Finite Group Databasebeta

← C2×C3×#65493 #14429431

Subgroups — C2×C3×#65493 #14429431

Order 1920.

896 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×#137C211yesmaximal, characteristic
2C2×C3×#4097C211yesmaximal, characteristic
3C2×C3×#4094C211yesmaximal, characteristic
4C22×C3×#689C211yesmaximal, characteristic, complemented
5C2×C3×#3968C211yesmaximal, characteristic, complemented
6C3×#65493C218yesmaximal, direct product factor
7C2×C3×#3956C211yesmaximal, characteristic, complemented
8C2×C3×#3954C211yesmaximal, characteristic, complemented
9C2×#65493C311yesmaximal, characteristic, direct product factor, Hall
10C2×C3×C5×#41C2211yescharacteristic
11C22×C3×C5×Q8C2211yescharacteristic
12C22×C3×Dic10C2211yescharacteristic
13C3×C5×#137—22no—
14C3×C5×#137C2214yes—
15C2×C3×C5×Q16—21no—
16C2×C3×C5×SD16—21no—
17C2×C3×C5×M16C2211yescharacteristic
18C3×#4097—22no—
19C3×#4097C2214yes—
20C3×#4094—22no—
21C3×#4094C2214yes—
22C2×C3×#689—21no—
23C2×C3×#689—22no—
24C2×C3×#689—21no—
25C2×C3×#689—21no—
26C2×C3×#689C2214yescomplemented
27C2×C3×#687—21no—
28C2×C3×#681—21no—
29C2×C3×#681C2211yescharacteristic
30C2×C3×#681C2211yescharacteristic
31C22×C3×#168—21no—
32C22×C3×#168C2211yescharacteristic
33C3×#3968C2214yescomplemented
34C3×#3956C2214yescomplemented
35C3×#3954C2214yescomplemented
36C2×C3×#473—51nomaximal, Hall
37C2×C5×#137C2×C311yescharacteristic
38C2×#4097C2×C311yescharacteristic
39C2×#4094C2×C311yescharacteristic
40C22×#689C2×C311yescharacteristic, complemented
41C2×#3968C2×C311yescharacteristic, complemented
42#65493C2×C318yesdirect product factor
43C2×#3956C2×C311yescharacteristic, complemented
44C2×#3954C2×C311yescharacteristic, complemented
45C3×C5×#41—21no—
46C3×C5×#41—21no—
47C3×C5×#41D412yes—
48C3×C5×#41C2312yes—
49C2×C3×C5×Q8—21no—
50C2×C3×C5×Q8—21no—
51C2×C3×C5×Q8—21no—
52C2×C3×C5×Q8—21no—
53C2×C3×C5×Q8—21no—
54C2×C3×C5×Q8D411yescharacteristic
55C2×C3×C5×Q8C2312yes—
56C2×C3×C5×D4—21no—
57C2×C3×C5×D4D411yescharacteristic
58C22×C3×C4×C5—21no—
59C22×C3×C4×C5—21no—
60C22×C3×C4×C5C2311yescharacteristic
61C22×C3×Dic5—21no—
62C2×C3×Dic10—21no—
63C2×C3×Dic10—21no—
64C2×C3×Dic10—21no—
65C2×C3×Dic10—21no—
66C2×C3×Dic10—21no—
67C2×C3×Dic10C2312yes—
68C3×C5×Q16—24no—
69C3×C5×SD16—24no—
70C3×C5×M16—21no—
71C3×C5×M16C2312yes—
72C2×C3×C5×C8—21no—
73C3×#689—24no—
74C3×#689—24no—
75C3×#689—24no—
76C3×#687—24no—
77C3×#681—21no—
78C3×#681—21no—
79C3×#681—24no—
80C3×#681C2312yes—
81C3×#681C2312yes—
82C2×C3×#168—21no—
83C2×C3×#168—21no—
84C2×C3×#168—24no—
85C2×C3×#168D412yes—
86C2×C3×#168D411yescharacteristic
87C2×C3×#168D411yescharacteristic
88C2×C3×#168C2312yes—
89C2×C3×#137—51no—
90C2×C3×#137D511yescharacteristic
91C22×C3×Q16—51no—
92C2×C3×#131—51no—
93C3×#473—58no—
94C2×C3×#108—51no—
95C2×C3×#101—51no—
96C2×C3×#101—51no—
97C2×C5×#41C22×C311yescharacteristic
98C22×C5×Q8C22×C311yescharacteristic
99C22×Dic10C22×C311yescharacteristic
100C5×#137—22no—
101C5×#137C22×C314yes—
102C2×C5×Q16—21no—
103C2×C5×SD16—21no—
104C2×C5×M16C22×C311yescharacteristic
105#4097—22no—
106#4097C22×C314yes—
107#4094—22no—
108#4094C22×C314yes—
109C2×#689—21no—
110C2×#689—21no—
111C2×#689—21no—
112C2×#689—22no—
113C2×#689C22×C314yescomplemented
114C2×#687—21no—
115C2×#681—21no—
116C2×#681C22×C311yescharacteristic
117C2×#681C22×C311yescharacteristic
118C22×#168—21no—
119C22×#168C22×C311yescharacteristic
120#3968C22×C314yescomplemented
121#3956C22×C314yescomplemented
122#3954C22×C314yescomplemented
123C2×#473—51noSylow
124C23×C3×C5—21no—
125C23×C3×C5#4111yescharacteristic
126C3×C5×Q8—21no—
127C3×C5×Q8—22no—
128C3×C5×Q8—22no—
129C3×C5×Q8—41no—
130C3×C5×Q8—41no—
131C3×C5×Q8C2×D412yes—
132C3×C5×D4—22no—
133C3×C5×D4—22no—
134C3×C5×D4—21no—
135C3×C5×D4C2×D412yes—
136C2×C3×C4×C5—21no—
137C2×C3×C4×C5—22no—
138C2×C3×C4×C5—22no—
139C2×C3×C4×C5—22no—
140C2×C3×C4×C5—41no—
141C2×C3×C4×C5—41no—
142C2×C3×C4×C5C2×D411yescharacteristic
143C2×C3×C4×C5C2×D411yescharacteristic
144C2×C3×C4×C5C2×D411yescharacteristic
145C2×C3×C4×C5C2×D411yescharacteristic
146C2×C3×C4×C5#4111yescharacteristic
147C2×C3×C4×C5C2411yescharacteristic
148C2×C3×Dic5—22no—
149C2×C3×Dic5—41no—
150C2×C3×Dic5—41no—
151C3×Dic10—22no—
152C3×Dic10—22no—
153C3×Dic10—41no—
154C3×Dic10—41no—
155C3×C5×C8—22no—
156C3×#168—22no—
157C3×#168—22no—
158C3×#168C2×D412yes—
159C3×#168C2×D412yes—
160C2×C3×#41C2×D511yescharacteristic
161C22×C3×Q8—51no—
162C22×C3×Q8C2×D511yescharacteristic
163C3×#137—22no—
164C3×#137—54no—
165C3×#137—102no—
166C3×#137C2×D514yes—
167C2×C3×Q16—21no—
168C2×C3×Q16—54no—
169C2×C3×Q16—101no—
170C2×C3×Q16—101no—
171C2×C3×Q16—102no—
172C2×C3×Q16—101no—
173C2×C3×SD16—21no—
174C2×C3×SD16—101no—
175C3×#131—54no—
176C3×#131—102no—
177C2×C3×M16—51no—
178C2×C3×M16—51no—
179C2×C3×M16—101no—
180C2×C3×M16C2×D511yescharacteristic
181C22×C3×C8—51no—
182C22×C3×C8—101no—
183C3×#108—54no—
184C3×#101—54no—
185C3×#101—54no—
186C5×#41—21no—
187C5×#41—21no—
188C5×#41C3×D412yes—
189C5×#41C23×C312yes—
190C2×C5×Q8—21no—
191C2×C5×Q8—21no—
192C2×C5×Q8—21no—
193C2×C5×Q8—21no—
194C2×C5×Q8—21no—
195C2×C5×Q8C3×D411yescharacteristic
196C2×C5×Q8C23×C312yes—
197C2×C5×D4—21no—
198C2×C5×D4C3×D411yescharacteristic
199C22×C4×C5—21no—
200C22×C4×C5—21no—
201C22×C4×C5C23×C311yescharacteristic
202C22×Dic5—21no—
203C2×Dic10—21no—
204C2×Dic10—21no—
205C2×Dic10—21no—
206C2×Dic10—21no—
207C2×Dic10—21no—
208C2×Dic10C23×C312yes—
209C5×Q16—24no—
210C5×SD16—24no—
211C5×M16—21no—
212C5×M16C23×C312yes—
213C2×C5×C8—21no—
214#689—24no—
215#689—24no—
216#689—24no—
217#687—24no—
218#681—21no—
219#681—21no—
220#681—24no—
221#681C23×C312yes—
222#681C23×C312yes—
223C2×#168—21no—
224C2×#168—21no—
225C2×#168—24no—
226C2×#168C3×D411yescharacteristic
227C2×#168C3×D411yescharacteristic
228C2×#168C3×D412yes—
229C2×#168C23×C312yes—
230C2×#137—51no—
231C2×#137C3×D511yescharacteristic
232C22×Q16—51no—
233C2×#131—51no—
234#473—58no—
235C2×#108—51no—
236C2×#101—51no—
237C2×#101—51no—
238C22×C3×C5—22no—
239C22×C3×C5—22no—
240C22×C3×C5—42no—
241C22×C3×C5#12111yescharacteristic
242C22×C3×C5#12111yescharacteristic
243C22×C3×C5C2×#4111yescharacteristic
244C3×C4×C5—22no—
245C3×C4×C5—42no—
246C3×C4×C5#12111yescharacteristic
247C3×C4×C5#12111yescharacteristic
248C3×C4×C5C22×D411yescharacteristic
249C3×C4×C5C22×D411yescharacteristic
250C3×Dic5—42no—
251C3×#41—21no—
252C3×#41—21no—
253C3×#41C22⋊D512yes—
254C3×#41C22×D512yes—
255C2×C3×Q8—21no—
256C2×C3×Q8—21no—
257C2×C3×Q8—21no—
258C2×C3×Q8—21no—
259C2×C3×Q8—21no—
260C2×C3×Q8—52no—
261C2×C3×Q8—101no—
262C2×C3×Q8—101no—
263C2×C3×Q8—101no—
264C2×C3×Q8—101no—
265C2×C3×Q8—101no—
266C2×C3×Q8C22⋊D511yescharacteristic
267C2×C3×Q8C22×D512yes—
268C2×C3×D4—21no—
269C2×C3×D4C22⋊D511yescharacteristic
270C22×C3×C4—21no—
271C22×C3×C4—21no—
272C22×C3×C4—101no—
273C22×C3×C4C22×D511yescharacteristic
274C3×Q16—24no—
275C3×Q16—104no—
276C3×Q16—104no—
277C3×Q16—104no—
278C3×SD16—24no—
279C3×SD16—104no—
280C3×M16—21no—
281C3×M16—52no—
282C3×M16—52no—
283C3×M16—104no—
284C3×M16—101no—
285C3×M16—101no—
286C3×M16C22×D512yes—
287C2×C3×C8—21no—
288C2×C3×C8—51no—
289C2×C3×C8—51no—
290C2×C3×C8—52no—
291C2×C3×C8—52no—
292C2×C3×C8—104no—
293C2×C3×C8—101no—
294C2×C3×C8—101no—
295C23×C5—21no—
296C23×C5C3×#4111yescharacteristic
297C5×Q8—21no—
298C5×Q8—22no—
299C5×Q8—22no—
300C5×Q8—41no—
301C5×Q8—41no—
302C5×Q8C2×C3×D412yes—
303C5×D4—22no—
304C5×D4—22no—
305C5×D4—21no—
306C5×D4C2×C3×D412yes—
307C2×C4×C5—21no—
308C2×C4×C5—22no—
309C2×C4×C5—22no—
310C2×C4×C5—22no—
311C2×C4×C5—41no—
312C2×C4×C5—41no—
313C2×C4×C5C2×C3×D411yescharacteristic
314C2×C4×C5C2×C3×D411yescharacteristic
315C2×C4×C5C2×C3×D411yescharacteristic
316C2×C4×C5C2×C3×D411yescharacteristic
317C2×C4×C5C3×#4111yescharacteristic
318C2×C4×C5C24×C311yescharacteristic
319C2×Dic5—22no—
320C2×Dic5—41no—
321C2×Dic5—41no—
322Dic10—22no—
323Dic10—22no—
324Dic10—41no—
325Dic10—41no—
326C5×C8—22no—
327#168—22no—
328#168—22no—
329#168C2×C3×D412yes—
330#168C2×C3×D412yes—
331C2×#41C2×C3×D511yescharacteristic
332C22×Q8—51no—
333C22×Q8C2×C3×D511yescharacteristic
334#137—22no—
335#137—54no—
336#137—102no—
337#137C2×C3×D514yes—
338C2×Q16—21no—
339C2×Q16—54no—
340C2×Q16—101no—
341C2×Q16—101no—
342C2×Q16—102no—
343C2×Q16—101no—
344C2×SD16—21no—
345C2×SD16—101no—
346#131—54no—
347#131—102no—
348C2×M16—51no—
349C2×M16—51no—
350C2×M16—101no—
351C2×M16C2×C3×D511yescharacteristic
352C22×C8—51no—
353C22×C8—101no—
354#108—54no—
355#101—54no—
356#101—54no—
357C2×C3×C5—21no—
358C2×C3×C5—21no—
359C2×C3×C5—42no—
360C2×C3×C5#47312yescomplemented
361C2×C3×C5C2×#12111yescharacteristic
362C23×C3—21no—
363C23×C3#71111yescharacteristic
364C3×Q8—21no—
365C3×Q8—22no—
366C3×Q8—22no—
367C3×Q8—41no—
368C3×Q8—41no—
369C3×Q8—102no—
370C3×Q8—102no—
371C3×Q8—201no—
372C3×Q8—201no—
373C3×Q8C2×C22⋊D512yes—
374C3×D4—22no—
375C3×D4—22no—
376C3×D4—21no—
377C3×D4C2×C22⋊D512yes—
378C2×C3×C4—21no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—22no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—102no—
385C2×C3×C4—201no—
386C2×C3×C4—201no—
387C2×C3×C4D4×D511yescharacteristic
388C2×C3×C4D4×D511yescharacteristic
389C2×C3×C4#71111yescharacteristic
390C2×C3×C4C2×C22⋊D511yescharacteristic
391C2×C3×C4C2×C22⋊D511yescharacteristic
392C2×C3×C4C23×D511yescharacteristic
393C3×C8—22no—
394C3×C8—52no—
395C3×C8—52no—
396C3×C8—102no—
397C3×C8—102no—
398C22×C5—22no—
399C22×C5—22no—
400C22×C5—42no—
401C22×C5C3×#12111yescharacteristic
402C22×C5C3×#12111yescharacteristic
403C22×C5C2×C3×#4111yescharacteristic
404C4×C5—22no—
405C4×C5—42no—
406C4×C5C3×#12111yescharacteristic
407C4×C5C3×#12111yescharacteristic
408C4×C5C22×C3×D411yescharacteristic
409C4×C5C22×C3×D411yescharacteristic
410Dic5—42no—
411#41—21no—
412#41—21no—
413#41C3×C22⋊D512yes—
414#41C22×C3×D512yes—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—52no—
421C2×Q8—101no—
422C2×Q8—101no—
423C2×Q8—101no—
424C2×Q8—101no—
425C2×Q8—101no—
426C2×Q8C3×C22⋊D511yescharacteristic
427C2×Q8C22×C3×D512yes—
428C2×D4—21no—
429C2×D4C3×C22⋊D511yescharacteristic
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—101no—
433C22×C4C22×C3×D511yescharacteristic
434Q16—24no—
435Q16—104no—
436Q16—104no—
437Q16—104no—
438SD16—24no—
439SD16—104no—
440M16—21no—
441M16—52no—
442M16—52no—
443M16—104no—
444M16—101no—
445M16—101no—
446M16C22×C3×D512yes—
447C2×C8—21no—
448C2×C8—51no—
449C2×C8—51no—
450C2×C8—52no—
451C2×C8—52no—
452C2×C8—104no—
453C2×C8—101no—
454C2×C8—101no—
455C3×C5C2×#47311yescharacteristic, complemented, Hall
456C22×C3—22no—
457C22×C3—22no—
458C22×C3—42no—
459C22×C3#408511yescharacteristic
460C22×C3#408511yescharacteristic
461C22×C3C2×#71111yescharacteristic
462C3×C4—22no—
463C3×C4—42no—
464C3×C4—202no—
465C3×C4#408511yescharacteristic
466C3×C4#408511yescharacteristic
467C3×C4C2×D4×D511yescharacteristic
468C3×C4C22×C22⋊D511yescharacteristic
469C2×C5—21no—
470C2×C5—21no—
471C2×C5—42no—
472C2×C5C3×#47312yescomplemented
473C2×C5C2×C3×#12111yescharacteristic
474C23—21no—
475C23C3×#71111yescharacteristic
476Q8—21no—
477Q8—22no—
478Q8—22no—
479Q8—41no—
480Q8—41no—
481Q8—102no—
482Q8—102no—
483Q8—201no—
484Q8—201no—
485Q8C2×C3×C22⋊D512yes—
486D4—22no—
487D4—22no—
488D4—21no—
489D4C2×C3×C22⋊D512yes—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—102no—
497C2×C4—201no—
498C2×C4—201no—
499C2×C4C3×D4×D511yescharacteristic
500C2×C4C3×D4×D511yescharacteristic
501C2×C4C3×#71111yescharacteristic
502C2×C4C2×C3×C22⋊D511yescharacteristic
503C2×C4C2×C3×C22⋊D511yescharacteristic
504C2×C4C23×C3×D511yescharacteristic
505C8—22no—
506C8—52no—
507C8—52no—
508C8—102no—
509C8—102no—
510C2×C3—21no—
511C2×C3—21no—
512C2×C3—42no—
513C2×C3#6549312yesdirect product factor
514C2×C3C2×#408511yescharacteristic
515C5C2×C3×#47311yescharacteristic, complemented, Sylow
516C22—22no—
517C22—22no—
518C22—42no—
519C22C3×#408511yescharacteristic
520C22C3×#408511yescharacteristic
521C22C2×C3×#71111yescharacteristic
522C4—22no—
523C4—42no—
524C4—202no—
525C4C3×#408511yescharacteristic
526C4C3×#408511yescharacteristic
527C4C2×C3×D4×D511yescharacteristic
528C4C22×C3×C22⋊D511yescharacteristic
529C3C2×#6549311yescharacteristic, direct product factor, Sylow
530C2—21no—
531C2—21no—
532C2—42no—
533C2C3×#6549312yesdirect product factor
534C2C2×C3×#408511yescharacteristic
subgroupG/H
centerC22×C3#4085
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC2×C3×C5×#137C2
socleC22×C3×C5#121
prime radicalC2×C3×C5×#41C22
hypercenterC2×C3×#137D5
hypocenterC5C2×C3×#473
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#137

Complements

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

NH ≅ G/Ntype
4. C22×C3×#689C2semidirect
5. C2×C3×#3968C2semidirect
6. C3×#65493C2direct (N × H)
7. C2×C3×#3956C2semidirect
8. C2×C3×#3954C2semidirect
9. C2×#65493C3direct (N × H)
26. C2×C3×#689C22semidirect
33. C3×#3968C22semidirect
34. C3×#3956C22semidirect
35. C3×#3954C22semidirect
40. C22×#689C2×C3semidirect
41. C2×#3968C2×C3semidirect
42. #65493C2×C3direct (N × H)
43. C2×#3956C2×C3semidirect
44. C2×#3954C2×C3semidirect
113. C2×#689C22×C3semidirect
120. #3968C22×C3semidirect
121. #3956C22×C3semidirect
122. #3954C22×C3semidirect
360. C2×C3×C5#473semidirect
455. C3×C5C2×#473semidirect
472. C2×C5C3×#473semidirect
513. C2×C3#65493direct (N × H)
515. C5C2×C3×#473semidirect
529. C3C2×#65493direct (N × H)
533. C2C3×#65493direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C5×#1372
C2×C3×#40972
C2×C3×#40942
C22×C3×#6892
C2×C3×#39682
C3×#654932
C2×C3×#39562
C2×C3×#39542
C2×#654933
C2×C3×#4735