Finite Group Databasebeta

← #14471460

Subgroups — #14471460

Order 1920.

852 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5553C211yesmaximal, characteristic, complemented
2#11752551C211yesmaximal, characteristic, complemented
3C4×#6715C211yesmaximal, characteristic, complemented
4C3×#65296C211yesmaximal, characteristic, complemented
5#11748987C212yesmaximal, complemented
6#11748941C212yesmaximal, complemented
7#11747416C212yesmaximal, complemented
8#11747230C212yesmaximal, complemented
9#11747045C211yesmaximal, characteristic, complemented
10#11746901C211yesmaximal, characteristic, complemented
11#11746857C211yesmaximal, characteristic, complemented
12#10615311—31nomaximal, Hall
13C2×C5×#235—21no—
14C2×D15⋊Q8—21no—
15C3×C4×C5×D4C2211yescharacteristic
16C2×#6715C2211yescharacteristic
17C4×D60C2211yescharacteristic
18C5×#940C2212yes—
19C5×#930—21no—
20C5×#893C2212yes—
21C5×#884—22no—
22C5×#874C2211yescharacteristic
23C4×C5×Dic6C2211yescharacteristic
24C3×#4015C2211yescharacteristic
25#91469C2212yes—
26#91429—22no—
27#91428C2212yes—
28#91390—21no—
29C2×#6654C2212yes—
30#91116C2211yescharacteristic
31#91093C2211yescharacteristic
32#91091C2211yescharacteristic
33C4×C3⋊#168C2211yescharacteristic
34C3×#3964C2212yes—
35C3×#3939C2212yes—
36C3×C4⋊#168C2211yescharacteristic
37Dic3⋊#168C2212yes—
38#90967C2212yes—
39#90966C2212yes—
40#90966C2212yes—
41Dic3×#168C2212yes—
42#80992—51nomaximal, Hall
43C5×#550—31no—
44Q8×D20—31no—
45#65361—32no—
46#65361—31no—
47#65359—31no—
48#65359—32no—
49Q8×#168—31no—
50C4×#686—32no—
51C4×#686—31no—
52#65296—32no—
53#65296D311yescharacteristic, complemented
54C2×C3×C5×D4C2311yescharacteristic
55C22×C3×C4×C5—21no—
56C2×D60C2311yescharacteristic
57C2×C4×D15—21no—
58C2×C5×#67—22no—
59C2×C5×#67—22no—
60C5×#235—44no—
61C2×C5×D12—21no—
62C2×C5×D12—21no—
63C2×C5×D12C2312yes—
64C2×C4×C5×D3—22no—
65C2×C5×Dic6D411yescharacteristic
66C2×C3×Dic10—21no—
67C2×#1204—22no—
68C2×#1200—22no—
69D15⋊Q8—44no—
70C3×C5×C4⋊C4C2311yescharacteristic
71C3×C5×C22⋊C4—21no—
72C3×C42×C5C2311yescharacteristic
73#6715—22no—
74#6714—21no—
75C4⋊Dic15C2311yescharacteristic
76C2×C3⋊#168C2311yescharacteristic
77C2×C3⋊#168C2311yescharacteristic
78C5×(C2×D3)⋊C4—22no—
79C5×C4⋊Dic3D411yescharacteristic
80C5×Dic3⋊C4D412yes—
81C4×C5×Dic3C2312yes—
82C3×C4⋊Dic5—21no—
83C3×C4⋊Dic5C2312yes—
84C2×C3×#168C2312yes—
85Dic3⋊Dic5—22no—
86#6667—22no—
87#6654—24no—
88#5727—51no—
89#5553D511yescharacteristic, complemented
90C3×#474—51no—
91#5088—51no—
92C4×#213—51no—
93#4890—52no—
94#4872—52no—
95#4843—51no—
96#4830—51no—
97#4786—52no—
98#4760—52no—
99C2×C5×#41—61no—
100C2×Q8×D5—61no—
101C5×#127—32no—
102C5×#127—31no—
103C5×#121—62no—
104C5×#121—61no—
105C4×C5×Q8—31no—
106C4×C5×D4—32no—
107C4×C5×D4C2×D311yescharacteristic
108C2×#686—31no—
109C2×#686—32no—
110#4043—62no—
111#4043—61no—
112C4×D20—31no—
113C4×D20—32no—
114#4015—32no—
115#4015C2×D311yescharacteristic
116#3964—31no—
117#3964—32no—
118#3964—32no—
119#3964—32no—
120#3964C2×D312yes—
121#3941—31no—
122#3939—31no—
123#3939C2×D312yes—
124C4⋊#168—32no—
125C4⋊#168C2×D311yescharacteristic
126C4×#168—31no—
127C4×#168—32no—
128#3358—151noSylow
129C23×C3×C5—21no—
130C22×D15—21no—
131C22×C5×D3—22no—
132C3×C5×D4—21no—
133C3×C5×D4—21no—
134C2×C3×C4×C5—21no—
135C2×C3×C4×C5—42no—
136C2×C3×C4×C5C2×D411yescharacteristic
137C2×C3×C4×C5C2×D411yescharacteristic
138C2×C3×C4×C5C2411yescharacteristic
139C2×Dic15—21no—
140D60—21no—
141D60#4112yescomplemented
142C4×D15—42no—
143C5×#67—44no—
144C5×#67—82no—
145C2×C5×Dic3C2×D412yes—
146C2×C5×Dic3C2×D412yes—
147C5×D12—22no—
148C5×D12—22no—
149C5×D12—42no—
150C5×D12—81no—
151C4×C5×D3—44no—
152C5×Dic6D814yescomplemented
153C2×C3×Dic5—22no—
154C3×Dic10—42no—
155#1204—44no—
156#1200—44no—
157C3⋊#168—21no—
158C3⋊#168—21no—
159C3×#168—22no—
160C2×D3×Q8—101no—
161C2×#235—21no—
162C3×C4⋊Q8—51no—
163C3×C4×D4C2×D511yescharacteristic
164#946—101no—
165#940C2×D512yes—
166C2×#213—51no—
167#930—21no—
168C2×D24—52no—
169#896—102no—
170#893—52no—
171#893C2×D512yes—
172#888—52no—
173#884—22no—
174#874C2×D511yescharacteristic
175C4×D12—51no—
176C4×Dic6C2×D511yescharacteristic
177C3×#107—52no—
178C3×C4⋊C8—51no—
179C3×#102—52no—
180#832—51no—
181#821—52no—
182Dic3⋊C8—52no—
183C8×Dic3—52no—
184#809—52no—
185#809—52no—
186#809—51no—
187#807—51no—
188C4×C3⋊C8—51no—
189C5×#41—124no—
190C2×C5×Q8—31no—
191C2×C5×D4—32no—
192C2×C5×D4—62no—
193C2×C5×D4—62no—
194C2×C5×D4—61no—
195C2×C5×D4—61no—
196C2×C5×D4C22×D311yescharacteristic
197C22×C4×C5—21no—
198C22×C4×C5—62no—
199Q8×D5—124no—
200C2×D20—31no—
201C2×C4×D5—62no—
202C2×C4×D5—61no—
203C2×Dic10—21no—
204C2×Dic10—62no—
205C5×C4⋊C4—32no—
206C5×C4⋊C4—31no—
207C5×C4⋊C4C22×D311yescharacteristic
208C5×C22⋊C4—21no—
209C5×C22⋊C4—62no—
210C42×C5—32no—
211C42×C5C22×D311yescharacteristic
212#686—64no—
213#686—62no—
214#685—62no—
215#685—61no—
216C4⋊Dic5—21no—
217C4⋊Dic5—31no—
218C4⋊Dic5—62no—
219C4⋊Dic5C22×D312yes—
220C2×#168—31no—
221C2×#168—31no—
222C2×#168C22×D312yes—
223#550—31no—
224D4×Q8—151no—
225#486—152no—
226#486—151no—
227#474—51no—
228#474—152no—
229#459—152no—
230#459—151no—
231C8×Q8—151no—
232C4×D8—152no—
233C4×D8—151no—
234C22×C3×C5—41no—
235C22×C3×C5—41no—
236C22×C3×C5—41no—
237C22×C3×C5C22×D411yescharacteristic
238C2×D15—22no—
239C2×D15—41no—
240C2×D15—41no—
241C2×C5×D3—42no—
242C2×C5×D3—42no—
243C2×C5×D3—42no—
244C3×C4×C5—21no—
245C3×C4×C5—41no—
246C3×C4×C5C2×D812yes—
247C3×C4×C5C2×#4111yescharacteristic
248C3×C4×C52−1+411yescharacteristic
249Dic15—41no—
250C3×Dic5—42no—
251C5×Dic3—22no—
252C5×Dic3C2×D814yes—
253C2×C3×Q8—101no—
254C2×C3×D4C22×D511yescharacteristic
255C22×C3×C4—21no—
256C2×#67—22no—
257C2×#67—22no—
258D3×Q8—204no—
259#235—44no—
260C2×D12—21no—
261C2×D12—21no—
262C2×D12—51no—
263C2×D12C22×D512yes—
264C2×C4×D3—22no—
265C2×C4×D3—101no—
266C2×C4×D3—102no—
267C2×Dic6—102no—
268C2×Dic6C22⋊D511yescharacteristic
269C2×C3×C8—52no—
270C3×C4⋊C4—52no—
271C3×C4⋊C4—101no—
272C3×C4⋊C4C22×D511yescharacteristic
273C3×C22⋊C4—21no—
274C3×C42C22×D511yescharacteristic
275#213—102no—
276(C2×D3)⋊C4—22no—
277(C2×D3)⋊C4—101no—
278(C2×D3)⋊C4—102no—
279C4⋊Dic3—51no—
280C4⋊Dic3C22⋊D511yescharacteristic
281Dic3⋊C4—102no—
282Dic3⋊C4C22⋊D512yes—
283C4×Dic3C22×D512yes—
284C2×C3⋊C8—51no—
285C2×C3⋊C8—51no—
286D24—104no—
287C23×C5—21no—
288C23×C5—62no—
289C22×D5—61no—
290C5×Q8—34no—
291C5×D4—21no—
292C5×D4—21no—
293C5×D4—62no—
294C5×D4—62no—
295C5×D4—124no—
296C5×D4—122no—
297C5×D4—241no—
298C5×D4—242no—
299C2×C4×C5—21no—
300C2×C4×C5—32no—
301C2×C4×C5—32no—
302C2×C4×C5—42no—
303C2×C4×C5—124no—
304C2×C4×C5D3×D411yescharacteristic
305C2×C4×C5D3×D411yescharacteristic
306C2×C4×C5C23×D311yescharacteristic
307C2×Dic5—22no—
308C2×Dic5—61no—
309D20—32no—
310D20—61no—
311C4×D5—124no—
312C4×D5—122no—
313Dic10—42no—
314Dic10—124no—
315#168—22no—
316#168—61no—
317#168—61no—
318C2×#41—61no—
319C22×Q8—301no—
320C2×D8—152no—
321C2×D8—151no—
322C4⋊Q8—51no—
323C4⋊Q8—152no—
324#127—32no—
325#127—31no—
326C22⋊Q8—302no—
327C22⋊Q8—301no—
328#121—62no—
329#121—61no—
330C4×Q8—31no—
331C4×D4—32no—
332C4×D4—152no—
333C4×D4—151no—
334C4×D4D3×D511yescharacteristic
335#107—52no—
336#107—151no—
337C4⋊C8—51no—
338C4⋊C8—152no—
339#102—52no—
340#102—152no—
341#102—152no—
342#102—152no—
343#102—151no—
344#102—151no—
345C4×C8—152no—
346C4×C8—151no—
347C2×C3×C5—81no—
348C2×C3×C5#54811yescharacteristic
349C2×C3×C5C22×D811yescharacteristic
350C2×C3×C5#57711yescharacteristic
351D15—42no—
352C5×D3—82no—
353C23×C3—21no—
354C22×D3—22no—
355C22×D3—101no—
356C3×Q8—202no—
357C3×D4—21no—
358C3×D4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—42no—
361C2×C3×C4—102no—
362C2×C3×C4C2×C22⋊D511yescharacteristic
363C2×C3×C4C2×C22⋊D511yescharacteristic
364C2×C3×C4C23×D511yescharacteristic
365#67—44no—
366#67—82no—
367C2×Dic3—101no—
368C2×Dic3C2×C22⋊D512yes—
369C2×Dic3C2×C22⋊D512yes—
370D12—22no—
371D12—22no—
372D12—42no—
373D12—52no—
374D12—81no—
375D12—101no—
376C4×D3—44no—
377C4×D3—202no—
378C4×D3—204no—
379Dic6—204no—
380Dic6#68614yescomplemented
381C3×C8—102no—
382C3⋊C8—101no—
383C3⋊C8—101no—
384C22×C5—41no—
385C22×C5—41no—
386C22×C5—41no—
387C22×C5—122no—
388C22×C5—122no—
389C22×C5—122no—
390C22×C5C2×D3×D411yescharacteristic
391C2×D5—62no—
392C2×D5—121no—
393C2×D5—121no—
394C4×C5—21no—
395C4×C5—34no—
396C4×C5—41no—
397C4×C5—62no—
398C4×C5D3×D812yes—
399C4×C5#100711yescharacteristic
400C4×C5D3×#4111yescharacteristic
401Dic5—42no—
402Dic5—121no—
403#41—124no—
404C2×Q8—31no—
405C2×Q8—101no—
406C2×Q8—302no—
407C2×Q8—604no—
408C2×D4—32no—
409C2×D4—62no—
410C2×D4—62no—
411C2×D4—61no—
412C2×D4—61no—
413C2×D4—151no—
414C2×D4C2×D3×D511yescharacteristic
415C22×C4—21no—
416C22×C4—62no—
417C22×C4—302no—
418C22×C4—301no—
419D8—304no—
420D8—302no—
421C2×C8—52no—
422C2×C8—151no—
423C2×C8—151no—
424C4⋊C4—32no—
425C4⋊C4—31no—
426C4⋊C4—52no—
427C4⋊C4—101no—
428C4⋊C4—151no—
429C4⋊C4—302no—
430C4⋊C4C2×D3×D511yescharacteristic
431C22⋊C4—21no—
432C22⋊C4—62no—
433C22⋊C4—302no—
434C22⋊C4—301no—
435C42—32no—
436C42C2×D3×D511yescharacteristic
437C3×C5#335811yescharacteristic, complemented, Hall
438C22×C3—41no—
439C22×C3—41no—
440C22×C3—41no—
441C22×C3C22×C22⋊D511yescharacteristic
442C2×D3—42no—
443C2×D3—42no—
444C2×D3—42no—
445C2×D3—102no—
446C2×D3—201no—
447C2×D3—201no—
448C3×C4—21no—
449C3×C4—41no—
450C3×C4—202no—
451C3×C4C2×#68612yes—
452C3×C4D5×#4111yescharacteristic
453C3×C4#415011yescharacteristic
454Dic3—22no—
455Dic3—201no—
456Dic3C2×#68614yes—
457C2×C5—81no—
458C2×C5—242no—
459C2×C5#563311yescharacteristic
460C2×C5C2×D3×D811yescharacteristic
461C2×C5#577811yescharacteristic
462D5—122no—
463C23—21no—
464C23—62no—
465C23—301no—
466Q8—34no—
467Q8—202no—
468Q8—604no—
469D4—21no—
470D4—21no—
471D4—62no—
472D4—62no—
473D4—124no—
474D4—122no—
475D4—152no—
476D4—241no—
477D4—242no—
478D4—301no—
479C2×C4—21no—
480C2×C4—32no—
481C2×C4—32no—
482C2×C4—42no—
483C2×C4—102no—
484C2×C4—124no—
485C2×C4—301no—
486C2×C4—604no—
487C2×C4—602no—
488C2×C4D3×C22⋊D511yescharacteristic
489C2×C4D3×C22⋊D511yescharacteristic
490C2×C4C22×D3×D511yescharacteristic
491C8—102no—
492C8—301no—
493C8—301no—
494C2×C3—81no—
495C2×C3C22×#68611yescharacteristic
496C2×C3#6616511yescharacteristic
497C2×C3#6617011yescharacteristic
498D3—82no—
499D3—202no—
500C5#8099211yescharacteristic, complemented, Sylow
501C22—41no—
502C22—41no—
503C22—41no—
504C22—122no—
505C22—122no—
506C22—122no—
507C22—302no—
508C22—601no—
509C22—601no—
510C22C2×D3×C22⋊D511yescharacteristic
511C4—21no—
512C4—34no—
513C4—41no—
514C4—62no—
515C4—202no—
516C4—601no—
517C4D3×#68612yes—
518C4#9200411yescharacteristic
519C4#9201611yescharacteristic
520C3#1061531111yescharacteristic, complemented, Sylow
521C2—81no—
522C2—242no—
523C2—602no—
524C2#1175454611yescharacteristic
525C2C2×D3×#68611yescharacteristic
526C2#1175534011yescharacteristic
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#3358
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×#5553C2semidirect
2. #11752551C2semidirect
3. C4×#6715C2semidirect
4. C3×#65296C2semidirect
5. #11748987C2semidirect
6. #11748941C2semidirect
7. #11747416C2semidirect
8. #11747230C2semidirect
9. #11747045C2semidirect
10. #11746901C2semidirect
11. #11746857C2semidirect
53. #65296D3semidirect
89. #5553D5semidirect
141. D60#41semidirect
152. C5×Dic6D8semidirect
380. Dic6#686semidirect
437. C3×C5#3358semidirect
500. C5#80992semidirect
520. C3#10615311semidirect

Maximal subgroups

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

subgroupindex
C5×#55532
#117525512
C4×#67152
C3×#652962
#117489872
#117489412
#117474162
#117472302
#117470452
#117469012
#117468572
#106153113
#809925