Finite Group Databasebeta

← #14327989

Subgroups — #14327989

Order 1920.

1120 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91782C211yesmaximal, characteristic
2C3×C4×C5×C4⋊C4C211yesmaximal, characteristic, complemented
3#11751089C211yesmaximal, characteristic
4#11751078C211yesmaximal, characteristic, complemented
5C4⋊C4×Dic15C211yesmaximal, characteristic, complemented
6#11751041C212yesmaximal, complemented
7#11751040C211yesmaximal, characteristic
8C4×#6714C212yesmaximal, complemented
9C4×#6712C211yesmaximal, characteristic, complemented
10#11750770C212yesmaximal, complemented
11#11750765C212yesmaximal, complemented
12#10605377—31nomaximal, Hall
13C22×D60C2211yescharacteristic
14C22×C4×D15C2212yes—
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C2×C3×C5×C4⋊C4C2211yescharacteristic
17C2×C3×C42×C5C2211yescharacteristic
18C2×C3×C42×C5C2212yes—
19C2×#6714C2212yes—
20C2×#6714C2212yes—
21C2×#6714C2212yes—
22C2×#6714C2212yes—
23C2×#6714C2212yes—
24C2×C4⋊Dic15C2211yescharacteristic
25C2×#6712C2211yescharacteristic
26C2×#6712C2211yescharacteristic
27C2×C4×Dic15C2211yescharacteristic
28C2×C4×Dic15C2212yes—
29C2×C4×Dic15C2211yescharacteristic
30#91782C418yescomplemented
31C3×C5×#95C2212yes—
32#91115C2211yescharacteristic
33#91115C2212yes—
34#91115C2212yes—
35#91115C2212yes—
36#91115C2211yescharacteristic
37#73646—51nomaximal, Hall
38C2×#4039—31no—
39C4×C5×C4⋊C4D311yescharacteristic, complemented
40#65277—31no—
41#65266—31no—
42C4⋊C4×Dic5—31no—
43#65229—32no—
44#65228—31no—
45C4×#685—32no—
46C4×#683—31no—
47#64958—32no—
48#64953—32no—
49C23×D15C2312yes—
50C22×C3×C4×C5C2311yescharacteristic
51C22×C3×C4×C5C2312yes—
52C22×C3×C4×C5C2312yes—
53C22×C3×C4×C5C2311yescharacteristic
54C22×C3×C4×C5C2311yescharacteristic
55C22×Dic15C2311yescharacteristic
56C22×Dic15C2312yes—
57C22×Dic15C2311yescharacteristic
58C22×Dic15C2311yescharacteristic
59C22×Dic15C2311yescharacteristic
60C2×D60—22no—
61C2×D60—22no—
62C2×D60C2×C414yes—
63C2×C4×D15—42no—
64C2×C4×D15—42no—
65C2×C4×D15C2×C418yes—
66C3×C5×C4⋊C4—22no—
67C3×C5×C4⋊C4C2×C414yes—
68C3×C42×C5—22no—
69C3×C42×C5—42no—
70#6714—24no—
71#6714—24no—
72#6714—42no—
73#6714—42no—
74#6714C2×C418yes—
75C4⋊Dic15—22no—
76#6712—22no—
77#6712—22no—
78C4×Dic15—22no—
79C4×Dic15—42no—
80C4×Dic15C2×C414yes—
81C2×#893—51no—
82C3×C4×C4⋊C4D511yescharacteristic, complemented
83#4988—51no—
84#4977—51no—
85Dic3×C4⋊C4—51no—
86#4940—52no—
87#4939—51no—
88C4×(C2×D3)⋊C4—52no—
89C4×Dic3⋊C4—51no—
90#4669—52no—
91#4664—52no—
92C22×D20—31no—
93C22×C4×D5—32no—
94C2×C5×C4⋊C4C2×D311yescharacteristic
95C2×C5×C4⋊C4C2×D311yescharacteristic
96C2×C42×C5C2×D311yescharacteristic
97C2×C42×C5C2×D312yes—
98C2×#685—32no—
99C2×#685—32no—
100C2×#685—32no—
101C2×#685—32no—
102C2×#685—32no—
103C2×C4⋊Dic5—31no—
104C2×#683—31no—
105C2×#683—31no—
106C2×C4×Dic5—31no—
107C2×C4×Dic5—31no—
108C2×C4×Dic5—32no—
109#4039—38no—
110C5×#95C2×D312yes—
111#3963—31no—
112#3963—32no—
113#3963—32no—
114#3963—32no—
115#3963—31no—
116#2357—151noSylow
117C23×C3×C5C2411yescharacteristic
118C22×D15—22no—
119C22×D15—22no—
120C22×D15—22no—
121C22×D15—22no—
122C22×D15—22no—
123C22×D15—22no—
124C22×D15C22×C414yes—
125C2×C3×C4×C5—21no—
126C2×C3×C4×C5—22no—
127C2×C3×C4×C5—21no—
128C2×C3×C4×C5—22no—
129C2×C3×C4×C5—21no—
130C2×C3×C4×C5—22no—
131C2×C3×C4×C5—21no—
132C2×C3×C4×C5—22no—
133C2×C3×C4×C5—21no—
134C2×C3×C4×C5—22no—
135C2×C3×C4×C5C22×C414yes—
136C2×C3×C4×C5C22×C412yes—
137C2×C3×C4×C5#4112yes—
138C2×C3×C4×C5#4112yes—
139C2×C3×C4×C5#4112yes—
140C2×Dic15—21no—
141C2×Dic15—21no—
142C2×Dic15—21no—
143C2×Dic15—21no—
144C2×Dic15—21no—
145C2×Dic15—21no—
146C2×Dic15—21no—
147C2×Dic15—21no—
148C2×Dic15—22no—
149C2×Dic15—22no—
150C2×Dic15—22no—
151C2×Dic15—21no—
152C2×Dic15C22×C412yes—
153C2×Dic15C22×C412yes—
154C2×Dic15#4112yes—
155D60—28no—
156C4×D15—48no—
157C22×D12—51no—
158C22×C4×D3—52no—
159C2×C3×C4⋊C4C2×D511yescharacteristic
160C2×C3×C4⋊C4C2×D511yescharacteristic
161C2×C3×C42C2×D511yescharacteristic
162C2×C3×C42C2×D512yes—
163C2×(C2×D3)⋊C4—52no—
164C2×(C2×D3)⋊C4—52no—
165C2×(C2×D3)⋊C4—52no—
166C2×(C2×D3)⋊C4—52no—
167C2×(C2×D3)⋊C4—52no—
168C2×C4⋊Dic3—51no—
169C2×Dic3⋊C4—51no—
170C2×Dic3⋊C4—51no—
171C2×C4×Dic3—51no—
172C2×C4×Dic3—51no—
173C2×C4×Dic3—52no—
174#893—58no—
175C3×#95C2×D512yes—
176C3⋊#95—51no—
177C3⋊#95—52no—
178C3⋊#95—52no—
179C3⋊#95—52no—
180C3⋊#95—51no—
181C23×D5—32no—
182C22×C4×C5C22×D312yes—
183C22×C4×C5C22×D311yescharacteristic
184C22×C4×C5C22×D312yes—
185C22×C4×C5C22×D311yescharacteristic
186C22×C4×C5C22×D311yescharacteristic
187C22×Dic5—31no—
188C22×Dic5—31no—
189C22×Dic5—31no—
190C22×Dic5—31no—
191C22×Dic5—32no—
192C2×D20—34no—
193C2×D20—62no—
194C2×D20—62no—
195C2×C4×D5—38no—
196C2×C4×D5—122no—
197C2×C4×D5—122no—
198C5×C4⋊C4—22no—
199C5×C4⋊C4C4×D314yes—
200C42×C5—22no—
201C42×C5—42no—
202#685—38no—
203#685—64no—
204#685—64no—
205#685—122no—
206#685—122no—
207C4⋊Dic5—62no—
208#683—62no—
209#683—62no—
210C4×Dic5—34no—
211C4×Dic5—62no—
212C4×Dic5—122no—
213C2×C4×D4—151no—
214#390—151no—
215#390—151no—
216#388—152no—
217#388—152no—
218#382—151no—
219#381—152no—
220C4×C4⋊C4—151no—
221C4×C4⋊C4—151no—
222C4×C4⋊C4D1511yescharacteristic, complemented
223C4×C22⋊C4—152no—
224C22×C3×C5C23×C411yescharacteristic
225C22×C3×C5C2×#4111yescharacteristic
226C22×C3×C5C2×#4111yescharacteristic
227C22×C3×C5C2×#4111yescharacteristic
228C22×C3×C5C2×#4111yescharacteristic
229C22×C3×C5H(5,𝔽2)11yescharacteristic
230C22×C3×C5H(5,𝔽2)11yescharacteristic
231C2×D15—24no—
232C2×D15—24no—
233C2×D15—24no—
234C2×D15—42no—
235C2×D15—42no—
236C2×D15—42no—
237C2×D15—42no—
238C3×C4×C5—24no—
239C3×C4×C5—22no—
240C3×C4×C5—41no—
241C3×C4×C5—42no—
242C3×C4×C5#11714yes—
243Dic15—22no—
244Dic15—22no—
245Dic15—22no—
246Dic15—41no—
247Dic15—42no—
248C23×D3—52no—
249C22×C3×C4C22×D512yes—
250C22×C3×C4C22×D511yescharacteristic
251C22×C3×C4C22×D512yes—
252C22×C3×C4C22×D511yescharacteristic
253C22×C3×C4C22×D511yescharacteristic
254C22×Dic3—51no—
255C22×Dic3—51no—
256C22×Dic3—51no—
257C22×Dic3—51no—
258C22×Dic3—52no—
259C2×D12—54no—
260C2×D12—102no—
261C2×D12—102no—
262C2×C4×D3—58no—
263C2×C4×D3—202no—
264C2×C4×D3—202no—
265C3×C4⋊C4—22no—
266C3×C4⋊C4C4×D514yes—
267C3×C42—22no—
268C3×C42—42no—
269(C2×D3)⋊C4—58no—
270(C2×D3)⋊C4—104no—
271(C2×D3)⋊C4—104no—
272(C2×D3)⋊C4—202no—
273(C2×D3)⋊C4—202no—
274C4⋊Dic3—102no—
275Dic3⋊C4—102no—
276Dic3⋊C4—102no—
277C4×Dic3—54no—
278C4×Dic3—102no—
279C4×Dic3—202no—
280C23×C5C23×D311yescharacteristic
281C22×D5—34no—
282C22×D5—62no—
283C22×D5—62no—
284C22×D5—62no—
285C22×D5—62no—
286C22×D5—62no—
287C22×D5—62no—
288C2×C4×C5—21no—
289C2×C4×C5—21no—
290C2×C4×C5—22no—
291C2×C4×C5—22no—
292C2×C4×C5—21no—
293C2×C4×C5—22no—
294C2×C4×C5—22no—
295C2×C4×C5—21no—
296C2×C4×C5—22no—
297C2×C4×C5—21no—
298C2×C4×C5C2×C4×D314yes—
299C2×C4×C5C2×C4×D312yes—
300C2×C4×C5#23512yes—
301C2×C4×C5#23512yes—
302C2×C4×C5#23912yes—
303C2×Dic5—32no—
304C2×Dic5—32no—
305C2×Dic5—32no—
306C2×Dic5—61no—
307C2×Dic5—61no—
308C2×Dic5—61no—
309C2×Dic5—61no—
310C2×Dic5—61no—
311C2×Dic5—61no—
312C2×Dic5—61no—
313C2×Dic5—62no—
314C2×Dic5—61no—
315C2×Dic5—61no—
316C2×Dic5—62no—
317C2×Dic5—62no—
318D20—68no—
319C4×D5—128no—
320C22×D4—151no—
321C23×C4—152no—
322C4×D4—158no—
323C2×C4⋊C4—151no—
324C2×C4⋊C4—151no—
325C2×C4⋊C4—151no—
326C2×C4⋊C4C2×D1511yescharacteristic
327C2×C4⋊C4C2×D1511yescharacteristic
328C2×C22⋊C4—152no—
329C2×C22⋊C4—152no—
330C2×C22⋊C4—152no—
331C2×C22⋊C4—152no—
332C2×C22⋊C4—152no—
333C2×C42—151no—
334C2×C42—151no—
335C2×C42—152no—
336C2×C42C2×D1511yescharacteristic
337C2×C42C2×D1512yes—
338#95—151no—
339#95—152no—
340#95—151no—
341#95—152no—
342#95—152no—
343#95C2×D1512yes—
344C2×C3×C5C2×#11711yescharacteristic
345C2×C3×C5C4×#4111yescharacteristic
346C2×C3×C5#51811yescharacteristic
347C2×C3×C5#55311yescharacteristic
348C2×C3×C5#55311yescharacteristic
349C2×C3×C5#55911yescharacteristic
350C2×C3×C5#55911yescharacteristic
351D15—44no—
352C23×C3C23×D511yescharacteristic
353C22×D3—54no—
354C22×D3—102no—
355C22×D3—102no—
356C22×D3—102no—
357C22×D3—102no—
358C22×D3—102no—
359C22×D3—102no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—21no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—21no—
368C2×C3×C4—22no—
369C2×C3×C4—21no—
370C2×C3×C4C2×C4×D514yes—
371C2×C3×C4C2×C4×D512yes—
372C2×C3×C4#70912yes—
373C2×C3×C4#70912yes—
374C2×C3×C4#71312yes—
375C2×Dic3—52no—
376C2×Dic3—52no—
377C2×Dic3—52no—
378C2×Dic3—101no—
379C2×Dic3—101no—
380C2×Dic3—101no—
381C2×Dic3—101no—
382C2×Dic3—101no—
383C2×Dic3—101no—
384C2×Dic3—101no—
385C2×Dic3—102no—
386C2×Dic3—101no—
387C2×Dic3—101no—
388C2×Dic3—102no—
389C2×Dic3—102no—
390D12—108no—
391C4×D3—208no—
392C22×C5C22×C4×D311yescharacteristic
393C22×C5C2×#23511yescharacteristic
394C22×C5C2×#23511yescharacteristic
395C22×C5#100411yescharacteristic
396C22×C5C2×#23911yescharacteristic
397C22×C5D3×#4111yescharacteristic
398C22×C5#100911yescharacteristic
399C2×D5—64no—
400C2×D5—64no—
401C2×D5—64no—
402C2×D5—122no—
403C2×D5—122no—
404C2×D5—122no—
405C2×D5—122no—
406C4×C5—22no—
407C4×C5—24no—
408C4×C5—41no—
409C4×C5—42no—
410C4×C5#87214yes—
411Dic5—62no—
412Dic5—62no—
413Dic5—62no—
414Dic5—121no—
415Dic5—122no—
416C24—152no—
417C2×D4—154no—
418C2×D4—302no—
419C2×D4—302no—
420C22×C4—151no—
421C22×C4—151no—
422C22×C4—151no—
423C22×C4—158no—
424C22×C4—151no—
425C22×C4—152no—
426C22×C4—602no—
427C22×C4—602no—
428C22×C4C22×D1512yes—
429C22×C4C22×D1511yescharacteristic
430C22×C4C22×D1512yes—
431C22×C4C22×D1511yescharacteristic
432C22×C4C22×D1511yescharacteristic
433C4⋊C4—22no—
434C4⋊C4—302no—
435C4⋊C4—302no—
436C4⋊C4—302no—
437C4⋊C4C4×D1514yes—
438C22⋊C4—158no—
439C22⋊C4—304no—
440C22⋊C4—304no—
441C22⋊C4—602no—
442C22⋊C4—602no—
443C42—22no—
444C42—42no—
445C42—154no—
446C42—302no—
447C42—602no—
448C3×C5#235711yescharacteristic, complemented, Hall
449C22×C3C22×C4×D511yescharacteristic
450C22×C3C2×#70911yescharacteristic
451C22×C3C2×#70911yescharacteristic
452C22×C3#414411yescharacteristic
453C22×C3C2×#71311yescharacteristic
454C22×C3D5×#4111yescharacteristic
455C22×C3#414911yescharacteristic
456C2×D3—104no—
457C2×D3—104no—
458C2×D3—104no—
459C2×D3—202no—
460C2×D3—202no—
461C2×D3—202no—
462C2×D3—202no—
463C3×C4—22no—
464C3×C4—24no—
465C3×C4—41no—
466C3×C4—42no—
467C3×C4#401814yes—
468Dic3—102no—
469Dic3—102no—
470Dic3—102no—
471Dic3—201no—
472Dic3—202no—
473C2×C5C2×#87211yescharacteristic
474C2×C5#551411yescharacteristic
475C2×C5#553011yescharacteristic
476C2×C5#554511yescharacteristic
477C2×C5#555911yescharacteristic
478C2×C5C4×#23911yescharacteristic
479C2×C5#558511yescharacteristic
480D5—124no—
481C23—154no—
482C23—302no—
483C23—302no—
484C23—302no—
485C23—302no—
486C23—302no—
487C23—302no—
488C23C23×D1511yescharacteristic
489D4—308no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—152no—
501C2×C4—152no—
502C2×C4—152no—
503C2×C4—301no—
504C2×C4—301no—
505C2×C4—301no—
506C2×C4—301no—
507C2×C4—301no—
508C2×C4—301no—
509C2×C4—301no—
510C2×C4—302no—
511C2×C4—301no—
512C2×C4—302no—
513C2×C4—301no—
514C2×C4—302no—
515C2×C4—608no—
516C2×C4C2×C4×D1514yes—
517C2×C4C2×C4×D1512yes—
518C2×C4#681712yes—
519C2×C4#681712yes—
520C2×C4#682112yes—
521C2×C3C2×#401811yescharacteristic
522C2×C3#6584811yescharacteristic
523C2×C3#6586411yescharacteristic
524C2×C3#6587911yescharacteristic
525C2×C3#6589311yescharacteristic
526C2×C3C4×#71311yescharacteristic
527C2×C3#6591911yescharacteristic
528D3—204no—
529C5#7364611yescharacteristic, complemented, Sylow
530C22—304no—
531C22—304no—
532C22—304no—
533C22—602no—
534C22—602no—
535C22—602no—
536C22—602no—
537C22C22×C4×D1511yescharacteristic
538C22C2×#681711yescharacteristic
539C22C2×#681711yescharacteristic
540C22#9209511yescharacteristic
541C22C2×#682111yescharacteristic
542C22#41×D1511yescharacteristic
543C22#9210011yescharacteristic
544C4—24no—
545C4—22no—
546C4—41no—
547C4—42no—
548C4—302no—
549C4—302no—
550C4—302no—
551C4—602no—
552C4—601no—
553C4#9176114yes—
554C3#1060537711yescharacteristic, complemented, Sylow
555C2—604no—
556C2C2×#9176111yescharacteristic
557C2#1175627311yescharacteristic
558C2#1175628911yescharacteristic
559C2#1175630411yescharacteristic
560C2#1175631811yescharacteristic
561C2C4×#682111yescharacteristic
562C2#1175634411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D15
commutatorC22×C3×C5C23×C4
FittingC3×C4×C5×C4⋊C4C2
socle, cosocleC23×C3×C5C24
prime radicalC22×D60C22
hypercenterC4×C4⋊C4D15
hypocenterC3×C5#2357
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×C4×C5×C4⋊C4C2semidirect
4. #11751078C2semidirect
5. C4⋊C4×Dic15C2semidirect
6. #11751041C2semidirect
8. C4×#6714C2semidirect
9. C4×#6712C2semidirect
10. #11750770C2semidirect
11. #11750765C2semidirect
30. #91782C4semidirect
39. C4×C5×C4⋊C4D3semidirect
82. C3×C4×C4⋊C4D5semidirect
222. C4×C4⋊C4D15semidirect
448. C3×C5#2357semidirect
529. C5#73646semidirect
554. C3#10605377semidirect

Maximal subgroups

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

subgroupindex
C2×#917822
C3×C4×C5×C4⋊C42
#117510892
#117510782
C4⋊C4×Dic152
#117510412
#117510402
C4×#67142
C4×#67122
#117507702
#117507652
#106053773
#736465