Finite Group Databasebeta

← #14460064

Subgroups — #14460064

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#11752483C211yesmaximal, characteristic, complemented
3C8×D60C211yesmaximal, characteristic, complemented
4C4×C5×#204C211yesmaximal, characteristic, complemented
5C3×#64970C211yesmaximal, characteristic, complemented
6#11748158C212yesmaximal, complemented
7#11748145C212yesmaximal, complemented
8#11747954C212yesmaximal, complemented
9#11747688C212yesmaximal, complemented
10#11747099C211yesmaximal, characteristic, complemented
11#11746900C211yesmaximal, characteristic, complemented
12#10611411—31nomaximal, Hall
13C2×#6768C2212yes—
14C2×C8×D15C2212yes—
15C4×D60C2211yescharacteristic
16C2×C5×#204C2211yescharacteristic
17C4×C5×D12C2211yescharacteristic
18C4×C5×Dic6C2211yescharacteristic
19C3×#4015C2211yescharacteristic
20C3×#4015C2211yescharacteristic
21#91460—21no—
22#91427—22no—
23D3×C4⋊Dic5—21no—
24#91390C2212yes—
25#91383C2212yes—
26#91374—21no—
27#91347C2212yes—
28#91345—22no—
29Dic3×Dic10—21no—
30C2×#6661C2212yes—
31C3×C4×C5×C8C2211yescharacteristic
32#91104C2212yes—
33C4⋊(C3⋊#168)C2211yescharacteristic
34C5×#821C2211yescharacteristic
35C5×#817C2211yescharacteristic
36C5×#816C2211yescharacteristic
37C3×#3948C2212yes—
38C3×#3948C2212yes—
39#90989C2212yes—
40#90970C2212yes—
41#90966C2212yes—
42#81614—51nomaximal, Hall
43#65918—31no—
44#65889—31no—
45C4×C5×SD16—31no—
46#65462—32no—
47#65320—31no—
48#65296—31no—
49#65156—32no—
50#65102—32no—
51#65061—32no—
52C8×D20—31no—
53#64970D311yescharacteristic, complemented
54C2×D60D411yescharacteristic
55C2×C4×D15C2312yes—
56C2×C5×D12C2311yescharacteristic
57C2×C4×C5×D3—21no—
58C2×C5×Dic6C2311yescharacteristic
59C2×C3×Dic10—21no—
60C2×C3×Dic10C2312yes—
61C2×#1203—22no—
62C2×D3×Dic5—22no—
63#6768—24no—
64#6768—42no—
65C2×C3×C5×C8C2311yescharacteristic
66C2×C3×C5×C8C2311yescharacteristic
67C3×C42×C5C2311yescharacteristic
68#6714D412yes—
69C4⋊Dic15D411yescharacteristic
70C2×C3⋊#168C2312yes—
71C8×D15—42no—
72C5×(C2×D3)⋊C4—21no—
73C5×C4⋊Dic3C2311yescharacteristic
74C5×C4⋊Dic3C2311yescharacteristic
75C5×Dic3⋊C4—21no—
76C4×C5×Dic3—21no—
77C5×#204—22no—
78C3×C4⋊Dic5—21no—
79C3×C4⋊Dic5—21no—
80C3×C4⋊Dic5—21no—
81C3×C4⋊Dic5C2312yes—
82Dic3⋊Dic5—22no—
83#6668—22no—
84#6667—22no—
85#6667—22no—
86#6667—22no—
87Dic3×Dic5—22no—
88#6661—42no—
89#5732—51no—
90#5726—51no—
91C3×#487—51no—
92#5090—51no—
93#5087—51no—
94#4815—52no—
95#4813—52no—
96#4808—52no—
97#4804—52no—
98C4×#204D511yescharacteristic, complemented
99C8×D12—51no—
100C2×#711—32no—
101C2×C5×SD16—31no—
102C4×C5×Q8—31no—
103C4×C5×D4—31no—
104#4092—32no—
105C2×#689—32no—
106#4076—61no—
107C2×C4⋊Dic5—61no—
108C2×C8×D5—32no—
109#4044—62no—
110#4043—32no—
111#4038—32no—
112#4032—62no—
113#4021—61no—
114C4×D20—31no—
115#4015C2×D311yescharacteristic
116#4015C2×D311yescharacteristic
117C4×Dic10—61no—
118C5×#106—31no—
119C5×#103—31no—
120C5×#102—31no—
121C4×C5×C8C2×D311yescharacteristic
122#3964—32no—
123#3952—32no—
124#3948C2×D312yes—
125#3948C2×D312yes—
126#3942—32no—
127#3939—32no—
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—21no—
140C2×C5×Dic3—21no—
141C2×C5×Dic3—21no—
142C5×D12—21no—
143C5×D12#4112yes—
144C4×C5×D3—42no—
145C5×Dic6—21no—
146C5×Dic6—21no—
147C2×C3×Dic5—22no—
148C2×C3×Dic5—22no—
149C3×Dic10—22no—
150C3×Dic10—22no—
151C3×Dic10—42no—
152#1203—44no—
153D3×Dic5—44no—
154C3×C5×C8—21no—
155C3×C5×C8—21no—
156C3⋊#168—22no—
157C2×#239—52no—
158C3×C4⋊Q8—51no—
159C3×C4⋊Q8—51no—
160#947—101no—
161#946—52no—
162Q8×Dic3—101no—
163C2×#216—52no—
164C2×#204C2×D511yescharacteristic
165C2×D3×C8—52no—
166#898—102no—
167#897—52no—
168#895—101no—
169#894—102no—
170#892—52no—
171D3×C4⋊C4—101no—
172C4×D12—51no—
173C4×D12C2×D511yescharacteristic
174C4×Dic6C2×D511yescharacteristic
175C3×#103—52no—
176C3×#103—52no—
177C3×C4×C8C2×D511yescharacteristic
178#835—52no—
179#821C2×D511yescharacteristic
180(C2×D3)⋊C8—52no—
181#817C2×D511yescharacteristic
182#816C2×D511yescharacteristic
183#809—52no—
184#807—52no—
185C4⋊(C3⋊C8)—51no—
186C2×C5×Q8—31no—
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×Dic10—21no—
196C2×Dic10C22×D312yes—
197C5×SD16—62no—
198C2×C5×C8C22×D311yescharacteristic
199C2×C5×C8C22×D311yescharacteristic
200C5×C4⋊C4—31no—
201C5×C4⋊C4—31no—
202C5×C4⋊C4—61no—
203C5×C22⋊C4—61no—
204C42×C5—61no—
205C42×C5C22×D311yescharacteristic
206#689—122no—
207#685—32no—
208#685—62no—
209#685—62no—
210#685—62no—
211C4⋊Dic5—21no—
212C4⋊Dic5—21no—
213C4⋊Dic5—21no—
214C4⋊Dic5—31no—
215C4⋊Dic5—62no—
216C4⋊Dic5C22×D312yes—
217#683—62no—
218C4×Dic5—62no—
219C2×#168—32no—
220C8×D5—122no—
221#556—151no—
222#547—151no—
223#487—51no—
224#485—152no—
225#474—151no—
226#467—152no—
227#462—151no—
228#452—152no—
229C4×SD16—31no—
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—42no—
246C5×Dic3—41no—
247C5×Dic3—41no—
248C5×Dic3—41no—
249C2×C3×Q8—52no—
250C2×C3×Q8—101no—
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×Dic6C22×D511yescharacteristic
260C2×C3×C8C22×D511yescharacteristic
261C2×C3×C8C22×D511yescharacteristic
262C3×C4⋊C4—52no—
263C3×C4⋊C4—101no—
264C3×C4⋊C4—101no—
265C3×C4⋊C4—101no—
266C3×C42C22×D511yescharacteristic
267#216—202no—
268(C2×D3)⋊C4—21no—
269(C2×D3)⋊C4—52no—
270(C2×D3)⋊C4—102no—
271(C2×D3)⋊C4—102no—
272(C2×D3)⋊C4—102no—
273C4⋊Dic3—51no—
274C4⋊Dic3—102no—
275C4⋊Dic3C22×D511yescharacteristic
276C4⋊Dic3C22×D511yescharacteristic
277Dic3⋊C4—21no—
278Dic3⋊C4—102no—
279C4×Dic3—21no—
280C4×Dic3—102no—
281C2×C3⋊C8—52no—
282#204—22no—
283D3×C8—202no—
284C23×C5—61no—
285C22×D5—32no—
286C5×Q8—61no—
287C5×Q8—61no—
288C5×D4—32no—
289C5×D4—61no—
290C2×C4×C5—61no—
291C2×C4×C5—61no—
292C2×C4×C5—61no—
293C2×C4×C5—122no—
294C2×C4×C5D3×D411yescharacteristic
295C2×C4×C5D3×D411yescharacteristic
296C2×C4×C5C23×D311yescharacteristic
297C22⋊D5—124no—
298C2×Dic5—22no—
299C2×Dic5—22no—
300C2×Dic5—32no—
301C2×Dic5—124no—
302D20—62no—
303C4×D5—62no—
304C4×D5—62no—
305Dic10—22no—
306Dic10—22no—
307Dic10—42no—
308C5×C8—21no—
309C5×C8—21no—
310#168—62no—
311C2×#41—152no—
312C2×Q16—152no—
313C2×SD16—31no—
314C22×C8—152no—
315C4⋊Q8—51no—
316C4⋊Q8—51no—
317#126—302no—
318#124—301no—
319#123—302no—
320C22⋊Q8—152no—
321C22⋊Q8—152no—
322#121—301no—
323C4×Q8—31no—
324C4×Q8—301no—
325C4×D4—31no—
326C4×D4—151no—
327#117—152no—
328C2×C4⋊C4—301no—
329#107—152no—
330#106—31no—
331C4⋊C8—151no—
332#103—31no—
333#103—52no—
334#103—52no—
335#103—152no—
336#102—31no—
337#102—152no—
338C22⋊C8—152no—
339C4×C8D3×D511yescharacteristic
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—102no—
348C3×Q8—102no—
349C3×Q8—202no—
350C2×C3×C4—102no—
351C2×C3×C4—102no—
352C2×C3×C4D4×D511yescharacteristic
353C2×C3×C4D4×D511yescharacteristic
354C2×C3×C4C23×D511yescharacteristic
355C2×Dic3—21no—
356C2×Dic3—21no—
357C2×Dic3—21no—
358C2×Dic3—52no—
359D12—21no—
360D12—102no—
361D12—204no—
362D12#70912yes—
363C4×D3—42no—
364C4×D3—102no—
365C4×D3—102no—
366C4×D3—204no—
367Dic6—21no—
368Dic6—21no—
369C3×C8—21no—
370C3×C8—21no—
371C3⋊C8—102no—
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—121no—
382C4×C5—121no—
383C4×C5#91612yes—
384C4×C5D3×#4111yescharacteristic
385C4×C5#100911yescharacteristic
386Dic5—42no—
387Dic5—42no—
388Dic5—62no—
389#41—304no—
390#41—602no—
391C2×Q8—31no—
392C2×Q8—52no—
393C2×Q8—101no—
394C2×D4—31no—
395C2×D4—151no—
396C2×D4—302no—
397C22×C4—61no—
398C22×C4—152no—
399C22×C4—302no—
400Q16—602no—
401SD16—62no—
402C2×C8—152no—
403C2×C8—602no—
404C2×C8C2×D3×D511yescharacteristic
405C2×C8C2×D3×D511yescharacteristic
406C4⋊C4—31no—
407C4⋊C4—31no—
408C4⋊C4—52no—
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—61no—
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—202no—
434C3×C4—202no—
435C3×C4#406212yes—
436C3×C4C2×#70911yescharacteristic
437C3×C4#414911yescharacteristic
438Dic3—41no—
439Dic3—41no—
440Dic3—41no—
441Dic3—102no—
442C2×C5—122no—
443C2×C5#567611yescharacteristic
444C2×C5C2×#91611yescharacteristic
445C2×C5#576611yescharacteristic
446D5—122no—
447C23—61no—
448C23—152no—
449Q8—61no—
450Q8—61no—
451Q8—102no—
452Q8—102no—
453Q8—202no—
454D4—32no—
455D4—61no—
456D4—302no—
457D4—604no—
458C2×C4—61no—
459C2×C4—61no—
460C2×C4—61no—
461C2×C4—102no—
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—21no—
472C8—21no—
473C8—302no—
474C2×C3#6588411yescharacteristic
475C2×C3C2×#406211yescharacteristic
476C2×C3#6610511yescharacteristic
477D3—42no—
478D3—202no—
479C5#8161411yescharacteristic, complemented, Sylow
480C22—62no—
481C22—121no—
482C22—121no—
483C22—302no—
484C22—302no—
485C22—302no—
486C22C2×#677711yescharacteristic
487C4—21no—
488C4—121no—
489C4—121no—
490C4—121no—
491C4—202no—
492C4—202no—
493C4—302no—
494C4#9127612yes—
495C4D3×#70911yescharacteristic
496C4#9201911yescharacteristic
497C3#1061141111yescharacteristic, complemented, Sylow
498C2—122no—
499C2—602no—
500C2#1175274911yescharacteristic
501C2C2×#9127611yescharacteristic
502C2#1175401111yescharacteristic
subgroupG/H
centerC22C2×#6777
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C4×C5×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. #11752483C2semidirect
3. C8×D60C2semidirect
4. C4×C5×#204C2semidirect
5. C3×#64970C2semidirect
6. #11748158C2semidirect
7. #11748145C2semidirect
8. #11747954C2semidirect
9. #11747688C2semidirect
10. #11747099C2semidirect
11. #11746900C2semidirect
53. #64970D3semidirect
98. C4×#204D5semidirect
424. C3×C5#3302semidirect
479. C5#81614semidirect
497. C3#10611411semidirect

Maximal subgroups

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

subgroupindex
#117525492
#117524832
C8×D602
C4×C5×#2042
C3×#649702
#117481582
#117481452
#117479542
#117476882
#117470992
#117469002
#106114113
#816145