Finite Group Databasebeta

← #14471746

Subgroups — #14471746

Order 1920.

724 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#559C211yesmaximal, characteristic, complemented
2#11756338C211yesmaximal, characteristic, complemented
3#11751173C211yesmaximal, characteristic
4#11751171C211yesmaximal, characteristic, complemented
5#11751158C212yesmaximal, complemented
6#11751157C212yesmaximal, complemented
7C4×#6717C211yesmaximal, characteristic, complemented
8Q8×C3⋊#168C211yesmaximal, characteristic, complemented
9#11751122C211yesmaximal, characteristic, complemented
10C4×#6715C212yesmaximal, complemented
11#11751110C212yesmaximal, complemented
12#10615444—31nomaximal, Hall
13C3×C5×#127C2211yescharacteristic
14C3×C5×#124—21no—
15C3×C5×#124C2212yes—
16C3×C5×#123—22no—
17C3×C4×C5×Q8C2211yescharacteristic
18C3×C4×C5×Q8C2211yescharacteristic
19C3×C4×C5×D4—21no—
20C3×C4×C5×D4C2212yes—
21#91835—21no—
22C2×#6717C2211yescharacteristic
23C2×#6715C2212yes—
24#91785—22no—
25C4⋊C4×D15—21no—
26C4×D60C2211yescharacteristic
27C4×D60C2212yes—
28#91759C2212yes—
29#91758C2211yescharacteristic
30#91119C2212yes—
31#91119C2211yescharacteristic
32#91116C2212yes—
33#91116C2212yes—
34#91116C2212yes—
35#91093C2211yescharacteristic
36#91092C2211yescharacteristic
37#91091C2212yes—
38C4⋊(C3⋊#168)C2211yescharacteristic
39C4⋊(C3⋊#168)C2212yes—
40C4×C3⋊#168C2211yescharacteristic
41C4×C3⋊#168C2212yes—
42#83713—51nomaximal, Hall
43C5×#559D311yescharacteristic, complemented
44#65913—31no—
45#65361—31no—
46#65359—31no—
47#65346—32no—
48#65345—32no—
49C4×#688—31no—
50Q8×#168—31no—
51#65310—31no—
52C4×#686—32no—
53#65298—32no—
54C2×C3×C5×Q8D411yescharacteristic
55C2×C3×C5×Q8C2311yescharacteristic
56C2×C3×C5×D4—21no—
57C2×C3×C5×D4—21no—
58C2×C3×C5×D4C2312yes—
59C22×C3×C4×C5—22no—
60C2×D60C2311yescharacteristic
61C2×C4×D15—21no—
62C2×C4×D15—22no—
63C2×Dic30—21no—
64C3×C5×C4⋊C4—21no—
65C3×C5×C4⋊C4D411yescharacteristic
66C3×C5×C4⋊C4D412yes—
67C3×C5×C4⋊C4C2311yescharacteristic
68C3×C5×C22⋊C4—22no—
69C3×C5×C22⋊C4—22no—
70C3×C5×C22⋊C4—22no—
71C3×C42×C5—21no—
72C3×C42×C5C2311yescharacteristic
73C3×C42×C5C2312yes—
74#6717—22no—
75#6715—24no—
76#6714—21no—
77#6714—22no—
78C4⋊Dic15—21no—
79C4⋊Dic15C2312yes—
80C4⋊Dic15C2311yescharacteristic
81#6712—22no—
82#6712—22no—
83C2×C3⋊#168C2312yes—
84C2×C3⋊#168C2311yescharacteristic
85C2×C3⋊#168C2311yescharacteristic
86C3×#559D511yescharacteristic, complemented
87#5579—51no—
88#5072—51no—
89#5070—51no—
90#5057—52no—
91#5056—52no—
92C4×#215—51no—
93Q8×C3⋊C8—51no—
94#5021—51no—
95C4×#213—52no—
96#5009—52no—
97C5×#127C2×D311yescharacteristic
98C5×#124—21no—
99C5×#124C2×D312yes—
100C5×#123—22no—
101C4×C5×Q8C2×D311yescharacteristic
102C4×C5×Q8C2×D311yescharacteristic
103C4×C5×D4—21no—
104C4×C5×D4C2×D312yes—
105#4092—61no—
106C2×#688—31no—
107C2×#686—32no—
108#4042—62no—
109D5×C4⋊C4—61no—
110C4×D20—32no—
111C4×D20—31no—
112#4016—32no—
113#4015—31no—
114#3967—32no—
115#3967—31no—
116#3964—32no—
117#3964—32no—
118#3964—32no—
119#3941—31no—
120#3940—31no—
121#3939—32no—
122C4⋊#168—32no—
123C4⋊#168—31no—
124C4×#168—32no—
125C4×#168—31no—
126#3361—151noSylow
127C23×C3×C5—22no—
128C22×D15—21no—
129C3×C5×Q8—22no—
130C3×C5×Q8—21no—
131C3×C5×Q8—21no—
132C3×C5×D4—22no—
133C3×C5×D4—22no—
134C3×C5×D4—42no—
135C3×C5×D4—81no—
136C2×C3×C4×C5—21no—
137C2×C3×C4×C5—21no—
138C2×C3×C4×C5—82no—
139C2×C3×C4×C5C2×D412yes—
140C2×C3×C4×C5C2×D412yes—
141C2×C3×C4×C5C2×D411yescharacteristic
142C2×C3×C4×C5C2×D411yescharacteristic
143C2×C3×C4×C5C2411yescharacteristic
144C2×Dic15—21no—
145C2×Dic15—22no—
146D60—21no—
147D60#4112yes—
148C4×D15—42no—
149C4×D15—82no—
150Dic30—81no—
151C3⋊#168—22no—
152C3⋊#168—21no—
153C3⋊#168—21no—
154C3×#127C2×D511yescharacteristic
155C3×#124—21no—
156C3×#124C2×D512yes—
157C3×#123—22no—
158C3×C4×Q8C2×D511yescharacteristic
159C3×C4×Q8C2×D511yescharacteristic
160C3×C4×D4—21no—
161C3×C4×D4C2×D512yes—
162#946—101no—
163C2×#215—51no—
164C2×#213—52no—
165#896—102no—
166D3×C4⋊C4—101no—
167C4×D12—52no—
168C4×D12—51no—
169#870—52no—
170#869—51no—
171#835—52no—
172#835—51no—
173#832—52no—
174#832—52no—
175#832—52no—
176#809—51no—
177#808—51no—
178#807—52no—
179C4⋊(C3⋊C8)—52no—
180C4⋊(C3⋊C8)—51no—
181C4×C3⋊C8—52no—
182C4×C3⋊C8—51no—
183C2×C5×Q8#6711yescharacteristic
184C2×C5×Q8C22×D311yescharacteristic
185C2×C5×D4—21no—
186C2×C5×D4—21no—
187C2×C5×D4C22×D312yes—
188C22×C4×C5—22no—
189C2×D20—31no—
190C2×C4×D5—61no—
191C2×C4×D5—62no—
192C2×Dic10—61no—
193C5×C4⋊C4—21no—
194C5×C4⋊C4#6711yescharacteristic
195C5×C4⋊C4#6712yes—
196C5×C4⋊C4C22×D311yescharacteristic
197C5×C22⋊C4—22no—
198C5×C22⋊C4—22no—
199C5×C22⋊C4—22no—
200C42×C5—21no—
201C42×C5C22×D312yes—
202C42×C5C22×D311yescharacteristic
203#688—62no—
204#686—64no—
205#685—61no—
206#685—62no—
207C4⋊Dic5—32no—
208C4⋊Dic5—31no—
209C4⋊Dic5—61no—
210#683—62no—
211#683—62no—
212C2×#168—31no—
213C2×#168—32no—
214C2×#168—31no—
215#559D1511yescharacteristic, complemented
216#554—151no—
217#488—152no—
218#486—151no—
219#478—152no—
220#477—151no—
221#463—152no—
222#459—151no—
223C8×Q8—151no—
224C4×SD16—151no—
225C4×D8—152no—
226C22×C3×C5—42no—
227C22×C3×C5—42no—
228C22×C3×C5—42no—
229C22×C3×C5C22×D411yescharacteristic
230C2×D15—22no—
231C2×D15—41no—
232C2×D15—41no—
233C3×C4×C5—22no—
234C3×C4×C5—21no—
235C3×C4×C5—22no—
236C3×C4×C5—41no—
237C3×C4×C5—41no—
238C3×C4×C5#13512yes—
239C3×C4×C5C2×#4111yescharacteristic
240C3×C4×C52−1+411yescharacteristic
241Dic15—41no—
242Dic15—42no—
243C2×C3×Q8C22⋊D511yescharacteristic
244C2×C3×Q8C22×D511yescharacteristic
245C2×C3×D4—21no—
246C2×C3×D4—21no—
247C2×C3×D4C22×D512yes—
248C22×C3×C4—22no—
249C2×D12—51no—
250C2×C4×D3—101no—
251C2×C4×D3—102no—
252C2×Dic6—101no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4C22⋊D511yescharacteristic
255C3×C4⋊C4C22⋊D512yes—
256C3×C4⋊C4C22×D511yescharacteristic
257C3×C22⋊C4—22no—
258C3×C22⋊C4—22no—
259C3×C22⋊C4—22no—
260C3×C42—21no—
261C3×C42C22×D511yescharacteristic
262C3×C42C22×D512yes—
263#215—102no—
264#213—104no—
265(C2×D3)⋊C4—101no—
266(C2×D3)⋊C4—102no—
267C4⋊Dic3—52no—
268C4⋊Dic3—51no—
269C4⋊Dic3—101no—
270Dic3⋊C4—102no—
271Dic3⋊C4—102no—
272C2×C3⋊C8—51no—
273C2×C3⋊C8—52no—
274C2×C3⋊C8—51no—
275C23×C5—22no—
276C22×D5—61no—
277C5×Q8—22no—
278C5×Q8—21no—
279C5×Q8—21no—
280C5×D4—22no—
281C5×D4—22no—
282C5×D4—42no—
283C5×D4—81no—
284C2×C4×C5—21no—
285C2×C4×C5—21no—
286C2×C4×C5—82no—
287C2×C4×C5C2×#6712yes—
288C2×C4×C5C2×#6712yes—
289C2×C4×C5C2×#6711yescharacteristic
290C2×C4×C5C2×#6711yescharacteristic
291C2×C4×C5C23×D311yescharacteristic
292C2×Dic5—61no—
293C2×Dic5—62no—
294D20—32no—
295D20—61no—
296C4×D5—122no—
297C4×D5—242no—
298Dic10—241no—
299#168—62no—
300#168—61no—
301#168—61no—
302C2×SD16—151no—
303C2×D8—152no—
304C4⋊Q8—151no—
305#127C2×D1511yescharacteristic
306#125—152no—
307#124—21no—
308#124C2×D1512yes—
309#123—22no—
310C22⋊Q8—301no—
311C22⋊Q8—302no—
312C4×Q8C2×D1511yescharacteristic
313C4×Q8C2×D1511yescharacteristic
314C4×D4—21no—
315C4×D4—152no—
316C4×D4—151no—
317C4×D4C2×D1512yes—
318C2×C4⋊C4—301no—
319#107—152no—
320#106—151no—
321C4⋊C8—151no—
322C4⋊C8—152no—
323#103—151no—
324#103—152no—
325#102—152no—
326#102—152no—
327#102—152no—
328#102—151no—
329C4×C8—152no—
330C4×C8—151no—
331C2×C3×C5—82no—
332C2×C3×C5#54811yescharacteristic
333C2×C3×C5C2×#13511yescharacteristic
334C2×C3×C5#57611yescharacteristic
335D15—42no—
336C23×C3—22no—
337C22×D3—101no—
338C3×Q8—22no—
339C3×Q8—21no—
340C3×Q8—21no—
341C3×D4—22no—
342C3×D4—22no—
343C3×D4—42no—
344C3×D4—81no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—82no—
348C2×C3×C4C2×C22⋊D512yes—
349C2×C3×C4C2×C22⋊D512yes—
350C2×C3×C4C2×C22⋊D511yescharacteristic
351C2×C3×C4C2×C22⋊D511yescharacteristic
352C2×C3×C4C23×D511yescharacteristic
353C2×Dic3—101no—
354C2×Dic3—102no—
355D12—52no—
356D12—101no—
357C4×D3—202no—
358C4×D3—402no—
359Dic6—401no—
360C3⋊C8—102no—
361C3⋊C8—101no—
362C3⋊C8—101no—
363C22×C5—42no—
364C22×C5—42no—
365C22×C5—42no—
366C22×C5C22×#6711yescharacteristic
367C2×D5—62no—
368C2×D5—121no—
369C2×D5—121no—
370C4×C5—22no—
371C4×C5—21no—
372C4×C5—22no—
373C4×C5—41no—
374C4×C5—41no—
375C4×C5#95012yes—
376C4×C5D3×#4111yescharacteristic
377C4×C5#101011yescharacteristic
378Dic5—121no—
379Dic5—122no—
380C2×Q8—301no—
381C2×Q8#122011yescharacteristic
382C2×Q8C22×D1511yescharacteristic
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—151no—
386C2×D4C22×D1512yes—
387C22×C4—22no—
388C22×C4—301no—
389C22×C4—302no—
390SD16—302no—
391D8—304no—
392C2×C8—151no—
393C2×C8—151no—
394C2×C8—152no—
395C4⋊C4—21no—
396C4⋊C4—151no—
397C4⋊C4—152no—
398C4⋊C4—302no—
399C4⋊C4—301no—
400C4⋊C4—302no—
401C4⋊C4#122011yescharacteristic
402C4⋊C4#122012yes—
403C4⋊C4C22×D1511yescharacteristic
404C22⋊C4—22no—
405C22⋊C4—22no—
406C22⋊C4—22no—
407C22⋊C4—301no—
408C22⋊C4—302no—
409C42—21no—
410C42C22×D1512yes—
411C42C22×D1511yescharacteristic
412C3×C5#336111yescharacteristic, complemented, Hall
413C22×C3—42no—
414C22×C3—42no—
415C22×C3—42no—
416C22×C3C22×C22⋊D511yescharacteristic
417C2×D3—102no—
418C2×D3—201no—
419C2×D3—201no—
420C3×C4—22no—
421C3×C4—21no—
422C3×C4—22no—
423C3×C4—41no—
424C3×C4—41no—
425C3×C4#409612yes—
426C3×C4D5×#4111yescharacteristic
427C3×C4#415011yescharacteristic
428Dic3—201no—
429Dic3—202no—
430C2×C5—82no—
431C2×C5C2×#95011yescharacteristic
432C2×C5#583111yescharacteristic
433C2×C5#583511yescharacteristic
434D5—122no—
435C23—22no—
436C23—301no—
437Q8—22no—
438Q8—21no—
439Q8—21no—
440Q8—1201no—
441D4—22no—
442D4—22no—
443D4—42no—
444D4—81no—
445D4—152no—
446D4—301no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—82no—
450C2×C4—301no—
451C2×C4—302no—
452C2×C4—602no—
453C2×C4—1202no—
454C2×C4C2×#122012yes—
455C2×C4C2×#122012yes—
456C2×C4C2×#122011yescharacteristic
457C2×C4C2×#122011yescharacteristic
458C2×C4C23×D1511yescharacteristic
459C8—302no—
460C8—301no—
461C8—301no—
462C2×C3—82no—
463C2×C3C2×#409611yescharacteristic
464C2×C3#6616511yescharacteristic
465C2×C3#6616911yescharacteristic
466D3—202no—
467C5#8371311yescharacteristic, complemented, Sylow
468C22—42no—
469C22—42no—
470C22—42no—
471C22—302no—
472C22—601no—
473C22—601no—
474C22C22×#122011yescharacteristic
475C4—22no—
476C4—22no—
477C4—21no—
478C4—41no—
479C4—41no—
480C4—601no—
481C4—602no—
482C4#9183912yes—
483C4#41×D1511yescharacteristic
484C4#9210111yescharacteristic
485C3#1061544411yescharacteristic, complemented, Sylow
486C2—82no—
487C2—602no—
488C2C2×#9183911yescharacteristic
489C2#1175659011yescharacteristic
490C2#1175659411yescharacteristic
subgroupG/H
centerC22C22×#1220
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C5×#559C2
socleC22×C3×C5C22×D4
prime radical#11751173C2
hypercenter#559D15
hypocenterC3×C5#3361
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #559

Complements

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

NH ≅ G/Ntype
1. C3×C5×#559C2semidirect
2. #11756338C2semidirect
4. #11751171C2semidirect
5. #11751158C2semidirect
6. #11751157C2semidirect
7. C4×#6717C2semidirect
8. Q8×C3⋊#168C2semidirect
9. #11751122C2semidirect
10. C4×#6715C2semidirect
11. #11751110C2semidirect
43. C5×#559D3semidirect
86. C3×#559D5semidirect
215. #559D15semidirect
412. C3×C5#3361semidirect
467. C5#83713semidirect
485. C3#10615444semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5592
#117563382
#117511732
#117511712
#117511582
#117511572
C4×#67172
Q8×C3⋊#1682
#117511222
C4×#67152
#117511102
#106154443
#837135