Finite Group Databasebeta

← #14299757

Subgroups — #14299757

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
1#11754492C211yesmaximal, characteristic
2C5×#5133C211yesmaximal, characteristic
3#11748866C211yesmaximal, characteristic, complemented
4#11748459C211yesmaximal, characteristic, complemented
5C2×#90978C211yesmaximal, characteristic
6#11746376C211yesmaximal, characteristic, complemented
7C3×#64774C211yesmaximal, characteristic, complemented
8#10601090—31nomaximal, Hall
9C2×C5×#235C2211yescharacteristic
10C2×C3×C5×M16C2211yescharacteristic
11#91815C2211yescharacteristic
12C2×C4×Dic15C2211yescharacteristic
13C2×C5×C4⋊Dic3C2211yescharacteristic
14C2×C3×C4⋊Dic5C2211yescharacteristic
15C2×C3×#681C2211yescharacteristic
16Dic5×#67—22no—
17C2×#6669—22no—
18#91435—21no—
19D3×C4⋊Dic5—21no—
20#91344—21no—
21#91338—21no—
22C5×#821—21no—
23C5×#816—21no—
24#90988—21no—
25#90985—21no—
26#90978C414yescomplemented
27#69002—51nomaximal, Hall
28#66161—31no—
29C5×#417—31no—
30C2×#3969—31no—
31#65265—31no—
32#65253—31no—
33#64775—31no—
34#64774D311yescharacteristic, complemented
35C22×C3×C4×C5C2311yescharacteristic
36C22×Dic15—21no—
37C2×C5×#67—21no—
38C22×C5×Dic3—21no—
39C5×#235—21no—
40C5×#235—21no—
41C5×#235C2×C412yes—
42C5×#235Q812yes—
43C2×C5×D12D411yescharacteristic
44C2×C4×C5×D3—21no—
45C2×C5×Dic6D411yescharacteristic
46C22×C3×Dic5D412yes—
47C2×D3×Dic5—22no—
48C3×C5×M16—21no—
49C3×C5×M16—21no—
50C2×C3×C5×C8—21no—
51#6719—21no—
52C4⋊Dic15D411yescharacteristic
53C4⋊Dic15D411yescharacteristic
54C4×Dic15—21no—
55C4×Dic15—21no—
56C4×Dic15C2×C412yescomplemented
57C5×C4⋊Dic3—21no—
58C5×C4⋊Dic3—21no—
59C3×C4⋊Dic5—21no—
60C3×C4⋊Dic5—21no—
61C3×#681—21no—
62C3×#681C2×C412yescomplemented
63C2×C3×#168—21no—
64Dic3⋊Dic5—22no—
65#6669—44no—
66(C2×D3)⋊Dic5—22no—
67Dic3×Dic5—22no—
68#5504—51no—
69C2×#825—51no—
70#5133D511yescharacteristic
71#5121—51no—
72#4975—51no—
73C3×#343—51no—
74#4555—51no—
75C2×C5×#41—31no—
76C2×C5×M16C2×D311yescharacteristic
77C2×C5×C4⋊C4—31no—
78Q8×Dic5—61no—
79D4×Dic5—61no—
80D4×Dic5—62no—
81#4072—31no—
82#4072—61no—
83C2×C4⋊Dic5—61no—
84C2×C4⋊Dic5C2×D311yescharacteristic
85C2×#683—62no—
86C2×C4×Dic5—31no—
87C2×#681C2×D311yescharacteristic
88C5×#103—61no—
89C5×#102—61no—
90#3969—34no—
91#3940—61no—
92#3939—61no—
93#1859—151noSylow
94C23×C3×C5C2×D411yescharacteristic
95C22×C5×D3—21no—
96C2×C3×C4×C5C22×C411yescharacteristic
97C2×C3×C4×C5C2×D411yescharacteristic
98C2×C3×C4×C5C2×D411yescharacteristic
99C2×C3×C4×C5C2×Q811yescharacteristic
100C2×C3×C4×C5#4111yescharacteristic
101C2×C3×C4×C5#4111yescharacteristic
102C2×Dic15—22no—
103C2×Dic15—22no—
104C2×Dic15—22no—
105C2×Dic15—21no—
106C2×Dic15—21no—
107C5×#67—22no—
108C5×#67—22no—
109C2×C5×Dic3—21no—
110C2×C5×Dic3—42no—
111C2×C5×Dic3—41no—
112C5×D12—21no—
113C5×D12C4⋊C412yescomplemented
114C4×C5×D3—22no—
115C4×C5×D3—22no—
116C5×Dic6—21no—
117C5×Dic6C4⋊C412yescomplemented
118C2×C3×Dic5—22no—
119C2×C3×Dic5—42no—
120D3×Dic5—82no—
121C3×C5×C8—41no—
122C3×#168—22no—
123C2×#235C2×D511yescharacteristic
124C2×C3×M16—51no—
125C2×C3×M16C2×D511yescharacteristic
126C2×C3×C4⋊C4—51no—
127C4×#67—102no—
128#926—51no—
129C2×C4⋊Dic3C2×D511yescharacteristic
130C2×Dic3⋊C4—102no—
131C2×C4×Dic3—51no—
132#893—101no—
133#892—101no—
134D3×C4⋊C4—101no—
135#887—101no—
136#825—54no—
137#821—21no—
138#818—101no—
139#817—101no—
140#816—21no—
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—61no—
149C22×C4×C5—61no—
150C22×C4×C5C22×D311yescharacteristic
151C22×Dic5—62no—
152C22×Dic5—61no—
153C22×Dic5D1212yes—
154C5×M16—21no—
155C5×M16—21no—
156C2×C5×C8—21no—
157C5×C4⋊C4—61no—
158C5×C4⋊C4—61no—
159C22⋊Dic5—62no—
160C22⋊Dic5—61no—
161C4⋊Dic5—21no—
162C4⋊Dic5—21no—
163C4⋊Dic5—31no—
164C4⋊Dic5—31no—
165C4⋊Dic5—62no—
166#683—124no—
167C4×Dic5—32no—
168C4×Dic5—62no—
169C4×Dic5—61no—
170C4×Dic5—61no—
171#681—21no—
172#681C4×D312yescomplemented
173C2×#168—21no—
174#520—151no—
175#428—151no—
176C2×#104—151no—
177#417—31no—
178#384—151no—
179#343—51no—
180#339—151no—
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×C5C2×C4⋊C411yescharacteristic
190C3×C4×C5C4×D411yescharacteristic
191C3×C4×C5C22⋊Q811yescharacteristic
192C3×C4×C5#12311yescharacteristic
193Dic15—24no—
194Dic15—41no—
195Dic15—41no—
196C3×Dic5—42no—
197C5×Dic3—22no—
198C5×Dic3—81no—
199C22×C3×C4—52no—
200C22×C3×C4C22×D511yescharacteristic
201C2×#67—21no—
202C22×Dic3—21no—
203C22×Dic3—101no—
204#235—21no—
205#235—21no—
206#235Dic1012yes—
207#235C4×D512yes—
208C2×D12C22⋊D511yescharacteristic
209C2×C4×D3—21no—
210C2×C4×D3—102no—
211C2×Dic6C22⋊D511yescharacteristic
212C3×M16—21no—
213C3×M16—21no—
214C3×M16—52no—
215C3×M16—101no—
216C2×C3×C8—21no—
217C2×C3×C8—101no—
218C3×C4⋊C4—101no—
219C3×C4⋊C4—101no—
220#217—101no—
221(C2×D3)⋊C4—102no—
222C4⋊Dic3—21no—
223C4⋊Dic3—21no—
224C4⋊Dic3—51no—
225C4⋊Dic3—51no—
226Dic3⋊C4—102no—
227Dic3⋊C4—204no—
228C4×Dic3—52no—
229C4×Dic3—102no—
230C4×Dic3—101no—
231C4×Dic3—101no—
232C23×C5—61no—
233C23×C5C2×D1211yescharacteristic
234C5×Q8—32no—
235C5×Q8—61no—
236C5×D4—32no—
237C5×D4—62no—
238C5×D4—62no—
239C5×D4—61no—
240C2×C4×C5—61no—
241C2×C4×C5—62no—
242C2×C4×C5—62no—
243C2×C4×C5—122no—
244C2×C4×C5—121no—
245C2×C4×C5C2×C4×D311yescharacteristic
246C2×C4×C5#23511yescharacteristic
247C2×C4×C5D3×D411yescharacteristic
248C2×C4×C5D3×D411yescharacteristic
249C2×C4×C5#23711yescharacteristic
250C2×C4×C5D3×Q811yescharacteristic
251C2×Dic5—22no—
252C2×Dic5—42no—
253C2×Dic5—62no—
254C2×Dic5—62no—
255C2×Dic5—62no—
256C2×Dic5—61no—
257C2×Dic5—61no—
258C2×Dic5—242no—
259C5×C8—41no—
260#168—22no—
261C2×#41—31no—
262C2×M16—51no—
263C2×M16D3×D511yescharacteristic
264C4×Q8—301no—
265C4×D4—301no—
266C4×D4—302no—
267#117—151no—
268#117—301no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—51no—
271C2×C4⋊C4—301no—
272C2×C4⋊C4—302no—
273C2×C42—151no—
274#107—301no—
275#106—301no—
276#104—154no—
277#103—61no—
278#102—61no—
279C2×C3×C5—21no—
280C2×C3×C5—21no—
281C2×C3×C5#38511yescharacteristic
282C2×C3×C5#45311yescharacteristic
283C2×C3×C5#45611yescharacteristic
284C5×D3—42no—
285C23×C3D4×D511yescharacteristic
286C22×D3—21no—
287C2×C3×C4—102no—
288C2×C3×C4—202no—
289C2×C3×C4C2×Dic1011yescharacteristic
290C2×C3×C4C2×C4×D511yescharacteristic
291C2×C3×C4D4×D511yescharacteristic
292C2×C3×C4#71111yescharacteristic
293C2×C3×C4#71111yescharacteristic
294C2×C3×C4C2×C22⋊D511yescharacteristic
295#67—22no—
296#67—22no—
297C2×Dic3—21no—
298C2×Dic3—42no—
299C2×Dic3—41no—
300C2×Dic3—102no—
301C2×Dic3—102no—
302C2×Dic3—102no—
303C2×Dic3—101no—
304C2×Dic3—101no—
305D12—21no—
306D12#68312yescomplemented
307C4×D3—22no—
308C4×D3—22no—
309C4×D3—402no—
310Dic6—21no—
311Dic6#68312yescomplemented
312C3×C8—41no—
313C3×C8—102no—
314C22×C5—21no—
315C22×C5—21no—
316C22×C5—62no—
317C22×C5—121no—
318C22×C5—121no—
319C22×C5C4×D1211yescharacteristic
320C22×C5#88211yescharacteristic
321C22×C5#89711yescharacteristic
322C4×C5—62no—
323C4×C5—241no—
324C4×C5#67⋊C411yescharacteristic
325C4×C5#88311yescharacteristic
326C4×C5D3×C4⋊C411yescharacteristic
327C4×C5#89611yescharacteristic
328Dic5—42no—
329Dic5—64no—
330Dic5—121no—
331Dic5—121no—
332#41—32no—
333#41—32no—
334#41—61no—
335#41—61no—
336C2×Q8—31no—
337C2×D4—31no—
338C2×D4—61no—
339C22×C4—52no—
340C22×C4—61no—
341C22×C4—61no—
342C22×C4—302no—
343C22×C4—301no—
344C22×C4C2×D3×D511yescharacteristic
345M16—21no—
346M16—21no—
347M16—52no—
348M16—101no—
349C2×C8—21no—
350C2×C8—101no—
351C4⋊C4—61no—
352C4⋊C4—61no—
353C4⋊C4—101no—
354C4⋊C4—101no—
355C4⋊C4—151no—
356C4⋊C4—151no—
357C4⋊C4—302no—
358C4⋊C4—604no—
359C22⋊C4—302no—
360C22⋊C4—301no—
361C42—152no—
362C42—302no—
363C42—301no—
364C42—301no—
365C3×C5#185911yescharacteristic, complemented, Hall
366C22×C3—21no—
367C22×C3—21no—
368C22×C3#402411yescharacteristic
369C22×C3#402711yescharacteristic
370C22×C3#408311yescharacteristic
371C2×D3—22no—
372C2×D3—41no—
373C2×D3—41no—
374C3×C4—202no—
375C3×C4#402411yescharacteristic
376C3×C4#402711yescharacteristic
377C3×C4C2×#68311yescharacteristic
378C3×C4#408111yescharacteristic
379Dic3—22no—
380Dic3—81no—
381Dic3—104no—
382Dic3—201no—
383Dic3—201no—
384C2×C5—21no—
385C2×C5—21no—
386C2×C5—122no—
387C2×C5#466811yescharacteristic
388C2×C5#474711yescharacteristic
389C2×C5#475211yescharacteristic
390C23—61no—
391C23D5×D1211yescharacteristic
392Q8—32no—
393Q8—61no—
394D4—32no—
395D4—62no—
396D4—62no—
397D4—61no—
398C2×C4—61no—
399C2×C4—62no—
400C2×C4—62no—
401C2×C4—102no—
402C2×C4—122no—
403C2×C4—121no—
404C2×C4—202no—
405C2×C4—302no—
406C2×C4—302no—
407C2×C4—302no—
408C2×C4—301no—
409C2×C4—301no—
410C2×C4—1202no—
411C2×C4D3×Dic1011yescharacteristic
412C2×C4C4×D3×D511yescharacteristic
413C2×C4#677711yescharacteristic
414C2×C4#678011yescharacteristic
415C2×C4#678211yescharacteristic
416C2×C4D3×C22⋊D511yescharacteristic
417C8—41no—
418C8—102no—
419C2×C3—21no—
420C2×C3—21no—
421C2×C3#6523511yescharacteristic
422C2×C3#6548711yescharacteristic
423C2×C3#6549411yescharacteristic
424D3—42no—
425C5#6900211yescharacteristic, complemented, Sylow
426C22—21no—
427C22—21no—
428C22—62no—
429C22—121no—
430C22—121no—
431C22#9140511yescharacteristic
432C22#9141711yescharacteristic
433C22#9147511yescharacteristic
434C4—62no—
435C4—202no—
436C4—241no—
437C4—304no—
438C4—601no—
439C4—601no—
440C4D3×#68311yescharacteristic
441C4#9143611yescharacteristic
442C4#9144211yescharacteristic
443C4#9155511yescharacteristic
444C3#1060109011yescharacteristic, complemented, Sylow
445C2—21no—
446C2—21no—
447C2—122no—
448C2#1174824611yescharacteristic
449C2#1174825811yescharacteristic
450C2#1174889311yescharacteristic
subgroupG/H
centerC22#91475
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC2×C3×C5×M16C22
socleC22×C3×C5#120
prime radicalC2×C5×#235C22
cosocleC22×C3×C4×C5C23
hypercenterC2×M16D3×D5
hypocenterC3×C5#1859
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
3. #11748866C2semidirect
4. #11748459C2semidirect
6. #11746376C2semidirect
7. C3×#64774C2semidirect
26. #90978C4semidirect
34. #64774D3semidirect
56. C4×Dic15C2×C4semidirect
62. C3×#681C2×C4semidirect
113. C5×D12C4⋊C4semidirect
117. C5×Dic6C4⋊C4semidirect
172. #681C4×D3semidirect
306. D12#683semidirect
311. Dic6#683semidirect
365. C3×C5#1859semidirect
425. C5#69002semidirect
444. C3#10601090semidirect

Maximal subgroups

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

subgroupindex
#117544922
C5×#51332
#117488662
#117484592
C2×#909782
#117463762
C3×#647742
#106010903
#690025