D4
prime radicalC22×C3×#713C2
hypercenterC22×C3×Q16D5
hypocenterC5C2×C3×#452
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C22×C3×Q16

Complements

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

NH ≅ G/Ntype
2. C22×C3×C5×Q16C2semidirect
3. C2×C3×#4092C2semidirect
4. C22×C3×#688C2semidirect
5. C2×C3×#3967C2semidirect
6. C3×#65476C2direct (N × H)
7. C2×C3×#3953C2semidirect
8. C2×C3×#3952C2semidirect
9. C2×#65476C3direct (N × H)
21. C2×C3×C5×Q16C22semidirect
24. C3×#4092C22semidirect
26. C2×C3×#688C22semidirect
31. C3×#3967C22semidirect
32. C3×#3953C22semidirect
33. C3×#3952C22semidirect
36. C22×C5×Q16C2×C3semidirect
37. C2×#4092C2×C3semidirect
38. C22×#688C2×C3semidirect
39. C2×#3967C2×C3semidirect
40. #65476C2×C3direct (N × H)
41. C2×#3953C2×C3semidirect
42. C2×#3952C2×C3semidirect
84. C22×C3×Q16D5semidirect
102. C2×C5×Q16C22×C3semidirect
105. #4092C22×C3semidirect
107. C2×#688C22×C3semidirect
112. #3967C22×C3semidirect
113. #3953C22×C3semidirect
114. #3952C22×C3semidirect
164. C2×C3×Q16C2×D5semidirect
218. C22×Q16C3×D5semidirect
332. C2×Q16C2×C3×D5semidirect
345. C2×C3×C5#452semidirect
447. C3×C5C2×#452semidirect
467. C2×C5C3×#452semidirect
512. C2×C3#65476direct (N × H)
516. C5C2×C3×#452semidirect
536. C3C2×#65476direct (N × H)
539. C2C3×#65476direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#7132
C22×C3×C5×Q162
C2×C3×#40922
C22×C3×#6882
C2×C3×#39672
C3×#654762
C2×C3×#39532
C2×C3×#39522
C2×#654763
C2×C3×#4525