Finite Group Databasebeta

← #28486243

Subgroups — #28486243

Order 15104.

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

normal subgroups only

This group has 620 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#21434546C211yesmaximal, characteristic
2#21422810C211yesmaximal, characteristic, complemented
3#21422642C212yesmaximal, complemented
4#21421960C212yesmaximal, complemented
5C59×#2380C211yesmaximal, characteristic, complemented
6C2×#14273753—22no—
7#16837952C2211yescharacteristic
8#16837789—22no—
9#16837778C412yescomplemented
10#16837774C412yescomplemented
11#16837459C2212yes—
12C4×#11745664C2211yescharacteristic
13#16837146—22no—
14#16837135—24no—
15C2×#127×C59C2211yescharacteristic
16C59×#386—22no—
17C59×#386—24no—
18C59×#383C2212yes—
19C2×#11745685—21no—
20#14273753—44no—
21#14273753—44no—
22C2×#11745670—24no—
23C2×#11745669—24no—
24C2×#11745664—21no—
25#14273727C2×C411yescharacteristic, complemented
26#14273724C2×C411yescharacteristic, complemented
27#14273715—21no—
28#14273701—22no—
29C4×D236—22no—
30C4⋊(C59⋊C8)—22no—
31C22⋊(C59⋊C8)—22no—
32#14273638—21no—
33#14273638—22no—
34C4×C59⋊C8C2×C412yescomplemented
35C22×D4×C59—22no—
36C22×D4×C59D412yes—
37C23×C4×C59—22no—
38#127×C59—22no—
39#127×C59C2×C412yes—
40#127×C59D412yes—
41C2×C22⋊C4×C59—24no—
42C2×C22⋊C4×C59—22no—
43C2×C42×C59C2311yescharacteristic
44#95×C59—22no—
45#95×C59—22no—
46#95×C59D414yes—
47C24×C59—22no—
48C2×D4×C59—24no—
49C2×D4×C59—24no—
50C2×D4×C59—22no—
51C2×D4×C59—42no—
52C2×D4×C59—42no—
53C2×D4×C59—82no—
54C2×D4×C59—82no—
55C2×D4×C59C22⋊C414yes—
56C22×C4×C59—24no—
57C22×C4×C59—84no—
58C22×C4×C59—82no—
59C22×C4×C59C2×D411yescharacteristic
60C22×C4×C59C2×D412yes—
61C2×C22⋊D59—41no—
62#11745685—42no—
63#11745685—42no—
64C2×D236—21no—
65C2×D236—22no—
66C2×C4×D59—41no—
67C2×Dic118—21no—
68C22⋊C4×C59—84no—
69C22⋊C4×C59—82no—
70C42×C59C22×C411yescharacteristic
71C42×C59C2×D411yescharacteristic
72C42×C59#4112yes—
73#11745670—48no—
74#11745669—48no—
75(C2×D59)⋊C4—42no—
76C4⋊Dic59—22no—
77#11745664—24no—
78C2×C59⋊C8—22no—
79C2×C59⋊C8—42no—
80C23×C59—24no—
81C23×C59—44no—
82C23×C59—44no—
83C23×C59—42no—
84C23×C59—42no—
85C23×C59#12011yescharacteristic
86C22×D59—41no—
87D4×C59—44no—
88D4×C59—82no—
89D4×C59—84no—
90D4×C59—84no—
91C2×C4×C59—22no—
92C2×C4×C59—42no—
93C2×C4×C59—48no—
94C2×C4×C59—44no—
95C2×C4×C59—88no—
96C2×C4×C59C2×C22⋊C411yescharacteristic
97C2×C4×C59C4×D412yes—
98C2×C4×C59#12112yes—
99C2×C4×C59#12311yescharacteristic
100C22⋊D59—82no—
101C2×Dic59—41no—
102D236—42no—
103D236—82no—
104C4×D59—81no—
105C4×D59—81no—
106Dic118—42no—
107C59⋊C8—24no—
108C59⋊C8—82no—
109#9428—591nomaximal, Sylow
110C22×C59—22no—
111C22×C59—44no—
112C22×C59—48no—
113C22×C59—82no—
114C22×C59—82no—
115C22×C59—84no—
116C22×C59#38611yescharacteristic
117C22×C59#45312yes—
118C2×D59—81no—
119C2×D59—82no—
120C4×C59—21no—
121C4×C59—21no—
122C4×C59—22no—
123C4×C59—22no—
124C4×C59—84no—
125Dic59—81no—
126#3151—591no—
127#2380D5911yescharacteristic, complemented
128#1555—591no—
129#1514—592no—
130#1288—592no—
131C2×C59—22no—
132C2×C59—84no—
133C2×C59#188711yescharacteristic
134C2×C59#190412yes—
135D59—161no—
136C2×AGL(1,ℤ/8)—1182no—
137#532—591no—
138C2×#127C2×D5911yescharacteristic
139#496—1182no—
140#493—592no—
141#492—592no—
142#432—592no—
143C4×M16—591no—
144#386—22no—
145#386—24no—
146#383C2×D5912yes—
147#331—1182no—
148#327—1184no—
149C59#942811yescharacteristic, complemented, Sylow
150C2×#41—1181no—
151C22×D4—22no—
152C22×D4C22⋊D5912yes—
153C23×C4—22no—
154AGL(1,ℤ/8)—2364no—
155AGL(1,ℤ/8)—2364no—
156C2×SD16—1184no—
157C2×D8—1184no—
158C2×M16—1181no—
159C4⋊Q8—591no—
160#127—22no—
161#127—591no—
162#127C4×D5912yes—
163#127D23612yes—
164#124—1181no—
165#121—1182no—
166C4×D4—1182no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—22no—
169C2×C42C22×D5911yescharacteristic
170C4⋊C8—1182no—
171C22⋊C8—1182no—
172#97—1181no—
173#97—1182no—
174C4×C8—592no—
175#95—22no—
176#95—22no—
177#95C22⋊D5914yes—
178C24—22no—
179#41—2362no—
180#41—2362no—
181C2×Q8—1181no—
182C2×D4—24no—
183C2×D4—24no—
184C2×D4—22no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—82no—
188C2×D4—82no—
189C2×D4—1181no—
190C2×D4—1182no—
191C2×D4—2361no—
192C2×D4(C2×D59)⋊C414yes—
193C22×C4—24no—
194C22×C4—84no—
195C22×C4—82no—
196C22×C4—2361no—
197C22×C4C2×C22⋊D5911yescharacteristic
198C22×C4C2×C22⋊D5912yes—
199SD16—2368no—
200D8—2368no—
201M16—1184no—
202C2×C8—1182no—
203C2×C8—2362no—
204C4⋊C4—1182no—
205C22⋊C4—84no—
206C22⋊C4—82no—
207C22⋊C4—2362no—
208C42C2×C4×D5911yescharacteristic
209C42C2×D23611yescharacteristic
210C42#1174568512yes—
211C23—24no—
212C23—44no—
213C23—44no—
214C23—42no—
215C23—42no—
216C23—2361no—
217C23#1427369611yescharacteristic
218Q8—2362no—
219D4—44no—
220D4—82no—
221D4—84no—
222D4—84no—
223D4—4722no—
224D4—4722no—
225D4—2362no—
226C2×C4—22no—
227C2×C4—42no—
228C2×C4—48no—
229C2×C4—44no—
230C2×C4—88no—
231C2×C4—4721no—
232C2×C4—4721no—
233C2×C4—2361no—
234C2×C4C2×(C2×D59)⋊C411yescharacteristic
235C2×C4C4×C22⋊D5912yes—
236C2×C4#1427370112yes—
237C2×C4#1427371011yescharacteristic
238C8—1184no—
239C8—4722no—
240C22—22no—
241C22—44no—
242C22—48no—
243C22—82no—
244C22—82no—
245C22—84no—
246C22—4721no—
247C22—4722no—
248C22#1683728111yescharacteristic
249C22#1683757012yes—
250C4—21no—
251C4—21no—
252C4—22no—
253C4—22no—
254C4—84no—
255C4—4721no—
256C2—22no—
257C2—84no—
258C2—9441no—
259C2#2142449411yescharacteristic
260C2#2142458312yes—
subgroupG/H
centerC22#16837281
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2380C2
socleC22×C59#386
prime radical#21434546C2
cosocleC2×C42×C59C23
hypercenter#2380D59
hypocenterC59#9428
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
2. #21422810C2semidirect
3. #21422642C2semidirect
4. #21421960C2semidirect
5. C59×#2380C2semidirect
9. #16837778C4semidirect
10. #16837774C4semidirect
25. #14273727C2×C4semidirect
26. #14273724C2×C4semidirect
34. C4×C59⋊C8C2×C4semidirect
127. #2380D59semidirect
149. C59#9428semidirect

Maximal subgroups

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

subgroupindex
#214345462
#214228102
#214226422
#214219602
C59×#23802
#942859