Finite Group Databasebeta

← C2×C3×#65343 #14426122

Subgroups — C2×C3×#65343 #14426122

Order 1920.

1000 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×#124C211yesmaximal, characteristic
2C22×C3×#689C211yesmaximal, characteristic, complemented
3C22×C3×#687C211yesmaximal, characteristic
4C2×C3×C4×Dic10C211yesmaximal, characteristic, complemented
5C2×C3×#3967C211yesmaximal, characteristic, complemented
6C2×C3×#3964C211yesmaximal, characteristic
7C3×#65343C218yesmaximal, direct product factor
8C2×C3×C4⋊#168C211yesmaximal, characteristic, complemented
9C2×#65343C311yesmaximal, characteristic, direct product factor, Hall
10C22×C3×C5×Q8C2211yescharacteristic
11C22×C3×C5×D4C2211yescharacteristic
12C22×C3×Dic10C2211yescharacteristic
13C3×C5×#124—22no—
14C3×C5×#124C2214yes—
15C2×C3×C5×C22⋊C4—21no—
16C2×C3×C5×C22⋊C4—21no—
17C2×C3×C42×C5C2211yescharacteristic
18C2×C3×#689—24no—
19C2×C3×#689C2214yescomplemented
20C2×C3×#687—24no—
21C2×C3×#687C2214yes—
22C2×C3×C4⋊Dic5C2211yescharacteristic
23C2×C3×#683—21no—
24C2×C3×C4×Dic5—21no—
25C22×C3×#168C2211yescharacteristic
26C22×C3×#168C2211yescharacteristic
27C3×C4×Dic10—22no—
28C3×C4×Dic10C2214yescomplemented
29C3×#3967C2214yescomplemented
30C3×#3964C2214yes—
31C3×C4⋊#168C2214yescomplemented
32C2×C3×#464—51nomaximal, Hall
33C2×C5×#124C2×C311yescharacteristic
34C22×#689C2×C311yescharacteristic, complemented
35C22×#687C2×C311yescharacteristic
36C2×C4×Dic10C2×C311yescharacteristic, complemented
37C2×#3967C2×C311yescharacteristic, complemented
38C2×#3964C2×C311yescharacteristic
39#65343C2×C318yesdirect product factor
40C2×C4⋊#168C2×C311yescharacteristic, complemented
41C24×C3×C5—21no—
42C2×C3×C5×Q8—21no—
43C2×C3×C5×Q8—22no—
44C2×C3×C5×Q8—22no—
45C2×C3×C5×Q8C2312yes—
46C2×C3×C5×D4—22no—
47C2×C3×C5×D4—21no—
48C2×C3×C5×D4—22no—
49C2×C3×C5×D4C2312yes—
50C22×C3×C4×C5—21no—
51C22×C3×C4×C5D411yescharacteristic
52C22×C3×C4×C5D411yescharacteristic
53C22×C3×C4×C5C2311yescharacteristic
54C22×C3×Dic5—21no—
55C22×C3×Dic5—21no—
56C2×C3×Dic10—21no—
57C2×C3×Dic10—22no—
58C2×C3×Dic10D414yes—
59C2×C3×Dic10C2312yes—
60C3×C5×C22⋊C4—24no—
61C3×C5×C22⋊C4—24no—
62C3×C42×C5D412yes—
63C3×C42×C5C2312yes—
64C3×#689—28no—
65C3×#687—28no—
66C3×C4⋊Dic5—21no—
67C3×C4⋊Dic5C2312yes—
68C3×#683—24no—
69C3×C4×Dic5—24no—
70C2×C3×#168—22no—
71C2×C3×#168—22no—
72C2×C3×#168C2312yes—
73C2×C3×#168C2312yes—
74C22×C3×Q16—51no—
75C22×C3×SD16—51no—
76C2×C3×#124D511yescharacteristic
77C2×C3×C4×Q8—51no—
78C3×#464—58no—
79C2×C3×C4⋊C8—51no—
80C2×C3×#103—51no—
81C2×C3×#102—51no—
82C22×C5×Q8C22×C311yescharacteristic
83C22×C5×D4C22×C311yescharacteristic
84C22×Dic10C22×C311yescharacteristic
85C5×#124—22no—
86C5×#124C22×C314yes—
87C2×C5×C22⋊C4—21no—
88C2×C5×C22⋊C4—21no—
89C2×C42×C5C22×C311yescharacteristic
90C2×#689—24no—
91C2×#689C22×C314yescomplemented
92C2×#687—24no—
93C2×#687C22×C314yes—
94C2×C4⋊Dic5C22×C311yescharacteristic
95C2×#683—21no—
96C2×C4×Dic5—21no—
97C22×#168C22×C311yescharacteristic
98C22×#168C22×C311yescharacteristic
99C4×Dic10—22no—
100C4×Dic10C22×C314yescomplemented
101#3967C22×C314yescomplemented
102#3964C22×C314yes—
103C4⋊#168C22×C314yescomplemented
104C2×#464—51noSylow
105C23×C3×C5—22no—
106C23×C3×C5—42no—
107C23×C3×C5—42no—
108C23×C3×C5—42no—
109C23×C3×C5C2×D411yescharacteristic
110C3×C5×Q8—22no—
111C3×C5×Q8—22no—
112C3×C5×Q8—42no—
113C3×C5×D4—22no—
114C3×C5×D4—22no—
115C3×C5×D4—42no—
116C2×C3×C4×C5—22no—
117C2×C3×C4×C5—22no—
118C2×C3×C4×C5—22no—
119C2×C3×C4×C5—42no—
120C2×C3×C4×C5C2×D412yes—
121C2×C3×C4×C5C2×D412yes—
122C2×C3×C4×C5C2×D411yescharacteristic
123C2×C3×C4×C5C2×D412yes—
124C2×C3×C4×C5#4112yes—
125C2×C3×C4×C5C2411yescharacteristic
126C2×C3×Dic5—24no—
127C2×C3×Dic5—22no—
128C2×C3×Dic5—22no—
129C2×C3×Dic5—42no—
130C3×Dic10—24no—
131C3×Dic10—22no—
132C3×Dic10C2×D414yes—
133C3×#168—22no—
134C3×#168—22no—
135C22×C3×Q8—51no—
136C22×C3×Q8C2×D511yescharacteristic
137C22×C3×D4C2×D511yescharacteristic
138C2×C3×Q16—54no—
139C2×C3×Q16—104no—
140C2×C3×SD16—54no—
141C2×C3×SD16—104no—
142C22×C3×C8—51no—
143C22×C3×C8—51no—
144C3×#124—22no—
145C3×#124C2×D514yes—
146C3×C4×Q8—54no—
147C3×C4×Q8—102no—
148C2×C3×C4⋊C4—51no—
149C2×C3×C4⋊C4—101no—
150C2×C3×C22⋊C4—21no—
151C2×C3×C22⋊C4—21no—
152C2×C3×C42—101no—
153C2×C3×C42C2×D511yescharacteristic
154C3×C4⋊C8—54no—
155C3×#103—54no—
156C3×#102—54no—
157C24×C5—21no—
158C2×C5×Q8—21no—
159C2×C5×Q8—22no—
160C2×C5×Q8—22no—
161C2×C5×Q8C23×C312yes—
162C2×C5×D4—22no—
163C2×C5×D4—21no—
164C2×C5×D4—22no—
165C2×C5×D4C23×C312yes—
166C22×C4×C5—21no—
167C22×C4×C5C3×D411yescharacteristic
168C22×C4×C5C3×D411yescharacteristic
169C22×C4×C5C23×C311yescharacteristic
170C22×Dic5—21no—
171C22×Dic5—21no—
172C2×Dic10—21no—
173C2×Dic10—22no—
174C2×Dic10C3×D414yes—
175C2×Dic10C23×C312yes—
176C5×C22⋊C4—24no—
177C5×C22⋊C4—24no—
178C42×C5C3×D412yes—
179C42×C5C23×C312yes—
180#689—28no—
181#687—28no—
182C4⋊Dic5—21no—
183C4⋊Dic5C23×C312yes—
184#683—24no—
185C4×Dic5—24no—
186C2×#168—22no—
187C2×#168—22no—
188C2×#168C23×C312yes—
189C2×#168C23×C312yes—
190C22×Q16—51no—
191C22×SD16—51no—
192C2×#124C3×D511yescharacteristic
193C2×C4×Q8—51no—
194#464—58no—
195C2×C4⋊C8—51no—
196C2×#103—51no—
197C2×#102—51no—
198C22×C3×C5—42no—
199C22×C3×C5—42no—
200C22×C3×C5—42no—
201C22×C3×C5—84no—
202C22×C3×C5#12112yes—
203C22×C3×C5#13512yes—
204C22×C3×C5#13712yes—
205C22×C3×C5C22×D411yescharacteristic
206C3×C4×C5—22no—
207C3×C4×C5—22no—
208C3×C4×C5—42no—
209C3×C4×C5#12112yes—
210C3×C4×C5C22×D411yescharacteristic
211C3×C4×C5C2×#4111yescharacteristic
212C3×Dic5—24no—
213C3×Dic5—42no—
214C24×C3—21no—
215C2×C3×Q8—21no—
216C2×C3×Q8—22no—
217C2×C3×Q8—22no—
218C2×C3×Q8—54no—
219C2×C3×Q8—52no—
220C2×C3×Q8—102no—
221C2×C3×Q8—101no—
222C2×C3×Q8C22×D512yes—
223C2×C3×D4—22no—
224C2×C3×D4—22no—
225C2×C3×D4—21no—
226C2×C3×D4C22×D512yes—
227C22×C3×C4—21no—
228C22×C3×C4—101no—
229C22×C3×C4—101no—
230C22×C3×C4C22⋊D511yescharacteristic
231C22×C3×C4C22⋊D511yescharacteristic
232C22×C3×C4C22×D511yescharacteristic
233C3×Q16—108no—
234C3×SD16—108no—
235C2×C3×C8—52no—
236C2×C3×C8—52no—
237C2×C3×C8—102no—
238C2×C3×C8—102no—
239C3×C4⋊C4—52no—
240C3×C4⋊C4—104no—
241C3×C4⋊C4—101no—
242C3×C22⋊C4—24no—
243C3×C22⋊C4—24no—
244C3×C42—104no—
245C3×C42C22⋊D512yes—
246C3×C42C22×D512yes—
247C23×C5—22no—
248C23×C5—42no—
249C23×C5—42no—
250C23×C5—42no—
251C23×C5C2×C3×D411yescharacteristic
252C5×Q8—22no—
253C5×Q8—22no—
254C5×Q8—42no—
255C5×D4—22no—
256C5×D4—22no—
257C5×D4—42no—
258C2×C4×C5—22no—
259C2×C4×C5—22no—
260C2×C4×C5—22no—
261C2×C4×C5—42no—
262C2×C4×C5C2×C3×D412yes—
263C2×C4×C5C2×C3×D412yes—
264C2×C4×C5C2×C3×D412yes—
265C2×C4×C5C2×C3×D411yescharacteristic
266C2×C4×C5C3×#4112yes—
267C2×C4×C5C24×C311yescharacteristic
268C2×Dic5—24no—
269C2×Dic5—22no—
270C2×Dic5—22no—
271C2×Dic5—42no—
272Dic10—24no—
273Dic10—22no—
274Dic10C2×C3×D414yes—
275#168—22no—
276#168—22no—
277C22×Q8—51no—
278C22×Q8C2×C3×D511yescharacteristic
279C22×D4C2×C3×D511yescharacteristic
280C2×Q16—54no—
281C2×Q16—104no—
282C2×SD16—54no—
283C2×SD16—104no—
284C22×C8—51no—
285C22×C8—51no—
286#124—22no—
287#124C2×C3×D514yes—
288C4×Q8—54no—
289C4×Q8—102no—
290C2×C4⋊C4—51no—
291C2×C4⋊C4—101no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C42—101no—
295C2×C42C2×C3×D511yescharacteristic
296C4⋊C8—54no—
297#103—54no—
298#102—54no—
299C2×C3×C5—82no—
300C2×C3×C5#46414yescomplemented
301C2×C3×C5C2×#12111yescharacteristic
302C2×C3×C5C2×#13511yescharacteristic
303C2×C3×C5C2×#13711yescharacteristic
304C23×C3—22no—
305C23×C3—42no—
306C23×C3—42no—
307C23×C3—42no—
308C23×C3C2×C22⋊D511yescharacteristic
309C3×Q8—22no—
310C3×Q8—22no—
311C3×Q8—42no—
312C3×Q8—54no—
313C3×Q8—102no—
314C3×Q8—104no—
315C3×D4—22no—
316C3×D4—22no—
317C3×D4—42no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—42no—
322C2×C3×C4—102no—
323C2×C3×C4—104no—
324C2×C3×C4—102no—
325C2×C3×C4—202no—
326C2×C3×C4D4×D512yes—
327C2×C3×C4#71112yes—
328C2×C3×C4C2×C22⋊D512yes—
329C2×C3×C4C2×C22⋊D512yes—
330C2×C3×C4C2×C22⋊D511yescharacteristic
331C2×C3×C4C23×D511yescharacteristic
332C3×C8—102no—
333C3×C8—102no—
334C22×C5—42no—
335C22×C5—42no—
336C22×C5—42no—
337C22×C5—84no—
338C22×C5C3×#12112yes—
339C22×C5C3×#13512yes—
340C22×C5C3×#13712yes—
341C22×C5C22×C3×D411yescharacteristic
342C4×C5—22no—
343C4×C5—22no—
344C4×C5—42no—
345C4×C5C3×#12112yes—
346C4×C5C22×C3×D411yescharacteristic
347C4×C5C2×C3×#4111yescharacteristic
348Dic5—24no—
349Dic5—42no—
350C24—21no—
351C2×Q8—21no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—54no—
355C2×Q8—52no—
356C2×Q8—102no—
357C2×Q8—101no—
358C2×Q8C22×C3×D512yes—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—21no—
362C2×D4C22×C3×D512yes—
363C22×C4—21no—
364C22×C4—101no—
365C22×C4—101no—
366C22×C4C3×C22⋊D511yescharacteristic
367C22×C4C3×C22⋊D511yescharacteristic
368C22×C4C22×C3×D511yescharacteristic
369Q16—108no—
370SD16—108no—
371C2×C8—52no—
372C2×C8—52no—
373C2×C8—102no—
374C2×C8—102no—
375C4⋊C4—52no—
376C4⋊C4—101no—
377C4⋊C4—104no—
378C22⋊C4—24no—
379C22⋊C4—24no—
380C42—104no—
381C42C3×C22⋊D512yes—
382C42C22×C3×D512yes—
383C3×C5C2×#46411yescharacteristic, complemented, Hall
384C22×C3—42no—
385C22×C3—42no—
386C22×C3—42no—
387C22×C3—84no—
388C22×C3#408412yes—
389C22×C3#409612yes—
390C22×C3#409712yes—
391C22×C3C22×C22⋊D511yescharacteristic
392C3×C4—22no—
393C3×C4—22no—
394C3×C4—42no—
395C3×C4—104no—
396C3×C4—202no—
397C3×C4#408412yes—
398C3×C4C2×D4×D511yescharacteristic
399C3×C4C2×#71111yescharacteristic
400C2×C5—82no—
401C2×C5C3×#46414yescomplemented
402C2×C5C2×C3×#12111yescharacteristic
403C2×C5C2×C3×#13511yescharacteristic
404C2×C5C2×C3×#13711yescharacteristic
405C23—22no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23C2×C3×C22⋊D511yescharacteristic
410Q8—22no—
411Q8—22no—
412Q8—42no—
413Q8—54no—
414Q8—104no—
415Q8—102no—
416D4—22no—
417D4—22no—
418D4—42no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—42no—
423C2×C4—104no—
424C2×C4—102no—
425C2×C4—102no—
426C2×C4—202no—
427C2×C4C3×D4×D512yes—
428C2×C4C3×#71112yes—
429C2×C4C2×C3×C22⋊D512yes—
430C2×C4C2×C3×C22⋊D512yes—
431C2×C4C2×C3×C22⋊D511yescharacteristic
432C2×C4C23×C3×D511yescharacteristic
433C8—102no—
434C8—102no—
435C2×C3—82no—
436C2×C3#6534314yesdirect product factor
437C2×C3C2×#408411yescharacteristic
438C2×C3C2×#409611yescharacteristic
439C2×C3C2×#409711yescharacteristic
440C5C2×C3×#46411yescharacteristic, complemented, Sylow
441C22—42no—
442C22—42no—
443C22—42no—
444C22—84no—
445C22C3×#408412yes—
446C22C3×#409612yes—
447C22C3×#409712yes—
448C22C22×C3×C22⋊D511yescharacteristic
449C4—22no—
450C4—22no—
451C4—42no—
452C4—104no—
453C4—202no—
454C4C3×#408412yes—
455C4C2×C3×D4×D511yescharacteristic
456C4C2×C3×#71111yescharacteristic
457C3C2×#6534311yescharacteristic, direct product factor, Sylow
458C2—82no—
459C2C3×#6534314yesdirect product factor
460C2C2×C3×#408411yescharacteristic
461C2C2×C3×#409611yescharacteristic
462C2C2×C3×#409711yescharacteristic
subgroupG/H
centerC23×C3C2×C22⋊D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC2×C3×C5×#124C2
socleC23×C3×C5C2×D4
prime radicalC22×C3×C5×D4C22
hypercenterC2×C3×#124D5
hypocenterC5C2×C3×#464
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ C2×C3×#124

Complements

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

NH ≅ G/Ntype
2. C22×C3×#689C2semidirect
4. C2×C3×C4×Dic10C2semidirect
5. C2×C3×#3967C2semidirect
7. C3×#65343C2direct (N × H)
8. C2×C3×C4⋊#168C2semidirect
9. C2×#65343C3direct (N × H)
19. C2×C3×#689C22semidirect
28. C3×C4×Dic10C22semidirect
29. C3×#3967C22semidirect
31. C3×C4⋊#168C22semidirect
34. C22×#689C2×C3semidirect
36. C2×C4×Dic10C2×C3semidirect
37. C2×#3967C2×C3semidirect
39. #65343C2×C3direct (N × H)
40. C2×C4⋊#168C2×C3semidirect
91. C2×#689C22×C3semidirect
100. C4×Dic10C22×C3semidirect
101. #3967C22×C3semidirect
103. C4⋊#168C22×C3semidirect
300. C2×C3×C5#464semidirect
383. C3×C5C2×#464semidirect
401. C2×C5C3×#464semidirect
436. C2×C3#65343direct (N × H)
440. C5C2×C3×#464semidirect
457. C3C2×#65343direct (N × H)
459. C2C3×#65343direct (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×#1242
C22×C3×#6892
C22×C3×#6872
C2×C3×C4×Dic102
C2×C3×#39672
C2×C3×#39642
C3×#653432
C2×C3×C4⋊#1682
C2×#653433
C2×C3×#4645