Finite Group Databasebeta

← #28486371

Subgroups — #28486371

Order 15104.

468 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 103 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#21433181C211yesmaximal, characteristic
2#21429081C211yesmaximal, characteristic
3#21422821C212yesmaximal
4#21421951C212yesmaximal
5C59×#1467C211yesmaximal, characteristic
6C2×#14273727C2211yescharacteristic
7C2×#14273725C412yes—
8#16837533—24no—
9#16837459—21no—
10#16837459C2212yes—
11#16837455—21no—
12#16837332—21no—
13#16837332—21no—
14#16837307C412yes—
15#16837300—21no—
16#16837300—21no—
17C4×C4⋊Dic59C2211yescharacteristic
18#16837150—24no—
19C2×C4×Q8×C59C2211yescharacteristic
20C59×#432C2212yes—
21C59×#324—21no—
22C59×#324—21no—
23C22×Dic118C2×C412yes—
24#14273727—22no—
25#14273727—22no—
26#14273725—44no—
27C2×C4⋊Dic59—21no—
28C2×C4⋊Dic59—21no—
29C2×C4⋊Dic59—21no—
30C2×Dic59⋊C4—22no—
31C2×Dic59⋊C4—22no—
32C2×C4×Dic59C2×C412yes—
33C4⋊(C59⋊C8)—22no—
34C4⋊(C59⋊C8)—22no—
35C4⋊(C59⋊C8)D414yes—
36C22⋊(C59⋊C8)—22no—
37#14273638—22no—
38#14273638—22no—
39C4×C59⋊C8—22no—
40C59⋊#95—22no—
41C59⋊#95—22no—
42C59⋊#95—21no—
43C22×Q8×C59C2×C411yescharacteristic
44C4×Q8×C59—22no—
45C4×Q8×C59—22no—
46C2×C4⋊C4×C59C2×C411yescharacteristic
47C2×C4⋊C4×C59D411yescharacteristic
48C2×C4⋊C4×C59D411yescharacteristic
49C2×C42×C59D411yescharacteristic
50C2×C42×C59D411yescharacteristic
51C2×C42×C59C2311yescharacteristic
52C4⋊C8×C59—22no—
53C22⋊C8×C59—22no—
54#97×C59—22no—
55C2×Q8×C59—22no—
56C2×Q8×C59—22no—
57C2×Q8×C59—41no—
58C22×C4×C59C22⋊C411yescharacteristic
59C22×C4×C59C22⋊C411yescharacteristic
60C22×C4×C59C22⋊C411yescharacteristic
61C22×C4×C59C22⋊C411yescharacteristic
62C22×C4×C59C22×C411yescharacteristic
63C22×C4×C59C2×D411yescharacteristic
64C22×C4×C59C2×D411yescharacteristic
65C22×Dic59—22no—
66C22×Dic59—22no—
67C2×Dic118—24no—
68C2×Dic118—42no—
69C2×Dic118—42no—
70C2×C8×C59—42no—
71C4⋊C4×C59—22no—
72C4⋊C4×C59—22no—
73C4⋊C4×C59—22no—
74C42×C59—21no—
75C42×C59—21no—
76C42×C59—21no—
77C42×C59—21no—
78C42×C59C2×D412yes—
79C42×C59#4112yes—
80C4⋊Dic59—24no—
81C4⋊Dic59—41no—
82C4⋊Dic59—42no—
83C4⋊Dic59—41no—
84Dic59⋊C4—44no—
85Dic59⋊C4—44no—
86C4×Dic59—42no—
87C2×C59⋊C8—24no—
88C23×C59C2×C22⋊C411yescharacteristic
89Q8×C59—44no—
90C2×C4×C59—21no—
91C2×C4×C59—21no—
92C2×C4×C59—21no—
93C2×C4×C59—21no—
94C2×C4×C59—22no—
95C2×C4×C59—21no—
96C2×C4×C59—21no—
97C2×C4×C59—21no—
98C2×C4×C59—21no—
99C2×C4×C59—21no—
100C2×C4×C59—41no—
101C2×C4×C59—41no—
102C2×C4×C59—41no—
103C2×C4×C59#10012yes—
104C2×C4×C59C4×D412yes—
105C2×C4×C59#12011yescharacteristic
106C2×C4×C59#12112yes—
107C2×C4×C59#12311yescharacteristic
108C2×Dic59—24no—
109C2×Dic59—42no—
110C2×Dic59—42no—
111C2×Dic59—42no—
112C2×Dic59—42no—
113C2×Dic59—42no—
114Dic118—84no—
115C8×C59—82no—
116C59⋊C8—44no—
117#9452—591nomaximal, Sylow
118C22×C59—21no—
119C22×C59—21no—
120C22×C59#38611yescharacteristic
121C22×C59C2×#10011yescharacteristic
122C22×C59#41311yescharacteristic
123C4×C59—21no—
124C4×C59—22no—
125C4×C59—22no—
126C4×C59—22no—
127C4×C59—41no—
128C4×C59—41no—
129C4×C59—41no—
130C4×C59#45612yes—
131Dic59—44no—
132Dic59—82no—
133#2983—591no—
134#2546—591no—
135#1558—592no—
136#1467D5911yescharacteristic
137#1285—592no—
138C2×C59—21no—
139C2×C59—21no—
140C2×C59#188611yescharacteristic
141C2×C59#190011yescharacteristic
142C2×C59#190911yescharacteristic
143C2×C4⋊Q8—592no—
144C2×C4⋊Q8—591no—
145C2×C4×Q8C2×D5911yescharacteristic
146#446—1184no—
147#432—592no—
148#432—1181no—
149#432C2×D5912yes—
150#431—1181no—
151#398—1181no—
152#398—1181no—
153#391—592no—
154#389—1181no—
155#389—1181no—
156C4×C4⋊C4—591no—
157#333—1184no—
158#324—21no—
159#324—21no—
160C59#945211yescharacteristic, complemented, Sylow
161C22×Q8—592no—
162C22×Q8C4×D5911yescharacteristic
163C4⋊Q8—1182no—
164C4⋊Q8—1182no—
165C4⋊Q8—2364no—
166C4×Q8—22no—
167C4×Q8—22no—
168C2×C4⋊C4—1181no—
169C2×C4⋊C4—1182no—
170C2×C4⋊C4—1182no—
171C2×C4⋊C4—1181no—
172C2×C4⋊C4—1181no—
173C2×C4⋊C4C4×D5911yescharacteristic
174C2×C4⋊C4C22⋊D5911yescharacteristic
175C2×C4⋊C4C22⋊D5911yescharacteristic
176C2×C42—592no—
177C2×C42D23611yescharacteristic
178C2×C42D23611yescharacteristic
179C2×C42C22×D5911yescharacteristic
180C4⋊C8—22no—
181C4⋊C8—594no—
182C4⋊C8—1182no—
183C4⋊C8—1182no—
184C22⋊C8—22no—
185C22⋊C8—1182no—
186#97—22no—
187#97—1182no—
188#97—1182no—
189C4×C8—1182no—
190#95—1182no—
191#95—1182no—
192#95—1181no—
193C2×Q8—22no—
194C2×Q8—22no—
195C2×Q8—41no—
196C2×Q8—1184no—
197C2×Q8—2362no—
198C2×Q8—2362no—
199C22×C4—1182no—
200C22×C4—1182no—
201C22×C4(C2×D59)⋊C411yescharacteristic
202C22×C4(C2×D59)⋊C411yescharacteristic
203C22×C4(C2×D59)⋊C411yescharacteristic
204C22×C4(C2×D59)⋊C411yescharacteristic
205C22×C4C2×C4×D5911yescharacteristic
206C22×C4C2×D23611yescharacteristic
207C22×C4C2×C22⋊D5911yescharacteristic
208C2×C8—42no—
209C2×C8—1184no—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—1184no—
214C4⋊C4—2364no—
215C4⋊C4—2364no—
216C4⋊C4—2361no—
217C4⋊C4—2362no—
218C4⋊C4—2361no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—2362no—
224C42D4×D5912yes—
225C42#1174568712yes—
226C23C2×(C2×D59)⋊C411yescharacteristic
227Q8—44no—
228Q8—4724no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—1184no—
243C2×C4—2362no—
244C2×C4—2362no—
245C2×C4—2362no—
246C2×C4—2362no—
247C2×C4—2362no—
248C2×C4#1427364312yes—
249C2×C4(C22⋊D59)⋊C412yes—
250C2×C4#1427369711yescharacteristic
251C2×C4#1427369912yes—
252C2×C4#1427371111yescharacteristic
253C8—82no—
254C8—2364no—
255C22—21no—
256C22—21no—
257C22#1683727911yescharacteristic
258C22C2×#1427364311yescharacteristic
259C22#1683738411yescharacteristic
260C4—21no—
261C4—22no—
262C4—22no—
263C4—22no—
264C4—41no—
265C4—41no—
266C4—41no—
267C4—4722no—
268C4—2364no—
269C4#1683758712yes—
270C2—21no—
271C2—21no—
272C2#2142448711yescharacteristic
273C2#2142455711yescharacteristic
274C2#2142461611yescharacteristic
subgroupG/H
centerC22#16837279
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1467C2
socleC22×C59#386
prime radicalC23×C59C2×C22⋊C4
cosocleC2×C42×C59C23
hypercenter#1467D59
hypocenterC59#9452
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1467

Complements

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

NH ≅ G/Ntype
160. C59#9452semidirect

Maximal subgroups

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

subgroupindex
#214331812
#214290812
#214228212
#214219512
C59×#14672
#945259