Finite Group Databasebeta

← #27295290

Subgroups — #27295290

Order 13568.

280 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 61 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#20285114C211yesmaximal, characteristic
2#20276525C212yesmaximal, complemented
3#20272749C212yesmaximal, complemented
4#20272748C211yesmaximal, characteristic, complemented
5C53×#1355C211yesmaximal, characteristic, complemented
6#16811171C411yescharacteristic
7#125×D53C411yescharacteristic
8#16810993C411yescharacteristic
9#16810992C2211yescharacteristic
10#16810987C411yescharacteristic
11#16810976—22no—
12#16810621—22no—
13#16810575C2211yescharacteristic
14#16810443—22no—
15#16810404—22no—
16#16810401C2212yes—
17C53×#490C2211yescharacteristic
18C53×#347C2212yes—
19C2×#11719947—21no—
20C2×#11719912D412yes—
21#13133217C2×C411yescharacteristic
22#13133209C2×C411yescharacteristic
23#13133208—21no—
24#13133207—41no—
25#13133204—21no—
26#13133188—41no—
27D4×Dic53—41no—
28C4⋊C4×D53—21no—
29C4⋊C4×D53—41no—
30C42×D53C2×C411yescharacteristic
31#13133157—22no—
32#13133142—22no—
33#13133140—22no—
34#13133138—22no—
35#13133128C2×C411yescharacteristic
36C8×Dic53D412yes—
37#127×C53C2×C411yescharacteristic
38#125×C53C2×C411yescharacteristic
39M32×C53—22no—
40#102×C53—22no—
41#97×C53C2311yescharacteristic
42C2×D4×C53—21no—
43C2×D4×C53—41no—
44C2×C22⋊D53—41no—
45C22×Dic53—41no—
46#11719947—44no—
47C2×C4×D53—21no—
48C2×C4×D53—41no—
49C2×C4×D53C22⋊C411yescharacteristic
50C2×Dic106—21no—
51C2×C8×C53C2×D412yes—
52C4⋊C4×C53—21no—
53C4⋊C4×C53—41no—
54C42×C53C22×C411yescharacteristic
55C22⋊Dic53—42no—
56C4⋊Dic53—21no—
57C4⋊Dic53—41no—
58C4⋊Dic53—41no—
59#11719919—41no—
60#11719919—41no—
61#11719919—41no—
62C4×Dic53—21no—
63C4×Dic53C22⋊C411yescharacteristic
64C2×#88322C22⋊C412yes—
65#11719912—42no—
66C16×C53—42no—
67#11719908—42no—
68C23×C53—41no—
69C22×D53#10011yescharacteristic
70D4×C53—42no—
71D4×C53—82no—
72C2×C4×C53—21no—
73C2×C4×C53—41no—
74C2×C4×C53C2×C22⋊C411yescharacteristic
75C22⋊D53—82no—
76C2×Dic53—21no—
77C2×Dic53—41no—
78C2×Dic53—41no—
79C2×Dic53—82no—
80C2×Dic53#10011yescharacteristic
81C4×D53—22no—
82C4×D53—81no—
83C4×D53—82no—
84Dic106—42no—
85C8×C53—22no—
86#88322—22no—
87#11917—531nomaximal, Sylow
88C22×C53—82no—
89C22×C53—81no—
90C22×C53C2×#10011yescharacteristic
91C2×D53—21no—
92C2×D53—41no—
93C4×C53—41no—
94C4×C53—81no—
95C4×C53#42112yes—
96Dic53—21no—
97Dic53—41no—
98Dic53—81no—
99Dic53—81no—
100#3142—531no—
101#2165—532no—
102#1356—532no—
103#1355—531no—
104#1355D5311yescharacteristic, complemented
105C2×C53—21no—
106C2×C53—161no—
107C2×C53#153311yescharacteristic
108D53—41no—
109#532—531no—
110C2×#125—531no—
111#491—531no—
112#491—531no—
113#490—531no—
114#490C2×D5311yescharacteristic
115#488—1062no—
116#417—1062no—
117C4×M16—531no—
118#364—1062no—
119#349—1062no—
120#347—532no—
121#347C2×D5312yes—
122C53#1191711yescharacteristic, complemented, Sylow
123C2×#41—1061no—
124C2×M16—532no—
125C4⋊Q8—531no—
126#127C4×D5311yescharacteristic
127#125—531no—
128#125—1061no—
129#125—2121no—
130#125C4×D5311yescharacteristic
131#124—1061no—
132#121—2121no—
133C4×D4—2121no—
134C2×C4⋊C4—1061no—
135C2×C4⋊C4—2121no—
136C2×C42—531no—
137M32—22no—
138M32—1062no—
139#103—1062no—
140#103—1062no—
141#102—22no—
142#102—1062no—
143#97—531no—
144#97C22×D5311yescharacteristic
145C4×C8—532no—
146#41—2124no—
147C2×Q8—1061no—
148C2×D4—21no—
149C2×D4—41no—
150C2×D4—2121no—
151C22×C4—531no—
152C22×C4—1061no—
153C22×C4—2121no—
154C22×C4—2121no—
155M16—2122no—
156C2×C8—532no—
157C2×C8D4×D5312yes—
158C4⋊C4—21no—
159C4⋊C4—41no—
160C4⋊C4—1061no—
161C4⋊C4—2121no—
162C4⋊C4—2121no—
163C4⋊C4—2121no—
164C4⋊C4—2121no—
165C4⋊C4—2121no—
166C22⋊C4—2122no—
167C42—531no—
168C42—1061no—
169C42C2×C4×D5311yescharacteristic
170C16—42no—
171C16—2122no—
172C23—41no—
173C23—531no—
174Q8—2122no—
175D4—42no—
176D4—82no—
177D4—4242no—
178C2×C4—21no—
179C2×C4—41no—
180C2×C4—531no—
181C2×C4—1062no—
182C2×C4—1061no—
183C2×C4—4241no—
184C2×C4—4242no—
185C2×C4—4242no—
186C2×C4—2121no—
187C2×C4—2121no—
188C2×C4C22⋊C4×D5311yescharacteristic
189C8—22no—
190C8—1062no—
191C22—82no—
192C22—81no—
193C22—1061no—
194C22—2121no—
195C22#100×D5311yescharacteristic
196C4—41no—
197C4—81no—
198C4—1061no—
199C4—4241no—
200C4—4241no—
201C4—2121no—
202C4#1681063812yes—
203C2—21no—
204C2—161no—
205C2—2121no—
206C2#2027333711yescharacteristic
subgroupG/H
centerC2#20273337
commutatorC42×C53C22×C4
Frattini#97C22×D53
FittingC53×#1355C2
socleC2×C53#1533
prime radical#16811171C4
cosocle#97×C53C23
hypercenter#1355D53
hypocenterC53#11917
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ #1355

Complements

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

NH ≅ G/Ntype
2. #20276525C2semidirect
3. #20272749C2semidirect
4. #20272748C2semidirect
5. C53×#1355C2semidirect
104. #1355D53semidirect
122. C53#11917semidirect

Maximal subgroups

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

subgroupindex
#202851142
#202765252
#202727492
#202727482
C53×#13552
#1191753