Finite Group Databasebeta

← C2×C3×#65075 #14425813

Subgroups — C2×C3×#65075 #14425813

Order 1920.

1296 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4086C211yesmaximal, characteristic, complemented
2C22×C3×#686C211yesmaximal, characteristic, complemented
3C22×C3×D40C211yesmaximal, characteristic, complemented
4C2×C3×#4039C211yesmaximal, characteristic, complemented
5C2×C3×C5×#102C211yesmaximal, characteristic, complemented
6C2×C3×#3946C211yesmaximal, characteristic, complemented
7C2×C3×#3941C211yesmaximal, characteristic, complemented
8C3×#65075C218yesmaximal, direct product factor
9C2×#65075C311yesmaximal, characteristic, direct product factor, Hall
10C22×C3×C5×D4C2211yescharacteristic
11C22×C3×C22⋊D5—21no—
12C22×C3×C22⋊D5—21no—
13C22×C3×D20C2211yescharacteristic
14C22×C3×D20C2211yescharacteristic
15C22×C3×C4×D5—21no—
16C22×C3×C5×C8C2211yescharacteristic
17C2×C3×C5×C4⋊C4C2211yescharacteristic
18C3×#4086—22no—
19C3×#4086C2214yescomplemented
20C2×C3×#686—24no—
21C2×C3×#686C2214yescomplemented
22C2×C3×#685—21no—
23C2×C3×C4×Dic5C2211yescharacteristic
24C22×C3×#168C2211yescharacteristic
25C2×C3×D40—24no—
26C2×C3×D40C2214yescomplemented
27C3×#4039—22no—
28C3×#4039C2214yescomplemented
29C3×C5×#102C2214yescomplemented
30C3×#3946C2214yescomplemented
31C3×#3941C2214yescomplemented
32C2×C3×#459—51nomaximal, Hall
33C2×#4086C2×C311yescharacteristic, complemented
34C22×#686C2×C311yescharacteristic, complemented
35C22×D40C2×C311yescharacteristic, complemented
36C2×#4039C2×C311yescharacteristic, complemented
37C2×C5×#102C2×C311yescharacteristic, complemented
38C2×#3946C2×C311yescharacteristic, complemented
39C2×#3941C2×C311yescharacteristic, complemented
40#65075C2×C318yesdirect product factor
41C24×C3×C5—21no—
42C23×C3×D5—21no—
43C23×C3×D5—21no—
44C2×C3×C5×D4—22no—
45C2×C3×C5×D4—21no—
46C2×C3×C5×D4—22no—
47C2×C3×C5×D4C2312yes—
48C22×C3×C4×C5—21no—
49C22×C3×C4×C5C2311yescharacteristic
50C2×C3×C22⋊D5—24no—
51C2×C3×C22⋊D5—24no—
52C2×C3×C22⋊D5—44no—
53C2×C3×C22⋊D5—44no—
54C22×C3×Dic5D411yescharacteristic
55C22×C3×Dic5D411yescharacteristic
56C2×C3×D20—21no—
57C2×C3×D20—22no—
58C2×C3×D20—22no—
59C2×C3×D20—22no—
60C2×C3×D20—21no—
61C2×C3×D20D414yescomplemented
62C2×C3×D20C2312yes—
63C2×C3×D20C2312yes—
64C2×C3×C4×D5—24no—
65C2×C3×C4×D5—44no—
66C2×C3×C5×C8—22no—
67C2×C3×C5×C8C2312yes—
68C3×C5×C4⋊C4—21no—
69C3×C5×C4⋊C4C2312yes—
70C3×#686—28no—
71C3×#685—24no—
72C3×C4×Dic5D412yes—
73C3×C4×Dic5C2312yes—
74C2×C3×#168—22no—
75C2×C3×#168C2312yes—
76C3×D40—28no—
77C22×C3×D8—51no—
78C22×C3×D8—51no—
79C2×C3×#127—51no—
80C2×C3×C4×D4—51no—
81C3×#459—58no—
82C2×C3×C4⋊C8—51no—
83C2×C3×#102—51no—
84C2×C3×#102D511yescharacteristic, complemented
85C22×C5×D4C22×C311yescharacteristic
86C22×C22⋊D5—21no—
87C22×C22⋊D5—21no—
88C22×D20C22×C311yescharacteristic
89C22×D20C22×C311yescharacteristic
90C22×C4×D5—21no—
91C22×C5×C8C22×C311yescharacteristic
92C2×C5×C4⋊C4C22×C311yescharacteristic
93#4086—22no—
94#4086C22×C314yescomplemented
95C2×#686—24no—
96C2×#686C22×C314yescomplemented
97C2×#685—21no—
98C2×C4×Dic5C22×C311yescharacteristic
99C22×#168C22×C311yescharacteristic
100C2×D40—24no—
101C2×D40C22×C314yescomplemented
102#4039—22no—
103#4039C22×C314yescomplemented
104C5×#102C22×C314yescomplemented
105#3946C22×C314yescomplemented
106#3941C22×C314yescomplemented
107C2×#459—51noSylow
108C23×C3×C5—22no—
109C23×C3×C5—42no—
110C23×C3×C5—42no—
111C23×C3×C5—42no—
112C23×C3×C5C2×D411yescharacteristic
113C22×C3×D5—24no—
114C22×C3×D5—22no—
115C22×C3×D5—22no—
116C22×C3×D5—42no—
117C22×C3×D5—42no—
118C22×C3×D5—42no—
119C22×C3×D5—42no—
120C22×C3×D5—42no—
121C3×C5×D4—22no—
122C3×C5×D4—22no—
123C3×C5×D4—42no—
124C2×C3×C4×C5—22no—
125C2×C3×C4×C5—42no—
126C2×C3×C4×C5C2×D411yescharacteristic
127C2×C3×C4×C5C2×D412yes—
128C2×C3×C4×C5#4112yes—
129C2×C3×C4×C5C2411yescharacteristic
130C3×C22⋊D5—48no—
131C3×C22⋊D5—48no—
132C2×C3×Dic5—22no—
133C2×C3×Dic5D814yescomplemented
134C2×C3×Dic5C2×D412yes—
135C2×C3×Dic5C2×D412yes—
136C3×D20—24no—
137C3×D20—22no—
138C3×D20—22no—
139C3×D20—22no—
140C3×D20—42no—
141C3×D20C2×D414yescomplemented
142C3×C4×D5—48no—
143C3×C5×C8—22no—
144C3×#168—22no—
145C22×C3×D4—51no—
146C22×C3×D4—51no—
147C22×C3×D4—101no—
148C22×C3×D4—101no—
149C22×C3×D4C2×D511yescharacteristic
150C23×C3×C4—101no—
151C2×C3×D8—54no—
152C2×C3×D8—54no—
153C2×C3×D8—104no—
154C2×C3×D8—104no—
155C22×C3×C8—51no—
156C22×C3×C8C2×D511yescharacteristic
157C3×#127—54no—
158C3×#127—102no—
159C3×C4×D4—54no—
160C3×C4×D4—102no—
161C2×C3×C4⋊C4C2×D511yescharacteristic
162C2×C3×C22⋊C4—101no—
163C2×C3×C42—51no—
164C3×C4⋊C8—54no—
165C3×#102—54no—
166C3×#102C2×D514yescomplemented
167C24×C5—21no—
168C23×D5—21no—
169C23×D5—21no—
170C2×C5×D4—22no—
171C2×C5×D4—21no—
172C2×C5×D4—22no—
173C2×C5×D4C23×C312yes—
174C22×C4×C5—21no—
175C22×C4×C5C23×C311yescharacteristic
176C2×C22⋊D5—24no—
177C2×C22⋊D5—24no—
178C2×C22⋊D5—44no—
179C2×C22⋊D5—44no—
180C22×Dic5C3×D411yescharacteristic
181C22×Dic5C3×D411yescharacteristic
182C2×D20—21no—
183C2×D20—22no—
184C2×D20—22no—
185C2×D20—22no—
186C2×D20—21no—
187C2×D20C3×D414yescomplemented
188C2×D20C23×C312yes—
189C2×D20C23×C312yes—
190C2×C4×D5—24no—
191C2×C4×D5—44no—
192C2×C5×C8—22no—
193C2×C5×C8C23×C312yes—
194C5×C4⋊C4—21no—
195C5×C4⋊C4C23×C312yes—
196#686—28no—
197#685—24no—
198C4×Dic5C3×D412yes—
199C4×Dic5C23×C312yes—
200C2×#168—22no—
201C2×#168C23×C312yes—
202D40—28no—
203C22×D8—51no—
204C22×D8—51no—
205C2×#127—51no—
206C2×C4×D4—51no—
207#459—58no—
208C2×C4⋊C8—51no—
209C2×#102—51no—
210C2×#102C3×D511yescharacteristic, complemented
211C22×C3×C5—42no—
212C22×C3×C5—42no—
213C22×C3×C5—42no—
214C22×C3×C5—84no—
215C22×C3×C5#12112yes—
216C22×C3×C5C2×D812yes—
217C22×C3×C5AGL(1,ℤ/8)12yes—
218C22×C3×C5C22×D411yescharacteristic
219C2×C3×D5—24no—
220C2×C3×D5—42no—
221C2×C3×D5—42no—
222C2×C3×D5—48no—
223C2×C3×D5—42no—
224C2×C3×D5—42no—
225C2×C3×D5—42no—
226C2×C3×D5—84no—
227C3×C4×C5—42no—
228C3×C4×C5#12112yes—
229C3×C4×C5C22×D411yescharacteristic
230C3×C4×C5C2×#4111yescharacteristic
231C3×Dic5—22no—
232C3×Dic5C2×D814yescomplemented
233C24×C3—21no—
234C24×C3—101no—
235C24×C3—101no—
236C2×C3×D4—22no—
237C2×C3×D4—22no—
238C2×C3×D4—21no—
239C2×C3×D4—54no—
240C2×C3×D4—52no—
241C2×C3×D4—52no—
242C2×C3×D4—104no—
243C2×C3×D4—104no—
244C2×C3×D4—102no—
245C2×C3×D4—102no—
246C2×C3×D4—101no—
247C2×C3×D4—102no—
248C2×C3×D4—101no—
249C2×C3×D4—204no—
250C2×C3×D4—204no—
251C2×C3×D4C22×D512yes—
252C22×C3×C4—21no—
253C22×C3×C4—51no—
254C22×C3×C4—51no—
255C22×C3×C4—104no—
256C22×C3×C4—204no—
257C22×C3×C4C22×D511yescharacteristic
258C3×D8—108no—
259C3×D8—108no—
260C2×C3×C8—22no—
261C2×C3×C8—52no—
262C2×C3×C8—102no—
263C2×C3×C8C22×D512yes—
264C3×C4⋊C4—21no—
265C3×C4⋊C4C22×D512yes—
266C3×C22⋊C4—104no—
267C3×C42—52no—
268C3×C42—52no—
269C23×C5—22no—
270C23×C5—42no—
271C23×C5—42no—
272C23×C5—42no—
273C23×C5C2×C3×D411yescharacteristic
274C22×D5—24no—
275C22×D5—22no—
276C22×D5—22no—
277C22×D5—42no—
278C22×D5—42no—
279C22×D5—42no—
280C22×D5—42no—
281C22×D5—42no—
282C5×D4—22no—
283C5×D4—22no—
284C5×D4—42no—
285C2×C4×C5—22no—
286C2×C4×C5—42no—
287C2×C4×C5C2×C3×D411yescharacteristic
288C2×C4×C5C2×C3×D412yes—
289C2×C4×C5C3×#4112yes—
290C2×C4×C5C24×C311yescharacteristic
291C22⋊D5—48no—
292C22⋊D5—48no—
293C2×Dic5—22no—
294C2×Dic5C3×D814yescomplemented
295C2×Dic5C2×C3×D412yes—
296C2×Dic5C2×C3×D412yes—
297D20—24no—
298D20—22no—
299D20—22no—
300D20—22no—
301D20—42no—
302D20C2×C3×D414yescomplemented
303C4×D5—48no—
304C5×C8—22no—
305#168—22no—
306C22×D4—51no—
307C22×D4—51no—
308C22×D4—101no—
309C22×D4—101no—
310C22×D4C2×C3×D511yescharacteristic
311C23×C4—101no—
312C2×D8—54no—
313C2×D8—54no—
314C2×D8—104no—
315C2×D8—104no—
316C22×C8—51no—
317C22×C8C2×C3×D511yescharacteristic
318#127—54no—
319#127—102no—
320C4×D4—54no—
321C4×D4—102no—
322C2×C4⋊C4C2×C3×D511yescharacteristic
323C2×C22⋊C4—101no—
324C2×C42—51no—
325C4⋊C8—54no—
326#102—54no—
327#102C2×C3×D514yescomplemented
328C2×C3×C5—82no—
329C2×C3×C5#45914yescomplemented
330C2×C3×C5C2×#12111yescharacteristic
331C2×C3×C5C22×D811yescharacteristic
332C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
333C3×D5—44no—
334C3×D5—82no—
335C23×C3—22no—
336C23×C3—42no—
337C23×C3—42no—
338C23×C3—42no—
339C23×C3—102no—
340C23×C3—104no—
341C23×C3—102no—
342C23×C3—202no—
343C23×C3—202no—
344C23×C3—202no—
345C23×C3—202no—
346C23×C3—202no—
347C23×C3D4×D511yescharacteristic
348C3×D4—22no—
349C3×D4—22no—
350C3×D4—42no—
351C3×D4—54no—
352C3×D4—102no—
353C3×D4—102no—
354C3×D4—102no—
355C3×D4—104no—
356C3×D4—208no—
357C3×D4—208no—
358C3×D4—202no—
359C2×C3×C4—22no—
360C2×C3×C4—42no—
361C2×C3×C4—52no—
362C2×C3×C4—54no—
363C2×C3×C4—52no—
364C2×C3×C4—102no—
365C2×C3×C4—208no—
366C2×C3×C4#70912yes—
367C2×C3×C4D4×D512yes—
368C2×C3×C4D4×D511yescharacteristic
369C2×C3×C4C23×D511yescharacteristic
370C3×C8—22no—
371C3×C8—102no—
372C22×C5—42no—
373C22×C5—42no—
374C22×C5—42no—
375C22×C5—84no—
376C22×C5C3×#12112yes—
377C22×C5C2×C3×D812yes—
378C22×C5C3×AGL(1,ℤ/8)12yes—
379C22×C5C22×C3×D411yescharacteristic
380C2×D5—24no—
381C2×D5—42no—
382C2×D5—42no—
383C2×D5—48no—
384C2×D5—42no—
385C2×D5—42no—
386C2×D5—42no—
387C2×D5—84no—
388C4×C5—42no—
389C4×C5C3×#12112yes—
390C4×C5C22×C3×D411yescharacteristic
391C4×C5C2×C3×#4111yescharacteristic
392Dic5—22no—
393Dic5C2×C3×D814yescomplemented
394C24—21no—
395C24—101no—
396C24—101no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—21no—
400C2×D4—54no—
401C2×D4—52no—
402C2×D4—52no—
403C2×D4—104no—
404C2×D4—104no—
405C2×D4—102no—
406C2×D4—102no—
407C2×D4—101no—
408C2×D4—102no—
409C2×D4—101no—
410C2×D4—204no—
411C2×D4—204no—
412C2×D4C22×C3×D512yes—
413C22×C4—21no—
414C22×C4—51no—
415C22×C4—51no—
416C22×C4—104no—
417C22×C4—204no—
418C22×C4C22×C3×D511yescharacteristic
419D8—108no—
420D8—108no—
421C2×C8—22no—
422C2×C8—52no—
423C2×C8—102no—
424C2×C8C22×C3×D512yes—
425C4⋊C4—21no—
426C4⋊C4C22×C3×D512yes—
427C22⋊C4—104no—
428C42—52no—
429C42—52no—
430C3×C5C2×#45911yescharacteristic, complemented, Hall
431C22×C3—42no—
432C22×C3—42no—
433C22×C3—42no—
434C22×C3—84no—
435C22×C3—104no—
436C22×C3—202no—
437C22×C3—202no—
438C22×C3—208no—
439C22×C3—202no—
440C22×C3—202no—
441C22×C3—202no—
442C22×C3—404no—
443C22×C3#403012yes—
444C22×C3D5×D812yes—
445C22×C3#406012yes—
446C22×C3C2×D4×D511yescharacteristic
447C3×C4—42no—
448C3×C4—54no—
449C3×C4—102no—
450C3×C4#403012yes—
451C3×C4C2×#70911yescharacteristic
452C3×C4C2×D4×D511yescharacteristic
453C2×C5—82no—
454C2×C5C3×#45914yescomplemented
455C2×C5C2×C3×#12111yescharacteristic
456C2×C5C22×C3×D811yescharacteristic
457C2×C5C2×C3×AGL(1,ℤ/8)11yescharacteristic
458D5—44no—
459D5—82no—
460C23—22no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—102no—
465C23—104no—
466C23—102no—
467C23—202no—
468C23—202no—
469C23—202no—
470C23—202no—
471C23—202no—
472C23C3×D4×D511yescharacteristic
473D4—22no—
474D4—22no—
475D4—42no—
476D4—54no—
477D4—102no—
478D4—102no—
479D4—102no—
480D4—104no—
481D4—208no—
482D4—208no—
483D4—202no—
484C2×C4—22no—
485C2×C4—42no—
486C2×C4—52no—
487C2×C4—54no—
488C2×C4—52no—
489C2×C4—102no—
490C2×C4—208no—
491C2×C4C3×#70912yes—
492C2×C4C3×D4×D512yes—
493C2×C4C3×D4×D511yescharacteristic
494C2×C4C23×C3×D511yescharacteristic
495C8—22no—
496C8—102no—
497C2×C3—82no—
498C2×C3—204no—
499C2×C3—402no—
500C2×C3#6507514yesdirect product factor
501C2×C3C2×#403011yescharacteristic
502C2×C3C2×D5×D811yescharacteristic
503C2×C3C2×#406011yescharacteristic
504C5C2×C3×#45911yescharacteristic, complemented, Sylow
505C22—42no—
506C22—42no—
507C22—42no—
508C22—84no—
509C22—104no—
510C22—208no—
511C22—202no—
512C22—202no—
513C22—202no—
514C22—202no—
515C22—202no—
516C22—404no—
517C22C3×#403012yes—
518C22C3×D5×D812yes—
519C22C3×#406012yes—
520C22C2×C3×D4×D511yescharacteristic
521C4—42no—
522C4—54no—
523C4—102no—
524C4C3×#403012yes—
525C4C2×C3×#70911yescharacteristic
526C4C2×C3×D4×D511yescharacteristic
527C3C2×#6507511yescharacteristic, direct product factor, Sylow
528C2—82no—
529C2—204no—
530C2—402no—
531C2C3×#6507514yesdirect product factor
532C2C2×C3×#403011yescharacteristic
533C2C2×C3×D5×D811yescharacteristic
534C2C2×C3×#406011yescharacteristic
subgroupG/H
centerC23×C3D4×D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC2×C3×C5×#102C2
socleC23×C3×C5C2×D4
hypercenterC2×C3×#102D5
hypocenterC5C2×C3×#459
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
1. C2×C3×#4086C2semidirect
2. C22×C3×#686C2semidirect
3. C22×C3×D40C2semidirect
4. C2×C3×#4039C2semidirect
5. C2×C3×C5×#102C2semidirect
6. C2×C3×#3946C2semidirect
7. C2×C3×#3941C2semidirect
8. C3×#65075C2direct (N × H)
9. C2×#65075C3direct (N × H)
19. C3×#4086C22semidirect
21. C2×C3×#686C22semidirect
26. C2×C3×D40C22semidirect
28. C3×#4039C22semidirect
29. C3×C5×#102C22semidirect
30. C3×#3946C22semidirect
31. C3×#3941C22semidirect
33. C2×#4086C2×C3semidirect
34. C22×#686C2×C3semidirect
35. C22×D40C2×C3semidirect
36. C2×#4039C2×C3semidirect
37. C2×C5×#102C2×C3semidirect
38. C2×#3946C2×C3semidirect
39. C2×#3941C2×C3semidirect
40. #65075C2×C3direct (N × H)
61. C2×C3×D20D4semidirect
84. C2×C3×#102D5semidirect
94. #4086C22×C3semidirect
96. C2×#686C22×C3semidirect
101. C2×D40C22×C3semidirect
103. #4039C22×C3semidirect
104. C5×#102C22×C3semidirect
105. #3946C22×C3semidirect
106. #3941C22×C3semidirect
133. C2×C3×Dic5D8semidirect
141. C3×D20C2×D4semidirect
166. C3×#102C2×D5semidirect
187. C2×D20C3×D4semidirect
210. C2×#102C3×D5semidirect
232. C3×Dic5C2×D8semidirect
294. C2×Dic5C3×D8semidirect
302. D20C2×C3×D4semidirect
327. #102C2×C3×D5semidirect
329. C2×C3×C5#459semidirect
393. Dic5C2×C3×D8semidirect
430. C3×C5C2×#459semidirect
454. C2×C5C3×#459semidirect
500. C2×C3#65075direct (N × H)
504. C5C2×C3×#459semidirect
527. C3C2×#65075direct (N × H)
531. C2C3×#65075direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#40862
C22×C3×#6862
C22×C3×D402
C2×C3×#40392
C2×C3×C5×#1022
C2×C3×#39462
C2×C3×#39412
C3×#650752
C2×#650753
C2×C3×#4595