Finite Group Databasebeta

← #28515460

Subgroups — #28515460

Order 15104.

1830 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic118C211yesmaximal, characteristic, complemented
2C2×#16837315C211yesmaximal, characteristic
3C2×#16837294C212yesmaximal
4C2×#16837293C211yesmaximal, characteristic
5#21422366C211yesmaximal, characteristic, complemented
6#21422361C211yesmaximal, characteristic, complemented
7#21422360C212yesmaximal, complemented
8#21422342C211yesmaximal, characteristic
9#21422340C212yesmaximal
10#21422320C212yesmaximal, complemented
11C59×#1434C211yesmaximal, characteristic, complemented
12C23×Dic118C2211yescharacteristic
13C2×C4×Dic118—28no—
14C2×C4×Dic118—24no—
15C22×C4⋊Dic59C2211yescharacteristic
16C22×C4⋊Dic59C2212yes—
17C22×Dic59⋊C4C2212yes—
18C22×Dic59⋊C4C2211yescharacteristic
19C22×(C2×D59)⋊C4C2211yescharacteristic
20C22×(C2×D59)⋊C4C2212yes—
21C22×(C2×D59)⋊C4C2211yescharacteristic
22C22×(C2×D59)⋊C4C2211yescharacteristic
23C22×(C2×D59)⋊C4C2212yes—
24C22×C4×Dic59C2212yes—
25#16837315C418yescomplemented
26#16837294—28no—
27#16837293—24no—
28C2×C59⋊#95C2212yes—
29C2×C59⋊#95C2212yes—
30C2×C59⋊#95C2211yescharacteristic
31C2×C59⋊#95C2211yescharacteristic
32C2×C59⋊#95C2212yes—
33C2×C59⋊#95C2212yes—
34C2×C59⋊#95C2212yes—
35C22×C42×C59C2211yescharacteristic
36C2×#95×C59C2211yescharacteristic
37C2×#95×C59C2212yes—
38C2×#95×C59C2212yes—
39C2×#95×C59C2211yescharacteristic
40C24×D59C2311yescharacteristic
41C22×Dic118—22no—
42C22×Dic118—22no—
43C22×Dic118—24no—
44C22×Dic118—24no—
45C22×Dic118C2×C414yes—
46C23×Dic59C2311yescharacteristic
47C23×Dic59C2311yescharacteristic
48C23×Dic59C2311yescharacteristic
49C23×Dic59C2312yes—
50C23×Dic59C2312yes—
51C4×Dic118—232no—
52C2×C4⋊Dic59—22no—
53C2×C4⋊Dic59—24no—
54C2×C4⋊Dic59—24no—
55C2×C4⋊Dic59—42no—
56C2×C4⋊Dic59—42no—
57C2×Dic59⋊C4—24no—
58C2×Dic59⋊C4—24no—
59C2×Dic59⋊C4—24no—
60C2×Dic59⋊C4—22no—
61C2×Dic59⋊C4—42no—
62C2×(C2×D59)⋊C4—22no—
63C2×(C2×D59)⋊C4—22no—
64C2×(C2×D59)⋊C4—24no—
65C2×(C2×D59)⋊C4—24no—
66C2×(C2×D59)⋊C4—22no—
67C2×(C2×D59)⋊C4—22no—
68C2×(C2×D59)⋊C4—24no—
69C2×(C2×D59)⋊C4—24no—
70C2×(C2×D59)⋊C4—42no—
71C2×(C2×D59)⋊C4—42no—
72C2×(C2×D59)⋊C4—42no—
73C2×(C2×D59)⋊C4C2×C414yes—
74C2×(C2×D59)⋊C4C2×C418yes—
75C2×C4×Dic59—24no—
76C2×C4×Dic59—24no—
77C2×C4×Dic59D418yes—
78C59⋊#95—22no—
79C59⋊#95—22no—
80C59⋊#95—24no—
81C59⋊#95—24no—
82C59⋊#95—42no—
83C59⋊#95—42no—
84C59⋊#95C2×C418yes—
85C23×C4×C59C2312yes—
86C23×C4×C59C2312yes—
87C23×C4×C59C2311yescharacteristic
88C23×C4×C59C2311yescharacteristic
89C23×C4×C59C2311yescharacteristic
90C2×C42×C59—24no—
91C2×C42×C59D414yes—
92#95×C59—24no—
93#95×C59—22no—
94#95×C59—42no—
95#95×C59C2×C414yes—
96C24×C59C2411yescharacteristic
97C23×D59—22no—
98C23×D59—21no—
99C23×D59—22no—
100C23×D59—21no—
101C23×D59—21no—
102C23×D59—22no—
103C23×D59—21no—
104C23×D59—21no—
105C23×D59—21no—
106C23×D59—22no—
107C23×D59C22×C412yes—
108C22×C4×C59—22no—
109C22×C4×C59—22no—
110C22×C4×C59—21no—
111C22×C4×C59—22no—
112C22×C4×C59—21no—
113C22×C4×C59—22no—
114C22×C4×C59—22no—
115C22×C4×C59—22no—
116C22×C4×C59—21no—
117C22×C4×C59—22no—
118C22×C4×C59—22no—
119C22×C4×C59—22no—
120C22×C4×C59—22no—
121C22×C4×C59—22no—
122C22×C4×C59—21no—
123C22×C4×C59—22no—
124C22×C4×C59—22no—
125C22×C4×C59—22no—
126C22×C4×C59—22no—
127C22×C4×C59—22no—
128C22×C4×C59—22no—
129C22×C4×C59—22no—
130C22×C4×C59C22×C414yes—
131C22×C4×C59C22×C412yes—
132C22×C4×C59C2×D412yes—
133C22×C4×C59C2×D412yes—
134C22×C4×C59C2×D412yes—
135C22×C4×C59C2×D414yes—
136C22×C4×C59#4112yes—
137C22×Dic59—22no—
138C22×Dic59—22no—
139C22×Dic59—22no—
140C22×Dic59—22no—
141C22×Dic59—21no—
142C22×Dic59—22no—
143C22×Dic59—22no—
144C22×Dic59—21no—
145C22×Dic59—22no—
146C22×Dic59—22no—
147C22×Dic59—22no—
148C22×Dic59—22no—
149C22×Dic59—22no—
150C22×Dic59—22no—
151C22×Dic59—21no—
152C22×Dic59—21no—
153C22×Dic59—22no—
154C22×Dic59—22no—
155C22×Dic59—22no—
156C22×Dic59—22no—
157C22×Dic59—21no—
158C22×Dic59—21no—
159C22×Dic59—21no—
160C22×Dic59—21no—
161C22×Dic59C22×C412yes—
162C22×Dic59C22×C414yes—
163C22×Dic59C2×D414yes—
164C22×Dic59C2×D414yes—
165C22×Dic59#4114yes—
166C2×Dic118—28no—
167C2×Dic118—216no—
168C2×Dic118—28no—
169C2×Dic118—28no—
170C2×Dic118—216no—
171C42×C59—28no—
172(C2×D59)⋊C4—28no—
173(C2×D59)⋊C4—216no—
174(C2×D59)⋊C4—48no—
175(C2×D59)⋊C4—44no—
176(C2×D59)⋊C4—44no—
177C4⋊Dic59—28no—
178C4⋊Dic59—48no—
179Dic59⋊C4—216no—
180Dic59⋊C4—44no—
181C4×Dic59—216no—
182C23×C59C23×C411yescharacteristic
183C23×C59C22×D412yes—
184C23×C59C22×D411yescharacteristic
185C23×C59C2×#4112yes—
186C23×C59C2×#4111yescharacteristic
187C23×C59H(5,𝔽2)11yescharacteristic
188C23×C59H(5,𝔽2)12yes—
189C23×C592−1+411yescharacteristic
190C23×C592−1+411yescharacteristic
191C23×C592−1+412yes—
192C23×C592−1+411yescharacteristic
193C22×D59—24no—
194C22×D59—22no—
195C22×D59—22no—
196C22×D59—22no—
197C22×D59—24no—
198C22×D59—41no—
199C22×D59—42no—
200C22×D59—41no—
201C22×D59—42no—
202C22×D59—41no—
203C22×D59—42no—
204C22×D59—42no—
205C22×D59—41no—
206C22×D59—41no—
207C22×D59—42no—
208C22×D59—41no—
209C22×D59—41no—
210C22×D59—42no—
211C22×D59—42no—
212C22×D59—41no—
213C22×D59—42no—
214C22×D59—42no—
215C22×D59—42no—
216C2×C4×C59—24no—
217C2×C4×C59—22no—
218C2×C4×C59—22no—
219C2×C4×C59—24no—
220C2×C4×C59—24no—
221C2×C4×C59—24no—
222C2×C4×C59—22no—
223C2×C4×C59—22no—
224C2×C4×C59—24no—
225C2×C4×C59—24no—
226C2×C4×C59—24no—
227C2×C4×C59—22no—
228C2×C4×C59—22no—
229C2×C4×C59—22no—
230C2×C4×C59—42no—
231C2×C4×C59—42no—
232C2×C4×C59—42no—
233C2×C4×C59—42no—
234C2×C4×C59—42no—
235C2×C4×C59—42no—
236C2×C4×C59—42no—
237C2×C4×C59—42no—
238C2×C4×C59—42no—
239C2×C4×C59—42no—
240C2×C4×C59—42no—
241C2×C4×C59—42no—
242C2×C4×C59—42no—
243C2×C4×C59C4×D414yes—
244C2×C4×C59#12014yes—
245C2×Dic59—24no—
246C2×Dic59—24no—
247C2×Dic59—24no—
248C2×Dic59—24no—
249C2×Dic59—24no—
250C2×Dic59—24no—
251C2×Dic59—22no—
252C2×Dic59—24no—
253C2×Dic59—24no—
254C2×Dic59—24no—
255C2×Dic59—24no—
256C2×Dic59—41no—
257C2×Dic59—42no—
258C2×Dic59—42no—
259C2×Dic59—41no—
260C2×Dic59—42no—
261C2×Dic59—42no—
262C2×Dic59—42no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—42no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—42no—
270C2×Dic59—41no—
271C2×Dic59—42no—
272C2×Dic59—42no—
273C2×Dic59C4×D418yes—
274Dic118—232no—
275#14301—591nomaximal, Sylow
276C22×C59C2×C4×D412yes—
277C22×C59C2×C4×D411yescharacteristic
278C22×C59#51911yescharacteristic
279C22×C59#52011yescharacteristic
280C22×C59#52012yes—
281C22×C59C2×#12011yescharacteristic
282C22×C59#52512yes—
283C22×C59#52511yescharacteristic
284C22×C59#53512yes—
285C22×C59#53511yescharacteristic
286C22×C59#53612yes—
287C22×C59#53612yes—
288C22×C59#53611yescharacteristic
289C22×C59#53611yescharacteristic
290C22×C59#53712yes—
291C22×C59#53711yescharacteristic
292C22×C59#54111yescharacteristic
293C22×C59#54112yes—
294C22×C59#54212yes—
295C22×C59#54211yescharacteristic
296C22×C59#54212yes—
297C22×C59#54212yes—
298C22×C59#54212yes—
299C2×D59—44no—
300C2×D59—42no—
301C2×D59—44no—
302C2×D59—42no—
303C2×D59—42no—
304C2×D59—81no—
305C2×D59—81no—
306C2×D59—81no—
307C2×D59—82no—
308C2×D59—81no—
309C2×D59—82no—
310C4×C59—24no—
311C4×C59—24no—
312C4×C59—44no—
313C4×C59—42no—
314C4×C59—82no—
315Dic59—28no—
316Dic59—42no—
317Dic59—44no—
318Dic59—81no—
319Dic59—81no—
320C22×C4×Q8—591no—
321C2×#393—591no—
322C2×#388—591no—
323C2×#388—592no—
324#1446—591no—
325#1444—592no—
326#1444—591no—
327#1439—592no—
328#1439—591no—
329#1434—592no—
330#1434D5911yescharacteristic, complemented
331C2×C59C4×#12011yescharacteristic
332C2×C59#232312yes—
333C2×C59#232311yescharacteristic
334C2×C59#232612yes—
335C2×C59#232611yescharacteristic
336C2×C59#261411yescharacteristic
337C2×C59#261511yescharacteristic
338C2×C59#263211yescharacteristic
339C2×C59#263212yes—
340C2×C59#263711yescharacteristic
341C2×C59#263712yes—
342D59—82no—
343C23×Q8—591no—
344C2×C4×Q8—1188no—
345C2×C4×Q8—1184no—
346C22×C4⋊C4—592no—
347C22×C4⋊C4—591no—
348C22×C4⋊C4—591no—
349C22×C4⋊C4—592no—
350C22×C22⋊C4—592no—
351C22×C22⋊C4—591no—
352C22×C22⋊C4—591no—
353C22×C22⋊C4—592no—
354C22×C22⋊C4—591no—
355C22×C42—592no—
356C22×C42C2×D5911yescharacteristic
357#393—598no—
358#388—1188no—
359#388—1184no—
360C2×#95—592no—
361C2×#95—591no—
362C2×#95—592no—
363C2×#95—592no—
364C2×#95—591no—
365C2×#95—592no—
366C2×#95—592no—
367C2×#95C2×D5912yes—
368C2×#95C2×D5911yescharacteristic
369C2×#95C2×D5912yes—
370C2×#95C2×D5911yescharacteristic
371C59#1430111yescharacteristic, complemented, Sylow
372C25—591no—
373C22×Q8—594no—
374C22×Q8—1184no—
375C22×Q8—1184no—
376C22×Q8—1182no—
377C22×Q8—1182no—
378C23×C4—592no—
379C23×C4—592no—
380C23×C4—591no—
381C23×C4—591no—
382C23×C4—591no—
383C23×C4C22×D5912yes—
384C23×C4C22×D5912yes—
385C23×C4C22×D5911yescharacteristic
386C23×C4C22×D5911yescharacteristic
387C23×C4C22×D5911yescharacteristic
388C4×Q8—11832no—
389C2×C4⋊C4—1184no—
390C2×C4⋊C4—1184no—
391C2×C4⋊C4—1184no—
392C2×C4⋊C4—1184no—
393C2×C4⋊C4—1182no—
394C2×C4⋊C4—1184no—
395C2×C4⋊C4—1182no—
396C2×C4⋊C4—2362no—
397C2×C4⋊C4—2362no—
398C2×C4⋊C4—2362no—
399C2×C22⋊C4—598no—
400C2×C22⋊C4—594no—
401C2×C22⋊C4—1184no—
402C2×C22⋊C4—1184no—
403C2×C22⋊C4—1182no—
404C2×C22⋊C4—1182no—
405C2×C22⋊C4—1184no—
406C2×C22⋊C4—1182no—
407C2×C22⋊C4—1182no—
408C2×C22⋊C4—1184no—
409C2×C22⋊C4—2362no—
410C2×C22⋊C4—2362no—
411C2×C22⋊C4—2362no—
412C2×C42—24no—
413C2×C42—598no—
414C2×C42—1184no—
415C2×C42—1184no—
416C2×C42D23614yes—
417#95—22no—
418#95—24no—
419#95—42no—
420#95—598no—
421#95—1184no—
422#95—1184no—
423#95—1182no—
424#95—1182no—
425#95—2362no—
426#95—2362no—
427#95C4×D5914yes—
428C24—592no—
429C24—1182no—
430C24—1181no—
431C24—1182no—
432C24—1181no—
433C24—1181no—
434C24—1182no—
435C24—1181no—
436C24—1181no—
437C24—1182no—
438C24—1181no—
439C24C23×D5911yescharacteristic
440C2×Q8—11816no—
441C2×Q8—1188no—
442C2×Q8—11816no—
443C2×Q8—1188no—
444C2×Q8—1188no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—594no—
468C22×C4—594no—
469C22×C4—594no—
470C22×C4—592no—
471C22×C4—594no—
472C22×C4—1182no—
473C22×C4—1182no—
474C22×C4—1182no—
475C22×C4—1182no—
476C22×C4—1182no—
477C22×C4—1181no—
478C22×C4—1182no—
479C22×C4—1181no—
480C22×C4—1182no—
481C22×C4—1182no—
482C22×C4—1182no—
483C22×C4—1182no—
484C22×C4—1182no—
485C22×C4—1182no—
486C22×C4—1182no—
487C22×C4—1181no—
488C22×C4—1181no—
489C22×C4—1182no—
490C22×C4—1181no—
491C22×C4—1182no—
492C22×C4—1182no—
493C22×C4—1181no—
494C22×C4—1181no—
495C22×C4—1181no—
496C22×C4C2×C4×D5914yes—
497C22×C4C2×C4×D5912yes—
498C22×C4C2×D23612yes—
499C22×C4C2×D23612yes—
500C22×C4C2×D23612yes—
501C22×C4#1174568512yes—
502C22×C4D4×D5914yes—
503C4⋊C4—11816no—
504C4⋊C4—1188no—
505C4⋊C4—2364no—
506C4⋊C4—2368no—
507C22⋊C4—11816no—
508C22⋊C4—1188no—
509C22⋊C4—2364no—
510C22⋊C4—2368no—
511C22⋊C4—2364no—
512C42—28no—
513C42—11816no—
514C23—1184no—
515C23—1182no—
516C23—1182no—
517C23—1182no—
518C23—1184no—
519C23—2361no—
520C23—2362no—
521C23—2361no—
522C23—2362no—
523C23—2361no—
524C23—2362no—
525C23—2362no—
526C23—2361no—
527C23—2361no—
528C23—2362no—
529C23—2361no—
530C23—2361no—
531C23—2362no—
532C23—2362no—
533C23—2361no—
534C23—2362no—
535C23—2362no—
536C23—2362no—
537C23C22×C4×D5911yescharacteristic
538C23C2×D4×D5912yes—
539C23C22×D23611yescharacteristic
540C23#41×D5912yes—
541C23C2×#1174568511yescharacteristic
542C23#1427377211yescharacteristic
543C23#1427377312yes—
544C23#1427377411yescharacteristic
545C23#1427377412yes—
546C23#1427377411yescharacteristic
547C23#1427377511yescharacteristic
548Q8—11832no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—598no—
577C2×C4—1184no—
578C2×C4—1184no—
579C2×C4—1184no—
580C2×C4—1184no—
581C2×C4—1184no—
582C2×C4—1184no—
583C2×C4—1184no—
584C2×C4—1184no—
585C2×C4—1184no—
586C2×C4—1182no—
587C2×C4—1184no—
588C2×C4—2361no—
589C2×C4—2361no—
590C2×C4—2362no—
591C2×C4—2361no—
592C2×C4—2361no—
593C2×C4—2361no—
594C2×C4—2362no—
595C2×C4—2361no—
596C2×C4—2362no—
597C2×C4—2362no—
598C2×C4—2362no—
599C2×C4—2361no—
600C2×C4—2362no—
601C2×C4—2361no—
602C2×C4—2362no—
603C2×C4—2362no—
604C2×C4—2362no—
605C2×C4C4×D23614yes—
606C2×C4#1427369714yes—
607C22—4721no—
608C22—4721no—
609C22—4721no—
610C22—4722no—
611C22—4721no—
612C22—4722no—
613C22—2364no—
614C22—2362no—
615C22—2364no—
616C22—2362no—
617C22—2362no—
618C22C4×D4×D5912yes—
619C22C2×C4×D23611yescharacteristic
620C22#1683787811yescharacteristic
621C22#1683788111yescharacteristic
622C22#1683788212yes—
623C22C2×#1427369711yescharacteristic
624C22#1683791112yes—
625C22#1683791411yescharacteristic
626C22#1683796812yes—
627C22#1683797111yescharacteristic
628C22#1683797412yes—
629C22#1683797412yes—
630C22#1683797811yescharacteristic
631C22#1683797811yescharacteristic
632C22#1683798212yes—
633C22#1683798311yescharacteristic
634C22#1683801011yescharacteristic
635C22#1683801112yes—
636C22#1683801612yes—
637C22#1683801711yescharacteristic
638C22#1683801812yes—
639C22#1683801812yes—
640C22#1683802312yes—
641C4—24no—
642C4—24no—
643C4—42no—
644C4—44no—
645C4—82no—
646C4—1188no—
647C4—4721no—
648C4—4721no—
649C4—2364no—
650C4—2362no—
651C2—4722no—
652C2C4×#1427369711yescharacteristic
653C2#2142677712yes—
654C2#2142677911yescharacteristic
655C2#2142680512yes—
656C2#2142680711yescharacteristic
657C2#2142981811yescharacteristic
658C2#2142982211yescharacteristic
659C2#2142995911yescharacteristic
660C2#2142996012yes—
661C2#2143000611yescharacteristic
662C2#2143001012yes—
subgroupG/H
center, FrattiniC24C23×D59
commutatorC23×C59C23×C4
FittingC59×#1434C2
socle, cosocleC24×C59C24
prime radicalC24×D59C23
hypercenter#1434D59
hypocenterC59#14301
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
1. C22×C4×Dic118C2semidirect
5. #21422366C2semidirect
6. #21422361C2semidirect
7. #21422360C2semidirect
10. #21422320C2semidirect
11. C59×#1434C2semidirect
25. #16837315C4semidirect
330. #1434D59semidirect
371. C59#14301semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic1182
C2×#168373152
C2×#168372942
C2×#168372932
#214223662
#214223612
#214223602
#214223422
#214223402
#214223202
C59×#14342
#1430159