Finite Group Databasebeta

← #14330630

Subgroups — #14330630

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
2C3×C5×#386C211yesmaximal, characteristic, complemented
3#11751322C211yesmaximal, characteristic, complemented
4#11751314C212yesmaximal, complemented
5#11751058C212yesmaximal, complemented
6#11751056C212yesmaximal, complemented
7#11751046C212yesmaximal, complemented
8#11750771C212yesmaximal, complemented
9#11750770C212yesmaximal, complemented
10#10606613—31nomaximal, Hall
11C24×D15C2211yescharacteristic
12C22×C3×C5×D4C2211yescharacteristic
13C23×C3×C4×C5C2211yescharacteristic, complemented
14C22×#1220C2212yes—
15C2×C3×C5×C22⋊C4C2212yes—
16C2×C3×C5×C22⋊C4C2211yescharacteristic, complemented
17C2×#6719C2212yes—
18C2×#6719C2211yescharacteristic
19C2×#6719C2212yescomplemented
20#91826C418yescomplemented
21C2×#6714C2212yes—
22C2×#6714C2212yes—
23C2×#6714C2212yes—
24C2×#6714C2212yes—
25C2×#6714C2212yes—
26C2×C4⋊Dic15C2211yescharacteristic, complemented
27C2×#6712C2212yescomplemented
28C2×C4×Dic15C2211yescharacteristic, complemented
29C2×C4×Dic15C2212yescomplemented
30C3×C5×#95C2212yescomplemented
31#91115C2212yescomplemented
32#91115C2212yescomplemented
33#74882—51nomaximal, Hall
34C2×#4083—31no—
35C5×#386D311yescharacteristic, complemented
36#65510—31no—
37#65502—32no—
38#65246—32no—
39#65244—32no—
40#65234—32no—
41#64959—32no—
42#64958—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×C5—41no—
55C22×C3×C4×C5—42no—
56C22×C3×C4×C5C2312yes—
57C22×C3×C4×C5C2312yes—
58C22×C3×C4×C5C2311yescharacteristic
59C2×#1220—42no—
60C2×#1220—42no—
61C2×#1220C2×C418yes—
62C22×Dic15C2312yes—
63C22×Dic15C2312yes—
64C22×Dic15C2312yes—
65C3×C5×C22⋊C4—42no—
66C3×C5×C22⋊C4—41no—
67#6719—42no—
68#6719—42no—
69#6719C2×C414yescomplemented
70#6714—42no—
71#6714—42no—
72#6714—42no—
73#6714—42no—
74#6714C2×C418yescomplemented
75C4⋊Dic15—41no—
76#6712—42no—
77C4×Dic15—42no—
78C4×Dic15—41no—
79C2×#937—51no—
80C3×#386D511yescharacteristic, complemented
81#5221—51no—
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
90C23×C4×C5C2×D311yescharacteristic, complemented
91C22×C22⋊D5—32no—
92C2×C5×C22⋊C4C2×D312yes—
93C2×C5×C22⋊C4C2×D311yescharacteristic, complemented
94C2×C22⋊Dic5—31no—
95C2×C22⋊Dic5—32no—
96C2×C22⋊Dic5—32no—
97#4083—38no—
98C2×#685—32no—
99C2×#685—32no—
100C2×#685—32no—
101C2×#685—32no—
102C2×#685—32no—
103C2×C4⋊Dic5—31no—
104C2×#683—32no—
105C2×C4×Dic5—31no—
106C2×C4×Dic5—32no—
107C5×#95C2×D312yescomplemented
108#3963—32no—
109#3963—32no—
110#2375—151noSylow
111C23×C3×C5—22no—
112C23×C3×C5—22no—
113C23×C3×C5—21no—
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—48no—
131C22×D15—44no—
132C22×D15—44no—
133C22×D15C22×C414yes—
134C3×C5×D4—44no—
135C2×C3×C4×C5—21no—
136C2×C3×C4×C5—21no—
137C2×C3×C4×C5—22no—
138C2×C3×C4×C5—22no—
139C2×C3×C4×C5—22no—
140C2×C3×C4×C5—22no—
141C2×C3×C4×C5—22no—
142C2×C3×C4×C5—22no—
143C2×C3×C4×C5—44no—
144C2×C3×C4×C5C22×C412yes—
145#1220—48no—
146C2×Dic15—22no—
147C2×Dic15—22no—
148C2×Dic15—22no—
149C2×Dic15—22no—
150C2×Dic15—22no—
151C2×Dic15—22no—
152C2×Dic15—22no—
153C2×Dic15—22no—
154C2×Dic15C22×C414yes—
155C24×D3—51no—
156C22×C3×D4C2×D511yescharacteristic
157C23×C3×C4C2×D511yescharacteristic, complemented
158C22×#67—52no—
159C2×C3×C22⋊C4C2×D512yes—
160C2×C3×C22⋊C4C2×D511yescharacteristic, complemented
161C2×#217—51no—
162C2×#217—52no—
163C2×#217—52no—
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⋊Dic3—51no—
171C2×Dic3⋊C4—52no—
172C2×C4×Dic3—51no—
173C2×C4×Dic3—52no—
174C3×#95C2×D512yescomplemented
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—41no—
189C22×C4×C5—42no—
190C22×C4×C5C22×D312yes—
191C22×C4×C5C22×D312yes—
192C22×C4×C5C22×D311yescharacteristic
193C2×C22⋊D5—38no—
194C2×C22⋊D5—122no—
195C2×C22⋊D5—122no—
196C22×Dic5—32no—
197C22×Dic5—32no—
198C22×Dic5—32no—
199C5×C22⋊C4—42no—
200C5×C22⋊C4—41no—
201C22⋊Dic5—34no—
202C22⋊Dic5—122no—
203C22⋊Dic5—122no—
204#685—38no—
205#685—122no—
206#685—122no—
207#685—122no—
208#685—122no—
209C4⋊Dic5—121no—
210#683—122no—
211C4×Dic5—122no—
212C4×Dic5—121no—
213C2×#120—151no—
214#390—151no—
215#390—152no—
216#388—152no—
217#388—152no—
218#388—152no—
219#386—152no—
220#386D1511yescharacteristic, complemented
221#379—152no—
222C22×C3×C5—24no—
223C22×C3×C5—22no—
224C22×C3×C5—22no—
225C22×C3×C5—24no—
226C22×C3×C5—42no—
227C22×C3×C5—41no—
228C22×C3×C5—42no—
229C22×C3×C5—41no—
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—
246C3×C4×C5—42no—
247C3×C4×C5—42no—
248Dic15—24no—
249Dic15—42no—
250Dic15—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—41no—
263C22×C3×C4—42no—
264C22×C3×C4C22×D512yes—
265C22×C3×C4C22×D512yes—
266C22×C3×C4C22×D511yescharacteristic
267C2×#67—58no—
268C2×#67—202no—
269C2×#67—202no—
270C22×Dic3—52no—
271C22×Dic3—52no—
272C22×Dic3—52no—
273C3×C22⋊C4—42no—
274C3×C22⋊C4—41no—
275#217—54no—
276#217—202no—
277#217—202no—
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—201no—
284Dic3⋊C4—202no—
285C4×Dic3—202no—
286C4×Dic3—201no—
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—22no—
314C2×C4×C5—22no—
315C2×C4×C5—22no—
316C2×C4×C5—22no—
317C2×C4×C5—22no—
318C2×C4×C5—22no—
319C2×C4×C5—44no—
320C2×C4×C5C2×C4×D312yes—
321C22⋊D5—128no—
322C2×Dic5—34no—
323C2×Dic5—62no—
324C2×Dic5—62no—
325C2×Dic5—62no—
326C2×Dic5—62no—
327C2×Dic5—62no—
328C2×Dic5—62no—
329C2×Dic5—62no—
330C2×Dic5—62no—
331C25—151no—
332C22×D4—152no—
333C22×D4C2×D1511yescharacteristic
334C23×C4C2×D1511yescharacteristic, complemented
335#120—158no—
336C2×C4⋊C4—152no—
337C2×C4⋊C4—151no—
338C2×C22⋊C4—151no—
339C2×C22⋊C4—152no—
340C2×C22⋊C4—152no—
341C2×C22⋊C4—152no—
342C2×C22⋊C4—152no—
343C2×C22⋊C4—152no—
344C2×C22⋊C4—152no—
345C2×C22⋊C4—152no—
346C2×C22⋊C4C2×D1512yes—
347C2×C22⋊C4C2×D1511yescharacteristic, complemented
348C2×C42—152no—
349C2×C42—151no—
350#95—152no—
351#95—152no—
352#95C2×D1512yescomplemented
353C2×C3×C5—42no—
354C2×C3×C5—42no—
355C2×C3×C5#51811yescharacteristic
356C2×C3×C5#51812yes—
357C2×C3×C5#56012yes—
358C2×C3×C5#56011yescharacteristic
359C2×C3×C5H(3,𝔽4)11yescharacteristic
360D15—44no—
361C23×C3—21no—
362C23×C3—22no—
363C23×C3—22no—
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—22no—
388C2×C3×C4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—22no—
393C2×C3×C4—44no—
394C2×C3×C4C2×C4×D512yes—
395#67—208no—
396C2×Dic3—54no—
397C2×Dic3—102no—
398C2×Dic3—102no—
399C2×Dic3—102no—
400C2×Dic3—102no—
401C2×Dic3—102no—
402C2×Dic3—102no—
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#100411yescharacteristic
417C22×C5#100411yescharacteristic
418C22×C5#100412yes—
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—42no—
430C4×C5—42no—
431Dic5—64no—
432Dic5—122no—
433Dic5—122no—
434C24—152no—
435C24—154no—
436C24—602no—
437C24—601no—
438C24—601no—
439C24—602no—
440C24C22×D1511yescharacteristic
441C24C22×D1511yescharacteristic
442C2×D4—41no—
443C2×D4—41no—
444C2×D4—158no—
445C2×D4—602no—
446C2×D4—602no—
447C2×D4C4×D1514yes—
448C22×C4—42no—
449C22×C4—41no—
450C22×C4—152no—
451C22×C4—152no—
452C22×C4—152no—
453C22×C4C22×D1512yes—
454C22×C4C22×D1512yes—
455C22×C4C22×D1511yescharacteristic
456C4⋊C4—602no—
457C4⋊C4—601no—
458C22⋊C4—42no—
459C22⋊C4—41no—
460C22⋊C4—154no—
461C22⋊C4—158no—
462C22⋊C4—602no—
463C22⋊C4—602no—
464C22⋊C4—602no—
465C22⋊C4—602no—
466C22⋊C4—602no—
467C22⋊C4—602no—
468C42—601no—
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#414411yescharacteristic
483C22×C3#414411yescharacteristic
484C22×C3#414412yes—
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—42no—
496C3×C4—42no—
497Dic3—104no—
498Dic3—202no—
499Dic3—202no—
500C2×C5—42no—
501C2×C5—42no—
502C2×C5#548611yescharacteristic
503C2×C5#554512yes—
504C2×C5#559411yescharacteristic
505C2×C5#562012yes—
506C2×C5#566611yescharacteristic
507D5—124no—
508C23—21no—
509C23—22no—
510C23—21no—
511C23—22no—
512C23—21no—
513C23—21no—
514C23—22no—
515C23—22no—
516C23—154no—
517C23—302no—
518C23—302no—
519C23—302no—
520C23—302no—
521C23—302no—
522C23—302no—
523C23—608no—
524C23—604no—
525C23—604no—
526C23—608no—
527C23—604no—
528C23C2×C4×D1512yes—
529C23C2×C4×D1512yes—
530C23C23×D1511yescharacteristic
531D4—44no—
532D4—608no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—44no—
542C2×C4—154no—
543C2×C4—302no—
544C2×C4—302no—
545C2×C4—302no—
546C2×C4—302no—
547C2×C4—302no—
548C2×C4—302no—
549C2×C4—302no—
550C2×C4—302no—
551C2×C4C2×C4×D1512yes—
552C2×C3—42no—
553C2×C3—42no—
554C2×C3#6582011yescharacteristic
555C2×C3#6587912yes—
556C2×C3#6592811yescharacteristic
557C2×C3#6595412yes—
558C2×C3#6600011yescharacteristic
559D3—204no—
560C5#7488211yescharacteristic, complemented, Sylow
561C22—24no—
562C22—24no—
563C22—22no—
564C22—22no—
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—
579C22C22×C4×D1511yescharacteristic
580C22#9209512yes—
581C22#9209511yescharacteristic
582C22#9209511yescharacteristic
583C22#9210012yes—
584C4—22no—
585C4—42no—
586C4—42no—
587C4—304no—
588C4—602no—
589C4—602no—
590C3#1060661311yescharacteristic, complemented, Sylow
591C2—42no—
592C2—42no—
593C2—604no—
594C2#1175624511yescharacteristic
595C2#1175630412yes—
596C2#1175635311yescharacteristic
597C2#1175637912yes—
598C2#1175642511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D15
commutatorC22×C3×C5C23×C4
FittingC3×C5×#386C2
socle, cosocleC23×C3×C5C24
prime radicalC2×#91826C2
hypercenter#386D15
hypocenterC3×C5#2375
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C3×C5×#386C2semidirect
3. #11751322C2semidirect
4. #11751314C2semidirect
5. #11751058C2semidirect
6. #11751056C2semidirect
7. #11751046C2semidirect
8. #11750771C2semidirect
9. #11750770C2semidirect
13. C23×C3×C4×C5C22semidirect
16. C2×C3×C5×C22⋊C4C22semidirect
19. C2×#6719C22semidirect
20. #91826C4semidirect
26. C2×C4⋊Dic15C22semidirect
27. C2×#6712C22semidirect
28. C2×C4×Dic15C22semidirect
29. C2×C4×Dic15C22semidirect
30. C3×C5×#95C22semidirect
31. #91115C22semidirect
32. #91115C22semidirect
35. C5×#386D3semidirect
49. C23×D15C2×C4semidirect
69. #6719C2×C4semidirect
74. #6714C2×C4semidirect
80. C3×#386D5semidirect
90. C23×C4×C5C2×D3semidirect
93. C2×C5×C22⋊C4C2×D3semidirect
107. C5×#95C2×D3semidirect
157. C23×C3×C4C2×D5semidirect
160. C2×C3×C22⋊C4C2×D5semidirect
174. C3×#95C2×D5semidirect
220. #386D15semidirect
334. C23×C4C2×D15semidirect
347. C2×C22⋊C4C2×D15semidirect
352. #95C2×D15semidirect
470. C3×C5#2375semidirect
560. C5#74882semidirect
590. C3#10606613semidirect

Maximal subgroups

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

subgroupindex
C2×#918262
C3×C5×#3862
#117513222
#117513142
#117510582
#117510562
#117510462
#117507712
#117507702
#106066133
#748825