Finite Group Databasebeta

← C2×C3×#65474 #14426137

Subgroups — C2×C3×#65474 #14426137

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
1C22×C3×C5×Q16C211yesmaximal, characteristic, complemented
2C2×C3×#4093C211yesmaximal, characteristic
3C2×C3×Q8×Dic5C211yesmaximal, characteristic, complemented
4C22×C3×#688C211yesmaximal, characteristic
5C2×C3×#3967C211yesmaximal, characteristic, complemented
6C3×#65474C218yesmaximal, direct product factor
7C2×C3×#3953C211yesmaximal, characteristic
8C2×C3×#3946C211yesmaximal, characteristic, complemented
9C2×#65474C311yesmaximal, characteristic, direct product factor, Hall
10C22×C3×C5×Q8C2211yescharacteristic
11C22×C3×C5×Q8C2211yescharacteristic
12C22×C3×D20C2211yescharacteristic
13C2×C3×C5×Q16—24no—
14C2×C3×C5×Q16C2214yescomplemented
15C22×C3×C5×C8C2211yescharacteristic
16C3×#4093—22no—
17C3×#4093C2214yes—
18C3×Q8×Dic5—22no—
19C3×Q8×Dic5C2214yescomplemented
20C2×C3×#688—24no—
21C2×C3×#688C2214yes—
22C2×C3×#685—21no—
23C2×C3×#685—21no—
24C2×C3×C4⋊Dic5—21no—
25C2×C3×C4⋊Dic5C2211yescharacteristic
26C2×C3×C4×Dic5—21no—
27C2×C3×C4×Dic5C2211yescharacteristic
28C22×C3×#168C2211yescharacteristic
29C3×#3967C2214yescomplemented
30C3×#3953C2214yes—
31C3×#3946C2214yescomplemented
32C2×C3×#464—51nomaximal, Hall
33C22×C5×Q16C2×C311yescharacteristic, complemented
34C2×#4093C2×C311yescharacteristic
35C2×Q8×Dic5C2×C311yescharacteristic, complemented
36C22×#688C2×C311yescharacteristic
37C2×#3967C2×C311yescharacteristic, complemented
38#65474C2×C318yesdirect product factor
39C2×#3953C2×C311yescharacteristic
40C2×#3946C2×C311yescharacteristic, complemented
41C23×C3×D5—21no—
42C2×C3×C5×Q8—21no—
43C2×C3×C5×Q8—21no—
44C2×C3×C5×Q8—22no—
45C2×C3×C5×Q8—22no—
46C2×C3×C5×Q8—22no—
47C2×C3×C5×Q8D414yes—
48C2×C3×C5×Q8C2312yes—
49C2×C3×C5×Q8C2312yes—
50C22×C3×C4×C5—21no—
51C22×C3×C4×C5—21no—
52C22×C3×C4×C5C2311yescharacteristic
53C22×C3×Dic5—21no—
54C22×C3×Dic5D411yescharacteristic
55C22×C3×Dic5D411yescharacteristic
56C2×C3×D20—22no—
57C2×C3×D20—22no—
58C2×C3×D20—21no—
59C2×C3×D20C2312yes—
60C3×C5×Q16—28no—
61C2×C3×C5×C8—22no—
62C2×C3×C5×C8C2312yes—
63C3×#688—28no—
64C3×#685—24no—
65C3×#685—24no—
66C3×C4⋊Dic5—21no—
67C3×C4⋊Dic5—24no—
68C3×C4⋊Dic5C2312yes—
69C3×C4×Dic5—24no—
70C3×C4×Dic5D412yes—
71C3×C4×Dic5C2312yes—
72C2×C3×#168—22no—
73C2×C3×#168C2312yes—
74C22×C3×Q16D511yescharacteristic, complemented
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×#102—51no—
82C22×C5×Q8C22×C311yescharacteristic
83C22×C5×Q8C22×C311yescharacteristic
84C22×D20C22×C311yescharacteristic
85C2×C5×Q16—24no—
86C2×C5×Q16C22×C314yescomplemented
87C22×C5×C8C22×C311yescharacteristic
88#4093—22no—
89#4093C22×C314yes—
90Q8×Dic5—22no—
91Q8×Dic5C22×C314yescomplemented
92C2×#688—24no—
93C2×#688C22×C314yes—
94C2×#685—21no—
95C2×#685—21no—
96C2×C4⋊Dic5—21no—
97C2×C4⋊Dic5C22×C311yescharacteristic
98C2×C4×Dic5—21no—
99C2×C4×Dic5C22×C311yescharacteristic
100C22×#168C22×C311yescharacteristic
101#3967C22×C314yescomplemented
102#3953C22×C314yes—
103#3946C22×C314yescomplemented
104C2×#464—51noSylow
105C23×C3×C5C2×D411yescharacteristic
106C22×C3×D5—22no—
107C22×C3×D5—42no—
108C22×C3×D5—42no—
109C22×C3×D5—42no—
110C3×C5×Q8—24no—
111C3×C5×Q8—22no—
112C3×C5×Q8—22no—
113C3×C5×Q8—22no—
114C3×C5×Q8—42no—
115C3×C5×Q8C2×D414yes—
116C2×C3×C4×C5—24no—
117C2×C3×C4×C5—22no—
118C2×C3×C4×C5—22no—
119C2×C3×C4×C5—42no—
120C2×C3×C4×C5C2×D411yescharacteristic
121C2×C3×C4×C5C2×D412yes—
122C2×C3×C4×C5#4112yes—
123C2×C3×C4×C5C2411yescharacteristic
124C2×C3×Dic5—22no—
125C2×C3×Dic5—22no—
126C2×C3×Dic5—22no—
127C2×C3×Dic5—42no—
128C2×C3×Dic5C2×D412yes—
129C2×C3×Dic5C2×D412yes—
130C3×D20—22no—
131C3×D20—22no—
132C3×D20—42no—
133C3×C5×C8—22no—
134C3×#168—22no—
135C22×C3×Q8C2×D511yescharacteristic
136C22×C3×Q8C2×D511yescharacteristic
137C22×C3×D4—51no—
138C2×C3×Q16—24no—
139C2×C3×Q16C2×D514yescomplemented
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—51no—
149C2×C3×C4⋊C4—101no—
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×#102—54no—
157C23×D5—21no—
158C2×C5×Q8—21no—
159C2×C5×Q8—21no—
160C2×C5×Q8—22no—
161C2×C5×Q8—22no—
162C2×C5×Q8—22no—
163C2×C5×Q8C3×D414yes—
164C2×C5×Q8C23×C312yes—
165C2×C5×Q8C23×C312yes—
166C22×C4×C5—21no—
167C22×C4×C5—21no—
168C22×C4×C5C23×C311yescharacteristic
169C22×Dic5—21no—
170C22×Dic5C3×D411yescharacteristic
171C22×Dic5C3×D411yescharacteristic
172C2×D20—22no—
173C2×D20—22no—
174C2×D20—21no—
175C2×D20C23×C312yes—
176C5×Q16—28no—
177C2×C5×C8—22no—
178C2×C5×C8C23×C312yes—
179#688—28no—
180#685—24no—
181#685—24no—
182C4⋊Dic5—21no—
183C4⋊Dic5—24no—
184C4⋊Dic5C23×C312yes—
185C4×Dic5—24no—
186C4×Dic5C3×D412yes—
187C4×Dic5C23×C312yes—
188C2×#168—22no—
189C2×#168C23×C312yes—
190C22×Q16C3×D511yescharacteristic, complemented
191C22×SD16—51no—
192C2×#124—51no—
193C2×C4×Q8—51no—
194#464—58no—
195C2×C4⋊C8—51no—
196C2×#103—51no—
197C2×#102—51no—
198C22×C3×C5#12112yes—
199C22×C3×C5#13512yes—
200C22×C3×C5#13712yes—
201C22×C3×C5C22×D411yescharacteristic
202C2×C3×D5—42no—
203C2×C3×D5—42no—
204C2×C3×D5—42no—
205C2×C3×D5—84no—
206C3×C4×C5—24no—
207C3×C4×C5—42no—
208C3×C4×C5#12112yes—
209C3×C4×C5C22×D411yescharacteristic
210C3×C4×C5C2×#4111yescharacteristic
211C3×Dic5—22no—
212C3×Dic5—22no—
213C3×Dic5—42no—
214C24×C3—101no—
215C2×C3×Q8—21no—
216C2×C3×Q8—21no—
217C2×C3×Q8—22no—
218C2×C3×Q8—22no—
219C2×C3×Q8—22no—
220C2×C3×Q8C22⋊D514yes—
221C2×C3×Q8C22×D512yes—
222C2×C3×Q8C22×D512yes—
223C2×C3×D4—52no—
224C2×C3×D4—101no—
225C2×C3×D4—102no—
226C2×C3×D4—102no—
227C22×C3×C4—21no—
228C22×C3×C4—21no—
229C22×C3×C4—51no—
230C22×C3×C4—51no—
231C22×C3×C4—101no—
232C22×C3×C4C22×D511yescharacteristic
233C3×Q16—28no—
234C3×SD16—108no—
235C2×C3×C8—22no—
236C2×C3×C8—52no—
237C2×C3×C8—102no—
238C2×C3×C8C22×D512yes—
239C3×C4⋊C4—52no—
240C3×C4⋊C4—104no—
241C3×C4⋊C4—101no—
242C3×C22⋊C4—104no—
243C3×C22⋊C4—104no—
244C3×C42—52no—
245C3×C42—52no—
246C3×C42—104no—
247C23×C5C2×C3×D411yescharacteristic
248C22×D5—22no—
249C22×D5—42no—
250C22×D5—42no—
251C22×D5—42no—
252C5×Q8—24no—
253C5×Q8—22no—
254C5×Q8—22no—
255C5×Q8—22no—
256C5×Q8—42no—
257C5×Q8C2×C3×D414yes—
258C2×C4×C5—24no—
259C2×C4×C5—22no—
260C2×C4×C5—22no—
261C2×C4×C5—42no—
262C2×C4×C5C2×C3×D411yescharacteristic
263C2×C4×C5C2×C3×D412yes—
264C2×C4×C5C3×#4112yes—
265C2×C4×C5C24×C311yescharacteristic
266C2×Dic5—22no—
267C2×Dic5—22no—
268C2×Dic5—22no—
269C2×Dic5—42no—
270C2×Dic5C2×C3×D412yes—
271C2×Dic5C2×C3×D412yes—
272D20—22no—
273D20—22no—
274D20—42no—
275C5×C8—22no—
276#168—22no—
277C22×Q8C2×C3×D511yescharacteristic
278C22×Q8C2×C3×D511yescharacteristic
279C22×D4—51no—
280C2×Q16—24no—
281C2×Q16C2×C3×D514yescomplemented
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—51no—
291C2×C4⋊C4—101no—
292C2×C22⋊C4—101no—
293C2×C22⋊C4—101no—
294C2×C42—51no—
295C2×C42—101no—
296C4⋊C8—54no—
297#103—54no—
298#102—54no—
299C2×C3×C5#46414yescomplemented
300C2×C3×C5C2×#12111yescharacteristic
301C2×C3×C5C2×#13511yescharacteristic
302C2×C3×C5C2×#13711yescharacteristic
303C3×D5—82no—
304C23×C3—102no—
305C23×C3—202no—
306C23×C3—202no—
307C23×C3—202no—
308C23×C3D4×D511yescharacteristic
309C3×Q8—22no—
310C3×Q8—24no—
311C3×Q8—22no—
312C3×Q8—22no—
313C3×Q8—42no—
314C3×Q8C2×C22⋊D514yes—
315C3×D4—102no—
316C3×D4—102no—
317C3×D4—202no—
318C2×C3×C4—22no—
319C2×C3×C4—24no—
320C2×C3×C4—22no—
321C2×C3×C4—42no—
322C2×C3×C4—52no—
323C2×C3×C4—52no—
324C2×C3×C4—102no—
325C2×C3×C4—102no—
326C2×C3×C4—102no—
327C2×C3×C4—202no—
328C2×C3×C4D4×D511yescharacteristic
329C2×C3×C4#71112yes—
330C2×C3×C4C2×C22⋊D512yes—
331C2×C3×C4C23×D511yescharacteristic
332C3×C8—22no—
333C3×C8—102no—
334C22×C5C3×#12112yes—
335C22×C5C3×#13512yes—
336C22×C5C3×#13712yes—
337C22×C5C22×C3×D411yescharacteristic
338C2×D5—42no—
339C2×D5—42no—
340C2×D5—42no—
341C2×D5—84no—
342C4×C5—24no—
343C4×C5—42no—
344C4×C5C3×#12112yes—
345C4×C5C22×C3×D411yescharacteristic
346C4×C5C2×C3×#4111yescharacteristic
347Dic5—22no—
348Dic5—22no—
349Dic5—42no—
350C24—101no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8C3×C22⋊D514yes—
357C2×Q8C22×C3×D512yes—
358C2×Q8C22×C3×D512yes—
359C2×D4—52no—
360C2×D4—101no—
361C2×D4—102no—
362C2×D4—102no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—51no—
366C22×C4—51no—
367C22×C4—101no—
368C22×C4C22×C3×D511yescharacteristic
369Q16—28no—
370SD16—108no—
371C2×C8—22no—
372C2×C8—52no—
373C2×C8—102no—
374C2×C8C22×C3×D512yes—
375C4⋊C4—52no—
376C4⋊C4—104no—
377C4⋊C4—101no—
378C22⋊C4—104no—
379C22⋊C4—104no—
380C42—52no—
381C42—52no—
382C42—104no—
383C3×C5C2×#46411yescharacteristic, complemented, Hall
384C22×C3—202no—
385C22×C3—202no—
386C22×C3—202no—
387C22×C3—404no—
388C22×C3#406412yes—
389C22×C3#406512yes—
390C22×C3#408512yes—
391C22×C3C2×D4×D511yescharacteristic
392C3×C4—24no—
393C3×C4—42no—
394C3×C4—102no—
395C3×C4—102no—
396C3×C4—202no—
397C3×C4#408512yes—
398C3×C4C2×#71111yescharacteristic
399C3×C4C22×C22⋊D511yescharacteristic
400C2×C5C3×#46414yescomplemented
401C2×C5C2×C3×#12111yescharacteristic
402C2×C5C2×C3×#13511yescharacteristic
403C2×C5C2×C3×#13711yescharacteristic
404D5—82no—
405C23—102no—
406C23—202no—
407C23—202no—
408C23—202no—
409C23C3×D4×D511yescharacteristic
410Q8—24no—
411Q8—22no—
412Q8—22no—
413Q8—22no—
414Q8—42no—
415Q8C2×C3×C22⋊D514yes—
416D4—102no—
417D4—102no—
418D4—202no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—42no—
423C2×C4—52no—
424C2×C4—52no—
425C2×C4—102no—
426C2×C4—102no—
427C2×C4—102no—
428C2×C4—202no—
429C2×C4C3×D4×D511yescharacteristic
430C2×C4C3×#71112yes—
431C2×C4C2×C3×C22⋊D512yes—
432C2×C4C23×C3×D511yescharacteristic
433C8—22no—
434C8—102no—
435C2×C3—402no—
436C2×C3#6547414yesdirect product factor
437C2×C3C2×#406411yescharacteristic
438C2×C3C2×#406511yescharacteristic
439C2×C3C2×#408511yescharacteristic
440C5C2×C3×#46411yescharacteristic, complemented, Sylow
441C22—202no—
442C22—202no—
443C22—202no—
444C22—404no—
445C22C3×#406412yes—
446C22C3×#406512yes—
447C22C3×#408512yes—
448C22C2×C3×D4×D511yescharacteristic
449C4—24no—
450C4—42no—
451C4—102no—
452C4—102no—
453C4—202no—
454C4C3×#408512yes—
455C4C2×C3×#71111yescharacteristic
456C4C22×C3×C22⋊D511yescharacteristic
457C3C2×#6547411yescharacteristic, direct product factor, Sylow
458C2—402no—
459C2C3×#6547414yesdirect product factor
460C2C2×C3×#406411yescharacteristic
461C2C2×C3×#406511yescharacteristic
462C2C2×C3×#408511yescharacteristic
subgroupG/H
centerC23×C3D4×D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC22×C3×C5×Q16C2
socleC23×C3×C5C2×D4
prime radicalC22×C3×D20C22
hypercenterC22×C3×Q16D5
hypocenterC5C2×C3×#464
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C22×C3×Q16

Complements

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

NH ≅ G/Ntype
1. C22×C3×C5×Q16C2semidirect
3. C2×C3×Q8×Dic5C2semidirect
5. C2×C3×#3967C2semidirect
6. C3×#65474C2direct (N × H)
8. C2×C3×#3946C2semidirect
9. C2×#65474C3direct (N × H)
14. C2×C3×C5×Q16C22semidirect
19. C3×Q8×Dic5C22semidirect
29. C3×#3967C22semidirect
31. C3×#3946C22semidirect
33. C22×C5×Q16C2×C3semidirect
35. C2×Q8×Dic5C2×C3semidirect
37. C2×#3967C2×C3semidirect
38. #65474C2×C3direct (N × H)
40. C2×#3946C2×C3semidirect
74. C22×C3×Q16D5semidirect
86. C2×C5×Q16C22×C3semidirect
91. Q8×Dic5C22×C3semidirect
101. #3967C22×C3semidirect
103. #3946C22×C3semidirect
139. C2×C3×Q16C2×D5semidirect
190. C22×Q16C3×D5semidirect
281. C2×Q16C2×C3×D5semidirect
299. C2×C3×C5#464semidirect
383. C3×C5C2×#464semidirect
400. C2×C5C3×#464semidirect
436. C2×C3#65474direct (N × H)
440. C5C2×C3×#464semidirect
457. C3C2×#65474direct (N × H)
459. C2C3×#65474direct (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×Q162
C2×C3×#40932
C2×C3×Q8×Dic52
C22×C3×#6882
C2×C3×#39672
C3×#654742
C2×C3×#39532
C2×C3×#39462
C2×#654743
C2×C3×#4645