Finite Group Databasebeta

← #27281716

Subgroups — #27281716

Order 13568.

434 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 99 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#20285264C211yesmaximal, characteristic
2#20273342C211yesmaximal, characteristic
3#20273226C212yesmaximal
4#20272608C212yesmaximal
5C53×#2371C211yesmaximal, characteristic
6#16811180C2211yescharacteristic
7#16811033C414yescomplemented
8#16811022—22no—
9#16810680C2212yes—
10#16810663—22no—
11C4×#11719917C2211yescharacteristic
12#16810365—22no—
13#16810359—22no—
14#16810359—22no—
15C2×#125×C53C2211yescharacteristic
16C53×#387—22no—
17C53×#387—22no—
18C4×C4⋊C4×C53—22no—
19C53×#376C2212yes—
20C2×#11719917—21no—
21#13133217C2×C412yes—
22#13133209—21no—
23#13133196—22no—
24C4×Dic106—22no—
25#13133176—21no—
26#13133150—24no—
27#13133148—24no—
28C4⋊#88322—22no—
29C22⋊#88322—22no—
30#13133128—22no—
31#13133128C2×C412yes—
32C4×#88322—21no—
33#125×C53—22no—
34#125×C53C2×C412yes—
35#125×C53D412yes—
36C2×C4⋊C4×C53—22no—
37C2×C4⋊C4×C53—22no—
38C2×C4⋊C4×C53D412yes—
39C2×C42×C53—22no—
40C2×C42×C53—22no—
41C2×C42×C53C2311yescharacteristic
42#95×C53—22no—
43#95×C53—22no—
44#95×C53—22no—
45#95×C53D412yes—
46#95×C53D412yes—
47C22×C4×C53—22no—
48C22×C4×C53—22no—
49C22×C4×C53—22no—
50C22×C4×C53C2×D411yescharacteristic
51C22×C4×C53C2×D412yes—
52C2×Dic106—22no—
53C4⋊C4×C53—24no—
54C4⋊C4×C53—24no—
55C4⋊C4×C53—22no—
56C4⋊C4×C53—44no—
57C4⋊C4×C53C22⋊C414yes—
58C42×C53—44no—
59C42×C53—44no—
60C42×C53C22×C411yescharacteristic
61C42×C53C2×D411yescharacteristic
62C42×C53#4112yes—
63C22⋊Dic53—41no—
64C4⋊Dic53—22no—
65C4⋊Dic53—22no—
66#11719919—42no—
67C4×Dic53—41no—
68#11719917—24no—
69C2×#88322—22no—
70C2×#88322—42no—
71C23×C53#12011yescharacteristic
72C2×C4×C53—22no—
73C2×C4×C53—24no—
74C2×C4×C53—24no—
75C2×C4×C53—42no—
76C2×C4×C53—42no—
77C2×C4×C53—42no—
78C2×C4×C53—42no—
79C2×C4×C53—42no—
80C2×C4×C53—42no—
81C2×C4×C53—42no—
82C2×C4×C53—42no—
83C2×C4×C53C2×C22⋊C411yescharacteristic
84C2×C4×C53C4×D412yes—
85C2×C4×C53#12112yes—
86C2×C4×C53#12311yescharacteristic
87C2×Dic53—42no—
88Dic106—82no—
89#88322—24no—
90#88322—82no—
91#9440—531nomaximal, Sylow
92C22×C53—22no—
93C22×C53#38611yescharacteristic
94C22×C53#45612yes—
95C4×C53—21no—
96C4×C53—21no—
97C4×C53—22no—
98C4×C53—22no—
99C4×C53—44no—
100C4×C53—44no—
101C4×C53—82no—
102Dic53—82no—
103#3166—531no—
104#2371D5311yescharacteristic
105#1534—531no—
106#1506—532no—
107#1300—532no—
108C2×C53—22no—
109C2×C53#188611yescharacteristic
110C2×C53#190612yes—
111#533—531no—
112C2×#125C2×D5311yescharacteristic
113#501—534no—
114#499—1062no—
115#432—532no—
116#428—1062no—
117C4×M16—531no—
118#387—22no—
119#387—22no—
120C4×C4⋊C4—22no—
121#376C2×D5312yes—
122#330—1062no—
123#328—1062no—
124#328—1062no—
125C53#944011yescharacteristic, complemented, Sylow
126C2×M16—1061no—
127C4⋊Q8—532no—
128#125—22no—
129#125—1061no—
130#125C4×D5312yes—
131#125D21212yes—
132C22⋊Q8—1062no—
133C4×Q8—1062no—
134#117—1061no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4C22⋊D5312yes—
138C2×C42—22no—
139C2×C42—22no—
140C2×C42C22×D5311yescharacteristic
141#107—1064no—
142#106—1064no—
143C4⋊C8—1062no—
144C22⋊C8—1062no—
145#97—532no—
146#97—1062no—
147C4×C8—1061no—
148#95—22no—
149#95—22no—
150#95—22no—
151#95C22⋊D5312yes—
152#95C22⋊D5312yes—
153C2×Q8—1062no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4C2×C22⋊D5311yescharacteristic
158C22×C4C2×C22⋊D5312yes—
159M16—1064no—
160C2×C8—1062no—
161C2×C8—2122no—
162C4⋊C4—24no—
163C4⋊C4—24no—
164C4⋊C4—22no—
165C4⋊C4—44no—
166C4⋊C4—1062no—
167C4⋊C4—1062no—
168C4⋊C4—2122no—
169C4⋊C4#1171992114yes—
170C22⋊C4—2121no—
171C42—44no—
172C42—44no—
173C42—2121no—
174C42C2×C4×D5311yescharacteristic
175C42C2×D21211yescharacteristic
176C42#1171994512yes—
177C23#1313318611yescharacteristic
178Q8—4242no—
179C2×C4—22no—
180C2×C4—24no—
181C2×C4—24no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—2122no—
191C2×C4C2×#1171992111yescharacteristic
192C2×C4C4×C22⋊D5312yes—
193C2×C4#1313319112yes—
194C2×C4#1313320011yescharacteristic
195C8—1064no—
196C8—4242no—
197C22—22no—
198C22#1681050211yescharacteristic
199C22#1681080612yes—
200C4—22no—
201C4—22no—
202C4—21no—
203C4—21no—
204C4—44no—
205C4—44no—
206C4—82no—
207C4—4242no—
208C2—22no—
209C2#2027510811yescharacteristic
210C2#2027521612yes—
subgroupG/H
centerC22#16810502
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2371C2
socleC22×C53#386
prime radicalC23×C53#120
cosocleC2×C42×C53C23
hypercenter#2371D53
hypocenterC53#9440
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2371

Complements

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

NH ≅ G/Ntype
7. #16811033C4semidirect
125. C53#9440semidirect

Maximal subgroups

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

subgroupindex
#202852642
#202733422
#202732262
#202726082
C53×#23712
#944053