Finite Group Databasebeta

← #14313407

Subgroups — #14313407

Order 1920.

516 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756591C211yesmaximal, characteristic
2C3×C5×#410C211yesmaximal, characteristic, complemented
3#11751341C211yesmaximal, characteristic, complemented
4#11746295C212yesmaximal, complemented
5#11746294C212yesmaximal, complemented
6#10600853—31nomaximal, Hall
7C2×C3×C5×#41C2211yescharacteristic
8C2×#6817C411yescharacteristic, complemented
9C3×C5×#117—21no—
10#91836C411yescharacteristic, complemented
11#91833C411yescharacteristic, complemented
12#91829C2211yescharacteristic
13#91828—21no—
14#91828—21no—
15#91825C2211yescharacteristic
16C4×#1220—21no—
17C4×#1220—21no—
18#91815—21no—
19C2×C4×Dic15C411yescharacteristic, complemented
20C3×C5×#99—21no—
21C3×C5×#99C2212yes—
22#91118—21no—
23#91118C2212yes—
24#70060—51nomaximal, Hall
25#66166—31no—
26C5×#410D311yescharacteristic, complemented
27#65529—31no—
28#64694—32no—
29#64693—32no—
30C3×C5×#41—41no—
31C3×C5×#41—41no—
32C2×C3×C5×Q8C2×C411yescharacteristic
33C2×C3×C5×D4D411yescharacteristic
34C2×C3×C5×D4D411yescharacteristic
35C2×C3×C5×D4C2311yescharacteristic
36C22×C3×C4×C5C2×C411yescharacteristic
37C22×C3×C4×C5D411yescharacteristic
38C22×C3×C4×C5D411yescharacteristic
39C2×#1220—21no—
40C2×#1220—21no—
41C2×#1220—21no—
42C22×Dic15—21no—
43#6817—41no—
44#6817—41no—
45C2×D60C2×C411yescharacteristic
46C2×C4×D15—21no—
47C2×Dic30C2×C411yescharacteristic
48C3×C5×C4⋊C4—41no—
49C3×C5×C22⋊C4—22no—
50C3×C42×C5—41no—
51#6719—21no—
52#6719—21no—
53#6719—22no—
54#6714—21no—
55#6714—21no—
56C4⋊Dic15—41no—
57#6712—21no—
58#6712—21no—
59C4×Dic15—21no—
60C4×Dic15—41no—
61C4×Dic15C2×C411yescharacteristic
62C4×Dic15C2×C411yescharacteristic
63#5832—51no—
64C3×#410D511yescharacteristic, complemented
65#5240—51no—
66#4474—52no—
67#4473—52no—
68C2×C5×#41C2×D311yescharacteristic
69C2×#709—31no—
70C5×#117—21no—
71#4093—31no—
72#4090—31no—
73#4086—31no—
74#4085—61no—
75#4085—61no—
76#4082—31no—
77C4×C22⋊D5—61no—
78C4×C22⋊D5—61no—
79#4072—61no—
80C2×C4×Dic5—31no—
81C5×#99—21no—
82C5×#99C2×D312yes—
83#3966—32no—
84#3966—61no—
85#2126—151noSylow
86C23×C3×C5C22⋊C411yescharacteristic
87C23×C3×C5C2×D411yescharacteristic
88C23×C3×C5C2×D411yescharacteristic
89C22×D15—21no—
90C3×C5×Q8—41no—
91C3×C5×D4—41no—
92C3×C5×D4—41no—
93C3×C5×D4—41no—
94C2×C3×C4×C5—21no—
95C2×C3×C4×C5—21no—
96C2×C3×C4×C5—21no—
97C2×C3×C4×C5—21no—
98C2×C3×C4×C5—21no—
99C2×C3×C4×C5—21no—
100C2×C3×C4×C5—42no—
101C2×C3×C4×C5C22⋊C411yescharacteristic
102C2×C3×C4×C5C22⋊C411yescharacteristic
103C2×C3×C4×C5C22⋊C411yescharacteristic
104C2×C3×C4×C5C22×C411yescharacteristic
105#1220—42no—
106#1220—42no—
107#1220—42no—
108C2×Dic15—21no—
109C2×Dic15—21no—
110C2×Dic15—21no—
111C2×Dic15—42no—
112C2×Dic15—42no—
113D60—41no—
114C4×D15—41no—
115C4×D15—41no—
116Dic30—41no—
117C2×C3×#41C2×D511yescharacteristic
118C2×#235—51no—
119C3×#117—21no—
120#947—51no—
121#944—51no—
122#940—51no—
123#939—101no—
124#939—101no—
125#936—51no—
126C4×#67—101no—
127C4×#67—101no—
128#926—101no—
129C2×C4×Dic3—51no—
130C3×#99—21no—
131C3×#99C2×D512yes—
132#834—52no—
133#834—101no—
134C5×#41—41no—
135C5×#41—41no—
136C2×C5×Q8C4×D311yescharacteristic
137C2×C5×D4D1211yescharacteristic
138C2×C5×D4#6711yescharacteristic
139C2×C5×D4C22×D311yescharacteristic
140C22×C4×C5C4×D311yescharacteristic
141C22×C4×C5D1211yescharacteristic
142C22×C4×C5#6711yescharacteristic
143C2×C22⋊D5—61no—
144C2×C22⋊D5—61no—
145C2×C22⋊D5—61no—
146C22×Dic5—61no—
147#709—121no—
148#709—121no—
149C2×D20—31no—
150C2×C4×D5—61no—
151C2×Dic10—31no—
152C5×C4⋊C4—41no—
153C5×C22⋊C4—22no—
154C42×C5—41no—
155C22⋊Dic5—61no—
156C22⋊Dic5—61no—
157C22⋊Dic5—62no—
158#685—61no—
159#685—61no—
160C4⋊Dic5—121no—
161#683—61no—
162#683—61no—
163C4×Dic5—31no—
164C4×Dic5—31no—
165C4×Dic5—61no—
166C4×Dic5—121no—
167#532—151no—
168#410—151no—
169#410D1511yescharacteristic, complemented
170#354—152no—
171#353—152no—
172C22×C3×C5—21no—
173C22×C3×C5—21no—
174C22×C3×C5—21no—
175C22×C3×C5—41no—
176C22×C3×C5—41no—
177C22×C3×C5—41no—
178C22×C3×C5C2×C22⋊C411yescharacteristic
179C2×D15—41no—
180C2×D15—41no—
181C2×D15—41no—
182C3×C4×C5—21no—
183C3×C4×C5—21no—
184C3×C4×C5—21no—
185C3×C4×C5—82no—
186C3×C4×C5#9911yescharacteristic
187C3×C4×C5#9911yescharacteristic
188Dic15—22no—
189Dic15—41no—
190Dic15—41no—
191Dic15—82no—
192C3×#41—41no—
193C3×#41—41no—
194C2×C3×Q8C4×D511yescharacteristic
195C2×C3×D4D2011yescharacteristic
196C2×C3×D4C22⋊D511yescharacteristic
197C2×C3×D4C22×D511yescharacteristic
198C22×C3×C4C4×D511yescharacteristic
199C22×C3×C4D2011yescharacteristic
200C22×C3×C4C22⋊D511yescharacteristic
201C2×#67—101no—
202C2×#67—101no—
203C2×#67—101no—
204C22×Dic3—101no—
205#235—201no—
206#235—201no—
207C2×D12—51no—
208C2×C4×D3—101no—
209C2×Dic6—51no—
210C3×C4⋊C4—41no—
211C3×C22⋊C4—22no—
212C3×C42—41no—
213#217—101no—
214#217—101no—
215#217—102no—
216(C2×D3)⋊C4—101no—
217(C2×D3)⋊C4—101no—
218C4⋊Dic3—201no—
219Dic3⋊C4—101no—
220Dic3⋊C4—101no—
221C4×Dic3—51no—
222C4×Dic3—51no—
223C4×Dic3—101no—
224C4×Dic3—201no—
225C23×C5(C2×D3)⋊C411yescharacteristic
226C23×C5C2×D1211yescharacteristic
227C23×C5C2×#6711yescharacteristic
228C22×D5—61no—
229C5×Q8—41no—
230C5×D4—41no—
231C5×D4—41no—
232C5×D4—41no—
233C2×C4×C5—21no—
234C2×C4×C5—21no—
235C2×C4×C5—21no—
236C2×C4×C5—21no—
237C2×C4×C5—21no—
238C2×C4×C5—21no—
239C2×C4×C5—42no—
240C2×C4×C5(C2×D3)⋊C411yescharacteristic
241C2×C4×C5(C2×D3)⋊C411yescharacteristic
242C2×C4×C5(C2×D3)⋊C411yescharacteristic
243C2×C4×C5C2×C4×D311yescharacteristic
244C22⋊D5—122no—
245C22⋊D5—122no—
246C22⋊D5—122no—
247C2×Dic5—61no—
248C2×Dic5—61no—
249C2×Dic5—61no—
250C2×Dic5—122no—
251C2×Dic5—122no—
252D20—121no—
253C4×D5—121no—
254C4×D5—121no—
255Dic10—121no—
256C2×#41—151no—
257C2×#41C2×D1511yescharacteristic
258C4⋊Q8—151no—
259#127—151no—
260#124—151no—
261#124—151no—
262#121—301no—
263#121—301no—
264C4×D4—301no—
265C4×D4—301no—
266#117—21no—
267#117—301no—
268C2×C42—151no—
269#99—21no—
270#99—152no—
271#99—301no—
272#99C2×D1512yes—
273C2×C3×C5—21no—
274C2×C3×C5—41no—
275C2×C3×C5—41no—
276C2×C3×C5—41no—
277C2×C3×C5C2×#9911yescharacteristic
278D15—81no—
279C23×C3#68511yescharacteristic
280C23×C3C2×D2011yescharacteristic
281C23×C3C2×C22⋊D511yescharacteristic
282C22×D3—101no—
283C3×Q8—41no—
284C3×D4—41no—
285C3×D4—41no—
286C3×D4—41no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—42no—
294C2×C3×C4#68511yescharacteristic
295C2×C3×C4#68511yescharacteristic
296C2×C3×C4#68511yescharacteristic
297C2×C3×C4C2×C4×D511yescharacteristic
298#67—202no—
299#67—202no—
300#67—202no—
301C2×Dic3—101no—
302C2×Dic3—101no—
303C2×Dic3—101no—
304C2×Dic3—202no—
305C2×Dic3—202no—
306D12—201no—
307C4×D3—201no—
308C4×D3—201no—
309Dic6—201no—
310C22×C5—21no—
311C22×C5—21no—
312C22×C5—21no—
313C22×C5—41no—
314C22×C5—41no—
315C22×C5—41no—
316C22×C5C2×(C2×D3)⋊C411yescharacteristic
317C2×D5—121no—
318C2×D5—121no—
319C2×D5—121no—
320C4×C5—21no—
321C4×C5—21no—
322C4×C5—21no—
323C4×C5—82no—
324C4×C5#80611yescharacteristic
325C4×C5#80611yescharacteristic
326Dic5—62no—
327Dic5—121no—
328Dic5—121no—
329Dic5—242no—
330#41—41no—
331#41—41no—
332#41—601no—
333#41—601no—
334C2×Q8—151no—
335C2×Q8C4×D1511yescharacteristic
336C2×D4—151no—
337C2×D4—301no—
338C2×D4—301no—
339C2×D4—301no—
340C2×D4D6011yescharacteristic
341C2×D4#122011yescharacteristic
342C2×D4C22×D1511yescharacteristic
343C22×C4—301no—
344C22×C4—301no—
345C22×C4C4×D1511yescharacteristic
346C22×C4D6011yescharacteristic
347C22×C4#122011yescharacteristic
348C4⋊C4—41no—
349C4⋊C4—301no—
350C4⋊C4—301no—
351C4⋊C4—601no—
352C22⋊C4—22no—
353C22⋊C4—301no—
354C22⋊C4—301no—
355C22⋊C4—301no—
356C22⋊C4—301no—
357C22⋊C4—302no—
358C42—41no—
359C42—151no—
360C42—151no—
361C42—301no—
362C42—601no—
363C3×C5#212611yescharacteristic, complemented, Hall
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—41no—
368C22×C3—41no—
369C22×C3—41no—
370C22×C3C2×#68511yescharacteristic
371C2×D3—201no—
372C2×D3—201no—
373C2×D3—201no—
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—21no—
377C3×C4—82no—
378C3×C4#393811yescharacteristic
379C3×C4#393811yescharacteristic
380Dic3—102no—
381Dic3—201no—
382Dic3—201no—
383Dic3—402no—
384C2×C5—21no—
385C2×C5—41no—
386C2×C5—41no—
387C2×C5—41no—
388C2×C5C2×#80611yescharacteristic
389D5—241no—
390C23—301no—
391C23#671411yescharacteristic
392C23C2×D6011yescharacteristic
393C23C2×#122011yescharacteristic
394Q8—41no—
395Q8—601no—
396D4—41no—
397D4—41no—
398D4—41no—
399D4—601no—
400D4—602no—
401D4—602no—
402D4—602no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—42no—
410C2×C4—301no—
411C2×C4—301no—
412C2×C4—301no—
413C2×C4—602no—
414C2×C4—601no—
415C2×C4—601no—
416C2×C4—602no—
417C2×C4#671411yescharacteristic
418C2×C4#671411yescharacteristic
419C2×C4#671411yescharacteristic
420C2×C4C2×C4×D1511yescharacteristic
421C2×C3—21no—
422C2×C3—41no—
423C2×C3—41no—
424C2×C3—41no—
425C2×C3C2×#393811yescharacteristic
426D3—401no—
427C5#7006011yescharacteristic, complemented, Sylow
428C22—21no—
429C22—21no—
430C22—21no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—601no—
435C22—601no—
436C22—601no—
437C22C2×#671411yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—82no—
442C4—302no—
443C4—601no—
444C4—601no—
445C4—1202no—
446C4#9109011yescharacteristic
447C4#9109011yescharacteristic
448C3#1060085311yescharacteristic, complemented, Sylow
449C2—21no—
450C2—41no—
451C2—41no—
452C2—41no—
453C2—1201no—
454C2C2×#9109011yescharacteristic
subgroupG/H
centerC4#91090
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×D4C22×D15
FittingC3×C5×#410C2
socleC2×C3×C5C2×#99
prime radical#11756591C2
cosocleC2×C3×C5×D4C23
hypercenter#410D15
hypocenterC3×C5#2126
LCSG ▷ C2×C3×C4×C5 ▷ C22×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
2. C3×C5×#410C2semidirect
3. #11751341C2semidirect
4. #11746295C2semidirect
5. #11746294C2semidirect
8. C2×#6817C4semidirect
10. #91836C4semidirect
11. #91833C4semidirect
19. C2×C4×Dic15C4semidirect
26. C5×#410D3semidirect
64. C3×#410D5semidirect
169. #410D15semidirect
363. C3×C5#2126semidirect
427. C5#70060semidirect
448. C3#10600853semidirect

Maximal subgroups

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

subgroupindex
#117565912
C3×C5×#4102
#117513412
#117462952
#117462942
#106008533
#700605