Finite Group Databasebeta

← #14331129

Subgroups — #14331129

Order 1920.

1040 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91764C211yesmaximal, characteristic
2C3×C5×#382C211yesmaximal, characteristic, complemented
3#11751091C212yesmaximal, complemented
4#11751090C212yesmaximal, complemented
5#11751037C211yesmaximal, characteristic, complemented
6#11750771C212yesmaximal, complemented
7#11750769C212yesmaximal, complemented
8#11750766C212yesmaximal
9#11750747C212yesmaximal, complemented
10#10605961—31nomaximal, Hall
11C22×D60C2211yescharacteristic
12C22×C4×D15C2212yes—
13C22×Dic30C2211yescharacteristic
14C2×C3×C5×C4⋊C4C2211yescharacteristic
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C2×C3×C42×C5C2211yescharacteristic
17C2×#6714C2212yes—
18C2×#6714C2212yes—
19C2×#6714C2212yes—
20C2×#6714C2212yes—
21C2×#6714C2212yes—
22C2×C4⋊Dic15C2211yescharacteristic
23C2×C4⋊Dic15C2211yescharacteristic
24C2×#6712C2212yes—
25C2×C4×Dic15C2212yes—
26#91764C418yescomplemented
27C3×C5×#95C2212yes—
28C3×C5×#95C2212yes—
29#91115C2212yes—
30#91115C2212yes—
31#91115C2212yes—
32#91115C2212yes—
33#74230—51nomaximal, Hall
34C2×#4021—31no—
35C5×#382D311yescharacteristic, complemented
36#65279—32no—
37#65278—32no—
38#65225—31no—
39#64959—32no—
40#64957—32no—
41#64954—32no—
42#64935—32no—
43C23×D15C2312yes—
44C22×C3×C4×C5C2312yes—
45C22×C3×C4×C5C2312yes—
46C22×C3×C4×C5C2311yescharacteristic
47C22×C3×C4×C5C2311yescharacteristic
48C22×C3×C4×C5C2311yescharacteristic
49C22×Dic15C2312yes—
50C22×Dic15C2312yes—
51C22×Dic15C2312yes—
52C2×D60—41no—
53C2×D60—41no—
54C2×D60C2×C414yes—
55C2×C4×D15—42no—
56C2×C4×D15—42no—
57C2×C4×D15—42no—
58C2×Dic30—41no—
59C2×Dic30—41no—
60C2×Dic30C2×C414yescomplemented
61C3×C5×C4⋊C4—41no—
62C3×C5×C4⋊C4—41no—
63C3×C42×C5C2×C414yescomplemented
64#6714—42no—
65#6714—42no—
66#6714—42no—
67#6714C2×C418yescomplemented
68#6714C2×C418yescomplemented
69C4⋊Dic15—41no—
70C4⋊Dic15—41no—
71#6712—42no—
72C4×Dic15—42no—
73C2×#875—51no—
74C3×#382D511yescharacteristic, complemented
75#4990—52no—
76#4989—52no—
77#4936—51no—
78#4670—52no—
79#4668—52no—
80#4665—52no—
81#4646—52no—
82C22×D20—31no—
83C22×C4×D5—32no—
84C22×Dic10—31no—
85C2×C5×C4⋊C4C2×D311yescharacteristic
86C2×C5×C4⋊C4C2×D311yescharacteristic
87C2×C42×C5C2×D311yescharacteristic
88C2×#685—32no—
89C2×#685—32no—
90C2×#685—32no—
91C2×#685—32no—
92C2×#685—32no—
93C2×C4⋊Dic5—31no—
94C2×C4⋊Dic5—31no—
95C2×#683—32no—
96C2×C4×Dic5—32no—
97#4021—38no—
98C5×#95C2×D312yes—
99C5×#95C2×D312yes—
100#3963—32no—
101#3963—32no—
102#3963—32no—
103#3963—32no—
104#2379—151noSylow
105C23×C3×C5C2411yescharacteristic
106C22×D15—22no—
107C22×D15—22no—
108C22×D15—22no—
109C22×D15—22no—
110C22×D15—22no—
111C22×D15—22no—
112C22×D15C22×C414yes—
113C2×C3×C4×C5—21no—
114C2×C3×C4×C5—21no—
115C2×C3×C4×C5—21no—
116C2×C3×C4×C5—21no—
117C2×C3×C4×C5—21no—
118C2×C3×C4×C5—21no—
119C2×C3×C4×C5—22no—
120C2×C3×C4×C5—22no—
121C2×C3×C4×C5—22no—
122C2×C3×C4×C5—22no—
123C2×C3×C4×C5—22no—
124C2×C3×C4×C5—22no—
125C2×C3×C4×C5C22×C412yes—
126C2×C3×C4×C5C22×C412yes—
127C2×C3×C4×C5C22×C412yes—
128C2×Dic15—22no—
129C2×Dic15—22no—
130C2×Dic15—22no—
131C2×Dic15—22no—
132C2×Dic15—22no—
133C2×Dic15—22no—
134C2×Dic15—22no—
135C2×Dic15—22no—
136C2×Dic15C22×C414yes—
137D60—44no—
138C4×D15—48no—
139Dic30—44no—
140C22×D12—51no—
141C22×C4×D3—52no—
142C22×Dic6—51no—
143C2×C3×C4⋊C4C2×D511yescharacteristic
144C2×C3×C4⋊C4C2×D511yescharacteristic
145C2×C3×C42C2×D511yescharacteristic
146C2×(C2×D3)⋊C4—52no—
147C2×(C2×D3)⋊C4—52no—
148C2×(C2×D3)⋊C4—52no—
149C2×(C2×D3)⋊C4—52no—
150C2×(C2×D3)⋊C4—52no—
151C2×C4⋊Dic3—51no—
152C2×C4⋊Dic3—51no—
153C2×Dic3⋊C4—52no—
154C2×C4×Dic3—52no—
155#875—58no—
156C3×#95C2×D512yes—
157C3×#95C2×D512yes—
158C3⋊#95—52no—
159C3⋊#95—52no—
160C3⋊#95—52no—
161C3⋊#95—52no—
162C23×D5—32no—
163C22×C4×C5C22×D312yes—
164C22×C4×C5C22×D312yes—
165C22×C4×C5C22×D311yescharacteristic
166C22×C4×C5C22×D311yescharacteristic
167C22×C4×C5C22×D311yescharacteristic
168C22×Dic5—32no—
169C22×Dic5—32no—
170C22×Dic5—32no—
171C2×D20—34no—
172C2×D20—121no—
173C2×D20—121no—
174C2×C4×D5—122no—
175C2×C4×D5—122no—
176C2×C4×D5—122no—
177C2×Dic10—34no—
178C2×Dic10—121no—
179C2×Dic10—121no—
180C5×C4⋊C4—41no—
181C5×C4⋊C4—41no—
182C42×C5C4×D314yescomplemented
183#685—38no—
184#685—38no—
185#685—122no—
186#685—122no—
187#685—122no—
188C4⋊Dic5—121no—
189C4⋊Dic5—121no—
190#683—122no—
191C4×Dic5—122no—
192C2×#124—151no—
193#391—152no—
194#388—152no—
195#388—152no—
196#386—152no—
197#385—152no—
198#385—152no—
199#382—151no—
200#382D1511yescharacteristic, complemented
201C22×C3×C5C23×C411yescharacteristic
202C22×C3×C5H(5,𝔽2)11yescharacteristic
203C22×C3×C5H(5,𝔽2)11yescharacteristic
204C22×C3×C5H(5,𝔽2)11yescharacteristic
205C22×C3×C5H(5,𝔽2)11yescharacteristic
206C22×C3×C52−1+411yescharacteristic
207C22×C3×C52−1+411yescharacteristic
208C2×D15—24no—
209C2×D15—24no—
210C2×D15—24no—
211C2×D15—42no—
212C2×D15—42no—
213C2×D15—42no—
214C2×D15—42no—
215C3×C4×C5—22no—
216C3×C4×C5—22no—
217C3×C4×C5—22no—
218C3×C4×C5—42no—
219C3×C4×C5—42no—
220Dic15—24no—
221Dic15—42no—
222Dic15—42no—
223C23×D3—52no—
224C22×C3×C4C22×D512yes—
225C22×C3×C4C22×D512yes—
226C22×C3×C4C22×D511yescharacteristic
227C22×C3×C4C22×D511yescharacteristic
228C22×C3×C4C22×D511yescharacteristic
229C22×Dic3—52no—
230C22×Dic3—52no—
231C22×Dic3—52no—
232C2×D12—54no—
233C2×D12—201no—
234C2×D12—201no—
235C2×C4×D3—202no—
236C2×C4×D3—202no—
237C2×C4×D3—202no—
238C2×Dic6—54no—
239C2×Dic6—201no—
240C2×Dic6—201no—
241C3×C4⋊C4—41no—
242C3×C4⋊C4—41no—
243C3×C42C4×D514yescomplemented
244(C2×D3)⋊C4—58no—
245(C2×D3)⋊C4—58no—
246(C2×D3)⋊C4—202no—
247(C2×D3)⋊C4—202no—
248(C2×D3)⋊C4—202no—
249C4⋊Dic3—201no—
250C4⋊Dic3—201no—
251Dic3⋊C4—202no—
252C4×Dic3—202no—
253C23×C5C23×D311yescharacteristic
254C22×D5—34no—
255C22×D5—62no—
256C22×D5—62no—
257C22×D5—62no—
258C22×D5—62no—
259C22×D5—62no—
260C22×D5—62no—
261C2×C4×C5—21no—
262C2×C4×C5—21no—
263C2×C4×C5—21no—
264C2×C4×C5—21no—
265C2×C4×C5—21no—
266C2×C4×C5—21no—
267C2×C4×C5—22no—
268C2×C4×C5—22no—
269C2×C4×C5—22no—
270C2×C4×C5—22no—
271C2×C4×C5—22no—
272C2×C4×C5—22no—
273C2×C4×C5C2×C4×D312yes—
274C2×C4×C5C2×C4×D312yes—
275C2×C4×C5C2×C4×D312yes—
276C2×Dic5—34no—
277C2×Dic5—62no—
278C2×Dic5—62no—
279C2×Dic5—62no—
280C2×Dic5—62no—
281C2×Dic5—62no—
282C2×Dic5—62no—
283C2×Dic5—62no—
284C2×Dic5—62no—
285D20—124no—
286C4×D5—128no—
287Dic10—124no—
288C22×Q8—151no—
289C22×D4—151no—
290C23×C4—152no—
291#124—158no—
292C2×C4⋊C4—151no—
293C2×C4⋊C4—151no—
294C2×C4⋊C4—152no—
295C2×C4⋊C4C2×D1511yescharacteristic
296C2×C4⋊C4C2×D1511yescharacteristic
297C2×C22⋊C4—152no—
298C2×C22⋊C4—152no—
299C2×C22⋊C4—152no—
300C2×C22⋊C4—152no—
301C2×C22⋊C4—152no—
302C2×C42—152no—
303C2×C42C2×D1511yescharacteristic
304#95—152no—
305#95—152no—
306#95—152no—
307#95—152no—
308#95C2×D1512yes—
309#95C2×D1512yes—
310C2×C3×C5#51811yescharacteristic
311C2×C3×C5#52011yescharacteristic
312C2×C3×C5#52011yescharacteristic
313C2×C3×C5H(3,𝔽4)11yescharacteristic
314C2×C3×C5#56211yescharacteristic
315C2×C3×C5#56211yescharacteristic
316C2×C3×C5#56311yescharacteristic
317D15—44no—
318C23×C3C23×D511yescharacteristic
319C22×D3—54no—
320C22×D3—102no—
321C22×D3—102no—
322C22×D3—102no—
323C22×D3—102no—
324C22×D3—102no—
325C22×D3—102no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4C2×C4×D512yes—
339C2×C3×C4C2×C4×D512yes—
340C2×C3×C4C2×C4×D512yes—
341C2×Dic3—54no—
342C2×Dic3—102no—
343C2×Dic3—102no—
344C2×Dic3—102no—
345C2×Dic3—102no—
346C2×Dic3—102no—
347C2×Dic3—102no—
348C2×Dic3—102no—
349C2×Dic3—102no—
350D12—204no—
351C4×D3—208no—
352Dic6—204no—
353C22×C5C22×C4×D311yescharacteristic
354C22×C5#100411yescharacteristic
355C22×C5#100711yescharacteristic
356C22×C5#100911yescharacteristic
357C22×C5#100911yescharacteristic
358C22×C5#100911yescharacteristic
359C22×C5#101011yescharacteristic
360C2×D5—64no—
361C2×D5—64no—
362C2×D5—64no—
363C2×D5—122no—
364C2×D5—122no—
365C2×D5—122no—
366C2×D5—122no—
367C4×C5—22no—
368C4×C5—22no—
369C4×C5—22no—
370C4×C5—42no—
371C4×C5—42no—
372Dic5—64no—
373Dic5—122no—
374Dic5—122no—
375C24—152no—
376C2×Q8—154no—
377C2×Q8—601no—
378C2×Q8—601no—
379C2×D4—154no—
380C2×D4—601no—
381C2×D4—601no—
382C22×C4—152no—
383C22×C4—152no—
384C22×C4—152no—
385C22×C4—602no—
386C22×C4—602no—
387C22×C4—602no—
388C22×C4C22×D1512yes—
389C22×C4C22×D1512yes—
390C22×C4C22×D1511yescharacteristic
391C22×C4C22×D1511yescharacteristic
392C22×C4C22×D1511yescharacteristic
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—602no—
396C4⋊C4—601no—
397C4⋊C4—601no—
398C22⋊C4—158no—
399C22⋊C4—158no—
400C22⋊C4—602no—
401C22⋊C4—602no—
402C22⋊C4—602no—
403C42—602no—
404C42C4×D1514yescomplemented
405C3×C5#237911yescharacteristic, complemented, Hall
406C22×C3C22×C4×D511yescharacteristic
407C22×C3#414411yescharacteristic
408C22×C3#414711yescharacteristic
409C22×C3#414911yescharacteristic
410C22×C3#414911yescharacteristic
411C22×C3#414911yescharacteristic
412C22×C3#415011yescharacteristic
413C2×D3—104no—
414C2×D3—104no—
415C2×D3—104no—
416C2×D3—202no—
417C2×D3—202no—
418C2×D3—202no—
419C2×D3—202no—
420C3×C4—22no—
421C3×C4—22no—
422C3×C4—22no—
423C3×C4—42no—
424C3×C4—42no—
425Dic3—104no—
426Dic3—202no—
427Dic3—202no—
428C2×C5#550411yescharacteristic
429C2×C5#552111yescharacteristic
430C2×C5#552311yescharacteristic
431C2×C5#568311yescharacteristic
432C2×C5#568411yescharacteristic
433C2×C5#569911yescharacteristic
434C2×C5#570011yescharacteristic
435D5—124no—
436C23—154no—
437C23—302no—
438C23—302no—
439C23—302no—
440C23—302no—
441C23—302no—
442C23—302no—
443C23C23×D1511yescharacteristic
444Q8—604no—
445D4—604no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—154no—
459C2×C4—302no—
460C2×C4—302no—
461C2×C4—302no—
462C2×C4—302no—
463C2×C4—302no—
464C2×C4—302no—
465C2×C4—302no—
466C2×C4—302no—
467C2×C4—608no—
468C2×C4C2×C4×D1512yes—
469C2×C4C2×C4×D1512yes—
470C2×C4C2×C4×D1512yes—
471C2×C3#6583811yescharacteristic
472C2×C3#6585511yescharacteristic
473C2×C3#6585711yescharacteristic
474C2×C3#6601711yescharacteristic
475C2×C3#6601811yescharacteristic
476C2×C3#6603311yescharacteristic
477C2×C3#6603411yescharacteristic
478D3—204no—
479C5#7423011yescharacteristic, complemented, Sylow
480C22—304no—
481C22—304no—
482C22—304no—
483C22—602no—
484C22—602no—
485C22—602no—
486C22—602no—
487C22C22×C4×D1511yescharacteristic
488C22#9209511yescharacteristic
489C22#9209811yescharacteristic
490C22#9210011yescharacteristic
491C22#9210011yescharacteristic
492C22#9210011yescharacteristic
493C22#9210111yescharacteristic
494C4—22no—
495C4—22no—
496C4—22no—
497C4—42no—
498C4—42no—
499C4—304no—
500C4—602no—
501C4—602no—
502C3#1060596111yescharacteristic, complemented, Sylow
503C2—604no—
504C2#1175626311yescharacteristic
505C2#1175628011yescharacteristic
506C2#1175628211yescharacteristic
507C2#1175644211yescharacteristic
508C2#1175644311yescharacteristic
509C2#1175645811yescharacteristic
510C2#1175645911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D15
commutatorC22×C3×C5C23×C4
FittingC3×C5×#382C2
socle, cosocleC23×C3×C5C24
prime radicalC22×D60C22
hypercenter#382D15
hypocenterC3×C5#2379
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
2. C3×C5×#382C2semidirect
3. #11751091C2semidirect
4. #11751090C2semidirect
5. #11751037C2semidirect
6. #11750771C2semidirect
7. #11750769C2semidirect
9. #11750747C2semidirect
26. #91764C4semidirect
35. C5×#382D3semidirect
60. C2×Dic30C2×C4semidirect
63. C3×C42×C5C2×C4semidirect
67. #6714C2×C4semidirect
68. #6714C2×C4semidirect
74. C3×#382D5semidirect
182. C42×C5C4×D3semidirect
200. #382D15semidirect
243. C3×C42C4×D5semidirect
404. C42C4×D15semidirect
405. C3×C5#2379semidirect
479. C5#74230semidirect
502. C3#10605961semidirect

Maximal subgroups

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

subgroupindex
C2×#917642
C3×C5×#3822
#117510912
#117510902
#117510372
#117507712
#117507692
#117507662
#117507472
#106059613
#742305