Finite Group Databasebeta

← #14427856

Subgroups — #14427856

Order 1920.

1040 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756283C211yesmaximal, characteristic, complemented
2#11754592C211yesmaximal, characteristic, complemented
3C2×#91303C211yesmaximal, characteristic, complemented
4C22×#6648C211yesmaximal, characteristic, complemented
5C5×#5002C211yesmaximal, characteristic, complemented
6C3×#65287C211yesmaximal, characteristic, complemented
7#11748411C211yesmaximal, characteristic, complemented
8#11747451C212yesmaximal, complemented
9#11747450C212yesmaximal, complemented
10#11747315C212yesmaximal, complemented
11#11747314C212yesmaximal, complemented
12#10610569—31nomaximal, Hall
13C2×#6817C2211yescharacteristic
14C22×C4×D15C2211yescharacteristic
15C2×C5×#235C2211yescharacteristic
16C3×C5×#117C2211yescharacteristic
17#91786C2212yes—
18#91784C2212yes—
19#91775—21no—
20#91771—21no—
21C4×D60—22no—
22C22×C5×C3⋊C8C2211yescharacteristic
23C3×#4072C2211yescharacteristic
24C22×C3×#168C2211yescharacteristic
25#91573—21no—
26#91543—21no—
27#91466C2212yes—
28C4×#1203—22no—
29#91383C2212yes—
30C2×#6663C2211yescharacteristic, complemented
31C2×#6660C2211yescharacteristic, complemented
32C2×#6659C2211yescharacteristic, complemented
33#91303—24no—
34C2×#6654C2211yescharacteristic, complemented
35C2×#6648—24no—
36C2×#6648C2212yes—
37C2×#6648C2212yes—
38C5×#810C2212yescomplemented
39C5×#809C2212yescomplemented
40C3×#3940C2212yescomplemented
41C3×#3939C2212yescomplemented
42#90974C2212yescomplemented
43#90968C2212yescomplemented
44#79180—51nomaximal, Hall
45#66162—31no—
46C2×#4096—31no—
47C22×C8×D5—31no—
48#65858—31no—
49C5×#416—31no—
50#65402—31no—
51#65287D311yescharacteristic, complemented
52#65102—32no—
53#65096—32no—
54#65067—32no—
55#65064—32no—
56C23×D15D411yescharacteristic
57C22×C3×C4×C5C2311yescharacteristic
58C2×#1220—21no—
59C22×Dic15D411yescharacteristic
60#6817—22no—
61#6817—22no—
62C2×D60—22no—
63C2×D60C2311yescharacteristic
64C2×C4×D15—22no—
65C2×C4×D15—22no—
66C2×C4×D15—21no—
67C2×C4×D15D412yes—
68C2×C4×D15C2312yes—
69C2×Dic30C2311yescharacteristic
70C2×C5×#67—21no—
71C5×#235—22no—
72C5×#235—22no—
73C2×C5×D12C2311yescharacteristic
74C2×C4×C5×D3—21no—
75C2×C5×Dic6C2311yescharacteristic
76C2×#1203—22no—
77C3×C5×C4⋊C4C2312yes—
78C3×C5×C22⋊C4—21no—
79C3×C42×C5—21no—
80#6714—22no—
81#6714—22no—
82C4⋊Dic15—22no—
83C2×C5×C3⋊C8—22no—
84C2×C5×C3⋊C8C2312yes—
85C3×C22⋊Dic5—21no—
86C3×C4⋊Dic5C2312yes—
87C3×C4×Dic5—21no—
88C2×C3×#168D414yescomplemented
89C2×C3×#168C2311yescharacteristic
90C2×C3×#168C2311yescharacteristic
91#6668—22no—
92#6667—22no—
93(C2×D3)⋊Dic5—22no—
94#6663—22no—
95#6660—22no—
96#6659—22no—
97#6654—22no—
98#6648—24no—
99#6648—24no—
100C2×#904—51no—
101C22×D3×C8—51no—
102#5524—51no—
103#5524—51no—
104C3×#427—51no—
105#5002—51no—
106#5002D511yescharacteristic, complemented
107#4887—52no—
108#4883—52no—
109#4864—52no—
110#4864—52no—
111C2×C5×#41—31no—
112C2×#709—31no—
113C22×C4×D5—31no—
114C22×C5×C8—31no—
115C5×#117C2×D311yescharacteristic
116#4096—64no—
117#4092—32no—
118C2×#689—31no—
119C2×#688—31no—
120#4084—32no—
121#4083—61no—
122#4081—61no—
123C2×#687—31no—
124C2×#686—31no—
125C4×C22⋊D5—62no—
126#4072C2×D311yescharacteristic
127C22×#168C2×D311yescharacteristic
128C2×C8×D5—32no—
129C2×C8×D5—32no—
130C2×C8×D5—64no—
131#4043—32no—
132#4041—32no—
133#4032—61no—
134#4028—61no—
135C4×D20—62no—
136C5×#103—32no—
137C5×#102—32no—
138#3953—32no—
139#3948—32no—
140#3940C2×D312yescomplemented
141#3939C2×D312yescomplemented
142#3050—151noSylow
143C23×C3×C5C2×D411yescharacteristic
144C22×D15—22no—
145C22×D15—24no—
146C22×D15—21no—
147C22×D15C2×D412yes—
148C22×C5×D3—21no—
149C2×C3×C4×C5—22no—
150C2×C3×C4×C5C2×D411yescharacteristic
151C2×C3×C4×C5C2×D412yes—
152C2×C3×C4×C5#4112yes—
153C2×C3×C4×C5C2411yescharacteristic
154#1220—42no—
155C2×Dic15—22no—
156C2×Dic15—21no—
157C2×Dic15C2×D412yes—
158D60—21no—
159D60—21no—
160D60—44no—
161C4×D15—24no—
162C4×D15—22no—
163C4×D15—22no—
164C4×D15—42no—
165Dic30—21no—
166Dic30—21no—
167C5×#67—42no—
168C2×C5×Dic3—21no—
169C5×D12—21no—
170C5×D12—21no—
171C4×C5×D3—42no—
172C5×Dic6—21no—
173C5×Dic6—21no—
174C2×C3×Dic5—22no—
175#1203—44no—
176C3×#168C2×D412yes—
177C3×#168C2×D412yes—
178C5×C3⋊C8—22no—
179C2×#235—51no—
180C2×#235C2×D511yescharacteristic
181C22×C4×D3—51no—
182C22×C3×C8—51no—
183C3×#117—51no—
184C3×#117C2×D511yescharacteristic
185C22×C3⋊C8C2×D511yescharacteristic
186C2×Dic12—51no—
187#904—104no—
188C2×D24—51no—
189C2×#204—51no—
190C2×#204—51no—
191C2×D3×C8—52no—
192C2×D3×C8—52no—
193C2×D3×C8—104no—
194#897—52no—
195#897—52no—
196#895—52no—
197#895—52no—
198#886—101no—
199#886—101no—
200#882—101no—
201#882—101no—
202C4×D12—102no—
203C4×D12—102no—
204C3×#107—52no—
205C3×#106—52no—
206#810—52no—
207#810C2×D512yescomplemented
208#809—52no—
209#809C2×D512yescomplemented
210C23×D5—31no—
211C5×#41—62no—
212C5×#41—62no—
213C2×C5×Q8—31no—
214C2×C5×D4—31no—
215C2×C5×D4—61no—
216C22×C4×C5—61no—
217C22×C4×C5C22×D311yescharacteristic
218C2×C22⋊D5—62no—
219C2×C22⋊D5—61no—
220C22×Dic5—31no—
221#709—62no—
222#709—62no—
223C2×D20—31no—
224C2×D20—62no—
225C2×C4×D5—32no—
226C2×C4×D5—32no—
227C2×C4×D5—62no—
228C2×C4×D5—62no—
229C2×C4×D5—61no—
230C2×Dic10—31no—
231C2×C5×C8—32no—
232C2×C5×C8—62no—
233C5×C4⋊C4C22×D312yes—
234C5×C22⋊C4—21no—
235C42×C5—21no—
236C22⋊Dic5—21no—
237C22⋊Dic5—62no—
238#689—62no—
239#688—62no—
240#687—62no—
241#686—62no—
242#685—62no—
243#685—62no—
244#685—62no—
245C4⋊Dic5—62no—
246C4⋊Dic5C22×D312yes—
247#683—62no—
248C4×Dic5—21no—
249C2×#168D1214yescomplemented
250C2×#168C22×D311yescharacteristic
251C2×#168C22×D311yescharacteristic
252C8×D5—64no—
253C8×D5—64no—
254C2×#135—151no—
255C23×C8—151no—
256#525—151no—
257#525—151no—
258#467—152no—
259#466—152no—
260#465—152no—
261#465—152no—
262#427—51no—
263#416—31no—
264#416—151no—
265C22×C3×C5—22no—
266C22×C3×C5#12112yes—
267C22×C3×C5C22×D411yescharacteristic
268C2×D15—28no—
269C2×D15—22no—
270C2×D15—22no—
271C2×D15—41no—
272C2×D15—42no—
273C2×D15#13514yescomplemented
274C2×C5×D3—41no—
275C2×C5×D3—42no—
276C3×C4×C5—42no—
277C3×C4×C5#12112yes—
278C3×C4×C5C22×D411yescharacteristic
279C3×C4×C5C2×#4111yescharacteristic
280Dic15—22no—
281Dic15—41no—
282C3×Dic5—42no—
283C5×Dic3—41no—
284C23×D3—51no—
285C22×C3×C4C22×D511yescharacteristic
286C2×#67—21no—
287C2×#67—101no—
288C22×Dic3—51no—
289#235—22no—
290#235—22no—
291#235—102no—
292#235—102no—
293C2×D12—51no—
294C2×D12—102no—
295C2×D12—102no—
296C2×D12C22×D511yescharacteristic
297C2×C4×D3—21no—
298C2×C4×D3—52no—
299C2×C4×D3—52no—
300C2×C4×D3—102no—
301C2×C4×D3—102no—
302C2×C4×D3—101no—
303C2×Dic6—51no—
304C2×Dic6C22×D511yescharacteristic
305C2×C3×C8—54no—
306C2×C3×C8—51no—
307C2×C3×C8—51no—
308C3×C4⋊C4—52no—
309C3×C4⋊C4C22×D512yes—
310C3×C22⋊C4—21no—
311C3×C22⋊C4—101no—
312C3×C42—21no—
313C3×C42—101no—
314(C2×D3)⋊C4—102no—
315(C2×D3)⋊C4—102no—
316(C2×D3)⋊C4—102no—
317(C2×D3)⋊C4—102no—
318C4⋊Dic3—102no—
319C4⋊Dic3—102no—
320C2×C3⋊C8—22no—
321C2×C3⋊C8C22×D512yes—
322Dic12—102no—
323D24—102no—
324#204—102no—
325#204—102no—
326D3×C8—104no—
327D3×C8—104no—
328C23×C5—61no—
329C23×C5D3×D411yescharacteristic
330C22×D5—32no—
331C22×D5—64no—
332C22×D5—62no—
333C22×D5—61no—
334C5×Q8—61no—
335C5×Q8—61no—
336C5×D4—61no—
337C5×D4—61no—
338C5×D4—122no—
339C2×C4×C5—22no—
340C2×C4×C5—61no—
341C2×C4×C5—122no—
342C2×C4×C5C2×D1212yes—
343C2×C4×C5D3×D411yescharacteristic
344C2×C4×C5#23912yes—
345C2×C4×C5C23×D311yescharacteristic
346C22⋊D5—124no—
347C22⋊D5—122no—
348C2×Dic5—22no—
349C2×Dic5—32no—
350C2×Dic5—62no—
351C2×Dic5—61no—
352D20—61no—
353D20—61no—
354D20—124no—
355C4×D5—64no—
356C4×D5—62no—
357C4×D5—62no—
358C4×D5—122no—
359Dic10—61no—
360Dic10—61no—
361C5×C8—62no—
362#168C2×D1212yes—
363#168C2×D1212yes—
364C2×#41—31no—
365C2×#41—151no—
366C23×C4—151no—
367#135—304no—
368C2×Q16—151no—
369C2×SD16—151no—
370C2×SD16—151no—
371C2×D8—151no—
372C22×C8—31no—
373C22×C8—51no—
374C22×C8—152no—
375C22×C8—152no—
376C22×C8—304no—
377#123—301no—
378#123—301no—
379C22⋊Q8—152no—
380C22⋊Q8—152no—
381#121—152no—
382#121—152no—
383#120—301no—
384#120—301no—
385C4×D4—302no—
386C4×D4—302no—
387#117—51no—
388#117D3×D511yescharacteristic
389#107—52no—
390#106—52no—
391#103—32no—
392#103—152no—
393#102—32no—
394#102—152no—
395C2×C3×C5—22no—
396C2×C3×C5C2×#12111yescharacteristic
397C2×C3×C5C2×#13512yes—
398D15—24no—
399D15—81no—
400C5×D3—81no—
401C23×C3D4×D511yescharacteristic
402C22×D3—21no—
403C22×D3—52no—
404C22×D3—104no—
405C22×D3—102no—
406C22×D3—101no—
407C2×C3×C4—22no—
408C2×C3×C4—102no—
409C2×C3×C4#70912yes—
410C2×C3×C4D4×D512yes—
411C2×C3×C4D4×D511yescharacteristic
412C2×C3×C4C23×D511yescharacteristic
413#67—42no—
414#67—202no—
415C2×Dic3—21no—
416C2×Dic3—52no—
417C2×Dic3—102no—
418C2×Dic3—101no—
419D12—21no—
420D12—21no—
421D12—101no—
422D12—101no—
423D12—204no—
424D12—204no—
425C4×D3—42no—
426C4×D3—102no—
427C4×D3—102no—
428C4×D3—104no—
429C4×D3—202no—
430Dic6—21no—
431Dic6—21no—
432Dic6—101no—
433Dic6—101no—
434C3×C8—52no—
435C3×C8—52no—
436C3⋊C8—22no—
437C22×C5—22no—
438C22×C5—121no—
439C22×C5—122no—
440C22×C5#89512yes—
441C22×C5C2×D3×D411yescharacteristic
442C2×D5—34no—
443C2×D5—68no—
444C2×D5—62no—
445C2×D5—62no—
446C2×D5—121no—
447C2×D5—122no—
448C4×C5—42no—
449C4×C5—121no—
450C4×C5#89512yes—
451C4×C5C22×D1211yescharacteristic
452C4×C5C2×#23911yescharacteristic
453Dic5—42no—
454Dic5—62no—
455Dic5—121no—
456C24—151no—
457#41—62no—
458#41—62no—
459#41—302no—
460#41—302no—
461C2×Q8—31no—
462C2×Q8—151no—
463C2×D4—31no—
464C2×D4—61no—
465C2×D4—151no—
466C2×D4—302no—
467C2×D4—302no—
468C2×D4—301no—
469C22×C4—61no—
470C22×C4—151no—
471C22×C4—152no—
472C22×C4—152no—
473C22×C4—302no—
474C22×C4—302no—
475C22×C4—301no—
476C22×C4C2×D3×D511yescharacteristic
477Q16—302no—
478SD16—302no—
479SD16—302no—
480D8—302no—
481C2×C8—32no—
482C2×C8—54no—
483C2×C8—51no—
484C2×C8—51no—
485C2×C8—62no—
486C2×C8—304no—
487C2×C8—304no—
488C4⋊C4—52no—
489C4⋊C4—302no—
490C4⋊C4—302no—
491C4⋊C4C2×D3×D512yes—
492C22⋊C4—21no—
493C22⋊C4—101no—
494C22⋊C4—302no—
495C22⋊C4—302no—
496C22⋊C4—302no—
497C22⋊C4—302no—
498C42—21no—
499C42—101no—
500C3×C5#305011yescharacteristic, complemented, Hall
501C22×C3—22no—
502C22×C3#403012yes—
503C22×C3C2×D4×D511yescharacteristic
504C2×D3—41no—
505C2×D3—42no—
506C2×D3—54no—
507C2×D3—108no—
508C2×D3—102no—
509C2×D3—102no—
510C2×D3—201no—
511C2×D3—202no—
512C3×C4—42no—
513C3×C4—202no—
514C3×C4#403012yes—
515C3×C4C2×#70911yescharacteristic
516C3×C4C2×D4×D511yescharacteristic
517Dic3—41no—
518Dic3—102no—
519Dic3—201no—
520C2×C5—22no—
521C2×C5—241no—
522C2×C5C2×#89511yescharacteristic
523C2×C5D3×#13512yes—
524D5—64no—
525D5—241no—
526C23—61no—
527C23—152no—
528C23—304no—
529C23—302no—
530C23—301no—
531C23#679011yescharacteristic
532Q8—61no—
533Q8—61no—
534Q8—301no—
535Q8—301no—
536D4—61no—
537D4—61no—
538D4—122no—
539D4—301no—
540D4—301no—
541D4—604no—
542D4—604no—
543D4—602no—
544C2×C4—22no—
545C2×C4—61no—
546C2×C4—102no—
547C2×C4—122no—
548C2×C4—152no—
549C2×C4—302no—
550C2×C4—304no—
551C2×C4—302no—
552C2×C4—302no—
553C2×C4—301no—
554C2×C4—602no—
555C2×C4#676912yes—
556C2×C4D5×D1212yes—
557C2×C4#679011yescharacteristic
558C2×C4C22×D3×D511yescharacteristic
559C8—52no—
560C8—52no—
561C8—62no—
562C2×C3—22no—
563C2×C3C2×#403011yescharacteristic
564C2×C3D5×#13512yes—
565D3—81no—
566D3—104no—
567D3—401no—
568C5#7918011yescharacteristic, complemented, Sylow
569C22—22no—
570C22—122no—
571C22—121no—
572C22—154no—
573C22—308no—
574C22—302no—
575C22—302no—
576C22—602no—
577C22—601no—
578C22#9142012yes—
579C22C2×#679011yescharacteristic
580C4—42no—
581C4—121no—
582C4—202no—
583C4—302no—
584C4—601no—
585C4#9142012yes—
586C4C2×#676911yescharacteristic
587C4C2×D5×D1211yescharacteristic
588C3#1061056911yescharacteristic, complemented, Sylow
589C2—22no—
590C2—241no—
591C2—304no—
592C2—1201no—
593C2C2×#9142011yescharacteristic
594C2#1175532812yes—
subgroupG/H
centerC2×C4#6790
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#117C22
socleC22×C3×C5C22×D4
hypercenter#117D3×D5
hypocenterC3×C5#3050
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. #11756283C2semidirect
2. #11754592C2semidirect
3. C2×#91303C2semidirect
4. C22×#6648C2semidirect
5. C5×#5002C2semidirect
6. C3×#65287C2semidirect
7. #11748411C2semidirect
8. #11747451C2semidirect
9. #11747450C2semidirect
10. #11747315C2semidirect
11. #11747314C2semidirect
30. C2×#6663C22semidirect
31. C2×#6660C22semidirect
32. C2×#6659C22semidirect
34. C2×#6654C22semidirect
38. C5×#810C22semidirect
39. C5×#809C22semidirect
40. C3×#3940C22semidirect
41. C3×#3939C22semidirect
42. #90974C22semidirect
43. #90968C22semidirect
51. #65287D3semidirect
88. C2×C3×#168D4semidirect
106. #5002D5semidirect
140. #3940C2×D3semidirect
141. #3939C2×D3semidirect
207. #810C2×D5semidirect
209. #809C2×D5semidirect
249. C2×#168D12semidirect
273. C2×D15#135semidirect
500. C3×C5#3050semidirect
568. C5#79180semidirect
588. C3#10610569semidirect

Maximal subgroups

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

subgroupindex
#117562832
#117545922
C2×#913032
C22×#66482
C5×#50022
C3×#652872
#117484112
#117474512
#117474502
#117473152
#117473142
#106105693
#791805