Finite Group Databasebeta

← #28501682

Subgroups — #28501682

Order 15104.

558 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 63 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#21425441C211yesmaximal, characteristic, complemented
2#21423035C212yesmaximal, complemented
3#21422099C212yesmaximal, complemented
4#21422042C211yesmaximal, characteristic, complemented
5C59×#2537C211yesmaximal, characteristic
6#16838033C2211yescharacteristic
7#16837553—22no—
8#16837442—21no—
9#16837171C2211yescharacteristic
10#16837162—21no—
11#16837138—22no—
12#16837128C2212yes—
13C2×#121×C59—22no—
14C2×C4×D4×C59—21no—
15C59×#394—21no—
16C59×#394C2212yes—
17C59×#384C2211yescharacteristic
18C59×#380—22no—
19C2×#11745670—42no—
20C2×#11745664—21no—
21#14273726—41no—
22#14273718—41no—
23#14273706—22no—
24#14273686—21no—
25#14273660—41no—
26#14273658—41no—
27#14273648—42no—
28C22⋊(C59⋊C8)—22no—
29C22×D4×C59—21no—
30C22×D4×C59—21no—
31C22×D4×C59D412yes—
32C23×C4×C59—22no—
33#121×C59—48no—
34C4×D4×C59—42no—
35C4×D4×C59—42no—
36C2×C4⋊C4×C59—21no—
37C2×C4⋊C4×C59—22no—
38C2×C4⋊C4×C59D411yescharacteristic
39C2×C4⋊C4×C59C2311yescharacteristic
40C2×C22⋊C4×C59—22no—
41C2×C22⋊C4×C59—22no—
42C2×C22⋊C4×C59—22no—
43C2×C42×C59D411yescharacteristic
44#95×C59—21no—
45#95×C59D412yes—
46C24×C59—22no—
47C2×D4×C59—24no—
48C2×D4×C59—42no—
49C2×D4×C59—42no—
50C2×D4×C59—42no—
51C2×D4×C59—44no—
52C2×D4×C59—44no—
53C2×D4×C59—81no—
54C22×C4×C59—21no—
55C22×C4×C59—22no—
56C22×C4×C59—21no—
57C22×C4×C59—44no—
58C22×C4×C59—44no—
59C22×C4×C59—44no—
60C22×C4×C59C2×D411yescharacteristic
61C22×C4×C59C2×D412yes—
62C2×Dic118—22no—
63C4⋊C4×C59—41no—
64C4⋊C4×C59—42no—
65C4⋊C4×C59—44no—
66C4⋊C4×C59—41no—
67C22⋊C4×C59—44no—
68C22⋊C4×C59—44no—
69C22⋊C4×C59—42no—
70C22⋊C4×C59—42no—
71C42×C59—21no—
72C42×C59SD1612yescomplemented
73C22⋊Dic59—41no—
74#11745670—84no—
75C4⋊Dic59—22no—
76C4⋊Dic59—21no—
77C4⋊Dic59—21no—
78Dic59⋊C4—42no—
79C4×Dic59—41no—
80#11745664—22no—
81#11745664—41no—
82C2×C59⋊C8—41no—
83C2×C59⋊C8—42no—
84C23×C59—24no—
85C23×C59—42no—
86C23×C59—42no—
87C23×C59—42no—
88C23×C59—42no—
89C23×C59—42no—
90C23×C59—42no—
91C23×C59#12011yescharacteristic
92D4×C59—48no—
93D4×C59—44no—
94D4×C59—82no—
95D4×C59—84no—
96C2×C4×C59—22no—
97C2×C4×C59—22no—
98C2×C4×C59—22no—
99C2×C4×C59—21no—
100C2×C4×C59—21no—
101C2×C4×C59—22no—
102C2×C4×C59—42no—
103C2×C4×C59—42no—
104C2×C4×C59—42no—
105C2×C4×C59—42no—
106C2×C4×C59—42no—
107C2×C4×C59—41no—
108C2×C4×C59—88no—
109C2×C4×C59C2×SD1611yescharacteristic
110C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
111C2×Dic59—41no—
112C2×Dic59—41no—
113Dic118—82no—
114C59⋊C8—42no—
115C59⋊C8—82no—
116#12246—591nomaximal, Sylow
117C22×C59—21no—
118C22×C59—21no—
119C22×C59—44no—
120C22×C59—44no—
121C22×C59—44no—
122C22×C59—82no—
123C22×C59—82no—
124C22×C59—82no—
125C22×C59—82no—
126C22×C59#45011yescharacteristic
127C22×C59#45611yescharacteristic
128C22×C59B(4,𝔽2)11yescharacteristic
129C4×C59—24no—
130C4×C59—21no—
131C4×C59—21no—
132C4×C59—42no—
133C4×C59—82no—
134C4×C59—81no—
135Dic59—81no—
136Dic59—81no—
137#2537D5911yescharacteristic
138#2060—591no—
139#1599—592no—
140#1345—592no—
141#1325—591no—
142C2×C59—21no—
143C2×C59—21no—
144C2×C59—84no—
145C2×C59#160411yescharacteristic
146C2×C59#219611yescharacteristic
147C2×C59#220411yescharacteristic
148#544—591no—
149C2×#121—22no—
150C2×C4×D4—21no—
151#450—1182no—
152#428—1181no—
153#394—21no—
154#394C2×D5912yes—
155#384C2×D5911yescharacteristic
156#380—22no—
157#343—591no—
158#339—1181no—
159#328—1182no—
160#324—592no—
161C59#1224611yescharacteristic, complemented, Sylow
162C22×D4—21no—
163C22×D4—21no—
164C22×D4C22⋊D5912yes—
165C23×C4—22no—
166C2×SD16—2362no—
167C2×M16—1181no—
168C4⋊Q8—2361no—
169#125—2361no—
170C22⋊Q8—1182no—
171#121—48no—
172C4×D4—42no—
173C4×D4—42no—
174#117—1181no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4C22⋊D5911yescharacteristic
178C2×C4⋊C4C22×D5911yescharacteristic
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C42C22⋊D5911yescharacteristic
183#107—2361no—
184#106—2361no—
185#102—2362no—
186C22⋊C8—1182no—
187#95—21no—
188#95C22⋊D5912yes—
189C24—22no—
190C2×Q8—1182no—
191C2×D4—24no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—44no—
196C2×D4—44no—
197C2×D4—81no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—44no—
204C22×C4C2×C22⋊D5911yescharacteristic
205C22×C4C2×C22⋊D5912yes—
206SD16—4724no—
207M16—1182no—
208M16—2361no—
209C2×C8—2362no—
210C2×C8—2361no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C4⋊C4—44no—
215C4⋊C4—1182no—
216C4⋊C4—1181no—
217C4⋊C4—1181no—
218C4⋊C4—2362no—
219C22⋊C4—44no—
220C22⋊C4—44no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—2361no—
224C42—21no—
225C42—2361no—
226C42#1174567012yescomplemented
227C23—24no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23#1427369611yescharacteristic
235Q8—4722no—
236D4—48no—
237D4—44no—
238D4—82no—
239D4—84no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—88no—
253C2×C4—2361no—
254C2×C4—2361no—
255C2×C4C2×#1174567011yescharacteristic
256C2×C4#1427375311yescharacteristic
257C8—4722no—
258C8—2362no—
259C22—21no—
260C22—21no—
261C22—44no—
262C22—44no—
263C22—44no—
264C22—82no—
265C22—82no—
266C22—82no—
267C22—82no—
268C22#1683755311yescharacteristic
269C22#1683758511yescharacteristic
270C22#1683758911yescharacteristic
271C4—24no—
272C4—21no—
273C4—21no—
274C4—42no—
275C4—82no—
276C4—81no—
277C4—4721no—
278C4—4721no—
279C2—21no—
280C2—21no—
281C2—84no—
282C2#2142306511yescharacteristic
283C2#2142608011yescharacteristic
284C2#2142612211yescharacteristic
subgroupG/H
centerC22#16837589
commutator, cosocleC2×C4⋊C4×C59C23
FrattiniC2×C4⋊C4C22×D59
Fitting, prime radicalC59×#2537C2
socleC22×C59B(4,𝔽2)
hypercenter#2537D59
hypocenterC59#12246
LCSG ▷ C2×C4⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2537

Complements

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

NH ≅ G/Ntype
1. #21425441C2semidirect
2. #21423035C2semidirect
3. #21422099C2semidirect
4. #21422042C2semidirect
72. C42×C59SD16semidirect
161. C59#12246semidirect
226. C42#11745670semidirect

Maximal subgroups

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

subgroupindex
#214254412
#214230352
#214220992
#214220422
C59×#25372
#1224659