Finite Group Databasebeta

← #28482227

Subgroups — #28482227

Order 15104.

394 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 83 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#21426698C211yesmaximal, characteristic
2#21422425C211yesmaximal, characteristic
3#21422425C211yesmaximal, characteristic
4#21422003C211yesmaximal, characteristic
5#21422002C211yesmaximal, characteristic
6#21422001C211yesmaximal, characteristic
7C59×#1305C211yesmaximal, characteristic
8C22×Dic59⋊C4C2211yescharacteristic
9#16837366C2211yescharacteristic
10#16837366C2211yescharacteristic
11#16837289—22no—
12#16837283—22no—
13#16837263C412yes—
14#16837262C2211yescharacteristic
15C22⋊C4×Dic59C412yes—
16#16837127—22no—
17#16837127—22no—
18C59×#407C2211yescharacteristic
19C59×#407C2211yescharacteristic
20C59×#380C2211yescharacteristic
21C23×Dic59C2×C411yescharacteristic
22C23×Dic59C2×C411yescharacteristic
23C2×#11745664—21no—
24C2×#11745664—21no—
25C2×C4⋊Dic59—21no—
26C2×Dic59⋊C4—21no—
27C2×Dic59⋊C4—21no—
28C2×Dic59⋊C4—21no—
29C2×Dic59⋊C4—21no—
30C2×Dic59⋊C4D412yes—
31C2×Dic59⋊C4D412yes—
32C2×C22⋊Dic59C2×C412yes—
33C2×C4×Dic59—22no—
34C22⋊(C59⋊C8)—21no—
35C22⋊(C59⋊C8)—21no—
36C22⋊(C59⋊C8)—21no—
37C22⋊(C59⋊C8)—21no—
38C59⋊#95—21no—
39C59⋊#95—22no—
40C59⋊#95—24no—
41C23×C4×C59C2311yescharacteristic
42C2×M16×C59—21no—
43C2×M16×C59—21no—
44C2×C4⋊C4×C59—21no—
45C2×C22⋊C4×C59C2×C412yes—
46C22⋊C8×C59—21no—
47C22⋊C8×C59—21no—
48C22⋊C8×C59—21no—
49C22⋊C8×C59—21no—
50#95×C59—21no—
51C24×C59C22×C411yescharacteristic
52C22×C4×C59—22no—
53C22×C4×C59—81no—
54C22×C4×C59C2×D411yescharacteristic
55C22×C4×C59C2×D411yescharacteristic
56C22×C4×C59#4111yescharacteristic
57C22×C4×C59#4111yescharacteristic
58C22×C4×C59#4111yescharacteristic
59C22×C4×C59#4111yescharacteristic
60C22×Dic59—21no—
61C22×Dic59—21no—
62C22×Dic59—21no—
63C22×Dic59—21no—
64C22×Dic59—22no—
65C22×Dic59—41no—
66C22×Dic59—41no—
67C22×Dic59—81no—
68C22×Dic59C22⋊C412yes—
69C22×Dic59C22⋊C412yes—
70M16×C59—81no—
71M16×C59—81no—
72C2×C8×C59—41no—
73C2×C8×C59—41no—
74C4⋊C4×C59—81no—
75C22⋊C4×C59—42no—
76C22⋊Dic59—42no—
77C4⋊Dic59—81no—
78Dic59⋊C4—22no—
79Dic59⋊C4—22no—
80Dic59⋊C4—22no—
81Dic59⋊C4—22no—
82C4×Dic59—82no—
83#11745664—81no—
84#11745664—81no—
85C2×C59⋊C8—41no—
86C2×C59⋊C8—41no—
87C23×C59—21no—
88C23×C59—21no—
89C23×C59—41no—
90C23×C59—41no—
91C23×C59C2×C22⋊C411yescharacteristic
92C23×C59#11711yescharacteristic
93C23×C59#11711yescharacteristic
94C2×C4×C59—41no—
95C2×C4×C59—41no—
96C2×C4×C59—41no—
97C2×C4×C59—41no—
98C2×C4×C59—41no—
99C2×C4×C59—41no—
100C2×C4×C59—42no—
101C2×C4×C59—42no—
102C2×C4×C59—42no—
103C2×C4×C59#12311yescharacteristic
104C2×C4×C59#12311yescharacteristic
105C2×C4×C59#12311yescharacteristic
106C2×C4×C59#12311yescharacteristic
107C2×Dic59—22no—
108C2×Dic59—22no—
109C2×Dic59—22no—
110C2×Dic59—22no—
111C2×Dic59—42no—
112C2×Dic59—42no—
113C2×Dic59—42no—
114C2×Dic59—42no—
115C2×Dic59—42no—
116C2×Dic59—42no—
117C2×Dic59—81no—
118C2×Dic59—81no—
119C8×C59—81no—
120C8×C59—81no—
121C59⋊C8—81no—
122C59⋊C8—81no—
123#8758—591nomaximal, Sylow
124C22×C59—21no—
125C22×C59—21no—
126C22×C59—21no—
127C22×C59—21no—
128C22×C59—21no—
129C22×C59—21no—
130C22×C59—21no—
131C22×C59—21no—
132C22×C59—42no—
133C22×C59—42no—
134C22×C59#38111yescharacteristic
135C22×C59#41311yescharacteristic
136C22×C59#41311yescharacteristic
137C4×C59—21no—
138C4×C59—21no—
139C4×C59—21no—
140C4×C59—21no—
141C4×C59—82no—
142Dic59—42no—
143Dic59—42no—
144Dic59—82no—
145#2313—591no—
146#1463—591no—
147#1463—591no—
148#1305—591no—
149#1305—591no—
150#1305—591no—
151#1305D5911yescharacteristic
152C2×C59—21no—
153C2×C59—21no—
154C2×C59—41no—
155C2×C59—41no—
156C2×C59#182211yescharacteristic
157C2×C59#182811yescharacteristic
158C2×C59#183011yescharacteristic
159C22×C4⋊C4—591no—
160#407—591no—
161#407—591no—
162#407C2×D5911yescharacteristic
163#407C2×D5911yescharacteristic
164#387—1182no—
165#387—1182no—
166#381—592no—
167#380—591no—
168#380C2×D5911yescharacteristic
169C4×C22⋊C4—592no—
170#324—1182no—
171#324—1182no—
172C59#875811yescharacteristic, complemented, Sylow
173C23×C4—591no—
174C23×C4—591no—
175C23×C4C22×D5911yescharacteristic
176C2×M16—21no—
177C2×M16—21no—
178C2×M16—1181no—
179C2×M16—1181no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—592no—
182C2×C4⋊C4—592no—
183C2×C4⋊C4—1181no—
184C2×C4⋊C4—1181no—
185C2×C4⋊C4—1181no—
186C2×C4⋊C4—1181no—
187C2×C4⋊C4—1181no—
188C2×C22⋊C4—592no—
189C2×C22⋊C4C4×D5912yes—
190C2×C42—1182no—
191C22⋊C8—21no—
192C22⋊C8—21no—
193C22⋊C8—21no—
194C22⋊C8—21no—
195C22⋊C8—1181no—
196C22⋊C8—1181no—
197C22⋊C8—1181no—
198C22⋊C8—1181no—
199#95—21no—
200#95—1182no—
201#95—1184no—
202#95—1181no—
203C24C2×C4×D5911yescharacteristic
204C22×C4—22no—
205C22×C4—81no—
206C22×C4—592no—
207C22×C4—592no—
208C22×C4—1181no—
209C22×C4—1181no—
210C22×C4—1181no—
211C22×C4—1181no—
212C22×C4—1182no—
213C22×C4—4721no—
214C22×C4—2361no—
215C22×C4—2361no—
216C22×C4#1174568511yescharacteristic
217C22×C4#1174568511yescharacteristic
218C22×C4D4×D5911yescharacteristic
219C22×C4D4×D5911yescharacteristic
220C22×C4#1174568711yescharacteristic
221C22×C4#1174568911yescharacteristic
222M16—81no—
223M16—81no—
224M16—4721no—
225M16—4721no—
226C2×C8—41no—
227C2×C8—41no—
228C2×C8—2361no—
229C2×C8—2361no—
230C4⋊C4—81no—
231C4⋊C4—1182no—
232C4⋊C4—1182no—
233C4⋊C4—1182no—
234C4⋊C4—1182no—
235C4⋊C4—4721no—
236C22⋊C4—42no—
237C22⋊C4—2362no—
238C42—4722no—
239C23—21no—
240C23—21no—
241C23—41no—
242C23—41no—
243C23C22⋊C4×D5911yescharacteristic
244C23#1427368411yescharacteristic
245C23#1427368511yescharacteristic
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—1182no—
256C2×C4—1182no—
257C2×C4—1182no—
258C2×C4—1182no—
259C2×C4—4721no—
260C2×C4—4721no—
261C2×C4—2362no—
262C2×C4—2362no—
263C2×C4—2362no—
264C2×C4—2362no—
265C2×C4—2362no—
266C2×C4—2362no—
267C2×C4#1427370911yescharacteristic
268C2×C4#1427370911yescharacteristic
269C2×C4#1427371211yescharacteristic
270C2×C4#1427371211yescharacteristic
271C8—81no—
272C8—81no—
273C8—4721no—
274C8—4721no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—42no—
284C22—42no—
285C22#1683726411yescharacteristic
286C22#1683738311yescharacteristic
287C22#1683738511yescharacteristic
288C4—21no—
289C4—21no—
290C4—21no—
291C4—21no—
292C4—82no—
293C4—4722no—
294C4—2362no—
295C4—2362no—
296C2—21no—
297C2—21no—
298C2—41no—
299C2—41no—
300C2#2142417811yescharacteristic
301C2#2142420311yescharacteristic
302C2#2142420911yescharacteristic
subgroupG/H
centerC22#16837264
commutator, prime radicalC24×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1305C2
socleC22×C59#381
cosocleC23×C4×C59C23
hypercenter#1305D59
hypocenterC59#8758
LCSG ▷ C24×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1305

Complements

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

NH ≅ G/Ntype
172. C59#8758semidirect

Maximal subgroups

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

subgroupindex
#214266982
#214224252
#214220032
#214220022
#214220012
C59×#13052
#875859