Finite Group Databasebeta

← #28506214

Subgroups — #28506214

Order 15104.

782 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C2×#16837376C211yesmaximal, characteristic, complemented
2C2×#16837195C212yesmaximal, complemented
3#21424014C211yesmaximal, characteristic, complemented
4#21422221C212yesmaximal, complemented
5C59×#2431C211yesmaximal, characteristic
6C2×#14273686—21no—
7#16837376—24no—
8C2×#14273642—21no—
9C2×#14273642—21no—
10C2×#14273642C2212yes—
11C2×#14273642C2212yes—
12#16837265—21no—
13#16837195—24no—
14C22×#41×C59C2211yescharacteristic
15C2×#127×C59C2211yescharacteristic
16C2×#124×C59C411yescharacteristic, complemented
17C2×#121×C59C411yescharacteristic, complemented
18C59×#392—21no—
19C59×#392—21no—
20C59×#391C411yescharacteristic, complemented
21C59×#390C2211yescharacteristic
22C59×#386—21no—
23C59×#386—21no—
24C4×C22⋊C4×C59C411yescharacteristic, complemented
25#14273686—24no—
26#14273686—42no—
27C2×C4⋊Dic59—41no—
28C2×C22⋊Dic59—22no—
29C2×C22⋊Dic59—22no—
30C2×C4×Dic59—41no—
31#14273642—24no—
32#14273642—22no—
33#14273642—22no—
34#14273642—22no—
35#14273642—42no—
36#14273642D414yescomplemented
37C59⋊#95—41no—
38C59⋊#95—41no—
39C2×#41×C59—22no—
40C2×#41×C59—22no—
41C2×#41×C59—41no—
42C2×#41×C59—41no—
43C2×#41×C59—42no—
44C22×Q8×C59C2×C411yescharacteristic
45C22×D4×C59—21no—
46C22×D4×C59—21no—
47C22×D4×C59—21no—
48C22×D4×C59C2×C411yescharacteristic
49C22×D4×C59D411yescharacteristic
50C22×D4×C59D411yescharacteristic
51C22×D4×C59C2311yescharacteristic
52C23×C4×C59C2×C411yescharacteristic
53C23×C4×C59D411yescharacteristic
54C23×C4×C59D411yescharacteristic
55#127×C59—22no—
56#127×C59—22no—
57#124×C59—42no—
58#121×C59—41no—
59#121×C59—41no—
60C2×C4⋊C4×C59C2×C411yescharacteristic
61C2×C22⋊C4×C59—21no—
62C2×C22⋊C4×C59—21no—
63C2×C22⋊C4×C59—21no—
64C2×C22⋊C4×C59—21no—
65C2×C22⋊C4×C59—21no—
66C2×C22⋊C4×C59—21no—
67C2×C42×C59C2×C411yescharacteristic
68C2×C42×C59C2×C411yescharacteristic
69#95×C59—21no—
70#95×C59—21no—
71#95×C59—21no—
72C24×C59—21no—
73C24×C59C22⋊C411yescharacteristic
74C24×C59C2×D411yescharacteristic
75C24×C59C2×D411yescharacteristic
76#41×C59—44no—
77#41×C59—44no—
78#41×C59—44no—
79#41×C59—44no—
80C2×Q8×C59—22no—
81C2×Q8×C59—22no—
82C2×Q8×C59—41no—
83C2×D4×C59—21no—
84C2×D4×C59—22no—
85C2×D4×C59—21no—
86C2×D4×C59—22no—
87C2×D4×C59—22no—
88C2×D4×C59—41no—
89C2×D4×C59—41no—
90C2×D4×C59—41no—
91C2×D4×C59—41no—
92C2×D4×C59—41no—
93C2×D4×C59—42no—
94C2×D4×C59—42no—
95C2×D4×C59—41no—
96C2×D4×C59—41no—
97C2×D4×C59—42no—
98C2×D4×C59—41no—
99C2×D4×C59—42no—
100C2×D4×C59—41no—
101C2×D4×C59—81no—
102C2×D4×C59—81no—
103C2×D4×C59—81no—
104C2×D4×C59—81no—
105C2×D4×C59C2×D412yes—
106C2×D4×C59#4112yes—
107C22×C4×C59—21no—
108C22×C4×C59—21no—
109C22×C4×C59—21no—
110C22×C4×C59—21no—
111C22×C4×C59—22no—
112C22×C4×C59—21no—
113C22×C4×C59—22no—
114C22×C4×C59—21no—
115C22×C4×C59—21no—
116C22×C4×C59—21no—
117C22×C4×C59—21no—
118C22×C4×C59—22no—
119C22×C4×C59—21no—
120C22×C4×C59—21no—
121C22×C4×C59—41no—
122C22×C4×C59—41no—
123C22×C4×C59C22⋊C411yescharacteristic
124C22×C4×C59C22⋊C411yescharacteristic
125C22×C4×C59C22⋊C411yescharacteristic
126C22×C4×C59C22×C411yescharacteristic
127C22×Dic59—42no—
128C22×Dic59—42no—
129C4⋊C4×C59—41no—
130C22⋊C4×C59—42no—
131C22⋊C4×C59—42no—
132C22⋊C4×C59—41no—
133C22⋊C4×C59—41no—
134C22⋊C4×C59—42no—
135C22⋊C4×C59—81no—
136C22⋊C4×C59—81no—
137C42×C59—22no—
138C42×C59—41no—
139C22⋊Dic59—24no—
140C22⋊Dic59—42no—
141C22⋊Dic59—42no—
142C22⋊Dic59—42no—
143C4⋊Dic59—44no—
144C4×Dic59—44no—
145C23×C59—21no—
146C23×C59—21no—
147C23×C59—21no—
148C23×C59—21no—
149C23×C59—21no—
150C23×C59—21no—
151C23×C59—21no—
152C23×C59—41no—
153C23×C59—41no—
154C23×C59—41no—
155C23×C59—41no—
156C23×C59—41no—
157C23×C59—41no—
158C23×C59—41no—
159C23×C59—41no—
160C23×C59—41no—
161C23×C59—41no—
162C23×C59—41no—
163C23×C59—41no—
164C23×C59—41no—
165C23×C59C2×C22⋊C411yescharacteristic
166C23×C59#12011yescharacteristic
167C23×C59#12112yes—
168C23×C59#12311yescharacteristic
169Q8×C59—44no—
170D4×C59—42no—
171D4×C59—42no—
172D4×C59—44no—
173D4×C59—42no—
174D4×C59—42no—
175D4×C59—82no—
176D4×C59—82no—
177D4×C59—84no—
178D4×C59—82no—
179D4×C59—84no—
180C2×C4×C59—22no—
181C2×C4×C59—22no—
182C2×C4×C59—22no—
183C2×C4×C59—22no—
184C2×C4×C59—21no—
185C2×C4×C59—21no—
186C2×C4×C59—21no—
187C2×C4×C59—21no—
188C2×C4×C59—21no—
189C2×C4×C59—21no—
190C2×C4×C59—22no—
191C2×C4×C59—21no—
192C2×C4×C59—21no—
193C2×C4×C59—21no—
194C2×C4×C59—42no—
195C2×C4×C59—42no—
196C2×C4×C59—42no—
197C2×C4×C59—42no—
198C2×C4×C59—41no—
199C2×C4×C59—41no—
200C2×C4×C59—41no—
201C2×C4×C59—41no—
202C2×C4×C59—41no—
203C2×C4×C59—41no—
204C2×C4×C59—41no—
205C2×C4×C59—41no—
206C2×C4×C59#9911yescharacteristic
207C2×C4×C59#9911yescharacteristic
208C2×C4×C59#9912yes—
209C2×C4×C59C4×D412yes—
210C2×Dic59—44no—
211C2×Dic59—82no—
212C2×Dic59—82no—
213C2×Dic59—82no—
214C2×Dic59—82no—
215C2×Dic59—82no—
216#13089—591nomaximal, Sylow
217C22×C59—21no—
218C22×C59—21no—
219C22×C59—21no—
220C22×C59—22no—
221C22×C59—21no—
222C22×C59—41no—
223C22×C59—42no—
224C22×C59—41no—
225C22×C59—41no—
226C22×C59—41no—
227C22×C59—41no—
228C22×C59—41no—
229C22×C59—41no—
230C22×C59—41no—
231C22×C59—41no—
232C22×C59—81no—
233C22×C59—81no—
234C22×C59—81no—
235C22×C59—81no—
236C22×C59—81no—
237C22×C59—81no—
238C22×C59—81no—
239C22×C59—81no—
240C22×C59—81no—
241C22×C59—81no—
242C22×C59—81no—
243C22×C59#38611yescharacteristic
244C22×C59C2×#9911yescharacteristic
245C22×C59C2×#9911yescharacteristic
246C4×C59—21no—
247C4×C59—21no—
248C4×C59—22no—
249C4×C59—22no—
250C4×C59—42no—
251C4×C59—41no—
252C4×C59—81no—
253C4×C59—81no—
254Dic59—84no—
255Dic59—162no—
256C2×#410—591no—
257#2431D5911yescharacteristic
258C2×#353—592no—
259#1782—591no—
260#1393—592no—
261C2×C59—21no—
262C2×C59—21no—
263C2×C59—41no—
264C2×C59—42no—
265C2×C59—41no—
266C2×C59—81no—
267C2×C59—161no—
268C2×C59#189811yescharacteristic
269C2×C59#212611yescharacteristic
270C2×C59#212811yescharacteristic
271C22×#41C2×D5911yescharacteristic
272C2×#127C2×D5911yescharacteristic
273C2×#124Dic5911yescharacteristic, complemented
274C2×#121Dic5911yescharacteristic, complemented
275C2×#117—1181no—
276#410—1184no—
277C2×#99—592no—
278C2×#99—592no—
279C2×#99—1181no—
280C2×#99—1181no—
281#392—21no—
282#392—21no—
283#391Dic5911yescharacteristic, complemented
284#390C2×D5911yescharacteristic
285#386—21no—
286#386—21no—
287#381—1181no—
288C4×C22⋊C4Dic5911yescharacteristic, complemented
289#353—1184no—
290C59#1308911yescharacteristic, complemented, Sylow
291C2×#41—22no—
292C2×#41—22no—
293C2×#41—41no—
294C2×#41—41no—
295C2×#41—42no—
296C22×Q8C2×Dic5911yescharacteristic
297C22×D4—21no—
298C22×D4—21no—
299C22×D4—21no—
300C22×D4C2×Dic5911yescharacteristic
301C22×D4C22⋊D5911yescharacteristic
302C22×D4C22⋊D5911yescharacteristic
303C22×D4C22×D5911yescharacteristic
304C23×C4C2×Dic5911yescharacteristic
305C23×C4C22⋊D5911yescharacteristic
306C23×C4C22⋊D5911yescharacteristic
307#127—22no—
308#127—22no—
309#124—42no—
310#121—41no—
311#121—41no—
312#117—1184no—
313#117—2362no—
314C2×C4⋊C4—2361no—
315C2×C4⋊C4C2×Dic5911yescharacteristic
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—1182no—
323C2×C22⋊C4—1182no—
324C2×C42—2361no—
325C2×C42C2×Dic5911yescharacteristic
326C2×C42C2×Dic5911yescharacteristic
327#99—594no—
328#99—1184no—
329#99—1182no—
330#99—1182no—
331#99—1182no—
332#99—2362no—
333#95—21no—
334#95—21no—
335#95—21no—
336#95—2361no—
337#95—2361no—
338C24—21no—
339C24C22⋊Dic5911yescharacteristic
340C24C2×C22⋊D5911yescharacteristic
341C24C2×C22⋊D5911yescharacteristic
342#41—44no—
343#41—44no—
344#41—44no—
345#41—44no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—41no—
349C2×D4—21no—
350C2×D4—22no—
351C2×D4—21no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—41no—
365C2×D4—42no—
366C2×D4—41no—
367C2×D4—81no—
368C2×D4—81no—
369C2×D4—81no—
370C2×D4—81no—
371C2×D4D4×D5912yes—
372C2×D4#1174568712yes—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—2362no—
390C22×C4—2362no—
391C22×C4C22⋊Dic5911yescharacteristic
392C22×C4C22⋊Dic5911yescharacteristic
393C22×C4C22⋊Dic5911yescharacteristic
394C22×C4C22×Dic5911yescharacteristic
395C4⋊C4—41no—
396C4⋊C4—2364no—
397C22⋊C4—42no—
398C22⋊C4—42no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—42no—
402C22⋊C4—81no—
403C22⋊C4—81no—
404C22⋊C4—1184no—
405C22⋊C4—2362no—
406C22⋊C4—2362no—
407C22⋊C4—2362no—
408C42—22no—
409C42—41no—
410C42—2364no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23C2×C22⋊Dic5911yescharacteristic
432C23#1427369511yescharacteristic
433C23#1427369912yes—
434C23#1427370811yescharacteristic
435Q8—44no—
436D4—42no—
437D4—42no—
438D4—44no—
439D4—42no—
440D4—42no—
441D4—82no—
442D4—82no—
443D4—84no—
444D4—82no—
445D4—84no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—4722no—
473C2×C4—4722no—
474C2×C4—4722no—
475C2×C4—4722no—
476C2×C4—4722no—
477C2×C4—2364no—
478C2×C4#1427364212yes—
479C2×C4#1427364211yescharacteristic
480C2×C4#1427364211yescharacteristic
481C2×C4D4×Dic5912yes—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—22no—
487C22—41no—
488C22—42no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—81no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22#1683728211yescharacteristic
509C22C2×#1427364211yescharacteristic
510C22C2×#1427364211yescharacteristic
511C4—21no—
512C4—21no—
513C4—22no—
514C4—22no—
515C4—42no—
516C4—41no—
517C4—81no—
518C4—81no—
519C4—9442no—
520C4—4724no—
521C2—21no—
522C2—21no—
523C2—42no—
524C2—41no—
525C2—41no—
526C2—81no—
527C2—161no—
528C2#2142455011yescharacteristic
529C2#2142576111yescharacteristic
530C2#2142576811yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC22×D4C22×D59
Fitting, prime radicalC59×#2431C2
socleC22×C59C2×#99
cosocleC22×D4×C59C23
hypercenter#2431D59
hypocenterC59#13089
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #2431

Complements

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

NH ≅ G/Ntype
1. C2×#16837376C2semidirect
2. C2×#16837195C2semidirect
3. #21424014C2semidirect
4. #21422221C2semidirect
16. C2×#124×C59C4semidirect
17. C2×#121×C59C4semidirect
20. C59×#391C4semidirect
24. C4×C22⋊C4×C59C4semidirect
36. #14273642D4semidirect
273. C2×#124Dic59semidirect
274. C2×#121Dic59semidirect
283. #391Dic59semidirect
288. C4×C22⋊C4Dic59semidirect
290. C59#13089semidirect

Maximal subgroups

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

subgroupindex
C2×#168373762
C2×#168371952
#214240142
#214222212
C59×#24312
#1308959