Finite Group Databasebeta

← #14459960

Subgroups — #14459960

Order 1920.

772 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756485C211yesmaximal, characteristic
2#11756343C211yesmaximal, characteristic, complemented
3C3×C5×#462C211yesmaximal, characteristic, complemented
4#11751289C212yesmaximal, complemented
5#11751288C212yesmaximal, complemented
6#11751180C211yesmaximal, characteristic, complemented
7C4×#6717C211yesmaximal, characteristic, complemented
8#11750947C211yesmaximal, characteristic, complemented
9#11750935C211yesmaximal, characteristic, complemented
10#11750915C212yesmaximal, complemented
11#11750905C212yesmaximal, complemented
12#10612223—31nomaximal, Hall
13C2×#6821C2212yes—
14C2×C3×C5×Q16C2212yes—
15C3×C5×C4⋊Q8C2211yescharacteristic
16C3×C4×C5×Q8C2211yescharacteristic
17#91835C2212yes—
18#91835C2212yes—
19C2×#6717C2211yescharacteristic
20C2×C8×D15C2212yes—
21#91787—22no—
22#91784—21no—
23#91783—22no—
24#91781C2212yes—
25C4⋊C4×D15—21no—
26#91764—21no—
27C4×D60C2211yescharacteristic
28C4×D60C2211yescharacteristic
29#91758C2211yescharacteristic
30C4×Dic30—21no—
31C3×C5×C4⋊C8C2211yescharacteristic
32C3×C5×#103C2212yes—
33#91119C2212yes—
34#91119C2212yes—
35#91119C2211yescharacteristic
36#91105C2212yes—
37#91104C2212yes—
38#91102C2212yes—
39#91093C2211yescharacteristic
40#91092C2211yescharacteristic
41C4×C3⋊#168C2211yescharacteristic
42#80492—51nomaximal, Hall
43#66060—31no—
44#65918—31no—
45C5×#462D311yescharacteristic, complemented
46#65477—32no—
47#65476—32no—
48#65370—31no—
49C4×#688—31no—
50#65135—31no—
51#65123—31no—
52#65103—32no—
53#65093—32no—
54C2×C3×C5×Q8C2311yescharacteristic
55C2×C3×C5×Q8C2312yes—
56#6821—24no—
57#6821—42no—
58C2×D60—22no—
59C2×D60D411yescharacteristic
60C2×D60C2311yescharacteristic
61C2×C4×D15—21no—
62C2×C4×D15—22no—
63C2×C4×D15C2312yes—
64C2×Dic30—21no—
65C3×C5×Q16—42no—
66C2×C3×C5×C8C2312yes—
67C3×C5×C4⋊C4—21no—
68C3×C5×C4⋊C4—21no—
69C3×C5×C4⋊C4—21no—
70C3×C5×C4⋊C4C2311yescharacteristic
71C3×C42×C5—21no—
72C3×C42×C5C2311yescharacteristic
73#6717—22no—
74#6714—22no—
75#6714—22no—
76#6714—22no—
77#6714—21no—
78#6714D412yes—
79C4⋊Dic15—21no—
80C4⋊Dic15D411yescharacteristic
81C4⋊Dic15C2312yes—
82C4⋊Dic15C2311yescharacteristic
83#6712—22no—
84#6712—22no—
85C4×Dic15—22no—
86C2×C3⋊#168C2311yescharacteristic
87C2×C3⋊#168C2311yescharacteristic
88C8×D15—42no—
89#5726—51no—
90#5584—51no—
91C3×#462D511yescharacteristic, complemented
92#5188—52no—
93#5187—52no—
94#5079—51no—
95C4×#215—51no—
96#4846—51no—
97#4834—51no—
98#4814—52no—
99#4804—52no—
100C2×#713—32no—
101C2×C5×Q16C2×D312yes—
102C5×C4⋊Q8C2×D311yescharacteristic
103C4×C5×Q8C2×D311yescharacteristic
104#4092—32no—
105#4092—32no—
106C2×#688—31no—
107C2×C8×D5—32no—
108#4044—62no—
109#4041—61no—
110#4040—62no—
111#4038—32no—
112D5×C4⋊C4—61no—
113#4021—61no—
114C4×D20—31no—
115C4×D20—31no—
116#4015—31no—
117C4×Dic10—61no—
118C5×C4⋊C8C2×D311yescharacteristic
119C5×#103C2×D312yes—
120#3967—31no—
121#3967—32no—
122#3967—32no—
123#3953—32no—
124#3952—32no—
125#3950—32no—
126#3941—31no—
127#3940—31no—
128C4×#168—31no—
129#3302—151noSylow
130C22×D15—21no—
131C22×D15C2×D412yes—
132C3×C5×Q8—21no—
133C3×C5×Q8—22no—
134C3×C5×Q8—21no—
135C3×C5×Q8—22no—
136C2×C3×C4×C5—21no—
137C2×C3×C4×C5—21no—
138C2×C3×C4×C5—22no—
139C2×C3×C4×C5C2×D411yescharacteristic
140C2×C3×C4×C5C2×D411yescharacteristic
141C2×C3×C4×C5C2411yescharacteristic
142C2×Dic15—22no—
143C2×Dic15—21no—
144C2×Dic15C2×D412yes—
145D60—22no—
146D60—21no—
147D60—44no—
148D60#4112yes—
149C4×D15—22no—
150C4×D15—22no—
151C4×D15—44no—
152C4×D15—42no—
153Dic30—42no—
154C3×C5×C8—22no—
155C3⋊#168—21no—
156C3⋊#168—21no—
157C2×#239—52no—
158C2×C3×Q16C2×D512yes—
159C3×C4⋊Q8C2×D511yescharacteristic
160C3×C4×Q8C2×D511yescharacteristic
161#946—52no—
162#946—52no—
163C2×#215—51no—
164C2×D3×C8—52no—
165#898—102no—
166#895—101no—
167#894—102no—
168#892—52no—
169D3×C4⋊C4—101no—
170#875—101no—
171C4×D12—51no—
172C4×D12—51no—
173#869—51no—
174C4×Dic6—101no—
175C3×C4⋊C8C2×D511yescharacteristic
176C3×#103C2×D512yes—
177#835—52no—
178#835—51no—
179#835—52no—
180#821—52no—
181(C2×D3)⋊C8—52no—
182#818—52no—
183#809—51no—
184#808—51no—
185C4×C3⋊C8—51no—
186C2×C5×Q8C22×D311yescharacteristic
187C2×C5×Q8C22×D312yes—
188#713—64no—
189#713—122no—
190C2×D20—31no—
191C2×D20—31no—
192C2×D20—62no—
193C2×C4×D5—32no—
194C2×C4×D5—62no—
195C2×C4×D5—61no—
196C2×Dic10—61no—
197C5×Q16—42no—
198C2×C5×C8C22×D312yes—
199C5×C4⋊C4—21no—
200C5×C4⋊C4—21no—
201C5×C4⋊C4—21no—
202C5×C4⋊C4C22×D311yescharacteristic
203C42×C5—21no—
204C42×C5C22×D311yescharacteristic
205#688—62no—
206#685—32no—
207#685—62no—
208#685—62no—
209#685—62no—
210#685—61no—
211C4⋊Dic5—31no—
212C4⋊Dic5—31no—
213C4⋊Dic5—32no—
214C4⋊Dic5—61no—
215#683—62no—
216#683—62no—
217C4×Dic5—62no—
218C2×#168—31no—
219C2×#168—31no—
220C8×D5—122no—
221#556—151no—
222#547—151no—
223#487—151no—
224#485—152no—
225#474—151no—
226#467—152no—
227#462D1511yescharacteristic, complemented
228#452—152no—
229C4×SD16—151no—
230C8×D4—151no—
231#416—152no—
232C22×C3×C5C22×D411yescharacteristic
233C2×D15—22no—
234C2×D15—22no—
235C2×D15—22no—
236C2×D15—22no—
237C2×D15—41no—
238C2×D15—41no—
239C3×C4×C5—21no—
240C3×C4×C5—42no—
241C3×C4×C5—41no—
242C3×C4×C5—41no—
243C3×C4×C5#13512yes—
244C3×C4×C5C2×#4111yescharacteristic
245C3×C4×C5H(5,𝔽2)11yescharacteristic
246Dic15—22no—
247Dic15—42no—
248Dic15—41no—
249C2×C3×Q8C22×D512yes—
250C2×C3×Q8C22×D511yescharacteristic
251#239—104no—
252#239—202no—
253C2×D12—51no—
254C2×D12—51no—
255C2×D12—102no—
256C2×C4×D3—52no—
257C2×C4×D3—102no—
258C2×C4×D3—101no—
259C2×Dic6—101no—
260C3×Q16—42no—
261C2×C3×C8C22×D512yes—
262C3×C4⋊C4—21no—
263C3×C4⋊C4—21no—
264C3×C4⋊C4—21no—
265C3×C4⋊C4C22×D511yescharacteristic
266C3×C42—21no—
267C3×C42C22×D511yescharacteristic
268#215—102no—
269(C2×D3)⋊C4—52no—
270(C2×D3)⋊C4—102no—
271(C2×D3)⋊C4—102no—
272(C2×D3)⋊C4—102no—
273(C2×D3)⋊C4—101no—
274C4⋊Dic3—51no—
275C4⋊Dic3—51no—
276C4⋊Dic3—52no—
277C4⋊Dic3—101no—
278Dic3⋊C4—102no—
279Dic3⋊C4—102no—
280C4×Dic3—102no—
281C2×C3⋊C8—51no—
282C2×C3⋊C8—51no—
283D3×C8—202no—
284C22×D5—32no—
285C22×D5—61no—
286C5×Q8—21no—
287C5×Q8—22no—
288C5×Q8—21no—
289C5×Q8—22no—
290C2×C4×C5—21no—
291C2×C4×C5—21no—
292C2×C4×C5—22no—
293C2×C4×C5D3×D411yescharacteristic
294C2×C4×C5D3×D411yescharacteristic
295C2×C4×C5C23×D311yescharacteristic
296C2×Dic5—32no—
297C2×Dic5—62no—
298C2×Dic5—61no—
299D20—32no—
300D20—62no—
301D20—61no—
302D20—124no—
303C4×D5—62no—
304C4×D5—62no—
305C4×D5—124no—
306C4×D5—122no—
307Dic10—122no—
308C5×C8—22no—
309#168—61no—
310#168—61no—
311C2×#41—152no—
312C2×Q16C2×D1512yes—
313C2×SD16—151no—
314C22×C8—152no—
315C4⋊Q8—151no—
316C4⋊Q8C2×D1511yescharacteristic
317#126—302no—
318#124—301no—
319#123—302no—
320C22⋊Q8—152no—
321C22⋊Q8—152no—
322#121—301no—
323C4×Q8—301no—
324C4×Q8C2×D1511yescharacteristic
325C4×D4—151no—
326C4×D4—151no—
327#117—152no—
328C2×C4⋊C4—301no—
329#107—152no—
330#106—151no—
331C4⋊C8C2×D1511yescharacteristic
332#103—152no—
333#103—151no—
334#103—152no—
335#103C2×D1512yes—
336#102—152no—
337#102—151no—
338C22⋊C8—152no—
339C4×C8—151no—
340C2×C3×C5#54611yescharacteristic
341C2×C3×C5C2×#13511yescharacteristic
342C2×C3×C5#57811yescharacteristic
343D15—42no—
344D15—42no—
345C22×D3—52no—
346C22×D3—101no—
347C3×Q8—21no—
348C3×Q8—22no—
349C3×Q8—21no—
350C3×Q8—22no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—22no—
354C2×C3×C4D4×D511yescharacteristic
355C2×C3×C4D4×D511yescharacteristic
356C2×C3×C4C23×D511yescharacteristic
357C2×Dic3—52no—
358C2×Dic3—102no—
359C2×Dic3—101no—
360D12—52no—
361D12—101no—
362D12—102no—
363D12—204no—
364C4×D3—102no—
365C4×D3—102no—
366C4×D3—204no—
367C4×D3—202no—
368Dic6—202no—
369C3×C8—22no—
370C3⋊C8—101no—
371C3⋊C8—101no—
372C22×C5C2×D3×D411yescharacteristic
373C2×D5—62no—
374C2×D5—62no—
375C2×D5—62no—
376C2×D5—62no—
377C2×D5—121no—
378C2×D5—121no—
379C4×C5—21no—
380C4×C5—41no—
381C4×C5—42no—
382C4×C5—41no—
383C4×C5#91912yes—
384C4×C5#100411yescharacteristic
385C4×C5D3×#4111yescharacteristic
386Dic5—62no—
387Dic5—122no—
388Dic5—121no—
389#41—304no—
390#41—602no—
391C2×Q8—301no—
392C2×Q8C22×D1512yes—
393C2×Q8C22×D1511yescharacteristic
394C2×D4—151no—
395C2×D4—151no—
396C2×D4—302no—
397C22×C4—152no—
398C22×C4—302no—
399C22×C4—301no—
400Q16—42no—
401SD16—302no—
402C2×C8—151no—
403C2×C8—151no—
404C2×C8—602no—
405C2×C8C22×D1512yes—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—151no—
410C4⋊C4—152no—
411C4⋊C4—151no—
412C4⋊C4—302no—
413C4⋊C4—302no—
414C4⋊C4—301no—
415C4⋊C4C22×D1511yescharacteristic
416C22⋊C4—152no—
417C22⋊C4—302no—
418C22⋊C4—302no—
419C22⋊C4—302no—
420C22⋊C4—301no—
421C42—21no—
422C42—302no—
423C42C22×D1511yescharacteristic
424C3×C5#330211yescharacteristic, complemented, Hall
425C22×C3C2×D4×D511yescharacteristic
426C2×D3—102no—
427C2×D3—102no—
428C2×D3—102no—
429C2×D3—102no—
430C2×D3—201no—
431C2×D3—201no—
432C3×C4—21no—
433C3×C4—41no—
434C3×C4—42no—
435C3×C4—41no—
436C3×C4#406512yes—
437C3×C4#414411yescharacteristic
438C3×C4D5×#4111yescharacteristic
439Dic3—102no—
440Dic3—202no—
441Dic3—201no—
442C2×C5#561211yescharacteristic
443C2×C5C2×#91911yescharacteristic
444C2×C5#577911yescharacteristic
445D5—122no—
446D5—122no—
447C23—152no—
448C23—301no—
449Q8—21no—
450Q8—22no—
451Q8—21no—
452Q8—22no—
453Q8—602no—
454D4—152no—
455D4—302no—
456D4—301no—
457D4—604no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—152no—
462C2×C4—302no—
463C2×C4—302no—
464C2×C4—302no—
465C2×C4—301no—
466C2×C4—604no—
467C2×C4—602no—
468C2×C4D4×D1511yescharacteristic
469C2×C4D4×D1511yescharacteristic
470C2×C4C23×D1511yescharacteristic
471C8—22no—
472C8—301no—
473C8—301no—
474C2×C3#6594611yescharacteristic
475C2×C3C2×#406511yescharacteristic
476C2×C3#6611311yescharacteristic
477D3—202no—
478D3—202no—
479C5#8049211yescharacteristic, complemented, Sylow
480C22—302no—
481C22—302no—
482C22—302no—
483C22—302no—
484C22—601no—
485C22—601no—
486C22C2×D4×D1511yescharacteristic
487C4—21no—
488C4—42no—
489C4—41no—
490C4—41no—
491C4—302no—
492C4—601no—
493C4—602no—
494C4#9180812yes—
495C4#9209511yescharacteristic
496C4#41×D1511yescharacteristic
497C3#1061222311yescharacteristic, complemented, Sylow
498C2—602no—
499C2—602no—
500C2#1175637111yescharacteristic
501C2C2×#9180811yescharacteristic
502C2#1175653811yescharacteristic
subgroupG/H
centerC22C2×D4×D15
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C5×#462C2
socleC22×C3×C5C22×D4
prime radical#11756485C2
hypercenter#462D15
hypocenterC3×C5#3302
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
2. #11756343C2semidirect
3. C3×C5×#462C2semidirect
4. #11751289C2semidirect
5. #11751288C2semidirect
6. #11751180C2semidirect
7. C4×#6717C2semidirect
8. #11750947C2semidirect
9. #11750935C2semidirect
10. #11750915C2semidirect
11. #11750905C2semidirect
45. C5×#462D3semidirect
91. C3×#462D5semidirect
227. #462D15semidirect
424. C3×C5#3302semidirect
479. C5#80492semidirect
497. C3#10612223semidirect

Maximal subgroups

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

subgroupindex
#117564852
#117563432
C3×C5×#4622
#117512892
#117512882
#117511802
C4×#67172
#117509472
#117509352
#117509152
#117509052
#106122233
#804925