Finite Group Databasebeta

← #28735638

Subgroups — #28735638

Order 15104.

984 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 984 conjugacy classes of subgroups — too many to draw. Only its 175 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#21432383C211yesmaximal, characteristic
2#21425527C213yesmaximal, complemented
3#21425299C213yesmaximal, complemented
4#21425296C213yesmaximal, complemented
5#21425104C213yesmaximal, complemented
6#21424006C211yesmaximal, characteristic
7C59×#2848C211yesmaximal, characteristic
8C22×#11745672C2211yescharacteristic, complemented
9C2×#14273706C2213yes—
10C2×#14273660C2213yescomplemented
11C2×#14273651C2213yescomplemented
12C2×#14273650C2213yescomplemented
13C2×C22⋊(C59⋊C8)C2213yes—
14C2×C22⋊(C59⋊C8)C2213yes—
15#16837275—23no—
16#16837262C2213yes—
17#16837258—21no—
18#16837164C2213yescomplemented
19#16837164C2213yescomplemented
20C24×C4×C59C2211yescharacteristic
21C2×C22⋊Q8×C59C2213yes—
22C59×#385—23no—
23C59×#380C2213yes—
24C59×#379—21no—
25C22×Dic118C2311yescharacteristic
26C2×#11745672—43no—
27C22×C59⋊C8C2313yes—
28C22×C59⋊C8C2311yescharacteristic
29#14273706—43no—
30#14273706—43no—
31C2×C4⋊Dic59C2313yes—
32C2×Dic59⋊C4—23no—
33C2×C22⋊Dic59—23no—
34C2×C22⋊Dic59—23no—
35#14273660—43no—
36#14273651—43no—
37#14273650—43no—
38C22⋊(C59⋊C8)—43no—
39C22⋊(C59⋊C8)—43no—
40C59⋊#95—23no—
41C25×C59D411yescharacteristic
42C22×Q8×C59C2311yescharacteristic
43C23×C4×C59—43no—
44C23×C4×C59—43no—
45C23×C4×C59D413yes—
46C23×C4×C59C2313yes—
47C22⋊Q8×C59—43no—
48C22⋊Q8×C59—43no—
49C2×C4⋊C4×C59—23no—
50C2×C4⋊C4×C59C2313yes—
51C2×C22⋊C4×C59—23no—
52C2×C22⋊C4×C59—23no—
53#95×C59—23no—
54C24×C59—43no—
55C24×C59—43no—
56C24×C59Q1614yescomplemented
57C24×C59C2×D413yes—
58C2×Q8×C59—23no—
59C2×Q8×C59—23no—
60C22×C4×C59—23no—
61C22×C4×C59—23no—
62C22×C4×C59—23no—
63C22×C4×C59—23no—
64C22×C4×C59—23no—
65C22×C4×C59—23no—
66C22×C4×C59—23no—
67C22×C4×C59—21no—
68C22×C4×C59—43no—
69C22×C4×C59—43no—
70C22×C4×C59—43no—
71C22×C4×C59—412no—
72C22×C4×C59—44no—
73C22×C4×C59C2×D413yes—
74C22×C4×C59C2411yescharacteristic
75C22×Dic59—23no—
76C22×Dic59—21no—
77C2×Dic118—23no—
78C2×Dic118—23no—
79C4⋊C4×C59—23no—
80C4⋊C4×C59—23no—
81C4⋊C4×C59—46no—
82C22⋊C4×C59—46no—
83C22⋊C4×C59—46no—
84C22⋊Dic59—46no—
85C22⋊Dic59—46no—
86#11745672—44no—
87C4⋊Dic59—23no—
88C4⋊Dic59—23no—
89Dic59⋊C4—46no—
90C2×C59⋊C8—23no—
91C2×C59⋊C8—23no—
92C2×C59⋊C8—23no—
93C2×C59⋊C8—23no—
94C23×C59—23no—
95C23×C59—23no—
96C23×C59—23no—
97C23×C59—23no—
98C23×C59—23no—
99C23×C59—23no—
100C23×C59—412no—
101C23×C59—44no—
102C23×C59—412no—
103C23×C59C2×Q1616yes—
104C23×C59C22×D411yescharacteristic
105Q8×C59—43no—
106Q8×C59—41no—
107C2×C4×C59—23no—
108C2×C4×C59—23no—
109C2×C4×C59—23no—
110C2×C4×C59—412no—
111C2×C4×C59—43no—
112C2×C4×C59—43no—
113C2×C4×C59—43no—
114C2×C4×C59—43no—
115C2×C4×C59—43no—
116C2×C4×C59—43no—
117C2×C4×C59—46no—
118C2×C4×C59H(5,𝔽2)13yes—
119C2×C4×C59H(5,𝔽2)13yes—
120C2×Dic59—43no—
121C2×Dic59—43no—
122C2×Dic59—46no—
123Dic118—43no—
124Dic118—41no—
125C59⋊C8—43no—
126C59⋊C8—41no—
127#32710—591nomaximal, Sylow
128C22×C59—26no—
129C22×C59—26no—
130C22×C59—26no—
131C22×C59—416no—
132C22×C59—43no—
133C22×C59—43no—
134C22×C59—43no—
135C22×C59—43no—
136C22×C59#53413yes—
137C22×C59C22×Q1611yescharacteristic
138C22×C59#57513yes—
139C4×C59—43no—
140C4×C59—83no—
141C4×C59—81no—
142C4×C59#56013yes—
143C4×C59H(3,𝔽4)11yescharacteristic
144Dic59—83no—
145Dic59—81no—
146#2848—591no—
147#2848D5911yescharacteristic
148#2076—593no—
149#2036—593no—
150#2036—593no—
151#2001—593no—
152#1779—591no—
153C2×C59—46no—
154C2×C59#281211yescharacteristic
155C2×C59#318913yes—
156C2×C59#319713yes—
157C24×C4C2×D5911yescharacteristic
158C22×Q16—591no—
159C2×C22⋊Q8—593no—
160C2×C22⋊Q8C2×D5913yes—
161C2×#107—593no—
162C2×#103—593no—
163C2×#103—593no—
164C2×C22⋊C8—593no—
165C2×C22⋊C8—593no—
166#385—23no—
167#385—1183no—
168#380—593no—
169#380C2×D5913yes—
170#379—21no—
171#379—1181no—
172#340—593no—
173#340—593no—
174C59#3271011yescharacteristic, complemented, Sylow
175C25C22⋊D5911yescharacteristic
176C22×Q8—591no—
177C22×Q8C22×D5911yescharacteristic
178C23×C4—43no—
179C23×C4—43no—
180C23×C4C22⋊D5913yes—
181C23×C4C22×D5913yes—
182C2×Q16—2363no—
183C22×C8—591no—
184C22×C8—593no—
185C22⋊Q8—43no—
186C22⋊Q8—43no—
187C22⋊Q8—2363no—
188C22⋊Q8—2363no—
189C2×C4⋊C4—23no—
190C2×C4⋊C4—593no—
191C2×C4⋊C4—1183no—
192C2×C4⋊C4C22×D5913yes—
193C2×C22⋊C4—23no—
194C2×C22⋊C4—23no—
195C2×C22⋊C4—1183no—
196C2×C22⋊C4—1183no—
197#107—2363no—
198#103—2363no—
199#103—2363no—
200C22⋊C8—2363no—
201C22⋊C8—2363no—
202#95—23no—
203#95—1183no—
204C24—43no—
205C24—43no—
206C24#1174567214yescomplemented
207C24C2×C22⋊D5913yes—
208C2×Q8—23no—
209C2×Q8—23no—
210C2×Q8—1183no—
211C2×Q8—1183no—
212C22×C4—23no—
213C22×C4—23no—
214C22×C4—23no—
215C22×C4—23no—
216C22×C4—23no—
217C22×C4—23no—
218C22×C4—23no—
219C22×C4—21no—
220C22×C4—43no—
221C22×C4—43no—
222C22×C4—43no—
223C22×C4—44no—
224C22×C4—412no—
225C22×C4—1183no—
226C22×C4—1181no—
227C22×C4C2×C22⋊D5913yes—
228C22×C4C23×D5911yescharacteristic
229Q16—2364no—
230C2×C8—1183no—
231C2×C8—1183no—
232C2×C8—1183no—
233C2×C8—1183no—
234C4⋊C4—23no—
235C4⋊C4—23no—
236C4⋊C4—46no—
237C4⋊C4—1183no—
238C4⋊C4—1183no—
239C4⋊C4—2366no—
240C22⋊C4—46no—
241C22⋊C4—46no—
242C22⋊C4—2366no—
243C22⋊C4—2366no—
244C23—23no—
245C23—23no—
246C23—23no—
247C23—23no—
248C23—23no—
249C23—23no—
250C23—412no—
251C23—44no—
252C23—412no—
253C23C2×#1174567216yes—
254C23C22×C22⋊D5911yescharacteristic
255Q8—41no—
256Q8—43no—
257Q8—2361no—
258Q8—2363no—
259C2×C4—23no—
260C2×C4—23no—
261C2×C4—23no—
262C2×C4—412no—
263C2×C4—43no—
264C2×C4—43no—
265C2×C4—43no—
266C2×C4—43no—
267C2×C4—46no—
268C2×C4—43no—
269C2×C4—43no—
270C2×C4—2366no—
271C2×C4—2363no—
272C2×C4—2363no—
273C2×C4#1427377213yes—
274C2×C4#1427377213yes—
275C8—2361no—
276C8—2363no—
277C22—26no—
278C22—26no—
279C22—26no—
280C22—416no—
281C22—43no—
282C22—43no—
283C22—43no—
284C22—43no—
285C22#1683796113yes—
286C22C22×#1174567211yescharacteristic
287C22#1683820013yes—
288C4—43no—
289C4—81no—
290C4—83no—
291C4—4721no—
292C4—4723no—
293C4#1683814113yes—
294C4#1683814511yescharacteristic
295C2—46no—
296C2#2143209111yescharacteristic
297C2#2143486813yes—
298C2#2143493313yes—
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2848C2
socleC23×C59C22×D4
prime radicalC25×C59D4
hypercenter#2848D59
hypocenterC59#32710
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2848

Complements

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

NH ≅ G/Ntype
2. #21425527C2semidirect
3. #21425299C2semidirect
4. #21425296C2semidirect
5. #21425104C2semidirect
8. C22×#11745672C22semidirect
10. C2×#14273660C22semidirect
11. C2×#14273651C22semidirect
12. C2×#14273650C22semidirect
18. #16837164C22semidirect
19. #16837164C22semidirect
56. C24×C59Q16semidirect
174. C59#32710semidirect
206. C24#11745672semidirect

Maximal subgroups

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

subgroupindex
#214323832
#214255272
#214252992
#214252962
#214251042
#214240062
C59×#28482
#3271059