Finite Group Databasebeta

← #28481048

Subgroups — #28481048

Order 15104.

692 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 131 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#21424929C211yesmaximal, characteristic
2#21422826C212yesmaximal, complemented
3#21422731C211yesmaximal, characteristic
4#21422608C212yesmaximal, complemented
5C59×#3550C211yesmaximal, characteristic
6#16837459C2212yes—
7#16837426C2211yescharacteristic
8C2×#14273654—22no—
9#16837248C2211yescharacteristic
10#16837151—22no—
11#16837148—22no—
12#16837142—22no—
13#16837139—24no—
14C59×#563—22no—
15C59×#559C412yescomplemented
16Q82×C59C412yescomplemented
17C59×#556—22no—
18D4×Q8×C59—22no—
19C59×#542—24no—
20C59×#536—22no—
21C59×#533C2212yes—
22C59×#532C2211yescharacteristic
23C2×#11745664—21no—
24C2×C4×Dic59—21no—
25C4⋊(C59⋊C8)—22no—
26C4⋊(C59⋊C8)—22no—
27#14273654—44no—
28C22⋊(C59⋊C8)—22no—
29#14273638—22no—
30C59⋊#95—22no—
31C2×#41×C59D412yes—
32C22×Q8×C59—22no—
33C4⋊Q8×C59—24no—
34C4⋊Q8×C59—24no—
35C4⋊Q8×C59C2×C411yescharacteristic
36C4⋊Q8×C59D412yes—
37C4⋊Q8×C59D412yes—
38#127×C59C2×C411yescharacteristic
39#126×C59—28no—
40#125×C59—22no—
41#124×C59—24no—
42#124×C59—24no—
43#124×C59D412yes—
44#123×C59—24no—
45#123×C59—24no—
46C22⋊Q8×C59—24no—
47C22⋊Q8×C59—28no—
48C22⋊Q8×C59—22no—
49C22⋊Q8×C59D414yes—
50#121×C59—22no—
51C4×Q8×C59—24no—
52C4×Q8×C59—22no—
53C4×Q8×C59C2×C414yes—
54C4×D4×C59—24no—
55C4×D4×C59—22no—
56#117×C59—22no—
57#117×C59—22no—
58C2×C42×C59C2311yescharacteristic
59#41×C59—42no—
60#41×C59—42no—
61C2×Q8×C59—24no—
62C2×Q8×C59—22no—
63C2×Q8×C59—82no—
64C2×Q8×C59—82no—
65C2×Q8×C59C22⋊C414yes—
66C2×Q8×C59C22⋊C412yes—
67C2×D4×C59—22no—
68C2×D4×C59—22no—
69C2×D4×C59C22⋊C412yes—
70C22×C4×C59—24no—
71C22×C4×C59—22no—
72C22×C4×C59C2×D411yescharacteristic
73C22×C4×C59C2×D412yes—
74C22×Dic59—41no—
75C4⋊C4×C59—28no—
76C4⋊C4×C59—22no—
77C4⋊C4×C59—24no—
78C4⋊C4×C59—22no—
79C4⋊C4×C59—22no—
80C4⋊C4×C59—22no—
81C4⋊C4×C59—24no—
82C4⋊C4×C59C22⋊C414yes—
83C22⋊C4×C59—24no—
84C22⋊C4×C59—28no—
85C22⋊C4×C59—24no—
86C22⋊C4×C59—24no—
87C42×C59—24no—
88C42×C59—24no—
89C42×C59C22×C411yescharacteristic
90C42×C59C2×D412yes—
91C42×C59C2×D411yescharacteristic
92C4×Dic59—22no—
93C4×Dic59—41no—
94#11745664—42no—
95C2×C59⋊C8—42no—
96C2×C59⋊C8—41no—
97C23×C59—22no—
98C23×C59#12011yescharacteristic
99Q8×C59—44no—
100Q8×C59—42no—
101Q8×C59—42no—
102Q8×C59—42no—
103Q8×C59—84no—
104D4×C59—42no—
105D4×C59—44no—
106D4×C59—82no—
107C2×C4×C59—21no—
108C2×C4×C59—21no—
109C2×C4×C59—24no—
110C2×C4×C59—22no—
111C2×C4×C59—24no—
112C2×C4×C59—42no—
113C2×C4×C59—42no—
114C2×C4×C59—42no—
115C2×C4×C59—84no—
116C2×C4×C59C2×C22⋊C412yes—
117C2×C4×C59C2×C22⋊C411yescharacteristic
118C2×C4×C59#12012yes—
119C2×C4×C59#12011yescharacteristic
120C2×Dic59—42no—
121C2×Dic59—82no—
122C59⋊C8—82no—
123C59⋊C8—81no—
124#8492—591nomaximal, Sylow
125C22×C59—21no—
126C22×C59—22no—
127C22×C59—42no—
128C22×C59—44no—
129C22×C59#37911yescharacteristic
130C4×C59—21no—
131C4×C59—22no—
132C4×C59—22no—
133C4×C59—42no—
134C4×C59—44no—
135C4×C59—44no—
136C4×C59#45311yescharacteristic
137C4×C59#45511yescharacteristic
138Dic59—82no—
139#3550D5911yescharacteristic
140#1963—591no—
141#1559—592no—
142#1536—591no—
143#1506—592no—
144C2×C59—41no—
145C2×C59—82no—
146C2×C59#179211yescharacteristic
147C2×C59#179612yes—
148#563—22no—
149#559Dic5912yescomplemented
150Q82Dic5912yescomplemented
151#556—22no—
152D4×Q8—22no—
153#542—24no—
154#536—22no—
155#533C2×D5912yes—
156#532C2×D5911yescharacteristic
157#432—592no—
158#423—591no—
159C2×#104—1182no—
160#376—591no—
161#333—1182no—
162#332—1182no—
163#329—1182no—
164#328—1184no—
165C59#849211yescharacteristic, complemented, Sylow
166C2×#41C22⋊D5912yes—
167C22×Q8—22no—
168C2×M16—1181no—
169C4⋊Q8—24no—
170C4⋊Q8—24no—
171C4⋊Q8C2×Dic5911yescharacteristic
172C4⋊Q8C22⋊D5912yes—
173C4⋊Q8C22⋊D5912yes—
174#127C2×Dic5911yescharacteristic
175#126—28no—
176#125—22no—
177#124—24no—
178#124—24no—
179#124C22⋊D5912yes—
180#123—24no—
181#123—24no—
182C22⋊Q8—24no—
183C22⋊Q8—28no—
184C22⋊Q8—22no—
185C22⋊Q8C22⋊D5914yes—
186#121—22no—
187C4×Q8—24no—
188C4×Q8—22no—
189C4×Q8C2×Dic5914yes—
190C4×D4—24no—
191C4×D4—22no—
192#117—22no—
193#117—22no—
194C2×C42—1181no—
195C2×C42C22×D5911yescharacteristic
196C4⋊C8—1182no—
197C4⋊C8—1182no—
198#104—2364no—
199C22⋊C8—1182no—
200#97—1182no—
201#95—1182no—
202#41—42no—
203#41—42no—
204C2×Q8—24no—
205C2×Q8—22no—
206C2×Q8—82no—
207C2×Q8—82no—
208C2×Q8C22⋊Dic5914yes—
209C2×Q8C22⋊Dic5912yes—
210C2×D4—22no—
211C2×D4—22no—
212C2×D4C22⋊Dic5912yes—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—2361no—
216C22×C4C2×C22⋊D5911yescharacteristic
217C22×C4C2×C22⋊D5912yes—
218M16—2362no—
219C2×C8—2362no—
220C2×C8—2361no—
221C4⋊C4—28no—
222C4⋊C4—22no—
223C4⋊C4—24no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—24no—
228C4⋊C4C22⋊Dic5914yes—
229C22⋊C4—24no—
230C22⋊C4—28no—
231C22⋊C4—24no—
232C22⋊C4—24no—
233C42—24no—
234C42—24no—
235C42—1182no—
236C42—2361no—
237C42C22×Dic5911yescharacteristic
238C42C2×C22⋊D5911yescharacteristic
239C42C2×C22⋊D5912yes—
240C23—22no—
241C23#1427369611yescharacteristic
242Q8—44no—
243Q8—42no—
244Q8—42no—
245Q8—42no—
246Q8—84no—
247D4—42no—
248D4—44no—
249D4—82no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—24no—
253C2×C4—22no—
254C2×C4—24no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—84no—
259C2×C4—4722no—
260C2×C4—2362no—
261C2×C4C2×C22⋊Dic5912yes—
262C2×C4C2×C22⋊Dic5911yescharacteristic
263C2×C4#1427369612yes—
264C2×C4#1427369611yescharacteristic
265C8—4722no—
266C8—4721no—
267C22—21no—
268C22—22no—
269C22—42no—
270C22—44no—
271C22#1683725811yescharacteristic
272C4—21no—
273C4—22no—
274C4—22no—
275C4—42no—
276C4—44no—
277C4—44no—
278C4—4722no—
279C4#1683757011yescharacteristic
280C4#1683758011yescharacteristic
281C2—41no—
282C2—82no—
283C2#2142405011yescharacteristic
284C2#2142407112yes—
subgroupG/H
centerC22#16837258
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#3550C2
socleC22×C59#379
prime radicalC59×#532C22
cosocleC2×C42×C59C23
hypercenter#3550D59
hypocenterC59#8492
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3550

Complements

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

NH ≅ G/Ntype
2. #21422826C2semidirect
4. #21422608C2semidirect
15. C59×#559C4semidirect
16. Q82×C59C4semidirect
149. #559Dic59semidirect
150. Q82Dic59semidirect
165. C59#8492semidirect

Maximal subgroups

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

subgroupindex
#214249292
#214228262
#214227312
#214226082
C59×#35502
#849259