Finite Group Databasebeta

← C2×C3×#65346 #14425921

Subgroups — C2×C3×#65346 #14425921

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
1C2×C3×C5×#124C211yesmaximal, characteristic, complemented
2C22×C3×#688C211yesmaximal, characteristic, complemented
3C22×C3×#686C211yesmaximal, characteristic
4C2×C3×C4×D20C211yesmaximal, characteristic, complemented
5C2×C3×#3967C211yesmaximal, characteristic, complemented
6C2×C3×#3964C211yesmaximal, characteristic, complemented
7C3×#65346C218yesmaximal, direct product factor
8C2×C3×C4⋊#168C211yesmaximal, characteristic, complemented
9C2×#65346C311yesmaximal, characteristic, direct product factor, Hall
10C22×C3×C5×Q8C2211yescharacteristic
11C22×C3×C5×D4C2211yescharacteristic
12C22×C3×D20C2211yescharacteristic
13C22×C3×C4×D5—21no—
14C3×C5×#124—22no—
15C3×C5×#124C2214yescomplemented
16C2×C3×C5×C22⋊C4—21no—
17C2×C3×C5×C22⋊C4—21no—
18C2×C3×C42×C5C2211yescharacteristic
19C2×C3×#688—24no—
20C2×C3×#688C2214yescomplemented
21C2×C3×#686—24no—
22C2×C3×#686C2214yes—
23C2×C3×#685—21no—
24C2×C3×C4⋊Dic5C2211yescharacteristic
25C22×C3×#168C2211yescharacteristic
26C22×C3×#168C2211yescharacteristic
27C3×C4×D20—22no—
28C3×C4×D20C2214yescomplemented
29C3×#3967C2214yescomplemented
30C3×#3964C2214yescomplemented
31C3×C4⋊#168C2214yescomplemented
32C2×C3×#463—51nomaximal, Hall
33C2×C5×#124C2×C311yescharacteristic, complemented
34C22×#688C2×C311yescharacteristic, complemented
35C22×#686C2×C311yescharacteristic
36C2×C4×D20C2×C311yescharacteristic, complemented
37C2×#3967C2×C311yescharacteristic, complemented
38C2×#3964C2×C311yescharacteristic, complemented
39#65346C2×C318yesdirect product factor
40C2×C4⋊#168C2×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—22no—
48C2×C3×C5×D4—21no—
49C2×C3×C5×D4—22no—
50C2×C3×C5×D4C2312yes—
51C22×C3×C4×C5—21no—
52C22×C3×C4×C5D411yescharacteristic
53C22×C3×C4×C5D411yescharacteristic
54C22×C3×C4×C5C2311yescharacteristic
55C22×C3×Dic5—21no—
56C2×C3×D20—21no—
57C2×C3×D20—22no—
58C2×C3×D20D414yes—
59C2×C3×D20C2312yes—
60C2×C3×C4×D5—24no—
61C2×C3×C4×D5—44no—
62C3×C5×C22⋊C4—24no—
63C3×C5×C22⋊C4—24no—
64C3×C42×C5D412yes—
65C3×C42×C5C2312yes—
66C3×#688—28no—
67C3×#686—28no—
68C3×#685—24no—
69C3×C4⋊Dic5—21no—
70C3×C4⋊Dic5C2312yes—
71C2×C3×#168—22no—
72C2×C3×#168—22no—
73C2×C3×#168C2312yes—
74C2×C3×#168C2312yes—
75C22×C3×SD16—51no—
76C22×C3×D8—51no—
77C2×C3×#124D511yescharacteristic, complemented
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
85C22×D20C22×C311yescharacteristic
86C22×C4×D5—21no—
87C5×#124—22no—
88C5×#124C22×C314yescomplemented
89C2×C5×C22⋊C4—21no—
90C2×C5×C22⋊C4—21no—
91C2×C42×C5C22×C311yescharacteristic
92C2×#688—24no—
93C2×#688C22×C314yescomplemented
94C2×#686—24no—
95C2×#686C22×C314yes—
96C2×#685—21no—
97C2×C4⋊Dic5C22×C311yescharacteristic
98C22×#168C22×C311yescharacteristic
99C22×#168C22×C311yescharacteristic
100C4×D20—22no—
101C4×D20C22×C314yescomplemented
102#3967C22×C314yescomplemented
103#3964C22×C314yescomplemented
104C4⋊#168C22×C314yescomplemented
105C2×#463—51noSylow
106C23×C3×C5—22no—
107C23×C3×C5—42no—
108C23×C3×C5—42no—
109C23×C3×C5—42no—
110C23×C3×C5C2×D411yescharacteristic
111C22×C3×D5—24no—
112C22×C3×D5—22no—
113C22×C3×D5—42no—
114C22×C3×D5—42no—
115C3×C5×Q8—22no—
116C3×C5×Q8—22no—
117C3×C5×Q8—42no—
118C3×C5×D4—22no—
119C3×C5×D4—22no—
120C3×C5×D4—42no—
121C2×C3×C4×C5—22no—
122C2×C3×C4×C5—22no—
123C2×C3×C4×C5—22no—
124C2×C3×C4×C5—42no—
125C2×C3×C4×C5C2×D412yes—
126C2×C3×C4×C5C2×D412yes—
127C2×C3×C4×C5C2×D411yescharacteristic
128C2×C3×C4×C5C2×D412yes—
129C2×C3×C4×C5#4112yes—
130C2×C3×C4×C5C2411yescharacteristic
131C2×C3×Dic5—22no—
132C2×C3×Dic5—42no—
133C3×D20—24no—
134C3×D20—22no—
135C3×D20C2×D414yes—
136C3×C4×D5—48no—
137C3×#168—22no—
138C3×#168—22no—
139C22×C3×Q8C2×D511yescharacteristic
140C22×C3×D4—51no—
141C22×C3×D4C2×D511yescharacteristic
142C23×C3×C4—101no—
143C2×C3×SD16—54no—
144C2×C3×SD16—104no—
145C2×C3×D8—54no—
146C2×C3×D8—104no—
147C22×C3×C8—51no—
148C22×C3×C8—51no—
149C3×#124—22no—
150C3×#124C2×D514yescomplemented
151C3×C4×D4—54no—
152C3×C4×D4—102no—
153C2×C3×C4⋊C4—51no—
154C2×C3×C22⋊C4—21no—
155C2×C3×C22⋊C4—21no—
156C2×C3×C22⋊C4—101no—
157C2×C3×C42C2×D511yescharacteristic
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—22no—
168C2×C5×D4—21no—
169C2×C5×D4—22no—
170C2×C5×D4C23×C312yes—
171C22×C4×C5—21no—
172C22×C4×C5C3×D411yescharacteristic
173C22×C4×C5C3×D411yescharacteristic
174C22×C4×C5C23×C311yescharacteristic
175C22×Dic5—21no—
176C2×D20—21no—
177C2×D20—22no—
178C2×D20C3×D414yes—
179C2×D20C23×C312yes—
180C2×C4×D5—24no—
181C2×C4×D5—44no—
182C5×C22⋊C4—24no—
183C5×C22⋊C4—24no—
184C42×C5C3×D412yes—
185C42×C5C23×C312yes—
186#688—28no—
187#686—28no—
188#685—24no—
189C4⋊Dic5—21no—
190C4⋊Dic5C23×C312yes—
191C2×#168—22no—
192C2×#168—22no—
193C2×#168C23×C312yes—
194C2×#168C23×C312yes—
195C22×SD16—51no—
196C22×D8—51no—
197C2×#124C3×D511yescharacteristic, complemented
198C2×C4×D4—51no—
199#463—58no—
200C2×C4⋊C8—51no—
201C2×#103—51no—
202C2×#102—51no—
203C22×C3×C5—42no—
204C22×C3×C5—42no—
205C22×C3×C5—42no—
206C22×C3×C5—84no—
207C22×C3×C5#12112yes—
208C22×C3×C5#13512yes—
209C22×C3×C5AGL(1,ℤ/8)12yes—
210C22×C3×C5C22×D411yescharacteristic
211C2×C3×D5—24no—
212C2×C3×D5—42no—
213C2×C3×D5—42no—
214C2×C3×D5—48no—
215C3×C4×C5—22no—
216C3×C4×C5—22no—
217C3×C4×C5—42no—
218C3×C4×C5#12112yes—
219C3×C4×C5C22×D411yescharacteristic
220C3×C4×C5C2×#4111yescharacteristic
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—22no—
229C2×C3×D4—22no—
230C2×C3×D4—21no—
231C2×C3×D4—54no—
232C2×C3×D4—52no—
233C2×C3×D4—102no—
234C2×C3×D4—101no—
235C2×C3×D4C22×D512yes—
236C22×C3×C4—21no—
237C22×C3×C4—101no—
238C22×C3×C4—104no—
239C22×C3×C4—204no—
240C22×C3×C4C22⋊D511yescharacteristic
241C22×C3×C4C22⋊D511yescharacteristic
242C22×C3×C4C22×D511yescharacteristic
243C3×SD16—108no—
244C3×D8—108no—
245C2×C3×C8—52no—
246C2×C3×C8—52no—
247C2×C3×C8—102no—
248C2×C3×C8—102no—
249C3×C4⋊C4—52no—
250C3×C4⋊C4—101no—
251C3×C22⋊C4—24no—
252C3×C22⋊C4—24no—
253C3×C22⋊C4—104no—
254C3×C42C22⋊D512yes—
255C3×C42C22×D512yes—
256C23×C5—22no—
257C23×C5—42no—
258C23×C5—42no—
259C23×C5—42no—
260C23×C5C2×C3×D411yescharacteristic
261C22×D5—24no—
262C22×D5—22no—
263C22×D5—42no—
264C22×D5—42no—
265C5×Q8—22no—
266C5×Q8—22no—
267C5×Q8—42no—
268C5×D4—22no—
269C5×D4—22no—
270C5×D4—42no—
271C2×C4×C5—22no—
272C2×C4×C5—22no—
273C2×C4×C5—22no—
274C2×C4×C5—42no—
275C2×C4×C5C2×C3×D412yes—
276C2×C4×C5C2×C3×D412yes—
277C2×C4×C5C2×C3×D412yes—
278C2×C4×C5C2×C3×D411yescharacteristic
279C2×C4×C5C3×#4112yes—
280C2×C4×C5C24×C311yescharacteristic
281C2×Dic5—22no—
282C2×Dic5—42no—
283D20—24no—
284D20—22no—
285D20C2×C3×D414yes—
286C4×D5—48no—
287#168—22no—
288#168—22no—
289C22×Q8C2×C3×D511yescharacteristic
290C22×D4—51no—
291C22×D4C2×C3×D511yescharacteristic
292C23×C4—101no—
293C2×SD16—54no—
294C2×SD16—104no—
295C2×D8—54no—
296C2×D8—104no—
297C22×C8—51no—
298C22×C8—51no—
299#124—22no—
300#124C2×C3×D514yescomplemented
301C4×D4—54no—
302C4×D4—102no—
303C2×C4⋊C4—51no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—101no—
307C2×C42C2×C3×D511yescharacteristic
308C4⋊C8—54no—
309#103—54no—
310#102—54no—
311C2×C3×C5—82no—
312C2×C3×C5#46314yescomplemented
313C2×C3×C5C2×#12111yescharacteristic
314C2×C3×C5C2×#13511yescharacteristic
315C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
316C3×D5—44no—
317C23×C3—22no—
318C23×C3—42no—
319C23×C3—42no—
320C23×C3—42no—
321C23×C3—102no—
322C23×C3—104no—
323C23×C3—202no—
324C23×C3—202no—
325C23×C3C2×C22⋊D511yescharacteristic
326C3×Q8—22no—
327C3×Q8—22no—
328C3×Q8—42no—
329C3×D4—22no—
330C3×D4—22no—
331C3×D4—42no—
332C3×D4—54no—
333C3×D4—102no—
334C3×D4—104no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—42no—
339C2×C3×C4—102no—
340C2×C3×C4—208no—
341C2×C3×C4—202no—
342C2×C3×C4D4×D512yes—
343C2×C3×C4#71112yes—
344C2×C3×C4C2×C22⋊D512yes—
345C2×C3×C4C2×C22⋊D512yes—
346C2×C3×C4C2×C22⋊D511yescharacteristic
347C2×C3×C4C23×D511yescharacteristic
348C3×C8—102no—
349C3×C8—102no—
350C22×C5—42no—
351C22×C5—42no—
352C22×C5—42no—
353C22×C5—84no—
354C22×C5C3×#12112yes—
355C22×C5C3×#13512yes—
356C22×C5C3×AGL(1,ℤ/8)12yes—
357C22×C5C22×C3×D411yescharacteristic
358C2×D5—24no—
359C2×D5—42no—
360C2×D5—42no—
361C2×D5—48no—
362C4×C5—22no—
363C4×C5—22no—
364C4×C5—42no—
365C4×C5C3×#12112yes—
366C4×C5C22×C3×D411yescharacteristic
367C4×C5C2×C3×#4111yescharacteristic
368Dic5—42no—
369C24—21no—
370C24—101no—
371C2×Q8—21no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8C22×C3×D512yes—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—21no—
378C2×D4—54no—
379C2×D4—52no—
380C2×D4—102no—
381C2×D4—101no—
382C2×D4C22×C3×D512yes—
383C22×C4—21no—
384C22×C4—101no—
385C22×C4—104no—
386C22×C4—204no—
387C22×C4C3×C22⋊D511yescharacteristic
388C22×C4C3×C22⋊D511yescharacteristic
389C22×C4C22×C3×D511yescharacteristic
390SD16—108no—
391D8—108no—
392C2×C8—52no—
393C2×C8—52no—
394C2×C8—102no—
395C2×C8—102no—
396C4⋊C4—52no—
397C4⋊C4—101no—
398C22⋊C4—24no—
399C22⋊C4—24no—
400C22⋊C4—104no—
401C42C3×C22⋊D512yes—
402C42C22×C3×D512yes—
403C3×C5C2×#46311yescharacteristic, complemented, Hall
404C22×C3—42no—
405C22×C3—42no—
406C22×C3—42no—
407C22×C3—84no—
408C22×C3—104no—
409C22×C3—208no—
410C22×C3—202no—
411C22×C3—202no—
412C22×C3#408412yes—
413C22×C3#409512yes—
414C22×C3#409612yes—
415C22×C3C22×C22⋊D511yescharacteristic
416C3×C4—22no—
417C3×C4—22no—
418C3×C4—42no—
419C3×C4—202no—
420C3×C4#408412yes—
421C3×C4C2×D4×D511yescharacteristic
422C3×C4C2×#71111yescharacteristic
423C2×C5—82no—
424C2×C5C3×#46314yescomplemented
425C2×C5C2×C3×#12111yescharacteristic
426C2×C5C2×C3×#13511yescharacteristic
427C2×C5C2×C3×AGL(1,ℤ/8)11yescharacteristic
428D5—44no—
429C23—22no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—104no—
434C23—102no—
435C23—202no—
436C23—202no—
437C23C2×C3×C22⋊D511yescharacteristic
438Q8—22no—
439Q8—22no—
440Q8—42no—
441D4—22no—
442D4—22no—
443D4—42no—
444D4—54no—
445D4—104no—
446D4—102no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—42no—
451C2×C4—102no—
452C2×C4—208no—
453C2×C4—202no—
454C2×C4C3×D4×D512yes—
455C2×C4C3×#71112yes—
456C2×C4C2×C3×C22⋊D512yes—
457C2×C4C2×C3×C22⋊D512yes—
458C2×C4C2×C3×C22⋊D511yescharacteristic
459C2×C4C23×C3×D511yescharacteristic
460C8—102no—
461C8—102no—
462C2×C3—82no—
463C2×C3—204no—
464C2×C3#6534614yesdirect product factor
465C2×C3C2×#408411yescharacteristic
466C2×C3C2×#409511yescharacteristic
467C2×C3C2×#409611yescharacteristic
468C5C2×C3×#46311yescharacteristic, complemented, Sylow
469C22—42no—
470C22—42no—
471C22—42no—
472C22—84no—
473C22—104no—
474C22—202no—
475C22—202no—
476C22—208no—
477C22C3×#408412yes—
478C22C3×#409512yes—
479C22C3×#409612yes—
480C22C22×C3×C22⋊D511yescharacteristic
481C4—22no—
482C4—22no—
483C4—42no—
484C4—202no—
485C4C3×#408412yes—
486C4C2×C3×D4×D511yescharacteristic
487C4C2×C3×#71111yescharacteristic
488C3C2×#6534611yescharacteristic, direct product factor, Sylow
489C2—82no—
490C2—204no—
491C2C3×#6534614yesdirect product factor
492C2C2×C3×#408411yescharacteristic
493C2C2×C3×#409511yescharacteristic
494C2C2×C3×#409611yescharacteristic
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×#686C2
hypercenterC2×C3×#124D5
hypocenterC5C2×C3×#463
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
1. C2×C3×C5×#124C2semidirect
2. C22×C3×#688C2semidirect
4. C2×C3×C4×D20C2semidirect
5. C2×C3×#3967C2semidirect
6. C2×C3×#3964C2semidirect
7. C3×#65346C2direct (N × H)
8. C2×C3×C4⋊#168C2semidirect
9. C2×#65346C3direct (N × H)
15. C3×C5×#124C22semidirect
20. C2×C3×#688C22semidirect
28. C3×C4×D20C22semidirect
29. C3×#3967C22semidirect
30. C3×#3964C22semidirect
31. C3×C4⋊#168C22semidirect
33. C2×C5×#124C2×C3semidirect
34. C22×#688C2×C3semidirect
36. C2×C4×D20C2×C3semidirect
37. C2×#3967C2×C3semidirect
38. C2×#3964C2×C3semidirect
39. #65346C2×C3direct (N × H)
40. C2×C4⋊#168C2×C3semidirect
77. C2×C3×#124D5semidirect
88. C5×#124C22×C3semidirect
93. C2×#688C22×C3semidirect
101. C4×D20C22×C3semidirect
102. #3967C22×C3semidirect
103. #3964C22×C3semidirect
104. C4⋊#168C22×C3semidirect
150. C3×#124C2×D5semidirect
197. C2×#124C3×D5semidirect
300. #124C2×C3×D5semidirect
312. C2×C3×C5#463semidirect
403. C3×C5C2×#463semidirect
424. C2×C5C3×#463semidirect
464. C2×C3#65346direct (N × H)
468. C5C2×C3×#463semidirect
488. C3C2×#65346direct (N × H)
491. C2C3×#65346direct (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×#6882
C22×C3×#6862
C2×C3×C4×D202
C2×C3×#39672
C2×C3×#39642
C3×#653462
C2×C3×C4⋊#1682
C2×#653463
C2×C3×#4635