Finite Group Databasebeta

← #28564809

Subgroups — #28564809

Order 15104.

982 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 982 conjugacy classes of subgroups — too many to draw. Only its 179 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#21434226C212yesmaximal, complemented
3#21423282C218yesmaximal, complemented
4#21422810C211yesmaximal, characteristic
5#21422543C212yesmaximal, complemented
6C2×C59×#559C211yesmaximal, characteristic
7#16837959C2211yescharacteristic
8#16837766—22no—
9#16837748C2214yes—
10#16837732—22no—
11#16837687C2214yescomplemented
12#16837612C2214yescomplemented
13#16837558—22no—
14#16837507—24no—
15#16837459C2212yes—
16#16837407—22no—
17C4×#11745664C2211yescharacteristic
18#16837146C2214yes—
19Q8⋊(C59⋊C8)C2218yescomplemented
20C59×#559—22no—
21C59×#559—24no—
22C59×#559C2214yes—
23C2×#127×C59C2211yescharacteristic
24C2×#124×C59—22no—
25C2×#123×C59—22no—
26C2×C4×Q8×C59C2212yes—
27C2×C4×D4×C59—22no—
28C2×#11745672—24no—
29C2×#11745664—21no—
30#14273727C2312yes—
31#14273719—21no—
32#14273706—22no—
33C4×Dic118—22no—
34#14273686—21no—
35#14273658—24no—
36C4⋊(C59⋊C8)C2314yes—
37#14273651—24no—
38#14273650—24no—
39#14273648—28no—
40C22⋊(C59⋊C8)—22no—
41#14273638—22no—
42#14273638—21no—
43C4×C59⋊C8C2312yes—
44C22×Q8×C59D412yes—
45C22×D4×C59—22no—
46C22×D4×C59D412yes—
47C23×C4×C59—22no—
48#127×C59—22no—
49#127×C59D412yes—
50#127×C59C2312yes—
51#124×C59—28no—
52#124×C59—28no—
53#123×C59—28no—
54#123×C59—28no—
55C4×Q8×C59—22no—
56C4×Q8×C59—22no—
57C4×Q8×C59D414yes—
58C4×Q8×C59C2314yes—
59C4×D4×C59—28no—
60C4×D4×C59—28no—
61C2×C4⋊C4×C59—22no—
62C2×C4⋊C4×C59D412yes—
63C2×C22⋊C4×C59—24no—
64C2×C22⋊C4×C59—22no—
65C2×C42×C59—22no—
66C2×C42×C59C2311yescharacteristic
67C24×C59—22no—
68C2×Q8×C59—22no—
69C2×Q8×C59—44no—
70C2×Q8×C59C2×D414yes—
71C2×D4×C59—24no—
72C2×D4×C59—22no—
73C2×D4×C59—24no—
74C2×D4×C59—48no—
75C2×D4×C59—44no—
76C2×D4×C59C2×D414yes—
77C22×C4×C59—22no—
78C22×C4×C59—22no—
79C22×C4×C59—28no—
80C22×C4×C59—84no—
81C22×C4×C59C2×D412yes—
82C22×C4×C59C2×D411yescharacteristic
83C2×Dic118—22no—
84C4⋊C4×C59—24no—
85C4⋊C4×C59—22no—
86C4⋊C4×C59—24no—
87C4⋊C4×C59C2×D414yes—
88C22⋊C4×C59—216no—
89C22⋊C4×C59—28no—
90C42×C59—24no—
91C42×C59—24no—
92C42×C59C2×D411yescharacteristic
93C42×C59C2×D412yes—
94C42×C59C2411yescharacteristic
95C22⋊Dic59—41no—
96#11745672—48no—
97C4⋊Dic59—22no—
98C4⋊Dic59—22no—
99Dic59⋊C4—42no—
100C4×Dic59—41no—
101#11745664—42no—
102C2×C59⋊C8—22no—
103C2×C59⋊C8—24no—
104C23×C59—24no—
105C23×C59—48no—
106C23×C59—44no—
107C23×C59#12011yescharacteristic
108Q8×C59—24no—
109Q8×C59—24no—
110Q8×C59—44no—
111D4×C59—44no—
112D4×C59—48no—
113D4×C59—48no—
114D4×C59—82no—
115C2×C4×C59—22no—
116C2×C4×C59—24no—
117C2×C4×C59—24no—
118C2×C4×C59—24no—
119C2×C4×C59—44no—
120C2×C4×C59—44no—
121C2×C4×C59—88no—
122C2×C4×C59#12011yescharacteristic
123C2×C4×C59#12012yes—
124C2×C4×C59C22×D411yescharacteristic
125C2×C4×C59C22×D412yes—
126C2×Dic59—42no—
127Dic118—44no—
128C59⋊C8—42no—
129C59⋊C8—44no—
130#18531—591nomaximal, Sylow
131C22×C59—22no—
132C22×C59—48no—
133C22×C59—44no—
134C22×C59—88no—
135C22×C59#45512yes—
136C22×C59C2×#12011yescharacteristic
137C4×C59—22no—
138C4×C59—21no—
139C4×C59—21no—
140C4×C59—44no—
141C4×C59—44no—
142C4×C59#57514yes—
143Dic59—82no—
144C2×#559D5911yescharacteristic
145#3138—591no—
146#3113—592no—
147#1640—598no—
148#1555—591no—
149#1491—592no—
150C2×C59—22no—
151C2×C59—84no—
152C2×C59#300512yes—
153C2×C59C2×#45511yescharacteristic
154#559—22no—
155#559—24no—
156#559C2×D5914yes—
157#533—591no—
158C2×#127C2×D5911yescharacteristic
159C2×#124—22no—
160C2×#123—22no—
161C2×C4×Q8C2×D5912yes—
162C2×C4×D4—22no—
163#490—1182no—
164#486—594no—
165#483—1182no—
166#475—594no—
167#462—594no—
168#451—1182no—
169#441—1184no—
170#432—592no—
171#418—1182no—
172C4×M16—591no—
173#331—594no—
174Q8⋊C8—598no—
175C59#1853111yescharacteristic, complemented, Sylow
176C22×Q8C22⋊D5912yes—
177C22×D4—22no—
178C22×D4C22⋊D5912yes—
179C23×C4—22no—
180C2×Q16—1184no—
181C2×M16—1181no—
182C4⋊Q8—592no—
183#127—22no—
184#127C22⋊D5912yes—
185#127C22×D5912yes—
186#125—1181no—
187#124—28no—
188#124—28no—
189#123—28no—
190#123—28no—
191C22⋊Q8—1182no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8—1182no—
195C4×Q8C22⋊D5914yes—
196C4×Q8C22×D5914yes—
197C4×D4—28no—
198C4×D4—28no—
199#117—1181no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4C22⋊D5912yes—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—22no—
204C2×C42—22no—
205C2×C42C22×D5911yescharacteristic
206#106—1184no—
207C4⋊C8—594no—
208#103—1184no—
209#103—1184no—
210#102—1188no—
211C22⋊C8—1182no—
212#97—1181no—
213#97—1182no—
214C4×C8—592no—
215C24—22no—
216C2×Q8—22no—
217C2×Q8—44no—
218C2×Q8—1182no—
219C2×Q8C2×C22⋊D5914yes—
220C2×D4—24no—
221C2×D4—22no—
222C2×D4—24no—
223C2×D4—48no—
224C2×D4—44no—
225C2×D4C2×C22⋊D5914yes—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—28no—
229C22×C4—84no—
230C22×C4C2×C22⋊D5912yes—
231C22×C4C2×C22⋊D5911yescharacteristic
232Q16—2368no—
233M16—2362no—
234C2×C8—1182no—
235C2×C8—1184no—
236C4⋊C4—24no—
237C4⋊C4—22no—
238C4⋊C4—24no—
239C4⋊C4—1182no—
240C4⋊C4—1182no—
241C4⋊C4—2362no—
242C4⋊C4C2×C22⋊D5914yes—
243C22⋊C4—216no—
244C22⋊C4—28no—
245C22⋊C4—2361no—
246C42—24no—
247C42—24no—
248C42—2361no—
249C42C2×C22⋊D5911yescharacteristic
250C42C2×C22⋊D5912yes—
251C42C23×D5911yescharacteristic
252C23—24no—
253C23—48no—
254C23—44no—
255C23#1427369611yescharacteristic
256Q8—24no—
257Q8—24no—
258Q8—44no—
259Q8—2364no—
260D4—44no—
261D4—48no—
262D4—48no—
263D4—82no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—88no—
271C2×C4—2362no—
272C2×C4#1427369612yes—
273C2×C4#1427369611yescharacteristic
274C2×C4C22×C22⋊D5911yescharacteristic
275C2×C4C22×C22⋊D5912yes—
276C8—2362no—
277C8—2364no—
278C22—22no—
279C22—44no—
280C22—48no—
281C22—88no—
282C22#1683758012yes—
283C22C2×#1427369611yescharacteristic
284C4—21no—
285C4—21no—
286C4—22no—
287C4—44no—
288C4—44no—
289C4—4722no—
290C4#1683820014yes—
291C2—22no—
292C2—84no—
293C2#2143333812yes—
294C2C2×#1683758011yescharacteristic
subgroupG/H
centerC22C2×#14273696
commutator, cosocleC42×C59C24
FrattiniC42C23×D59
FittingC2×C59×#559C2
socleC22×C59C2×#120
prime radicalC2×#127×C59C22
hypercenterC2×#559D59
hypocenterC59#18531
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#559

Complements

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

NH ≅ G/Ntype
2. #21434226C2semidirect
3. #21423282C2semidirect
5. #21422543C2semidirect
11. #16837687C22semidirect
12. #16837612C22semidirect
19. Q8⋊(C59⋊C8)C22semidirect
175. C59#18531semidirect

Maximal subgroups

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

subgroupindex
#214344702
#214342262
#214232822
#214228102
#214225432
C2×C59×#5592
#1853159