Finite Group Databasebeta

← #28486039

Subgroups — #28486039

Order 15104.

446 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 99 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#21429794C211yesmaximal, characteristic
2#21422697C212yesmaximal, complemented
3#21422430C211yesmaximal, characteristic
4#21421990C212yesmaximal, complemented
5C59×#2921C211yesmaximal, characteristic
6C2×#14273719C2211yescharacteristic
7#16837459C2212yes—
8#16837341C411yescharacteristic
9#16837338C411yescharacteristic
10#16837332C411yescharacteristic
11#16837327C411yescharacteristic
12#16837297—21no—
13#16837296—21no—
14#16837286—21no—
15#16837275—21no—
16#16837248C2211yescharacteristic
17#16837143—24no—
18#16837128—21no—
19#16837126—21no—
20C2×#117×C59C2211yescharacteristic
21C59×#433—24no—
22C59×#423—21no—
23C59×#407—21no—
24C4×M16×C59C2212yes—
25#14273719—22no—
26#14273719—22no—
27C2×C4⋊Dic59C2×C411yescharacteristic
28C2×C4⋊Dic59C2×C411yescharacteristic
29C2×Dic59⋊C4—21no—
30C2×Dic59⋊C4—21no—
31C2×Dic59⋊C4—21no—
32C2×Dic59⋊C4—21no—
33C2×C22⋊Dic59—21no—
34C2×C22⋊Dic59—21no—
35C2×C4×Dic59—21no—
36C4⋊(C59⋊C8)—22no—
37C22⋊(C59⋊C8)—22no—
38#14273638—22no—
39C59⋊#95—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C59⋊#95—21no—
44C59⋊#95C2×C411yescharacteristic
45C59⋊#95C2×C411yescharacteristic
46C23×C4×C59D411yescharacteristic
47C2×M16×C59—22no—
48#117×C59—22no—
49#117×C59—22no—
50C2×C4⋊C4×C59C2×C411yescharacteristic
51C2×C4⋊C4×C59D411yescharacteristic
52C2×C22⋊C4×C59C2×C411yescharacteristic
53C2×C22⋊C4×C59D411yescharacteristic
54C2×C42×C59D411yescharacteristic
55C2×C42×C59C2311yescharacteristic
56C4⋊C8×C59—24no—
57C22⋊C8×C59—24no—
58#97×C59D414yes—
59C4×C8×C59—22no—
60C24×C59C22⋊C411yescharacteristic
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—41no—
64C22×C4×C59—41no—
65C22×C4×C59C22⋊C411yescharacteristic
66C22×C4×C59C22⋊C411yescharacteristic
67C22×C4×C59C22⋊C411yescharacteristic
68C22×C4×C59C22×C411yescharacteristic
69C22×C4×C59C2×D411yescharacteristic
70C22×C4×C59C2×D411yescharacteristic
71C22×Dic59—21no—
72C22×Dic59—21no—
73C22×Dic59—21no—
74C22×Dic59—21no—
75M16×C59—42no—
76M16×C59—42no—
77C2×C8×C59—24no—
78C4⋊C4×C59—22no—
79C4⋊C4×C59—22no—
80C22⋊C4×C59—22no—
81C22⋊C4×C59—22no—
82C42×C59—21no—
83C42×C59—21no—
84C42×C59C2×D412yes—
85C42×C59#4112yes—
86C22⋊Dic59—42no—
87C22⋊Dic59—42no—
88C4⋊Dic59—22no—
89C4⋊Dic59—22no—
90Dic59⋊C4—42no—
91Dic59⋊C4—42no—
92Dic59⋊C4—41no—
93Dic59⋊C4—42no—
94Dic59⋊C4—41no—
95C4×Dic59—42no—
96C2×C59⋊C8—42no—
97C23×C59—21no—
98C23×C59—21no—
99C23×C59—21no—
100C23×C59—41no—
101C23×C59—41no—
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—41no—
111C2×C4×C59—41no—
112C2×C4×C59—42no—
113C2×C4×C59—42no—
114C2×C4×C59—41no—
115C2×C4×C59—41no—
116C2×C4×C59C4×D412yes—
117C2×C4×C59#12011yescharacteristic
118C2×C4×C59#12112yes—
119C2×C4×C59#12311yescharacteristic
120C2×Dic59—22no—
121C2×Dic59—41no—
122C2×Dic59—41no—
123C2×Dic59—41no—
124C2×Dic59—41no—
125C2×Dic59—41no—
126C2×Dic59—41no—
127C2×Dic59—41no—
128C2×Dic59—41no—
129C2×Dic59—41no—
130C2×Dic59—41no—
131C2×Dic59—41no—
132C8×C59—44no—
133C59⋊C8—82no—
134#9390—591nomaximal, Sylow
135C22×C59—21no—
136C22×C59—21no—
137C22×C59—42no—
138C22×C59—41no—
139C22×C59—41no—
140C22×C59—41no—
141C22×C59—81no—
142C22×C59#38611yescharacteristic
143C22×C59#42111yescharacteristic
144C22×C59#42111yescharacteristic
145C4×C59—22no—
146C4×C59—21no—
147C4×C59—21no—
148C4×C59—22no—
149C4×C59—41no—
150C4×C59—41no—
151C4×C59—41no—
152Dic59—42no—
153Dic59—81no—
154Dic59—81no—
155Dic59—81no—
156#2921D5911yescharacteristic
157#2611—591no—
158#1528—592no—
159#1464—591no—
160#1300—592no—
161C2×C59—21no—
162C2×C59—21no—
163C2×C59—81no—
164C2×C59#188411yescharacteristic
165C2×C59#189711yescharacteristic
166C2×C59#191511yescharacteristic
167C2×#125—591no—
168C2×#117C2×D5911yescharacteristic
169#433—24no—
170#432—592no—
171#423—21no—
172#407—21no—
173C4×M16C2×D5912yes—
174#400—591no—
175#399—591no—
176#398—591no—
177#397—591no—
178#389—1181no—
179#388—1181no—
180#387—1181no—
181#385—1181no—
182#376—591no—
183#330—1184no—
184#324—1181no—
185#323—1181no—
186C59#939011yescharacteristic, complemented, Sylow
187C23×C4C22⋊D5911yescharacteristic
188C2×M16—22no—
189#125—1182no—
190#125—1182no—
191#117—22no—
192#117—22no—
193C2×C4⋊C4—591no—
194C2×C4⋊C4—591no—
195C2×C4⋊C4—1181no—
196C2×C4⋊C4—1181no—
197C2×C4⋊C4—1181no—
198C2×C4⋊C4—1181no—
199C2×C4⋊C4C4×D5911yescharacteristic
200C2×C4⋊C4D23611yescharacteristic
201C2×C22⋊C4—1181no—
202C2×C22⋊C4—1181no—
203C2×C22⋊C4C4×D5911yescharacteristic
204C2×C22⋊C4D23611yescharacteristic
205C2×C42—1181no—
206C2×C42C22⋊D5911yescharacteristic
207C2×C42C22×D5911yescharacteristic
208C4⋊C8—24no—
209C4⋊C8—1182no—
210C22⋊C8—24no—
211C22⋊C8—1182no—
212#97—1182no—
213#97C22⋊D5914yes—
214C4×C8—22no—
215#95—591no—
216#95—591no—
217#95—1181no—
218#95—1181no—
219#95—1181no—
220#95—1181no—
221#95—1181no—
222C24(C2×D59)⋊C411yescharacteristic
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—1181no—
228C22×C4—1181no—
229C22×C4—1181no—
230C22×C4—1181no—
231C22×C4(C2×D59)⋊C411yescharacteristic
232C22×C4(C2×D59)⋊C411yescharacteristic
233C22×C4(C2×D59)⋊C411yescharacteristic
234C22×C4C2×C4×D5911yescharacteristic
235C22×C4C2×D23611yescharacteristic
236C22×C4C2×C22⋊D5911yescharacteristic
237M16—42no—
238M16—42no—
239C2×C8—24no—
240C2×C8—2362no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—1182no—
244C4⋊C4—1182no—
245C4⋊C4—2362no—
246C4⋊C4—2362no—
247C4⋊C4—2361no—
248C4⋊C4—2362no—
249C4⋊C4—2361no—
250C22⋊C4—22no—
251C22⋊C4—22no—
252C22⋊C4—2362no—
253C22⋊C4—2362no—
254C42—21no—
255C42—21no—
256C42—2362no—
257C42#1174568512yes—
258C42C2×C22⋊D5912yes—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—41no—
263C23—41no—
264C23C2×(C2×D59)⋊C411yescharacteristic
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—1182no—
279C2×C4—2361no—
280C2×C4—2361no—
281C2×C4—2361no—
282C2×C4—2361no—
283C2×C4—2361no—
284C2×C4—2361no—
285C2×C4—2361no—
286C2×C4—2361no—
287C2×C4—2361no—
288C2×C4—2361no—
289C2×C4—2361no—
290C2×C4C4×C22⋊D5912yes—
291C2×C4#1427369611yescharacteristic
292C2×C4#1427370112yes—
293C2×C4#1427371011yescharacteristic
294C8—44no—
295C8—4722no—
296C22—21no—
297C22—21no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22#1683728111yescharacteristic
304C22#1683742011yescharacteristic
305C22#1683742011yescharacteristic
306C4—22no—
307C4—22no—
308C4—21no—
309C4—21no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—4721no—
314C4—4721no—
315C4—4721no—
316C4—2362no—
317C2—21no—
318C2—21no—
319C2—81no—
320C2#2142447911yescharacteristic
321C2#2142454411yescharacteristic
322C2#2142464811yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2921C2
socleC22×C59#386
prime radicalC24×C59C22⋊C4
cosocleC2×C42×C59C23
hypercenter#2921D59
hypocenterC59#9390
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2921

Complements

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

NH ≅ G/Ntype
2. #21422697C2semidirect
4. #21421990C2semidirect
186. C59#9390semidirect

Maximal subgroups

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

subgroupindex
#214297942
#214226972
#214224302
#214219902
C59×#29212
#939059