Finite Group Databasebeta

← C3×#10615854 #14425531

Subgroups — C3×#10615854 #14425531

Order 1920.

1408 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#585C211yesmaximal, characteristic
2C3×#66159C211yesmaximal, characteristic, complemented
3C2×C3×#4097C212yesmaximal, complemented
4C3×#65535C214yesmaximal, complemented
5C3×#65531C214yesmaximal, complemented
6C3×#65526C211yesmaximal, characteristic, complemented
7C3×#65525C212yesmaximal, complemented
8#10615854C311yesmaximal, characteristic, direct product factor, Hall
9C3×C5×2−1+4—22no—
10C3×C5×2−1+4C2212yes—
11C3×C5×H(5,𝔽2)—22no—
12C3×C5×H(5,𝔽2)—22no—
13C3×C5×H(5,𝔽2)C2212yes—
14C2×C3×C5×#41—24no—
15C2×C3×C5×#41—22no—
16C2×C3×C5×#41C2212yes—
17C2×C3×C5×#41C2211yescharacteristic
18C22×C3×Dic10C2211yescharacteristic, complemented
19C3×#4097—24no—
20C3×#4097C2218yescomplemented
21C3×#4090C2212yescomplemented
22C2×C3×#689—22no—
23C3×#4082C2212yescomplemented
24C3×#4081—22no—
25C2×C3×#687—22no—
26C3×#4072C2211yescharacteristic, complemented
27C3×#4070—22no—
28C2×C3×#681C2212yescomplemented
29C3×#3969C2218yescomplemented
30C3×#3968C2212yescomplemented
31C3×#3965C2212yescomplemented
32C3×#3022—51nomaximal, Hall
33C5×#585C2×C311yescharacteristic
34#66159C2×C311yescharacteristic, complemented
35C2×#4097C2×C312yescomplemented
36#65535C2×C314yescomplemented
37#65531C2×C314yescomplemented
38#65526C2×C311yescharacteristic, complemented
39#65525C2×C312yescomplemented
40C3×C5×#41—24no—
41C3×C5×#41—28no—
42C3×C5×#41—24no—
43C3×C5×#41—24no—
44C3×C5×#41—24no—
45C3×C5×#41—22no—
46C3×C5×#41—22no—
47C3×C5×#41—24no—
48C3×C5×#41—22no—
49C3×C5×#41—22no—
50C3×C5×#41D414yes—
51C3×C5×#41C2314yes—
52C2×C3×C5×Q8—22no—
53C2×C3×C5×Q8—24no—
54C2×C3×C5×Q8D412yes—
55C2×C3×C5×Q8C2311yescharacteristic
56C2×C3×C5×D4—28no—
57C2×C3×C5×D4—22no—
58C2×C3×C5×D4—24no—
59C2×C3×C5×D4—22no—
60C2×C3×C5×D4—24no—
61C2×C3×C5×D4D412yes—
62C2×C3×C5×D4D412yes—
63C2×C3×C5×D4C2311yescharacteristic
64C22×C3×C4×C5—24no—
65C22×C3×C4×C5—22no—
66C22×C3×C4×C5D412yes—
67C22×C3×C4×C5C2311yescharacteristic
68C22×C3×Dic5—21no—
69C2×C3×Dic10—22no—
70C2×C3×Dic10—22no—
71C2×C3×Dic10—21no—
72C2×C3×Dic10C2312yes—
73C3×C22⋊Dic5—21no—
74C3×C22⋊Dic5—24no—
75C3×#689—28no—
76C3×#687—28no—
77C3×C4⋊Dic5—21no—
78C3×#683—24no—
79C3×C4×Dic5C2312yes—
80C3×#681—22no—
81C3×#681C2314yes—
82C2×C3×#168—22no—
83C3×#585D511yescharacteristic
84C2×C3×#137—52no—
85C3×#536—51no—
86C3×#456—54no—
87C3×#455—54no—
88C3×#421—52no—
89C3×#413—51no—
90C5×2−1+4—22no—
91C5×2−1+4C22×C312yes—
92C5×H(5,𝔽2)—22no—
93C5×H(5,𝔽2)—22no—
94C5×H(5,𝔽2)C22×C312yes—
95C2×C5×#41—24no—
96C2×C5×#41—22no—
97C2×C5×#41C22×C312yes—
98C2×C5×#41C22×C311yescharacteristic
99C22×Dic10C22×C311yescharacteristic, complemented
100#4097—24no—
101#4097C22×C318yescomplemented
102#4090C22×C312yescomplemented
103C2×#689—22no—
104#4082C22×C312yescomplemented
105#4081—22no—
106C2×#687—22no—
107#4072C22×C311yescharacteristic, complemented
108#4070—22no—
109C2×#681C22×C312yescomplemented
110#3969C22×C318yescomplemented
111#3968C22×C312yescomplemented
112#3965C22×C312yescomplemented
113#3022—51noSylow
114C23×C3×C5—24no—
115C23×C3×C5—22no—
116C23×C3×C5C2×D412yes—
117C23×C3×C5C2×D411yescharacteristic
118C3×C5×Q8—24no—
119C3×C5×Q8—22no—
120C3×C5×Q8—22no—
121C3×C5×Q8C2×D414yes—
122C3×C5×D4—28no—
123C3×C5×D4—24no—
124C3×C5×D4—24no—
125C3×C5×D4—24no—
126C3×C5×D4—24no—
127C3×C5×D4—22no—
128C3×C5×D4—22no—
129C3×C5×D4C2×D414yes—
130C2×C3×C4×C5—24no—
131C2×C3×C4×C5—28no—
132C2×C3×C4×C5—22no—
133C2×C3×C4×C5—24no—
134C2×C3×C4×C5—22no—
135C2×C3×C4×C5—22no—
136C2×C3×C4×C5—24no—
137C2×C3×C4×C5C2×D412yes—
138C2×C3×C4×C5C2×D411yescharacteristic
139C2×C3×C4×C5C2×D412yes—
140C2×C3×C4×C5C2×D412yes—
141C2×C3×C4×C5C2411yescharacteristic
142C2×C3×Dic5—22no—
143C2×C3×Dic5—22no—
144C2×C3×Dic5—42no—
145C3×Dic10—24no—
146C3×Dic10—41no—
147C3×Dic10—41no—
148C3×#168—24no—
149C3×2−1+4—22no—
150C3×2−1+4C2×D512yes—
151C3×H(5,𝔽2)—22no—
152C3×H(5,𝔽2)—22no—
153C3×H(5,𝔽2)C2×D512yes—
154C2×C3×#41—24no—
155C2×C3×#41—22no—
156C2×C3×#41C2×D512yes—
157C2×C3×#41C2×D511yescharacteristic
158C22×C3×Q8—51no—
159C3×#137—58no—
160C3×#137—104no—
161C2×C3×Q16—102no—
162C2×C3×SD16—102no—
163C2×C3×M16—52no—
164C3×C4⋊Q8—52no—
165C3×#124—52no—
166C3×#123—102no—
167C3×C22⋊Q8—102no—
168C3×#117—51no—
169C3×#104—58no—
170C3×#101—52no—
171C3×#100—52no—
172C5×#41—24no—
173C5×#41—28no—
174C5×#41—24no—
175C5×#41—24no—
176C5×#41—24no—
177C5×#41—22no—
178C5×#41—22no—
179C5×#41—24no—
180C5×#41—22no—
181C5×#41—22no—
182C5×#41C3×D414yes—
183C5×#41C23×C314yes—
184C2×C5×Q8—22no—
185C2×C5×Q8—24no—
186C2×C5×Q8C3×D412yes—
187C2×C5×Q8C23×C311yescharacteristic
188C2×C5×D4—28no—
189C2×C5×D4—22no—
190C2×C5×D4—24no—
191C2×C5×D4—22no—
192C2×C5×D4—24no—
193C2×C5×D4C3×D412yes—
194C2×C5×D4C3×D412yes—
195C2×C5×D4C23×C311yescharacteristic
196C22×C4×C5—24no—
197C22×C4×C5—22no—
198C22×C4×C5C3×D412yes—
199C22×C4×C5C23×C311yescharacteristic
200C22×Dic5—21no—
201C2×Dic10—22no—
202C2×Dic10—22no—
203C2×Dic10—21no—
204C2×Dic10C23×C312yes—
205C22⋊Dic5—24no—
206C22⋊Dic5—21no—
207#689—28no—
208#687—28no—
209C4⋊Dic5—21no—
210#683—24no—
211C4×Dic5C23×C312yes—
212#681—22no—
213#681C23×C314yes—
214C2×#168—22no—
215#585C3×D511yescharacteristic
216C2×#137—52no—
217#536—51no—
218#456—54no—
219#455—54no—
220#421—52no—
221#413—51no—
222C22×C3×C5—22no—
223C22×C3×C5—24no—
224C22×C3×C5—41no—
225C22×C3×C5—42no—
226C22×C3×C5—82no—
227C22×C3×C5—84no—
228C22×C3×C5#12012yes—
229C22×C3×C5C22×D411yescharacteristic
230C3×C4×C5—24no—
231C3×C4×C5—22no—
232C3×C4×C5#12011yescharacteristic
233C3×C4×C5#12011yescharacteristic
234C3×C4×C5C22×D412yes—
235C3×Dic5—42no—
236C3×Dic5—42no—
237C3×#41—24no—
238C3×#41—28no—
239C3×#41—24no—
240C3×#41—24no—
241C3×#41—24no—
242C3×#41—22no—
243C3×#41—22no—
244C3×#41—24no—
245C3×#41—22no—
246C3×#41—22no—
247C3×#41C22⋊D514yes—
248C3×#41C22×D514yes—
249C2×C3×Q8—22no—
250C2×C3×Q8—24no—
251C2×C3×Q8—52no—
252C2×C3×Q8—101no—
253C2×C3×Q8—102no—
254C2×C3×Q8—102no—
255C2×C3×Q8C22⋊D512yes—
256C2×C3×Q8C22×D511yescharacteristic
257C2×C3×D4—28no—
258C2×C3×D4—22no—
259C2×C3×D4—24no—
260C2×C3×D4—22no—
261C2×C3×D4—24no—
262C2×C3×D4C22⋊D512yes—
263C2×C3×D4C22⋊D512yes—
264C2×C3×D4C22×D511yescharacteristic
265C22×C3×C4—24no—
266C22×C3×C4—22no—
267C22×C3×C4—101no—
268C22×C3×C4C22⋊D512yes—
269C22×C3×C4C22×D511yescharacteristic
270C3×Q16—108no—
271C3×SD16—108no—
272C3×M16—54no—
273C3×M16—102no—
274C2×C3×C8—102no—
275C3×C4⋊C4—104no—
276C3×C4⋊C4—101no—
277C3×C22⋊C4—101no—
278C3×C22⋊C4—104no—
279C3×C42—52no—
280C23×C5—24no—
281C23×C5—22no—
282C23×C5C2×C3×D412yes—
283C23×C5C2×C3×D411yescharacteristic
284C5×Q8—24no—
285C5×Q8—22no—
286C5×Q8—22no—
287C5×Q8C2×C3×D414yes—
288C5×D4—28no—
289C5×D4—24no—
290C5×D4—24no—
291C5×D4—24no—
292C5×D4—24no—
293C5×D4—22no—
294C5×D4—22no—
295C5×D4C2×C3×D414yes—
296C2×C4×C5—24no—
297C2×C4×C5—28no—
298C2×C4×C5—22no—
299C2×C4×C5—24no—
300C2×C4×C5—22no—
301C2×C4×C5—22no—
302C2×C4×C5—24no—
303C2×C4×C5C2×C3×D412yes—
304C2×C4×C5C2×C3×D411yescharacteristic
305C2×C4×C5C2×C3×D412yes—
306C2×C4×C5C2×C3×D412yes—
307C2×C4×C5C24×C311yescharacteristic
308C2×Dic5—22no—
309C2×Dic5—22no—
310C2×Dic5—42no—
311Dic10—24no—
312Dic10—41no—
313Dic10—41no—
314#168—24no—
3152−1+4—22no—
3162−1+4C2×C3×D512yes—
317H(5,𝔽2)—22no—
318H(5,𝔽2)—22no—
319H(5,𝔽2)C2×C3×D512yes—
320C2×#41—24no—
321C2×#41—22no—
322C2×#41C2×C3×D512yes—
323C2×#41C2×C3×D511yescharacteristic
324C22×Q8—51no—
325#137—58no—
326#137—104no—
327C2×Q16—102no—
328C2×SD16—102no—
329C2×M16—52no—
330C4⋊Q8—52no—
331#124—52no—
332#123—102no—
333C22⋊Q8—102no—
334#117—51no—
335#104—58no—
336#101—52no—
337#100—52no—
338C2×C3×C5—21no—
339C2×C3×C5—22no—
340C2×C3×C5—42no—
341C2×C3×C5—44no—
342C2×C3×C5C2×#12011yescharacteristic
343C23×C3—24no—
344C23×C3—22no—
345C23×C3C2×C22⋊D512yes—
346C23×C3C2×C22⋊D511yescharacteristic
347C3×Q8—24no—
348C3×Q8—22no—
349C3×Q8—22no—
350C3×Q8—104no—
351C3×Q8—201no—
352C3×Q8—201no—
353C3×Q8C2×C22⋊D514yes—
354C3×D4—28no—
355C3×D4—24no—
356C3×D4—24no—
357C3×D4—24no—
358C3×D4—24no—
359C3×D4—22no—
360C3×D4—22no—
361C3×D4C2×C22⋊D514yes—
362C2×C3×C4—24no—
363C2×C3×C4—28no—
364C2×C3×C4—22no—
365C2×C3×C4—24no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—24no—
369C2×C3×C4—102no—
370C2×C3×C4—102no—
371C2×C3×C4—202no—
372C2×C3×C4C2×C22⋊D512yes—
373C2×C3×C4C2×C22⋊D511yescharacteristic
374C2×C3×C4C2×C22⋊D512yes—
375C2×C3×C4C2×C22⋊D512yes—
376C2×C3×C4C23×D511yescharacteristic
377C3×C8—104no—
378C22×C5—22no—
379C22×C5—24no—
380C22×C5—41no—
381C22×C5—42no—
382C22×C5—82no—
383C22×C5—84no—
384C22×C5C3×#12012yes—
385C22×C5C22×C3×D411yescharacteristic
386C4×C5—24no—
387C4×C5—22no—
388C4×C5C3×#12011yescharacteristic
389C4×C5C3×#12011yescharacteristic
390C4×C5C22×C3×D412yes—
391Dic5—42no—
392Dic5—42no—
393#41—24no—
394#41—28no—
395#41—24no—
396#41—24no—
397#41—24no—
398#41—22no—
399#41—22no—
400#41—24no—
401#41—22no—
402#41—22no—
403#41C3×C22⋊D514yes—
404#41C22×C3×D514yes—
405C2×Q8—22no—
406C2×Q8—24no—
407C2×Q8—52no—
408C2×Q8—101no—
409C2×Q8—102no—
410C2×Q8—102no—
411C2×Q8C3×C22⋊D512yes—
412C2×Q8C22×C3×D511yescharacteristic
413C2×D4—28no—
414C2×D4—22no—
415C2×D4—24no—
416C2×D4—22no—
417C2×D4—24no—
418C2×D4C3×C22⋊D512yes—
419C2×D4C3×C22⋊D512yes—
420C2×D4C22×C3×D511yescharacteristic
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—101no—
424C22×C4C3×C22⋊D512yes—
425C22×C4C22×C3×D511yescharacteristic
426Q16—108no—
427SD16—108no—
428M16—54no—
429M16—102no—
430C2×C8—102no—
431C4⋊C4—101no—
432C4⋊C4—104no—
433C22⋊C4—101no—
434C22⋊C4—104no—
435C42—52no—
436C3×C5#302211yescharacteristic, complemented, Hall
437C22×C3—22no—
438C22×C3—24no—
439C22×C3—41no—
440C22×C3—42no—
441C22×C3—82no—
442C22×C3—84no—
443C22×C3#409912yes—
444C22×C3C22×C22⋊D511yescharacteristic
445C3×C4—24no—
446C3×C4—22no—
447C3×C4—202no—
448C3×C4—202no—
449C3×C4#409911yescharacteristic
450C3×C4#409911yescharacteristic
451C3×C4C22×C22⋊D512yes—
452C2×C5—21no—
453C2×C5—22no—
454C2×C5—42no—
455C2×C5—44no—
456C2×C5C2×C3×#12011yescharacteristic
457C23—24no—
458C23—22no—
459C23C2×C3×C22⋊D512yes—
460C23C2×C3×C22⋊D511yescharacteristic
461Q8—24no—
462Q8—22no—
463Q8—22no—
464Q8—104no—
465Q8—201no—
466Q8—201no—
467Q8C2×C3×C22⋊D514yes—
468D4—28no—
469D4—24no—
470D4—24no—
471D4—24no—
472D4—24no—
473D4—22no—
474D4—22no—
475D4C2×C3×C22⋊D514yes—
476C2×C4—24no—
477C2×C4—28no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—102no—
484C2×C4—102no—
485C2×C4—202no—
486C2×C4C2×C3×C22⋊D512yes—
487C2×C4C2×C3×C22⋊D511yescharacteristic
488C2×C4C2×C3×C22⋊D512yes—
489C2×C4C2×C3×C22⋊D512yes—
490C2×C4C23×C3×D511yescharacteristic
491C8—104no—
492C2×C3—21no—
493C2×C3—22no—
494C2×C3—42no—
495C2×C3—44no—
496C2×C3C2×#409911yescharacteristic
497C5C3×#302211yescharacteristic, complemented, Sylow
498C22—22no—
499C22—24no—
500C22—41no—
501C22—42no—
502C22—82no—
503C22—84no—
504C22C3×#409912yes—
505C22C22×C3×C22⋊D511yescharacteristic
506C4—24no—
507C4—22no—
508C4—202no—
509C4—202no—
510C4C3×#409911yescharacteristic
511C4C3×#409911yescharacteristic
512C4C22×C3×C22⋊D512yes—
513C3#1061585411yescharacteristic, direct product factor, Sylow
514C2—21no—
515C2—22no—
516C2—42no—
517C2—44no—
518C2C2×C3×#409911yescharacteristic
subgroupG/H
centerC2×C3C2×#4099
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
Fitting, prime radicalC3×C5×#585C2
socleC2×C3×C5C2×#120
hypercenterC3×#585D5
hypocenterC5C3×#3022
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C3×#585

Complements

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

NH ≅ G/Ntype
2. C3×#66159C2semidirect
3. C2×C3×#4097C2semidirect
4. C3×#65535C2semidirect
5. C3×#65531C2semidirect
6. C3×#65526C2semidirect
7. C3×#65525C2semidirect
8. #10615854C3direct (N × H)
18. C22×C3×Dic10C22semidirect
20. C3×#4097C22semidirect
21. C3×#4090C22semidirect
23. C3×#4082C22semidirect
26. C3×#4072C22semidirect
28. C2×C3×#681C22semidirect
29. C3×#3969C22semidirect
30. C3×#3968C22semidirect
31. C3×#3965C22semidirect
34. #66159C2×C3semidirect
35. C2×#4097C2×C3semidirect
36. #65535C2×C3semidirect
37. #65531C2×C3semidirect
38. #65526C2×C3semidirect
39. #65525C2×C3semidirect
99. C22×Dic10C22×C3semidirect
101. #4097C22×C3semidirect
102. #4090C22×C3semidirect
104. #4082C22×C3semidirect
107. #4072C22×C3semidirect
109. C2×#681C22×C3semidirect
110. #3969C22×C3semidirect
111. #3968C22×C3semidirect
112. #3965C22×C3semidirect
436. C3×C5#3022semidirect
497. C5C3×#3022semidirect
513. C3#10615854direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5852
C3×#661592
C2×C3×#40972
C3×#655352
C3×#655312
C3×#655262
C3×#655252
#106158543
C3×#30225