Finite Group Databasebeta

← #28499232

Subgroups — #28499232

Order 15104.

316 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 65 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#21434741C211yesmaximal, characteristic
2#21425909C211yesmaximal, characteristic, complemented
3#21423852C211yesmaximal, characteristic, complemented
4#21422086C211yesmaximal, characteristic, complemented
5#21422082C211yesmaximal, characteristic, complemented
6#21422081C211yesmaximal, characteristic, complemented
7C59×#1341C211yesmaximal, characteristic, complemented
8#16838005—21no—
9#16837802C411yescharacteristic, complemented
10#16837777C411yescharacteristic, complemented
11#16837765—21no—
12#16837655C2211yescharacteristic
13#16837636—21no—
14#16837514C411yescharacteristic, complemented
15#16837513C411yescharacteristic, complemented
16#16837482—21no—
17#16837360—21no—
18#16837360C2211yescharacteristic
19#16837222—21no—
20#16837184C2211yescharacteristic
21#16837179—21no—
22#16837159—21no—
23#16837159C2211yescharacteristic
24C59×#469C2211yescharacteristic
25C59×#349C2211yescharacteristic
26C59×#338C2211yescharacteristic
27C2×D472C2×C411yescharacteristic
28C2×#11745669—21no—
29C22×C59⋊C8D411yescharacteristic
30#14273723—21no—
31#14273718—21no—
32#14273711—41no—
33#14273709—41no—
34#14273701—21no—
35#14273699—41no—
36#14273698—41no—
37D4×Dic59—21no—
38#14273689—21no—
39#14273686—21no—
40#14273686D411yescharacteristic
41#14273667—21no—
42#14273660—21no—
43#14273659C2×C411yescharacteristic
44#14273649—21no—
45#14273648—21no—
46#14273647—21no—
47#14273637—22no—
48#14273637C2×C411yescharacteristic
49#14273637D411yescharacteristic
50C8×Dic59—22no—
51C8×Dic59C2×C411yescharacteristic
52C8×Dic59D411yescharacteristic
53C2×D8×C59C2×C411yescharacteristic
54C2×M16×C59—21no—
55C2×M16×C59C2311yescharacteristic
56#121×C59—21no—
57M32×C59—21no—
58#106×C59C2×C411yescharacteristic
59#102×C59—21no—
60C2×D4×C59—21no—
61C2×D4×C59—41no—
62C22×C4×C59C2×D411yescharacteristic
63C2×C22⋊D59—41no—
64C2×C22⋊D59—41no—
65C2×C22⋊D59—41no—
66C22×Dic59—41no—
67C2×D236—21no—
68C2×C4×D59—41no—
69D8×C59—41no—
70M16×C59—41no—
71M16×C59—41no—
72M16×C59—41no—
73C2×C8×C59—22no—
74C2×C8×C59C22×C411yescharacteristic
75C2×C8×C59C2×D411yescharacteristic
76C4⋊C4×C59—21no—
77C22⋊C4×C59—41no—
78C22⋊Dic59—21no—
79C22⋊Dic59—41no—
80C22⋊Dic59—41no—
81#11745669—41no—
82#11745669—41no—
83(C2×D59)⋊C4—41no—
84(C2×D59)⋊C4—41no—
85C4⋊Dic59—21no—
86C4⋊Dic59—21no—
87C4⋊Dic59—41no—
88C4⋊Dic59—41no—
89Dic59⋊C4—41no—
90C4×Dic59—22no—
91C4×Dic59C22⋊C411yescharacteristic
92C4×Dic59C22⋊C411yescharacteristic
93C2×C59⋊C8—21no—
94C2×C59⋊C8—21no—
95C2×C59⋊C8C22⋊C411yescharacteristic
96C2×C59⋊C8C22⋊C411yescharacteristic
97D472—41no—
98C16×C59—41no—
99C59⋊C16—41no—
100C23×C59—41no—
101C23×C59AGL(1,ℤ/8)11yescharacteristic
102C22×D59—41no—
103D4×C59—41no—
104D4×C59—41no—
105D4×C59—161no—
106C2×C4×C59—21no—
107C2×C4×C59—21no—
108C2×C4×C59—41no—
109C2×C4×C59C2×C22⋊C411yescharacteristic
110C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
111C22⋊D59—161no—
112C22⋊D59—161no—
113C22⋊D59—161no—
114C2×Dic59—21no—
115C2×Dic59—21no—
116C2×Dic59—42no—
117C2×Dic59—41no—
118C2×Dic59—161no—
119D236—41no—
120D236—41no—
121C4×D59—161no—
122C8×C59—21no—
123C8×C59—21no—
124C8×C59—42no—
125C59⋊C8—21no—
126C59⋊C8#10411yescharacteristic
127C59⋊C8#10411yescharacteristic
128#11770—591nomaximal, Sylow
129C22×C59—21no—
130C22×C59—41no—
131C22×C59—81no—
132C22×C59—81no—
133C22×C59—81no—
134C22×C59#41811yescharacteristic
135C2×D59—81no—
136C2×D59—81no—
137C2×D59—81no—
138C4×C59—21no—
139C4×C59—81no—
140C4×C59#41011yescharacteristic
141C4×C59C2×#10411yescharacteristic
142Dic59—41no—
143Dic59—41no—
144Dic59—81no—
145Dic59—82no—
146#3176—591no—
147#2166—591no—
148#1747—591no—
149#1341—591no—
150#1341D5911yescharacteristic, complemented
151#1339—591no—
152#1339—591no—
153C2×C59—21no—
154C2×C59—41no—
155C2×C59—161no—
156C2×C59#150511yescharacteristic
157D59—161no—
158#540—1181no—
159#499—591no—
160#493—591no—
161#490—1181no—
162#469—591no—
163#469C2×D5911yescharacteristic
164#466—1181no—
165#442—591no—
166#442—591no—
167C4×D8—1181no—
168#405—591no—
169#405—1181no—
170#364—1181no—
171#349—591no—
172#349C2×D5911yescharacteristic
173#347—1181no—
174#338C2×D5911yescharacteristic
175H(3,ℤ/4)—591no—
176H(3,ℤ/4)—1181no—
177C59#1177011yescharacteristic, complemented, Sylow
178C2×D8—591no—
179C2×D8—1181no—
180C2×D8C4×D5911yescharacteristic
181C2×M16—21no—
182C2×M16C22×D5911yescharacteristic
183C22×C8—591no—
184#127—1181no—
185#125—1181no—
186#123—2361no—
187#123—2361no—
188#121—21no—
189#121—1181no—
190#121—2361no—
191#121—2361no—
192C4×D4—1181no—
193C4×D4—1181no—
194#117—591no—
195#117—1181no—
196M32—21no—
197M32—1181no—
198#107—1181no—
199#106—591no—
200#106C4×D5911yescharacteristic
201#102—21no—
202#102—1181no—
203#102—1181no—
204#102—1181no—
205#97—591no—
206#97—591no—
207#97—1182no—
208C4×C8—591no—
209C4×C8—591no—
210C4×C8—1182no—
211C2×D4—21no—
212C2×D4—41no—
213C2×D4—1181no—
214C2×D4—2361no—
215C2×D4—2361no—
216C2×D4—2361no—
217C22×C4—2361no—
218C22×C4—2361no—
219C22×C4D4×D5911yescharacteristic
220D8—41no—
221D8—2361no—
222D8—2361no—
223D8—2361no—
224M16—41no—
225M16—41no—
226M16—41no—
227C2×C8—22no—
228C2×C8—591no—
229C2×C8—591no—
230C2×C8—1181no—
231C2×C8—1181no—
232C2×C8C2×C4×D5911yescharacteristic
233C2×C8D4×D5911yescharacteristic
234C4⋊C4—21no—
235C4⋊C4—1181no—
236C4⋊C4—1181no—
237C4⋊C4—2361no—
238C4⋊C4—2361no—
239C4⋊C4—2361no—
240C22⋊C4—41no—
241C22⋊C4—1181no—
242C22⋊C4—2361no—
243C22⋊C4—2361no—
244C22⋊C4—2361no—
245C22⋊C4—2361no—
246C42—591no—
247C42—591no—
248C42—1182no—
249C16—41no—
250C16—2361no—
251C23—41no—
252C23—2361no—
253C23#1427375411yescharacteristic
254D4—41no—
255D4—41no—
256D4—161no—
257D4—9441no—
258D4—9441no—
259D4—9441no—
260D4—2361no—
261D4—2361no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—41no—
265C2×C4—1181no—
266C2×C4—1181no—
267C2×C4—9441no—
268C2×C4—9441no—
269C2×C4—2362no—
270C2×C4—2361no—
271C2×C4C22⋊C4×D5911yescharacteristic
272C2×C4#1427375111yescharacteristic
273C8—21no—
274C8—21no—
275C8—42no—
276C8—591no—
277C8—591no—
278C8—1181no—
279C22—21no—
280C22—41no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22—4721no—
285C22—4721no—
286C22—4721no—
287C22#1683740511yescharacteristic
288C4—21no—
289C4—81no—
290C4—4721no—
291C4—4722no—
292C4—2361no—
293C4—2361no—
294C4#1683737411yescharacteristic
295C4#104×D5911yescharacteristic
296C2—21no—
297C2—41no—
298C2—161no—
299C2—9441no—
300C2#2142260211yescharacteristic
subgroupG/H
centerC2#21422602
commutatorC2×C8×C59C22×C4
FrattiniC2×M16C22×D59
FittingC59×#1341C2
socleC2×C59#1505
prime radical#21434741C2
cosocleC2×M16×C59C23
hypercenter#1341D59
hypocenterC59#11770
LCSG ▷ C2×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #1341

Complements

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

NH ≅ G/Ntype
2. #21425909C2semidirect
3. #21423852C2semidirect
4. #21422086C2semidirect
5. #21422082C2semidirect
6. #21422081C2semidirect
7. C59×#1341C2semidirect
9. #16837802C4semidirect
10. #16837777C4semidirect
14. #16837514C4semidirect
15. #16837513C4semidirect
150. #1341D59semidirect
177. C59#11770semidirect

Maximal subgroups

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

subgroupindex
#214347412
#214259092
#214238522
#214220862
#214220822
#214220812
C59×#13412
#1177059