Finite Group Databasebeta

← #28499747

Subgroups — #28499747

Order 15104.

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

normal subgroups only

This group has 408 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#21424092C211yesmaximal, characteristic
2#21422420C211yesmaximal, characteristic
3#21422102C214yesmaximal
4C59×#2546C211yesmaximal, characteristic
5#16837455—21no—
6#16837426—21no—
7C2×#14273646—21no—
8#16837171C2212yes—
9#16837128—21no—
10#16837128C2212yes—
11C2×C4⋊Q8×C59—21no—
12C2×C4⋊Q8×C59C411yescharacteristic
13C2×C4×Q8×C59C2211yescharacteristic
14C59×#398C412yes—
15C59×#398C2212yes—
16C59×#391—21no—
17C59×#389—21no—
18C59×#389—21no—
19C4×C4⋊C4×C59C411yescharacteristic
20C2×#11745664—22no—
21C4⋊(C59⋊C8)—41no—
22C4⋊(C59⋊C8)—41no—
23C4⋊(C59⋊C8)—41no—
24#14273646—42no—
25C22⋊(C59⋊C8)—22no—
26C4×C59⋊C8—41no—
27C22×Q8×C59—21no—
28C22×Q8×C59D411yescharacteristic
29C4⋊Q8×C59—41no—
30C4⋊Q8×C59—41no—
31C4⋊Q8×C59—42no—
32C4⋊Q8×C59—42no—
33C4×Q8×C59—41no—
34C4×Q8×C59—41no—
35C2×C4⋊C4×C59—21no—
36C2×C4⋊C4×C59—21no—
37C2×C4⋊C4×C59—22no—
38C2×C4⋊C4×C59—22no—
39C2×C4⋊C4×C59C2×C412yes—
40C2×C4⋊C4×C59C2×C411yescharacteristic
41C2×C4⋊C4×C59D411yescharacteristic
42C2×C4⋊C4×C59C2311yescharacteristic
43C2×C42×C59—21no—
44C2×C42×C59C2×C411yescharacteristic
45C2×C42×C59D411yescharacteristic
46C2×C42×C59D411yescharacteristic
47#95×C59—22no—
48#95×C59—22no—
49#95×C59C2×C412yes—
50C2×Q8×C59—41no—
51C2×Q8×C59—41no—
52C2×Q8×C59—42no—
53C2×Q8×C59—42no—
54C2×Q8×C59—42no—
55C2×Q8×C59D812yes—
56C2×Q8×C59SD1612yes—
57C22×C4×C59—22no—
58C22×C4×C59—22no—
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×C4×C59C2×D411yescharacteristic
66C4⋊C4×C59—24no—
67C4⋊C4×C59—41no—
68C4⋊C4×C59—41no—
69C4⋊C4×C59—41no—
70C4⋊C4×C59—42no—
71C4⋊C4×C59—41no—
72C4⋊C4×C59—41no—
73C4⋊C4×C59—44no—
74C4⋊C4×C59—42no—
75C4⋊C4×C59—42no—
76C42×C59—21no—
77C42×C59—21no—
78C42×C59—21no—
79C42×C59—21no—
80C42×C59—21no—
81C42×C59—21no—
82C42×C59—42no—
83#11745664—42no—
84#11745664—42no—
85C2×C59⋊C8—42no—
86C2×C59⋊C8—42no—
87C23×C59C2×C22⋊C411yescharacteristic
88Q8×C59—42no—
89Q8×C59—42no—
90Q8×C59—84no—
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—22no—
98C2×C4×C59—24no—
99C2×C4×C59—21no—
100C2×C4×C59—21no—
101C2×C4×C59—21no—
102C2×C4×C59—42no—
103C2×C4×C59—42no—
104C2×C4×C59—41no—
105C2×C4×C59—41no—
106C2×C4×C59—41no—
107C2×C4×C59—42no—
108C2×C4×C59—42no—
109C2×C4×C59—42no—
110C2×C4×C59#9911yescharacteristic
111C2×C4×C59#9911yescharacteristic
112C2×C4×C59#10212yes—
113C2×C4×C59#10212yes—
114C2×C4×C59C2×D811yescharacteristic
115C2×C4×C59C2×SD1611yescharacteristic
116C59⋊C8—82no—
117C59⋊C8—82no—
118#11883—591nomaximal, Sylow
119C22×C59—21no—
120C22×C59—21no—
121C22×C59C2×#9911yescharacteristic
122C22×C59C2×#10211yescharacteristic
123C22×C59#42111yescharacteristic
124C4×C59—21no—
125C4×C59—21no—
126C4×C59—22no—
127C4×C59—22no—
128C4×C59—21no—
129C4×C59—44no—
130C4×C59—41no—
131C4×C59—41no—
132C4×C59—41no—
133C4×C59—82no—
134C4×C59#35612yes—
135#2546D5911yescharacteristic
136#1801—591no—
137#1461—591no—
138#1346—594no—
139C2×C59—21no—
140C2×C59—21no—
141C2×C59#151911yescharacteristic
142C2×C59#212311yescharacteristic
143C2×C59C2×#35611yescharacteristic
144C2×C4⋊Q8—21no—
145C2×C4⋊Q8Dic5911yescharacteristic
146C2×C4×Q8C2×D5911yescharacteristic
147#431—1181no—
148#423—1181no—
149C2×#101—1181no—
150#398Dic5912yes—
151#398C2×D5912yes—
152#391—21no—
153#389—21no—
154#389—21no—
155C4×C4⋊C4Dic5911yescharacteristic
156#343—592no—
157#324—592no—
158#324—1181no—
159C59#1188311yescharacteristic, complemented, Sylow
160C22×Q8—21no—
161C22×Q8C22⋊D5911yescharacteristic
162C2×M16—1182no—
163C4⋊Q8—41no—
164C4⋊Q8—41no—
165C4⋊Q8—42no—
166C4⋊Q8—42no—
167C4×Q8—41no—
168C4×Q8—41no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4C2×Dic5912yes—
174C2×C4⋊C4C2×Dic5911yescharacteristic
175C2×C4⋊C4C22⋊D5911yescharacteristic
176C2×C4⋊C4C22×D5911yescharacteristic
177C2×C42—21no—
178C2×C42C2×Dic5911yescharacteristic
179C2×C42C22⋊D5911yescharacteristic
180C2×C42C22⋊D5911yescharacteristic
181C4⋊C8—2361no—
182C4⋊C8—2361no—
183C4⋊C8—2361no—
184#101—2362no—
185C22⋊C8—1182no—
186C4×C8—2361no—
187#95—22no—
188#95—22no—
189#95C2×Dic5912yes—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8—42no—
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8#1174566912yes—
196C2×Q8#1174567012yes—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4C22⋊Dic5911yescharacteristic
200C22×C4C22⋊Dic5911yescharacteristic
201C22×C4C22⋊Dic5911yescharacteristic
202C22×C4C22⋊Dic5911yescharacteristic
203C22×C4C22×Dic5911yescharacteristic
204C22×C4C2×C22⋊D5911yescharacteristic
205C22×C4C2×C22⋊D5911yescharacteristic
206M16—2362no—
207M16—2362no—
208C2×C8—2362no—
209C2×C8—2362no—
210C4⋊C4—24no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—42no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—44no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—42no—
227C23C2×C22⋊Dic5911yescharacteristic
228Q8—42no—
229Q8—42no—
230Q8—84no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—24no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4#1427364211yescharacteristic
251C2×C4#1427364211yescharacteristic
252C2×C4#1427364812yes—
253C2×C4#1427364812yes—
254C2×C4C2×#1174566911yescharacteristic
255C2×C4C2×#1174567011yescharacteristic
256C8—4722no—
257C8—4722no—
258C22—21no—
259C22—21no—
260C22C2×#1427364211yescharacteristic
261C22C2×#1427364811yescharacteristic
262C22#1683741911yescharacteristic
263C4—21no—
264C4—22no—
265C4—22no—
266C4—21no—
267C4—21no—
268C4—44no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—82no—
273C4#1683720112yes—
274C2—21no—
275C2—21no—
276C2#2142266411yescharacteristic
277C2#2142575011yescharacteristic
278C2C2×#1683720111yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2546C2
socleC22×C59C2×#99
prime radicalC23×C59C2×C22⋊C4
cosocleC2×C4⋊C4×C59C23
hypercenter#2546D59
hypocenterC59#11883
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2546

Complements

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

NH ≅ G/Ntype
159. C59#11883semidirect

Maximal subgroups

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

subgroupindex
#214240922
#214224202
#214221022
C59×#25462
#1188359