Finite Group Databasebeta

← #28482198

Subgroups — #28482198

Order 15104.

538 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 91 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#21426694C211yesmaximal, characteristic
2#21424170C211yesmaximal, characteristic
3C59⋊#1304C212yesmaximal
4C59⋊#1304C212yesmaximal
5C59×#1819C211yesmaximal, characteristic
6#16837295—22no—
7#16837295—22no—
8#16837262C412yescomplemented
9#16837258C2212yes—
10#16837258C2211yescharacteristic
11#16837258C2211yescharacteristic
12C4×C22⋊Dic59C412yescomplemented
13#16837169—22no—
14C22×C22⋊C4×C59C2211yescharacteristic
15C59×#379C2212yes—
16C59×#342—22no—
17C2×C4⋊Dic59—22no—
18C2×C22⋊Dic59—22no—
19C2×C22⋊Dic59—21no—
20C2×C22⋊Dic59—21no—
21C2×C22⋊Dic59—21no—
22C2×C22⋊Dic59—21no—
23C2×C22⋊Dic59—22no—
24C2×C22⋊Dic59—22no—
25C2×C22⋊Dic59C2×C412yes—
26C2×C22⋊Dic59C2×C412yes—
27C2×C4×Dic59—22no—
28C59⋊#95—22no—
29C59⋊#95—22no—
30C25×C59C2311yescharacteristic
31C23×C4×C59C2×C411yescharacteristic
32C23×C4×C59C2×C411yescharacteristic
33C2×C22⋊C4×C59—22no—
34C2×C22⋊C4×C59—22no—
35C2×C22⋊C4×C59—22no—
36C2×C22⋊C4×C59—22no—
37C2×C22⋊C4×C59—22no—
38C2×C22⋊C4×C59D412yes—
39C2×C22⋊C4×C59D412yes—
40C24×C59—42no—
41C24×C59—81no—
42C24×C59—81no—
43C24×C59C22×C411yescharacteristic
44C24×C59C2×D411yescharacteristic
45C24×C59C2×D411yescharacteristic
46C24×C59#4112yes—
47C24×C59#4112yes—
48C22×C4×C59—21no—
49C22×C4×C59—21no—
50C22×C4×C59—21no—
51C22×C4×C59—21no—
52C22×C4×C59—42no—
53C22×C4×C59—81no—
54C22×C4×C59—81no—
55C22×C4×C59C22⋊C412yes—
56C22×C4×C59C22⋊C412yes—
57C22×Dic59—22no—
58C22×Dic59—22no—
59C22×Dic59—42no—
60C22⋊C4×C59—24no—
61C22⋊C4×C59—24no—
62C22⋊C4×C59—44no—
63C22⋊C4×C59—82no—
64C22⋊C4×C59—82no—
65C22⋊Dic59—42no—
66C22⋊Dic59—42no—
67C22⋊Dic59—44no—
68C22⋊Dic59—81no—
69C22⋊Dic59—81no—
70C22⋊Dic59—81no—
71C22⋊Dic59—81no—
72C22⋊Dic59—82no—
73C22⋊Dic59—82no—
74C4⋊Dic59—82no—
75C4×Dic59—82no—
76C23×C59—22no—
77C23×C59—22no—
78C23×C59—22no—
79C23×C59—41no—
80C23×C59—41no—
81C23×C59—41no—
82C23×C59—41no—
83C23×C59—41no—
84C23×C59—41no—
85C23×C59—42no—
86C23×C59—42no—
87C23×C59—42no—
88C23×C59—42no—
89C23×C59—42no—
90C23×C59—42no—
91C23×C59—88no—
92C23×C59C2×C22⋊C411yescharacteristic
93C23×C59#11711yescharacteristic
94C23×C59#11711yescharacteristic
95C23×C59#12312yes—
96C23×C59#12312yes—
97C2×C4×C59—22no—
98C2×C4×C59—22no—
99C2×C4×C59—42no—
100C2×C4×C59—42no—
101C2×C4×C59—81no—
102C2×C4×C59—81no—
103C2×C4×C59—81no—
104C2×C4×C59—81no—
105C2×C4×C59—82no—
106C2×C4×C59—82no—
107C2×C4×C59—82no—
108C2×Dic59—42no—
109C2×Dic59—42no—
110C2×Dic59—42no—
111C2×Dic59—42no—
112C2×Dic59—42no—
113C2×Dic59—42no—
114C2×Dic59—82no—
115C2×Dic59—82no—
116C2×Dic59—82no—
117#8750—591nomaximal, Sylow
118C22×C59—21no—
119C22×C59—21no—
120C22×C59—22no—
121C22×C59—22no—
122C22×C59—21no—
123C22×C59—21no—
124C22×C59—22no—
125C22×C59—22no—
126C22×C59—21no—
127C22×C59—21no—
128C22×C59—21no—
129C22×C59—21no—
130C22×C59—22no—
131C22×C59—22no—
132C22×C59—44no—
133C22×C59—82no—
134C22×C59—82no—
135C22×C59—84no—
136C22×C59—84no—
137C22×C59#38111yescharacteristic
138C22×C59#41012yes—
139C4×C59—42no—
140C4×C59—42no—
141C4×C59—162no—
142Dic59—82no—
143Dic59—82no—
144Dic59—162no—
145#2312—591no—
146#1819—591no—
147#1819D5911yescharacteristic
148#1304—592no—
149#1304—592no—
150C2×C59—22no—
151C2×C59—42no—
152C2×C59—42no—
153C2×C59—41no—
154C2×C59—41no—
155C2×C59#182011yescharacteristic
156C2×C59#182811yescharacteristic
157C2×C59#182811yescharacteristic
158C22×C22⋊C4C2×D5911yescharacteristic
159#388—1182no—
160#388—1182no—
161#380—592no—
162#379—591no—
163#379—591no—
164#379—592no—
165#379C2×D5912yes—
166C4×C22⋊C4—592no—
167#342—22no—
168#342—1182no—
169C59#875011yescharacteristic, complemented, Sylow
170C25C22×D5911yescharacteristic
171C23×C4C4×D5911yescharacteristic
172C23×C4C4×D5911yescharacteristic
173C2×C4⋊C4—1182no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—592no—
180C2×C22⋊C4—592no—
181C2×C22⋊C4—1181no—
182C2×C22⋊C4—1181no—
183C2×C22⋊C4—1181no—
184C2×C22⋊C4—1181no—
185C2×C22⋊C4—1182no—
186C2×C22⋊C4—1182no—
187C2×C22⋊C4—1182no—
188C2×C22⋊C4D23612yes—
189C2×C22⋊C4C22⋊D5912yes—
190C2×C42—1182no—
191#95—1182no—
192#95—1182no—
193C24—42no—
194C24—81no—
195C24—81no—
196C24C2×C4×D5911yescharacteristic
197C24C2×D23611yescharacteristic
198C24#1174568712yes—
199C24#1174568712yes—
200C24C2×C22⋊D5911yescharacteristic
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—42no—
206C22×C4—81no—
207C22×C4—81no—
208C22×C4—1182no—
209C22×C4—1182no—
210C22×C4—2362no—
211C22×C4(C2×D59)⋊C412yes—
212C22×C4(C2×D59)⋊C412yes—
213C4⋊C4—4722no—
214C22⋊C4—24no—
215C22⋊C4—24no—
216C22⋊C4—44no—
217C22⋊C4—82no—
218C22⋊C4—82no—
219C22⋊C4—4722no—
220C22⋊C4—4722no—
221C22⋊C4—4721no—
222C22⋊C4—4721no—
223C22⋊C4—4721no—
224C22⋊C4—4721no—
225C22⋊C4—2364no—
226C22⋊C4—2362no—
227C22⋊C4—2362no—
228C42—4722no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—41no—
233C23—41no—
234C23—42no—
235C23—42no—
236C23—41no—
237C23—41no—
238C23—42no—
239C23—42no—
240C23—41no—
241C23—41no—
242C23—42no—
243C23—42no—
244C23—88no—
245C23C2×(C2×D59)⋊C411yescharacteristic
246C23#1427368311yescharacteristic
247C23#1427368311yescharacteristic
248C23#1427370812yes—
249C23#1427371112yes—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—4722no—
262C2×C4—4722no—
263C2×C4—4722no—
264C2×C4—2362no—
265C2×C4—2362no—
266C2×C4—2362no—
267C2×C4—2362no—
268C2×C4—2362no—
269C2×C4—2362no—
270C22—21no—
271C22—21no—
272C22—22no—
273C22—22no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—22no—
281C22—22no—
282C22—22no—
283C22—22no—
284C22—44no—
285C22—82no—
286C22—82no—
287C22—84no—
288C22—84no—
289C22#1683726311yescharacteristic
290C22#1683737512yes—
291C4—42no—
292C4—42no—
293C4—162no—
294C4—9442no—
295C4—4722no—
296C4—4722no—
297C2—22no—
298C2—42no—
299C2—42no—
300C2—41no—
301C2—41no—
302C2#2142417111yescharacteristic
303C2#2142420211yescharacteristic
304C2#2142420211yescharacteristic
subgroupG/H
centerC22#16837263
commutatorC24×C59C22×C4
FrattiniC25C22×D59
FittingC59×#1819C2
socleC22×C59#381
prime radical, cosocleC25×C59C23
hypercenter#1819D59
hypocenterC59#8750
LCSG ▷ C24×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C25 ◁ #1819

Complements

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

NH ≅ G/Ntype
8. #16837262C4semidirect
12. C4×C22⋊Dic59C4semidirect
169. C59#8750semidirect

Maximal subgroups

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

subgroupindex
#214266942
#214241702
C59⋊#13042
C59×#18192
#875059