Finite Group Databasebeta

← #28480519

Subgroups — #28480519

Order 15104.

1040 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21424414C211yesmaximal, characteristic, complemented
2#21424404C211yesmaximal, characteristic, complemented
3C2×#16837139C211yesmaximal, characteristic, complemented
4C2×#16837136C211yesmaximal, characteristic, complemented
5C2×#16837126C211yesmaximal, characteristic, complemented
6#21422341C211yesmaximal, characteristic, complemented
7C2×C59×#538C211yesmaximal, characteristic
8C22×C4⋊Dic59C2211yescharacteristic
9C22×C22⋊Dic59—21no—
10C2×#14273651—21no—
11C2×#14273648—21no—
12C2×C22⋊(C59⋊C8)C2211yescharacteristic
13C2×C22⋊(C59⋊C8)C2211yescharacteristic
14C2×C59⋊#95—21no—
15C2×C59⋊#95C2211yescharacteristic
16#16837164—21no—
17#16837164—21no—
18#16837139—22no—
19#16837136—22no—
20#16837126—22no—
21C59×#538—22no—
22C59×#538—22no—
23C59×#538—22no—
24C59×#538C414yescomplemented
25C2×#126×C59—21no—
26C2×#124×C59—21no—
27C2×#123×C59—21no—
28C2×C22⋊Q8×C59C2211yescharacteristic
29C2×#121×C59—21no—
30C2×#121×C59C2211yescharacteristic
31C2×#120×C59—21no—
32C2×C4×D4×C59—21no—
33C22×C22⋊C4×C59C2211yescharacteristic
34C23×Dic59—21no—
35C23×Dic59—21no—
36C22×C59⋊C8—21no—
37C22×C59⋊C8—21no—
38C2×C4⋊Dic59—21no—
39C2×C4⋊Dic59—21no—
40C2×C4⋊Dic59—42no—
41C2×C22⋊Dic59—42no—
42C2×C22⋊Dic59—42no—
43C2×C22⋊Dic59—42no—
44#14273651—42no—
45#14273648—42no—
46C22⋊(C59⋊C8)—21no—
47C22⋊(C59⋊C8)—21no—
48C22⋊(C59⋊C8)—21no—
49C22⋊(C59⋊C8)—21no—
50C59⋊#95—21no—
51C59⋊#95—21no—
52C59⋊#95—42no—
53C25×C59D411yescharacteristic
54C22×Q8×C59D411yescharacteristic
55C22×D4×C59—21no—
56C22×D4×C59—21no—
57C22×D4×C59—21no—
58C22×D4×C59D411yescharacteristic
59C23×C4×C59—21no—
60C23×C4×C59D411yescharacteristic
61C23×C4×C59C2311yescharacteristic
62#126×C59—24no—
63#126×C59—24no—
64#124×C59—24no—
65#124×C59—24no—
66#123×C59—24no—
67#123×C59—24no—
68C22⋊Q8×C59—21no—
69C22⋊Q8×C59—21no—
70C22⋊Q8×C59C2×C412yes—
71C22⋊Q8×C59D412yes—
72#121×C59—24no—
73#121×C59—24no—
74#121×C59—21no—
75#121×C59—21no—
76#121×C59C2×C412yes—
77#121×C59D412yes—
78#120×C59—24no—
79#120×C59—24no—
80C4×D4×C59—24no—
81C4×D4×C59—24no—
82C2×C4⋊C4×C59—21no—
83C2×C4⋊C4×C59—21no—
84C2×C4⋊C4×C59D411yescharacteristic
85C2×C4⋊C4×C59D411yescharacteristic
86C2×C22⋊C4×C59—21no—
87C2×C22⋊C4×C59—21no—
88C2×C22⋊C4×C59—21no—
89C2×C22⋊C4×C59—21no—
90C2×C22⋊C4×C59—21no—
91C2×C22⋊C4×C59—21no—
92C2×C22⋊C4×C59—21no—
93C2×C22⋊C4×C59—21no—
94C2×C22⋊C4×C59—21no—
95C2×C22⋊C4×C59C2×C412yes—
96C2×C22⋊C4×C59D412yes—
97C2×C42×C59—21no—
98C24×C59—21no—
99C24×C59—21no—
100C24×C59—21no—
101C24×C59—21no—
102C24×C59—21no—
103C24×C59—21no—
104C24×C59—21no—
105C24×C59—41no—
106C24×C59—41no—
107C24×C59—41no—
108C24×C59—41no—
109C24×C59C22⋊C412yes—
110C24×C59C2×D411yescharacteristic
111C2×Q8×C59—21no—
112C2×Q8×C59—41no—
113C2×Q8×C59—41no—
114C2×Q8×C59C22⋊C412yescomplemented
115C2×D4×C59—22no—
116C2×D4×C59—22no—
117C2×D4×C59—24no—
118C2×D4×C59—24no—
119C2×D4×C59—21no—
120C2×D4×C59—41no—
121C2×D4×C59—41no—
122C2×D4×C59—41no—
123C2×D4×C59—41no—
124C2×D4×C59—41no—
125C2×D4×C59—41no—
126C2×D4×C59—81no—
127C2×D4×C59—81no—
128C2×D4×C59—81no—
129C2×D4×C59—81no—
130C2×D4×C59C22⋊C412yescomplemented
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—24no—
136C22×C4×C59—21no—
137C22×C4×C59—41no—
138C22×C4×C59—41no—
139C22×C4×C59—41no—
140C22×C4×C59—41no—
141C22×C4×C59—81no—
142C22×C4×C59—81no—
143C22×C4×C59C22⋊C412yes—
144C22×C4×C59C22×C411yescharacteristic
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2×D411yescharacteristic
150C22×Dic59—22no—
151C22×Dic59—22no—
152C22×Dic59—41no—
153C22×Dic59—41no—
154C22×Dic59—41no—
155C22×Dic59—41no—
156C22×Dic59—41no—
157C22×Dic59—41no—
158C22×Dic59—41no—
159C22×Dic59—41no—
160C22×Dic59—41no—
161C22×Dic59—41no—
162C22×Dic59—41no—
163C22×Dic59—41no—
164C4⋊C4×C59—24no—
165C4⋊C4×C59—22no—
166C4⋊C4×C59—22no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59C22⋊C412yescomplemented
170C4⋊C4×C59C22⋊C412yescomplemented
171C22⋊C4×C59—24no—
172C22⋊C4×C59—24no—
173C22⋊C4×C59—24no—
174C22⋊C4×C59—22no—
175C22⋊C4×C59—22no—
176C22⋊C4×C59—22no—
177C22⋊C4×C59—22no—
178C22⋊C4×C59—22no—
179C22⋊C4×C59—22no—
180C22⋊C4×C59—22no—
181C22⋊C4×C59—22no—
182C42×C59—24no—
183C22⋊Dic59—48no—
184C4⋊Dic59—22no—
185C4⋊Dic59—22no—
186C4⋊Dic59—41no—
187C4⋊Dic59—41no—
188C2×C59⋊C8—41no—
189C2×C59⋊C8—41no—
190C2×C59⋊C8—41no—
191C2×C59⋊C8—41no—
192C2×C59⋊C8—41no—
193C2×C59⋊C8—41no—
194C23×C59—21no—
195C23×C59—21no—
196C23×C59—21no—
197C23×C59—21no—
198C23×C59—22no—
199C23×C59—22no—
200C23×C59—22no—
201C23×C59—22no—
202C23×C59—42no—
203C23×C59—42no—
204C23×C59—42no—
205C23×C59—42no—
206C23×C59—42no—
207C23×C59—42no—
208C23×C59—41no—
209C23×C59—41no—
210C23×C59—41no—
211C23×C59—41no—
212C23×C59—41no—
213C23×C59—41no—
214C23×C59—41no—
215C23×C59—41no—
216C23×C59—41no—
217C23×C59—41no—
218C23×C59—41no—
219C23×C59—41no—
220C23×C59—41no—
221C23×C59—41no—
222C23×C59—81no—
223C23×C59—81no—
224C23×C59—81no—
225C23×C59—81no—
226C23×C59—81no—
227C23×C59—81no—
228C23×C59C2×C22⋊C411yescharacteristic
229C23×C59#12011yescharacteristic
230C23×C59#12011yescharacteristic
231C23×C59#13511yescharacteristic
232C23×C59#13511yescharacteristic
233C23×C59AGL(1,ℤ/8)11yescharacteristic
234C23×C59#13711yescharacteristic
235Q8×C59—42no—
236Q8×C59—81no—
237D4×C59—42no—
238D4×C59—42no—
239D4×C59—42no—
240D4×C59—42no—
241D4×C59—42no—
242D4×C59—81no—
243D4×C59—84no—
244D4×C59—84no—
245C2×C4×C59—21no—
246C2×C4×C59—21no—
247C2×C4×C59—21no—
248C2×C4×C59—21no—
249C2×C4×C59—22no—
250C2×C4×C59—22no—
251C2×C4×C59—22no—
252C2×C4×C59—22no—
253C2×C4×C59—41no—
254C2×C4×C59—41no—
255C2×C4×C59—41no—
256C2×C4×C59—41no—
257C2×C4×C59—41no—
258C2×C4×C59—41no—
259C2×C4×C59—41no—
260C2×C4×C59—41no—
261C2×C4×C59—41no—
262C2×C4×C59—41no—
263C2×C4×C59—41no—
264C2×C4×C59—41no—
265C2×C4×C59—42no—
266C2×C4×C59—81no—
267C2×C4×C59—84no—
268C2×C4×C59C2×C22⋊C411yescharacteristic
269C2×C4×C59C2×C22⋊C411yescharacteristic
270C2×C4×C59#12011yescharacteristic
271C2×C4×C59#12011yescharacteristic
272C2×Dic59—42no—
273C2×Dic59—42no—
274C2×Dic59—42no—
275C2×Dic59—42no—
276C2×Dic59—42no—
277C2×Dic59—42no—
278C2×Dic59—81no—
279C2×Dic59—81no—
280C2×Dic59—81no—
281C2×Dic59—81no—
282C2×Dic59—81no—
283C2×Dic59—81no—
284C2×Dic59—81no—
285C2×Dic59—81no—
286C59⋊C8—81no—
287C59⋊C8—81no—
288#8408—591nomaximal, Sylow
289C22×C59—21no—
290C22×C59—21no—
291C22×C59—21no—
292C22×C59—21no—
293C22×C59—21no—
294C22×C59—21no—
295C22×C59—21no—
296C22×C59—21no—
297C22×C59—21no—
298C22×C59—21no—
299C22×C59—21no—
300C22×C59—21no—
301C22×C59—44no—
302C22×C59—42no—
303C22×C59—42no—
304C22×C59—42no—
305C22×C59—42no—
306C22×C59—42no—
307C22×C59—42no—
308C22×C59—82no—
309C22×C59—82no—
310C22×C59—82no—
311C22×C59—81no—
312C22×C59—81no—
313C22×C59—81no—
314C22×C59—81no—
315C22×C59—81no—
316C22×C59—81no—
317C22×C59—81no—
318C22×C59—81no—
319C22×C59#37911yescharacteristic
320C22×C59#41611yescharacteristic
321C22×C59#41711yescharacteristic
322C22×C59#44911yescharacteristic
323C22×C59#44911yescharacteristic
324C22×C59#45211yescharacteristic
325C22×C59#45211yescharacteristic
326C22×C59#45311yescharacteristic
327C22×C59#45511yescharacteristic
328C22×C59B(4,𝔽2)11yescharacteristic
329C22×C59#45811yescharacteristic
330C4×C59—21no—
331C4×C59—21no—
332C4×C59—21no—
333C4×C59—21no—
334C4×C59—42no—
335C4×C59—42no—
336C4×C59—42no—
337C4×C59—42no—
338Dic59—82no—
339Dic59—82no—
340C2×#538D5911yescharacteristic
341#1874—591no—
342#1872—591no—
343C2×#328—591no—
344C2×#327—591no—
345C2×#323—591no—
346#1439—591no—
347C2×C59—21no—
348C2×C59—21no—
349C2×C59—21no—
350C2×C59—21no—
351C2×C59—82no—
352C2×C59—82no—
353C2×C59#160011yescharacteristic
354C2×C59#160611yescharacteristic
355C2×C59#161211yescharacteristic
356C2×C59#161811yescharacteristic
357C2×C59#178811yescharacteristic
358C2×C59#179011yescharacteristic
359C2×C59#179211yescharacteristic
360#538—22no—
361#538—22no—
362#538—22no—
363#538Dic5914yescomplemented
364C2×#126—21no—
365C2×#124—21no—
366C2×#123—21no—
367C2×C22⋊Q8C2×D5911yescharacteristic
368C2×#121—21no—
369C2×#121C2×D5911yescharacteristic
370C2×#120—21no—
371C2×C4×D4—21no—
372C22×C4⋊C4—591no—
373C22×C22⋊C4—1181no—
374C22×C22⋊C4C2×D5911yescharacteristic
375C2×#103—1181no—
376C2×#102—1181no—
377C2×C22⋊C8—591no—
378C2×C22⋊C8—591no—
379C2×#95—591no—
380C2×#95—1181no—
381#340—1181no—
382#340—1181no—
383#328—1182no—
384#327—1182no—
385#323—1182no—
386C59#840811yescharacteristic, complemented, Sylow
387C25C22⋊D5911yescharacteristic
388C22×Q8C22⋊D5911yescharacteristic
389C22×D4—21no—
390C22×D4—21no—
391C22×D4—21no—
392C22×D4C22⋊D5911yescharacteristic
393C23×C4—21no—
394C23×C4—1181no—
395C23×C4—1181no—
396C23×C4C22⋊D5911yescharacteristic
397C23×C4C22×D5911yescharacteristic
398C22×C8—1181no—
399C22×C8—1181no—
400#126—24no—
401#126—24no—
402#124—24no—
403#124—24no—
404#123—24no—
405#123—24no—
406C22⋊Q8—21no—
407C22⋊Q8—21no—
408C22⋊Q8C2×Dic5912yes—
409C22⋊Q8C22⋊D5912yes—
410#121—24no—
411#121—24no—
412#121—21no—
413#121—21no—
414#121C2×Dic5912yes—
415#121C22⋊D5912yes—
416#120—24no—
417#120—24no—
418C4×D4—24no—
419C4×D4—24no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—1181no—
423C2×C4⋊C4—1181no—
424C2×C4⋊C4—2362no—
425C2×C4⋊C4C22⋊D5911yescharacteristic
426C2×C4⋊C4C22⋊D5911yescharacteristic
427C2×C22⋊C4—21no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—2362no—
437C2×C22⋊C4—2362no—
438C2×C22⋊C4—2362no—
439C2×C22⋊C4C2×Dic5912yes—
440C2×C22⋊C4C22⋊D5912yes—
441C2×C42—21no—
442#103—2362no—
443#102—2362no—
444C22⋊C8—1181no—
445C22⋊C8—1181no—
446C22⋊C8—1181no—
447C22⋊C8—1181no—
448#95—1181no—
449#95—1181no—
450#95—2362no—
451C24—21no—
452C24—21no—
453C24—21no—
454C24—21no—
455C24—21no—
456C24—21no—
457C24—21no—
458C24—41no—
459C24—41no—
460C24—41no—
461C24—41no—
462C24C22⋊Dic5912yes—
463C24C2×C22⋊D5911yescharacteristic
464C2×Q8—21no—
465C2×Q8—41no—
466C2×Q8—41no—
467C2×Q8C22⋊Dic5912yescomplemented
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—24no—
471C2×D4—24no—
472C2×D4—21no—
473C2×D4—41no—
474C2×D4—41no—
475C2×D4—41no—
476C2×D4—41no—
477C2×D4—41no—
478C2×D4—41no—
479C2×D4—81no—
480C2×D4—81no—
481C2×D4—81no—
482C2×D4—81no—
483C2×D4C22⋊Dic5912yescomplemented
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—24no—
489C22×C4—21no—
490C22×C4—41no—
491C22×C4—41no—
492C22×C4—41no—
493C22×C4—41no—
494C22×C4—81no—
495C22×C4—81no—
496C22×C4—1182no—
497C22×C4—1182no—
498C22×C4—2361no—
499C22×C4—2361no—
500C22×C4—2361no—
501C22×C4—2361no—
502C22×C4—2361no—
503C22×C4—2361no—
504C22×C4—2361no—
505C22×C4—2361no—
506C22×C4—2361no—
507C22×C4—2361no—
508C22×C4—2361no—
509C22×C4—2361no—
510C22×C4C22⋊Dic5912yes—
511C22×C4C22×Dic5911yescharacteristic
512C22×C4C2×C22⋊D5911yescharacteristic
513C22×C4C2×C22⋊D5911yescharacteristic
514C22×C4C2×C22⋊D5911yescharacteristic
515C22×C4C2×C22⋊D5911yescharacteristic
516C22×C4C2×C22⋊D5911yescharacteristic
517C2×C8—2361no—
518C2×C8—2361no—
519C2×C8—2361no—
520C2×C8—2361no—
521C2×C8—2361no—
522C2×C8—2361no—
523C4⋊C4—24no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—1182no—
529C4⋊C4—1182no—
530C4⋊C4—2361no—
531C4⋊C4—2361no—
532C4⋊C4C22⋊Dic5912yescomplemented
533C4⋊C4C22⋊Dic5912yescomplemented
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—24no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—2368no—
546C42—24no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—81no—
576C23—81no—
577C23—81no—
578C23—81no—
579C23—81no—
580C23—81no—
581C23C2×C22⋊Dic5911yescharacteristic
582C23#1427369611yescharacteristic
583C23#1427369611yescharacteristic
584C23#1427374711yescharacteristic
585C23#1427374711yescharacteristic
586C23#1427375211yescharacteristic
587C23#1427375711yescharacteristic
588Q8—42no—
589Q8—81no—
590D4—42no—
591D4—42no—
592D4—42no—
593D4—42no—
594D4—42no—
595D4—81no—
596D4—84no—
597D4—84no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—81no—
620C2×C4—84no—
621C2×C4—4721no—
622C2×C4—4721no—
623C2×C4—4721no—
624C2×C4—4721no—
625C2×C4—4721no—
626C2×C4—4721no—
627C2×C4—4721no—
628C2×C4—4721no—
629C2×C4—2362no—
630C2×C4—2362no—
631C2×C4—2362no—
632C2×C4—2362no—
633C2×C4—2362no—
634C2×C4—2362no—
635C2×C4C2×C22⋊Dic5911yescharacteristic
636C2×C4C2×C22⋊Dic5911yescharacteristic
637C2×C4#1427369611yescharacteristic
638C2×C4#1427369611yescharacteristic
639C8—4721no—
640C8—4721no—
641C22—21no—
642C22—21no—
643C22—21no—
644C22—21no—
645C22—21no—
646C22—21no—
647C22—21no—
648C22—21no—
649C22—21no—
650C22—21no—
651C22—21no—
652C22—21no—
653C22—44no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—81no—
664C22—81no—
665C22—81no—
666C22—81no—
667C22—81no—
668C22—81no—
669C22—81no—
670C22—81no—
671C22#1683725811yescharacteristic
672C22#1683739811yescharacteristic
673C22#1683740311yescharacteristic
674C22#1683754711yescharacteristic
675C22#1683754711yescharacteristic
676C22#1683756411yescharacteristic
677C22#1683756411yescharacteristic
678C22#1683757011yescharacteristic
679C22#1683758011yescharacteristic
680C22#1683758911yescharacteristic
681C22#1683759211yescharacteristic
682C4—21no—
683C4—21no—
684C4—21no—
685C4—21no—
686C4—42no—
687C4—42no—
688C4—42no—
689C4—42no—
690C4—4722no—
691C4—4722no—
692C2—21no—
693C2—21no—
694C2—21no—
695C2—21no—
696C2—82no—
697C2—82no—
698C2#2142304211yescharacteristic
699C2#2142307611yescharacteristic
700C2#2142311011yescharacteristic
701C2#2142314411yescharacteristic
702C2#2142403611yescharacteristic
703C2#2142404411yescharacteristic
704C2#2142405011yescharacteristic
subgroupG/H
centerC23#14273696
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
Fitting, prime radicalC2×C59×#538C2
socleC23×C59#120
cosocleC23×C4×C59C23
hypercenterC2×#538D59
hypocenterC59#8408
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#538

Complements

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

NH ≅ G/Ntype
1. #21424414C2semidirect
2. #21424404C2semidirect
3. C2×#16837139C2semidirect
4. C2×#16837136C2semidirect
5. C2×#16837126C2semidirect
6. #21422341C2semidirect
24. C59×#538C4semidirect
114. C2×Q8×C59C22⋊C4semidirect
130. C2×D4×C59C22⋊C4semidirect
169. C4⋊C4×C59C22⋊C4semidirect
170. C4⋊C4×C59C22⋊C4semidirect
363. #538Dic59semidirect
386. C59#8408semidirect
467. C2×Q8C22⋊Dic59semidirect
483. C2×D4C22⋊Dic59semidirect
532. C4⋊C4C22⋊Dic59semidirect
533. C4⋊C4C22⋊Dic59semidirect

Maximal subgroups

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

subgroupindex
#214244142
#214244042
C2×#168371392
C2×#168371362
C2×#168371262
#214223412
C2×C59×#5382
#840859