Finite Group Databasebeta

← #28588429

Subgroups — #28588429

Order 15104.

1602 conjugacy classes of subgroups; only the 275 normal subgroups are shown.

normal subgroups only

This group has 1602 conjugacy classes of subgroups — too many to draw. Only its 275 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×#16837875C211yesmaximal, characteristic
2#21426672C212yesmaximal
3#21424172C214yesmaximal, complemented
4#21424170C211yesmaximal, characteristic, complemented
5#21424168C212yesmaximal, complemented
6#342×D59C214yesmaximal, complemented
7C59×#1819C211yesmaximal, characteristic, complemented
8C23×C22⋊D59C2211yescharacteristic
9#16837875—28no—
10C2×C4×C22⋊D59C414yescomplemented
11C2×(C22⋊D59)⋊C4C414yescomplemented
12C2×#14273683C2212yes—
13C22×C22⋊Dic59C2211yescharacteristic
14C2×C22⋊C4×D59C2214yes—
15C2×C22⋊C4×D59C2212yes—
16#16837278—28no—
17#16837263C2214yes—
18#16837258C2212yes—
19#16837169C2214yes—
20#16837168C2218yes—
21C22×C22⋊C4×C59C2211yescharacteristic
22C59×#379C2212yes—
23C59×#342C2214yes—
24C24×D59C2311yescharacteristic
25C22×C22⋊D59—28no—
26C22×C22⋊D59C2×C412yes—
27C22×C22⋊D59C2×C412yes—
28C22×C22⋊D59D418yes—
29C23×Dic59C2×C412yescomplemented
30C23×Dic59C2311yescharacteristic
31C22×C4×D59—24no—
32C22×C4×D59C2×C414yes—
33C4×C22⋊D59—28no—
34C4×C22⋊D59—28no—
35(C22⋊D59)⋊C4—216no—
36#14273683—28no—
37C2×Dic59⋊C4C2×C414yescomplemented
38C2×C22⋊Dic59—24no—
39C2×C22⋊Dic59—24no—
40C2×C22⋊Dic59C2×C412yescomplemented
41C2×C22⋊Dic59C2314yes—
42C2×C22⋊Dic59C2312yes—
43C2×(C2×D59)⋊C4—24no—
44C2×(C2×D59)⋊C4C2×C414yes—
45C22⋊C4×D59—28no—
46C22⋊C4×D59—216no—
47C2×C4×Dic59C2×C414yescomplemented
48C59⋊#95—28no—
49C25×C59C2311yescharacteristic
50C23×C4×C59C2×C412yescomplemented
51C2×C22⋊C4×C59—24no—
52C2×C22⋊C4×C59—24no—
53C2×C22⋊C4×C59C2×C412yescomplemented
54C2×C22⋊C4×C59C2314yes—
55C2×C22⋊C4×C59C2312yes—
56C24×C59—24no—
57C24×C59—82no—
58C24×C59C22×C411yescharacteristic
59C24×C59C22×C411yescharacteristic
60C24×C59C2×D414yes—
61C24×C59C2411yescharacteristic
62C23×D59—82no—
63C23×D59C22×C412yes—
64C23×D59C2×D414yes—
65C23×D59#4118yes—
66C22×C4×C59—22no—
67C22×C4×C59—22no—
68C22×C4×C59—24no—
69C22×C4×C59—82no—
70C22×C4×C59C22×C414yes—
71C2×C22⋊D59—28no—
72C2×C22⋊D59—28no—
73C2×C22⋊D59—216no—
74C2×C22⋊D59—216no—
75C2×C22⋊D59C22⋊C418yes—
76C2×C22⋊D59C22⋊C418yes—
77C22×Dic59—24no—
78C22×Dic59—24no—
79C22×Dic59—24no—
80C22×Dic59—82no—
81C22×Dic59C22×C412yes—
82C22×Dic59C22×C414yes—
83C22×Dic59C2×D414yes—
84C2×C4×D59—28no—
85C2×C4×D59—48no—
86C2×C4×D59—88no—
87C2×C4×D59—84no—
88C22⋊C4×C59—24no—
89C22⋊C4×C59—24no—
90C22⋊C4×C59—28no—
91C22⋊C4×C59—84no—
92C22⋊Dic59—24no—
93C22⋊Dic59—24no—
94C22⋊Dic59—28no—
95C22⋊Dic59—84no—
96(C2×D59)⋊C4—28no—
97(C2×D59)⋊C4—48no—
98Dic59⋊C4—28no—
99C4×Dic59—28no—
100C23×C59—22no—
101C23×C59—22no—
102C23×C59—28no—
103C23×C59—42no—
104C23×C59—42no—
105C23×C59—42no—
106C23×C59—44no—
107C23×C59—44no—
108C23×C59—88no—
109C23×C59C2×C22⋊C412yes—
110C23×C59C2×C22⋊C412yes—
111C23×C59C23×C411yescharacteristic
112C23×C59C22×D412yes—
113C23×C59C2×#4114yes—
114C22×D59—44no—
115C22×D59—48no—
116C22×D59—88no—
117C22×D59C2×C22⋊C414yes—
118C22×D59#11718yes—
119C22×D59#12318yes—
120C22×D59#12318yes—
121C2×C4×C59—24no—
122C2×C4×C59—44no—
123C2×C4×C59—48no—
124C2×C4×C59—44no—
125C2×C4×C59—82no—
126C2×C4×C59—82no—
127C22⋊D59—216no—
128C22⋊D59—216no—
129C2×Dic59—24no—
130C2×Dic59—28no—
131C2×Dic59—24no—
132C2×Dic59—44no—
133C2×Dic59—48no—
134C2×Dic59—44no—
135C2×Dic59—84no—
136C2×Dic59C2×C22⋊C414yes—
137C4×D59—88no—
138C4×D59—816no—
139#20689—591nomaximal, Sylow
140C22×C59—22no—
141C22×C59—24no—
142C22×C59—24no—
143C22×C59—22no—
144C22×C59—22no—
145C22×C59—22no—
146C22×C59—22no—
147C22×C59—44no—
148C22×C59—44no—
149C22×C59—44no—
150C22×C59—88no—
151C22×C59C22×C22⋊C411yescharacteristic
152C22×C59C2×#11712yes—
153C22×C59C2×#12314yes—
154C2×D59—28no—
155C2×D59—48no—
156C2×D59—48no—
157C2×D59—48no—
158C2×D59#38118yes—
159C4×C59—44no—
160C4×C59—84no—
161Dic59—24no—
162Dic59—44no—
163Dic59—84no—
164C2×#518—591no—
165#2308—592no—
166#1820—594no—
167#1819—592no—
168#1819—591no—
169#1819D5911yescharacteristic, complemented
170C2×#342—594no—
171C2×C59—24no—
172C2×C59—22no—
173C2×C59—42no—
174C2×C59—42no—
175C2×C59C2×#38111yescharacteristic
176C2×C59#288312yes—
177D59—28no—
178C23×D4—591no—
179#518—1188no—
180C2×C4×D4—594no—
181C2×C4×D4—594no—
182C2×#117—592no—
183C22×C22⋊C4—591no—
184C22×C22⋊C4—592no—
185C22×C22⋊C4—594no—
186C22×C22⋊C4C2×D5911yescharacteristic
187#386—1188no—
188#381—594no—
189#379—592no—
190#379C2×D5912yes—
191#342—598no—
192#342—594no—
193#342C2×D5914yes—
194C59#2068911yescharacteristic, complemented, Sylow
195C25—591no—
196C25C22×D5911yescharacteristic
197C22×D4—592no—
198C22×D4—592no—
199C22×D4—598no—
200C22×D4—1188no—
201C23×C4—591no—
202C23×C4—592no—
203C23×C4—594no—
204C23×C4—1184no—
205C23×C4C4×D5912yescomplemented
206C4×D4—11816no—
207C4×D4—1188no—
208C4×D4—1188no—
209#117—1188no—
210C2×C4⋊C4—594no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—592no—
214C2×C22⋊C4—592no—
215C2×C22⋊C4—594no—
216C2×C22⋊C4—594no—
217C2×C22⋊C4—1188no—
218C2×C22⋊C4—11816no—
219C2×C22⋊C4—1184no—
220C2×C22⋊C4—1184no—
221C2×C22⋊C4—1184no—
222C2×C22⋊C4C4×D5912yescomplemented
223C2×C22⋊C4C22×D5914yes—
224C2×C22⋊C4C22×D5912yes—
225C2×C42—594no—
226#95—1188no—
227C24—24no—
228C24—82no—
229C24—598no—
230C24—592no—
231C24—594no—
232C24—4722no—
233C24C2×C4×D5911yescharacteristic
234C24C2×C4×D5911yescharacteristic
235C24D4×D5914yes—
236C24C23×D5911yescharacteristic
237C2×D4—598no—
238C2×D4—598no—
239C2×D4—11816no—
240C2×D4—11816no—
241C2×D4—1188no—
242C2×D4—1188no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—82no—
247C22×C4—592no—
248C22×C4—594no—
249C22×C4—594no—
250C22×C4—1188no—
251C22×C4—1184no—
252C22×C4—1184no—
253C22×C4—1184no—
254C22×C4—4728no—
255C22×C4—4724no—
256C22×C4—4722no—
257C22×C4—2368no—
258C22×C4C2×C4×D5914yes—
259C4⋊C4—1188no—
260C22⋊C4—24no—
261C22⋊C4—24no—
262C22⋊C4—28no—
263C22⋊C4—84no—
264C22⋊C4—1188no—
265C22⋊C4—1184no—
266C22⋊C4—1184no—
267C22⋊C4—1188no—
268C22⋊C4—4724no—
269C22⋊C4—2368no—
270C42—1188no—
271C23—22no—
272C23—22no—
273C23—28no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—44no—
278C23—44no—
279C23—88no—
280C23—594no—
281C23—598no—
282C23—598no—
283C23—598no—
284C23—4728no—
285C23—2364no—
286C23—2368no—
287C23C22⋊C4×D5912yes—
288C23C22⋊C4×D5912yes—
289C23C22×C4×D5911yescharacteristic
290C23C2×D4×D5912yes—
291C23#41×D5914yes—
292D4—11816no—
293D4—11816no—
294C2×C4—24no—
295C2×C4—44no—
296C2×C4—48no—
297C2×C4—44no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—594no—
301C2×C4—1188no—
302C2×C4—1184no—
303C2×C4—1184no—
304C2×C4—4728no—
305C2×C4—47216no—
306C2×C4—4724no—
307C2×C4—2368no—
308C2×C4—2364no—
309C2×C4—2364no—
310C22—22no—
311C22—24no—
312C22—22no—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—24no—
317C22—44no—
318C22—44no—
319C22—44no—
320C22—88no—
321C22—598no—
322C22—1188no—
323C22—2368no—
324C22—2368no—
325C22—2368no—
326C22C2×C22⋊C4×D5911yescharacteristic
327C22#117×D5912yes—
328C22#123×D5914yes—
329C4—44no—
330C4—84no—
331C4—1184no—
332C4—4724no—
333C4—2364no—
334C2—24no—
335C2—22no—
336C2—42no—
337C2—42no—
338C2—1188no—
339C2#381×D5911yescharacteristic
340C2#2143255412yes—
subgroupG/H
centerC22C2×C22⋊C4×D59
commutatorC23×C59C23×C4
FrattiniC24C23×D59
FittingC59×#1819C2
socleC22×C59C22×C22⋊C4
prime radicalC23×C22⋊D59C22
cosocleC24×C59C24
hypercenter#1819D59
hypocenterC59#20689
LCSG ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C25 ◁ #1819

Complements

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

NH ≅ G/Ntype
3. #21424172C2semidirect
4. #21424170C2semidirect
5. #21424168C2semidirect
6. #342×D59C2semidirect
7. C59×#1819C2semidirect
10. C2×C4×C22⋊D59C4semidirect
11. C2×(C22⋊D59)⋊C4C4semidirect
29. C23×Dic59C2×C4semidirect
37. C2×Dic59⋊C4C2×C4semidirect
40. C2×C22⋊Dic59C2×C4semidirect
47. C2×C4×Dic59C2×C4semidirect
50. C23×C4×C59C2×C4semidirect
53. C2×C22⋊C4×C59C2×C4semidirect
169. #1819D59semidirect
194. C59#20689semidirect
205. C23×C4C4×D59semidirect
222. C2×C22⋊C4C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168378752
#214266722
#214241722
#214241702
#214241682
#342×D592
C59×#18192
#2068959