Finite Group Databasebeta

← C2×C3×#65072 #14423503

Subgroups — C2×C3×#65072 #14423503

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×#711C211yesmaximal, characteristic
2C22×C3×#687C211yesmaximal, characteristic, complemented
3C22×C3×Dic20C211yesmaximal, characteristic, complemented
4C2×C3×#4043C211yesmaximal, characteristic, complemented
5C2×C3×C5×#102C211yesmaximal, characteristic, complemented
6C2×C3×#3952C211yesmaximal, characteristic, complemented
7C2×C3×#3942C211yesmaximal, characteristic, complemented
8C3×#65072C218yesmaximal, direct product factor
9C2×#65072C311yesmaximal, 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×#711C2214yes—
17C22×C3×C4×D5C2211yescharacteristic
18C22×C3×Dic10C2211yescharacteristic
19C22×C3×Dic10C2211yescharacteristic
20C22×C3×C5×C8C2211yescharacteristic
21C2×C3×C5×C4⋊C4C2211yescharacteristic
22C2×C3×#687—24no—
23C2×C3×#687C2214yescomplemented
24C2×C3×#685—21no—
25C2×C3×C4⋊Dic5—21no—
26C22×C3×#168C2211yescharacteristic
27C2×C3×Dic20—24no—
28C2×C3×Dic20C2214yescomplemented
29C3×#4043—22no—
30C3×#4043C2214yescomplemented
31C3×C5×#102C2214yescomplemented
32C3×#3952C2214yescomplemented
33C3×#3942C2214yescomplemented
34C2×C3×#452—51nomaximal, Hall
35C22×#711C2×C311yescharacteristic
36C22×#687C2×C311yescharacteristic, complemented
37C22×Dic20C2×C311yescharacteristic, complemented
38C2×#4043C2×C311yescharacteristic, complemented
39C2×C5×#102C2×C311yescharacteristic, complemented
40C2×#3952C2×C311yescharacteristic, complemented
41C2×#3942C2×C311yescharacteristic, complemented
42#65072C2×C318yesdirect product factor
43C24×C3×C5—21no—
44C23×C3×D5D411yescharacteristic
45C2×C3×C5×D4—21no—
46C2×C3×C5×D4—22no—
47C2×C3×C5×D4D414yes—
48C2×C3×C5×D4C2312yes—
49C22×C3×C4×C5—21no—
50C22×C3×C4×C5C2311yescharacteristic
51C2×C3×C22⋊D5—28no—
52C2×C3×C22⋊D5—24no—
53C22×C3×Dic5—28no—
54C22×C3×Dic5—21no—
55C22×C3×Dic5—24no—
56C22×C3×Dic5—21no—
57C22×C3×Dic5D411yescharacteristic
58C3×#711—216no—
59C3×#711—28no—
60C3×#711—28no—
61C2×C3×C4×D5—22no—
62C2×C3×C4×D5—22no—
63C2×C3×C4×D5D412yes—
64C2×C3×C4×D5C2312yes—
65C2×C3×Dic10—21no—
66C2×C3×Dic10—22no—
67C2×C3×Dic10—22no—
68C2×C3×Dic10—22no—
69C2×C3×Dic10—21no—
70C2×C3×Dic10D414yes—
71C2×C3×Dic10C2312yes—
72C2×C3×Dic10C2312yes—
73C2×C3×C5×C8—22no—
74C2×C3×C5×C8C2312yes—
75C3×C5×C4⋊C4—21no—
76C3×C5×C4⋊C4C2312yes—
77C3×#687—28no—
78C3×#685—24no—
79C3×C4⋊Dic5—24no—
80C2×C3×#168—22no—
81C2×C3×#168C2312yes—
82C3×Dic20—28no—
83C22×C3×#41—51no—
84C22×C3×Q16—51no—
85C22×C3×SD16—51no—
86C2×C3×C22⋊Q8—51no—
87C3×#452—58no—
88C2×C3×#103—51no—
89C2×C3×#102D511yescharacteristic, complemented
90C2×C3×C22⋊C8—51no—
91C22×C5×D4C22×C311yescharacteristic
92C22×C22⋊D5—21no—
93C23×Dic5—21no—
94C2×#711—24no—
95C2×#711—22no—
96C2×#711—24no—
97C2×#711C22×C314yes—
98C22×C4×D5C22×C311yescharacteristic
99C22×Dic10C22×C311yescharacteristic
100C22×Dic10C22×C311yescharacteristic
101C22×C5×C8C22×C311yescharacteristic
102C2×C5×C4⋊C4C22×C311yescharacteristic
103C2×#687—24no—
104C2×#687C22×C314yescomplemented
105C2×#685—21no—
106C2×C4⋊Dic5—21no—
107C22×#168C22×C311yescharacteristic
108C2×Dic20—24no—
109C2×Dic20C22×C314yescomplemented
110#4043—22no—
111#4043C22×C314yescomplemented
112C5×#102C22×C314yescomplemented
113#3952C22×C314yescomplemented
114#3942C22×C314yescomplemented
115C2×#452—51noSylow
116C23×C3×C5—24no—
117C23×C3×C5—22no—
118C23×C3×C5—42no—
119C23×C3×C5—42no—
120C23×C3×C5C2×D411yescharacteristic
121C22×C3×D5—22no—
122C22×C3×D5—22no—
123C22×C3×D5—22no—
124C22×C3×D5C2×D412yes—
125C3×C5×D4—24no—
126C3×C5×D4—22no—
127C3×C5×D4C2×D414yes—
128C2×C3×C4×C5—22no—
129C2×C3×C4×C5—42no—
130C2×C3×C4×C5C2×D411yescharacteristic
131C2×C3×C4×C5C2×D412yes—
132C2×C3×C4×C5C2×D412yes—
133C2×C3×C4×C5C2411yescharacteristic
134C3×C22⋊D5—216no—
135C2×C3×Dic5—216no—
136C2×C3×Dic5—24no—
137C2×C3×Dic5—22no—
138C2×C3×Dic5—22no—
139C2×C3×Dic5—22no—
140C2×C3×Dic5—42no—
141C2×C3×Dic5C2×D412yes—
142C3×C4×D5—24no—
143C3×C4×D5—22no—
144C3×C4×D5—22no—
145C3×Dic10—24no—
146C3×Dic10—22no—
147C3×Dic10—22no—
148C3×Dic10—22no—
149C3×Dic10—42no—
150C3×Dic10C2×D414yes—
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×Q8—51no—
158C22×C3×Q8—51no—
159C22×C3×D4—101no—
160C22×C3×D4C2×D511yescharacteristic
161C23×C3×C4—51no—
162C23×C3×C4—101no—
163C2×C3×Q16—54no—
164C2×C3×Q16—104no—
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—101no—
172C2×C3×C4⋊C4C2×D511yescharacteristic
173C2×C3×C22⋊C4—101no—
174C3×#103—54no—
175C3×#102C2×D514yescomplemented
176C3×C22⋊C8—54no—
177C24×C5—21no—
178C23×D5C3×D411yescharacteristic
179C2×C5×D4—21no—
180C2×C5×D4—22no—
181C2×C5×D4C3×D414yes—
182C2×C5×D4C23×C312yes—
183C22×C4×C5—21no—
184C22×C4×C5C23×C311yescharacteristic
185C2×C22⋊D5—28no—
186C2×C22⋊D5—24no—
187C22×Dic5—28no—
188C22×Dic5—21no—
189C22×Dic5—24no—
190C22×Dic5—21no—
191C22×Dic5C3×D411yescharacteristic
192#711—216no—
193#711—28no—
194#711—28no—
195C2×C4×D5—22no—
196C2×C4×D5—22no—
197C2×C4×D5C3×D412yes—
198C2×C4×D5C23×C312yes—
199C2×Dic10—22no—
200C2×Dic10—21no—
201C2×Dic10—22no—
202C2×Dic10—22no—
203C2×Dic10—21no—
204C2×Dic10C3×D414yes—
205C2×Dic10C23×C312yes—
206C2×Dic10C23×C312yes—
207C2×C5×C8—22no—
208C2×C5×C8C23×C312yes—
209C5×C4⋊C4—21no—
210C5×C4⋊C4C23×C312yes—
211#687—28no—
212#685—24no—
213C4⋊Dic5—24no—
214C2×#168—22no—
215C2×#168C23×C312yes—
216Dic20—28no—
217C22×#41—51no—
218C22×Q16—51no—
219C22×SD16—51no—
220C2×C22⋊Q8—51no—
221#452—58no—
222C2×#103—51no—
223C2×#102C3×D511yescharacteristic, complemented
224C2×C22⋊C8—51no—
225C22×C3×C5—24no—
226C22×C3×C5—42no—
227C22×C3×C5—42no—
228C22×C3×C5—48no—
229C22×C3×C5#12012yes—
230C22×C3×C5#13512yes—
231C22×C3×C5#13712yes—
232C22×C3×C5C22×D411yescharacteristic
233C2×C3×D5—22no—
234C2×C3×D5—22no—
235C2×C3×D5—22no—
236C2×C3×D5—44no—
237C3×C4×C5—42no—
238C3×C4×C5#12012yes—
239C3×C4×C5C22×D411yescharacteristic
240C3×C4×C5C22×D411yescharacteristic
241C3×Dic5—24no—
242C3×Dic5—22no—
243C3×Dic5—42no—
244C24×C3—21no—
245C24×C3—51no—
246C3×#41—108no—
247C3×#41—1016no—
248C3×#41—108no—
249C2×C3×Q8—54no—
250C2×C3×Q8—52no—
251C2×C3×Q8—52no—
252C2×C3×Q8—102no—
253C2×C3×Q8—101no—
254C2×C3×Q8—101no—
255C2×C3×Q8—102no—
256C2×C3×Q8—102no—
257C2×C3×D4—22no—
258C2×C3×D4—21no—
259C2×C3×D4—104no—
260C2×C3×D4—108no—
261C2×C3×D4D2014yes—
262C2×C3×D4C22×D512yes—
263C22×C3×C4—21no—
264C22×C3×C4—51no—
265C22×C3×C4—52no—
266C22×C3×C4—52no—
267C22×C3×C4—104no—
268C22×C3×C4—108no—
269C22×C3×C4—101no—
270C22×C3×C4—102no—
271C22×C3×C4—102no—
272C22×C3×C4—101no—
273C22×C3×C4C22×D511yescharacteristic
274C3×Q16—108no—
275C3×SD16—108no—
276C2×C3×C8—22no—
277C2×C3×C8—52no—
278C2×C3×C8—102no—
279C2×C3×C8C22×D512yes—
280C3×C4⋊C4—21no—
281C3×C4⋊C4—104no—
282C3×C4⋊C4C22×D512yes—
283C3×C22⋊C4—104no—
284C23×C5—24no—
285C23×C5—22no—
286C23×C5—42no—
287C23×C5—42no—
288C23×C5C2×C3×D411yescharacteristic
289C22×D5—22no—
290C22×D5—22no—
291C22×D5—22no—
292C22×D5C2×C3×D412yes—
293C5×D4—24no—
294C5×D4—22no—
295C5×D4C2×C3×D414yes—
296C2×C4×C5—22no—
297C2×C4×C5—42no—
298C2×C4×C5C2×C3×D412yes—
299C2×C4×C5C2×C3×D411yescharacteristic
300C2×C4×C5C2×C3×D412yes—
301C2×C4×C5C24×C311yescharacteristic
302C22⋊D5—216no—
303C2×Dic5—216no—
304C2×Dic5—24no—
305C2×Dic5—22no—
306C2×Dic5—22no—
307C2×Dic5—22no—
308C2×Dic5—42no—
309C2×Dic5C2×C3×D412yes—
310C4×D5—24no—
311C4×D5—22no—
312C4×D5—22no—
313Dic10—24no—
314Dic10—22no—
315Dic10—22no—
316Dic10—22no—
317Dic10—42no—
318Dic10C2×C3×D414yes—
319C5×C8—22no—
320#168—22no—
321C2×#41—54no—
322C2×#41—102no—
323C2×#41—104no—
324C2×#41—104no—
325C22×Q8—51no—
326C22×Q8—51no—
327C22×D4—101no—
328C22×D4C2×C3×D511yescharacteristic
329C23×C4—51no—
330C23×C4—101no—
331C2×Q16—54no—
332C2×Q16—104no—
333C2×SD16—54no—
334C2×SD16—104no—
335C22×C8—51no—
336C22×C8C2×C3×D511yescharacteristic
337C22⋊Q8—54no—
338C22⋊Q8—102no—
339C2×C4⋊C4—101no—
340C2×C4⋊C4C2×C3×D511yescharacteristic
341C2×C22⋊C4—101no—
342#103—54no—
343#102C2×C3×D514yescomplemented
344C22⋊C8—54no—
345C2×C3×C5—44no—
346C2×C3×C5#45214yescomplemented
347C2×C3×C5C2×#12011yescharacteristic
348C2×C3×C5C2×#13511yescharacteristic
349C2×C3×C5C2×#13711yescharacteristic
350C3×D5—42no—
351C23×C3—22no—
352C23×C3—24no—
353C23×C3—42no—
354C23×C3—42no—
355C23×C3—52no—
356C23×C3—102no—
357C23×C3—102no—
358C23×C3—102no—
359C23×C3D4×D511yescharacteristic
360C3×Q8—54no—
361C3×Q8—102no—
362C3×Q8—104no—
363C3×Q8—102no—
364C3×Q8—102no—
365C3×Q8—202no—
366C3×D4—22no—
367C3×D4—24no—
368C3×D4—1016no—
369C3×D4C2×D2014yes—
370C2×C3×C4—22no—
371C2×C3×C4—42no—
372C2×C3×C4—52no—
373C2×C3×C4—1016no—
374C2×C3×C4—102no—
375C2×C3×C4—102no—
376C2×C3×C4—104no—
377C2×C3×C4—102no—
378C2×C3×C4—102no—
379C2×C3×C4—104no—
380C2×C3×C4—102no—
381C2×C3×C4—202no—
382C2×C3×C4C2×D2012yes—
383C2×C3×C4D4×D512yes—
384C2×C3×C4D4×D511yescharacteristic
385C2×C3×C4C23×D511yescharacteristic
386C3×C8—22no—
387C3×C8—102no—
388C22×C5—24no—
389C22×C5—42no—
390C22×C5—42no—
391C22×C5—48no—
392C22×C5C3×#12012yes—
393C22×C5C3×#13512yes—
394C22×C5C3×#13712yes—
395C22×C5C22×C3×D411yescharacteristic
396C2×D5—22no—
397C2×D5—22no—
398C2×D5—22no—
399C2×D5—44no—
400C4×C5—42no—
401C4×C5C3×#12012yes—
402C4×C5C22×C3×D411yescharacteristic
403C4×C5C22×C3×D411yescharacteristic
404Dic5—24no—
405Dic5—22no—
406Dic5—42no—
407C24—21no—
408C24—51no—
409#41—108no—
410#41—1016no—
411#41—108no—
412C2×Q8—54no—
413C2×Q8—52no—
414C2×Q8—52no—
415C2×Q8—102no—
416C2×Q8—101no—
417C2×Q8—101no—
418C2×Q8—102no—
419C2×Q8—102no—
420C2×D4—22no—
421C2×D4—21no—
422C2×D4—104no—
423C2×D4—108no—
424C2×D4C3×D2014yes—
425C2×D4C22×C3×D512yes—
426C22×C4—21no—
427C22×C4—51no—
428C22×C4—52no—
429C22×C4—52no—
430C22×C4—104no—
431C22×C4—108no—
432C22×C4—101no—
433C22×C4—102no—
434C22×C4—102no—
435C22×C4—101no—
436C22×C4C22×C3×D511yescharacteristic
437Q16—108no—
438SD16—108no—
439C2×C8—22no—
440C2×C8—52no—
441C2×C8—102no—
442C2×C8C22×C3×D512yes—
443C4⋊C4—21no—
444C4⋊C4—104no—
445C4⋊C4C22×C3×D512yes—
446C22⋊C4—104no—
447C3×C5C2×#45211yescharacteristic, complemented, Hall
448C22×C3—24no—
449C22×C3—42no—
450C22×C3—42no—
451C22×C3—48no—
452C22×C3—102no—
453C22×C3—102no—
454C22×C3—102no—
455C22×C3—204no—
456C22×C3#402812yes—
457C22×C3#405812yes—
458C22×C3#406112yes—
459C22×C3C2×D4×D511yescharacteristic
460C3×C4—42no—
461C3×C4—102no—
462C3×C4—104no—
463C3×C4—202no—
464C3×C4#402812yes—
465C3×C4C22×D2011yescharacteristic
466C3×C4C2×D4×D511yescharacteristic
467C2×C5—44no—
468C2×C5C3×#45214yescomplemented
469C2×C5C2×C3×#12011yescharacteristic
470C2×C5C2×C3×#13511yescharacteristic
471C2×C5C2×C3×#13711yescharacteristic
472D5—42no—
473C23—22no—
474C23—24no—
475C23—42no—
476C23—42no—
477C23—52no—
478C23—102no—
479C23—102no—
480C23—102no—
481C23C3×D4×D511yescharacteristic
482Q8—54no—
483Q8—102no—
484Q8—104no—
485Q8—102no—
486Q8—102no—
487Q8—202no—
488D4—22no—
489D4—24no—
490D4—1016no—
491D4C2×C3×D2014yes—
492C2×C4—22no—
493C2×C4—42no—
494C2×C4—52no—
495C2×C4—1016no—
496C2×C4—102no—
497C2×C4—102no—
498C2×C4—104no—
499C2×C4—102no—
500C2×C4—102no—
501C2×C4—104no—
502C2×C4—102no—
503C2×C4—202no—
504C2×C4C2×C3×D2012yes—
505C2×C4C3×D4×D512yes—
506C2×C4C3×D4×D511yescharacteristic
507C2×C4C23×C3×D511yescharacteristic
508C8—22no—
509C8—102no—
510C2×C3—44no—
511C2×C3—202no—
512C2×C3#6507214yesdirect product factor
513C2×C3C2×#402811yescharacteristic
514C2×C3C2×#405811yescharacteristic
515C2×C3C2×#406111yescharacteristic
516C5C2×C3×#45211yescharacteristic, complemented, Sylow
517C22—24no—
518C22—48no—
519C22—42no—
520C22—42no—
521C22—102no—
522C22—102no—
523C22—102no—
524C22—204no—
525C22C3×#402812yes—
526C22C3×#405812yes—
527C22C3×#406112yes—
528C22C2×C3×D4×D511yescharacteristic
529C4—42no—
530C4—102no—
531C4—104no—
532C4—202no—
533C4C3×#402812yes—
534C4C22×C3×D2011yescharacteristic
535C4C2×C3×D4×D511yescharacteristic
536C3C2×#6507211yescharacteristic, direct product factor, Sylow
537C2—44no—
538C2—202no—
539C2C3×#6507214yesdirect product factor
540C2C2×C3×#402811yescharacteristic
541C2C2×C3×#405811yescharacteristic
542C2C2×C3×#406111yescharacteristic
subgroupG/H
centerC23×C3D4×D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC2×C3×C5×#102C2
socleC23×C3×C5C2×D4
prime radicalC22×C3×#711C2
hypercenterC2×C3×#102D5
hypocenterC5C2×C3×#452
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
2. C22×C3×#687C2semidirect
3. C22×C3×Dic20C2semidirect
4. C2×C3×#4043C2semidirect
5. C2×C3×C5×#102C2semidirect
6. C2×C3×#3952C2semidirect
7. C2×C3×#3942C2semidirect
8. C3×#65072C2direct (N × H)
9. C2×#65072C3direct (N × H)
23. C2×C3×#687C22semidirect
28. C2×C3×Dic20C22semidirect
30. C3×#4043C22semidirect
31. C3×C5×#102C22semidirect
32. C3×#3952C22semidirect
33. C3×#3942C22semidirect
36. C22×#687C2×C3semidirect
37. C22×Dic20C2×C3semidirect
38. C2×#4043C2×C3semidirect
39. C2×C5×#102C2×C3semidirect
40. C2×#3952C2×C3semidirect
41. C2×#3942C2×C3semidirect
42. #65072C2×C3direct (N × H)
89. C2×C3×#102D5semidirect
104. C2×#687C22×C3semidirect
109. C2×Dic20C22×C3semidirect
111. #4043C22×C3semidirect
112. C5×#102C22×C3semidirect
113. #3952C22×C3semidirect
114. #3942C22×C3semidirect
175. C3×#102C2×D5semidirect
223. C2×#102C3×D5semidirect
343. #102C2×C3×D5semidirect
346. C2×C3×C5#452semidirect
447. C3×C5C2×#452semidirect
468. C2×C5C3×#452semidirect
512. C2×C3#65072direct (N × H)
516. C5C2×C3×#452semidirect
536. C3C2×#65072direct (N × H)
539. C2C3×#65072direct (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×#6872
C22×C3×Dic202
C2×C3×#40432
C2×C3×C5×#1022
C2×C3×#39522
C2×C3×#39422
C3×#650722
C2×#650723
C2×C3×#4525