Finite Group Databasebeta

← #14330653

Subgroups — #14330653

Order 1920.

1376 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91826C211yesmaximal, characteristic
2C5×#5221C211yesmaximal, characteristic, complemented
3C3×#65509C211yesmaximal, characteristic, complemented
4#11749502C212yesmaximal, complemented
5#11749494C212yesmaximal, complemented
6#11749483C212yesmaximal, complemented
7#11749480C212yesmaximal, complemented
8#11748854C212yesmaximal, complemented
9#11748685C212yesmaximal, complemented
10#10608165—31nomaximal, Hall
11C24×D15C2211yescharacteristic
12C22×C3×C5×D4C2211yescharacteristic
13C22×#1220C2212yes—
14C23×C3×Dic5C2211yescharacteristic, complemented
15C2×#6719C2211yescharacteristic
16#91826C418yescomplemented
17C2×#6714C2212yes—
18C2×C5×#217C2212yes—
19C2×C5×#217C2212yescomplemented
20C2×C5×C4⋊Dic3C2211yescharacteristic, complemented
21C2×C4×C5×Dic3C2211yescharacteristic, complemented
22C2×C3×C22⋊Dic5C2212yes—
23C2×C3×C22⋊Dic5C2211yescharacteristic, complemented
24C2×#6669C2212yescomplemented
25C2×#6667C2212yes—
26C2×#6667C2212yes—
27C2×#6667C2212yes—
28C2×#6667C2212yes—
29C2×Dic3×Dic5C2212yescomplemented
30C3×#3963C2212yescomplemented
31#90994C2212yescomplemented
32#90994C2212yescomplemented
33#74882—51nomaximal, Hall
34C2×#4083—31no—
35C5×#390—31no—
36#65509D311yescharacteristic, complemented
37#65279—32no—
38#65247—32no—
39#65246—32no—
40#65244—32no—
41#65233—32no—
42#65229—32no—
43C24×C3×C5C2311yescharacteristic
44C24×C3×C5C2311yescharacteristic
45C23×D15—41no—
46C23×D15—41no—
47C23×D15—42no—
48C23×D15—42no—
49C23×D15C2×C414yescomplemented
50C23×D15C2312yes—
51C2×C3×C5×D4—41no—
52C2×C3×C5×D4—41no—
53C2×C3×C5×D4C2×C414yes—
54C22×C3×C4×C5C2311yescharacteristic
55C2×#1220—42no—
56C2×#1220—42no—
57C2×#1220C2×C418yes—
58C22×Dic15C2312yes—
59C22×C5×Dic3C2312yes—
60C22×C5×Dic3C2312yes—
61C22×C3×Dic5—41no—
62C22×C3×Dic5—42no—
63C22×C3×Dic5C2312yes—
64C22×C3×Dic5C2312yes—
65#6719C2×C414yescomplemented
66#6714C2×C418yescomplemented
67C5×#217—42no—
68C5×#217—42no—
69C5×C4⋊Dic3—41no—
70C4×C5×Dic3—41no—
71C3×C22⋊Dic5—42no—
72C3×C22⋊Dic5—41no—
73#6669—42no—
74#6667—42no—
75#6667—42no—
76#6667—42no—
77#6667—42no—
78Dic3×Dic5—42no—
79C2×#937—51no—
80C3×#386—51no—
81#5221D511yescharacteristic, complemented
82#5213—52no—
83#4957—52no—
84#4955—52no—
85#4945—52no—
86#4670—52no—
87#4669—52no—
88C24×D5—31no—
89C22×C5×D4C2×D311yescharacteristic
90C22×C22⋊D5—32no—
91C23×Dic5C2×D311yescharacteristic, complemented
92C2×C5×C4⋊C4—31no—
93C2×C5×C22⋊C4—32no—
94C2×C5×C22⋊C4—32no—
95C2×C42×C5—31no—
96C2×C22⋊Dic5—31no—
97C2×C22⋊Dic5C2×D312yes—
98C2×C22⋊Dic5C2×D311yescharacteristic, complemented
99#4083—38no—
100C2×#685—32no—
101C2×#685—32no—
102C2×#685—32no—
103C2×#685—32no—
104C2×#685—32no—
105C2×#683—32no—
106C2×C4×Dic5—32no—
107#3963—32no—
108#3963—32no—
109#3963C2×D312yescomplemented
110#2375—151noSylow
111C23×C3×C5—21no—
112C23×C3×C5—22no—
113C23×C3×C5—22no—
114C23×C3×C5—22no—
115C23×C3×C5—22no—
116C23×C3×C5—21no—
117C23×C3×C5—21no—
118C23×C3×C5—21no—
119C23×C3×C5C22×C412yes—
120C23×C3×C5C22×C412yes—
121C23×C3×C5C2411yescharacteristic
122C22×D15—22no—
123C22×D15—22no—
124C22×D15—22no—
125C22×D15—22no—
126C22×D15—22no—
127C22×D15—22no—
128C22×D15—48no—
129C22×D15—44no—
130C22×D15—44no—
131C22×D15—48no—
132C22×D15—44no—
133C22×D15C22×C414yes—
134C3×C5×D4—44no—
135C2×C3×C4×C5—21no—
136C2×C3×C4×C5—21no—
137C2×C3×C4×C5C22×C412yes—
138#1220—48no—
139C2×Dic15—22no—
140C2×Dic15—22no—
141C2×Dic15C22×C414yes—
142C2×C5×Dic3—22no—
143C2×C5×Dic3—22no—
144C2×C5×Dic3—22no—
145C2×C5×Dic3—22no—
146C2×C5×Dic3—22no—
147C2×C5×Dic3—22no—
148C2×C3×Dic5—22no—
149C2×C3×Dic5—22no—
150C2×C3×Dic5—22no—
151C2×C3×Dic5—22no—
152C2×C3×Dic5—22no—
153C2×C3×Dic5—22no—
154C2×C3×Dic5—44no—
155C24×D3—51no—
156C22×C3×D4C2×D511yescharacteristic
157C23×C3×C4—51no—
158C22×#67—52no—
159C2×C3×C22⋊C4—52no—
160C2×C3×C22⋊C4—51no—
161C2×#217—51no—
162C2×#217C2×D512yes—
163C2×#217C2×D512yescomplemented
164#937—58no—
165C2×(C2×D3)⋊C4—52no—
166C2×(C2×D3)⋊C4—52no—
167C2×(C2×D3)⋊C4—52no—
168C2×(C2×D3)⋊C4—52no—
169C2×(C2×D3)⋊C4—52no—
170C2×C4⋊Dic3C2×D511yescharacteristic, complemented
171C2×Dic3⋊C4—52no—
172C2×C4×Dic3—52no—
173C2×C4×Dic3C2×D511yescharacteristic, complemented
174C3×#95—52no—
175C3⋊#95—52no—
176C3⋊#95—52no—
177C24×C5C22×D311yescharacteristic
178C24×C5C22×D311yescharacteristic
179C23×D5—32no—
180C23×D5—34no—
181C23×D5—121no—
182C23×D5—121no—
183C23×D5—122no—
184C23×D5—122no—
185C2×C5×D4—41no—
186C2×C5×D4—41no—
187C2×C5×D4C4×D314yes—
188C22×C4×C5—32no—
189C22×C4×C5—32no—
190C22×C4×C5C22×D311yescharacteristic
191C2×C22⋊D5—38no—
192C2×C22⋊D5—122no—
193C2×C22⋊D5—122no—
194C22×Dic5—32no—
195C22×Dic5—41no—
196C22×Dic5—42no—
197C22×Dic5C22×D312yes—
198C22×Dic5C22×D312yes—
199C5×C4⋊C4—121no—
200C5×C22⋊C4—122no—
201C5×C22⋊C4—122no—
202C42×C5—121no—
203C22⋊Dic5—34no—
204C22⋊Dic5—42no—
205C22⋊Dic5—41no—
206#685—38no—
207#685—122no—
208#685—122no—
209#685—122no—
210#685—122no—
211#683—122no—
212C4×Dic5—122no—
213C2×#120—151no—
214#390—31no—
215#390—152no—
216#388—152no—
217#388—152no—
218#388—152no—
219#386—51no—
220#386—152no—
221#379—152no—
222C22×C3×C5—24no—
223C22×C3×C5—22no—
224C22×C3×C5—22no—
225C22×C3×C5—24no—
226C22×C3×C5—41no—
227C22×C3×C5—42no—
228C22×C3×C5—41no—
229C22×C3×C5—42no—
230C22×C3×C5—41no—
231C22×C3×C5—41no—
232C22×C3×C5C23×C411yescharacteristic
233C22×C3×C5H(5,𝔽2)11yescharacteristic
234C22×C3×C5H(5,𝔽2)11yescharacteristic
235C22×C3×C5H(5,𝔽2)12yes—
236C22×C3×C5H(5,𝔽2)12yes—
237C2×D15—24no—
238C2×D15—24no—
239C2×D15—24no—
240C2×D15—416no—
241C2×D15—42no—
242C2×D15—42no—
243C2×D15—42no—
244C2×D15—42no—
245C3×C4×C5—22no—
246Dic15—24no—
247C3×Dic5—42no—
248C3×Dic5—42no—
249C5×Dic3—42no—
250C5×Dic3—42no—
251C24×C3C22×D511yescharacteristic
252C24×C3C22×D511yescharacteristic
253C23×D3—52no—
254C23×D3—54no—
255C23×D3—201no—
256C23×D3—201no—
257C23×D3—202no—
258C23×D3—202no—
259C2×C3×D4—41no—
260C2×C3×D4—41no—
261C2×C3×D4C4×D514yes—
262C22×C3×C4—52no—
263C22×C3×C4—52no—
264C22×C3×C4—201no—
265C22×C3×C4—202no—
266C22×C3×C4C22×D511yescharacteristic
267C2×#67—58no—
268C2×#67—202no—
269C2×#67—202no—
270C22×Dic3—52no—
271C22×Dic3C22×D512yes—
272C22×Dic3C22×D512yes—
273C3×C22⋊C4—201no—
274C3×C22⋊C4—202no—
275#217—42no—
276#217—42no—
277#217—54no—
278(C2×D3)⋊C4—58no—
279(C2×D3)⋊C4—202no—
280(C2×D3)⋊C4—202no—
281(C2×D3)⋊C4—202no—
282(C2×D3)⋊C4—202no—
283C4⋊Dic3—41no—
284Dic3⋊C4—202no—
285C4×Dic3—41no—
286C4×Dic3—202no—
287C23×C5—21no—
288C23×C5—22no—
289C23×C5—22no—
290C23×C5—21no—
291C23×C5—21no—
292C23×C5—21no—
293C23×C5—22no—
294C23×C5—22no—
295C23×C5C2×C4×D312yes—
296C23×C5C2×C4×D312yes—
297C23×C5C23×D311yescharacteristic
298C22×D5—34no—
299C22×D5—62no—
300C22×D5—62no—
301C22×D5—62no—
302C22×D5—62no—
303C22×D5—62no—
304C22×D5—62no—
305C22×D5—124no—
306C22×D5—128no—
307C22×D5—124no—
308C22×D5—124no—
309C22×D5—128no—
310C5×D4—44no—
311C2×C4×C5—21no—
312C2×C4×C5—21no—
313C2×C4×C5—62no—
314C2×C4×C5—62no—
315C2×C4×C5—62no—
316C2×C4×C5—62no—
317C2×C4×C5—62no—
318C2×C4×C5—62no—
319C2×C4×C5C2×C4×D312yes—
320C22⋊D5—128no—
321C2×Dic5—22no—
322C2×Dic5—22no—
323C2×Dic5—22no—
324C2×Dic5—22no—
325C2×Dic5—22no—
326C2×Dic5—22no—
327C2×Dic5—34no—
328C2×Dic5—44no—
329C2×Dic5—62no—
330C2×Dic5—62no—
331C25—151no—
332C22×D4—152no—
333C22×D4D3×D511yescharacteristic
334C23×C4—51no—
335#120—158no—
336C2×C4⋊C4—31no—
337C2×C4⋊C4—152no—
338C2×C22⋊C4—32no—
339C2×C22⋊C4—32no—
340C2×C22⋊C4—52no—
341C2×C22⋊C4—51no—
342C2×C22⋊C4—151no—
343C2×C22⋊C4—152no—
344C2×C22⋊C4—152no—
345C2×C22⋊C4—152no—
346C2×C22⋊C4—152no—
347C2×C22⋊C4—152no—
348C2×C42—31no—
349C2×C42—152no—
350#95—52no—
351#95—152no—
352#95—152no—
353C2×C3×C5—42no—
354C2×C3×C5—42no—
355C2×C3×C5#51811yescharacteristic
356C2×C3×C5#51812yes—
357C2×C3×C5#56011yescharacteristic
358C2×C3×C5#56012yes—
359C2×C3×C5H(3,𝔽4)11yescharacteristic
360D15—44no—
361C23×C3—22no—
362C23×C3—22no—
363C23×C3—21no—
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—21no—
367C23×C3—22no—
368C23×C3—22no—
369C23×C3C2×C4×D512yes—
370C23×C3C2×C4×D512yes—
371C23×C3C23×D511yescharacteristic
372C22×D3—54no—
373C22×D3—102no—
374C22×D3—102no—
375C22×D3—102no—
376C22×D3—102no—
377C22×D3—102no—
378C22×D3—102no—
379C22×D3—204no—
380C22×D3—208no—
381C22×D3—204no—
382C22×D3—208no—
383C22×D3—204no—
384C3×D4—44no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—102no—
388C2×C3×C4—102no—
389C2×C3×C4—102no—
390C2×C3×C4—102no—
391C2×C3×C4—102no—
392C2×C3×C4—102no—
393C2×C3×C4—204no—
394C2×C3×C4C2×C4×D512yes—
395#67—208no—
396C2×Dic3—22no—
397C2×Dic3—22no—
398C2×Dic3—22no—
399C2×Dic3—22no—
400C2×Dic3—22no—
401C2×Dic3—22no—
402C2×Dic3—54no—
403C2×Dic3—102no—
404C2×Dic3—102no—
405C22×C5—22no—
406C22×C5—22no—
407C22×C5—24no—
408C22×C5—24no—
409C22×C5—41no—
410C22×C5—41no—
411C22×C5—41no—
412C22×C5—41no—
413C22×C5—42no—
414C22×C5—42no—
415C22×C5C22×C4×D311yescharacteristic
416C22×C5#100412yes—
417C22×C5#100411yescharacteristic
418C22×C5#100411yescharacteristic
419C22×C5#100912yes—
420C2×D5—64no—
421C2×D5—64no—
422C2×D5—64no—
423C2×D5—1216no—
424C2×D5—122no—
425C2×D5—122no—
426C2×D5—122no—
427C2×D5—122no—
428C4×C5—22no—
429C4×C5—122no—
430C4×C5—122no—
431Dic5—42no—
432Dic5—42no—
433Dic5—64no—
434C24—152no—
435C24—154no—
436C24—602no—
437C24—602no—
438C24—601no—
439C24—601no—
440C24C2×D3×D511yescharacteristic
441C24C2×D3×D511yescharacteristic
442C2×D4—41no—
443C2×D4—41no—
444C2×D4—158no—
445C2×D4—602no—
446C2×D4—602no—
447C2×D4#120014yes—
448C22×C4—32no—
449C22×C4—32no—
450C22×C4—52no—
451C22×C4—52no—
452C22×C4—152no—
453C22×C4—202no—
454C22×C4—201no—
455C22×C4C2×D3×D511yescharacteristic
456C4⋊C4—121no—
457C4⋊C4—602no—
458C22⋊C4—122no—
459C22⋊C4—122no—
460C22⋊C4—154no—
461C22⋊C4—158no—
462C22⋊C4—202no—
463C22⋊C4—201no—
464C22⋊C4—602no—
465C22⋊C4—602no—
466C22⋊C4—602no—
467C22⋊C4—602no—
468C42—121no—
469C42—602no—
470C3×C5#237511yescharacteristic, complemented, Hall
471C22×C3—22no—
472C22×C3—22no—
473C22×C3—24no—
474C22×C3—24no—
475C22×C3—41no—
476C22×C3—41no—
477C22×C3—42no—
478C22×C3—42no—
479C22×C3—41no—
480C22×C3—41no—
481C22×C3C22×C4×D511yescharacteristic
482C22×C3#414412yes—
483C22×C3#414411yescharacteristic
484C22×C3#414411yescharacteristic
485C22×C3#414912yes—
486C2×D3—104no—
487C2×D3—104no—
488C2×D3—104no—
489C2×D3—2016no—
490C2×D3—202no—
491C2×D3—202no—
492C2×D3—202no—
493C2×D3—202no—
494C3×C4—22no—
495C3×C4—202no—
496C3×C4—202no—
497Dic3—42no—
498Dic3—42no—
499Dic3—104no—
500C2×C5—42no—
501C2×C5—42no—
502C2×C5#548611yescharacteristic
503C2×C5#554512yes—
504C2×C5#559411yescharacteristic
505C2×C5#562012yes—
506C2×C5#566611yescharacteristic
507D5—124no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—154no—
517C23—302no—
518C23—302no—
519C23—302no—
520C23—302no—
521C23—302no—
522C23—302no—
523C23—604no—
524C23—608no—
525C23—604no—
526C23—608no—
527C23—604no—
528C23C2×#120012yes—
529C23C2×#120012yes—
530C23C22×D3×D511yescharacteristic
531D4—44no—
532D4—608no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—62no—
536C2×C4—62no—
537C2×C4—62no—
538C2×C4—62no—
539C2×C4—62no—
540C2×C4—62no—
541C2×C4—102no—
542C2×C4—102no—
543C2×C4—102no—
544C2×C4—102no—
545C2×C4—102no—
546C2×C4—102no—
547C2×C4—154no—
548C2×C4—204no—
549C2×C4—302no—
550C2×C4—302no—
551C2×C4C2×#120012yes—
552C2×C3—42no—
553C2×C3—42no—
554C2×C3#6582011yescharacteristic
555C2×C3#6587912yes—
556C2×C3#6595412yes—
557C2×C3#6601711yescharacteristic
558C2×C3#6605411yescharacteristic
559D3—204no—
560C5#7488211yescharacteristic, complemented, Sylow
561C22—22no—
562C22—22no—
563C22—24no—
564C22—24no—
565C22—41no—
566C22—41no—
567C22—42no—
568C22—42no—
569C22—41no—
570C22—41no—
571C22—304no—
572C22—304no—
573C22—304no—
574C22—6016no—
575C22—602no—
576C22—602no—
577C22—602no—
578C22—602no—
579C22#9202612yes—
580C22#9202712yes—
581C22C22×#120011yescharacteristic
582C22#9204911yescharacteristic
583C22#9204911yescharacteristic
584C4—22no—
585C4—122no—
586C4—122no—
587C4—202no—
588C4—202no—
589C4—304no—
590C3#1060816511yescharacteristic, complemented, Sylow
591C2—42no—
592C2—42no—
593C2—604no—
594C2#1175492212yes—
595C2#1175499311yescharacteristic
596C2#1175499712yes—
597C2#1175500211yescharacteristic
598C2#1175541311yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC22×C3×C5×D4C22
socle, cosocleC23×C3×C5C24
prime radicalC2×#91826C2
hypercenterC22×D4D3×D5
hypocenterC3×C5#2375
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
2. C5×#5221C2semidirect
3. C3×#65509C2semidirect
4. #11749502C2semidirect
5. #11749494C2semidirect
6. #11749483C2semidirect
7. #11749480C2semidirect
8. #11748854C2semidirect
9. #11748685C2semidirect
14. C23×C3×Dic5C22semidirect
16. #91826C4semidirect
19. C2×C5×#217C22semidirect
20. C2×C5×C4⋊Dic3C22semidirect
21. C2×C4×C5×Dic3C22semidirect
23. C2×C3×C22⋊Dic5C22semidirect
24. C2×#6669C22semidirect
29. C2×Dic3×Dic5C22semidirect
30. C3×#3963C22semidirect
31. #90994C22semidirect
32. #90994C22semidirect
36. #65509D3semidirect
49. C23×D15C2×C4semidirect
65. #6719C2×C4semidirect
66. #6714C2×C4semidirect
81. #5221D5semidirect
91. C23×Dic5C2×D3semidirect
98. C2×C22⋊Dic5C2×D3semidirect
109. #3963C2×D3semidirect
163. C2×#217C2×D5semidirect
170. C2×C4⋊Dic3C2×D5semidirect
173. C2×C4×Dic3C2×D5semidirect
470. C3×C5#2375semidirect
560. C5#74882semidirect
590. C3#10608165semidirect

Maximal subgroups

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

subgroupindex
C2×#918262
C5×#52212
C3×#655092
#117495022
#117494942
#117494832
#117494802
#117488542
#117486852
#106081653
#748825