Finite Group Databasebeta

← #28503842

Subgroups — #28503842

Order 15104.

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

normal subgroups only

This group has 470 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#21432793C211yesmaximal, characteristic
2#21425738C212yesmaximal
3#21423899C212yesmaximal, complemented
4#21422217C211yesmaximal, characteristic, complemented
5C59×#1392C211yesmaximal, characteristic, complemented
6C2×#14273734C2211yescharacteristic
7C22×C8×D59C411yescharacteristic, complemented
8#16837909C2211yescharacteristic
9#16837476—22no—
10(C22⋊D59)⋊C8—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—41no—
23#14273734—41no—
24C2×#11745664C2×C411yescharacteristic
25C2×#11745664D412yes—
26C22×C59⋊C8C2×C411yescharacteristic
27C22×C59⋊C8D412yes—
28C2×C8×D59—21no—
29C2×C8×D59—21no—
30C2×C8×D59—41no—
31C2×C8×D59—41no—
32#14273708C2×C412yes—
33#14273703—21no—
34#14273698—21no—
35#14273695C2×C412yes—
36C4×C22⋊D59—22no—
37#14273686—22no—
38#14273686C2×C411yescharacteristic
39Dic59⋊C8—22no—
40#14273642—22no—
41#14273642C2×C414yes—
42(C2×D59)⋊C8—22no—
43#14273637—21no—
44#14273637—42no—
45C8×Dic59—21no—
46C8×Dic59—42no—
47C2×#41×C59C2311yescharacteristic
48C2×M16×C59—22no—
49C2×M16×C59C2×C411yescharacteristic
50C22×C8×C59C2×C411yescharacteristic
51#101×C59—22no—
52#100×C59C2×C414yes—
53C22⋊C8×C59—22no—
54C23×D59C4211yescharacteristic
55#41×C59—41no—
56#41×C59—41no—
57C2×Q8×C59#4111yescharacteristic
58C2×D4×C59C22×C412yes—
59C2×D4×C59#4111yescharacteristic
60C22×C4×C59C22×C411yescharacteristic
61C22×C4×C59C2×D412yes—
62C2×C22⋊D59—22no—
63C22×Dic59C4211yescharacteristic
64C2×C4×D59—21no—
65C2×C4×D59—21no—
66C2×C4×D59—41no—
67C2×C4×D59—41no—
68M16×C59—24no—
69M16×C59—42no—
70M16×C59—41no—
71C2×C8×C59—21no—
72C2×C8×C59—21no—
73C2×C8×C59—42no—
74C2×C8×C59—41no—
75C22⋊Dic59—22no—
76C22⋊Dic59—24no—
77C22⋊Dic59C4212yes—
78(C2×D59)⋊C4—22no—
79C4⋊Dic59—21no—
80C4⋊Dic59—42no—
81Dic59⋊C4—22no—
82C4×Dic59—21no—
83C4×Dic59—42no—
84#11745664—41no—
85#11745664—42no—
86C2×C59⋊C8—22no—
87C2×C59⋊C8—22no—
88C2×C59⋊C8—21no—
89C2×C59⋊C8—21no—
90C2×C59⋊C8C22⋊C411yescharacteristic
91C2×C59⋊C8C22⋊C412yes—
92C2×C59⋊C8C22⋊C411yescharacteristic
93C8×D59—42no—
94C8×D59—41no—
95C8×D59—41no—
96C23×C59C2×C4211yescharacteristic
97C23×C59C4×D412yes—
98C22×D59—21no—
99C22×D59—42no—
100C22×D59—41no—
101Q8×C59—41no—
102D4×C59—42no—
103D4×C59—41no—
104C2×C4×C59—22no—
105C2×C4×C59—21no—
106C2×C4×C59—22no—
107C2×C4×C59—21no—
108C2×C4×C59C2×C22⋊C411yescharacteristic
109C2×C4×C59#11711yescharacteristic
110C2×C4×C59C4×D412yes—
111C22⋊D59—44no—
112C2×Dic59—22no—
113C2×Dic59—21no—
114C2×Dic59—41no—
115C2×Dic59—44no—
116C4×D59—42no—
117C4×D59—41no—
118C4×D59—41no—
119C8×C59—44no—
120C8×C59—41no—
121C8×C59—41no—
122C59⋊C8—21no—
123C59⋊C8—22no—
124C59⋊C8#9911yescharacteristic
125C59⋊C8#9911yescharacteristic
126#12623—591nomaximal, Sylow
127C22×C59—22no—
128C22×C59—21no—
129C22×C59—42no—
130C22×C59—41no—
131C22×C59C4×C22⋊C411yescharacteristic
132C2×D59—22no—
133C2×D59—41no—
134C2×D59—42no—
135C2×D59—41no—
136C2×D59—42no—
137C4×C59—22no—
138C4×C59—21no—
139C4×C59C2×#9911yescharacteristic
140C4×C59#41011yescharacteristic
141Dic59—42no—
142Dic59—41no—
143Dic59—84no—
144#2929—591no—
145#2120—592no—
146#1757—592no—
147#1392D5911yescharacteristic, complemented
148#1391—591no—
149C2×C59—21no—
150C2×C59—42no—
151C2×C59—41no—
152C2×C59C4×#9911yescharacteristic
153D59—42no—
154C2×#131—591no—
155C23×C8—591no—
156#525—591no—
157#435—1182no—
158C8×D4—1182no—
159#424—591no—
160#413C2×D5912yes—
161#410—592no—
162#408C2×D5911yescharacteristic
163#407—591no—
164#405—591no—
165#405—1182no—
166#352—594no—
167AΣL(1,𝔽16)—594no—
168#344—1182no—
169C59#1262311yescharacteristic, complemented, Sylow
170C2×#41C22×D5911yescharacteristic
171C23×C4—591no—
172#131—2361no—
173#131—2361no—
174C2×M16—22no—
175C2×M16—591no—
176C2×M16—592no—
177C2×M16C4×D5911yescharacteristic
178C22×C8—591no—
179C22×C8—592no—
180C22×C8—1181no—
181C22×C8—1181no—
182C22×C8—2361no—
183C22×C8—2361no—
184C22×C8C4×D5911yescharacteristic
185#123—592no—
186C22⋊Q8—1181no—
187#121—1181no—
188#120—592no—
189C4×D4—1182no—
190#117—591no—
191#117—1182no—
192C4⋊C8—1182no—
193#101—22no—
194#100C4×D5914yes—
195#99—594no—
196#99—1182no—
197C22⋊C8—22no—
198C22⋊C8—1182no—
199#97—1181no—
200#97—2362no—
201C4×C8—1181no—
202C4×C8—2362no—
203C24—591no—
204#41—41no—
205#41—41no—
206C2×Q8#1174568711yescharacteristic
207C2×D4—1182no—
208C2×D4C2×C4×D5912yes—
209C2×D4#1174568911yescharacteristic
210C22×C4—591no—
211C22×C4—1181no—
212C22×C4—1181no—
213C22×C4—2361no—
214C22×C4—2361no—
215C22×C4C2×C4×D5911yescharacteristic
216C22×C4D4×D5912yes—
217M16—24no—
218M16—42no—
219M16—41no—
220M16—2362no—
221M16—2361no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—42no—
225C2×C8—41no—
226C2×C8—591no—
227C2×C8—591no—
228C2×C8—592no—
229C2×C8—1182no—
230C2×C8—1182no—
231C2×C8—1181no—
232C2×C8—1181no—
233C2×C8—2361no—
234C2×C8—2361no—
235C2×C8—2362no—
236C4⋊C4—1181no—
237C4⋊C4—1182no—
238C4⋊C4—2362no—
239C22⋊C4—592no—
240C22⋊C4—1182no—
241C22⋊C4—1182no—
242C22⋊C4—1184no—
243C42—1181no—
244C42—2362no—
245C23—1181no—
246C23—2361no—
247C23—2362no—
248C23C42×D5911yescharacteristic
249C23#1427368912yes—
250Q8—41no—
251D4—42no—
252D4—41no—
253D4—2364no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—1182no—
259C2×C4—1181no—
260C2×C4—2362no—
261C2×C4—2364no—
262C2×C4—2361no—
263C2×C4—2361no—
264C2×C4—2361no—
265C2×C4C22⋊C4×D5911yescharacteristic
266C2×C4#1427368511yescharacteristic
267C2×C4(C22⋊D59)⋊C412yes—
268C8—44no—
269C8—41no—
270C8—41no—
271C8—591no—
272C8—591no—
273C8—1181no—
274C8—1182no—
275C22—22no—
276C22—21no—
277C22—42no—
278C22—41no—
279C22—1182no—
280C22—2361no—
281C22—2362no—
282C22—2362no—
283C22—2361no—
284C22#1683724911yescharacteristic
285C4—22no—
286C4—21no—
287C4—4724no—
288C4—2362no—
289C4—2361no—
290C4#99×D5911yescharacteristic
291C4#1683737411yescharacteristic
292C2—21no—
293C2—42no—
294C2—41no—
295C2—2362no—
296C2#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#12623
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×C8×D59C4semidirect
11. #16837428C4semidirect
13. #16837365C4semidirect
15. #16837360C4semidirect
16. #16837193C4semidirect
17. #16837190C4semidirect
147. #1392D59semidirect
169. C59#12623semidirect

Maximal subgroups

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

subgroupindex
#214327932
#214257382
#214238992
#214222172
C59×#13922
#1262359