Finite Group Databasebeta

← #28483204

Subgroups — #28483204

Order 15104.

356 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 97 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#21433197C211yesmaximal, characteristic
2#21425073C211yesmaximal, characteristic
3#21422885C211yesmaximal, characteristic
4#21422884C211yesmaximal, characteristic
5#21422760C211yesmaximal, characteristic
6#21421929C211yesmaximal, characteristic
7C59×#1276C211yesmaximal, characteristic
8#16837959C2211yescharacteristic
9#16837532—21no—
10#16837529—21no—
11C8×Dic118—21no—
12#16837526—21no—
13#16837460—22no—
14#16837455C2211yescharacteristic
15#16837455C2211yescharacteristic
16#16837426C2211yescharacteristic
17#16837162—22no—
18#16837155—21no—
19#16837154—21no—
20#16837153—21no—
21#16837152—21no—
22#16837149C411yescharacteristic, complemented
23#16837149C411yescharacteristic, complemented
24#16837147C411yescharacteristic, complemented
25#16837147C411yescharacteristic, complemented
26C59×#431C2211yescharacteristic
27C59×#431C2211yescharacteristic
28C43×C59C2211yescharacteristic
29C2×#11745664—21no—
30#14273727C2×C411yescharacteristic
31#14273727C2×C411yescharacteristic
32#14273719Q812yes—
33#14273706—21no—
34#14273706—21no—
35C4×Dic118—21no—
36C4×Dic118—21no—
37#14273686D412yes—
38C4⋊(C59⋊C8)—21no—
39C4⋊(C59⋊C8)—21no—
40C4⋊(C59⋊C8)C2×C411yescharacteristic
41C4⋊(C59⋊C8)C2×C411yescharacteristic
42C4⋊(C59⋊C8)D411yescharacteristic
43C4⋊(C59⋊C8)D411yescharacteristic
44Dic59⋊C8—22no—
45Dic59⋊C8—22no—
46C22⋊(C59⋊C8)—21no—
47C22⋊(C59⋊C8)—21no—
48#14273637—22no—
49C4×C59⋊C8—21no—
50C4×C59⋊C8—21no—
51C8×Dic59—22no—
52C2×C42×C59—21no—
53C2×C42×C59—21no—
54C2×C42×C59—21no—
55C2×C42×C59C2311yescharacteristic
56C4⋊C8×C59C2×C411yescharacteristic
57C4⋊C8×C59C2×C411yescharacteristic
58C4⋊C8×C59D411yescharacteristic
59C4⋊C8×C59D411yescharacteristic
60C22⋊C8×C59—21no—
61C22⋊C8×C59—21no—
62C4×C8×C59—21no—
63C4×C8×C59—21no—
64C22×C4×C59—41no—
65C22×C4×C59C2×D411yescharacteristic
66C22×C4×C59#4111yescharacteristic
67C22×C4×C59#4111yescharacteristic
68C2×Dic118—21no—
69C2×Dic118—21no—
70C2×C8×C59—21no—
71C2×C8×C59—21no—
72C2×C8×C59—21no—
73C2×C8×C59—21no—
74C42×C59—22no—
75C42×C59—22no—
76C42×C59—22no—
77C42×C59—41no—
78C42×C59—41no—
79C42×C59—41no—
80C42×C59C22×C411yescharacteristic
81C42×C59C2×D411yescharacteristic
82C42×C59C2×D411yescharacteristic
83C42×C59C2×Q811yescharacteristic
84C22⋊Dic59—22no—
85C4⋊Dic59—21no—
86C4⋊Dic59—21no—
87C4⋊Dic59C4⋊C412yes—
88C4⋊Dic59C4⋊C412yes—
89Dic59⋊C4—22no—
90Dic59⋊C4—22no—
91C4×Dic59—22no—
92#11745664—42no—
93C2×C59⋊C8—21no—
94C2×C59⋊C8—21no—
95C2×C59⋊C8—21no—
96C2×C59⋊C8—21no—
97C2×C59⋊C8—41no—
98C23×C59#12311yescharacteristic
99C2×C4×C59—21no—
100C2×C4×C59—21no—
101C2×C4×C59—21no—
102C2×C4×C59—21no—
103C2×C4×C59—21no—
104C2×C4×C59—21no—
105C2×C4×C59—42no—
106C2×C4×C59—42no—
107C2×C4×C59—42no—
108C2×C4×C59C2×C4⋊C411yescharacteristic
109C2×C4×C59C4×D411yescharacteristic
110C2×C4×C59C4×D411yescharacteristic
111C2×C4×C59#12011yescharacteristic
112C2×C4×C59C22⋊Q811yescharacteristic
113C2×C4×C59C22⋊Q811yescharacteristic
114C2×Dic59—22no—
115C2×Dic59—22no—
116Dic118—41no—
117Dic118—41no—
118Dic118—41no—
119Dic118—41no—
120C8×C59—41no—
121C8×C59—41no—
122C8×C59—41no—
123C8×C59—41no—
124C59⋊C8—41no—
125C59⋊C8—41no—
126C59⋊C8—41no—
127C59⋊C8—41no—
128C59⋊C8—81no—
129#8936—591nomaximal, Sylow
130C22×C59—21no—
131C22×C59—21no—
132C22×C59—21no—
133C22×C59#38511yescharacteristic
134C4×C59—21no—
135C4×C59—21no—
136C4×C59—21no—
137C4×C59—44no—
138C4×C59#44311yescharacteristic
139C4×C59#44311yescharacteristic
140C4×C59#44411yescharacteristic
141C4×C59#44411yescharacteristic
142C4×C59#45411yescharacteristic
143C4×C59#45611yescharacteristic
144Dic59—42no—
145Dic59—42no—
146#2986—591no—
147#1994—591no—
148#1574—591no—
149#1574—591no—
150#1544—591no—
151#1276—591no—
152#1276D5911yescharacteristic
153C2×C59—41no—
154C2×C59#185711yescharacteristic
155C2×C59#186611yescharacteristic
156C2×C59#186611yescharacteristic
157#533—591no—
158#446—1181no—
159#446—1181no—
160C8×Q8—1181no—
161C8×Q8—1181no—
162#433—1182no—
163#431—591no—
164#431—591no—
165#431C2×D5911yescharacteristic
166#431C2×D5911yescharacteristic
167#423—591no—
168C43C2×D5911yescharacteristic
169#339—1182no—
170#335—1181no—
171#335—1181no—
172#334—1181no—
173#334—1181no—
174#332—591no—
175#332—591no—
176#332—591no—
177#332—591no—
178C59#893611yescharacteristic, complemented, Sylow
179C2×M16—1181no—
180C4⋊Q8—591no—
181C4⋊Q8—591no—
182#125—592no—
183C22⋊Q8—1181no—
184C22⋊Q8—1181no—
185C4×Q8—1181no—
186C4×Q8—1181no—
187#117—592no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42C22×D5911yescharacteristic
192C4⋊C8—591no—
193C4⋊C8—591no—
194C4⋊C8—591no—
195C4⋊C8—591no—
196C4⋊C8—1182no—
197C4⋊C8—1182no—
198C4⋊C8—1181no—
199C4⋊C8—1181no—
200C4⋊C8C4×D5911yescharacteristic
201C4⋊C8C4×D5911yescharacteristic
202C4⋊C8D23611yescharacteristic
203C4⋊C8D23611yescharacteristic
204C22⋊C8—21no—
205C22⋊C8—21no—
206C22⋊C8—1181no—
207C22⋊C8—1181no—
208#97—1182no—
209C4×C8—21no—
210C4×C8—21no—
211C4×C8—1182no—
212C4×C8—1181no—
213C4×C8—1181no—
214C2×Q8—1181no—
215C2×Q8—1181no—
216C22×C4—41no—
217C22×C4#1174568511yescharacteristic
218C22×C4D4×D5911yescharacteristic
219C22×C4#1174568711yescharacteristic
220M16—2362no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—1181no—
226C2×C8—1181no—
227C2×C8—1181no—
228C2×C8—1181no—
229C2×C8—2361no—
230C4⋊C4—592no—
231C4⋊C4—592no—
232C4⋊C4—1181no—
233C4⋊C4—1181no—
234C4⋊C4—1182no—
235C4⋊C4—1182no—
236C22⋊C4—1182no—
237C42—22no—
238C42—22no—
239C42—22no—
240C42—41no—
241C42—41no—
242C42—41no—
243C42—1182no—
244C42C2×C4×D5911yescharacteristic
245C42C2×D23611yescharacteristic
246C42D4×D5911yescharacteristic
247C42Q8×D5911yescharacteristic
248C23#1427370911yescharacteristic
249Q8—2361no—
250Q8—2361no—
251Q8—2361no—
252Q8—2361no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—1182no—
263C2×C4—1182no—
264C2×C4C4⋊C4×D5911yescharacteristic
265C2×C4(C22⋊D59)⋊C411yescharacteristic
266C2×C4C4×D23611yescharacteristic
267C2×C4#1427369711yescharacteristic
268C2×C4#1427370511yescharacteristic
269C2×C4#1427370711yescharacteristic
270C8—41no—
271C8—41no—
272C8—41no—
273C8—41no—
274C8—4721no—
275C8—2361no—
276C8—2361no—
277C8—2361no—
278C8—2361no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22#1683727611yescharacteristic
283C4—21no—
284C4—21no—
285C4—21no—
286C4—44no—
287C4—2362no—
288C4—2362no—
289C4#1683751611yescharacteristic
290C4#1683752011yescharacteristic
291C4#1683752111yescharacteristic
292C4#1683752511yescharacteristic
293C4#1683757611yescharacteristic
294C4#1683758611yescharacteristic
295C2—41no—
296C2#2142431211yescharacteristic
297C2#2142437211yescharacteristic
298C2#2142437411yescharacteristic
subgroupG/H
centerC2×C4#14273697
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1276C2
socleC22×C59#385
prime radicalC23×C59#123
cosocleC2×C42×C59C23
hypercenter#1276D59
hypocenterC59#8936
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
22. #16837149C4semidirect
23. #16837149C4semidirect
24. #16837147C4semidirect
25. #16837147C4semidirect
178. C59#8936semidirect

Maximal subgroups

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

subgroupindex
#214331972
#214250732
#214228852
#214228842
#214227602
#214219292
C59×#12762
#893659