Finite Group Databasebeta

← #28506199

Subgroups — #28506199

Order 15104.

1118 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 1118 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14273642C211yesmaximal, characteristic, complemented
2C2×#16837189C212yesmaximal, complemented
3#21424017C211yesmaximal, characteristic, complemented
4#21422214C212yesmaximal, complemented
5C59×#2403C211yesmaximal, characteristic
6C22×C22⋊Dic59—21no—
7C2×#14273644—21no—
8C2×#14273644C2212yes—
9C2×#14273642—24no—
10C2×#14273642—21no—
11C2×#14273642C2212yes—
12#16837366—21no—
13#16837189—24no—
14C23×D4×C59C411yescharacteristic, complemented
15C23×D4×C59C2211yescharacteristic
16C2×#120×C59—21no—
17C2×#120×C59C411yescharacteristic, complemented
18C2×#120×C59C2211yescharacteristic
19C59×#392—21no—
20C59×#392—21no—
21C59×#386—21no—
22C59×#386C411yescharacteristic, complemented
23C59×#386C2211yescharacteristic
24C59×#379C411yescharacteristic, complemented
25C23×Dic59—41no—
26C2×#11745664—22no—
27C2×C22⋊Dic59—24no—
28C2×C22⋊Dic59—22no—
29C2×C22⋊Dic59—41no—
30C2×C22⋊Dic59—42no—
31#14273644—22no—
32#14273644—22no—
33#14273644—42no—
34#14273642—24no—
35#14273642—22no—
36#14273642D414yescomplemented
37C22⋊(C59⋊C8)—41no—
38C22⋊(C59⋊C8)—41no—
39C25×C59C2×C411yescharacteristic
40C25×C59C2×C411yescharacteristic, complemented
41C25×C59D411yescharacteristic
42C22×D4×C59—21no—
43C22×D4×C59—22no—
44C22×D4×C59—22no—
45C22×D4×C59—21no—
46C22×D4×C59—21no—
47C22×D4×C59—21no—
48C22×D4×C59—41no—
49C22×D4×C59—41no—
50C22×D4×C59—42no—
51C22×D4×C59—44no—
52C22×D4×C59—41no—
53C22×D4×C59—41no—
54C22×D4×C59C2×C411yescharacteristic
55C22×D4×C59D411yescharacteristic
56C22×D4×C59D411yescharacteristic
57C22×D4×C59C2311yescharacteristic
58C23×C4×C59C2×C411yescharacteristic, complemented
59C23×C4×C59D411yescharacteristic
60#120×C59—22no—
61#120×C59—22no—
62#120×C59—44no—
63#120×C59—42no—
64C2×C22⋊C4×C59—21no—
65C2×C22⋊C4×C59—21no—
66C2×C22⋊C4×C59—21no—
67C2×C22⋊C4×C59—21no—
68C2×C22⋊C4×C59—21no—
69C2×C22⋊C4×C59—21no—
70C2×C22⋊C4×C59C2×C411yescharacteristic, complemented
71C2×C22⋊C4×C59C2×C411yescharacteristic, complemented
72#95×C59—21no—
73#95×C59—21no—
74C24×C59—21no—
75C24×C59—22no—
76C24×C59—21no—
77C24×C59—21no—
78C24×C59—21no—
79C24×C59—21no—
80C24×C59—21no—
81C24×C59—21no—
82C24×C59—22no—
83C24×C59—22no—
84C24×C59—41no—
85C24×C59—42no—
86C24×C59—41no—
87C24×C59—42no—
88C24×C59—42no—
89C24×C59—41no—
90C24×C59—81no—
91C24×C59—81no—
92C24×C59—81no—
93C24×C59C22⋊C411yescharacteristic
94C24×C59C22⋊C411yescharacteristic
95C24×C59C22⋊C411yescharacteristic
96C24×C59C22×C411yescharacteristic
97C24×C59C2×D411yescharacteristic
98C2×D4×C59—24no—
99C2×D4×C59—22no—
100C2×D4×C59—22no—
101C2×D4×C59—22no—
102C2×D4×C59—21no—
103C2×D4×C59—21no—
104C2×D4×C59—22no—
105C2×D4×C59—41no—
106C2×D4×C59—41no—
107C2×D4×C59—41no—
108C2×D4×C59—41no—
109C2×D4×C59—41no—
110C2×D4×C59—41no—
111C2×D4×C59—42no—
112C2×D4×C59—44no—
113C2×D4×C59—44no—
114C2×D4×C59—44no—
115C2×D4×C59—44no—
116C2×D4×C59—42no—
117C2×D4×C59—42no—
118C2×D4×C59—42no—
119C2×D4×C59—44no—
120C2×D4×C59—44no—
121C2×D4×C59—44no—
122C2×D4×C59—41no—
123C2×D4×C59—41no—
124C2×D4×C59—44no—
125C2×D4×C59—42no—
126C2×D4×C59—42no—
127C2×D4×C59—42no—
128C2×D4×C59—41no—
129C2×D4×C59—41no—
130C2×D4×C59—81no—
131C2×D4×C59—81no—
132C2×D4×C59C2×D412yes—
133C2×D4×C59#4112yes—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59—22no—
138C22×C4×C59—21no—
139C22×C4×C59—21no—
140C22×C4×C59—41no—
141C22×C4×C59—42no—
142C22×C4×C59—41no—
143C22×C4×C59C22⋊C411yescharacteristic
144C22×C4×C59C2×D411yescharacteristic
145C22×Dic59—44no—
146C22×Dic59—42no—
147C22×Dic59—82no—
148C22×Dic59—82no—
149C22⋊C4×C59—22no—
150C22⋊C4×C59—42no—
151C22⋊C4×C59—42no—
152C22⋊C4×C59—42no—
153C22⋊C4×C59—42no—
154C22⋊C4×C59—42no—
155C22⋊C4×C59—41no—
156C22⋊C4×C59—81no—
157C22⋊Dic59—24no—
158C22⋊Dic59—44no—
159C22⋊Dic59—42no—
160#11745664—42no—
161#11745664—42no—
162C2×C59⋊C8—42no—
163C23×C59—22no—
164C23×C59—22no—
165C23×C59—22no—
166C23×C59—22no—
167C23×C59—22no—
168C23×C59—21no—
169C23×C59—21no—
170C23×C59—21no—
171C23×C59—21no—
172C23×C59—21no—
173C23×C59—21no—
174C23×C59—21no—
175C23×C59—21no—
176C23×C59—22no—
177C23×C59—21no—
178C23×C59—21no—
179C23×C59—21no—
180C23×C59—41no—
181C23×C59—42no—
182C23×C59—42no—
183C23×C59—41no—
184C23×C59—41no—
185C23×C59—42no—
186C23×C59—41no—
187C23×C59—41no—
188C23×C59—41no—
189C23×C59—41no—
190C23×C59—41no—
191C23×C59—42no—
192C23×C59—41no—
193C23×C59—41no—
194C23×C59—41no—
195C23×C59—41no—
196C23×C59—42no—
197C23×C59—42no—
198C23×C59—42no—
199C23×C59—42no—
200C23×C59—41no—
201C23×C59—41no—
202C23×C59—41no—
203C23×C59—81no—
204C23×C59—81no—
205C23×C59—81no—
206C23×C59—81no—
207C23×C59—82no—
208C23×C59—84no—
209C23×C59—82no—
210C23×C59—82no—
211C23×C59—84no—
212C23×C59—84no—
213C23×C59—84no—
214C23×C59—84no—
215C23×C59—82no—
216C23×C59#9911yescharacteristic
217C23×C59#9911yescharacteristic
218C23×C59#9912yes—
219C23×C59C2×C22⋊C411yescharacteristic
220C23×C59C4×D412yes—
221C23×C59#12011yescharacteristic
222C23×C59#12311yescharacteristic
223D4×C59—42no—
224D4×C59—42no—
225D4×C59—44no—
226D4×C59—44no—
227D4×C59—44no—
228D4×C59—42no—
229D4×C59—42no—
230D4×C59—48no—
231D4×C59—44no—
232D4×C59—84no—
233D4×C59—82no—
234D4×C59—82no—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
237C2×C4×C59—22no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—21no—
241C2×C4×C59—42no—
242C2×C4×C59—42no—
243C2×C4×C59—41no—
244C2×C4×C59—42no—
245C2×C4×C59—41no—
246C2×C4×C59—42no—
247C2×C4×C59—41no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59—41no—
252C2×C4×C59—41no—
253C2×C4×C59#12112yes—
254C2×Dic59—44no—
255C2×Dic59—88no—
256C2×Dic59—82no—
257C2×Dic59—82no—
258C59⋊C8—82no—
259#13086—591nomaximal, Sylow
260C22×C59—21no—
261C22×C59—21no—
262C22×C59—21no—
263C22×C59—21no—
264C22×C59—21no—
265C22×C59—21no—
266C22×C59—21no—
267C22×C59—21no—
268C22×C59—22no—
269C22×C59—21no—
270C22×C59—22no—
271C22×C59—42no—
272C22×C59—42no—
273C22×C59—42no—
274C22×C59—41no—
275C22×C59—41no—
276C22×C59—42no—
277C22×C59—44no—
278C22×C59—44no—
279C22×C59—42no—
280C22×C59—42no—
281C22×C59—42no—
282C22×C59—41no—
283C22×C59—41no—
284C22×C59—41no—
285C22×C59—41no—
286C22×C59—41no—
287C22×C59—41no—
288C22×C59—42no—
289C22×C59—41no—
290C22×C59—41no—
291C22×C59—41no—
292C22×C59—81no—
293C22×C59—81no—
294C22×C59—84no—
295C22×C59—81no—
296C22×C59—81no—
297C22×C59—81no—
298C22×C59—81no—
299C22×C59—82no—
300C22×C59—82no—
301C22×C59—84no—
302C22×C59—84no—
303C22×C59—84no—
304C22×C59—81no—
305C22×C59—81no—
306C22×C59AΣL(1,𝔽16)12yescomplemented
307C22×C59#38611yescharacteristic
308C22×C59C2×#9911yescharacteristic
309C22×C59C2×#9911yescharacteristic
310C4×C59—22no—
311C4×C59—42no—
312C4×C59—41no—
313C4×C59—81no—
314C4×C59—81no—
315Dic59—84no—
316C22×#99—591no—
317#2403D5911yescharacteristic
318C2×AΣL(1,𝔽16)—592no—
319#1783—591no—
320#1390—592no—
321C2×C59—21no—
322C2×C59—21no—
323C2×C59—22no—
324C2×C59—41no—
325C2×C59—41no—
326C2×C59—42no—
327C2×C59—41no—
328C2×C59—42no—
329C2×C59—82no—
330C2×C59—81no—
331C2×C59#189811yescharacteristic
332C2×C59C2×AΣL(1,𝔽16)11yescharacteristic
333C2×C59#212111yescharacteristic
334C23×D4Dic5911yescharacteristic, complemented
335C23×D4C2×D5911yescharacteristic
336C2×#120—21no—
337C2×#120Dic5911yescharacteristic, complemented
338C2×#120C2×D5911yescharacteristic
339C22×C22⋊C4—1181no—
340C2×#100—592no—
341C2×#100—1181no—
342C2×#99—592no—
343C2×#99—1184no—
344C2×#99—1181no—
345#407—1181no—
346#392—21no—
347#392—21no—
348#386—21no—
349#386Dic5911yescharacteristic, complemented
350#386C2×D5911yescharacteristic
351#379Dic5911yescharacteristic, complemented
352AΣL(1,𝔽16)—1184no—
353C59#1308611yescharacteristic, complemented, Sylow
354C25C2×Dic5911yescharacteristic
355C25C2×Dic5911yescharacteristic, complemented
356C25C22⋊D5911yescharacteristic
357C22×D4—21no—
358C22×D4—22no—
359C22×D4—22no—
360C22×D4—21no—
361C22×D4—21no—
362C22×D4—21no—
363C22×D4—41no—
364C22×D4—41no—
365C22×D4—42no—
366C22×D4—44no—
367C22×D4—41no—
368C22×D4—41no—
369C22×D4C2×Dic5911yescharacteristic
370C22×D4C22⋊D5911yescharacteristic
371C22×D4C22⋊D5911yescharacteristic
372C22×D4C22×D5911yescharacteristic
373C23×C4—2361no—
374C23×C4C2×Dic5911yescharacteristic, complemented
375C23×C4C22⋊D5911yescharacteristic
376C2×M16—1182no—
377#120—22no—
378#120—22no—
379#120—44no—
380#120—42no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—1184no—
388C2×C22⋊C4—1182no—
389C2×C22⋊C4—2361no—
390C2×C22⋊C4—2362no—
391C2×C22⋊C4C2×Dic5911yescharacteristic, complemented
392C2×C22⋊C4C2×Dic5911yescharacteristic, complemented
393#100—1182no—
394#100—1182no—
395#100—2362no—
396#99—594no—
397#99—1184no—
398#99—1182no—
399C22⋊C8—2361no—
400C22⋊C8—2361no—
401#95—21no—
402#95—21no—
403C24—21no—
404C24—22no—
405C24—21no—
406C24—21no—
407C24—21no—
408C24—21no—
409C24—21no—
410C24—21no—
411C24—22no—
412C24—22no—
413C24—41no—
414C24—42no—
415C24—41no—
416C24—42no—
417C24—42no—
418C24—41no—
419C24—81no—
420C24—81no—
421C24—81no—
422C24C22⋊Dic5911yescharacteristic
423C24C22⋊Dic5911yescharacteristic
424C24C22⋊Dic5911yescharacteristic
425C24C22×Dic5911yescharacteristic
426C24C2×C22⋊D5911yescharacteristic
427C2×D4—24no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—22no—
434C2×D4—41no—
435C2×D4—41no—
436C2×D4—41no—
437C2×D4—41no—
438C2×D4—41no—
439C2×D4—41no—
440C2×D4—42no—
441C2×D4—44no—
442C2×D4—44no—
443C2×D4—44no—
444C2×D4—44no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—44no—
449C2×D4—44no—
450C2×D4—44no—
451C2×D4—41no—
452C2×D4—41no—
453C2×D4—44no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—41no—
458C2×D4—41no—
459C2×D4—81no—
460C2×D4—81no—
461C2×D4D4×D5912yes—
462C2×D4#1174568712yes—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—41no—
470C22×C4—42no—
471C22×C4—41no—
472C22×C4—4722no—
473C22×C4—4722no—
474C22×C4—2364no—
475C22×C4—2362no—
476C22×C4C22⋊Dic5911yescharacteristic
477C22×C4C2×C22⋊D5911yescharacteristic
478M16—2362no—
479M16—2362no—
480C2×C8—2362no—
481C22⋊C4—22no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—41no—
488C22⋊C4—81no—
489C22⋊C4—1184no—
490C22⋊C4—2364no—
491C22⋊C4—2362no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—22no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—41no—
510C23—42no—
511C23—42no—
512C23—41no—
513C23—41no—
514C23—42no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—42no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—81no—
533C23—81no—
534C23—81no—
535C23—81no—
536C23—82no—
537C23—84no—
538C23—82no—
539C23—82no—
540C23—84no—
541C23—84no—
542C23—84no—
543C23—84no—
544C23—82no—
545C23#1427364211yescharacteristic
546C23#1427364211yescharacteristic
547C23#1427364212yes—
548C23C2×C22⋊Dic5911yescharacteristic
549C23D4×Dic5912yes—
550C23#1427369511yescharacteristic
551C23#1427370811yescharacteristic
552D4—42no—
553D4—42no—
554D4—44no—
555D4—44no—
556D4—44no—
557D4—42no—
558D4—42no—
559D4—48no—
560D4—44no—
561D4—84no—
562D4—82no—
563D4—82no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—4728no—
583C2×C4—4722no—
584C2×C4—4722no—
585C2×C4—2364no—
586C2×C4#1427369912yes—
587C8—4722no—
588C22—21no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—21no—
596C22—22no—
597C22—21no—
598C22—22no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—41no—
603C22—41no—
604C22—42no—
605C22—44no—
606C22—42no—
607C22—44no—
608C22—42no—
609C22—42no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—42no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—81no—
621C22—81no—
622C22—84no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—82no—
628C22—82no—
629C22—84no—
630C22—84no—
631C22—84no—
632C22—81no—
633C22—81no—
634C22#1683718912yescomplemented
635C22#1683728211yescharacteristic
636C22C2×#1427364211yescharacteristic
637C22C2×#1427364211yescharacteristic
638C4—22no—
639C4—42no—
640C4—41no—
641C4—81no—
642C4—81no—
643C4—4724no—
644C2—21no—
645C2—21no—
646C2—22no—
647C2—41no—
648C2—41no—
649C2—42no—
650C2—41no—
651C2—42no—
652C2—82no—
653C2—81no—
654C2#2142455011yescharacteristic
655C2C2×#1683718911yescharacteristic
656C2#2142574211yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC24×C59C22×C4
FrattiniC22×D4C22×D59
Fitting, prime radicalC59×#2403C2
socleC22×C59C2×#99
cosocleC22×D4×C59C23
hypercenter#2403D59
hypocenterC59#13086
LCSG ▷ C24×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #2403

Complements

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

NH ≅ G/Ntype
1. C22×#14273642C2semidirect
2. C2×#16837189C2semidirect
3. #21424017C2semidirect
4. #21422214C2semidirect
14. C23×D4×C59C4semidirect
17. C2×#120×C59C4semidirect
22. C59×#386C4semidirect
24. C59×#379C4semidirect
36. #14273642D4semidirect
40. C25×C59C2×C4semidirect
58. C23×C4×C59C2×C4semidirect
70. C2×C22⋊C4×C59C2×C4semidirect
71. C2×C22⋊C4×C59C2×C4semidirect
306. C22×C59AΣL(1,𝔽16)semidirect
334. C23×D4Dic59semidirect
337. C2×#120Dic59semidirect
349. #386Dic59semidirect
351. #379Dic59semidirect
353. C59#13086semidirect
355. C25C2×Dic59semidirect
374. C23×C4C2×Dic59semidirect
391. C2×C22⋊C4C2×Dic59semidirect
392. C2×C22⋊C4C2×Dic59semidirect
634. C22#16837189semidirect

Maximal subgroups

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

subgroupindex
C22×#142736422
C2×#168371892
#214240172
#214222142
C59×#24032
#1308659