Finite Group Databasebeta

← #28486496

Subgroups — #28486496

Order 15104.

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

normal subgroups only

This group has 734 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#21428370C211yesmaximal, characteristic
2#21426449C211yesmaximal, characteristic
3#21424193C212yesmaximal, complemented
4#21421950C212yesmaximal, complemented
5C59×#1462C211yesmaximal, characteristic
6C2×#14273686C2211yescharacteristic
7C22×C22⋊Dic59—21no—
8#16837338C412yescomplemented
9#16837327C412yescomplemented
10#16837290—22no—
11#16837275—21no—
12#16837265C2211yescharacteristic
13#16837258—21no—
14C22⋊C4×Dic59—24no—
15C4×C22⋊Dic59—21no—
16C4×C22⋊Dic59C2212yes—
17#16837163—24no—
18C22×C22⋊C4×C59C2211yescharacteristic
19C59×#407C2212yes—
20C59×#323—22no—
21C23×Dic59—22no—
22#14273686—22no—
23#14273686—22no—
24C2×C4⋊Dic59C2×C411yescharacteristic, complemented
25C2×C4⋊Dic59C2×C411yescharacteristic, complemented
26C2×Dic59⋊C4—22no—
27C2×C22⋊Dic59—22no—
28C2×C22⋊Dic59—28no—
29C2×C22⋊Dic59—22no—
30C2×C22⋊Dic59—22no—
31C2×C22⋊Dic59—21no—
32C2×C22⋊Dic59—21no—
33C2×C22⋊Dic59—22no—
34C2×C4×Dic59—21no—
35C2×C4×Dic59—22no—
36C2×C4×Dic59D414yes—
37C59⋊#95—22no—
38C59⋊#95—22no—
39C59⋊#95—22no—
40C59⋊#95—21no—
41C59⋊#95C2×C412yescomplemented
42C25×C59D411yescharacteristic
43C23×C4×C59D411yescharacteristic
44C23×C4×C59C2311yescharacteristic
45C2×M16×C59—22no—
46C2×C22⋊C4×C59—22no—
47C2×C22⋊C4×C59—22no—
48C2×C22⋊C4×C59C2×C412yes—
49C2×C22⋊C4×C59D412yes—
50C22⋊C8×C59—22no—
51C22⋊C8×C59—22no—
52C24×C59—21no—
53C24×C59—24no—
54C24×C59—21no—
55C24×C59—42no—
56C24×C59—81no—
57C24×C59C22⋊C412yes—
58C24×C59C2×D411yescharacteristic
59C22×C4×C59—21no—
60C22×C4×C59—21no—
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—24no—
69C22×Dic59—28no—
70C22×Dic59—22no—
71C22×Dic59—21no—
72C22×Dic59—21no—
73C22×Dic59—82no—
74C22×Dic59—82no—
75M16×C59—44no—
76C2×C8×C59—42no—
77C22⋊C4×C59—24no—
78C22⋊C4×C59—24no—
79C22⋊Dic59—28no—
80C22⋊Dic59—28no—
81C22⋊Dic59—42no—
82C22⋊Dic59—44no—
83C22⋊Dic59—41no—
84C22⋊Dic59—41no—
85C22⋊Dic59—82no—
86C4⋊Dic59—22no—
87C4⋊Dic59—22no—
88Dic59⋊C4—82no—
89C4×Dic59—42no—
90C4×Dic59—44no—
91C4×Dic59—82no—
92C23×C59—22no—
93C23×C59—22no—
94C23×C59—24no—
95C23×C59—22no—
96C23×C59—22no—
97C23×C59—22no—
98C23×C59—22no—
99C23×C59—41no—
100C23×C59—42no—
101C23×C59—42no—
102C23×C59—42no—
103C23×C59—42no—
104C23×C59—81no—
105C23×C59—81no—
106C23×C59—88no—
107C23×C59#10414yes—
108C23×C59C2×C22⋊C411yescharacteristic
109C23×C59#12011yescharacteristic
110C23×C59#12311yescharacteristic
111C2×C4×C59—24no—
112C2×C4×C59—22no—
113C2×C4×C59—41no—
114C2×C4×C59—41no—
115C2×C4×C59—42no—
116C2×C4×C59—42no—
117C2×C4×C59—42no—
118C2×C4×C59—81no—
119C2×C4×C59C4×D412yes—
120C2×C4×C59#12112yes—
121C2×Dic59—28no—
122C2×Dic59—44no—
123C2×Dic59—44no—
124C2×Dic59—42no—
125C2×Dic59—42no—
126C2×Dic59—42no—
127C2×Dic59—41no—
128C2×Dic59—41no—
129C2×Dic59—42no—
130C2×Dic59—42no—
131C2×Dic59—88no—
132C8×C59—82no—
133#9477—591nomaximal, Sylow
134C22×C59—21no—
135C22×C59—21no—
136C22×C59—21no—
137C22×C59—21no—
138C22×C59—21no—
139C22×C59—24no—
140C22×C59—21no—
141C22×C59—21no—
142C22×C59—44no—
143C22×C59—44no—
144C22×C59—44no—
145C22×C59—44no—
146C22×C59—42no—
147C22×C59—42no—
148C22×C59—82no—
149C22×C59—84no—
150C22×C59#38611yescharacteristic
151C22×C59C2×#10412yes—
152C22×C59#45812yes—
153C4×C59—22no—
154C4×C59—22no—
155C4×C59—42no—
156Dic59—48no—
157Dic59—82no—
158Dic59—81no—
159Dic59—81no—
160#2482—591no—
161#2273—591no—
162#1826—592no—
163#1462D5911yescharacteristic
164#1284—592no—
165C2×C59—22no—
166C2×C59—22no—
167C2×C59—41no—
168C2×C59—44no—
169C2×C59#189011yescharacteristic
170C2×C59#189611yescharacteristic
171C2×C59#191111yescharacteristic
172C2×#117—591no—
173C22×C22⋊C4—1181no—
174C22×C22⋊C4C2×D5911yescharacteristic
175#407C2×D5912yes—
176#399—592no—
177#397—592no—
178#388—1182no—
179#385—1181no—
180#381—591no—
181#379—1181no—
182C4×C22⋊C4—592no—
183C4×C22⋊C4—1184no—
184C4×C22⋊C4—1181no—
185#339—1184no—
186#323—22no—
187C59#947711yescharacteristic, complemented, Sylow
188C25D23611yescharacteristic
189C23×C4—1182no—
190C23×C4D23611yescharacteristic
191C23×C4C22×D5911yescharacteristic
192C2×M16—22no—
193#117—1182no—
194#117—1182no—
195C2×C4⋊C4—591no—
196C2×C4⋊C4—591no—
197C2×C4⋊C4—1182no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—1182no—
201C2×C22⋊C4—1188no—
202C2×C22⋊C4—1182no—
203C2×C22⋊C4—1182no—
204C2×C22⋊C4—1182no—
205C2×C22⋊C4—1181no—
206C2×C22⋊C4—1181no—
207C2×C22⋊C4C4×D5912yes—
208C2×C22⋊C4C22⋊D5912yes—
209C2×C42—594no—
210C2×C42—1182no—
211C2×C42—1181no—
212C22⋊C8—22no—
213C22⋊C8—22no—
214#95—592no—
215#95—1182no—
216#95—1182no—
217#95—1182no—
218#95—1181no—
219C24—21no—
220C24—24no—
221C24—21no—
222C24—42no—
223C24—81no—
224C24(C2×D59)⋊C412yes—
225C24C2×D23611yescharacteristic
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—42no—
229C22×C4—81no—
230C22×C4—1188no—
231C22×C4—1184no—
232C22×C4—1182no—
233C22×C4—1181no—
234C22×C4—1181no—
235C22×C4—4722no—
236C22×C4—4722no—
237C22×C4(C2×D59)⋊C412yes—
238C22×C4C2×C4×D5911yescharacteristic
239C22×C4D4×D5912yes—
240C22×C4#1174568712yes—
241C22×C4C2×C22⋊D5911yescharacteristic
242M16—44no—
243C2×C8—42no—
244C4⋊C4—1182no—
245C4⋊C4—1182no—
246C4⋊C4—4722no—
247C22⋊C4—24no—
248C22⋊C4—24no—
249C22⋊C4—1188no—
250C22⋊C4—1188no—
251C22⋊C4—4722no—
252C22⋊C4—2362no—
253C22⋊C4—2364no—
254C22⋊C4—2361no—
255C22⋊C4—2361no—
256C42—4722no—
257C42—2364no—
258C42—2362no—
259C23—22no—
260C23—22no—
261C23—24no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—41no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—81no—
272C23—81no—
273C23—88no—
274C23#1427365314yes—
275C23C2×(C2×D59)⋊C411yescharacteristic
276C23#1427369711yescharacteristic
277C23#1427371111yescharacteristic
278C2×C4—24no—
279C2×C4—22no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—81no—
286C2×C4—1188no—
287C2×C4—4728no—
288C2×C4—2362no—
289C2×C4—2362no—
290C2×C4—2362no—
291C2×C4—2362no—
292C2×C4—2362no—
293C2×C4—2361no—
294C2×C4—2361no—
295C2×C4—2364no—
296C2×C4—2364no—
297C2×C4(C22⋊D59)⋊C412yes—
298C2×C4#1427369912yes—
299C8—82no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—24no—
306C22—21no—
307C22—21no—
308C22—44no—
309C22—44no—
310C22—44no—
311C22—44no—
312C22—42no—
313C22—42no—
314C22—82no—
315C22—84no—
316C22#1683727911yescharacteristic
317C22C2×#1427365312yes—
318C22#1683759312yes—
319C4—22no—
320C4—22no—
321C4—42no—
322C4—4722no—
323C4—4721no—
324C4—4721no—
325C4—2368no—
326C2—22no—
327C2—22no—
328C2—41no—
329C2—44no—
330C2#2142450711yescharacteristic
331C2#2142453711yescharacteristic
332C2#2142462611yescharacteristic
subgroupG/H
centerC22#16837279
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1462C2
socleC22×C59#386
prime radicalC25×C59D4
cosocleC23×C4×C59C23
hypercenter#1462D59
hypocenterC59#9477
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1462

Complements

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

NH ≅ G/Ntype
3. #21424193C2semidirect
4. #21421950C2semidirect
8. #16837338C4semidirect
9. #16837327C4semidirect
24. C2×C4⋊Dic59C2×C4semidirect
25. C2×C4⋊Dic59C2×C4semidirect
41. C59⋊#95C2×C4semidirect
187. C59#9477semidirect

Maximal subgroups

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

subgroupindex
#214283702
#214264492
#214241932
#214219502
C59×#14622
#947759