Finite Group Databasebeta

← #14458397

Subgroups — #14458397

Order 1920.

988 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D12×D20C211yesmaximal, characteristic, complemented
2#11752549C211yesmaximal, characteristic, complemented
3#11750944C211yesmaximal, characteristic, complemented
4C5×#4834C211yesmaximal, characteristic, complemented
5C3×#65135C211yesmaximal, characteristic, complemented
6#11748121C212yesmaximal, complemented
7D3×#3953C212yesmaximal, complemented
8#11747917C212yesmaximal, complemented
9#11747627C212yesmaximal, complemented
10C4×#6653C211yesmaximal, characteristic, complemented
11#11746910C211yesmaximal, characteristic, complemented
12#10611169—31nomaximal, Hall
13C2×D3×D20C2212yes—
14C2×D5×D12—21no—
15C2×#6768—21no—
16C2×D120C2212yescomplemented
17#91763C2211yescharacteristic
18C4×D60C2211yescharacteristic, complemented
19C2×C5×D3×C8C2212yescomplemented
20C4×C5×D12C2211yescharacteristic
21C3×C4×D20C2211yescharacteristic
22C3×#4015C2211yescharacteristic, complemented
23#91476—22no—
24#91460C2212yes—
25#91460C2212yescomplemented
26#91450—21no—
27#91427—22no—
28D3×C4⋊Dic5C2212yescomplemented
29#91390—21no—
30C2×#6653C2211yescharacteristic
31C3×C5×C4⋊C8C2211yescharacteristic, complemented
32#91105C2212yescomplemented
33C5×(C2×D3)⋊C8C2212yescomplemented
34C4×C5×C3⋊C8C2211yescharacteristic, complemented
35C3×#3953C2212yes—
36C3×#3950C2212yescomplemented
37#90990C2211yescharacteristic, complemented
38#90968C2212yescomplemented
39#90968C2212yescomplemented
40#90968C2211yescharacteristic, complemented
41#90965C2211yescharacteristic
42#80980—51nomaximal, Hall
43#65889—31no—
44D4×D20—31no—
45C5×C8×D4—31no—
46#65404—32no—
47C2×#3953—32no—
48#65135D311yescharacteristic, complemented
49#65132—31no—
50#65025—32no—
51#65021—32no—
52#64983—31no—
53C4×D40—31no—
54C22×D3×D5—22no—
55C2×D60—21no—
56C2×D60—21no—
57C2×D60C2312yes—
58C2×D60C2311yescharacteristic
59C2×C4×D15—21no—
60C2×C5×D12D411yescharacteristic
61C2×C4×C5×D3C2312yes—
62C2×C3×D20C2311yescharacteristic
63C2×C3×C4×D5—21no—
64C2×C3×Dic10—21no—
65C2×#1203—22no—
66C2×#1203—22no—
67C2×#1202—22no—
68C2×D3×Dic5—22no—
69D3×D20—24no—
70D3×D20—42no—
71D5×D12—44no—
72#6768—44no—
73C2×C3×C5×C8C2312yes—
74C3×C42×C5C2311yescharacteristic
75#6714—21no—
76C4⋊Dic15C2311yescharacteristic
77D120—42no—
78C5×(C2×D3)⋊C4D412yes—
79C5×C4⋊Dic3D411yescharacteristic
80C2×C5×C3⋊C8C2311yescharacteristic
81C2×C5×C3⋊C8C2311yescharacteristic
82C5×D3×C8—42no—
83C3×#685—21no—
84C3×C4⋊Dic5—21no—
85C3×C4⋊Dic5C2312yes—
86C3×C4⋊Dic5C2311yescharacteristic
87Dic3⋊Dic5—22no—
88#6667—22no—
89#6667—22no—
90#6653—22no—
91#5726—51no—
92D4×D12—51no—
93C3×#474—51no—
94#5088—51no—
95C4×#213—51no—
96#4883—52no—
97#4882—52no—
98#4843—51no—
99#4834D511yescharacteristic, complemented
100#4773—52no—
101D3×#102—52no—
102C2×#711—61no—
103C2×D4×D5—61no—
104C22×D20—32no—
105C22×C5×C8—32no—
106C4×C5×D4—31no—
107#4076—32no—
108#4076—32no—
109C2×C4⋊Dic5—32no—
110C2×D40—31no—
111C2×D40—32no—
112#4043—61no—
113#4041—61no—
114#4032—62no—
115#4028—62no—
116#4020—31no—
117C4×D20—31no—
118C4×D20C2×D311yescharacteristic
119#4015C2×D311yescharacteristic, complemented
120C5×C4⋊C8C2×D311yescharacteristic, complemented
121C5×C22⋊C8—32no—
122C4×C5×C8—31no—
123#3953—31no—
124#3953—32no—
125#3953—32no—
126#3953—32no—
127#3953—31no—
128#3953C2×D312yes—
129#3950—31no—
130#3950C2×D312yescomplemented
131#3294—151noSylow
132C22×D15—21no—
133C22×D15—22no—
134C22×C5×D3C2×D412yes—
135C22×C3×D5—21no—
136C2×D3×D5—44no—
137C2×D3×D5—44no—
138C2×D3×D5—82no—
139C2×C3×C4×C5C2×D411yescharacteristic
140C2×C3×C4×C5C2×D411yescharacteristic
141C2×C3×C4×C5C2411yescharacteristic
142C2×Dic15—21no—
143D60—22no—
144D60—21no—
145D60—21no—
146D60—22no—
147D60—42no—
148D60—81no—
149C4×D15—42no—
150C2×C5×Dic3C2×D412yes—
151C5×D12D814yescomplemented
152C4×C5×D3—22no—
153C4×C5×D3—22no—
154C2×C3×Dic5—21no—
155C2×C3×Dic5—22no—
156C3×D20—21no—
157C3×D20#4112yescomplemented
158C3×C4×D5—42no—
159C3×Dic10—42no—
160#1203—44no—
161#1203—44no—
162#1202—44no—
163D3×Dic5—44no—
164C3×C5×C8—22no—
165C5×C3⋊C8—21no—
166C5×C3⋊C8—21no—
167C2×#239—101no—
168C2×D3×D4—52no—
169C22×D12—101no—
170C3×C4⋊Q8—51no—
171C3×C4×D4—51no—
172#946—101no—
173C2×#213—51no—
174#930—101no—
175C2×D24—52no—
176C2×D3×C8C2×D512yescomplemented
177#895—52no—
178#895—52no—
179#894—102no—
180D3×C4⋊C4—52no—
181#882—102no—
182#874—51no—
183C4×D12—51no—
184C4×D12C2×D511yescharacteristic
185C3×#107—52no—
186C3×C4⋊C8C2×D511yescharacteristic, complemented
187C3×#102—52no—
188#832—51no—
189#821—52no—
190(C2×D3)⋊C8C2×D512yescomplemented
191#809—52no—
192#809—51no—
193#809—52no—
194#807—51no—
195C4×C3⋊C8C2×D511yescharacteristic, complemented
196C23×D5—62no—
197C2×C5×D4—31no—
198C22×C4×C5—32no—
199C2×C22⋊D5—62no—
200C2×C22⋊D5—62no—
201C22×Dic5—62no—
202#711—124no—
203D4×D5—124no—
204C2×D20—32no—
205C2×D20—31no—
206C2×D20—64no—
207C2×D20—62no—
208C2×D20—61no—
209C2×D20—61no—
210C2×D20—122no—
211C2×D20C22×D311yescharacteristic
212C2×C4×D5—21no—
213C2×C4×D5—61no—
214C2×Dic10—21no—
215C2×C5×C8—31no—
216C2×C5×C8—31no—
217C2×C5×C8—122no—
218C2×C5×C8C22×D312yes—
219C5×C4⋊C4—31no—
220C5×C22⋊C4—32no—
221C42×C5C22×D311yescharacteristic
222#685—21no—
223#685—62no—
224#685—62no—
225#685—61no—
226C4⋊Dic5—21no—
227C4⋊Dic5—31no—
228C4⋊Dic5—62no—
229C4⋊Dic5C22×D311yescharacteristic
230C4⋊Dic5C22×D312yes—
231D40—62no—
232D40—122no—
233#547—151no—
234D42—151no—
235#486—151no—
236#480—152no—
237#474—51no—
238#466—152no—
239#459—151no—
240#447—152no—
241C4×D8—151no—
242C8×D4—31no—
243C2×#102—152no—
244C22×C3×C5C22×D411yescharacteristic
245C2×D15—42no—
246C2×D15—41no—
247C2×D15—41no—
248C2×D15—42no—
249C2×D15—42no—
250C2×D15—41no—
251C2×C5×D3—22no—
252C2×C5×D3—22no—
253C2×C5×D3C2×D814yes—
254C2×C3×D5—22no—
255C2×C3×D5—41no—
256C2×C3×D5—41no—
257D3×D5—88no—
258C3×C4×C5—21no—
259C3×C4×C5C2×D812yes—
260C3×C4×C5C2×#4111yescharacteristic
261C3×C4×C5H(5,𝔽2)11yescharacteristic
262Dic15—41no—
263C3×Dic5—42no—
264C3×Dic5—41no—
265C5×Dic3—22no—
266C23×D3—102no—
267C2×C3×Q8—101no—
268C2×C3×D4—51no—
269C22×C3×C4—101no—
270C2×#67—102no—
271#239—204no—
272D3×D4—104no—
273D3×D4—202no—
274C2×D12—52no—
275C2×D12—51no—
276C2×D12—102no—
277C2×D12—102no—
278C2×D12—101no—
279C2×D12—101no—
280C2×D12—204no—
281C2×D12D2011yescharacteristic
282C2×C4×D3—102no—
283C2×C4×D3—101no—
284C2×C4×D3C22×D512yes—
285C2×C3×C8C22×D512yes—
286C3×C4⋊C4—51no—
287C3×C4⋊C4—52no—
288C3×C4⋊C4—101no—
289C3×C22⋊C4—101no—
290C3×C42C22×D511yescharacteristic
291#213—102no—
292(C2×D3)⋊C4—101no—
293(C2×D3)⋊C4—102no—
294(C2×D3)⋊C4—102no—
295(C2×D3)⋊C4D2012yes—
296C4⋊Dic3—51no—
297C4⋊Dic3D2011yescharacteristic
298Dic3⋊C4—102no—
299C2×C3⋊C8C22×D511yescharacteristic
300C2×C3⋊C8C22×D511yescharacteristic
301D24—202no—
302D3×C8—42no—
303C23×C5—32no—
304C22×D5—21no—
305C22×D5—61no—
306C22×D5—62no—
307C22×D5—124no—
308C22×D5—124no—
309C22×D5—242no—
310C5×D4—34no—
311C2×C4×C5—32no—
312C2×C4×C5—62no—
313C2×C4×C5—62no—
314C2×C4×C5D3×D411yescharacteristic
315C2×C4×C5D3×D411yescharacteristic
316C2×C4×C5C23×D311yescharacteristic
317C22⋊D5—124no—
318C22⋊D5—124no—
319C2×Dic5—22no—
320C2×Dic5—21no—
321C2×Dic5—61no—
322C2×Dic5—124no—
323D20—21no—
324D20—61no—
325D20—62no—
326D20—61no—
327D20—62no—
328D20—124no—
329D20—122no—
330D20—241no—
331D20#23912yescomplemented
332C4×D5—42no—
333C4×D5—122no—
334Dic10—42no—
335C5×C8—22no—
336C5×C8—61no—
337C5×C8—61no—
338C2×#41—301no—
339C22×D4—152no—
340C22×D4—301no—
341C2×D8—151no—
342C2×D8—152no—
343C22×C8—32no—
344C4⋊Q8—51no—
345#127—151no—
346#123—302no—
347C22⋊Q8—301no—
348#121—152no—
349#121—152no—
350#121—301no—
351#120—302no—
352C4×D4—31no—
353C4×D4—51no—
354C4×D4—151no—
355C2×C4⋊C4—152no—
356#107—52no—
357#107—151no—
358C4⋊C8D3×D511yescharacteristic, complemented
359#102—52no—
360#102—152no—
361#102—151no—
362#102—152no—
363#102—151no—
364#102—152no—
365C22⋊C8—32no—
366C4×C8—31no—
367C2×C3×C5#54611yescharacteristic
368C2×C3×C5C22×D811yescharacteristic
369C2×C3×C5#57711yescharacteristic
370D15—82no—
371D15—81no—
372C3×D5—42no—
373C5×D3—24no—
374C23×C3—101no—
375C22×D3—101no—
376C22×D3—102no—
377C22×D3—204no—
378C22×D3—204no—
379C22×D3—402no—
380C22×D3C2×D2012yes—
381C3×Q8—202no—
382C3×D4—52no—
383C3×D4—101no—
384C2×C3×C4—102no—
385C2×C3×C4—101no—
386C2×C3×C4—202no—
387C2×C3×C4C2×D2011yescharacteristic
388C2×C3×C4C2×D2011yescharacteristic
389C2×C3×C4C23×D511yescharacteristic
390#67—204no—
391C2×Dic3—101no—
392C2×Dic3C2×D2012yes—
393D12—101no—
394D12—102no—
395D12—101no—
396D12—102no—
397D12—204no—
398D12—204no—
399D12—202no—
400D12—401no—
401D12D4014yescomplemented
402C4×D3—22no—
403C4×D3—22no—
404C4×D3—204no—
405C4×D3—202no—
406C3×C8—22no—
407C3⋊C8—21no—
408C3⋊C8—21no—
409C22×C5—34no—
410C22×C5—62no—
411C22×C5—62no—
412C22×C5C2×D3×D411yescharacteristic
413C2×D5—22no—
414C2×D5—41no—
415C2×D5—41no—
416C2×D5—121no—
417C2×D5—122no—
418C2×D5—122no—
419C2×D5—121no—
420C2×D5—122no—
421C2×D5—121no—
422C2×D5—248no—
423C4×C5—21no—
424C4×C5—62no—
425C4×C5D3×D812yes—
426C4×C5C2×#23911yescharacteristic
427C4×C5#100911yescharacteristic
428Dic5—42no—
429Dic5—41no—
430Dic5—121no—
431C24—302no—
432#41—604no—
433C2×Q8—101no—
434C2×D4—31no—
435C2×D4—51no—
436C2×D4—152no—
437C2×D4—151no—
438C2×D4—302no—
439C2×D4—304no—
440C2×D4—302no—
441C2×D4—302no—
442C2×D4—301no—
443C2×D4—301no—
444C2×D4—602no—
445C2×D4—604no—
446C22×C4—32no—
447C22×C4—101no—
448C22×C4—302no—
449C22×C4—301no—
450D8—302no—
451D8—602no—
452C2×C8—31no—
453C2×C8—31no—
454C2×C8—122no—
455C2×C8C2×D3×D512yes—
456C4⋊C4—31no—
457C4⋊C4—52no—
458C4⋊C4—51no—
459C4⋊C4—101no—
460C4⋊C4—151no—
461C4⋊C4—302no—
462C22⋊C4—32no—
463C22⋊C4—101no—
464C22⋊C4—302no—
465C22⋊C4—302no—
466C22⋊C4—301no—
467C42C2×D3×D511yescharacteristic
468C3×C5#329411yescharacteristic, complemented, Hall
469C22×C3—102no—
470C22×C3—201no—
471C22×C3—201no—
472C22×C3C22×D2011yescharacteristic
473C2×D3—22no—
474C2×D3—22no—
475C2×D3—201no—
476C2×D3—202no—
477C2×D3—202no—
478C2×D3—201no—
479C2×D3—202no—
480C2×D3—201no—
481C2×D3—408no—
482C2×D3C2×D4014yes—
483C3×C4—21no—
484C3×C4—202no—
485C3×C4—201no—
486C3×C4C2×D4012yes—
487C3×C4#414411yescharacteristic
488C3×C4D5×#4111yescharacteristic
489Dic3—22no—
490Dic3—201no—
491C2×C5—64no—
492C2×C5#563011yescharacteristic
493C2×C5C2×D3×D811yescharacteristic
494C2×C5#577811yescharacteristic
495D5—42no—
496D5—242no—
497D5—241no—
498C23—32no—
499C23—101no—
500C23—301no—
501C23—302no—
502C23—604no—
503C23—604no—
504C23—1202no—
505Q8—202no—
506D4—34no—
507D4—52no—
508D4—101no—
509D4—301no—
510D4—302no—
511D4—301no—
512D4—302no—
513D4—604no—
514D4—604no—
515D4—604no—
516D4—602no—
517D4—1201no—
518C2×C4—32no—
519C2×C4—62no—
520C2×C4—62no—
521C2×C4—101no—
522C2×C4—102no—
523C2×C4—202no—
524C2×C4—301no—
525C2×C4—604no—
526C2×C4—602no—
527C2×C4D3×D2011yescharacteristic
528C2×C4D3×D2011yescharacteristic
529C2×C4C22×D3×D511yescharacteristic
530C8—22no—
531C8—61no—
532C8—61no—
533C2×C3—202no—
534C2×C3#6588811yescharacteristic
535C2×C3C22×D4011yescharacteristic
536C2×C3#6608511yescharacteristic
537D3—24no—
538D3—402no—
539D3—401no—
540C5#8098011yescharacteristic, complemented, Sylow
541C22—34no—
542C22—62no—
543C22—62no—
544C22—102no—
545C22—201no—
546C22—201no—
547C22—602no—
548C22—601no—
549C22—602no—
550C22—601no—
551C22—601no—
552C22—602no—
553C22—1208no—
554C22C2×D3×D2011yescharacteristic
555C4—21no—
556C4—62no—
557C4—201no—
558C4—202no—
559C4—601no—
560C4D3×D4012yes—
561C4#9202711yescharacteristic
562C4D5×#23911yescharacteristic
563C3#1061116911yescharacteristic, complemented, Sylow
564C2—64no—
565C2—202no—
566C2—1202no—
567C2—1201no—
568C2#1175388111yescharacteristic
569C2C2×D3×D4011yescharacteristic
570C2#1175403211yescharacteristic
subgroupG/H
centerC22C2×D3×D20
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×C4⋊C8C22
socleC22×C3×C5C22×D4
hypercenterC4⋊C8D3×D5
hypocenterC3×C5#3294
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
1. D12×D20C2semidirect
2. #11752549C2semidirect
3. #11750944C2semidirect
4. C5×#4834C2semidirect
5. C3×#65135C2semidirect
6. #11748121C2semidirect
7. D3×#3953C2semidirect
8. #11747917C2semidirect
9. #11747627C2semidirect
10. C4×#6653C2semidirect
11. #11746910C2semidirect
16. C2×D120C22semidirect
18. C4×D60C22semidirect
19. C2×C5×D3×C8C22semidirect
22. C3×#4015C22semidirect
25. #91460C22semidirect
28. D3×C4⋊Dic5C22semidirect
31. C3×C5×C4⋊C8C22semidirect
32. #91105C22semidirect
33. C5×(C2×D3)⋊C8C22semidirect
34. C4×C5×C3⋊C8C22semidirect
36. C3×#3950C22semidirect
37. #90990C22semidirect
38. #90968C22semidirect
39. #90968C22semidirect
40. #90968C22semidirect
48. #65135D3semidirect
99. #4834D5semidirect
119. #4015C2×D3semidirect
120. C5×C4⋊C8C2×D3semidirect
130. #3950C2×D3semidirect
151. C5×D12D8semidirect
157. C3×D20#41semidirect
176. C2×D3×C8C2×D5semidirect
186. C3×C4⋊C8C2×D5semidirect
190. (C2×D3)⋊C8C2×D5semidirect
195. C4×C3⋊C8C2×D5semidirect
331. D20#239semidirect
358. C4⋊C8D3×D5semidirect
401. D12D40semidirect
468. C3×C5#3294semidirect
540. C5#80980semidirect
563. C3#10611169semidirect

Maximal subgroups

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

subgroupindex
D12×D202
#117525492
#117509442
C5×#48342
C3×#651352
#117481212
D3×#39532
#117479172
#117476272
C4×#66532
#117469102
#106111693
#809805