Finite Group Databasebeta

← #28482230

Subgroups — #28482230

Order 15104.

474 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 83 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#21426693C211yesmaximal, characteristic
2#21422422C211yesmaximal, characteristic
3#21422422C211yesmaximal, characteristic
4#21422003C211yesmaximal, characteristic, complemented
5#21422002C211yesmaximal, characteristic, complemented
6#21422001C211yesmaximal, characteristic, complemented
7C59×#1305C211yesmaximal, characteristic, complemented
8C22×(C2×D59)⋊C4C2211yescharacteristic
9#16837366C2211yescharacteristic
10#16837366C2211yescharacteristic
11#16837294—22no—
12#16837292—22no—
13#16837262C2211yescharacteristic
14#16837257C412yes—
15C22⋊C4×Dic59C412yes—
16#16837125—22no—
17#16837125—22no—
18C59×#407C2211yescharacteristic
19C59×#407C2211yescharacteristic
20C59×#380C2211yescharacteristic
21C24×D59C2×C411yescharacteristic
22C23×Dic59C2×C411yescharacteristic
23C2×#11745664—21no—
24C2×#11745664—21no—
25C2×C4⋊Dic59—21no—
26C2×C22⋊Dic59C2×C412yes—
27C2×(C2×D59)⋊C4—21no—
28C2×(C2×D59)⋊C4—21no—
29C2×(C2×D59)⋊C4—21no—
30C2×(C2×D59)⋊C4—21no—
31C2×(C2×D59)⋊C4—22no—
32C2×(C2×D59)⋊C4—22no—
33C2×(C2×D59)⋊C4D412yes—
34C2×(C2×D59)⋊C4D412yes—
35C2×C4×Dic59—22no—
36C22⋊(C59⋊C8)—21no—
37C22⋊(C59⋊C8)—21no—
38C22⋊(C59⋊C8)—21no—
39C22⋊(C59⋊C8)—21no—
40C59⋊#95—21no—
41C59⋊#95—22no—
42C23×C4×C59C2311yescharacteristic
43C2×M16×C59—21no—
44C2×M16×C59—21no—
45C2×C4⋊C4×C59—21no—
46C2×C22⋊C4×C59C2×C412yes—
47C22⋊C8×C59—21no—
48C22⋊C8×C59—21no—
49C22⋊C8×C59—21no—
50C22⋊C8×C59—21no—
51#95×C59—21no—
52C24×C59C22×C411yescharacteristic
53C23×D59—21no—
54C23×D59—21no—
55C23×D59—41no—
56C23×D59—41no—
57C23×D59—81no—
58C23×D59—81no—
59C23×D59C22⋊C412yes—
60C22×C4×C59—22no—
61C22×C4×C59—81no—
62C22×C4×C59C2×D411yescharacteristic
63C22×C4×C59C2×D411yescharacteristic
64C22×C4×C59#4111yescharacteristic
65C22×C4×C59#4111yescharacteristic
66C22×C4×C59#4111yescharacteristic
67C22×C4×C59#4111yescharacteristic
68C22×Dic59—21no—
69C22×Dic59—21no—
70C22×Dic59—22no—
71C22×Dic59—81no—
72C22×Dic59C22⋊C412yes—
73M16×C59—81no—
74M16×C59—81no—
75C2×C8×C59—41no—
76C2×C8×C59—41no—
77C4⋊C4×C59—81no—
78C22⋊C4×C59—42no—
79C22⋊Dic59—42no—
80(C2×D59)⋊C4—22no—
81(C2×D59)⋊C4—22no—
82(C2×D59)⋊C4—22no—
83(C2×D59)⋊C4—22no—
84(C2×D59)⋊C4—82no—
85(C2×D59)⋊C4—82no—
86C4⋊Dic59—81no—
87C4×Dic59—82no—
88#11745664—81no—
89#11745664—81no—
90C2×C59⋊C8—41no—
91C2×C59⋊C8—41no—
92C23×C59—21no—
93C23×C59—21no—
94C23×C59—41no—
95C23×C59—41no—
96C23×C59C2×C22⋊C411yescharacteristic
97C23×C59#11711yescharacteristic
98C23×C59#11711yescharacteristic
99C22×D59—22no—
100C22×D59—22no—
101C22×D59—22no—
102C22×D59—42no—
103C22×D59—42no—
104C22×D59—42no—
105C22×D59—42no—
106C22×D59—81no—
107C22×D59—81no—
108C22×D59—81no—
109C22×D59—81no—
110C22×D59—84no—
111C22×D59—84no—
112C2×C4×C59—41no—
113C2×C4×C59—41no—
114C2×C4×C59—41no—
115C2×C4×C59—41no—
116C2×C4×C59—41no—
117C2×C4×C59—41no—
118C2×C4×C59—42no—
119C2×C4×C59—42no—
120C2×C4×C59—42no—
121C2×C4×C59#12311yescharacteristic
122C2×C4×C59#12311yescharacteristic
123C2×C4×C59#12311yescharacteristic
124C2×C4×C59#12311yescharacteristic
125C2×Dic59—22no—
126C2×Dic59—42no—
127C2×Dic59—42no—
128C2×Dic59—42no—
129C2×Dic59—42no—
130C2×Dic59—81no—
131C2×Dic59—81no—
132C8×C59—81no—
133C8×C59—81no—
134C59⋊C8—81no—
135C59⋊C8—81no—
136#8759—591nomaximal, Sylow
137C22×C59—21no—
138C22×C59—21no—
139C22×C59—21no—
140C22×C59—21no—
141C22×C59—21no—
142C22×C59—21no—
143C22×C59—21no—
144C22×C59—21no—
145C22×C59—42no—
146C22×C59—42no—
147C22×C59#38111yescharacteristic
148C22×C59#41311yescharacteristic
149C22×C59#41311yescharacteristic
150C2×D59—42no—
151C2×D59—42no—
152C2×D59—42no—
153C2×D59—84no—
154C2×D59—84no—
155C2×D59—82no—
156C2×D59—82no—
157C4×C59—21no—
158C4×C59—21no—
159C4×C59—21no—
160C4×C59—21no—
161C4×C59—82no—
162Dic59—42no—
163Dic59—82no—
164#2312—591no—
165#1462—591no—
166#1462—591no—
167#1305—591no—
168#1305—591no—
169#1305—591no—
170#1305D5911yescharacteristic, complemented
171C2×C59—21no—
172C2×C59—21no—
173C2×C59—41no—
174C2×C59—41no—
175C2×C59#182211yescharacteristic
176C2×C59#182811yescharacteristic
177C2×C59#182911yescharacteristic
178D59—82no—
179C22×C22⋊C4—591no—
180#407—591no—
181#407—591no—
182#407C2×D5911yescharacteristic
183#407C2×D5911yescharacteristic
184#388—1182no—
185#388—1182no—
186#380—591no—
187#380C2×D5911yescharacteristic
188#379—592no—
189C4×C22⋊C4—592no—
190#323—1182no—
191#323—1182no—
192C59#875911yescharacteristic, complemented, Sylow
193C25—591no—
194C23×C4—591no—
195C23×C4C22×D5911yescharacteristic
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—1181no—
199C2×M16—1181no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—1181no—
202C2×C22⋊C4—592no—
203C2×C22⋊C4—592no—
204C2×C22⋊C4—592no—
205C2×C22⋊C4—1181no—
206C2×C22⋊C4—1181no—
207C2×C22⋊C4—1181no—
208C2×C22⋊C4—1181no—
209C2×C22⋊C4—1182no—
210C2×C22⋊C4—1182no—
211C2×C22⋊C4C4×D5912yes—
212C2×C42—1182no—
213C22⋊C8—21no—
214C22⋊C8—21no—
215C22⋊C8—21no—
216C22⋊C8—21no—
217C22⋊C8—1181no—
218C22⋊C8—1181no—
219C22⋊C8—1181no—
220C22⋊C8—1181no—
221#95—21no—
222#95—1182no—
223#95—1181no—
224C24—592no—
225C24—1181no—
226C24—1181no—
227C24—4721no—
228C24—4721no—
229C24—2361no—
230C24—2361no—
231C24C2×C4×D5911yescharacteristic
232C22×C4—22no—
233C22×C4—81no—
234C22×C4—592no—
235C22×C4—1181no—
236C22×C4—1181no—
237C22×C4—1182no—
238C22×C4—4721no—
239C22×C4#1174568511yescharacteristic
240C22×C4#1174568511yescharacteristic
241C22×C4D4×D5911yescharacteristic
242C22×C4D4×D5911yescharacteristic
243C22×C4#1174568711yescharacteristic
244C22×C4#1174568911yescharacteristic
245M16—81no—
246M16—81no—
247M16—4721no—
248M16—4721no—
249C2×C8—41no—
250C2×C8—41no—
251C2×C8—2361no—
252C2×C8—2361no—
253C4⋊C4—81no—
254C4⋊C4—4721no—
255C22⋊C4—42no—
256C22⋊C4—1182no—
257C22⋊C4—1182no—
258C22⋊C4—1182no—
259C22⋊C4—1182no—
260C22⋊C4—4722no—
261C22⋊C4—4722no—
262C22⋊C4—2362no—
263C42—4722no—
264C23—21no—
265C23—21no—
266C23—41no—
267C23—41no—
268C23—1182no—
269C23—1182no—
270C23—1182no—
271C23—4721no—
272C23—4721no—
273C23—4721no—
274C23—4721no—
275C23—4724no—
276C23—4724no—
277C23—2362no—
278C23—2362no—
279C23—2362no—
280C23—2362no—
281C23C22⋊C4×D5911yescharacteristic
282C23#1427368411yescharacteristic
283C23#1427368511yescharacteristic
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—1182no—
294C2×C4—4721no—
295C2×C4—4721no—
296C2×C4—2362no—
297C2×C4—2362no—
298C2×C4—2362no—
299C2×C4—2362no—
300C2×C4#1427370911yescharacteristic
301C2×C4#1427370911yescharacteristic
302C2×C4#1427371211yescharacteristic
303C2×C4#1427371211yescharacteristic
304C8—81no—
305C8—81no—
306C8—4721no—
307C8—4721no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—42no—
317C22—42no—
318C22—4722no—
319C22—4722no—
320C22—4724no—
321C22—4724no—
322C22—2362no—
323C22—2362no—
324C22—2362no—
325C22#1683726411yescharacteristic
326C22#1683738311yescharacteristic
327C22#1683738511yescharacteristic
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—82no—
333C4—4722no—
334C4—2362no—
335C2—21no—
336C2—21no—
337C2—41no—
338C2—41no—
339C2—4722no—
340C2#2142417811yescharacteristic
341C2#2142420311yescharacteristic
342C2#2142420611yescharacteristic
subgroupG/H
centerC22#16837264
commutatorC24×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1305C2
socleC22×C59#381
prime radicalC24×D59C2×C4
cosocleC23×C4×C59C23
hypercenter#1305D59
hypocenterC59#8759
LCSG ▷ C24×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1305

Complements

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

NH ≅ G/Ntype
4. #21422003C2semidirect
5. #21422002C2semidirect
6. #21422001C2semidirect
7. C59×#1305C2semidirect
170. #1305D59semidirect
192. C59#8759semidirect

Maximal subgroups

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

subgroupindex
#214266932
#214224222
#214220032
#214220022
#214220012
C59×#13052
#875959