Finite Group Databasebeta

← #28500223

Subgroups — #28500223

Order 15104.

360 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 79 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#21434555C211yesmaximal, characteristic
2#21425908C211yesmaximal, characteristic, complemented
3#21422136C212yesmaximal, complemented
4#21422134C212yesmaximal, complemented
5C59×#2165C211yesmaximal, characteristic, complemented
6C2×#14273760—21no—
7C2×#14273760—21no—
8#16837952C411yescharacteristic
9C2×#14273727C411yescharacteristic
10#16837796—21no—
11#16837796—21no—
12#16837782C411yescharacteristic
13#16837782C2211yescharacteristic
14#16837775C411yescharacteristic
15#16837775C2211yescharacteristic
16#16837223C2212yes—
17#16837184—21no—
18#16837180—21no—
19C4×M16×C59C2211yescharacteristic
20C59×#364C2212yes—
21C59×#349—21no—
22C59×#347—21no—
23C2×#11745685—21no—
24C22×Dic118—21no—
25#14273760—44no—
26#14273760—44no—
27C2×Dic236—21no—
28C2×Dic236—21no—
29C2×Dic236—21no—
30C2×Dic236—21no—
31C2×#11745660—21no—
32C2×#11745660—21no—
33C2×#11745660—21no—
34C2×#11745660—21no—
35#14273727—21no—
36#14273727—41no—
37#14273727C2×C411yescharacteristic
38#14273727C2×C411yescharacteristic
39#14273727C2×C411yescharacteristic
40#14273724C2×C411yescharacteristic
41#14273715—21no—
42#14273701—41no—
43C4×D236—41no—
44C2×C4⋊Dic59—41no—
45#14273667—22no—
46C2×M16×C59D411yescharacteristic
47C2×M16×C59D411yescharacteristic
48C2×C42×C59C2×C411yescharacteristic
49M32×C59—22no—
50#97×C59D411yescharacteristic
51#97×C59D411yescharacteristic
52C4×C8×C59C2×C411yescharacteristic
53C4×C8×C59C2311yescharacteristic
54C22×C4×C59—21no—
55C22×C4×C59C22⋊C411yescharacteristic
56C2×C22⋊D59—41no—
57C22×Dic59—41no—
58#11745685—42no—
59#11745685—42no—
60C2×D236—21no—
61C2×D236—41no—
62C2×C4×D59—41no—
63C2×Dic118—21no—
64C2×Dic118—21no—
65C2×Dic118—21no—
66C2×Dic118—42no—
67C2×Dic118—42no—
68C2×Dic118—41no—
69M16×C59—22no—
70M16×C59—22no—
71C2×C8×C59C22⋊C411yescharacteristic
72C2×C8×C59C22⋊C411yescharacteristic
73C2×C8×C59C2×D411yescharacteristic
74C2×C8×C59C2×D411yescharacteristic
75C42×C59—21no—
76C42×C59C22⋊C411yescharacteristic
77C42×C59C22×C411yescharacteristic
78(C2×D59)⋊C4—42no—
79C4⋊Dic59—42no—
80C4⋊Dic59—41no—
81C4⋊Dic59—41no—
82C4⋊Dic59—41no—
83Dic236—42no—
84Dic236—42no—
85Dic236—42no—
86Dic236—42no—
87#11745660—42no—
88#11745660—42no—
89#11745660—42no—
90#11745660—42no—
91C16×C59—42no—
92C59⋊C16—42no—
93C23×C59#10011yescharacteristic
94C22×D59—41no—
95C2×C4×C59—21no—
96C2×C4×C59—21no—
97C2×C4×C59—21no—
98C2×C4×C59—21no—
99C2×C4×C59—42no—
100C2×C4×C59#10011yescharacteristic
101C2×C4×C59C2×C22⋊C411yescharacteristic
102C22⋊D59—82no—
103C2×Dic59—41no—
104C2×Dic59—41no—
105C2×Dic59—41no—
106C2×Dic59—82no—
107D236—41no—
108D236—41no—
109D236—82no—
110C4×D59—82no—
111Dic118—41no—
112Dic118—41no—
113Dic118—41no—
114Dic118—41no—
115Dic118—41no—
116Dic118—41no—
117Dic118—82no—
118C8×C59—21no—
119C8×C59—21no—
120C8×C59AGL(1,ℤ/8)12yes—
121C8×C59AGL(1,ℤ/8)12yes—
122#11993—591nomaximal, Sylow
123C22×C59—21no—
124C22×C59—41no—
125C22×C59C2×#10011yescharacteristic
126C2×D59—82no—
127C2×D59—81no—
128C4×C59—22no—
129C4×C59—21no—
130C4×C59—41no—
131C4×C59#41811yescharacteristic
132C4×C59#41811yescharacteristic
133Dic59—81no—
134Dic59—81no—
135Dic59—81no—
136#3152—591no—
137#2165—591no—
138#2165D5911yescharacteristic, complemented
139#1359—592no—
140#1358—592no—
141C2×C59—21no—
142C2×C59—41no—
143C2×C59#155511yescharacteristic
144D59—161no—
145C2×#137—1181no—
146C2×#137—1181no—
147#532—591no—
148C2×C4⋊Q8—591no—
149#497—1181no—
150#497—1181no—
151#494—591no—
152#494—591no—
153#492—591no—
154#492—591no—
155C4×M16C2×D5911yescharacteristic
156#364—592no—
157#364C2×D5912yes—
158#349—21no—
159#349—1181no—
160#347—21no—
161#347—1181no—
162C59#1199311yescharacteristic, complemented, Sylow
163C2×#41—1181no—
164C22×Q8—1181no—
165#137—2364no—
166#137—2364no—
167C2×Q16—1181no—
168C2×Q16—1181no—
169C2×Q16—1181no—
170C2×Q16—1181no—
171C2×SD16—1181no—
172C2×SD16—1181no—
173C2×SD16—1181no—
174C2×SD16—1181no—
175C2×M16D23611yescharacteristic
176C2×M16C22⋊D5911yescharacteristic
177C4⋊Q8—591no—
178C4⋊Q8—591no—
179C4⋊Q8—591no—
180C4⋊Q8—1181no—
181C4⋊Q8—2361no—
182#127—591no—
183#124—1181no—
184#121—2361no—
185C4×D4—2361no—
186C2×C4⋊C4—2361no—
187C2×C42C4×D5911yescharacteristic
188M32—22no—
189M32—1182no—
190#97D23611yescharacteristic
191#97C22⋊D5911yescharacteristic
192C4×C8C4×D5911yescharacteristic
193C4×C8C22×D5911yescharacteristic
194#41—2362no—
195#41—2362no—
196C2×Q8—1181no—
197C2×Q8—1181no—
198C2×Q8—1181no—
199C2×Q8—2362no—
200C2×Q8—2362no—
201C2×Q8—2361no—
202C2×D4—1181no—
203C2×D4—2361no—
204C2×D4—2361no—
205C22×C4—21no—
206C22×C4—2361no—
207C22×C4—2361no—
208C22×C4(C2×D59)⋊C411yescharacteristic
209Q16—2362no—
210Q16—2362no—
211Q16—2362no—
212Q16—2362no—
213SD16—2362no—
214SD16—2362no—
215SD16—2362no—
216SD16—2362no—
217M16—22no—
218M16—22no—
219C2×C8(C2×D59)⋊C411yescharacteristic
220C2×C8(C2×D59)⋊C411yescharacteristic
221C2×C8C2×D23611yescharacteristic
222C2×C8C2×C22⋊D5911yescharacteristic
223C4⋊C4—2361no—
224C4⋊C4—2361no—
225C4⋊C4—2361no—
226C4⋊C4—2362no—
227C22⋊C4—2362no—
228C42—21no—
229C42(C2×D59)⋊C411yescharacteristic
230C42C2×C4×D5911yescharacteristic
231C16—42no—
232C16—2362no—
233C23—2361no—
234C23#1427364311yescharacteristic
235Q8—4722no—
236Q8—2361no—
237Q8—2361no—
238Q8—2361no—
239Q8—2361no—
240Q8—2361no—
241Q8—2361no—
242D4—4722no—
243D4—4722no—
244D4—2361no—
245D4—2361no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—42no—
251C2×C4—4722no—
252C2×C4—4722no—
253C2×C4—2361no—
254C2×C4—2361no—
255C2×C4—2361no—
256C2×C4#1427364311yescharacteristic
257C2×C4C2×(C2×D59)⋊C411yescharacteristic
258C8—21no—
259C8—21no—
260C8#1427375212yes—
261C8#1427375512yes—
262C22—21no—
263C22—41no—
264C22—4722no—
265C22—4721no—
266C22C2×#1427364311yescharacteristic
267C4—22no—
268C4—21no—
269C4—41no—
270C4—4721no—
271C4—4721no—
272C4—4721no—
273C4#1683740611yescharacteristic
274C4#1683740811yescharacteristic
275C2—21no—
276C2—41no—
277C2—9441no—
278C2#2142280911yescharacteristic
subgroupG/H
centerC2#21422809
commutatorC42×C59C22×C4
FrattiniC4×C8C22×D59
FittingC59×#2165C2
socleC2×C59#1555
prime radical#16837952C4
cosocleC4×C8×C59C23
hypercenter#2165D59
hypocenterC59#11993
LCSG ▷ C42×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #2165

Complements

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

NH ≅ G/Ntype
2. #21425908C2semidirect
3. #21422136C2semidirect
4. #21422134C2semidirect
5. C59×#2165C2semidirect
138. #2165D59semidirect
162. C59#11993semidirect

Maximal subgroups

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

subgroupindex
#214345552
#214259082
#214221362
#214221342
C59×#21652
#1199359