Finite Group Databasebeta

← C2×C3×#65328 #14427979

Subgroups — C2×C3×#65328 #14427979

Order 1920.

1192 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×#121C211yesmaximal, characteristic, complemented
2C22×C3×#686C211yesmaximal, characteristic, complemented
3C2×C3×#4076C211yesmaximal, characteristic, complemented
4C22×C3×#681C211yesmaximal, characteristic, complemented
5C2×C3×#3964C211yesmaximal, characteristic, complemented
6C3×#65328C218yesmaximal, direct product factor
7C2×C3×#3941C211yesmaximal, characteristic, complemented
8C2×C3×#3940C211yesmaximal, characteristic, complemented
9C2×#65328C311yesmaximal, characteristic, direct product factor, Hall
10C22×C3×C5×D4—21no—
11C22×C3×C5×D4C2211yescharacteristic
12C23×C3×C4×C5C2211yescharacteristic
13C22×C3×C22⋊D5—21no—
14C22×C3×D20C2211yescharacteristic
15C3×C5×#121—22no—
16C3×C5×#121C2214yescomplemented
17C2×C3×C5×C4⋊C4C2211yescharacteristic
18C2×C3×C5×C22⋊C4—21no—
19C2×C3×#686—24no—
20C2×C3×#686C2214yescomplemented
21C3×#4076—22no—
22C3×#4076C2214yescomplemented
23C2×C3×#685—21no—
24C2×C3×C4⋊Dic5C2211yescharacteristic
25C2×C3×#681—24no—
26C2×C3×#681C2214yescomplemented
27C22×C3×#168C2211yescharacteristic
28C22×C3×#168C2211yescharacteristic
29C3×#3964C2214yescomplemented
30C3×#3941C2214yescomplemented
31C3×#3940C2214yescomplemented
32C2×C3×#469—51nomaximal, Hall
33C2×C5×#121C2×C311yescharacteristic, complemented
34C22×#686C2×C311yescharacteristic, complemented
35C2×#4076C2×C311yescharacteristic, complemented
36C22×#681C2×C311yescharacteristic, complemented
37C2×#3964C2×C311yescharacteristic, complemented
38#65328C2×C318yesdirect product factor
39C2×#3941C2×C311yescharacteristic, complemented
40C2×#3940C2×C311yescharacteristic, complemented
41C24×C3×C5—21no—
42C24×C3×C5D411yescharacteristic
43C23×C3×D5—21no—
44C2×C3×C5×D4—24no—
45C2×C3×C5×D4—22no—
46C2×C3×C5×D4—21no—
47C2×C3×C5×D4—22no—
48C2×C3×C5×D4—44no—
49C2×C3×C5×D4C2312yes—
50C22×C3×C4×C5—22no—
51C22×C3×C4×C5—22no—
52C22×C3×C4×C5—21no—
53C22×C3×C4×C5D411yescharacteristic
54C22×C3×C4×C5D412yes—
55C22×C3×C4×C5C2312yes—
56C22×C3×C4×C5C2311yescharacteristic
57C2×C3×C22⋊D5—24no—
58C2×C3×C22⋊D5—44no—
59C22×C3×Dic5—21no—
60C2×C3×D20—22no—
61C2×C3×D20—22no—
62C2×C3×D20—21no—
63C2×C3×D20C2312yes—
64C3×C5×C4⋊C4—21no—
65C3×C5×C4⋊C4C2312yes—
66C3×C5×C22⋊C4—24no—
67C3×#686—28no—
68C3×#685—24no—
69C3×C4⋊Dic5—21no—
70C3×C4⋊Dic5C2312yes—
71C3×#681—28no—
72C2×C3×#168—22no—
73C2×C3×#168D414yescomplemented
74C2×C3×#168C2312yes—
75C2×C3×#168C2312yes—
76C22×C3×D8—51no—
77C22×C3×M16—51no—
78C2×C3×#121—51no—
79C2×C3×#121D511yescharacteristic, complemented
80C3×#469—58no—
81C2×C3×#106—51no—
82C2×C3×#102—51no—
83C2×C3×#102—51no—
84C22×C5×D4—21no—
85C22×C5×D4C22×C311yescharacteristic
86C23×C4×C5C22×C311yescharacteristic
87C22×C22⋊D5—21no—
88C22×D20C22×C311yescharacteristic
89C5×#121—22no—
90C5×#121C22×C314yescomplemented
91C2×C5×C4⋊C4C22×C311yescharacteristic
92C2×C5×C22⋊C4—21no—
93C2×#686—24no—
94C2×#686C22×C314yescomplemented
95#4076—22no—
96#4076C22×C314yescomplemented
97C2×#685—21no—
98C2×C4⋊Dic5C22×C311yescharacteristic
99C2×#681—24no—
100C2×#681C22×C314yescomplemented
101C22×#168C22×C311yescharacteristic
102C22×#168C22×C311yescharacteristic
103#3964C22×C314yescomplemented
104#3941C22×C314yescomplemented
105#3940C22×C314yescomplemented
106C2×#469—51noSylow
107C23×C3×C5—22no—
108C23×C3×C5—22no—
109C23×C3×C5—22no—
110C23×C3×C5—22no—
111C23×C3×C5—42no—
112C23×C3×C5—42no—
113C23×C3×C5—42no—
114C23×C3×C5C2×D412yes—
115C23×C3×C5C2×D411yescharacteristic
116C22×C3×D5—22no—
117C22×C3×D5—42no—
118C22×C3×D5—42no—
119C22×C3×D5—42no—
120C3×C5×D4—22no—
121C3×C5×D4—22no—
122C3×C5×D4—48no—
123C3×C5×D4—42no—
124C2×C3×C4×C5—22no—
125C2×C3×C4×C5—24no—
126C2×C3×C4×C5—22no—
127C2×C3×C4×C5—22no—
128C2×C3×C4×C5—22no—
129C2×C3×C4×C5—42no—
130C2×C3×C4×C5C2×D412yes—
131C2×C3×C4×C5C2×D411yescharacteristic
132C2×C3×C4×C5C2×D412yes—
133C2×C3×C4×C5#4112yes—
134C2×C3×C4×C5C2411yescharacteristic
135C3×C22⋊D5—48no—
136C2×C3×Dic5—22no—
137C2×C3×Dic5—42no—
138C3×D20—22no—
139C3×D20—22no—
140C3×D20—42no—
141C3×#168—22no—
142C3×#168C2×D414yescomplemented
143C22×C3×D4—21no—
144C22×C3×D4—51no—
145C22×C3×D4—101no—
146C22×C3×D4C2×D511yescharacteristic
147C23×C3×C4C2×D511yescharacteristic
148C2×C3×D8—54no—
149C2×C3×D8—104no—
150C2×C3×M16—54no—
151C2×C3×M16—104no—
152C22×C3×C8—51no—
153C22×C3×C8—51no—
154C3×#121—22no—
155C3×#121—54no—
156C3×#121—102no—
157C3×#121C2×D514yescomplemented
158C2×C3×C4⋊C4—51no—
159C2×C3×C4⋊C4C2×D511yescharacteristic
160C2×C3×C22⋊C4—21no—
161C2×C3×C22⋊C4—101no—
162C3×#106—54no—
163C3×#102—54no—
164C3×#102—54no—
165C24×C5—21no—
166C24×C5C3×D411yescharacteristic
167C23×D5—21no—
168C2×C5×D4—24no—
169C2×C5×D4—22no—
170C2×C5×D4—21no—
171C2×C5×D4—22no—
172C2×C5×D4—44no—
173C2×C5×D4C23×C312yes—
174C22×C4×C5—22no—
175C22×C4×C5—22no—
176C22×C4×C5—21no—
177C22×C4×C5C3×D411yescharacteristic
178C22×C4×C5C3×D412yes—
179C22×C4×C5C23×C312yes—
180C22×C4×C5C23×C311yescharacteristic
181C2×C22⋊D5—24no—
182C2×C22⋊D5—44no—
183C22×Dic5—21no—
184C2×D20—22no—
185C2×D20—22no—
186C2×D20—21no—
187C2×D20C23×C312yes—
188C5×C4⋊C4—21no—
189C5×C4⋊C4C23×C312yes—
190C5×C22⋊C4—24no—
191#686—28no—
192#685—24no—
193C4⋊Dic5—21no—
194C4⋊Dic5C23×C312yes—
195#681—28no—
196C2×#168—22no—
197C2×#168C3×D414yescomplemented
198C2×#168C23×C312yes—
199C2×#168C23×C312yes—
200C22×D8—51no—
201C22×M16—51no—
202C2×#121—51no—
203C2×#121C3×D511yescharacteristic, complemented
204#469—58no—
205C2×#106—51no—
206C2×#102—51no—
207C2×#102—51no—
208C22×C3×C5—22no—
209C22×C3×C5—22no—
210C22×C3×C5—22no—
211C22×C3×C5—44no—
212C22×C3×C5—42no—
213C22×C3×C5—42no—
214C22×C3×C5—42no—
215C22×C3×C5—84no—
216C22×C3×C5#12112yes—
217C22×C3×C5AGL(1,ℤ/8)12yes—
218C22×C3×C5AGL(1,ℤ/8)12yes—
219C22×C3×C5C22×D411yescharacteristic
220C2×C3×D5—42no—
221C2×C3×D5—42no—
222C2×C3×D5—42no—
223C2×C3×D5—84no—
224C3×C4×C5—22no—
225C3×C4×C5—42no—
226C3×C4×C5#12112yes—
227C3×C4×C5C22×D411yescharacteristic
228C3×C4×C5C2×#4111yescharacteristic
229C3×Dic5—42no—
230C24×C3—21no—
231C24×C3—101no—
232C24×C3C22⋊D511yescharacteristic
233C2×C3×D4—24no—
234C2×C3×D4—22no—
235C2×C3×D4—22no—
236C2×C3×D4—21no—
237C2×C3×D4—44no—
238C2×C3×D4—52no—
239C2×C3×D4—104no—
240C2×C3×D4—102no—
241C2×C3×D4—102no—
242C2×C3×D4—101no—
243C2×C3×D4—204no—
244C2×C3×D4C22×D512yes—
245C22×C3×C4—22no—
246C22×C3×C4—22no—
247C22×C3×C4—21no—
248C22×C3×C4—101no—
249C22×C3×C4C22⋊D512yes—
250C22×C3×C4C22⋊D511yescharacteristic
251C22×C3×C4C22×D512yes—
252C22×C3×C4C22×D511yescharacteristic
253C3×D8—108no—
254C3×M16—108no—
255C2×C3×C8—54no—
256C2×C3×C8—52no—
257C2×C3×C8—52no—
258C2×C3×C8—102no—
259C3×C4⋊C4—21no—
260C3×C4⋊C4—52no—
261C3×C4⋊C4—101no—
262C3×C4⋊C4C22×D512yes—
263C3×C22⋊C4—24no—
264C3×C22⋊C4—104no—
265C23×C5—22no—
266C23×C5—22no—
267C23×C5—22no—
268C23×C5—22no—
269C23×C5—42no—
270C23×C5—42no—
271C23×C5—42no—
272C23×C5C2×C3×D412yes—
273C23×C5C2×C3×D411yescharacteristic
274C22×D5—22no—
275C22×D5—42no—
276C22×D5—42no—
277C22×D5—42no—
278C5×D4—22no—
279C5×D4—22no—
280C5×D4—48no—
281C5×D4—42no—
282C2×C4×C5—22no—
283C2×C4×C5—24no—
284C2×C4×C5—22no—
285C2×C4×C5—22no—
286C2×C4×C5—22no—
287C2×C4×C5—42no—
288C2×C4×C5C2×C3×D412yes—
289C2×C4×C5C2×C3×D412yes—
290C2×C4×C5C2×C3×D411yescharacteristic
291C2×C4×C5C3×#4112yes—
292C2×C4×C5C24×C311yescharacteristic
293C22⋊D5—48no—
294C2×Dic5—22no—
295C2×Dic5—42no—
296D20—22no—
297D20—22no—
298D20—42no—
299#168—22no—
300#168C2×C3×D414yescomplemented
301C22×D4—21no—
302C22×D4—51no—
303C22×D4—101no—
304C22×D4C2×C3×D511yescharacteristic
305C23×C4C2×C3×D511yescharacteristic
306C2×D8—54no—
307C2×D8—104no—
308C2×M16—54no—
309C2×M16—104no—
310C22×C8—51no—
311C22×C8—51no—
312#121—22no—
313#121—54no—
314#121—102no—
315#121C2×C3×D514yescomplemented
316C2×C4⋊C4—51no—
317C2×C4⋊C4C2×C3×D511yescharacteristic
318C2×C22⋊C4—21no—
319C2×C22⋊C4—101no—
320#106—54no—
321#102—54no—
322#102—54no—
323C2×C3×C5—42no—
324C2×C3×C5—82no—
325C2×C3×C5#46914yescomplemented
326C2×C3×C5C2×#12111yescharacteristic
327C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
328C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
329C3×D5—82no—
330C23×C3—22no—
331C23×C3—22no—
332C23×C3—22no—
333C23×C3—22no—
334C23×C3—42no—
335C23×C3—42no—
336C23×C3—42no—
337C23×C3—102no—
338C23×C3—202no—
339C23×C3—202no—
340C23×C3—202no—
341C23×C3C2×C22⋊D512yes—
342C23×C3C2×C22⋊D511yescharacteristic
343C3×D4—22no—
344C3×D4—22no—
345C3×D4—48no—
346C3×D4—42no—
347C3×D4—102no—
348C3×D4—102no—
349C3×D4—208no—
350C3×D4—202no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—24no—
355C2×C3×C4—22no—
356C2×C3×C4—42no—
357C2×C3×C4—102no—
358C2×C3×C4—202no—
359C2×C3×C4D4×D512yes—
360C2×C3×C4#71112yes—
361C2×C3×C4C2×C22⋊D512yes—
362C2×C3×C4C2×C22⋊D511yescharacteristic
363C2×C3×C4C23×D511yescharacteristic
364C3×C8—54no—
365C3×C8—102no—
366C22×C5—22no—
367C22×C5—22no—
368C22×C5—22no—
369C22×C5—44no—
370C22×C5—42no—
371C22×C5—42no—
372C22×C5—42no—
373C22×C5—84no—
374C22×C5C3×#12112yes—
375C22×C5C3×AGL(1,ℤ/8)12yes—
376C22×C5C3×AGL(1,ℤ/8)12yes—
377C22×C5C22×C3×D411yescharacteristic
378C2×D5—42no—
379C2×D5—42no—
380C2×D5—42no—
381C2×D5—84no—
382C4×C5—22no—
383C4×C5—42no—
384C4×C5C3×#12112yes—
385C4×C5C22×C3×D411yescharacteristic
386C4×C5C2×C3×#4111yescharacteristic
387Dic5—42no—
388C24—21no—
389C24—101no—
390C24C3×C22⋊D511yescharacteristic
391C2×D4—24no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—21no—
395C2×D4—44no—
396C2×D4—52no—
397C2×D4—104no—
398C2×D4—102no—
399C2×D4—102no—
400C2×D4—101no—
401C2×D4—204no—
402C2×D4C22×C3×D512yes—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—101no—
407C22×C4C3×C22⋊D511yescharacteristic
408C22×C4C3×C22⋊D512yes—
409C22×C4C22×C3×D512yes—
410C22×C4C22×C3×D511yescharacteristic
411D8—108no—
412M16—108no—
413C2×C8—54no—
414C2×C8—52no—
415C2×C8—52no—
416C2×C8—102no—
417C4⋊C4—21no—
418C4⋊C4—52no—
419C4⋊C4—101no—
420C4⋊C4C22×C3×D512yes—
421C22⋊C4—24no—
422C22⋊C4—104no—
423C3×C5C2×#46911yescharacteristic, complemented, Hall
424C22×C3—22no—
425C22×C3—22no—
426C22×C3—22no—
427C22×C3—44no—
428C22×C3—42no—
429C22×C3—42no—
430C22×C3—42no—
431C22×C3—84no—
432C22×C3—202no—
433C22×C3—202no—
434C22×C3—202no—
435C22×C3—404no—
436C22×C3#407812yes—
437C22×C3#408512yes—
438C22×C3#409512yes—
439C22×C3C22×C22⋊D511yescharacteristic
440C3×C4—22no—
441C3×C4—42no—
442C3×C4—202no—
443C3×C4#408512yes—
444C3×C4C2×D4×D511yescharacteristic
445C3×C4C2×#71111yescharacteristic
446C2×C5—42no—
447C2×C5—82no—
448C2×C5C3×#46914yescomplemented
449C2×C5C2×C3×#12111yescharacteristic
450C2×C5C2×C3×AGL(1,ℤ/8)11yescharacteristic
451C2×C5C2×C3×AGL(1,ℤ/8)11yescharacteristic
452D5—82no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—102no—
461C23—202no—
462C23—202no—
463C23—202no—
464C23C2×C3×C22⋊D512yes—
465C23C2×C3×C22⋊D511yescharacteristic
466D4—22no—
467D4—22no—
468D4—48no—
469D4—42no—
470D4—102no—
471D4—102no—
472D4—208no—
473D4—202no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—42no—
480C2×C4—102no—
481C2×C4—202no—
482C2×C4C3×D4×D512yes—
483C2×C4C3×#71112yes—
484C2×C4C2×C3×C22⋊D512yes—
485C2×C4C2×C3×C22⋊D511yescharacteristic
486C2×C4C23×C3×D511yescharacteristic
487C8—54no—
488C8—102no—
489C2×C3—42no—
490C2×C3—82no—
491C2×C3—402no—
492C2×C3#6532814yesdirect product factor
493C2×C3C2×#407811yescharacteristic
494C2×C3C2×#408511yescharacteristic
495C2×C3C2×#409511yescharacteristic
496C5C2×C3×#46911yescharacteristic, complemented, Sylow
497C22—22no—
498C22—22no—
499C22—22no—
500C22—44no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—84no—
505C22—202no—
506C22—202no—
507C22—202no—
508C22—404no—
509C22C3×#407812yes—
510C22C3×#408512yes—
511C22C3×#409512yes—
512C22C22×C3×C22⋊D511yescharacteristic
513C4—22no—
514C4—42no—
515C4—202no—
516C4C3×#408512yes—
517C4C2×C3×D4×D511yescharacteristic
518C4C2×C3×#71111yescharacteristic
519C3C2×#6532811yescharacteristic, direct product factor, Sylow
520C2—42no—
521C2—82no—
522C2—402no—
523C2C3×#6532814yesdirect product factor
524C2C2×C3×#407811yescharacteristic
525C2C2×C3×#408511yescharacteristic
526C2C2×C3×#409511yescharacteristic
subgroupG/H
centerC23×C3C2×C22⋊D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC2×C3×C5×#121C2
socleC23×C3×C5C2×D4
hypercenterC2×C3×#121D5
hypocenterC5C2×C3×#469
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ C2×C3×#121

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×#121C2semidirect
2. C22×C3×#686C2semidirect
3. C2×C3×#4076C2semidirect
4. C22×C3×#681C2semidirect
5. C2×C3×#3964C2semidirect
6. C3×#65328C2direct (N × H)
7. C2×C3×#3941C2semidirect
8. C2×C3×#3940C2semidirect
9. C2×#65328C3direct (N × H)
16. C3×C5×#121C22semidirect
20. C2×C3×#686C22semidirect
22. C3×#4076C22semidirect
26. C2×C3×#681C22semidirect
29. C3×#3964C22semidirect
30. C3×#3941C22semidirect
31. C3×#3940C22semidirect
33. C2×C5×#121C2×C3semidirect
34. C22×#686C2×C3semidirect
35. C2×#4076C2×C3semidirect
36. C22×#681C2×C3semidirect
37. C2×#3964C2×C3semidirect
38. #65328C2×C3direct (N × H)
39. C2×#3941C2×C3semidirect
40. C2×#3940C2×C3semidirect
73. C2×C3×#168D4semidirect
79. C2×C3×#121D5semidirect
90. C5×#121C22×C3semidirect
94. C2×#686C22×C3semidirect
96. #4076C22×C3semidirect
100. C2×#681C22×C3semidirect
103. #3964C22×C3semidirect
104. #3941C22×C3semidirect
105. #3940C22×C3semidirect
142. C3×#168C2×D4semidirect
157. C3×#121C2×D5semidirect
197. C2×#168C3×D4semidirect
203. C2×#121C3×D5semidirect
300. #168C2×C3×D4semidirect
315. #121C2×C3×D5semidirect
325. C2×C3×C5#469semidirect
423. C3×C5C2×#469semidirect
448. C2×C5C3×#469semidirect
492. C2×C3#65328direct (N × H)
496. C5C2×C3×#469semidirect
519. C3C2×#65328direct (N × H)
523. C2C3×#65328direct (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×#1212
C22×C3×#6862
C2×C3×#40762
C22×C3×#6812
C2×C3×#39642
C3×#653282
C2×C3×#39412
C2×C3×#39402
C2×#653283
C2×C3×#4695