Finite Group Databasebeta

← #28485337

Subgroups — #28485337

Order 15104.

410 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 113 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#21435244C211yesmaximal, characteristic
2#21433037C211yesmaximal, characteristic
3#21425068C211yesmaximal, characteristic
4#21422776C211yesmaximal, characteristic
5#21421954C211yesmaximal, characteristic
6#21421926C211yesmaximal, characteristic
7C59×#1518C211yesmaximal, characteristic
8C2×#14273727C2211yescharacteristic
9#16837805C412yescomplemented
10#16837752C412yescomplemented
11#16837735—22no—
12#16837697—22no—
13#16837533—22no—
14#16837533—22no—
15#16837455C2211yescharacteristic
16#16837429—21no—
17#16837426C2211yescharacteristic
18#16837358—21no—
19C2×#14273638C2211yescharacteristic
20#16837164—21no—
21#16837164—21no—
22#16837150—22no—
23C59×#533C2211yescharacteristic
24C59×#431C2211yescharacteristic
25C59×#382C2211yescharacteristic
26C59×#328—22no—
27C22×Dic118—21no—
28C2×#11745664—21no—
29C22×C59⋊C8D411yescharacteristic
30C22×C59⋊C8D411yescharacteristic
31#14273727—21no—
32#14273727—21no—
33#14273727—21no—
34#14273727C2×C412yes—
35C2×C4⋊Dic59—21no—
36C2×C4⋊Dic59—21no—
37#14273660—24no—
38C4⋊(C59⋊C8)—24no—
39C4⋊(C59⋊C8)—21no—
40C4⋊(C59⋊C8)—21no—
41C4⋊(C59⋊C8)D412yes—
42#14273651—24no—
43C22⋊(C59⋊C8)—21no—
44#14273638—22no—
45#14273638—21no—
46#14273638—21no—
47C4×C59⋊C8—22no—
48C4×C59⋊C8C2×C412yes—
49C4⋊Q8×C59C2×C411yescharacteristic
50C4⋊Q8×C59C2×C411yescharacteristic
51#125×C59—21no—
52C22⋊Q8×C59—21no—
53C22⋊Q8×C59D412yes—
54C4×Q8×C59—21no—
55C4×Q8×C59D412yes—
56#117×C59—21no—
57C2×C4⋊C4×C59—21no—
58C2×C42×C59C2311yescharacteristic
59C4⋊C8×C59—21no—
60C22⋊C8×C59—21no—
61C4×C8×C59—21no—
62#95×C59—21no—
63#95×C59—21no—
64C2×Q8×C59—21no—
65C2×Q8×C59C22⋊C412yes—
66C22×C4×C59—41no—
67C22×C4×C59C2×D411yescharacteristic
68C22×C4×C59C2×D411yescharacteristic
69C22×C4×C59#4111yescharacteristic
70C22×Dic59—41no—
71C2×Dic118—22no—
72C2×Dic118—41no—
73C2×Dic118—82no—
74C2×C8×C59—41no—
75C4⋊C4×C59—21no—
76C4⋊C4×C59—21no—
77C4⋊C4×C59—21no—
78C4⋊C4×C59—22no—
79C4⋊C4×C59—22no—
80C4⋊C4×C59—41no—
81C4⋊C4×C59—41no—
82C4⋊C4×C59C22⋊C412yes—
83C22⋊C4×C59—22no—
84C42×C59—22no—
85C42×C59C22×C411yescharacteristic
86C42×C59C2×D411yescharacteristic
87C42×C59C2×D411yescharacteristic
88C42×C59#4111yescharacteristic
89C4⋊Dic59—22no—
90C4⋊Dic59—22no—
91C4⋊Dic59—41no—
92C4⋊Dic59—41no—
93#11745664—41no—
94#11745664—41no—
95C2×C59⋊C8—21no—
96C2×C59⋊C8—21no—
97C2×C59⋊C8—22no—
98C2×C59⋊C8—22no—
99C2×C59⋊C8—21no—
100C2×C59⋊C8—21no—
101C2×C59⋊C8—21no—
102C2×C59⋊C8Q1612yes—
103C23×C59#12111yescharacteristic
104Q8×C59—42no—
105Q8×C59—81no—
106C2×C4×C59—22no—
107C2×C4×C59—22no—
108C2×C4×C59—41no—
109C2×C4×C59—41no—
110C2×C4×C59—81no—
111C2×C4×C59—81no—
112C2×C4×C59—81no—
113C2×C4×C59C2×C22⋊C411yescharacteristic
114C2×C4×C59C4×D411yescharacteristic
115C2×C4×C59C4×D411yescharacteristic
116C2×C4×C59#12011yescharacteristic
117C2×C4×C59#12111yescharacteristic
118C2×C4×C59#12311yescharacteristic
119C2×C4×C59C2×Q1611yescharacteristic
120C2×C4×C59#13511yescharacteristic
121C2×C4×C59#13711yescharacteristic
122C2×C4×C59#13711yescharacteristic
123C2×Dic59—42no—
124C2×Dic59—82no—
125Dic118—82no—
126Dic118—82no—
127C8×C59—81no—
128C59⋊C8—22no—
129C59⋊C8—42no—
130C59⋊C8—41no—
131C59⋊C8—42no—
132#9288—591nomaximal, Sylow
133C22×C59—21no—
134C22×C59—21no—
135C22×C59#38611yescharacteristic
136C22×C59#46211yescharacteristic
137C22×C59#46411yescharacteristic
138C4×C59—21no—
139C4×C59—21no—
140C4×C59—21no—
141C4×C59—21no—
142C4×C59—42no—
143C4×C59—42no—
144C4×C59—161no—
145C4×C59C4×Q1611yescharacteristic
146C4×C59#44111yescharacteristic
147C4×C59#45111yescharacteristic
148C4×C59#45211yescharacteristic
149Dic59—82no—
150#3230—591no—
151#2961—591no—
152#1993—591no—
153#1548—591no—
154#1518D5911yescharacteristic
155#1286—591no—
156#1275—591no—
157C2×C59—21no—
158C2×C59—21no—
159C2×C59#187811yescharacteristic
160C2×C59#190311yescharacteristic
161C2×C59#190911yescharacteristic
162#533C2×D5911yescharacteristic
163C2×C4⋊Q8—591no—
164#500—592no—
165#487—592no—
166#484—1182no—
167#477—1182no—
168#446—1182no—
169#446—1182no—
170#431—591no—
171#431C2×D5911yescharacteristic
172#424—1181no—
173#423—591no—
174#405—1181no—
175C2×#97—591no—
176#382C2×D5911yescharacteristic
177#340—1181no—
178#340—1181no—
179#333—1182no—
180#328—22no—
181C59#928811yescharacteristic, complemented, Sylow
182C22×Q8—1181no—
183C2×M16—1181no—
184C22×C8—591no—
185C22×C8—591no—
186C4⋊Q8—592no—
187C4⋊Q8—1181no—
188C4⋊Q8—1181no—
189C4⋊Q8—1181no—
190C4⋊Q8C4×D5911yescharacteristic
191C4⋊Q8C4×D5911yescharacteristic
192#125—21no—
193C22⋊Q8—21no—
194C22⋊Q8C22⋊D5912yes—
195C4×Q8—21no—
196C4×Q8D23612yes—
197#117—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—1181no—
200C2×C4⋊C4—1181no—
201C2×C42C22×D5911yescharacteristic
202#107—1184no—
203C4⋊C8—21no—
204C4⋊C8—592no—
205C4⋊C8—1184no—
206C4⋊C8—1181no—
207C4⋊C8—1181no—
208#103—1184no—
209C22⋊C8—21no—
210C22⋊C8—1181no—
211#97—1182no—
212#97—1181no—
213#97—1181no—
214C4×C8—21no—
215C4×C8—592no—
216C4×C8—1182no—
217#95—21no—
218#95—21no—
219C2×Q8—21no—
220C2×Q8—1182no—
221C2×Q8—4722no—
222C2×Q8—2361no—
223C2×Q8(C2×D59)⋊C412yes—
224C22×C4—41no—
225C22×C4—2361no—
226C22×C4D4×D5911yescharacteristic
227C22×C4#1174568711yescharacteristic
228C22×C4C2×C22⋊D5911yescharacteristic
229M16—2361no—
230M16—2361no—
231C2×C8—41no—
232C2×C8—592no—
233C2×C8—1181no—
234C2×C8—1181no—
235C2×C8—1182no—
236C2×C8—1182no—
237C2×C8—1181no—
238C2×C8—1181no—
239C2×C8—1181no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—1182no—
248C4⋊C4—1182no—
249C4⋊C4—2361no—
250C4⋊C4—2361no—
251C4⋊C4(C2×D59)⋊C412yes—
252C22⋊C4—22no—
253C42—22no—
254C42C2×C4×D5911yescharacteristic
255C42C2×D23611yescharacteristic
256C42D4×D5911yescharacteristic
257C42#1174568711yescharacteristic
258C23#1427369911yescharacteristic
259Q8—42no—
260Q8—81no—
261Q8—4722no—
262Q8—4722no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—4722no—
271C2×C4—2362no—
272C2×C4C2×(C2×D59)⋊C411yescharacteristic
273C2×C4(C22⋊D59)⋊C411yescharacteristic
274C2×C4(C22⋊D59)⋊C411yescharacteristic
275C2×C4#1427369711yescharacteristic
276C2×C4#1427369911yescharacteristic
277C2×C4#1427371111yescharacteristic
278C2×C4Q16×D5911yescharacteristic
279C2×C4#1427374811yescharacteristic
280C2×C4#1427375611yescharacteristic
281C2×C4#1427375911yescharacteristic
282C8—81no—
283C8—1182no—
284C8—2362no—
285C8—2361no—
286C8—2362no—
287C22—21no—
288C22—21no—
289C22#1683727911yescharacteristic
290C22#1683761011yescharacteristic
291C22#1683762211yescharacteristic
292C4—21no—
293C4—21no—
294C4—21no—
295C4—21no—
296C4—42no—
297C4—42no—
298C4—161no—
299C4—4722no—
300C4#1683749411yescharacteristic
301C4#1683750611yescharacteristic
302C4#1683755911yescharacteristic
303C4#1683756711yescharacteristic
304C2—21no—
305C2—21no—
306C2#2142443511yescharacteristic
307C2#2142457411yescharacteristic
308C2#2142461611yescharacteristic
subgroupG/H
centerC22#16837279
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1518C2
socleC22×C59#386
prime radicalC23×C59#121
cosocleC2×C42×C59C23
hypercenter#1518D59
hypocenterC59#9288
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1518

Complements

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

NH ≅ G/Ntype
9. #16837805C4semidirect
10. #16837752C4semidirect
181. C59#9288semidirect

Maximal subgroups

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

subgroupindex
#214352442
#214330372
#214250682
#214227762
#214219542
#214219262
C59×#15182
#928859