Finite Group Databasebeta

← #28486414

Subgroups — #28486414

Order 15104.

512 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 107 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#21426699C211yesmaximal, characteristic
2#21424176C211yesmaximal, characteristic
3#21422629C212yesmaximal
4#21421949C212yesmaximal
5C59×#2503C211yesmaximal, characteristic
6#16837366C2212yes—
7#16837288—24no—
8#16837265C2212yes—
9#16837262—21no—
10C4×C22⋊Dic59—21no—
11#16837171—22no—
12#16837139—24no—
13C2×C22⋊Q8×C59C2211yescharacteristic
14C22×C4⋊C4×C59C2211yescharacteristic
15C59×#400—22no—
16C59×#391C412yescomplemented
17C59×#382C412yescomplemented
18C59×#381—21no—
19C59×#381—21no—
20C4×C22⋊C4×C59C2211yescharacteristic
21C2×#11745664—22no—
22C2×C4⋊Dic59—22no—
23C2×C22⋊Dic59—22no—
24C2×C4×Dic59—22no—
25C22⋊(C59⋊C8)—22no—
26C22⋊(C59⋊C8)—22no—
27C59⋊#95—22no—
28C59⋊#95—22no—
29C59⋊#95—22no—
30C59⋊#95D414yes—
31C22×Q8×C59C2×C411yescharacteristic
32C23×C4×C59D411yescharacteristic
33C23×C4×C59D411yescharacteristic
34C23×C4×C59C2311yescharacteristic
35C22⋊Q8×C59—22no—
36C22⋊Q8×C59—22no—
37C2×C4⋊C4×C59—22no—
38C2×C4⋊C4×C59—22no—
39C2×C4⋊C4×C59—21no—
40C2×C4⋊C4×C59C2×C411yescharacteristic
41C2×C4⋊C4×C59C2×C412yes—
42C2×C4⋊C4×C59D412yes—
43C2×C22⋊C4×C59—21no—
44C2×C22⋊C4×C59—21no—
45C2×C42×C59C2×C412yes—
46#95×C59—22no—
47#95×C59—22no—
48#95×C59—22no—
49#95×C59—22no—
50#95×C59—21no—
51C24×C59C2×D411yescharacteristic
52C2×Q8×C59—22no—
53C2×Q8×C59—42no—
54C22×C4×C59—21no—
55C22×C4×C59—21no—
56C22×C4×C59—21no—
57C22×C4×C59—21no—
58C22×C4×C59—22no—
59C22×C4×C59—21no—
60C22×C4×C59—21no—
61C22×C4×C59—42no—
62C22×C4×C59—42no—
63C22×C4×C59—42no—
64C22×C4×C59C22⋊C412yes—
65C22×C4×C59C22⋊C412yes—
66C22×C4×C59C22×C411yescharacteristic
67C22×C4×C59C2×D411yescharacteristic
68C22×C4×C59C2×D412yes—
69C22×C4×C59#4112yes—
70C22×Dic59—24no—
71C4⋊C4×C59—24no—
72C4⋊C4×C59—24no—
73C4⋊C4×C59—22no—
74C4⋊C4×C59—42no—
75C22⋊C4×C59—42no—
76C22⋊C4×C59—41no—
77C22⋊C4×C59—41no—
78C42×C59—42no—
79C22⋊Dic59—44no—
80C4⋊Dic59—82no—
81C4×Dic59—82no—
82#11745664—44no—
83C2×C59⋊C8—42no—
84C23×C59—22no—
85C23×C59—22no—
86C23×C59—41no—
87C23×C59C2×C22⋊C411yescharacteristic
88C23×C59#12011yescharacteristic
89C23×C59#12311yescharacteristic
90Q8×C59—82no—
91C2×C4×C59—22no—
92C2×C4×C59—22no—
93C2×C4×C59—22no—
94C2×C4×C59—22no—
95C2×C4×C59—42no—
96C2×C4×C59—42no—
97C2×C4×C59—42no—
98C2×C4×C59—42no—
99C2×C4×C59—42no—
100C2×C4×C59—42no—
101C2×C4×C59—42no—
102C2×C4×C59—42no—
103C2×C4×C59—42no—
104C2×C4×C59—41no—
105C2×C4×C59—41no—
106C2×C4×C59—42no—
107C2×C4×C59—42no—
108C2×C4×C59—81no—
109C2×C4×C59—81no—
110C2×C4×C59C4×D412yes—
111C2×C4×C59#12112yes—
112C2×Dic59—44no—
113C2×Dic59—44no—
114C2×Dic59—44no—
115C59⋊C8—82no—
116#9460—591nomaximal, Sylow
117C22×C59—21no—
118C22×C59—21no—
119C22×C59—21no—
120C22×C59—21no—
121C22×C59—21no—
122C22×C59—21no—
123C22×C59—21no—
124C22×C59—44no—
125C22×C59#38611yescharacteristic
126C22×C59#41012yes—
127C22×C59#45512yes—
128C4×C59—22no—
129C4×C59—22no—
130C4×C59—42no—
131C4×C59—42no—
132C4×C59—82no—
133C4×C59—81no—
134C4×C59—81no—
135Dic59—84no—
136#2503D5911yescharacteristic
137#2313—591no—
138#1821—591no—
139#1511—592no—
140#1284—592no—
141C2×C59—22no—
142C2×C59—22no—
143C2×C59—41no—
144C2×C59#188511yescharacteristic
145C2×C59#189911yescharacteristic
146C2×C59#191211yescharacteristic
147C2×C22⋊Q8C2×D5911yescharacteristic
148C22×C4⋊C4C2×D5911yescharacteristic
149#407—592no—
150#400—22no—
151#391Dic5912yescomplemented
152#387—1184no—
153#382Dic5912yescomplemented
154#381—21no—
155#381—21no—
156#381—592no—
157#380—1181no—
158C4×C22⋊C4—1181no—
159C4×C22⋊C4C2×D5911yescharacteristic
160#343—1182no—
161#328—1184no—
162C59#946011yescharacteristic, complemented, Sylow
163C22×Q8C2×Dic5911yescharacteristic
164C23×C4C22⋊D5911yescharacteristic
165C23×C4C22⋊D5911yescharacteristic
166C23×C4C22×D5911yescharacteristic
167C2×M16—1182no—
168C22⋊Q8—22no—
169C22⋊Q8—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—1182no—
174C2×C4⋊C4C2×Dic5911yescharacteristic
175C2×C4⋊C4C2×Dic5912yes—
176C2×C4⋊C4C22⋊D5912yes—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—1182no—
180C2×C42—1182no—
181C2×C42C2×Dic5912yes—
182C22⋊C8—1182no—
183C22⋊C8—1182no—
184#95—22no—
185#95—22no—
186#95—22no—
187#95—22no—
188#95—21no—
189#95—594no—
190#95—1182no—
191#95—1182no—
192#95—1182no—
193C24C2×C22⋊D5911yescharacteristic
194C2×Q8—22no—
195C2×Q8—42no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—1184no—
207C22×C4C22⋊Dic5912yes—
208C22×C4C22⋊Dic5912yes—
209C22×C4D4×D5912yes—
210C22×C4#1174568712yes—
211C22×C4C22×Dic5911yescharacteristic
212C22×C4C2×C22⋊D5911yescharacteristic
213M16—2364no—
214C2×C8—2362no—
215C4⋊C4—24no—
216C4⋊C4—24no—
217C4⋊C4—22no—
218C4⋊C4—42no—
219C4⋊C4—4722no—
220C22⋊C4—42no—
221C22⋊C4—41no—
222C22⋊C4—41no—
223C22⋊C4—2364no—
224C42—42no—
225C42—4722no—
226C23—22no—
227C23—22no—
228C23—41no—
229C23C2×C22⋊Dic5911yescharacteristic
230C23#1427369511yescharacteristic
231C23#1427370811yescharacteristic
232Q8—82no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—2364no—
253C2×C4—2364no—
254C2×C4—2364no—
255C2×C4D4×Dic5912yes—
256C2×C4#1427369912yes—
257C8—4722no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—44no—
266C22#1683728211yescharacteristic
267C22#1683737612yes—
268C22#1683757912yes—
269C4—22no—
270C4—22no—
271C4—42no—
272C4—42no—
273C4—82no—
274C4—81no—
275C4—81no—
276C4—4724no—
277C2—22no—
278C2—22no—
279C2—41no—
280C2#2142448511yescharacteristic
281C2#2142455511yescharacteristic
282C2#2142463411yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#2503C2
socleC22×C59#386
prime radicalC24×C59C2×D4
cosocleC23×C4×C59C23
hypercenter#2503D59
hypocenterC59#9460
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2503

Complements

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

NH ≅ G/Ntype
16. C59×#391C4semidirect
17. C59×#382C4semidirect
151. #391Dic59semidirect
153. #382Dic59semidirect
162. C59#9460semidirect

Maximal subgroups

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

subgroupindex
#214266992
#214241762
#214226292
#214219492
C59×#25032
#946059