Finite Group Databasebeta

← #28499367

Subgroups — #28499367

Order 15104.

402 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 87 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#21424173C211yesmaximal, characteristic
2#21422430C211yesmaximal, characteristic
3#21422096C214yesmaximal
4C59×#2621C211yesmaximal, characteristic
5#16837459—21no—
6#16837265—21no—
7#16837169—21no—
8#16837169C2212yes—
9#16837128—21no—
10#16837126C2212yes—
11C2×C4⋊Q8×C59C411yescharacteristic, complemented
12C2×C22⋊Q8×C59—21no—
13C2×#117×C59C2211yescharacteristic
14C59×#398—21no—
15C59×#398C412yescomplemented
16C59×#397—21no—
17C59×#397C2212yes—
18C59×#391—21no—
19C59×#382C411yescharacteristic, complemented
20C2×C22⋊Dic59—22no—
21C4⋊(C59⋊C8)—41no—
22C22⋊(C59⋊C8)—22no—
23#14273638—41no—
24C59⋊#95—41no—
25C59⋊#95—41no—
26C22×Q8×C59—21no—
27C23×C4×C59D411yescharacteristic
28C4⋊Q8×C59—41no—
29C4⋊Q8×C59—41no—
30C22⋊Q8×C59—44no—
31#117×C59—41no—
32#117×C59—41no—
33C2×C4⋊C4×C59—21no—
34C2×C4⋊C4×C59—21no—
35C2×C4⋊C4×C59—22no—
36C2×C4⋊C4×C59C2×C412yes—
37C2×C4⋊C4×C59C2×C411yescharacteristic
38C2×C4⋊C4×C59D411yescharacteristic
39C2×C22⋊C4×C59—22no—
40C2×C22⋊C4×C59D411yescharacteristic
41C2×C22⋊C4×C59C2311yescharacteristic
42C2×C42×C59C2×C411yescharacteristic
43C2×C42×C59D411yescharacteristic
44#95×C59—21no—
45#95×C59—22no—
46#95×C59—22no—
47#95×C59C2×C412yes—
48C24×C59C2×D411yescharacteristic
49C2×Q8×C59—42no—
50C2×Q8×C59—42no—
51C2×Q8×C59—41no—
52C2×Q8×C59—41no—
53C22×C4×C59—22no—
54C22×C4×C59—21no—
55C22×C4×C59—21no—
56C22×C4×C59—22no—
57C22×C4×C59—22no—
58C22×C4×C59—41no—
59C22×C4×C59C22⋊C411yescharacteristic
60C22×C4×C59C22⋊C411yescharacteristic
61C22×C4×C59C22⋊C411yescharacteristic
62C22×C4×C59C22⋊C411yescharacteristic
63C22×C4×C59C22×C411yescharacteristic
64C22×C4×C59C2×D411yescharacteristic
65C22×Dic59—42no—
66C4⋊C4×C59—41no—
67C4⋊C4×C59—41no—
68C4⋊C4×C59—42no—
69C4⋊C4×C59—44no—
70C4⋊C4×C59—42no—
71C4⋊C4×C59—42no—
72C22⋊C4×C59—41no—
73C22⋊C4×C59—41no—
74C22⋊C4×C59—44no—
75C42×C59—21no—
76C42×C59—21no—
77C42×C59—21no—
78C42×C59—21no—
79C22⋊Dic59—42no—
80C22⋊Dic59—42no—
81C2×C59⋊C8—42no—
82C23×C59—21no—
83C23×C59—21no—
84C23×C59—41no—
85C23×C59—41no—
86C23×C59C2×C22⋊C411yescharacteristic
87C23×C59AGL(1,ℤ/8)12yes—
88Q8×C59—82no—
89Q8×C59—82no—
90C2×C4×C59—21no—
91C2×C4×C59—21no—
92C2×C4×C59—21no—
93C2×C4×C59—21no—
94C2×C4×C59—21no—
95C2×C4×C59—21no—
96C2×C4×C59—21no—
97C2×C4×C59—21no—
98C2×C4×C59—21no—
99C2×C4×C59—42no—
100C2×C4×C59—42no—
101C2×C4×C59—41no—
102C2×C4×C59—41no—
103C2×C4×C59—41no—
104C2×C4×C59—41no—
105C2×C4×C59—42no—
106C2×C4×C59—42no—
107C2×C4×C59—42no—
108C2×C4×C59—42no—
109C2×C4×C59—42no—
110C2×C4×C59—42no—
111C2×C4×C59#9911yescharacteristic
112C2×C4×C59#9911yescharacteristic
113C2×Dic59—82no—
114C2×Dic59—82no—
115C2×Dic59—82no—
116C59⋊C8—82no—
117#11801—591nomaximal, Sylow
118C22×C59—21no—
119C22×C59—21no—
120C22×C59—22no—
121C22×C59—42no—
122C22×C59—41no—
123C22×C59—41no—
124C22×C59—81no—
125C22×C59C2×#9911yescharacteristic
126C22×C59#41811yescharacteristic
127C22×C59#42111yescharacteristic
128C4×C59—21no—
129C4×C59—21no—
130C4×C59—41no—
131C4×C59—41no—
132C4×C59—41no—
133C4×C59—41no—
134C4×C59—41no—
135C4×C59—82no—
136C4×C59—82no—
137Dic59—162no—
138#2621D5911yescharacteristic
139#1820—591no—
140#1464—591no—
141#1344—594no—
142C2×C59—21no—
143C2×C59—21no—
144C2×C59—42no—
145C2×C59#151011yescharacteristic
146C2×C59#212311yescharacteristic
147C2×C59#212911yescharacteristic
148C2×C4⋊Q8Dic5911yescharacteristic, complemented
149C2×C22⋊Q8—21no—
150C2×#117C2×D5911yescharacteristic
151#432—1181no—
152#398—21no—
153#398Dic5912yescomplemented
154#397—21no—
155#397C2×D5912yes—
156#391—21no—
157#382Dic5911yescharacteristic, complemented
158#381—1181no—
159#342—592no—
160#342—1181no—
161#324—1181no—
162#323—592no—
163C59#1180111yescharacteristic, complemented, Sylow
164C22×Q8—21no—
165C23×C4C22⋊D5911yescharacteristic
166C4⋊Q8—41no—
167C4⋊Q8—41no—
168C22⋊Q8—44no—
169#117—41no—
170#117—41no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4C2×Dic5912yes—
175C2×C4⋊C4C2×Dic5911yescharacteristic
176C2×C4⋊C4C22⋊D5911yescharacteristic
177C2×C22⋊C4—22no—
178C2×C22⋊C4—1182no—
179C2×C22⋊C4C22⋊D5911yescharacteristic
180C2×C22⋊C4C22×D5911yescharacteristic
181C2×C42C2×Dic5911yescharacteristic
182C2×C42C22⋊D5911yescharacteristic
183C4⋊C8—2361no—
184C22⋊C8—1182no—
185#97—2361no—
186#95—21no—
187#95—22no—
188#95—22no—
189#95—2361no—
190#95—2361no—
191#95C2×Dic5912yes—
192C24C2×C22⋊D5911yescharacteristic
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8—41no—
196C2×Q8—41no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—41no—
203C22×C4—2362no—
204C22×C4C22⋊Dic5911yescharacteristic
205C22×C4C22⋊Dic5911yescharacteristic
206C22×C4C22⋊Dic5911yescharacteristic
207C22×C4C22⋊Dic5911yescharacteristic
208C22×C4C22×Dic5911yescharacteristic
209C22×C4C2×C22⋊D5911yescharacteristic
210C2×C8—2362no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—44no—
216C4⋊C4—42no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—44no—
220C22⋊C4—2362no—
221C22⋊C4—2362no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C23—21no—
227C23—21no—
228C23—41no—
229C23—41no—
230C23C2×C22⋊Dic5911yescharacteristic
231C23#1427375312yes—
232Q8—82no—
233Q8—82no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—4722no—
256C2×C4—4722no—
257C2×C4—4722no—
258C2×C4#1427364211yescharacteristic
259C2×C4#1427364211yescharacteristic
260C8—4722no—
261C22—21no—
262C22—21no—
263C22—22no—
264C22—42no—
265C22—41no—
266C22—41no—
267C22—81no—
268C22C2×#1427364211yescharacteristic
269C22#1683740711yescharacteristic
270C22#1683741911yescharacteristic
271C4—21no—
272C4—21no—
273C4—41no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—41no—
278C4—82no—
279C4—82no—
280C4—9442no—
281C2—21no—
282C2—21no—
283C2—42no—
284C2#2142262511yescharacteristic
285C2#2142575011yescharacteristic
286C2#2142577111yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#2621C2
socleC22×C59C2×#99
prime radicalC24×C59C2×D4
cosocleC2×C22⋊C4×C59C23
hypercenter#2621D59
hypocenterC59#11801
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2621

Complements

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

NH ≅ G/Ntype
11. C2×C4⋊Q8×C59C4semidirect
15. C59×#398C4semidirect
19. C59×#382C4semidirect
148. C2×C4⋊Q8Dic59semidirect
153. #398Dic59semidirect
157. #382Dic59semidirect
163. C59#11801semidirect

Maximal subgroups

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

subgroupindex
#214241732
#214224302
#214220962
C59×#26212
#1180159