Finite Group Databasebeta

← #28510306

Subgroups — #28510306

Order 15104.

2878 conjugacy classes of subgroups; only the 427 normal subgroups are shown.

normal subgroups only

This group has 2878 conjugacy classes of subgroups — too many to draw. Only its 427 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
1C2×#16837257C213yesmaximal
2#21422334C212yesmaximal, complemented
3#21422333C213yesmaximal, complemented
4#21422332C216yesmaximal, complemented
5C59×#1437C211yesmaximal, characteristic, complemented
6C25×D59C2211yescharacteristic
7C22×C22⋊Dic59C2213yes—
8C22×(C2×D59)⋊C4C2216yes—
9C22×(C2×D59)⋊C4C2216yes—
10C22×C4×Dic59C2213yescomplemented
11#16837257C4124yescomplemented
12C2×C59⋊#95C2213yescomplemented
13C2×C59⋊#95C2216yescomplemented
14C22×C22⋊C4×C59C2213yes—
15C22×C42×C59C2211yescharacteristic, complemented
16C2×#95×C59C2213yescomplemented
17C24×D59—43no—
18C24×D59—43no—
19C24×D59—46no—
20C24×D59C2×C4112yes—
21C24×D59C2312yes—
22C23×Dic59C2316yes—
23C2×C22⋊Dic59—46no—
24C2×C22⋊Dic59C2×C4112yes—
25C2×(C2×D59)⋊C4—46no—
26C2×(C2×D59)⋊C4—46no—
27C2×(C2×D59)⋊C4—46no—
28C2×(C2×D59)⋊C4—46no—
29C2×(C2×D59)⋊C4C2×C4124yes—
30C2×(C2×D59)⋊C4C2×C4124yes—
31C2×C4×Dic59—43no—
32C2×C4×Dic59—46no—
33C59⋊#95—43no—
34C59⋊#95—46no—
35C25×C59C2311yescharacteristic
36C23×C4×C59C2313yes—
37C23×C4×C59C2313yes—
38C2×C22⋊C4×C59—43no—
39C2×C22⋊C4×C59—43no—
40C2×C22⋊C4×C59C2×C4112yes—
41C2×C42×C59—43no—
42#95×C59—43no—
43C24×C59—23no—
44C24×C59—23no—
45C24×C59—26no—
46C24×C59C22×C416yes—
47C24×C59C2411yescharacteristic
48C23×D59—26no—
49C23×D59—26no—
50C23×D59—26no—
51C23×D59—26no—
52C23×D59—424no—
53C23×D59—424no—
54C23×D59—412no—
55C23×D59—412no—
56C23×D59—44no—
57C23×D59—424no—
58C23×D59—412no—
59C23×D59—48no—
60C23×D59—412no—
61C23×D59—44no—
62C23×D59C42116yescomplemented
63C23×D59C22×C4112yes—
64C22×C4×C59—23no—
65C22×C4×C59—26no—
66C22×C4×C59—23no—
67C22×C4×C59—26no—
68C22×C4×C59—23no—
69C22×C4×C59—23no—
70C22×C4×C59—26no—
71C22×C4×C59—26no—
72C22×C4×C59C22×C416yes—
73C22×C4×C59C22×C416yes—
74C22×Dic59—26no—
75C22×Dic59—26no—
76C22×Dic59—26no—
77C22×Dic59—26no—
78C22×Dic59—26no—
79C22×Dic59—26no—
80C22×Dic59C22×C4112yes—
81C22⋊C4×C59—412no—
82C42×C59—44no—
83C22⋊Dic59—412no—
84(C2×D59)⋊C4—424no—
85(C2×D59)⋊C4—424no—
86C4×Dic59—412no—
87C23×C59—26no—
88C23×C59—26no—
89C23×C59—212no—
90C23×C59—212no—
91C23×C59—41no—
92C23×C59—46no—
93C23×C59—43no—
94C23×C59—42no—
95C23×C59—43no—
96C23×C59—41no—
97C23×C59C2×C4214yes—
98C23×C59C23×C413yes—
99C23×C59H(5,𝔽2)13yes—
100C23×C59H(5,𝔽2)13yes—
101C23×C59H(5,𝔽2)16yes—
102C22×D59—212no—
103C22×D59—212no—
104C22×D59—212no—
105C22×D59—212no—
106C22×D59—212no—
107C22×D59—212no—
108C22×D59—448no—
109C22×D59—448no—
110C22×D59—448no—
111C22×D59—496no—
112C22×D59—42no—
113C22×D59—46no—
114C22×D59—46no—
115C22×D59—46no—
116C22×D59—42no—
117C22×D59—42no—
118C22×D59—46no—
119C22×D59—42no—
120C22×D59C2×C4218yes—
121C2×C4×C59—212no—
122C2×C4×C59—212no—
123C2×C4×C59—26no—
124C2×C4×C59—26no—
125C2×C4×C59—43no—
126C2×C4×C59—46no—
127C2×C4×C59—46no—
128C2×C4×C59—43no—
129C2×C4×C59—43no—
130C2×C4×C59—43no—
131C2×Dic59—212no—
132C2×Dic59—212no—
133C2×Dic59—212no—
134C2×Dic59—46no—
135C2×Dic59—46no—
136C2×Dic59—46no—
137C2×Dic59—46no—
138#13827—591nomaximal, Sylow
139C22×C59—212no—
140C22×C59—46no—
141C22×C59—46no—
142C22×C59—412no—
143C22×C59C22×C4211yescharacteristic
144C22×C59#51816yes—
145C22×C59#51816yes—
146C22×C59#51813yes—
147C22×C59#51813yes—
148C22×C59#56011yescharacteristic
149C22×C59#56016yes—
150C22×C59#56013yes—
151C22×C59#56012yes—
152C22×C59H(3,𝔽4)13yes—
153C22×C59H(3,𝔽4)11yescharacteristic
154C2×D59—224no—
155C2×D59—464no—
156C2×D59—412no—
157C2×D59—412no—
158C2×D59—412no—
159C2×D59—412no—
160C4×C59—46no—
161C4×C59—46no—
162Dic59—412no—
163C2×#379—593no—
164#1437—592no—
165#1437—596no—
166#1437—593no—
167#1437D5911yescharacteristic, complemented
168C2×C59—44no—
169C2×C59#227313yes—
170C2×C59#237513yes—
171C2×C59#237513yes—
172C2×C59#237516yes—
173D59—48no—
174C26—591no—
175C22×C22⋊C4—593no—
176C22×C22⋊C4—596no—
177C22×C22⋊C4—596no—
178C22×C22⋊C4C2×D5913yes—
179C22×C42—593no—
180C22×C42C2×D5911yescharacteristic, complemented
181#379—5924no—
182C2×#95—596no—
183C2×#95—593no—
184C2×#95C2×D5913yescomplemented
185C59#1382711yescharacteristic, complemented, Sylow
186C25—592no—
187C25—5912no—
188C25—2363no—
189C25—2363no—
190C25—2366no—
191C25C22×D5911yescharacteristic
192C23×C4—596no—
193C23×C4C22×D5913yes—
194C23×C4C22×D5913yes—
195C2×C22⋊C4—43no—
196C2×C22⋊C4—43no—
197C2×C22⋊C4—5912no—
198C2×C22⋊C4—5924no—
199C2×C22⋊C4—5924no—
200C2×C22⋊C4—2366no—
201C2×C22⋊C4—2366no—
202C2×C22⋊C4—2366no—
203C2×C22⋊C4—2366no—
204C2×C22⋊C4—2366no—
205C2×C22⋊C4C4×D59112yes—
206C2×C42—43no—
207C2×C42—2363no—
208C2×C42—2366no—
209#95—43no—
210#95—2363no—
211#95—2366no—
212C24—23no—
213C24—23no—
214C24—26no—
215C24—5912no—
216C24—5916no—
217C24—1186no—
218C24—1186no—
219C24—1186no—
220C24—1186no—
221C24—23624no—
222C24—23624no—
223C24—23612no—
224C24—23612no—
225C24—23612no—
226C24—2364no—
227C24—2364no—
228C24—23624no—
229C24—23612no—
230C24—2368no—
231C24C2×C4×D5916yes—
232C24C23×D5911yescharacteristic
233C22×C4—23no—
234C22×C4—26no—
235C22×C4—23no—
236C22×C4—23no—
237C22×C4—26no—
238C22×C4—23no—
239C22×C4—26no—
240C22×C4—26no—
241C22×C4—5912no—
242C22×C4—1186no—
243C22×C4—1186no—
244C22×C4—1186no—
245C22×C4—1186no—
246C22×C4—1186no—
247C22×C4—1186no—
248C22×C4C2×C4×D5916yes—
249C22×C4C2×C4×D5916yes—
250C22⋊C4—412no—
251C22⋊C4—23624no—
252C22⋊C4—23624no—
253C22⋊C4—23612no—
254C42—44no—
255C42—23612no—
256C23—212no—
257C23—212no—
258C23—26no—
259C23—26no—
260C23—43no—
261C23—41no—
262C23—41no—
263C23—46no—
264C23—43no—
265C23—42no—
266C23—598no—
267C23—11812no—
268C23—11812no—
269C23—11812no—
270C23—11812no—
271C23—11812no—
272C23—11812no—
273C23—23648no—
274C23—23648no—
275C23—23648no—
276C23—23696no—
277C23—2366no—
278C23—2362no—
279C23—2362no—
280C23—2366no—
281C23—2366no—
282C23—2366no—
283C23—2362no—
284C23—2362no—
285C23C42×D5914yes—
286C23C22×C4×D5913yes—
287C23#1427377213yes—
288C23#1427377213yes—
289C23#1427377316yes—
290C2×C4—212no—
291C2×C4—212no—
292C2×C4—26no—
293C2×C4—26no—
294C2×C4—43no—
295C2×C4—43no—
296C2×C4—43no—
297C2×C4—46no—
298C2×C4—46no—
299C2×C4—43no—
300C2×C4—11812no—
301C2×C4—11812no—
302C2×C4—11812no—
303C2×C4—2366no—
304C2×C4—2366no—
305C2×C4—2366no—
306C2×C4—2366no—
307C22—212no—
308C22—46no—
309C22—46no—
310C22—412no—
311C22—11824no—
312C22—23664no—
313C22—23612no—
314C22—23612no—
315C22—23612no—
316C22—23612no—
317C22C2×C42×D5911yescharacteristic
318C22#1683787416yes—
319C22#1683787416yes—
320C22#1683787513yes—
321C22#1683787613yes—
322C22#1683813911yescharacteristic
323C22#1683814016yes—
324C22#1683814213yes—
325C22#1683814312yes—
326C22#1683814413yes—
327C22#1683814511yescharacteristic
328C4—46no—
329C4—46no—
330C4—23612no—
331C2—44no—
332C2—2368no—
333C2#2142644813yes—
334C2#2142723313yes—
335C2#2142723413yes—
336C2#2142723616yes—
subgroupG/H
center, FrattiniC24C23×D59
commutatorC22×C59C22×C42
FittingC59×#1437C2
socle, cosocleC24×C59C24
prime radicalC25×D59C22
hypercenter#1437D59
hypocenterC59#13827
LCSG ▷ C22×C59 ▷ C59
UCSC1 ◁ C24 ◁ #1437

Complements

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

NH ≅ G/Ntype
2. #21422334C2semidirect
3. #21422333C2semidirect
4. #21422332C2semidirect
5. C59×#1437C2semidirect
10. C22×C4×Dic59C22semidirect
11. #16837257C4semidirect
12. C2×C59⋊#95C22semidirect
13. C2×C59⋊#95C22semidirect
15. C22×C42×C59C22semidirect
16. C2×#95×C59C22semidirect
62. C23×D59C42semidirect
167. #1437D59semidirect
180. C22×C42C2×D59semidirect
184. C2×#95C2×D59semidirect
185. C59#13827semidirect

Maximal subgroups

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

subgroupindex
C2×#168372572
#214223342
#214223332
#214223322
C59×#14372
#1382759