Finite Group Databasebeta

← #28482226

Subgroups — #28482226

Order 15104.

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

normal subgroups only

This group has 394 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#21426699C211yesmaximal, characteristic
2#21422426C211yesmaximal, characteristic
3#21422003C212yesmaximal
4#21422001C212yesmaximal
5C59×#1463C211yesmaximal, characteristic
6#16837366C2212yes—
7#16837288—22no—
8#16837288—22no—
9#16837265C412yes—
10#16837262C2211yescharacteristic
11#16837262C2211yescharacteristic
12C4×C22⋊Dic59C412yes—
13#16837128—22no—
14C22×C4⋊C4×C59C2211yescharacteristic
15C59×#407C2212yes—
16C59×#324—22no—
17C2×#11745664—22no—
18C2×C4⋊Dic59—21no—
19C2×C4⋊Dic59—21no—
20C2×C22⋊Dic59C2×C412yes—
21C2×C22⋊Dic59C2×C412yes—
22C2×C4×Dic59—22no—
23C22⋊(C59⋊C8)—22no—
24C22⋊(C59⋊C8)—22no—
25C59⋊#95—22no—
26C59⋊#95—24no—
27C59⋊#95—21no—
28C59⋊#95—21no—
29C23×C4×C59C2×C411yescharacteristic
30C23×C4×C59C2×C411yescharacteristic
31C23×C4×C59C2311yescharacteristic
32C2×M16×C59—22no—
33C2×C4⋊C4×C59—22no—
34C2×C4⋊C4×C59—22no—
35C2×C4⋊C4×C59D412yes—
36C2×C4⋊C4×C59D412yes—
37C22⋊C8×C59—22no—
38C22⋊C8×C59—22no—
39C24×C59C22×C411yescharacteristic
40C22×C4×C59—21no—
41C22×C4×C59—21no—
42C22×C4×C59—21no—
43C22×C4×C59—21no—
44C22×C4×C59—41no—
45C22×C4×C59—41no—
46C22×C4×C59—81no—
47C22×C4×C59—81no—
48C22×C4×C59C22⋊C412yes—
49C22×C4×C59C22⋊C412yes—
50C22×C4×C59C2×D411yescharacteristic
51C22×C4×C59C2×D411yescharacteristic
52C22×C4×C59#4112yes—
53C22×C4×C59#4112yes—
54C22×Dic59—22no—
55C22×Dic59—22no—
56M16×C59—82no—
57C2×C8×C59—42no—
58C4⋊C4×C59—24no—
59C4⋊C4×C59—24no—
60C22⋊Dic59—42no—
61C22⋊Dic59—42no—
62C4⋊Dic59—81no—
63C4⋊Dic59—81no—
64C4×Dic59—82no—
65#11745664—82no—
66C2×C59⋊C8—42no—
67C23×C59—21no—
68C23×C59—21no—
69C23×C59—41no—
70C23×C59—41no—
71C23×C59C2×C22⋊C411yescharacteristic
72C23×C59#11711yescharacteristic
73C23×C59#11711yescharacteristic
74C2×C4×C59—22no—
75C2×C4×C59—22no—
76C2×C4×C59—22no—
77C2×C4×C59—22no—
78C2×C4×C59—42no—
79C2×C4×C59—41no—
80C2×C4×C59—41no—
81C2×C4×C59—42no—
82C2×C4×C59—42no—
83C2×C4×C59—42no—
84C2×C4×C59—42no—
85C2×C4×C59—81no—
86C2×C4×C59—81no—
87C2×C4×C59#12312yes—
88C2×C4×C59#12312yes—
89C2×Dic59—42no—
90C2×Dic59—42no—
91C2×Dic59—42no—
92C2×Dic59—42no—
93C2×Dic59—42no—
94C2×Dic59—42no—
95C8×C59—82no—
96C59⋊C8—82no—
97#8758—591nomaximal, Sylow
98C22×C59—21no—
99C22×C59—21no—
100C22×C59—21no—
101C22×C59—21no—
102C22×C59—21no—
103C22×C59—21no—
104C22×C59—21no—
105C22×C59—21no—
106C22×C59—42no—
107C22×C59—42no—
108C22×C59#38111yescharacteristic
109C22×C59#41311yescharacteristic
110C22×C59#41311yescharacteristic
111C4×C59—22no—
112C4×C59—22no—
113C4×C59—42no—
114C4×C59—42no—
115Dic59—82no—
116Dic59—82no—
117#2313—591no—
118#1463—591no—
119#1463D5911yescharacteristic
120#1305—592no—
121#1305—592no—
122C2×C59—21no—
123C2×C59—21no—
124C2×C59—41no—
125C2×C59—41no—
126C2×C59#182211yescharacteristic
127C2×C59#182811yescharacteristic
128C2×C59#183011yescharacteristic
129C22×C4⋊C4C2×D5911yescharacteristic
130#407—592no—
131#407C2×D5912yes—
132#387—1182no—
133#387—1182no—
134#381—592no—
135#380—591no—
136#380—591no—
137C4×C22⋊C4—592no—
138#324—22no—
139#324—1182no—
140C59#875811yescharacteristic, complemented, Sylow
141C23×C4C4×D5911yescharacteristic
142C23×C4C4×D5911yescharacteristic
143C23×C4C22×D5911yescharacteristic
144C2×M16—22no—
145C2×M16—1182no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—1181no—
149C2×C4⋊C4—1181no—
150C2×C4⋊C4D23612yes—
151C2×C4⋊C4C22⋊D5912yes—
152C2×C22⋊C4—592no—
153C2×C22⋊C4—592no—
154C2×C42—1182no—
155C22⋊C8—22no—
156C22⋊C8—22no—
157C22⋊C8—1182no—
158C22⋊C8—1182no—
159#95—1182no—
160#95—1184no—
161#95—1181no—
162#95—1181no—
163C24C2×C4×D5911yescharacteristic
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—81no—
171C22×C4—81no—
172C22×C4—1182no—
173C22×C4—1182no—
174C22×C4(C2×D59)⋊C412yes—
175C22×C4(C2×D59)⋊C412yes—
176C22×C4C2×D23611yescharacteristic
177C22×C4#1174568712yes—
178C22×C4#1174568712yes—
179C22×C4C2×C22⋊D5911yescharacteristic
180M16—82no—
181M16—4722no—
182C2×C8—42no—
183C2×C8—2362no—
184C4⋊C4—24no—
185C4⋊C4—24no—
186C4⋊C4—4721no—
187C4⋊C4—4721no—
188C22⋊C4—2362no—
189C22⋊C4—2362no—
190C42—4722no—
191C23—21no—
192C23—21no—
193C23—41no—
194C23—41no—
195C23C2×(C2×D59)⋊C411yescharacteristic
196C23#1427368311yescharacteristic
197C23#1427368311yescharacteristic
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—2362no—
212C2×C4—2362no—
213C2×C4—2362no—
214C2×C4—2362no—
215C2×C4—2362no—
216C2×C4—2362no—
217C2×C4#1427370812yes—
218C2×C4#1427371112yes—
219C8—82no—
220C8—4722no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—42no—
230C22—42no—
231C22#1683726311yescharacteristic
232C22#1683738411yescharacteristic
233C22#1683738411yescharacteristic
234C4—22no—
235C4—22no—
236C4—42no—
237C4—42no—
238C4—4722no—
239C4—4722no—
240C2—21no—
241C2—21no—
242C2—41no—
243C2—41no—
244C2#2142417711yescharacteristic
245C2#2142420211yescharacteristic
246C2#2142420811yescharacteristic
subgroupG/H
centerC22#16837263
commutator, prime radicalC24×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1463C2
socleC22×C59#381
cosocleC23×C4×C59C23
hypercenter#1463D59
hypocenterC59#8758
LCSG ▷ C24×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
140. C59#8758semidirect

Maximal subgroups

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

subgroupindex
#214266992
#214224262
#214220032
#214220012
C59×#14632
#875859