Finite Group Databasebeta

← #28498288

Subgroups — #28498288

Order 15104.

446 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 95 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#21422437C211yesmaximal, characteristic
2#21422414C211yesmaximal, characteristic
3#21422026C214yesmaximal, complemented
4C59×#2486C211yesmaximal, characteristic
5#16837459—21no—
6C2×C22⋊(C59⋊C8)—21no—
7#16837128—21no—
8#16837126—21no—
9#16837126C2212yes—
10#16837126C2212yes—
11C2×#123×C59C412yes—
12C2×C4×D4×C59C2211yescharacteristic
13C22×C4⋊C4×C59C411yescharacteristic
14C59×#398—21no—
15C59×#397—21no—
16C59×#389—21no—
17C59×#381C2212yes—
18C59×#380—21no—
19C59×#380C411yescharacteristic
20C22×C59⋊C8—41no—
21C4⋊(C59⋊C8)—41no—
22C22⋊(C59⋊C8)—22no—
23C22⋊(C59⋊C8)—22no—
24C22⋊(C59⋊C8)—41no—
25#14273638—41no—
26C22×D4×C59C2×C411yescharacteristic
27C23×C4×C59C2×C412yes—
28C23×C4×C59C2×C411yescharacteristic
29C23×C4×C59D411yescharacteristic
30#123×C59—44no—
31C4×D4×C59—41no—
32C4×D4×C59—41no—
33C2×C4⋊C4×C59—21no—
34C2×C4⋊C4×C59—21no—
35C2×C4⋊C4×C59—22no—
36C2×C4⋊C4×C59—21no—
37C2×C4⋊C4×C59—41no—
38C2×C4⋊C4×C59—41no—
39C2×C4⋊C4×C59D411yescharacteristic
40C2×C22⋊C4×C59—22no—
41C2×C22⋊C4×C59C2×C412yes—
42C2×C22⋊C4×C59D411yescharacteristic
43C2×C22⋊C4×C59C2311yescharacteristic
44C2×C42×C59D411yescharacteristic
45#95×C59—21no—
46#95×C59—22no—
47#95×C59—22no—
48C24×C59C22⋊C411yescharacteristic
49C24×C59C22×C411yescharacteristic
50C2×D4×C59—22no—
51C2×D4×C59—22no—
52C2×D4×C59—41no—
53C22×C4×C59—21no—
54C22×C4×C59—21no—
55C22×C4×C59—21no—
56C22×C4×C59—21no—
57C22×C4×C59—22no—
58C22×C4×C59—22no—
59C22×C4×C59—24no—
60C22×C4×C59—42no—
61C22×C4×C59—42no—
62C22×C4×C59—42no—
63C22×C4×C59—42no—
64C22×C4×C59C22⋊C411yescharacteristic
65C22×C4×C59C22⋊C411yescharacteristic
66C22×C4×C59C22⋊C411yescharacteristic
67C22×C4×C59C2×D411yescharacteristic
68C22×C4×C59C2×D411yescharacteristic
69C4⋊C4×C59—41no—
70C4⋊C4×C59—42no—
71C4⋊C4×C59—42no—
72C4⋊C4×C59—44no—
73C4⋊C4×C59—41no—
74C4⋊C4×C59—41no—
75C22⋊C4×C59—41no—
76C22⋊C4×C59—41no—
77C22⋊C4×C59—44no—
78C22⋊C4×C59—42no—
79C42×C59—21no—
80C42×C59—21no—
81C2×C59⋊C8—42no—
82C2×C59⋊C8—42no—
83C2×C59⋊C8—82no—
84C23×C59—21no—
85C23×C59—21no—
86C23×C59—21no—
87C23×C59—21no—
88C23×C59—41no—
89C23×C59—41no—
90C23×C59—41no—
91C23×C59—41no—
92C23×C59#10012yes—
93C23×C59C2×C22⋊C411yescharacteristic
94C23×C59#13112yes—
95D4×C59—44no—
96C2×C4×C59—22no—
97C2×C4×C59—21no—
98C2×C4×C59—21no—
99C2×C4×C59—21no—
100C2×C4×C59—21no—
101C2×C4×C59—21no—
102C2×C4×C59—21no—
103C2×C4×C59—42no—
104C2×C4×C59—42no—
105C2×C4×C59—42no—
106C2×C4×C59—41no—
107C2×C4×C59—41no—
108C2×C4×C59—41no—
109C2×C4×C59—42no—
110C2×C4×C59—42no—
111C2×C4×C59—42no—
112C2×C4×C59—42no—
113C2×C4×C59—42no—
114C2×C4×C59—42no—
115C2×C4×C59—81no—
116C2×C4×C59—84no—
117C2×C4×C59#9911yescharacteristic
118C2×C4×C59#9911yescharacteristic
119C59⋊C8—82no—
120C59⋊C8—82no—
121#11558—591nomaximal, Sylow
122C22×C59—21no—
123C22×C59—21no—
124C22×C59—22no—
125C22×C59—22no—
126C22×C59—42no—
127C22×C59—42no—
128C22×C59—42no—
129C22×C59—41no—
130C22×C59—41no—
131C22×C59—41no—
132C22×C59—41no—
133C22×C59—81no—
134C22×C59#40811yescharacteristic
135C22×C59C2×#9911yescharacteristic
136C22×C59C2×#10011yescharacteristic
137C4×C59—21no—
138C4×C59—41no—
139C4×C59—41no—
140C4×C59—41no—
141C4×C59—41no—
142C4×C59—82no—
143C4×C59—82no—
144C4×C59#35212yes—
145#2486D5911yescharacteristic
146#1466—591no—
147#1459—591no—
148#1317—594no—
149C2×C59—21no—
150C2×C59—21no—
151C2×C59—42no—
152C2×C59—42no—
153C2×C59#147111yescharacteristic
154C2×C59C2×#35211yescharacteristic
155C2×C59#212311yescharacteristic
156C2×#123Dic5912yes—
157C2×C4×D4C2×D5911yescharacteristic
158C22×C4⋊C4Dic5911yescharacteristic
159#432—1181no—
160C2×C22⋊C8—1181no—
161#398—21no—
162#397—21no—
163#389—21no—
164#381C2×D5912yes—
165#380—21no—
166#380Dic5911yescharacteristic
167#324—1181no—
168#323—592no—
169#323—592no—
170#323—1181no—
171C59#1155811yescharacteristic, complemented, Sylow
172C22×D4C2×Dic5911yescharacteristic
173C23×C4C2×Dic5912yes—
174C23×C4C2×Dic5911yescharacteristic
175C23×C4C22⋊D5911yescharacteristic
176C22×C8—2361no—
177#123—44no—
178C4×D4—41no—
179C4×D4—41no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4C22⋊D5911yescharacteristic
187C2×C22⋊C4—22no—
188C2×C22⋊C4C2×Dic5912yes—
189C2×C22⋊C4C22⋊D5911yescharacteristic
190C2×C22⋊C4C22×D5911yescharacteristic
191C2×C42C22⋊D5911yescharacteristic
192C4⋊C8—2361no—
193C22⋊C8—1182no—
194C22⋊C8—1182no—
195C22⋊C8—2361no—
196#97—2361no—
197#95—21no—
198#95—22no—
199#95—22no—
200C24C22⋊Dic5911yescharacteristic
201C24C22×Dic5911yescharacteristic
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—41no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4C22⋊Dic5911yescharacteristic
217C22×C4C22⋊Dic5911yescharacteristic
218C22×C4C22⋊Dic5911yescharacteristic
219C22×C4C2×C22⋊D5911yescharacteristic
220C22×C4C2×C22⋊D5911yescharacteristic
221C2×C8—4722no—
222C2×C8—2362no—
223C2×C8—2362no—
224C4⋊C4—41no—
225C4⋊C4—42no—
226C4⋊C4—44no—
227C4⋊C4—42no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C22⋊C4—41no—
231C22⋊C4—41no—
232C22⋊C4—44no—
233C22⋊C4—42no—
234C42—21no—
235C42—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23#1427364412yes—
245C23C2×C22⋊Dic5911yescharacteristic
246C23#1427373412yes—
247D4—44no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—81no—
268C2×C4—84no—
269C2×C4#1427364211yescharacteristic
270C2×C4#1427364211yescharacteristic
271C8—4722no—
272C8—4722no—
273C22—21no—
274C22—21no—
275C22—22no—
276C22—22no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—81no—
285C22#1683736911yescharacteristic
286C22C2×#1427364211yescharacteristic
287C22C2×#1427364411yescharacteristic
288C4—21no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—82no—
294C4—82no—
295C4#1683719212yes—
296C2—21no—
297C2—21no—
298C2—42no—
299C2—42no—
300C2#2142245611yescharacteristic
301C2C2×#1683719211yescharacteristic
302C2#2142575011yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC24×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#2486C2
socleC22×C59C2×#99
prime radicalC22×D4×C59C2×C4
cosocleC2×C22⋊C4×C59C23
hypercenter#2486D59
hypocenterC59#11558
LCSG ▷ C24×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2486

Complements

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

NH ≅ G/Ntype
3. #21422026C2semidirect
171. C59#11558semidirect

Maximal subgroups

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

subgroupindex
#214224372
#214224142
#214220262
C59×#24862
#1155859