Finite Group Databasebeta

← #14424605

Subgroups — #14424605

Order 1920.

1400 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×#92001C211yesmaximal, characteristic
2C22×Dic60C211yesmaximal, characteristic, complemented
3#11754252C211yesmaximal, characteristic, complemented
4C2×#91322C211yesmaximal, characteristic, complemented
5C5×#5113C211yesmaximal, characteristic, complemented
6C3×#65399C211yesmaximal, characteristic, complemented
7#11748417C211yesmaximal, characteristic, complemented
8#11748164C214yesmaximal, complemented
9#11748130C214yesmaximal, complemented
10#10610568—31nomaximal, Hall
11C22×Dic30C2211yescharacteristic
12C22×Dic30C2211yescharacteristic
13C2×C5×#235C2211yescharacteristic
14C2×C3×#709C2211yescharacteristic
15C2×#6780—22no—
16C2×#6772C2212yes—
17C2×D3×Dic10—21no—
18#92001—24no—
19#92001—24no—
20C2×#6765—21no—
21C2×D5×Dic6C2212yes—
22C22×C3×C5×C8C2211yescharacteristic
23C2×Dic60—24no—
24C2×Dic60C2214yes—
25C5×#926C2211yescharacteristic
26C2×C3×#681C2211yescharacteristic
27#91537—22no—
28#91389—21no—
29#91364—21no—
30#91360C2214yes—
31#91322—24no—
32C2×#6663C2212yes—
33C2×#6659C2212yes—
34C5×#821C2212yes—
35C5×#816C2212yes—
36C3×#3952C2214yes—
37#90975C2214yes—
38#78679—51nomaximal, Hall
39C2×#4150—31no—
40C2×#4097—31no—
41C22×Dic20—31no—
42#65860—31no—
43C5×#416—31no—
44#65399D311yescharacteristic, complemented
45#65293—31no—
46#65097—34no—
47#65072—34no—
48C22×C3×C4×C5C2311yescharacteristic
49C22×Dic15—21no—
50C22×Dic15—21no—
51C2×Dic30—21no—
52C2×Dic30—22no—
53C2×Dic30—22no—
54C2×Dic30—22no—
55C2×Dic30—21no—
56C2×Dic30D414yes—
57C2×Dic30C2312yes—
58C2×Dic30C2312yes—
59C2×C5×#67—21no—
60C5×#235—22no—
61C5×#235D414yes—
62C2×C5×D12C2311yescharacteristic
63C2×C4×C5×D3—21no—
64C2×C5×Dic6C2311yescharacteristic
65C2×C3×C22⋊D5D412yes—
66C3×#709—22no—
67C3×#709—22no—
68C2×C3×D20D411yescharacteristic
69C2×C3×C4×D5C2312yes—
70C2×C3×Dic10D411yescharacteristic
71C2×#1204—22no—
72C2×#1201—22no—
73C2×D3×Dic5—22no—
74#6780—216no—
75C2×D5×Dic3—22no—
76#6772—24no—
77#6772—24no—
78D3×Dic10—28no—
79#6765—28no—
80D5×Dic6—24no—
81D5×Dic6—24no—
82C2×C3×C5×C8—22no—
83C2×C3×C5×C8C2312yes—
84Dic60—28no—
85C5×#217—21no—
86C5×C4⋊Dic3C2312yes—
87C4×C5×Dic3—21no—
88C3×#681—22no—
89C2×C3×#168C2312yes—
90Dic3⋊Dic5—24no—
91#6665—24no—
92#6663—24no—
93#6659—24no—
94C2×#1010—51no—
95#5825—51no—
96C2×#909—51no—
97C22×Dic12—51no—
98C3×#408—51no—
99#5120—51no—
100#5113D511yescharacteristic, complemented
101#4739—54no—
102#4733—54no—
103C2×C5×#41—31no—
104#4150—64no—
105#4150—64no—
106C2×Q8×D5—32no—
107C2×#711—32no—
108C2×#711—62no—
109C2×#709—61no—
110C2×#709C2×D311yescharacteristic
111C22×Dic10—31no—
112C22×Dic10—31no—
113C22×Dic10—61no—
114C22×C5×C8C2×D311yescharacteristic
115C5×#117—31no—
116#4097—64no—
117C2×#689—32no—
118C2×#687—32no—
119C2×#681C2×D311yescharacteristic
120C2×Dic20—34no—
121C2×Dic20—64no—
122#4043—34no—
123#4032—62no—
124#4021—61no—
125#4015—61no—
126C5×#103—32no—
127C5×#102—32no—
128#3952C2×D314yes—
129#3942—34no—
130#3011—151noSylow
131C23×C3×C5C2×D411yescharacteristic
132C22×C5×D3—21no—
133C22×C3×D5C2×D412yes—
134C2×C3×C4×C5C2×D411yescharacteristic
135C2×C3×C4×C5C2×D412yes—
136C2×C3×C4×C5C2×D412yes—
137C2×C3×C4×C5C2411yescharacteristic
138C2×Dic15—24no—
139C2×Dic15—22no—
140C2×Dic15—22no—
141C2×Dic15—42no—
142Dic30—24no—
143Dic30—22no—
144Dic30—22no—
145Dic30—22no—
146Dic30—42no—
147Dic30C2×D414yes—
148C5×#67—24no—
149C2×C5×Dic3—21no—
150C2×C5×Dic3—22no—
151C5×D12—21no—
152C5×D12C2×D412yes—
153C4×C5×D3—24no—
154C5×Dic6—21no—
155C5×Dic6C2×D412yes—
156C3×C22⋊D5—24no—
157C2×C3×Dic5C2×D412yes—
158C3×D20—22no—
159C3×C4×D5—22no—
160C3×C4×D5—22no—
161C3×Dic10—22no—
162#1204—28no—
163#1201—28no—
164D3×Dic5—28no—
165D5×Dic3—28no—
166C3×C5×C8—22no—
167C3×#168—22no—
168C2×C3×#41—51no—
169#1010—104no—
170#1010—104no—
171C2×D3×Q8—101no—
172C2×#237—101no—
173C2×#237—102no—
174C2×#235—52no—
175C2×#235C2×D511yescharacteristic
176C22×Dic6—51no—
177C22×Dic6—51no—
178C22×Dic6—52no—
179C2×C3×M16—51no—
180C22×C3×C8C2×D511yescharacteristic
181#944—101no—
182#936—101no—
183#935—102no—
184#926C2×D511yescharacteristic
185#924—54no—
186#909—104no—
187C2×Dic12—52no—
188C2×Dic12—54no—
189C2×Dic12—104no—
190C2×#204—52no—
191C3×C22⋊C8—54no—
192#821C2×D512yes—
193#816—54no—
194#816C2×D512yes—
195C5×#41—34no—
196C5×#41—62no—
197C2×C5×Q8—31no—
198C2×C5×D4—31no—
199C2×C5×D4—61no—
200C22×C4×C5—61no—
201C22×C4×C5C22×D311yescharacteristic
202C2×C22⋊D5—62no—
203C2×C22⋊D5D1212yes—
204C22×Dic5—61no—
205C22×Dic5—62no—
206C22×Dic5—61no—
207Q8×D5—64no—
208Q8×D5—64no—
209#711—64no—
210#711—64no—
211#711—616no—
212#709—22no—
213#709—22no—
214#709—68no—
215C2×D20D1211yescharacteristic
216C2×C4×D5—62no—
217C2×C4×D5C22×D312yes—
218C2×Dic10—34no—
219C2×Dic10—32no—
220C2×Dic10—32no—
221C2×Dic10—68no—
222C2×Dic10—61no—
223C2×Dic10—62no—
224C2×Dic10—62no—
225C2×Dic10—62no—
226C2×Dic10—62no—
227C2×Dic10—61no—
228C2×Dic10D1211yescharacteristic
229C2×C5×C8—22no—
230C2×C5×C8C22×D312yes—
231C5×C4⋊C4—32no—
232C5×C22⋊C4—61no—
233C42×C5—61no—
234#689—64no—
235#687—64no—
236#685—64no—
237C4⋊Dic5—64no—
238#681—22no—
239C2×#168C22×D312yes—
240Dic20—68no—
241C2×2−1+4—151no—
242C2×#137—151no—
243C22×Q16—151no—
244#536—151no—
245#452—154no—
246#451—154no—
247#419—151no—
248#416—31no—
249#408—51no—
250C22×C3×C5—22no—
251C22×C3×C5#12012yes—
252C22×C3×C5C22×D411yescharacteristic
253C2×C5×D3—22no—
254C2×C5×D3—42no—
255C2×C3×D5—22no—
256C2×C3×D5—24no—
257C3×C4×C5#12012yes—
258C3×C4×C5C22×D411yescharacteristic
259C3×C4×C5C22×D411yescharacteristic
260Dic15—24no—
261Dic15—42no—
262C3×Dic5—22no—
263C5×Dic3—22no—
264C5×Dic3—42no—
265C3×#41—102no—
266C3×#41—102no—
267C2×C3×Q8—51no—
268C2×C3×D4—52no—
269C2×C3×D4—51no—
270C22×C3×C4—52no—
271C22×C3×C4C22×D511yescharacteristic
272C2×#67—21no—
273C2×#67—102no—
274C22×Dic3—102no—
275C22×Dic3—101no—
276C22×Dic3—101no—
277D3×Q8—108no—
278#237—108no—
279#237—1016no—
280#235—22no—
281#235—104no—
282#235—104no—
283#235D2014yes—
284C2×D12C22×D511yescharacteristic
285C2×C4×D3—21no—
286C2×C4×D3—102no—
287C2×Dic6—54no—
288C2×Dic6—52no—
289C2×Dic6—52no—
290C2×Dic6—104no—
291C2×Dic6—102no—
292C2×Dic6—102no—
293C2×Dic6—101no—
294C2×Dic6—104no—
295C2×Dic6—102no—
296C2×Dic6—102no—
297C2×Dic6—101no—
298C2×Dic6C22×D511yescharacteristic
299C3×M16—102no—
300C2×C3×C8—22no—
301C2×C3×C8—52no—
302C2×C3×C8C22×D512yes—
303#217—21no—
304#217—104no—
305C4⋊Dic3C22×D512yes—
306Dic3⋊C4—104no—
307C4×Dic3—21no—
308Dic12—108no—
309Dic12—104no—
310#204—104no—
311C23×C5—61no—
312C23×C5C2×D1211yescharacteristic
313C22×D5C2×D1212yes—
314C5×Q8—32no—
315C5×Q8—61no—
316C5×D4—32no—
317C5×D4—64no—
318C5×D4—61no—
319C2×C4×C5—61no—
320C2×C4×C5—64no—
321C2×C4×C5—62no—
322C2×C4×C5C2×D1211yescharacteristic
323C2×C4×C5D3×D412yes—
324C2×C4×C5D3×D412yes—
325C2×C4×C5C23×D311yescharacteristic
326C22⋊D5—24no—
327C22⋊D5—68no—
328C2×Dic5—68no—
329C2×Dic5—64no—
330C2×Dic5—62no—
331C2×Dic5—62no—
332C2×Dic5—122no—
333C2×Dic5C2×D1212yes—
334D20—22no—
335C4×D5—22no—
336C4×D5—22no—
337C4×D5—68no—
338Dic10—22no—
339Dic10—34no—
340Dic10—68no—
341Dic10—64no—
342Dic10—62no—
343Dic10—62no—
344Dic10—62no—
345Dic10—122no—
346C5×C8—22no—
347#168—22no—
3482−1+4—304no—
3492−1+4—304no—
350C2×#41—31no—
351C2×#41—51no—
352C2×#41—152no—
353C2×#41—301no—
354C2×#41—302no—
355C22×Q8—152no—
356C22×Q8—151no—
357C22×Q8—151no—
358C22×Q8—301no—
359#137—304no—
360C2×Q16—152no—
361C2×Q16—154no—
362C2×Q16—304no—
363C2×SD16—152no—
364C2×M16—51no—
365C22×C8D3×D511yescharacteristic
366C4⋊Q8—301no—
367#124—301no—
368#123—302no—
369C22⋊Q8—154no—
370#117—31no—
371#103—32no—
372#103—154no—
373#102—32no—
374C22⋊C8—54no—
375C2×C3×C5—22no—
376C2×C3×C5C2×#12011yescharacteristic
377C2×C3×C5#57812yes—
378C3×D5—42no—
379C5×D3—42no—
380C23×C3—52no—
381C23×C3D4×D511yescharacteristic
382C22×D3—21no—
383C3×Q8—102no—
384C3×D4—104no—
385C3×D4—102no—
386C2×C3×C4—52no—
387C2×C3×C4—102no—
388C2×C3×C4—102no—
389C2×C3×C4C2×D2012yes—
390C2×C3×C4D4×D511yescharacteristic
391C2×C3×C4D4×D512yes—
392C2×C3×C4C23×D511yescharacteristic
393#67—24no—
394#67—108no—
395C2×Dic3—21no—
396C2×Dic3—22no—
397C2×Dic3—108no—
398C2×Dic3—102no—
399C2×Dic3—104no—
400C2×Dic3—102no—
401C2×Dic3—202no—
402D12—21no—
403D12C2×D2012yes—
404C4×D3—24no—
405C4×D3—108no—
406Dic6—21no—
407Dic6—54no—
408Dic6—108no—
409Dic6—102no—
410Dic6—104no—
411Dic6—102no—
412Dic6—102no—
413Dic6—202no—
414Dic6C2×D2012yes—
415C3×C8—22no—
416C3×C8—102no—
417C22×C5—22no—
418C22×C5—62no—
419C22×C5—122no—
420C22×C5#88212yes—
421C22×C5C22×D1211yescharacteristic
422C2×D5—22no—
423C2×D5—24no—
424C4×C5—62no—
425C4×C5—122no—
426C4×C5#88212yes—
427C4×C5C2×D3×D411yescharacteristic
428C4×C5C2×D3×D411yescharacteristic
429Dic5—22no—
430Dic5—64no—
431Dic5—122no—
432#41—34no—
433#41—62no—
434#41—102no—
435#41—102no—
436#41—308no—
437#41—304no—
438#41—304no—
439#41—3016no—
440C2×Q8—31no—
441C2×Q8—51no—
442C2×Q8—154no—
443C2×Q8—152no—
444C2×Q8—152no—
445C2×Q8—302no—
446C2×Q8—308no—
447C2×Q8—304no—
448C2×Q8—304no—
449C2×Q8—301no—
450C2×Q8—301no—
451C2×Q8—302no—
452C2×Q8—302no—
453C2×Q8—302no—
454C2×D4—31no—
455C2×D4—52no—
456C2×D4—51no—
457C2×D4—61no—
458C2×D4—302no—
459C22×C4—52no—
460C22×C4—61no—
461C22×C4—302no—
462C22×C4—302no—
463C22×C4—301no—
464C22×C4—301no—
465C22×C4C2×D3×D511yescharacteristic
466Q16—308no—
467Q16—304no—
468SD16—304no—
469M16—102no—
470C2×C8—22no—
471C2×C8—52no—
472C2×C8C2×D3×D512yes—
473C4⋊C4—32no—
474C4⋊C4—304no—
475C22⋊C4—61no—
476C22⋊C4—304no—
477C42—61no—
478C3×C5#301111yescharacteristic, complemented, Hall
479C22×C3—22no—
480C22×C3—102no—
481C22×C3—104no—
482C22×C3#402812yes—
483C22×C3C2×D4×D511yescharacteristic
484C2×D3—22no—
485C2×D3—42no—
486C3×C4—102no—
487C3×C4#402812yes—
488C3×C4C22×D2011yescharacteristic
489C3×C4C2×D4×D511yescharacteristic
490Dic3—22no—
491Dic3—42no—
492Dic3—104no—
493Dic3—202no—
494C2×C5—22no—
495C2×C5—122no—
496C2×C5C2×#88211yescharacteristic
497C2×C5#575312yes—
498D5—42no—
499C23—52no—
500C23—61no—
501C23D5×D1211yescharacteristic
502Q8—32no—
503Q8—61no—
504Q8—102no—
505Q8—154no—
506Q8—308no—
507Q8—304no—
508Q8—302no—
509Q8—302no—
510Q8—302no—
511Q8—602no—
512D4—32no—
513D4—64no—
514D4—61no—
515D4—104no—
516D4—102no—
517D4—308no—
518C2×C4—52no—
519C2×C4—61no—
520C2×C4—64no—
521C2×C4—62no—
522C2×C4—102no—
523C2×C4—102no—
524C2×C4—308no—
525C2×C4—308no—
526C2×C4—304no—
527C2×C4—302no—
528C2×C4—302no—
529C2×C4—602no—
530C2×C4D5×D1211yescharacteristic
531C2×C4D3×D2012yes—
532C2×C4#679012yes—
533C2×C4C22×D3×D511yescharacteristic
534C8—22no—
535C8—102no—
536C2×C3—22no—
537C2×C3—202no—
538C2×C3C2×#402811yescharacteristic
539C2×C3#6610512yes—
540D3—42no—
541C5#7867911yescharacteristic, complemented, Sylow
542C22—22no—
543C22—62no—
544C22—104no—
545C22—102no—
546C22—122no—
547C22#9147612yes—
548C22C2×D5×D1211yescharacteristic
549C4—62no—
550C4—102no—
551C4—122no—
552C4—304no—
553C4—602no—
554C4#9147612yes—
555C4C2×D3×D2011yescharacteristic
556C4C2×#679011yescharacteristic
557C3#1061056811yescharacteristic, complemented, Sylow
558C2—22no—
559C2—122no—
560C2—202no—
561C2#1175398812yes—
562C2C2×#9147611yescharacteristic
subgroupG/H
centerC22C2×D5×D12
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC22×C3×C5×C8C22
socleC22×C3×C5C22×D4
prime radicalC2×#92001C2
hypercenterC22×C8D3×D5
hypocenterC3×C5#3011
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
2. C22×Dic60C2semidirect
3. #11754252C2semidirect
4. C2×#91322C2semidirect
5. C5×#5113C2semidirect
6. C3×#65399C2semidirect
7. #11748417C2semidirect
8. #11748164C2semidirect
9. #11748130C2semidirect
44. #65399D3semidirect
100. #5113D5semidirect
478. C3×C5#3011semidirect
541. C5#78679semidirect
557. C3#10610568semidirect

Maximal subgroups

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

subgroupindex
C2×#920012
C22×Dic602
#117542522
C2×#913222
C5×#51132
C3×#653992
#117484172
#117481642
#117481302
#106105683
#786795