Finite Group Databasebeta

← #28498977

Subgroups — #28498977

Order 15104.

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

normal subgroups only

This group has 448 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#21429917C211yesmaximal, characteristic
2#21422410C211yesmaximal, characteristic
3#21422116C214yesmaximal
4C59×#1740C211yesmaximal, characteristic
5C2×#14273727C411yescharacteristic, complemented
6#16837455—21no—
7#16837341—21no—
8#16837338—21no—
9#16837322C412yescomplemented
10#16837314C2212yes—
11#16837300—21no—
12#16837268C411yescharacteristic, complemented
13#16837262—21no—
14#16837128—21no—
15#16837126C2212yes—
16C2×#117×C59—21no—
17C2×#117×C59C2211yescharacteristic
18C59×#342—21no—
19C59×#342C2212yes—
20C22×Dic118C2×C412yes—
21#14273727—41no—
22#14273727—41no—
23C2×C4⋊Dic59—21no—
24C2×C4⋊Dic59—21no—
25C2×C4⋊Dic59—21no—
26C2×Dic59⋊C4—22no—
27C2×C22⋊Dic59—22no—
28C4⋊(C59⋊C8)—41no—
29C22⋊(C59⋊C8)—22no—
30C4×C59⋊C8—41no—
31C59⋊#95—22no—
32C59⋊#95—21no—
33C59⋊#95—22no—
34C59⋊#95C2×C412yes—
35C23×C4×C59D411yescharacteristic
36#117×C59—24no—
37#117×C59—41no—
38#117×C59—41no—
39#117×C59—42no—
40C2×C4⋊C4×C59—41no—
41C2×C4⋊C4×C59C2×C411yescharacteristic
42C2×C4⋊C4×C59D411yescharacteristic
43C2×C22⋊C4×C59—22no—
44C2×C22⋊C4×C59D411yescharacteristic
45C2×C22⋊C4×C59C2311yescharacteristic
46C2×C42×C59—41no—
47C2×C42×C59C2×C411yescharacteristic
48C2×C42×C59D411yescharacteristic
49C24×C59C2×D411yescharacteristic
50C22×C4×C59—21no—
51C22×C4×C59—21no—
52C22×C4×C59—22no—
53C22×C4×C59—42no—
54C22×C4×C59C22⋊C411yescharacteristic
55C22×C4×C59C22⋊C411yescharacteristic
56C22×C4×C59C22⋊C411yescharacteristic
57C22×C4×C59C22⋊C411yescharacteristic
58C22×C4×C59SD1612yes—
59C22×C4×C59Q1612yes—
60C22×C4×C59C22×C411yescharacteristic
61C22×C4×C59C2×D411yescharacteristic
62C22×Dic59—22no—
63C22×Dic59—22no—
64C2×Dic118—42no—
65C2×Dic118—82no—
66C4⋊C4×C59—41no—
67C4⋊C4×C59—41no—
68C4⋊C4×C59—44no—
69C22⋊C4×C59—24no—
70C22⋊C4×C59—41no—
71C22⋊C4×C59—41no—
72C22⋊C4×C59—42no—
73C42×C59—21no—
74C42×C59—21no—
75C42×C59—21no—
76C42×C59—21no—
77C42×C59—44no—
78C22⋊Dic59—82no—
79C4⋊Dic59—42no—
80C4⋊Dic59—41no—
81C4⋊Dic59—41no—
82C4⋊Dic59—42no—
83Dic59⋊C4—82no—
84C2×C59⋊C8—42no—
85C23×C59—21no—
86C23×C59—21no—
87C23×C59—41no—
88C23×C59—41no—
89C23×C59C2×C22⋊C411yescharacteristic
90C23×C59C2×SD1611yescharacteristic
91C23×C59C2×Q1611yescharacteristic
92C2×C4×C59—21no—
93C2×C4×C59—21no—
94C2×C4×C59—22no—
95C2×C4×C59—22no—
96C2×C4×C59—22no—
97C2×C4×C59—22no—
98C2×C4×C59—21no—
99C2×C4×C59—21no—
100C2×C4×C59—21no—
101C2×C4×C59—21no—
102C2×C4×C59—41no—
103C2×C4×C59—41no—
104C2×C4×C59—41no—
105C2×C4×C59—41no—
106C2×C4×C59—44no—
107C2×C4×C59—82no—
108C2×C4×C59—82no—
109C2×C4×C59#9911yescharacteristic
110C2×C4×C59#9911yescharacteristic
111C2×C4×C59#10312yes—
112C2×C4×C59#10312yes—
113C2×C4×C59#10412yes—
114C2×Dic59—42no—
115C2×Dic59—42no—
116C2×Dic59—42no—
117C2×Dic59—42no—
118C2×Dic59—42no—
119C2×Dic59—42no—
120Dic118—84no—
121C59⋊C8—82no—
122#11717—591nomaximal, Sylow
123C22×C59—21no—
124C22×C59—21no—
125C22×C59—21no—
126C22×C59—21no—
127C22×C59—41no—
128C22×C59—41no—
129C22×C59—41no—
130C22×C59—41no—
131C22×C59—81no—
132C22×C59C2×#9911yescharacteristic
133C22×C59C2×#10311yescharacteristic
134C22×C59C2×#10411yescharacteristic
135C4×C59—22no—
136C4×C59—41no—
137C4×C59—41no—
138C4×C59—41no—
139C4×C59—41no—
140C4×C59—84no—
141C4×C59#32812yes—
142C4×C59#32812yes—
143Dic59—82no—
144Dic59—82no—
145#2628—591no—
146#1740D5911yescharacteristic
147#1458—591no—
148#1351—594no—
149C2×C59—21no—
150C2×C59—21no—
151C2×C59—41no—
152C2×C59—41no—
153C2×C59C2×#32811yescharacteristic
154C2×C59#212211yescharacteristic
155C2×C59#212911yescharacteristic
156C2×C4⋊Q8—591no—
157C2×#117—21no—
158C2×#117C2×D5911yescharacteristic
159#431—1181no—
160#400—1181no—
161#399—1181no—
162#395—592no—
163#393—592no—
164#389—1181no—
165#382—591no—
166#380—1181no—
167#342—21no—
168#342C2×D5912yes—
169#324—1181no—
170#323—592no—
171C59#1171711yescharacteristic, complemented, Sylow
172C22×Q8—592no—
173C23×C4C22⋊D5911yescharacteristic
174C4⋊Q8—2361no—
175C4⋊Q8—2361no—
176#117—24no—
177#117—41no—
178#117—41no—
179#117—42no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—1182no—
182C2×C4⋊C4—1181no—
183C2×C4⋊C4—1181no—
184C2×C4⋊C4—1181no—
185C2×C4⋊C4C4×D5911yescharacteristic
186C2×C4⋊C4D23611yescharacteristic
187C2×C22⋊C4—22no—
188C2×C22⋊C4—1182no—
189C2×C22⋊C4C22⋊D5911yescharacteristic
190C2×C22⋊C4C22×D5911yescharacteristic
191C2×C42—41no—
192C2×C42C4×D5911yescharacteristic
193C2×C42D23611yescharacteristic
194C4⋊C8—2361no—
195C22⋊C8—1182no—
196C4×C8—2361no—
197#95—592no—
198#95—1182no—
199#95—1181no—
200#95—1182no—
201C24C2×C22⋊D5911yescharacteristic
202C2×Q8—4722no—
203C2×Q8—2362no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—42no—
208C22×C4—1182no—
209C22×C4—1182no—
210C22×C4(C2×D59)⋊C411yescharacteristic
211C22×C4(C2×D59)⋊C411yescharacteristic
212C22×C4(C2×D59)⋊C411yescharacteristic
213C22×C4(C2×D59)⋊C411yescharacteristic
214C22×C4#1174567012yes—
215C22×C4#1174567212yes—
216C22×C4C2×C4×D5911yescharacteristic
217C22×C4C2×D23611yescharacteristic
218C2×C8—2362no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—44no—
222C4⋊C4—4722no—
223C4⋊C4—2362no—
224C4⋊C4—2361no—
225C4⋊C4—2361no—
226C4⋊C4—2362no—
227C22⋊C4—24no—
228C22⋊C4—41no—
229C22⋊C4—41no—
230C22⋊C4—42no—
231C22⋊C4—4722no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—44no—
237C23—21no—
238C23—21no—
239C23—41no—
240C23—41no—
241C23C2×(C2×D59)⋊C411yescharacteristic
242C23C2×#1174567011yescharacteristic
243C23C2×#1174567211yescharacteristic
244Q8—4724no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—44no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—2362no—
263C2×C4—2362no—
264C2×C4—2362no—
265C2×C4—2362no—
266C2×C4—2362no—
267C2×C4—2362no—
268C2×C4#1427364111yescharacteristic
269C2×C4#1427364111yescharacteristic
270C2×C4#1427365012yes—
271C2×C4#1427365012yes—
272C2×C4#1427365512yes—
273C8—4722no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—81no—
283C22C2×#1427364111yescharacteristic
284C22C2×#1427365011yescharacteristic
285C22C2×#1427365511yescharacteristic
286C4—22no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—84no—
292C4—4722no—
293C4—4722no—
294C4#1683713812yes—
295C4#1683713812yes—
296C2—21no—
297C2—21no—
298C2—41no—
299C2—41no—
300C2C2×#1683713811yescharacteristic
301C2#2142574511yescharacteristic
302C2#2142577011yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#1740C2
socleC22×C59C2×#99
prime radicalC24×C59C2×D4
cosocleC2×C22⋊C4×C59C23
hypercenter#1740D59
hypocenterC59#11717
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1740

Complements

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

NH ≅ G/Ntype
5. C2×#14273727C4semidirect
9. #16837322C4semidirect
12. #16837268C4semidirect
171. C59#11717semidirect

Maximal subgroups

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

subgroupindex
#214299172
#214224102
#214221162
C59×#17402
#1171759