Finite Group Databasebeta

← #14471487

Subgroups — #14471487

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×#5727C211yesmaximal, characteristic, complemented
2#11752677C211yesmaximal, characteristic, complemented
3#11751189C211yesmaximal, characteristic, complemented
4C3×#65376C211yesmaximal, characteristic, complemented
5#11747403C212yesmaximal, complemented
6#11747402C212yesmaximal, complemented
7#11747246C212yesmaximal, complemented
8#11747230C212yesmaximal, complemented
9#11747037C211yesmaximal, characteristic, complemented
10C4×#6652C211yesmaximal, characteristic, complemented
11#11746857C211yesmaximal, characteristic, complemented
12#10615293—31nomaximal, Hall
13C2×C5×D3×Q8—21no—
14C2×#6773—21no—
15C3×C5×C4⋊Q8C2211yescharacteristic
16#91763C2211yescharacteristic
17C5×#946—21no—
18C5×#896—22no—
19C5×#893C2212yes—
20C5×#888C2212yes—
21C4×C5×D12C2211yescharacteristic
22C4×C5×Dic6C2211yescharacteristic
23C3×C4×D20C2211yescharacteristic
24#91459—22no—
25#91458C2212yes—
26#91450—21no—
27#91428C2212yes—
28C2×#6654C2212yes—
29C2×#6652C2211yescharacteristic
30#91093C2212yes—
31#91093C2212yes—
32C4×C3⋊#168C2211yescharacteristic
33C3×#3941C2212yes—
34C3×#3939C2212yes—
35C3×C4⋊#168C2211yescharacteristic
36#90989C2211yescharacteristic
37Dic3⋊#168C2212yes—
38#90969C2212yes—
39#90966C2211yescharacteristic
40#90964C2211yescharacteristic
41Dic3×#168C2212yes—
42#80992—51nomaximal, Hall
43C5×D4×Q8—31no—
44#65911—31no—
45#65377—32no—
46#65377—31no—
47#65376—32no—
48#65376D311yescharacteristic, complemented
49Q8×#168—31no—
50#65304—32no—
51#65304—31no—
52C4×#686—31no—
53C4×#686—32no—
54C2×C3×C5×Q8—21no—
55C2×D60—21no—
56C2×D60—21no—
57C2×D60C2312yes—
58C5×D3×Q8—44no—
59C2×C5×D12C2311yescharacteristic
60C2×C4×C5×D3—21no—
61C2×C4×C5×D3—22no—
62C2×C5×Dic6—22no—
63C2×C5×Dic6D411yescharacteristic
64C2×C3×D20C2311yescharacteristic
65C2×C3×C4×D5—21no—
66C2×#1202—22no—
67C2×#1202—22no—
68C2×#1200—22no—
69#6773—44no—
70C3×C5×C4⋊C4—21no—
71C3×C5×C4⋊C4C2312yes—
72C3×C42×C5C2311yescharacteristic
73C2×C3⋊#168C2311yescharacteristic
74C2×C3⋊#168C2311yescharacteristic
75C5×(C2×D3)⋊C4—21no—
76C5×(C2×D3)⋊C4—22no—
77C5×C4⋊Dic3D411yescharacteristic
78C5×C4⋊Dic3C2311yescharacteristic
79C5×Dic3⋊C4—22no—
80C5×Dic3⋊C4D412yes—
81C4×C5×Dic3C2312yes—
82C3×#685—21no—
83C3×C4⋊Dic5C2311yescharacteristic
84C2×C3×#168C2312yes—
85#6667—22no—
86#6654—24no—
87#6652—22no—
88#5727D511yescharacteristic, complemented
89#5553—51no—
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—
99C22×C5×Q8—61no—
100C2×#713—61no—
101C5×C4⋊Q8—32no—
102C5×C4⋊Q8C2×D311yescharacteristic
103C5×C22⋊Q8—62no—
104C5×C22⋊Q8—61no—
105C4×C5×Q8—31no—
106C4×C5×D4—32no—
107C4×C5×D4—31no—
108C2×#686—31no—
109C2×#686—32no—
110#4041—62no—
111#4041—61no—
112#4020—32no—
113#4020—31no—
114C4×D20—32no—
115C4×D20C2×D311yescharacteristic
116#3964—31no—
117#3941—31no—
118#3941—32no—
119#3941—32no—
120#3941—32no—
121#3941C2×D312yes—
122#3939—31no—
123#3939C2×D312yes—
124C4⋊#168—32no—
125C4⋊#168C2×D311yescharacteristic
126C4×#168—31no—
127C4×#168—32no—
128#3358—151noSylow
129C22×D15—22no—
130C22×C5×D3—21no—
131C22×C3×D5—21no—
132C3×C5×Q8—42no—
133C2×C3×C4×C5—22no—
134C2×C3×C4×C5C2×D411yescharacteristic
135C2×C3×C4×C5C2×D411yescharacteristic
136C2×C3×C4×C5C2411yescharacteristic
137D60—22no—
138D60—22no—
139D60—42no—
140D60—81no—
141C2×C5×Dic3—21no—
142C2×C5×Dic3C2×D412yes—
143C2×C5×Dic3C2×D412yes—
144C5×D12—21no—
145C5×D12#4112yescomplemented
146C4×C5×D3—42no—
147C4×C5×D3—44no—
148C5×Dic6—44no—
149C5×Dic6D814yescomplemented
150C2×C3×Dic5—21no—
151C3×D20—21no—
152C3×D20—21no—
153C3×C4×D5—42no—
154#1202—44no—
155#1202—82no—
156#1200—44no—
157C3⋊#168—21no—
158C3⋊#168—21no—
159C3×#168—22no—
160C2×D3×Q8—21no—
161C2×#235—101no—
162C3×C4⋊Q8C2×D511yescharacteristic
163C3×C4×D4—51no—
164#946—21no—
165#940—52no—
166C2×#213—51no—
167#930—101no—
168C2×D24—52no—
169#896—22no—
170#893—52no—
171#893C2×D512yes—
172#888C2×D512yes—
173#884—102no—
174#874—51no—
175C4×D12C2×D511yescharacteristic
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—51no—
185#809—52no—
186#809—52no—
187#807—51no—
188C4×C3⋊C8—51no—
189C2×C5×Q8—21no—
190C2×C5×Q8—31no—
191C2×C5×Q8—62no—
192C2×C5×Q8—124no—
193C2×C5×D4—31no—
194C22×C4×C5—62no—
195C22×C4×C5—61no—
196#713—124no—
197C2×D20—32no—
198C2×D20—62no—
199C2×D20—61no—
200C2×D20—62no—
201C2×D20—61no—
202C2×D20C22×D311yescharacteristic
203C2×C4×D5—21no—
204C2×C4×D5—62no—
205C5×C4⋊C4—21no—
206C5×C4⋊C4—32no—
207C5×C4⋊C4—31no—
208C5×C4⋊C4—31no—
209C5×C4⋊C4—62no—
210C5×C4⋊C4C22×D312yes—
211C5×C22⋊C4—62no—
212C5×C22⋊C4—61no—
213C42×C5—32no—
214C42×C5C22×D311yescharacteristic
215#686—64no—
216#686—62no—
217#685—21no—
218#685—62no—
219C4⋊Dic5C22×D311yescharacteristic
220C2×#168—31no—
221C2×#168—31no—
222C2×#168C22×D312yes—
223#550—151no—
224D4×Q8—31no—
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×C5C22×D411yescharacteristic
235C2×D15—42no—
236C2×D15—42no—
237C2×D15—42no—
238C2×C5×D3—22no—
239C2×C5×D3—41no—
240C2×C5×D3—41no—
241C2×C3×D5—41no—
242C2×C3×D5—41no—
243C2×C3×D5—41no—
244C3×C4×C5—21no—
245C3×C4×C5—42no—
246C3×C4×C5C2×D812yes—
247C3×C4×C5C2×#4111yescharacteristic
248C3×C4×C52−1+411yescharacteristic
249C3×Dic5—41no—
250C5×Dic3—22no—
251C5×Dic3—41no—
252C5×Dic3C2×D814yes—
253C2×C3×Q8—21no—
254C2×C3×D4—51no—
255C22×C3×C4—101no—
256C2×#67—102no—
257C2×#67—102no—
258D3×Q8—44no—
259#235—204no—
260C2×D12—52no—
261C2×D12—101no—
262C2×D12—101no—
263C2×D12C22×D511yescharacteristic
264C2×C4×D3—21no—
265C2×C4×D3—22no—
266C2×C4×D3—102no—
267C2×Dic6—22no—
268C2×Dic6C22⋊D511yescharacteristic
269C2×C3×C8—52no—
270C3×C4⋊C4—21no—
271C3×C4⋊C4—51no—
272C3×C4⋊C4C22×D512yes—
273C3×C22⋊C4—101no—
274C3×C42C22×D511yescharacteristic
275#213—102no—
276(C2×D3)⋊C4—21no—
277(C2×D3)⋊C4—22no—
278(C2×D3)⋊C4—102no—
279C4⋊Dic3C22⋊D511yescharacteristic
280C4⋊Dic3C22×D511yescharacteristic
281Dic3⋊C4—22no—
282Dic3⋊C4C22⋊D512yes—
283C4×Dic3C22×D512yes—
284C2×C3⋊C8—51no—
285C2×C3⋊C8—51no—
286D24—104no—
287C23×C5—61no—
288C22×D5—21no—
289C22×D5—62no—
290C5×Q8—34no—
291C5×Q8—42no—
292C5×Q8—124no—
293C5×D4—32no—
294C5×D4—61no—
295C2×C4×C5—22no—
296C2×C4×C5—32no—
297C2×C4×C5—32no—
298C2×C4×C5—61no—
299C2×C4×C5—124no—
300C2×C4×C5—122no—
301C2×C4×C5D3×D411yescharacteristic
302C2×C4×C5D3×D411yescharacteristic
303C2×C4×C5C23×D311yescharacteristic
304C2×Dic5—21no—
305D20—21no—
306D20—21no—
307D20—62no—
308D20—62no—
309D20—124no—
310D20—122no—
311D20—241no—
312D20—242no—
313C4×D5—42no—
314C4×D5—124no—
315#168—22no—
316#168—61no—
317#168—61no—
318C2×#41—301no—
319C22×Q8—61no—
320C2×D8—152no—
321C2×D8—151no—
322C4⋊Q8—32no—
323C4⋊Q8D3×D511yescharacteristic
324#127—152no—
325#127—151no—
326C22⋊Q8—62no—
327C22⋊Q8—61no—
328#121—302no—
329#121—301no—
330C4×Q8—31no—
331C4×D4—32no—
332C4×D4—31no—
333C4×D4—51no—
334C4×D4—152no—
335#107—52no—
336#107—151no—
337C4⋊C8—51no—
338C4⋊C8—152no—
339#102—52no—
340#102—152no—
341#102—151no—
342#102—151no—
343#102—152no—
344#102—152no—
345C4×C8—152no—
346C4×C8—151no—
347C2×C3×C5#54811yescharacteristic
348C2×C3×C5C22×D811yescharacteristic
349C2×C3×C5#57711yescharacteristic
350D15—82no—
351C3×D5—81no—
352C5×D3—42no—
353C23×C3—101no—
354C22×D3—21no—
355C22×D3—102no—
356C3×Q8—42no—
357C3×D4—101no—
358C3×D4—101no—
359C2×C3×C4—22no—
360C2×C3×C4—101no—
361C2×C3×C4—202no—
362C2×C3×C4C2×C22⋊D511yescharacteristic
363C2×C3×C4C2×C22⋊D511yescharacteristic
364C2×C3×C4C23×D511yescharacteristic
365#67—204no—
366#67—402no—
367C2×Dic3—21no—
368C2×Dic3C2×C22⋊D512yes—
369C2×Dic3C2×C22⋊D512yes—
370D12—21no—
371D12—102no—
372D12—102no—
373D12—202no—
374D12—401no—
375D12#71312yescomplemented
376C4×D3—42no—
377C4×D3—44no—
378C4×D3—204no—
379Dic6—44no—
380Dic6#68614yescomplemented
381C3×C8—102no—
382C3⋊C8—101no—
383C3⋊C8—101no—
384C22×C5—62no—
385C22×C5—121no—
386C22×C5—121no—
387C22×C5C2×D3×D411yescharacteristic
388C2×D5—41no—
389C2×D5—41no—
390C2×D5—41no—
391C2×D5—122no—
392C2×D5—122no—
393C2×D5—122no—
394C4×C5—21no—
395C4×C5—34no—
396C4×C5—42no—
397C4×C5—62no—
398C4×C5—121no—
399C4×C5D3×D812yes—
400C4×C5#100711yescharacteristic
401C4×C5D3×#4111yescharacteristic
402Dic5—41no—
403#41—604no—
404C2×Q8—21no—
405C2×Q8—31no—
406C2×Q8—62no—
407C2×Q8—124no—
408C2×D4—31no—
409C2×D4—51no—
410C2×D4—152no—
411C2×D4—302no—
412C2×D4—302no—
413C2×D4—301no—
414C2×D4—301no—
415C22×C4—62no—
416C22×C4—61no—
417C22×C4—101no—
418C22×C4—302no—
419D8—304no—
420D8—302no—
421C2×C8—52no—
422C2×C8—151no—
423C2×C8—151no—
424C4⋊C4—21no—
425C4⋊C4—32no—
426C4⋊C4—31no—
427C4⋊C4—31no—
428C4⋊C4—51no—
429C4⋊C4—62no—
430C4⋊C4C2×D3×D512yes—
431C22⋊C4—62no—
432C22⋊C4—61no—
433C22⋊C4—101no—
434C22⋊C4—302no—
435C42—32no—
436C42C2×D3×D511yescharacteristic
437C3×C5#335811yescharacteristic, complemented, Hall
438C22×C3—201no—
439C22×C3—201no—
440C22×C3—201no—
441C22×C3C22×C22⋊D511yescharacteristic
442C2×D3—22no—
443C2×D3—41no—
444C2×D3—41no—
445C2×D3—202no—
446C2×D3—202no—
447C2×D3—202no—
448C3×C4—21no—
449C3×C4—42no—
450C3×C4—201no—
451C3×C4C2×#68612yes—
452C3×C4C2×#71311yescharacteristic
453C3×C4#414711yescharacteristic
454Dic3—22no—
455Dic3—41no—
456Dic3C2×#68614yes—
457C2×C5—122no—
458C2×C5#563311yescharacteristic
459C2×C5C2×D3×D811yescharacteristic
460C2×C5#577811yescharacteristic
461D5—81no—
462D5—242no—
463C23—61no—
464C23—101no—
465C23—302no—
466Q8—34no—
467Q8—42no—
468Q8—124no—
469D4—32no—
470D4—61no—
471D4—101no—
472D4—101no—
473D4—302no—
474D4—302no—
475D4—604no—
476D4—602no—
477D4—1201no—
478D4—1202no—
479C2×C4—22no—
480C2×C4—32no—
481C2×C4—32no—
482C2×C4—61no—
483C2×C4—101no—
484C2×C4—124no—
485C2×C4—122no—
486C2×C4—202no—
487C2×C4—604no—
488C2×C4D3×C22⋊D511yescharacteristic
489C2×C4D3×C22⋊D511yescharacteristic
490C2×C4C22×D3×D511yescharacteristic
491C8—102no—
492C8—301no—
493C8—301no—
494C2×C3—401no—
495C2×C3C22×#68611yescharacteristic
496C2×C3#6615111yescharacteristic
497C2×C3#6617111yescharacteristic
498D3—42no—
499D3—402no—
500C5#8099211yescharacteristic, complemented, Sylow
501C22—62no—
502C22—121no—
503C22—121no—
504C22—201no—
505C22—201no—
506C22—201no—
507C22—602no—
508C22—602no—
509C22—602no—
510C22C2×D3×C22⋊D511yescharacteristic
511C4—21no—
512C4—34no—
513C4—42no—
514C4—62no—
515C4—121no—
516C4—201no—
517C4D3×#68612yes—
518C4#9202911yescharacteristic
519C4D3×#71311yescharacteristic
520C3#1061529311yescharacteristic, complemented, Sylow
521C2—122no—
522C2—401no—
523C2—1202no—
524C2C2×D3×#68611yescharacteristic
525C2#1175509211yescharacteristic
526C2#1175523411yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊D5
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×C4⋊Q8C22
socleC22×C3×C5C22×D4
hypercenterC4⋊Q8D3×D5
hypocenterC3×C5#3358
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C5×#5727C2semidirect
2. #11752677C2semidirect
3. #11751189C2semidirect
4. C3×#65376C2semidirect
5. #11747403C2semidirect
6. #11747402C2semidirect
7. #11747246C2semidirect
8. #11747230C2semidirect
9. #11747037C2semidirect
10. C4×#6652C2semidirect
11. #11746857C2semidirect
48. #65376D3semidirect
88. #5727D5semidirect
145. C5×D12#41semidirect
149. C5×Dic6D8semidirect
375. D12#713semidirect
380. Dic6#686semidirect
437. C3×C5#3358semidirect
500. C5#80992semidirect
520. C3#10615293semidirect

Maximal subgroups

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

subgroupindex
C5×#57272
#117526772
#117511892
C3×#653762
#117474032
#117474022
#117472462
#117472302
#117470372
C4×#66522
#117468572
#106152933
#809925