Finite Group Databasebeta

← #28735646

Subgroups — #28735646

Order 15104.

984 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21432383C212yesmaximal
2#21425529C212yesmaximal, complemented
3#21425519C211yesmaximal, characteristic, complemented
4#21425300C212yesmaximal, complemented
5#21425293C212yesmaximal, complemented
6#21425293C212yesmaximal, complemented
7#21425106C211yesmaximal, characteristic, complemented
8#21425106C212yesmaximal, complemented
9C59×#1779C211yesmaximal, characteristic
10C22×Dic236C2211yescharacteristic, complemented
11C2×#14273706C2212yes—
12C2×#14273706C2212yes—
13C2×#14273706C2212yes—
14C2×#14273661C2211yescharacteristic, complemented
15C2×#14273659C2212yescomplemented
16C2×#14273652C2212yescomplemented
17C2×#14273652C2212yescomplemented
18C2×#14273652C2212yescomplemented
19#16837275—22no—
20#16837275—22no—
21#16837275—22no—
22#16837262C2212yes—
23#16837262C2212yes—
24#16837262C2212yes—
25#16837258—22no—
26#16837165C2212yescomplemented
27#16837165C2211yescharacteristic, complemented
28#16837165C2211yescharacteristic, complemented
29#16837165C2212yescomplemented
30C24×C4×C59C2211yescharacteristic
31C2×C22⋊C8×C59C2211yescharacteristic
32C2×C22⋊C8×C59C2212yes—
33C2×C22⋊C8×C59C2211yescharacteristic
34C2×C22⋊C8×C59C2212yes—
35C22×Dic118C2312yes—
36C2×Dic236—42no—
37C2×Dic236—41no—
38#14273706—42no—
39#14273706—42no—
40#14273706—42no—
41#14273706—42no—
42#14273706—42no—
43#14273706—42no—
44C2×C4⋊Dic59C2312yes—
45C2×C4⋊Dic59C2312yes—
46C2×C4⋊Dic59C2312yes—
47C2×Dic59⋊C4—22no—
48C2×Dic59⋊C4—22no—
49C2×Dic59⋊C4—22no—
50C2×C22⋊Dic59—22no—
51C2×C22⋊Dic59—22no—
52C2×C22⋊Dic59—22no—
53C2×C22⋊Dic59—22no—
54C2×C22⋊Dic59—22no—
55C2×C22⋊Dic59—22no—
56#14273661—41no—
57#14273659—42no—
58#14273652—42no—
59#14273652—42no—
60#14273652—42no—
61C59⋊#95—22no—
62C59⋊#95—22no—
63C59⋊#95—22no—
64C25×C59D411yescharacteristic
65C23×C4×C59—42no—
66C23×C4×C59—41no—
67C23×C4×C59—42no—
68C23×C4×C59—41no—
69C23×C4×C59D411yescharacteristic
70C23×C4×C59D412yes—
71C23×C4×C59C2311yescharacteristic
72C23×C4×C59C2312yes—
73C22×C8×C59C2312yes—
74C22×C8×C59C2311yescharacteristic
75C22×C8×C59C2311yescharacteristic
76C22⋊C8×C59—41no—
77C22⋊C8×C59—42no—
78C22⋊C8×C59—42no—
79C22⋊C8×C59—41no—
80C24×C59—22no—
81C24×C59—24no—
82C24×C59—42no—
83C24×C59—41no—
84C24×C59—41no—
85C24×C59C2×D411yescharacteristic
86C24×C59C2×D412yes—
87C22×C4×C59—22no—
88C22×C4×C59—21no—
89C22×C4×C59—22no—
90C22×C4×C59—21no—
91C22×C4×C59—22no—
92C22×C4×C59—22no—
93C22×C4×C59—22no—
94C22×C4×C59—22no—
95C22×C4×C59—22no—
96C22×C4×C59—22no—
97C22×C4×C59—42no—
98C22×C4×C59—41no—
99C22×C4×C59—42no—
100C22×C4×C59—42no—
101C22×C4×C59—42no—
102C22×C4×C59—48no—
103C22×C4×C59—44no—
104C22×C4×C59—44no—
105C22×C4×C59C2×D411yescharacteristic
106C22×C4×C59C2×D412yes—
107C22×C4×C59C2411yescharacteristic
108C22×Dic59—22no—
109C22×Dic59—22no—
110C22×Dic59—22no—
111C22×Dic59—22no—
112C2×Dic118—22no—
113C2×Dic118—22no—
114C2×Dic118—22no—
115C2×Dic118—22no—
116C2×Dic118—22no—
117C2×Dic118—22no—
118C2×C8×C59—22no—
119C2×C8×C59—22no—
120C2×C8×C59—22no—
121C2×C8×C59—22no—
122C2×C8×C59—21no—
123C2×C8×C59—22no—
124C2×C8×C59—21no—
125C22⋊Dic59—44no—
126C22⋊Dic59—44no—
127C22⋊Dic59—44no—
128C22⋊Dic59—44no—
129C22⋊Dic59—44no—
130C22⋊Dic59—44no—
131C4⋊Dic59—22no—
132C4⋊Dic59—22no—
133C4⋊Dic59—22no—
134C4⋊Dic59—22no—
135C4⋊Dic59—22no—
136C4⋊Dic59—22no—
137Dic59⋊C4—44no—
138Dic59⋊C4—44no—
139Dic59⋊C4—44no—
140Dic236—44no—
141C23×C59—22no—
142C23×C59—22no—
143C23×C59—22no—
144C23×C59—22no—
145C23×C59—21no—
146C23×C59—22no—
147C23×C59—22no—
148C23×C59—22no—
149C23×C59—22no—
150C23×C59—21no—
151C23×C59—48no—
152C23×C59—44no—
153C23×C59—44no—
154C23×C59—48no—
155C23×C59—44no—
156C23×C59#13512yes—
157C23×C59#13714yes—
158C23×C59C22×D411yescharacteristic
159C2×C4×C59—22no—
160C2×C4×C59—22no—
161C2×C4×C59—22no—
162C2×C4×C59—22no—
163C2×C4×C59—21no—
164C2×C4×C59—48no—
165C2×C4×C59—44no—
166C2×C4×C59—42no—
167C2×C4×C59—42no—
168C2×C4×C59—42no—
169C2×C4×C59—41no—
170C2×C4×C59—42no—
171C2×C4×C59—41no—
172C2×C4×C59—42no—
173C2×C4×C59H(5,𝔽2)12yes—
174C2×C4×C59H(5,𝔽2)11yescharacteristic
175C2×C4×C59H(5,𝔽2)12yes—
176C2×C4×C59H(5,𝔽2)11yescharacteristic
177C2×Dic59—42no—
178C2×Dic59—42no—
179C2×Dic59—42no—
180C2×Dic59—42no—
181C2×Dic59—42no—
182C2×Dic59—42no—
183C2×Dic59—42no—
184C2×Dic59—42no—
185C2×Dic59—42no—
186C2×Dic59—42no—
187C2×Dic59—42no—
188C2×Dic59—42no—
189Dic118—42no—
190Dic118—42no—
191Dic118—42no—
192Dic118—42no—
193C8×C59—42no—
194C8×C59—41no—
195C8×C59—41no—
196#32711—591nomaximal, Sylow
197C22×C59—24no—
198C22×C59—24no—
199C22×C59—22no—
200C22×C59—24no—
201C22×C59—24no—
202C22×C59—416no—
203C22×C59—42no—
204C22×C59—42no—
205C22×C59—41no—
206C22×C59—42no—
207C22×C59—42no—
208C22×C59—41no—
209C22×C59—42no—
210C22×C59#53412yes—
211C22×C59#53411yescharacteristic
212C22×C59C2×#13511yescharacteristic
213C22×C59C2×#13712yes—
214C22×C59#57511yescharacteristic
215C4×C59—41no—
216C4×C59—42no—
217C4×C59#56012yes—
218C4×C59#56011yescharacteristic
219C4×C59H(3,𝔽4)11yescharacteristic
220Dic59—82no—
221Dic59—82no—
222Dic59—82no—
223Dic59—82no—
224#2848—592no—
225#2077—592no—
226#2075—591no—
227#2036—592no—
228#2035—592no—
229#2035—592no—
230#2001—591no—
231#2001—592no—
232#1779D5911yescharacteristic
233C2×C59—42no—
234C2×C59—44no—
235C2×C59#281211yescharacteristic
236C2×C59#319012yes—
237C2×C59#319111yescharacteristic
238C2×C59#319511yescharacteristic
239C2×C59#319612yes—
240C24×C4C2×D5911yescharacteristic
241C22×Q16—591no—
242C2×C22⋊Q8—592no—
243C2×C22⋊Q8—592no—
244C2×C22⋊Q8—592no—
245C2×#107—591no—
246C2×#106—592no—
247C2×#103—592no—
248C2×#103—592no—
249C2×#103—592no—
250C2×C22⋊C8C2×D5911yescharacteristic
251C2×C22⋊C8C2×D5912yes—
252C2×C22⋊C8C2×D5911yescharacteristic
253C2×C22⋊C8C2×D5912yes—
254#385—1182no—
255#385—1182no—
256#385—1182no—
257#380—592no—
258#380—592no—
259#380—592no—
260#379—1182no—
261#340—592no—
262#340—591no—
263#340—591no—
264#340—592no—
265C59#3271111yescharacteristic, complemented, Sylow
266C25D23611yescharacteristic
267C22×Q8—592no—
268C23×C4—41no—
269C23×C4—42no—
270C23×C4—41no—
271C23×C4—42no—
272C23×C4D23611yescharacteristic
273C23×C4D23612yes—
274C23×C4C22×D5911yescharacteristic
275C23×C4C22×D5912yes—
276C2×Q16—2361no—
277C2×Q16—2362no—
278C22×C8C22×D5911yescharacteristic
279C22×C8C22×D5912yes—
280C22×C8C22×D5911yescharacteristic
281C22⋊Q8—2362no—
282C22⋊Q8—2362no—
283C22⋊Q8—2362no—
284C22⋊Q8—2362no—
285C22⋊Q8—2362no—
286C22⋊Q8—2362no—
287C2×C4⋊C4—592no—
288C2×C4⋊C4—592no—
289C2×C4⋊C4—592no—
290C2×C4⋊C4—1182no—
291C2×C4⋊C4—1182no—
292C2×C4⋊C4—1182no—
293C2×C22⋊C4—1182no—
294C2×C22⋊C4—1182no—
295C2×C22⋊C4—1182no—
296C2×C22⋊C4—1182no—
297C2×C22⋊C4—1182no—
298C2×C22⋊C4—1182no—
299#107—2361no—
300#106—2362no—
301#103—2362no—
302#103—2362no—
303#103—2362no—
304C22⋊C8—41no—
305C22⋊C8—42no—
306C22⋊C8—42no—
307C22⋊C8—41no—
308#95—1182no—
309#95—1182no—
310#95—1182no—
311C24—22no—
312C24—24no—
313C24—41no—
314C24—42no—
315C24—41no—
316C24C2×D23611yescharacteristic
317C24C2×D23612yes—
318C2×Q8—1182no—
319C2×Q8—1182no—
320C2×Q8—1182no—
321C2×Q8—1182no—
322C2×Q8—1182no—
323C2×Q8—1182no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—44no—
340C22×C4—48no—
341C22×C4—44no—
342C22×C4—1182no—
343C22×C4—1182no—
344C22×C4—1182no—
345C22×C4—1182no—
346C22×C4C2×D23612yes—
347C22×C4C2×D23611yescharacteristic
348C22×C4C23×D5911yescharacteristic
349Q16—2364no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—22no—
357C4⋊C4—1182no—
358C4⋊C4—1182no—
359C4⋊C4—1182no—
360C4⋊C4—1182no—
361C4⋊C4—1182no—
362C4⋊C4—1182no—
363C4⋊C4—2364no—
364C4⋊C4—2364no—
365C4⋊C4—2364no—
366C22⋊C4—2364no—
367C22⋊C4—2364no—
368C22⋊C4—2364no—
369C22⋊C4—2364no—
370C22⋊C4—2364no—
371C22⋊C4—2364no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—21no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—21no—
381C23—22no—
382C23—44no—
383C23—48no—
384C23—44no—
385C23—48no—
386C23—44no—
387C23#1427374912yes—
388C23#1427376014yes—
389C23C22×D23611yescharacteristic
390Q8—2362no—
391Q8—2362no—
392Q8—2362no—
393Q8—2362no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—44no—
400C2×C4—48no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—2362no—
409C2×C4—2362no—
410C2×C4—2362no—
411C2×C4—2362no—
412C2×C4—2362no—
413C2×C4—2362no—
414C2×C4—2362no—
415C2×C4—2362no—
416C2×C4—2362no—
417C2×C4—2362no—
418C2×C4—2362no—
419C2×C4—2362no—
420C2×C4#1427377211yescharacteristic
421C2×C4#1427377212yes—
422C2×C4#1427377211yescharacteristic
423C2×C4#1427377212yes—
424C8—41no—
425C8—42no—
426C8—41no—
427C22—22no—
428C22—24no—
429C22—24no—
430C22—24no—
431C22—24no—
432C22—416no—
433C22—42no—
434C22—41no—
435C22—42no—
436C22—41no—
437C22—42no—
438C22—42no—
439C22—42no—
440C22#1683796411yescharacteristic
441C22#1683796412yes—
442C22C2×#1427374911yescharacteristic
443C22C2×#1427376012yes—
444C22#1683820211yescharacteristic
445C4—42no—
446C4—41no—
447C4—4722no—
448C4—4722no—
449C4—4722no—
450C4—4722no—
451C4#1683814112yes—
452C4#1683814111yescharacteristic
453C4#1683814511yescharacteristic
454C2—42no—
455C2—44no—
456C2#2143209311yescharacteristic
457C2#2143487912yes—
458C2#2143488911yescharacteristic
459C2#2143492011yescharacteristic
460C2#2143492812yes—
subgroupG/H
centerC23C22×D236
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#1779C2
socleC23×C59C22×D4
prime radicalC25×C59D4
hypercenter#1779D59
hypocenterC59#32711
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C24×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
2. #21425529C2semidirect
3. #21425519C2semidirect
4. #21425300C2semidirect
5. #21425293C2semidirect
6. #21425293C2semidirect
7. #21425106C2semidirect
8. #21425106C2semidirect
10. C22×Dic236C22semidirect
14. C2×#14273661C22semidirect
15. C2×#14273659C22semidirect
16. C2×#14273652C22semidirect
17. C2×#14273652C22semidirect
18. C2×#14273652C22semidirect
26. #16837165C22semidirect
27. #16837165C22semidirect
28. #16837165C22semidirect
29. #16837165C22semidirect
265. C59#32711semidirect

Maximal subgroups

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

subgroupindex
#214323832
#214255292
#214255192
#214253002
#214252932
#214251062
C59×#17792
#3271159