Finite Group Databasebeta

← #72230108

Subgroups — #72230108

Order 2880.

1588 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D5×#638C211yesmaximal, characteristic
2C2×C3×#6763C212yesmaximal, complemented
3#13088746C212yesmaximal, complemented
4#13088745C218yesmaximal, complemented
5#13088735C212yesmaximal, complemented
6#11756871—32nomaximal
7C23×C32×D5C2211yescharacteristic
8C22×D5×C3⋊D3C2211yescharacteristic
9C22×C3×ΓL(1,𝔽16)C2214yescomplemented
10C2×ΓL(1,𝔽16)⋊D3C2214yescomplemented
11C2×#67184C411yescharacteristic, complemented
12C2×C5×#638C411yescharacteristic, complemented
13D5×#638C414yescomplemented
14D5×#638C2214yes—
15C2×#67139C411yescharacteristic, complemented
16C2×#67138C411yescharacteristic, complemented
17C2×D5×C3⋊Dic3C2211yescharacteristic
18C3×#6763C2214yes—
19C3×#6763C2214yescomplemented
20#10623979C2214yescomplemented
21#10623978C2214yescomplemented
22ΓL(1,𝔽16)⋊Dic3C2214yescomplemented
23#10595510—51nomaximal, Hall
24C2×D5×#67—32no—
25C2×D5×#67—61no—
26C2×#6763D312yescomplemented
27C2×C3×C22⋊AGL(1,𝔽5)—32no—
28#67⋊AGL(1,𝔽5)—38no—
29D3×C22⋊AGL(1,𝔽5)—34no—
30AGL(1,𝔽5)×#67—38no—
31#91923—34no—
32C23×C32×C5C2×C411yescharacteristic, complemented
33C22×C3⋊D15C2×C411yescharacteristic, complemented
34C22×C5×C3⋊D3C2×C411yescharacteristic, complemented
35C22×C32×D5—42no—
36C22×C32×D5C2×C411yescharacteristic, complemented
37C22×C32×D5C2×C412yescomplemented
38C22×C32×D5C2311yescharacteristic
39C22×C32×D5C2312yes—
40C2×D5×C3⋊D3—42no—
41C2×D5×C3⋊D3C2×C412yescomplemented
42C2×D5×C3⋊D3C2312yes—
43C2×C3×ΓL(1,𝔽16)—44no—
44C2×C3×ΓL(1,𝔽16)C2318yes—
45ΓL(1,𝔽16)⋊D3—44no—
46#67184C2×C414yescomplemented
47C2×C3⋊Dic15C2×C411yescharacteristic, complemented
48C5×#638C2×C414yescomplemented
49C2×C5×C3⋊Dic3C2×C411yescharacteristic, complemented
50#67139C2×C414yescomplemented
51#67138C2×C414yescomplemented
52D5×C3⋊Dic3C2×C412yescomplemented
53D5×C3⋊Dic3C2312yes—
54#66258—91noHall
55C22×#638—51no—
56C2×C3×#217—52no—
57#63974—52no—
58#63957—52no—
59#63952—58no—
60C23×C3×D5—21no—
61C23×C3×D5C2×D312yes—
62C22×D3×D5—32no—
63C22×D3×D5—61no—
64C22×ΓL(1,𝔽16)C2×D314yescomplemented
65C22×C3×AGL(1,𝔽5)—34no—
66C2×D3×AGL(1,𝔽5)—38no—
67C2×#1220—32no—
68C2×#1220—61no—
69C2×C5×#67—32no—
70C2×C5×#67—61no—
71D5×#67—34no—
72D5×#67—34no—
73D5×#67—34no—
74D5×#67—34no—
75D5×#67—64no—
76D5×#67—64no—
77C2×#1202—32no—
78C2×#1202—61no—
79C2×#1201—32no—
80C2×#1201—61no—
81C2×D5×Dic3—32no—
82C2×D5×Dic3—61no—
83#6763C2×D314yes—
84#6763C2×D314yescomplemented
85C3×C22⋊AGL(1,𝔽5)—34no—
86C3×C22⋊AGL(1,𝔽5)—34no—
87Dic3⋊AGL(1,𝔽5)—38no—
88(C2×D3)⋊AGL(1,𝔽5)—38no—
89Dic3×AGL(1,𝔽5)—38no—
90#5486—152no—
91C22×C32×C5—41no—
92C22×C32×C5C22×C411yescharacteristic
93C22×C32×C5C22×C412yes—
94C2×C3⋊D15—41no—
95C2×C3⋊D15C22×C412yes—
96C2×C5×C3⋊D3—41no—
97C2×C5×C3⋊D3C22×C412yes—
98C2×C32×D5—22no—
99C2×C32×D5—24no—
100C2×C32×D5—42no—
101C2×C32×D5—41no—
102C2×C32×D5C22×C411yescharacteristic
103C2×C32×D5C22×C412yes—
104C2×C32×D5C2411yescharacteristic
105D5×C3⋊D3—24no—
106D5×C3⋊D3—42no—
107C3×ΓL(1,𝔽16)—28no—
108C3⋊Dic15C22×C412yes—
109C5×C3⋊Dic3C22×C412yes—
110C2×D4×D5—91no—
111C2×C22⋊AGL(1,𝔽5)—32no—
112C2×C22⋊AGL(1,𝔽5)—92no—
113D4×AGL(1,𝔽5)—98no—
114#4130—92no—
115C24×C32—51no—
116C23×C3⋊D3—51no—
117C2×#638—51no—
118C2×#638—51no—
119C2×#638—54no—
120C2×#638—54no—
121C2×#638—51no—
122C2×#638AGL(1,𝔽5)11yescharacteristic, complemented
123C22×C3⋊Dic3—51no—
124C22×C3×Dic3—54no—
125C2×#624—54no—
126C3×#217—54no—
127C3×#217—54no—
128#3724—54no—
129#3722—54no—
130Dic32—54no—
131C23×C3×C5—21no—
132C23×C3×C5C4×D312yescomplemented
133C22×D15—32no—
134C22×D15—61no—
135C22×C5×D3—32no—
136C22×C5×D3—61no—
137C22×C3×D5—21no—
138C22×C3×D5—22no—
139C22×C3×D5—21no—
140C22×C3×D5—22no—
141C22×C3×D5—44no—
142C22×C3×D5—42no—
143C22×C3×D5—42no—
144C22×C3×D5C4×D312yescomplemented
145C22×C3×D5C4×D312yescomplemented
146C22×C3×D5C4×D312yescomplemented
147C22×C3×D5C22×D312yes—
148C22×C3×D5C22×D312yes—
149C22×C3×D5C22×D312yes—
150C2×D3×D5—34no—
151C2×D3×D5—34no—
152C2×D3×D5—62no—
153C2×D3×D5—62no—
154C2×D3×D5—124no—
155C2×D3×D5—122no—
156C2×D3×D5—122no—
157C2×ΓL(1,𝔽16)—44no—
158C2×ΓL(1,𝔽16)C22×D318yes—
159C2×C3×AGL(1,𝔽5)—38no—
160C2×C3×AGL(1,𝔽5)—124no—
161D3×AGL(1,𝔽5)—128no—
162#1220—34no—
163#1220—34no—
164#1220—64no—
165C2×Dic15—32no—
166C2×Dic15—61no—
167C5×#67—34no—
168C5×#67—34no—
169C5×#67—64no—
170C2×C5×Dic3—32no—
171C2×C5×Dic3—61no—
172#1202—34no—
173#1202—34no—
174#1202—64no—
175#1201—34no—
176#1201—34no—
177#1201—64no—
178D5×Dic3—34no—
179D5×Dic3—34no—
180D5×Dic3—62no—
181D5×Dic3—62no—
182C22×#67—152no—
183C22×#67—301no—
184C2×C3×C22⋊C4—152no—
185C2×#217—52no—
186C4×#67—158no—
187#67⋊C4—158no—
188D3×C22⋊C4—154no—
189#877—154no—
190C2×C32×C5—21no—
191C2×C32×C5—22no—
192C2×C32×C5C23×C411yescharacteristic
193C3⋊D15—22no—
194C5×C3⋊D3—22no—
195C32×D5—21no—
196C32×D5—22no—
197C32×D5H(5,𝔽2)12yes—
198C23×D5—91no—
199C23×D5D3211yescharacteristic
200C22×AGL(1,𝔽5)—34no—
201C22×AGL(1,𝔽5)—94no—
202C2×C5×D4—91no—
203C2×C22⋊D5—91no—
204C2×C22⋊D5—91no—
205D4×D5—94no—
206D4×D5—94no—
207C2×D20—91no—
208C2×C4×D5—91no—
209C22⋊AGL(1,𝔽5)—34no—
210C22⋊AGL(1,𝔽5)—34no—
211C22⋊AGL(1,𝔽5)—94no—
212C4⋊AGL(1,𝔽5)—94no—
213C4×AGL(1,𝔽5)—94no—
214C23×C32—51no—
215C23×C32—52no—
216C23×C32—52no—
217C23×C32—51no—
218C23×C32—202no—
219C23×C32C2×AGL(1,𝔽5)11yescharacteristic, complemented
220C22×C3⋊D3—51no—
221C22×C3⋊D3—52no—
222C22×C3⋊D3—52no—
223C22×C3⋊D3—202no—
224C22×C3⋊D3C2×AGL(1,𝔽5)11yescharacteristic, complemented
225#638—54no—
226#638—54no—
227#638—54no—
228#638C2×AGL(1,𝔽5)14yescomplemented
229C2×C3⋊Dic3—51no—
230C2×C3⋊Dic3—52no—
231C2×C3⋊Dic3—52no—
232C2×C3⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic, complemented
233C2×C3×Dic3—58no—
234C2×C3×Dic3—204no—
235#624—204no—
236#518—451noSylow
237C22×C3×C5—21no—
238C22×C3×C5—22no—
239C22×C3×C5—42no—
240C22×C3×C5—41no—
241C22×C3×C5—41no—
242C22×C3×C5C2×C4×D312yes—
243C22×C3×C5C2×C4×D312yes—
244C22×C3×C5C2×C4×D312yes—
245C2×D15—34no—
246C2×D15—62no—
247C2×D15—122no—
248C2×D15—121no—
249C2×D15—121no—
250C2×C5×D3—34no—
251C2×C5×D3—62no—
252C2×C5×D3—122no—
253C2×C5×D3—121no—
254C2×C5×D3—121no—
255C2×C3×D5—22no—
256C2×C3×D5—22no—
257C2×C3×D5—24no—
258C2×C3×D5—24no—
259C2×C3×D5—24no—
260C2×C3×D5—21no—
261C2×C3×D5—21no—
262C2×C3×D5—22no—
263C2×C3×D5—44no—
264C2×C3×D5—42no—
265C2×C3×D5—42no—
266C2×C3×D5—42no—
267C2×C3×D5—42no—
268C2×C3×D5—42no—
269C2×C3×D5—41no—
270C2×C3×D5—41no—
271C2×C3×D5C2×C4×D312yes—
272C2×C3×D5C2×C4×D312yes—
273C2×C3×D5C2×C4×D312yes—
274C2×C3×D5C23×D312yes—
275D3×D5—68no—
276D3×D5—124no—
277D3×D5—124no—
278D3×D5—122no—
279D3×D5—122no—
280ΓL(1,𝔽16)—28no—
281C3×AGL(1,𝔽5)—68no—
282Dic15—34no—
283Dic15—62no—
284C5×Dic3—34no—
285C5×Dic3—62no—
286C24×C3—52no—
287C24×C3—101no—
288C23×D3—152no—
289C23×D3—301no—
290C22×C3×C4—154no—
291C2×#67—32no—
292C2×#67—61no—
293C2×#67—152no—
294C2×#67—154no—
295C2×#67—154no—
296C2×#67—154no—
297C2×#67—154no—
298C2×#67—152no—
299C2×#67—152no—
300C2×#67—301no—
301C2×#67—304no—
302C2×#67—304no—
303C2×#67—301no—
304C2×#67—301no—
305C22×Dic3—54no—
306C22×Dic3—152no—
307C22×Dic3—301no—
308C2×C4×D3—158no—
309C3×C22⋊C4—154no—
310C3×C22⋊C4—154no—
311#217—54no—
312#217—54no—
313(C2×D3)⋊C4—158no—
314Dic3⋊C4—158no—
315C4×Dic3—158no—
316C32×C5#51811yescharacteristic, complemented, Hall
317C23×C5—91no—
318C23×C5#62411yescharacteristic, complemented
319C22×D5—42no—
320C22×D5—91no—
321C22×D5—92no—
322C22×D5—92no—
323C22×D5—362no—
324C22×D5#62411yescharacteristic, complemented
325C22×D5#62412yescomplemented
326C22×D5C2×D3211yescharacteristic
327C22×D5C2×D3212yes—
328C2×AGL(1,𝔽5)—38no—
329C2×AGL(1,𝔽5)—124no—
330C2×AGL(1,𝔽5)—364no—
331C5×D4—94no—
332C2×C4×C5—91no—
333C22⋊D5—94no—
334C22⋊D5—94no—
335C2×Dic5—91no—
336D20—94no—
337C4×D5—92no—
338C4×D5—92no—
339C22×C32—41no—
340C22×C32—51no—
341C22×C32—51no—
342C22×C32—52no—
343C22×C32—102no—
344C22×C32—104no—
345C22×C32—201no—
346C22×C32—202no—
347C22×C32C22×AGL(1,𝔽5)11yescharacteristic
348C22×C32C22×AGL(1,𝔽5)12yes—
349C2×C3⋊D3—41no—
350C2×C3⋊D3—52no—
351C2×C3⋊D3—104no—
352C2×C3⋊D3—202no—
353C2×C3⋊D3—201no—
354C2×C3⋊D3C22×AGL(1,𝔽5)12yes—
355C3⋊Dic3—52no—
356C3⋊Dic3C22×AGL(1,𝔽5)12yes—
357C3×Dic3—108no—
358C22×D4—451no—
359C4×D4—458no—
360#117—452no—
361C2×C22⋊C4—152no—
362C2×C22⋊C4—452no—
363C2×C3×C5—21no—
364C2×C3×C5—21no—
365C2×C3×C5—22no—
366C2×C3×C5—22no—
367C2×C3×C5—22no—
368C2×C3×C5—21no—
369C2×C3×C5—42no—
370C2×C3×C5C22×C4×D312yes—
371D15—64no—
372D15—122no—
373C3×D5—21no—
374C3×D5—21no—
375C3×D5—22no—
376C3×D5—22no—
377C3×D5—22no—
378C3×D5—22no—
379C3×D5—42no—
380C3×D5#100414yes—
381C5×D3—64no—
382C5×D3—122no—
383C23×C3—21no—
384C23×C3—52no—
385C23×C3—52no—
386C23×C3—52no—
387C23×C3—52no—
388C23×C3—52no—
389C23×C3—52no—
390C23×C3—101no—
391C23×C3—101no—
392C23×C3—102no—
393C23×C3—102no—
394C23×C3—204no—
395C23×C3—202no—
396C23×C3—202no—
397C23×C3D3×AGL(1,𝔽5)12yescomplemented
398C22×D3—32no—
399C22×D3—61no—
400C22×D3—154no—
401C22×D3—154no—
402C22×D3—152no—
403C22×D3—302no—
404C22×D3—302no—
405C22×D3—301no—
406C22×D3—604no—
407C22×D3—602no—
408C22×D3—602no—
409C2×C3×C4—158no—
410C2×C3×C4—604no—
411#67—34no—
412#67—34no—
413#67—64no—
414#67—154no—
415#67—154no—
416#67—154no—
417#67—154no—
418#67—154no—
419#67—154no—
420#67—304no—
421#67—304no—
422#67—304no—
423C2×Dic3—32no—
424C2×Dic3—58no—
425C2×Dic3—61no—
426C2×Dic3—154no—
427C2×Dic3—154no—
428C2×Dic3—152no—
429C2×Dic3—204no—
430C2×Dic3—302no—
431C2×Dic3—302no—
432C2×Dic3—301no—
433C4×D3—608no—
434C22×C5—41no—
435C22×C5—92no—
436C22×C5—361no—
437C22×C5C2×#62411yescharacteristic
438C22×C5C2×#62412yes—
439C2×D5—22no—
440C2×D5—24no—
441C2×D5—42no—
442C2×D5—41no—
443C2×D5—92no—
444C2×D5—184no—
445C2×D5—362no—
446C2×D5—361no—
447C2×D5C2×#62411yescharacteristic
448C2×D5C2×#62412yes—
449C2×D5C22×D3211yescharacteristic
450AGL(1,𝔽5)—68no—
451C4×C5—92no—
452Dic5—92no—
453C2×C32—21no—
454C2×C32—22no—
455C2×C32—52no—
456C2×C32—101no—
457C2×C32—102no—
458C2×C32C23×AGL(1,𝔽5)11yescharacteristic
459C3⋊D3—22no—
460C3⋊D3—102no—
461C24—51no—
462C24—451no—
463C2×D4—91no—
464C2×D4—451no—
465C2×D4—454no—
466C2×D4—454no—
467C2×D4—451no—
468C2×D4—451no—
469C22×C4—154no—
470C22×C4—451no—
471C22×C4—454no—
472C4⋊C4—454no—
473C22⋊C4—154no—
474C22⋊C4—154no—
475C22⋊C4—454no—
476C42—454no—
477C3×C5—21no—
478C3×C5#548612yescomplemented
479C22×C3—21no—
480C22×C3—22no—
481C22×C3—42no—
482C22×C3—41no—
483C22×C3—41no—
484C22×C3—52no—
485C22×C3—52no—
486C22×C3—52no—
487C22×C3—52no—
488C22×C3—101no—
489C22×C3—101no—
490C22×C3—102no—
491C22×C3—104no—
492C22×C3—104no—
493C22×C3—104no—
494C22×C3—102no—
495C22×C3—102no—
496C22×C3—204no—
497C22×C3—202no—
498C22×C3—202no—
499C22×C3—202no—
500C22×C3—201no—
501C22×C3—201no—
502C22×C3—202no—
503C22×C3—202no—
504C22×C3C2×D3×AGL(1,𝔽5)12yes—
505C22×C3C2×D3×AGL(1,𝔽5)12yes—
506C22×C3C2×D3×AGL(1,𝔽5)12yes—
507C2×D3—34no—
508C2×D3—62no—
509C2×D3—122no—
510C2×D3—121no—
511C2×D3—121no—
512C2×D3—154no—
513C2×D3—302no—
514C2×D3—308no—
515C2×D3—604no—
516C2×D3—604no—
517C2×D3—602no—
518C2×D3—602no—
519C2×D3—602no—
520C2×D3—601no—
521C2×D3—601no—
522C3×C4—308no—
523Dic3—34no—
524Dic3—62no—
525Dic3—108no—
526Dic3—154no—
527Dic3—302no—
528C2×C5—21no—
529C2×C5—22no—
530C2×C5—182no—
531C2×C5C22×#62411yescharacteristic
532D5—21no—
533D5—22no—
534D5—182no—
535D5#6433012yes—
536C32#6625811yescharacteristic, complemented, Sylow
537C23—51no—
538C23—52no—
539C23—52no—
540C23—51no—
541C23—91no—
542C23—202no—
543C23—452no—
544C23—452no—
545C23—451no—
546C23—1802no—
547C23ΓL(1,𝔽16)⋊D311yescharacteristic, complemented
548D4—94no—
549D4—454no—
550D4—454no—
551D4—454no—
552C2×C4—91no—
553C2×C4—158no—
554C2×C4—452no—
555C2×C4—452no—
556C2×C4—451no—
557C2×C4—604no—
558C2×C4—1804no—
559C2×C3—21no—
560C2×C3—21no—
561C2×C3—22no—
562C2×C3—22no—
563C2×C3—22no—
564C2×C3—21no—
565C2×C3—42no—
566C2×C3—54no—
567C2×C3—102no—
568C2×C3—101no—
569C2×C3—101no—
570C2×C3—102no—
571C2×C3—102no—
572C2×C3—102no—
573C2×C3—202no—
574C2×C3C22×D3×AGL(1,𝔽5)12yes—
575D3—64no—
576D3—122no—
577D3—304no—
578D3—602no—
579C5#1059551011yescharacteristic, complemented, Sylow
580C22—41no—
581C22—51no—
582C22—51no—
583C22—52no—
584C22—92no—
585C22—104no—
586C22—102no—
587C22—201no—
588C22—202no—
589C22—361no—
590C22—452no—
591C22—904no—
592C22—1801no—
593C22—1802no—
594C22C2×ΓL(1,𝔽16)⋊D311yescharacteristic
595C22C2×ΓL(1,𝔽16)⋊D312yes—
596C4—92no—
597C4—308no—
598C4—452no—
599C3—21no—
600C3#1175687112yescomplemented
601C2—21no—
602C2—22no—
603C2—52no—
604C2—101no—
605C2—102no—
606C2—182no—
607C2—902no—
608C2C22×ΓL(1,𝔽16)⋊D311yescharacteristic
subgroupG/H
center, FrattiniC2C22×ΓL(1,𝔽16)⋊D3
commutator, socleC2×C32×C5C23×C4
FittingC23×C32×C5C2×C4
prime radicalC2×D5×#638C2
cosocleC2×C32×D5C24
hypercenterC23ΓL(1,𝔽16)⋊D3
hypocenterC32×C5#518
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×C3×#6763C2semidirect
3. #13088746C2semidirect
4. #13088745C2semidirect
5. #13088735C2semidirect
9. C22×C3×ΓL(1,𝔽16)C22semidirect
10. C2×ΓL(1,𝔽16)⋊D3C22semidirect
11. C2×#67184C4semidirect
12. C2×C5×#638C4semidirect
13. D5×#638C4semidirect
15. C2×#67139C4semidirect
16. C2×#67138C4semidirect
19. C3×#6763C22semidirect
20. #10623979C22semidirect
21. #10623978C22semidirect
22. ΓL(1,𝔽16)⋊Dic3C22semidirect
26. C2×#6763D3semidirect
32. C23×C32×C5C2×C4semidirect
33. C22×C3⋊D15C2×C4semidirect
34. C22×C5×C3⋊D3C2×C4semidirect
36. C22×C32×D5C2×C4semidirect
37. C22×C32×D5C2×C4semidirect
41. C2×D5×C3⋊D3C2×C4semidirect
46. #67184C2×C4semidirect
47. C2×C3⋊Dic15C2×C4semidirect
48. C5×#638C2×C4semidirect
49. C2×C5×C3⋊Dic3C2×C4semidirect
50. #67139C2×C4semidirect
51. #67138C2×C4semidirect
52. D5×C3⋊Dic3C2×C4semidirect
64. C22×ΓL(1,𝔽16)C2×D3semidirect
84. #6763C2×D3semidirect
122. C2×#638AGL(1,𝔽5)semidirect
132. C23×C3×C5C4×D3semidirect
144. C22×C3×D5C4×D3semidirect
145. C22×C3×D5C4×D3semidirect
146. C22×C3×D5C4×D3semidirect
219. C23×C32C2×AGL(1,𝔽5)semidirect
224. C22×C3⋊D3C2×AGL(1,𝔽5)semidirect
228. #638C2×AGL(1,𝔽5)semidirect
232. C2×C3⋊Dic3C2×AGL(1,𝔽5)semidirect
316. C32×C5#518semidirect
318. C23×C5#624semidirect
324. C22×D5#624semidirect
325. C22×D5#624semidirect
397. C23×C3D3×AGL(1,𝔽5)semidirect
478. C3×C5#5486semidirect
536. C32#66258semidirect
547. C23ΓL(1,𝔽16)⋊D3semidirect
579. C5#10595510semidirect
600. C3#11756871semidirect

Maximal subgroups

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

subgroupindex
C2×D5×#6382
C2×C3×#67632
#130887462
#130887452
#130887352
#117568713
#105955105