Finite Group Databasebeta

← #28482004

Subgroups — #28482004

Order 15104.

384 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 107 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#21433071C211yesmaximal, characteristic
2#21422859C211yesmaximal, characteristic, complemented
3#21422619C212yesmaximal
4#21421952C212yesmaximal, complemented
5C4×#106×C59C211yesmaximal, characteristic, complemented
6#16837936C2211yescharacteristic
7#16837532C411yescharacteristic, complemented
8C8×Dic118C411yescharacteristic, complemented
9#16837481C411yescharacteristic, complemented
10#16837473—21no—
11C8×D236C411yescharacteristic, complemented
12C8×C22⋊D59—21no—
13#16837461—21no—
14#16837459C2212yes—
15#16837459C2211yescharacteristic
16#16837356—21no—
17#16837155—21no—
18#16837153—21no—
19#16837138—22no—
20#16837135—22no—
21C2×#106×C59—21no—
22C2×C4×C8×C59C2211yescharacteristic
23C4×C4⋊C4×C59C2212yes—
24C59×#340—21no—
25C2×#11745659—21no—
26C2×C8×D59—21no—
27#14273721—21no—
28#14273719Q811yescharacteristic
29#14273715Q811yescharacteristic
30#14273710—21no—
31#14273706D411yescharacteristic
32#14273701D411yescharacteristic
33C4×Dic118C2×C411yescharacteristic
34C4×D236C2×C411yescharacteristic
35C4×C22⋊D59—21no—
36#14273684—21no—
37C4⋊(C59⋊C8)—22no—
38C4⋊(C59⋊C8)C2×C411yescharacteristic
39C4⋊(C59⋊C8)C2×C411yescharacteristic
40Dic59⋊C8—21no—
41Dic59⋊C8—21no—
42C22⋊(C59⋊C8)—21no—
43C22⋊(C59⋊C8)—22no—
44(C2×D59)⋊C8—21no—
45(C2×D59)⋊C8—21no—
46#14273638—21no—
47#14273638—22no—
48#14273637—21no—
49C8×Dic59—21no—
50C22×C8×C59D411yescharacteristic
51C22×C8×C59D411yescharacteristic
52C2×C4⋊C4×C59—22no—
53C2×C42×C59—22no—
54C2×C42×C59C2311yescharacteristic
55#106×C59—24no—
56C4×C8×C59C2×C411yescharacteristic
57C4×C8×C59C2×C411yescharacteristic
58C4×C8×C59D411yescharacteristic
59C4×C8×C59D411yescharacteristic
60#95×C59—22no—
61C22×C4×C59—42no—
62C22×C4×C59C2×D411yescharacteristic
63C22×C4×C59C2×D411yescharacteristic
64C22×C4×C59#4111yescharacteristic
65C2×C22⋊D59—21no—
66C2×D236C4⋊C411yescharacteristic
67C2×C4×D59—21no—
68C2×Dic118C4⋊C411yescharacteristic
69C2×C8×C59—22no—
70C2×C8×C59—41no—
71C2×C8×C59C22⋊C411yescharacteristic
72C2×C8×C59C22⋊C411yescharacteristic
73C2×C8×C59C22⋊C411yescharacteristic
74C2×C8×C59C22⋊C411yescharacteristic
75C4⋊C4×C59—24no—
76C4⋊C4×C59—42no—
77C42×C59—82no—
78C42×C59C22×C411yescharacteristic
79C42×C59C2×D411yescharacteristic
80C42×C59C2×Q811yescharacteristic
81C42×C59#4111yescharacteristic
82C22⋊Dic59—21no—
83(C2×D59)⋊C4—21no—
84(C2×D59)⋊C4—21no—
85(C2×D59)⋊C4—21no—
86(C2×D59)⋊C4—21no—
87C4⋊Dic59C4⋊C411yescharacteristic
88C4⋊Dic59C4⋊C411yescharacteristic
89Dic59⋊C4—21no—
90Dic59⋊C4—21no—
91Dic59⋊C4—21no—
92Dic59⋊C4—21no—
93C4×Dic59—21no—
94C2×C59⋊C8—21no—
95C2×C59⋊C8—21no—
96C2×C59⋊C8—42no—
97#11745659—42no—
98C8×D59—42no—
99C23×C59#12111yescharacteristic
100C22×D59—21no—
101C2×C4×C59—22no—
102C2×C4×C59—21no—
103C2×C4×C59—21no—
104C2×C4×C59—41no—
105C2×C4×C59—44no—
106C2×C4×C59—82no—
107C2×C4×C59—82no—
108C2×C4×C59C2×C22⋊C411yescharacteristic
109C2×C4×C59C2×C4⋊C411yescharacteristic
110C2×C4×C59#11711yescharacteristic
111C2×C4×C59#12111yescharacteristic
112C2×C4×C59C22⋊Q811yescharacteristic
113C2×C4×C59C22⋊Q811yescharacteristic
114C22⋊D59—42no—
115C2×Dic59—21no—
116C2×Dic59—21no—
117C2×Dic59—21no—
118D236—41no—
119C4×D59—41no—
120C4×D59—41no—
121Dic118—41no—
122C8×C59—21no—
123C8×C59—21no—
124C8×C59—21no—
125C8×C59#10412yes—
126C59⋊C8—41no—
127C59⋊C8—41no—
128C59⋊C8—82no—
129#8696—591nomaximal, Sylow
130C22×C59—21no—
131C22×C59—21no—
132C22×C59#38011yescharacteristic
133C22×C59#47112yes—
134C2×D59—41no—
135C2×D59—41no—
136C2×D59—41no—
137C4×C59—21no—
138C4×C59—21no—
139C4×C59—21no—
140C4×C59—21no—
141C4×C59—21no—
142C4×C59—84no—
143C4×C59C2×#10411yescharacteristic
144C4×C59#42111yescharacteristic
145Dic59—41no—
146Dic59—41no—
147Dic59—41no—
148#2966—591no—
149C4×#106D5911yescharacteristic, complemented
150#1568—591no—
151#1509—592no—
152#1285—592no—
153C2×C59—22no—
154C2×C59#180611yescharacteristic
155C2×C59#181211yescharacteristic
156C2×C59#181511yescharacteristic
157D59—81no—
158#529—591no—
159#446—591no—
160C8×Q8—591no—
161#436—591no—
162#435—1181no—
163C8×D4—591no—
164C8×D4—1181no—
165#433—1181no—
166#432—591no—
167#432—592no—
168C2×#106—21no—
169#405—1181no—
170C2×C4×C8C2×D5911yescharacteristic
171C4×C4⋊C4C2×D5912yes—
172#340—21no—
173#335—1181no—
174#334—1181no—
175#328—1182no—
176#327—1182no—
177C59#869611yescharacteristic, complemented, Sylow
178C2×M16—1181no—
179C22×C8—1181no—
180C22×C8C22⋊D5911yescharacteristic
181C22×C8C22⋊D5911yescharacteristic
182#126—1181no—
183#125—591no—
184#124—591no—
185#123—1181no—
186C22⋊Q8—591no—
187#121—591no—
188C4×Q8—591no—
189C4×D4—591no—
190C4×D4—1181no—
191#117—1181no—
192C2×C4⋊C4—22no—
193C2×C42—22no—
194C2×C42C22×D5911yescharacteristic
195#106—24no—
196C4⋊C8—591no—
197C4⋊C8—591no—
198C4⋊C8—1181no—
199C4⋊C8—1181no—
200C4⋊C8—1182no—
201C22⋊C8—1181no—
202C22⋊C8—1181no—
203C22⋊C8—1181no—
204C22⋊C8—1182no—
205#97—1181no—
206#97—1181no—
207#97—1182no—
208C4×C8—1181no—
209C4×C8C4×D5911yescharacteristic
210C4×C8C4×D5911yescharacteristic
211C4×C8D23611yescharacteristic
212C4×C8D23611yescharacteristic
213#95—22no—
214C2×Q8—591no—
215C2×D4—591no—
216C2×D4—1181no—
217C22×C4—42no—
218C22×C4—1181no—
219C22×C4D4×D5911yescharacteristic
220C22×C4#1174568711yescharacteristic
221C22×C4C2×C22⋊D5911yescharacteristic
222M16—2362no—
223C2×C8—22no—
224C2×C8—41no—
225C2×C8—1181no—
226C2×C8—1181no—
227C2×C8—2362no—
228C2×C8—2362no—
229C2×C8(C2×D59)⋊C411yescharacteristic
230C2×C8(C2×D59)⋊C411yescharacteristic
231C2×C8(C2×D59)⋊C411yescharacteristic
232C2×C8(C2×D59)⋊C411yescharacteristic
233C4⋊C4—24no—
234C4⋊C4—42no—
235C4⋊C4—591no—
236C4⋊C4—591no—
237C4⋊C4—1181no—
238C4⋊C4—1181no—
239C4⋊C4—1181no—
240C4⋊C4—1181no—
241C22⋊C4—1181no—
242C22⋊C4—1181no—
243C22⋊C4—1181no—
244C22⋊C4—1181no—
245C22⋊C4—1181no—
246C42—82no—
247C42—1181no—
248C42C2×C4×D5911yescharacteristic
249C42C2×D23611yescharacteristic
250C42Q8×D5911yescharacteristic
251C42#1174568911yescharacteristic
252C23—1181no—
253C23#1427369811yescharacteristic
254Q8—2361no—
255D4—2361no—
256D4—2362no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—41no—
261C2×C4—44no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—1181no—
265C2×C4—1181no—
266C2×C4—1181no—
267C2×C4—2361no—
268C2×C4—2361no—
269C2×C4C2×(C2×D59)⋊C411yescharacteristic
270C2×C4C4⋊C4×D5911yescharacteristic
271C2×C4#1427368511yescharacteristic
272C2×C4#1427370211yescharacteristic
273C2×C4#1427370311yescharacteristic
274C2×C4#1427370711yescharacteristic
275C8—21no—
276C8—21no—
277C8—21no—
278C8—4722no—
279C8—2361no—
280C8—2361no—
281C8#1427365512yes—
282C22—21no—
283C22—21no—
284C22—2361no—
285C22—2361no—
286C22—2361no—
287C22#1683725911yescharacteristic
288C22#1683766212yes—
289C4—21no—
290C4—21no—
291C4—21no—
292C4—21no—
293C4—21no—
294C4—84no—
295C4—2361no—
296C4—2361no—
297C4—2361no—
298C4C2×#1427365511yescharacteristic
299C4#1683742011yescharacteristic
300C2—22no—
301C2—4721no—
302C2#2142411711yescharacteristic
303C2#2142413911yescharacteristic
304C2#2142415611yescharacteristic
subgroupG/H
centerC22#16837259
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC4×#106×C59C2
socleC22×C59#380
prime radical#14273701D4
cosocleC2×C42×C59C23
hypercenterC4×#106D59
hypocenterC59#8696
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#106

Complements

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

NH ≅ G/Ntype
2. #21422859C2semidirect
4. #21421952C2semidirect
5. C4×#106×C59C2semidirect
7. #16837532C4semidirect
8. C8×Dic118C4semidirect
9. #16837481C4semidirect
11. C8×D236C4semidirect
149. C4×#106D59semidirect
177. C59#8696semidirect

Maximal subgroups

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

subgroupindex
#214330712
#214228592
#214226192
#214219522
C4×#106×C592
#869659