Finite Group Databasebeta

← #28482229

Subgroups — #28482229

Order 15104.

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

normal subgroups only

This group has 474 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#21422423C211yesmaximal, characteristic
3#21422003C212yesmaximal, complemented
4#21422001C212yesmaximal, complemented
5C59×#1462C211yesmaximal, characteristic
6#16837366C2212yes—
7#16837295—22no—
8#16837295—22no—
9#16837262C2211yescharacteristic
10#16837262C2211yescharacteristic
11#16837258C412yes—
12C4×C22⋊Dic59C412yes—
13#16837126—22no—
14C22×C22⋊C4×C59C2211yescharacteristic
15C59×#407C2212yes—
16C59×#323—22no—
17C2×#11745664—22no—
18C2×C4⋊Dic59—21no—
19C2×C4⋊Dic59—21no—
20C2×C22⋊Dic59—22no—
21C2×C22⋊Dic59—22no—
22C2×C22⋊Dic59C2×C412yes—
23C2×C22⋊Dic59C2×C412yes—
24C2×C4×Dic59—22no—
25C22⋊(C59⋊C8)—22no—
26C22⋊(C59⋊C8)—22no—
27C59⋊#95—22no—
28C59⋊#95—21no—
29C59⋊#95—21no—
30C25×C59C2×C411yescharacteristic
31C23×C4×C59C2×C411yescharacteristic
32C23×C4×C59C2311yescharacteristic
33C2×M16×C59—22no—
34C2×C22⋊C4×C59—22no—
35C2×C22⋊C4×C59—22no—
36C2×C22⋊C4×C59D412yes—
37C2×C22⋊C4×C59D412yes—
38C22⋊C8×C59—22no—
39C22⋊C8×C59—22no—
40C24×C59—21no—
41C24×C59—21no—
42C24×C59—41no—
43C24×C59—41no—
44C24×C59—81no—
45C24×C59—81no—
46C24×C59C22⋊C412yes—
47C24×C59C22×C411yescharacteristic
48C22×C4×C59—21no—
49C22×C4×C59—21no—
50C22×C4×C59—81no—
51C22×C4×C59—81no—
52C22×C4×C59C22⋊C412yes—
53C22×C4×C59C2×D411yescharacteristic
54C22×C4×C59C2×D411yescharacteristic
55C22×C4×C59#4112yes—
56C22×C4×C59#4112yes—
57C22×Dic59—22no—
58C22×Dic59—22no—
59M16×C59—82no—
60C2×C8×C59—42no—
61C22⋊C4×C59—24no—
62C22⋊C4×C59—24no—
63C22⋊Dic59—42no—
64C22⋊Dic59—42no—
65C22⋊Dic59—82no—
66C22⋊Dic59—82no—
67C4⋊Dic59—81no—
68C4⋊Dic59—81no—
69C4×Dic59—82no—
70#11745664—82no—
71C2×C59⋊C8—42no—
72C23×C59—21no—
73C23×C59—21no—
74C23×C59—22no—
75C23×C59—22no—
76C23×C59—22no—
77C23×C59—41no—
78C23×C59—41no—
79C23×C59—42no—
80C23×C59—42no—
81C23×C59—42no—
82C23×C59—42no—
83C23×C59—81no—
84C23×C59—81no—
85C23×C59—81no—
86C23×C59—81no—
87C23×C59—84no—
88C23×C59—84no—
89C23×C59C2×C22⋊C411yescharacteristic
90C23×C59#11711yescharacteristic
91C23×C59#11711yescharacteristic
92C2×C4×C59—22no—
93C2×C4×C59—42no—
94C2×C4×C59—41no—
95C2×C4×C59—41no—
96C2×C4×C59—42no—
97C2×C4×C59—42no—
98C2×C4×C59—81no—
99C2×C4×C59—81no—
100C2×C4×C59#12312yes—
101C2×C4×C59#12312yes—
102C2×Dic59—42no—
103C2×Dic59—42no—
104C2×Dic59—42no—
105C2×Dic59—42no—
106C2×Dic59—42no—
107C2×Dic59—42no—
108C8×C59—82no—
109C59⋊C8—82no—
110#8759—591nomaximal, Sylow
111C22×C59—21no—
112C22×C59—21no—
113C22×C59—21no—
114C22×C59—21no—
115C22×C59—21no—
116C22×C59—21no—
117C22×C59—21no—
118C22×C59—21no—
119C22×C59—42no—
120C22×C59—42no—
121C22×C59—42no—
122C22×C59—42no—
123C22×C59—42no—
124C22×C59—82no—
125C22×C59—82no—
126C22×C59—84no—
127C22×C59—84no—
128C22×C59#38111yescharacteristic
129C22×C59#41311yescharacteristic
130C22×C59#41311yescharacteristic
131C4×C59—22no—
132C4×C59—22no—
133C4×C59—42no—
134Dic59—82no—
135Dic59—82no—
136#2312—591no—
137#1462—591no—
138#1462D5911yescharacteristic
139#1305—592no—
140#1305—592no—
141C2×C59—21no—
142C2×C59—21no—
143C2×C59—41no—
144C2×C59—41no—
145C2×C59—82no—
146C2×C59#182211yescharacteristic
147C2×C59#182811yescharacteristic
148C2×C59#182911yescharacteristic
149C22×C22⋊C4C2×D5911yescharacteristic
150#407—592no—
151#407C2×D5912yes—
152#388—1182no—
153#388—1182no—
154#380—591no—
155#380—591no—
156#379—592no—
157C4×C22⋊C4—592no—
158#323—22no—
159#323—1182no—
160C59#875911yescharacteristic, complemented, Sylow
161C25C4×D5911yescharacteristic
162C23×C4C4×D5911yescharacteristic
163C23×C4C22×D5911yescharacteristic
164C2×M16—22no—
165C2×M16—1182no—
166C2×C4⋊C4—1181no—
167C2×C4⋊C4—1181no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—592no—
171C2×C22⋊C4—592no—
172C2×C22⋊C4—1182no—
173C2×C22⋊C4—1182no—
174C2×C22⋊C4D23612yes—
175C2×C22⋊C4C22⋊D5912yes—
176C2×C42—1182no—
177C22⋊C8—22no—
178C22⋊C8—22no—
179C22⋊C8—1182no—
180C22⋊C8—1182no—
181#95—1182no—
182#95—1181no—
183#95—1181no—
184C24—21no—
185C24—21no—
186C24—41no—
187C24—41no—
188C24—81no—
189C24—81no—
190C24(C2×D59)⋊C412yes—
191C24C2×C4×D5911yescharacteristic
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—81no—
195C22×C4—81no—
196C22×C4—1182no—
197C22×C4—1182no—
198C22×C4(C2×D59)⋊C412yes—
199C22×C4C2×D23611yescharacteristic
200C22×C4#1174568712yes—
201C22×C4#1174568712yes—
202C22×C4C2×C22⋊D5911yescharacteristic
203M16—82no—
204M16—4722no—
205C2×C8—42no—
206C2×C8—2362no—
207C4⋊C4—4721no—
208C4⋊C4—4721no—
209C22⋊C4—24no—
210C22⋊C4—24no—
211C22⋊C4—4722no—
212C22⋊C4—4722no—
213C22⋊C4—2362no—
214C22⋊C4—2362no—
215C42—4722no—
216C23—21no—
217C23—21no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—41no—
222C23—41no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23—81no—
228C23—81no—
229C23—81no—
230C23—81no—
231C23—84no—
232C23—84no—
233C23C2×(C2×D59)⋊C411yescharacteristic
234C23#1427368311yescharacteristic
235C23#1427368311yescharacteristic
236C2×C4—22no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—2362no—
245C2×C4—2362no—
246C2×C4—2362no—
247C2×C4—2362no—
248C2×C4—2362no—
249C2×C4—2362no—
250C2×C4#1427370812yes—
251C2×C4#1427371112yes—
252C8—82no—
253C8—4722no—
254C22—21no—
255C22—21no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—42no—
267C22—82no—
268C22—82no—
269C22—84no—
270C22—84no—
271C22#1683726311yescharacteristic
272C22#1683738411yescharacteristic
273C22#1683738411yescharacteristic
274C4—22no—
275C4—22no—
276C4—42no—
277C4—4722no—
278C4—4722no—
279C2—21no—
280C2—21no—
281C2—41no—
282C2—41no—
283C2—82no—
284C2#2142417711yescharacteristic
285C2#2142420211yescharacteristic
286C2#2142420511yescharacteristic
subgroupG/H
centerC22#16837263
commutatorC24×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1462C2
socleC22×C59#381
prime radicalC25×C59C2×C4
cosocleC23×C4×C59C23
hypercenter#1462D59
hypocenterC59#8759
LCSG ▷ C24×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1462

Complements

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

NH ≅ G/Ntype
3. #21422003C2semidirect
4. #21422001C2semidirect
160. C59#8759semidirect

Maximal subgroups

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

subgroupindex
#214266942
#214224232
#214220032
#214220012
C59×#14622
#875959