Finite Group Databasebeta

← #14458404

Subgroups — #14458404

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×#5551C211yesmaximal, characteristic, complemented
2#11752745C211yesmaximal, characteristic, complemented
3C4×#6715C211yesmaximal, characteristic, complemented
4C3×#65304C211yesmaximal, characteristic, complemented
5#11749152C212yesmaximal, complemented
6#11749028C212yesmaximal, complemented
7D3×#3941C212yesmaximal, complemented
8#11747318C212yesmaximal, complemented
9#11746998C211yesmaximal, characteristic, complemented
10#11746966C211yesmaximal, characteristic, complemented
11#11746895C211yesmaximal, characteristic, complemented
12#10615256—31nomaximal, Hall
13C2×C5×#235—21no—
14C2×#6777—21no—
15C2×D3×D20C2212yes—
16C3×C4×C5×D4C2211yescharacteristic, complemented
17C2×#6715C2211yescharacteristic
18C4×D60C2211yescharacteristic
19C5×#938C2212yescomplemented
20C5×#924—21no—
21C5×D3×C4⋊C4C2212yescomplemented
22C5×#883—22no—
23C4×C5×D12C2211yescharacteristic
24C5×#869C2211yescharacteristic, complemented
25C2×C3×#686C2212yescomplemented
26C3×#4020C2211yescharacteristic
27#91474—22no—
28#91465—21no—
29#91456C2212yes—
30C2×D3×#168C2212yescomplemented
31#91116C2211yescharacteristic, complemented
32#91093C2211yescharacteristic
33#91091C2211yescharacteristic, complemented
34C4×C3⋊#168C2211yescharacteristic, complemented
35C3×#3941C2212yescomplemented
36C3×C4⋊#168C2211yescharacteristic, complemented
37#90988C2212yescomplemented
38#90969C2212yes—
39#90964C2212yescomplemented
40#90964C2212yescomplemented
41(C2×D3)⋊#168C2212yescomplemented
42#80352—51nomaximal, Hall
43C5×#547—31no—
44D4×D20—31no—
45#65377—31no—
46#65376—31no—
47#65327—32no—
48#65325—32no—
49#65304D311yescharacteristic, complemented
50C4×#686—31no—
51D4×#168—31no—
52#65256—32no—
53C2×#3941—32no—
54C22×D3×D5—22no—
55C2×C3×C5×D4C2311yescharacteristic
56C22×C3×C4×C5—21no—
57C2×D60C2311yescharacteristic
58C2×C4×D15—21no—
59C2×C5×#67—22no—
60C5×#235—44no—
61C2×C5×D12D411yescharacteristic
62C2×C4×C5×D3—22no—
63C2×C4×C5×D3C2312yes—
64C2×C5×Dic6—21no—
65C2×C3×D20—21no—
66C2×C3×D20—21no—
67C2×C3×D20C2312yes—
68C2×#1202—22no—
69C2×#1201—22no—
70#6777—44no—
71D3×D20—24no—
72D3×D20—42no—
73C3×C5×C4⋊C4C2311yescharacteristic
74C3×C5×C22⋊C4—21no—
75C3×C42×C5C2311yescharacteristic
76#6715—22no—
77#6714—21no—
78C4⋊Dic15C2311yescharacteristic
79C2×C3⋊#168C2311yescharacteristic
80C2×C3⋊#168C2311yescharacteristic
81C5×#217—22no—
82C5×(C2×D3)⋊C4D412yes—
83C5×C4⋊Dic3—21no—
84C5×C4⋊Dic3D411yescharacteristic
85C5×C4⋊Dic3C2312yes—
86C5×Dic3⋊C4—22no—
87C3×#686—42no—
88C2×C3×#168C2312yes—
89#6665—22no—
90D3×#168—42no—
91#5717—51no—
92#5551D511yescharacteristic, complemented
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—
102C2×C5×#41—61no—
103C2×D4×D5—61no—
104C22×D20—32no—
105C5×C4⋊Q8—31no—
106C5×#123—62no—
107C5×C22⋊Q8—61no—
108C5×#121—32no—
109C4×C5×D4—31no—
110C4×C5×D4C2×D311yescharacteristic, complemented
111C2×C5×C4⋊C4—32no—
112C2×#686—31no—
113C2×#686C2×D312yescomplemented
114#4076—32no—
115C22×#168—32no—
116#4041—61no—
117#4028—62no—
118#4020C2×D311yescharacteristic
119C4×D20—31no—
120#3964—31no—
121C22⋊#168—32no—
122#3941—32no—
123#3941—32no—
124#3941—31no—
125#3941—32no—
126#3941C2×D312yescomplemented
127#3939—31no—
128#3939—32no—
129C4⋊#168C2×D311yescharacteristic, complemented
130C4×#168—31no—
131#3294—151noSylow
132C23×C3×C5—21no—
133C22×D15—21no—
134C22×C5×D3C2×D412yes—
135C22×C3×D5—22no—
136C2×D3×D5—44no—
137C2×D3×D5—44no—
138C2×D3×D5—82no—
139C3×C5×D4—21no—
140C3×C5×D4—21no—
141C2×C3×C4×C5—21no—
142C2×C3×C4×C5—42no—
143C2×C3×C4×C5C2×D411yescharacteristic
144C2×C3×C4×C5C2×D411yescharacteristic
145C2×C3×C4×C5C2411yescharacteristic
146C2×Dic15—21no—
147D60—21no—
148D60#4112yescomplemented
149C4×D15—42no—
150C5×#67—44no—
151C2×C5×Dic3—22no—
152C2×C5×Dic3C2×D412yes—
153C5×D12D814yescomplemented
154C4×C5×D3—22no—
155C4×C5×D3—22no—
156C4×C5×D3—44no—
157C5×Dic6—42no—
158C3×D20—22no—
159C3×D20—22no—
160C3×D20—42no—
161C3×D20—81no—
162#1202—44no—
163#1201—44no—
164C3⋊#168—21no—
165C3⋊#168—21no—
166C3×#168—22no—
167C2×D3×D4—52no—
168C2×D3×D4—101no—
169C2×#235—21no—
170C2×C3×D8—52no—
171C3×#127—51no—
172C3×C4×D4C2×D511yescharacteristic, complemented
173#938—52no—
174#938—101no—
175#938C2×D512yescomplemented
176#937—102no—
177C2×#213—51no—
178#924—21no—
179C2×D3×C8—52no—
180D3×C4⋊C4C2×D512yescomplemented
181#883—22no—
182C4×D12—51no—
183C4×D12C2×D511yescharacteristic
184#869C2×D511yescharacteristic, complemented
185C3×C4⋊C8—51no—
186C3×#102—52no—
187#832—51no—
188#832—52no—
189#832—52no—
190#821—52no—
191(C2×D3)⋊C8—52no—
192#818—52no—
193#809—51no—
194#807—51no—
195C4×C3⋊C8—51no—
196C23×D5—62no—
197C5×#41—124no—
198C2×C5×Q8—61no—
199C2×C5×D4—31no—
200C2×C5×D4—62no—
201C2×C5×D4C22×D311yescharacteristic
202C22×C4×C5—21no—
203C22×C4×C5—32no—
204C22×C4×C5—62no—
205C2×C22⋊D5—62no—
206D4×D5—124no—
207C2×D20—21no—
208C2×D20—21no—
209C2×D20—31no—
210C2×D20—64no—
211C2×D20—62no—
212C2×D20—122no—
213C2×D20C22×D312yes—
214C2×C4×D5—61no—
215C5×C4⋊C4—31no—
216C5×C4⋊C4—32no—
217C5×C4⋊C4—62no—
218C5×C4⋊C4—61no—
219C5×C4⋊C4C22×D311yescharacteristic
220C5×C22⋊C4—21no—
221C5×C22⋊C4—32no—
222C5×C22⋊C4—62no—
223C42×C5C22×D311yescharacteristic
224#686—42no—
225#686—62no—
226#685—62no—
227#685—61no—
228C4⋊Dic5—31no—
229C2×#168—31no—
230C2×#168—31no—
231C2×#168—122no—
232C2×#168C22×D312yes—
233#547—31no—
234D42—151no—
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—41no—
245C22×C3×C5—41no—
246C22×C3×C5—41no—
247C22×C3×C5C22×D411yescharacteristic
248C2×D15—22no—
249C2×D15—41no—
250C2×D15—41no—
251C2×C5×D3—22no—
252C2×C5×D3—22no—
253C2×C5×D3C2×D814yes—
254C2×C3×D5—42no—
255C2×C3×D5—42no—
256C2×C3×D5—42no—
257D3×D5—88no—
258C3×C4×C5—21no—
259C3×C4×C5—41no—
260C3×C4×C5C2×D812yes—
261C3×C4×C5C2×#4111yescharacteristic
262C3×C4×C5H(5,𝔽2)11yescharacteristic
263Dic15—41no—
264C5×Dic3—22no—
265C5×Dic3—42no—
266C23×D3—102no—
267C2×C3×D4—52no—
268C2×C3×D4—101no—
269C2×C3×D4—101no—
270C2×C3×D4C22×D511yescharacteristic
271C22×C3×C4—21no—
272C2×#67—22no—
273C2×#67—102no—
274C2×#67—102no—
275D3×D4—104no—
276D3×D4—202no—
277D3×D4—204no—
278#235—44no—
279C2×D12—51no—
280C2×D12C22⋊D511yescharacteristic
281C2×C4×D3—22no—
282C2×C4×D3—101no—
283C2×C4×D3C22×D512yes—
284C2×Dic6—21no—
285C3×D8—202no—
286C2×C3×C8—52no—
287C3×C4⋊C4C22×D511yescharacteristic
288C3×C22⋊C4—21no—
289C3×C42C22×D511yescharacteristic
290#217—22no—
291#217—102no—
292#213—102no—
293(C2×D3)⋊C4—101no—
294(C2×D3)⋊C4C22⋊D512yes—
295C4⋊Dic3—21no—
296C4⋊Dic3—51no—
297C4⋊Dic3C22⋊D511yescharacteristic
298C4⋊Dic3C22×D512yes—
299Dic3⋊C4—22no—
300C2×C3⋊C8—51no—
301C2×C3⋊C8—51no—
302D3×C8—202no—
303C23×C5—21no—
304C23×C5—32no—
305C22×D5—22no—
306C22×D5—61no—
307C22×D5—124no—
308C22×D5—124no—
309C22×D5—242no—
310C5×Q8—122no—
311C5×D4—21no—
312C5×D4—21no—
313C5×D4—34no—
314C5×D4—124no—
315C2×C4×C5—21no—
316C2×C4×C5—32no—
317C2×C4×C5—42no—
318C2×C4×C5—62no—
319C2×C4×C5—62no—
320C2×C4×C5—62no—
321C2×C4×C5—124no—
322C2×C4×C5D3×D411yescharacteristic
323C2×C4×C5D3×D411yescharacteristic
324C2×C4×C5C23×D311yescharacteristic
325C22⋊D5—124no—
326C2×Dic5—61no—
327D20—22no—
328D20—22no—
329D20—32no—
330D20—42no—
331D20—61no—
332D20—81no—
333D20—124no—
334C4×D5—122no—
335#168—22no—
336#168—61no—
337#168—61no—
338C2×#41—61no—
339C22×D4—152no—
340C22×D4—301no—
341C2×D8—52no—
342C2×D8—151no—
343C22×C8—152no—
344C4⋊Q8—31no—
345#127—51no—
346#123—62no—
347C22⋊Q8—61no—
348#121—32no—
349#121—152no—
350#121—301no—
351#120—302no—
352C4×D4—31no—
353C4×D4—151no—
354C4×D4D3×D511yescharacteristic, complemented
355C2×C4⋊C4—32no—
356#107—152no—
357#107—151no—
358C4⋊C8—51no—
359#102—52no—
360#102—152no—
361#102—152no—
362#102—152no—
363#102—151no—
364#102—151no—
365C22⋊C8—152no—
366C4×C8—151no—
367C2×C3×C5—81no—
368C2×C3×C5#54611yescharacteristic
369C2×C3×C5C22×D811yescharacteristic
370C2×C3×C5#57711yescharacteristic
371D15—42no—
372C3×D5—82no—
373C5×D3—24no—
374C23×C3—21no—
375C23×C3—102no—
376C22×D3—101no—
377C22×D3—204no—
378C22×D3—204no—
379C22×D3—402no—
380C22×D3C2×C22⋊D512yes—
381C3×D4—21no—
382C3×D4—21no—
383C3×D4—102no—
384C3×D4—102no—
385C3×D4—202no—
386C3×D4—401no—
387C2×C3×C4—21no—
388C2×C3×C4—42no—
389C2×C3×C4C2×C22⋊D511yescharacteristic
390C2×C3×C4C2×C22⋊D511yescharacteristic
391C2×C3×C4C23×D511yescharacteristic
392#67—44no—
393#67—204no—
394#67—204no—
395C2×Dic3—22no—
396C2×Dic3—101no—
397C2×Dic3C2×C22⋊D512yes—
398D12—52no—
399D12—101no—
400D12#68614yescomplemented
401C4×D3—22no—
402C4×D3—22no—
403C4×D3—44no—
404C4×D3—202no—
405Dic6—42no—
406C3×C8—102no—
407C3⋊C8—101no—
408C3⋊C8—101no—
409C22×C5—34no—
410C22×C5—41no—
411C22×C5—41no—
412C22×C5—41no—
413C22×C5—62no—
414C22×C5—62no—
415C22×C5C2×D3×D411yescharacteristic
416C2×D5—42no—
417C2×D5—42no—
418C2×D5—42no—
419C2×D5—62no—
420C2×D5—121no—
421C2×D5—121no—
422C2×D5—248no—
423C4×C5—21no—
424C4×C5—41no—
425C4×C5—62no—
426C4×C5—122no—
427C4×C5D3×D812yes—
428C4×C5#100411yescharacteristic
429C4×C5D3×#4111yescharacteristic
430Dic5—121no—
431C24—302no—
432#41—124no—
433C2×Q8—61no—
434C2×D4—31no—
435C2×D4—52no—
436C2×D4—62no—
437C2×D4—101no—
438C2×D4—101no—
439C2×D4—151no—
440C2×D4—302no—
441C2×D4—304no—
442C2×D4—302no—
443C2×D4—602no—
444C2×D4—604no—
445C2×D4C2×D3×D511yescharacteristic
446C22×C4—21no—
447C22×C4—32no—
448C22×C4—62no—
449C22×C4—301no—
450D8—202no—
451D8—302no—
452C2×C8—52no—
453C2×C8—151no—
454C2×C8—151no—
455C2×C8—602no—
456C4⋊C4—31no—
457C4⋊C4—32no—
458C4⋊C4—62no—
459C4⋊C4—61no—
460C4⋊C4—151no—
461C4⋊C4C2×D3×D511yescharacteristic
462C22⋊C4—21no—
463C22⋊C4—32no—
464C22⋊C4—62no—
465C22⋊C4—302no—
466C22⋊C4—301no—
467C42C2×D3×D511yescharacteristic
468C3×C5#329411yescharacteristic, complemented, Hall
469C22×C3—41no—
470C22×C3—41no—
471C22×C3—41no—
472C22×C3—202no—
473C22×C3—202no—
474C22×C3—202no—
475C22×C3C22×C22⋊D511yescharacteristic
476C2×D3—22no—
477C2×D3—22no—
478C2×D3—102no—
479C2×D3—201no—
480C2×D3—201no—
481C2×D3—408no—
482C2×D3C2×#68614yes—
483C3×C4—21no—
484C3×C4—41no—
485C3×C4C2×#68612yes—
486C3×C4D5×#4111yescharacteristic
487C3×C4#414911yescharacteristic
488Dic3—22no—
489Dic3—42no—
490Dic3—201no—
491C2×C5—64no—
492C2×C5—81no—
493C2×C5#561211yescharacteristic
494C2×C5C2×D3×D811yescharacteristic
495C2×C5#577511yescharacteristic
496D5—82no—
497D5—122no—
498C23—21no—
499C23—32no—
500C23—102no—
501C23—301no—
502C23—604no—
503C23—604no—
504C23—1202no—
505Q8—122no—
506D4—21no—
507D4—21no—
508D4—34no—
509D4—102no—
510D4—102no—
511D4—124no—
512D4—152no—
513D4—202no—
514D4—301no—
515D4—401no—
516D4—604no—
517D4—604no—
518C2×C4—21no—
519C2×C4—32no—
520C2×C4—42no—
521C2×C4—62no—
522C2×C4—62no—
523C2×C4—62no—
524C2×C4—124no—
525C2×C4—301no—
526C2×C4—602no—
527C2×C4D3×C22⋊D511yescharacteristic
528C2×C4D3×C22⋊D511yescharacteristic
529C2×C4C22×D3×D511yescharacteristic
530C8—102no—
531C8—301no—
532C8—301no—
533C2×C3—81no—
534C2×C3—402no—
535C2×C3C22×#68611yescharacteristic
536C2×C3#6616311yescharacteristic
537C2×C3#6617111yescharacteristic
538D3—24no—
539D3—202no—
540C5#8035211yescharacteristic, complemented, Sylow
541C22—34no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—62no—
546C22—62no—
547C22—202no—
548C22—202no—
549C22—202no—
550C22—302no—
551C22—601no—
552C22—601no—
553C22—1208no—
554C22C2×D3×C22⋊D511yescharacteristic
555C4—21no—
556C4—41no—
557C4—62no—
558C4—122no—
559C4—601no—
560C4D3×#68612yes—
561C4#9201611yescharacteristic
562C4#9201711yescharacteristic
563C3#1061525611yescharacteristic, complemented, Sylow
564C2—64no—
565C2—81no—
566C2—402no—
567C2—602no—
568C2#1175465811yescharacteristic
569C2C2×D3×#68611yescharacteristic
570C2#1175532411yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊D5
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C4×C5×D4C22
socleC22×C3×C5C22×D4
hypercenterC4×D4D3×D5
hypocenterC3×C5#3294
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C5×#5551C2semidirect
2. #11752745C2semidirect
3. C4×#6715C2semidirect
4. C3×#65304C2semidirect
5. #11749152C2semidirect
6. #11749028C2semidirect
7. D3×#3941C2semidirect
8. #11747318C2semidirect
9. #11746998C2semidirect
10. #11746966C2semidirect
11. #11746895C2semidirect
16. C3×C4×C5×D4C22semidirect
19. C5×#938C22semidirect
21. C5×D3×C4⋊C4C22semidirect
24. C5×#869C22semidirect
25. C2×C3×#686C22semidirect
30. C2×D3×#168C22semidirect
31. #91116C22semidirect
33. #91091C22semidirect
34. C4×C3⋊#168C22semidirect
35. C3×#3941C22semidirect
36. C3×C4⋊#168C22semidirect
37. #90988C22semidirect
39. #90964C22semidirect
40. #90964C22semidirect
41. (C2×D3)⋊#168C22semidirect
49. #65304D3semidirect
92. #5551D5semidirect
110. C4×C5×D4C2×D3semidirect
113. C2×#686C2×D3semidirect
126. #3941C2×D3semidirect
129. C4⋊#168C2×D3semidirect
148. D60#41semidirect
153. C5×D12D8semidirect
172. C3×C4×D4C2×D5semidirect
175. #938C2×D5semidirect
180. D3×C4⋊C4C2×D5semidirect
184. #869C2×D5semidirect
354. C4×D4D3×D5semidirect
400. D12#686semidirect
468. C3×C5#3294semidirect
540. C5#80352semidirect
563. C3#10615256semidirect

Maximal subgroups

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

subgroupindex
C5×#55512
#117527452
C4×#67152
C3×#653042
#117491522
#117490282
D3×#39412
#117473182
#117469982
#117469662
#117468952
#106152563
#803525