Finite Group Databasebeta

← #28562529

Subgroups — #28562529

Order 15104.

1408 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1408 conjugacy classes of subgroups — too many to draw. Only its 187 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×#16837964C211yesmaximal, characteristic
2#21434849C212yesmaximal, complemented
3#21423115C214yesmaximal, complemented
4#21423047C214yesmaximal, complemented
5#21422663C212yesmaximal, complemented
6C2×#16837126C211yesmaximal, characteristic, complemented
7C59×#2626C211yesmaximal, characteristic, complemented
8C23×C22⋊D59—21no—
9#16837964—24no—
10#16837964—22no—
11#16837964C2214yes—
12C2×#14273711—22no—
13C2×#14273701—21no—
14C2×#14273701C2212yes—
15C2×#14273697—22no—
16#16837730—22no—
17#16837720C2214yescomplemented
18#16837633—24no—
19#16837552—22no—
20#16837541C2214yescomplemented
21C2×C22⋊(C59⋊C8)C2212yescomplemented
22#16837135C2218yescomplemented
23#16837126C2214yescomplemented
24C2×C22⋊Q8×C59C2211yescharacteristic, complemented
25C22×C22⋊C4×C59C2211yescharacteristic
26C59×#399C2212yescomplemented
27C59×#396—22no—
28C59×#393C2212yescomplemented
29C59×#385—21no—
30C59×#380C2211yescharacteristic, complemented
31C59×#379—21no—
32C24×D59—21no—
33C22×D236—21no—
34C22×D236D412yes—
35C22×C22⋊D59—22no—
36C22×C22⋊D59—24no—
37C22×C22⋊D59—22no—
38C22×C22⋊D59—24no—
39C22×C22⋊D59—44no—
40C22×C22⋊D59—44no—
41C23×Dic59—21no—
42C2×#11745671—24no—
43C22×C59⋊C8—22no—
44#14273711—28no—
45#14273711—28no—
46#14273701—24no—
47#14273701—24no—
48#14273701—22no—
49#14273701—22no—
50#14273701D414yes—
51#14273701C2314yes—
52#14273697—28no—
53#14273697—28no—
54C2×C4⋊Dic59—21no—
55C2×C4⋊Dic59D412yes—
56C2×(C2×D59)⋊C4—22no—
57C2×(C2×D59)⋊C4—22no—
58#14273658—24no—
59#14273651—24no—
60#14273647—24no—
61C22⋊(C59⋊C8)—22no—
62C22⋊(C59⋊C8)C2314yescomplemented
63C25×C59D411yescharacteristic
64C22×Q8×C59C2311yescharacteristic, complemented
65C23×C4×C59D411yescharacteristic
66C23×C4×C59C2311yescharacteristic
67C22⋊Q8×C59—22no—
68C22⋊Q8×C59—22no—
69C2×C4⋊C4×C59—21no—
70C2×C4⋊C4×C59—21no—
71C2×C4⋊C4×C59—22no—
72C2×C4⋊C4×C59C2311yescharacteristic, complemented
73C2×C22⋊C4×C59—22no—
74C2×C22⋊C4×C59—22no—
75C2×C22⋊C4×C59—21no—
76C2×C22⋊C4×C59—21no—
77C2×C22⋊C4×C59—22no—
78C2×C22⋊C4×C59—22no—
79C2×C22⋊C4×C59D412yes—
80C2×C22⋊C4×C59C2312yes—
81#95×C59—22no—
82#95×C59C2312yescomplemented
83C24×C59—21no—
84C24×C59—21no—
85C24×C59—22no—
86C24×C59—81no—
87C24×C59—81no—
88C24×C59SD1614yescomplemented
89C24×C59C2×D412yes—
90C24×C59C2×D411yescharacteristic
91C23×D59—22no—
92C23×D59—24no—
93C23×D59—44no—
94C23×D59—82no—
95C23×D59—82no—
96C2×Q8×C59—22no—
97C2×Q8×C59—81no—
98C22×C4×C59—21no—
99C22×C4×C59—21no—
100C22×C4×C59—21no—
101C22×C4×C59—22no—
102C22×C4×C59—21no—
103C22×C4×C59—81no—
104C22×C4×C59—81no—
105C22×C4×C59C2×D412yes—
106C22×C4×C59C2×D411yescharacteristic
107C22×C4×C59C2×D412yes—
108C22×C4×C59C2×D412yes—
109C22×C4×C59C2411yescharacteristic
110C2×C22⋊D59—28no—
111C2×C22⋊D59—24no—
112C2×C22⋊D59—24no—
113C2×C22⋊D59—48no—
114C2×C22⋊D59—416no—
115C2×C22⋊D59—48no—
116C2×C22⋊D59—48no—
117C2×C22⋊D59—44no—
118C2×C22⋊D59—44no—
119C22×Dic59—24no—
120C22×Dic59—22no—
121C22×Dic59—44no—
122C2×D236—24no—
123C2×D236—22no—
124C2×D236—82no—
125C2×D236—82no—
126C2×D236C2×D414yes—
127C4⋊C4×C59—22no—
128C4⋊C4×C59—42no—
129C4⋊C4×C59—81no—
130C4⋊C4×C59—82no—
131C22⋊C4×C59—24no—
132C22⋊C4×C59—24no—
133C22⋊C4×C59—41no—
134C22⋊C4×C59—41no—
135C22⋊C4×C59—42no—
136C22⋊C4×C59—82no—
137C22⋊C4×C59—82no—
138#11745671—84no—
139(C2×D59)⋊C4—28no—
140(C2×D59)⋊C4—24no—
141(C2×D59)⋊C4—24no—
142C4⋊Dic59—24no—
143C4⋊Dic59—22no—
144C4⋊Dic59C2×D414yes—
145C2×C59⋊C8—24no—
146C2×C59⋊C8—82no—
147C23×C59—22no—
148C23×C59—24no—
149C23×C59—22no—
150C23×C59—24no—
151C23×C59—41no—
152C23×C59—41no—
153C23×C59—42no—
154C23×C59—42no—
155C23×C59—42no—
156C23×C59—42no—
157C23×C59—81no—
158C23×C59—81no—
159C23×C59—84no—
160C23×C59—84no—
161C23×C59#12011yescharacteristic
162C23×C59#12011yescharacteristic
163C23×C59C2×SD1614yes—
164C23×C59C2×SD1612yes—
165C23×C59AGL(1,ℤ/8)12yes—
166C23×C59C22×D411yescharacteristic
167C22×D59—24no—
168C22×D59—44no—
169C22×D59—44no—
170C22×D59—48no—
171C22×D59—44no—
172C22×D59—84no—
173C22×D59—84no—
174C22×D59—816no—
175Q8×C59—44no—
176C2×C4×C59—22no—
177C2×C4×C59—42no—
178C2×C4×C59—41no—
179C2×C4×C59—42no—
180C2×C4×C59—41no—
181C2×C4×C59—42no—
182C2×C4×C59—42no—
183C2×C4×C59—42no—
184C2×C4×C59—42no—
185C2×C4×C59—42no—
186C2×C4×C59—42no—
187C2×C4×C59—41no—
188C2×C4×C59—41no—
189C2×C4×C59—42no—
190C2×C4×C59—81no—
191C2×C4×C59#12012yes—
192C2×C4×C59C22×D412yes—
193C22⋊D59—416no—
194C22⋊D59—44no—
195C22⋊D59—44no—
196C22⋊D59—48no—
197C2×Dic59—24no—
198C2×Dic59—48no—
199C2×Dic59—44no—
200D236—44no—
201D236—82no—
202D236—84no—
203C59⋊C8—44no—
204#18285—591nomaximal, Sylow
205C22×C59—21no—
206C22×C59—21no—
207C22×C59—21no—
208C22×C59—21no—
209C22×C59—21no—
210C22×C59—21no—
211C22×C59—24no—
212C22×C59—42no—
213C22×C59—42no—
214C22×C59—44no—
215C22×C59—48no—
216C22×C59—42no—
217C22×C59—42no—
218C22×C59—82no—
219C22×C59—84no—
220C22×C59#45012yes—
221C22×C59#45012yes—
222C22×C59C2×#12011yescharacteristic
223C22×C59C22×SD1611yescharacteristic
224C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
225C2×D59—44no—
226C2×D59—44no—
227C2×D59—44no—
228C2×D59—88no—
229C2×D59—816no—
230C4×C59—22no—
231C4×C59—22no—
232C4×C59—42no—
233C4×C59—82no—
234C4×C59—81no—
235C4×C59—81no—
236Dic59—44no—
237C2×#534—591no—
238#3187—592no—
239#2626D5911yescharacteristic, complemented
240#1613—594no—
241#1601—594no—
242#1519—592no—
243C2×#323—591no—
244C2×C59—21no—
245C2×C59—22no—
246C2×C59—21no—
247C2×C59—22no—
248C2×C59—44no—
249C2×C59C2×#45011yescharacteristic
250C2×C59#301911yescharacteristic
251C2×C59#302611yescharacteristic
252D59—84no—
253C23×D4—1181no—
254#534—594no—
255#534—1184no—
256#534—1182no—
257C2×#123—1182no—
258C2×C22⋊Q8C2×D5911yescharacteristic, complemented
259C2×#121—592no—
260C2×#121—1181no—
261C2×#120—1182no—
262C22×C22⋊C4C2×D5911yescharacteristic
263#483—1182no—
264#481—594no—
265#465—1184no—
266#450—1182no—
267#448—594no—
268C2×C22⋊C8—592no—
269#399C2×D5912yescomplemented
270#396—22no—
271#393C2×D5912yescomplemented
272#385—21no—
273#380C2×D5911yescharacteristic, complemented
274#379—21no—
275#327—598no—
276#323—594no—
277C59#1828511yescharacteristic, complemented, Sylow
278C25—1181no—
279C25C22⋊D5911yescharacteristic
280C22×Q8C22×D5911yescharacteristic, complemented
281C22×D4—592no—
282C22×D4—1181no—
283C22×D4—1182no—
284C22×D4—1184no—
285C22×D4—1182no—
286C22×D4—1184no—
287C22×D4—2364no—
288C22×D4—2364no—
289C23×C4—1181no—
290C23×C4C22⋊D5911yescharacteristic
291C23×C4C22×D5911yescharacteristic
292C2×SD16—1184no—
293C22×C8—1182no—
294#123—1188no—
295#123—1188no—
296C22⋊Q8—22no—
297C22⋊Q8—22no—
298#121—594no—
299#121—594no—
300#121—1184no—
301#121—1182no—
302#121—1182no—
303#121—1184no—
304#120—1188no—
305#120—1188no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—592no—
310C2×C4⋊C4—1181no—
311C2×C4⋊C4C22×D5911yescharacteristic, complemented
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—1182no—
319C2×C22⋊C4—1182no—
320C2×C22⋊C4C22⋊D5912yes—
321C2×C22⋊C4C22×D5912yes—
322#106—1184no—
323#103—1184no—
324#102—1184no—
325C22⋊C8—594no—
326C22⋊C8—1182no—
327#95—22no—
328#95C22×D5912yescomplemented
329C24—21no—
330C24—21no—
331C24—22no—
332C24—81no—
333C24—81no—
334C24—1184no—
335C24—1182no—
336C24—4722no—
337C24—4722no—
338C24—2364no—
339C24#1174567114yescomplemented
340C24C2×C22⋊D5912yes—
341C24C2×C22⋊D5911yescharacteristic
342C2×Q8—22no—
343C2×Q8—81no—
344C2×D4—594no—
345C2×D4—1184no—
346C2×D4—1188no—
347C2×D4—1184no—
348C2×D4—1184no—
349C2×D4—1182no—
350C2×D4—4722no—
351C2×D4—4722no—
352C2×D4—2368no—
353C2×D4—23616no—
354C2×D4—2368no—
355C2×D4—2368no—
356C2×D4—2364no—
357C2×D4—2364no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—81no—
364C22×C4—81no—
365C22×C4—1184no—
366C22×C4—1182no—
367C22×C4—2364no—
368C22×C4D4×D5912yes—
369C22×C4D4×D5912yes—
370C22×C4C2×C22⋊D5912yes—
371C22×C4C2×C22⋊D5911yescharacteristic
372C22×C4C23×D5911yescharacteristic
373SD16—4724no—
374C2×C8—1184no—
375C2×C8—4722no—
376C4⋊C4—22no—
377C4⋊C4—42no—
378C4⋊C4—81no—
379C4⋊C4—82no—
380C4⋊C4—594no—
381C4⋊C4—1184no—
382C4⋊C4—1182no—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C22⋊C4—41no—
386C22⋊C4—41no—
387C22⋊C4—42no—
388C22⋊C4—82no—
389C22⋊C4—82no—
390C22⋊C4—1188no—
391C22⋊C4—1184no—
392C22⋊C4—1184no—
393C23—24no—
394C23—22no—
395C23—22no—
396C23—24no—
397C23—41no—
398C23—41no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—81no—
404C23—81no—
405C23—84no—
406C23—84no—
407C23—1184no—
408C23—47216no—
409C23—4724no—
410C23—4724no—
411C23—2364no—
412C23—2364no—
413C23—2368no—
414C23—2364no—
415C23#1427369511yescharacteristic
416C23#1427369511yescharacteristic
417C23C2×#1174567114yes—
418C23C2×#1174567112yes—
419C23#1427375212yes—
420C23C22×C22⋊D5911yescharacteristic
421Q8—44no—
422D4—4722no—
423D4—4724no—
424D4—2364no—
425D4—23616no—
426D4—2364no—
427D4—2364no—
428D4—2368no—
429C2×C4—22no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—81no—
444C2×C4—1184no—
445C2×C4—2368no—
446C2×C4—2364no—
447C2×C4#1427369512yes—
448C2×C4C2×D4×D5912yes—
449C8—2364no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—24no—
457C22—42no—
458C22—42no—
459C22—44no—
460C22—48no—
461C22—42no—
462C22—42no—
463C22—82no—
464C22—84no—
465C22—4728no—
466C22—47216no—
467C22—2364no—
468C22—2364no—
469C22—2364no—
470C22#1683755212yes—
471C22#1683755212yes—
472C22C2×#1427369511yescharacteristic
473C22C22×#1174567111yescharacteristic
474C22C2×#1427375211yescharacteristic
475C4—22no—
476C4—22no—
477C4—42no—
478C4—82no—
479C4—81no—
480C4—81no—
481C4—2364no—
482C2—21no—
483C2—22no—
484C2—21no—
485C2—22no—
486C2—44no—
487C2—4724no—
488C2C2×#1683755211yescharacteristic
489C2#2143345211yescharacteristic
490C2#2143349211yescharacteristic
subgroupG/H
centerC22C2×#14273695
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2626C2
socleC22×C59C2×#120
prime radicalC2×#16837964C2
hypercenter#2626D59
hypocenterC59#18285
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2626

Complements

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

NH ≅ G/Ntype
2. #21434849C2semidirect
3. #21423115C2semidirect
4. #21423047C2semidirect
5. #21422663C2semidirect
6. C2×#16837126C2semidirect
7. C59×#2626C2semidirect
17. #16837720C22semidirect
20. #16837541C22semidirect
21. C2×C22⋊(C59⋊C8)C22semidirect
22. #16837135C22semidirect
23. #16837126C22semidirect
24. C2×C22⋊Q8×C59C22semidirect
26. C59×#399C22semidirect
28. C59×#393C22semidirect
30. C59×#380C22semidirect
62. C22⋊(C59⋊C8)C23semidirect
64. C22×Q8×C59C23semidirect
72. C2×C4⋊C4×C59C23semidirect
82. #95×C59C23semidirect
88. C24×C59SD16semidirect
239. #2626D59semidirect
258. C2×C22⋊Q8C2×D59semidirect
269. #399C2×D59semidirect
271. #393C2×D59semidirect
273. #380C2×D59semidirect
277. C59#18285semidirect
280. C22×Q8C22×D59semidirect
311. C2×C4⋊C4C22×D59semidirect
328. #95C22×D59semidirect
339. C24#11745671semidirect

Maximal subgroups

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

subgroupindex
C2×#168379642
#214348492
#214231152
#214230472
#214226632
C2×#168371262
C59×#26262
#1828559