Finite Group Databasebeta

← #14417090

Subgroups — #14417090

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
2C5×#5523C211yesmaximal, characteristic, complemented
3C2×#91305C211yesmaximal, characteristic
4C3×#65286C211yesmaximal, characteristic, complemented
5#11748459C211yesmaximal, characteristic, complemented
6C2×#90969C211yesmaximal, characteristic, complemented
7#11748388C211yesmaximal, characteristic, complemented
8#11747058C212yesmaximal, complemented
9C4×#6660C212yesmaximal, complemented
10#11747025C212yesmaximal, complemented
11C4×#6654C212yesmaximal, complemented
12#10614295—31nomaximal, Hall
13C22×D60C2211yescharacteristic
14C2×C5×#235C2211yescharacteristic
15C3×C5×#117C2211yescharacteristic
16#91815C2211yescharacteristic
17#91782—21no—
18#91782—21no—
19C22⋊C4×D15—22no—
20C4×D60C2212yes—
21C4×D60C2212yes—
22C2×C5×C4⋊Dic3C2211yescharacteristic
23C5×#892—21no—
24C5×D3×C4⋊C4—21no—
25C5×#67⋊C4—22no—
26C4×C5×D12C2212yescomplemented
27C4×C5×Dic6C2212yescomplemented
28C2×C3×#681C2211yescharacteristic
29C22×C3×#168C2211yescharacteristic
30C2×#6660C2212yes—
31#91305C418yescomplemented
32C2×#6654C2212yes—
33C3×#3935C2212yescomplemented
34C3×C4×#168C2212yescomplemented
35#90988C2212yescomplemented
36#90985C2212yescomplemented
37#90973C2211yescharacteristic, complemented
38#90973C2211yescharacteristic, complemented
39#90969C2212yes—
40#90969C2212yes—
41#90967C2211yescharacteristic, complemented
42#90967C2211yescharacteristic, complemented
43#78560—51nomaximal, Hall
44C5×#520—31no—
45C2×#4095—31no—
46#65855—31no—
47#65522—31no—
48#65315—32no—
49C4×#688—32no—
50#65303—32no—
51C4×#686—32no—
52#65286D311yescharacteristic, complemented
53C2×#3941—31no—
54#65253—31no—
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×C5×#67—21no—
64C22×C5×Dic3—21no—
65C5×#235—22no—
66C5×#235C2×C414yescomplemented
67C2×C5×D12C2311yescharacteristic
68C2×C4×C5×D3—21no—
69C2×C4×C5×D3—22no—
70C2×C5×Dic6C2311yescharacteristic
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×(C2×D3)⋊C4—22no—
82C5×C4⋊Dic3—21no—
83C5×C4⋊Dic3C2312yes—
84C5×Dic3⋊C4—22no—
85C4×C5×Dic3—22no—
86C3×#681C2×C414yescomplemented
87C2×C3×#168—22no—
88C2×C3×#168C2311yescharacteristic
89C2×C3×#168C2311yescharacteristic
90C2×C3×#168C2312yes—
91#6660C2×C418yescomplemented
92#6654C2×C418yescomplemented
93C2×#908—51no—
94#5523D511yescharacteristic, complemented
95#5521—51no—
96C3×#405—51no—
97#5121—51no—
98#5113—51no—
99C2×#821—51no—
100#4711—52no—
101#4710—52no—
102C4×D24—52no—
103C4×#204—52no—
104C2×C5×#41—31no—
105C22×D20—31no—
106C4×C5×Q8—32no—
107C4×C5×D4—32no—
108C4×C5×D4—62no—
109C5×#117—61no—
110C5×#117C2×D311yescharacteristic
111C2×C5×C4⋊C4—31no—
112C2×C5×C4⋊C4—61no—
113#4095—38no—
114C2×#688—32no—
115C2×#686—32no—
116#4072—31no—
117C2×#681C2×D311yescharacteristic
118C22×#168C2×D311yescharacteristic
119#4039—61no—
120#4039—61no—
121D5×C22⋊C4—62no—
122C4×D20—32no—
123C4×D20—32no—
124#3967—31no—
125#3967—31no—
126#3964—31no—
127#3964—31no—
128#3941—32no—
129#3941—32no—
130#3940—32no—
131#3939—32no—
132#3935C2×D312yescomplemented
133C4×#168C2×D312yescomplemented
134#2946—151noSylow
135C23×C3×C5C2×D411yescharacteristic
136C22×D15—22no—
137C22×D15—24no—
138C22×D15—81no—
139C22×D15—81no—
140C22×C5×D3—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—
154C5×#67—24no—
155C2×C5×Dic3—21no—
156C2×C5×Dic3—22no—
157C2×C5×Dic3—81no—
158C5×D12—21no—
159C5×D12C22×C412yes—
160C4×C5×D3—24no—
161C4×C5×D3—82no—
162C5×Dic6—21no—
163C5×Dic6C22×C412yes—
164C3×#168—21no—
165C3×#168—21no—
166C3×#168C22×C414yes—
167C2×#235C2×D511yescharacteristic
168C22×D12—51no—
169C2×C3×M16—51no—
170C22×C3×C8—51no—
171C3×#117C2×D511yescharacteristic
172#926—51no—
173C2×C4⋊Dic3C2×D511yescharacteristic
174#908—58no—
175C2×D24—52no—
176C2×#204—52no—
177#893—101no—
178#893—101no—
179#892—21no—
180D3×C4⋊C4—21no—
181#67⋊C4—22no—
182D3×C22⋊C4—102no—
183C4×D12—52no—
184C4×D12—52no—
185C4×D12C2×D512yescomplemented
186C4×Dic6C2×D512yescomplemented
187C3×#97—52no—
188C3×C4×C8—52no—
189#821—51no—
190#821—51no—
191#821—52no—
192#821—52no—
193#818—52no—
194#817—52no—
195#816—51no—
196#816—51no—
197C23×D5—61no—
198C5×#41—34no—
199C5×#41—62no—
200C2×C5×Q8—31no—
201C2×C5×D4—31no—
202C2×C5×D4—61no—
203C22×C4×C5—61no—
204C22×C4×C5—61no—
205C22×C4×C5—62no—
206C22×C4×C5C22×D311yescharacteristic
207C2×D20—34no—
208C2×D20—32no—
209C2×D20—61no—
210C2×D20—62no—
211C2×C4×D5—62no—
212C2×C4×D5—62no—
213C5×C4⋊C4—32no—
214C5×C4⋊C4—61no—
215C5×C4⋊C4—62no—
216C5×C4⋊C4D1211yescharacteristic
217C5×C4⋊C4D1211yescharacteristic
218C5×C22⋊C4—62no—
219C5×C22⋊C4D1212yes—
220C42×C5—62no—
221C42×C5C22×D312yes—
222C22⋊Dic5—61no—
223#688—38no—
224#686—38no—
225#685—62no—
226#685—62no—
227C4⋊Dic5—31no—
228C4⋊Dic5—31no—
229C4×Dic5—61no—
230#681C4×D314yescomplemented
231C2×#168—22no—
232C2×#168C22×D311yescharacteristic
233C2×#168C22×D311yescharacteristic
234C2×#168C22×D312yes—
235C2×AGL(1,ℤ/8)—151no—
236#520—31no—
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—51no—
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×C5×D3—22no—
255C2×C5×D3—41no—
256C2×C5×D3—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—
264C5×Dic3—22no—
265C5×Dic3—42no—
266C23×D3—101no—
267C22×C3×C4C22×D511yescharacteristic
268C2×#67—21no—
269C22×Dic3—21no—
270#235—22no—
271#235C4×D514yescomplemented
272C2×D12—54no—
273C2×D12—52no—
274C2×D12—102no—
275C2×D12—101no—
276C2×D12C22×D511yescharacteristic
277C2×C4×D3—21no—
278C2×C4×D3—22no—
279C2×C4×D3—102no—
280C2×C4×D3—102no—
281C2×Dic6C22×D511yescharacteristic
282C3×M16—54no—
283C2×C3×C8—52no—
284C2×C3×C8—51no—
285C2×C3×C8—51no—
286C2×C3×C8—102no—
287C3×C4⋊C4C22⋊D511yescharacteristic
288C3×C4⋊C4C22⋊D511yescharacteristic
289C3×C22⋊C4C22⋊D512yes—
290C3×C42C22×D512yes—
291#217—101no—
292(C2×D3)⋊C4—22no—
293(C2×D3)⋊C4—102no—
294(C2×D3)⋊C4—102no—
295C4⋊Dic3—21no—
296C4⋊Dic3—51no—
297C4⋊Dic3—51no—
298C4⋊Dic3C22×D512yes—
299Dic3⋊C4—22no—
300C4×Dic3—22no—
301C4×Dic3—101no—
302D24—58no—
303#204—58no—
304C23×C5—61no—
305C23×C5C2×D1211yescharacteristic
306C22×D5—62no—
307C22×D5—64no—
308C22×D5—241no—
309C22×D5—241no—
310C5×Q8—32no—
311C5×Q8—61no—
312C5×D4—32no—
313C5×D4—64no—
314C5×D4—61no—
315C2×C4×C5—61no—
316C2×C4×C5—64no—
317C2×C4×C5—62no—
318C2×C4×C5—242no—
319C2×C4×C5—241no—
320C2×C4×C5C2×C4×D312yes—
321C2×C4×C5C2×D1212yes—
322C2×C4×C5C2×D1212yes—
323C2×C4×C5C2×D1211yescharacteristic
324C2×C4×C5#23512yes—
325C2×C4×C5C23×D311yescharacteristic
326C2×Dic5—61no—
327C2×Dic5—61no—
328D20—34no—
329D20—64no—
330D20—62no—
331C4×D5—242no—
332C4×D5—242no—
333#168—21no—
334#168—21no—
335#168C2×C4×D314yes—
336C2×#41—31no—
337C22×D4—151no—
338AGL(1,ℤ/8)—158no—
339C2×SD16—152no—
340C2×D8—152no—
341C2×M16—51no—
342C22×C8—51no—
343C4×Q8—32no—
344C4×D4—32no—
345C4×D4—62no—
346C4×D4—152no—
347C4×D4—152no—
348C4×D4—301no—
349C4×D4—301no—
350#117—61no—
351#117—151no—
352#117D3×D511yescharacteristic
353C2×C4⋊C4—31no—
354C2×C4⋊C4—61no—
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—52no—
365C4×C8—52no—
366C2×C3×C5—22no—
367C2×C3×C5C2×C4×D411yescharacteristic
368C2×C3×C5#57611yescharacteristic
369C2×C3×C5#57711yescharacteristic
370D15—44no—
371C5×D3—42no—
372C23×C3C2×C22⋊D511yescharacteristic
373C22×D3—21no—
374C22×D3—102no—
375C22×D3—104no—
376C22×D3—401no—
377C22×D3—401no—
378C2×C3×C4C2×C4×D512yes—
379C2×C3×C4#70912yes—
380C2×C3×C4C2×C22⋊D512yes—
381C2×C3×C4C2×C22⋊D512yes—
382C2×C3×C4C2×C22⋊D511yescharacteristic
383C2×C3×C4C23×D511yescharacteristic
384#67—24no—
385C2×Dic3—21no—
386C2×Dic3—22no—
387C2×Dic3—81no—
388C2×Dic3—101no—
389C2×Dic3—101no—
390D12—21no—
391D12—54no—
392D12—102no—
393D12—104no—
394D12C2×C4×D512yes—
395C4×D3—24no—
396C4×D3—82no—
397C4×D3—402no—
398C4×D3—402no—
399Dic6—21no—
400Dic6C2×C4×D512yes—
401C3×C8—54no—
402C3×C8—101no—
403C3×C8—101no—
404C22×C5—21no—
405C22×C5—21no—
406C22×C5—62no—
407C22×C5—121no—
408C22×C5—121no—
409C22×C5C4×D1212yes—
410C22×C5C22×D1211yescharacteristic
411C2×D5—64no—
412C2×D5—122no—
413C2×D5—122no—
414C2×D5—244no—
415C4×C5—22no—
416C4×C5—22no—
417C4×C5—62no—
418C4×C5—122no—
419C4×C5C4×D1212yes—
420C4×C5C22×C4×D311yescharacteristic
421C4×C5C2×#23511yescharacteristic
422Dic5—121no—
423Dic5—121no—
424C24—301no—
425#41—34no—
426#41—62no—
427C2×Q8—31no—
428C2×D4—31no—
429C2×D4—61no—
430C2×D4—154no—
431C2×D4—152no—
432C2×D4—302no—
433C2×D4—301no—
434C22×C4—61no—
435C22×C4—61no—
436C22×C4—62no—
437C22×C4—302no—
438C22×C4—302no—
439C22×C4C2×D3×D511yescharacteristic
440SD16—158no—
441D8—158no—
442M16—54no—
443C2×C8—52no—
444C2×C8—51no—
445C2×C8—51no—
446C2×C8—102no—
447C4⋊C4—32no—
448C4⋊C4—61no—
449C4⋊C4—62no—
450C4⋊C4—151no—
451C4⋊C4—151no—
452C4⋊C4#120311yescharacteristic
453C4⋊C4#120311yescharacteristic
454C22⋊C4—62no—
455C22⋊C4—301no—
456C22⋊C4—302no—
457C22⋊C4—302no—
458C22⋊C4#120312yes—
459C42—62no—
460C42—301no—
461C42C2×D3×D512yes—
462C3×C5#294611yescharacteristic, complemented, Hall
463C22×C3—21no—
464C22×C3—21no—
465C22×C3C4×C22⋊D512yes—
466C22×C3C22×C22⋊D511yescharacteristic
467C2×D3—22no—
468C2×D3—41no—
469C2×D3—41no—
470C2×D3—104no—
471C2×D3—202no—
472C2×D3—202no—
473C2×D3—404no—
474C3×C4—22no—
475C3×C4—22no—
476C3×C4C4×C22⋊D512yes—
477C3×C4C22×C4×D511yescharacteristic
478C3×C4C2×#70911yescharacteristic
479Dic3—22no—
480Dic3—42no—
481Dic3—201no—
482Dic3—201no—
483C2×C5—22no—
484C2×C5—122no—
485C2×C5C2×C4×D1211yescharacteristic
486C2×C5#575111yescharacteristic
487C2×C5#575211yescharacteristic
488D5—124no—
489C23—61no—
490C23—302no—
491C23—304no—
492C23—1201no—
493C23—1201no—
494C23C2×#120311yescharacteristic
495Q8—32no—
496Q8—61no—
497D4—32no—
498D4—64no—
499D4—61no—
500D4—154no—
501D4—304no—
502D4—302no—
503C2×C4—61no—
504C2×C4—64no—
505C2×C4—62no—
506C2×C4—242no—
507C2×C4—241no—
508C2×C4—301no—
509C2×C4—301no—
510C2×C4—1202no—
511C2×C4—1202no—
512C2×C4#677112yes—
513C2×C4C4×D3×D512yes—
514C2×C4C2×#120312yes—
515C2×C4C2×#120312yes—
516C2×C4C2×#120311yescharacteristic
517C2×C4C22×D3×D511yescharacteristic
518C8—54no—
519C8—101no—
520C8—101no—
521C2×C3—22no—
522C2×C3C2×C4×C22⋊D511yescharacteristic
523C2×C3#6616911yescharacteristic
524C2×C3#6617111yescharacteristic
525D3—42no—
526D3—204no—
527C5#7856011yescharacteristic, complemented, Sylow
528C22—21no—
529C22—21no—
530C22—62no—
531C22—121no—
532C22—121no—
533C22—304no—
534C22—602no—
535C22—602no—
536C22—1204no—
537C22C4×#120312yes—
538C22C22×#120311yescharacteristic
539C4—22no—
540C4—22no—
541C4—62no—
542C4—122no—
543C4—601no—
544C4—601no—
545C4C4×#120312yes—
546C4C2×#677111yescharacteristic
547C4C2×C4×D3×D511yescharacteristic
548C3#1061429511yescharacteristic, complemented, Sylow
549C2—22no—
550C2—122no—
551C2—604no—
552C2C2×C4×#120311yescharacteristic
553C2#1175530511yescharacteristic
554C2#1175530811yescharacteristic
subgroupG/H
centerC22C22×#1203
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#117C22
socleC22×C3×C5C22×D4
prime radicalC2×#91305C2
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. C5×#5523C2semidirect
4. C3×#65286C2semidirect
5. #11748459C2semidirect
6. C2×#90969C2semidirect
7. #11748388C2semidirect
8. #11747058C2semidirect
9. C4×#6660C2semidirect
10. #11747025C2semidirect
11. C4×#6654C2semidirect
26. C4×C5×D12C22semidirect
27. C4×C5×Dic6C22semidirect
31. #91305C4semidirect
33. C3×#3935C22semidirect
34. C3×C4×#168C22semidirect
35. #90988C22semidirect
36. #90985C22semidirect
37. #90973C22semidirect
38. #90973C22semidirect
41. #90967C22semidirect
42. #90967C22semidirect
52. #65286D3semidirect
59. C2×D60C2×C4semidirect
66. C5×#235C2×C4semidirect
86. C3×#681C2×C4semidirect
91. #6660C2×C4semidirect
92. #6654C2×C4semidirect
94. #5523D5semidirect
132. #3935C2×D3semidirect
133. C4×#168C2×D3semidirect
185. C4×D12C2×D5semidirect
186. C4×Dic6C2×D5semidirect
230. #681C4×D3semidirect
271. #235C4×D5semidirect
462. C3×C5#2946semidirect
527. C5#78560semidirect
548. C3#10614295semidirect

Maximal subgroups

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

subgroupindex
#117562802
C5×#55232
C2×#913052
C3×#652862
#117484592
C2×#909692
#117483882
#117470582
C4×#66602
#117470252
C4×#66542
#106142953
#785605