Finite Group Databasebeta

← #14430788

Subgroups — #14430788

Order 1920.

796 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756557C211yesmaximal, characteristic
2#11756278C211yesmaximal, characteristic, complemented
3C3×C5×#418C211yesmaximal, characteristic
4C2×#91116C211yesmaximal, characteristic
5#11751222C211yesmaximal, characteristic, complemented
6#11751220C211yesmaximal, characteristic, complemented
7#11751099C211yesmaximal, characteristic, complemented
8#11750867C212yesmaximal, complemented
9#11750864C212yesmaximal, complemented
10#11750862C212yesmaximal, complemented
11#11750860C212yesmaximal, complemented
12#10610649—31nomaximal, Hall
13C22×C3×C5×D4C2211yescharacteristic
14C2×C3×C5×M16C2211yescharacteristic
15C3×C5×#117C2211yescharacteristic
16C2×#6719—22no—
17D4×Dic15—21no—
18D4×Dic15—21no—
19D4×Dic15C2212yes—
20D4×Dic15C2212yes—
21#91815C2211yescharacteristic
22#91815C2211yescharacteristic
23C2×C4⋊Dic15C2211yescharacteristic
24C22×C3⋊#168C2211yescharacteristic
25#91779C2212yescomplemented
26#91777C2212yescomplemented
27#91767—22no—
28#91759—21no—
29#91758—21no—
30C3×C5×#102C2212yes—
31C3×C5×#102C2212yes—
32#91116C2212yes—
33#91116C2212yes—
34#91102C2212yescomplemented
35#91101C2212yescomplemented
36#91098C2212yescomplemented
37#91098C2212yescomplemented
38#91092C2211yescharacteristic, complemented
39#91092C2211yescharacteristic, complemented
40#91091C2211yescharacteristic, complemented
41#91091C2211yescharacteristic, complemented
42#78918—51nomaximal, Hall
43#66132—31no—
44#65853—31no—
45C5×#418D311yescharacteristic
46C2×#3964—31no—
47#65410—31no—
48#65408—31no—
49#65287—31no—
50#65055—32no—
51#65052—32no—
52#65050—32no—
53#65048—32no—
54C24×C3×C5—21no—
55C2×C3×C5×D4—21no—
56C2×C3×C5×D4—22no—
57C2×C3×C5×D4Q814yescomplemented
58C2×C3×C5×D4C2312yes—
59C22×C3×C4×C5C2311yescharacteristic
60C22×Dic15—21no—
61C22×Dic15—22no—
62C22×Dic15—22no—
63C2×Dic30—22no—
64C3×C5×M16—22no—
65C2×C3×C5×C8C2312yes—
66C3×C5×C4⋊C4C2311yescharacteristic
67C3×C5×C4⋊C4C2311yescharacteristic
68C3×C5×C22⋊C4—21no—
69C3×C42×C5—21no—
70#6719—21no—
71#6719—22no—
72#6719—22no—
73#6719D412yes—
74C4⋊Dic15—21no—
75C4⋊Dic15—22no—
76C4⋊Dic15D411yescharacteristic
77C4⋊Dic15D411yescharacteristic
78C4⋊Dic15C2312yes—
79C4⋊Dic15C2311yescharacteristic
80C4⋊Dic15C2311yescharacteristic
81#6712—22no—
82#6712—22no—
83C4×Dic15—21no—
84C4×Dic15C2312yes—
85C2×C3⋊#168—22no—
86C2×C3⋊#168C2311yescharacteristic
87C2×C3⋊#168C2311yescharacteristic
88#5798—51no—
89#5519—51no—
90C3×#418D511yescharacteristic
91C2×#832—51no—
92#5121—51no—
93#5119—51no—
94#4998—51no—
95#4766—52no—
96#4763—52no—
97#4761—52no—
98#4759—52no—
99C22×C5×D4C2×D311yescharacteristic
100C2×C5×M16C2×D311yescharacteristic
101C5×#117C2×D311yescharacteristic
102C2×C22⋊Dic5—62no—
103D4×Dic5—32no—
104D4×Dic5—32no—
105D4×Dic5—61no—
106D4×Dic5—61no—
107#4072—31no—
108#4072—31no—
109C2×C4⋊Dic5—31no—
110C22×#168—31no—
111#4036—32no—
112#4034—32no—
113#4024—62no—
114#4016—61no—
115#4015—61no—
116C5×#102C2×D312yes—
117C5×#102C2×D312yes—
118#3964—32no—
119#3964—32no—
120#3950—32no—
121#3949—32no—
122#3946—32no—
123#3946—32no—
124#3940—31no—
125#3940—31no—
126#3939—31no—
127#3939—31no—
128#3081—151noSylow
129C23×C3×C5—22no—
130C23×C3×C5—24no—
131C23×C3×C5—81no—
132C23×C3×C5—81no—
133C23×C3×C5C2×D411yescharacteristic
134C3×C5×D4—24no—
135C3×C5×D4—22no—
136C3×C5×D4C2×Q814yes—
137C2×C3×C4×C5—21no—
138C2×C3×C4×C5—21no—
139C2×C3×C4×C5C2×D411yescharacteristic
140C2×C3×C4×C5C2×Q812yes—
141C2×C3×C4×C5#4112yes—
142C2×C3×C4×C5C2411yescharacteristic
143C2×Dic15—21no—
144C2×Dic15—21no—
145C2×Dic15—22no—
146C2×Dic15—82no—
147C2×Dic15—82no—
148C2×Dic15—81no—
149C2×Dic15C2×D412yes—
150C2×Dic15C2×D412yes—
151Dic30—82no—
152C3×C5×C8—22no—
153C3⋊#168—21no—
154C3⋊#168—21no—
155C22×C3×D4C2×D511yescharacteristic
156C2×C3×M16C2×D511yescharacteristic
157C3×#117C2×D511yescharacteristic
158C2×#217—102no—
159D4×Dic3—52no—
160D4×Dic3—52no—
161D4×Dic3—101no—
162D4×Dic3—101no—
163#926—51no—
164#926—51no—
165C2×C4⋊Dic3—51no—
166C22×C3⋊C8—51no—
167#890—52no—
168#888—52no—
169#878—102no—
170#870—101no—
171#869—101no—
172C3×#102C2×D512yes—
173C3×#102C2×D512yes—
174#832—52no—
175#832—52no—
176#818—52no—
177#817—52no—
178Dic3⋊C8—52no—
179Dic3⋊C8—52no—
180#808—51no—
181#808—51no—
182#807—51no—
183#807—51no—
184C24×C5—21no—
185C2×C5×D4—21no—
186C2×C5×D4—22no—
187C2×C5×D4Dic614yescomplemented
188C2×C5×D4C22×D312yes—
189C22×C4×C5C22×D311yescharacteristic
190C22×Dic5—62no—
191C22×Dic5—62no—
192C22×Dic5—61no—
193C2×Dic10—62no—
194C5×M16—22no—
195C2×C5×C8C22×D312yes—
196C5×C4⋊C4C22×D311yescharacteristic
197C5×C4⋊C4C22×D311yescharacteristic
198C5×C22⋊C4—21no—
199C42×C5—21no—
200C22⋊Dic5—32no—
201C22⋊Dic5—62no—
202C22⋊Dic5—62no—
203C22⋊Dic5—61no—
204C4⋊Dic5—31no—
205C4⋊Dic5—31no—
206C4⋊Dic5—31no—
207C4⋊Dic5—31no—
208C4⋊Dic5—32no—
209C4⋊Dic5—62no—
210C4⋊Dic5—61no—
211#683—62no—
212#683—62no—
213C4×Dic5—32no—
214C4×Dic5—61no—
215C2×#168—31no—
216C2×#168—31no—
217C2×#168—62no—
218#544—151no—
219#518—151no—
220#478—152no—
221#478—152no—
222#476—152no—
223#474—152no—
224#428—151no—
225#427—151no—
226#424—151no—
227#418D1511yescharacteristic
228C2×#102—151no—
229C22×C3×C5—21no—
230C22×C3×C5—21no—
231C22×C3×C5—24no—
232C22×C3×C5—42no—
233C22×C3×C5—42no—
234C22×C3×C5—84no—
235C22×C3×C5C22⋊Q812yes—
236C22×C3×C5C22×D411yescharacteristic
237C3×C4×C5—41no—
238C3×C4×C5—41no—
239C3×C4×C5C22⋊Q812yes—
240C3×C4×C5C22×Q811yescharacteristic
241C3×C4×C5C2×#4111yescharacteristic
242Dic15—22no—
243Dic15—22no—
244Dic15—42no—
245Dic15—41no—
246Dic15—41no—
247C24×C3—21no—
248C2×C3×D4—21no—
249C2×C3×D4—22no—
250C2×C3×D4Dic1014yescomplemented
251C2×C3×D4C22×D512yes—
252C22×C3×C4C22×D511yescharacteristic
253C22×Dic3—102no—
254C22×Dic3—102no—
255C22×Dic3—101no—
256C2×Dic6—102no—
257C3×M16—22no—
258C2×C3×C8C22×D512yes—
259C3×C4⋊C4C22×D511yescharacteristic
260C3×C4⋊C4C22×D511yescharacteristic
261C3×C22⋊C4—21no—
262C3×C42—21no—
263#217—52no—
264#217—102no—
265#217—102no—
266#217—101no—
267C4⋊Dic3—51no—
268C4⋊Dic3—51no—
269C4⋊Dic3—51no—
270C4⋊Dic3—51no—
271C4⋊Dic3—52no—
272C4⋊Dic3—102no—
273C4⋊Dic3—101no—
274Dic3⋊C4—102no—
275Dic3⋊C4—102no—
276C4×Dic3—52no—
277C4×Dic3—101no—
278C2×C3⋊C8—51no—
279C2×C3⋊C8—51no—
280C2×C3⋊C8—102no—
281C23×C5—22no—
282C23×C5—24no—
283C23×C5—81no—
284C23×C5—81no—
285C23×C5D3×D411yescharacteristic
286C5×D4—24no—
287C5×D4—22no—
288C5×D4C2×Dic614yes—
289C2×C4×C5—21no—
290C2×C4×C5—21no—
291C2×C4×C5C2×Dic612yes—
292C2×C4×C5D3×D411yescharacteristic
293C2×C4×C5#23712yes—
294C2×C4×C5C23×D311yescharacteristic
295C2×Dic5—32no—
296C2×Dic5—32no—
297C2×Dic5—61no—
298C2×Dic5—61no—
299C2×Dic5—62no—
300C2×Dic5—242no—
301C2×Dic5—242no—
302C2×Dic5—241no—
303Dic10—242no—
304C5×C8—22no—
305#168—61no—
306#168—61no—
307C22×D4C2×D1511yescharacteristic
308C2×M16C2×D1511yescharacteristic
309C22×C8—151no—
310C4⋊Q8—152no—
311C4⋊Q8—301no—
312#125—152no—
313#125—301no—
314C22⋊Q8—302no—
315C4×D4—152no—
316C4×D4—152no—
317C4×D4—301no—
318C4×D4—301no—
319#117—151no—
320#117—151no—
321#117C2×D1511yescharacteristic
322C2×C4⋊C4—151no—
323C2×C22⋊C4—302no—
324#107—152no—
325#107—151no—
326#107—151no—
327#106—152no—
328#106—151no—
329#106—151no—
330C4⋊C8—152no—
331C4⋊C8—152no—
332#102—152no—
333#102—152no—
334#102C2×D1512yes—
335#102C2×D1512yes—
336C2×C3×C5—22no—
337C2×C3×C5—44no—
338C2×C3×C5C2×C22⋊Q811yescharacteristic
339C2×C3×C5#57611yescharacteristic
340C2×C3×C5#57711yescharacteristic
341C23×C3—22no—
342C23×C3—24no—
343C23×C3—81no—
344C23×C3—81no—
345C23×C3D4×D511yescharacteristic
346C3×D4—22no—
347C3×D4—24no—
348C3×D4C2×Dic1014yes—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4C2×Dic1012yes—
352C2×C3×C4D4×D511yescharacteristic
353C2×C3×C4#71112yes—
354C2×C3×C4C23×D511yescharacteristic
355C2×Dic3—52no—
356C2×Dic3—52no—
357C2×Dic3—101no—
358C2×Dic3—101no—
359C2×Dic3—102no—
360C2×Dic3—402no—
361C2×Dic3—402no—
362C2×Dic3—401no—
363Dic6—402no—
364C3×C8—22no—
365C3⋊C8—101no—
366C3⋊C8—101no—
367C22×C5—21no—
368C22×C5—21no—
369C22×C5—24no—
370C22×C5—42no—
371C22×C5—42no—
372C22×C5—84no—
373C22×C5#87812yes—
374C22×C5C2×D3×D411yescharacteristic
375C4×C5—41no—
376C4×C5—41no—
377C4×C5#87812yes—
378C4×C5C22×Dic611yescharacteristic
379C4×C5C2×#23711yescharacteristic
380Dic5—62no—
381Dic5—62no—
382Dic5—122no—
383Dic5—121no—
384Dic5—121no—
385C24—21no—
386C2×Q8—302no—
387C2×D4—21no—
388C2×D4—22no—
389C2×D4Dic3014yescomplemented
390C2×D4C22×D1512yes—
391C22×C4—302no—
392C22×C4—302no—
393C22×C4—301no—
394C22×C4C22×D1511yescharacteristic
395M16—22no—
396C2×C8—151no—
397C2×C8—151no—
398C2×C8—302no—
399C2×C8C22×D1512yes—
400C4⋊C4—151no—
401C4⋊C4—151no—
402C4⋊C4—152no—
403C4⋊C4—151no—
404C4⋊C4—151no—
405C4⋊C4—302no—
406C4⋊C4—302no—
407C4⋊C4—302no—
408C4⋊C4—301no—
409C4⋊C4C22×D1511yescharacteristic
410C4⋊C4C22×D1511yescharacteristic
411C22⋊C4—21no—
412C22⋊C4—152no—
413C22⋊C4—302no—
414C22⋊C4—302no—
415C22⋊C4—301no—
416C42—21no—
417C42—152no—
418C42—301no—
419C3×C5#308111yescharacteristic, complemented, Hall
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—24no—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—84no—
426C22×C3#402412yes—
427C22×C3C2×D4×D511yescharacteristic
428C3×C4—41no—
429C3×C4—41no—
430C3×C4#402412yes—
431C3×C4C22×Dic1011yescharacteristic
432C3×C4C2×#71111yescharacteristic
433Dic3—102no—
434Dic3—102no—
435Dic3—202no—
436Dic3—201no—
437Dic3—201no—
438C2×C5—22no—
439C2×C5—44no—
440C2×C5C2×#87811yescharacteristic
441C2×C5#577411yescharacteristic
442C2×C5#577511yescharacteristic
443C23—22no—
444C23—24no—
445C23—81no—
446C23—81no—
447C23D4×D1511yescharacteristic
448Q8—1202no—
449D4—24no—
450D4—22no—
451D4C2×Dic3014yes—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—152no—
455C2×C4—152no—
456C2×C4—301no—
457C2×C4—301no—
458C2×C4—302no—
459C2×C4—1202no—
460C2×C4—1202no—
461C2×C4—1201no—
462C2×C4C2×Dic3012yes—
463C2×C4D4×D1511yescharacteristic
464C2×C4#681912yes—
465C2×C4C23×D1511yescharacteristic
466C8—22no—
467C8—301no—
468C8—301no—
469C2×C3—22no—
470C2×C3—44no—
471C2×C3C2×#402411yescharacteristic
472C2×C3#6610811yescharacteristic
473C2×C3#6610911yescharacteristic
474C5#7891811yescharacteristic, complemented, Sylow
475C22—21no—
476C22—21no—
477C22—24no—
478C22—42no—
479C22—42no—
480C22—84no—
481C22#9176712yes—
482C22C2×D4×D1511yescharacteristic
483C4—41no—
484C4—41no—
485C4—302no—
486C4—302no—
487C4—601no—
488C4—602no—
489C4—601no—
490C4#9176712yes—
491C4C22×Dic3011yescharacteristic
492C4C2×#681911yescharacteristic
493C3#1061064911yescharacteristic, complemented, Sylow
494C2—22no—
495C2—44no—
496C2C2×#9176711yescharacteristic
497C2#1175653311yescharacteristic
498C2#1175653411yescharacteristic
subgroupG/H
centerC22C2×D4×D15
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C5×#418C2
socleC22×C3×C5C22×D4
prime radicalC22×C3×C5×D4C22
hypercenter#418D15
hypocenterC3×C5#3081
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. #11756278C2semidirect
5. #11751222C2semidirect
6. #11751220C2semidirect
7. #11751099C2semidirect
8. #11750867C2semidirect
9. #11750864C2semidirect
10. #11750862C2semidirect
11. #11750860C2semidirect
25. #91779C22semidirect
26. #91777C22semidirect
34. #91102C22semidirect
35. #91101C22semidirect
36. #91098C22semidirect
37. #91098C22semidirect
38. #91092C22semidirect
39. #91092C22semidirect
40. #91091C22semidirect
41. #91091C22semidirect
57. C2×C3×C5×D4Q8semidirect
187. C2×C5×D4Dic6semidirect
250. C2×C3×D4Dic10semidirect
389. C2×D4Dic30semidirect
419. C3×C5#3081semidirect
474. C5#78918semidirect
493. C3#10610649semidirect

Maximal subgroups

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

subgroupindex
#117565572
#117562782
C3×C5×#4182
C2×#911162
#117512222
#117512202
#117510992
#117508672
#117508642
#117508622
#117508602
#106106493
#789185