Finite Group Databasebeta

← #28486518

Subgroups — #28486518

Order 15104.

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

normal subgroups only

This group has 614 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#21428370C211yesmaximal, characteristic
2#21424196C212yesmaximal, complemented
3#21422423C211yesmaximal, characteristic
4#21421949C212yesmaximal, complemented
5C59×#2312C211yesmaximal, characteristic
6C2×#14273686C2211yescharacteristic
7#16837366C2212yes—
8#16837338C412yescomplemented
9#16837327C412yescomplemented
10#16837290—22no—
11#16837275—21no—
12#16837265C2211yescharacteristic
13#16837258—21no—
14#16837162—24no—
15#16837126—22no—
16C22×C22⋊C4×C59C2211yescharacteristic
17C59×#388—24no—
18C59×#380—21no—
19C59×#379—21no—
20C4×C22⋊C4×C59C2212yes—
21C2×#11745664—22no—
22#14273686—22no—
23#14273686—22no—
24C2×C4⋊Dic59C2×C411yescharacteristic
25C2×C4⋊Dic59C2×C411yescharacteristic
26C2×Dic59⋊C4—22no—
27C2×C22⋊Dic59—22no—
28C2×C22⋊Dic59—21no—
29C2×C22⋊Dic59—21no—
30C2×C22⋊Dic59—22no—
31C2×C4×Dic59—21no—
32C22⋊(C59⋊C8)—22no—
33C22⋊(C59⋊C8)—22no—
34C59⋊#95—22no—
35C59⋊#95—21no—
36C59⋊#95C2×C412yes—
37C25×C59D411yescharacteristic
38C23×C4×C59D411yescharacteristic
39C23×C4×C59C2311yescharacteristic
40C2×C4⋊C4×C59—22no—
41C2×C22⋊C4×C59—22no—
42C2×C22⋊C4×C59—22no—
43C2×C22⋊C4×C59—22no—
44C2×C22⋊C4×C59—22no—
45C2×C22⋊C4×C59—22no—
46C2×C22⋊C4×C59C2×C412yes—
47C2×C22⋊C4×C59D412yes—
48C2×C42×C59D414yes—
49#95×C59—22no—
50#95×C59—22no—
51C24×C59—21no—
52C24×C59—21no—
53C24×C59—42no—
54C24×C59—42no—
55C24×C59—81no—
56C24×C59C22⋊C412yes—
57C24×C59C2×D411yescharacteristic
58C22×C4×C59—21no—
59C22×C4×C59—21no—
60C22×C4×C59—24no—
61C22×C4×C59—42no—
62C22×C4×C59—81no—
63C22×C4×C59C22⋊C412yes—
64C22×C4×C59C22×C411yescharacteristic
65C22×C4×C59C2×D412yes—
66C22×C4×C59C2×D411yescharacteristic
67C22×C4×C59#4112yes—
68C22×Dic59—22no—
69C22×Dic59—21no—
70C22×Dic59—21no—
71C4⋊C4×C59—82no—
72C22⋊C4×C59—24no—
73C22⋊C4×C59—24no—
74C22⋊C4×C59—44no—
75C22⋊C4×C59—82no—
76C22⋊C4×C59—82no—
77C42×C59—44no—
78C22⋊Dic59—42no—
79C22⋊Dic59—44no—
80C22⋊Dic59—41no—
81C22⋊Dic59—41no—
82C22⋊Dic59—82no—
83C4⋊Dic59—22no—
84C4⋊Dic59—22no—
85Dic59⋊C4—82no—
86C4×Dic59—42no—
87#11745664—44no—
88C2×C59⋊C8—42no—
89C23×C59—22no—
90C23×C59—22no—
91C23×C59—22no—
92C23×C59—22no—
93C23×C59—22no—
94C23×C59—41no—
95C23×C59—42no—
96C23×C59—42no—
97C23×C59—42no—
98C23×C59—42no—
99C23×C59—42no—
100C23×C59—42no—
101C23×C59—81no—
102C23×C59—81no—
103C23×C59—88no—
104C23×C59C2×C22⋊C411yescharacteristic
105C23×C59#12011yescharacteristic
106C23×C59#12311yescharacteristic
107C2×C4×C59—24no—
108C2×C4×C59—22no—
109C2×C4×C59—41no—
110C2×C4×C59—41no—
111C2×C4×C59—42no—
112C2×C4×C59—42no—
113C2×C4×C59—42no—
114C2×C4×C59—44no—
115C2×C4×C59—44no—
116C2×C4×C59—44no—
117C2×C4×C59—81no—
118C2×C4×C59C4×D412yes—
119C2×C4×C59#12112yes—
120C2×Dic59—42no—
121C2×Dic59—42no—
122C2×Dic59—42no—
123C2×Dic59—41no—
124C2×Dic59—41no—
125C2×Dic59—42no—
126C2×Dic59—42no—
127C59⋊C8—82no—
128#9481—591nomaximal, Sylow
129C22×C59—21no—
130C22×C59—21no—
131C22×C59—21no—
132C22×C59—21no—
133C22×C59—21no—
134C22×C59—21no—
135C22×C59—21no—
136C22×C59—44no—
137C22×C59—44no—
138C22×C59—42no—
139C22×C59—42no—
140C22×C59—42no—
141C22×C59—82no—
142C22×C59—84no—
143C22×C59—84no—
144C22×C59#38611yescharacteristic
145C22×C59#42112yes—
146C22×C59#45812yes—
147C4×C59—22no—
148C4×C59—22no—
149C4×C59—42no—
150C4×C59—84no—
151Dic59—82no—
152Dic59—81no—
153Dic59—81no—
154#2482—591no—
155#2312D5911yescharacteristic
156#1826—592no—
157#1462—591no—
158#1284—592no—
159C2×C59—22no—
160C2×C59—22no—
161C2×C59—41no—
162C2×C59—82no—
163C2×C59#189011yescharacteristic
164C2×C59#189711yescharacteristic
165C2×C59#191111yescharacteristic
166C2×#117—591no—
167C22×C22⋊C4C2×D5911yescharacteristic
168#407—592no—
169#399—592no—
170#397—592no—
171#388—24no—
172#388—1182no—
173#385—1181no—
174#381—591no—
175#380—21no—
176#379—21no—
177#379—1181no—
178C4×C22⋊C4C2×D5912yes—
179#339—1184no—
180#323—1182no—
181C59#948111yescharacteristic, complemented, Sylow
182C25C22⋊D5911yescharacteristic
183C23×C4C22⋊D5911yescharacteristic
184C23×C4C22×D5911yescharacteristic
185C2×M16—1182no—
186#117—1182no—
187#117—1182no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—591no—
190C2×C4⋊C4—591no—
191C2×C4⋊C4—1182no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—1182no—
198C2×C22⋊C4—1182no—
199C2×C22⋊C4—1181no—
200C2×C22⋊C4—1181no—
201C2×C22⋊C4C4×D5912yes—
202C2×C22⋊C4D23612yes—
203C2×C42—1181no—
204C2×C42C22⋊D5914yes—
205C22⋊C8—1182no—
206C22⋊C8—1182no—
207#95—22no—
208#95—22no—
209#95—592no—
210#95—1182no—
211#95—1181no—
212C24—21no—
213C24—21no—
214C24—42no—
215C24—42no—
216C24—81no—
217C24(C2×D59)⋊C412yes—
218C24C2×C22⋊D5911yescharacteristic
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—24no—
222C22×C4—42no—
223C22×C4—81no—
224C22×C4—1182no—
225C22×C4—1181no—
226C22×C4—1181no—
227C22×C4(C2×D59)⋊C412yes—
228C22×C4C2×C4×D5911yescharacteristic
229C22×C4C2×D23611yescharacteristic
230C22×C4#1174568512yes—
231C22×C4C2×C22⋊D5912yes—
232M16—2364no—
233C2×C8—2362no—
234C4⋊C4—82no—
235C4⋊C4—1182no—
236C4⋊C4—1182no—
237C4⋊C4—4722no—
238C22⋊C4—24no—
239C22⋊C4—24no—
240C22⋊C4—44no—
241C22⋊C4—82no—
242C22⋊C4—82no—
243C22⋊C4—4722no—
244C22⋊C4—2362no—
245C22⋊C4—2364no—
246C22⋊C4—2361no—
247C22⋊C4—2361no—
248C42—44no—
249C42—2362no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—41no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—81no—
263C23—81no—
264C23—88no—
265C23C2×(C2×D59)⋊C411yescharacteristic
266C23#1427369611yescharacteristic
267C23#1427371011yescharacteristic
268C2×C4—24no—
269C2×C4—22no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—81no—
279C2×C4—2362no—
280C2×C4—2362no—
281C2×C4—2362no—
282C2×C4—2362no—
283C2×C4—2362no—
284C2×C4—2361no—
285C2×C4—2361no—
286C2×C4C4×C22⋊D5912yes—
287C2×C4#1427370112yes—
288C8—4722no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—44no—
297C22—44no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—82no—
302C22—84no—
303C22—84no—
304C22#1683728111yescharacteristic
305C22#1683742012yes—
306C22#1683759212yes—
307C4—22no—
308C4—22no—
309C4—42no—
310C4—84no—
311C4—4722no—
312C4—4721no—
313C4—4721no—
314C2—22no—
315C2—22no—
316C2—41no—
317C2—82no—
318C2#2142450911yescharacteristic
319C2#2142454411yescharacteristic
320C2#2142462811yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#2312C2
socleC22×C59#386
prime radicalC25×C59D4
cosocleC23×C4×C59C23
hypercenter#2312D59
hypocenterC59#9481
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2312

Complements

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

NH ≅ G/Ntype
2. #21424196C2semidirect
4. #21421949C2semidirect
8. #16837338C4semidirect
9. #16837327C4semidirect
181. C59#9481semidirect

Maximal subgroups

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

subgroupindex
#214283702
#214241962
#214224232
#214219492
C59×#23122
#948159