Finite Group Databasebeta

← #14417093

Subgroups — #14417093

Order 1920.

1012 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756280C211yesmaximal, characteristic, complemented
2C3×#65857C211yesmaximal, characteristic, complemented
3C2×#91301C211yesmaximal, characteristic
4C5×#4997C211yesmaximal, characteristic, complemented
5#11748453C211yesmaximal, characteristic, complemented
6C2×#90968C211yesmaximal, characteristic, complemented
7#11748365C211yesmaximal, characteristic, complemented
8#11747054C212yesmaximal, complemented
9C4×#6658C212yesmaximal, complemented
10#11747023C212yesmaximal, complemented
11C4×#6653C212yesmaximal, complemented
12#10610291—31nomaximal, Hall
13C22×D60C2211yescharacteristic
14C2×C3×#709C2211yescharacteristic
15C3×C5×#117C2211yescharacteristic
16#91815C2211yescharacteristic
17#91782—21no—
18#91782—21no—
19C22⋊C4×D15—22no—
20C4×D60C2212yes—
21C4×D60C2212yes—
22C2×C5×#208C2211yescharacteristic
23C22×C5×C3⋊C8C2211yescharacteristic
24C2×C3×C4⋊Dic5C2211yescharacteristic
25C3×#4038—21no—
26C3×D5×C4⋊C4—21no—
27C3×#4027—22no—
28C3×C4×D20C2212yescomplemented
29C3×C4×Dic10C2212yescomplemented
30C2×#6658C2212yes—
31#91301C418yescomplemented
32C2×#6653C2212yes—
33C5×#803C2212yescomplemented
34C4×C5×C3⋊C8C2212yescomplemented
35#90990C2212yescomplemented
36#90987C2212yescomplemented
37#90971C2211yescharacteristic, complemented
38#90971C2211yescharacteristic, complemented
39#90968C2212yes—
40#90968C2212yes—
41#90965C2211yescharacteristic, complemented
42#90965C2211yescharacteristic, complemented
43#82564—51nomaximal, Hall
44C2×#4054—31no—
45#65857D311yescharacteristic, complemented
46#65855—31no—
47C5×#405—31no—
48#65410—31no—
49#65402—31no—
50C2×#3953—31no—
51#65000—32no—
52#64999—32no—
53C4×D40—32no—
54C4×#677—32no—
55C23×D15—21no—
56C22×C3×C4×C5C2311yescharacteristic
57C2×D60—21no—
58C2×D60—22no—
59C2×D60C2×C414yescomplemented
60C2×D60C2312yes—
61C2×C4×D15—22no—
62C2×C4×D15—22no—
63C2×C3×C22⋊D5—21no—
64C22×C3×Dic5—21no—
65C3×#709—22no—
66C3×#709C2×C414yescomplemented
67C2×C3×D20C2311yescharacteristic
68C2×C3×C4×D5—21no—
69C2×C3×C4×D5—22no—
70C2×C3×Dic10C2311yescharacteristic
71C3×C5×C4⋊C4D411yescharacteristic
72C3×C5×C4⋊C4D411yescharacteristic
73C3×C5×C22⋊C4D412yes—
74C3×C42×C5C2312yes—
75#6719—21no—
76#6714—22no—
77#6714—22no—
78C4⋊Dic15C2311yescharacteristic
79C4⋊Dic15C2311yescharacteristic
80C4×Dic15—21no—
81C5×#208C2×C414yescomplemented
82C2×C5×C3⋊C8—22no—
83C2×C5×C3⋊C8C2311yescharacteristic
84C2×C5×C3⋊C8C2311yescharacteristic
85C2×C5×C3⋊C8C2312yes—
86C3×#685—22no—
87C3×C4⋊Dic5—21no—
88C3×C4⋊Dic5C2312yes—
89C3×#683—22no—
90C3×C4×Dic5—22no—
91#6658C2×C418yescomplemented
92#6653C2×C418yescomplemented
93C3×#520—51no—
94C2×#949—51no—
95#5521—51no—
96#5233—51no—
97#5026—52no—
98C4×#215—52no—
99#5014—52no—
100C4×#213—52no—
101#4997D511yescharacteristic, complemented
102C2×#809—51no—
103#4964—51no—
104C2×#709C2×D311yescharacteristic
105C22×D20—31no—
106C2×C5×M16—31no—
107C22×C5×C8—31no—
108C5×#117C2×D311yescharacteristic
109#4072—31no—
110C2×C4⋊Dic5C2×D311yescharacteristic
111#4054—38no—
112C2×D40—32no—
113C2×#677—32no—
114#4039—61no—
115#4039—61no—
116#4038—21no—
117D5×C4⋊C4—21no—
118#4027—22no—
119D5×C22⋊C4—62no—
120C4×D20—32no—
121C4×D20—32no—
122C4×D20C2×D312yescomplemented
123C4×Dic10C2×D312yescomplemented
124C5×#97—32no—
125C4×C5×C8—32no—
126#3953—31no—
127#3953—31no—
128#3953—32no—
129#3953—32no—
130#3950—32no—
131#3949—32no—
132#3948—31no—
133#3948—31no—
134#2946—151noSylow
135C23×C3×C5C2×D411yescharacteristic
136C22×D15—22no—
137C22×D15—24no—
138C22×D15—81no—
139C22×D15—81no—
140C22×C3×D5—21no—
141C2×C3×C4×C5C22×C412yes—
142C2×C3×C4×C5C2×D412yes—
143C2×C3×C4×C5C2×D412yes—
144C2×C3×C4×C5C2×D411yescharacteristic
145C2×C3×C4×C5#4112yes—
146C2×C3×C4×C5C2411yescharacteristic
147C2×Dic15—21no—
148C2×Dic15—21no—
149D60—24no—
150D60—22no—
151D60C22×C414yes—
152C4×D15—82no—
153C4×D15—82no—
154C3×C22⋊D5—24no—
155C2×C3×Dic5—21no—
156C2×C3×Dic5—22no—
157C2×C3×Dic5—81no—
158C3×D20—21no—
159C3×D20C22×C412yes—
160C3×C4×D5—24no—
161C3×C4×D5—82no—
162C3×Dic10—21no—
163C3×Dic10C22×C412yes—
164C5×C3⋊C8—21no—
165C5×C3⋊C8—21no—
166C5×C3⋊C8C22×C414yes—
167C2×C3×#41—51no—
168C22×D12—51no—
169C3×C4×Q8—52no—
170C3×C4×D4—52no—
171C3×C4×D4—102no—
172C3×#117—101no—
173C3×#117C2×D511yescharacteristic
174C2×C3×C4⋊C4—51no—
175C2×C3×C4⋊C4—101no—
176#949—58no—
177C2×#215—52no—
178C2×#213—52no—
179#926—51no—
180C2×#208C2×D511yescharacteristic
181C22×C3⋊C8C2×D511yescharacteristic
182#893—101no—
183#893—101no—
184D3×C22⋊C4—102no—
185C4×D12—52no—
186C4×D12—52no—
187#835—51no—
188#835—51no—
189#832—51no—
190#832—51no—
191#809—52no—
192#809—52no—
193#808—52no—
194#807—52no—
195#803C2×D512yescomplemented
196C4×C3⋊C8C2×D512yescomplemented
197C23×D5—61no—
198C22×C4×C5C22×D311yescharacteristic
199C2×C22⋊D5—21no—
200C22×Dic5—21no—
201#709—22no—
202#709C4×D314yescomplemented
203C2×D20—34no—
204C2×D20—32no—
205C2×D20—62no—
206C2×D20—61no—
207C2×D20C22×D311yescharacteristic
208C2×C4×D5—21no—
209C2×C4×D5—22no—
210C2×C4×D5—62no—
211C2×C4×D5—62no—
212C2×Dic10C22×D311yescharacteristic
213C5×M16—34no—
214C2×C5×C8—32no—
215C2×C5×C8—31no—
216C2×C5×C8—31no—
217C2×C5×C8—62no—
218C5×C4⋊C4#6711yescharacteristic
219C5×C4⋊C4#6711yescharacteristic
220C5×C22⋊C4#6712yes—
221C42×C5C22×D312yes—
222C22⋊Dic5—61no—
223#685—22no—
224#685—62no—
225#685—62no—
226C4⋊Dic5—21no—
227C4⋊Dic5—31no—
228C4⋊Dic5—31no—
229C4⋊Dic5C22×D312yes—
230#683—22no—
231C4×Dic5—22no—
232C4×Dic5—61no—
233D40—38no—
234#677—38no—
235C2×AGL(1,ℤ/8)—151no—
236#520—51no—
237#518—151no—
238#442—152no—
239#440—152no—
240C4×SD16—152no—
241C4×D8—152no—
242#428—151no—
243#416—151no—
244C2×#102—151no—
245#405—31no—
246C22×C3×C5—21no—
247C22×C3×C5—21no—
248C22×C3×C5C4×D412yes—
249C22×C3×C5C22×D411yescharacteristic
250C2×D15—24no—
251C2×D15—42no—
252C2×D15—42no—
253C2×D15—84no—
254C2×C3×D5—22no—
255C2×C3×D5—41no—
256C2×C3×D5—41no—
257C3×C4×C5—22no—
258C3×C4×C5—22no—
259C3×C4×C5C4×D412yes—
260C3×C4×C5C23×C411yescharacteristic
261C3×C4×C5C2×#4111yescharacteristic
262Dic15—41no—
263Dic15—41no—
264C3×Dic5—22no—
265C3×Dic5—42no—
266C23×D3—101no—
267C3×#41—54no—
268C3×#41—102no—
269C2×C3×Q8—51no—
270C2×C3×D4—51no—
271C2×C3×D4—101no—
272C22×C3×C4—101no—
273C22×C3×C4—101no—
274C22×C3×C4—102no—
275C22×C3×C4C22×D511yescharacteristic
276C2×D12—54no—
277C2×D12—52no—
278C2×D12—101no—
279C2×D12—102no—
280C2×C4×D3—102no—
281C2×C4×D3—102no—
282C3×C4⋊C4—52no—
283C3×C4⋊C4—102no—
284C3×C4⋊C4—101no—
285C3×C4⋊C4D2011yescharacteristic
286C3×C4⋊C4D2011yescharacteristic
287C3×C22⋊C4—102no—
288C3×C22⋊C4D2012yes—
289C3×C42—102no—
290C3×C42C22×D512yes—
291#217—101no—
292#215—58no—
293#213—58no—
294(C2×D3)⋊C4—102no—
295(C2×D3)⋊C4—102no—
296C4⋊Dic3—51no—
297C4⋊Dic3—51no—
298C4×Dic3—101no—
299#208C4×D514yescomplemented
300C2×C3⋊C8—22no—
301C2×C3⋊C8C22×D511yescharacteristic
302C2×C3⋊C8C22×D511yescharacteristic
303C2×C3⋊C8C22×D512yes—
304C23×C5C2×#6711yescharacteristic
305C22×D5—21no—
306C22×D5—62no—
307C22×D5—64no—
308C22×D5—241no—
309C22×D5—241no—
310C2×C4×C5C2×C4×D312yes—
311C2×C4×C5#23512yes—
312C2×C4×C5C2×#6712yes—
313C2×C4×C5C2×#6712yes—
314C2×C4×C5C2×#6711yescharacteristic
315C2×C4×C5C23×D311yescharacteristic
316C22⋊D5—24no—
317C2×Dic5—21no—
318C2×Dic5—22no—
319C2×Dic5—61no—
320C2×Dic5—61no—
321C2×Dic5—81no—
322D20—21no—
323D20—34no—
324D20—64no—
325D20—62no—
326D20C2×C4×D312yes—
327C4×D5—24no—
328C4×D5—82no—
329C4×D5—242no—
330C4×D5—242no—
331Dic10—21no—
332Dic10C2×C4×D312yes—
333C5×C8—34no—
334C5×C8—61no—
335C5×C8—61no—
336C2×#41—51no—
337C22×D4—151no—
338AGL(1,ℤ/8)—158no—
339C2×SD16—152no—
340C2×D8—152no—
341C2×M16—31no—
342C22×C8—31no—
343C4×Q8—52no—
344C4×D4—52no—
345C4×D4—102no—
346C4×D4—152no—
347C4×D4—152no—
348C4×D4—301no—
349C4×D4—301no—
350#117—101no—
351#117—151no—
352#117D3×D511yescharacteristic
353C2×C4⋊C4—51no—
354C2×C4⋊C4—101no—
355C2×C22⋊C4—302no—
356#107—152no—
357#106—152no—
358#103—151no—
359#103—151no—
360#102—152no—
361#102—152no—
362#102—151no—
363#102—151no—
364#97—32no—
365C4×C8—32no—
366C2×C3×C5—22no—
367C2×C3×C5C2×C4×D411yescharacteristic
368C2×C3×C5#57611yescharacteristic
369C2×C3×C5#57711yescharacteristic
370D15—44no—
371C3×D5—42no—
372C23×C3—101no—
373C23×C3C2×D2011yescharacteristic
374C22×D3—102no—
375C22×D3—104no—
376C22×D3—401no—
377C22×D3—401no—
378C3×Q8—52no—
379C3×Q8—101no—
380C3×D4—52no—
381C3×D4—104no—
382C3×D4—101no—
383C2×C3×C4—101no—
384C2×C3×C4—104no—
385C2×C3×C4—102no—
386C2×C3×C4—402no—
387C2×C3×C4—401no—
388C2×C3×C4C2×C4×D512yes—
389C2×C3×C4C2×D2012yes—
390C2×C3×C4C2×D2012yes—
391C2×C3×C4C2×D2011yescharacteristic
392C2×C3×C4#70912yes—
393C2×C3×C4C23×D511yescharacteristic
394C2×Dic3—101no—
395C2×Dic3—101no—
396D12—54no—
397D12—102no—
398D12—104no—
399C4×D3—402no—
400C4×D3—402no—
401C3⋊C8—21no—
402C3⋊C8—21no—
403C3⋊C8C2×C4×D514yes—
404C22×C5—21no—
405C22×C5—21no—
406C22×C5C4×#6712yes—
407C22×C5C22×#6711yescharacteristic
408C2×D5—22no—
409C2×D5—41no—
410C2×D5—41no—
411C2×D5—64no—
412C2×D5—122no—
413C2×D5—122no—
414C2×D5—244no—
415C4×C5—22no—
416C4×C5—22no—
417C4×C5C4×#6712yes—
418C4×C5C22×C4×D311yescharacteristic
419C4×C5C2×#23511yescharacteristic
420Dic5—22no—
421Dic5—42no—
422Dic5—121no—
423Dic5—121no—
424C24—301no—
425#41—54no—
426#41—102no—
427C2×Q8—51no—
428C2×D4—51no—
429C2×D4—101no—
430C2×D4—154no—
431C2×D4—152no—
432C2×D4—302no—
433C2×D4—301no—
434C22×C4—101no—
435C22×C4—101no—
436C22×C4—102no—
437C22×C4—302no—
438C22×C4—302no—
439C22×C4C2×D3×D511yescharacteristic
440SD16—158no—
441D8—158no—
442M16—34no—
443C2×C8—32no—
444C2×C8—31no—
445C2×C8—31no—
446C2×C8—62no—
447C4⋊C4—52no—
448C4⋊C4—101no—
449C4⋊C4—102no—
450C4⋊C4—151no—
451C4⋊C4—151no—
452C4⋊C4#120211yescharacteristic
453C4⋊C4#120211yescharacteristic
454C22⋊C4—102no—
455C22⋊C4—301no—
456C22⋊C4—302no—
457C22⋊C4—302no—
458C22⋊C4#120212yes—
459C42—102no—
460C42—301no—
461C42C2×D3×D512yes—
462C3×C5#294611yescharacteristic, complemented, Hall
463C22×C3—21no—
464C22×C3—21no—
465C22×C3—102no—
466C22×C3—201no—
467C22×C3—201no—
468C22×C3C4×D2012yes—
469C22×C3C22×D2011yescharacteristic
470C2×D3—104no—
471C2×D3—202no—
472C2×D3—202no—
473C2×D3—404no—
474C3×C4—22no—
475C3×C4—22no—
476C3×C4—102no—
477C3×C4—202no—
478C3×C4C4×D2012yes—
479C3×C4C22×C4×D511yescharacteristic
480C3×C4C2×#70911yescharacteristic
481Dic3—201no—
482Dic3—201no—
483C2×C5—22no—
484C2×C5C2×C4×#6711yescharacteristic
485C2×C5#583511yescharacteristic
486C2×C5#583711yescharacteristic
487D5—42no—
488D5—124no—
489C23—101no—
490C23—302no—
491C23—304no—
492C23—1201no—
493C23—1201no—
494C23C2×#120211yescharacteristic
495Q8—52no—
496Q8—101no—
497D4—52no—
498D4—104no—
499D4—101no—
500D4—154no—
501D4—304no—
502D4—302no—
503C2×C4—101no—
504C2×C4—104no—
505C2×C4—102no—
506C2×C4—301no—
507C2×C4—301no—
508C2×C4—402no—
509C2×C4—401no—
510C2×C4—1202no—
511C2×C4—1202no—
512C2×C4#677112yes—
513C2×C4C4×D3×D512yes—
514C2×C4C2×#120212yes—
515C2×C4C2×#120212yes—
516C2×C4C2×#120211yescharacteristic
517C2×C4C22×D3×D511yescharacteristic
518C8—34no—
519C8—61no—
520C8—61no—
521C2×C3—22no—
522C2×C3—202no—
523C2×C3C2×C4×D2011yescharacteristic
524C2×C3#6608511yescharacteristic
525C2×C3#6608611yescharacteristic
526D3—204no—
527C5#8256411yescharacteristic, complemented, Sylow
528C22—21no—
529C22—21no—
530C22—102no—
531C22—201no—
532C22—201no—
533C22—304no—
534C22—602no—
535C22—602no—
536C22—1204no—
537C22C4×#120212yes—
538C22C22×#120211yescharacteristic
539C4—22no—
540C4—22no—
541C4—102no—
542C4—202no—
543C4—601no—
544C4—601no—
545C4C4×#120212yes—
546C4C2×#677111yescharacteristic
547C4C2×C4×D3×D511yescharacteristic
548C3#1061029111yescharacteristic, complemented, Sylow
549C2—22no—
550C2—202no—
551C2—604no—
552C2C2×C4×#120211yescharacteristic
553C2#1175531011yescharacteristic
554C2#1175531111yescharacteristic
subgroupG/H
centerC22C22×#1202
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#117C22
socleC22×C3×C5C22×D4
prime radicalC2×#91301C2
cosocleC2×C3×C4×C5C24
hypercenter#117D3×D5
hypocenterC3×C5#2946
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ #117

Complements

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

NH ≅ G/Ntype
1. #11756280C2semidirect
2. C3×#65857C2semidirect
4. C5×#4997C2semidirect
5. #11748453C2semidirect
6. C2×#90968C2semidirect
7. #11748365C2semidirect
8. #11747054C2semidirect
9. C4×#6658C2semidirect
10. #11747023C2semidirect
11. C4×#6653C2semidirect
28. C3×C4×D20C22semidirect
29. C3×C4×Dic10C22semidirect
31. #91301C4semidirect
33. C5×#803C22semidirect
34. C4×C5×C3⋊C8C22semidirect
35. #90990C22semidirect
36. #90987C22semidirect
37. #90971C22semidirect
38. #90971C22semidirect
41. #90965C22semidirect
42. #90965C22semidirect
45. #65857D3semidirect
59. C2×D60C2×C4semidirect
66. C3×#709C2×C4semidirect
81. C5×#208C2×C4semidirect
91. #6658C2×C4semidirect
92. #6653C2×C4semidirect
101. #4997D5semidirect
122. C4×D20C2×D3semidirect
123. C4×Dic10C2×D3semidirect
195. #803C2×D5semidirect
196. C4×C3⋊C8C2×D5semidirect
202. #709C4×D3semidirect
299. #208C4×D5semidirect
462. C3×C5#2946semidirect
527. C5#82564semidirect
548. C3#10610291semidirect

Maximal subgroups

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

subgroupindex
#117562802
C3×#658572
C2×#913012
C5×#49972
#117484532
C2×#909682
#117483652
#117470542
C4×#66582
#117470232
C4×#66532
#106102913
#825645