Finite Group Databasebeta

← #14460052

Subgroups — #14460052

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#11752549C211yesmaximal, characteristic
2#11752486C211yesmaximal, characteristic, complemented
3#11750935C211yesmaximal, characteristic, complemented
4C5×#4846C211yesmaximal, characteristic, complemented
5C3×#65138C211yesmaximal, characteristic, complemented
6#11748151C212yesmaximal, complemented
7#11748145C212yesmaximal, complemented
8#11747958C212yesmaximal, complemented
9#11747692C212yesmaximal, complemented
10C4×#6656C211yesmaximal, characteristic, complemented
11#11746924C211yesmaximal, characteristic, complemented
12#10612712—31nomaximal, Hall
13C2×#6768C2212yes—
14C2×C8×D15C2212yes—
15C4×D60C2211yescharacteristic
16C4×C5×D12C2211yescharacteristic
17C5×#869C2211yescharacteristic
18C2×C3×Dic20C2212yes—
19C3×#4015C2211yescharacteristic
20C3×C4×Dic10C2211yescharacteristic
21#91460—21no—
22#91427—22no—
23D3×C4⋊Dic5—21no—
24#91390C2212yes—
25#91390C2212yes—
26#91375—21no—
27#91347C2212yes—
28#91337—22no—
29Dic5×Dic6—21no—
30C2×#6656C2211yescharacteristic
31C3×C5×C4⋊C8C2211yescharacteristic
32#91104C2212yes—
33C4×C3⋊#168C2211yescharacteristic
34C5×#821C2212yes—
35C5×#818C2212yes—
36C3×#3948C2212yes—
37#90986C2211yescharacteristic
38#90970C2212yes—
39#90970C2212yes—
40#90970C2211yescharacteristic
41#90966C2211yescharacteristic
42#80492—51nomaximal, Hall
43#66066—31no—
44#65889—31no—
45C5×#474—31no—
46#65382—31no—
47C4×#687—31no—
48#65177—32no—
49#65176—32no—
50#65138D311yescharacteristic, complemented
51#65123—31no—
52#65072—32no—
53#65061—32no—
54C2×D60D411yescharacteristic
55C2×C4×D15C2312yes—
56C2×C5×D12C2311yescharacteristic
57C2×C4×C5×D3—21no—
58C2×C5×Dic6—21no—
59C2×C3×Dic10C2312yes—
60C2×C3×Dic10C2311yescharacteristic
61C2×#1203—22no—
62C2×D3×Dic5—22no—
63#6768—24no—
64#6768—42no—
65C2×C3×C5×C8C2312yes—
66C3×C42×C5C2311yescharacteristic
67#6714D412yes—
68C4⋊Dic15D411yescharacteristic
69C2×C3⋊#168C2311yescharacteristic
70C2×C3⋊#168C2311yescharacteristic
71C8×D15—42no—
72C5×(C2×D3)⋊C4—21no—
73C5×C4⋊Dic3—21no—
74C5×C4⋊Dic3C2312yes—
75C5×C4⋊Dic3C2311yescharacteristic
76C3×C4⋊Dic5—21no—
77C3×C4⋊Dic5—21no—
78C3×C4⋊Dic5C2311yescharacteristic
79C3×#683—21no—
80C3×C4×Dic5—21no—
81C3×Dic20—42no—
82Dic3⋊Dic5—22no—
83Dic3⋊Dic5—22no—
84#6667—22no—
85#6667—22no—
86#6667—22no—
87Dic3×Dic5—22no—
88#6656—22no—
89#5726—51no—
90#5584—51no—
91C3×#462—51no—
92#5188—52no—
93#5187—52no—
94#5079—51no—
95C4×#215—51no—
96#4846D511yescharacteristic, complemented
97#4834—51no—
98#4814—52no—
99#4804—52no—
100C2×#711—32no—
101C5×C4⋊Q8—31no—
102C4×C5×D4—31no—
103#4093—61no—
104Q8×Dic5—61no—
105C2×#687—31no—
106#4076—61no—
107C2×C4⋊Dic5—61no—
108C2×Dic20C2×D312yes—
109C2×C8×D5—32no—
110#4044—62no—
111#4043—32no—
112#4043—32no—
113#4038—32no—
114#4032—62no—
115C4×D20—31no—
116#4015C2×D311yescharacteristic
117C4×Dic10C2×D311yescharacteristic
118C5×#107—32no—
119C5×C4⋊C8C2×D311yescharacteristic
120C5×#102—32no—
121#3964—31no—
122#3952—32no—
123#3948C2×D312yes—
124#3942—31no—
125#3942—32no—
126#3942—32no—
127#3940—31no—
128C4×#168—31no—
129#3302—151noSylow
130C22×D15C2×D412yes—
131C22×C5×D3—21no—
132C2×C3×C4×C5C2×D411yescharacteristic
133C2×C3×C4×C5C2×D411yescharacteristic
134C2×C3×C4×C5C2411yescharacteristic
135C2×Dic15C2×D412yes—
136D60—22no—
137C4×D15—22no—
138C4×D15—22no—
139C2×C5×Dic3—22no—
140C2×C5×Dic3—21no—
141C5×D12—21no—
142C5×D12#4112yes—
143C4×C5×D3—42no—
144C5×Dic6—42no—
145C2×C3×Dic5—21no—
146C2×C3×Dic5—21no—
147C2×C3×Dic5—22no—
148C3×Dic10—21no—
149C3×Dic10—22no—
150C3×Dic10—21no—
151C3×Dic10—22no—
152#1203—44no—
153D3×Dic5—44no—
154C3×C5×C8—22no—
155C3⋊#168—21no—
156C3⋊#168—21no—
157C2×#239—52no—
158C2×C3×Q16—52no—
159C3×C4⋊Q8—51no—
160C3×C4×Q8—51no—
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×D12C2×D511yescharacteristic
173#869C2×D511yescharacteristic
174C4×Dic6—101no—
175C3×C4⋊C8C2×D511yescharacteristic
176C3×#103—52no—
177#835—52no—
178#835—51no—
179#835—52no—
180#821C2×D512yes—
181(C2×D3)⋊C8—52no—
182#818C2×D512yes—
183#809—51no—
184#808—51no—
185C4×C3⋊C8—51no—
186C2×C5×Q8—61no—
187C2×C5×D4—31no—
188C22×C4×C5—61no—
189C2×C22⋊D5—62no—
190C22×Dic5—62no—
191#711—64no—
192#711—122no—
193C2×D20—31no—
194C2×C4×D5—32no—
195C2×Dic10C22×D312yes—
196C2×Dic10C22×D311yescharacteristic
197C2×C5×C8C22×D312yes—
198C5×C4⋊C4—31no—
199C5×C4⋊C4—32no—
200C5×C4⋊C4—61no—
201C5×C22⋊C4—61no—
202C42×C5C22×D311yescharacteristic
203#687—62no—
204#685—32no—
205#685—62no—
206#685—62no—
207#685—62no—
208C4⋊Dic5—21no—
209C4⋊Dic5—21no—
210C4⋊Dic5—31no—
211C4⋊Dic5—62no—
212C4⋊Dic5—62no—
213C4⋊Dic5C22×D311yescharacteristic
214#683—21no—
215C4×Dic5—21no—
216C4×Dic5—62no—
217C2×#168—31no—
218C2×#168—31no—
219Dic20—42no—
220C8×D5—122no—
221#556—151no—
222#547—151no—
223#487—151no—
224#485—152no—
225#474—31no—
226#467—152no—
227#462—51no—
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×C5×D3—22no—
237C2×C5×D3—41no—
238C2×C5×D3—41no—
239C3×C4×C5—21no—
240C3×C4×C5#13512yes—
241C3×C4×C5C2×#4111yescharacteristic
242C3×C4×C5H(5,𝔽2)11yescharacteristic
243Dic15—22no—
244C3×Dic5—42no—
245C3×Dic5—41no—
246C3×Dic5—41no—
247C5×Dic3—42no—
248C5×Dic3—41no—
249C2×C3×Q8—52no—
250C2×C3×Q8—51no—
251#239—104no—
252#239—202no—
253C2×D12—51no—
254C2×D12—102no—
255C2×D12C22×D511yescharacteristic
256C2×C4×D3—21no—
257C2×C4×D3—52no—
258C2×C4×D3—102no—
259C2×Dic6—21no—
260C3×Q16—202no—
261C2×C3×C8C22×D512yes—
262C3×C4⋊C4—51no—
263C3×C4⋊C4—101no—
264C3×C4⋊C4—101no—
265C3×C4⋊C4—101no—
266C3×C42—101no—
267C3×C42C22×D511yescharacteristic
268#215—102no—
269(C2×D3)⋊C4—21no—
270(C2×D3)⋊C4—52no—
271(C2×D3)⋊C4—102no—
272(C2×D3)⋊C4—102no—
273(C2×D3)⋊C4—102no—
274C4⋊Dic3—21no—
275C4⋊Dic3—51no—
276C4⋊Dic3C22×D511yescharacteristic
277C4⋊Dic3C22×D512yes—
278Dic3⋊C4—102no—
279Dic3⋊C4—102no—
280C4×Dic3—102no—
281C2×C3⋊C8—51no—
282C2×C3⋊C8—51no—
283D3×C8—202no—
284C23×C5—61no—
285C22×D5—32no—
286C5×Q8—122no—
287C5×D4—32no—
288C5×D4—61no—
289C2×C4×C5—62no—
290C2×C4×C5—61no—
291C2×C4×C5—122no—
292C2×C4×C5D3×D411yescharacteristic
293C2×C4×C5D3×D411yescharacteristic
294C2×C4×C5C23×D311yescharacteristic
295C22⋊D5—124no—
296C2×Dic5—21no—
297C2×Dic5—21no—
298C2×Dic5—22no—
299C2×Dic5—32no—
300C2×Dic5—124no—
301D20—62no—
302C4×D5—62no—
303C4×D5—62no—
304Dic10—21no—
305Dic10—22no—
306Dic10—21no—
307Dic10—22no—
308C5×C8—22no—
309#168—61no—
310#168—61no—
311C2×#41—152no—
312C2×Q16—52no—
313C2×SD16—151no—
314C22×C8—152no—
315C4⋊Q8—31no—
316C4⋊Q8—51no—
317#126—302no—
318#124—301no—
319#123—302no—
320C22⋊Q8—152no—
321C22⋊Q8—152no—
322#121—301no—
323C4×Q8—51no—
324C4×Q8—301no—
325C4×D4—31no—
326C4×D4—151no—
327#117—152no—
328C2×C4⋊C4—301no—
329#107—32no—
330#106—151no—
331C4⋊C8D3×D511yescharacteristic
332#103—52no—
333#103—152no—
334#103—151no—
335#103—152no—
336#102—32no—
337#102—151no—
338C22⋊C8—152no—
339C4×C8—151no—
340C2×C3×C5#54611yescharacteristic
341C2×C3×C5C2×#13511yescharacteristic
342C2×C3×C5#57811yescharacteristic
343D15—42no—
344C5×D3—42no—
345C22×D3—21no—
346C22×D3—52no—
347C3×Q8—101no—
348C3×Q8—102no—
349C3×Q8—101no—
350C3×Q8—102no—
351C2×C3×C4—101no—
352C2×C3×C4—101no—
353C2×C3×C4—102no—
354C2×C3×C4D4×D511yescharacteristic
355C2×C3×C4D4×D511yescharacteristic
356C2×C3×C4C23×D511yescharacteristic
357C2×Dic3—22no—
358C2×Dic3—21no—
359C2×Dic3—52no—
360D12—21no—
361D12—102no—
362D12—204no—
363D12#71312yes—
364C4×D3—42no—
365C4×D3—102no—
366C4×D3—102no—
367C4×D3—204no—
368Dic6—42no—
369C3×C8—22no—
370C3⋊C8—101no—
371C3⋊C8—101no—
372C22×C5—62no—
373C22×C5—121no—
374C22×C5—121no—
375C22×C5C2×D3×D411yescharacteristic
376C2×D5—62no—
377C2×D5—62no—
378C2×D5—62no—
379C4×C5—21no—
380C4×C5—121no—
381C4×C5—122no—
382C4×C5#91912yes—
383C4×C5#100411yescharacteristic
384C4×C5D3×#4111yescharacteristic
385Dic5—41no—
386Dic5—41no—
387Dic5—42no—
388Dic5—62no—
389#41—304no—
390#41—602no—
391C2×Q8—52no—
392C2×Q8—51no—
393C2×Q8—61no—
394C2×D4—31no—
395C2×D4—151no—
396C2×D4—302no—
397C22×C4—61no—
398C22×C4—152no—
399C22×C4—302no—
400Q16—202no—
401SD16—302no—
402C2×C8—151no—
403C2×C8—151no—
404C2×C8—602no—
405C2×C8C2×D3×D512yes—
406C4⋊C4—32no—
407C4⋊C4—31no—
408C4⋊C4—51no—
409C4⋊C4—61no—
410C4⋊C4—101no—
411C4⋊C4—101no—
412C4⋊C4—101no—
413C4⋊C4—151no—
414C4⋊C4—302no—
415C4⋊C4—302no—
416C22⋊C4—61no—
417C22⋊C4—152no—
418C22⋊C4—302no—
419C22⋊C4—302no—
420C22⋊C4—302no—
421C42—101no—
422C42—302no—
423C42C2×D3×D511yescharacteristic
424C3×C5#330211yescharacteristic, complemented, Hall
425C22×C3C2×D4×D511yescharacteristic
426C2×D3—22no—
427C2×D3—41no—
428C2×D3—41no—
429C2×D3—102no—
430C2×D3—102no—
431C2×D3—102no—
432C3×C4—21no—
433C3×C4—201no—
434C3×C4—202no—
435C3×C4—201no—
436C3×C4#405812yes—
437C3×C4C2×#71311yescharacteristic
438C3×C4#414911yescharacteristic
439Dic3—42no—
440Dic3—41no—
441Dic3—102no—
442C2×C5—122no—
443C2×C5#561211yescharacteristic
444C2×C5C2×#91911yescharacteristic
445C2×C5#577911yescharacteristic
446D5—122no—
447C23—61no—
448C23—152no—
449Q8—101no—
450Q8—102no—
451Q8—101no—
452Q8—102no—
453Q8—122no—
454D4—32no—
455D4—61no—
456D4—302no—
457D4—604no—
458C2×C4—62no—
459C2×C4—61no—
460C2×C4—101no—
461C2×C4—101no—
462C2×C4—102no—
463C2×C4—122no—
464C2×C4—152no—
465C2×C4—302no—
466C2×C4—302no—
467C2×C4—604no—
468C2×C4#677711yescharacteristic
469C2×C4#677711yescharacteristic
470C2×C4C22×D3×D511yescharacteristic
471C8—22no—
472C8—301no—
473C8—301no—
474C2×C3#6596411yescharacteristic
475C2×C3C2×#405811yescharacteristic
476C2×C3#6611311yescharacteristic
477D3—42no—
478D3—202no—
479C5#8049211yescharacteristic, complemented, Sylow
480C22—62no—
481C22—121no—
482C22—121no—
483C22—302no—
484C22—302no—
485C22—302no—
486C22C2×#677711yescharacteristic
487C4—21no—
488C4—121no—
489C4—122no—
490C4—201no—
491C4—202no—
492C4—201no—
493C4—302no—
494C4#9127912yes—
495C4#9202611yescharacteristic
496C4D3×#71311yescharacteristic
497C3#1061271211yescharacteristic, complemented, Sylow
498C2—122no—
499C2—602no—
500C2#1175388811yescharacteristic
501C2C2×#9127911yescharacteristic
502C2#1175404311yescharacteristic
subgroupG/H
centerC22C2×#6777
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×C4⋊C8C22
socleC22×C3×C5C22×D4
prime radical#11752549C2
hypercenterC4⋊C8D3×D5
hypocenterC3×C5#3302
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
2. #11752486C2semidirect
3. #11750935C2semidirect
4. C5×#4846C2semidirect
5. C3×#65138C2semidirect
6. #11748151C2semidirect
7. #11748145C2semidirect
8. #11747958C2semidirect
9. #11747692C2semidirect
10. C4×#6656C2semidirect
11. #11746924C2semidirect
50. #65138D3semidirect
96. #4846D5semidirect
424. C3×C5#3302semidirect
479. C5#80492semidirect
497. C3#10612712semidirect

Maximal subgroups

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

subgroupindex
#117525492
#117524862
#117509352
C5×#48462
C3×#651382
#117481512
#117481452
#117479582
#117476922
C4×#66562
#117469242
#106127123
#804925