Finite Group Databasebeta

← #14299739

Subgroups — #14299739

Order 1920.

604 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5523C211yesmaximal, characteristic
2#11751096C211yesmaximal, characteristic, complemented
3#11748834C211yesmaximal, characteristic, complemented
4C2×#90979C211yesmaximal, characteristic
5#11748390C211yesmaximal, characteristic
6C3×#64752C211yesmaximal, characteristic, complemented
7#11745986C211yesmaximal, characteristic, complemented
8#10602457—31nomaximal, Hall
9C2×C5×#235C2211yescharacteristic
10C3×C5×#117C2211yescharacteristic
11C2×C4⋊Dic15C2211yescharacteristic
12C2×#6710C2211yescharacteristic
13C2×C5×C4⋊Dic3C2211yescharacteristic
14C5×#892—21no—
15C5×D3×C4⋊C4—21no—
16C5×#67⋊C4—22no—
17C4×C5×D12—21no—
18C4×C5×Dic6—21no—
19C2×C3×C4×Dic5C2211yescharacteristic
20C2×C3×#681C2211yescharacteristic
21C2×#6668—22no—
22#91092—21no—
23#91091—21no—
24#90979C414yescomplemented
25#90973—21no—
26#90967—21no—
27#68771—51nomaximal, Hall
28C5×#520—31no—
29C2×#3969—31no—
30#65521—31no—
31#65284—31no—
32#65265—31no—
33#64752D311yescharacteristic, complemented
34#64750—31no—
35C22×C3×C4×C5C2311yescharacteristic
36C22×Dic15—21no—
37C2×C5×#67—21no—
38C22×C5×Dic3D412yes—
39C5×#235—21no—
40C5×#235—21no—
41C5×#235C2×C412yes—
42C5×#235Q812yes—
43C2×C5×D12D411yescharacteristic
44C2×C4×C5×D3—21no—
45C2×C4×C5×D3—22no—
46C2×C5×Dic6D411yescharacteristic
47C22×C3×Dic5—21no—
48C3×C5×C4⋊C4D411yescharacteristic
49C3×C5×C4⋊C4D411yescharacteristic
50C3×C5×C22⋊C4—21no—
51C3×C42×C5—21no—
52C4⋊Dic15—21no—
53C4⋊Dic15—21no—
54#6710—21no—
55#6710C2×C412yescomplemented
56C2×C3⋊#168—21no—
57C5×(C2×D3)⋊C4—22no—
58C5×C4⋊Dic3—21no—
59C5×C4⋊Dic3—21no—
60C5×Dic3⋊C4—22no—
61C4×C5×Dic3—22no—
62C3×C4×Dic5—21no—
63C3×C4×Dic5C2×C412yescomplemented
64C3×#681—21no—
65C3×#681—21no—
66C2×C3×#168—21no—
67#6668—44no—
68#5523D511yescharacteristic
69#5133—51no—
70#4995—51no—
71#4935—51no—
72C2×#805—51no—
73C3×#339—51no—
74#4554—51no—
75C2×C5×#41—31no—
76C4×C5×Q8—61no—
77C4×C5×D4—61no—
78C4×C5×D4—62no—
79C5×#117—61no—
80C5×#117C2×D311yescharacteristic
81C2×C5×C4⋊C4—31no—
82C2×C5×C4⋊C4—61no—
83C2×C4⋊Dic5—31no—
84C2×#683—62no—
85C2×C4×Dic5C2×D311yescharacteristic
86C2×#681—31no—
87C2×#681C2×D311yescharacteristic
88#3969—34no—
89#3967—61no—
90#3964—61no—
91#3940—61no—
92#3939—61no—
93#1859—151noSylow
94C23×C3×C5C2×D411yescharacteristic
95C22×C5×D3—21no—
96C2×C3×C4×C5—21no—
97C2×C3×C4×C5—21no—
98C2×C3×C4×C5C22×C411yescharacteristic
99C2×C3×C4×C5C2×D411yescharacteristic
100C2×C3×C4×C5C2×D411yescharacteristic
101C2×C3×C4×C5C2×Q811yescharacteristic
102C2×C3×C4×C5#4111yescharacteristic
103C2×C3×C4×C5#4111yescharacteristic
104C2×Dic15—42no—
105C2×Dic15—41no—
106C5×#67—22no—
107C5×#67—22no—
108C2×C5×Dic3—21no—
109C2×C5×Dic3—22no—
110C2×C5×Dic3—42no—
111C5×D12—21no—
112C5×D12C4⋊C412yescomplemented
113C4×C5×D3—22no—
114C4×C5×D3—22no—
115C4×C5×D3—82no—
116C5×Dic6—21no—
117C5×Dic6C4⋊C412yescomplemented
118C2×C3×Dic5—22no—
119C2×C3×Dic5—22no—
120C2×C3×Dic5—22no—
121C3⋊#168—22no—
122C3×#168—41no—
123C2×#235C2×D511yescharacteristic
124C2×C3×M16—51no—
125C3×#117C2×D511yescharacteristic
126C2×C3×C42—51no—
127C2×C4⋊Dic3—51no—
128C2×C4⋊Dic3—102no—
129C2×C4⋊Dic3C2×D511yescharacteristic
130C2×#208—51no—
131#892—21no—
132D3×C4⋊C4—21no—
133#67⋊C4—22no—
134C4×D12—21no—
135C4×Dic6—21no—
136#821—101no—
137#816—101no—
138#808—101no—
139#807—101no—
140#805—54no—
141C5×#41—32no—
142C5×#41—32no—
143C5×#41—61no—
144C5×#41—61no—
145C2×C5×Q8—31no—
146C2×C5×D4—31no—
147C2×C5×D4—61no—
148C22×C4×C5—32no—
149C22×C4×C5—62no—
150C22×C4×C5—61no—
151C22×C4×C5C22×D311yescharacteristic
152C22×Dic5—21no—
153C22×Dic5—61no—
154C5×C4⋊C4—62no—
155C5×C4⋊C4—61no—
156C5×C4⋊C4—61no—
157C5×C4⋊C4D1211yescharacteristic
158C5×C4⋊C4D1211yescharacteristic
159C5×C22⋊C4—21no—
160C5×C22⋊C4—62no—
161C42×C5—21no—
162C42×C5—62no—
163C4⋊Dic5—61no—
164C4⋊Dic5—61no—
165#683—124no—
166C4×Dic5—21no—
167C4×Dic5C4×D312yescomplemented
168#681—21no—
169#681—21no—
170#681—32no—
171#681—61no—
172C2×#168—21no—
173C2×#168—61no—
174#520—31no—
175#428—151no—
176C2×#104—151no—
177#417—151no—
178#384—151no—
179#343—151no—
180#339—51no—
181C22×C3×C5—21no—
182C22×C3×C5—21no—
183C22×C3×C5C4×D411yescharacteristic
184C22×C3×C5#12011yescharacteristic
185C22×C3×C5C22⋊Q811yescharacteristic
186C2×C5×D3—22no—
187C2×C5×D3—41no—
188C2×C5×D3—41no—
189C3×C4×C5—41no—
190C3×C4×C5—41no—
191C3×C4×C5C2×C4⋊C411yescharacteristic
192C3×C4×C5C4×D411yescharacteristic
193C3×C4×C5C22⋊Q811yescharacteristic
194C3×C4×C5#12311yescharacteristic
195Dic15—81no—
196C3×Dic5—24no—
197C5×Dic3—22no—
198C5×Dic3—42no—
199C22×C3×C4—101no—
200C22×C3×C4C22×D511yescharacteristic
201C2×#67—21no—
202C22×Dic3—101no—
203C22×Dic3C22⋊D512yes—
204#235—21no—
205#235—21no—
206#235Dic1012yes—
207#235C4×D512yes—
208C2×D12C22⋊D511yescharacteristic
209C2×C4×D3—21no—
210C2×C4×D3—22no—
211C2×Dic6C22⋊D511yescharacteristic
212C3×M16—101no—
213C3×M16—101no—
214C2×C3×C8—101no—
215C3×C4⋊C4C22⋊D511yescharacteristic
216C3×C4⋊C4C22⋊D511yescharacteristic
217C3×C22⋊C4—21no—
218C3×C42—21no—
219C3×C42—52no—
220C3×C42—101no—
221(C2×D3)⋊C4—22no—
222C4⋊Dic3—21no—
223C4⋊Dic3—21no—
224C4⋊Dic3—101no—
225C4⋊Dic3—101no—
226C4⋊Dic3—204no—
227Dic3⋊C4—22no—
228C4×Dic3—22no—
229#208—52no—
230#208—101no—
231C2×C3⋊C8—101no—
232C23×C5—61no—
233C23×C5D3×D411yescharacteristic
234C5×Q8—32no—
235C5×Q8—61no—
236C5×D4—32no—
237C5×D4—62no—
238C5×D4—62no—
239C5×D4—61no—
240C2×C4×C5—21no—
241C2×C4×C5—21no—
242C2×C4×C5—61no—
243C2×C4×C5—62no—
244C2×C4×C5—62no—
245C2×C4×C5—62no—
246C2×C4×C5—122no—
247C2×C4×C5—242no—
248C2×C4×C5C2×C4×D311yescharacteristic
249C2×C4×C5C2×D1211yescharacteristic
250C2×C4×C5#23511yescharacteristic
251C2×C4×C5D3×D411yescharacteristic
252C2×C4×C5#23711yescharacteristic
253C2×C4×C5D3×Q811yescharacteristic
254C2×Dic5—22no—
255C2×Dic5—22no—
256C2×Dic5—22no—
257C2×Dic5—122no—
258C2×Dic5—121no—
259#168—41no—
260#168—62no—
261C2×#41—31no—
262C2×M16—51no—
263C2×M16—151no—
264C4×Q8—61no—
265C4×D4—61no—
266C4×D4—62no—
267#117—61no—
268#117D3×D511yescharacteristic
269C2×C4⋊C4—31no—
270C2×C4⋊C4—61no—
271C2×C4⋊C4—151no—
272C2×C4⋊C4—302no—
273C2×C42—51no—
274#107—301no—
275#106—301no—
276#104—154no—
277#103—301no—
278#102—301no—
279C2×C3×C5—21no—
280C2×C3×C5—21no—
281C2×C3×C5#38511yescharacteristic
282C2×C3×C5#45311yescharacteristic
283C2×C3×C5#45611yescharacteristic
284C5×D3—42no—
285C23×C3C2×C22⋊D511yescharacteristic
286C22×D3—21no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—102no—
290C2×C3×C4—102no—
291C2×C3×C4—102no—
292C2×C3×C4C2×Dic1011yescharacteristic
293C2×C3×C4C2×C4×D511yescharacteristic
294C2×C3×C4#70911yescharacteristic
295C2×C3×C4#70911yescharacteristic
296C2×C3×C4C2×C22⋊D511yescharacteristic
297C2×C3×C4C2×C22⋊D511yescharacteristic
298#67—22no—
299#67—22no—
300C2×Dic3—21no—
301C2×Dic3—22no—
302C2×Dic3—42no—
303C2×Dic3—202no—
304C2×Dic3—201no—
305D12—21no—
306D12#68312yescomplemented
307C4×D3—22no—
308C4×D3—22no—
309C4×D3—82no—
310Dic6—21no—
311Dic6#68312yescomplemented
312C3×C8—201no—
313C3⋊C8—102no—
314C22×C5—21no—
315C22×C5—21no—
316C22×C5—62no—
317C22×C5—121no—
318C22×C5—121no—
319C22×C5#67⋊C411yescharacteristic
320C22×C5#88211yescharacteristic
321C22×C5#89611yescharacteristic
322C4×C5—41no—
323C4×C5—41no—
324C4×C5—62no—
325C4×C5—122no—
326C4×C5C4×D1211yescharacteristic
327C4×C5#88311yescharacteristic
328C4×C5D3×C4⋊C411yescharacteristic
329C4×C5#89711yescharacteristic
330Dic5—24no—
331Dic5—241no—
332#41—32no—
333#41—32no—
334#41—61no—
335#41—61no—
336C2×Q8—31no—
337C2×D4—31no—
338C2×D4—61no—
339C22×C4—32no—
340C22×C4—62no—
341C22×C4—61no—
342C22×C4—101no—
343C22×C4—301no—
344C22×C4C2×D3×D511yescharacteristic
345M16—101no—
346M16—101no—
347M16—152no—
348M16—301no—
349C2×C8—101no—
350C2×C8—301no—
351C4⋊C4—62no—
352C4⋊C4—61no—
353C4⋊C4—61no—
354C4⋊C4—301no—
355C4⋊C4—301no—
356C4⋊C4—604no—
357C4⋊C4#120311yescharacteristic
358C4⋊C4#120311yescharacteristic
359C22⋊C4—21no—
360C22⋊C4—62no—
361C42—21no—
362C42—52no—
363C42—62no—
364C42—101no—
365C3×C5#185911yescharacteristic, complemented, Hall
366C22×C3—21no—
367C22×C3—21no—
368C22×C3#407011yescharacteristic
369C22×C3C4×C22⋊D511yescharacteristic
370C22×C3#409911yescharacteristic
371C2×D3—22no—
372C2×D3—41no—
373C2×D3—41no—
374C3×C4—41no—
375C3×C4—41no—
376C3×C4—104no—
377C3×C4C2×#68311yescharacteristic
378C3×C4#407011yescharacteristic
379C3×C4C4×C22⋊D511yescharacteristic
380C3×C4#407511yescharacteristic
381Dic3—22no—
382Dic3—42no—
383Dic3—401no—
384C2×C5—21no—
385C2×C5—21no—
386C2×C5—122no—
387C2×C5#466811yescharacteristic
388C2×C5#482211yescharacteristic
389C2×C5#482611yescharacteristic
390C23—61no—
391C23D3×C22⋊D511yescharacteristic
392Q8—32no—
393Q8—61no—
394D4—32no—
395D4—62no—
396D4—62no—
397D4—61no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—61no—
401C2×C4—62no—
402C2×C4—62no—
403C2×C4—62no—
404C2×C4—102no—
405C2×C4—102no—
406C2×C4—102no—
407C2×C4—122no—
408C2×C4—242no—
409C2×C4—602no—
410C2×C4—601no—
411C2×C4#676511yescharacteristic
412C2×C4D3×Dic1011yescharacteristic
413C2×C4#677111yescharacteristic
414C2×C4C4×D3×D511yescharacteristic
415C2×C4C2×#120311yescharacteristic
416C2×C4D3×C22⋊D511yescharacteristic
417C8—201no—
418C8—302no—
419C2×C3—21no—
420C2×C3—21no—
421C2×C3#6549911yescharacteristic
422C2×C3#6553011yescharacteristic
423C2×C3#6553511yescharacteristic
424D3—42no—
425C5#6877111yescharacteristic, complemented, Sylow
426C22—21no—
427C22—21no—
428C22—62no—
429C22—121no—
430C22—121no—
431C22#9135211yescharacteristic
432C22#9144111yescharacteristic
433C22#9157011yescharacteristic
434C4—41no—
435C4—41no—
436C4—62no—
437C4—104no—
438C4—122no—
439C4—1201no—
440C4#9136511yescharacteristic
441C4D3×#68311yescharacteristic
442C4C4×#120311yescharacteristic
443C4#9145211yescharacteristic
444C3#1060245711yescharacteristic, complemented, Sylow
445C2—21no—
446C2—21no—
447C2—122no—
448C2#1174872211yescharacteristic
449C2#1174942011yescharacteristic
450C2#1174942311yescharacteristic
subgroupG/H
centerC22#91570
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC3×C5×#117C22
socleC22×C3×C5#120
prime radicalC2×C5×#235C22
cosocleC22×C3×C4×C5C23
hypercenter#117D3×D5
hypocenterC3×C5#1859
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
2. #11751096C2semidirect
3. #11748834C2semidirect
6. C3×#64752C2semidirect
7. #11745986C2semidirect
24. #90979C4semidirect
33. #64752D3semidirect
55. #6710C2×C4semidirect
63. C3×C4×Dic5C2×C4semidirect
112. C5×D12C4⋊C4semidirect
117. C5×Dic6C4⋊C4semidirect
167. C4×Dic5C4×D3semidirect
306. D12#683semidirect
311. Dic6#683semidirect
365. C3×C5#1859semidirect
425. C5#68771semidirect
444. C3#10602457semidirect

Maximal subgroups

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

subgroupindex
C5×#55232
#117510962
#117488342
C2×#909792
#117483902
C3×#647522
#117459862
#106024573
#687715