Finite Group Databasebeta

← #28486145

Subgroups — #28486145

Order 15104.

606 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 107 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#21426694C211yesmaximal, characteristic
2#21422616C212yesmaximal, complemented
3#21422423C211yesmaximal, characteristic
4#21421982C212yesmaximal, complemented
5C59×#2561C211yesmaximal, characteristic
6#16837366C2212yes—
7#16837295—24no—
8#16837262—21no—
9#16837258—21no—
10C4×C22⋊Dic59C2212yes—
11#16837139—24no—
12#16837126—22no—
13C2×C22⋊Q8×C59C2211yescharacteristic
14C22×C22⋊C4×C59C2211yescharacteristic
15C59×#397C412yescomplemented
16C59×#393C412yescomplemented
17C59×#388—22no—
18C59×#381—21no—
19C59×#379—21no—
20C4×C22⋊C4×C59C2211yescharacteristic
21C2×#11745664—22no—
22C2×C4⋊Dic59—22no—
23C2×C22⋊Dic59—22no—
24C2×C22⋊Dic59—22no—
25C2×C22⋊Dic59—22no—
26C2×C4×Dic59—22no—
27C22⋊(C59⋊C8)—22no—
28C22⋊(C59⋊C8)—22no—
29C59⋊#95—22no—
30C59⋊#95D414yes—
31C25×C59D411yescharacteristic
32C22×Q8×C59C2×C411yescharacteristic
33C23×C4×C59D411yescharacteristic
34C23×C4×C59C2311yescharacteristic
35C22⋊Q8×C59—22no—
36C22⋊Q8×C59—22no—
37C2×C4⋊C4×C59—21no—
38C2×C4⋊C4×C59C2×C411yescharacteristic
39C2×C22⋊C4×C59—22no—
40C2×C22⋊C4×C59—22no—
41C2×C22⋊C4×C59—22no—
42C2×C22⋊C4×C59—21no—
43C2×C22⋊C4×C59—21no—
44C2×C22⋊C4×C59—22no—
45C2×C22⋊C4×C59C2×C412yes—
46C2×C22⋊C4×C59D412yes—
47C2×C42×C59—21no—
48#95×C59—22no—
49#95×C59—22no—
50#95×C59C2×C412yes—
51C24×C59—21no—
52C24×C59—21no—
53C24×C59—42no—
54C24×C59—42no—
55C24×C59—81no—
56C24×C59C22⋊C412yes—
57C24×C59C2×D411yescharacteristic
58C2×Q8×C59—22no—
59C2×Q8×C59—81no—
60C22×C4×C59—21no—
61C22×C4×C59—21no—
62C22×C4×C59—22no—
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—42no—
66C22×C4×C59—81no—
67C22×C4×C59C22⋊C412yes—
68C22×C4×C59C22×C411yescharacteristic
69C22×C4×C59C2×D412yes—
70C22×C4×C59C2×D411yescharacteristic
71C22×C4×C59#4112yes—
72C22×Dic59—24no—
73C4⋊C4×C59—22no—
74C4⋊C4×C59—42no—
75C22⋊C4×C59—24no—
76C22⋊C4×C59—24no—
77C22⋊C4×C59—42no—
78C22⋊C4×C59—44no—
79C22⋊C4×C59—41no—
80C22⋊C4×C59—41no—
81C22⋊C4×C59—82no—
82C42×C59—81no—
83C22⋊Dic59—44no—
84C22⋊Dic59—82no—
85C22⋊Dic59—82no—
86C4⋊Dic59—82no—
87C4×Dic59—82no—
88#11745664—82no—
89C2×C59⋊C8—42no—
90C23×C59—22no—
91C23×C59—22no—
92C23×C59—22no—
93C23×C59—22no—
94C23×C59—22no—
95C23×C59—41no—
96C23×C59—42no—
97C23×C59—42no—
98C23×C59—42no—
99C23×C59—42no—
100C23×C59—42no—
101C23×C59—42no—
102C23×C59—81no—
103C23×C59—81no—
104C23×C59—88no—
105C23×C59C2×C22⋊C411yescharacteristic
106C23×C59#12011yescharacteristic
107C23×C59#12311yescharacteristic
108Q8×C59—82no—
109C2×C4×C59—22no—
110C2×C4×C59—42no—
111C2×C4×C59—42no—
112C2×C4×C59—41no—
113C2×C4×C59—41no—
114C2×C4×C59—42no—
115C2×C4×C59—42no—
116C2×C4×C59—42no—
117C2×C4×C59—42no—
118C2×C4×C59—42no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59—42no—
122C2×C4×C59—42no—
123C2×C4×C59—81no—
124C2×C4×C59C4×D412yes—
125C2×C4×C59#12112yes—
126C2×Dic59—44no—
127C2×Dic59—44no—
128C2×Dic59—44no—
129C59⋊C8—82no—
130#9411—591nomaximal, Sylow
131C22×C59—21no—
132C22×C59—21no—
133C22×C59—21no—
134C22×C59—21no—
135C22×C59—21no—
136C22×C59—21no—
137C22×C59—21no—
138C22×C59—44no—
139C22×C59—44no—
140C22×C59—42no—
141C22×C59—42no—
142C22×C59—42no—
143C22×C59—82no—
144C22×C59—84no—
145C22×C59—84no—
146C22×C59#38611yescharacteristic
147C22×C59#42112yes—
148C22×C59#45512yes—
149C4×C59—22no—
150C4×C59—22no—
151C4×C59—42no—
152C4×C59—82no—
153C4×C59—81no—
154C4×C59—81no—
155Dic59—84no—
156#2561D5911yescharacteristic
157#2312—591no—
158#1508—592no—
159#1462—591no—
160#1297—592no—
161C2×C59—22no—
162C2×C59—22no—
163C2×C59—41no—
164C2×C59—82no—
165C2×C59#188511yescharacteristic
166C2×C59#189711yescharacteristic
167C2×C59#191311yescharacteristic
168C2×C22⋊Q8C2×D5911yescharacteristic
169C22×C22⋊C4C2×D5911yescharacteristic
170#407—592no—
171#397Dic5912yescomplemented
172#393Dic5912yescomplemented
173#388—22no—
174#388—1184no—
175#381—21no—
176#380—1181no—
177#379—21no—
178#379—1181no—
179C4×C22⋊C4—592no—
180C4×C22⋊C4C2×D5911yescharacteristic
181#328—1184no—
182#323—1182no—
183C59#941111yescharacteristic, complemented, Sylow
184C25C22⋊D5911yescharacteristic
185C22×Q8C2×Dic5911yescharacteristic
186C23×C4C22⋊D5911yescharacteristic
187C23×C4C22×D5911yescharacteristic
188C2×M16—1182no—
189C22⋊Q8—22no—
190C22⋊Q8—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—1182no—
193C2×C4⋊C4C2×Dic5911yescharacteristic
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—1182no—
201C2×C22⋊C4—1182no—
202C2×C22⋊C4—1182no—
203C2×C22⋊C4C2×Dic5912yes—
204C2×C22⋊C4C22⋊D5912yes—
205C2×C42—21no—
206C2×C42—1182no—
207C22⋊C8—1182no—
208C22⋊C8—1182no—
209#95—22no—
210#95—22no—
211#95—594no—
212#95—1182no—
213#95C2×Dic5912yes—
214C24—21no—
215C24—21no—
216C24—42no—
217C24—42no—
218C24—81no—
219C24C22⋊Dic5912yes—
220C24C2×C22⋊D5911yescharacteristic
221C2×Q8—22no—
222C2×Q8—81no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—42no—
229C22×C4—81no—
230C22×C4—1184no—
231C22×C4C22⋊Dic5912yes—
232C22×C4D4×D5912yes—
233C22×C4#1174568712yes—
234C22×C4C22×Dic5911yescharacteristic
235C22×C4C2×C22⋊D5911yescharacteristic
236M16—4722no—
237C2×C8—2362no—
238C4⋊C4—22no—
239C4⋊C4—42no—
240C4⋊C4—4722no—
241C22⋊C4—24no—
242C22⋊C4—24no—
243C22⋊C4—42no—
244C22⋊C4—44no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—82no—
248C22⋊C4—4722no—
249C22⋊C4—4722no—
250C22⋊C4—2364no—
251C42—81no—
252C42—4722no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—41no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—81no—
266C23—81no—
267C23—88no—
268C23C2×C22⋊Dic5911yescharacteristic
269C23#1427369511yescharacteristic
270C23#1427370811yescharacteristic
271Q8—82no—
272C2×C4—22no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—81no—
287C2×C4—2364no—
288C2×C4—2364no—
289C2×C4—2364no—
290C2×C4D4×Dic5912yes—
291C2×C4#1427369912yes—
292C8—4722no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—44no—
301C22—44no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—82no—
306C22—84no—
307C22—84no—
308C22#1683728211yescharacteristic
309C22#1683741912yes—
310C22#1683757912yes—
311C4—22no—
312C4—22no—
313C4—42no—
314C4—82no—
315C4—81no—
316C4—81no—
317C4—4724no—
318C2—22no—
319C2—22no—
320C2—41no—
321C2—82no—
322C2#2142448511yescharacteristic
323C2#2142454511yescharacteristic
324C2#2142463911yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#2561C2
socleC22×C59#386
prime radicalC25×C59D4
cosocleC23×C4×C59C23
hypercenter#2561D59
hypocenterC59#9411
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2561

Complements

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

NH ≅ G/Ntype
2. #21422616C2semidirect
4. #21421982C2semidirect
15. C59×#397C4semidirect
16. C59×#393C4semidirect
171. #397Dic59semidirect
172. #393Dic59semidirect
183. C59#9411semidirect

Maximal subgroups

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

subgroupindex
#214266942
#214226162
#214224232
#214219822
C59×#25612
#941159