Finite Group Databasebeta

← #14460119

Subgroups — #14460119

Order 1920.

796 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756485C211yesmaximal, characteristic, complemented
2#11756322C211yesmaximal, characteristic, complemented
3C3×C5×#461C211yesmaximal, characteristic, complemented
4#11751273C212yesmaximal, complemented
5#11751272C212yesmaximal, complemented
6#11751180C211yesmaximal, characteristic, complemented
7C4×#6715C211yesmaximal, characteristic, complemented
8#11750943C211yesmaximal, characteristic, complemented
9#11750935C211yesmaximal, characteristic, complemented
10#11750916C212yesmaximal, complemented
11#11750905C212yesmaximal, complemented
12#10612182—31nomaximal, Hall
13C2×#6821C2212yes—
14C2×C3×C5×SD16C2212yes—
15C3×C5×C4⋊Q8C2211yescharacteristic
16C3×C4×C5×D4C2211yescharacteristic
17#91835C2212yes—
18#91827C2212yes—
19C2×#6715C2211yescharacteristic
20#91819—21no—
21#91815—21no—
22C2×C8×D15C2212yes—
23#91784—21no—
24#91783—22no—
25#91781C2212yes—
26C4⋊C4×D15—21no—
27#91774—22no—
28C4×D60C2211yescharacteristic
29C4×D60C2211yescharacteristic
30#91758C2211yescharacteristic
31C3×C5×C4⋊C8C2211yescharacteristic
32C3×C5×#103C2212yes—
33#91119C2212yes—
34#91119C2212yes—
35#91116C2211yescharacteristic
36#91105C2212yes—
37#91104C2212yes—
38#91101C2212yes—
39#91093C2211yescharacteristic
40#91091C2211yescharacteristic
41C4×C3⋊#168C2211yescharacteristic
42#80451—51nomaximal, Hall
43#66060—31no—
44#65897—31no—
45C5×#461D311yescharacteristic, complemented
46#65461—32no—
47#65460—32no—
48#65370—31no—
49C4×#686—31no—
50#65131—31no—
51#65123—31no—
52#65104—32no—
53#65093—32no—
54C2×C3×C5×Q8C2312yes—
55C2×C3×C5×D4C2311yescharacteristic
56C22×C3×C4×C5—21no—
57C2×#1220—22no—
58#6821—24no—
59#6821—42no—
60C2×D60—22no—
61C2×D60D411yescharacteristic
62C2×D60C2311yescharacteristic
63C2×C4×D15—21no—
64C2×C4×D15—22no—
65C2×C4×D15C2312yes—
66C2×Dic30—21no—
67C3×C5×SD16—42no—
68C2×C3×C5×C8C2312yes—
69C3×C5×C4⋊C4—21no—
70C3×C5×C4⋊C4—21no—
71C3×C5×C4⋊C4C2311yescharacteristic
72C3×C5×C22⋊C4—21no—
73C3×C42×C5C2311yescharacteristic
74#6719—22no—
75#6715—22no—
76#6714—22no—
77#6714—22no—
78#6714—21no—
79#6714D412yes—
80C4⋊Dic15—21no—
81C4⋊Dic15D411yescharacteristic
82C4⋊Dic15C2312yes—
83C4⋊Dic15C2311yescharacteristic
84#6712—22no—
85C4×Dic15—22no—
86C2×C3⋊#168C2311yescharacteristic
87C2×C3⋊#168C2311yescharacteristic
88C8×D15—42no—
89#5726—51no—
90#5563—51no—
91C3×#461D511yescharacteristic, complemented
92#5172—52no—
93#5171—52no—
94#5079—51no—
95C4×#213—51no—
96#4842—51no—
97#4834—51no—
98#4815—52no—
99#4804—52no—
100C2×#713—32no—
101C2×C5×SD16C2×D312yes—
102C5×C4⋊Q8C2×D311yescharacteristic
103C4×C5×D4C2×D311yescharacteristic
104#4092—32no—
105#4084—32no—
106C2×#686—31no—
107#4076—61no—
108#4072—61no—
109C2×C8×D5—32no—
110#4041—61no—
111#4040—62no—
112#4038—32no—
113D5×C4⋊C4—61no—
114#4031—62no—
115C4×D20—31no—
116C4×D20—31no—
117#4015—31no—
118C5×C4⋊C8C2×D311yescharacteristic
119C5×#103C2×D312yes—
120#3967—32no—
121#3967—32no—
122#3964—31no—
123#3953—32no—
124#3952—32no—
125#3949—32no—
126#3941—31no—
127#3939—31no—
128C4×#168—31no—
129#3303—151noSylow
130C23×C3×C5—21no—
131C22×D15—21no—
132C22×D15C2×D412yes—
133C3×C5×Q8—22no—
134C3×C5×Q8—22no—
135C3×C5×D4—21no—
136C3×C5×D4—21no—
137C2×C3×C4×C5—21no—
138C2×C3×C4×C5—22no—
139C2×C3×C4×C5—42no—
140C2×C3×C4×C5C2×D411yescharacteristic
141C2×C3×C4×C5C2×D411yescharacteristic
142C2×C3×C4×C5C2411yescharacteristic
143#1220—82no—
144C2×Dic15—22no—
145C2×Dic15—21no—
146C2×Dic15C2×D412yes—
147D60—22no—
148D60—21no—
149D60—44no—
150D60#4112yes—
151C4×D15—22no—
152C4×D15—22no—
153C4×D15—44no—
154C4×D15—42no—
155Dic30—81no—
156C3×C5×C8—22no—
157C3⋊#168—21no—
158C3⋊#168—21no—
159C2×#239—52no—
160C2×C3×SD16C2×D512yes—
161C3×C4⋊Q8C2×D511yescharacteristic
162C3×C4×D4C2×D511yescharacteristic
163#946—52no—
164#938—52no—
165C2×#213—51no—
166#930—101no—
167#926—101no—
168C2×D3×C8—52no—
169#895—101no—
170#894—102no—
171#892—52no—
172D3×C4⋊C4—101no—
173#885—102no—
174C4×D12—51no—
175C4×D12—51no—
176#869—51no—
177C3×C4⋊C8C2×D511yescharacteristic
178C3×#103C2×D512yes—
179#835—52no—
180#835—52no—
181#832—51no—
182#821—52no—
183(C2×D3)⋊C8—52no—
184#817—52no—
185#809—51no—
186#807—51no—
187C4×C3⋊C8—51no—
188C2×C5×Q8C22×D312yes—
189C2×C5×D4C22×D311yescharacteristic
190C22×C4×C5—21no—
191C2×C22⋊D5—62no—
192#713—64no—
193#713—122no—
194C2×D20—31no—
195C2×D20—31no—
196C2×D20—62no—
197C2×C4×D5—32no—
198C2×C4×D5—62no—
199C2×C4×D5—61no—
200C2×Dic10—61no—
201C5×SD16—42no—
202C2×C5×C8C22×D312yes—
203C5×C4⋊C4—21no—
204C5×C4⋊C4—21no—
205C5×C4⋊C4C22×D311yescharacteristic
206C5×C22⋊C4—21no—
207C42×C5C22×D311yescharacteristic
208C22⋊Dic5—62no—
209#686—62no—
210#685—32no—
211#685—62no—
212#685—62no—
213#685—61no—
214C4⋊Dic5—31no—
215C4⋊Dic5—31no—
216C4⋊Dic5—32no—
217C4⋊Dic5—61no—
218#683—62no—
219C4×Dic5—62no—
220C2×#168—31no—
221C2×#168—31no—
222C8×D5—122no—
223#555—151no—
224#547—151no—
225#487—151no—
226#485—152no—
227#476—151no—
228#465—152no—
229#461D1511yescharacteristic, complemented
230#449—152no—
231C4×D8—151no—
232C8×D4—151no—
233#416—152no—
234C22×C3×C5—41no—
235C22×C3×C5—41no—
236C22×C3×C5—41no—
237C22×C3×C5C22×D411yescharacteristic
238C2×D15—22no—
239C2×D15—22no—
240C2×D15—22no—
241C2×D15—22no—
242C2×D15—41no—
243C2×D15—41no—
244C3×C4×C5—21no—
245C3×C4×C5—42no—
246C3×C4×C5—41no—
247C3×C4×C5#13512yes—
248C3×C4×C5C2×#4111yescharacteristic
249C3×C4×C5H(5,𝔽2)11yescharacteristic
250Dic15—22no—
251Dic15—41no—
252Dic15—42no—
253C2×C3×Q8C22×D512yes—
254C2×C3×D4C22×D511yescharacteristic
255C22×C3×C4—21no—
256C2×#67—102no—
257#239—104no—
258#239—202no—
259C2×D12—51no—
260C2×D12—51no—
261C2×D12—102no—
262C2×C4×D3—52no—
263C2×C4×D3—102no—
264C2×C4×D3—101no—
265C2×Dic6—101no—
266C3×SD16—42no—
267C2×C3×C8C22×D512yes—
268C3×C4⋊C4—21no—
269C3×C4⋊C4—21no—
270C3×C4⋊C4C22×D511yescharacteristic
271C3×C22⋊C4—21no—
272C3×C42C22×D511yescharacteristic
273#217—102no—
274#213—102no—
275(C2×D3)⋊C4—52no—
276(C2×D3)⋊C4—102no—
277(C2×D3)⋊C4—102no—
278(C2×D3)⋊C4—101no—
279C4⋊Dic3—51no—
280C4⋊Dic3—51no—
281C4⋊Dic3—52no—
282C4⋊Dic3—101no—
283Dic3⋊C4—102no—
284C4×Dic3—102no—
285C2×C3⋊C8—51no—
286C2×C3⋊C8—51no—
287D3×C8—202no—
288C23×C5—21no—
289C22×D5—32no—
290C22×D5—61no—
291C5×Q8—22no—
292C5×Q8—22no—
293C5×D4—21no—
294C5×D4—21no—
295C2×C4×C5—21no—
296C2×C4×C5—22no—
297C2×C4×C5—42no—
298C2×C4×C5D3×D411yescharacteristic
299C2×C4×C5D3×D411yescharacteristic
300C2×C4×C5C23×D311yescharacteristic
301C22⋊D5—242no—
302C2×Dic5—32no—
303C2×Dic5—62no—
304C2×Dic5—61no—
305D20—32no—
306D20—62no—
307D20—61no—
308D20—124no—
309C4×D5—62no—
310C4×D5—62no—
311C4×D5—124no—
312C4×D5—122no—
313Dic10—241no—
314C5×C8—22no—
315#168—61no—
316#168—61no—
317C2×#41—152no—
318C2×SD16C2×D1512yes—
319C2×D8—151no—
320C22×C8—152no—
321C4⋊Q8—151no—
322C4⋊Q8C2×D1511yescharacteristic
323#124—302no—
324#123—302no—
325C22⋊Q8—152no—
326#121—152no—
327#121—301no—
328#121—301no—
329C4×D4—151no—
330C4×D4—151no—
331C4×D4C2×D1511yescharacteristic
332#117—152no—
333#117—301no—
334C2×C4⋊C4—301no—
335#107—151no—
336#106—152no—
337C4⋊C8C2×D1511yescharacteristic
338#103—152no—
339#103—152no—
340#103C2×D1512yes—
341#102—151no—
342#102—152no—
343#102—151no—
344C22⋊C8—152no—
345C4×C8—151no—
346C2×C3×C5—81no—
347C2×C3×C5#54611yescharacteristic
348C2×C3×C5C2×#13511yescharacteristic
349C2×C3×C5#57711yescharacteristic
350D15—42no—
351D15—42no—
352C23×C3—21no—
353C22×D3—52no—
354C22×D3—101no—
355C3×Q8—22no—
356C3×Q8—22no—
357C3×D4—21no—
358C3×D4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—22no—
361C2×C3×C4—42no—
362C2×C3×C4D4×D511yescharacteristic
363C2×C3×C4D4×D511yescharacteristic
364C2×C3×C4C23×D511yescharacteristic
365#67—402no—
366C2×Dic3—52no—
367C2×Dic3—102no—
368C2×Dic3—101no—
369D12—52no—
370D12—101no—
371D12—102no—
372D12—204no—
373C4×D3—102no—
374C4×D3—102no—
375C4×D3—204no—
376C4×D3—202no—
377Dic6—401no—
378C3×C8—22no—
379C3⋊C8—101no—
380C3⋊C8—101no—
381C22×C5—41no—
382C22×C5—41no—
383C22×C5—41no—
384C22×C5C2×D3×D411yescharacteristic
385C2×D5—62no—
386C2×D5—62no—
387C2×D5—62no—
388C2×D5—62no—
389C2×D5—121no—
390C2×D5—121no—
391C4×C5—21no—
392C4×C5—41no—
393C4×C5—42no—
394C4×C5#91612yes—
395C4×C5#100411yescharacteristic
396C4×C5D3×#4111yescharacteristic
397Dic5—62no—
398Dic5—121no—
399Dic5—122no—
400#41—304no—
401#41—602no—
402C2×Q8—301no—
403C2×Q8C22×D1512yes—
404C2×D4—151no—
405C2×D4—151no—
406C2×D4—302no—
407C2×D4—302no—
408C2×D4C22×D1511yescharacteristic
409C22×C4—21no—
410C22×C4—152no—
411C22×C4—302no—
412C22×C4—301no—
413SD16—42no—
414D8—302no—
415C2×C8—151no—
416C2×C8—151no—
417C2×C8—602no—
418C2×C8C22×D1512yes—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—151no—
422C4⋊C4—152no—
423C4⋊C4—151no—
424C4⋊C4—302no—
425C4⋊C4—301no—
426C4⋊C4C22×D1511yescharacteristic
427C22⋊C4—21no—
428C22⋊C4—152no—
429C22⋊C4—302no—
430C22⋊C4—302no—
431C22⋊C4—302no—
432C22⋊C4—301no—
433C42—302no—
434C42C22×D1511yescharacteristic
435C3×C5#330311yescharacteristic, complemented, Hall
436C22×C3—41no—
437C22×C3—41no—
438C22×C3—41no—
439C22×C3C2×D4×D511yescharacteristic
440C2×D3—102no—
441C2×D3—102no—
442C2×D3—102no—
443C2×D3—102no—
444C2×D3—201no—
445C2×D3—201no—
446C3×C4—21no—
447C3×C4—41no—
448C3×C4—42no—
449C3×C4#406212yes—
450C3×C4#414411yescharacteristic
451C3×C4D5×#4111yescharacteristic
452Dic3—102no—
453Dic3—201no—
454Dic3—202no—
455C2×C5—81no—
456C2×C5#561211yescharacteristic
457C2×C5C2×#91611yescharacteristic
458C2×C5#577811yescharacteristic
459D5—122no—
460D5—122no—
461C23—21no—
462C23—152no—
463C23—301no—
464Q8—22no—
465Q8—22no—
466Q8—1201no—
467D4—21no—
468D4—21no—
469D4—152no—
470D4—302no—
471D4—301no—
472D4—604no—
473D4—1202no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—42no—
477C2×C4—152no—
478C2×C4—302no—
479C2×C4—302no—
480C2×C4—302no—
481C2×C4—301no—
482C2×C4—604no—
483C2×C4—602no—
484C2×C4D4×D1511yescharacteristic
485C2×C4D4×D1511yescharacteristic
486C2×C4C23×D1511yescharacteristic
487C8—22no—
488C8—301no—
489C8—301no—
490C2×C3—81no—
491C2×C3#6594611yescharacteristic
492C2×C3C2×#406211yescharacteristic
493C2×C3#6611211yescharacteristic
494D3—202no—
495D3—202no—
496C5#8045111yescharacteristic, complemented, Sylow
497C22—41no—
498C22—41no—
499C22—41no—
500C22—302no—
501C22—302no—
502C22—302no—
503C22—302no—
504C22—601no—
505C22—601no—
506C22C2×D4×D1511yescharacteristic
507C4—21no—
508C4—42no—
509C4—41no—
510C4—302no—
511C4—601no—
512C4—602no—
513C4#9180512yes—
514C4#9209511yescharacteristic
515C4#41×D1511yescharacteristic
516C3#1061218211yescharacteristic, complemented, Sylow
517C2—81no—
518C2—602no—
519C2—602no—
520C2#1175637111yescharacteristic
521C2C2×#9180511yescharacteristic
522C2#1175653711yescharacteristic
subgroupG/H
centerC22C2×D4×D15
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C5×#461C2
socleC22×C3×C5C22×D4
hypercenter#461D15
hypocenterC3×C5#3303
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #11756485C2semidirect
2. #11756322C2semidirect
3. C3×C5×#461C2semidirect
4. #11751273C2semidirect
5. #11751272C2semidirect
6. #11751180C2semidirect
7. C4×#6715C2semidirect
8. #11750943C2semidirect
9. #11750935C2semidirect
10. #11750916C2semidirect
11. #11750905C2semidirect
45. C5×#461D3semidirect
91. C3×#461D5semidirect
229. #461D15semidirect
435. C3×C5#3303semidirect
496. C5#80451semidirect
516. C3#10612182semidirect

Maximal subgroups

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

subgroupindex
#117564852
#117563222
C3×C5×#4612
#117512732
#117512722
#117511802
C4×#67152
#117509432
#117509352
#117509162
#117509052
#106121823
#804515