Finite Group Databasebeta

← #28564771

Subgroups — #28564771

Order 15104.

1068 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 179 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#21434546C211yesmaximal, characteristic
2#21434218C212yesmaximal, complemented
3#21423217C214yesmaximal, complemented
4#21423155C214yesmaximal, complemented
5#21422810C211yesmaximal, characteristic, complemented
6#21422543C212yesmaximal, complemented
7C2×C59×#559C211yesmaximal, characteristic, complemented
8C2×#14273753—22no—
9#16837952C2211yescharacteristic
10#16837789—22no—
11#16837778C2212yes—
12#16837774C2212yes—
13#16837715—22no—
14#16837697C2214yescomplemented
15#16837602C2214yescomplemented
16#16837541—22no—
17#16837500—24no—
18#16837459C2212yes—
19C4×#11745664C2211yescharacteristic
20#16837146C2214yes—
21Q8⋊(C59⋊C8)C2218yescomplemented
22C59×#559—22no—
23C59×#559—24no—
24C59×#559C2214yes—
25C2×#127×C59C2211yescharacteristic
26C2×#124×C59—22no—
27C2×#123×C59—22no—
28C2×C4×Q8×C59C2212yes—
29C2×C4×D4×C59—22no—
30C2×#11745685—21no—
31#14273753—48no—
32C2×#11745670—24no—
33C2×#11745671—24no—
34C2×#11745669—24no—
35C2×#11745664—21no—
36#14273727C2311yescharacteristic
37#14273724C2311yescharacteristic
38#14273715—21no—
39#14273701—22no—
40C4×D236—22no—
41#14273660—24no—
42C4⋊(C59⋊C8)C2314yes—
43#14273651—24no—
44#14273647—24no—
45C22⋊(C59⋊C8)—22no—
46#14273638—22no—
47#14273638—21no—
48C4×C59⋊C8C2312yes—
49C22×Q8×C59D412yes—
50C22×D4×C59—22no—
51C22×D4×C59D412yes—
52C23×C4×C59—22no—
53#127×C59—22no—
54#127×C59D412yes—
55#127×C59C2312yes—
56#124×C59—28no—
57#124×C59—28no—
58#123×C59—28no—
59#123×C59—28no—
60C4×Q8×C59—22no—
61C4×Q8×C59—22no—
62C4×Q8×C59D414yes—
63C4×Q8×C59C2314yes—
64C4×D4×C59—28no—
65C4×D4×C59—28no—
66C2×C4⋊C4×C59—22no—
67C2×C4⋊C4×C59D412yes—
68C2×C22⋊C4×C59—24no—
69C2×C22⋊C4×C59—22no—
70C2×C42×C59—22no—
71C2×C42×C59C2311yescharacteristic
72C24×C59—22no—
73C2×Q8×C59—22no—
74C2×Q8×C59—44no—
75C2×Q8×C59C2×D414yes—
76C2×D4×C59—24no—
77C2×D4×C59—22no—
78C2×D4×C59—24no—
79C2×D4×C59—48no—
80C2×D4×C59—44no—
81C2×D4×C59C2×D414yes—
82C22×C4×C59—22no—
83C22×C4×C59—22no—
84C22×C4×C59—28no—
85C22×C4×C59—84no—
86C22×C4×C59C2×D412yes—
87C22×C4×C59C2×D411yescharacteristic
88C2×C22⋊D59—41no—
89#11745685—44no—
90C2×D236—22no—
91C2×D236—21no—
92C2×C4×D59—41no—
93C2×Dic118—21no—
94C4⋊C4×C59—24no—
95C4⋊C4×C59—22no—
96C4⋊C4×C59—24no—
97C4⋊C4×C59C2×D414yes—
98C22⋊C4×C59—216no—
99C22⋊C4×C59—28no—
100C42×C59—24no—
101C42×C59—24no—
102C42×C59C2×D411yescharacteristic
103C42×C59C2×D412yes—
104C42×C59C2411yescharacteristic
105#11745671—48no—
106#11745670—48no—
107#11745669—48no—
108(C2×D59)⋊C4—42no—
109C4⋊Dic59—22no—
110#11745664—24no—
111C2×C59⋊C8—22no—
112C2×C59⋊C8—24no—
113C23×C59—24no—
114C23×C59—48no—
115C23×C59—44no—
116C23×C59#12011yescharacteristic
117C22×D59—41no—
118Q8×C59—24no—
119Q8×C59—24no—
120Q8×C59—44no—
121D4×C59—44no—
122D4×C59—48no—
123D4×C59—48no—
124D4×C59—82no—
125C2×C4×C59—22no—
126C2×C4×C59—24no—
127C2×C4×C59—24no—
128C2×C4×C59—24no—
129C2×C4×C59—44no—
130C2×C4×C59—44no—
131C2×C4×C59—88no—
132C2×C4×C59#12012yes—
133C2×C4×C59#12011yescharacteristic
134C2×C4×C59C22×D411yescharacteristic
135C2×C4×C59C22×D412yes—
136C22⋊D59—82no—
137C2×Dic59—41no—
138D236—42no—
139D236—44no—
140C4×D59—82no—
141Dic118—42no—
142C59⋊C8—24no—
143C59⋊C8—44no—
144#18526—591nomaximal, Sylow
145C22×C59—22no—
146C22×C59—48no—
147C22×C59—44no—
148C22×C59—88no—
149C22×C59#45312yes—
150C22×C59C2×#12011yescharacteristic
151C2×D59—82no—
152C2×D59—81no—
153C4×C59—22no—
154C4×C59—21no—
155C4×C59—21no—
156C4×C59—44no—
157C4×C59—44no—
158C4×C59#57514yes—
159Dic59—81no—
160C2×#559D5911yescharacteristic, complemented
161#3151—591no—
162#3112—592no—
163#1630—594no—
164#1620—594no—
165#1555—591no—
166#1491—592no—
167C2×C59—22no—
168C2×C59—84no—
169C2×C59#300512yes—
170C2×C59C2×#45311yescharacteristic
171D59—161no—
172C2×AGL(1,ℤ/8)—1182no—
173#559—22no—
174#559—24no—
175#559C2×D5914yes—
176#532—591no—
177C2×#127C2×D5911yescharacteristic
178C2×#124—22no—
179C2×#123—22no—
180C2×C4×Q8C2×D5912yes—
181C2×C4×D4—22no—
182#496—1182no—
183#493—592no—
184#492—592no—
185#480—1182no—
186#477—594no—
187#460—594no—
188#448—1182no—
189#440—1184no—
190#432—592no—
191C4×M16—591no—
192#331—594no—
193Q8⋊C8—598no—
194C59#1852611yescharacteristic, complemented, Sylow
195C2×#41—1181no—
196C22×Q8C22⋊D5912yes—
197C22×D4—22no—
198C22×D4C22⋊D5912yes—
199C23×C4—22no—
200AGL(1,ℤ/8)—2368no—
201C2×SD16—1184no—
202C2×SD16—1184no—
203C2×D8—1184no—
204C2×M16—1181no—
205C4⋊Q8—591no—
206#127—22no—
207#127—591no—
208#127C22⋊D5912yes—
209#127C22×D5912yes—
210#124—28no—
211#124—28no—
212#124—1181no—
213#123—28no—
214#123—28no—
215#121—1182no—
216C4×Q8—22no—
217C4×Q8—22no—
218C4×Q8C22⋊D5914yes—
219C4×Q8C22×D5914yes—
220C4×D4—28no—
221C4×D4—28no—
222C4×D4—1182no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4C22⋊D5912yes—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—22no—
227C2×C42—22no—
228C2×C42C22×D5911yescharacteristic
229#107—1184no—
230C4⋊C8—594no—
231#103—1184no—
232#102—1184no—
233C22⋊C8—1182no—
234#97—1181no—
235#97—1182no—
236C4×C8—592no—
237C24—22no—
238#41—2364no—
239C2×Q8—22no—
240C2×Q8—44no—
241C2×Q8—1181no—
242C2×Q8C2×C22⋊D5914yes—
243C2×D4—24no—
244C2×D4—22no—
245C2×D4—24no—
246C2×D4—48no—
247C2×D4—44no—
248C2×D4—1182no—
249C2×D4—1181no—
250C2×D4—2361no—
251C2×D4C2×C22⋊D5914yes—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—28no—
255C22×C4—84no—
256C22×C4—2361no—
257C22×C4C2×C22⋊D5912yes—
258C22×C4C2×C22⋊D5911yescharacteristic
259SD16—2368no—
260SD16—2368no—
261D8—2368no—
262M16—1184no—
263C2×C8—1182no—
264C2×C8—1184no—
265C4⋊C4—24no—
266C4⋊C4—22no—
267C4⋊C4—24no—
268C4⋊C4—1182no—
269C4⋊C4C2×C22⋊D5914yes—
270C22⋊C4—216no—
271C22⋊C4—28no—
272C22⋊C4—2362no—
273C42—24no—
274C42—24no—
275C42C2×C22⋊D5912yes—
276C42C2×C22⋊D5911yescharacteristic
277C42C23×D5911yescharacteristic
278C23—24no—
279C23—48no—
280C23—44no—
281C23—2361no—
282C23#1427369611yescharacteristic
283Q8—24no—
284Q8—24no—
285Q8—44no—
286Q8—2362no—
287D4—44no—
288D4—48no—
289D4—48no—
290D4—82no—
291D4—4722no—
292D4—2362no—
293D4—2364no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—88no—
301C2×C4—4722no—
302C2×C4—2361no—
303C2×C4#1427369612yes—
304C2×C4#1427369611yescharacteristic
305C2×C4C22×C22⋊D5911yescharacteristic
306C2×C4C22×C22⋊D5912yes—
307C8—1184no—
308C8—2364no—
309C22—22no—
310C22—44no—
311C22—48no—
312C22—88no—
313C22—4721no—
314C22—4722no—
315C22#1683757012yes—
316C22C2×#1427369611yescharacteristic
317C4—21no—
318C4—21no—
319C4—22no—
320C4—44no—
321C4—44no—
322C4—4721no—
323C4#1683820014yes—
324C2—22no—
325C2—84no—
326C2—9441no—
327C2#2143333812yes—
328C2C2×#1683757011yescharacteristic
subgroupG/H
centerC22C2×#14273696
commutator, cosocleC42×C59C24
FrattiniC42C23×D59
FittingC2×C59×#559C2
socleC22×C59C2×#120
prime radical#21434546C2
hypercenterC2×#559D59
hypocenterC59#18526
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#559

Complements

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

NH ≅ G/Ntype
2. #21434218C2semidirect
3. #21423217C2semidirect
4. #21423155C2semidirect
5. #21422810C2semidirect
6. #21422543C2semidirect
7. C2×C59×#559C2semidirect
14. #16837697C22semidirect
15. #16837602C22semidirect
21. Q8⋊(C59⋊C8)C22semidirect
160. C2×#559D59semidirect
194. C59#18526semidirect

Maximal subgroups

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

subgroupindex
#214345462
#214342182
#214232172
#214231552
#214228102
#214225432
C2×C59×#5592
#1852659