Finite Group Databasebeta

← #14412036

Subgroups — #14412036

Order 1920.

1432 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#92020C211yesmaximal, characteristic
2#11756812C212yesmaximal, complemented
3#11752430C211yesmaximal, characteristic, complemented
4C3×#65771C211yesmaximal, characteristic, complemented
5#11752031C214yesmaximal, complemented
6#11752030C214yesmaximal, complemented
7#11751815C212yesmaximal, complemented
8#10617539—31nomaximal, Hall
9C2×#6819C411yescharacteristic, complemented
10C2×C5×#237C411yescharacteristic, complemented
11C2×C3×#711C2211yescharacteristic
12C22×#1204C412yescomplemented
13C2×#6782C412yescomplemented
14C2×#6780C2212yes—
15C2×#6779C2212yes—
16#92020—24no—
17#92020—24no—
18C2×#6771C2211yescharacteristic
19C2×D15⋊Q8C411yescharacteristic, complemented
20C2×D3×Dic10C411yescharacteristic, complemented
21C2×D5×Dic6C2211yescharacteristic
22#91989C2212yes—
23C3×#4130C2212yes—
24#91923C2218yes—
25ΓL(1,𝔽16)⋊Q8—24no—
26AGL(1,𝔽5)×Dic6—24no—
27#91240C2214yes—
28C3×#4011C2214yes—
29#91173C2214yes—
30#91172C2214yes—
31#78689—51nomaximal, Hall
32C2×#4150—31no—
33#66262—32no—
34#65771—34no—
35#65771—31no—
36#65771D311yescharacteristic, complemented
37#65745—34no—
38#65745—32no—
39C2×C3×C5×D4C2×C411yescharacteristic
40C2×#1220C2×C412yes—
41C22×Dic15C2×C412yes—
42#6819—24no—
43C2×C4×D15C2×C411yescharacteristic
44C2×Dic30C2×C411yescharacteristic
45C2×C5×#67C2×C412yes—
46C22×C5×Dic3C2×C412yes—
47C5×#237—24no—
48C2×C4×C5×D3C2×C411yescharacteristic
49C2×C5×Dic6C2×C411yescharacteristic
50C2×C3×C22⋊D5C2312yes—
51C22×C3×Dic5C2×C412yes—
52C3×#711—22no—
53C3×#711—22no—
54C2×C3×C4×D5C2311yescharacteristic
55C2×C3×Dic10C2×C411yescharacteristic
56C2×#1204—24no—
57C2×#1204—24no—
58C2×#1204C2×C412yes—
59C2×#1204C2×C412yes—
60C2×#1204C2×C411yescharacteristic
61C2×#1204C2×C412yes—
62C2×#1203C2×C411yescharacteristic
63C2×#1202C2311yescharacteristic
64C2×#1201C2311yescharacteristic
65C2×#1200C2×C412yes—
66#6782—24no—
67#6782—24no—
68C2×D3×Dic5C2×C412yes—
69#6780—28no—
70#6779—28no—
71C2×D5×Dic3C2312yes—
72#6771—22no—
73#6771—22no—
74D15⋊Q8—24no—
75D3×Dic10—24no—
76D5×Dic6D414yes—
77D5×Dic6D414yes—
78#6763C2314yes—
79C4⋊ΓL(1,𝔽16)—22no—
80C4×ΓL(1,𝔽16)—22no—
81C3×C22⋊AGL(1,𝔽5)C2314yes—
82C3×C4⋊AGL(1,𝔽5)—22no—
83C3×C4×AGL(1,𝔽5)—22no—
84Dic3⋊AGL(1,𝔽5)—28no—
85Dic3×AGL(1,𝔽5)—28no—
86C2×#1010—51no—
87#5516—52no—
88C3×#410—51no—
89#5240—51no—
90#5006—52no—
91#4740—54no—
92#4740—54no—
93C2×C5×#41—31no—
94#4150—64no—
95#4150—64no—
96C2×Q8×D5—31no—
97C2×Q8×D5—31no—
98C2×#711—31no—
99C2×#711—32no—
100C2×#711—32no—
101C2×#711C2×D311yescharacteristic
102C2×#709—32no—
103C2×#709—31no—
104C22×Dic10—32no—
105C22×Dic10—31no—
106Q8×AGL(1,𝔽5)—64no—
107Q8×AGL(1,𝔽5)—64no—
108#4130—32no—
109#4130—38no—
110#4130C2×D312yes—
111#4011—34no—
112#4011—34no—
113#4011C2×D314yes—
114#3999—34no—
115#2884—151noSylow
116C23×C3×C5C22×C412yes—
117C22×D15C22×C411yescharacteristic
118C22×C5×D3C22×C411yescharacteristic
119C22×C3×D5C2411yescharacteristic
120C2×ΓL(1,𝔽16)—24no—
121C2×C3×AGL(1,𝔽5)—24no—
122C3×C5×D4—22no—
123C2×C3×C4×C5C22×C411yescharacteristic
124#1220—24no—
125C2×Dic15—24no—
126C2×Dic15C22×C411yescharacteristic
127C2×Dic15C22×C412yes—
128C4×D15—22no—
129Dic30C22⋊C414yes—
130C5×#67—24no—
131C2×C5×Dic3—24no—
132C2×C5×Dic3C22×C411yescharacteristic
133C2×C5×Dic3C22×C412yes—
134C4×C5×D3—22no—
135C5×Dic6C22⋊C414yes—
136C3×C22⋊D5—24no—
137C2×C3×Dic5—22no—
138C2×C3×Dic5—22no—
139C2×C3×Dic5C22×C412yes—
140C2×C3×Dic5C22×C411yescharacteristic
141C3×C4×D5C2×D412yes—
142C3×C4×D5C2×D412yes—
143C3×Dic10—22no—
144#1204—24no—
145#1204—21no—
146#1204—21no—
147#1204—24no—
148#1204C22⋊C414yes—
149#1204C22⋊C414yes—
150#1203—21no—
151#1203—21no—
152#1202—22no—
153#1201—22no—
154#1200—24no—
155D3×Dic5—24no—
156D5×Dic3C2×D418yes—
157C2×C3×#41—51no—
158#1010—104no—
159#1010—104no—
160C2×D3×Q8—51no—
161C2×D3×Q8—51no—
162C2×#237—51no—
163C2×#237—52no—
164C2×#237—52no—
165C2×#237AGL(1,𝔽5)11yescharacteristic, complemented
166C2×#235—52no—
167C2×#235—51no—
168C22×Dic6—51no—
169C22×Dic6—52no—
170C3×#117—52no—
171#926—52no—
172#887—104no—
173#877—58no—
174C4×Dic6—104no—
175C3×#99—54no—
176#834—54no—
177#806—54no—
178#806—54no—
179C5×#41—64no—
180C2×C5×Q8—31no—
181C2×C5×D4—32no—
182C2×C5×D4C4×D311yescharacteristic
183C22×C4×C5—32no—
184C22×C4×C5—31no—
185C2×C22⋊D5—32no—
186C2×C22⋊D5—31no—
187C2×C22⋊D5—31no—
188C2×C22⋊D5C22×D312yes—
189C22×Dic5—32no—
190C22×Dic5—32no—
191C22×Dic5C4×D312yes—
192Q8×D5—34no—
193Q8×D5—34no—
194Q8×D5—64no—
195#711—22no—
196#711—22no—
197#711—64no—
198#711—64no—
199#711—64no—
200#711—68no—
201#709—68no—
202#709—62no—
203#709—62no—
204C2×D20—31no—
205C2×C4×D5—32no—
206C2×C4×D5—31no—
207C2×C4×D5—32no—
208C2×C4×D5C22×D311yescharacteristic
209C2×Dic10—31no—
210C2×Dic10—32no—
211C2×Dic10—32no—
212C2×Dic10—31no—
213C2×Dic10—32no—
214C2×Dic10—64no—
215C2×Dic10—64no—
216C2×Dic10—64no—
217C2×Dic10C4×D311yescharacteristic
218C22⋊AGL(1,𝔽5)—34no—
219C22⋊AGL(1,𝔽5)C22×D314yes—
220C4⋊AGL(1,𝔽5)—22no—
221C4⋊AGL(1,𝔽5)—68no—
222C4⋊AGL(1,𝔽5)—62no—
223C4×AGL(1,𝔽5)—22no—
224C4×AGL(1,𝔽5)—68no—
225C4×AGL(1,𝔽5)—62no—
226C2×2−1+4—151no—
227#519—152no—
228#410—51no—
229#410—154no—
230#410—154no—
231#410—152no—
232#410—151no—
233C22×C3×C5—22no—
234C22×C3×C5—42no—
235C22×C3×C5C23×C411yescharacteristic
236C2×D15—21no—
237C2×D15—41no—
238C2×C5×D3—21no—
239C2×C5×D3—41no—
240C2×C3×D5—41no—
241C2×C3×D5C22×D412yes—
242ΓL(1,𝔽16)—44no—
243C3×AGL(1,𝔽5)—44no—
244C3×C4×C5C2×C22⋊C412yes—
245Dic15—21no—
246Dic15C2×C22⋊C414yes—
247C3×Dic5—22no—
248C3×Dic5C2×C22⋊C411yescharacteristic
249C3×Dic5C2×C22⋊C411yescharacteristic
250C5×Dic3—21no—
251C5×Dic3C2×C22⋊C414yes—
252C3×#41—102no—
253C3×#41—102no—
254C2×C3×Q8—51no—
255C2×C3×D4—52no—
256C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
257C22×C3×C4—51no—
258C22×C3×C4—52no—
259C2×#67—52no—
260C2×#67—51no—
261C2×#67—51no—
262C2×#67C2×AGL(1,𝔽5)12yes—
263C22×Dic3—52no—
264C22×Dic3—52no—
265C22×Dic3C2×AGL(1,𝔽5)12yes—
266D3×Q8—104no—
267D3×Q8—104no—
268#237—24no—
269#237—104no—
270#237—108no—
271#237—108no—
272#235—104no—
273#235—104no—
274#235—102no—
275#235—102no—
276C2×D12—51no—
277C2×C4×D3—52no—
278C2×C4×D3—51no—
279C2×C4×D3—52no—
280C2×C4×D3C2×AGL(1,𝔽5)11yescharacteristic
281C2×Dic6—51no—
282C2×Dic6—52no—
283C2×Dic6—54no—
284C2×Dic6—54no—
285C2×Dic6—51no—
286C2×Dic6—52no—
287C2×Dic6—52no—
288C2×Dic6—104no—
289C2×Dic6—104no—
290C2×Dic6C2×AGL(1,𝔽5)11yescharacteristic
291C3×C4⋊C4—102no—
292C3×C22⋊C4—54no—
293C3×C42—102no—
294#217—54no—
295C4⋊Dic3—102no—
296Dic3⋊C4—108no—
297C4×Dic3—108no—
298C4×Dic3—102no—
299C23×C5—31no—
300C23×C5C2×C4×D312yes—
301C22×D5—31no—
302C22×D5C23×D311yescharacteristic
303C2×AGL(1,𝔽5)—24no—
304C2×AGL(1,𝔽5)—64no—
305C5×Q8—34no—
306C5×D4—22no—
307C5×D4—64no—
308C2×C4×C5—32no—
309C2×C4×C5—31no—
310C2×C4×C5—64no—
311C2×C4×C5—62no—
312C2×C4×C5C2×C4×D311yescharacteristic
313C22⋊D5—24no—
314C22⋊D5—64no—
315C22⋊D5—62no—
316C22⋊D5—61no—
317C22⋊D5—61no—
318C2×Dic5—22no—
319C2×Dic5—22no—
320C2×Dic5—32no—
321C2×Dic5—31no—
322C2×Dic5—64no—
323C2×Dic5—64no—
324C2×Dic5C2×C4×D312yes—
325C2×Dic5C2×C4×D311yescharacteristic
326D20—62no—
327C4×D5—38no—
328C4×D5—64no—
329C4×D5—62no—
330C4×D5D3×D412yes—
331C4×D5D3×D412yes—
332Dic10—22no—
333Dic10—34no—
334Dic10—34no—
335Dic10—34no—
336Dic10—64no—
337Dic10—64no—
338Dic10—61no—
339Dic10—61no—
3402−1+4—304no—
3412−1+4—304no—
342C2×#41—31no—
343C2×#41—51no—
344C2×#41—151no—
345C2×#41—152no—
346C2×#41—152no—
347C2×#41—152no—
348C2×#41—151no—
349C22×Q8—152no—
350C22×Q8—151no—
351C22×Q8—151no—
352C22×Q8—151no—
353C4×Q8—304no—
354C4×Q8—304no—
355#117—52no—
356#117—158no—
357#117—152no—
358#99—54no—
359#99—154no—
360#99—154no—
361#99—154no—
362C2×C3×C5—21no—
363C2×C3×C5—42no—
364C2×C3×C5C22×C22⋊C411yescharacteristic
365D15—41no—
366C3×D5—22no—
367C5×D3—41no—
368C23×C3—51no—
369C23×C3C22×AGL(1,𝔽5)12yes—
370C22×D3—51no—
371C22×D3C22×AGL(1,𝔽5)11yescharacteristic
372C3×Q8—102no—
373C3×D4—22no—
374C3×D4—104no—
375C2×C3×C4—51no—
376C2×C3×C4—52no—
377C2×C3×C4—52no—
378C2×C3×C4—52no—
379C2×C3×C4—102no—
380C2×C3×C4—102no—
381C2×C3×C4—104no—
382C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
383#67—24no—
384#67—102no—
385#67—104no—
386#67—102no—
387C2×Dic3—24no—
388C2×Dic3—52no—
389C2×Dic3—58no—
390C2×Dic3—51no—
391C2×Dic3—104no—
392C2×Dic3—104no—
393C2×Dic3C22×AGL(1,𝔽5)11yescharacteristic
394C2×Dic3C22×AGL(1,𝔽5)12yes—
395D12—101no—
396D12—101no—
397C4×D3—22no—
398C4×D3—104no—
399C4×D3—102no—
400C4×D3—104no—
401Dic6—54no—
402Dic6—54no—
403Dic6—54no—
404Dic6—104no—
405Dic6—101no—
406Dic6—101no—
407Dic6—104no—
408Dic6C22⋊AGL(1,𝔽5)14yes—
409C22×C5—22no—
410C22×C5—42no—
411C22×C5—61no—
412C22×C5—121no—
413C22×C5C22×C4×D311yescharacteristic
414C2×D5—41no—
415C2×D5—61no—
416C2×D5—121no—
417C2×D5C2×D3×D412yes—
418AGL(1,𝔽5)—44no—
419AGL(1,𝔽5)—124no—
420C4×C5—34no—
421C4×C5—61no—
422C4×C5D3×C22⋊C412yes—
423Dic5—22no—
424Dic5—34no—
425Dic5—61no—
426Dic5D3×C22⋊C411yescharacteristic
427Dic5D3×C22⋊C411yescharacteristic
428#41—64no—
429#41—102no—
430#41—102no—
431#41—302no—
432#41—302no—
433#41—308no—
434#41—304no—
435#41—304no—
436#41—308no—
437#41—304no—
438C2×Q8—31no—
439C2×Q8—51no—
440C2×Q8—152no—
441C2×Q8—152no—
442C2×Q8—151no—
443C2×Q8—152no—
444C2×Q8—154no—
445C2×Q8—154no—
446C2×Q8—151no—
447C2×Q8—304no—
448C2×Q8—304no—
449C2×Q8—304no—
450C2×Q8—304no—
451C2×D4—32no—
452C2×D4—52no—
453C2×D4—151no—
454C2×D4—151no—
455C2×D4—151no—
456C2×D4—152no—
457C2×D4D3×AGL(1,𝔽5)11yescharacteristic
458C22×C4—32no—
459C22×C4—31no—
460C22×C4—51no—
461C22×C4—52no—
462C22×C4—152no—
463C22×C4—152no—
464C22×C4—152no—
465C22×C4—152no—
466C22×C4—151no—
467C4⋊C4—102no—
468C4⋊C4—302no—
469C4⋊C4—308no—
470C22⋊C4—54no—
471C22⋊C4—154no—
472C42—102no—
473C42—302no—
474C42—308no—
475C3×C5#288411yescharacteristic, complemented, Hall
476C22×C3—22no—
477C22×C3—42no—
478C22×C3—52no—
479C22×C3—201no—
480C22×C3C23×AGL(1,𝔽5)11yescharacteristic
481C2×D3—21no—
482C2×D3—41no—
483C2×D3—101no—
484C2×D3—201no—
485C3×C4—51no—
486C3×C4—51no—
487C3×C4—102no—
488C3×C4—204no—
489C3×C4C2×C22⋊AGL(1,𝔽5)12yes—
490Dic3—21no—
491Dic3—54no—
492Dic3—101no—
493Dic3—204no—
494Dic3C2×C22⋊AGL(1,𝔽5)14yes—
495C2×C5—21no—
496C2×C5—42no—
497C2×C5—121no—
498C2×C5C2×D3×C22⋊C411yescharacteristic
499D5—22no—
500D5—121no—
501C23—31no—
502C23—51no—
503C23—151no—
504C23C2×D3×AGL(1,𝔽5)12yes—
505Q8—34no—
506Q8—102no—
507Q8—154no—
508Q8—154no—
509Q8—154no—
510Q8—304no—
511Q8—301no—
512Q8—301no—
513Q8—304no—
514D4—22no—
515D4—64no—
516D4—104no—
517D4—301no—
518D4—301no—
519D4—302no—
520D4—304no—
521D4—302no—
522C2×C4—32no—
523C2×C4—31no—
524C2×C4—51no—
525C2×C4—52no—
526C2×C4—52no—
527C2×C4—52no—
528C2×C4—64no—
529C2×C4—62no—
530C2×C4—102no—
531C2×C4—102no—
532C2×C4—104no—
533C2×C4—158no—
534C2×C4—152no—
535C2×C4—151no—
536C2×C4—304no—
537C2×C4—304no—
538C2×C4—304no—
539C2×C4—302no—
540C2×C4—304no—
541C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
542C2×C3—21no—
543C2×C3—42no—
544C2×C3—102no—
545C2×C3C22×C22⋊AGL(1,𝔽5)11yescharacteristic
546D3—41no—
547D3—201no—
548C5#7868911yescharacteristic, complemented, Sylow
549C22—22no—
550C22—42no—
551C22—52no—
552C22—61no—
553C22—121no—
554C22—201no—
555C22—301no—
556C22—601no—
557C22C22×D3×AGL(1,𝔽5)11yescharacteristic
558C4—34no—
559C4—51no—
560C4—51no—
561C4—61no—
562C4—102no—
563C4—154no—
564C4—204no—
565C4—301no—
566C4—604no—
567C4D3×C22⋊AGL(1,𝔽5)12yes—
568C3#1061753911yescharacteristic, complemented, Sylow
569C2—21no—
570C2—42no—
571C2—102no—
572C2—121no—
573C2—601no—
574C2C2×D3×C22⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC2C2×D3×C22⋊AGL(1,𝔽5)
commutatorC22×C3×C5C23×C4
FrattiniC22C22×D3×AGL(1,𝔽5)
FittingC2×C3×C5×D4C2×C4
socleC2×C3×C5C22×C22⋊C4
prime radicalC2×#92020C2
cosocleC22×C3×D5C24
hypercenterC2×D4D3×AGL(1,𝔽5)
hypocenterC3×C5#2884
LCSG ▷ C22×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11756812C2semidirect
3. #11752430C2semidirect
4. C3×#65771C2semidirect
5. #11752031C2semidirect
6. #11752030C2semidirect
7. #11751815C2semidirect
9. C2×#6819C4semidirect
10. C2×C5×#237C4semidirect
12. C22×#1204C4semidirect
13. C2×#6782C4semidirect
19. C2×D15⋊Q8C4semidirect
20. C2×D3×Dic10C4semidirect
36. #65771D3semidirect
165. C2×#237AGL(1,𝔽5)semidirect
475. C3×C5#2884semidirect
548. C5#78689semidirect
568. C3#10617539semidirect

Maximal subgroups

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

subgroupindex
C2×#920202
#117568122
#117524302
C3×#657712
#117520312
#117520302
#117518152
#106175393
#786895