Finite Group Databasebeta

← #14323221

Subgroups — #14323221

Order 1920.

1080 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4039C211yesmaximal, characteristic
2#11751085C211yesmaximal, characteristic, complemented
3C5×#4984C211yesmaximal, characteristic, complemented
4#11748796C211yesmaximal, characteristic
5#11748785C211yesmaximal, characteristic
6#11748772C212yesmaximal, complemented
7#11748771C212yesmaximal, complemented
8#11748759C212yesmaximal, complemented
9#11748737C212yesmaximal, complemented
10Dic5×C4⋊Dic3C211yesmaximal, characteristic, complemented
11#11748611C211yesmaximal, characteristic, complemented
12#10607700—31nomaximal, Hall
13C22×C3×D20C2211yescharacteristic
14C22×C3×C4×D5C2212yes—
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C2×C4⋊Dic15C2211yescharacteristic
17C2×C4⋊Dic15C2212yes—
18C2×C4×Dic15C2211yescharacteristic
19C2×C5×C4⋊Dic3C2211yescharacteristic
20C2×C5×C4⋊Dic3C2212yes—
21C2×C4×C5×Dic3C2211yescharacteristic
22C2×C3×#685C2212yes—
23C2×C3×C4×Dic5C2211yescharacteristic
24C3×#4039C418yescomplemented
25C2×#6665C2212yes—
26C2×#6665C2212yes—
27C2×#6665C2212yes—
28C2×#6665C2212yes—
29C2×Dic3×Dic5C2212yes—
30#91115C2212yes—
31C5×C3⋊#95C2212yes—
32#90994C2211yescharacteristic
33#90994C2211yescharacteristic
34#90994C2211yescharacteristic
35#90994C2211yescharacteristic
36#90994C2212yes—
37#82325—51nomaximal, Hall
38C2×#4039D311yescharacteristic
39C5×#389—31no—
40#65279—32no—
41#65277—31no—
42#65276—32no—
43#65274—31no—
44C4⋊C4×Dic5—31no—
45#65228—31no—
46#65221—31no—
47#64959—32no—
48#64953—32no—
49C23×C3×D5C2312yes—
50C22×C3×C4×C5C2312yes—
51C22×C3×C4×C5C2311yescharacteristic
52C22×Dic15C2311yescharacteristic
53C22×Dic15C2311yescharacteristic
54C22×Dic15C2312yes—
55C22×C5×Dic3C2311yescharacteristic
56C22×C5×Dic3C2311yescharacteristic
57C22×C5×Dic3C2312yes—
58C22×C3×Dic5C2311yescharacteristic
59C22×C3×Dic5C2311yescharacteristic
60C2×C3×D20—22no—
61C2×C3×D20—22no—
62C2×C3×D20C2×C414yes—
63C2×C3×C4×D5—42no—
64C2×C3×C4×D5—42no—
65C2×C3×C4×D5C2×C418yes—
66C3×C5×C4⋊C4C2×C414yes—
67C4⋊Dic15—22no—
68C4⋊Dic15—42no—
69C4×Dic15—22no—
70C5×C4⋊Dic3—22no—
71C5×C4⋊Dic3—42no—
72C4×C5×Dic3—22no—
73C3×#685C2×C418yes—
74C3×C4×Dic5C2×C414yes—
75#6665—42no—
76#6665—42no—
77#6665—42no—
78#6665—42no—
79Dic3×Dic5—42no—
80C2×C3×C4×D4—51no—
81#5221—51no—
82#5221—51no—
83#5212—52no—
84#5211—52no—
85#4984—51no—
86#4984D511yescharacteristic, complemented
87#4951—52no—
88#4951—52no—
89C4×C4⋊Dic3—51no—
90#4932—51no—
91C22×D20C2×D311yescharacteristic
92C22×C4×D5C2×D312yes—
93C2×C5×C4⋊C4—31no—
94C2×C5×C4⋊C4—32no—
95C2×C5×C4⋊C4C2×D311yescharacteristic
96C2×C42×C5—31no—
97C2×#685—32no—
98C2×#685—32no—
99C2×#685—32no—
100C2×#685—32no—
101C2×#685C2×D312yes—
102C2×C4⋊Dic5—31no—
103C2×C4⋊Dic5—32no—
104C2×C4×Dic5—31no—
105C2×C4×Dic5—32no—
106C2×C4×Dic5C2×D311yescharacteristic
107#4039Dic318yescomplemented
108C5×#95—32no—
109#3963—32no—
110#3963—31no—
111#3963—32no—
112#3963—31no—
113#3963—31no—
114#3963—31no—
115#2319—151noSylow
116C23×C3×C5C2411yescharacteristic
117C22×C3×D5—22no—
118C22×C3×D5—22no—
119C22×C3×D5—22no—
120C22×C3×D5—22no—
121C22×C3×D5—22no—
122C22×C3×D5—22no—
123C22×C3×D5C22×C414yes—
124C2×C3×C4×C5—22no—
125C2×C3×C4×C5—22no—
126C2×C3×C4×C5C22×C414yes—
127C2×C3×C4×C5C22×C412yes—
128C2×C3×C4×C5#4112yes—
129C2×C3×C4×C5#4112yes—
130C2×Dic15—21no—
131C2×Dic15—21no—
132C2×Dic15—22no—
133C2×Dic15—22no—
134C2×Dic15—21no—
135C2×Dic15—21no—
136C2×Dic15—22no—
137C2×Dic15—21no—
138C2×Dic15—21no—
139C2×C5×Dic3—21no—
140C2×C5×Dic3—21no—
141C2×C5×Dic3—22no—
142C2×C5×Dic3—22no—
143C2×C5×Dic3—21no—
144C2×C5×Dic3—21no—
145C2×C5×Dic3—22no—
146C2×C5×Dic3—21no—
147C2×C5×Dic3—21no—
148C2×C3×Dic5—21no—
149C2×C3×Dic5—21no—
150C2×C3×Dic5—21no—
151C2×C3×Dic5—21no—
152C2×C3×Dic5C22×C412yes—
153C2×C3×Dic5C22×C412yes—
154C3×D20—28no—
155C3×C4×D5—48no—
156C22×C3×D4—51no—
157C23×C3×C4—52no—
158C3×C4×D4—58no—
159C2×C3×C4⋊C4C2×D511yescharacteristic
160C2×C3×C22⋊C4—52no—
161C2×C3×C42—51no—
162C2×#217—52no—
163C2×#217—52no—
164C2×#217—52no—
165C2×#217—52no—
166C2×C4⋊Dic3—51no—
167C2×C4⋊Dic3—52no—
168C2×C4⋊Dic3C2×D511yescharacteristic
169C2×C4⋊Dic3C2×D512yes—
170C2×C4×Dic3—51no—
171C2×C4×Dic3—52no—
172C2×C4×Dic3C2×D511yescharacteristic
173C3⋊#95—52no—
174C3⋊#95—51no—
175C3⋊#95—51no—
176C3⋊#95—51no—
177C3⋊#95—51no—
178C3⋊#95—52no—
179C3⋊#95C2×D512yes—
180C23×D5C22×D312yes—
181C22×C4×C5—31no—
182C22×C4×C5—31no—
183C22×C4×C5—32no—
184C22×C4×C5C22×D312yes—
185C22×C4×C5C22×D311yescharacteristic
186C22×Dic5—31no—
187C22×Dic5—31no—
188C22×Dic5—32no—
189C22×Dic5C22×D311yescharacteristic
190C22×Dic5C22×D311yescharacteristic
191C2×D20—22no—
192C2×D20—22no—
193C2×D20C2×Dic314yes—
194C2×C4×D5—42no—
195C2×C4×D5—42no—
196C2×C4×D5C2×Dic318yes—
197C5×C4⋊C4—62no—
198C5×C4⋊C4—122no—
199C5×C4⋊C4C2×Dic314yes—
200C42×C5—62no—
201#685—122no—
202#685—122no—
203#685—122no—
204#685—122no—
205#685C2×Dic318yes—
206C4⋊Dic5—62no—
207C4⋊Dic5—122no—
208C4×Dic5—62no—
209C4×Dic5—122no—
210C4×Dic5C2×Dic314yes—
211C2×C4×D4—51no—
212#390—151no—
213#390—151no—
214#389—31no—
215#389—151no—
216#388—152no—
217#388—152no—
218#381—152no—
219#380—152no—
220C4×C4⋊C4—151no—
221#376—151no—
222C22×C3×C5C23×C411yescharacteristic
223C22×C3×C5C2×#4111yescharacteristic
224C22×C3×C5C2×#4111yescharacteristic
225C22×C3×C5H(5,𝔽2)11yescharacteristic
226C22×C3×C5H(5,𝔽2)11yescharacteristic
227C22×C3×C5H(5,𝔽2)11yescharacteristic
228C22×C3×C52−1+411yescharacteristic
229C2×C3×D5—24no—
230C2×C3×D5—24no—
231C2×C3×D5—24no—
232C2×C3×D5—42no—
233C2×C3×D5—42no—
234C2×C3×D5—42no—
235C2×C3×D5—42no—
236C3×C4×C5—24no—
237C3×C4×C5#11714yes—
238Dic15—41no—
239Dic15—42no—
240Dic15—41no—
241C3×Dic5—22no—
242C3×Dic5—22no—
243C5×Dic3—41no—
244C5×Dic3—42no—
245C5×Dic3—41no—
246C24×C3—52no—
247C2×C3×D4—54no—
248C2×C3×D4—102no—
249C2×C3×D4—102no—
250C22×C3×C4—51no—
251C22×C3×C4—51no—
252C22×C3×C4—58no—
253C22×C3×C4—202no—
254C22×C3×C4—202no—
255C22×C3×C4C22×D512yes—
256C22×C3×C4C22×D511yescharacteristic
257C22×Dic3—51no—
258C22×Dic3—51no—
259C22×Dic3—52no—
260C22×Dic3C22×D511yescharacteristic
261C22×Dic3C22×D511yescharacteristic
262C22×Dic3C22×D512yes—
263C3×C4⋊C4C4×D514yes—
264C3×C22⋊C4—58no—
265C3×C42—54no—
266#217—202no—
267#217—202no—
268#217—202no—
269#217—202no—
270C4⋊Dic3—22no—
271C4⋊Dic3—42no—
272C4⋊Dic3—102no—
273C4⋊Dic3—202no—
274C4×Dic3—22no—
275C4×Dic3—102no—
276C4×Dic3—202no—
277C23×C5C23×D311yescharacteristic
278C22×D5—22no—
279C22×D5—22no—
280C22×D5—22no—
281C22×D5—22no—
282C22×D5—22no—
283C22×D5—22no—
284C22×D5C22×Dic314yes—
285C2×C4×C5—22no—
286C2×C4×C5—22no—
287C2×C4×C5—61no—
288C2×C4×C5—61no—
289C2×C4×C5—62no—
290C2×C4×C5—61no—
291C2×C4×C5—62no—
292C2×C4×C5—61no—
293C2×C4×C5—62no—
294C2×C4×C5—61no—
295C2×C4×C5—61no—
296C2×C4×C5#23512yes—
297C2×C4×C5#23512yes—
298C2×C4×C5C22×Dic314yes—
299C2×C4×C5C22×Dic312yes—
300C2×Dic5—21no—
301C2×Dic5—21no—
302C2×Dic5—21no—
303C2×Dic5—21no—
304C2×Dic5—61no—
305C2×Dic5—61no—
306C2×Dic5—62no—
307C2×Dic5—61no—
308C2×Dic5—62no—
309C2×Dic5—61no—
310C2×Dic5—62no—
311C2×Dic5—61no—
312C2×Dic5—61no—
313C2×Dic5C22×Dic312yes—
314C2×Dic5C22×Dic312yes—
315D20—28no—
316C4×D5—48no—
317C22×D4—51no—
318C23×C4—52no—
319C4×D4—58no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—32no—
322C2×C4⋊C4—151no—
323C2×C4⋊C4—152no—
324C2×C4⋊C4D3×D511yescharacteristic
325C2×C22⋊C4—52no—
326C2×C22⋊C4—152no—
327C2×C22⋊C4—152no—
328C2×C22⋊C4—152no—
329C2×C22⋊C4—152no—
330C2×C42—31no—
331C2×C42—51no—
332C2×C42—151no—
333C2×C42—152no—
334#95—32no—
335#95—151no—
336#95—151no—
337#95—151no—
338#95—151no—
339#95—152no—
340#95—152no—
341C2×C3×C5C2×#11711yescharacteristic
342C2×C3×C5#51811yescharacteristic
343C2×C3×C5#52011yescharacteristic
344C2×C3×C5#54011yescharacteristic
345C2×C3×C5#54011yescharacteristic
346C2×C3×C5#54211yescharacteristic
347C2×C3×C5#54211yescharacteristic
348C3×D5—44no—
349C23×C3—54no—
350C23×C3—102no—
351C23×C3—102no—
352C23×C3—102no—
353C23×C3—102no—
354C23×C3—102no—
355C23×C3—102no—
356C23×C3C23×D511yescharacteristic
357C3×D4—108no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—52no—
361C2×C3×C4—52no—
362C2×C3×C4—101no—
363C2×C3×C4—101no—
364C2×C3×C4—101no—
365C2×C3×C4—101no—
366C2×C3×C4—208no—
367C2×C3×C4C2×C4×D514yes—
368C2×C3×C4C2×C4×D512yes—
369C2×C3×C4#71112yes—
370C2×C3×C4#71312yes—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—22no—
374C2×Dic3—21no—
375C2×Dic3—22no—
376C2×Dic3—21no—
377C2×Dic3—22no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—101no—
381C2×Dic3—101no—
382C2×Dic3—101no—
383C2×Dic3—102no—
384C2×Dic3—101no—
385C2×Dic3—102no—
386C2×Dic3—101no—
387C2×Dic3—102no—
388C2×Dic3—101no—
389C22×C5C2×#23511yescharacteristic
390C22×C5C2×#23511yescharacteristic
391C22×C5#100411yescharacteristic
392C22×C5#100411yescharacteristic
393C22×C5#100911yescharacteristic
394C22×C5#101011yescharacteristic
395C22×C5C23×Dic311yescharacteristic
396C2×D5—24no—
397C2×D5—24no—
398C2×D5—24no—
399C2×D5—42no—
400C2×D5—42no—
401C2×D5—42no—
402C2×D5—42no—
403C4×C5—24no—
404C4×C5—121no—
405C4×C5—121no—
406C4×C5—122no—
407C4×C5#92614yes—
408Dic5—22no—
409Dic5—22no—
410Dic5—121no—
411Dic5—121no—
412Dic5—122no—
413C24—52no—
414C2×D4—54no—
415C2×D4—102no—
416C2×D4—102no—
417C22×C4—31no—
418C22×C4—32no—
419C22×C4—31no—
420C22×C4—51no—
421C22×C4—51no—
422C22×C4—58no—
423C22×C4—151no—
424C22×C4—151no—
425C22×C4—152no—
426C22×C4—202no—
427C22×C4—202no—
428C22×C4C2×D3×D512yes—
429C22×C4C2×D3×D511yescharacteristic
430C4⋊C4—62no—
431C4⋊C4—122no—
432C4⋊C4—302no—
433C4⋊C4—602no—
434C4⋊C4D5×Dic314yes—
435C22⋊C4—58no—
436C22⋊C4—602no—
437C22⋊C4—602no—
438C22⋊C4—602no—
439C22⋊C4—602no—
440C42—54no—
441C42—62no—
442C42—302no—
443C42—602no—
444C3×C5#231911yescharacteristic, complemented, Hall
445C22×C3—104no—
446C22×C3—104no—
447C22×C3—104no—
448C22×C3—202no—
449C22×C3—202no—
450C22×C3—202no—
451C22×C3—202no—
452C22×C3C22×C4×D511yescharacteristic
453C22×C3C2×#71111yescharacteristic
454C22×C3#414411yescharacteristic
455C22×C3C2×#71311yescharacteristic
456C22×C3#414711yescharacteristic
457C22×C3#414911yescharacteristic
458C22×C3#414911yescharacteristic
459C3×C4—24no—
460C3×C4—102no—
461C3×C4—102no—
462C3×C4#403814yes—
463Dic3—41no—
464Dic3—42no—
465Dic3—41no—
466Dic3—201no—
467Dic3—201no—
468Dic3—202no—
469C2×C5#556111yescharacteristic
470C2×C5#556111yescharacteristic
471C2×C5#556211yescharacteristic
472C2×C5#556211yescharacteristic
473C2×C5C2×#92611yescharacteristic
474C2×C5#579811yescharacteristic
475C2×C5#582711yescharacteristic
476D5—44no—
477C23—54no—
478C23—102no—
479C23—102no—
480C23—102no—
481C23—102no—
482C23—102no—
483C23—102no—
484C23C22×D3×D511yescharacteristic
485D4—108no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—52no—
489C2×C4—52no—
490C2×C4—61no—
491C2×C4—61no—
492C2×C4—62no—
493C2×C4—62no—
494C2×C4—61no—
495C2×C4—61no—
496C2×C4—61no—
497C2×C4—62no—
498C2×C4—61no—
499C2×C4—101no—
500C2×C4—101no—
501C2×C4—101no—
502C2×C4—101no—
503C2×C4—208no—
504C2×C4—301no—
505C2×C4—301no—
506C2×C4—302no—
507C2×C4—301no—
508C2×C4—302no—
509C2×C4—301no—
510C2×C4—301no—
511C2×C4—302no—
512C2×C4—301no—
513C2×C4#677212yes—
514C2×C4#677312yes—
515C2×C4C2×D5×Dic314yes—
516C2×C4C2×D5×Dic312yes—
517C2×C3—204no—
518C2×C3C2×#403811yescharacteristic
519C2×C3#6587911yescharacteristic
520C2×C3#6590811yescharacteristic
521C2×C3#6595311yescharacteristic
522C2×C3#6597711yescharacteristic
523C2×C3#6603211yescharacteristic
524C2×C3#6606711yescharacteristic
525C5#8232511yescharacteristic, complemented, Sylow
526C22—104no—
527C22—104no—
528C22—104no—
529C22—202no—
530C22—202no—
531C22—202no—
532C22—202no—
533C22C2×#677211yescharacteristic
534C22C2×#677311yescharacteristic
535C22#9202511yescharacteristic
536C22#9202611yescharacteristic
537C22#9202811yescharacteristic
538C22#9203511yescharacteristic
539C22C22×D5×Dic311yescharacteristic
540C4—24no—
541C4—102no—
542C4—102no—
543C4—121no—
544C4—122no—
545C4—121no—
546C4—601no—
547C4—602no—
548C4—601no—
549C4#9135814yes—
550C3#1060770011yescharacteristic, complemented, Sylow
551C2—204no—
552C2#1175336511yescharacteristic
553C2#1175381511yescharacteristic
554C2#1175381911yescharacteristic
555C2#1175382111yescharacteristic
556C2C2×#9135811yescharacteristic
557C2#1175491011yescharacteristic
558C2#1175521711yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C5×C4⋊C4C22
socle, cosocleC23×C3×C5C24
prime radicalC22×C3×D20C22
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5#2319
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11751085C2semidirect
3. C5×#4984C2semidirect
6. #11748772C2semidirect
7. #11748771C2semidirect
8. #11748759C2semidirect
9. #11748737C2semidirect
10. Dic5×C4⋊Dic3C2semidirect
11. #11748611C2semidirect
24. C3×#4039C4semidirect
86. #4984D5semidirect
107. #4039Dic3semidirect
444. C3×C5#2319semidirect
525. C5#82325semidirect
550. C3#10607700semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#40392
#117510852
C5×#49842
#117487962
#117487852
#117487722
#117487712
#117487592
#117487372
Dic5×C4⋊Dic32
#117486112
#106077003
#823255