Finite Group Databasebeta

← #28485354

Subgroups — #28485354

Order 15104.

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

normal subgroups only

This group has 422 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#21435046C211yesmaximal, characteristic
2#21433033C211yesmaximal, characteristic
3#21425068C211yesmaximal, characteristic
4#21422782C211yesmaximal, characteristic
5#21422659C211yesmaximal, characteristic
6#21421926C211yesmaximal, characteristic
7C59×#1286C211yesmaximal, characteristic
8#16837959C2211yescharacteristic
9C2×#14273727C2211yescharacteristic
10#16837782C412yescomplemented
11#16837753C412yescomplemented
12#16837613—22no—
13#16837560—22no—
14#16837531—22no—
15#16837531—22no—
16#16837455C2211yescharacteristic
17#16837430—21no—
18#16837360—21no—
19C2×#14273638C2211yescharacteristic
20#16837164—21no—
21#16837164—21no—
22#16837147—22no—
23#16837138—22no—
24C59×#431C2211yescharacteristic
25C59×#423C2211yescharacteristic
26C59×#382C2211yescharacteristic
27C22×Dic118—21no—
28C2×Dic236—24no—
29C22×C59⋊C8D411yescharacteristic
30C22×C59⋊C8D411yescharacteristic
31#14273727—21no—
32#14273727—21no—
33#14273727—21no—
34#14273727C2×C412yes—
35#14273727C2×C411yescharacteristic
36#14273727C2×C411yescharacteristic
37#14273719—21no—
38#14273706—21no—
39#14273706D412yes—
40C4×Dic118—21no—
41C4×Dic118D412yes—
42#14273686—21no—
43C2×C4⋊Dic59—21no—
44C2×C4⋊Dic59—21no—
45C4⋊(C59⋊C8)—21no—
46Dic59⋊C8—24no—
47#14273652—24no—
48C22⋊(C59⋊C8)—21no—
49#14273638—21no—
50#14273638—21no—
51#14273637—22no—
52C4×C59⋊C8—21no—
53C8×Dic59—22no—
54C2×M16×C59—21no—
55C2×C4⋊C4×C59—21no—
56C2×C42×C59C2311yescharacteristic
57C4⋊C8×C59—21no—
58C4⋊C8×C59—21no—
59C4⋊C8×C59D412yes—
60C22⋊C8×C59—21no—
61C4×C8×C59C2×C412yes—
62#95×C59—21no—
63#95×C59—21no—
64C22×C4×C59—41no—
65C22×C4×C59C2×D411yescharacteristic
66C22×C4×C59C2×D411yescharacteristic
67C22×C4×C59#4111yescharacteristic
68C22×Dic59—41no—
69C2×Dic118—22no—
70C2×Dic118—21no—
71C2×Dic118—41no—
72C2×Dic118—82no—
73C2×Dic118C22⋊C412yes—
74M16×C59—41no—
75M16×C59—41no—
76C2×C8×C59—22no—
77C2×C8×C59—22no—
78C4⋊C4×C59—41no—
79C4⋊C4×C59—41no—
80C42×C59C22×C411yescharacteristic
81C42×C59C2×D411yescharacteristic
82C42×C59C2×D411yescharacteristic
83C42×C59#4111yescharacteristic
84C22⋊Dic59—22no—
85C4⋊Dic59—21no—
86C4⋊Dic59—21no—
87C4⋊Dic59—21no—
88C4⋊Dic59—22no—
89C4⋊Dic59—22no—
90C4⋊Dic59—41no—
91C4⋊Dic59—41no—
92C4⋊Dic59C22⋊C412yes—
93Dic59⋊C4—22no—
94Dic59⋊C4—22no—
95C4×Dic59—22no—
96C2×C59⋊C8—21no—
97C2×C59⋊C8—21no—
98C2×C59⋊C8—21no—
99C2×C59⋊C8—21no—
100C2×C59⋊C8—21no—
101C2×C59⋊C8—41no—
102C2×C59⋊C8Q1612yescomplemented
103Dic236—84no—
104C23×C59#12111yescharacteristic
105C2×C4×C59—41no—
106C2×C4×C59—41no—
107C2×C4×C59—81no—
108C2×C4×C59—81no—
109C2×C4×C59—81no—
110C2×C4×C59C2×C22⋊C411yescharacteristic
111C2×C4×C59C4×D411yescharacteristic
112C2×C4×C59C4×D411yescharacteristic
113C2×C4×C59#12011yescharacteristic
114C2×C4×C59#12111yescharacteristic
115C2×C4×C59#12311yescharacteristic
116C2×C4×C59C2×Q1611yescharacteristic
117C2×C4×C59#13511yescharacteristic
118C2×C4×C59#13711yescharacteristic
119C2×C4×C59#13711yescharacteristic
120C2×Dic59—22no—
121C2×Dic59—22no—
122C2×Dic59—42no—
123C2×Dic59—82no—
124Dic118—42no—
125Dic118—44no—
126Dic118—81no—
127Dic118—82no—
128C8×C59—42no—
129C8×C59—42no—
130C59⋊C8—22no—
131C59⋊C8—41no—
132C59⋊C8—81no—
133#9290—591nomaximal, Sylow
134C22×C59—21no—
135C22×C59—21no—
136C22×C59#38611yescharacteristic
137C22×C59#46211yescharacteristic
138C22×C59#46411yescharacteristic
139C4×C59—21no—
140C4×C59—21no—
141C4×C59—21no—
142C4×C59—21no—
143C4×C59—161no—
144C4×C59C4×Q1611yescharacteristic
145C4×C59#44111yescharacteristic
146C4×C59#45111yescharacteristic
147C4×C59#45211yescharacteristic
148Dic59—42no—
149Dic59—42no—
150Dic59—82no—
151#3209—591no—
152#2961—591no—
153#1993—591no—
154#1549—591no—
155#1518—591no—
156#1286D5911yescharacteristic
157#1275—591no—
158C2×C59—21no—
159C2×C59—21no—
160C2×C59#187811yescharacteristic
161C2×C59#190211yescharacteristic
162C2×C59#191011yescharacteristic
163#533—591no—
164C2×C4⋊Q8—591no—
165#494—592no—
166#487—592no—
167#462—1182no—
168#451—1182no—
169#446—1182no—
170#446—1182no—
171#431—591no—
172#431C2×D5911yescharacteristic
173#424—1181no—
174#423C2×D5911yescharacteristic
175#405—1181no—
176C2×#97—591no—
177#382C2×D5911yescharacteristic
178#340—1181no—
179#340—1181no—
180#332—1182no—
181#328—1182no—
182C59#929011yescharacteristic, complemented, Sylow
183C22×Q8—1181no—
184C2×Q16—1184no—
185C2×M16—21no—
186C22×C8—591no—
187C22×C8—591no—
188C4⋊Q8—591no—
189C4⋊Q8—591no—
190C4⋊Q8—592no—
191C4⋊Q8—1181no—
192C4⋊Q8—1181no—
193C4⋊Q8—1181no—
194#125—1181no—
195C22⋊Q8—592no—
196C22⋊Q8—1181no—
197C4×Q8—592no—
198C4×Q8—1181no—
199#117—1181no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—1181no—
202C2×C4⋊C4—1181no—
203C2×C42C22×D5911yescharacteristic
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—1184no—
207C4⋊C8—1181no—
208C4⋊C8D23612yes—
209#103—1184no—
210C22⋊C8—21no—
211C22⋊C8—1181no—
212#97—1182no—
213#97—1181no—
214#97—1181no—
215C4×C8—1182no—
216C4×C8—1181no—
217C4×C8C4×D5912yes—
218#95—21no—
219#95—21no—
220C2×Q8—592no—
221C2×Q8—1181no—
222C2×Q8—1182no—
223C2×Q8—4722no—
224C2×Q8—2361no—
225C22×C4—41no—
226C22×C4—2361no—
227C22×C4#1174568511yescharacteristic
228C22×C4D4×D5911yescharacteristic
229C22×C4D4×D5911yescharacteristic
230Q16—4724no—
231M16—41no—
232M16—41no—
233C2×C8—22no—
234C2×C8—22no—
235C2×C8—592no—
236C2×C8—1181no—
237C2×C8—1181no—
238C2×C8—1181no—
239C2×C8—1181no—
240C2×C8—1181no—
241C2×C8—2361no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—592no—
245C4⋊C4—1181no—
246C4⋊C4—1181no—
247C4⋊C4—1181no—
248C4⋊C4—1182no—
249C4⋊C4—1182no—
250C4⋊C4—1182no—
251C4⋊C4—1182no—
252C4⋊C4—2361no—
253C4⋊C4—2361no—
254C22⋊C4—1182no—
255C42—1182no—
256C42C2×C4×D5911yescharacteristic
257C42C2×D23611yescharacteristic
258C42D4×D5911yescharacteristic
259C42#1174568911yescharacteristic
260C23#1427370011yescharacteristic
261Q8—4722no—
262Q8—4721no—
263Q8—2364no—
264Q8—2362no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—1182no—
271C2×C4—1182no—
272C2×C4—4722no—
273C2×C4—2362no—
274C2×C4C22⋊C4×D5911yescharacteristic
275C2×C4#1427368911yescharacteristic
276C2×C4C4×D23611yescharacteristic
277C2×C4#1427369711yescharacteristic
278C2×C4#1427370211yescharacteristic
279C2×C4#1427371211yescharacteristic
280C2×C4Q16×D5911yescharacteristic
281C2×C4#1427374611yescharacteristic
282C2×C4#1427375611yescharacteristic
283C2×C4#1427375611yescharacteristic
284C8—42no—
285C8—42no—
286C8—1182no—
287C8—4721no—
288C8—2361no—
289C22—21no—
290C22—21no—
291C22#1683728011yescharacteristic
292C22#1683761111yescharacteristic
293C22#1683762311yescharacteristic
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—161no—
299C4—4722no—
300C4—2362no—
301C4—2362no—
302C4#1683749611yescharacteristic
303C4#1683750811yescharacteristic
304C4#1683755911yescharacteristic
305C4#1683756511yescharacteristic
306C2—21no—
307C2—21no—
308C2#2142443811yescharacteristic
309C2#2142456811yescharacteristic
310C2#2142462211yescharacteristic
subgroupG/H
centerC22#16837280
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1286C2
socleC22×C59#386
prime radicalC23×C59#121
cosocleC2×C42×C59C23
hypercenter#1286D59
hypocenterC59#9290
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1286

Complements

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

NH ≅ G/Ntype
10. #16837782C4semidirect
11. #16837753C4semidirect
102. C2×C59⋊C8Q16semidirect
182. C59#9290semidirect

Maximal subgroups

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

subgroupindex
#214350462
#214330332
#214250682
#214227822
#214226592
#214219262
C59×#12862
#929059