Finite Group Databasebeta

← #28564794

Subgroups — #28564794

Order 15104.

1630 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 1630 conjugacy classes of subgroups — too many to draw. Only its 195 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#21434470C211yesmaximal, characteristic, complemented
2#21434195C212yesmaximal, complemented
3#21423275C218yesmaximal, complemented
4#21422810C211yesmaximal, characteristic, complemented
5#21422539C212yesmaximal, complemented
6C2×D42×C59C211yesmaximal, characteristic
7#16837959C2211yescharacteristic, complemented
8#16837766—22no—
9#16837748C2214yescomplemented
10#16837737—22no—
11#16837682C2214yescomplemented
12#16837607C2214yescomplemented
13#16837553—22no—
14#16837502—24no—
15#16837459C2212yescomplemented
16#16837407—22no—
17C4×#11745664C2211yescharacteristic, complemented
18#16837146C2214yescomplemented
19D4⋊(C59⋊C8)C2218yescomplemented
20C23×D4×C59—22no—
21D42×C59—22no—
22D42×C59—24no—
23D42×C59C2214yes—
24C2×#127×C59C2211yescharacteristic
25C2×#121×C59—22no—
26C2×#120×C59—22no—
27C2×C4×D4×C59C2212yes—
28C2×#11745670—24no—
29C2×#11745664—21no—
30#14273727C2312yescomplemented
31#14273719—21no—
32#14273706—22no—
33C4×Dic118—22no—
34#14273686—21no—
35#14273660—24no—
36C4⋊(C59⋊C8)C2314yescomplemented
37#14273650—24no—
38#14273648—28no—
39#14273648—24no—
40C22⋊(C59⋊C8)—22no—
41#14273638—22no—
42#14273638—21no—
43C4×C59⋊C8C2312yescomplemented
44C25×C59—22no—
45C22×D4×C59—24no—
46C22×D4×C59—28no—
47C22×D4×C59—28no—
48C22×D4×C59—22no—
49C22×D4×C59—416no—
50C22×D4×C59D412yes—
51C22×D4×C59D412yes—
52C23×C4×C59—22no—
53#127×C59—22no—
54#127×C59D412yes—
55#127×C59C2312yes—
56#121×C59—28no—
57#121×C59—28no—
58#120×C59—28no—
59#120×C59—28no—
60C4×D4×C59—22no—
61C4×D4×C59—22no—
62C4×D4×C59D414yes—
63C4×D4×C59C2314yes—
64C2×C4⋊C4×C59D412yes—
65C2×C22⋊C4×C59—22no—
66C2×C22⋊C4×C59—22no—
67C2×C42×C59C2311yescharacteristic
68C24×C59—22no—
69C24×C59—28no—
70C24×C59—22no—
71C24×C59—416no—
72C24×C59—84no—
73C2×D4×C59—28no—
74C2×D4×C59—216no—
75C2×D4×C59—22no—
76C2×D4×C59—24no—
77C2×D4×C59—22no—
78C2×D4×C59—24no—
79C2×D4×C59—48no—
80C2×D4×C59—44no—
81C2×D4×C59—432no—
82C2×D4×C59—416no—
83C2×D4×C59—416no—
84C2×D4×C59—416no—
85C2×D4×C59C2×D414yes—
86C2×D4×C59C2×D414yes—
87C22×C4×C59—22no—
88C22×C4×C59—24no—
89C22×C4×C59—24no—
90C22×C4×C59—48no—
91C22×C4×C59C2×D411yescharacteristic
92C22×C4×C59C2×D412yes—
93C2×Dic118—22no—
94C4⋊C4×C59—22no—
95C4⋊C4×C59C2×D414yes—
96C22⋊C4×C59—28no—
97C22⋊C4×C59—24no—
98C22⋊C4×C59—24no—
99C42×C59C2×D411yescharacteristic
100C42×C59C2×D412yes—
101C42×C59C2411yescharacteristic
102C22⋊Dic59—41no—
103#11745670—48no—
104C4⋊Dic59—22no—
105C4⋊Dic59—22no—
106Dic59⋊C4—42no—
107C4×Dic59—41no—
108#11745664—42no—
109C2×C59⋊C8—22no—
110C2×C59⋊C8—24no—
111C23×C59—28no—
112C23×C59—24no—
113C23×C59—24no—
114C23×C59—44no—
115C23×C59—48no—
116C23×C59—432no—
117C23×C59—44no—
118C23×C59—44no—
119C23×C59—88no—
120C23×C59—832no—
121C23×C59#12011yescharacteristic
122D4×C59—28no—
123D4×C59—24no—
124D4×C59—44no—
125D4×C59—48no—
126D4×C59—432no—
127D4×C59—48no—
128D4×C59—82no—
129D4×C59AGL(1,ℤ/8)18yes—
130C2×C4×C59—22no—
131C2×C4×C59—24no—
132C2×C4×C59—24no—
133C2×C4×C59—48no—
134C2×C4×C59—44no—
135C2×C4×C59—48no—
136C2×C4×C59#12011yescharacteristic
137C2×C4×C59#12012yes—
138C2×C4×C59C22×D411yescharacteristic
139C2×C4×C59C22×D412yes—
140C2×Dic59—42no—
141Dic118—44no—
142C59⋊C8—42no—
143C59⋊C8—44no—
144#18529—591nomaximal, Sylow
145C22×C59—22no—
146C22×C59—28no—
147C22×C59—416no—
148C22×C59—44no—
149C22×C59—44no—
150C22×C59—48no—
151C22×C59—44no—
152C22×C59—88no—
153C22×C59—832no—
154C22×C59#45512yes—
155C22×C59C2×#12011yescharacteristic
156C4×C59—22no—
157C4×C59—21no—
158C4×C59—21no—
159C4×C59—44no—
160C4×C59C2×AGL(1,ℤ/8)14yes—
161Dic59—82no—
162C2×D42D5911yescharacteristic
163#3138—591no—
164#3110—592no—
165#1639—598no—
166#1555—591no—
167#1490—592no—
168C2×C59—22no—
169C2×C59—48no—
170C2×C59—84no—
171C2×C59#300312yes—
172C2×C59C2×#45511yescharacteristic
173C23×D4—22no—
174D42—22no—
175D42—24no—
176D42C2×D5914yes—
177#533—591no—
178C2×#127C2×D5911yescharacteristic
179C2×#121—22no—
180C2×#120—22no—
181C2×C4×D4C2×D5912yes—
182#490—1182no—
183#486—594no—
184#484—1182no—
185#474—594no—
186#461—594no—
187#450—1182no—
188#440—1184no—
189#432—592no—
190#418—1182no—
191C4×M16—591no—
192#331—594no—
193D4⋊C8—598no—
194C59#1852911yescharacteristic, complemented, Sylow
195C25—22no—
196C22×D4—24no—
197C22×D4—28no—
198C22×D4—28no—
199C22×D4—22no—
200C22×D4—416no—
201C22×D4C22⋊D5912yes—
202C22×D4C22⋊D5912yes—
203C23×C4—22no—
204C2×SD16—1184no—
205C2×M16—1181no—
206C4⋊Q8—592no—
207#127—22no—
208#127C22⋊D5912yes—
209#127C22×D5912yes—
210#125—1181no—
211C22⋊Q8—1182no—
212#121—28no—
213#121—28no—
214#120—28no—
215#120—28no—
216C4×Q8—1182no—
217C4×D4—22no—
218C4×D4—22no—
219C4×D4C22⋊D5914yes—
220C4×D4C22×D5914yes—
221#117—1181no—
222C2×C4⋊C4C22⋊D5912yes—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C42C22×D5911yescharacteristic
226#107—1184no—
227C4⋊C8—594no—
228#103—1184no—
229#102—1188no—
230#102—1184no—
231C22⋊C8—1182no—
232#97—1181no—
233#97—1182no—
234C4×C8—592no—
235C24—22no—
236C24—28no—
237C24—22no—
238C24—416no—
239C24—84no—
240C2×Q8—1182no—
241C2×D4—28no—
242C2×D4—216no—
243C2×D4—22no—
244C2×D4—24no—
245C2×D4—22no—
246C2×D4—24no—
247C2×D4—48no—
248C2×D4—44no—
249C2×D4—432no—
250C2×D4—416no—
251C2×D4—416no—
252C2×D4—416no—
253C2×D4C2×C22⋊D5914yes—
254C2×D4C2×C22⋊D5914yes—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—48no—
259C22×C4C2×C22⋊D5912yes—
260C22×C4C2×C22⋊D5911yescharacteristic
261SD16—2368no—
262M16—2362no—
263C2×C8—1182no—
264C2×C8—1184no—
265C4⋊C4—22no—
266C4⋊C4—1182no—
267C4⋊C4—1182no—
268C4⋊C4—2362no—
269C4⋊C4C2×C22⋊D5914yes—
270C22⋊C4—28no—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—2361no—
274C42—2361no—
275C42C2×C22⋊D5911yescharacteristic
276C42C2×C22⋊D5912yes—
277C42C23×D5911yescharacteristic
278C23—28no—
279C23—24no—
280C23—24no—
281C23—44no—
282C23—48no—
283C23—432no—
284C23—44no—
285C23—44no—
286C23—88no—
287C23—832no—
288C23#1427369611yescharacteristic
289Q8—2364no—
290D4—28no—
291D4—24no—
292D4—44no—
293D4—48no—
294D4—432no—
295D4—48no—
296D4—82no—
297D4#1427375318yes—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—48no—
302C2×C4—44no—
303C2×C4—48no—
304C2×C4—2362no—
305C2×C4#1427369611yescharacteristic
306C2×C4#1427369612yes—
307C2×C4C22×C22⋊D5911yescharacteristic
308C2×C4C22×C22⋊D5912yes—
309C8—2362no—
310C8—2364no—
311C22—22no—
312C22—28no—
313C22—416no—
314C22—44no—
315C22—44no—
316C22—44no—
317C22—48no—
318C22—88no—
319C22—832no—
320C22#1683758012yes—
321C22C2×#1427369611yescharacteristic
322C4—21no—
323C4—21no—
324C4—22no—
325C4—44no—
326C4—4722no—
327C4C2×#1427375314yes—
328C2—22no—
329C2—48no—
330C2—84no—
331C2#2143332312yes—
332C2C2×#1683758011yescharacteristic
subgroupG/H
centerC22C2×#14273696
commutator, cosocleC42×C59C24
FrattiniC42C23×D59
Fitting, prime radicalC2×D42×C59C2
socleC22×C59C2×#120
hypercenterC2×D42D59
hypocenterC59#18529
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×D42

Complements

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

NH ≅ G/Ntype
1. #21434470C2semidirect
2. #21434195C2semidirect
3. #21423275C2semidirect
4. #21422810C2semidirect
5. #21422539C2semidirect
7. #16837959C22semidirect
9. #16837748C22semidirect
11. #16837682C22semidirect
12. #16837607C22semidirect
15. #16837459C22semidirect
17. C4×#11745664C22semidirect
18. #16837146C22semidirect
19. D4⋊(C59⋊C8)C22semidirect
30. #14273727C23semidirect
36. C4⋊(C59⋊C8)C23semidirect
43. C4×C59⋊C8C23semidirect
194. C59#18529semidirect

Maximal subgroups

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

subgroupindex
#214344702
#214341952
#214232752
#214228102
#214225392
C2×D42×C592
#1852959