Finite Group Databasebeta

← #28499435

Subgroups — #28499435

Order 15104.

386 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 79 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#21429909C211yesmaximal, characteristic
2#21422426C211yesmaximal, characteristic
3#21422125C212yesmaximal, complemented
4#21422122C212yesmaximal, complemented
5C59×#1821C211yesmaximal, characteristic
6C2×#14273706C411yescharacteristic
7#16837366—21no—
8#16837341C411yescharacteristic
9#16837341C2211yescharacteristic
10#16837335—21no—
11#16837335—21no—
12#16837322C411yescharacteristic
13#16837322C2211yescharacteristic
14#16837275—21no—
15#16837265—21no—
16#16837262C411yescharacteristic
17#16837128—21no—
18#16837128C2212yes—
19C22×C4⋊C4×C59C2211yescharacteristic
20C59×#407—21no—
21C59×#343—21no—
22C59×#343C2212yes—
23C22×Dic118C2×C411yescharacteristic
24C2×#11745664—41no—
25#14273706—41no—
26#14273706—41no—
27C2×C4⋊Dic59—21no—
28C2×C4⋊Dic59C2×C411yescharacteristic
29C2×Dic59⋊C4—21no—
30C2×Dic59⋊C4—21no—
31C2×Dic59⋊C4—21no—
32C2×C22⋊Dic59—21no—
33C2×C22⋊Dic59—21no—
34C22⋊(C59⋊C8)—22no—
35C22⋊(C59⋊C8)—41no—
36C59⋊#95—21no—
37C59⋊#95—21no—
38C59⋊#95—21no—
39C59⋊#95—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95C2×C411yescharacteristic
43C59⋊#95C2×C411yescharacteristic
44C23×C4×C59C2×C411yescharacteristic
45C23×C4×C59D411yescharacteristic
46C23×C4×C59D411yescharacteristic
47C2×M16×C59—22no—
48C2×C4⋊C4×C59—41no—
49C2×C4⋊C4×C59—41no—
50C2×C4⋊C4×C59C2×C411yescharacteristic
51C2×C4⋊C4×C59D411yescharacteristic
52C2×C4⋊C4×C59D411yescharacteristic
53C2×C4⋊C4×C59C2311yescharacteristic
54C22⋊C8×C59—41no—
55C22⋊C8×C59—41no—
56C24×C59C22⋊C411yescharacteristic
57C22×C4×C59—21no—
58C22×C4×C59—21no—
59C22×C4×C59—21no—
60C22×C4×C59—21no—
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—22no—
64C22×C4×C59—41no—
65C22×C4×C59—81no—
66C22×C4×C59—81no—
67C22×C4×C59C22⋊C411yescharacteristic
68C22×C4×C59C22⋊C411yescharacteristic
69C22×C4×C59C22⋊C411yescharacteristic
70C22×C4×C59C22×C411yescharacteristic
71C22×C4×C59C2×D411yescharacteristic
72C22×C4×C59C2×D411yescharacteristic
73C22×Dic59—21no—
74C22×Dic59—21no—
75C22×Dic59—21no—
76C22×Dic59—21no—
77C2×Dic118—41no—
78C2×Dic118—81no—
79M16×C59—42no—
80M16×C59—42no—
81C2×C8×C59—42no—
82C4⋊C4×C59—41no—
83C4⋊C4×C59—41no—
84C4⋊C4×C59—41no—
85C4⋊C4×C59—41no—
86C22⋊Dic59—42no—
87C22⋊Dic59—41no—
88C22⋊Dic59—41no—
89C4⋊Dic59—41no—
90C4⋊Dic59—81no—
91Dic59⋊C4—42no—
92Dic59⋊C4—81no—
93Dic59⋊C4—81no—
94#11745664—82no—
95C2×C59⋊C8—42no—
96C23×C59—21no—
97C23×C59—21no—
98C23×C59—41no—
99C23×C59—41no—
100C23×C59#9911yescharacteristic
101C23×C59#9911yescharacteristic
102C23×C59C2×C22⋊C411yescharacteristic
103C2×C4×C59—21no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—21no—
107C2×C4×C59—21no—
108C2×C4×C59—21no—
109C2×C4×C59—21no—
110C2×C4×C59—21no—
111C2×C4×C59—21no—
112C2×C4×C59—42no—
113C2×C4×C59—42no—
114C2×C4×C59—42no—
115C2×C4×C59—41no—
116C2×C4×C59—41no—
117C2×C4×C59—41no—
118C2×C4×C59—41no—
119C2×C4×C59—81no—
120C2×C4×C59—81no—
121C2×C4×C59—81no—
122C2×C4×C59#13712yes—
123C2×Dic59—41no—
124C2×Dic59—41no—
125C2×Dic59—41no—
126C2×Dic59—41no—
127C2×Dic59—41no—
128C2×Dic59—41no—
129C2×Dic59—42no—
130C2×Dic59—41no—
131C2×Dic59—41no—
132C2×Dic59—41no—
133C2×Dic59—41no—
134Dic118—82no—
135C8×C59—82no—
136C59⋊C8—82no—
137#11817—591nomaximal, Sylow
138C22×C59—21no—
139C22×C59—21no—
140C22×C59—21no—
141C22×C59—21no—
142C22×C59—21no—
143C22×C59—21no—
144C22×C59—21no—
145C22×C59—21no—
146C22×C59—42no—
147C22×C59—42no—
148C22×C59C2×#9911yescharacteristic
149C22×C59#41911yescharacteristic
150C22×C59#42111yescharacteristic
151C4×C59—22no—
152C4×C59—41no—
153C4×C59—41no—
154C4×C59—41no—
155C4×C59—41no—
156C4×C59—41no—
157Dic59—81no—
158Dic59—81no—
159Dic59—81no—
160Dic59—81no—
161#2627—591no—
162#1821D5911yescharacteristic
163#1463—591no—
164#1354—592no—
165#1353—592no—
166C2×C59—21no—
167C2×C59—21no—
168C2×C59—41no—
169C2×C59—41no—
170C2×C59#151111yescharacteristic
171C2×C59#212111yescharacteristic
172C2×C59#213311yescharacteristic
173C2×C22⋊Q8—591no—
174C22×C4⋊C4C2×D5911yescharacteristic
175#407—21no—
176#407—1181no—
177#400—591no—
178#400—591no—
179#398—1181no—
180#398—1181no—
181#395—591no—
182#395—591no—
183#385—1181no—
184#381—1181no—
185#380—591no—
186#343—21no—
187#343C2×D5912yes—
188#324—592no—
189#324—1181no—
190C59#1181711yescharacteristic, complemented, Sylow
191C22×Q8—591no—
192C23×C4C4×D5911yescharacteristic
193C23×C4D23611yescharacteristic
194C23×C4C22⋊D5911yescharacteristic
195C2×M16—22no—
196C2×M16—2361no—
197C22⋊Q8—2361no—
198C22⋊Q8—2361no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—591no—
202C2×C4⋊C4—1181no—
203C2×C4⋊C4—1181no—
204C2×C4⋊C4—1181no—
205C2×C4⋊C4—1181no—
206C2×C4⋊C4C4×D5911yescharacteristic
207C2×C4⋊C4D23611yescharacteristic
208C2×C4⋊C4C22⋊D5911yescharacteristic
209C2×C4⋊C4C22×D5911yescharacteristic
210C2×C22⋊C4—1181no—
211C2×C22⋊C4—1181no—
212C22⋊C8—41no—
213C22⋊C8—41no—
214C22⋊C8—1182no—
215C22⋊C8—2361no—
216#95—591no—
217#95—591no—
218#95—1181no—
219#95—1181no—
220#95—1181no—
221#95—1181no—
222#95—1181no—
223#95—1181no—
224C24(C2×D59)⋊C411yescharacteristic
225C2×Q8—4721no—
226C2×Q8—2361no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—41no—
235C22×C4—81no—
236C22×C4—81no—
237C22×C4—1181no—
238C22×C4—1181no—
239C22×C4—1181no—
240C22×C4—1181no—
241C22×C4(C2×D59)⋊C411yescharacteristic
242C22×C4(C2×D59)⋊C411yescharacteristic
243C22×C4(C2×D59)⋊C411yescharacteristic
244C22×C4C2×C4×D5911yescharacteristic
245C22×C4C2×D23611yescharacteristic
246C22×C4C2×C22⋊D5911yescharacteristic
247M16—42no—
248M16—42no—
249M16—4722no—
250C2×C8—42no—
251C2×C8—2362no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—4721no—
257C4⋊C4—4721no—
258C4⋊C4—4721no—
259C4⋊C4—2361no—
260C4⋊C4—2362no—
261C22⋊C4—2362no—
262C22⋊C4—2361no—
263C22⋊C4—2361no—
264C23—21no—
265C23—21no—
266C23—41no—
267C23—41no—
268C23#1427364111yescharacteristic
269C23#1427364111yescharacteristic
270C23C2×(C2×D59)⋊C411yescharacteristic
271Q8—4722no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—2361no—
292C2×C4—2361no—
293C2×C4—2361no—
294C2×C4—2361no—
295C2×C4—2361no—
296C2×C4—2361no—
297C2×C4—2362no—
298C2×C4—2361no—
299C2×C4—2361no—
300C2×C4—2361no—
301C2×C4—2361no—
302C2×C4#1427375712yes—
303C8—82no—
304C8—4722no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—42no—
314C22—42no—
315C22C2×#1427364111yescharacteristic
316C22#1683741011yescharacteristic
317C22#1683742011yescharacteristic
318C4—22no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—4721no—
325C4—4721no—
326C4—4721no—
327C4—4721no—
328C2—21no—
329C2—21no—
330C2—41no—
331C2—41no—
332C2#2142262811yescharacteristic
333C2#2142574311yescharacteristic
334C2#2142578311yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1821C2
socleC22×C59C2×#99
prime radicalC24×C59C22⋊C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1821D59
hypocenterC59#11817
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
3. #21422125C2semidirect
4. #21422122C2semidirect
190. C59#11817semidirect

Maximal subgroups

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

subgroupindex
#214299092
#214224262
#214221252
#214221222
C59×#18212
#1181759