Finite Group Databasebeta

← #27489996

Subgroups — #27489996

Order 13568.

974 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 974 conjugacy classes of subgroups — too many to draw. Only its 187 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#20278998C213yesmaximal
2#20276067C214yesmaximal
3#20275881C214yesmaximal
4#20274792C213yesmaximal
5C53×#3458C211yesmaximal, characteristic
6C22×#11719919C2211yescharacteristic
7#16810561—23no—
8#16810554—23no—
9#16810507—26no—
10#16810495C2216yes—
11#16810486C2213yes—
12#16810484C2216yes—
13#16810390C22112yes—
14C53×#544—23no—
15C53×#544—23no—
16C53×#543C2214yes—
17C53×#533—26no—
18C2×#117×C53C2213yes—
19C23×Dic53C2311yescharacteristic
20C23×Dic53C2311yescharacteristic
21C2×#11719919—23no—
22C2×#11719919—26no—
23C2×#11719919—23no—
24C2×C22⋊Dic53C2316yes—
25#13133124—23no—
26#13133124—26no—
27#13133124—26no—
28#13133124—26no—
29#13133124—23no—
30C23×C4×C53C2311yescharacteristic
31C4⋊Q8×C53—26no—
32C4⋊Q8×C53—26no—
33#125×C53—212no—
34C22⋊Q8×C53—212no—
35C22⋊Q8×C53—212no—
36C4×Q8×C53—24no—
37C4×Q8×C53—24no—
38#117×C53—26no—
39#117×C53—26no—
40C2×C4⋊C4×C53D413yes—
41C2×C4⋊C4×C53D413yes—
42C2×C22⋊C4×C53C2316yes—
43C2×C42×C53D416yes—
44C24×C53C2411yescharacteristic
45C2×Q8×C53—24no—
46C2×Q8×C53—24no—
47C22×C4×C53—24no—
48C22×C4×C53C2×D413yes—
49C22×C4×C53C2×D413yes—
50C22×C4×C53C2×D416yes—
51C22×C4×C53C2×D416yes—
52C22×Dic53—23no—
53C22×Dic53—26no—
54C22×Dic53—23no—
55C22×Dic53—24no—
56C22×Dic53—42no—
57C4⋊C4×C53—212no—
58C4⋊C4×C53—212no—
59C4⋊C4×C53—26no—
60C4⋊C4×C53—26no—
61C22⋊C4×C53—212no—
62C42×C53—26no—
63C42×C53—26no—
64C22⋊Dic53—212no—
65#11719919—412no—
66#11719919—41no—
67#11719919—43no—
68C23×C53—41no—
69C23×C53—41no—
70C23×C53C22×D413yes—
71C23×C532−1+414yes—
72Q8×C53—84no—
73Q8×C53—84no—
74C2×C4×C53—26no—
75C2×C4×C53—26no—
76C2×C4×C53—26no—
77C2×C4×C53—26no—
78C2×C4×C53—46no—
79C2×C4×C53—46no—
80C2×C4×C53#12011yescharacteristic
81C2×C4×C53#12013yes—
82C2×Dic53—26no—
83C2×Dic53—43no—
84C2×Dic53—41no—
85C2×Dic53—46no—
86C2×Dic53—46no—
87C2×Dic53—41no—
88C2×Dic53—46no—
89C2×Dic53—46no—
90#29659—531nomaximal, Sylow
91C22×C53—23no—
92C22×C53—23no—
93C22×C53—44no—
94C22×C53C2×#12011yescharacteristic
95C22×C53#53616yes—
96C4×C53—21no—
97C4×C53—23no—
98C4×C53—46no—
99C4×C53—46no—
100Dic53—42no—
101Dic53—86no—
102Dic53—81no—
103#3458D5311yescharacteristic
104#2484—533no—
105#2062—534no—
106#2030—534no—
107#1820—533no—
108C2×C53—26no—
109C2×C53#261411yescharacteristic
110C2×C53#302511yescharacteristic
111C2×C53#302811yescharacteristic
112#544—23no—
113#544—23no—
114#543C2×D5314yes—
115#533—26no—
116C2×#117C2×D5313yes—
117C22×C4⋊C4—531no—
118#400—1063no—
119#398—1063no—
120#387—1066no—
121#385—536no—
122#381—533no—
123#381—536no—
124#342—5312no—
125C53#2965911yescharacteristic, complemented, Sylow
126C23×C4—531no—
127C23×C4—531no—
128C23×C4C22×D5311yescharacteristic
129C4⋊Q8—26no—
130C4⋊Q8—26no—
131#125—212no—
132C22⋊Q8—212no—
133C22⋊Q8—212no—
134C4×Q8—24no—
135C4×Q8—24no—
136#117—26no—
137#117—26no—
138C2×C4⋊C4—1063no—
139C2×C4⋊C4—1063no—
140C2×C4⋊C4—1066no—
141C2×C4⋊C4C22⋊D5313yes—
142C2×C4⋊C4C22⋊D5313yes—
143C2×C22⋊C4—536no—
144C2×C22⋊C4C22×D5316yes—
145C2×C42C22⋊D5316yes—
146#95—1063no—
147#95—1066no—
148#95—1066no—
149#95—1063no—
150#95—1066no—
151C24C23×D5311yescharacteristic
152C2×Q8—24no—
153C2×Q8—24no—
154C22×C4—24no—
155C22×C4—1063no—
156C22×C4—1066no—
157C22×C4—1063no—
158C22×C4—1064no—
159C22×C4—2122no—
160C22×C4C2×C22⋊D5313yes—
161C22×C4C2×C22⋊D5313yes—
162C22×C4C2×C22⋊D5316yes—
163C22×C4C2×C22⋊D5316yes—
164C4⋊C4—212no—
165C4⋊C4—212no—
166C4⋊C4—26no—
167C4⋊C4—26no—
168C4⋊C4—21212no—
169C4⋊C4—2121no—
170C4⋊C4—2123no—
171C22⋊C4—212no—
172C22⋊C4—10612no—
173C42—26no—
174C42—26no—
175C23—41no—
176C23—41no—
177C23C22×C22⋊D5313yes—
178C23#1313326514yes—
179Q8—84no—
180Q8—84no—
181C2×C4—26no—
182C2×C4—26no—
183C2×C4—26no—
184C2×C4—26no—
185C2×C4—46no—
186C2×C4—46no—
187C2×C4—1066no—
188C2×C4—2126no—
189C2×C4—2126no—
190C2×C4—2121no—
191C2×C4—2126no—
192C2×C4—2126no—
193C2×C4—2123no—
194C2×C4—2121no—
195C2×C4#1313318611yescharacteristic
196C2×C4#1313318613yes—
197C22—23no—
198C22—23no—
199C22—44no—
200C22C2×#1313318611yescharacteristic
201C22#1681119716yes—
202C4—21no—
203C4—23no—
204C4—46no—
205C4—46no—
206C4—4246no—
207C4—4241no—
208C4—2122no—
209C2—26no—
210C2#2028043411yescharacteristic
211C2#2028410711yescharacteristic
212C2#2028412111yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, prime radical, cosocleC24×C53C24
FrattiniC24C23×D53
FittingC53×#3458C2
socleC22×C53C2×#120
hypercenter#3458D53
hypocenterC53#29659
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3458

Complements

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

NH ≅ G/Ntype
125. C53#29659semidirect

Maximal subgroups

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

subgroupindex
#202789982
#202760672
#202758812
#202747922
C53×#34582
#2965953