Finite Group Databasebeta

← #28735640

Subgroups — #28735640

Order 15104.

984 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 984 conjugacy classes of subgroups — too many to draw. Only its 175 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#21432383C212yesmaximal
2#21425524C213yesmaximal, complemented
3#21425300C216yesmaximal, complemented
4#21425106C213yesmaximal, complemented
5C59×#1779C211yesmaximal, characteristic
6C22×Dic236C2211yescharacteristic, complemented
7C2×#14273706C2216yes—
8C2×#14273661C2213yescomplemented
9C2×#14273652C2216yescomplemented
10#16837275—26no—
11#16837262C2216yes—
12#16837258—22no—
13#16837165C2213yescomplemented
14#16837165C2213yescomplemented
15C24×C4×C59C2211yescharacteristic
16C2×C22⋊C8×C59C2213yes—
17C2×C22⋊C8×C59C2213yes—
18C22×Dic118C2312yes—
19C2×Dic236—43no—
20#14273706—46no—
21#14273706—46no—
22C2×C4⋊Dic59C2316yes—
23C2×Dic59⋊C4—26no—
24C2×C22⋊Dic59—26no—
25C2×C22⋊Dic59—26no—
26#14273661—43no—
27#14273652—46no—
28C59⋊#95—26no—
29C25×C59D411yescharacteristic
30C23×C4×C59—43no—
31C23×C4×C59—43no—
32C23×C4×C59D413yes—
33C23×C4×C59C2313yes—
34C22×C8×C59C2313yes—
35C22×C8×C59C2311yescharacteristic
36C22⋊C8×C59—43no—
37C22⋊C8×C59—43no—
38C24×C59—43no—
39C24×C59—43no—
40C24×C59Q1614yescomplemented
41C24×C59C2×D413yes—
42C22×C4×C59—23no—
43C22×C4×C59—23no—
44C22×C4×C59—26no—
45C22×C4×C59—26no—
46C22×C4×C59—43no—
47C22×C4×C59—46no—
48C22×C4×C59—412no—
49C22×C4×C59—44no—
50C22×C4×C59C2×D413yes—
51C22×C4×C59C2411yescharacteristic
52C22×Dic59—26no—
53C22×Dic59—22no—
54C2×Dic118—26no—
55C2×Dic118—26no—
56C2×C8×C59—26no—
57C2×C8×C59—23no—
58C2×C8×C59—23no—
59C22⋊Dic59—412no—
60C22⋊Dic59—412no—
61C4⋊Dic59—26no—
62C4⋊Dic59—26no—
63Dic59⋊C4—412no—
64Dic236—44no—
65C23×C59—26no—
66C23×C59—23no—
67C23×C59—26no—
68C23×C59—23no—
69C23×C59—412no—
70C23×C59—44no—
71C23×C59—412no—
72C23×C59C2×Q1616yes—
73C23×C59C22×D411yescharacteristic
74C2×C4×C59—26no—
75C2×C4×C59—23no—
76C2×C4×C59—412no—
77C2×C4×C59—46no—
78C2×C4×C59—43no—
79C2×C4×C59—43no—
80C2×C4×C59H(5,𝔽2)13yes—
81C2×C4×C59H(5,𝔽2)13yes—
82C2×Dic59—46no—
83C2×Dic59—46no—
84C2×Dic59—412no—
85Dic118—46no—
86Dic118—42no—
87C8×C59—43no—
88C8×C59—41no—
89#32710—591nomaximal, Sylow
90C22×C59—212no—
91C22×C59—26no—
92C22×C59—416no—
93C22×C59—46no—
94C22×C59—43no—
95C22×C59—43no—
96C22×C59#53413yes—
97C22×C59C22×Q1611yescharacteristic
98C22×C59#57513yes—
99C4×C59—43no—
100C4×C59#56013yes—
101C4×C59H(3,𝔽4)11yescharacteristic
102Dic59—86no—
103Dic59—82no—
104#2848—592no—
105#2076—593no—
106#2036—596no—
107#2001—593no—
108#1779D5911yescharacteristic
109C2×C59—46no—
110C2×C59#281211yescharacteristic
111C2×C59#318913yes—
112C2×C59#319713yes—
113C24×C4C2×D5911yescharacteristic
114C22×Q16—591no—
115C2×C22⋊Q8—596no—
116C2×#107—593no—
117C2×#103—596no—
118C2×C22⋊C8C2×D5913yes—
119C2×C22⋊C8C2×D5913yes—
120#385—1186no—
121#380—596no—
122#379—1182no—
123#340—593no—
124#340—593no—
125C59#3271011yescharacteristic, complemented, Sylow
126C25D23611yescharacteristic
127C22×Q8—592no—
128C23×C4—43no—
129C23×C4—43no—
130C23×C4D23613yes—
131C23×C4C22×D5913yes—
132C2×Q16—2363no—
133C22×C8C22×D5911yescharacteristic
134C22×C8C22×D5913yes—
135C22⋊Q8—2366no—
136C22⋊Q8—2366no—
137C2×C4⋊C4—596no—
138C2×C4⋊C4—1186no—
139C2×C22⋊C4—1186no—
140C2×C22⋊C4—1186no—
141#107—2363no—
142#103—2366no—
143C22⋊C8—43no—
144C22⋊C8—43no—
145#95—1186no—
146C24—43no—
147C24—43no—
148C24Dic23614yescomplemented
149C24C2×D23613yes—
150C2×Q8—1186no—
151C2×Q8—1186no—
152C22×C4—23no—
153C22×C4—23no—
154C22×C4—26no—
155C22×C4—26no—
156C22×C4—43no—
157C22×C4—46no—
158C22×C4—44no—
159C22×C4—412no—
160C22×C4—1186no—
161C22×C4—1182no—
162C22×C4C2×D23613yes—
163C22×C4C23×D5911yescharacteristic
164Q16—2364no—
165C2×C8—26no—
166C2×C8—23no—
167C2×C8—23no—
168C4⋊C4—1186no—
169C4⋊C4—1186no—
170C4⋊C4—23612no—
171C22⋊C4—23612no—
172C22⋊C4—23612no—
173C23—26no—
174C23—23no—
175C23—26no—
176C23—23no—
177C23—412no—
178C23—44no—
179C23—412no—
180C23C2×Dic23616yes—
181C23C22×D23611yescharacteristic
182Q8—2362no—
183Q8—2366no—
184C2×C4—26no—
185C2×C4—23no—
186C2×C4—412no—
187C2×C4—43no—
188C2×C4—43no—
189C2×C4—46no—
190C2×C4—23612no—
191C2×C4—2366no—
192C2×C4—2366no—
193C2×C4#1427377213yes—
194C2×C4#1427377213yes—
195C8—41no—
196C8—43no—
197C22—212no—
198C22—26no—
199C22—416no—
200C22—43no—
201C22—43no—
202C22—46no—
203C22#1683796413yes—
204C22C22×Dic23611yescharacteristic
205C22#1683820213yes—
206C4—43no—
207C4—4722no—
208C4—4726no—
209C4#1683814113yes—
210C4#1683814511yescharacteristic
211C2—46no—
212C2#2143209311yescharacteristic
213C2#2143487113yes—
214C2#2143493813yes—
subgroupG/H
centerC23C22×D236
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#1779C2
socleC23×C59C22×D4
prime radicalC25×C59D4
hypercenter#1779D59
hypocenterC59#32710
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C24×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
2. #21425524C2semidirect
3. #21425300C2semidirect
4. #21425106C2semidirect
6. C22×Dic236C22semidirect
8. C2×#14273661C22semidirect
9. C2×#14273652C22semidirect
13. #16837165C22semidirect
14. #16837165C22semidirect
40. C24×C59Q16semidirect
125. C59#32710semidirect
148. C24Dic236semidirect

Maximal subgroups

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

subgroupindex
#214323832
#214255242
#214253002
#214251062
C59×#17792
#3271059