Finite Group Databasebeta

← #28503847

Subgroups — #28503847

Order 15104.

456 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 109 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#21432802C211yesmaximal, characteristic
2#21425738C212yesmaximal
3#21423899C212yesmaximal, complemented
4#21422217C211yesmaximal, characteristic, complemented
5C59×#1392C211yesmaximal, characteristic, complemented
6C2×#14273734C2211yescharacteristic
7C22×#11745659C411yescharacteristic, complemented
8#16837909C2211yescharacteristic
9#16837477—22no—
10#16837470—22no—
11#16837428C411yescharacteristic, complemented
12#16837376C2212yes—
13#16837365C411yescharacteristic, complemented
14#16837360—22no—
15#16837360C411yescharacteristic, complemented
16#16837193C414yescomplemented
17#16837190C414yescomplemented
18#16837174—22no—
19C59×#413C2212yes—
20C59×#408C2211yescharacteristic
21C22×C4×D59C2×C411yescharacteristic
22#14273734—42no—
23C2×#11745664C2×C411yescharacteristic
24C2×#11745664D412yes—
25C2×#11745659—21no—
26C2×#11745659—21no—
27C2×#11745659—42no—
28C22×C59⋊C8C2×C411yescharacteristic
29C22×C59⋊C8D412yes—
30#14273708C2×C412yes—
31#14273703—21no—
32#14273698—21no—
33#14273695C2×C412yes—
34C4×C22⋊D59—22no—
35#14273686—22no—
36#14273686C2×C411yescharacteristic
37Dic59⋊C8—22no—
38#14273642—22no—
39#14273642C2×C414yes—
40(C2×D59)⋊C8—22no—
41#14273637—21no—
42#14273637—42no—
43C8×Dic59—21no—
44C8×Dic59—42no—
45C2×#41×C59C2311yescharacteristic
46C2×M16×C59—22no—
47C2×M16×C59C2×C411yescharacteristic
48C22×C8×C59C2×C411yescharacteristic
49#101×C59—22no—
50#100×C59C2×C414yes—
51C22⋊C8×C59—22no—
52C23×D59C4211yescharacteristic
53#41×C59—41no—
54#41×C59—41no—
55C2×Q8×C59#4111yescharacteristic
56C2×D4×C59C22×C412yes—
57C2×D4×C59#4111yescharacteristic
58C22×C4×C59C22×C411yescharacteristic
59C22×C4×C59C2×D412yes—
60C2×C22⋊D59—22no—
61C22×Dic59C4211yescharacteristic
62C2×C4×D59—21no—
63C2×C4×D59—21no—
64C2×C4×D59—41no—
65C2×C4×D59—41no—
66M16×C59—24no—
67M16×C59—42no—
68M16×C59—41no—
69C2×C8×C59—21no—
70C2×C8×C59—21no—
71C2×C8×C59—42no—
72C2×C8×C59—41no—
73C22⋊Dic59—22no—
74C22⋊Dic59—24no—
75C22⋊Dic59C4212yes—
76(C2×D59)⋊C4—22no—
77C4⋊Dic59—21no—
78C4⋊Dic59—42no—
79Dic59⋊C4—22no—
80C4×Dic59—21no—
81C4×Dic59—42no—
82#11745664—41no—
83#11745664—42no—
84C2×C59⋊C8—22no—
85C2×C59⋊C8—22no—
86C2×C59⋊C8—22no—
87C2×C59⋊C8C22⋊C411yescharacteristic
88C2×C59⋊C8C22⋊C412yes—
89C2×C59⋊C8C22⋊C411yescharacteristic
90#11745659—42no—
91#11745659—42no—
92C23×C59C2×C4211yescharacteristic
93C23×C59C4×D412yes—
94C22×D59—21no—
95C22×D59—41no—
96C22×D59—81no—
97Q8×C59—41no—
98D4×C59—42no—
99D4×C59—41no—
100C2×C4×C59—22no—
101C2×C4×C59—21no—
102C2×C4×C59—22no—
103C2×C4×C59—21no—
104C2×C4×C59C2×C22⋊C411yescharacteristic
105C2×C4×C59#11711yescharacteristic
106C2×C4×C59C4×D412yes—
107C22⋊D59—82no—
108C2×Dic59—22no—
109C2×Dic59—21no—
110C2×Dic59—41no—
111C2×Dic59—44no—
112C4×D59—42no—
113C4×D59—41no—
114C4×D59—41no—
115C8×C59—44no—
116C8×C59—41no—
117C8×C59—41no—
118C59⋊C8—21no—
119C59⋊C8—22no—
120C59⋊C8#9912yes—
121#12624—591nomaximal, Sylow
122C22×C59—22no—
123C22×C59—21no—
124C22×C59—42no—
125C22×C59—41no—
126C22×C59C4×C22⋊C411yescharacteristic
127C2×D59—41no—
128C2×D59—41no—
129C2×D59—41no—
130C2×D59—82no—
131C4×C59—22no—
132C4×C59—21no—
133C4×C59C2×#9911yescharacteristic
134C4×C59#41011yescharacteristic
135Dic59—42no—
136Dic59—41no—
137Dic59—84no—
138#2930—591no—
139#2120—592no—
140#1757—592no—
141#1392D5911yescharacteristic, complemented
142#1391—591no—
143C2×C59—21no—
144C2×C59—42no—
145C2×C59—41no—
146C2×C59C4×#9911yescharacteristic
147D59—81no—
148C2×#131—591no—
149C22×M16—591no—
150#525—591no—
151#436—1182no—
152#435—1182no—
153#424—591no—
154#413C2×D5912yes—
155#410—592no—
156#408C2×D5911yescharacteristic
157#407—591no—
158#405—591no—
159#405—1182no—
160#352—594no—
161AΣL(1,𝔽16)—594no—
162#344—1182no—
163C59#1262411yescharacteristic, complemented, Sylow
164C2×#41C22×D5911yescharacteristic
165C23×C4—591no—
166#131—2362no—
167C2×M16—22no—
168C2×M16—591no—
169C2×M16—592no—
170C2×M16—1181no—
171C2×M16—1181no—
172C2×M16—2362no—
173C2×M16C4×D5911yescharacteristic
174C22×C8—591no—
175C22×C8—592no—
176C22×C8C4×D5911yescharacteristic
177#123—592no—
178C22⋊Q8—1181no—
179#121—1181no—
180#120—592no—
181C4×D4—1182no—
182#117—591no—
183#117—1182no—
184C4⋊C8—1182no—
185#101—22no—
186#100C4×D5914yes—
187#99—594no—
188#99—1182no—
189C22⋊C8—22no—
190C22⋊C8—1182no—
191#97—1181no—
192#97—2362no—
193C4×C8—1181no—
194C4×C8—2362no—
195C24—591no—
196#41—41no—
197#41—41no—
198C2×Q8#1174568711yescharacteristic
199C2×D4—1182no—
200C2×D4C2×C4×D5912yes—
201C2×D4#1174568911yescharacteristic
202C22×C4—591no—
203C22×C4—1181no—
204C22×C4—1181no—
205C22×C4—2361no—
206C22×C4—2361no—
207C22×C4C2×C4×D5911yescharacteristic
208C22×C4D4×D5912yes—
209M16—24no—
210M16—42no—
211M16—41no—
212M16—2362no—
213M16—2362no—
214M16—2361no—
215M16—2362no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—42no—
219C2×C8—41no—
220C2×C8—591no—
221C2×C8—591no—
222C2×C8—592no—
223C2×C8—1182no—
224C2×C8—1182no—
225C2×C8—1182no—
226C4⋊C4—1181no—
227C4⋊C4—1182no—
228C4⋊C4—2362no—
229C22⋊C4—592no—
230C22⋊C4—1182no—
231C22⋊C4—1182no—
232C22⋊C4—1184no—
233C42—1181no—
234C42—2362no—
235C23—1181no—
236C23—4721no—
237C23—2361no—
238C23C42×D5911yescharacteristic
239C23#1427368912yes—
240Q8—41no—
241D4—42no—
242D4—41no—
243D4—4722no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—1182no—
249C2×C4—1181no—
250C2×C4—2362no—
251C2×C4—2364no—
252C2×C4—2361no—
253C2×C4—2361no—
254C2×C4—2361no—
255C2×C4C22⋊C4×D5911yescharacteristic
256C2×C4#1427368511yescharacteristic
257C2×C4(C22⋊D59)⋊C412yes—
258C8—44no—
259C8—41no—
260C8—41no—
261C8—592no—
262C8—1182no—
263C8—1181no—
264C22—22no—
265C22—21no—
266C22—42no—
267C22—41no—
268C22—4722no—
269C22—2361no—
270C22—2361no—
271C22—2361no—
272C22#1683724911yescharacteristic
273C4—22no—
274C4—21no—
275C4—4724no—
276C4—2362no—
277C4—2361no—
278C4#99×D5911yescharacteristic
279C4#1683737411yescharacteristic
280C2—21no—
281C2—42no—
282C2—41no—
283C2—4721no—
284C2#2142388311yescharacteristic
subgroupG/H
centerC4#99×D59
commutatorC23×C59C2×C42
FrattiniC2×#41C22×D59
FittingC59×#1392C2
socleC2×C59C4×#99
prime radical#16837909C22
cosocleC2×#41×C59C23
hypercenter#1392D59
hypocenterC59#12624
LCSG ▷ C23×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #1392

Complements

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

NH ≅ G/Ntype
3. #21423899C2semidirect
4. #21422217C2semidirect
5. C59×#1392C2semidirect
7. C22×#11745659C4semidirect
11. #16837428C4semidirect
13. #16837365C4semidirect
15. #16837360C4semidirect
16. #16837193C4semidirect
17. #16837190C4semidirect
141. #1392D59semidirect
163. C59#12624semidirect

Maximal subgroups

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

subgroupindex
#214328022
#214257382
#214238992
#214222172
C59×#13922
#1262459