Finite Group Databasebeta

← #28500193

Subgroups — #28500193

Order 15104.

300 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 83 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#21434680C211yesmaximal, characteristic
2#21425850C211yesmaximal, characteristic
3#21422858C211yesmaximal, characteristic
4#21422155C211yesmaximal, characteristic, complemented
5#21422153C211yesmaximal, characteristic, complemented
6#21422145C211yesmaximal, characteristic, complemented
7C59×#1362C211yesmaximal, characteristic, complemented
8#16837957—21no—
9C2×#14273666—21no—
10#16837807C2211yescharacteristic
11#16837752—21no—
12#16837640—21no—
13#16837640C412yes—
14Q16×Dic59—21no—
15Q16×Dic59C412yes—
16#16837458—21no—
17C4×C8×D59C2211yescharacteristic
18C4⋊(C59⋊C16)C2211yescharacteristic
19Dic59⋊C16—21no—
20#16837155C2211yescharacteristic
21#16837152—21no—
22C59×#494C2211yescharacteristic
23C59×C4⋊C16C2211yescharacteristic
24C59×#335C2211yescharacteristic
25C2×C8×D59C2×C411yescharacteristic
26C2×C8×D59D411yescharacteristic
27#14273727—21no—
28#14273718—41no—
29#14273707—41no—
30#14273703—22no—
31Q8×Dic59—22no—
32#14273692—41no—
33#14273685—41no—
34C42×D59D411yescharacteristic
35#14273666—42no—
36C2×C59⋊C16—21no—
37#14273661—21no—
38#14273661C2×C412yes—
39C4⋊(C59⋊C8)—21no—
40#14273651—22no—
41#14273651—22no—
42(C2×D59)⋊C8—41no—
43#14273637—41no—
44C4×C59⋊C8D411yescharacteristic
45C8×Dic59C2×C411yescharacteristic
46C8×Dic59D411yescharacteristic
47C2×Q16×C59—21no—
48C2×Q16×C59C2×C412yes—
49C4⋊Q8×C59—21no—
50C2×C16×C59—21no—
51C4⋊C8×C59—21no—
52C4×C8×C59C2311yescharacteristic
53C2×Q8×C59—22no—
54C2×C4×D59C22⋊C411yescharacteristic
55C2×C4×D59D811yescharacteristic
56C2×C4×D59SD1611yescharacteristic
57C2×Dic118—41no—
58Q16×C59—42no—
59Q16×C59—41no—
60Q16×C59—41no—
61C2×C8×C59—41no—
62C2×C8×C59C22×C411yescharacteristic
63C2×C8×C59C2×D411yescharacteristic
64C4⋊C4×C59—41no—
65C4⋊C4×C59—41no—
66C42×C59C2×D411yescharacteristic
67(C2×D59)⋊C4—42no—
68C4⋊Dic59—22no—
69C4⋊Dic59—41no—
70C4⋊Dic59—42no—
71Dic59⋊C4—42no—
72C4×Dic59—42no—
73C4×Dic59C22⋊C411yescharacteristic
74C4×Dic59D811yescharacteristic
75C4×Dic59SD1611yescharacteristic
76C2×C59⋊C8—41no—
77C2×C59⋊C8C22⋊C411yescharacteristic
78C2×C59⋊C8C22⋊C411yescharacteristic
79C8×D59—21no—
80C8×D59—21no—
81C8×D59—41no—
82C16×C59—41no—
83C59⋊C16—41no—
84C22×D59#10211yescharacteristic
85Q8×C59—42no—
86Q8×C59—42no—
87C2×C4×C59—42no—
88C2×C4×C59C2×C22⋊C411yescharacteristic
89C2×C4×C59C2×D811yescharacteristic
90C2×C4×C59C2×SD1611yescharacteristic
91C2×Dic59—42no—
92C2×Dic59#10211yescharacteristic
93C2×Dic59#10211yescharacteristic
94C2×Dic59#10211yescharacteristic
95C4×D59—21no—
96C4×D59—21no—
97C4×D59—21no—
98C4×D59—21no—
99C4×D59—41no—
100Dic118—82no—
101C8×C59—21no—
102C8×C59—81no—
103C8×C59AGL(1,ℤ/8)11yescharacteristic
104C8×C59#13711yescharacteristic
105C59⋊C8—21no—
106C59⋊C8—21no—
107C59⋊C8—81no—
108#11987—591nomaximal, Sylow
109C22×C59C2×#10211yescharacteristic
110C2×D59—21no—
111C2×D59—21no—
112C2×D59—21no—
113C4×C59—21no—
114C4×C59—82no—
115C4×C59#41311yescharacteristic
116C4×C59#41711yescharacteristic
117C4×C59#51011yescharacteristic
118C4×C59#51011yescharacteristic
119Dic59—21no—
120Dic59—21no—
121Dic59—21no—
122Dic59—82no—
123#3170—591no—
124#2150—591no—
125#1568—591no—
126#1365—591no—
127#1365—591no—
128#1362—591no—
129#1362D5911yescharacteristic, complemented
130C2×C59#154711yescharacteristic
131C2×C59#214111yescharacteristic
132C2×C59#214811yescharacteristic
133D59—41no—
134#533—1181no—
135C2×M32—1181no—
136#500—591no—
137#494C2×D5911yescharacteristic
138#487—1181no—
139#467—592no—
140#467—1181no—
141C4×Q16—592no—
142C4×Q16—1181no—
143#432—1181no—
144C2×C4×C8—591no—
145C4⋊C16—591no—
146C4⋊C16—1181no—
147C4⋊C16C2×D5911yescharacteristic
148#335—591no—
149#335C2×D5911yescharacteristic
150#334—1181no—
151C59#1198711yescharacteristic, complemented, Sylow
152C2×Q16—21no—
153C2×Q16C4×D5912yes—
154C22×C8—591no—
155C22×C8—591no—
156C4⋊Q8—21no—
157C4⋊Q8—1181no—
158#125—2361no—
159C22⋊Q8—1182no—
160C22⋊Q8—2361no—
161C4×Q8—1182no—
162C4×Q8—2361no—
163#117—2361no—
164C2×C42—591no—
165M32—2362no—
166C2×C16—21no—
167C2×C16—1181no—
168#107—592no—
169#107—1181no—
170C4⋊C8—21no—
171C4⋊C8—1181no—
172#103—1182no—
173#103—1182no—
174C22⋊C8—2361no—
175#97—2361no—
176C4×C8—591no—
177C4×C8—591no—
178C4×C8—591no—
179C4×C8C22×D5911yescharacteristic
180C2×Q8—22no—
181C2×Q8—2361no—
182C22×C4—591no—
183C22×C4—591no—
184C22×C4—591no—
185Q16—42no—
186Q16—41no—
187Q16—41no—
188C2×C8—41no—
189C2×C8—591no—
190C2×C8—591no—
191C2×C8—1181no—
192C2×C8—1181no—
193C2×C8—2361no—
194C2×C8—2361no—
195C2×C8C2×C4×D5911yescharacteristic
196C2×C8D4×D5911yescharacteristic
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—1182no—
200C4⋊C4—2362no—
201C4⋊C4—2362no—
202C4⋊C4—2361no—
203C22⋊C4—2362no—
204C42—591no—
205C42—591no—
206C42—591no—
207C42—2362no—
208C42D4×D5911yescharacteristic
209C16—41no—
210C16—2361no—
211C23—591no—
212Q8—42no—
213Q8—42no—
214Q8—4722no—
215C2×C4—42no—
216C2×C4—591no—
217C2×C4—591no—
218C2×C4—591no—
219C2×C4—1181no—
220C2×C4—1181no—
221C2×C4—1181no—
222C2×C4—1181no—
223C2×C4—2361no—
224C2×C4—2362no—
225C2×C4C22⋊C4×D5911yescharacteristic
226C2×C4D8×D5911yescharacteristic
227C2×C4SD16×D5911yescharacteristic
228C8—21no—
229C8—81no—
230C8—1181no—
231C8—1181no—
232C8—4721no—
233C8#1427375111yescharacteristic
234C8#1427375611yescharacteristic
235C22—1181no—
236C22—1181no—
237C22—1181no—
238C22#102×D5911yescharacteristic
239C4—21no—
240C4—82no—
241C4—1181no—
242C4—1181no—
243C4—1181no—
244C4—4722no—
245C4#1683738311yescharacteristic
246C4#1683740011yescharacteristic
247C4#1683784111yescharacteristic
248C4#1683784411yescharacteristic
249C2—2361no—
250C2#2142277011yescharacteristic
251C2#2142581411yescharacteristic
252C2#2142584411yescharacteristic
subgroupG/H
centerC22#102×D59
commutatorC2×C8×C59C22×C4
FrattiniC4×C8C22×D59
FittingC59×#1362C2
socleC22×C59C2×#102
prime radicalC22×D59#102
cosocleC4×C8×C59C23
hypercenter#1362D59
hypocenterC59#11987
LCSG ▷ C2×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1362

Complements

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

NH ≅ G/Ntype
4. #21422155C2semidirect
5. #21422153C2semidirect
6. #21422145C2semidirect
7. C59×#1362C2semidirect
129. #1362D59semidirect
151. C59#11987semidirect

Maximal subgroups

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

subgroupindex
#214346802
#214258502
#214228582
#214221552
#214221532
#214221452
C59×#13622
#1198759