Finite Group Databasebeta

← #28484454

Subgroups — #28484454

Order 15104.

586 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 131 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#21432435C211yesmaximal, characteristic
2#21424164C211yesmaximal, characteristic
3C2×#16837138C212yesmaximal
4#21421975C212yesmaximal
5C2×C59×#383C211yesmaximal, characteristic
6C2×#14273706C2211yescharacteristic
7C2×C22⋊(C59⋊C8)C2212yes—
8#16837308C412yescomplemented
9#16837300C412yescomplemented
10#16837286—22no—
11#16837275—21no—
12#16837265—21no—
13#16837262C2211yescharacteristic
14#16837157—22no—
15#16837138—24no—
16C22×C42×C59C2211yescharacteristic
17C59×#383—24no—
18C2×#95×C59—21no—
19C2×#95×C59—21no—
20C2×#95×C59C2212yes—
21C22×Dic118C2×C411yescharacteristic
22C22×C59⋊C8—22no—
23#14273706—22no—
24#14273706—22no—
25C2×C4⋊Dic59C2×C411yescharacteristic
26C2×C4⋊Dic59C2×C412yes—
27C2×Dic59⋊C4—22no—
28C2×Dic59⋊C4—21no—
29C2×C22⋊Dic59—21no—
30C2×C22⋊Dic59—21no—
31C22⋊(C59⋊C8)—22no—
32C22⋊(C59⋊C8)—22no—
33C59⋊#95—22no—
34C59⋊#95—22no—
35C59⋊#95—22no—
36C59⋊#95—21no—
37C23×C4×C59—22no—
38C23×C4×C59D411yescharacteristic
39C23×C4×C59D411yescharacteristic
40C23×C4×C59C2311yescharacteristic
41C2×C42×C59—22no—
42C2×C42×C59—22no—
43C2×C42×C59C2×C412yes—
44C2×C42×C59D412yes—
45#95×C59—24no—
46#95×C59—24no—
47#95×C59—22no—
48#95×C59D414yes—
49C24×C59C2×D411yescharacteristic
50C22×C4×C59—21no—
51C22×C4×C59—21no—
52C22×C4×C59—21no—
53C22×C4×C59—21no—
54C22×C4×C59—22no—
55C22×C4×C59—22no—
56C22×C4×C59—24no—
57C22×C4×C59—42no—
58C22×C4×C59—42no—
59C22×C4×C59—44no—
60C22×C4×C59—44no—
61C22×C4×C59—42no—
62C22×C4×C59—42no—
63C22×C4×C59C22⋊C412yes—
64C22×C4×C59C22⋊C412yes—
65C22×C4×C59C22×C411yescharacteristic
66C22×C4×C59C2×D411yescharacteristic
67C22×C4×C59C2×D412yes—
68C22×C4×C59#4112yes—
69C22×Dic59—22no—
70C22×Dic59—21no—
71C22×Dic59—21no—
72C2×Dic118—22no—
73C2×Dic118—42no—
74C42×C59—24no—
75C42×C59—24no—
76C22⋊Dic59—41no—
77C22⋊Dic59—41no—
78C22⋊Dic59—42no—
79C4⋊Dic59—22no—
80C4⋊Dic59—42no—
81Dic59⋊C4—44no—
82Dic59⋊C4—42no—
83C2×C59⋊C8—44no—
84C2×C59⋊C8—42no—
85C23×C59—22no—
86C23×C59—22no—
87C23×C59C2×C22⋊C411yescharacteristic
88C23×C59#12011yescharacteristic
89C23×C59#12311yescharacteristic
90C23×C59#13512yes—
91C23×C59#13712yes—
92C2×C4×C59—22no—
93C2×C4×C59—22no—
94C2×C4×C59—22no—
95C2×C4×C59—22no—
96C2×C4×C59—22no—
97C2×C4×C59—22no—
98C2×C4×C59—22no—
99C2×C4×C59—22no—
100C2×C4×C59—42no—
101C2×C4×C59—42no—
102C2×C4×C59—42no—
103C2×C4×C59—42no—
104C2×C4×C59—42no—
105C2×C4×C59—42no—
106C2×C4×C59—44no—
107C2×C4×C59—44no—
108C2×C4×C59—44no—
109C2×C4×C59—84no—
110C2×C4×C59—84no—
111C2×C4×C59C4×D412yes—
112C2×C4×C59#12112yes—
113C2×Dic59—41no—
114C2×Dic59—41no—
115C2×Dic59—42no—
116C2×Dic59—42no—
117C2×Dic59—42no—
118C2×Dic59—42no—
119C2×Dic59—42no—
120Dic118—82no—
121C59⋊C8—82no—
122#9153—591nomaximal, Sylow
123C22×C59—21no—
124C22×C59—21no—
125C22×C59—21no—
126C22×C59—21no—
127C22×C59—24no—
128C22×C59—24no—
129C22×C59#38611yescharacteristic
130C22×C59#41012yes—
131C22×C59#41611yescharacteristic
132C22×C59#41911yescharacteristic
133C22×C59#42112yes—
134C22×C59#45212yes—
135C22×C59#48512yes—
136C4×C59—22no—
137C4×C59—22no—
138C4×C59—42no—
139C4×C59—42no—
140C4×C59—84no—
141Dic59—81no—
142Dic59—81no—
143Dic59—82no—
144#2855—591no—
145C2×#383D5911yescharacteristic
146#1817—591no—
147C2×#328—592no—
148#1294—592no—
149C2×C59—22no—
150C2×C59—22no—
151C2×C59#151512yes—
152C2×C59#152112yes—
153C2×C59#187311yescharacteristic
154C2×C59#189711yescharacteristic
155C2×C59#189911yescharacteristic
156C2×C22⋊Q8—591no—
157C22×C42C2×D5911yescharacteristic
158C2×C22⋊C8—592no—
159#391—592no—
160#389—592no—
161#387—1182no—
162#385—1181no—
163#383—24no—
164#381—1181no—
165#380—591no—
166C2×#95—21no—
167C2×#95—21no—
168C2×#95C2×D5912yes—
169#336—1182no—
170#328—1184no—
171C59#915311yescharacteristic, complemented, Sylow
172C22×Q8—591no—
173C23×C4—22no—
174C23×C4C22⋊D5911yescharacteristic
175C23×C4C22⋊D5911yescharacteristic
176C23×C4C22×D5911yescharacteristic
177C22×C8—1182no—
178C22⋊Q8—1182no—
179C22⋊Q8—1182no—
180C2×C4⋊C4—591no—
181C2×C4⋊C4—592no—
182C2×C4⋊C4—1182no—
183C2×C4⋊C4—1181no—
184C2×C22⋊C4—1181no—
185C2×C22⋊C4—1181no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42C4×D5912yes—
189C2×C42D23612yes—
190C22⋊C8—1182no—
191C22⋊C8—1182no—
192#95—24no—
193#95—24no—
194#95—22no—
195#95—1182no—
196#95—1182no—
197#95—1182no—
198#95—1181no—
199#95C22⋊D5914yes—
200C24C2×C22⋊D5911yescharacteristic
201C2×Q8—1182no—
202C2×Q8—2362no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—1182no—
217C22×C4—1181no—
218C22×C4—1181no—
219C22×C4(C2×D59)⋊C412yes—
220C22×C4(C2×D59)⋊C412yes—
221C22×C4C2×C4×D5911yescharacteristic
222C22×C4C2×D23611yescharacteristic
223C22×C4#1174568512yes—
224C22×C4C2×C22⋊D5912yes—
225C2×C8—2364no—
226C2×C8—2362no—
227C4⋊C4—1182no—
228C4⋊C4—2362no—
229C4⋊C4—2364no—
230C4⋊C4—2362no—
231C22⋊C4—2361no—
232C22⋊C4—2361no—
233C22⋊C4—2362no—
234C42—24no—
235C42—24no—
236C23—22no—
237C23—22no—
238C23C2×(C2×D59)⋊C411yescharacteristic
239C23#1427369611yescharacteristic
240C23#1427371011yescharacteristic
241C23#1427374712yes—
242C23#1427375712yes—
243Q8—4722no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—84no—
262C2×C4—84no—
263C2×C4—2362no—
264C2×C4—2362no—
265C2×C4—2362no—
266C2×C4—2361no—
267C2×C4—2361no—
268C2×C4—2362no—
269C2×C4—2362no—
270C2×C4C4×C22⋊D5912yes—
271C2×C4#1427370112yes—
272C8—4722no—
273C22—24no—
274C22—24no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22#1683728111yescharacteristic
280C22#1683737512yes—
281C22#1683739611yescharacteristic
282C22#1683741011yescharacteristic
283C22#1683742012yes—
284C22#1683756412yes—
285C22#1683774312yes—
286C4—22no—
287C4—22no—
288C4—42no—
289C4—42no—
290C4—84no—
291C4—4722no—
292C4—4721no—
293C4—4721no—
294C2—22no—
295C2—22no—
296C2#2142264612yes—
297C2#2142267112yes—
298C2#2142440811yescharacteristic
299C2#2142454411yescharacteristic
300C2#2142455411yescharacteristic
subgroupG/H
centerC23C2×(C2×D59)⋊C4
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC2×C59×#383C2
socleC23×C59C2×C22⋊C4
prime radicalC24×C59C2×D4
cosocleC23×C4×C59C23
hypercenterC2×#383D59
hypocenterC59#9153
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#383

Complements

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

NH ≅ G/Ntype
8. #16837308C4semidirect
9. #16837300C4semidirect
171. C59#9153semidirect

Maximal subgroups

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

subgroupindex
#214324352
#214241642
C2×#168371382
#214219752
C2×C59×#3832
#915359