Finite Group Databasebeta

← C2×C3×#65454 #14425936

Subgroups — C2×C3×#65454 #14425936

Order 1920.

1128 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C5×SD16C211yesmaximal, characteristic, complemented
2C2×C3×#4093C211yesmaximal, characteristic, complemented
3C2×C3×D4×Dic5C211yesmaximal, characteristic, complemented
4C22×C3×#686C211yesmaximal, characteristic
5C2×C3×#3967C211yesmaximal, characteristic, complemented
6C3×#65454C218yesmaximal, direct product factor
7C2×C3×#3953C211yesmaximal, characteristic, complemented
8C2×C3×#3946C211yesmaximal, characteristic, complemented
9C2×#65454C311yesmaximal, characteristic, direct product factor, Hall
10C22×C3×C5×Q8C2211yescharacteristic
11C22×C3×C5×D4C2211yescharacteristic
12C23×C3×Dic5—21no—
13C22×C3×D20C2211yescharacteristic
14C2×C3×C5×SD16—24no—
15C2×C3×C5×SD16C2214yescomplemented
16C22×C3×C5×C8C2211yescharacteristic
17C2×C3×C22⋊Dic5—21no—
18C3×#4093—22no—
19C3×#4093C2214yescomplemented
20C3×D4×Dic5—22no—
21C3×D4×Dic5C2214yescomplemented
22C2×C3×#686—24no—
23C2×C3×#686C2214yes—
24C2×C3×#685—21no—
25C2×C3×#685—21no—
26C2×C3×C4⋊Dic5C2211yescharacteristic
27C2×C3×C4×Dic5C2211yescharacteristic
28C22×C3×#168C2211yescharacteristic
29C3×#3967C2214yescomplemented
30C3×#3953C2214yescomplemented
31C3×#3946C2214yescomplemented
32C2×C3×#463—51nomaximal, Hall
33C22×C5×SD16C2×C311yescharacteristic, complemented
34C2×#4093C2×C311yescharacteristic, complemented
35C2×D4×Dic5C2×C311yescharacteristic, complemented
36C22×#686C2×C311yescharacteristic
37C2×#3967C2×C311yescharacteristic, complemented
38#65454C2×C318yesdirect product factor
39C2×#3953C2×C311yescharacteristic, complemented
40C2×#3946C2×C311yescharacteristic, complemented
41C24×C3×C5—21no—
42C23×C3×D5—21no—
43C2×C3×C5×Q8—21no—
44C2×C3×C5×Q8—22no—
45C2×C3×C5×Q8—22no—
46C2×C3×C5×Q8C2312yes—
47C2×C3×C5×D4—21no—
48C2×C3×C5×D4—22no—
49C2×C3×C5×D4D414yes—
50C2×C3×C5×D4C2312yes—
51C22×C3×C4×C5—21no—
52C22×C3×C4×C5C2311yescharacteristic
53C22×C3×Dic5—21no—
54C22×C3×Dic5—24no—
55C22×C3×Dic5—44no—
56C22×C3×Dic5D411yescharacteristic
57C22×C3×Dic5D411yescharacteristic
58C2×C3×D20—22no—
59C2×C3×D20—22no—
60C2×C3×D20—21no—
61C2×C3×D20C2312yes—
62C3×C5×SD16—28no—
63C2×C3×C5×C8—22no—
64C2×C3×C5×C8C2312yes—
65C3×C22⋊Dic5—24no—
66C3×#686—28no—
67C3×#685—24no—
68C3×#685—24no—
69C3×C4⋊Dic5—21no—
70C3×C4⋊Dic5C2312yes—
71C3×C4×Dic5D412yes—
72C3×C4×Dic5C2312yes—
73C2×C3×#168—22no—
74C2×C3×#168C2312yes—
75C22×C3×SD16D511yescharacteristic, complemented
76C22×C3×D8—51no—
77C2×C3×#124—51no—
78C2×C3×C4×D4—51no—
79C3×#463—58no—
80C2×C3×C4⋊C8—51no—
81C2×C3×#103—51no—
82C2×C3×#102—51no—
83C22×C5×Q8C22×C311yescharacteristic
84C22×C5×D4C22×C311yescharacteristic
85C23×Dic5—21no—
86C22×D20C22×C311yescharacteristic
87C2×C5×SD16—24no—
88C2×C5×SD16C22×C314yescomplemented
89C22×C5×C8C22×C311yescharacteristic
90C2×C22⋊Dic5—21no—
91#4093—22no—
92#4093C22×C314yescomplemented
93D4×Dic5—22no—
94D4×Dic5C22×C314yescomplemented
95C2×#686—24no—
96C2×#686C22×C314yes—
97C2×#685—21no—
98C2×#685—21no—
99C2×C4⋊Dic5C22×C311yescharacteristic
100C2×C4×Dic5C22×C311yescharacteristic
101C22×#168C22×C311yescharacteristic
102#3967C22×C314yescomplemented
103#3953C22×C314yescomplemented
104#3946C22×C314yescomplemented
105C2×#463—51noSylow
106C23×C3×C5—24no—
107C23×C3×C5—22no—
108C23×C3×C5—42no—
109C23×C3×C5—42no—
110C23×C3×C5C2×D411yescharacteristic
111C22×C3×D5—22no—
112C22×C3×D5—42no—
113C22×C3×D5—42no—
114C22×C3×D5—42no—
115C3×C5×Q8—22no—
116C3×C5×Q8—22no—
117C3×C5×Q8—42no—
118C3×C5×D4—24no—
119C3×C5×D4—22no—
120C3×C5×D4C2×D414yes—
121C2×C3×C4×C5—22no—
122C2×C3×C4×C5—42no—
123C2×C3×C4×C5C2×D411yescharacteristic
124C2×C3×C4×C5C2×D412yes—
125C2×C3×C4×C5#4112yes—
126C2×C3×C4×C5C2411yescharacteristic
127C2×C3×Dic5—22no—
128C2×C3×Dic5—22no—
129C2×C3×Dic5—22no—
130C2×C3×Dic5—48no—
131C2×C3×Dic5—42no—
132C2×C3×Dic5C2×D412yes—
133C2×C3×Dic5C2×D412yes—
134C3×D20—22no—
135C3×D20—22no—
136C3×D20—42no—
137C3×C5×C8—22no—
138C3×#168—22no—
139C22×C3×Q8C2×D511yescharacteristic
140C22×C3×D4—51no—
141C22×C3×D4C2×D511yescharacteristic
142C23×C3×C4—101no—
143C2×C3×SD16—24no—
144C2×C3×SD16C2×D514yescomplemented
145C2×C3×D8—54no—
146C2×C3×D8—104no—
147C22×C3×C8—51no—
148C22×C3×C8C2×D511yescharacteristic
149C3×#124—54no—
150C3×#124—102no—
151C3×C4×D4—54no—
152C3×C4×D4—102no—
153C2×C3×C4⋊C4—51no—
154C2×C3×C22⋊C4—101no—
155C2×C3×C22⋊C4—101no—
156C2×C3×C22⋊C4—101no—
157C2×C3×C42—51no—
158C3×C4⋊C8—54no—
159C3×#103—54no—
160C3×#102—54no—
161C24×C5—21no—
162C23×D5—21no—
163C2×C5×Q8—21no—
164C2×C5×Q8—22no—
165C2×C5×Q8—22no—
166C2×C5×Q8C23×C312yes—
167C2×C5×D4—21no—
168C2×C5×D4—22no—
169C2×C5×D4C3×D414yes—
170C2×C5×D4C23×C312yes—
171C22×C4×C5—21no—
172C22×C4×C5C23×C311yescharacteristic
173C22×Dic5—21no—
174C22×Dic5—24no—
175C22×Dic5—44no—
176C22×Dic5C3×D411yescharacteristic
177C22×Dic5C3×D411yescharacteristic
178C2×D20—22no—
179C2×D20—22no—
180C2×D20—21no—
181C2×D20C23×C312yes—
182C5×SD16—28no—
183C2×C5×C8—22no—
184C2×C5×C8C23×C312yes—
185C22⋊Dic5—24no—
186#686—28no—
187#685—24no—
188#685—24no—
189C4⋊Dic5—21no—
190C4⋊Dic5C23×C312yes—
191C4×Dic5C3×D412yes—
192C4×Dic5C23×C312yes—
193C2×#168—22no—
194C2×#168C23×C312yes—
195C22×SD16C3×D511yescharacteristic, complemented
196C22×D8—51no—
197C2×#124—51no—
198C2×C4×D4—51no—
199#463—58no—
200C2×C4⋊C8—51no—
201C2×#103—51no—
202C2×#102—51no—
203C22×C3×C5—24no—
204C22×C3×C5—42no—
205C22×C3×C5—42no—
206C22×C3×C5—48no—
207C22×C3×C5#12112yes—
208C22×C3×C5#13512yes—
209C22×C3×C5AGL(1,ℤ/8)12yes—
210C22×C3×C5C22×D411yescharacteristic
211C2×C3×D5—42no—
212C2×C3×D5—42no—
213C2×C3×D5—42no—
214C2×C3×D5—84no—
215C3×C4×C5—42no—
216C3×C4×C5#12112yes—
217C3×C4×C5C22×D411yescharacteristic
218C3×C4×C5C2×#4111yescharacteristic
219C3×Dic5—22no—
220C3×Dic5—22no—
221C3×Dic5—42no—
222C24×C3—21no—
223C24×C3—101no—
224C2×C3×Q8—21no—
225C2×C3×Q8—22no—
226C2×C3×Q8—22no—
227C2×C3×Q8C22×D512yes—
228C2×C3×D4—21no—
229C2×C3×D4—22no—
230C2×C3×D4—52no—
231C2×C3×D4—101no—
232C2×C3×D4—102no—
233C2×C3×D4—102no—
234C2×C3×D4C22⋊D514yes—
235C2×C3×D4C22×D512yes—
236C22×C3×C4—21no—
237C22×C3×C4—51no—
238C22×C3×C4—51no—
239C22×C3×C4—104no—
240C22×C3×C4—101no—
241C22×C3×C4—204no—
242C22×C3×C4C22×D511yescharacteristic
243C3×SD16—28no—
244C3×D8—108no—
245C2×C3×C8—22no—
246C2×C3×C8—52no—
247C2×C3×C8—102no—
248C2×C3×C8C22×D512yes—
249C3×C4⋊C4—52no—
250C3×C4⋊C4—101no—
251C3×C22⋊C4—104no—
252C3×C22⋊C4—104no—
253C3×C22⋊C4—104no—
254C3×C42—52no—
255C3×C42—52no—
256C23×C5—24no—
257C23×C5—22no—
258C23×C5—42no—
259C23×C5—42no—
260C23×C5C2×C3×D411yescharacteristic
261C22×D5—22no—
262C22×D5—42no—
263C22×D5—42no—
264C22×D5—42no—
265C5×Q8—22no—
266C5×Q8—22no—
267C5×Q8—42no—
268C5×D4—24no—
269C5×D4—22no—
270C5×D4C2×C3×D414yes—
271C2×C4×C5—22no—
272C2×C4×C5—42no—
273C2×C4×C5C2×C3×D411yescharacteristic
274C2×C4×C5C2×C3×D412yes—
275C2×C4×C5C3×#4112yes—
276C2×C4×C5C24×C311yescharacteristic
277C2×Dic5—22no—
278C2×Dic5—22no—
279C2×Dic5—22no—
280C2×Dic5—48no—
281C2×Dic5—42no—
282C2×Dic5C2×C3×D412yes—
283C2×Dic5C2×C3×D412yes—
284D20—22no—
285D20—22no—
286D20—42no—
287C5×C8—22no—
288#168—22no—
289C22×Q8C2×C3×D511yescharacteristic
290C22×D4—51no—
291C22×D4C2×C3×D511yescharacteristic
292C23×C4—101no—
293C2×SD16—24no—
294C2×SD16C2×C3×D514yescomplemented
295C2×D8—54no—
296C2×D8—104no—
297C22×C8—51no—
298C22×C8C2×C3×D511yescharacteristic
299#124—54no—
300#124—102no—
301C4×D4—54no—
302C4×D4—102no—
303C2×C4⋊C4—51no—
304C2×C22⋊C4—101no—
305C2×C22⋊C4—101no—
306C2×C22⋊C4—101no—
307C2×C42—51no—
308C4⋊C8—54no—
309#103—54no—
310#102—54no—
311C2×C3×C5—44no—
312C2×C3×C5#46314yescomplemented
313C2×C3×C5C2×#12111yescharacteristic
314C2×C3×C5C2×#13511yescharacteristic
315C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
316C3×D5—82no—
317C23×C3—22no—
318C23×C3—24no—
319C23×C3—42no—
320C23×C3—42no—
321C23×C3—102no—
322C23×C3—202no—
323C23×C3—202no—
324C23×C3—202no—
325C23×C3D4×D511yescharacteristic
326C3×Q8—22no—
327C3×Q8—22no—
328C3×Q8—42no—
329C3×D4—22no—
330C3×D4—24no—
331C3×D4—102no—
332C3×D4—102no—
333C3×D4—202no—
334C3×D4C2×C22⋊D514yes—
335C2×C3×C4—22no—
336C2×C3×C4—42no—
337C2×C3×C4—52no—
338C2×C3×C4—52no—
339C2×C3×C4—102no—
340C2×C3×C4—102no—
341C2×C3×C4—102no—
342C2×C3×C4—208no—
343C2×C3×C4—202no—
344C2×C3×C4D4×D511yescharacteristic
345C2×C3×C4#71112yes—
346C2×C3×C4C2×C22⋊D512yes—
347C2×C3×C4C23×D511yescharacteristic
348C3×C8—22no—
349C3×C8—102no—
350C22×C5—24no—
351C22×C5—42no—
352C22×C5—42no—
353C22×C5—48no—
354C22×C5C3×#12112yes—
355C22×C5C3×#13512yes—
356C22×C5C3×AGL(1,ℤ/8)12yes—
357C22×C5C22×C3×D411yescharacteristic
358C2×D5—42no—
359C2×D5—42no—
360C2×D5—42no—
361C2×D5—84no—
362C4×C5—42no—
363C4×C5C3×#12112yes—
364C4×C5C22×C3×D411yescharacteristic
365C4×C5C2×C3×#4111yescharacteristic
366Dic5—22no—
367Dic5—22no—
368Dic5—42no—
369C24—21no—
370C24—101no—
371C2×Q8—21no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8C22×C3×D512yes—
375C2×D4—21no—
376C2×D4—22no—
377C2×D4—52no—
378C2×D4—101no—
379C2×D4—102no—
380C2×D4—102no—
381C2×D4C3×C22⋊D514yes—
382C2×D4C22×C3×D512yes—
383C22×C4—21no—
384C22×C4—51no—
385C22×C4—51no—
386C22×C4—104no—
387C22×C4—101no—
388C22×C4—204no—
389C22×C4C22×C3×D511yescharacteristic
390SD16—28no—
391D8—108no—
392C2×C8—22no—
393C2×C8—52no—
394C2×C8—102no—
395C2×C8C22×C3×D512yes—
396C4⋊C4—52no—
397C4⋊C4—101no—
398C22⋊C4—104no—
399C22⋊C4—104no—
400C22⋊C4—104no—
401C42—52no—
402C42—52no—
403C3×C5C2×#46311yescharacteristic, complemented, Hall
404C22×C3—24no—
405C22×C3—42no—
406C22×C3—42no—
407C22×C3—48no—
408C22×C3—202no—
409C22×C3—202no—
410C22×C3—202no—
411C22×C3—404no—
412C22×C3#406012yes—
413C22×C3#406212yes—
414C22×C3#408512yes—
415C22×C3C2×D4×D511yescharacteristic
416C3×C4—42no—
417C3×C4—102no—
418C3×C4—102no—
419C3×C4—202no—
420C3×C4#408512yes—
421C3×C4C2×#71111yescharacteristic
422C3×C4C22×C22⋊D511yescharacteristic
423C2×C5—44no—
424C2×C5C3×#46314yescomplemented
425C2×C5C2×C3×#12111yescharacteristic
426C2×C5C2×C3×#13511yescharacteristic
427C2×C5C2×C3×AGL(1,ℤ/8)11yescharacteristic
428D5—82no—
429C23—24no—
430C23—22no—
431C23—42no—
432C23—42no—
433C23—102no—
434C23—202no—
435C23—202no—
436C23—202no—
437C23C3×D4×D511yescharacteristic
438Q8—22no—
439Q8—22no—
440Q8—42no—
441D4—24no—
442D4—22no—
443D4—102no—
444D4—102no—
445D4—202no—
446D4C2×C3×C22⋊D514yes—
447C2×C4—22no—
448C2×C4—42no—
449C2×C4—52no—
450C2×C4—52no—
451C2×C4—102no—
452C2×C4—102no—
453C2×C4—102no—
454C2×C4—208no—
455C2×C4—202no—
456C2×C4C3×D4×D511yescharacteristic
457C2×C4C3×#71112yes—
458C2×C4C2×C3×C22⋊D512yes—
459C2×C4C23×C3×D511yescharacteristic
460C8—22no—
461C8—102no—
462C2×C3—44no—
463C2×C3—402no—
464C2×C3#6545414yesdirect product factor
465C2×C3C2×#406011yescharacteristic
466C2×C3C2×#406211yescharacteristic
467C2×C3C2×#408511yescharacteristic
468C5C2×C3×#46311yescharacteristic, complemented, Sylow
469C22—24no—
470C22—48no—
471C22—42no—
472C22—42no—
473C22—202no—
474C22—202no—
475C22—202no—
476C22—404no—
477C22C3×#406012yes—
478C22C3×#406212yes—
479C22C3×#408512yes—
480C22C2×C3×D4×D511yescharacteristic
481C4—42no—
482C4—102no—
483C4—102no—
484C4—202no—
485C4C3×#408512yes—
486C4C2×C3×#71111yescharacteristic
487C4C22×C3×C22⋊D511yescharacteristic
488C3C2×#6545411yescharacteristic, direct product factor, Sylow
489C2—44no—
490C2—402no—
491C2C3×#6545414yesdirect product factor
492C2C2×C3×#406011yescharacteristic
493C2C2×C3×#406211yescharacteristic
494C2C2×C3×#408511yescharacteristic
subgroupG/H
centerC23×C3D4×D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC22×C3×C5×SD16C2
socleC23×C3×C5C2×D4
prime radicalC22×C3×#686C2
hypercenterC22×C3×SD16D5
hypocenterC5C2×C3×#463
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C22×C3×SD16

Complements

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

NH ≅ G/Ntype
1. C22×C3×C5×SD16C2semidirect
2. C2×C3×#4093C2semidirect
3. C2×C3×D4×Dic5C2semidirect
5. C2×C3×#3967C2semidirect
6. C3×#65454C2direct (N × H)
7. C2×C3×#3953C2semidirect
8. C2×C3×#3946C2semidirect
9. C2×#65454C3direct (N × H)
15. C2×C3×C5×SD16C22semidirect
19. C3×#4093C22semidirect
21. C3×D4×Dic5C22semidirect
29. C3×#3967C22semidirect
30. C3×#3953C22semidirect
31. C3×#3946C22semidirect
33. C22×C5×SD16C2×C3semidirect
34. C2×#4093C2×C3semidirect
35. C2×D4×Dic5C2×C3semidirect
37. C2×#3967C2×C3semidirect
38. #65454C2×C3direct (N × H)
39. C2×#3953C2×C3semidirect
40. C2×#3946C2×C3semidirect
75. C22×C3×SD16D5semidirect
88. C2×C5×SD16C22×C3semidirect
92. #4093C22×C3semidirect
94. D4×Dic5C22×C3semidirect
102. #3967C22×C3semidirect
103. #3953C22×C3semidirect
104. #3946C22×C3semidirect
144. C2×C3×SD16C2×D5semidirect
195. C22×SD16C3×D5semidirect
294. C2×SD16C2×C3×D5semidirect
312. C2×C3×C5#463semidirect
403. C3×C5C2×#463semidirect
424. C2×C5C3×#463semidirect
464. C2×C3#65454direct (N × H)
468. C5C2×C3×#463semidirect
488. C3C2×#65454direct (N × H)
491. C2C3×#65454direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C5×SD162
C2×C3×#40932
C2×C3×D4×Dic52
C22×C3×#6862
C2×C3×#39672
C3×#654542
C2×C3×#39532
C2×C3×#39462
C2×#654543
C2×C3×#4635