Finite Group Databasebeta

← #14471539

Subgroups — #14471539

Order 1920.

740 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5567C211yesmaximal, characteristic, complemented
2#11752551C211yesmaximal, characteristic
3C4×#6717C211yesmaximal, characteristic
4C3×#65309C211yesmaximal, characteristic, complemented
5#11749233C212yesmaximal
6#11749187C212yesmaximal, complemented
7#11747417C212yesmaximal, complemented
8#11747232C212yesmaximal
9#11747087C211yesmaximal, characteristic
10#11746923C211yesmaximal, characteristic, complemented
11#11746857C211yesmaximal, characteristic, complemented
12#10615419—31nomaximal, Hall
13C2×D15⋊Q8—21no—
14C3×C4×C5×Q8C2211yescharacteristic
15C2×#6717C2211yescharacteristic
16C4×D60C2211yescharacteristic
17C5×#944C2212yes—
18C5×#889—22no—
19C5×#887C2212yes—
20C5×#870—21no—
21C5×#869C2211yescharacteristic
22C4×C5×Dic6—21no—
23C4×C5×Dic6C2211yescharacteristic
24C3×#4015C2211yescharacteristic
25#91469C2212yes—
26#91429—22no—
27#91428C2212yes—
28#91390—21no—
29C2×#6660C2212yes—
30#91119C2211yescharacteristic
31#91093C2211yescharacteristic
32#91092C2211yescharacteristic
33C4×C3⋊#168C2211yescharacteristic
34C3×#3967C2212yes—
35C3×#3940C2212yes—
36C3×C4⋊#168C2211yescharacteristic
37Dic3⋊#168C2212yes—
38#90973C2212yes—
39#90972C2212yes—
40#90972C2212yes—
41Dic3×#168C2212yes—
42#81037—51nomaximal, Hall
43C5×#557—31no—
44Q8×D20—31no—
45#65372—32no—
46#65372—31no—
47#65370—31no—
48#65370—32no—
49C4×#688—32no—
50C4×#688—31no—
51Q8×#168—31no—
52#65309—32no—
53#65309D311yescharacteristic, complemented
54C2×C3×C5×Q8C2311yescharacteristic
55C2×D60C2311yescharacteristic
56C2×C4×D15—21no—
57C2×C5×Dic6D411yescharacteristic
58C2×C5×Dic6C2312yes—
59C2×C3×Dic10—21no—
60C2×#1204—22no—
61C2×#1200—22no—
62D15⋊Q8—44no—
63C3×C5×C4⋊C4—21no—
64C3×C5×C4⋊C4C2311yescharacteristic
65C3×C42×C5—21no—
66C3×C42×C5C2311yescharacteristic
67#6717—22no—
68#6714—21no—
69C4⋊Dic15C2311yescharacteristic
70C2×C3⋊#168C2311yescharacteristic
71C2×C3⋊#168C2311yescharacteristic
72C5×C4⋊Dic3—21no—
73C5×C4⋊Dic3—21no—
74C5×C4⋊Dic3D411yescharacteristic
75C5×Dic3⋊C4—22no—
76C5×Dic3⋊C4—22no—
77C5×Dic3⋊C4—22no—
78C5×Dic3⋊C4D412yes—
79C4×C5×Dic3—22no—
80C4×C5×Dic3C2312yes—
81C3×C4⋊Dic5—21no—
82C3×C4⋊Dic5C2312yes—
83C2×C3×#168C2312yes—
84Dic3⋊Dic5—22no—
85#6667—22no—
86#6660—24no—
87#5727—51no—
88#5567D511yescharacteristic, complemented
89C3×#475—51no—
90#5093—51no—
91C4×#215—51no—
92#4854—52no—
93#4852—52no—
94#4841—51no—
95#4830—51no—
96#4818—52no—
97#4792—52no—
98C2×Q8×D5—61no—
99C5×C4⋊Q8—32no—
100C5×C4⋊Q8—31no—
101C5×#125—61no—
102C5×#125—62no—
103C4×C5×Q8—31no—
104C4×C5×Q8—32no—
105C4×C5×Q8—61no—
106C4×C5×Q8C2×D311yescharacteristic
107C2×#688—31no—
108C2×#688—32no—
109#4043—62no—
110#4043—61no—
111C4×D20—31no—
112C4×D20—32no—
113#4015—32no—
114#4015C2×D311yescharacteristic
115#3967—31no—
116#3967—32no—
117#3967—32no—
118#3967—32no—
119#3967C2×D312yes—
120#3941—31no—
121#3940—31no—
122#3940C2×D312yes—
123C4⋊#168—32no—
124C4⋊#168C2×D311yescharacteristic
125C4×#168—31no—
126C4×#168—32no—
127#3359—151noSylow
128C22×D15—21no—
129C3×C5×Q8—21no—
130C3×C5×Q8—21no—
131C2×C3×C4×C5—21no—
132C2×C3×C4×C5—21no—
133C2×C3×C4×C5C2×D411yescharacteristic
134C2×C3×C4×C5C2×D411yescharacteristic
135C2×C3×C4×C5C2411yescharacteristic
136C2×Dic15—21no—
137D60—21no—
138D60#4112yes—
139C4×D15—42no—
140C2×C5×Dic3—22no—
141C2×C5×Dic3C2×D412yes—
142C2×C5×Dic3C2×D412yes—
143C5×Dic6—22no—
144C5×Dic6—22no—
145C5×Dic6SD1614yescomplemented
146C2×C3×Dic5—22no—
147C3×Dic10—42no—
148#1204—44no—
149#1200—44no—
150C3⋊#168—21no—
151C3⋊#168—21no—
152C3×#168—22no—
153C2×D3×Q8—101no—
154C3×C4⋊Q8—51no—
155C3×C4×Q8C2×D511yescharacteristic
156#946—101no—
157#944C2×D512yes—
158C2×#215—51no—
159C2×#204—52no—
160#896—102no—
161#893—52no—
162#889—22no—
163#888—52no—
164#887C2×D512yes—
165C4×D12—51no—
166#870—21no—
167#869C2×D511yescharacteristic
168C4×Dic6—21no—
169C4×Dic6C2×D511yescharacteristic
170C3×#106—52no—
171C3×C4⋊C8—51no—
172C3×#103—52no—
173#835—51no—
174#816—52no—
175Dic3⋊C8—52no—
176C8×Dic3—52no—
177#810—52no—
178#810—52no—
179#809—51no—
180#808—51no—
181C4×C3⋊C8—51no—
182C2×C5×Q8—31no—
183C2×C5×Q8—32no—
184C2×C5×Q8C22×D311yescharacteristic
185Q8×D5—124no—
186C2×D20—31no—
187C2×C4×D5—62no—
188C2×C4×D5—61no—
189C2×Dic10—21no—
190C2×Dic10—62no—
191C5×C4⋊C4—21no—
192C5×C4⋊C4—32no—
193C5×C4⋊C4—31no—
194C5×C4⋊C4—62no—
195C5×C4⋊C4—62no—
196C5×C4⋊C4—62no—
197C5×C4⋊C4—61no—
198C5×C4⋊C4—61no—
199C5×C4⋊C4C22×D311yescharacteristic
200C42×C5—21no—
201C42×C5—32no—
202C42×C5—62no—
203C42×C5C22×D311yescharacteristic
204#688—64no—
205#688—62no—
206#685—62no—
207#685—61no—
208C4⋊Dic5—21no—
209C4⋊Dic5—31no—
210C4⋊Dic5—62no—
211C4⋊Dic5C22×D312yes—
212C2×#168—31no—
213C2×#168—31no—
214C2×#168C22×D312yes—
215#557—31no—
216D4×Q8—151no—
217#487—152no—
218#487—151no—
219#475—51no—
220#475—152no—
221#461—152no—
222#461—151no—
223C8×Q8—151no—
224C4×SD16—152no—
225C4×SD16—151no—
226C22×C3×C5C22×D411yescharacteristic
227C2×D15—22no—
228C2×D15—41no—
229C2×D15—41no—
230C3×C4×C5—21no—
231C3×C4×C5—41no—
232C3×C4×C5—41no—
233C3×C4×C5C2×SD1612yes—
234C3×C4×C5C2×#4111yescharacteristic
235C3×C4×C52−1+411yescharacteristic
236Dic15—41no—
237C3×Dic5—42no—
238C5×Dic3—22no—
239C5×Dic3—42no—
240C5×Dic3C2×SD1614yes—
241C2×C3×Q8—101no—
242C2×C3×Q8C22×D511yescharacteristic
243D3×Q8—204no—
244C2×D12—51no—
245C2×C4×D3—101no—
246C2×C4×D3—102no—
247C2×Dic6—102no—
248C2×Dic6C22⋊D511yescharacteristic
249C2×Dic6C22×D512yes—
250C2×C3×C8—52no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—52no—
253C3×C4⋊C4—101no—
254C3×C4⋊C4C22×D511yescharacteristic
255C3×C42—21no—
256C3×C42C22×D511yescharacteristic
257#215—102no—
258(C2×D3)⋊C4—101no—
259(C2×D3)⋊C4—102no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—51no—
263C4⋊Dic3C22⋊D511yescharacteristic
264Dic3⋊C4—22no—
265Dic3⋊C4—22no—
266Dic3⋊C4—22no—
267Dic3⋊C4—102no—
268Dic3⋊C4C22⋊D512yes—
269C4×Dic3—22no—
270C4×Dic3C22×D512yes—
271C2×C3⋊C8—51no—
272C2×C3⋊C8—51no—
273#204—104no—
274C22×D5—61no—
275C5×Q8—21no—
276C5×Q8—21no—
277C5×Q8—34no—
278C5×Q8—62no—
279C5×Q8—62no—
280C2×C4×C5—21no—
281C2×C4×C5—21no—
282C2×C4×C5—32no—
283C2×C4×C5—32no—
284C2×C4×C5—62no—
285C2×C4×C5D3×D411yescharacteristic
286C2×C4×C5D3×D411yescharacteristic
287C2×C4×C5C23×D311yescharacteristic
288C2×Dic5—22no—
289C2×Dic5—61no—
290D20—32no—
291D20—61no—
292C4×D5—124no—
293C4×D5—122no—
294Dic10—42no—
295Dic10—124no—
296#168—22no—
297#168—61no—
298#168—61no—
299C22×Q8—301no—
300C2×SD16—152no—
301C2×SD16—151no—
302C4⋊Q8—32no—
303C4⋊Q8—31no—
304C4⋊Q8—51no—
305C4⋊Q8—152no—
306#125—61no—
307#125—62no—
308C22⋊Q8—302no—
309C22⋊Q8—301no—
310C4×Q8—31no—
311C4×Q8—32no—
312C4×Q8—61no—
313C4×Q8D3×D511yescharacteristic
314C4×D4—152no—
315C4×D4—151no—
316#106—52no—
317#106—151no—
318C4⋊C8—51no—
319C4⋊C8—152no—
320#103—52no—
321#103—152no—
322#103—152no—
323#103—152no—
324#103—151no—
325#102—151no—
326C4×C8—152no—
327C4×C8—151no—
328C2×C3×C5#54811yescharacteristic
329C2×C3×C5C22×SD1611yescharacteristic
330C2×C3×C5#57811yescharacteristic
331D15—42no—
332C22×D3—101no—
333C3×Q8—21no—
334C3×Q8—21no—
335C3×Q8—202no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—102no—
339C2×C3×C4C2×C22⋊D511yescharacteristic
340C2×C3×C4C2×C22⋊D511yescharacteristic
341C2×C3×C4C23×D511yescharacteristic
342C2×Dic3—22no—
343C2×Dic3—101no—
344C2×Dic3C2×C22⋊D512yes—
345C2×Dic3C2×C22⋊D512yes—
346D12—52no—
347D12—101no—
348C4×D3—202no—
349C4×D3—204no—
350Dic6—22no—
351Dic6—22no—
352Dic6—204no—
353Dic6#68814yescomplemented
354C3×C8—102no—
355C3⋊C8—101no—
356C3⋊C8—101no—
357C22×C5C2×D3×D411yescharacteristic
358C2×D5—62no—
359C2×D5—121no—
360C2×D5—121no—
361C4×C5—21no—
362C4×C5—34no—
363C4×C5—41no—
364C4×C5—41no—
365C4×C5—62no—
366C4×C5—122no—
367C4×C5D3×SD1612yes—
368C4×C5#100711yescharacteristic
369C4×C5D3×#4111yescharacteristic
370Dic5—42no—
371Dic5—121no—
372C2×Q8—31no—
373C2×Q8—32no—
374C2×Q8—101no—
375C2×Q8—302no—
376C2×Q8—604no—
377C2×Q8C2×D3×D511yescharacteristic
378C2×D4—151no—
379C22×C4—302no—
380C22×C4—301no—
381SD16—304no—
382SD16—302no—
383C2×C8—52no—
384C2×C8—151no—
385C2×C8—151no—
386C4⋊C4—21no—
387C4⋊C4—32no—
388C4⋊C4—31no—
389C4⋊C4—52no—
390C4⋊C4—62no—
391C4⋊C4—62no—
392C4⋊C4—62no—
393C4⋊C4—61no—
394C4⋊C4—61no—
395C4⋊C4—101no—
396C4⋊C4—151no—
397C4⋊C4—302no—
398C4⋊C4C2×D3×D511yescharacteristic
399C22⋊C4—302no—
400C22⋊C4—301no—
401C42—21no—
402C42—32no—
403C42—62no—
404C42C2×D3×D511yescharacteristic
405C3×C5#335911yescharacteristic, complemented, Hall
406C22×C3C22×C22⋊D511yescharacteristic
407C2×D3—102no—
408C2×D3—201no—
409C2×D3—201no—
410C3×C4—21no—
411C3×C4—41no—
412C3×C4—41no—
413C3×C4—202no—
414C3×C4C2×#68812yes—
415C3×C4D5×#4111yescharacteristic
416C3×C4#415011yescharacteristic
417Dic3—22no—
418Dic3—42no—
419Dic3—201no—
420Dic3C2×#68814yes—
421C2×C5#563311yescharacteristic
422C2×C5C2×D3×SD1611yescharacteristic
423C2×C5#577911yescharacteristic
424D5—122no—
425C23—301no—
426Q8—21no—
427Q8—21no—
428Q8—34no—
429Q8—62no—
430Q8—62no—
431Q8—202no—
432Q8—604no—
433D4—152no—
434D4—301no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—32no—
438C2×C4—32no—
439C2×C4—62no—
440C2×C4—102no—
441C2×C4—301no—
442C2×C4—604no—
443C2×C4—602no—
444C2×C4D3×C22⋊D511yescharacteristic
445C2×C4D3×C22⋊D511yescharacteristic
446C2×C4C22×D3×D511yescharacteristic
447C8—102no—
448C8—301no—
449C8—301no—
450C2×C3C22×#68811yescharacteristic
451C2×C3#6616511yescharacteristic
452C2×C3#6617211yescharacteristic
453D3—202no—
454C5#8103711yescharacteristic, complemented, Sylow
455C22—302no—
456C22—601no—
457C22—601no—
458C22C2×D3×C22⋊D511yescharacteristic
459C4—21no—
460C4—34no—
461C4—41no—
462C4—41no—
463C4—62no—
464C4—122no—
465C4—202no—
466C4—601no—
467C4D3×#68812yes—
468C4#9200411yescharacteristic
469C4#9201611yescharacteristic
470C3#1061541911yescharacteristic, complemented, Sylow
471C2—602no—
472C2#1175454611yescharacteristic
473C2C2×D3×#68811yescharacteristic
474C2#1175534111yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊D5
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C4×C5×Q8C22
socleC22×C3×C5C22×D4
prime radicalC2×D60C23
hypercenterC4×Q8D3×D5
hypocenterC3×C5#3359
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. C5×#5567C2semidirect
4. C3×#65309C2semidirect
6. #11749187C2semidirect
7. #11747417C2semidirect
10. #11746923C2semidirect
11. #11746857C2semidirect
53. #65309D3semidirect
88. #5567D5semidirect
145. C5×Dic6SD16semidirect
353. Dic6#688semidirect
405. C3×C5#3359semidirect
454. C5#81037semidirect
470. C3#10615419semidirect

Maximal subgroups

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

subgroupindex
C5×#55672
#117525512
C4×#67172
C3×#653092
#117492332
#117491872
#117474172
#117472322
#117470872
#117469232
#117468572
#106154193
#810375