Finite Group Databasebeta

← #14412723

Subgroups — #14412723

Order 1920.

1452 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×D5×2−1+4C211yesmaximal, characteristic
2#11756995C211yesmaximal, characteristic, complemented
3D5⋊#948C211yesmaximal, characteristic, complemented
4#11752453C213yesmaximal, complemented
5#11752452C213yesmaximal, complemented
6#11752441C213yesmaximal, complemented
7C2×#91238C213yesmaximal, complemented
8#10617561—31nomaximal, Hall
9C3×C5×2−1+4C411yescharacteristic, complemented
10C3×#4150—23no—
11C3×#4150C413yescomplemented
12C3×#4150C411yescharacteristic, complemented
13C3×D5×#41—23no—
14C3×D5×#41C2213yes—
15C3×D5×#41C2211yescharacteristic
16C3×#4147—21no—
17C3×#4147C413yescomplemented
18C2×C3×Q8×D5—21no—
19C2×C3×Q8×D5C2213yes—
20Q8×ΓL(1,𝔽16)—21no—
21#91992—21no—
22D4×ΓL(1,𝔽16)—23no—
23#91990—23no—
24#91989C2213yes—
25C2×C4⋊ΓL(1,𝔽16)C2213yes—
26D5⋊#208C2213yes—
27C2×D5⋊(C3⋊C8)C2213yes—
28#91238C2216yes—
29#91238C2213yes—
30#91238C2213yes—
31#91236C2213yes—
32#91236C2211yescharacteristic
33#84111—51nomaximal, Hall
34D5×2−1+4D311yescharacteristic
35#66266—31no—
36D5⋊#131—31no—
37#65794—33no—
38#65793—33no—
39#65782—33no—
40C2×#4009—33no—
41C22×ΓL(1,𝔽16)—23no—
42C3×C5×#41—23no—
43C3×C5×#41C2×C411yescharacteristic
44C3×C5×#41C2×C413yescomplemented
45C2×C3×C5×Q8—21no—
46C2×C3×C5×Q8C2×C413yescomplemented
47C3×#713—23no—
48C3×#713C2×C413yescomplemented
49C3×#713D411yescharacteristic
50C3×Q8×D5—21no—
51C3×Q8×D5—23no—
52C3×Q8×D5—23no—
53C3×Q8×D5—23no—
54C3×Q8×D5C2×C416yescomplemented
55C3×Q8×D5D411yescharacteristic
56C3×Q8×D5C2313yes—
57C3×#711—23no—
58C3×#711—26no—
59C3×#711—23no—
60C3×#711C2×C413yescomplemented
61C3×#711D413yes—
62C3×D4×D5—23no—
63C3×D4×D5D413yes—
64C3×D4×D5C2311yescharacteristic
65C3×#709—26no—
66C3×#709—23no—
67C3×#709—23no—
68C3×#709C2×C413yes—
69C3×#709C2×C413yescomplemented
70C2×C3×C4×D5—23no—
71C2×C3×C4×D5—23no—
72C2×C3×C4×D5C2313yes—
73C2×C3×Dic10—23no—
74C2×C3×Dic10—23no—
75C2×C3×Dic10C2×C413yescomplemented
76#6763—23no—
77#6762—23no—
78C2×C3⋊#170—23no—
79C4⋊ΓL(1,𝔽16)—23no—
80C4⋊ΓL(1,𝔽16)C2313yes—
81C4⋊ΓL(1,𝔽16)C2311yescharacteristic
82C4×ΓL(1,𝔽16)—23no—
83#6758—23no—
84D5⋊(C3⋊C8)—23no—
85D5⋊(C3⋊C8)C2311yescharacteristic
86D5⋊(C3⋊C8)C2313yes—
87C2×C3×2−1+4—51no—
88#5827—51no—
89C2×#948—51no—
90#5233—53no—
91#5232—53no—
92#5225—53no—
93C2×#835—53no—
94C5×2−1+4Dic311yescharacteristic, complemented
95#4150—23no—
96#4150Dic313yescomplemented
97#4150Dic311yescharacteristic, complemented
98D5×#41—23no—
99D5×#41C2×D313yes—
100D5×#41C2×D311yescharacteristic
101#4147—21no—
102#4147Dic313yescomplemented
103C2×Q8×D5—21no—
104C2×Q8×D5C2×D313yes—
105Q8×AGL(1,𝔽5)—61no—
106#4133—61no—
107D4×AGL(1,𝔽5)—63no—
108#4131—63no—
109#4130—33no—
110C2×C4⋊AGL(1,𝔽5)—33no—
111D5⋊M16—33no—
112C2×D5⋊C8—33no—
113#4009—36no—
114#4009—33no—
115#4009—33no—
116#4007—33no—
117#4007—31no—
118#2898—151noSylow
119C22×C3×D5—21no—
120C22×C3×D5C2×D413yes—
121C2×ΓL(1,𝔽16)—21no—
122C2×ΓL(1,𝔽16)—23no—
123C2×ΓL(1,𝔽16)—83no—
124C3×C5×Q8—23no—
125C3×C5×Q8C22⋊C411yescharacteristic
126C3×C5×Q8C22×C413yes—
127C3×C5×D4—23no—
128C3×C5×D4C22⋊C413yes—
129C3×C5×D4C22×C411yescharacteristic
130C2×C3×C4×C5—23no—
131C2×C3×C4×C5—23no—
132C2×C3×C4×C5C22×C413yes—
133C3×C22⋊D5—23no—
134C3×C22⋊D5—23no—
135C3×C22⋊D5—21no—
136C3×C22⋊D5C22⋊C416yes—
137C2×C3×Dic5—23no—
138C2×C3×Dic5—23no—
139C2×C3×Dic5C2×D413yes—
140C3×D20—23no—
141C3×D20C22⋊C413yes—
142C3×D20C22×C411yescharacteristic
143C3×C4×D5—26no—
144C3×C4×D5—23no—
145C3×C4×D5—23no—
146C3×C4×D5—23no—
147C3×C4×D5C22×C413yes—
148C3×C4×D5C2×D413yes—
149C3×C4×D5C2×D413yes—
150C3×C4×D5C2411yescharacteristic
151C3×Dic10—23no—
152C3×Dic10—26no—
153C3×Dic10—23no—
154C3×Dic10C22⋊C413yes—
155C3×Dic10C22×C413yes—
156C3⋊#170—21no—
157C3⋊#170—23no—
158C3×2−1+4—51no—
159C3×2−1+4—53no—
160C3×2−1+4—53no—
161C3×2−1+4—101no—
162C3×2−1+4—103no—
163C3×2−1+4AGL(1,𝔽5)11yescharacteristic, complemented
164C2×C3×#41—51no—
165C2×C3×#41—53no—
166C2×C3×#41—103no—
167C22×C3×Q8—53no—
168C22×C3×Q8—101no—
169#948—103no—
170#948—101no—
171Q8×Dic3—101no—
172D4×Dic3—103no—
173#926—53no—
174C2×C4⋊Dic3—53no—
175C2×#208—53no—
176C22×C3⋊C8—53no—
177#835—53no—
178#835—56no—
179#835—53no—
180#832—53no—
181#832—51no—
182C22×AGL(1,𝔽5)—63no—
183C5×#41—23no—
184C5×#41C2×Dic311yescharacteristic
185C5×#41C2×Dic313yescomplemented
186C2×C5×Q8—21no—
187C2×C5×Q8C2×Dic313yescomplemented
188#713—23no—
189#713C2×Dic313yescomplemented
190#713#6711yescharacteristic
191Q8×D5—21no—
192Q8×D5—23no—
193Q8×D5—23no—
194Q8×D5—23no—
195Q8×D5C2×Dic316yescomplemented
196Q8×D5#6711yescharacteristic
197Q8×D5C22×D313yes—
198#711—23no—
199#711—26no—
200#711—23no—
201#711C2×Dic313yescomplemented
202#711#6713yes—
203D4×D5—23no—
204D4×D5#6713yes—
205D4×D5C22×D311yescharacteristic
206#709—26no—
207#709—23no—
208#709—23no—
209#709C2×Dic313yes—
210#709C2×Dic313yescomplemented
211C2×C4×D5—23no—
212C2×C4×D5—23no—
213C2×C4×D5C22×D313yes—
214C2×Dic10—23no—
215C2×Dic10—23no—
216C2×Dic10C2×Dic313yescomplemented
217C22⋊AGL(1,𝔽5)—63no—
218#704—63no—
219C2×#170—63no—
220C4⋊AGL(1,𝔽5)—33no—
221C4⋊AGL(1,𝔽5)—31no—
222C4⋊AGL(1,𝔽5)—63no—
223C4×AGL(1,𝔽5)—63no—
224#700—63no—
225D5⋊C8—31no—
226D5⋊C8—33no—
227D5⋊C8—63no—
228C2×2−1+4—51no—
229C2×#131—151no—
230#520—151no—
231#419—153no—
232#417—153no—
233#416—153no—
234C2×#103—153no—
235C22×C3×C5—21no—
236C22×C3×C5C2×C22⋊C413yes—
237C2×C3×D5—23no—
238C2×C3×D5—23no—
239C2×C3×D5—21no—
240C2×C3×D5—41no—
241C2×C3×D5—41no—
242C2×C3×D5C2×C22⋊C413yes—
243C2×C3×D5C22×D411yescharacteristic
244ΓL(1,𝔽16)—43no—
245ΓL(1,𝔽16)—41no—
246C3×C4×C5—23no—
247C3×C4×C5C2×C22⋊C413yes—
248C3×C4×C5C23×C411yescharacteristic
249C3×Dic5—23no—
250C3×Dic5C2×C22⋊C413yes—
251C3×Dic5C22×D411yescharacteristic
252C3×#41—23no—
253C3×#41—51no—
254C3×#41—53no—
255C3×#41—53no—
256C3×#41—53no—
257C3×#41—53no—
258C3×#41—53no—
259C3×#41—103no—
260C3×#41—106no—
261C3×#41—103no—
262C3×#41—103no—
263C3×#41—106no—
264C3×#41—103no—
265C3×#41—103no—
266C3×#41C2×AGL(1,𝔽5)11yescharacteristic
267C3×#41C2×AGL(1,𝔽5)13yescomplemented
268C2×C3×Q8—21no—
269C2×C3×Q8—51no—
270C2×C3×Q8—53no—
271C2×C3×Q8—56no—
272C2×C3×Q8—53no—
273C2×C3×Q8—103no—
274C2×C3×Q8—103no—
275C2×C3×Q8—103no—
276C2×C3×Q8—103no—
277C2×C3×Q8—103no—
278C2×C3×Q8—101no—
279C2×C3×Q8C2×AGL(1,𝔽5)13yescomplemented
280C2×C3×D4—53no—
281C2×C3×D4—51no—
282C2×C3×D4—103no—
283C22×C3×C4—53no—
284C22×C3×C4—103no—
285C22×C3×C4—103no—
286C22×Dic3—103no—
287#217—103no—
288C4⋊Dic3—53no—
289C4⋊Dic3—51no—
290C4⋊Dic3—103no—
291C4×Dic3—103no—
292#208—103no—
293#208—103no—
294C2×C3⋊C8—51no—
295C2×C3⋊C8—53no—
296C2×C3⋊C8—103no—
297C2×C3⋊C8—103no—
298C22×D5—21no—
299C22×D5C2×#6713yes—
300C2×AGL(1,𝔽5)—61no—
301C2×AGL(1,𝔽5)—63no—
302C2×AGL(1,𝔽5)—243no—
303C5×Q8—23no—
304C5×Q8#21711yescharacteristic
305C5×Q8C22×Dic313yes—
306C5×D4—23no—
307C5×D4#21713yes—
308C5×D4C22×Dic311yescharacteristic
309C2×C4×C5—23no—
310C2×C4×C5—23no—
311C2×C4×C5C22×Dic313yes—
312C22⋊D5—23no—
313C22⋊D5—23no—
314C22⋊D5—21no—
315C22⋊D5#21716yes—
316C2×Dic5—23no—
317C2×Dic5—23no—
318C2×Dic5C2×#6713yes—
319D20—23no—
320D20#21713yes—
321D20C22×Dic311yescharacteristic
322C4×D5—26no—
323C4×D5—23no—
324C4×D5—23no—
325C4×D5—23no—
326C4×D5C22×Dic313yes—
327C4×D5C2×#6713yes—
328C4×D5C2×#6713yes—
329C4×D5C23×D311yescharacteristic
330Dic10—23no—
331Dic10—26no—
332Dic10—23no—
333Dic10#21713yes—
334Dic10C22×Dic313yes—
335#170—61no—
336#170—63no—
3372−1+4—51no—
3382−1+4—53no—
3392−1+4—53no—
3402−1+4—101no—
3412−1+4—103no—
3422−1+4ΓL(1,𝔽16)11yescharacteristic, complemented
343C2×#41—51no—
344C2×#41—53no—
345C2×#41—103no—
346C22×Q8—53no—
347C22×Q8—101no—
348#131—303no—
349#131—301no—
350C2×M16—153no—
351C22×C8—153no—
352C4×Q8—301no—
353C4×D4—303no—
354#117—153no—
355C2×C4⋊C4—153no—
356#103—153no—
357#103—156no—
358#103—153no—
359#102—151no—
360#102—153no—
361C2×C3×C5—23no—
362C2×C3×C5—41no—
363C2×C3×C5C22×C22⋊C411yescharacteristic
364C3×D5—23no—
365C3×D5—41no—
366C3×D5#57711yescharacteristic
367C3×D5#57811yescharacteristic
368C23×C3—53no—
369C23×C3—101no—
370C3×Q8—23no—
371C3×Q8—53no—
372C3×Q8—53no—
373C3×Q8—106no—
374C3×Q8—103no—
375C3×Q8—103no—
376C3×Q8C22⋊AGL(1,𝔽5)11yescharacteristic
377C3×Q8C22×AGL(1,𝔽5)13yes—
378C3×D4—23no—
379C3×D4—56no—
380C3×D4—53no—
381C3×D4—51no—
382C3×D4—103no—
383C3×D4—101no—
384C3×D4—103no—
385C3×D4—103no—
386C3×D4C22⋊AGL(1,𝔽5)13yes—
387C3×D4C22×AGL(1,𝔽5)11yescharacteristic
388C2×C3×C4—23no—
389C2×C3×C4—23no—
390C2×C3×C4—53no—
391C2×C3×C4—53no—
392C2×C3×C4—53no—
393C2×C3×C4—53no—
394C2×C3×C4—51no—
395C2×C3×C4—103no—
396C2×C3×C4—103no—
397C2×C3×C4—106no—
398C2×C3×C4—103no—
399C2×C3×C4—103no—
400C2×C3×C4—103no—
401C2×C3×C4C22×AGL(1,𝔽5)13yes—
402C2×Dic3—101no—
403C2×Dic3—103no—
404C2×Dic3—403no—
405C3⋊C8—101no—
406C3⋊C8—103no—
407C22×C5—21no—
408C22×C5C2×#21713yes—
409C2×D5—23no—
410C2×D5—23no—
411C2×D5—21no—
412C2×D5—41no—
413C2×D5—41no—
414C2×D5C2×#21713yes—
415C2×D5C22×#6711yescharacteristic
416AGL(1,𝔽5)—123no—
417AGL(1,𝔽5)—121no—
418C4×C5—23no—
419C4×C5C2×#21713yes—
420C4×C5C23×Dic311yescharacteristic
421Dic5—23no—
422Dic5C2×#21713yes—
423Dic5C22×#6711yescharacteristic
424#41—23no—
425#41—51no—
426#41—53no—
427#41—53no—
428#41—53no—
429#41—53no—
430#41—53no—
431#41—103no—
432#41—106no—
433#41—103no—
434#41—103no—
435#41—106no—
436#41—103no—
437#41—103no—
438#41C2×ΓL(1,𝔽16)11yescharacteristic
439#41C2×ΓL(1,𝔽16)13yescomplemented
440C2×Q8—21no—
441C2×Q8—51no—
442C2×Q8—53no—
443C2×Q8—56no—
444C2×Q8—53no—
445C2×Q8—103no—
446C2×Q8—103no—
447C2×Q8—103no—
448C2×Q8—103no—
449C2×Q8—103no—
450C2×Q8—101no—
451C2×Q8C2×ΓL(1,𝔽16)13yescomplemented
452C2×D4—53no—
453C2×D4—51no—
454C2×D4—103no—
455C22×C4—53no—
456C22×C4—103no—
457C22×C4—103no—
458C22×C4—303no—
459M16—303no—
460M16—303no—
461C2×C8—151no—
462C2×C8—153no—
463C2×C8—303no—
464C2×C8—303no—
465C4⋊C4—153no—
466C4⋊C4—151no—
467C4⋊C4—303no—
468C22⋊C4—303no—
469C42—303no—
470C3×C5#289811yescharacteristic, complemented, Hall
471C22×C3—21no—
472C22×C3—51no—
473C22×C3—53no—
474C22×C3—103no—
475C22×C3—103no—
476C22×C3—101no—
477C22×C3—201no—
478C22×C3—201no—
479C22×C3C2×C22⋊AGL(1,𝔽5)13yes—
480C3×C4—23no—
481C3×C4—53no—
482C3×C4—51no—
483C3×C4—103no—
484C3×C4C2×C22⋊AGL(1,𝔽5)13yes—
485C3×C4C23×AGL(1,𝔽5)11yescharacteristic
486Dic3—201no—
487Dic3—203no—
488C2×C5—23no—
489C2×C5—41no—
490C2×C5C22×#21711yescharacteristic
491D5—23no—
492D5—41no—
493D5#583711yescharacteristic
494D5#583811yescharacteristic
495C23—53no—
496C23—101no—
497Q8—23no—
498Q8—53no—
499Q8—53no—
500Q8—103no—
501Q8—106no—
502Q8—103no—
503Q8#676311yescharacteristic
504Q8C22×ΓL(1,𝔽16)13yes—
505D4—23no—
506D4—56no—
507D4—53no—
508D4—51no—
509D4—103no—
510D4—101no—
511D4—103no—
512D4—103no—
513D4#676313yes—
514D4C22×ΓL(1,𝔽16)11yescharacteristic
515C2×C4—23no—
516C2×C4—23no—
517C2×C4—53no—
518C2×C4—53no—
519C2×C4—53no—
520C2×C4—51no—
521C2×C4—53no—
522C2×C4—103no—
523C2×C4—103no—
524C2×C4—106no—
525C2×C4—103no—
526C2×C4—103no—
527C2×C4—103no—
528C2×C4—301no—
529C2×C4—303no—
530C2×C4—1203no—
531C2×C4C22×ΓL(1,𝔽16)13yes—
532C8—301no—
533C8—303no—
534C2×C3—23no—
535C2×C3—41no—
536C2×C3—51no—
537C2×C3—51no—
538C2×C3—103no—
539C2×C3—201no—
540C2×C3C22×C22⋊AGL(1,𝔽5)11yescharacteristic
541C5#8411111yescharacteristic, complemented, Sylow
542C22—21no—
543C22—51no—
544C22—53no—
545C22—103no—
546C22—103no—
547C22—101no—
548C22—201no—
549C22—201no—
550C22C2×#676313yes—
551C4—23no—
552C4—51no—
553C4—53no—
554C4—103no—
555C4—603no—
556C4—601no—
557C4C2×#676313yes—
558C4C23×ΓL(1,𝔽16)11yescharacteristic
559C3#1061756111yescharacteristic, complemented, Sylow
560C2—23no—
561C2—41no—
562C2—51no—
563C2—51no—
564C2—103no—
565C2—201no—
566C2C22×#676311yescharacteristic
subgroupG/H
centerC2C22×#6763
commutatorC3×C4×C5C23×C4
FrattiniC4C23×ΓL(1,𝔽16)
FittingC3×C5×2−1+4C4
socleC2×C3×C5C22×C22⋊C4
prime radicalC3×D5×2−1+4C2
cosocleC3×C4×D5C24
hypercenter2−1+4ΓL(1,𝔽16)
hypocenterC3×C5#2898
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ #41 ◁ 2−1+4

Complements

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

NH ≅ G/Ntype
2. #11756995C2semidirect
3. D5⋊#948C2semidirect
4. #11752453C2semidirect
5. #11752452C2semidirect
6. #11752441C2semidirect
7. C2×#91238C2semidirect
9. C3×C5×2−1+4C4semidirect
11. C3×#4150C4semidirect
12. C3×#4150C4semidirect
17. C3×#4147C4semidirect
44. C3×C5×#41C2×C4semidirect
46. C2×C3×C5×Q8C2×C4semidirect
48. C3×#713C2×C4semidirect
54. C3×Q8×D5C2×C4semidirect
60. C3×#711C2×C4semidirect
69. C3×#709C2×C4semidirect
75. C2×C3×Dic10C2×C4semidirect
94. C5×2−1+4Dic3semidirect
96. #4150Dic3semidirect
97. #4150Dic3semidirect
102. #4147Dic3semidirect
163. C3×2−1+4AGL(1,𝔽5)semidirect
185. C5×#41C2×Dic3semidirect
187. C2×C5×Q8C2×Dic3semidirect
189. #713C2×Dic3semidirect
195. Q8×D5C2×Dic3semidirect
201. #711C2×Dic3semidirect
210. #709C2×Dic3semidirect
216. C2×Dic10C2×Dic3semidirect
267. C3×#41C2×AGL(1,𝔽5)semidirect
279. C2×C3×Q8C2×AGL(1,𝔽5)semidirect
342. 2−1+4ΓL(1,𝔽16)semidirect
439. #41C2×ΓL(1,𝔽16)semidirect
451. C2×Q8C2×ΓL(1,𝔽16)semidirect
470. C3×C5#2898semidirect
541. C5#84111semidirect
559. C3#10617561semidirect

Maximal subgroups

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

subgroupindex
C3×D5×2−1+42
#117569952
D5⋊#9482
#117524532
#117524522
#117524412
C2×#912382
#106175613
#841115