Finite Group Databasebeta

← C2×C3×#65056 #14426128

Subgroups — C2×C3×#65056 #14426128

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×#4082C211yesmaximal, characteristic
2C22×C3×#687C211yesmaximal, characteristic
3C22×C3×Dic20C211yesmaximal, characteristic, complemented
4C2×C3×#4033C211yesmaximal, characteristic, complemented
5C2×C3×C5×#102C211yesmaximal, characteristic
6C2×C3×#3946C211yesmaximal, characteristic, complemented
7C2×C3×#3942C211yesmaximal, characteristic, complemented
8C3×#65056C218yesmaximal, direct product factor
9C2×#65056C311yesmaximal, characteristic, direct product factor, Hall
10C22×C3×C5×D4C2211yescharacteristic
11C22×C3×Dic10C2211yescharacteristic
12C22×C3×Dic10C2211yescharacteristic
13C22×C3×C5×C8C2211yescharacteristic
14C2×C3×C5×C4⋊C4C2211yescharacteristic
15C2×C3×C22⋊Dic5—21no—
16C2×C3×C22⋊Dic5—21no—
17C3×#4082—22no—
18C3×#4082C2214yes—
19C2×C3×#687—24no—
20C2×C3×#687C2214yes—
21C2×C3×#683—21no—
22C2×C3×C4×Dic5—21no—
23C2×C3×C4×Dic5C2211yescharacteristic
24C22×C3×#168C2211yescharacteristic
25C2×C3×Dic20—24no—
26C2×C3×Dic20C2214yescomplemented
27C3×#4033—22no—
28C3×#4033C2214yescomplemented
29C3×C5×#102C2214yes—
30C3×#3946C2214yescomplemented
31C3×#3942C2214yescomplemented
32C2×C3×#464—51nomaximal, Hall
33C2×#4082C2×C311yescharacteristic
34C22×#687C2×C311yescharacteristic
35C22×Dic20C2×C311yescharacteristic, complemented
36C2×#4033C2×C311yescharacteristic, complemented
37C2×C5×#102C2×C311yescharacteristic
38C2×#3946C2×C311yescharacteristic, complemented
39C2×#3942C2×C311yescharacteristic, complemented
40#65056C2×C318yesdirect product factor
41C24×C3×C5—21no—
42C2×C3×C5×D4—22no—
43C2×C3×C5×D4—21no—
44C2×C3×C5×D4—22no—
45C2×C3×C5×D4C2312yes—
46C22×C3×C4×C5—21no—
47C22×C3×C4×C5C2311yescharacteristic
48C22×C3×Dic5—21no—
49C22×C3×Dic5—21no—
50C22×C3×Dic5D411yescharacteristic
51C22×C3×Dic5D411yescharacteristic
52C2×C3×Dic10—21no—
53C2×C3×Dic10—22no—
54C2×C3×Dic10—22no—
55C2×C3×Dic10—22no—
56C2×C3×Dic10—21no—
57C2×C3×Dic10D414yes—
58C2×C3×Dic10C2312yes—
59C2×C3×Dic10C2312yes—
60C2×C3×C5×C8—22no—
61C2×C3×C5×C8C2312yes—
62C3×C5×C4⋊C4—21no—
63C3×C5×C4⋊C4C2312yes—
64C3×C22⋊Dic5—24no—
65C3×C22⋊Dic5—24no—
66C3×#687—28no—
67C3×#683—24no—
68C3×C4×Dic5—24no—
69C3×C4×Dic5D412yes—
70C3×C4×Dic5C2312yes—
71C2×C3×#168—22no—
72C2×C3×#168C2312yes—
73C3×Dic20—28no—
74C22×C3×Q16—51no—
75C22×C3×SD16—51no—
76C2×C3×#124—51no—
77C2×C3×C4×Q8—51no—
78C3×#464—58no—
79C2×C3×C4⋊C8—51no—
80C2×C3×#103—51no—
81C2×C3×#102D511yescharacteristic
82C22×C5×D4C22×C311yescharacteristic
83C22×Dic10C22×C311yescharacteristic
84C22×Dic10C22×C311yescharacteristic
85C22×C5×C8C22×C311yescharacteristic
86C2×C5×C4⋊C4C22×C311yescharacteristic
87C2×C22⋊Dic5—21no—
88C2×C22⋊Dic5—21no—
89#4082—22no—
90#4082C22×C314yes—
91C2×#687—24no—
92C2×#687C22×C314yes—
93C2×#683—21no—
94C2×C4×Dic5—21no—
95C2×C4×Dic5C22×C311yescharacteristic
96C22×#168C22×C311yescharacteristic
97C2×Dic20—24no—
98C2×Dic20C22×C314yescomplemented
99#4033—22no—
100#4033C22×C314yescomplemented
101C5×#102C22×C314yes—
102#3946C22×C314yescomplemented
103#3942C22×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×D4—22no—
111C3×C5×D4—22no—
112C3×C5×D4—42no—
113C2×C3×C4×C5—22no—
114C2×C3×C4×C5—42no—
115C2×C3×C4×C5C2×D411yescharacteristic
116C2×C3×C4×C5C2×D412yes—
117C2×C3×C4×C5#4112yes—
118C2×C3×C4×C5C2411yescharacteristic
119C2×C3×Dic5—24no—
120C2×C3×Dic5—22no—
121C2×C3×Dic5—22no—
122C2×C3×Dic5—22no—
123C2×C3×Dic5—22no—
124C2×C3×Dic5—42no—
125C2×C3×Dic5C2×D412yes—
126C2×C3×Dic5C2×D412yes—
127C3×Dic10—24no—
128C3×Dic10—22no—
129C3×Dic10—22no—
130C3×Dic10—22no—
131C3×Dic10—42no—
132C3×Dic10C2×D414yes—
133C3×C5×C8—22no—
134C3×#168—22no—
135C22×C3×Q8—51no—
136C22×C3×Q8—51no—
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×C8C2×D511yescharacteristic
144C3×#124—54no—
145C3×#124—102no—
146C3×C4×Q8—54no—
147C3×C4×Q8—102no—
148C2×C3×C4⋊C4—101no—
149C2×C3×C4⋊C4C2×D511yescharacteristic
150C2×C3×C22⋊C4—101no—
151C2×C3×C22⋊C4—101no—
152C2×C3×C42—51no—
153C2×C3×C42—101no—
154C3×C4⋊C8—54no—
155C3×#103—54no—
156C3×#102C2×D514yes—
157C24×C5—21no—
158C2×C5×D4—22no—
159C2×C5×D4—21no—
160C2×C5×D4—22no—
161C2×C5×D4C23×C312yes—
162C22×C4×C5—21no—
163C22×C4×C5C23×C311yescharacteristic
164C22×Dic5—21no—
165C22×Dic5—21no—
166C22×Dic5C3×D411yescharacteristic
167C22×Dic5C3×D411yescharacteristic
168C2×Dic10—21no—
169C2×Dic10—22no—
170C2×Dic10—22no—
171C2×Dic10—22no—
172C2×Dic10—21no—
173C2×Dic10C3×D414yes—
174C2×Dic10C23×C312yes—
175C2×Dic10C23×C312yes—
176C2×C5×C8—22no—
177C2×C5×C8C23×C312yes—
178C5×C4⋊C4—21no—
179C5×C4⋊C4C23×C312yes—
180C22⋊Dic5—24no—
181C22⋊Dic5—24no—
182#687—28no—
183#683—24no—
184C4×Dic5—24no—
185C4×Dic5C3×D412yes—
186C4×Dic5C23×C312yes—
187C2×#168—22no—
188C2×#168C23×C312yes—
189Dic20—28no—
190C22×Q16—51no—
191C22×SD16—51no—
192C2×#124—51no—
193C2×C4×Q8—51no—
194#464—58no—
195C2×C4⋊C8—51no—
196C2×#103—51no—
197C2×#102C3×D511yescharacteristic
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—42no—
207C3×C4×C5#12112yes—
208C3×C4×C5C22×D411yescharacteristic
209C3×C4×C5C2×#4111yescharacteristic
210C3×Dic5—24no—
211C3×Dic5—22no—
212C3×Dic5—22no—
213C3×Dic5—42no—
214C24×C3—21no—
215C2×C3×Q8—54no—
216C2×C3×Q8—52no—
217C2×C3×Q8—52no—
218C2×C3×Q8—102no—
219C2×C3×Q8—101no—
220C2×C3×Q8—101no—
221C2×C3×Q8—102no—
222C2×C3×Q8—102no—
223C2×C3×D4—22no—
224C2×C3×D4—22no—
225C2×C3×D4—21no—
226C2×C3×D4C22×D512yes—
227C22×C3×C4—21no—
228C22×C3×C4—51no—
229C22×C3×C4—51no—
230C22×C3×C4—101no—
231C22×C3×C4—101no—
232C22×C3×C4C22×D511yescharacteristic
233C3×Q16—108no—
234C3×SD16—108no—
235C2×C3×C8—22no—
236C2×C3×C8—52no—
237C2×C3×C8—102no—
238C2×C3×C8C22×D512yes—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—104no—
241C3×C4⋊C4C22×D512yes—
242C3×C22⋊C4—104no—
243C3×C22⋊C4—104no—
244C3×C42—52no—
245C3×C42—52no—
246C3×C42—104no—
247C23×C5—22no—
248C23×C5—42no—
249C23×C5—42no—
250C23×C5—42no—
251C23×C5C2×C3×D411yescharacteristic
252C5×D4—22no—
253C5×D4—22no—
254C5×D4—42no—
255C2×C4×C5—22no—
256C2×C4×C5—42no—
257C2×C4×C5C2×C3×D411yescharacteristic
258C2×C4×C5C2×C3×D412yes—
259C2×C4×C5C3×#4112yes—
260C2×C4×C5C24×C311yescharacteristic
261C2×Dic5—24no—
262C2×Dic5—22no—
263C2×Dic5—22no—
264C2×Dic5—22no—
265C2×Dic5—22no—
266C2×Dic5—42no—
267C2×Dic5C2×C3×D412yes—
268C2×Dic5C2×C3×D412yes—
269Dic10—24no—
270Dic10—22no—
271Dic10—22no—
272Dic10—22no—
273Dic10—42no—
274Dic10C2×C3×D414yes—
275C5×C8—22no—
276#168—22no—
277C22×Q8—51no—
278C22×Q8—51no—
279C22×D4C2×C3×D511yescharacteristic
280C2×Q16—54no—
281C2×Q16—104no—
282C2×SD16—54no—
283C2×SD16—104no—
284C22×C8—51no—
285C22×C8C2×C3×D511yescharacteristic
286#124—54no—
287#124—102no—
288C4×Q8—54no—
289C4×Q8—102no—
290C2×C4⋊C4—101no—
291C2×C4⋊C4C2×C3×D511yescharacteristic
292C2×C22⋊C4—101no—
293C2×C22⋊C4—101no—
294C2×C42—51no—
295C2×C42—101no—
296C4⋊C8—54no—
297#103—54no—
298#102C2×C3×D514yes—
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×C3D4×D511yescharacteristic
309C3×Q8—54no—
310C3×Q8—102no—
311C3×Q8—104no—
312C3×Q8—102no—
313C3×Q8—102no—
314C3×Q8—202no—
315C3×D4—22no—
316C3×D4—22no—
317C3×D4—42no—
318C2×C3×C4—22no—
319C2×C3×C4—42no—
320C2×C3×C4—52no—
321C2×C3×C4—52no—
322C2×C3×C4—102no—
323C2×C3×C4—104no—
324C2×C3×C4—102no—
325C2×C3×C4—102no—
326C2×C3×C4—102no—
327C2×C3×C4—202no—
328C2×C3×C4#70912yes—
329C2×C3×C4D4×D512yes—
330C2×C3×C4D4×D511yescharacteristic
331C2×C3×C4C23×D511yescharacteristic
332C3×C8—22no—
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—42no—
343C4×C5C3×#12112yes—
344C4×C5C22×C3×D411yescharacteristic
345C4×C5C2×C3×#4111yescharacteristic
346Dic5—22no—
347Dic5—22no—
348Dic5—24no—
349Dic5—42no—
350C24—21no—
351C2×Q8—54no—
352C2×Q8—52no—
353C2×Q8—52no—
354C2×Q8—102no—
355C2×Q8—101no—
356C2×Q8—101no—
357C2×Q8—102no—
358C2×Q8—102no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—21no—
362C2×D4C22×C3×D512yes—
363C22×C4—21no—
364C22×C4—51no—
365C22×C4—51no—
366C22×C4—101no—
367C22×C4—101no—
368C22×C4C22×C3×D511yescharacteristic
369Q16—108no—
370SD16—108no—
371C2×C8—22no—
372C2×C8—52no—
373C2×C8—102no—
374C2×C8C22×C3×D512yes—
375C4⋊C4—21no—
376C4⋊C4—104no—
377C4⋊C4C22×C3×D512yes—
378C22⋊C4—104no—
379C22⋊C4—104no—
380C42—52no—
381C42—52no—
382C42—104no—
383C3×C5C2×#46411yescharacteristic, complemented, Hall
384C22×C3—42no—
385C22×C3—42no—
386C22×C3—42no—
387C22×C3—84no—
388C22×C3#403012yes—
389C22×C3#405812yes—
390C22×C3#406112yes—
391C22×C3C2×D4×D511yescharacteristic
392C3×C4—42no—
393C3×C4—104no—
394C3×C4—102no—
395C3×C4—102no—
396C3×C4—202no—
397C3×C4#403012yes—
398C3×C4C2×#70911yescharacteristic
399C3×C4C2×D4×D511yescharacteristic
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—
409C23C3×D4×D511yescharacteristic
410Q8—54no—
411Q8—102no—
412Q8—104no—
413Q8—102no—
414Q8—102no—
415Q8—202no—
416D4—22no—
417D4—22no—
418D4—42no—
419C2×C4—22no—
420C2×C4—42no—
421C2×C4—52no—
422C2×C4—52no—
423C2×C4—102no—
424C2×C4—104no—
425C2×C4—102no—
426C2×C4—102no—
427C2×C4—102no—
428C2×C4—202no—
429C2×C4C3×#70912yes—
430C2×C4C3×D4×D512yes—
431C2×C4C3×D4×D511yescharacteristic
432C2×C4C23×C3×D511yescharacteristic
433C8—22no—
434C8—102no—
435C2×C3—82no—
436C2×C3#6505614yesdirect product factor
437C2×C3C2×#403011yescharacteristic
438C2×C3C2×#405811yescharacteristic
439C2×C3C2×#406111yescharacteristic
440C5C2×C3×#46411yescharacteristic, complemented, Sylow
441C22—42no—
442C22—42no—
443C22—42no—
444C22—84no—
445C22C3×#403012yes—
446C22C3×#405812yes—
447C22C3×#406112yes—
448C22C2×C3×D4×D511yescharacteristic
449C4—42no—
450C4—104no—
451C4—102no—
452C4—102no—
453C4—202no—
454C4C3×#403012yes—
455C4C2×C3×#70911yescharacteristic
456C4C2×C3×D4×D511yescharacteristic
457C3C2×#6505611yescharacteristic, direct product factor, Sylow
458C2—82no—
459C2C3×#6505614yesdirect product factor
460C2C2×C3×#403011yescharacteristic
461C2C2×C3×#405811yescharacteristic
462C2C2×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×C5×D4C22
hypercenterC2×C3×#102D5
hypocenterC5C2×C3×#464
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
3. C22×C3×Dic20C2semidirect
4. C2×C3×#4033C2semidirect
6. C2×C3×#3946C2semidirect
7. C2×C3×#3942C2semidirect
8. C3×#65056C2direct (N × H)
9. C2×#65056C3direct (N × H)
26. C2×C3×Dic20C22semidirect
28. C3×#4033C22semidirect
30. C3×#3946C22semidirect
31. C3×#3942C22semidirect
35. C22×Dic20C2×C3semidirect
36. C2×#4033C2×C3semidirect
38. C2×#3946C2×C3semidirect
39. C2×#3942C2×C3semidirect
40. #65056C2×C3direct (N × H)
98. C2×Dic20C22×C3semidirect
100. #4033C22×C3semidirect
102. #3946C22×C3semidirect
103. #3942C22×C3semidirect
300. C2×C3×C5#464semidirect
383. C3×C5C2×#464semidirect
401. C2×C5C3×#464semidirect
436. C2×C3#65056direct (N × H)
440. C5C2×C3×#464semidirect
457. C3C2×#65056direct (N × H)
459. C2C3×#65056direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#40822
C22×C3×#6872
C22×C3×Dic202
C2×C3×#40332
C2×C3×C5×#1022
C2×C3×#39462
C2×C3×#39422
C3×#650562
C2×#650563
C2×C3×#4645