Finite Group Databasebeta

← #28499262

Subgroups — #28499262

Order 15104.

480 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 87 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#21429088C211yesmaximal, characteristic
2#21422434C211yesmaximal, characteristic
3#21422095C214yesmaximal, complemented
4C4×#99×C59C211yesmaximal, characteristic
5C2×#14273727C411yescharacteristic, complemented
6C2×#14273714—21no—
7#16837366—21no—
8#16837327C412yescomplemented
9#16837327C2212yes—
10#16837307—21no—
11#16837262—21no—
12#16837262—21no—
13C4×C4⋊Dic59C411yescharacteristic, complemented
14#16837128—21no—
15#16837126C2212yes—
16C2×C4×D4×C59C2211yescharacteristic
17C2×#99×C59—21no—
18C4×C22⋊C4×C59—21no—
19C59×#342C2212yes—
20C22×Dic118—21no—
21C2×#11745664—41no—
22#14273727—41no—
23#14273727—41no—
24#14273714—44no—
25C2×C4⋊Dic59—21no—
26C2×C4⋊Dic59—21no—
27C2×C4⋊Dic59C2×C412yes—
28C2×C22⋊Dic59—22no—
29C2×C22⋊Dic59—22no—
30C2×C4×Dic59—21no—
31C22⋊(C59⋊C8)—22no—
32C22⋊(C59⋊C8)—41no—
33C59⋊#95—22no—
34C59⋊#95—22no—
35C59⋊#95C2×C412yes—
36C22×D4×C59D411yescharacteristic
37C23×C4×C59D411yescharacteristic
38C23×C4×C59D411yescharacteristic
39C4×D4×C59—41no—
40C4×D4×C59—41no—
41C2×C4⋊C4×C59D411yescharacteristic
42C2×C22⋊C4×C59—22no—
43C2×C22⋊C4×C59C2×C411yescharacteristic
44C2×C22⋊C4×C59C2311yescharacteristic
45C2×C42×C59—41no—
46C2×C42×C59C2×C411yescharacteristic
47#99×C59—24no—
48#95×C59—41no—
49C24×C59C22⋊C411yescharacteristic
50C24×C59C2×D411yescharacteristic
51C2×D4×C59—22no—
52C2×D4×C59—22no—
53C2×D4×C59—41no—
54C22×C4×C59—21no—
55C22×C4×C59—21no—
56C22×C4×C59—21no—
57C22×C4×C59—21no—
58C22×C4×C59—42no—
59C22×C4×C59—42no—
60C22×C4×C59—42no—
61C22×C4×C59C22⋊C411yescharacteristic
62C22×C4×C59C22⋊C411yescharacteristic
63C22×C4×C59C22⋊C411yescharacteristic
64C22×C4×C59C22×C411yescharacteristic
65C22×C4×C59C2×D411yescharacteristic
66C22×Dic59—22no—
67C22×Dic59—22no—
68C2×Dic118—42no—
69C2×Dic118—42no—
70C2×Dic118—42no—
71C4⋊C4×C59—41no—
72C22⋊C4×C59—24no—
73C22⋊C4×C59—41no—
74C22⋊C4×C59—41no—
75C22⋊C4×C59—42no—
76C42×C59—21no—
77C42×C59—21no—
78C42×C59—82no—
79C22⋊Dic59—44no—
80C22⋊Dic59—44no—
81C4⋊Dic59—42no—
82C4⋊Dic59—41no—
83C4⋊Dic59—41no—
84C4⋊Dic59—42no—
85C4×Dic59—42no—
86#11745664—82no—
87C2×C59⋊C8—42no—
88C23×C59—21no—
89C23×C59—21no—
90C23×C59—21no—
91C23×C59—21no—
92C23×C59—21no—
93C23×C59—41no—
94C23×C59—41no—
95C23×C59—41no—
96C23×C59—41no—
97C23×C59C2×C22⋊C411yescharacteristic
98C23×C59AGL(1,ℤ/8)12yes—
99D4×C59—44no—
100C2×C4×C59—21no—
101C2×C4×C59—21no—
102C2×C4×C59—22no—
103C2×C4×C59—21no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—42no—
107C2×C4×C59—42no—
108C2×C4×C59—41no—
109C2×C4×C59—41no—
110C2×C4×C59—41no—
111C2×C4×C59—44no—
112C2×C4×C59—81no—
113C2×C4×C59—81no—
114C2×C4×C59—82no—
115C2×C4×C59—82no—
116C2×C4×C59#9911yescharacteristic
117C2×C4×C59#9911yescharacteristic
118C2×C4×C59#10412yes—
119C2×Dic59—42no—
120C2×Dic59—42no—
121C2×Dic59—42no—
122C2×Dic59—42no—
123C2×Dic59—42no—
124C2×Dic59—42no—
125Dic118—84no—
126C59⋊C8—82no—
127#11777—591nomaximal, Sylow
128C22×C59—21no—
129C22×C59—21no—
130C22×C59—22no—
131C22×C59—42no—
132C22×C59—42no—
133C22×C59—41no—
134C22×C59—41no—
135C22×C59—41no—
136C22×C59—41no—
137C22×C59—41no—
138C22×C59—81no—
139C22×C59—81no—
140C22×C59C2×#9911yescharacteristic
141C22×C59#41811yescharacteristic
142C22×C59C2×#10411yescharacteristic
143C4×C59—21no—
144C4×C59—22no—
145C4×C59—41no—
146C4×C59—41no—
147C4×C59—41no—
148C4×C59—84no—
149C4×C59#35412yes—
150Dic59—82no—
151Dic59—82no—
152#2547—591no—
153C4×#99D5911yescharacteristic
154#1465—591no—
155#1344—594no—
156C2×C59—21no—
157C2×C59—21no—
158C2×C59—42no—
159C2×C59—81no—
160C2×C59#150711yescharacteristic
161C2×C59#212311yescharacteristic
162C2×C59C2×#35411yescharacteristic
163C2×C4⋊Q8—591no—
164C2×#124—1181no—
165C2×C4×D4C2×D5911yescharacteristic
166C2×#99—21no—
167#407—1181no—
168#397—592no—
169#397—592no—
170#391—1181no—
171#380—1181no—
172#380—1181no—
173C4×C4⋊C4—591no—
174C4×C22⋊C4—21no—
175#342C2×D5912yes—
176#324—1181no—
177#323—592no—
178C59#1177711yescharacteristic, complemented, Sylow
179C22×Q8—1181no—
180C22×D4C22⋊D5911yescharacteristic
181C23×C4D23611yescharacteristic
182C23×C4C22⋊D5911yescharacteristic
183C2×M16—2361no—
184C4⋊Q8—2361no—
185C4⋊Q8—2361no—
186#124—2364no—
187C4×D4—41no—
188C4×D4—41no—
189C2×C4⋊C4—592no—
190C2×C4⋊C4—1181no—
191C2×C4⋊C4—1181no—
192C2×C4⋊C4D23611yescharacteristic
193C2×C22⋊C4—22no—
194C2×C22⋊C4—1182no—
195C2×C22⋊C4—1182no—
196C2×C22⋊C4C4×D5911yescharacteristic
197C2×C22⋊C4C22×D5911yescharacteristic
198C2×C42—41no—
199C2×C42—1181no—
200C2×C42C4×D5911yescharacteristic
201#99—24no—
202C22⋊C8—1182no—
203C22⋊C8—2361no—
204#95—41no—
205#95—592no—
206#95—1182no—
207#95—1182no—
208C24(C2×D59)⋊C411yescharacteristic
209C24C2×C22⋊D5911yescharacteristic
210C2×Q8—2362no—
211C2×Q8—2362no—
212C2×Q8—2362no—
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—41no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—1182no—
224C22×C4—1182no—
225C22×C4(C2×D59)⋊C411yescharacteristic
226C22×C4(C2×D59)⋊C411yescharacteristic
227C22×C4(C2×D59)⋊C411yescharacteristic
228C22×C4C2×C4×D5911yescharacteristic
229C22×C4C2×D23611yescharacteristic
230M16—4722no—
231C2×C8—2362no—
232C4⋊C4—41no—
233C4⋊C4—2362no—
234C4⋊C4—2362no—
235C4⋊C4—2361no—
236C4⋊C4—2361no—
237C22⋊C4—24no—
238C22⋊C4—41no—
239C22⋊C4—41no—
240C22⋊C4—42no—
241C22⋊C4—2364no—
242C22⋊C4—2364no—
243C42—21no—
244C42—21no—
245C42—82no—
246C42—2362no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23C2×(C2×D59)⋊C411yescharacteristic
257C23#1427375212yes—
258Q8—4724no—
259D4—44no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—44no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—2362no—
277C2×C4—2362no—
278C2×C4—2362no—
279C2×C4—2362no—
280C2×C4—2362no—
281C2×C4—2362no—
282C2×C4#1427364111yescharacteristic
283C2×C4#1427364111yescharacteristic
284C2×C4#1427365512yes—
285C8—4722no—
286C22—21no—
287C22—21no—
288C22—22no—
289C22—42no—
290C22—42no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—81no—
297C22—81no—
298C22C2×#1427364111yescharacteristic
299C22#1683740611yescharacteristic
300C22C2×#1427365511yescharacteristic
301C4—22no—
302C4—21no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—84no—
307C4—4722no—
308C4—4722no—
309C4#1683719612yes—
310C2—21no—
311C2—21no—
312C2—42no—
313C2—81no—
314C2#2142261111yescharacteristic
315C2#2142574911yescharacteristic
316C2C2×#1683719611yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC4×#99×C59C2
socleC22×C59C2×#99
prime radicalC22×D4×C59D4
cosocleC2×C22⋊C4×C59C23
hypercenterC4×#99D59
hypocenterC59#11777
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
3. #21422095C2semidirect
5. C2×#14273727C4semidirect
8. #16837327C4semidirect
13. C4×C4⋊Dic59C4semidirect
178. C59#11777semidirect

Maximal subgroups

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

subgroupindex
#214290882
#214224342
#214220952
C4×#99×C592
#1177759