Finite Group Databasebeta

← C2×C3×#65152 #14431699

Subgroups — C2×C3×#65152 #14431699

Order 1920.

1024 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4041C211yesmaximal, characteristic
2C2×C3×D5×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×C3×C5×#106C211yesmaximal, characteristic, complemented
4C2×C3×#3953C211yesmaximal, characteristic, complemented
5C2×C3×#3952C211yesmaximal, characteristic, complemented
6C2×C3×#3941C211yesmaximal, characteristic, complemented
7C2×C3×#3940C211yesmaximal, characteristic, complemented
8C3×#65152C218yesmaximal, direct product factor
9C2×#65152C311yesmaximal, characteristic, direct product factor, Hall
10C22×C3×D20—21no—
11C22×C3×D20C2211yescharacteristic
12C22×C3×C4×D5—21no—
13C22×C3×C4×D5C2211yescharacteristic
14C22×C3×C5×C8C2211yescharacteristic
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C2×C3×C5×C4⋊C4C2211yescharacteristic
17C2×C3×#685—21no—
18C2×C3×C4⋊Dic5C2211yescharacteristic
19C2×C3×#683—21no—
20C22×C3×#168C2211yescharacteristic
21C3×#4041—22no—
22C3×#4041C2214yes—
23C3×D5×C4⋊C4—22no—
24C3×D5×C4⋊C4C2214yescomplemented
25C3×C5×#106C2214yescomplemented
26C3×#3953C2214yescomplemented
27C3×#3952C2214yescomplemented
28C3×#3941C2214yescomplemented
29C3×#3940C2214yescomplemented
30C2×C3×#481—51nomaximal, Hall
31C2×#4041C2×C311yescharacteristic
32C2×D5×C4⋊C4C2×C311yescharacteristic, complemented
33C2×C5×#106C2×C311yescharacteristic, complemented
34C2×#3953C2×C311yescharacteristic, complemented
35C2×#3952C2×C311yescharacteristic, complemented
36C2×#3941C2×C311yescharacteristic, complemented
37C2×#3940C2×C311yescharacteristic, complemented
38#65152C2×C318yesdirect product factor
39C23×C3×D5—21no—
40C23×C3×D5D411yescharacteristic
41C22×C3×C4×C5—21no—
42C22×C3×C4×C5—21no—
43C22×C3×C4×C5C2311yescharacteristic
44C22×C3×Dic5—21no—
45C22×C3×Dic5D411yescharacteristic
46C2×C3×D20—22no—
47C2×C3×D20—22no—
48C2×C3×D20—24no—
49C2×C3×D20—21no—
50C2×C3×D20—44no—
51C2×C3×D20C2312yes—
52C2×C3×C4×D5—22no—
53C2×C3×C4×D5—22no—
54C2×C3×C4×D5—24no—
55C2×C3×C4×D5—44no—
56C2×C3×C4×D5D412yes—
57C2×C3×C4×D5C2312yes—
58C2×C3×C5×C8—22no—
59C2×C3×C5×C8C2312yes—
60C3×C5×C4⋊C4—21no—
61C3×C5×C4⋊C4—21no—
62C3×C5×C4⋊C4C2312yes—
63C3×C5×C4⋊C4C2312yes—
64C3×#685—24no—
65C3×C4⋊Dic5—21no—
66C3×C4⋊Dic5C2312yes—
67C3×#683—24no—
68C2×C3×#168—22no—
69C2×C3×#168C2312yes—
70C2×C3×#121—51no—
71C22×C3×C4⋊C4—51no—
72C3×#481—58no—
73C2×C3×#106—51no—
74C2×C3×#106D511yescharacteristic, complemented
75C2×C3×#102—51no—
76C2×C3×#102—51no—
77C2×C3×C22⋊C8—51no—
78C22×D20—21no—
79C22×D20C22×C311yescharacteristic
80C22×C4×D5—21no—
81C22×C4×D5C22×C311yescharacteristic
82C22×C5×C8C22×C311yescharacteristic
83C2×C5×C4⋊C4C22×C311yescharacteristic
84C2×C5×C4⋊C4C22×C311yescharacteristic
85C2×#685—21no—
86C2×C4⋊Dic5C22×C311yescharacteristic
87C2×#683—21no—
88C22×#168C22×C311yescharacteristic
89#4041—22no—
90#4041C22×C314yes—
91D5×C4⋊C4—22no—
92D5×C4⋊C4C22×C314yescomplemented
93C5×#106C22×C314yescomplemented
94#3953C22×C314yescomplemented
95#3952C22×C314yescomplemented
96#3941C22×C314yescomplemented
97#3940C22×C314yescomplemented
98C2×#481—51noSylow
99C23×C3×C5C2×D411yescharacteristic
100C22×C3×D5—22no—
101C22×C3×D5—22no—
102C22×C3×D5—22no—
103C22×C3×D5—42no—
104C22×C3×D5—42no—
105C22×C3×D5—42no—
106C22×C3×D5SD1614yes—
107C22×C3×D5C2×D412yes—
108C2×C3×C4×C5—22no—
109C2×C3×C4×C5—22no—
110C2×C3×C4×C5—42no—
111C2×C3×C4×C5—42no—
112C2×C3×C4×C5C2×D411yescharacteristic
113C2×C3×C4×C5#4112yes—
114C2×C3×C4×C5#4112yes—
115C2×C3×C4×C5C2411yescharacteristic
116C2×C3×Dic5—22no—
117C2×C3×Dic5—22no—
118C2×C3×Dic5—42no—
119C2×C3×Dic5C2×D412yes—
120C3×D20—22no—
121C3×D20—22no—
122C3×D20—48no—
123C3×D20—42no—
124C3×C4×D5—24no—
125C3×C4×D5—22no—
126C3×C4×D5—22no—
127C3×C4×D5—48no—
128C3×C5×C8—22no—
129C3×#168—22no—
130C22×C3×D4—51no—
131C22×C3×D4—101no—
132C23×C3×C4—51no—
133C23×C3×C4—101no—
134C22×C3×C8—51no—
135C22×C3×C8C2×D511yescharacteristic
136C3×#121—54no—
137C3×#121—102no—
138C2×C3×C4⋊C4—51no—
139C2×C3×C4⋊C4—54no—
140C2×C3×C4⋊C4—102no—
141C2×C3×C4⋊C4—101no—
142C2×C3×C4⋊C4C2×D511yescharacteristic
143C2×C3×C4⋊C4C2×D511yescharacteristic
144C2×C3×C22⋊C4—101no—
145C3×#106—54no—
146C3×#106C2×D514yescomplemented
147C3×#102—54no—
148C3×#102—54no—
149C3×C22⋊C8—54no—
150C23×D5—21no—
151C23×D5C3×D411yescharacteristic
152C22×C4×C5—21no—
153C22×C4×C5—21no—
154C22×C4×C5C23×C311yescharacteristic
155C22×Dic5—21no—
156C22×Dic5C3×D411yescharacteristic
157C2×D20—22no—
158C2×D20—22no—
159C2×D20—24no—
160C2×D20—21no—
161C2×D20—44no—
162C2×D20C23×C312yes—
163C2×C4×D5—22no—
164C2×C4×D5—22no—
165C2×C4×D5—24no—
166C2×C4×D5—44no—
167C2×C4×D5C3×D412yes—
168C2×C4×D5C23×C312yes—
169C2×C5×C8—22no—
170C2×C5×C8C23×C312yes—
171C5×C4⋊C4—21no—
172C5×C4⋊C4—21no—
173C5×C4⋊C4C23×C312yes—
174C5×C4⋊C4C23×C312yes—
175#685—24no—
176C4⋊Dic5—21no—
177C4⋊Dic5C23×C312yes—
178#683—24no—
179C2×#168—22no—
180C2×#168C23×C312yes—
181C2×#121—51no—
182C22×C4⋊C4—51no—
183#481—58no—
184C2×#106—51no—
185C2×#106C3×D511yescharacteristic, complemented
186C2×#102—51no—
187C2×#102—51no—
188C2×C22⋊C8—51no—
189C22×C3×C5#12312yes—
190C22×C3×C5C2×SD1612yes—
191C22×C3×C5AGL(1,ℤ/8)12yes—
192C22×C3×C5C22×D411yescharacteristic
193C2×C3×D5—22no—
194C2×C3×D5—22no—
195C2×C3×D5—28no—
196C2×C3×D5—42no—
197C2×C3×D5—42no—
198C2×C3×D5—42no—
199C2×C3×D5—84no—
200C2×C3×D5C2×SD1614yes—
201C3×C4×C5—42no—
202C3×C4×C5—42no—
203C3×C4×C5#12312yes—
204C3×C4×C5C2×#4111yescharacteristic
205C3×C4×C5C2×#4111yescharacteristic
206C3×Dic5—22no—
207C3×Dic5—42no—
208C24×C3—51no—
209C24×C3—101no—
210C2×C3×D4—52no—
211C2×C3×D4—104no—
212C2×C3×D4—102no—
213C2×C3×D4—101no—
214C2×C3×D4—102no—
215C2×C3×D4—204no—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—51no—
219C22×C3×C4—52no—
220C22×C3×C4—52no—
221C22×C3×C4—104no—
222C22×C3×C4—102no—
223C22×C3×C4—102no—
224C22×C3×C4—101no—
225C22×C3×C4—204no—
226C22×C3×C4C22×D511yescharacteristic
227C2×C3×C8—22no—
228C2×C3×C8—52no—
229C2×C3×C8—102no—
230C2×C3×C8C22×D512yes—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—52no—
234C3×C4⋊C4—104no—
235C3×C4⋊C4—101no—
236C3×C4⋊C4C22×D512yes—
237C3×C4⋊C4C22×D512yes—
238C3×C22⋊C4—104no—
239C23×C5C2×C3×D411yescharacteristic
240C22×D5—22no—
241C22×D5—22no—
242C22×D5—22no—
243C22×D5—42no—
244C22×D5—42no—
245C22×D5—42no—
246C22×D5C3×SD1614yes—
247C22×D5C2×C3×D412yes—
248C2×C4×C5—22no—
249C2×C4×C5—22no—
250C2×C4×C5—42no—
251C2×C4×C5—42no—
252C2×C4×C5C2×C3×D411yescharacteristic
253C2×C4×C5C3×#4112yes—
254C2×C4×C5C3×#4112yes—
255C2×C4×C5C24×C311yescharacteristic
256C2×Dic5—22no—
257C2×Dic5—22no—
258C2×Dic5—42no—
259C2×Dic5C2×C3×D412yes—
260D20—22no—
261D20—22no—
262D20—48no—
263D20—42no—
264C4×D5—24no—
265C4×D5—22no—
266C4×D5—22no—
267C4×D5—48no—
268C5×C8—22no—
269#168—22no—
270C22×D4—51no—
271C22×D4—101no—
272C23×C4—51no—
273C23×C4—101no—
274C22×C8—51no—
275C22×C8C2×C3×D511yescharacteristic
276#121—54no—
277#121—102no—
278C2×C4⋊C4—51no—
279C2×C4⋊C4—54no—
280C2×C4⋊C4—102no—
281C2×C4⋊C4—101no—
282C2×C4⋊C4C2×C3×D511yescharacteristic
283C2×C4⋊C4C2×C3×D511yescharacteristic
284C2×C22⋊C4—101no—
285#106—54no—
286#106C2×C3×D514yescomplemented
287#102—54no—
288#102—54no—
289C22⋊C8—54no—
290C2×C3×C5#48114yescomplemented
291C2×C3×C5C2×#12311yescharacteristic
292C2×C3×C5C22×SD1611yescharacteristic
293C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
294C3×D5—24no—
295C3×D5—82no—
296C23×C3—52no—
297C23×C3—54no—
298C23×C3—102no—
299C23×C3—102no—
300C23×C3—102no—
301C23×C3—202no—
302C23×C3—202no—
303C23×C3—202no—
304C23×C3D4×D511yescharacteristic
305C3×D4—102no—
306C3×D4—102no—
307C3×D4—208no—
308C3×D4—202no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—52no—
314C2×C3×C4—102no—
315C2×C3×C4—102no—
316C2×C3×C4—102no—
317C2×C3×C4—104no—
318C2×C3×C4—102no—
319C2×C3×C4—208no—
320C2×C3×C4—202no—
321C2×C3×C4#70912yes—
322C2×C3×C4D4×D511yescharacteristic
323C2×C3×C4#71312yes—
324C2×C3×C4C23×D511yescharacteristic
325C3×C8—22no—
326C3×C8—102no—
327C22×C5C3×#12312yes—
328C22×C5C2×C3×SD1612yes—
329C22×C5C3×AGL(1,ℤ/8)12yes—
330C22×C5C22×C3×D411yescharacteristic
331C2×D5—22no—
332C2×D5—22no—
333C2×D5—28no—
334C2×D5—42no—
335C2×D5—42no—
336C2×D5—42no—
337C2×D5—84no—
338C2×D5C2×C3×SD1614yes—
339C4×C5—42no—
340C4×C5—42no—
341C4×C5C3×#12312yes—
342C4×C5C2×C3×#4111yescharacteristic
343C4×C5C2×C3×#4111yescharacteristic
344Dic5—22no—
345Dic5—42no—
346C24—51no—
347C24—101no—
348C2×D4—52no—
349C2×D4—104no—
350C2×D4—102no—
351C2×D4—101no—
352C2×D4—102no—
353C2×D4—204no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—51no—
357C22×C4—52no—
358C22×C4—52no—
359C22×C4—104no—
360C22×C4—102no—
361C22×C4—102no—
362C22×C4—101no—
363C22×C4—204no—
364C22×C4C22×C3×D511yescharacteristic
365C2×C8—22no—
366C2×C8—52no—
367C2×C8—102no—
368C2×C8C22×C3×D512yes—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—52no—
372C4⋊C4—104no—
373C4⋊C4—101no—
374C4⋊C4C22×C3×D512yes—
375C4⋊C4C22×C3×D512yes—
376C22⋊C4—104no—
377C3×C5C2×#48111yescharacteristic, complemented, Hall
378C22×C3—54no—
379C22×C3—102no—
380C22×C3—102no—
381C22×C3—108no—
382C22×C3—202no—
383C22×C3—202no—
384C22×C3—202no—
385C22×C3—404no—
386C22×C3#404012yes—
387C22×C3D5×SD1612yes—
388C22×C3#406012yes—
389C22×C3C2×D4×D511yescharacteristic
390C3×C4—42no—
391C3×C4—42no—
392C3×C4—102no—
393C3×C4—202no—
394C3×C4#404012yes—
395C3×C4C2×#70911yescharacteristic
396C3×C4C2×#71311yescharacteristic
397C2×C5C3×#48114yescomplemented
398C2×C5C2×C3×#12311yescharacteristic
399C2×C5C22×C3×SD1611yescharacteristic
400C2×C5C2×C3×AGL(1,ℤ/8)11yescharacteristic
401D5—24no—
402D5—82no—
403C23—52no—
404C23—54no—
405C23—102no—
406C23—102no—
407C23—102no—
408C23—202no—
409C23—202no—
410C23—202no—
411C23C3×D4×D511yescharacteristic
412D4—102no—
413D4—102no—
414D4—208no—
415D4—202no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—52no—
421C2×C4—102no—
422C2×C4—102no—
423C2×C4—102no—
424C2×C4—104no—
425C2×C4—102no—
426C2×C4—208no—
427C2×C4—202no—
428C2×C4C3×#70912yes—
429C2×C4C3×D4×D511yescharacteristic
430C2×C4C3×#71312yes—
431C2×C4C23×C3×D511yescharacteristic
432C8—22no—
433C8—102no—
434C2×C3—104no—
435C2×C3—402no—
436C2×C3#6515214yesdirect product factor
437C2×C3C2×#404011yescharacteristic
438C2×C3C2×D5×SD1611yescharacteristic
439C2×C3C2×#406011yescharacteristic
440C5C2×C3×#48111yescharacteristic, complemented, Sylow
441C22—54no—
442C22—108no—
443C22—102no—
444C22—102no—
445C22—202no—
446C22—202no—
447C22—202no—
448C22—404no—
449C22C3×#404012yes—
450C22C3×D5×SD1612yes—
451C22C3×#406012yes—
452C22C2×C3×D4×D511yescharacteristic
453C4—42no—
454C4—42no—
455C4—102no—
456C4—202no—
457C4C3×#404012yes—
458C4C2×C3×#70911yescharacteristic
459C4C2×C3×#71311yescharacteristic
460C3C2×#6515211yescharacteristic, direct product factor, Sylow
461C2—104no—
462C2—402no—
463C2C3×#6515214yesdirect product factor
464C2C2×C3×#404011yescharacteristic
465C2C2×C3×D5×SD1611yescharacteristic
466C2C2×C3×#406011yescharacteristic
subgroupG/H
centerC23×C3D4×D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC2×C3×C5×#106C2
socleC23×C3×C5C2×D4
prime radicalC2×C3×#4041C2
hypercenterC2×C3×#106D5
hypocenterC5C2×C3×#481
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C2×C3×#106

Complements

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

NH ≅ G/Ntype
2. C2×C3×D5×C4⋊C4C2semidirect
3. C2×C3×C5×#106C2semidirect
4. C2×C3×#3953C2semidirect
5. C2×C3×#3952C2semidirect
6. C2×C3×#3941C2semidirect
7. C2×C3×#3940C2semidirect
8. C3×#65152C2direct (N × H)
9. C2×#65152C3direct (N × H)
24. C3×D5×C4⋊C4C22semidirect
25. C3×C5×#106C22semidirect
26. C3×#3953C22semidirect
27. C3×#3952C22semidirect
28. C3×#3941C22semidirect
29. C3×#3940C22semidirect
32. C2×D5×C4⋊C4C2×C3semidirect
33. C2×C5×#106C2×C3semidirect
34. C2×#3953C2×C3semidirect
35. C2×#3952C2×C3semidirect
36. C2×#3941C2×C3semidirect
37. C2×#3940C2×C3semidirect
38. #65152C2×C3direct (N × H)
74. C2×C3×#106D5semidirect
92. D5×C4⋊C4C22×C3semidirect
93. C5×#106C22×C3semidirect
94. #3953C22×C3semidirect
95. #3952C22×C3semidirect
96. #3941C22×C3semidirect
97. #3940C22×C3semidirect
146. C3×#106C2×D5semidirect
185. C2×#106C3×D5semidirect
286. #106C2×C3×D5semidirect
290. C2×C3×C5#481semidirect
377. C3×C5C2×#481semidirect
397. C2×C5C3×#481semidirect
436. C2×C3#65152direct (N × H)
440. C5C2×C3×#481semidirect
460. C3C2×#65152direct (N × H)
463. C2C3×#65152direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#40412
C2×C3×D5×C4⋊C42
C2×C3×C5×#1062
C2×C3×#39532
C2×C3×#39522
C2×C3×#39412
C2×C3×#39402
C3×#651522
C2×#651523
C2×C3×#4815