Finite Group Databasebeta

← #28486418

Subgroups — #28486418

Order 15104.

528 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 103 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#21428570C211yesmaximal, characteristic
2#21424176C211yesmaximal, characteristic
3#21422628C212yesmaximal
4#21421948C212yesmaximal
5C59×#2313C211yesmaximal, characteristic
6C2×#14273706C2211yescharacteristic
7#16837366C2212yes—
8#16837333—22no—
9#16837307C412yescomplemented
10#16837275—21no—
11#16837275—21no—
12#16837271C412yescomplemented
13C4×C22⋊Dic59C2211yescharacteristic
14#16837171—22no—
15#16837138—24no—
16C22×C4⋊C4×C59C2211yescharacteristic
17C59×#387—24no—
18C59×#381C2212yes—
19C59×#380—21no—
20C4×C22⋊C4×C59—21no—
21C22×Dic118C2×C411yescharacteristic
22C2×#11745664—22no—
23#14273706—22no—
24#14273706—22no—
25C2×C4⋊Dic59C2×C411yescharacteristic
26C2×Dic59⋊C4—21no—
27C2×Dic59⋊C4—22no—
28C2×Dic59⋊C4—22no—
29C2×C22⋊Dic59—21no—
30C2×C22⋊Dic59—21no—
31C2×C4×Dic59C2×C412yes—
32C22⋊(C59⋊C8)—22no—
33C22⋊(C59⋊C8)—22no—
34C59⋊#95—22no—
35C59⋊#95—22no—
36C59⋊#95—21no—
37C23×C4×C59D411yescharacteristic
38C23×C4×C59D411yescharacteristic
39C23×C4×C59C2311yescharacteristic
40C2×C4⋊C4×C59—22no—
41C2×C4⋊C4×C59—22no—
42C2×C4⋊C4×C59—22no—
43C2×C4⋊C4×C59C2×C412yes—
44C2×C4⋊C4×C59D412yes—
45C2×C22⋊C4×C59—22no—
46C2×C42×C59—22no—
47#95×C59—22no—
48#95×C59—22no—
49#95×C59—22no—
50#95×C59D414yes—
51C24×C59C2×D411yescharacteristic
52C22×C4×C59—21no—
53C22×C4×C59—21no—
54C22×C4×C59—21no—
55C22×C4×C59—21no—
56C22×C4×C59—24no—
57C22×C4×C59—42no—
58C22×C4×C59—81no—
59C22×C4×C59—81no—
60C22×C4×C59C22⋊C412yes—
61C22×C4×C59C22⋊C412yes—
62C22×C4×C59C22×C411yescharacteristic
63C22×C4×C59C2×D411yescharacteristic
64C22×C4×C59C2×D412yes—
65C22×C4×C59#4112yes—
66C22×Dic59—22no—
67C22×Dic59—21no—
68C22×Dic59—21no—
69C2×Dic118—22no—
70C2×Dic118—42no—
71C4⋊C4×C59—24no—
72C4⋊C4×C59—24no—
73C4⋊C4×C59—82no—
74C22⋊C4×C59—44no—
75C42×C59—82no—
76C22⋊Dic59—24no—
77C22⋊Dic59—41no—
78C22⋊Dic59—41no—
79C4⋊Dic59—22no—
80Dic59⋊C4—42no—
81Dic59⋊C4—82no—
82Dic59⋊C4—82no—
83C4×Dic59—42no—
84#11745664—44no—
85C2×C59⋊C8—42no—
86C23×C59—22no—
87C23×C59—22no—
88C23×C59—41no—
89C23×C59C2×C22⋊C411yescharacteristic
90C23×C59#12011yescharacteristic
91C23×C59#12311yescharacteristic
92C2×C4×C59—22no—
93C2×C4×C59—22no—
94C2×C4×C59—22no—
95C2×C4×C59—22no—
96C2×C4×C59—42no—
97C2×C4×C59—42no—
98C2×C4×C59—42no—
99C2×C4×C59—42no—
100C2×C4×C59—42no—
101C2×C4×C59—42no—
102C2×C4×C59—44no—
103C2×C4×C59—44no—
104C2×C4×C59—44no—
105C2×C4×C59—81no—
106C2×C4×C59—81no—
107C2×C4×C59C4×D412yes—
108C2×C4×C59#12112yes—
109C2×Dic59—24no—
110C2×Dic59—42no—
111C2×Dic59—42no—
112C2×Dic59—42no—
113C2×Dic59—41no—
114C2×Dic59—41no—
115C2×Dic59—42no—
116Dic118—82no—
117C59⋊C8—82no—
118#9461—591nomaximal, Sylow
119C22×C59—21no—
120C22×C59—21no—
121C22×C59—21no—
122C22×C59—21no—
123C22×C59—21no—
124C22×C59—21no—
125C22×C59—21no—
126C22×C59—44no—
127C22×C59#38611yescharacteristic
128C22×C59#41012yes—
129C22×C59#45612yes—
130C4×C59—22no—
131C4×C59—22no—
132C4×C59—42no—
133C4×C59—42no—
134C4×C59—84no—
135Dic59—44no—
136Dic59—81no—
137Dic59—81no—
138#2502—591no—
139#2313D5911yescharacteristic
140#1821—591no—
141#1511—592no—
142#1284—592no—
143C2×C59—22no—
144C2×C59—22no—
145C2×C59—41no—
146C2×C59#188611yescharacteristic
147C2×C59#189911yescharacteristic
148C2×C59#191111yescharacteristic
149C2×C22⋊Q8—591no—
150C22×C4⋊C4C2×D5911yescharacteristic
151#407—592no—
152#398—1182no—
153#391—592no—
154#387—24no—
155#385—1181no—
156#385—1181no—
157#384—592no—
158#381C2×D5912yes—
159#380—21no—
160C4×C22⋊C4—21no—
161C4×C22⋊C4—591no—
162#343—1182no—
163#328—1184no—
164C59#946111yescharacteristic, complemented, Sylow
165C22×Q8—591no—
166C23×C4C22⋊D5911yescharacteristic
167C23×C4C22⋊D5911yescharacteristic
168C23×C4C22×D5911yescharacteristic
169C2×M16—1182no—
170C22⋊Q8—1182no—
171C22⋊Q8—1182no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—591no—
176C2×C4⋊C4—1182no—
177C2×C4⋊C4—1182no—
178C2×C4⋊C4—1181no—
179C2×C4⋊C4C4×D5912yes—
180C2×C4⋊C4D23612yes—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—1181no—
183C2×C22⋊C4—1181no—
184C2×C42—22no—
185C2×C42—592no—
186C22⋊C8—1182no—
187C22⋊C8—1182no—
188#95—22no—
189#95—22no—
190#95—22no—
191#95—1182no—
192#95—1182no—
193#95—1181no—
194#95C22⋊D5914yes—
195C24C2×C22⋊D5911yescharacteristic
196C2×Q8—1182no—
197C2×Q8—2362no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—24no—
203C22×C4—42no—
204C22×C4—81no—
205C22×C4—81no—
206C22×C4—1182no—
207C22×C4—1181no—
208C22×C4—1181no—
209C22×C4(C2×D59)⋊C412yes—
210C22×C4(C2×D59)⋊C412yes—
211C22×C4C2×C4×D5911yescharacteristic
212C22×C4C2×D23611yescharacteristic
213C22×C4#1174568512yes—
214C22×C4C2×C22⋊D5912yes—
215M16—2364no—
216C2×C8—2362no—
217C4⋊C4—24no—
218C4⋊C4—24no—
219C4⋊C4—82no—
220C4⋊C4—1182no—
221C4⋊C4—4722no—
222C4⋊C4—4722no—
223C4⋊C4—2362no—
224C22⋊C4—44no—
225C22⋊C4—1184no—
226C22⋊C4—2361no—
227C22⋊C4—2361no—
228C42—82no—
229C42—2362no—
230C23—22no—
231C23—22no—
232C23—41no—
233C23C2×(C2×D59)⋊C411yescharacteristic
234C23#1427369611yescharacteristic
235C23#1427371011yescharacteristic
236Q8—4722no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—1184no—
253C2×C4—2362no—
254C2×C4—2361no—
255C2×C4—2361no—
256C2×C4—2362no—
257C2×C4—2362no—
258C2×C4—2362no—
259C2×C4C4×C22⋊D5912yes—
260C2×C4#1427370112yes—
261C8—4722no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—44no—
270C22#1683728111yescharacteristic
271C22#1683737512yes—
272C22#1683758512yes—
273C4—22no—
274C4—22no—
275C4—42no—
276C4—42no—
277C4—84no—
278C4—4721no—
279C4—4721no—
280C4—2364no—
281C2—22no—
282C2—22no—
283C2—41no—
284C2#2142448911yescharacteristic
285C2#2142455411yescharacteristic
286C2#2142462811yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#2313C2
socleC22×C59#386
prime radicalC24×C59C2×D4
cosocleC23×C4×C59C23
hypercenter#2313D59
hypocenterC59#9461
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2313

Complements

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

NH ≅ G/Ntype
9. #16837307C4semidirect
12. #16837271C4semidirect
164. C59#9461semidirect

Maximal subgroups

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

subgroupindex
#214285702
#214241762
#214226282
#214219482
C59×#23132
#946159