Finite Group Databasebeta

← #14459120

Subgroups — #14459120

Order 1920.

840 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756486C211yesmaximal, characteristic, complemented
2#11756485C211yesmaximal, characteristic
3C3×C5×#487C211yesmaximal, characteristic, complemented
4#11751195C211yesmaximal, characteristic, complemented
5#11751184C211yesmaximal, characteristic, complemented
6#11750908C212yesmaximal, complemented
7#11750907C212yesmaximal, complemented
8#11750906C212yesmaximal, complemented
9D15×#103C212yesmaximal, complemented
10C4×#6706C211yesmaximal, characteristic, complemented
11C8×D60C211yesmaximal, characteristic, complemented
12#10613343—31nomaximal, Hall
13C2×#6821—21no—
14C2×Q8×D15C2212yes—
15C3×C5×C4⋊Q8C2211yescharacteristic
16C3×C5×C4⋊Q8C2211yescharacteristic
17#91835—21no—
18#91835C2212yes—
19C2×#6717C2212yes—
20C2×#6706C2211yescharacteristic
21C2×C8×D15C2212yes—
22#91785—22no—
23#91784C2212yes—
24#91783—22no—
25#91782—21no—
26C4⋊C4×D15C2212yes—
27#91777—21no—
28C4×D60C2211yescharacteristic
29C4×D60C2211yescharacteristic
30C4×Dic30C2211yescharacteristic
31C3×C5×#103C2212yes—
32C3×C5×#103C2212yes—
33C3×C4×C5×C8C2211yescharacteristic
34#91119C2212yes—
35#91105C2211yescharacteristic
36#91104C2212yes—
37#91101C2211yescharacteristic
38#91100C2211yescharacteristic
39#91094C2212yes—
40#91092C2212yes—
41C4⋊(C3⋊#168)C2211yescharacteristic
42#81612—51nomaximal, Hall
43#66061—31no—
44#66060—31no—
45C5×#487D311yescharacteristic, complemented
46#65383—31no—
47#65372—31no—
48#65096—32no—
49#65095—32no—
50#65094—32no—
51D5×#103—32no—
52C4×#677—31no—
53C8×D20—31no—
54C2×C3×C5×Q8—21no—
55C2×C3×C5×Q8C2312yes—
56#6821—44no—
57Q8×D15—24no—
58Q8×D15—42no—
59C2×D60—22no—
60C2×D60D411yescharacteristic
61C2×D60C2311yescharacteristic
62C2×C4×D15—22no—
63C2×C4×D15—22no—
64C2×C4×D15—21no—
65C2×C4×D15C2312yes—
66C2×Dic30—22no—
67C2×Dic30C2311yescharacteristic
68C2×C3×C5×C8C2311yescharacteristic
69C2×C3×C5×C8C2311yescharacteristic
70C3×C5×C4⋊C4—21no—
71C3×C5×C4⋊C4—21no—
72C3×C5×C4⋊C4—21no—
73C3×C5×C4⋊C4C2312yes—
74C3×C42×C5C2311yescharacteristic
75#6717—42no—
76#6714—22no—
77#6714—21no—
78#6714—22no—
79#6714D412yes—
80C4⋊Dic15D411yescharacteristic
81C4⋊Dic15C2311yescharacteristic
82C4⋊Dic15C2311yescharacteristic
83#6712—21no—
84#6712—22no—
85#6712—22no—
86C4×Dic15—21no—
87C2×C3⋊#168C2312yes—
88#6706—22no—
89C8×D15—42no—
90#5727—51no—
91#5726—51no—
92C3×#487D511yescharacteristic, complemented
93#5094—51no—
94#5083—51no—
95#4807—52no—
96#4806—52no—
97#4805—52no—
98D3×#103—52no—
99C4×#204—51no—
100C8×D12—51no—
101C2×#713—61no—
102C2×Q8×D5—32no—
103C5×C4⋊Q8C2×D311yescharacteristic
104C5×C4⋊Q8C2×D311yescharacteristic
105#4092—32no—
106#4092—61no—
107C2×#688—32no—
108C2×#677—31no—
109C2×C8×D5—32no—
110#4042—62no—
111#4041—32no—
112#4040—62no—
113#4039—61no—
114D5×C4⋊C4—32no—
115#4034—61no—
116C4×D20—31no—
117C4×D20—31no—
118C4×Dic10—31no—
119C5×#103C2×D312yes—
120C5×#103C2×D312yes—
121C4×C5×C8C2×D311yescharacteristic
122#3967—32no—
123#3953—31no—
124#3952—32no—
125#3949—31no—
126#3948—31no—
127#3942—32no—
128#3940—32no—
129C4⋊#168—31no—
130#3298—151noSylow
131C22×D15—21no—
132C22×D15C2×D412yes—
133C3×C5×Q8—22no—
134C3×C5×Q8—22no—
135C3×C5×Q8—42no—
136C2×C3×C4×C5—22no—
137C2×C3×C4×C5—22no—
138C2×C3×C4×C5C2×D411yescharacteristic
139C2×C3×C4×C5C2×D411yescharacteristic
140C2×C3×C4×C5C2411yescharacteristic
141C2×Dic15—21no—
142C2×Dic15—21no—
143C2×Dic15—21no—
144C2×Dic15C2×D412yes—
145D60—21no—
146D60—21no—
147D60—44no—
148D60SD1614yescomplemented
149C4×D15—22no—
150C4×D15—22no—
151C4×D15—44no—
152C4×D15—44no—
153C4×D15—42no—
154Dic30—21no—
155Dic30—44no—
156Dic30#4112yescomplemented
157C3×C5×C8—21no—
158C3×C5×C8—21no—
159C3⋊#168—22no—
160C2×#239—101no—
161C2×D3×Q8—52no—
162C3×C4⋊Q8C2×D511yescharacteristic
163C3×C4⋊Q8C2×D511yescharacteristic
164#946—52no—
165#946—101no—
166C2×#215—52no—
167C2×#204—51no—
168C2×D3×C8—52no—
169#896—102no—
170#895—52no—
171#894—102no—
172#893—101no—
173D3×C4⋊C4—52no—
174#888—101no—
175C4×D12—51no—
176C4×D12—51no—
177C4×Dic6—51no—
178C3×#103C2×D512yes—
179C3×#103C2×D512yes—
180C3×C4×C8C2×D511yescharacteristic
181#835—52no—
182#821—51no—
183(C2×D3)⋊C8—52no—
184#817—51no—
185#816—51no—
186#810—52no—
187#808—52no—
188C4⋊(C3⋊C8)—51no—
189C2×C5×Q8—21no—
190C2×C5×Q8C22×D312yes—
191#713—124no—
192Q8×D5—64no—
193Q8×D5—122no—
194C2×D20—31no—
195C2×D20—31no—
196C2×D20—62no—
197C2×C4×D5—32no—
198C2×C4×D5—62no—
199C2×C4×D5—62no—
200C2×C4×D5—61no—
201C2×Dic10—31no—
202C2×Dic10—62no—
203C2×C5×C8C22×D311yescharacteristic
204C2×C5×C8C22×D311yescharacteristic
205C5×C4⋊C4—21no—
206C5×C4⋊C4—21no—
207C5×C4⋊C4—21no—
208C5×C4⋊C4C22×D312yes—
209C42×C5C22×D311yescharacteristic
210#688—122no—
211#685—32no—
212#685—62no—
213#685—62no—
214#685—61no—
215C4⋊Dic5—31no—
216C4⋊Dic5—31no—
217C4⋊Dic5—31no—
218#683—62no—
219#683—62no—
220#683—61no—
221C4×Dic5—61no—
222C2×#168—32no—
223#677—62no—
224C8×D5—122no—
225D4×Q8—151no—
226#547—151no—
227#487D1511yescharacteristic, complemented
228#483—152no—
229#475—151no—
230#465—152no—
231#461—151no—
232#448—152no—
233C4×SD16—151no—
234C8×D4—151no—
235C2×#103—152no—
236C22×C3×C5C22×D411yescharacteristic
237C2×D15—22no—
238C2×D15—22no—
239C2×D15—41no—
240C2×D15—41no—
241C2×D15—41no—
242C2×D15C2×SD1614yes—
243C3×C4×C5—21no—
244C3×C4×C5—42no—
245C3×C4×C5—42no—
246C3×C4×C5C2×SD1612yes—
247C3×C4×C5C2×#4111yescharacteristic
248C3×C4×C5H(5,𝔽2)11yescharacteristic
249Dic15—22no—
250Dic15—22no—
251Dic15—41no—
252Dic15—41no—
253C2×C3×Q8—21no—
254C2×C3×Q8C22×D512yes—
255#239—204no—
256D3×Q8—104no—
257D3×Q8—202no—
258C2×D12—51no—
259C2×D12—51no—
260C2×D12—102no—
261C2×C4×D3—52no—
262C2×C4×D3—102no—
263C2×C4×D3—102no—
264C2×C4×D3—101no—
265C2×Dic6—51no—
266C2×Dic6—102no—
267C2×C3×C8C22×D511yescharacteristic
268C2×C3×C8C22×D511yescharacteristic
269C3×C4⋊C4—21no—
270C3×C4⋊C4—21no—
271C3×C4⋊C4—21no—
272C3×C4⋊C4C22×D512yes—
273C3×C42C22×D511yescharacteristic
274#215—202no—
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—51no—
282Dic3⋊C4—102no—
283Dic3⋊C4—102no—
284Dic3⋊C4—101no—
285C4×Dic3—101no—
286C2×C3⋊C8—52no—
287#204—102no—
288D3×C8—202no—
289C22×D5—32no—
290C22×D5—61no—
291C5×Q8—22no—
292C5×Q8—22no—
293C5×Q8—42no—
294C2×C4×C5—22no—
295C2×C4×C5—22no—
296C2×C4×C5D3×D411yescharacteristic
297C2×C4×C5D3×D411yescharacteristic
298C2×C4×C5C23×D311yescharacteristic
299C2×Dic5—32no—
300C2×Dic5—61no—
301C2×Dic5—61no—
302C2×Dic5—61no—
303D20—34no—
304D20—61no—
305D20—61no—
306D20—124no—
307C4×D5—62no—
308C4×D5—62no—
309C4×D5—124no—
310C4×D5—124no—
311C4×D5—122no—
312Dic10—32no—
313Dic10—61no—
314Dic10—124no—
315C5×C8—21no—
316C5×C8—21no—
317#168—62no—
318C2×#41—301no—
319C22×Q8—152no—
320C2×SD16—152no—
321C2×SD16—151no—
322C22×C8—152no—
323C4⋊Q8—301no—
324C4⋊Q8C2×D1511yescharacteristic
325C4⋊Q8C2×D1511yescharacteristic
326#123—302no—
327C22⋊Q8—152no—
328C22⋊Q8—302no—
329C22⋊Q8—301no—
330#121—152no—
331C4×Q8—151no—
332C4×D4—151no—
333C4×D4—151no—
334C4×D4—301no—
335C2×C4⋊C4—152no—
336#106—151no—
337#106—152no—
338C4⋊C8—151no—
339#103—152no—
340#103—151no—
341#103—152no—
342#103C2×D1512yes—
343#103C2×D1512yes—
344#102—151no—
345C22⋊C8—152no—
346C4×C8C2×D1511yescharacteristic
347C2×C3×C5#54611yescharacteristic
348C2×C3×C5C22×SD1611yescharacteristic
349C2×C3×C5#57811yescharacteristic
350D15—24no—
351D15—81no—
352C22×D3—52no—
353C22×D3—101no—
354C3×Q8—22no—
355C3×Q8—22no—
356C3×Q8—42no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4D4×D511yescharacteristic
360C2×C3×C4D4×D511yescharacteristic
361C2×C3×C4C23×D511yescharacteristic
362C2×Dic3—52no—
363C2×Dic3—101no—
364C2×Dic3—101no—
365C2×Dic3—101no—
366D12—54no—
367D12—101no—
368D12—101no—
369D12—204no—
370C4×D3—102no—
371C4×D3—102no—
372C4×D3—204no—
373C4×D3—204no—
374C4×D3—202no—
375Dic6—52no—
376Dic6—101no—
377Dic6—204no—
378C3×C8—21no—
379C3×C8—21no—
380C3⋊C8—102no—
381C22×C5C2×D3×D411yescharacteristic
382C2×D5—34no—
383C2×D5—62no—
384C2×D5—62no—
385C2×D5—121no—
386C2×D5—121no—
387C2×D5—121no—
388C4×C5—21no—
389C4×C5—42no—
390C4×C5—42no—
391C4×C5D3×SD1612yes—
392C4×C5D3×#4111yescharacteristic
393C4×C5#100911yescharacteristic
394Dic5—62no—
395Dic5—62no—
396Dic5—121no—
397Dic5—121no—
398#41—604no—
399C2×Q8—21no—
400C2×Q8—151no—
401C2×Q8—302no—
402C2×Q8—304no—
403C2×Q8—602no—
404C2×Q8C22×D1512yes—
405C2×D4—151no—
406C2×D4—151no—
407C2×D4—302no—
408C22×C4—152no—
409C22×C4—302no—
410C22×C4—302no—
411C22×C4—301no—
412SD16—302no—
413SD16—602no—
414C2×C8—152no—
415C2×C8—602no—
416C2×C8C22×D1511yescharacteristic
417C2×C8C22×D1511yescharacteristic
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—151no—
422C4⋊C4—151no—
423C4⋊C4—151no—
424C4⋊C4—302no—
425C4⋊C4—302no—
426C4⋊C4—301no—
427C4⋊C4C22×D1512yes—
428C22⋊C4—152no—
429C22⋊C4—302no—
430C22⋊C4—302no—
431C22⋊C4—301no—
432C42—301no—
433C42C22×D1511yescharacteristic
434C3×C5#329811yescharacteristic, complemented, Hall
435C22×C3C2×D4×D511yescharacteristic
436C2×D3—54no—
437C2×D3—102no—
438C2×D3—102no—
439C2×D3—201no—
440C2×D3—201no—
441C2×D3—201no—
442C3×C4—21no—
443C3×C4—42no—
444C3×C4—42no—
445C3×C4D5×SD1612yes—
446C3×C4D5×#4111yescharacteristic
447C3×C4#414911yescharacteristic
448Dic3—102no—
449Dic3—102no—
450Dic3—201no—
451Dic3—201no—
452C2×C5#567611yescharacteristic
453C2×C5C2×D3×SD1611yescharacteristic
454C2×C5#576611yescharacteristic
455D5—64no—
456D5—241no—
457C23—152no—
458C23—301no—
459Q8—22no—
460Q8—22no—
461Q8—42no—
462Q8—152no—
463Q8—301no—
464Q8—604no—
465D4—154no—
466D4—301no—
467D4—301no—
468D4—604no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—152no—
472C2×C4—301no—
473C2×C4—302no—
474C2×C4—302no—
475C2×C4—301no—
476C2×C4—301no—
477C2×C4—604no—
478C2×C4—604no—
479C2×C4—602no—
480C2×C4D4×D1511yescharacteristic
481C2×C4D4×D1511yescharacteristic
482C2×C4C23×D1511yescharacteristic
483C8—21no—
484C8—21no—
485C8—302no—
486C2×C3#6601011yescharacteristic
487C2×C3C2×D5×SD1611yescharacteristic
488C2×C3#6610011yescharacteristic
489D3—104no—
490D3—401no—
491C5#8161211yescharacteristic, complemented, Sylow
492C22—154no—
493C22—302no—
494C22—302no—
495C22—601no—
496C22—601no—
497C22—601no—
498C22C2×D4×D1511yescharacteristic
499C4—21no—
500C4—42no—
501C4—42no—
502C4—302no—
503C4—302no—
504C4—601no—
505C4—601no—
506C4SD16×D1512yes—
507C4#41×D1511yescharacteristic
508C4#9210011yescharacteristic
509C3#1061334311yescharacteristic, complemented, Sylow
510C2—304no—
511C2—1201no—
512C2#1175643511yescharacteristic
513C2C2×SD16×D1511yescharacteristic
514C2#1175652511yescharacteristic
subgroupG/H
centerC22C2×D4×D15
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C5×#487C2
socleC22×C3×C5C22×D4
prime radical#11756485C2
hypercenter#487D15
hypocenterC3×C5#3298
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #11756486C2semidirect
3. C3×C5×#487C2semidirect
4. #11751195C2semidirect
5. #11751184C2semidirect
6. #11750908C2semidirect
7. #11750907C2semidirect
8. #11750906C2semidirect
9. D15×#103C2semidirect
10. C4×#6706C2semidirect
11. C8×D60C2semidirect
45. C5×#487D3semidirect
92. C3×#487D5semidirect
148. D60SD16semidirect
156. Dic30#41semidirect
227. #487D15semidirect
434. C3×C5#3298semidirect
491. C5#81612semidirect
509. C3#10613343semidirect

Maximal subgroups

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

subgroupindex
#117564862
#117564852
C3×C5×#4872
#117511952
#117511842
#117509082
#117509072
#117509062
D15×#1032
C4×#67062
C8×D602
#106133433
#816125