Finite Group Databasebeta

← #14299727

Subgroups — #14299727

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#11756263C211yesmaximal, characteristic
2C2×#91109C211yesmaximal, characteristic
3#11751234C211yesmaximal, characteristic
4#11751222C211yesmaximal, characteristic, complemented
5#11751076C211yesmaximal, characteristic, complemented
6C3×C5×#343C211yesmaximal, characteristic, complemented
7#11746376C211yesmaximal, characteristic, complemented
8#10599795—31nomaximal, Hall
9C2×#6817C2211yescharacteristic
10C2×C3×C5×M16C2211yescharacteristic
11C2×C3×C5×M16C2211yescharacteristic
12C2×C3×C5×C4⋊C4C2211yescharacteristic
13C4×#1220—22no—
14#91815C2211yescharacteristic
15C2×C4⋊Dic15C2211yescharacteristic
16C2×#6712—22no—
17C2×C4×Dic15C2211yescharacteristic
18#91782—21no—
19#91781—21no—
20C4⋊C4×D15—21no—
21#91776—21no—
22#91109C414yescomplemented
23#91105—21no—
24#91102—21no—
25#91101—21no—
26#91100—21no—
27#69002—51nomaximal, Hall
28#65838—31no—
29C2×#3957—31no—
30#65422—31no—
31#65410—31no—
32#65265—31no—
33C5×#343D311yescharacteristic, complemented
34#64775—31no—
35C22×C3×C4×C5D412yes—
36C22×C3×C4×C5C2311yescharacteristic
37C2×#1220—21no—
38C22×Dic15—21no—
39C22×Dic15—21no—
40#6817—21no—
41#6817—21no—
42#6817C2×C412yes—
43#6817Q812yes—
44C2×D60D411yescharacteristic
45C2×C4×D15—21no—
46C2×C4×D15—22no—
47C2×Dic30D411yescharacteristic
48C3×C5×M16—21no—
49C3×C5×M16—21no—
50C3×C5×M16—21no—
51C3×C5×M16C2×C412yescomplemented
52C2×C3×C5×C8—21no—
53C2×C3×C5×C8—21no—
54C3×C5×C4⋊C4—21no—
55C3×C5×C4⋊C4—21no—
56#6719—21no—
57#6714—22no—
58C4⋊Dic15—21no—
59C4⋊Dic15—21no—
60C4⋊Dic15D411yescharacteristic
61C4⋊Dic15D411yescharacteristic
62#6712—22no—
63#6712—44no—
64C4×Dic15—22no—
65C4×Dic15—21no—
66C4×Dic15—21no—
67C4×Dic15C2×C412yescomplemented
68#5504—51no—
69C2×#825—51no—
70#5133—51no—
71#5121—51no—
72#4975—51no—
73C3×#343D511yescharacteristic, complemented
74#4555—51no—
75C2×#709—31no—
76C2×C5×M16C2×D311yescharacteristic
77C2×C5×M16C2×D311yescharacteristic
78C2×C5×C4⋊C4C2×D311yescharacteristic
79C4×C22⋊D5—62no—
80#4072—31no—
81C2×C4⋊Dic5—31no—
82C2×#683—62no—
83C2×C4×Dic5—31no—
84#4039—61no—
85#4038—61no—
86D5×C4⋊C4—61no—
87#4033—61no—
88#3957—34no—
89#3953—61no—
90#3950—61no—
91#3949—61no—
92#3948—61no—
93#1859—151noSylow
94C23×C3×C5C2×D411yescharacteristic
95C22×D15—21no—
96C2×C3×C4×C5—22no—
97C2×C3×C4×C5—42no—
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
104#1220—22no—
105#1220—22no—
106C2×Dic15—22no—
107C2×Dic15—22no—
108C2×Dic15—22no—
109C2×Dic15—21no—
110C2×Dic15—21no—
111C2×Dic15—21no—
112C2×Dic15—42no—
113C2×Dic15—41no—
114D60—21no—
115D60C4⋊C412yescomplemented
116C4×D15—22no—
117C4×D15—22no—
118C4×D15—82no—
119Dic30—21no—
120Dic30C4⋊C412yescomplemented
121C3×C5×C8—22no—
122C3×C5×C8—41no—
123C2×#235—51no—
124C2×C3×M16C2×D511yescharacteristic
125C2×C3×M16C2×D511yescharacteristic
126C2×C3×C4⋊C4C2×D511yescharacteristic
127C4×#67—102no—
128#926—51no—
129C2×C4⋊Dic3—51no—
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—101no—
138#818—101no—
139#817—101no—
140#816—101no—
141C22×C4×C5D1212yes—
142C22×C4×C5C22×D311yescharacteristic
143C2×C22⋊D5—61no—
144C22×Dic5—61no—
145C22×Dic5—61no—
146#709—32no—
147#709—32no—
148#709—61no—
149#709—61no—
150C2×D20—31no—
151C2×C4×D5—61no—
152C2×C4×D5—62no—
153C2×Dic10—31no—
154C5×M16—21no—
155C5×M16—21no—
156C5×M16—21no—
157C5×M16C4×D312yescomplemented
158C2×C5×C8—21no—
159C2×C5×C8—21no—
160C5×C4⋊C4—21no—
161C5×C4⋊C4—21no—
162C22⋊Dic5—61no—
163#685—62no—
164C4⋊Dic5—31no—
165C4⋊Dic5—31no—
166C4⋊Dic5—61no—
167C4⋊Dic5—61no—
168#683—62no—
169#683—124no—
170C4×Dic5—32no—
171C4×Dic5—62no—
172C4×Dic5—61no—
173C4×Dic5—61no—
174#520—151no—
175#428—151no—
176C2×#104—151no—
177#417—151no—
178#384—151no—
179#343D1511yescharacteristic, complemented
180#339—151no—
181C22×C3×C5—21no—
182C22×C3×C5—21no—
183C22×C3×C5C4×D411yescharacteristic
184C22×C3×C5#12011yescharacteristic
185C22×C3×C5C22⋊Q811yescharacteristic
186C2×D15—22no—
187C2×D15—41no—
188C2×D15—41no—
189C3×C4×C5—42no—
190C3×C4×C5C2×C4⋊C411yescharacteristic
191C3×C4×C5C4×D411yescharacteristic
192C3×C4×C5C22⋊Q811yescharacteristic
193C3×C4×C5#12311yescharacteristic
194Dic15—24no—
195Dic15—22no—
196Dic15—41no—
197Dic15—41no—
198Dic15—81no—
199C22×C3×C4D2012yes—
200C22×C3×C4C22×D511yescharacteristic
201C2×#67—101no—
202C22×Dic3—101no—
203C22×Dic3—101no—
204#235—52no—
205#235—52no—
206#235—101no—
207#235—101no—
208C2×D12—51no—
209C2×C4×D3—101no—
210C2×C4×D3—102no—
211C2×Dic6—51no—
212C3×M16—21no—
213C3×M16—21no—
214C3×M16—21no—
215C3×M16C4×D512yescomplemented
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220#217—101no—
221(C2×D3)⋊C4—102no—
222C4⋊Dic3—51no—
223C4⋊Dic3—51no—
224C4⋊Dic3—101no—
225C4⋊Dic3—101no—
226Dic3⋊C4—102no—
227Dic3⋊C4—204no—
228C4×Dic3—52no—
229C4×Dic3—102no—
230C4×Dic3—101no—
231C4×Dic3—101no—
232C23×C5C2×D1211yescharacteristic
233C22×D5—61no—
234C2×C4×C5—22no—
235C2×C4×C5—42no—
236C2×C4×C5C2×C4×D311yescharacteristic
237C2×C4×C5#23511yescharacteristic
238C2×C4×C5D3×D411yescharacteristic
239C2×C4×C5D3×D411yescharacteristic
240C2×C4×C5#23711yescharacteristic
241C2×C4×C5D3×Q811yescharacteristic
242C22⋊D5—62no—
243C22⋊D5—62no—
244C2×Dic5—62no—
245C2×Dic5—62no—
246C2×Dic5—62no—
247C2×Dic5—61no—
248C2×Dic5—61no—
249C2×Dic5—61no—
250C2×Dic5—122no—
251C2×Dic5—121no—
252D20—32no—
253D20—61no—
254C4×D5—62no—
255C4×D5—62no—
256C4×D5—242no—
257Dic10—32no—
258Dic10—61no—
259C5×C8—22no—
260C5×C8—41no—
261C2×#41—151no—
262C2×M16C2×D1511yescharacteristic
263C2×M16C2×D1511yescharacteristic
264C4×Q8—301no—
265C4×D4—301no—
266C4×D4—302no—
267#117—151no—
268#117—301no—
269C2×C4⋊C4—151no—
270C2×C4⋊C4—301no—
271C2×C4⋊C4—302no—
272C2×C4⋊C4C2×D1511yescharacteristic
273C2×C42—151no—
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
284D15—42no—
285C23×C3C2×D2011yescharacteristic
286C22×D3—101no—
287C2×C3×C4—22no—
288C2×C3×C4—42no—
289C2×C3×C4C2×C4×D511yescharacteristic
290C2×C3×C4#70911yescharacteristic
291C2×C3×C4D4×D511yescharacteristic
292C2×C3×C4D4×D511yescharacteristic
293C2×C3×C4#71111yescharacteristic
294C2×C3×C4Q8×D511yescharacteristic
295#67—102no—
296#67—102no—
297C2×Dic3—102no—
298C2×Dic3—102no—
299C2×Dic3—102no—
300C2×Dic3—101no—
301C2×Dic3—101no—
302C2×Dic3—101no—
303C2×Dic3—202no—
304C2×Dic3—201no—
305D12—52no—
306D12—101no—
307C4×D3—102no—
308C4×D3—102no—
309C4×D3—402no—
310Dic6—52no—
311Dic6—101no—
312C3×C8—22no—
313C3×C8—41no—
314C22×C5—21no—
315C22×C5—21no—
316C22×C5C4×D1211yescharacteristic
317C22×C5#88211yescharacteristic
318C22×C5#89711yescharacteristic
319C2×D5—62no—
320C2×D5—121no—
321C2×D5—121no—
322C4×C5—42no—
323C4×C5#67⋊C411yescharacteristic
324C4×C5#88311yescharacteristic
325C4×C5D3×C4⋊C411yescharacteristic
326C4×C5#89611yescharacteristic
327Dic5—64no—
328Dic5—62no—
329Dic5—121no—
330Dic5—121no—
331Dic5—241no—
332#41—152no—
333#41—152no—
334#41—301no—
335#41—301no—
336C2×Q8—151no—
337C2×D4—151no—
338C2×D4—301no—
339C22×C4—302no—
340C22×C4—301no—
341C22×C4—301no—
342C22×C4—301no—
343C22×C4D6012yes—
344C22×C4C22×D1511yescharacteristic
345M16—21no—
346M16—21no—
347M16—21no—
348M16C4×D1512yescomplemented
349C2×C8—21no—
350C2×C8—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—151no—
354C4⋊C4—151no—
355C4⋊C4—302no—
356C4⋊C4—301no—
357C4⋊C4—301no—
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×C3C4×D2011yescharacteristic
369C22×C3#402811yescharacteristic
370C22×C3#404311yescharacteristic
371C2×D3—102no—
372C2×D3—201no—
373C2×D3—201no—
374C3×C4—42no—
375C3×C4#402711yescharacteristic
376C3×C4#402911yescharacteristic
377C3×C4D5×C4⋊C411yescharacteristic
378C3×C4#404211yescharacteristic
379Dic3—104no—
380Dic3—102no—
381Dic3—201no—
382Dic3—201no—
383Dic3—401no—
384C2×C5—21no—
385C2×C5—21no—
386C2×C5#466811yescharacteristic
387C2×C5#474711yescharacteristic
388C2×C5#475211yescharacteristic
389D5—122no—
390C23—301no—
391C23C2×D6011yescharacteristic
392Q8—152no—
393Q8—301no—
394D4—152no—
395D4—302no—
396D4—302no—
397D4—301no—
398C2×C4—22no—
399C2×C4—42no—
400C2×C4—302no—
401C2×C4—302no—
402C2×C4—302no—
403C2×C4—301no—
404C2×C4—302no—
405C2×C4—302no—
406C2×C4—301no—
407C2×C4—301no—
408C2×C4—602no—
409C2×C4—601no—
410C2×C4—1202no—
411C2×C4C2×C4×D1511yescharacteristic
412C2×C4#681711yescharacteristic
413C2×C4D4×D1511yescharacteristic
414C2×C4D4×D1511yescharacteristic
415C2×C4#681911yescharacteristic
416C2×C4Q8×D1511yescharacteristic
417C8—22no—
418C8—41no—
419C2×C3—21no—
420C2×C3—21no—
421C2×C3#6495711yescharacteristic
422C2×C3#6503611yescharacteristic
423C2×C3#6504111yescharacteristic
424D3—202no—
425C5#6900211yescharacteristic, complemented, Sylow
426C22—21no—
427C22—21no—
428C22—302no—
429C22—601no—
430C22—601no—
431C22C4×D6011yescharacteristic
432C22#9177111yescharacteristic
433C22#9178611yescharacteristic
434C4—42no—
435C4—304no—
436C4—302no—
437C4—601no—
438C4—601no—
439C4—1201no—
440C4#9177011yescharacteristic
441C4#9177211yescharacteristic
442C4C4⋊C4×D1511yescharacteristic
443C4#9178511yescharacteristic
444C3#1059979511yescharacteristic, complemented, Sylow
445C2—21no—
446C2—21no—
447C2—602no—
448C2#1175076911yescharacteristic
449C2#1175084811yescharacteristic
450C2#1175085311yescharacteristic
subgroupG/H
centerC22#91771
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C22×D15
FittingC3×C5×#343C2
socleC22×C3×C5#120
prime radicalC2×#6817C22
cosocleC22×C3×C4×C5C23
hypercenter#343D15
hypocenterC3×C5#1859
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
4. #11751222C2semidirect
5. #11751076C2semidirect
6. C3×C5×#343C2semidirect
7. #11746376C2semidirect
22. #91109C4semidirect
33. C5×#343D3semidirect
51. C3×C5×M16C2×C4semidirect
67. C4×Dic15C2×C4semidirect
73. C3×#343D5semidirect
115. D60C4⋊C4semidirect
120. Dic30C4⋊C4semidirect
157. C5×M16C4×D3semidirect
179. #343D15semidirect
215. C3×M16C4×D5semidirect
348. M16C4×D15semidirect
365. C3×C5#1859semidirect
425. C5#69002semidirect
444. C3#10599795semidirect

Maximal subgroups

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

subgroupindex
#117562632
C2×#911092
#117512342
#117512222
#117510762
C3×C5×#3432
#117463762
#105997953
#690025