Finite Group Databasebeta

← #14299577

Subgroups — #14299577

Order 1920.

648 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756280C211yesmaximal, characteristic
2C2×#91107C211yesmaximal, characteristic
3#11751222C211yesmaximal, characteristic, complemented
4#11751096C211yesmaximal, characteristic, complemented
5#11751040C211yesmaximal, characteristic
6C3×C5×#339C211yesmaximal, characteristic, complemented
7#11746344C211yesmaximal, characteristic, complemented
8#10599551—31nomaximal, Hall
9C22×D60C2211yescharacteristic
10C2×C3×C5×M16C2211yescharacteristic
11C3×C5×#117C2211yescharacteristic
12C2×C3×C42×C5C2211yescharacteristic
13C2×#6714—22no—
14#91815C2211yescharacteristic
15C2×C4⋊Dic15C2211yescharacteristic
16C2×#6710C2211yescharacteristic
17#91782—21no—
18#91782—21no—
19C22⋊C4×D15—22no—
20C4×D60—21no—
21C4×D60—21no—
22#91107C414yescomplemented
23#91102—21no—
24#91101—21no—
25#91092—21no—
26#91091—21no—
27#68758—51nomaximal, Hall
28#65855—31no—
29C2×#3955—31no—
30#65410—31no—
31#65284—31no—
32#65228—31no—
33C5×#339D311yescharacteristic, complemented
34#64743—31no—
35C23×D15D412yes—
36C22×C3×C4×C5—21no—
37C22×C3×C4×C5C2311yescharacteristic
38C22×Dic15—21no—
39C2×D60—21no—
40C2×D60—21no—
41C2×D60—21no—
42C2×D60—21no—
43C2×D60C2×C412yes—
44C2×D60Q812yes—
45C2×C4×D15—22no—
46C2×C4×D15—22no—
47C3×C5×M16—21no—
48C3×C5×M16C2×C412yescomplemented
49C2×C3×C5×C8—21no—
50C3×C5×C4⋊C4D411yescharacteristic
51C3×C5×C4⋊C4D411yescharacteristic
52C3×C5×C22⋊C4—21no—
53C3×C42×C5—21no—
54C3×C42×C5—21no—
55C3×C42×C5—21no—
56#6719—21no—
57#6714—22no—
58#6714—22no—
59#6714—44no—
60C4⋊Dic15—21no—
61C4⋊Dic15—21no—
62C4⋊Dic15D411yescharacteristic
63C4⋊Dic15D411yescharacteristic
64C4×Dic15—21no—
65#6710—21no—
66#6710C2×C412yescomplemented
67C2×C3⋊#168—21no—
68#5521—51no—
69C2×#823—51no—
70#5121—51no—
71#4995—51no—
72#4939—51no—
73C3×#339D511yescharacteristic, complemented
74#4523—51no—
75C22×D20—31no—
76C2×C5×M16C2×D311yescharacteristic
77C5×#117C2×D311yescharacteristic
78C2×C42×C5C2×D311yescharacteristic
79C2×#685—62no—
80#4072—31no—
81C2×C4⋊Dic5—31no—
82C2×#681—31no—
83#4039—61no—
84#4039—61no—
85D5×C22⋊C4—62no—
86C4×D20—61no—
87C4×D20—61no—
88#3955—34no—
89#3950—61no—
90#3949—61no—
91#3940—61no—
92#3939—61no—
93#1856—151noSylow
94C23×C3×C5C2×D411yescharacteristic
95C22×D15—22no—
96C22×D15—22no—
97C22×D15—42no—
98C22×D15—42no—
99C22×D15C4⋊C414yes—
100C2×C3×C4×C5—22no—
101C2×C3×C4×C5—21no—
102C2×C3×C4×C5—21no—
103C2×C3×C4×C5—41no—
104C2×C3×C4×C5—41no—
105C2×C3×C4×C5C22×C411yescharacteristic
106C2×C3×C4×C5C2×D411yescharacteristic
107C2×C3×C4×C5C2×D411yescharacteristic
108C2×C3×C4×C5C2×Q811yescharacteristic
109C2×C3×C4×C5#4111yescharacteristic
110C2×C3×C4×C5#4111yescharacteristic
111C2×Dic15—21no—
112C2×Dic15—21no—
113C2×Dic15—42no—
114C2×Dic15—41no—
115D60—22no—
116D60—22no—
117D60—22no—
118D60—22no—
119C4×D15—82no—
120C4×D15—82no—
121C3×C5×C8—22no—
122C3⋊#168—22no—
123C22×D12—51no—
124C2×C3×M16C2×D511yescharacteristic
125C3×#117C2×D511yescharacteristic
126C2×C3×C42C2×D511yescharacteristic
127C2×(C2×D3)⋊C4—102no—
128#926—51no—
129C2×C4⋊Dic3—51no—
130C2×#208—51no—
131#893—101no—
132#893—101no—
133D3×C22⋊C4—102no—
134C4×D12—101no—
135C4×D12—101no—
136#823—54no—
137#818—101no—
138#817—101no—
139#808—101no—
140#807—101no—
141C23×D5—32no—
142C22×C4×C5—21no—
143C22×C4×C5C22×D311yescharacteristic
144C22×Dic5—61no—
145C2×D20—32no—
146C2×D20—32no—
147C2×D20—61no—
148C2×D20—61no—
149C2×D20—61no—
150C2×D20—61no—
151C2×C4×D5—62no—
152C2×C4×D5—62no—
153C5×M16—21no—
154C5×M16C4×D312yescomplemented
155C2×C5×C8—21no—
156C5×C4⋊C4D1211yescharacteristic
157C5×C4⋊C4D1211yescharacteristic
158C5×C22⋊C4—21no—
159C42×C5—21no—
160C42×C5—21no—
161C42×C5—21no—
162C22⋊Dic5—61no—
163#685—62no—
164#685—62no—
165#685—124no—
166C4⋊Dic5—31no—
167C4⋊Dic5—31no—
168C4⋊Dic5—61no—
169C4⋊Dic5—61no—
170C4×Dic5—61no—
171#681—32no—
172#681—61no—
173C2×#168—61no—
174#518—151no—
175#428—151no—
176#428—151no—
177C2×#100—151no—
178#390—151no—
179#339—151no—
180#339D1511yescharacteristic, complemented
181C22×C3×C5—21no—
182C22×C3×C5—21no—
183C22×C3×C5C2×C4⋊C411yescharacteristic
184C22×C3×C5#12011yescharacteristic
185C22×C3×C5#12311yescharacteristic
186C2×D15—24no—
187C2×D15—42no—
188C2×D15—42no—
189C2×D15—44no—
190C2×D15—82no—
191C3×C4×C5—42no—
192C3×C4×C5—41no—
193C3×C4×C5—41no—
194C3×C4×C5C4×D411yescharacteristic
195C3×C4×C5C4×D411yescharacteristic
196C3×C4×C5C22⋊Q811yescharacteristic
197C3×C4×C5C22⋊Q811yescharacteristic
198Dic15—41no—
199Dic15—41no—
200Dic15—81no—
201C23×D3—52no—
202C22×C3×C4—21no—
203C22×C3×C4C22×D511yescharacteristic
204C22×Dic3—101no—
205C2×D12—52no—
206C2×D12—52no—
207C2×D12—101no—
208C2×D12—101no—
209C2×D12—101no—
210C2×D12—101no—
211C2×C4×D3—102no—
212C2×C4×D3—102no—
213C3×M16—21no—
214C3×M16C4×D512yescomplemented
215C2×C3×C8—21no—
216C3×C4⋊C4D2011yescharacteristic
217C3×C4⋊C4D2011yescharacteristic
218C3×C22⋊C4—21no—
219C3×C42—21no—
220C3×C42—21no—
221C3×C42—21no—
222#217—101no—
223(C2×D3)⋊C4—102no—
224(C2×D3)⋊C4—102no—
225(C2×D3)⋊C4—204no—
226C4⋊Dic3—51no—
227C4⋊Dic3—51no—
228C4⋊Dic3—101no—
229C4⋊Dic3—101no—
230C4×Dic3—101no—
231#208—52no—
232#208—101no—
233C2×C3⋊C8—101no—
234C23×C5D3×D411yescharacteristic
235C22×D5—34no—
236C22×D5—62no—
237C22×D5—62no—
238C22×D5—122no—
239C22×D5—122no—
240C2×C4×C5—22no—
241C2×C4×C5—21no—
242C2×C4×C5—21no—
243C2×C4×C5—41no—
244C2×C4×C5—41no—
245C2×C4×C5C2×C4×D311yescharacteristic
246C2×C4×C5C2×D1211yescharacteristic
247C2×C4×C5#23511yescharacteristic
248C2×C4×C5D3×D411yescharacteristic
249C2×C4×C5#23711yescharacteristic
250C2×C4×C5D3×Q811yescharacteristic
251C2×Dic5—61no—
252C2×Dic5—61no—
253C2×Dic5—122no—
254C2×Dic5—121no—
255D20—62no—
256D20—62no—
257D20—62no—
258D20—62no—
259C4×D5—242no—
260C4×D5—242no—
261C5×C8—22no—
262#168—62no—
263C22×D4—151no—
264C2×M16—151no—
265C2×M16C2×D1511yescharacteristic
266C4×D4—301no—
267C4×D4—301no—
268C4×D4—301no—
269C4×D4—301no—
270#117—151no—
271#117C2×D1511yescharacteristic
272C2×C4⋊C4—151no—
273C2×C22⋊C4—302no—
274C2×C22⋊C4—302no—
275C2×C42C2×D1511yescharacteristic
276#107—301no—
277#107—301no—
278#106—301no—
279#106—301no—
280#100—154no—
281C2×C3×C5—21no—
282C2×C3×C5—21no—
283C2×C3×C5#38511yescharacteristic
284C2×C3×C5#45311yescharacteristic
285C2×C3×C5#45511yescharacteristic
286D15—44no—
287C23×C3D4×D511yescharacteristic
288C22×D3—54no—
289C22×D3—102no—
290C22×D3—102no—
291C22×D3—202no—
292C22×D3—202no—
293C2×C3×C4—22no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4C2×C4×D511yescharacteristic
299C2×C3×C4C2×D2011yescharacteristic
300C2×C3×C4#70911yescharacteristic
301C2×C3×C4D4×D511yescharacteristic
302C2×C3×C4#71111yescharacteristic
303C2×C3×C4Q8×D511yescharacteristic
304C2×Dic3—101no—
305C2×Dic3—101no—
306C2×Dic3—202no—
307C2×Dic3—201no—
308D12—102no—
309D12—102no—
310D12—102no—
311D12—102no—
312C4×D3—402no—
313C4×D3—402no—
314C3×C8—22no—
315C3⋊C8—102no—
316C22×C5—21no—
317C22×C5—21no—
318C22×C5#88211yescharacteristic
319C22×C5#88311yescharacteristic
320C22×C5D3×C4⋊C411yescharacteristic
321C2×D5—64no—
322C2×D5—122no—
323C2×D5—122no—
324C2×D5—124no—
325C2×D5—242no—
326C4×C5—42no—
327C4×C5—41no—
328C4×C5—41no—
329C4×C5C4×D1211yescharacteristic
330C4×C5#67⋊C411yescharacteristic
331C4×C5#89611yescharacteristic
332C4×C5#89711yescharacteristic
333Dic5—121no—
334Dic5—121no—
335Dic5—241no—
336C24—152no—
337C2×D4—152no—
338C2×D4—152no—
339C2×D4—301no—
340C2×D4—301no—
341C2×D4—301no—
342C2×D4—301no—
343C22×C4—21no—
344C22×C4—302no—
345C22×C4—302no—
346C22×C4—301no—
347C22×C4C22×D1511yescharacteristic
348M16—21no—
349M16—152no—
350M16—301no—
351M16C4×D1512yescomplemented
352C2×C8—21no—
353C2×C8—301no—
354C4⋊C4—151no—
355C4⋊C4—151no—
356C4⋊C4—301no—
357C4⋊C4—301no—
358C4⋊C4D6011yescharacteristic
359C4⋊C4D6011yescharacteristic
360C22⋊C4—21no—
361C22⋊C4—302no—
362C22⋊C4—302no—
363C22⋊C4—301no—
364C22⋊C4—604no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—301no—
369C3×C5#185611yescharacteristic, complemented, Hall
370C22×C3—21no—
371C22×C3—21no—
372C22×C3#402811yescharacteristic
373C22×C3#402911yescharacteristic
374C22×C3D5×C4⋊C411yescharacteristic
375C2×D3—104no—
376C2×D3—202no—
377C2×D3—202no—
378C2×D3—204no—
379C2×D3—402no—
380C3×C4—42no—
381C3×C4—41no—
382C3×C4—41no—
383C3×C4C4×D2011yescharacteristic
384C3×C4#402711yescharacteristic
385C3×C4#404211yescharacteristic
386C3×C4#404311yescharacteristic
387Dic3—201no—
388Dic3—201no—
389Dic3—401no—
390C2×C5—21no—
391C2×C5—21no—
392C2×C5#466811yescharacteristic
393C2×C5#482211yescharacteristic
394C2×C5#482511yescharacteristic
395D5—124no—
396C23—154no—
397C23—302no—
398C23—302no—
399C23—602no—
400C23—602no—
401C23D4×D1511yescharacteristic
402D4—302no—
403D4—302no—
404D4—302no—
405D4—302no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—301no—
412C2×C4—301no—
413C2×C4—602no—
414C2×C4—601no—
415C2×C4—1202no—
416C2×C4—1202no—
417C2×C4C2×C4×D1511yescharacteristic
418C2×C4C2×D6011yescharacteristic
419C2×C4#681711yescharacteristic
420C2×C4D4×D1511yescharacteristic
421C2×C4#681911yescharacteristic
422C2×C4Q8×D1511yescharacteristic
423C8—22no—
424C8—302no—
425C2×C3—21no—
426C2×C3—21no—
427C2×C3#6495711yescharacteristic
428C2×C3#6511111yescharacteristic
429C2×C3#6511411yescharacteristic
430D3—204no—
431C5#6875811yescharacteristic, complemented, Sylow
432C22—21no—
433C22—21no—
434C22—304no—
435C22—604no—
436C22—602no—
437C22—602no—
438C22—1202no—
439C22#9177111yescharacteristic
440C22#9177211yescharacteristic
441C22C4⋊C4×D1511yescharacteristic
442C4—42no—
443C4—41no—
444C4—41no—
445C4—601no—
446C4—601no—
447C4—1201no—
448C4C4×D6011yescharacteristic
449C4#9177011yescharacteristic
450C4#9178511yescharacteristic
451C4#9178611yescharacteristic
452C3#1059955111yescharacteristic, complemented, Sylow
453C2—21no—
454C2—21no—
455C2—604no—
456C2#1175076911yescharacteristic
457C2#1175092311yescharacteristic
458C2#1175092611yescharacteristic
subgroupG/H
centerC22#91771
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C22×D15
FittingC3×C5×#339C2
socleC22×C3×C5#120
prime radicalC22×D60C22
cosocleC22×C3×C4×C5C23
hypercenter#339D15
hypocenterC3×C5#1856
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
3. #11751222C2semidirect
4. #11751096C2semidirect
6. C3×C5×#339C2semidirect
7. #11746344C2semidirect
22. #91107C4semidirect
33. C5×#339D3semidirect
48. C3×C5×M16C2×C4semidirect
66. #6710C2×C4semidirect
73. C3×#339D5semidirect
154. C5×M16C4×D3semidirect
180. #339D15semidirect
214. C3×M16C4×D5semidirect
351. M16C4×D15semidirect
369. C3×C5#1856semidirect
431. C5#68758semidirect
452. C3#10599551semidirect

Maximal subgroups

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

subgroupindex
#117562802
C2×#911072
#117512222
#117510962
#117510402
C3×C5×#3392
#117463442
#105995513
#687585