Finite Group Databasebeta

← #28486339

Subgroups — #28486339

Order 15104.

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

normal subgroups only

This group has 770 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
1#21432541C211yesmaximal, characteristic, complemented
2#21424014C211yesmaximal, characteristic, complemented
3#21422625C212yesmaximal, complemented
4#21421949C212yesmaximal, complemented
5C59×#2431C211yesmaximal, characteristic
6C22×#11745664—21no—
7C2×#14273642—21no—
8C2×#14273642—21no—
9#16837369—24no—
10#16837366—21no—
11#16837366C2212yes—
12#16837265C2212yes—
13#16837136—24no—
14C22×#41×C59C2211yescharacteristic
15C2×#127×C59C411yescharacteristic, complemented
16C2×#124×C59C411yescharacteristic, complemented
17C2×#121×C59C2211yescharacteristic
18C59×#392—21no—
19C59×#392—21no—
20C59×#391C411yescharacteristic, complemented
21C59×#390C411yescharacteristic, complemented
22C59×#386—21no—
23C59×#386—21no—
24C4×C22⋊C4×C59C2211yescharacteristic
25C2×#11745664—22no—
26C2×#11745664—22no—
27C2×#11745664—42no—
28C2×#11745664—42no—
29C22×C59⋊C8—22no—
30C2×C22⋊Dic59—22no—
31#14273642—42no—
32#14273642—42no—
33C22⋊(C59⋊C8)—24no—
34C22⋊(C59⋊C8)—22no—
35C22⋊(C59⋊C8)D414yescomplemented
36C59⋊#95—22no—
37C59⋊#95—22no—
38C2×#41×C59—22no—
39C2×#41×C59—22no—
40C2×#41×C59—42no—
41C2×#41×C59—42no—
42C22×Q8×C59C2×C411yescharacteristic
43C22×D4×C59—21no—
44C22×D4×C59—21no—
45C22×D4×C59—21no—
46C22×D4×C59C2×C411yescharacteristic
47C22×D4×C59C2×C411yescharacteristic
48C22×D4×C59D411yescharacteristic
49C22×D4×C59D411yescharacteristic
50C23×C4×C59D411yescharacteristic
51C23×C4×C59D411yescharacteristic
52C23×C4×C59C2311yescharacteristic
53#127×C59—42no—
54#124×C59—42no—
55#121×C59—22no—
56#121×C59—22no—
57C2×C4⋊C4×C59C2×C411yescharacteristic
58C2×C22⋊C4×C59—21no—
59C2×C22⋊C4×C59—21no—
60C2×C22⋊C4×C59—21no—
61C2×C22⋊C4×C59—21no—
62C2×C22⋊C4×C59—21no—
63C2×C22⋊C4×C59—21no—
64C2×C42×C59C2×C411yescharacteristic
65C2×C42×C59C2×C411yescharacteristic
66#95×C59—21no—
67#95×C59—21no—
68#95×C59—21no—
69C24×C59—21no—
70C24×C59C22⋊C411yescharacteristic
71C24×C59C22⋊C411yescharacteristic
72C24×C59C2×D411yescharacteristic
73#41×C59—44no—
74#41×C59—44no—
75#41×C59—48no—
76C2×Q8×C59—22no—
77C2×Q8×C59—22no—
78C2×Q8×C59—41no—
79C2×D4×C59—22no—
80C2×D4×C59—22no—
81C2×D4×C59—22no—
82C2×D4×C59—22no—
83C2×D4×C59—22no—
84C2×D4×C59—22no—
85C2×D4×C59—22no—
86C2×D4×C59—41no—
87C2×D4×C59—41no—
88C2×D4×C59—41no—
89C2×D4×C59—42no—
90C2×D4×C59—42no—
91C2×D4×C59—41no—
92C2×D4×C59—41no—
93C2×D4×C59—42no—
94C2×D4×C59—42no—
95C2×D4×C59—42no—
96C2×D4×C59—81no—
97C2×D4×C59—81no—
98C2×D4×C59—81no—
99C2×D4×C59—81no—
100C22×C4×C59—22no—
101C22×C4×C59—22no—
102C22×C4×C59—21no—
103C22×C4×C59—21no—
104C22×C4×C59—21no—
105C22×C4×C59—21no—
106C22×C4×C59—21no—
107C22×C4×C59—21no—
108C22×C4×C59—21no—
109C22×C4×C59—41no—
110C22×C4×C59—41no—
111C22×C4×C59—41no—
112C22×C4×C59—41no—
113C22×C4×C59C22⋊C411yescharacteristic
114C22×C4×C59C22⋊C411yescharacteristic
115C22×C4×C59C22×C411yescharacteristic
116C22×C4×C59C2×D411yescharacteristic
117C22×C4×C59C2×D412yes—
118C22×C4×C59#4112yes—
119C22×Dic59—42no—
120C4⋊C4×C59—22no—
121C22⋊C4×C59—42no—
122C22⋊C4×C59—42no—
123C22⋊C4×C59—42no—
124C22⋊C4×C59—41no—
125C22⋊C4×C59—41no—
126C22⋊C4×C59—81no—
127C22⋊C4×C59—81no—
128C42×C59—41no—
129C42×C59—41no—
130C22⋊Dic59—42no—
131C22⋊Dic59—42no—
132#11745664—44no—
133#11745664—42no—
134#11745664—42no—
135C2×C59⋊C8—24no—
136C2×C59⋊C8—44no—
137C23×C59—21no—
138C23×C59—21no—
139C23×C59—21no—
140C23×C59—21no—
141C23×C59—21no—
142C23×C59—21no—
143C23×C59—21no—
144C23×C59—41no—
145C23×C59—41no—
146C23×C59—41no—
147C23×C59—41no—
148C23×C59—41no—
149C23×C59—41no—
150C23×C59—41no—
151C23×C59—41no—
152C23×C59—41no—
153C23×C59—41no—
154C23×C59—41no—
155C23×C59—41no—
156C23×C59—41no—
157C23×C59#9912yes—
158C23×C59C2×C22⋊C411yescharacteristic
159C23×C59#12112yes—
160Q8×C59—44no—
161D4×C59—44no—
162D4×C59—44no—
163D4×C59—44no—
164D4×C59—82no—
165D4×C59—84no—
166D4×C59—84no—
167D4×C59—84no—
168C2×C4×C59—22no—
169C2×C4×C59—22no—
170C2×C4×C59—24no—
171C2×C4×C59—21no—
172C2×C4×C59—21no—
173C2×C4×C59—21no—
174C2×C4×C59—21no—
175C2×C4×C59—21no—
176C2×C4×C59—21no—
177C2×C4×C59—22no—
178C2×C4×C59—21no—
179C2×C4×C59—21no—
180C2×C4×C59—21no—
181C2×C4×C59—42no—
182C2×C4×C59—42no—
183C2×C4×C59—42no—
184C2×C4×C59—42no—
185C2×C4×C59—41no—
186C2×C4×C59—41no—
187C2×C4×C59—41no—
188C2×C4×C59—41no—
189C2×C4×C59—41no—
190C2×C4×C59—41no—
191C2×C4×C59—41no—
192C2×C4×C59—41no—
193C2×C4×C59#9912yes—
194C2×C4×C59C4×D412yes—
195C2×C4×C59#12011yescharacteristic
196C2×C4×C59#12311yescharacteristic
197C2×Dic59—82no—
198C2×Dic59—82no—
199C2×Dic59—82no—
200C59⋊C8—44no—
201#9445—591nomaximal, Sylow
202C22×C59—21no—
203C22×C59—21no—
204C22×C59—22no—
205C22×C59—22no—
206C22×C59—22no—
207C22×C59—22no—
208C22×C59—42no—
209C22×C59—42no—
210C22×C59—41no—
211C22×C59—41no—
212C22×C59—41no—
213C22×C59—41no—
214C22×C59—41no—
215C22×C59—81no—
216C22×C59—81no—
217C22×C59—81no—
218C22×C59—81no—
219C22×C59—81no—
220C22×C59—81no—
221C22×C59—81no—
222C22×C59—81no—
223C22×C59—81no—
224C22×C59—81no—
225C22×C59—81no—
226C22×C59#38611yescharacteristic
227C22×C59C2×#9911yescharacteristic
228C22×C59C2×#9911yescharacteristic
229C4×C59—21no—
230C4×C59—21no—
231C4×C59—22no—
232C4×C59—22no—
233C4×C59—42no—
234C4×C59—41no—
235C4×C59—81no—
236C4×C59—81no—
237Dic59—162no—
238#2879—591no—
239#2431D5911yescharacteristic
240#1782—591no—
241#1510—592no—
242#1284—592no—
243C2×C59—21no—
244C2×C59—21no—
245C2×C59—42no—
246C2×C59—42no—
247C2×C59—81no—
248C2×C59—161no—
249C2×C59#188411yescharacteristic
250C2×C59#189811yescharacteristic
251C2×C59#191011yescharacteristic
252C22×#41C2×D5911yescharacteristic
253C22×M16—1181no—
254C2×#127Dic5911yescharacteristic, complemented
255C2×#124Dic5911yescharacteristic, complemented
256C2×#121C2×D5911yescharacteristic
257C2×#99—1181no—
258C2×#99—1181no—
259#408—1184no—
260#407—592no—
261#407—1181no—
262#392—21no—
263#392—21no—
264#391Dic5911yescharacteristic, complemented
265#390Dic5911yescharacteristic, complemented
266#386—21no—
267#386—21no—
268#381—592no—
269C4×C22⋊C4C2×D5911yescharacteristic
270#327—1184no—
271C59#944511yescharacteristic, complemented, Sylow
272C2×#41—22no—
273C2×#41—22no—
274C2×#41—42no—
275C2×#41—42no—
276C22×Q8C2×Dic5911yescharacteristic
277C22×D4—21no—
278C22×D4—21no—
279C22×D4—21no—
280C22×D4C2×Dic5911yescharacteristic
281C22×D4C2×Dic5911yescharacteristic
282C22×D4C22⋊D5911yescharacteristic
283C22×D4C22⋊D5911yescharacteristic
284C23×C4C22⋊D5911yescharacteristic
285C23×C4C22⋊D5911yescharacteristic
286C23×C4C22×D5911yescharacteristic
287C2×M16—1182no—
288C2×M16—1182no—
289C2×M16—2362no—
290C2×M16—2362no—
291C22×C8—1182no—
292#127—42no—
293#124—42no—
294#121—22no—
295#121—22no—
296C2×C4⋊C4C2×Dic5911yescharacteristic
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—1182no—
304C2×C42C2×Dic5911yescharacteristic
305C2×C42C2×Dic5911yescharacteristic
306#99—2362no—
307#99—2362no—
308C22⋊C8—594no—
309C22⋊C8—1184no—
310C22⋊C8—1182no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—1182no—
315#95—1182no—
316C24—21no—
317C24C22⋊Dic5911yescharacteristic
318C24C22⋊Dic5911yescharacteristic
319C24C2×C22⋊D5911yescharacteristic
320#41—44no—
321#41—44no—
322#41—48no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—41no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—81no—
344C2×D4—81no—
345C2×D4—81no—
346C2×D4—81no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—2362no—
361C22×C4C22⋊Dic5911yescharacteristic
362C22×C4C22⋊Dic5911yescharacteristic
363C22×C4D4×D5912yes—
364C22×C4#1174568712yes—
365C22×C4C22×Dic5911yescharacteristic
366C22×C4C2×C22⋊D5911yescharacteristic
367M16—2364no—
368M16—2362no—
369M16—2362no—
370C2×C8—1184no—
371C2×C8—2364no—
372C4⋊C4—22no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C22⋊C4—81no—
379C22⋊C4—81no—
380C22⋊C4—2362no—
381C22⋊C4—2362no—
382C42—41no—
383C42—41no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23#1427364212yes—
405C23C2×C22⋊Dic5911yescharacteristic
406C23#1427369912yes—
407Q8—44no—
408D4—44no—
409D4—44no—
410D4—44no—
411D4—82no—
412D4—84no—
413D4—84no—
414D4—84no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—4722no—
441C2×C4—4722no—
442C2×C4—4722no—
443C2×C4#1427364212yes—
444C2×C4D4×Dic5912yes—
445C2×C4#1427369511yescharacteristic
446C2×C4#1427370811yescharacteristic
447C8—2364no—
448C22—21no—
449C22—21no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—42no—
455C22—42no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22#1683728211yescharacteristic
473C22C2×#1427364211yescharacteristic
474C22C2×#1427364211yescharacteristic
475C4—21no—
476C4—21no—
477C4—22no—
478C4—22no—
479C4—42no—
480C4—41no—
481C4—81no—
482C4—81no—
483C4—9442no—
484C2—21no—
485C2—21no—
486C2—42no—
487C2—42no—
488C2—81no—
489C2—161no—
490C2#2142448011yescharacteristic
491C2#2142455011yescharacteristic
492C2#2142462411yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
Fitting, prime radicalC59×#2431C2
socleC22×C59#386
cosocleC23×C4×C59C23
hypercenter#2431D59
hypocenterC59#9445
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2431

Complements

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

NH ≅ G/Ntype
1. #21432541C2semidirect
2. #21424014C2semidirect
3. #21422625C2semidirect
4. #21421949C2semidirect
15. C2×#127×C59C4semidirect
16. C2×#124×C59C4semidirect
20. C59×#391C4semidirect
21. C59×#390C4semidirect
35. C22⋊(C59⋊C8)D4semidirect
254. C2×#127Dic59semidirect
255. C2×#124Dic59semidirect
264. #391Dic59semidirect
265. #390Dic59semidirect
271. C59#9445semidirect

Maximal subgroups

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

subgroupindex
#214325412
#214240142
#214226252
#214219492
C59×#24312
#944559