Finite Group Databasebeta

← #28499582

Subgroups — #28499582

Order 15104.

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

normal subgroups only

This group has 382 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#21424173C211yesmaximal, characteristic
2#21422410C211yesmaximal, characteristic
3#21422120C212yesmaximal
4#21422096C212yesmaximal
5C59×#2611C211yesmaximal, characteristic
6#16837455—21no—
7#16837265—21no—
8#16837169—21no—
9#16837169C2212yes—
10#16837128—21no—
11#16837126C2212yes—
12C2×#125×C59C411yescharacteristic, complemented
13C2×#117×C59C2211yescharacteristic
14C59×#400C411yescharacteristic, complemented
15C59×#399C2211yescharacteristic
16C59×#398C411yescharacteristic, complemented
17C59×#397C2211yescharacteristic
18C59×#389—21no—
19C59×#388—21no—
20C59×#387—21no—
21C59×#385—21no—
22C59×#376C411yescharacteristic, complemented
23C2×C22⋊Dic59—22no—
24C4⋊(C59⋊C8)—41no—
25C22⋊(C59⋊C8)—22no—
26C4×C59⋊C8—41no—
27C59⋊#95—41no—
28C59⋊#95—41no—
29C23×C4×C59D411yescharacteristic
30#125×C59—41no—
31#125×C59—41no—
32#117×C59—41no—
33#117×C59—41no—
34C2×C4⋊C4×C59—21no—
35C2×C4⋊C4×C59—21no—
36C2×C4⋊C4×C59—21no—
37C2×C4⋊C4×C59—21no—
38C2×C4⋊C4×C59C2×C411yescharacteristic
39C2×C4⋊C4×C59C2×C411yescharacteristic
40C2×C4⋊C4×C59C2×C411yescharacteristic
41C2×C4⋊C4×C59D411yescharacteristic
42C2×C22⋊C4×C59—21no—
43C2×C22⋊C4×C59—21no—
44C2×C22⋊C4×C59D411yescharacteristic
45C2×C22⋊C4×C59C2311yescharacteristic
46C2×C42×C59—21no—
47C2×C42×C59C2×C411yescharacteristic
48C2×C42×C59D411yescharacteristic
49#95×C59—21no—
50#95×C59—21no—
51#95×C59—21no—
52#95×C59—21no—
53#95×C59—21no—
54#95×C59C2×C411yescharacteristic
55#95×C59C2×C411yescharacteristic
56C24×C59C2×D411yescharacteristic
57C22×C4×C59—21no—
58C22×C4×C59—22no—
59C22×C4×C59—21no—
60C22×C4×C59—21no—
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—21no—
64C22×C4×C59—41no—
65C22×C4×C59C22⋊C411yescharacteristic
66C22×C4×C59C22⋊C411yescharacteristic
67C22×C4×C59C22⋊C411yescharacteristic
68C22×C4×C59C22⋊C411yescharacteristic
69C22×C4×C59C22×C411yescharacteristic
70C22×C4×C59C2×D411yescharacteristic
71C22×Dic59—42no—
72C4⋊C4×C59—41no—
73C4⋊C4×C59—41no—
74C4⋊C4×C59—42no—
75C4⋊C4×C59—42no—
76C4⋊C4×C59—42no—
77C4⋊C4×C59—41no—
78C4⋊C4×C59—41no—
79C4⋊C4×C59—41no—
80C4⋊C4×C59—41no—
81C22⋊C4×C59—41no—
82C22⋊C4×C59—41no—
83C22⋊C4×C59—42no—
84C22⋊C4×C59—42no—
85C42×C59—21no—
86C42×C59—21no—
87C42×C59—21no—
88C42×C59—21no—
89C42×C59—42no—
90C22⋊Dic59—42no—
91C22⋊Dic59—42no—
92C2×C59⋊C8—42no—
93C23×C59—21no—
94C23×C59—21no—
95C23×C59—41no—
96C23×C59—41no—
97C23×C59C2×C22⋊C411yescharacteristic
98C23×C59#13512yes—
99C2×C4×C59—21no—
100C2×C4×C59—21no—
101C2×C4×C59—22no—
102C2×C4×C59—21no—
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—42no—
110C2×C4×C59—42no—
111C2×C4×C59—41no—
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—41no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59—41no—
125C2×C4×C59—41no—
126C2×C4×C59#9911yescharacteristic
127C2×C4×C59#9911yescharacteristic
128C2×Dic59—82no—
129C2×Dic59—82no—
130C2×Dic59—82no—
131C59⋊C8—82no—
132#11846—591nomaximal, Sylow
133C22×C59—21no—
134C22×C59—21no—
135C22×C59—22no—
136C22×C59—42no—
137C22×C59—41no—
138C22×C59—41no—
139C22×C59—81no—
140C22×C59C2×#9911yescharacteristic
141C22×C59#41611yescharacteristic
142C22×C59#42111yescharacteristic
143C4×C59—21no—
144C4×C59—21no—
145C4×C59—42no—
146C4×C59—41no—
147C4×C59—41no—
148C4×C59—41no—
149C4×C59—41no—
150C4×C59—41no—
151C4×C59—81no—
152C4×C59—81no—
153C4×C59—81no—
154Dic59—162no—
155#2611D5911yescharacteristic
156#1820—591no—
157#1458—591no—
158#1352—592no—
159#1344—592no—
160C2×C59—21no—
161C2×C59—21no—
162C2×C59—42no—
163C2×C59#151611yescharacteristic
164C2×C59#212311yescharacteristic
165C2×C59#212611yescharacteristic
166C2×#125Dic5911yescharacteristic, complemented
167C2×#117C2×D5911yescharacteristic
168#431—1181no—
169#400Dic5911yescharacteristic, complemented
170#399C2×D5911yescharacteristic
171#398Dic5911yescharacteristic, complemented
172#397C2×D5911yescharacteristic
173#389—21no—
174#388—21no—
175#387—21no—
176#385—21no—
177#381—1181no—
178#376Dic5911yescharacteristic, complemented
179#342—592no—
180#342—1181no—
181#324—1181no—
182#323—592no—
183C59#1184611yescharacteristic, complemented, Sylow
184C23×C4C22⋊D5911yescharacteristic
185#125—41no—
186#125—41no—
187#117—41no—
188#117—41no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4C2×Dic5911yescharacteristic
194C2×C4⋊C4C2×Dic5911yescharacteristic
195C2×C4⋊C4C2×Dic5911yescharacteristic
196C2×C4⋊C4C22⋊D5911yescharacteristic
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—1182no—
200C2×C22⋊C4C22⋊D5911yescharacteristic
201C2×C22⋊C4C22×D5911yescharacteristic
202C2×C42—21no—
203C2×C42C2×Dic5911yescharacteristic
204C2×C42C22⋊D5911yescharacteristic
205C4⋊C8—2361no—
206C22⋊C8—1182no—
207C4×C8—2361no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—2361no—
214#95—2361no—
215#95C2×Dic5911yescharacteristic
216#95C2×Dic5911yescharacteristic
217C24C2×C22⋊D5911yescharacteristic
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—41no—
226C22×C4—2362no—
227C22×C4C22⋊Dic5911yescharacteristic
228C22×C4C22⋊Dic5911yescharacteristic
229C22×C4C22⋊Dic5911yescharacteristic
230C22×C4C22⋊Dic5911yescharacteristic
231C22×C4C22×Dic5911yescharacteristic
232C22×C4C2×C22⋊D5911yescharacteristic
233C2×C8—2362no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C22⋊C4—41no—
244C22⋊C4—41no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—2362no—
248C22⋊C4—2362no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—42no—
254C23—21no—
255C23—21no—
256C23—41no—
257C23—41no—
258C23C2×C22⋊Dic5911yescharacteristic
259C23#1427374712yes—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—4722no—
288C2×C4—4722no—
289C2×C4—4722no—
290C2×C4#1427364211yescharacteristic
291C2×C4#1427364211yescharacteristic
292C8—4722no—
293C22—21no—
294C22—21no—
295C22—22no—
296C22—42no—
297C22—41no—
298C22—41no—
299C22—81no—
300C22C2×#1427364211yescharacteristic
301C22#1683739811yescharacteristic
302C22#1683741911yescharacteristic
303C4—21no—
304C4—21no—
305C4—42no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4—9442no—
315C2—21no—
316C2—21no—
317C2—42no—
318C2#2142265211yescharacteristic
319C2#2142575011yescharacteristic
320C2#2142576111yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#2611C2
socleC22×C59C2×#99
prime radicalC24×C59C2×D4
cosocleC2×C22⋊C4×C59C23
hypercenter#2611D59
hypocenterC59#11846
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2611

Complements

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

NH ≅ G/Ntype
12. C2×#125×C59C4semidirect
14. C59×#400C4semidirect
16. C59×#398C4semidirect
22. C59×#376C4semidirect
166. C2×#125Dic59semidirect
169. #400Dic59semidirect
171. #398Dic59semidirect
178. #376Dic59semidirect
183. C59#11846semidirect

Maximal subgroups

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

subgroupindex
#214241732
#214224102
#214221202
#214220962
C59×#26112
#1184659