Finite Group Databasebeta

← #14319729

Subgroups — #14319729

Order 1920.

1560 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91444C212yesmaximal
2C2×C4×#1202C211yesmaximal, characteristic
3#11750767C211yesmaximal, characteristic, complemented
4C5×#4649C211yesmaximal, characteristic, complemented
5C3×D5×#95C211yesmaximal, characteristic, complemented
6#11748888C211yesmaximal, characteristic, complemented
7#11748881C212yesmaximal, complemented
8#11748702C211yesmaximal, characteristic, complemented
9C4×#6665C212yesmaximal, complemented
10#11748651C212yesmaximal, complemented
11#11748643C211yesmaximal, characteristic, complemented
12#10605482—31nomaximal, Hall
13C22×C4×D15C2211yescharacteristic
14C22×C3×C4×D5C2211yescharacteristic
15C22×C3×C4×D5C2212yes—
16C22×#1202C2211yescharacteristic
17C22×#1200C2212yes—
18C2×#6714C2212yes—
19C2×C4×Dic15C2212yes—
20C2×C5×Dic3⋊C4C2212yes—
21C2×C4×C5×Dic3C2212yes—
22C2×C4×C5×Dic3C2211yescharacteristic
23C2×Dic3⋊Dic5C2211yescharacteristic
24C2×#6667C2211yescharacteristic
25C2×#6665C2211yescharacteristic
26C2×#6665C2212yes—
27C2×Dic3×Dic5C2212yes—
28C2×Dic3×Dic5C2212yes—
29#91444C4116yescomplemented
30C4×#1202C418yescomplemented
31C3×C5×#95C2211yescharacteristic
32#91115C2211yescharacteristic
33C5×C3⋊#95C2211yescharacteristic
34C3×#3963C2212yes—
35C3×#3963C2211yescharacteristic
36#90994C2212yes—
37#90994C2212yes—
38#73322—51nomaximal, Hall
39C2×#4039—32no—
40C2×C4×D20—31no—
41C4×C5×C4⋊C4—31no—
42C4⋊C4×Dic5—32no—
43C4×#685—32no—
44C4×#685—32no—
45#65221—31no—
46#64955—31no—
47#64955—31no—
48D5×#95—31no—
49D5×#95D311yescharacteristic, complemented
50C23×D15C2311yescharacteristic
51C23×C3×D5C2311yescharacteristic
52C22×C3×C4×C5C2312yes—
53C22×C3×C4×C5C2311yescharacteristic
54C22×Dic15C2311yescharacteristic
55C22×Dic15C2312yes—
56C2×C4×D15—42no—
57C2×C4×D15C2×C414yescomplemented
58C22×C5×Dic3C2311yescharacteristic
59C22×C5×Dic3C2312yes—
60C22×C5×Dic3C2311yescharacteristic
61C22×C3×Dic5C2312yes—
62C22×C3×Dic5C2311yescharacteristic
63C2×C3×C4×D5—42no—
64C2×C3×C4×D5—42no—
65C2×C3×C4×D5—42no—
66C2×C3×C4×D5C2×C414yescomplemented
67C2×C3×C4×D5C2×C418yescomplemented
68C2×#1202C2×C414yes—
69C2×#1202C2×C418yes—
70C2×#1200—42no—
71C2×#1200—42no—
72C2×#1200C2×C418yescomplemented
73#6714C2×C418yescomplemented
74C4×Dic15—42no—
75C5×Dic3⋊C4C2×C418yescomplemented
76C4×C5×Dic3—42no—
77C4×C5×Dic3C2×C414yescomplemented
78Dic3⋊Dic5C2×C414yescomplemented
79#6667C2×C414yescomplemented
80#6665C2×C414yescomplemented
81#6665C2×C418yescomplemented
82Dic3×Dic5—42no—
83Dic3×Dic5C2×C418yescomplemented
84C2×C4×#67—52no—
85C2×C4×#67—51no—
86C2×C3×#95—51no—
87C4×#217—52no—
88C4×#217—51no—
89#4666—52no—
90#4666—51no—
91D3×#95—51no—
92#4649—52no—
93#4649D511yescharacteristic, complemented
94#4648—51no—
95C22×D20—31no—
96C22×C4×D5—31no—
97C22×C4×D5—32no—
98C22×C4×D5C2×D311yescharacteristic
99C22×C4×D5C2×D312yes—
100C2×C5×C4⋊C4—32no—
101C2×C42×C5—31no—
102C2×C42×C5—32no—
103C2×#685—31no—
104C2×#685—32no—
105C2×#685—32no—
106C2×#685—31no—
107C2×C4⋊Dic5—31no—
108C2×C4×Dic5—32no—
109C2×C4×Dic5—32no—
110C2×C4×Dic5—32no—
111#4039—316no—
112C4×D20—38no—
113C5×#95—31no—
114C5×#95C2×D311yescharacteristic
115#3963—31no—
116#3963—32no—
117#3963—32no—
118#3963C2×D311yescharacteristic
119#3963C2×D312yes—
120#2275—151noSylow
121C23×C3×C5C2411yescharacteristic
122C22×D15—21no—
123C22×D15—22no—
124C22×D15—21no—
125C22×D15C22×C412yes—
126C22×D15C22×C414yes—
127C22×C3×D5—21no—
128C22×C3×D5—22no—
129C22×C3×D5—21no—
130C22×C3×D5C22×C412yes—
131C22×C3×D5C22×C414yes—
132C2×C3×C4×C5—22no—
133C2×C3×C4×C5—22no—
134C2×C3×C4×C5—22no—
135C2×C3×C4×C5C22×C412yes—
136C2×C3×C4×C5C22×C414yes—
137C2×Dic15—22no—
138C2×Dic15—22no—
139C2×Dic15—22no—
140C2×Dic15C22×C414yes—
141C2×Dic15C22×C412yes—
142C4×D15—44no—
143C2×C5×Dic3—22no—
144C2×C5×Dic3—21no—
145C2×C5×Dic3—22no—
146C2×C5×Dic3—21no—
147C2×C5×Dic3C22×C412yes—
148C2×C5×Dic3C22×C414yes—
149C2×C5×Dic3C22×C414yes—
150C2×C5×Dic3C22×C412yes—
151C2×C3×Dic5—22no—
152C2×C3×Dic5—22no—
153C2×C3×Dic5—22no—
154C2×C3×Dic5C22×C414yes—
155C2×C3×Dic5C22×C412yes—
156C3×C4×D5—48no—
157C3×C4×D5—44no—
158#1202C42116yescomplemented
159#1200—48no—
160C23×C3×C4—52no—
161C23×C3×C4—51no—
162C22×#67—51no—
163C22×C4×D3—51no—
164C22×C4×D3—52no—
165C2×#217—52no—
166C2×#217—51no—
167C4×#67—516no—
168C4×#67—58no—
169C2×(C2×D3)⋊C4—52no—
170C2×(C2×D3)⋊C4—51no—
171C2×Dic3⋊C4—51no—
172C2×Dic3⋊C4C2×D512yes—
173C2×C4×Dic3—52no—
174C2×C4×Dic3—52no—
175C2×C4×Dic3—52no—
176C2×C4×Dic3C2×D512yes—
177C2×C4×Dic3C2×D511yescharacteristic
178C3×#95—51no—
179C3×#95—52no—
180C3×#95C2×D511yescharacteristic
181C3⋊#95—51no—
182C3⋊#95—52no—
183C3⋊#95—52no—
184C3⋊#95C2×D511yescharacteristic
185C23×D5—31no—
186C23×D5C22×D311yescharacteristic
187C22×C4×C5—31no—
188C22×C4×C5—31no—
189C22×C4×C5—32no—
190C22×C4×C5C22×D312yes—
191C22×C4×C5C22×D311yescharacteristic
192C22×Dic5—31no—
193C22×Dic5—32no—
194C22×Dic5C22×D312yes—
195C22×Dic5C22×D311yescharacteristic
196C2×D20—34no—
197C2×D20—38no—
198C2×C4×D5—34no—
199C2×C4×D5—38no—
200C2×C4×D5—42no—
201C2×C4×D5—42no—
202C2×C4×D5—42no—
203C2×C4×D5—122no—
204C2×C4×D5—122no—
205C2×C4×D5—122no—
206C2×C4×D5C4×D314yescomplemented
207C2×C4×D5C4×D318yescomplemented
208C5×C4⋊C4—38no—
209C42×C5—34no—
210C42×C5—122no—
211#685—34no—
212#685—38no—
213#685—38no—
214#685—34no—
215C4⋊Dic5—34no—
216C4×Dic5—38no—
217C4×Dic5—122no—
218C4×Dic5—122no—
219C2×C4×D4—151no—
220C2×C4×D4—152no—
221C4×C4⋊C4—31no—
222C4×C4⋊C4—152no—
223C4×C22⋊C4—151no—
224C4×C22⋊C4—152no—
225C4×C22⋊C4—152no—
226C4×C22⋊C4—151no—
227#376—151no—
228C2×#95—51no—
229C2×#95—151no—
230C22×C3×C5C23×C411yescharacteristic
231C22×C3×C5C23×C412yes—
232C22×C3×C5H(5,𝔽2)11yescharacteristic
233C22×C3×C5H(5,𝔽2)12yes—
234C22×C3×C52−1+411yescharacteristic
235C2×D15—24no—
236C2×D15—22no—
237C2×D15—22no—
238C2×D15—24no—
239C2×D15C2×C4214yes—
240C2×C3×D5—22no—
241C2×C3×D5—24no—
242C2×C3×D5—22no—
243C2×C3×D5—24no—
244C2×C3×D5C2×C4214yes—
245C3×C4×C5—22no—
246C3×C4×C5—24no—
247Dic15—22no—
248Dic15—24no—
249C3×Dic5—24no—
250C3×Dic5—22no—
251C5×Dic3—24no—
252C5×Dic3—22no—
253C5×Dic3C2×C4214yes—
254C24×C3—51no—
255C23×D3—51no—
256C22×C3×C4—52no—
257C22×C3×C4—51no—
258C22×C3×C4—58no—
259C22×C3×C4—54no—
260C22×C3×C4—202no—
261C22×C3×C4—202no—
262C22×C3×C4—202no—
263C22×C3×C4C22×D511yescharacteristic
264C22×C3×C4C22×D512yes—
265C2×#67—58no—
266C2×#67—54no—
267C22×Dic3—52no—
268C22×Dic3—51no—
269C22×Dic3C22×D511yescharacteristic
270C22×Dic3C22×D512yes—
271C22×Dic3C22×D511yescharacteristic
272C2×C4×D3—54no—
273C2×C4×D3—58no—
274C2×C4×D3—202no—
275C2×C4×D3—202no—
276C2×C4×D3—202no—
277#217—58no—
278#217—54no—
279(C2×D3)⋊C4—58no—
280(C2×D3)⋊C4—54no—
281Dic3⋊C4—54no—
282Dic3⋊C4C4×D518yescomplemented
283C4×Dic3—42no—
284C4×Dic3—58no—
285C4×Dic3—202no—
286C4×Dic3—202no—
287C4×Dic3C4×D514yescomplemented
288C23×C5C23×D311yescharacteristic
289C22×D5—21no—
290C22×D5—22no—
291C22×D5—21no—
292C22×D5—34no—
293C22×D5—32no—
294C22×D5—61no—
295C22×D5—62no—
296C22×D5—61no—
297C22×D5C2×C4×D312yes—
298C22×D5C2×C4×D314yes—
299C2×C4×C5—22no—
300C2×C4×C5—22no—
301C2×C4×C5—22no—
302C2×C4×C5—34no—
303C2×C4×C5—32no—
304C2×C4×C5—32no—
305C2×C4×C5—34no—
306C2×C4×C5—62no—
307C2×C4×C5—61no—
308C2×C4×C5—62no—
309C2×C4×C5—61no—
310C2×C4×C5C2×C4×D314yes—
311C2×C4×C5C2×C4×D312yes—
312C2×Dic5—22no—
313C2×Dic5—22no—
314C2×Dic5—22no—
315C2×Dic5—34no—
316C2×Dic5—32no—
317C2×Dic5—62no—
318C2×Dic5—62no—
319C2×Dic5—62no—
320C2×Dic5C2×C4×D314yes—
321C2×Dic5C2×C4×D312yes—
322D20—316no—
323C4×D5—48no—
324C4×D5—44no—
325C4×D5—128no—
326C4×D5—124no—
327C22×D4—151no—
328C23×C4—52no—
329C23×C4—51no—
330C23×C4—152no—
331C23×C4—151no—
332C4×D4—158no—
333C4×D4—1516no—
334C2×C4⋊C4—32no—
335C2×C4⋊C4—151no—
336C2×C22⋊C4—151no—
337C2×C22⋊C4—152no—
338C2×C22⋊C4—151no—
339C2×C22⋊C4—152no—
340C2×C42—31no—
341C2×C42—32no—
342C2×C42—152no—
343C2×C42—152no—
344C2×C42—152no—
345#95—31no—
346#95—51no—
347#95—52no—
348#95—152no—
349#95—152no—
350#95—151no—
351#95D3×D511yescharacteristic
352C2×C3×C5C22×C4211yescharacteristic
353C2×C3×C5#51811yescharacteristic
354C2×C3×C5#51812yes—
355C2×C3×C5#52011yescharacteristic
356C2×C3×C5#52012yes—
357D15—24no—
358C3×D5—24no—
359C23×C3—52no—
360C23×C3—54no—
361C23×C3—101no—
362C23×C3—101no—
363C23×C3—102no—
364C23×C3C23×D511yescharacteristic
365C22×D3—52no—
366C22×D3—54no—
367C22×D3—101no—
368C22×D3—101no—
369C22×D3—102no—
370C2×C3×C4—22no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—54no—
374C2×C3×C4—52no—
375C2×C3×C4—102no—
376C2×C3×C4—102no—
377C2×C3×C4—102no—
378C2×C3×C4—204no—
379C2×C3×C4—208no—
380C2×C3×C4C2×C4×D512yes—
381C2×C3×C4C2×C4×D514yes—
382#67—516no—
383C2×Dic3—21no—
384C2×Dic3—22no—
385C2×Dic3—21no—
386C2×Dic3—22no—
387C2×Dic3—54no—
388C2×Dic3—52no—
389C2×Dic3—102no—
390C2×Dic3—102no—
391C2×Dic3—102no—
392C2×Dic3C2×C4×D512yes—
393C2×Dic3C2×C4×D514yes—
394C2×Dic3C2×C4×D514yes—
395C2×Dic3C2×C4×D512yes—
396C4×D3—208no—
397C4×D3—204no—
398C22×C5C22×C4×D311yescharacteristic
399C22×C5C22×C4×D312yes—
400C22×C5#100411yescharacteristic
401C22×C5#100412yes—
402C22×C5#100711yescharacteristic
403C2×D5—22no—
404C2×D5—24no—
405C2×D5—24no—
406C2×D5—22no—
407C2×D5—34no—
408C2×D5—62no—
409C2×D5—64no—
410C2×D5—64no—
411C2×D5—62no—
412C2×D5C42×D314yes—
413C4×C5—22no—
414C4×C5—24no—
415C4×C5—34no—
416C4×C5—64no—
417C4×C5—62no—
418Dic5—24no—
419Dic5—22no—
420Dic5—62no—
421Dic5—64no—
422C24—51no—
423C24—151no—
424C2×D4—154no—
425C2×D4—158no—
426C22×C4—31no—
427C22×C4—32no—
428C22×C4—31no—
429C22×C4—52no—
430C22×C4—51no—
431C22×C4—58no—
432C22×C4—54no—
433C22×C4—158no—
434C22×C4—152no—
435C22×C4—151no—
436C22×C4—154no—
437C22×C4—202no—
438C22×C4—202no—
439C22×C4—202no—
440C22×C4—602no—
441C22×C4—602no—
442C22×C4—602no—
443C22×C4C2×D3×D511yescharacteristic
444C22×C4C2×D3×D512yes—
445C4⋊C4—38no—
446C4⋊C4—154no—
447C22⋊C4—154no—
448C22⋊C4—154no—
449C22⋊C4—158no—
450C22⋊C4—158no—
451C42—34no—
452C42—122no—
453C42—158no—
454C42—602no—
455C42—602no—
456C3×C5#227511yescharacteristic, complemented, Hall
457C22×C3—54no—
458C22×C3—102no—
459C22×C3—102no—
460C22×C3—104no—
461C22×C3—104no—
462C22×C3C22×C4×D511yescharacteristic
463C22×C3C22×C4×D512yes—
464C22×C3#414411yescharacteristic
465C22×C3#414711yescharacteristic
466C22×C3#414912yes—
467C2×D3—54no—
468C2×D3—102no—
469C2×D3—102no—
470C2×D3—104no—
471C2×D3—104no—
472C3×C4—22no—
473C3×C4—24no—
474C3×C4—102no—
475C3×C4—104no—
476Dic3—22no—
477Dic3—24no—
478Dic3—102no—
479Dic3—104no—
480Dic3C42×D514yes—
481C2×C5C2×C42×D311yescharacteristic
482C2×C5#548611yescharacteristic
483C2×C5#548612yes—
484C2×C5#550412yes—
485C2×C5#550411yescharacteristic
486D5—24no—
487D5—64no—
488C23—52no—
489C23—54no—
490C23—101no—
491C23—102no—
492C23—101no—
493C23—152no—
494C23—154no—
495C23—301no—
496C23—302no—
497C23—301no—
498C23C22×D3×D511yescharacteristic
499D4—1516no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—32no—
504C2×C4—34no—
505C2×C4—32no—
506C2×C4—34no—
507C2×C4—54no—
508C2×C4—52no—
509C2×C4—61no—
510C2×C4—62no—
511C2×C4—62no—
512C2×C4—61no—
513C2×C4—102no—
514C2×C4—102no—
515C2×C4—102no—
516C2×C4—154no—
517C2×C4—152no—
518C2×C4—204no—
519C2×C4—208no—
520C2×C4—302no—
521C2×C4—302no—
522C2×C4—302no—
523C2×C4—604no—
524C2×C4—608no—
525C2×C4C4×D3×D512yes—
526C2×C4C4×D3×D514yes—
527C2×C3—104no—
528C2×C3C2×C42×D511yescharacteristic
529C2×C3#6583811yescharacteristic
530C2×C3#6585511yescharacteristic
531C2×C3#6587912yes—
532C2×C3#6590812yes—
533D3—104no—
534C5#7332211yescharacteristic, complemented, Sylow
535C22—54no—
536C22—102no—
537C22—104no—
538C22—102no—
539C22—104no—
540C22—154no—
541C22—302no—
542C22—304no—
543C22—302no—
544C22—304no—
545C22C2×C4×D3×D511yescharacteristic
546C22C2×C4×D3×D512yes—
547C22#9202511yescharacteristic
548C22#9202612yes—
549C22#9202911yescharacteristic
550C4—22no—
551C4—24no—
552C4—34no—
553C4—62no—
554C4—64no—
555C4—102no—
556C4—104no—
557C4—302no—
558C4—304no—
559C3#1060548211yescharacteristic, complemented, Sylow
560C2—104no—
561C2—304no—
562C2C42×D3×D511yescharacteristic
563C2#1175306611yescharacteristic
564C2#1175320012yes—
565C2#1175333612yes—
566C2#1175366711yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC2×C3×C5C22×C42
FittingC3×C5×#95C22
socle, cosocleC23×C3×C5C24
prime radicalC22×#1202C22
hypercenter#95D3×D5
hypocenterC3×C5#2275
LCSG ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
3. #11750767C2semidirect
4. C5×#4649C2semidirect
5. C3×D5×#95C2semidirect
6. #11748888C2semidirect
7. #11748881C2semidirect
8. #11748702C2semidirect
9. C4×#6665C2semidirect
10. #11748651C2semidirect
11. #11748643C2semidirect
29. #91444C4semidirect
30. C4×#1202C4semidirect
49. D5×#95D3semidirect
57. C2×C4×D15C2×C4semidirect
66. C2×C3×C4×D5C2×C4semidirect
67. C2×C3×C4×D5C2×C4semidirect
72. C2×#1200C2×C4semidirect
73. #6714C2×C4semidirect
75. C5×Dic3⋊C4C2×C4semidirect
77. C4×C5×Dic3C2×C4semidirect
78. Dic3⋊Dic5C2×C4semidirect
79. #6667C2×C4semidirect
80. #6665C2×C4semidirect
81. #6665C2×C4semidirect
83. Dic3×Dic5C2×C4semidirect
93. #4649D5semidirect
158. #1202C42semidirect
206. C2×C4×D5C4×D3semidirect
207. C2×C4×D5C4×D3semidirect
282. Dic3⋊C4C4×D5semidirect
287. C4×Dic3C4×D5semidirect
456. C3×C5#2275semidirect
534. C5#73322semidirect
559. C3#10605482semidirect

Maximal subgroups

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

subgroupindex
C2×#914442
C2×C4×#12022
#117507672
C5×#46492
C3×D5×#952
#117488882
#117488812
#117487022
C4×#66652
#117486512
#117486432
#106054823
#733225