Finite Group Databasebeta

← #28486361

Subgroups — #28486361

Order 15104.

424 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 99 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#21429283C211yesmaximal, characteristic
3#21422722C212yesmaximal
4#21421953C212yesmaximal
5C59×#1467C211yesmaximal, characteristic
6C2×#14273719C2211yescharacteristic
7#16837533—24no—
8#16837459—21no—
9#16837459C2212yes—
10#16837455—21no—
11#16837335—21no—
12#16837335—21no—
13#16837309C411yescharacteristic
14#16837309C411yescharacteristic
15#16837286—21no—
16#16837286—21no—
17#16837271C411yescharacteristic
18#16837267C411yescharacteristic
19C4×C4⋊Dic59C2211yescharacteristic
20#16837143—24no—
21C2×C4×Q8×C59C2211yescharacteristic
22C59×#432C2212yes—
23C59×#324—21no—
24C59×#324—21no—
25#14273719—22no—
26#14273719—22no—
27C2×C4⋊Dic59—21no—
28C2×C4⋊Dic59C2×C411yescharacteristic
29C2×C4⋊Dic59C2×C411yescharacteristic
30C2×Dic59⋊C4—21no—
31C2×Dic59⋊C4—21no—
32C2×Dic59⋊C4—21no—
33C2×Dic59⋊C4—21no—
34C2×C4×Dic59C2×C411yescharacteristic
35C2×C4×Dic59C2×C411yescharacteristic
36C4⋊(C59⋊C8)—22no—
37C4⋊(C59⋊C8)—22no—
38C4⋊(C59⋊C8)—22no—
39C22⋊(C59⋊C8)—22no—
40#14273638—22no—
41#14273638D414yes—
42C4×C59⋊C8—22no—
43C59⋊#95—21no—
44C59⋊#95—21no—
45C59⋊#95—21no—
46C59⋊#95—21no—
47C59⋊#95—21no—
48C59⋊#95—21no—
49C59⋊#95—21no—
50C22×Q8×C59C2×C411yescharacteristic
51C4×Q8×C59—22no—
52C4×Q8×C59—22no—
53C2×C4⋊C4×C59C2×C411yescharacteristic
54C2×C4⋊C4×C59D411yescharacteristic
55C2×C4⋊C4×C59D411yescharacteristic
56C2×C42×C59D411yescharacteristic
57C2×C42×C59D411yescharacteristic
58C2×C42×C59C2311yescharacteristic
59C4⋊C8×C59—22no—
60C22⋊C8×C59—22no—
61#97×C59—22no—
62C2×Q8×C59—22no—
63C2×Q8×C59—22no—
64C2×Q8×C59—41no—
65C22×C4×C59C22⋊C411yescharacteristic
66C22×C4×C59C22⋊C411yescharacteristic
67C22×C4×C59C22⋊C411yescharacteristic
68C22×C4×C59C22⋊C411yescharacteristic
69C22×C4×C59C22×C411yescharacteristic
70C22×C4×C59C2×D411yescharacteristic
71C22×C4×C59C2×D411yescharacteristic
72C22×Dic59—21no—
73C22×Dic59—21no—
74C22×Dic59—21no—
75C22×Dic59—21no—
76C2×C8×C59—42no—
77C4⋊C4×C59—22no—
78C4⋊C4×C59—22no—
79C4⋊C4×C59—22no—
80C42×C59—21no—
81C42×C59—21no—
82C42×C59—21no—
83C42×C59—21no—
84C42×C59C2×D412yes—
85C42×C59#4112yes—
86C4⋊Dic59—22no—
87C4⋊Dic59—22no—
88C4⋊Dic59—42no—
89Dic59⋊C4—42no—
90Dic59⋊C4—42no—
91Dic59⋊C4—41no—
92Dic59⋊C4—42no—
93Dic59⋊C4—41no—
94C4×Dic59—41no—
95C4×Dic59—41no—
96C2×C59⋊C8—24no—
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—21no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—21no—
107C2×C4×C59—21no—
108C2×C4×C59—41no—
109C2×C4×C59—41no—
110C2×C4×C59—41no—
111C2×C4×C59—41no—
112C2×C4×C59#10112yes—
113C2×C4×C59C4×D412yes—
114C2×C4×C59#12011yescharacteristic
115C2×C4×C59#12112yes—
116C2×C4×C59#12311yescharacteristic
117C2×Dic59—22no—
118C2×Dic59—41no—
119C2×Dic59—41no—
120C2×Dic59—41no—
121C2×Dic59—41no—
122C2×Dic59—41no—
123C2×Dic59—41no—
124C2×Dic59—42no—
125C2×Dic59—41no—
126C2×Dic59—41no—
127C2×Dic59—41no—
128C8×C59—82no—
129C59⋊C8—44no—
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—41no—
142C4×C59—41no—
143C4×C59—41no—
144Dic59—42no—
145Dic59—81no—
146Dic59—81no—
147Dic59—81no—
148#2983—591no—
149#2565—591no—
150#1534—592no—
151#1467D5911yescharacteristic
152#1285—592no—
153C2×C59—21no—
154C2×C59—21no—
155C2×C59#188411yescharacteristic
156C2×C59#190111yescharacteristic
157C2×C59#191211yescharacteristic
158C2×#125—591no—
159C2×C4×Q8C2×D5911yescharacteristic
160#446—1184no—
161#432—592no—
162#432—1181no—
163#432C2×D5912yes—
164#431—1181no—
165#398—1181no—
166#398—1181no—
167#391—591no—
168#391—591no—
169#387—1181no—
170#387—1181no—
171#384—591no—
172#382—591no—
173C4×C4⋊C4—591no—
174#330—1184no—
175#324—21no—
176#324—21no—
177C59#945011yescharacteristic, complemented, Sylow
178C22×Q8C4×D5911yescharacteristic
179#125—1182no—
180#125—1182no—
181C4×Q8—22no—
182C4×Q8—22no—
183C2×C4⋊C4—591no—
184C2×C4⋊C4—591no—
185C2×C4⋊C4—1181no—
186C2×C4⋊C4—1181no—
187C2×C4⋊C4—1181no—
188C2×C4⋊C4—1181no—
189C2×C4⋊C4—1181no—
190C2×C4⋊C4C4×D5911yescharacteristic
191C2×C4⋊C4C22⋊D5911yescharacteristic
192C2×C4⋊C4C22⋊D5911yescharacteristic
193C2×C42—591no—
194C2×C42—591no—
195C2×C42D23611yescharacteristic
196C2×C42D23611yescharacteristic
197C2×C42C22×D5911yescharacteristic
198C4⋊C8—22no—
199C4⋊C8—1182no—
200C4⋊C8—1182no—
201C4⋊C8—1182no—
202C22⋊C8—22no—
203C22⋊C8—1182no—
204#97—22no—
205#97—594no—
206#97—1182no—
207C4×C8—1182no—
208#95—1181no—
209#95—1181no—
210#95—1181no—
211#95—1181no—
212#95—1181no—
213#95—1181no—
214#95—1181no—
215C2×Q8—22no—
216C2×Q8—22no—
217C2×Q8—41no—
218C22×C4—1181no—
219C22×C4—1181no—
220C22×C4—1181no—
221C22×C4—1181no—
222C22×C4(C2×D59)⋊C411yescharacteristic
223C22×C4(C2×D59)⋊C411yescharacteristic
224C22×C4(C2×D59)⋊C411yescharacteristic
225C22×C4(C2×D59)⋊C411yescharacteristic
226C22×C4C2×C4×D5911yescharacteristic
227C22×C4C2×D23611yescharacteristic
228C22×C4C2×C22⋊D5911yescharacteristic
229C2×C8—42no—
230C2×C8—1184no—
231C4⋊C4—22no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—1182no—
235C4⋊C4—1182no—
236C4⋊C4—2362no—
237C4⋊C4—2362no—
238C4⋊C4—2362no—
239C4⋊C4—2361no—
240C4⋊C4—2362no—
241C4⋊C4—2361no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—2361no—
247C42—2361no—
248C42D4×D5912yes—
249C42#1174568712yes—
250C23C2×(C2×D59)⋊C411yescharacteristic
251Q8—44no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—1182no—
266C2×C4—2361no—
267C2×C4—2361no—
268C2×C4—2361no—
269C2×C4—2361no—
270C2×C4—2362no—
271C2×C4—2361no—
272C2×C4—2361no—
273C2×C4—2361no—
274C2×C4—2361no—
275C2×C4—2361no—
276C2×C4#1427364512yes—
277C2×C4(C22⋊D59)⋊C412yes—
278C2×C4#1427369711yescharacteristic
279C2×C4#1427369912yes—
280C2×C4#1427371111yescharacteristic
281C8—82no—
282C8—2364no—
283C22—21no—
284C22—21no—
285C22#1683727911yescharacteristic
286C22C2×#1427364511yescharacteristic
287C22#1683738411yescharacteristic
288C4—22no—
289C4—22no—
290C4—22no—
291C4—21no—
292C4—21no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—4721no—
297C4—4721no—
298C4—4721no—
299C4—2362no—
300C2—21no—
301C2—21no—
302C2#2142447711yescharacteristic
303C2#2142456211yescharacteristic
304C2#2142463111yescharacteristic
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#9450
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
177. C59#9450semidirect

Maximal subgroups

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

subgroupindex
#214331812
#214292832
#214227222
#214219532
C59×#14672
#945059