Finite Group Databasebeta

← C3×#10615835 #14425303

Subgroups — C3×#10615835 #14425303

Order 1920.

1524 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×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C3×#66156C212yesmaximal, complemented
3C3×#66139C211yesmaximal, characteristic, complemented
4C3×#65531C214yesmaximal, complemented
5C3×#65530C214yesmaximal, complemented
6C3×#65525C212yesmaximal, complemented
7C2×C3×#3965C211yesmaximal, characteristic, complemented
8#10615835C311yesmaximal, characteristic, direct product factor, Hall
9C3×C5×H(5,𝔽2)—24no—
10C3×C5×H(5,𝔽2)—22no—
11C3×C5×H(5,𝔽2)C2214yes—
12C2×C3×C5×#41—22no—
13C2×C3×C5×#41C2212yes—
14C22×C3×C5×D4—22no—
15C22×C3×C5×D4—22no—
16C22×C3×C5×D4C2211yescharacteristic
17C2×C3×#709C2211yescharacteristic, complemented
18C3×#4099—22no—
19C3×#4097C2214yescomplemented
20C3×#4096—24no—
21C3×#4095C2214yescomplemented
22C3×#4088—22no—
23C3×#4086C2212yescomplemented
24C3×#4084—22no—
25C3×#4082C2212yescomplemented
26C3×#4078—22no—
27C3×#4072C2211yescharacteristic, complemented
28C2×C3×#681C2212yescomplemented
29C3×#3969C2218yescomplemented
30C3×#3965C2214yescomplemented
31C3×#3019—51nomaximal, Hall
32C2×C5×H(5,𝔽2)C2×C311yescharacteristic, complemented
33#66156C2×C312yescomplemented
34#66139C2×C311yescharacteristic, complemented
35#65531C2×C314yescomplemented
36#65530C2×C314yescomplemented
37#65525C2×C312yescomplemented
38C2×#3965C2×C311yescharacteristic, complemented
39C24×C3×C5—22no—
40C24×C3×C5D412yes—
41C3×C5×#41—28no—
42C3×C5×#41—28no—
43C3×C5×#41—22no—
44C3×C5×#41—22no—
45C3×C5×#41D414yescomplemented
46C3×C5×#41C2314yes—
47C2×C3×C5×Q8D412yescomplemented
48C2×C3×C5×D4—24no—
49C2×C3×C5×D4—28no—
50C2×C3×C5×D4—28no—
51C2×C3×C5×D4—22no—
52C2×C3×C5×D4—28no—
53C2×C3×C5×D4—28no—
54C2×C3×C5×D4—22no—
55C2×C3×C5×D4—22no—
56C2×C3×C5×D4D412yes—
57C2×C3×C5×D4D412yescomplemented
58C2×C3×C5×D4C2312yes—
59C22×C3×C4×C5—22no—
60C22×C3×C4×C5—22no—
61C22×C3×C4×C5C2311yescharacteristic
62C2×C3×C22⋊D5—21no—
63C2×C3×C22⋊D5—24no—
64C3×#709—22no—
65C3×#709—22no—
66C2×C3×D20C2311yescharacteristic, complemented
67C2×C3×C4×D5—21no—
68C2×C3×Dic10C2311yescharacteristic, complemented
69C3×C22⋊Dic5—24no—
70C3×C22⋊Dic5—21no—
71C3×#689—24no—
72C3×#688—24no—
73C3×#687—24no—
74C3×#686—24no—
75C3×C4⋊Dic5—21no—
76C3×C4×Dic5C2312yescomplemented
77C3×#681—22no—
78C3×#681C2314yescomplemented
79C2×C3×#168—22no—
80C2×C3×H(5,𝔽2)D511yescharacteristic, complemented
81C3×#575—52no—
82C3×#535—51no—
83C3×#455—54no—
84C3×#453—54no—
85C3×#421—52no—
86C2×C3×#100—51no—
87C5×H(5,𝔽2)—24no—
88C5×H(5,𝔽2)—22no—
89C5×H(5,𝔽2)C22×C314yes—
90C2×C5×#41—22no—
91C2×C5×#41C22×C312yes—
92C22×C5×D4—22no—
93C22×C5×D4—22no—
94C22×C5×D4C22×C311yescharacteristic
95C2×#709C22×C311yescharacteristic, complemented
96#4099—22no—
97#4097C22×C314yescomplemented
98#4096—24no—
99#4095C22×C314yescomplemented
100#4088—22no—
101#4086C22×C312yescomplemented
102#4084—22no—
103#4082C22×C312yescomplemented
104#4078—22no—
105#4072C22×C311yescharacteristic, complemented
106C2×#681C22×C312yescomplemented
107#3969C22×C318yescomplemented
108#3965C22×C314yescomplemented
109#3019—51noSylow
110C23×C3×C5—28no—
111C23×C3×C5—22no—
112C23×C3×C5—22no—
113C23×C3×C5—22no—
114C23×C3×C5—82no—
115C23×C3×C5—84no—
116C23×C3×C5C2×D414yes—
117C23×C3×C5C2×D411yescharacteristic
118C22×C3×D5—21no—
119C3×C5×Q8—22no—
120C3×C5×Q8C2×D414yes—
121C3×C5×D4—216no—
122C3×C5×D4—24no—
123C3×C5×D4—24no—
124C3×C5×D4—24no—
125C3×C5×D4—22no—
126C3×C5×D4—24no—
127C3×C5×D4C2×D414yes—
128C2×C3×C4×C5—28no—
129C2×C3×C4×C5—22no—
130C2×C3×C4×C5—24no—
131C2×C3×C4×C5—24no—
132C2×C3×C4×C5C2×D411yescharacteristic
133C2×C3×C4×C5C2×D412yes—
134C2×C3×C4×C5C2×D412yes—
135C2×C3×C4×C5C2411yescharacteristic
136C3×C22⋊D5—41no—
137C3×C22⋊D5—41no—
138C3×C22⋊D5—84no—
139C2×C3×Dic5—22no—
140C2×C3×Dic5—21no—
141C3×D20—22no—
142C3×C4×D5—42no—
143C3×Dic10—22no—
144C3×#168—24no—
145C3×H(5,𝔽2)—24no—
146C3×H(5,𝔽2)—22no—
147C3×H(5,𝔽2)C2×D514yes—
148C2×C3×#41—22no—
149C2×C3×#41—51no—
150C2×C3×#41C2×D512yes—
151C22×C3×D4—22no—
152C22×C3×D4—22no—
153C22×C3×D4C2×D511yescharacteristic
154C3×#137—54no—
155C3×#137—102no—
156C3×AGL(1,ℤ/8)—54no—
157C3×AGL(1,ℤ/8)—102no—
158C3×#135—104no—
159C2×C3×M16—52no—
160C3×#127—52no—
161C3×#124—52no—
162C3×#121—102no—
163C3×#120—102no—
164C3×#117—51no—
165C3×#104—58no—
166C3×#100—54no—
167C24×C5—22no—
168C24×C5C3×D412yes—
169C5×#41—28no—
170C5×#41—28no—
171C5×#41—22no—
172C5×#41—22no—
173C5×#41C3×D414yescomplemented
174C5×#41C23×C314yes—
175C2×C5×Q8C3×D412yescomplemented
176C2×C5×D4—24no—
177C2×C5×D4—28no—
178C2×C5×D4—28no—
179C2×C5×D4—22no—
180C2×C5×D4—28no—
181C2×C5×D4—28no—
182C2×C5×D4—22no—
183C2×C5×D4—22no—
184C2×C5×D4C3×D412yes—
185C2×C5×D4C3×D412yescomplemented
186C2×C5×D4C23×C312yes—
187C22×C4×C5—22no—
188C22×C4×C5—22no—
189C22×C4×C5C23×C311yescharacteristic
190C2×C22⋊D5—21no—
191C2×C22⋊D5—24no—
192#709—22no—
193#709—22no—
194C2×D20C23×C311yescharacteristic, complemented
195C2×C4×D5—21no—
196C2×Dic10C23×C311yescharacteristic, complemented
197C22⋊Dic5—21no—
198C22⋊Dic5—24no—
199#689—24no—
200#688—24no—
201#687—24no—
202#686—24no—
203C4⋊Dic5—21no—
204C4×Dic5C23×C312yescomplemented
205#681—22no—
206#681C23×C314yescomplemented
207C2×#168—22no—
208C2×H(5,𝔽2)C3×D511yescharacteristic, complemented
209#575—52no—
210#535—51no—
211#455—54no—
212#453—54no—
213#421—52no—
214C2×#100—51no—
215C22×C3×C5—24no—
216C22×C3×C5—24no—
217C22×C3×C5—41no—
218C22×C3×C5—44no—
219C22×C3×C5—44no—
220C22×C3×C5—44no—
221C22×C3×C5—82no—
222C22×C3×C5—88no—
223C22×C3×C5#12011yescharacteristic
224C22×C3×C5#12011yescharacteristic
225C22×C3×C5C22×D411yescharacteristic
226C2×C3×D5—42no—
227C2×C3×D5—41no—
228C3×C4×C5—24no—
229C3×C4×C5#12012yes—
230C3×C4×C5C22×D412yes—
231C3×Dic5—42no—
232C3×Dic5—41no—
233C24×C3—22no—
234C24×C3C22⋊D512yes—
235C3×#41—28no—
236C3×#41—28no—
237C3×#41—22no—
238C3×#41—22no—
239C3×#41—102no—
240C3×#41—102no—
241C3×#41C22⋊D514yescomplemented
242C3×#41C22×D514yes—
243C2×C3×Q8—51no—
244C2×C3×Q8C22⋊D512yescomplemented
245C2×C3×D4—24no—
246C2×C3×D4—28no—
247C2×C3×D4—28no—
248C2×C3×D4—22no—
249C2×C3×D4—28no—
250C2×C3×D4—28no—
251C2×C3×D4—22no—
252C2×C3×D4—22no—
253C2×C3×D4—51no—
254C2×C3×D4—101no—
255C2×C3×D4—104no—
256C2×C3×D4C22⋊D512yes—
257C2×C3×D4C22⋊D512yescomplemented
258C2×C3×D4C22×D512yes—
259C22×C3×C4—22no—
260C22×C3×C4—22no—
261C22×C3×C4—101no—
262C22×C3×C4C22×D511yescharacteristic
263C3×Q16—104no—
264C3×SD16—104no—
265C3×SD16—104no—
266C3×D8—104no—
267C3×M16—54no—
268C3×M16—102no—
269C2×C3×C8—102no—
270C3×C4⋊C4—101no—
271C3×C22⋊C4—101no—
272C3×C22⋊C4—104no—
273C3×C42—52no—
274C23×C5—28no—
275C23×C5—22no—
276C23×C5—22no—
277C23×C5—22no—
278C23×C5—82no—
279C23×C5—84no—
280C23×C5C2×C3×D414yes—
281C23×C5C2×C3×D411yescharacteristic
282C22×D5—21no—
283C5×Q8—22no—
284C5×Q8C2×C3×D414yes—
285C5×D4—216no—
286C5×D4—24no—
287C5×D4—24no—
288C5×D4—24no—
289C5×D4—22no—
290C5×D4—24no—
291C5×D4C2×C3×D414yes—
292C2×C4×C5—28no—
293C2×C4×C5—22no—
294C2×C4×C5—24no—
295C2×C4×C5—24no—
296C2×C4×C5C2×C3×D411yescharacteristic
297C2×C4×C5C2×C3×D412yes—
298C2×C4×C5C2×C3×D412yes—
299C2×C4×C5C24×C311yescharacteristic
300C22⋊D5—41no—
301C22⋊D5—41no—
302C22⋊D5—84no—
303C2×Dic5—22no—
304C2×Dic5—21no—
305D20—22no—
306C4×D5—42no—
307Dic10—22no—
308#168—24no—
309H(5,𝔽2)—24no—
310H(5,𝔽2)—22no—
311H(5,𝔽2)C2×C3×D514yes—
312C2×#41—22no—
313C2×#41—51no—
314C2×#41C2×C3×D512yes—
315C22×D4—22no—
316C22×D4—22no—
317C22×D4C2×C3×D511yescharacteristic
318#137—54no—
319#137—102no—
320AGL(1,ℤ/8)—54no—
321AGL(1,ℤ/8)—102no—
322#135—104no—
323C2×M16—52no—
324#127—52no—
325#124—52no—
326#121—102no—
327#120—102no—
328#117—51no—
329#104—58no—
330#100—54no—
331C2×C3×C5—21no—
332C2×C3×C5—21no—
333C2×C3×C5—21no—
334C2×C3×C5—44no—
335C2×C3×C5—44no—
336C2×C3×C5C2×#12011yescharacteristic
337C3×D5—81no—
338C23×C3—28no—
339C23×C3—22no—
340C23×C3—22no—
341C23×C3—22no—
342C23×C3—82no—
343C23×C3—84no—
344C23×C3—101no—
345C23×C3C2×C22⋊D514yes—
346C23×C3C2×C22⋊D511yescharacteristic
347C3×Q8—22no—
348C3×Q8—102no—
349C3×Q8C2×C22⋊D514yes—
350C3×D4—216no—
351C3×D4—24no—
352C3×D4—24no—
353C3×D4—24no—
354C3×D4—22no—
355C3×D4—24no—
356C3×D4—102no—
357C3×D4—201no—
358C3×D4—201no—
359C3×D4—404no—
360C3×D4C2×C22⋊D514yes—
361C2×C3×C4—28no—
362C2×C3×C4—22no—
363C2×C3×C4—24no—
364C2×C3×C4—24no—
365C2×C3×C4—102no—
366C2×C3×C4—101no—
367C2×C3×C4—202no—
368C2×C3×C4C2×C22⋊D511yescharacteristic
369C2×C3×C4C2×C22⋊D512yes—
370C2×C3×C4C2×C22⋊D512yes—
371C2×C3×C4C23×D511yescharacteristic
372C3×C8—104no—
373C22×C5—24no—
374C22×C5—24no—
375C22×C5—41no—
376C22×C5—44no—
377C22×C5—44no—
378C22×C5—44no—
379C22×C5—82no—
380C22×C5—88no—
381C22×C5C3×#12011yescharacteristic
382C22×C5C3×#12011yescharacteristic
383C22×C5C22×C3×D411yescharacteristic
384C2×D5—42no—
385C2×D5—41no—
386C4×C5—24no—
387C4×C5C3×#12012yes—
388C4×C5C22×C3×D412yes—
389Dic5—42no—
390Dic5—41no—
391C24—22no—
392C24C3×C22⋊D512yes—
393#41—28no—
394#41—28no—
395#41—22no—
396#41—22no—
397#41—102no—
398#41—102no—
399#41C3×C22⋊D514yescomplemented
400#41C22×C3×D514yes—
401C2×Q8—51no—
402C2×Q8C3×C22⋊D512yescomplemented
403C2×D4—24no—
404C2×D4—28no—
405C2×D4—28no—
406C2×D4—22no—
407C2×D4—28no—
408C2×D4—28no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—51no—
412C2×D4—101no—
413C2×D4—104no—
414C2×D4C3×C22⋊D512yes—
415C2×D4C3×C22⋊D512yescomplemented
416C2×D4C22×C3×D512yes—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—101no—
420C22×C4C22×C3×D511yescharacteristic
421Q16—104no—
422SD16—104no—
423SD16—104no—
424D8—104no—
425M16—54no—
426M16—102no—
427C2×C8—102no—
428C4⋊C4—101no—
429C22⋊C4—101no—
430C22⋊C4—104no—
431C42—52no—
432C3×C5#301911yescharacteristic, complemented, Hall
433C22×C3—24no—
434C22×C3—24no—
435C22×C3—41no—
436C22×C3—44no—
437C22×C3—44no—
438C22×C3—44no—
439C22×C3—82no—
440C22×C3—88no—
441C22×C3—202no—
442C22×C3—201no—
443C22×C3#409911yescharacteristic
444C22×C3#409911yescharacteristic
445C22×C3C22×C22⋊D511yescharacteristic
446C3×C4—24no—
447C3×C4—202no—
448C3×C4—201no—
449C3×C4#409912yes—
450C3×C4C22×C22⋊D512yes—
451C2×C5—21no—
452C2×C5—21no—
453C2×C5—21no—
454C2×C5—44no—
455C2×C5—44no—
456C2×C5C2×C3×#12011yescharacteristic
457D5—81no—
458C23—28no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—82no—
463C23—84no—
464C23—101no—
465C23C2×C3×C22⋊D514yes—
466C23C2×C3×C22⋊D511yescharacteristic
467Q8—22no—
468Q8—102no—
469Q8C2×C3×C22⋊D514yes—
470D4—216no—
471D4—24no—
472D4—24no—
473D4—24no—
474D4—22no—
475D4—24no—
476D4—102no—
477D4—201no—
478D4—201no—
479D4—404no—
480D4C2×C3×C22⋊D514yes—
481C2×C4—28no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—102no—
486C2×C4—101no—
487C2×C4—202no—
488C2×C4C2×C3×C22⋊D511yescharacteristic
489C2×C4C2×C3×C22⋊D512yes—
490C2×C4C2×C3×C22⋊D512yes—
491C2×C4C23×C3×D511yescharacteristic
492C8—104no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—44no—
497C2×C3—44no—
498C2×C3—401no—
499C2×C3C2×#409911yescharacteristic
500C5C3×#301911yescharacteristic, complemented, Sylow
501C22—24no—
502C22—24no—
503C22—41no—
504C22—44no—
505C22—44no—
506C22—44no—
507C22—82no—
508C22—88no—
509C22—202no—
510C22—201no—
511C22C3×#409911yescharacteristic
512C22C3×#409911yescharacteristic
513C22C22×C3×C22⋊D511yescharacteristic
514C4—24no—
515C4—202no—
516C4—201no—
517C4C3×#409912yes—
518C4C22×C3×C22⋊D512yes—
519C3#1061583511yescharacteristic, direct product factor, Sylow
520C2—21no—
521C2—21no—
522C2—21no—
523C2—44no—
524C2—44no—
525C2—401no—
526C2C2×C3×#409911yescharacteristic
subgroupG/H
centerC2×C3C2×#4099
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC2×C3×C5×H(5,𝔽2)C2
socleC2×C3×C5C2×#120
hypercenterC2×C3×H(5,𝔽2)D5
hypocenterC5C3×#3019
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×H(5,𝔽2)C2semidirect
2. C3×#66156C2semidirect
3. C3×#66139C2semidirect
4. C3×#65531C2semidirect
5. C3×#65530C2semidirect
6. C3×#65525C2semidirect
7. C2×C3×#3965C2semidirect
8. #10615835C3direct (N × H)
17. C2×C3×#709C22semidirect
19. C3×#4097C22semidirect
21. C3×#4095C22semidirect
23. C3×#4086C22semidirect
25. C3×#4082C22semidirect
27. C3×#4072C22semidirect
28. C2×C3×#681C22semidirect
29. C3×#3969C22semidirect
30. C3×#3965C22semidirect
32. C2×C5×H(5,𝔽2)C2×C3semidirect
33. #66156C2×C3semidirect
34. #66139C2×C3semidirect
35. #65531C2×C3semidirect
36. #65530C2×C3semidirect
37. #65525C2×C3semidirect
38. C2×#3965C2×C3semidirect
45. C3×C5×#41D4semidirect
47. C2×C3×C5×Q8D4semidirect
57. C2×C3×C5×D4D4semidirect
66. C2×C3×D20C23semidirect
68. C2×C3×Dic10C23semidirect
76. C3×C4×Dic5C23semidirect
78. C3×#681C23semidirect
80. C2×C3×H(5,𝔽2)D5semidirect
95. C2×#709C22×C3semidirect
97. #4097C22×C3semidirect
99. #4095C22×C3semidirect
101. #4086C22×C3semidirect
103. #4082C22×C3semidirect
105. #4072C22×C3semidirect
106. C2×#681C22×C3semidirect
107. #3969C22×C3semidirect
108. #3965C22×C3semidirect
173. C5×#41C3×D4semidirect
175. C2×C5×Q8C3×D4semidirect
185. C2×C5×D4C3×D4semidirect
194. C2×D20C23×C3semidirect
196. C2×Dic10C23×C3semidirect
204. C4×Dic5C23×C3semidirect
206. #681C23×C3semidirect
208. C2×H(5,𝔽2)C3×D5semidirect
241. C3×#41C22⋊D5semidirect
244. C2×C3×Q8C22⋊D5semidirect
257. C2×C3×D4C22⋊D5semidirect
399. #41C3×C22⋊D5semidirect
402. C2×Q8C3×C22⋊D5semidirect
415. C2×D4C3×C22⋊D5semidirect
432. C3×C5#3019semidirect
500. C5C3×#3019semidirect
519. C3#10615835direct (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×H(5,𝔽2)2
C3×#661562
C3×#661392
C3×#655312
C3×#655302
C3×#655252
C2×C3×#39652
#106158353
C3×#30195