Finite Group Databasebeta

← #28486364

Subgroups — #28486364

Order 15104.

424 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 107 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#21422723C212yesmaximal
3#21422440C211yesmaximal, characteristic
4#21421952C212yesmaximal
5C59×#2565C211yesmaximal, characteristic
6#16837533—24no—
7#16837459—21no—
8#16837459C2212yes—
9#16837459C2212yes—
10#16837455—21no—
11#16837144—24no—
12#16837128—21no—
13#16837128—21no—
14C2×#125×C59C2211yescharacteristic
15C2×C4×Q8×C59C2211yescharacteristic
16C59×#398—21no—
17C59×#398—21no—
18C59×#391C411yescharacteristic
19C59×#391C411yescharacteristic
20C59×#387—21no—
21C59×#387—21no—
22C59×#384C411yescharacteristic
23C59×#382C411yescharacteristic
24C4×C4⋊C4×C59C2211yescharacteristic
25C4⋊(C59⋊C8)—22no—
26C4⋊(C59⋊C8)—22no—
27C4⋊(C59⋊C8)—22no—
28C4⋊(C59⋊C8)—22no—
29C22⋊(C59⋊C8)—22no—
30C22⋊(C59⋊C8)—22no—
31#14273638—22no—
32#14273638—22no—
33#14273638D414yes—
34C4×C59⋊C8—22no—
35C22×Q8×C59C2×C411yescharacteristic
36#125×C59—22no—
37#125×C59—22no—
38C4×Q8×C59—22no—
39C4×Q8×C59—22no—
40C2×C4⋊C4×C59—21no—
41C2×C4⋊C4×C59—21no—
42C2×C4⋊C4×C59—21no—
43C2×C4⋊C4×C59—21no—
44C2×C4⋊C4×C59—21no—
45C2×C4⋊C4×C59C2×C411yescharacteristic
46C2×C4⋊C4×C59C2×C411yescharacteristic
47C2×C4⋊C4×C59C2×C411yescharacteristic
48C2×C4⋊C4×C59D411yescharacteristic
49C2×C4⋊C4×C59D411yescharacteristic
50C2×C42×C59C2×C411yescharacteristic
51C2×C42×C59C2×C411yescharacteristic
52C2×C42×C59D411yescharacteristic
53C2×C42×C59D411yescharacteristic
54C2×C42×C59C2311yescharacteristic
55#95×C59—21no—
56#95×C59—21no—
57#95×C59—21no—
58#95×C59—21no—
59#95×C59—21no—
60#95×C59—21no—
61#95×C59—21no—
62C2×Q8×C59—22no—
63C2×Q8×C59—22no—
64C2×Q8×C59—41no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59C22⋊C411yescharacteristic
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59C22⋊C411yescharacteristic
72C22×C4×C59C22⋊C411yescharacteristic
73C22×C4×C59C22×C411yescharacteristic
74C22×C4×C59C2×D411yescharacteristic
75C22×C4×C59C2×D411yescharacteristic
76C4⋊C4×C59—22no—
77C4⋊C4×C59—22no—
78C4⋊C4×C59—22no—
79C4⋊C4×C59—22no—
80C4⋊C4×C59—22no—
81C4⋊C4×C59—42no—
82C4⋊C4×C59—42no—
83C4⋊C4×C59—42no—
84C4⋊C4×C59—41no—
85C4⋊C4×C59—42no—
86C4⋊C4×C59—41no—
87C42×C59—21no—
88C42×C59—21no—
89C42×C59—21no—
90C42×C59—21no—
91C42×C59—41no—
92C42×C59—41no—
93C42×C59C2×D412yes—
94C42×C59#4112yes—
95C2×C59⋊C8—24no—
96C2×C59⋊C8—42no—
97C23×C59C2×C22⋊C411yescharacteristic
98Q8×C59—44no—
99C2×C4×C59—21no—
100C2×C4×C59—21no—
101C2×C4×C59—21no—
102C2×C4×C59—21no—
103C2×C4×C59—22no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—21no—
107C2×C4×C59—21no—
108C2×C4×C59—21no—
109C2×C4×C59—41no—
110C2×C4×C59—41no—
111C2×C4×C59—41no—
112C2×C4×C59—41no—
113C2×C4×C59—41no—
114C2×C4×C59—41no—
115C2×C4×C59—41no—
116C2×C4×C59—41no—
117C2×C4×C59—41no—
118C2×C4×C59—41no—
119C2×C4×C59—42no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
122C2×C4×C59—41no—
123C2×C4×C59#10112yes—
124C2×C4×C59C4×D412yes—
125C2×C4×C59#12011yescharacteristic
126C2×C4×C59#12112yes—
127C2×C4×C59#12311yescharacteristic
128C59⋊C8—44no—
129C59⋊C8—82no—
130#9450—591nomaximal, Sylow
131C22×C59—21no—
132C22×C59—21no—
133C22×C59#38611yescharacteristic
134C22×C59C2×#10111yescharacteristic
135C22×C59#41311yescharacteristic
136C4×C59—22no—
137C4×C59—22no—
138C4×C59—22no—
139C4×C59—21no—
140C4×C59—21no—
141C4×C59—42no—
142C4×C59—41no—
143C4×C59—41no—
144C4×C59—41no—
145C4×C59—81no—
146C4×C59—81no—
147C4×C59—81no—
148#2983—591no—
149#2565D5911yescharacteristic
150#1534—592no—
151#1467—591no—
152#1285—592no—
153C2×C59—21no—
154C2×C59—21no—
155C2×C59#188411yescharacteristic
156C2×C59#190111yescharacteristic
157C2×C59#191211yescharacteristic
158C2×#125C2×D5911yescharacteristic
159C2×C4×Q8C2×D5911yescharacteristic
160#446—1184no—
161#432—592no—
162#432—592no—
163#432—1181no—
164#431—1181no—
165#398—21no—
166#398—21no—
167#391Dic5911yescharacteristic
168#391Dic5911yescharacteristic
169#387—21no—
170#387—21no—
171#384Dic5911yescharacteristic
172#382Dic5911yescharacteristic
173C4×C4⋊C4C2×D5911yescharacteristic
174#330—1184no—
175#324—1181no—
176#324—1181no—
177C59#945011yescharacteristic, complemented, Sylow
178C22×Q8C2×Dic5911yescharacteristic
179#125—22no—
180#125—22no—
181C4×Q8—22no—
182C4×Q8—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4C2×Dic5911yescharacteristic
189C2×C4⋊C4C2×Dic5911yescharacteristic
190C2×C4⋊C4C2×Dic5911yescharacteristic
191C2×C4⋊C4C22⋊D5911yescharacteristic
192C2×C4⋊C4C22⋊D5911yescharacteristic
193C2×C42C2×Dic5911yescharacteristic
194C2×C42C2×Dic5911yescharacteristic
195C2×C42C22⋊D5911yescharacteristic
196C2×C42C22⋊D5911yescharacteristic
197C2×C42C22×D5911yescharacteristic
198C4⋊C8—1182no—
199C4⋊C8—1182no—
200C4⋊C8—1182no—
201C4⋊C8—1182no—
202C22⋊C8—1182no—
203C22⋊C8—1182no—
204#97—594no—
205#97—1182no—
206#97—1182no—
207C4×C8—1182no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—21no—
215C2×Q8—22no—
216C2×Q8—22no—
217C2×Q8—41no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4C22⋊Dic5911yescharacteristic
223C22×C4C22⋊Dic5911yescharacteristic
224C22×C4C22⋊Dic5911yescharacteristic
225C22×C4C22⋊Dic5911yescharacteristic
226C22×C4C22×Dic5911yescharacteristic
227C22×C4C2×C22⋊D5911yescharacteristic
228C22×C4C2×C22⋊D5911yescharacteristic
229C2×C8—1184no—
230C2×C8—2362no—
231C4⋊C4—22no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—41no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—41no—
247C42—41no—
248C42D4×D5912yes—
249C42#1174568712yes—
250C23C2×C22⋊Dic5911yescharacteristic
251Q8—44no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4#1427364612yes—
277C2×C4D4×Dic5912yes—
278C2×C4#1427369511yescharacteristic
279C2×C4#1427369912yes—
280C2×C4#1427370811yescharacteristic
281C8—4722no—
282C8—2364no—
283C22—21no—
284C22—21no—
285C22#1683728211yescharacteristic
286C22C2×#1427364611yescharacteristic
287C22#1683738611yescharacteristic
288C4—22no—
289C4—22no—
290C4—22no—
291C4—21no—
292C4—21no—
293C4—42no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—81no—
298C4—81no—
299C4—81no—
300C2—21no—
301C2—21no—
302C2#2142448011yescharacteristic
303C2#2142456511yescharacteristic
304C2#2142463411yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2565C2
socleC22×C59#386
prime radicalC23×C59C2×C22⋊C4
cosocleC2×C42×C59C23
hypercenter#2565D59
hypocenterC59#9450
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2565

Complements

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

NH ≅ G/Ntype
177. C59#9450semidirect

Maximal subgroups

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

subgroupindex
#214331812
#214227232
#214224402
#214219522
C59×#25652
#945059