Finite Group Databasebeta

← C2×C3×#65107 #14425870

Subgroups — C2×C3×#65107 #14425870

Order 1920.

1088 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4090C211yesmaximal, characteristic, complemented
2C22×C3×#688C211yesmaximal, characteristic
3C22×C3×#677C211yesmaximal, characteristic
4C2×C3×#4039C211yesmaximal, characteristic
5C2×C3×C5×#103C211yesmaximal, characteristic, complemented
6C2×C3×#3946C211yesmaximal, characteristic, complemented
7C2×C3×#3942C211yesmaximal, characteristic, complemented
8C3×#65107C218yesmaximal, direct product factor
9C2×#65107C311yesmaximal, characteristic, direct product factor, Hall
10C22×C3×C5×Q8C2211yescharacteristic
11C22×C3×D20C2211yescharacteristic
12C22×C3×C4×D5—21no—
13C22×C3×Dic10C2211yescharacteristic
14C22×C3×C5×C8C2211yescharacteristic
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C3×#4090—22no—
17C3×#4090C2214yescomplemented
18C2×C3×#688—24no—
19C2×C3×#688C2214yes—
20C2×C3×#685—21no—
21C2×C3×#683—21no—
22C2×C3×#683—21no—
23C2×C3×C4×Dic5C2211yescharacteristic
24C22×C3×#168C2211yescharacteristic
25C2×C3×#677—24no—
26C2×C3×#677C2214yes—
27C3×#4039—22no—
28C3×#4039C2214yes—
29C3×C5×#103C2214yescomplemented
30C3×#3946C2214yescomplemented
31C3×#3942C2214yescomplemented
32C2×C3×#461—51nomaximal, Hall
33C2×#4090C2×C311yescharacteristic, complemented
34C22×#688C2×C311yescharacteristic
35C22×#677C2×C311yescharacteristic
36C2×#4039C2×C311yescharacteristic
37C2×C5×#103C2×C311yescharacteristic, complemented
38C2×#3946C2×C311yescharacteristic, complemented
39C2×#3942C2×C311yescharacteristic, complemented
40#65107C2×C318yesdirect product factor
41C23×C3×D5—21no—
42C2×C3×C5×Q8—21no—
43C2×C3×C5×Q8—22no—
44C2×C3×C5×Q8—22no—
45C2×C3×C5×Q8C2312yes—
46C22×C3×C4×C5—21no—
47C22×C3×C4×C5—21no—
48C22×C3×C4×C5C2311yescharacteristic
49C22×C3×Dic5—21no—
50C22×C3×Dic5D411yescharacteristic
51C22×C3×Dic5D411yescharacteristic
52C2×C3×D20—21no—
53C2×C3×D20—22no—
54C2×C3×D20D414yes—
55C2×C3×D20C2312yes—
56C2×C3×C4×D5—24no—
57C2×C3×C4×D5—44no—
58C2×C3×Dic10—22no—
59C2×C3×Dic10—22no—
60C2×C3×Dic10—21no—
61C2×C3×Dic10C2312yes—
62C2×C3×C5×C8—22no—
63C2×C3×C5×C8C2312yes—
64C3×C5×C4⋊C4—21no—
65C3×C5×C4⋊C4C2312yes—
66C3×#688—28no—
67C3×#685—24no—
68C3×#683—24no—
69C3×#683—24no—
70C3×C4×Dic5D412yes—
71C3×C4×Dic5C2312yes—
72C2×C3×#168—22no—
73C2×C3×#168C2312yes—
74C3×#677—28no—
75C22×C3×SD16—51no—
76C22×C3×SD16—51no—
77C2×C3×C4⋊Q8—51no—
78C2×C3×C4×D4—51no—
79C3×#461—58no—
80C2×C3×C4⋊C8—51no—
81C2×C3×#103—51no—
82C2×C3×#103D511yescharacteristic, complemented
83C22×C5×Q8C22×C311yescharacteristic
84C22×D20C22×C311yescharacteristic
85C22×C4×D5—21no—
86C22×Dic10C22×C311yescharacteristic
87C22×C5×C8C22×C311yescharacteristic
88C2×C5×C4⋊C4C22×C311yescharacteristic
89#4090—22no—
90#4090C22×C314yescomplemented
91C2×#688—24no—
92C2×#688C22×C314yes—
93C2×#685—21no—
94C2×#683—21no—
95C2×#683—21no—
96C2×C4×Dic5C22×C311yescharacteristic
97C22×#168C22×C311yescharacteristic
98C2×#677—24no—
99C2×#677C22×C314yes—
100#4039—22no—
101#4039C22×C314yes—
102C5×#103C22×C314yescomplemented
103#3946C22×C314yescomplemented
104#3942C22×C314yescomplemented
105C2×#461—51noSylow
106C23×C3×C5C2×D411yescharacteristic
107C22×C3×D5—24no—
108C22×C3×D5—22no—
109C22×C3×D5—42no—
110C22×C3×D5—42no—
111C3×C5×Q8—22no—
112C3×C5×Q8—22no—
113C3×C5×Q8—42no—
114C2×C3×C4×C5—22no—
115C2×C3×C4×C5—22no—
116C2×C3×C4×C5—42no—
117C2×C3×C4×C5—42no—
118C2×C3×C4×C5C2×D411yescharacteristic
119C2×C3×C4×C5C2×D412yes—
120C2×C3×C4×C5#4112yes—
121C2×C3×C4×C5C2411yescharacteristic
122C2×C3×Dic5—22no—
123C2×C3×Dic5—22no—
124C2×C3×Dic5—42no—
125C2×C3×Dic5SD1614yescomplemented
126C2×C3×Dic5C2×D412yes—
127C2×C3×Dic5C2×D412yes—
128C3×D20—24no—
129C3×D20—22no—
130C3×D20C2×D414yes—
131C3×C4×D5—48no—
132C3×Dic10—22no—
133C3×Dic10—22no—
134C3×Dic10—42no—
135C3×C5×C8—22no—
136C3×#168—22no—
137C22×C3×Q8—51no—
138C22×C3×Q8C2×D511yescharacteristic
139C22×C3×D4—51no—
140C23×C3×C4—101no—
141C2×C3×SD16—54no—
142C2×C3×SD16—54no—
143C2×C3×SD16—104no—
144C2×C3×SD16—104no—
145C22×C3×C8—51no—
146C22×C3×C8C2×D511yescharacteristic
147C3×C4⋊Q8—54no—
148C3×C4⋊Q8—102no—
149C3×C4×D4—54no—
150C3×C4×D4—102no—
151C2×C3×C4⋊C4—101no—
152C2×C3×C4⋊C4—101no—
153C2×C3×C4⋊C4C2×D511yescharacteristic
154C2×C3×C22⋊C4—101no—
155C2×C3×C42—51no—
156C3×C4⋊C8—54no—
157C3×#103—54no—
158C3×#103C2×D514yescomplemented
159C23×D5—21no—
160C2×C5×Q8—21no—
161C2×C5×Q8—22no—
162C2×C5×Q8—22no—
163C2×C5×Q8C23×C312yes—
164C22×C4×C5—21no—
165C22×C4×C5—21no—
166C22×C4×C5C23×C311yescharacteristic
167C22×Dic5—21no—
168C22×Dic5C3×D411yescharacteristic
169C22×Dic5C3×D411yescharacteristic
170C2×D20—21no—
171C2×D20—22no—
172C2×D20C3×D414yes—
173C2×D20C23×C312yes—
174C2×C4×D5—24no—
175C2×C4×D5—44no—
176C2×Dic10—22no—
177C2×Dic10—22no—
178C2×Dic10—21no—
179C2×Dic10C23×C312yes—
180C2×C5×C8—22no—
181C2×C5×C8C23×C312yes—
182C5×C4⋊C4—21no—
183C5×C4⋊C4C23×C312yes—
184#688—28no—
185#685—24no—
186#683—24no—
187#683—24no—
188C4×Dic5C3×D412yes—
189C4×Dic5C23×C312yes—
190C2×#168—22no—
191C2×#168C23×C312yes—
192#677—28no—
193C22×SD16—51no—
194C22×SD16—51no—
195C2×C4⋊Q8—51no—
196C2×C4×D4—51no—
197#461—58no—
198C2×C4⋊C8—51no—
199C2×#103—51no—
200C2×#103C3×D511yescharacteristic, complemented
201C22×C3×C5#12112yes—
202C22×C3×C5C2×SD1612yes—
203C22×C3×C5#13712yes—
204C22×C3×C5C22×D411yescharacteristic
205C2×C3×D5—24no—
206C2×C3×D5—42no—
207C2×C3×D5—42no—
208C2×C3×D5—48no—
209C3×C4×C5—42no—
210C3×C4×C5—42no—
211C3×C4×C5#12112yes—
212C3×C4×C5C22×D411yescharacteristic
213C3×C4×C5C2×#4111yescharacteristic
214C3×Dic5—22no—
215C3×Dic5—42no—
216C3×Dic5C2×SD1614yescomplemented
217C24×C3—101no—
218C2×C3×Q8—21no—
219C2×C3×Q8—22no—
220C2×C3×Q8—22no—
221C2×C3×Q8—52no—
222C2×C3×Q8—101no—
223C2×C3×Q8—102no—
224C2×C3×Q8—102no—
225C2×C3×Q8C22×D512yes—
226C2×C3×D4—54no—
227C2×C3×D4—52no—
228C2×C3×D4—102no—
229C2×C3×D4—101no—
230C22×C3×C4—21no—
231C22×C3×C4—21no—
232C22×C3×C4—51no—
233C22×C3×C4—51no—
234C22×C3×C4—104no—
235C22×C3×C4—101no—
236C22×C3×C4—204no—
237C22×C3×C4C22×D511yescharacteristic
238C3×SD16—108no—
239C3×SD16—108no—
240C2×C3×C8—22no—
241C2×C3×C8—52no—
242C2×C3×C8—102no—
243C2×C3×C8C22×D512yes—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—104no—
246C3×C4⋊C4—104no—
247C3×C4⋊C4C22×D512yes—
248C3×C22⋊C4—104no—
249C3×C42—52no—
250C3×C42—52no—
251C23×C5C2×C3×D411yescharacteristic
252C22×D5—24no—
253C22×D5—22no—
254C22×D5—42no—
255C22×D5—42no—
256C5×Q8—22no—
257C5×Q8—22no—
258C5×Q8—42no—
259C2×C4×C5—22no—
260C2×C4×C5—22no—
261C2×C4×C5—42no—
262C2×C4×C5—42no—
263C2×C4×C5C2×C3×D411yescharacteristic
264C2×C4×C5C2×C3×D412yes—
265C2×C4×C5C3×#4112yes—
266C2×C4×C5C24×C311yescharacteristic
267C2×Dic5—22no—
268C2×Dic5—22no—
269C2×Dic5—42no—
270C2×Dic5C3×SD1614yescomplemented
271C2×Dic5C2×C3×D412yes—
272C2×Dic5C2×C3×D412yes—
273D20—24no—
274D20—22no—
275D20C2×C3×D414yes—
276C4×D5—48no—
277Dic10—22no—
278Dic10—22no—
279Dic10—42no—
280C5×C8—22no—
281#168—22no—
282C22×Q8—51no—
283C22×Q8C2×C3×D511yescharacteristic
284C22×D4—51no—
285C23×C4—101no—
286C2×SD16—54no—
287C2×SD16—54no—
288C2×SD16—104no—
289C2×SD16—104no—
290C22×C8—51no—
291C22×C8C2×C3×D511yescharacteristic
292C4⋊Q8—54no—
293C4⋊Q8—102no—
294C4×D4—54no—
295C4×D4—102no—
296C2×C4⋊C4—101no—
297C2×C4⋊C4—101no—
298C2×C4⋊C4C2×C3×D511yescharacteristic
299C2×C22⋊C4—101no—
300C2×C42—51no—
301C4⋊C8—54no—
302#103—54no—
303#103C2×C3×D514yescomplemented
304C2×C3×C5#46114yescomplemented
305C2×C3×C5C2×#12111yescharacteristic
306C2×C3×C5C22×SD1611yescharacteristic
307C2×C3×C5C2×#13711yescharacteristic
308C3×D5—44no—
309C23×C3—102no—
310C23×C3—104no—
311C23×C3—202no—
312C23×C3—202no—
313C23×C3D4×D511yescharacteristic
314C3×Q8—22no—
315C3×Q8—22no—
316C3×Q8—42no—
317C3×Q8—102no—
318C3×Q8—102no—
319C3×Q8—202no—
320C3×D4—54no—
321C3×D4—102no—
322C3×D4—104no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—52no—
328C2×C3×C4—54no—
329C2×C3×C4—52no—
330C2×C3×C4—102no—
331C2×C3×C4—102no—
332C2×C3×C4—208no—
333C2×C3×C4—202no—
334C2×C3×C4#70912yes—
335C2×C3×C4D4×D512yes—
336C2×C3×C4D4×D511yescharacteristic
337C2×C3×C4C23×D511yescharacteristic
338C3×C8—22no—
339C3×C8—102no—
340C22×C5C3×#12112yes—
341C22×C5C2×C3×SD1612yes—
342C22×C5C3×#13712yes—
343C22×C5C22×C3×D411yescharacteristic
344C2×D5—24no—
345C2×D5—42no—
346C2×D5—42no—
347C2×D5—48no—
348C4×C5—42no—
349C4×C5—42no—
350C4×C5C3×#12112yes—
351C4×C5C22×C3×D411yescharacteristic
352C4×C5C2×C3×#4111yescharacteristic
353Dic5—22no—
354Dic5—42no—
355Dic5C2×C3×SD1614yescomplemented
356C24—101no—
357C2×Q8—21no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—52no—
361C2×Q8—101no—
362C2×Q8—102no—
363C2×Q8—102no—
364C2×Q8C22×C3×D512yes—
365C2×D4—54no—
366C2×D4—52no—
367C2×D4—102no—
368C2×D4—101no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—51no—
372C22×C4—51no—
373C22×C4—104no—
374C22×C4—101no—
375C22×C4—204no—
376C22×C4C22×C3×D511yescharacteristic
377SD16—108no—
378SD16—108no—
379C2×C8—22no—
380C2×C8—52no—
381C2×C8—102no—
382C2×C8C22×C3×D512yes—
383C4⋊C4—21no—
384C4⋊C4—104no—
385C4⋊C4—104no—
386C4⋊C4C22×C3×D512yes—
387C22⋊C4—104no—
388C42—52no—
389C42—52no—
390C3×C5C2×#46111yescharacteristic, complemented, Hall
391C22×C3—104no—
392C22×C3—202no—
393C22×C3—202no—
394C22×C3—208no—
395C22×C3#403012yes—
396C22×C3D5×SD1612yes—
397C22×C3#406412yes—
398C22×C3C2×D4×D511yescharacteristic
399C3×C4—42no—
400C3×C4—42no—
401C3×C4—54no—
402C3×C4—102no—
403C3×C4—202no—
404C3×C4#403012yes—
405C3×C4C2×#70911yescharacteristic
406C3×C4C2×D4×D511yescharacteristic
407C2×C5C3×#46114yescomplemented
408C2×C5C2×C3×#12111yescharacteristic
409C2×C5C22×C3×SD1611yescharacteristic
410C2×C5C2×C3×#13711yescharacteristic
411D5—44no—
412C23—102no—
413C23—104no—
414C23—202no—
415C23—202no—
416C23C3×D4×D511yescharacteristic
417Q8—22no—
418Q8—22no—
419Q8—42no—
420Q8—102no—
421Q8—102no—
422Q8—202no—
423D4—54no—
424D4—102no—
425D4—104no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—52no—
431C2×C4—54no—
432C2×C4—52no—
433C2×C4—102no—
434C2×C4—102no—
435C2×C4—208no—
436C2×C4—202no—
437C2×C4C3×#70912yes—
438C2×C4C3×D4×D512yes—
439C2×C4C3×D4×D511yescharacteristic
440C2×C4C23×C3×D511yescharacteristic
441C8—22no—
442C8—102no—
443C2×C3—204no—
444C2×C3#6510714yesdirect product factor
445C2×C3C2×#403011yescharacteristic
446C2×C3C2×D5×SD1611yescharacteristic
447C2×C3C2×#406411yescharacteristic
448C5C2×C3×#46111yescharacteristic, complemented, Sylow
449C22—104no—
450C22—208no—
451C22—202no—
452C22—202no—
453C22C3×#403012yes—
454C22C3×D5×SD1612yes—
455C22C3×#406412yes—
456C22C2×C3×D4×D511yescharacteristic
457C4—42no—
458C4—42no—
459C4—54no—
460C4—102no—
461C4—202no—
462C4C3×#403012yes—
463C4C2×C3×#70911yescharacteristic
464C4C2×C3×D4×D511yescharacteristic
465C3C2×#6510711yescharacteristic, direct product factor, Sylow
466C2—204no—
467C2C3×#6510714yesdirect product factor
468C2C2×C3×#403011yescharacteristic
469C2C2×C3×D5×SD1611yescharacteristic
470C2C2×C3×#406411yescharacteristic
subgroupG/H
centerC23×C3D4×D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC2×C3×C5×#103C2
socleC23×C3×C5C2×D4
prime radicalC22×C3×D20C22
hypercenterC2×C3×#103D5
hypocenterC5C2×C3×#461
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C2×C3×#103

Complements

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

NH ≅ G/Ntype
1. C2×C3×#4090C2semidirect
5. C2×C3×C5×#103C2semidirect
6. C2×C3×#3946C2semidirect
7. C2×C3×#3942C2semidirect
8. C3×#65107C2direct (N × H)
9. C2×#65107C3direct (N × H)
17. C3×#4090C22semidirect
29. C3×C5×#103C22semidirect
30. C3×#3946C22semidirect
31. C3×#3942C22semidirect
33. C2×#4090C2×C3semidirect
37. C2×C5×#103C2×C3semidirect
38. C2×#3946C2×C3semidirect
39. C2×#3942C2×C3semidirect
40. #65107C2×C3direct (N × H)
82. C2×C3×#103D5semidirect
90. #4090C22×C3semidirect
102. C5×#103C22×C3semidirect
103. #3946C22×C3semidirect
104. #3942C22×C3semidirect
125. C2×C3×Dic5SD16semidirect
158. C3×#103C2×D5semidirect
200. C2×#103C3×D5semidirect
216. C3×Dic5C2×SD16semidirect
270. C2×Dic5C3×SD16semidirect
303. #103C2×C3×D5semidirect
304. C2×C3×C5#461semidirect
355. Dic5C2×C3×SD16semidirect
390. C3×C5C2×#461semidirect
407. C2×C5C3×#461semidirect
444. C2×C3#65107direct (N × H)
448. C5C2×C3×#461semidirect
465. C3C2×#65107direct (N × H)
467. C2C3×#65107direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#40902
C22×C3×#6882
C22×C3×#6772
C2×C3×#40392
C2×C3×C5×#1032
C2×C3×#39462
C2×C3×#39422
C3×#651072
C2×#651073
C2×C3×#4615