Finite Group Databasebeta

← #28587502

Subgroups — #28587502

Order 15104.

1006 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837459C211yesmaximal, characteristic
2C4×#14273686C211yesmaximal, characteristic
3#21423973C214yesmaximal, complemented
4#21423951C212yesmaximal, complemented
5C4⋊C8×Dic59C212yesmaximal, complemented
6#21423872C212yesmaximal
7C8×C22⋊Dic59C212yesmaximal
8C22×C4⋊C8×C59C211yesmaximal, characteristic
9C2×#14273686C412yes—
10#16837459C418yescomplemented
11C2×C4⋊(C59⋊C8)C2212yes—
12C2×C22⋊(C59⋊C8)C2212yes—
13C2×#14273638C2212yes—
14C2×#14273637C2214yes—
15C2×C8×Dic59C2214yes—
16C4×C4⋊Dic59C412yes—
17C4×C4⋊Dic59C2212yes—
18C4×C22⋊Dic59C412yes—
19C4×C22⋊Dic59C2211yescharacteristic
20C4×C22⋊Dic59C2211yescharacteristic
21#16837248C412yes—
22C42×Dic59C2212yes—
23#16837156C2214yes—
24#16837156C2214yes—
25C23×C8×C59C2212yes—
26C22×C42×C59C2211yescharacteristic
27C2×C4⋊C8×C59C418yescomplemented
28C2×C4⋊C8×C59C2211yescharacteristic
29C2×C4⋊C8×C59C2211yescharacteristic
30C2×C4⋊C8×C59C2212yes—
31C22×C59⋊C8C2314yes—
32#14273686—28no—
33C2×C4⋊Dic59C2×C414yes—
34C2×C22⋊Dic59C2×C412yes—
35C2×C22⋊Dic59C2×C412yes—
36C2×C4×Dic59C2×C414yes—
37C2×C4×Dic59C2×C414yes—
38C2×C4×Dic59C2312yes—
39C2×C4×Dic59C2312yes—
40C4⋊(C59⋊C8)C2×C418yes—
41C22⋊(C59⋊C8)C2×C418yes—
42#14273638C2×C418yes—
43#14273637—44no—
44C8×Dic59—44no—
45C59⋊#95C2×C414yes—
46C59⋊#95C2×C412yes—
47C59⋊#95C2×C412yes—
48C23×C4×C59C2×C411yescharacteristic
49C23×C4×C59C2×C411yescharacteristic
50C23×C4×C59C2311yescharacteristic
51C22×C8×C59—44no—
52C22×C8×C59C2×C418yes—
53C22×C8×C59C2312yes—
54C22×C8×C59C2312yes—
55C2×C42×C59C2×C411yescharacteristic
56C2×C42×C59C2×C411yescharacteristic
57C2×C42×C59C2×C414yes—
58C2×C42×C59C2×C414yes—
59C2×C42×C59C2311yescharacteristic
60C2×C42×C59C2311yescharacteristic
61C4⋊C8×C59C2×C414yes—
62C4⋊C8×C59C2×C414yes—
63C4⋊C8×C59C2×C418yes—
64C24×C59C22×C411yescharacteristic
65C22×C4×C59—24no—
66C22×C4×C59—24no—
67C22×C4×C59—22no—
68C22×C4×C59—22no—
69C22×C4×C59C4214yes—
70C22×C4×C59C4214yes—
71C22×C4×C59C22×C411yescharacteristic
72C22×C4×C59C22×C411yescharacteristic
73C22×C4×C59C22×C411yescharacteristic
74C22×C4×C59C22×C411yescharacteristic
75C22×C4×C59C22×C412yes—
76C22×C4×C59C22×C412yes—
77C22×C4×C59C22×C411yescharacteristic
78C22×C4×C59C2411yescharacteristic
79C22×Dic59C22×C414yes—
80C22×Dic59C22×C414yes—
81C2×C8×C59—24no—
82C2×C8×C59—24no—
83C2×C8×C59—48no—
84C2×C8×C59C22×C414yes—
85C2×C8×C59C22×C414yes—
86C42×C59C4214yes—
87C42×C59C4214yes—
88C42×C59C22×C412yes—
89C42×C59C22×C412yes—
90C42×C59C22×C412yes—
91C42×C59C22×C412yes—
92C22⋊Dic59—24no—
93C22⋊Dic59M1618yes—
94C4⋊Dic59—28no—
95C4×Dic59—28no—
96C4×Dic59—22no—
97C4×Dic59—22no—
98C4×Dic59—22no—
99C4×Dic59—22no—
100C4×Dic59—28no—
101C2×C59⋊C8—24no—
102C2×C59⋊C8—24no—
103C2×C59⋊C8C22×C418yes—
104C23×C59—22no—
105C23×C59—22no—
106C23×C59C2×C4212yes—
107C23×C59C2×M1614yes—
108C23×C59C23×C411yescharacteristic
109C2×C4×C59—22no—
110C2×C4×C59—22no—
111C2×C4×C59—22no—
112C2×C4×C59—22no—
113C2×C4×C59—28no—
114C2×C4×C59—22no—
115C2×C4×C59—22no—
116C2×C4×C59—24no—
117C2×C4×C59—28no—
118C2×C4×C59C2×C4212yes—
119C2×C4×C59C2×C4212yes—
120C2×C4×C59C2×C4212yes—
121C2×C4×C59C2×C4212yes—
122C2×C4×C59C2×C4212yes—
123C2×C4×C59#13114yes—
124C2×C4×C59C23×C411yescharacteristic
125C2×C4×C59C23×C411yescharacteristic
126C2×C4×C59H(5,𝔽2)12yes—
127C2×C4×C592−1+412yes—
128C2×Dic59—24no—
129C2×Dic59—24no—
130C2×Dic59—24no—
131C2×Dic59—24no—
132C2×Dic59—24no—
133C2×Dic59C2×M1618yes—
134C8×C59—24no—
135C8×C59—24no—
136C59⋊C8—28no—
137#20546—591nomaximal, Sylow
138C22×C59—22no—
139C22×C59—28no—
140C22×C59—22no—
141C22×C59C4×M1614yes—
142C22×C59C22×C4211yescharacteristic
143C22×C59#52012yes—
144C22×C59C22×M1612yes—
145C22×C59C2×#13112yes—
146C4×C59—22no—
147C4×C59—22no—
148C4×C59—22no—
149C4×C59—22no—
150C4×C59#40514yes—
151C4×C59#51811yescharacteristic
152C4×C59#51911yescharacteristic
153C4×C59#52012yes—
154Dic59—28no—
155Dic59—44no—
156C2×#432—591no—
157C22×C4⋊C8D5911yescharacteristic
158C4×#117—591no—
159#1771—594no—
160#1767—592no—
161C4×C4⋊C8—592no—
162#1752—592no—
163C8×C22⋊C4—592no—
164C2×C59—24no—
165C2×C59#227411yescharacteristic
166C2×C59C2×C4×M1611yescharacteristic
167C2×C59C2×#40511yescharacteristic
168C2×C59#299014yes—
169C23×C8C2×D5912yes—
170C2×#117—592no—
171C22×C42C2×D5911yescharacteristic
172#432—598no—
173C2×C4⋊C8—592no—
174C2×C4⋊C8Dic5918yescomplemented
175C2×C4⋊C8C2×D5911yescharacteristic
176C2×C4⋊C8C2×D5911yescharacteristic
177C2×C4⋊C8C2×D5912yes—
178C2×C22⋊C8—592no—
179C2×#97—592no—
180C2×#97—594no—
181C2×C4×C8—594no—
182C4×C4⋊C4—592no—
183C4×C4⋊C4—592no—
184C4×C22⋊C4—592no—
185C4×C22⋊C4—591no—
186C4×C22⋊C4—591no—
187#376—592no—
188C43—592no—
189#336—594no—
190#336—594no—
191C59#2054611yescharacteristic, complemented, Sylow
192C23×C4C4×D5911yescharacteristic
193C23×C4C4×D5911yescharacteristic
194C23×C4C22×D5911yescharacteristic
195C22×C8—44no—
196C22×C8—594no—
197C22×C8C2×Dic5918yes—
198C22×C8C22×D5912yes—
199C22×C8C22×D5912yes—
200#117—1188no—
201C2×C4⋊C4—594no—
202C2×C22⋊C4—592no—
203C2×C22⋊C4—592no—
204C2×C42—594no—
205C2×C42—594no—
206C2×C42—592no—
207C2×C42—592no—
208C2×C42C4×D5911yescharacteristic
209C2×C42C4×D5911yescharacteristic
210C2×C42C4×D5914yes—
211C2×C42C2×Dic5914yes—
212C2×C42C22×D5911yescharacteristic
213C2×C42C22×D5911yescharacteristic
214C4⋊C8—598no—
215C4⋊C8C2×Dic5914yes—
216C4⋊C8C2×Dic5914yes—
217C4⋊C8C2×Dic5918yes—
218C22⋊C8—598no—
219#97—598no—
220#97—2364no—
221C4×C8—2364no—
222#95—592no—
223#95—594no—
224#95—592no—
225C24C2×C4×D5911yescharacteristic
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—594no—
231C22×C4—594no—
232C22×C4C4×Dic5914yes—
233C22×C4C4×Dic5914yes—
234C22×C4C2×C4×D5911yescharacteristic
235C22×C4C2×C4×D5911yescharacteristic
236C22×C4C2×C4×D5911yescharacteristic
237C22×C4C2×C4×D5911yescharacteristic
238C22×C4C2×C4×D5912yes—
239C22×C4C2×C4×D5911yescharacteristic
240C22×C4C22×Dic5912yes—
241C22×C4C23×D5911yescharacteristic
242C2×C8—24no—
243C2×C8—24no—
244C2×C8—48no—
245C2×C8—598no—
246C2×C8—1184no—
247C2×C8—1184no—
248C2×C8C22×Dic5914yes—
249C2×C8C22×Dic5914yes—
250C4⋊C4—1188no—
251C22⋊C4—598no—
252C22⋊C4—1184no—
253C42—1188no—
254C42—1188no—
255C42—1182no—
256C42—1182no—
257C42—1182no—
258C42—1182no—
259C42C4×Dic5914yes—
260C42C4×Dic5914yes—
261C42C2×C4×D5912yes—
262C42C2×C4×D5912yes—
263C42C22×Dic5912yes—
264C42C22×Dic5912yes—
265C23—22no—
266C23—22no—
267C23C2×C4×Dic5912yes—
268C23M16×D5914yes—
269C23C22×C4×D5911yescharacteristic
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—28no—
274C2×C4—22no—
275C2×C4—28no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—24no—
279C2×C4—598no—
280C2×C4—1184no—
281C2×C4—1184no—
282C2×C4—1184no—
283C2×C4—1184no—
284C2×C4—1184no—
285C2×C4C2×C4×Dic5912yes—
286C2×C4C2×C4×Dic5912yes—
287C2×C4C2×C4×Dic5912yes—
288C2×C4C2×C4×Dic5912yes—
289C2×C4C2×C4×Dic5912yes—
290C2×C4#1427373314yes—
291C2×C4C22×C4×D5911yescharacteristic
292C2×C4C23×Dic5911yescharacteristic
293C2×C4#1427377212yes—
294C2×C4#1427377512yes—
295C8—24no—
296C8—24no—
297C8—1188no—
298C22—22no—
299C22—28no—
300C22—22no—
301C22M16×Dic5914yes—
302C22C22×C4×Dic5911yescharacteristic
303C22#1683788312yes—
304C22C2×M16×D5912yes—
305C22C2×#1427373312yes—
306C4—22no—
307C4—22no—
308C4—22no—
309C4—22no—
310C4—1188no—
311C4—2364no—
312C4#1683735714yes—
313C4#1683787711yescharacteristic
314C4#1683788011yescharacteristic
315C4#1683788312yes—
316C2—24no—
317C2#2142645311yescharacteristic
318C2C2×M16×Dic5911yescharacteristic
319C2C2×#1683735711yescharacteristic
320C2#2143321814yes—
subgroupG/H
center, FrattiniC22×C4C23×D59
commutatorC22×C59C22×C42
FittingC22×C4⋊C8×C59C2
socleC23×C59C23×C4
prime radicalC24×C59C22×C4
cosocleC22×C4×C59C24
hypercenterC22×C4⋊C8D59
hypocenterC59#20546
LCSG ▷ C22×C59 ▷ C59
UCSC1 ◁ C22×C4 ◁ C22×C4⋊C8

Complements

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

NH ≅ G/Ntype
3. #21423973C2semidirect
4. #21423951C2semidirect
5. C4⋊C8×Dic59C2semidirect
10. #16837459C4semidirect
27. C2×C4⋊C8×C59C4semidirect
174. C2×C4⋊C8Dic59semidirect
191. C59#20546semidirect

Maximal subgroups

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

subgroupindex
C2×#168374592
C4×#142736862
#214239732
#214239512
C4⋊C8×Dic592
#214238722
C8×C22⋊Dic592
C22×C4⋊C8×C592
#2054659