Finite Group Databasebeta

← #28498119

Subgroups — #28498119

Order 15104.

330 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 75 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#21432727C211yesmaximal, characteristic
2#21425908C211yesmaximal, characteristic, complemented
3#21422046C212yesmaximal, complemented
4#21422031C212yesmaximal
5C59×#2165C211yesmaximal, characteristic, complemented
6C2×C42×D59C411yescharacteristic
7#16837457—21no—
8#16837457—21no—
9#16837453—21no—
10#16837453—21no—
11#16837425C411yescharacteristic
12#16837425C2211yescharacteristic
13#16837365C411yescharacteristic
14#16837365C2211yescharacteristic
15C4×#11745664C411yescharacteristic
16#16837223—21no—
17#16837184C2212yes—
18#16837180—21no—
19C4×M16×C59C2211yescharacteristic
20C59×#364—21no—
21C59×#349C2212yes—
22C59×#347—21no—
23C22×C4×D59—21no—
24C22×C4×D59C2×C411yescharacteristic
25C2×#11745664C2×C411yescharacteristic
26C2×#11745664C2×C411yescharacteristic
27C2×C4×Dic59—21no—
28C2×C4×Dic59C2×C411yescharacteristic
29C42×D59—21no—
30C42×D59—21no—
31C42×D59—21no—
32C42×D59—21no—
33#14273667—22no—
34Dic59⋊C8—21no—
35Dic59⋊C8—21no—
36Dic59⋊C8—21no—
37Dic59⋊C8—21no—
38(C2×D59)⋊C8—21no—
39(C2×D59)⋊C8—21no—
40(C2×D59)⋊C8—21no—
41(C2×D59)⋊C8—21no—
42#14273638—21no—
43C4×C59⋊C8—21no—
44C2×M16×C59C2×C411yescharacteristic
45C2×M16×C59C2311yescharacteristic
46C2×C42×C59C2×C411yescharacteristic
47M32×C59—22no—
48#97×C59D411yescharacteristic
49#97×C59D411yescharacteristic
50C4×C8×C59D411yescharacteristic
51C4×C8×C59D411yescharacteristic
52C23×D59M1611yescharacteristic
53C22×C4×C59—21no—
54C22×C4×C59C22×C411yescharacteristic
55C22×Dic59—21no—
56C22×Dic59M1611yescharacteristic
57C2×C4×D59—21no—
58C2×C4×D59—21no—
59C2×C4×D59—21no—
60C2×C4×D59—21no—
61C2×C4×D59—21no—
62C2×C4×D59—21no—
63C2×C4×D59—44no—
64C2×C4×D59—41no—
65C2×C4×D59—41no—
66M16×C59—41no—
67M16×C59—41no—
68C2×C8×C59C22⋊C411yescharacteristic
69C2×C8×C59C22⋊C411yescharacteristic
70C2×C8×C59C2×D411yescharacteristic
71C2×C8×C59C2×D411yescharacteristic
72C42×C59—21no—
73C42×C59C22⋊C411yescharacteristic
74C42×C59C22⋊C411yescharacteristic
75C4×Dic59—21no—
76C4×Dic59—21no—
77C4×Dic59—21no—
78C4×Dic59—21no—
79C4×Dic59—21no—
80C4×Dic59M1611yescharacteristic
81C4×Dic59M1611yescharacteristic
82#11745664—41no—
83#11745664—41no—
84C2×C59⋊C8—21no—
85C2×C59⋊C8—21no—
86C16×C59—42no—
87C59⋊C16—42no—
88C23×C59C2×M1611yescharacteristic
89C22×D59—21no—
90C22×D59—42no—
91C22×D59—41no—
92C2×C4×C59—21no—
93C2×C4×C59—21no—
94C2×C4×C59—21no—
95C2×C4×C59—21no—
96C2×C4×C59—42no—
97C2×C4×C59C2×C22⋊C411yescharacteristic
98C2×C4×C59C2×M1611yescharacteristic
99C2×Dic59—21no—
100C2×Dic59—21no—
101C2×Dic59—21no—
102C2×Dic59—42no—
103C2×Dic59—41no—
104C4×D59—24no—
105C4×D59—42no—
106C4×D59—42no—
107C4×D59—42no—
108C4×D59—42no—
109C4×D59—41no—
110C4×D59—41no—
111C8×C59—21no—
112C8×C59—21no—
113C8×C59—21no—
114C8×C59—21no—
115C59⋊C8—41no—
116C59⋊C8—41no—
117#11521—591nomaximal, Sylow
118C22×C59—21no—
119C22×C59—41no—
120C22×C59#40711yescharacteristic
121C2×D59—22no—
122C2×D59—41no—
123C2×D59—42no—
124C2×D59—41no—
125C2×D59—42no—
126C4×C59—22no—
127C4×C59—21no—
128C4×C59—41no—
129C4×C59#41011yescharacteristic
130C4×C59#41311yescharacteristic
131Dic59—22no—
132Dic59—41no—
133Dic59—41no—
134#2920—591no—
135#2165—591no—
136#2165D5911yescharacteristic, complemented
137#1327—592no—
138#1320—592no—
139C2×C59—21no—
140C2×C59—41no—
141C2×C59#146411yescharacteristic
142D59—42no—
143C22×C42—591no—
144#432—1181no—
145#432—1181no—
146#431—1181no—
147#431—1181no—
148#423—591no—
149#423—591no—
150#407—591no—
151#407—591no—
152C4×M16—591no—
153C4×M16C2×D5911yescharacteristic
154#364—21no—
155#364—1181no—
156#349—592no—
157#349C2×D5912yes—
158#347—21no—
159#347—1181no—
160C59#1152111yescharacteristic, complemented, Sylow
161C23×C4—591no—
162C23×C4—1181no—
163C2×M16—591no—
164C2×M16—591no—
165C2×M16C4×D5911yescharacteristic
166C2×M16C22×D5911yescharacteristic
167C2×C42—591no—
168C2×C42—1181no—
169C2×C42—1181no—
170C2×C42—1181no—
171C2×C42—1181no—
172C2×C42—1181no—
173C2×C42C4×D5911yescharacteristic
174M32—22no—
175M32—1182no—
176C4⋊C8—1181no—
177C4⋊C8—1181no—
178C4⋊C8—1181no—
179C4⋊C8—1181no—
180C22⋊C8—1181no—
181C22⋊C8—1181no—
182C22⋊C8—1181no—
183C22⋊C8—1181no—
184#97—1181no—
185#97D23611yescharacteristic
186#97C22⋊D5911yescharacteristic
187C4×C8—1181no—
188C4×C8D23611yescharacteristic
189C4×C8C22⋊D5911yescharacteristic
190C24—591no—
191C22×C4—21no—
192C22×C4—591no—
193C22×C4—1181no—
194C22×C4—1181no—
195C22×C4—1181no—
196C22×C4—1181no—
197C22×C4—1181no—
198C22×C4—1181no—
199C22×C4—1181no—
200C22×C4—2361no—
201C22×C4—2361no—
202C22×C4—2364no—
203C22×C4C2×C4×D5911yescharacteristic
204M16—41no—
205M16—41no—
206M16—2361no—
207M16—2361no—
208C2×C8—1181no—
209C2×C8—1181no—
210C2×C8(C2×D59)⋊C411yescharacteristic
211C2×C8(C2×D59)⋊C411yescharacteristic
212C2×C8C2×D23611yescharacteristic
213C2×C8C2×C22⋊D5911yescharacteristic
214C42—21no—
215C42—591no—
216C42—591no—
217C42—1181no—
218C42—1181no—
219C42—1181no—
220C42—1181no—
221C42—1181no—
222C42(C2×D59)⋊C411yescharacteristic
223C42(C2×D59)⋊C411yescharacteristic
224C16—42no—
225C16—2362no—
226C23—1181no—
227C23—2361no—
228C23—2362no—
229C23M16×D5911yescharacteristic
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—1184no—
236C2×C4—1181no—
237C2×C4—1181no—
238C2×C4—1181no—
239C2×C4—2362no—
240C2×C4—2362no—
241C2×C4—2362no—
242C2×C4—2362no—
243C2×C4—2362no—
244C2×C4—2361no—
245C2×C4—2361no—
246C2×C4—2361no—
247C2×C4C2×(C2×D59)⋊C411yescharacteristic
248C2×C4M16×D5911yescharacteristic
249C8—21no—
250C8—21no—
251C8—21no—
252C8—21no—
253C8—2361no—
254C8—2361no—
255C22—21no—
256C22—41no—
257C22—1182no—
258C22—2361no—
259C22—2362no—
260C22—2362no—
261C22—2361no—
262C22#1683736511yescharacteristic
263C4—22no—
264C4—21no—
265C4—41no—
266C4—1182no—
267C4—2361no—
268C4—2361no—
269C4#1683737511yescharacteristic
270C4#1683738411yescharacteristic
271C2—21no—
272C2—41no—
273C2—2362no—
274C2#2142242811yescharacteristic
subgroupG/H
centerC4#16837375
commutatorC22×C4×C59C22×C4
FrattiniC2×M16C22×D59
FittingC59×#2165C2
socleC2×C59#1464
prime radicalC23×D59M16
cosocleC2×M16×C59C23
hypercenter#2165D59
hypocenterC59#11521
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C42 ◁ C4×M16 ◁ #2165

Complements

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

NH ≅ G/Ntype
2. #21425908C2semidirect
3. #21422046C2semidirect
5. C59×#2165C2semidirect
136. #2165D59semidirect
160. C59#11521semidirect

Maximal subgroups

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

subgroupindex
#214327272
#214259082
#214220462
#214220312
C59×#21652
#1152159