Finite Group Databasebeta

← #28486125

Subgroups — #28486125

Order 15104.

726 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 115 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#21426449C211yesmaximal, characteristic
2#21422616C212yesmaximal, complemented
3#21422423C211yesmaximal, characteristic
4#21421982C212yesmaximal, complemented
5C59×#2561C211yesmaximal, characteristic
6C22×C22⋊Dic59—21no—
7#16837366C2212yes—
8C22⋊C4×Dic59—24no—
9C4×C22⋊Dic59—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
21C23×Dic59—22no—
22C2×#11745664—22no—
23C2×C22⋊Dic59—22no—
24C2×C22⋊Dic59—28no—
25C2×C22⋊Dic59—22no—
26C2×C4×Dic59—22no—
27C2×C4×Dic59—22no—
28C22⋊(C59⋊C8)—22no—
29C22⋊(C59⋊C8)—22no—
30C59⋊#95—22no—
31C59⋊#95D414yes—
32C25×C59D411yescharacteristic
33C22×Q8×C59C2×C411yescharacteristic, complemented
34C23×C4×C59D411yescharacteristic
35C23×C4×C59C2311yescharacteristic
36C22⋊Q8×C59—22no—
37C22⋊Q8×C59—22no—
38C2×C4⋊C4×C59—21no—
39C2×C4⋊C4×C59C2×C411yescharacteristic, complemented
40C2×C22⋊C4×C59—22no—
41C2×C22⋊C4×C59—22no—
42C2×C22⋊C4×C59—22no—
43C2×C22⋊C4×C59—21no—
44C2×C22⋊C4×C59—21no—
45C2×C22⋊C4×C59—22no—
46C2×C22⋊C4×C59C2×C412yes—
47C2×C22⋊C4×C59D412yes—
48C2×C42×C59—21no—
49#95×C59—22no—
50#95×C59—22no—
51#95×C59C2×C412yescomplemented
52C24×C59—21no—
53C24×C59—24no—
54C24×C59—21no—
55C24×C59—42no—
56C24×C59—81no—
57C24×C59C22⋊C412yes—
58C24×C59C2×D411yescharacteristic
59C2×Q8×C59—22no—
60C2×Q8×C59—81no—
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—22no—
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—42no—
67C22×C4×C59—81no—
68C22×C4×C59C22⋊C412yes—
69C22×C4×C59C22×C411yescharacteristic
70C22×C4×C59C2×D412yes—
71C22×C4×C59C2×D411yescharacteristic
72C22×C4×C59#4112yes—
73C22×Dic59—24no—
74C22×Dic59—28no—
75C22×Dic59—82no—
76C22×Dic59—82no—
77C4⋊C4×C59—22no—
78C4⋊C4×C59—42no—
79C22⋊C4×C59—24no—
80C22⋊C4×C59—24no—
81C22⋊C4×C59—42no—
82C22⋊C4×C59—44no—
83C22⋊C4×C59—41no—
84C22⋊C4×C59—41no—
85C22⋊C4×C59—82no—
86C42×C59—81no—
87C22⋊Dic59—28no—
88C22⋊Dic59—28no—
89C4×Dic59—82no—
90C4×Dic59—82no—
91#11745664—82no—
92C2×C59⋊C8—42no—
93C23×C59—22no—
94C23×C59—22no—
95C23×C59—24no—
96C23×C59—22no—
97C23×C59—22no—
98C23×C59—22no—
99C23×C59—22no—
100C23×C59—41no—
101C23×C59—42no—
102C23×C59—42no—
103C23×C59—42no—
104C23×C59—42no—
105C23×C59—81no—
106C23×C59—81no—
107C23×C59—88no—
108C23×C59#10414yes—
109C23×C59C2×C22⋊C411yescharacteristic
110C23×C59#12011yescharacteristic
111C23×C59#12311yescharacteristic
112Q8×C59—82no—
113C2×C4×C59—22no—
114C2×C4×C59—42no—
115C2×C4×C59—42no—
116C2×C4×C59—41no—
117C2×C4×C59—41no—
118C2×C4×C59—42no—
119C2×C4×C59—42no—
120C2×C4×C59—42no—
121C2×C4×C59—42no—
122C2×C4×C59—42no—
123C2×C4×C59—41no—
124C2×C4×C59—41no—
125C2×C4×C59—42no—
126C2×C4×C59—42no—
127C2×C4×C59—81no—
128C2×C4×C59C4×D412yes—
129C2×C4×C59#12112yes—
130C2×Dic59—28no—
131C2×Dic59—44no—
132C2×Dic59—44no—
133C2×Dic59—88no—
134C59⋊C8—82no—
135#9407—591nomaximal, Sylow
136C22×C59—21no—
137C22×C59—21no—
138C22×C59—21no—
139C22×C59—21no—
140C22×C59—21no—
141C22×C59—24no—
142C22×C59—21no—
143C22×C59—21no—
144C22×C59—44no—
145C22×C59—44no—
146C22×C59—44no—
147C22×C59—44no—
148C22×C59—42no—
149C22×C59—42no—
150C22×C59—82no—
151C22×C59—84no—
152C22×C59#38611yescharacteristic
153C22×C59C2×#10412yes—
154C22×C59#45512yes—
155C4×C59—22no—
156C4×C59—22no—
157C4×C59—42no—
158C4×C59—82no—
159C4×C59—81no—
160C4×C59—81no—
161Dic59—48no—
162#2561D5911yescharacteristic
163#2273—591no—
164#1508—592no—
165#1462—591no—
166#1297—592no—
167C2×C59—22no—
168C2×C59—22no—
169C2×C59—41no—
170C2×C59—44no—
171C2×C59#188511yescharacteristic
172C2×C59#189611yescharacteristic
173C2×C59#191311yescharacteristic
174C2×C22⋊Q8C2×D5911yescharacteristic
175C22×C22⋊C4—1181no—
176C22×C22⋊C4C2×D5911yescharacteristic
177#407—592no—
178#397Dic5912yescomplemented
179#393Dic5912yescomplemented
180#388—22no—
181#381—21no—
182#379—21no—
183C4×C22⋊C4—592no—
184C4×C22⋊C4—1184no—
185C4×C22⋊C4—1181no—
186C4×C22⋊C4C2×D5911yescharacteristic
187#328—1184no—
188#323—1182no—
189C59#940711yescharacteristic, complemented, Sylow
190C25C22⋊D5911yescharacteristic
191C22×Q8C2×Dic5911yescharacteristic, complemented
192C23×C4—1182no—
193C23×C4C22⋊D5911yescharacteristic
194C23×C4C22×D5911yescharacteristic
195C2×M16—1182no—
196C22⋊Q8—22no—
197C22⋊Q8—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4C2×Dic5911yescharacteristic, complemented
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—1182no—
207C2×C22⋊C4—1188no—
208C2×C22⋊C4—1182no—
209C2×C22⋊C4C2×Dic5912yes—
210C2×C22⋊C4C22⋊D5912yes—
211C2×C42—21no—
212C2×C42—1182no—
213C2×C42—1182no—
214C22⋊C8—1182no—
215C22⋊C8—1182no—
216#95—22no—
217#95—22no—
218#95—594no—
219#95—1182no—
220#95C2×Dic5912yescomplemented
221C24—21no—
222C24—24no—
223C24—21no—
224C24—42no—
225C24—81no—
226C24C22⋊Dic5912yes—
227C24C2×C22⋊D5911yescharacteristic
228C2×Q8—22no—
229C2×Q8—81no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—42no—
236C22×C4—81no—
237C22×C4—1188no—
238C22×C4—1184no—
239C22×C4—4722no—
240C22×C4—4722no—
241C22×C4C22⋊Dic5912yes—
242C22×C4D4×D5912yes—
243C22×C4#1174568712yes—
244C22×C4C22×Dic5911yescharacteristic
245C22×C4C2×C22⋊D5911yescharacteristic
246M16—4722no—
247C2×C8—2362no—
248C4⋊C4—22no—
249C4⋊C4—42no—
250C22⋊C4—24no—
251C22⋊C4—24no—
252C22⋊C4—42no—
253C22⋊C4—44no—
254C22⋊C4—41no—
255C22⋊C4—41no—
256C22⋊C4—82no—
257C22⋊C4—1188no—
258C22⋊C4—1188no—
259C42—81no—
260C42—4722no—
261C42—4722no—
262C23—22no—
263C23—22no—
264C23—24no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—22no—
269C23—41no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—81no—
275C23—81no—
276C23—88no—
277C23#1427365414yes—
278C23C2×C22⋊Dic5911yescharacteristic
279C23#1427369511yescharacteristic
280C23#1427370811yescharacteristic
281Q8—82no—
282C2×C4—22no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—81no—
297C2×C4—1188no—
298C2×C4—4728no—
299C2×C4—2364no—
300C2×C4—2364no—
301C2×C4D4×Dic5912yes—
302C2×C4#1427369912yes—
303C8—4722no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—24no—
310C22—21no—
311C22—21no—
312C22—44no—
313C22—44no—
314C22—44no—
315C22—44no—
316C22—42no—
317C22—42no—
318C22—82no—
319C22—84no—
320C22#1683728211yescharacteristic
321C22C2×#1427365412yes—
322C22#1683757912yes—
323C4—22no—
324C4—22no—
325C4—42no—
326C4—82no—
327C4—81no—
328C4—81no—
329C4—2368no—
330C2—22no—
331C2—22no—
332C2—41no—
333C2—44no—
334C2#2142448511yescharacteristic
335C2#2142454011yescharacteristic
336C2#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#9407
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
33. C22×Q8×C59C2×C4semidirect
39. C2×C4⋊C4×C59C2×C4semidirect
51. #95×C59C2×C4semidirect
178. #397Dic59semidirect
179. #393Dic59semidirect
189. C59#9407semidirect
191. C22×Q8C2×Dic59semidirect
199. C2×C4⋊C4C2×Dic59semidirect
220. #95C2×Dic59semidirect

Maximal subgroups

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

subgroupindex
#214264492
#214226162
#214224232
#214219822
C59×#25612
#940759