Finite Group Databasebeta

← #28506209

Subgroups — #28506209

Order 15104.

1040 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 1040 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×#16837195C212yesmaximal, complemented
3#21424011C211yesmaximal, characteristic, complemented
4#21422221C212yesmaximal, complemented
5C59×#2433C211yesmaximal, characteristic
6C22×C22⋊Dic59—21no—
7C2×#14273642—24no—
8C2×#14273642—21no—
9C2×#14273642—21no—
10C2×#14273642C2212yes—
11C2×#14273642C2212yes—
12#16837258—21no—
13#16837195—24no—
14C23×D4×C59C2211yescharacteristic
15C2×#127×C59C411yescharacteristic, complemented
16C2×#127×C59C2211yescharacteristic
17C2×#121×C59C411yescharacteristic, complemented
18C2×#120×C59—21no—
19C2×#120×C59—21no—
20C59×#392—21no—
21C59×#392—21no—
22C59×#390C411yescharacteristic, complemented
23C59×#390C2211yescharacteristic
24C4×C22⋊C4×C59C411yescharacteristic, complemented
25C23×Dic59—41no—
26C2×C22⋊Dic59—24no—
27C2×C22⋊Dic59—22no—
28C2×C22⋊Dic59—22no—
29C2×C22⋊Dic59—41no—
30C2×C22⋊Dic59—42no—
31C2×C22⋊Dic59—41no—
32C2×C22⋊Dic59—41no—
33#14273642—24no—
34#14273642—22no—
35#14273642—22no—
36#14273642—22no—
37#14273642—42no—
38#14273642D414yescomplemented
39C25×C59D411yescharacteristic
40C25×C59D411yescharacteristic
41C22×D4×C59—22no—
42C22×D4×C59—22no—
43C22×D4×C59—21no—
44C22×D4×C59—21no—
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×C59C2×C411yescharacteristic
52C22×D4×C59C2×C411yescharacteristic, complemented
53C22×D4×C59D411yescharacteristic
54C22×D4×C59D411yescharacteristic
55C22×D4×C59C2311yescharacteristic
56C23×C4×C59C2×C411yescharacteristic
57#127×C59—22no—
58#127×C59—22no—
59#127×C59—42no—
60#121×C59—41no—
61#121×C59—41no—
62#120×C59—44no—
63#120×C59—44no—
64C2×C4⋊C4×C59C2×C411yescharacteristic, complemented
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×C59—21no—
71C2×C42×C59C2×C411yescharacteristic, complemented
72C2×C42×C59C2×C411yescharacteristic, complemented
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—22no—
82C24×C59—21no—
83C24×C59—41no—
84C24×C59—42no—
85C24×C59—42no—
86C24×C59—81no—
87C24×C59—81no—
88C24×C59C22⋊C411yescharacteristic
89C24×C59C22⋊C411yescharacteristic
90C24×C59C22⋊C411yescharacteristic
91C24×C59C2×D411yescharacteristic
92C24×C59C2×D411yescharacteristic
93C2×D4×C59—22no—
94C2×D4×C59—22no—
95C2×D4×C59—21no—
96C2×D4×C59—22no—
97C2×D4×C59—21no—
98C2×D4×C59—22no—
99C2×D4×C59—22no—
100C2×D4×C59—41no—
101C2×D4×C59—41no—
102C2×D4×C59—41no—
103C2×D4×C59—41no—
104C2×D4×C59—41no—
105C2×D4×C59—41no—
106C2×D4×C59—42no—
107C2×D4×C59—42no—
108C2×D4×C59—42no—
109C2×D4×C59—42no—
110C2×D4×C59—41no—
111C2×D4×C59—41no—
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—41no—
120C2×D4×C59—41no—
121C2×D4×C59—42no—
122C2×D4×C59—41no—
123C2×D4×C59—42no—
124C2×D4×C59—41no—
125C2×D4×C59—81no—
126C2×D4×C59—81no—
127C2×D4×C59—81no—
128C2×D4×C59—81no—
129C2×D4×C59C2×D412yes—
130C2×D4×C59#4112yes—
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—22no—
136C22×C4×C59—21no—
137C22×C4×C59—41no—
138C22×C4×C59C22⋊C411yescharacteristic
139C22×C4×C59C22×C411yescharacteristic
140C22×Dic59—44no—
141C22×Dic59—42no—
142C22×Dic59—42no—
143C22×Dic59—82no—
144C22×Dic59—82no—
145C4⋊C4×C59—41no—
146C22⋊C4×C59—24no—
147C22⋊C4×C59—42no—
148C22⋊C4×C59—42no—
149C22⋊C4×C59—42no—
150C22⋊C4×C59—42no—
151C22⋊C4×C59—42no—
152C42×C59—22no—
153C42×C59—41no—
154C22⋊Dic59—24no—
155C22⋊Dic59—44no—
156C22⋊Dic59—42no—
157C22⋊Dic59—42no—
158C22⋊Dic59—42no—
159C22⋊Dic59—82no—
160C22⋊Dic59—82no—
161C23×C59—22no—
162C23×C59—22no—
163C23×C59—22no—
164C23×C59—22no—
165C23×C59—22no—
166C23×C59—21no—
167C23×C59—21no—
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—21no—
177C23×C59—41no—
178C23×C59—42no—
179C23×C59—42no—
180C23×C59—42no—
181C23×C59—41no—
182C23×C59—41no—
183C23×C59—41no—
184C23×C59—41no—
185C23×C59—41no—
186C23×C59—41no—
187C23×C59—41no—
188C23×C59—41no—
189C23×C59—42no—
190C23×C59—41no—
191C23×C59—41no—
192C23×C59—41no—
193C23×C59—41no—
194C23×C59—42no—
195C23×C59—42no—
196C23×C59—41no—
197C23×C59—41no—
198C23×C59—41no—
199C23×C59—81no—
200C23×C59—81no—
201C23×C59—81no—
202C23×C59—81no—
203C23×C59—84no—
204C23×C59—82no—
205C23×C59—82no—
206C23×C59—84no—
207C23×C59—84no—
208C23×C59#9911yescharacteristic
209C23×C59#9911yescharacteristic
210C23×C59#9912yes—
211C23×C59C2×C22⋊C411yescharacteristic
212C23×C59#12011yescharacteristic
213C23×C59#12112yes—
214C23×C59#12311yescharacteristic
215D4×C59—42no—
216D4×C59—42no—
217D4×C59—44no—
218D4×C59—44no—
219D4×C59—42no—
220D4×C59—42no—
221D4×C59—82no—
222D4×C59—84no—
223D4×C59—82no—
224D4×C59—84no—
225D4×C59—82no—
226D4×C59—84no—
227D4×C59—82no—
228D4×C59—82no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—22no—
232C2×C4×C59—22no—
233C2×C4×C59—21no—
234C2×C4×C59—21no—
235C2×C4×C59—21no—
236C2×C4×C59—42no—
237C2×C4×C59—42no—
238C2×C4×C59—41no—
239C2×C4×C59—41no—
240C2×C4×C59—41no—
241C2×C4×C59—41no—
242C2×C4×C59—41no—
243C2×C4×C59—41no—
244C2×C4×C59—41no—
245C2×C4×C59C4×D412yes—
246C2×Dic59—44no—
247C2×Dic59—88no—
248C2×Dic59—82no—
249C2×Dic59—82no—
250C2×Dic59—82no—
251C2×Dic59—82no—
252C2×Dic59—82no—
253#13088—591nomaximal, Sylow
254C22×C59—21no—
255C22×C59—21no—
256C22×C59—21no—
257C22×C59—21no—
258C22×C59—21no—
259C22×C59—21no—
260C22×C59—21no—
261C22×C59—22no—
262C22×C59—21no—
263C22×C59—22no—
264C22×C59—21no—
265C22×C59—42no—
266C22×C59—42no—
267C22×C59—42no—
268C22×C59—41no—
269C22×C59—42no—
270C22×C59—41no—
271C22×C59—44no—
272C22×C59—44no—
273C22×C59—42no—
274C22×C59—41no—
275C22×C59—41no—
276C22×C59—41no—
277C22×C59—41no—
278C22×C59—41no—
279C22×C59—41no—
280C22×C59—41no—
281C22×C59—41no—
282C22×C59—41no—
283C22×C59—81no—
284C22×C59—81no—
285C22×C59—81no—
286C22×C59—84no—
287C22×C59—81no—
288C22×C59—81no—
289C22×C59—81no—
290C22×C59—81no—
291C22×C59—81no—
292C22×C59—82no—
293C22×C59—82no—
294C22×C59—84no—
295C22×C59—81no—
296C22×C59—81no—
297C22×C59—81no—
298C22×C59#35312yescomplemented
299C22×C59#38611yescharacteristic
300C22×C59C2×#9911yescharacteristic
301C22×C59C2×#9911yescharacteristic
302C4×C59—22no—
303C4×C59—42no—
304C4×C59—42no—
305C4×C59—41no—
306C4×C59—81no—
307Dic59—84no—
308Dic59—162no—
309C22×#99—591no—
310#2433D5911yescharacteristic
311C2×#353—592no—
312#1781—591no—
313#1393—592no—
314C2×C59—21no—
315C2×C59—21no—
316C2×C59—22no—
317C2×C59—41no—
318C2×C59—42no—
319C2×C59—41no—
320C2×C59—42no—
321C2×C59—41no—
322C2×C59—81no—
323C2×C59—161no—
324C2×C59#189811yescharacteristic
325C2×C59C2×#35311yescharacteristic
326C2×C59#212711yescharacteristic
327C23×D4C2×D5911yescharacteristic
328C2×#127Dic5911yescharacteristic, complemented
329C2×#127C2×D5911yescharacteristic
330C2×#121Dic5911yescharacteristic, complemented
331C2×#120—21no—
332C2×#120—21no—
333C22×C22⋊C4—1181no—
334C2×#99—592no—
335C2×#99—592no—
336C2×#99—1184no—
337C2×#99—1181no—
338C2×#99—1181no—
339#392—21no—
340#392—21no—
341#390Dic5911yescharacteristic, complemented
342#390C2×D5911yescharacteristic
343#379—1181no—
344C4×C22⋊C4Dic5911yescharacteristic, complemented
345#353—1184no—
346C59#1308811yescharacteristic, complemented, Sylow
347C25C22⋊D5911yescharacteristic
348C25C22⋊D5911yescharacteristic
349C22×D4—22no—
350C22×D4—22no—
351C22×D4—21no—
352C22×D4—21no—
353C22×D4—21no—
354C22×D4—21no—
355C22×D4—21no—
356C22×D4—41no—
357C22×D4—41no—
358C22×D4—42no—
359C22×D4C2×Dic5911yescharacteristic
360C22×D4C2×Dic5911yescharacteristic, complemented
361C22×D4C22⋊D5911yescharacteristic
362C22×D4C22⋊D5911yescharacteristic
363C22×D4C22×D5911yescharacteristic
364C23×C4—2361no—
365C23×C4C2×Dic5911yescharacteristic
366#127—22no—
367#127—22no—
368#127—42no—
369#121—41no—
370#121—41no—
371#120—44no—
372#120—44no—
373C2×C4⋊C4C2×Dic5911yescharacteristic, complemented
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—1184no—
381C2×C22⋊C4—1182no—
382C2×C22⋊C4—1182no—
383C2×C22⋊C4—2361no—
384C2×C22⋊C4—2361no—
385C2×C22⋊C4—2361no—
386C2×C22⋊C4—2362no—
387C2×C42C2×Dic5911yescharacteristic, complemented
388C2×C42C2×Dic5911yescharacteristic, complemented
389#99—594no—
390#99—1184no—
391#99—1182no—
392#99—1182no—
393#99—1182no—
394#99—2362no—
395#95—21no—
396C24—21no—
397C24—22no—
398C24—21no—
399C24—21no—
400C24—21no—
401C24—21no—
402C24—21no—
403C24—22no—
404C24—21no—
405C24—41no—
406C24—42no—
407C24—42no—
408C24—81no—
409C24—81no—
410C24C22⋊Dic5911yescharacteristic
411C24C22⋊Dic5911yescharacteristic
412C24C22⋊Dic5911yescharacteristic
413C24C2×C22⋊D5911yescharacteristic
414C24C2×C22⋊D5911yescharacteristic
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—21no—
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—41no—
423C2×D4—41no—
424C2×D4—41no—
425C2×D4—41no—
426C2×D4—41no—
427C2×D4—41no—
428C2×D4—41no—
429C2×D4—41no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—44no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—41no—
442C2×D4—41no—
443C2×D4—42no—
444C2×D4—41no—
445C2×D4—42no—
446C2×D4—41no—
447C2×D4—81no—
448C2×D4—81no—
449C2×D4—81no—
450C2×D4—81no—
451C2×D4D4×D5912yes—
452C2×D4#1174568712yes—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—41no—
460C22×C4—4722no—
461C22×C4—4722no—
462C22×C4—2362no—
463C22×C4—2364no—
464C22×C4—2362no—
465C22×C4C22⋊Dic5911yescharacteristic
466C22×C4C22×Dic5911yescharacteristic
467C4⋊C4—41no—
468C22⋊C4—24no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—1184no—
475C22⋊C4—4722no—
476C22⋊C4—4722no—
477C22⋊C4—2362no—
478C22⋊C4—2362no—
479C22⋊C4—2364no—
480C22⋊C4—2362no—
481C42—22no—
482C42—41no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—41no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—42no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—42no—
517C23—42no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—81no—
522C23—81no—
523C23—81no—
524C23—81no—
525C23—84no—
526C23—82no—
527C23—82no—
528C23—84no—
529C23—84no—
530C23#1427364211yescharacteristic
531C23#1427364211yescharacteristic
532C23#1427364212yes—
533C23C2×C22⋊Dic5911yescharacteristic
534C23#1427369511yescharacteristic
535C23#1427369912yes—
536C23#1427370811yescharacteristic
537D4—42no—
538D4—42no—
539D4—44no—
540D4—44no—
541D4—42no—
542D4—42no—
543D4—82no—
544D4—84no—
545D4—82no—
546D4—84no—
547D4—82no—
548D4—84no—
549D4—82no—
550D4—82no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—4722no—
568C2×C4—4722no—
569C2×C4—4722no—
570C2×C4—4728no—
571C2×C4—4722no—
572C2×C4—4722no—
573C2×C4—2364no—
574C2×C4D4×Dic5912yes—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—22no—
583C22—21no—
584C22—22no—
585C22—21no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—41no—
590C22—42no—
591C22—41no—
592C22—44no—
593C22—44no—
594C22—42no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—84no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—82no—
614C22—82no—
615C22—84no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22#1683719512yescomplemented
620C22#1683728211yescharacteristic
621C22C2×#1427364211yescharacteristic
622C22C2×#1427364211yescharacteristic
623C4—22no—
624C4—42no—
625C4—42no—
626C4—41no—
627C4—81no—
628C4—9442no—
629C4—4724no—
630C2—21no—
631C2—21no—
632C2—22no—
633C2—42no—
634C2—41no—
635C2—41no—
636C2—42no—
637C2—41no—
638C2—81no—
639C2—161no—
640C2#2142455011yescharacteristic
641C2C2×#1683719511yescharacteristic
642C2#2142576511yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC22×D4C22×D59
Fitting, prime radicalC59×#2433C2
socleC22×C59C2×#99
cosocleC22×D4×C59C23
hypercenter#2433D59
hypocenterC59#13088
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #2433

Complements

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

NH ≅ G/Ntype
1. C22×#14273642C2semidirect
2. C2×#16837195C2semidirect
3. #21424011C2semidirect
4. #21422221C2semidirect
15. C2×#127×C59C4semidirect
17. C2×#121×C59C4semidirect
22. C59×#390C4semidirect
24. C4×C22⋊C4×C59C4semidirect
38. #14273642D4semidirect
52. C22×D4×C59C2×C4semidirect
64. C2×C4⋊C4×C59C2×C4semidirect
71. C2×C42×C59C2×C4semidirect
72. C2×C42×C59C2×C4semidirect
298. C22×C59#353semidirect
328. C2×#127Dic59semidirect
330. C2×#121Dic59semidirect
341. #390Dic59semidirect
344. C4×C22⋊C4Dic59semidirect
346. C59#13088semidirect
360. C22×D4C2×Dic59semidirect
373. C2×C4⋊C4C2×Dic59semidirect
387. C2×C42C2×Dic59semidirect
388. C2×C42C2×Dic59semidirect
619. C22#16837195semidirect

Maximal subgroups

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

subgroupindex
C22×#142736422
C2×#168371952
#214240112
#214222212
C59×#24332
#1308859