Finite Group Databasebeta

← #28499533

Subgroups — #28499533

Order 15104.

286 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 63 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#21434826C211yesmaximal, characteristic
2#21425905C211yesmaximal, characteristic, complemented
3#21424246C211yesmaximal, characteristic, complemented
4#21422086C211yesmaximal, characteristic
5#21422084C211yesmaximal, characteristic, complemented
6#21422082C211yesmaximal, characteristic
7C59×#1342C211yesmaximal, characteristic, complemented
8#16837938—21no—
9#16837802C411yescharacteristic, complemented
10#16837787C411yescharacteristic, complemented
11#16837758—21no—
12#16837655C2211yescharacteristic
13#16837650—21no—
14#16837513C411yescharacteristic, complemented
15#16837509C411yescharacteristic, complemented
16#16837505—21no—
17M16×Dic59C2211yescharacteristic
18#16837222—21no—
19#16837184C2211yescharacteristic
20#16837179—21no—
21#16837174—21no—
22#16837163—21no—
23#16837159C2211yescharacteristic
24C59×#470C2211yescharacteristic
25C59×#349C2211yescharacteristic
26C59×#338C2211yescharacteristic
27C2×#11745671—21no—
28C2×D472C2×C411yescharacteristic
29C2×#11745664D411yescharacteristic
30#14273722—41no—
31#14273718—21no—
32#14273716—21no—
33#14273711—41no—
34#14273701—21no—
35Q8×Dic59—21no—
36#14273689—21no—
37(C22⋊D59)⋊C4—41no—
38#14273686—21no—
39#14273685—41no—
40C2×C4×Dic59D411yescharacteristic
41#14273667—21no—
42#14273660—21no—
43#14273659C2×C411yescharacteristic
44#14273651—21no—
45#14273649—21no—
46#14273647—21no—
47#14273637C2×C411yescharacteristic
48#14273637D411yescharacteristic
49C8×Dic59C2×C411yescharacteristic
50C8×Dic59D411yescharacteristic
51C2×Q16×C59C2×C411yescharacteristic
52C2×M16×C59—21no—
53C2×M16×C59C2311yescharacteristic
54C22⋊Q8×C59—21no—
55M32×C59—21no—
56#106×C59C2×C411yescharacteristic
57#103×C59—21no—
58C2×Q8×C59—21no—
59C22×C4×C59C2×D411yescharacteristic
60C2×C22⋊D59—41no—
61C22×Dic59—21no—
62C2×D236—21no—
63C2×C4×D59—41no—
64Q16×C59—41no—
65M16×C59—41no—
66M16×C59—41no—
67M16×C59—41no—
68C2×C8×C59—41no—
69C2×C8×C59C22×C411yescharacteristic
70C2×C8×C59C2×D411yescharacteristic
71C4⋊C4×C59—21no—
72C4⋊C4×C59—41no—
73C22⋊C4×C59—41no—
74C22⋊Dic59—41no—
75#11745671—42no—
76(C2×D59)⋊C4—41no—
77(C2×D59)⋊C4—41no—
78(C2×D59)⋊C4—41no—
79(C2×D59)⋊C4—41no—
80C4⋊Dic59—21no—
81C4⋊Dic59—41no—
82C4⋊Dic59—41no—
83C4⋊Dic59—41no—
84Dic59⋊C4—41no—
85C4×Dic59—21no—
86C4×Dic59—41no—
87C4×Dic59—41no—
88C4×Dic59C22⋊C411yescharacteristic
89C4×Dic59C22⋊C411yescharacteristic
90#11745664—41no—
91C2×C59⋊C8C22⋊C411yescharacteristic
92C2×C59⋊C8C22⋊C411yescharacteristic
93D472—41no—
94C16×C59—41no—
95C59⋊C16—41no—
96C23×C59AGL(1,ℤ/8)11yescharacteristic
97C22×D59—41no—
98Q8×C59—41no—
99Q8×C59—41no—
100C2×C4×C59—21no—
101C2×C4×C59—21no—
102C2×C4×C59—41no—
103C2×C4×C59—41no—
104C2×C4×C59C2×C22⋊C411yescharacteristic
105C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
106C22⋊D59—82no—
107C2×Dic59—21no—
108C2×Dic59—21no—
109C2×Dic59—41no—
110C2×Dic59—81no—
111C2×Dic59—81no—
112D236—41no—
113D236—41no—
114C4×D59—82no—
115C8×C59—21no—
116C8×C59—21no—
117C8×C59—81no—
118C59⋊C8—21no—
119C59⋊C8—21no—
120#11836—591nomaximal, Sylow
121C22×C59—21no—
122C22×C59—41no—
123C22×C59#41811yescharacteristic
124C2×D59—81no—
125C2×D59—81no—
126C2×D59—81no—
127C4×C59—21no—
128C4×C59—81no—
129C4×C59—81no—
130C4×C59#41011yescharacteristic
131C4×C59#42111yescharacteristic
132Dic59—41no—
133Dic59—41no—
134Dic59—81no—
135Dic59—161no—
136#3183—591no—
137#2165—591no—
138#1841—591no—
139#1342D5911yescharacteristic, complemented
140#1341—591no—
141#1340—591no—
142#1339—591no—
143C2×C59—21no—
144C2×C59—41no—
145C2×C59#151411yescharacteristic
146D59—161no—
147#529—1181no—
148#499—591no—
149#495—591no—
150#488—1181no—
151#470C2×D5911yescharacteristic
152#469—591no—
153#468—1181no—
154#442—591no—
155#441—591no—
156#440—1181no—
157C4×M16—591no—
158#364—1181no—
159#349—591no—
160#349C2×D5911yescharacteristic
161#347—1181no—
162#344—1181no—
163#339—1181no—
164#338C2×D5911yescharacteristic
165H(3,ℤ/4)—591no—
166C59#1183611yescharacteristic, complemented, Sylow
167C2×Q16C4×D5911yescharacteristic
168C2×SD16—1181no—
169C2×D8—591no—
170C2×M16—21no—
171C2×M16—591no—
172C2×M16C22×D5911yescharacteristic
173#126—2361no—
174#125—1181no—
175#124—1181no—
176#123—2361no—
177C22⋊Q8—21no—
178#121—1181no—
179C4×Q8—1181no—
180C4×D4—1181no—
181C4×D4—2361no—
182#117—1181no—
183#117—2361no—
184C2×C42—591no—
185M32—21no—
186M32—1181no—
187#107—1181no—
188#106—591no—
189#106C4×D5911yescharacteristic
190#103—21no—
191#103—1181no—
192#102—1181no—
193#102—1181no—
194#97—591no—
195#97—591no—
196C4×C8—591no—
197C4×C8—591no—
198C2×Q8—21no—
199C2×D4—1181no—
200C2×D4—2361no—
201C22×C4—1181no—
202C22×C4—2361no—
203C22×C4D4×D5911yescharacteristic
204Q16—41no—
205SD16—2362no—
206D8—2361no—
207M16—41no—
208M16—41no—
209M16—41no—
210M16—2361no—
211C2×C8—41no—
212C2×C8—591no—
213C2×C8—591no—
214C2×C8C2×C4×D5911yescharacteristic
215C2×C8D4×D5911yescharacteristic
216C4⋊C4—21no—
217C4⋊C4—41no—
218C4⋊C4—1181no—
219C4⋊C4—2361no—
220C4⋊C4—2361no—
221C4⋊C4—2361no—
222C4⋊C4—2361no—
223C22⋊C4—41no—
224C22⋊C4—2361no—
225C22⋊C4—2361no—
226C22⋊C4—2361no—
227C22⋊C4—2361no—
228C22⋊C4—2361no—
229C42—591no—
230C42—591no—
231C42—1181no—
232C42—2361no—
233C42—2361no—
234C16—41no—
235C16—2361no—
236C23—2361no—
237C23#1427375411yescharacteristic
238Q8—41no—
239Q8—41no—
240D4—4722no—
241D4—2361no—
242D4—2361no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—1181no—
248C2×C4—1181no—
249C2×C4—4722no—
250C2×C4—4721no—
251C2×C4—4721no—
252C2×C4—2361no—
253C2×C4C22⋊C4×D5911yescharacteristic
254C2×C4#1427375111yescharacteristic
255C8—21no—
256C8—21no—
257C8—81no—
258C8—1181no—
259C8—1181no—
260C22—21no—
261C22—41no—
262C22—4721no—
263C22—4721no—
264C22—4721no—
265C22#1683740511yescharacteristic
266C4—21no—
267C4—81no—
268C4—81no—
269C4—9441no—
270C4—4721no—
271C4—2361no—
272C4—2361no—
273C4#1683737411yescharacteristic
274C4#1683741711yescharacteristic
275C2—21no—
276C2—41no—
277C2—9441no—
278C2#2142264111yescharacteristic
subgroupG/H
centerC2#21422641
commutatorC2×C8×C59C22×C4
FrattiniC2×M16C22×D59
FittingC59×#1342C2
socleC2×C59#1514
prime radical#16837655C22
cosocleC2×M16×C59C23
hypercenter#1342D59
hypocenterC59#11836
LCSG ▷ C2×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #1342

Complements

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

NH ≅ G/Ntype
2. #21425905C2semidirect
3. #21424246C2semidirect
5. #21422084C2semidirect
7. C59×#1342C2semidirect
9. #16837802C4semidirect
10. #16837787C4semidirect
14. #16837513C4semidirect
15. #16837509C4semidirect
139. #1342D59semidirect
166. C59#11836semidirect

Maximal subgroups

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

subgroupindex
#214348262
#214259052
#214242462
#214220862
#214220842
#214220822
C59×#13422
#1183659