Finite Group Databasebeta

← #14417108

Subgroups — #14417108

Order 1920.

1012 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756280C211yesmaximal, characteristic, complemented
2#11754499C211yesmaximal, characteristic, complemented
3C2×#91301C211yesmaximal, characteristic
4C5×#4995C211yesmaximal, characteristic, complemented
5C3×#65291C211yesmaximal, characteristic, complemented
6C2×#90969C211yesmaximal, characteristic, complemented
7#11748324C211yesmaximal, characteristic, complemented
8#11747273C212yesmaximal, complemented
9#11747272C212yesmaximal, complemented
10#11747229C212yesmaximal, complemented
11#11747226C212yesmaximal, complemented
12#10610932—31nomaximal, Hall
13C22×D60C2211yescharacteristic
14C2×C3×#709C2211yescharacteristic
15C3×C5×#117C2211yescharacteristic
16#91815C2211yescharacteristic
17#91782C2212yes—
18#91782C2212yes—
19C22⋊C4×D15—22no—
20C4×D60—21no—
21C4×D60—21no—
22C2×C5×C4⋊Dic3C2211yescharacteristic
23C2×C5×#208C2211yescharacteristic
24C22×C3×#168C2211yescharacteristic
25Dic3×C22⋊D5—22no—
26#91434C2212yescomplemented
27D5×C4⋊Dic3—21no—
28#91358—21no—
29#91331C2212yescomplemented
30C2×#6658C2212yes—
31#91301C418yescomplemented
32C2×#6653C2212yes—
33C5×#808C2212yescomplemented
34C5×#807C2212yescomplemented
35C3×#3942C2211yescharacteristic, complemented
36C3×#3942C2211yescharacteristic, complemented
37C3×#3941C2211yescharacteristic, complemented
38C3×#3941C2211yescharacteristic, complemented
39#90969C2212yes—
40#90969C2212yes—
41#90954C2212yescomplemented
42#90951C2212yescomplemented
43#78820—51nomaximal, Hall
44C2×#4054—31no—
45#65855—31no—
46#65838—31no—
47C5×#428—31no—
48#65411—31no—
49#65291D311yescharacteristic, complemented
50C2×#3941—31no—
51#65178—32no—
52#65161—32no—
53#65158—32no—
54#65141—32no—
55C23×D15—21no—
56C22×C3×C4×C5C2311yescharacteristic
57C2×D60—21no—
58C2×D60—22no—
59C2×D60C2×C414yescomplemented
60C2×D60C2312yes—
61C2×C4×D15—22no—
62C2×C4×D15—22no—
63C22×C5×Dic3—21no—
64C2×C3×C22⋊D5—21no—
65C3×#709—22no—
66C3×#709C2×C414yescomplemented
67C2×C3×D20C2311yescharacteristic
68C2×C3×C4×D5—21no—
69C2×C3×Dic10C2311yescharacteristic
70C2×D5×Dic3—22no—
71C3×C5×C4⋊C4C2311yescharacteristic
72C3×C5×C4⋊C4C2311yescharacteristic
73C3×C5×C22⋊C4—21no—
74C3×C42×C5—21no—
75#6719D412yes—
76#6714—22no—
77#6714—22no—
78C4⋊Dic15D411yescharacteristic
79C4⋊Dic15D411yescharacteristic
80C4×Dic15C2312yes—
81C5×C4⋊Dic3—21no—
82C5×C4⋊Dic3C2312yes—
83C5×#208C2×C414yescomplemented
84C2×C5×C3⋊C8C2312yes—
85C2×C3×#168—22no—
86C2×C3×#168C2311yescharacteristic
87C2×C3×#168C2311yescharacteristic
88#6668—22no—
89#6665—22no—
90Dic3×Dic5—22no—
91#6658C2×C418yescomplemented
92#6653C2×C418yescomplemented
93#5827—51no—
94C2×#949—51no—
95#5521—51no—
96C3×#416—51no—
97C2×#821—51no—
98#5101—51no—
99#4995D511yescharacteristic, complemented
100#4817—52no—
101#4788—52no—
102#4785—52no—
103#4756—52no—
104C2×#709C2×D311yescharacteristic
105C22×D20—31no—
106C2×C5×M16—31no—
107C5×#117C2×D311yescharacteristic
108C2×C5×C4⋊C4—31no—
109C4×C22⋊D5—62no—
110#4072—31no—
111C22×#168C2×D311yescharacteristic
112#4054—38no—
113C2×D40—32no—
114C2×#677—32no—
115#4039—32no—
116#4039—32no—
117#4039—32no—
118#4038—61no—
119D5×C4⋊C4—61no—
120#4033—32no—
121D5×C22⋊C4—62no—
122C4×D20—61no—
123C4×D20—61no—
124C5×#107—32no—
125C5×#106—32no—
126#3947—32no—
127C8×Dic5—32no—
128#3942C2×D311yescharacteristic, complemented
129#3942C2×D311yescharacteristic, complemented
130#3941—32no—
131#3941—32no—
132#3941C2×D311yescharacteristic, complemented
133#3941C2×D311yescharacteristic, complemented
134#2946—151noSylow
135C23×C3×C5C2×D411yescharacteristic
136C22×D15—22no—
137C22×D15—24no—
138C22×D15—81no—
139C22×D15—81no—
140C22×C3×D5—21no—
141C2×C3×C4×C5—21no—
142C2×C3×C4×C5—21no—
143C2×C3×C4×C5C22×C412yes—
144C2×C3×C4×C5C2×D411yescharacteristic
145C2×C3×C4×C5#4112yes—
146C2×C3×C4×C5C2411yescharacteristic
147C2×Dic15C2×D412yes—
148C2×Dic15C2×D412yes—
149D60—24no—
150D60—22no—
151D60C22×C414yes—
152C4×D15—82no—
153C4×D15—82no—
154C2×C5×Dic3—22no—
155C2×C5×Dic3—81no—
156C3×C22⋊D5—24no—
157C2×C3×Dic5—21no—
158C3×D20—21no—
159C3×D20C22×C412yes—
160C3×C4×D5—24no—
161C3×Dic10—21no—
162C3×Dic10C22×C412yes—
163D5×Dic3—82no—
164C3×#168—21no—
165C3×#168—21no—
166C5×C3⋊C8C22×C414yes—
167C2×C3×#41—51no—
168C22×D12—51no—
169C22×C3×C8—51no—
170C3×#117C2×D511yescharacteristic
171#949—58no—
172Q8×Dic3—52no—
173C2×#215—52no—
174D4×Dic3—52no—
175D4×Dic3—102no—
176C2×#213—52no—
177#926—51no—
178#926—101no—
179C2×C4⋊Dic3—101no—
180C2×C4⋊Dic3C2×D511yescharacteristic
181C2×#208C2×D511yescharacteristic
182#893—52no—
183#893—52no—
184D3×C22⋊C4—102no—
185C4×D12—101no—
186C4×D12—101no—
187C3×#103—51no—
188C3×#103—51no—
189C3×#102—51no—
190C3×#102—51no—
191#821—52no—
192#821—52no—
193#815—52no—
194C8×Dic3—52no—
195#808C2×D512yescomplemented
196#807C2×D512yescomplemented
197C23×D5—61no—
198C22×C4×C5—61no—
199C22×C4×C5C22×D311yescharacteristic
200C2×C22⋊D5—21no—
201#709—22no—
202#709C4×D314yescomplemented
203C2×D20—34no—
204C2×D20—32no—
205C2×D20—61no—
206C2×D20—62no—
207C2×D20C22×D311yescharacteristic
208C2×C4×D5—21no—
209C2×C4×D5—62no—
210C2×C4×D5—62no—
211C2×C4×D5—62no—
212C2×Dic10C22×D311yescharacteristic
213C5×M16—34no—
214C2×C5×C8—32no—
215C5×C4⋊C4—32no—
216C5×C4⋊C4—61no—
217C5×C4⋊C4C22×D311yescharacteristic
218C5×C4⋊C4C22×D311yescharacteristic
219C5×C22⋊C4—21no—
220C42×C5—21no—
221C22⋊Dic5—32no—
222#685—62no—
223#685—62no—
224#685—62no—
225C4⋊Dic5—31no—
226C4⋊Dic5—31no—
227#683—62no—
228C4×Dic5—32no—
229C4×Dic5—62no—
230C2×#168—22no—
231C2×#168C22×D311yescharacteristic
232C2×#168C22×D311yescharacteristic
233D40—38no—
234#677—38no—
235C2×AGL(1,ℤ/8)—151no—
236#520—151no—
237#518—151no—
238#442—152no—
239#440—152no—
240C4×SD16—152no—
241C4×D8—152no—
242#428—31no—
243#416—51no—
244C2×#102—151no—
245#405—151no—
246C22×C3×C5—21no—
247C22×C3×C5—21no—
248C22×C3×C5C4×D412yes—
249C22×C3×C5C22×D411yescharacteristic
250C2×D15—24no—
251C2×D15—42no—
252C2×D15—42no—
253C2×D15—84no—
254C2×C3×D5—22no—
255C2×C3×D5—41no—
256C2×C3×D5—41no—
257C3×C4×C5—41no—
258C3×C4×C5—41no—
259C3×C4×C5C4×D412yes—
260C3×C4×C5C23×C411yescharacteristic
261C3×C4×C5C2×#4111yescharacteristic
262Dic15—22no—
263Dic15—22no—
264C3×Dic5—22no—
265C5×Dic3—42no—
266C23×D3—101no—
267C3×#41—54no—
268C3×#41—102no—
269C2×C3×Q8—51no—
270C2×C3×D4—51no—
271C2×C3×D4—101no—
272C22×C3×C4—101no—
273C22×C3×C4C22×D511yescharacteristic
274C22×Dic3—21no—
275C22×Dic3—102no—
276C2×D12—54no—
277C2×D12—52no—
278C2×D12—101no—
279C2×D12—102no—
280C2×C4×D3—102no—
281C2×C4×D3—102no—
282C2×C3×C8—51no—
283C2×C3×C8—51no—
284C2×C3×C8—102no—
285C3×C4⋊C4C22×D511yescharacteristic
286C3×C4⋊C4C22×D511yescharacteristic
287C3×C22⋊C4—21no—
288C3×C42—21no—
289#217—52no—
290#217—102no—
291#215—58no—
292#213—58no—
293(C2×D3)⋊C4—102no—
294(C2×D3)⋊C4—102no—
295C4⋊Dic3—21no—
296C4⋊Dic3—51no—
297C4⋊Dic3—51no—
298C4⋊Dic3—102no—
299C4⋊Dic3C22×D512yes—
300C4×Dic3—52no—
301C4×Dic3—102no—
302#208C4×D514yescomplemented
303C2×C3⋊C8C22×D512yes—
304C23×C5D3×D411yescharacteristic
305C22×D5—21no—
306C22×D5—62no—
307C22×D5—64no—
308C22×D5—241no—
309C22×D5—241no—
310C2×C4×C5—21no—
311C2×C4×C5—21no—
312C2×C4×C5—62no—
313C2×C4×C5—241no—
314C2×C4×C5C2×C4×D312yes—
315C2×C4×C5D3×D411yescharacteristic
316C2×C4×C5#23712yes—
317C2×C4×C5C23×D311yescharacteristic
318C22⋊D5—24no—
319C2×Dic5—21no—
320C2×Dic5—32no—
321C2×Dic5—32no—
322D20—21no—
323D20—34no—
324D20—64no—
325D20—62no—
326D20C2×C4×D312yes—
327C4×D5—24no—
328C4×D5—242no—
329C4×D5—242no—
330C4×D5—242no—
331Dic10—21no—
332Dic10C2×C4×D312yes—
333C5×C8—34no—
334#168—21no—
335#168—21no—
336C2×#41—51no—
337C22×D4—151no—
338AGL(1,ℤ/8)—158no—
339C2×SD16—152no—
340C2×D8—152no—
341C2×M16—31no—
342C22×C8—51no—
343C4×Q8—152no—
344C4×D4—152no—
345C4×D4—152no—
346C4×D4—152no—
347C4×D4—301no—
348C4×D4—301no—
349C4×D4—302no—
350#117—151no—
351#117—301no—
352#117D3×D511yescharacteristic
353C2×C4⋊C4—31no—
354C2×C4⋊C4—301no—
355C2×C22⋊C4—302no—
356#107—32no—
357#106—32no—
358#103—51no—
359#103—51no—
360#102—51no—
361#102—51no—
362#102—152no—
363#102—152no—
364#97—152no—
365C4×C8—152no—
366C2×C3×C5—22no—
367C2×C3×C5C2×C4×D411yescharacteristic
368C2×C3×C5#57611yescharacteristic
369C2×C3×C5#57711yescharacteristic
370D15—44no—
371C3×D5—42no—
372C23×C3—101no—
373C23×C3D4×D511yescharacteristic
374C22×D3—102no—
375C22×D3—104no—
376C22×D3—401no—
377C22×D3—401no—
378C3×Q8—52no—
379C3×Q8—101no—
380C3×D4—52no—
381C3×D4—104no—
382C3×D4—101no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—101no—
386C2×C3×C4—104no—
387C2×C3×C4C2×C4×D512yes—
388C2×C3×C4D4×D511yescharacteristic
389C2×C3×C4#71312yes—
390C2×C3×C4C23×D511yescharacteristic
391C2×Dic3—22no—
392C2×Dic3—52no—
393C2×Dic3—52no—
394C2×Dic3—81no—
395C2×Dic3—402no—
396D12—54no—
397D12—102no—
398D12—104no—
399C4×D3—402no—
400C4×D3—402no—
401C3×C8—101no—
402C3×C8—101no—
403C3⋊C8C2×C4×D514yes—
404C22×C5—21no—
405C22×C5—21no—
406C22×C5#67⋊C412yes—
407C22×C5C2×D3×D411yescharacteristic
408C2×D5—22no—
409C2×D5—41no—
410C2×D5—41no—
411C2×D5—64no—
412C2×D5—122no—
413C2×D5—122no—
414C2×D5—244no—
415C4×C5—41no—
416C4×C5—41no—
417C4×C5—122no—
418C4×C5#67⋊C412yes—
419C4×C5C22×C4×D311yescharacteristic
420C4×C5C2×#23711yescharacteristic
421Dic5—22no—
422Dic5—62no—
423Dic5—62no—
424C24—301no—
425#41—54no—
426#41—102no—
427C2×Q8—51no—
428C2×D4—51no—
429C2×D4—101no—
430C2×D4—154no—
431C2×D4—152no—
432C2×D4—302no—
433C2×D4—301no—
434C22×C4—61no—
435C22×C4—101no—
436C22×C4—302no—
437C22×C4—302no—
438C22×C4—302no—
439C22×C4C2×D3×D511yescharacteristic
440SD16—158no—
441D8—158no—
442M16—34no—
443C2×C8—32no—
444C2×C8—51no—
445C2×C8—51no—
446C2×C8—102no—
447C4⋊C4—32no—
448C4⋊C4—61no—
449C4⋊C4—151no—
450C4⋊C4—151no—
451C4⋊C4—302no—
452C4⋊C4C2×D3×D511yescharacteristic
453C4⋊C4C2×D3×D511yescharacteristic
454C22⋊C4—21no—
455C22⋊C4—152no—
456C22⋊C4—302no—
457C22⋊C4—302no—
458C22⋊C4—302no—
459C42—21no—
460C42—152no—
461C42—302no—
462C3×C5#294611yescharacteristic, complemented, Hall
463C22×C3—21no—
464C22×C3—21no—
465C22×C3—102no—
466C22×C3—201no—
467C22×C3—201no—
468C22×C3#403912yes—
469C22×C3C2×D4×D511yescharacteristic
470C2×D3—104no—
471C2×D3—202no—
472C2×D3—202no—
473C2×D3—404no—
474C3×C4—41no—
475C3×C4—41no—
476C3×C4—102no—
477C3×C4#403912yes—
478C3×C4C22×C4×D511yescharacteristic
479C3×C4C2×#71311yescharacteristic
480Dic3—42no—
481Dic3—102no—
482Dic3—102no—
483C2×C5—22no—
484C2×C5C2×#67⋊C411yescharacteristic
485C2×C5#576911yescharacteristic
486C2×C5#577011yescharacteristic
487D5—42no—
488D5—124no—
489C23—101no—
490C23—302no—
491C23—304no—
492C23—1201no—
493C23—1201no—
494C23#679011yescharacteristic
495Q8—52no—
496Q8—101no—
497D4—52no—
498D4—104no—
499D4—101no—
500D4—154no—
501D4—304no—
502D4—302no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—62no—
506C2×C4—101no—
507C2×C4—104no—
508C2×C4—152no—
509C2×C4—152no—
510C2×C4—241no—
511C2×C4—1202no—
512C2×C4—1202no—
513C2×C4—1202no—
514C2×C4#676612yes—
515C2×C4C4×D3×D512yes—
516C2×C4#679011yescharacteristic
517C2×C4C22×D3×D511yescharacteristic
518C8—34no—
519C8—101no—
520C8—101no—
521C2×C3—22no—
522C2×C3—202no—
523C2×C3C2×#403911yescharacteristic
524C2×C3#6610811yescharacteristic
525C2×C3#6611211yescharacteristic
526D3—204no—
527C5#7882011yescharacteristic, complemented, Sylow
528C22—21no—
529C22—21no—
530C22—102no—
531C22—201no—
532C22—201no—
533C22—304no—
534C22—602no—
535C22—602no—
536C22—1204no—
537C22#9139612yes—
538C22C2×#679011yescharacteristic
539C4—41no—
540C4—41no—
541C4—102no—
542C4—122no—
543C4—302no—
544C4—302no—
545C4#9139612yes—
546C4C2×#676611yescharacteristic
547C4C2×C4×D3×D511yescharacteristic
548C3#1061093211yescharacteristic, complemented, Sylow
549C2—22no—
550C2—202no—
551C2—604no—
552C2C2×#9139611yescharacteristic
553C2#1175533211yescharacteristic
554C2#1175535111yescharacteristic
subgroupG/H
centerC22C2×#6790
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#117C22
socleC22×C3×C5C22×D4
prime radicalC2×#91301C2
cosocleC2×C3×C4×C5C24
hypercenter#117D3×D5
hypocenterC3×C5#2946
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. #11756280C2semidirect
2. #11754499C2semidirect
4. C5×#4995C2semidirect
5. C3×#65291C2semidirect
6. C2×#90969C2semidirect
7. #11748324C2semidirect
8. #11747273C2semidirect
9. #11747272C2semidirect
10. #11747229C2semidirect
11. #11747226C2semidirect
26. #91434C22semidirect
29. #91331C22semidirect
31. #91301C4semidirect
33. C5×#808C22semidirect
34. C5×#807C22semidirect
35. C3×#3942C22semidirect
36. C3×#3942C22semidirect
37. C3×#3941C22semidirect
38. C3×#3941C22semidirect
41. #90954C22semidirect
42. #90951C22semidirect
49. #65291D3semidirect
59. C2×D60C2×C4semidirect
66. C3×#709C2×C4semidirect
83. C5×#208C2×C4semidirect
91. #6658C2×C4semidirect
92. #6653C2×C4semidirect
99. #4995D5semidirect
128. #3942C2×D3semidirect
129. #3942C2×D3semidirect
132. #3941C2×D3semidirect
133. #3941C2×D3semidirect
195. #808C2×D5semidirect
196. #807C2×D5semidirect
202. #709C4×D3semidirect
302. #208C4×D5semidirect
462. C3×C5#2946semidirect
527. C5#78820semidirect
548. C3#10610932semidirect

Maximal subgroups

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

subgroupindex
#117562802
#117544992
C2×#913012
C5×#49952
C3×#652912
C2×#909692
#117483242
#117472732
#117472722
#117472292
#117472262
#106109323
#788205