Finite Group Databasebeta

← #28486226

Subgroups — #28486226

Order 15104.

534 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 99 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#21434470C211yesmaximal, characteristic
2#21422810C211yesmaximal, characteristic
3#21422646C212yesmaximal, complemented
4#21421960C212yesmaximal, complemented
5C59×#2380C211yesmaximal, characteristic
6#16837959C2211yescharacteristic
7#16837766—22no—
8#16837748C414yescomplemented
9#16837459C2212yes—
10#16837407—22no—
11C4×#11745664C2211yescharacteristic
12#16837146—22no—
13#16837138—24no—
14C2×#127×C59C2211yescharacteristic
15C59×#386—22no—
16C59×#386—24no—
17C59×#383C2212yes—
18C2×#11745664—21no—
19#14273727C2×C412yescomplemented
20#14273719—21no—
21#14273706—22no—
22C4×Dic118—22no—
23#14273686—21no—
24C4⋊(C59⋊C8)—22no—
25#14273648—28no—
26C22⋊(C59⋊C8)—22no—
27#14273638—21no—
28#14273638—22no—
29C4×C59⋊C8C2×C412yescomplemented
30C22×D4×C59—22no—
31C22×D4×C59D412yes—
32C23×C4×C59—22no—
33#127×C59—22no—
34#127×C59C2×C412yes—
35#127×C59D412yes—
36C2×C22⋊C4×C59—24no—
37C2×C22⋊C4×C59—22no—
38C2×C42×C59C2311yescharacteristic
39#95×C59—22no—
40#95×C59—22no—
41#95×C59D414yes—
42C24×C59—22no—
43C2×D4×C59—24no—
44C2×D4×C59—24no—
45C2×D4×C59—22no—
46C2×D4×C59—42no—
47C2×D4×C59—42no—
48C2×D4×C59—82no—
49C2×D4×C59—82no—
50C2×D4×C59C22⋊C414yes—
51C22×C4×C59—24no—
52C22×C4×C59—84no—
53C22×C4×C59—82no—
54C22×C4×C59C2×D411yescharacteristic
55C22×C4×C59C2×D412yes—
56C2×Dic118—22no—
57C22⋊C4×C59—84no—
58C22⋊C4×C59—82no—
59C42×C59C22×C411yescharacteristic
60C42×C59C2×D411yescharacteristic
61C42×C59#4112yes—
62C22⋊Dic59—41no—
63C4⋊Dic59—22no—
64C4⋊Dic59—22no—
65Dic59⋊C4—42no—
66C4×Dic59—41no—
67#11745664—42no—
68C2×C59⋊C8—22no—
69C2×C59⋊C8—42no—
70C23×C59—24no—
71C23×C59—44no—
72C23×C59—44no—
73C23×C59—42no—
74C23×C59—42no—
75C23×C59#12011yescharacteristic
76D4×C59—44no—
77D4×C59—82no—
78D4×C59—84no—
79D4×C59—84no—
80C2×C4×C59—22no—
81C2×C4×C59—42no—
82C2×C4×C59—48no—
83C2×C4×C59—44no—
84C2×C4×C59—88no—
85C2×C4×C59C2×C22⋊C411yescharacteristic
86C2×C4×C59C4×D412yes—
87C2×C4×C59#12112yes—
88C2×C4×C59#12311yescharacteristic
89C2×Dic59—42no—
90Dic118—82no—
91C59⋊C8—42no—
92C59⋊C8—82no—
93#9425—591nomaximal, Sylow
94C22×C59—22no—
95C22×C59—44no—
96C22×C59—48no—
97C22×C59—82no—
98C22×C59—82no—
99C22×C59—84no—
100C22×C59#38611yescharacteristic
101C22×C59#45512yes—
102C4×C59—21no—
103C4×C59—21no—
104C4×C59—22no—
105C4×C59—22no—
106C4×C59—84no—
107Dic59—82no—
108#3138—591no—
109#2380D5911yescharacteristic
110#1555—591no—
111#1515—592no—
112#1288—592no—
113C2×C59—22no—
114C2×C59—84no—
115C2×C59#188511yescharacteristic
116C2×C59#190412yes—
117#533—591no—
118C2×#127C2×D5911yescharacteristic
119#490—1182no—
120#486—594no—
121#432—592no—
122#418—1182no—
123C4×M16—591no—
124#386—22no—
125#386—24no—
126#383C2×D5912yes—
127#331—1182no—
128#328—1184no—
129C59#942511yescharacteristic, complemented, Sylow
130C22×D4—22no—
131C22×D4C22⋊D5912yes—
132C23×C4—22no—
133C2×M16—1181no—
134C4⋊Q8—592no—
135#127—22no—
136#127C4×D5912yes—
137#127D23612yes—
138#125—1181no—
139C22⋊Q8—1182no—
140C4×Q8—1182no—
141#117—1181no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—22no—
144C2×C42C22×D5911yescharacteristic
145C4⋊C8—1182no—
146#102—1188no—
147C22⋊C8—1182no—
148#97—1181no—
149#97—1182no—
150C4×C8—592no—
151#95—22no—
152#95—22no—
153#95C22⋊D5914yes—
154C24—22no—
155C2×Q8—1182no—
156C2×D4—24no—
157C2×D4—24no—
158C2×D4—22no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—82no—
162C2×D4—82no—
163C2×D4(C2×D59)⋊C414yes—
164C22×C4—24no—
165C22×C4—84no—
166C22×C4—82no—
167C22×C4C2×C22⋊D5911yescharacteristic
168C22×C4C2×C22⋊D5912yes—
169M16—2362no—
170C2×C8—1182no—
171C2×C8—2362no—
172C4⋊C4—1182no—
173C4⋊C4—1182no—
174C4⋊C4—2362no—
175C22⋊C4—84no—
176C22⋊C4—82no—
177C22⋊C4—2361no—
178C42—2361no—
179C42C2×C4×D5911yescharacteristic
180C42C2×D23611yescharacteristic
181C42#1174568512yes—
182C23—24no—
183C23—44no—
184C23—44no—
185C23—42no—
186C23—42no—
187C23#1427369611yescharacteristic
188Q8—4722no—
189D4—44no—
190D4—82no—
191D4—84no—
192D4—84no—
193C2×C4—22no—
194C2×C4—42no—
195C2×C4—48no—
196C2×C4—44no—
197C2×C4—88no—
198C2×C4—2362no—
199C2×C4C2×(C2×D59)⋊C411yescharacteristic
200C2×C4C4×C22⋊D5912yes—
201C2×C4#1427370112yes—
202C2×C4#1427371011yescharacteristic
203C8—4722no—
204C8—2362no—
205C22—22no—
206C22—44no—
207C22—48no—
208C22—82no—
209C22—82no—
210C22—84no—
211C22#1683728111yescharacteristic
212C22#1683758012yes—
213C4—21no—
214C4—21no—
215C4—22no—
216C4—22no—
217C4—84no—
218C4—4722no—
219C2—22no—
220C2—84no—
221C2#2142448411yescharacteristic
222C2#2142458312yes—
subgroupG/H
centerC22#16837281
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2380C2
socleC22×C59#386
prime radicalC2×#127×C59C22
cosocleC2×C42×C59C23
hypercenter#2380D59
hypocenterC59#9425
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2380

Complements

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

NH ≅ G/Ntype
3. #21422646C2semidirect
4. #21421960C2semidirect
8. #16837748C4semidirect
19. #14273727C2×C4semidirect
29. C4×C59⋊C8C2×C4semidirect
129. C59#9425semidirect

Maximal subgroups

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

subgroupindex
#214344702
#214228102
#214226462
#214219602
C59×#23802
#942559