Finite Group Databasebeta

← #14425263

Subgroups — #14425263

Order 1920.

1448 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5964C211yesmaximal, characteristic, complemented
2#11756591C211yesmaximal, characteristic, complemented
3#11754145C212yesmaximal, complemented
4C3×#65526C211yesmaximal, characteristic, complemented
5#11749394C212yesmaximal, complemented
6#11749387C212yesmaximal, complemented
7#11749148C212yesmaximal, complemented
8#11749141C212yesmaximal, complemented
9C2×#90978C212yesmaximal, complemented
10#10615852—31nomaximal, Hall
11C2×C3×C5×#41C2211yescharacteristic
12C2×#6817C2211yescharacteristic, complemented
13C5×#1010—22no—
14C5×#1009—22no—
15C5×D3×#41—24no—
16C5×#1007C2212yes—
17C5×#1004—22no—
18C5×#1004C2212yes—
19C2×C5×#235—22no—
20C2×C5×#235C2212yes—
21#91836C2211yescharacteristic, complemented
22#91833C2211yescharacteristic, complemented
23#91829C2211yescharacteristic, complemented
24#91828—22no—
25#91825C2211yescharacteristic, complemented
26C4×#1220—22no—
27C2×C4×Dic15C2211yescharacteristic, complemented
28C2×C3×#681C2212yescomplemented
29#91322C2214yescomplemented
30#91321—22no—
31#91305C2214yescomplemented
32#91304—22no—
33#91303—24no—
34C3×#3968C2212yescomplemented
35C3×#3965C2212yescomplemented
36#90978C2214yescomplemented
37#90978C2214yescomplemented
38#78716—51nomaximal, Hall
39C5×#585—31no—
40#66166—31no—
41#66156—32no—
42#65535—32no—
43#65534—32no—
44#65531—32no—
45#65530—32no—
46#65526D311yescharacteristic, complemented
47C2×#3969—32no—
48C3×C5×#41—22no—
49C3×C5×#41—22no—
50C2×C3×C5×Q8C2311yescharacteristic
51C2×C3×C5×D4D412yes—
52C2×C3×C5×D4C2311yescharacteristic
53C22×C3×C4×C5D412yes—
54C22×C3×C4×C5C2311yescharacteristic
55C2×#1220—22no—
56C2×#1220—21no—
57C22×Dic15—21no—
58#6817—22no—
59#6817—22no—
60C2×D60C2311yescharacteristic
61C2×C4×D15—21no—
62C2×Dic30C2311yescharacteristic
63C2×C5×#67—24no—
64C2×C5×#67—22no—
65C5×#239—24no—
66C5×D3×Q8—24no—
67C5×#237—28no—
68C5×#237—24no—
69C5×D3×D4—28no—
70C5×D3×D4—24no—
71C5×#235—24no—
72C5×#235—24no—
73C5×#235—22no—
74C5×#235—24no—
75C5×#235—22no—
76C5×#235D414yescomplemented
77C5×#235C2314yes—
78C2×C5×D12—22no—
79C2×C5×D12D412yescomplemented
80C2×C4×C5×D3—24no—
81C2×C4×C5×D3—22no—
82C2×C5×Dic6—22no—
83C2×C5×Dic6D412yescomplemented
84#6719—22no—
85#6714—22no—
86#6712—22no—
87C4×Dic15—21no—
88C4×Dic15C2311yescharacteristic
89C4×Dic15C2311yescharacteristic
90C3×#681—22no—
91C3×#681C2314yes—
92C2×C3×#168—22no—
93#6663—24no—
94#6660—24no—
95#6659—24no—
96#6654—24no—
97#5964D511yescharacteristic, complemented
98#5832—51no—
99#5748—52no—
100C3×#413—51no—
101C2×#825—52no—
102#4753—52no—
103#4752—52no—
104#4747—52no—
105#4746—52no—
106C5×2−1+4—32no—
107C5×2−1+4—62no—
108C5×H(5,𝔽2)—32no—
109C5×H(5,𝔽2)—62no—
110C5×H(5,𝔽2)—62no—
111C2×C5×#41—32no—
112C2×C5×#41—64no—
113C2×C5×#41—62no—
114C2×C5×#41C2×D311yescharacteristic
115C2×#709—31no—
116#4097—34no—
117#4096—64no—
118#4095—34no—
119#4093—31no—
120#4090—31no—
121#4088—62no—
122#4086—31no—
123#4085—62no—
124#4082—31no—
125#4078—62no—
126C4×C22⋊D5—62no—
127C2×C4×Dic5—31no—
128C2×#681C2×D312yescomplemented
129#3969—34no—
130#3969—34no—
131#3968C2×D312yescomplemented
132#3965C2×D312yescomplemented
133#3018—151noSylow
134C23×C3×C5C2×D412yes—
135C23×C3×C5C2×D411yescharacteristic
136C22×D15—21no—
137C22×C5×D3—24no—
138C22×C5×D3—22no—
139C3×C5×Q8—22no—
140C3×C5×D4—24no—
141C3×C5×D4—22no—
142C2×C3×C4×C5—22no—
143C2×C3×C4×C5—22no—
144C2×C3×C4×C5C2×D412yes—
145C2×C3×C4×C5C2×D411yescharacteristic
146C2×C3×C4×C5C2×D412yes—
147C2×C3×C4×C5C2×D412yes—
148C2×C3×C4×C5C2411yescharacteristic
149#1220—44no—
150#1220—42no—
151C2×Dic15—21no—
152C2×Dic15—21no—
153C2×Dic15—21no—
154C2×Dic15—42no—
155D60—22no—
156C4×D15—41no—
157C4×D15—41no—
158Dic30—22no—
159C5×#67—28no—
160C5×#67—24no—
161C5×#67—24no—
162C2×C5×Dic3—24no—
163C2×C5×Dic3—22no—
164C5×D12—24no—
165C5×D12—22no—
166C5×D12C2×D414yes—
167C4×C5×D3—28no—
168C4×C5×D3—24no—
169C4×C5×D3—24no—
170C5×Dic6—24no—
171C5×Dic6—22no—
172C5×Dic6C2×D414yes—
173C3×#168—24no—
174C2×C3×#41C2×D511yescharacteristic
175#1010—22no—
176#1009—22no—
177D3×#41—24no—
178#1007C2×D512yes—
179#1004—22no—
180#1004C2×D512yes—
181C2×#235—22no—
182C2×#235—51no—
183C2×#235C2×D512yes—
184C2×C3×M16—52no—
185#947—51no—
186#944—51no—
187#940—51no—
188#939—102no—
189#936—51no—
190C4×#67—102no—
191C2×C4×Dic3—51no—
192#909—54no—
193#909—102no—
194#908—54no—
195#908—102no—
196#904—104no—
197C3×#101—52no—
198C3×#100—52no—
199#825—54no—
200#825—54no—
201C5×#41—22no—
202C5×#41—22no—
203C5×#41—34no—
204C5×#41—34no—
205C5×#41—64no—
206C5×#41—64no—
207C5×#41—64no—
208C5×#41—68no—
209C5×#41—64no—
210C5×#41—62no—
211C5×#41—64no—
212C5×#41—62no—
213C2×C5×Q8—32no—
214C2×C5×Q8—62no—
215C2×C5×Q8—64no—
216C2×C5×Q8C22×D311yescharacteristic
217C2×C5×D4—32no—
218C2×C5×D4—62no—
219C2×C5×D4—64no—
220C2×C5×D4—68no—
221C2×C5×D4—62no—
222C2×C5×D4—64no—
223C2×C5×D4D1212yes—
224C2×C5×D4C22×D311yescharacteristic
225C22×C4×C5—64no—
226C22×C4×C5—62no—
227C22×C4×C5D1212yes—
228C22×C4×C5C22×D311yescharacteristic
229C2×C22⋊D5—61no—
230C2×C22⋊D5—62no—
231C22×Dic5—61no—
232#709—62no—
233#709—62no—
234C2×D20—31no—
235C2×C4×D5—61no—
236C2×Dic10—31no—
237C22⋊Dic5—62no—
238#689—64no—
239#688—64no—
240#687—64no—
241#686—64no—
242#685—62no—
243#683—62no—
244C4×Dic5—31no—
245C4×Dic5—31no—
246C4×Dic5—61no—
247#681—22no—
248#681C22×D314yes—
249C2×#168—22no—
250#585—31no—
251#575—152no—
252#532—151no—
253#456—152no—
254#455—152no—
255#454—152no—
256#453—152no—
257C2×#104—152no—
258#413—51no—
259C22×C3×C5—22no—
260C22×C3×C5—41no—
261C22×C3×C5—42no—
262C22×C3×C5#12012yes—
263C22×C3×C5C22×D411yescharacteristic
264C2×D15—42no—
265C2×D15—41no—
266C2×C5×D3—24no—
267C2×C5×D3—82no—
268C2×C5×D3—84no—
269C3×C4×C5—22no—
270C3×C4×C5#12011yescharacteristic
271C3×C4×C5#12011yescharacteristic
272C3×C4×C5C22×D412yes—
273Dic15—22no—
274Dic15—41no—
275Dic15—41no—
276C5×Dic3—24no—
277C3×#41—22no—
278C3×#41—22no—
279C2×C3×Q8C22×D511yescharacteristic
280C2×C3×D4C22⋊D512yes—
281C2×C3×D4C22×D511yescharacteristic
282C22×C3×C4C22⋊D512yes—
283C22×C3×C4C22×D511yescharacteristic
284C2×#67—24no—
285C2×#67—22no—
286C2×#67—102no—
287C2×#67—101no—
288C22×Dic3—101no—
289#239—24no—
290D3×Q8—24no—
291#237—28no—
292#237—24no—
293D3×D4—28no—
294D3×D4—24no—
295#235—24no—
296#235—24no—
297#235—22no—
298#235—24no—
299#235—22no—
300#235—102no—
301#235—102no—
302#235C22⋊D514yescomplemented
303#235C22×D514yes—
304C2×D12—22no—
305C2×D12—51no—
306C2×D12C22⋊D512yescomplemented
307C2×C4×D3—24no—
308C2×C4×D3—22no—
309C2×C4×D3—101no—
310C2×Dic6—22no—
311C2×Dic6—51no—
312C2×Dic6C22⋊D512yescomplemented
313C3×M16—54no—
314C3×M16—102no—
315C2×C3×C8—102no—
316#217—102no—
317(C2×D3)⋊C4—102no—
318Dic3⋊C4—102no—
319C4×Dic3—51no—
320C4×Dic3—51no—
321C4×Dic3—101no—
322Dic12—104no—
323D24—104no—
324#204—104no—
325#204—104no—
326C23×C5—64no—
327C23×C5—62no—
328C23×C5C2×D1212yes—
329C23×C5C2×D1211yescharacteristic
330C22×D5—61no—
331C5×Q8—22no—
332C5×Q8—34no—
333C5×Q8—64no—
334C5×Q8—62no—
335C5×D4—24no—
336C5×D4—22no—
337C5×D4—34no—
338C5×D4—64no—
339C5×D4—68no—
340C5×D4—64no—
341C5×D4—64no—
342C5×D4—62no—
343C2×C4×C5—22no—
344C2×C4×C5—22no—
345C2×C4×C5—68no—
346C2×C4×C5—64no—
347C2×C4×C5—62no—
348C2×C4×C5—64no—
349C2×C4×C5—64no—
350C2×C4×C5C2×D1212yes—
351C2×C4×C5C2×D1211yescharacteristic
352C2×C4×C5D3×D412yes—
353C2×C4×C5D3×D412yes—
354C2×C4×C5C23×D311yescharacteristic
355C22⋊D5—124no—
356C22⋊D5—122no—
357C2×Dic5—61no—
358C2×Dic5—61no—
359C2×Dic5—61no—
360C2×Dic5—122no—
361D20—62no—
362C4×D5—121no—
363C4×D5—121no—
364Dic10—62no—
365#168—24no—
3662−1+4—32no—
3672−1+4—62no—
368H(5,𝔽2)—32no—
369H(5,𝔽2)—62no—
370H(5,𝔽2)—62no—
371C2×#41—32no—
372C2×#41—64no—
373C2×#41—62no—
374C2×#41—151no—
375C2×#41D3×D511yescharacteristic
376#137—154no—
377#137—302no—
378AGL(1,ℤ/8)—154no—
379AGL(1,ℤ/8)—302no—
380#135—304no—
381C2×M16—52no—
382C4⋊Q8—151no—
383#127—151no—
384#124—151no—
385#124—151no—
386#121—302no—
387C4×D4—302no—
388C2×C42—151no—
389#104—154no—
390#104—154no—
391#101—52no—
392#100—52no—
393C2×C3×C5—21no—
394C2×C3×C5—22no—
395C2×C3×C5—42no—
396C2×C3×C5C2×#12011yescharacteristic
397D15—81no—
398C5×D3—44no—
399C23×C3C2×C22⋊D512yes—
400C23×C3C2×C22⋊D511yescharacteristic
401C22×D3—24no—
402C22×D3—22no—
403C22×D3—101no—
404C3×Q8—22no—
405C3×D4—24no—
406C3×D4—22no—
407C2×C3×C4—22no—
408C2×C3×C4—22no—
409C2×C3×C4C2×C22⋊D512yes—
410C2×C3×C4C2×C22⋊D511yescharacteristic
411C2×C3×C4C2×C22⋊D512yes—
412C2×C3×C4C2×C22⋊D512yes—
413C2×C3×C4C23×D511yescharacteristic
414#67—28no—
415#67—24no—
416#67—24no—
417#67—204no—
418#67—202no—
419C2×Dic3—24no—
420C2×Dic3—22no—
421C2×Dic3—101no—
422C2×Dic3—101no—
423C2×Dic3—101no—
424C2×Dic3—202no—
425D12—24no—
426D12—22no—
427D12—102no—
428D12C2×C22⋊D514yes—
429C4×D3—28no—
430C4×D3—24no—
431C4×D3—24no—
432C4×D3—201no—
433C4×D3—201no—
434Dic6—24no—
435Dic6—22no—
436Dic6—102no—
437Dic6C2×C22⋊D514yes—
438C3×C8—104no—
439C22×C5—22no—
440C22×C5—41no—
441C22×C5—42no—
442C22×C5—64no—
443C22×C5—242no—
444C22×C5—244no—
445C22×C5#88212yes—
446C22×C5C22×D1211yescharacteristic
447C2×D5—122no—
448C2×D5—121no—
449C4×C5—22no—
450C4×C5—64no—
451C4×C5#88211yescharacteristic
452C4×C5#88211yescharacteristic
453C4×C5C2×D3×D412yes—
454Dic5—62no—
455Dic5—121no—
456Dic5—121no—
457#41—22no—
458#41—22no—
459#41—34no—
460#41—34no—
461#41—64no—
462#41—64no—
463#41—64no—
464#41—68no—
465#41—64no—
466#41—62no—
467#41—64no—
468#41—62no—
469#41—302no—
470#41—302no—
471C2×Q8—32no—
472C2×Q8—62no—
473C2×Q8—64no—
474C2×Q8—151no—
475C2×Q8C2×D3×D511yescharacteristic
476C2×D4—32no—
477C2×D4—62no—
478C2×D4—64no—
479C2×D4—68no—
480C2×D4—62no—
481C2×D4—64no—
482C2×D4—151no—
483C2×D4—302no—
484C2×D4—301no—
485C2×D4#120312yes—
486C2×D4C2×D3×D511yescharacteristic
487C22×C4—64no—
488C22×C4—62no—
489C22×C4—301no—
490C22×C4—301no—
491C22×C4#120312yes—
492C22×C4C2×D3×D511yescharacteristic
493Q16—304no—
494SD16—304no—
495SD16—304no—
496D8—304no—
497M16—54no—
498M16—102no—
499C2×C8—102no—
500C4⋊C4—302no—
501C22⋊C4—302no—
502C22⋊C4—302no—
503C42—151no—
504C42—151no—
505C42—301no—
506C3×C5#301811yescharacteristic, complemented, Hall
507C22×C3—22no—
508C22×C3—41no—
509C22×C3—42no—
510C22×C3#409912yes—
511C22×C3C22×C22⋊D511yescharacteristic
512C2×D3—24no—
513C2×D3—82no—
514C2×D3—84no—
515C2×D3—202no—
516C2×D3—201no—
517C3×C4—22no—
518C3×C4#409911yescharacteristic
519C3×C4#409911yescharacteristic
520C3×C4C22×C22⋊D512yes—
521Dic3—24no—
522Dic3—102no—
523Dic3—201no—
524Dic3—201no—
525C2×C5—21no—
526C2×C5—22no—
527C2×C5—42no—
528C2×C5—124no—
529C2×C5C2×#88211yescharacteristic
530D5—241no—
531C23—64no—
532C23—62no—
533C23—301no—
534C23C2×#120312yes—
535C23C2×#120311yescharacteristic
536Q8—22no—
537Q8—34no—
538Q8—64no—
539Q8—62no—
540Q8—302no—
541D4—24no—
542D4—22no—
543D4—34no—
544D4—64no—
545D4—68no—
546D4—64no—
547D4—64no—
548D4—62no—
549D4—302no—
550D4—604no—
551D4—602no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—68no—
555C2×C4—64no—
556C2×C4—62no—
557C2×C4—64no—
558C2×C4—64no—
559C2×C4—301no—
560C2×C4—301no—
561C2×C4—301no—
562C2×C4—602no—
563C2×C4—601no—
564C2×C4—601no—
565C2×C4C2×#120312yes—
566C2×C4C2×#120311yescharacteristic
567C2×C4D3×C22⋊D512yes—
568C2×C4D3×C22⋊D512yes—
569C2×C4C22×D3×D511yescharacteristic
570C8—104no—
571C2×C3—21no—
572C2×C3—22no—
573C2×C3—42no—
574C2×C3C2×#409911yescharacteristic
575D3—44no—
576D3—401no—
577C5#7871611yescharacteristic, complemented, Sylow
578C22—22no—
579C22—41no—
580C22—42no—
581C22—64no—
582C22—242no—
583C22—244no—
584C22—602no—
585C22—601no—
586C22#9157012yes—
587C22C22×#120311yescharacteristic
588C4—22no—
589C4—64no—
590C4—302no—
591C4—601no—
592C4—601no—
593C4#9157011yescharacteristic
594C4#9157011yescharacteristic
595C4C2×D3×C22⋊D512yes—
596C3#1061585211yescharacteristic, complemented, Sylow
597C2—21no—
598C2—22no—
599C2—42no—
600C2—124no—
601C2—1201no—
602C2C2×#9157011yescharacteristic
subgroupG/H
centerC4#91570
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×#41C22
socleC2×C3×C5C2×#120
hypercenterC2×#41D3×D5
hypocenterC3×C5#3018
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C5×#5964C2semidirect
2. #11756591C2semidirect
3. #11754145C2semidirect
4. C3×#65526C2semidirect
5. #11749394C2semidirect
6. #11749387C2semidirect
7. #11749148C2semidirect
8. #11749141C2semidirect
9. C2×#90978C2semidirect
12. C2×#6817C22semidirect
21. #91836C22semidirect
22. #91833C22semidirect
23. #91829C22semidirect
25. #91825C22semidirect
27. C2×C4×Dic15C22semidirect
28. C2×C3×#681C22semidirect
29. #91322C22semidirect
31. #91305C22semidirect
34. C3×#3968C22semidirect
35. C3×#3965C22semidirect
36. #90978C22semidirect
37. #90978C22semidirect
46. #65526D3semidirect
76. C5×#235D4semidirect
79. C2×C5×D12D4semidirect
83. C2×C5×Dic6D4semidirect
97. #5964D5semidirect
128. C2×#681C2×D3semidirect
131. #3968C2×D3semidirect
132. #3965C2×D3semidirect
302. #235C22⋊D5semidirect
306. C2×D12C22⋊D5semidirect
312. C2×Dic6C22⋊D5semidirect
506. C3×C5#3018semidirect
577. C5#78716semidirect
596. C3#10615852semidirect

Maximal subgroups

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

subgroupindex
C5×#59642
#117565912
#117541452
C3×#655262
#117493942
#117493872
#117491482
#117491412
C2×#909782
#106158523
#787165