Finite Group Databasebeta

← #14458416

Subgroups — #14458416

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
1C5×#5717C211yesmaximal, characteristic, complemented
2#11752676C211yesmaximal, characteristic, complemented
3#11751171C211yesmaximal, characteristic, complemented
4C3×#65361C211yesmaximal, characteristic, complemented
5#11749032C212yesmaximal, complemented
6#11749028C212yesmaximal, complemented
7#11748954C212yesmaximal, complemented
8D3×#3964C212yesmaximal, complemented
9C4×#6652C211yesmaximal, characteristic, complemented
10#11746998C211yesmaximal, characteristic, complemented
11#11746964C211yesmaximal, characteristic, complemented
12#10615213—31nomaximal, Hall
13C2×C5×D3×D4—21no—
14C2×C5×D3×D4C2212yes—
15C2×#6772—21no—
16C3×C5×#127C2211yescharacteristic
17#91758C2211yescharacteristic, complemented
18C5×#938—21no—
19C5×#938C2212yes—
20C5×#937—22no—
21C4×C5×D12C2211yescharacteristic
22C4×C5×D12C2211yescharacteristic
23C2×C3×#686C2212yescomplemented
24C3×C4×D20C2211yescharacteristic, complemented
25#91457—22no—
26#91456C2212yescomplemented
27D3×C4⋊Dic5C2212yescomplemented
28#91360—21no—
29C2×#6652C2211yescharacteristic
30C2×D3×#168C2212yescomplemented
31#91116C2212yescomplemented
32#91116C2212yescomplemented
33C4×C3⋊#168C2211yescharacteristic, complemented
34C3×#3964C2212yescomplemented
35C3×C4⋊#168C2211yescharacteristic, complemented
36#90989C2211yescharacteristic, complemented
37#90988C2212yescomplemented
38#90967C2212yes—
39#90966C2211yescharacteristic
40#90964C2211yescharacteristic, complemented
41(C2×D3)⋊#168C2212yescomplemented
42#80352—51nomaximal, Hall
43C5×D42—31no—
44#65889—31no—
45#65506—32no—
46C2×#3964—32no—
47#65361D311yescharacteristic, complemented
48#65359—31no—
49#65327—32no—
50#65322—32no—
51C4×#686—31no—
52D4×#168—31no—
53#65296—31no—
54C23×C5×D3—22no—
55C2×C3×C5×D4—21no—
56C2×C3×C5×D4—21no—
57C2×C3×C5×D4C2312yes—
58C2×Dic30—21no—
59C2×C5×#67—22no—
60C2×C5×#67—22no—
61C5×D3×D4—24no—
62C5×D3×D4—42no—
63C5×D3×D4—44no—
64C2×C5×D12D411yescharacteristic
65C2×C5×D12C2311yescharacteristic
66C2×C4×C5×D3—21no—
67C2×C4×C5×D3C2312yes—
68C2×C3×D20C2311yescharacteristic
69C2×C3×C4×D5—21no—
70C2×#1201—22no—
71C2×D3×Dic5—22no—
72#6772—44no—
73C3×C42×C5C2311yescharacteristic
74C4⋊Dic15—21no—
75C4⋊Dic15C2312yes—
76C2×C3⋊#168C2311yescharacteristic
77C2×C3⋊#168C2311yescharacteristic
78C5×#217—22no—
79C5×(C2×D3)⋊C4—21no—
80C5×(C2×D3)⋊C4D412yes—
81C5×C4⋊Dic3D411yescharacteristic
82C5×C4⋊Dic3C2311yescharacteristic
83C3×#686—42no—
84C3×#685—21no—
85C3×C4⋊Dic5C2311yescharacteristic
86C2×C3×#168C2312yes—
87Dic3⋊Dic5—22no—
88#6665—22no—
89#6652—22no—
90D3×#168—42no—
91#5717D511yescharacteristic, complemented
92#5551—51no—
93C3×#459—51no—
94#5157—52no—
95#5156—52no—
96#5070—51no—
97C4×#213—51no—
98#4846—51no—
99#4834—51no—
100#4774—52no—
101D3×#102—52no—
102C22×C5×D4—32no—
103C22×C5×D4—61no—
104C2×#711—61no—
105C5×#127C2×D311yescharacteristic
106C5×#121—32no—
107C5×#121—61no—
108C5×#120—62no—
109C4×C5×D4—31no—
110C4×C5×D4—31no—
111C2×#686—31no—
112C2×#686C2×D312yescomplemented
113#4076—32no—
114C2×C4⋊Dic5—32no—
115C22×#168—32no—
116#4043—61no—
117#4032—62no—
118C4×D20C2×D311yescharacteristic, complemented
119#4015—31no—
120#3964—32no—
121#3964—32no—
122#3964—31no—
123#3964—32no—
124#3964C2×D312yescomplemented
125C22⋊#168—32no—
126#3941—31no—
127#3939—31no—
128#3939—32no—
129C4⋊#168C2×D311yescharacteristic, complemented
130C4×#168—31no—
131#3294—151noSylow
132C23×C3×C5—22no—
133C22×C5×D3—21no—
134C22×C5×D3—44no—
135C22×C5×D3—44no—
136C22×C5×D3—82no—
137C22×C5×D3C2×D412yes—
138C22×C3×D5—21no—
139C3×C5×D4—22no—
140C3×C5×D4—22no—
141C3×C5×D4—42no—
142C3×C5×D4—81no—
143C2×C3×C4×C5C2×D411yescharacteristic
144C2×C3×C4×C5C2×D411yescharacteristic
145C2×C3×C4×C5C2411yescharacteristic
146C2×Dic15—22no—
147Dic30—42no—
148C5×#67—44no—
149C5×#67—44no—
150C2×C5×Dic3—21no—
151C2×C5×Dic3C2×D412yes—
152C5×D12—21no—
153C5×D12D814yescomplemented
154C5×D12#4112yescomplemented
155C4×C5×D3—22no—
156C4×C5×D3—22no—
157C4×C5×D3—42no—
158C2×C3×Dic5—21no—
159C3×D20—21no—
160C3×D20—21no—
161C3×C4×D5—42no—
162#1201—44no—
163D3×Dic5—44no—
164C3⋊#168—21no—
165C3⋊#168—21no—
166C3×#168—22no—
167C2×D3×D4—21no—
168C2×D3×D4C2×D512yes—
169C2×#235—101no—
170C2×C3×D8—52no—
171C3×#127C2×D511yescharacteristic
172C3×C4×D4—51no—
173#938—21no—
174#938—52no—
175#938C2×D512yes—
176#937—22no—
177C2×#213—51no—
178#924—101no—
179C2×D3×C8—52no—
180D3×C4⋊C4—52no—
181#883—102no—
182C4×D12C2×D511yescharacteristic
183C4×D12C2×D511yescharacteristic
184#869—51no—
185C3×C4⋊C8—51no—
186C3×#102—52no—
187#832—52no—
188#832—52no—
189#832—51no—
190#821—52no—
191(C2×D3)⋊C8—52no—
192#818—52no—
193#809—51no—
194#807—51no—
195C4×C3⋊C8—51no—
196C24×C5—62no—
197C2×C5×D4—21no—
198C2×C5×D4—21no—
199C2×C5×D4—31no—
200C2×C5×D4—31no—
201C2×C5×D4—62no—
202C2×C5×D4—64no—
203C2×C5×D4—62no—
204C2×C5×D4—122no—
205C2×C5×D4—124no—
206C2×C5×D4C22×D312yes—
207C22×C4×C5—32no—
208C22×C4×C5—61no—
209C2×C22⋊D5—62no—
210C22×Dic5—62no—
211#711—124no—
212C2×D20C22×D311yescharacteristic
213C2×C4×D5—21no—
214C2×Dic10—61no—
215C5×C4⋊C4—31no—
216C5×C4⋊C4—31no—
217C5×C22⋊C4—32no—
218C5×C22⋊C4—62no—
219C5×C22⋊C4—61no—
220C42×C5C22×D311yescharacteristic
221#686—42no—
222#686—62no—
223#685—21no—
224#685—62no—
225C4⋊Dic5—32no—
226C4⋊Dic5—62no—
227C4⋊Dic5—61no—
228C4⋊Dic5C22×D311yescharacteristic
229C2×#168—31no—
230C2×#168—31no—
231C2×#168—122no—
232C2×#168C22×D312yes—
233#547—151no—
234D42—31no—
235#486—151no—
236#480—152no—
237#474—151no—
238#466—152no—
239#459—51no—
240#447—152no—
241C4×D8—151no—
242C8×D4—151no—
243C2×#102—152no—
244C22×C3×C5—42no—
245C22×C3×C5—42no—
246C22×C3×C5—42no—
247C22×C3×C5C22×D411yescharacteristic
248C2×C5×D3—22no—
249C2×C5×D3—22no—
250C2×C5×D3—22no—
251C2×C5×D3—41no—
252C2×C5×D3—41no—
253C2×C5×D3—88no—
254C2×C5×D3C2×D814yes—
255C2×C3×D5—41no—
256C2×C3×D5—41no—
257C2×C3×D5—41no—
258C3×C4×C5—21no—
259C3×C4×C5C2×D812yes—
260C3×C4×C5C2×#4111yescharacteristic
261C3×C4×C5H(5,𝔽2)11yescharacteristic
262Dic15—42no—
263C3×Dic5—41no—
264C5×Dic3—22no—
265C5×Dic3—41no—
266C23×D3—22no—
267C2×C3×D4—21no—
268C2×C3×D4—21no—
269C2×C3×D4—51no—
270C2×C3×D4C22×D512yes—
271C22×C3×C4—101no—
272C2×#67—22no—
273C2×#67—22no—
274C2×#67—102no—
275D3×D4—24no—
276D3×D4—42no—
277D3×D4—44no—
278#235—204no—
279C2×D12C22⋊D511yescharacteristic
280C2×D12C22×D511yescharacteristic
281C2×C4×D3—21no—
282C2×C4×D3—102no—
283C2×C4×D3C22×D512yes—
284C2×Dic6—101no—
285C3×D8—202no—
286C2×C3×C8—52no—
287C3×C4⋊C4—51no—
288C3×C22⋊C4—101no—
289C3×C42C22×D511yescharacteristic
290#217—22no—
291#217—102no—
292#213—102no—
293(C2×D3)⋊C4—21no—
294(C2×D3)⋊C4C22⋊D512yes—
295C4⋊Dic3—52no—
296C4⋊Dic3—101no—
297C4⋊Dic3C22⋊D511yescharacteristic
298C4⋊Dic3C22×D511yescharacteristic
299Dic3⋊C4—102no—
300C2×C3⋊C8—51no—
301C2×C3⋊C8—51no—
302D3×C8—202no—
303C23×C5—22no—
304C23×C5—32no—
305C23×C5—61no—
306C23×C5—124no—
307C23×C5—124no—
308C23×C5—242no—
309C22×D5—21no—
310C5×D4—22no—
311C5×D4—22no—
312C5×D4—34no—
313C5×D4—32no—
314C5×D4—42no—
315C5×D4—61no—
316C5×D4—81no—
317C5×D4—124no—
318C5×D4—124no—
319C2×C4×C5—32no—
320C2×C4×C5—62no—
321C2×C4×C5—62no—
322C2×C4×C5—61no—
323C2×C4×C5—122no—
324C2×C4×C5D3×D411yescharacteristic
325C2×C4×C5D3×D411yescharacteristic
326C2×C4×C5C23×D311yescharacteristic
327C22⋊D5—124no—
328C2×Dic5—21no—
329C2×Dic5—62no—
330C2×Dic5—124no—
331D20—21no—
332D20—21no—
333C4×D5—42no—
334Dic10—122no—
335#168—22no—
336#168—61no—
337#168—61no—
338C2×#41—301no—
339C22×D4—32no—
340C22×D4—61no—
341C2×D8—52no—
342C2×D8—151no—
343C22×C8—152no—
344C4⋊Q8—151no—
345#127D3×D511yescharacteristic
346#123—302no—
347C22⋊Q8—301no—
348#121—32no—
349#121—61no—
350#121—152no—
351#120—62no—
352C4×D4—31no—
353C4×D4—31no—
354C4×D4—51no—
355C2×C4⋊C4—152no—
356#107—151no—
357#107—152no—
358C4⋊C8—51no—
359#102—52no—
360#102—151no—
361#102—151no—
362#102—152no—
363#102—152no—
364#102—152no—
365C22⋊C8—152no—
366C4×C8—151no—
367C2×C3×C5—82no—
368C2×C3×C5#54611yescharacteristic
369C2×C3×C5C22×D811yescharacteristic
370C2×C3×C5#57711yescharacteristic
371C3×D5—81no—
372C5×D3—24no—
373C5×D3—42no—
374C23×C3—22no—
375C23×C3—101no—
376C22×D3—21no—
377C22×D3—44no—
378C22×D3—44no—
379C22×D3—82no—
380C22×D3C2×C22⋊D512yes—
381C3×D4—22no—
382C3×D4—22no—
383C3×D4—42no—
384C3×D4—81no—
385C3×D4—101no—
386C3×D4—101no—
387C2×C3×C4—101no—
388C2×C3×C4—202no—
389C2×C3×C4C2×C22⋊D511yescharacteristic
390C2×C3×C4C2×C22⋊D511yescharacteristic
391C2×C3×C4C23×D511yescharacteristic
392#67—44no—
393#67—44no—
394#67—204no—
395C2×Dic3—21no—
396C2×Dic3—102no—
397C2×Dic3C2×C22⋊D512yes—
398D12—21no—
399D12#68614yescomplemented
400D12#71112yescomplemented
401C4×D3—22no—
402C4×D3—22no—
403C4×D3—42no—
404C4×D3—204no—
405Dic6—202no—
406C3×C8—102no—
407C3⋊C8—101no—
408C3⋊C8—101no—
409C22×C5—34no—
410C22×C5—42no—
411C22×C5—42no—
412C22×C5—42no—
413C22×C5—62no—
414C22×C5—62no—
415C22×C5—62no—
416C22×C5—121no—
417C22×C5—121no—
418C22×C5—248no—
419C22×C5C2×D3×D411yescharacteristic
420C2×D5—41no—
421C2×D5—41no—
422C2×D5—41no—
423C4×C5—21no—
424C4×C5—62no—
425C4×C5—121no—
426C4×C5D3×D812yes—
427C4×C5#100411yescharacteristic
428C4×C5D3×#4111yescharacteristic
429Dic5—41no—
430Dic5—122no—
431C24—62no—
432#41—604no—
433C2×Q8—301no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—31no—
437C2×D4—31no—
438C2×D4—51no—
439C2×D4—62no—
440C2×D4—64no—
441C2×D4—62no—
442C2×D4—122no—
443C2×D4—124no—
444C2×D4—302no—
445C2×D4C2×D3×D512yes—
446C22×C4—32no—
447C22×C4—61no—
448C22×C4—101no—
449C22×C4—302no—
450D8—202no—
451D8—302no—
452C2×C8—52no—
453C2×C8—151no—
454C2×C8—151no—
455C2×C8—602no—
456C4⋊C4—31no—
457C4⋊C4—31no—
458C4⋊C4—51no—
459C4⋊C4—152no—
460C4⋊C4—302no—
461C4⋊C4—301no—
462C22⋊C4—32no—
463C22⋊C4—62no—
464C22⋊C4—61no—
465C22⋊C4—101no—
466C22⋊C4—302no—
467C42C2×D3×D511yescharacteristic
468C3×C5#329411yescharacteristic, complemented, Hall
469C22×C3—42no—
470C22×C3—42no—
471C22×C3—42no—
472C22×C3—201no—
473C22×C3—201no—
474C22×C3—201no—
475C22×C3C22×C22⋊D511yescharacteristic
476C2×D3—22no—
477C2×D3—22no—
478C2×D3—22no—
479C2×D3—41no—
480C2×D3—41no—
481C2×D3—88no—
482C2×D3C2×#68614yes—
483C3×C4—21no—
484C3×C4—201no—
485C3×C4C2×#68612yes—
486C3×C4C2×#71111yescharacteristic
487C3×C4#414411yescharacteristic
488Dic3—22no—
489Dic3—41no—
490Dic3—202no—
491C2×C5—64no—
492C2×C5—82no—
493C2×C5—122no—
494C2×C5#561211yescharacteristic
495C2×C5C2×D3×D811yescharacteristic
496C2×C5#577511yescharacteristic
497D5—81no—
498C23—22no—
499C23—32no—
500C23—61no—
501C23—101no—
502C23—124no—
503C23—124no—
504C23—242no—
505Q8—602no—
506D4—22no—
507D4—22no—
508D4—34no—
509D4—32no—
510D4—42no—
511D4—61no—
512D4—81no—
513D4—101no—
514D4—101no—
515D4—124no—
516D4—124no—
517D4—604no—
518C2×C4—32no—
519C2×C4—62no—
520C2×C4—62no—
521C2×C4—61no—
522C2×C4—101no—
523C2×C4—122no—
524C2×C4—202no—
525C2×C4—302no—
526C2×C4—604no—
527C2×C4D3×C22⋊D511yescharacteristic
528C2×C4D3×C22⋊D511yescharacteristic
529C2×C4C22×D3×D511yescharacteristic
530C8—102no—
531C8—301no—
532C8—301no—
533C2×C3—82no—
534C2×C3—401no—
535C2×C3C22×#68611yescharacteristic
536C2×C3#6614011yescharacteristic
537C2×C3#6617011yescharacteristic
538D3—24no—
539D3—42no—
540C5#8035211yescharacteristic, complemented, Sylow
541C22—34no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—62no—
546C22—62no—
547C22—62no—
548C22—121no—
549C22—121no—
550C22—201no—
551C22—201no—
552C22—201no—
553C22—248no—
554C22C2×D3×C22⋊D511yescharacteristic
555C4—21no—
556C4—62no—
557C4—121no—
558C4—201no—
559C4—602no—
560C4D3×#68612yes—
561C4D3×#71111yescharacteristic
562C4#9202511yescharacteristic
563C3#1061521311yescharacteristic, complemented, Sylow
564C2—64no—
565C2—82no—
566C2—122no—
567C2—401no—
568C2C2×D3×#68611yescharacteristic
569C2#1175472011yescharacteristic
570C2#1175494611yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊D5
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#127C22
socleC22×C3×C5C22×D4
hypercenter#127D3×D5
hypocenterC3×C5#3294
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
1. C5×#5717C2semidirect
2. #11752676C2semidirect
3. #11751171C2semidirect
4. C3×#65361C2semidirect
5. #11749032C2semidirect
6. #11749028C2semidirect
7. #11748954C2semidirect
8. D3×#3964C2semidirect
9. C4×#6652C2semidirect
10. #11746998C2semidirect
11. #11746964C2semidirect
17. #91758C22semidirect
23. C2×C3×#686C22semidirect
24. C3×C4×D20C22semidirect
26. #91456C22semidirect
27. D3×C4⋊Dic5C22semidirect
30. C2×D3×#168C22semidirect
31. #91116C22semidirect
32. #91116C22semidirect
33. C4×C3⋊#168C22semidirect
34. C3×#3964C22semidirect
35. C3×C4⋊#168C22semidirect
36. #90989C22semidirect
37. #90988C22semidirect
40. #90964C22semidirect
41. (C2×D3)⋊#168C22semidirect
47. #65361D3semidirect
91. #5717D5semidirect
112. C2×#686C2×D3semidirect
118. C4×D20C2×D3semidirect
124. #3964C2×D3semidirect
129. C4⋊#168C2×D3semidirect
153. C5×D12D8semidirect
154. C5×D12#41semidirect
399. D12#686semidirect
400. D12#711semidirect
468. C3×C5#3294semidirect
540. C5#80352semidirect
563. C3#10615213semidirect

Maximal subgroups

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

subgroupindex
C5×#57172
#117526762
#117511712
C3×#653612
#117490322
#117490282
#117489542
D3×#39642
C4×#66522
#117469982
#117469642
#106152133
#803525