Finite Group Databasebeta

← #28309602

Subgroups — #28309602

Order 13568.

296 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 62 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#20291090C211yesmaximal, characteristic, complemented
2#20290937C211yesmaximal, characteristic, complemented
3#20289717C214yesmaximal, complemented
4#20285362C211yesmaximal, characteristic
5D53⋊M32—21no—
6#16811545—21no—
7#16811520—21no—
8#16811516—21no—
9#16811515C2212yes—
10#16811515C2212yes—
11#16811227C412yes—
12#16811025C411yescharacteristic
13#16810987C2212yes—
14#16810873—21no—
15#16810856—21no—
16#16810735—21no—
17D8×Dic53—21no—
18#16810577C2211yescharacteristic
19C53×#493C411yescharacteristic
20#13133292—22no—
21#13133291—22no—
22#13133291—22no—
23D53⋊C16—22no—
24C2×#11719912D411yescharacteristic
25C2×C8×D53D411yescharacteristic
26#13133209C2×C412yes—
27#13133198—42no—
28#13133189—42no—
29#13133188—22no—
30D4×Dic53—22no—
31#13133174C2×C411yescharacteristic
32#13133151—21no—
33#13133149—21no—
34#13133138—22no—
35#13133138—22no—
36#13133128C2311yescharacteristic
37#13133127D411yescharacteristic
38C8×Dic53D411yescharacteristic
39C2×D8×C53—21no—
40C2×D8×C53—21no—
41#127×C53C2×C412yes—
42C4×C8×C53C2×C411yescharacteristic
43C2×D4×C53—22no—
44C2×D4×C53—42no—
45C2×C22⋊D53—42no—
46C22×Dic53—42no—
47C2×C4×D53C22⋊C411yescharacteristic
48D8×C53—42no—
49D8×C53—41no—
50D8×C53—41no—
51C2×C8×C53C22⋊C411yescharacteristic
52C2×C8×C53C22⋊C411yescharacteristic
53C42×C53C22×C411yescharacteristic
54C22⋊Dic53—42no—
55C22⋊Dic53—42no—
56#11719921—21no—
57C4⋊Dic53—22no—
58#11719919—21no—
59#11719919—42no—
60C4×Dic53C22⋊C411yescharacteristic
61C2×#88322C2×D411yescharacteristic
62C2×#88322C2×D411yescharacteristic
63#11719912—41no—
64C8×D53—21no—
65C8×D53—21no—
66#11719910—24no—
67#11719910—42no—
68C23×C53—42no—
69C22×D53#10011yescharacteristic
70D4×C53—42no—
71D4×C53—42no—
72D4×C53—162no—
73C2×C4×C53—21no—
74C2×C4×C53C2×C22⋊C411yescharacteristic
75C22⋊D53—162no—
76C2×Dic53—21no—
77C2×Dic53—42no—
78C2×Dic53—162no—
79C2×Dic53#10011yescharacteristic
80C4×D53—21no—
81C4×D53—21no—
82C8×C53—21no—
83C8×C53#10411yescharacteristic
84C8×C53#10411yescharacteristic
85#88322—21no—
86#88322—21no—
87#11913—531nomaximal, Sylow
88C22×C53—82no—
89C22×C53—82no—
90C22×C53—82no—
91C22×C53C2×#10011yescharacteristic
92C2×D53—21no—
93C2×D53—41no—
94C4×C53—41no—
95C4×C53C2×#10411yescharacteristic
96C4×C53#42111yescharacteristic
97Dic53—21no—
98Dic53—41no—
99Dic53—82no—
100#3176—531no—
101#2166—531no—
102#2110—531no—
103#1355—534no—
104C2×C53—21no—
105C2×C53—162no—
106C2×C53#153011yescharacteristic
107D53—41no—
108#540—532no—
109C2×M32—1061no—
110#499—531no—
111#493#615711yescharacteristic
112#490—532no—
113#469—1061no—
114#466—1061no—
115#442—1061no—
116C4×D8—1061no—
117#405—531no—
118#364—1061no—
119#349—1061no—
120#347—532no—
121#347—532no—
122#347—1061no—
123C53#1191311yescharacteristic, complemented, Sylow
124C2×D8—21no—
125C2×D8—21no—
126C2×M16—531no—
127C22×C8—531no—
128#127C2×#615712yes—
129#125—532no—
130#123—2122no—
131#121—1062no—
132#121—2122no—
133C4×D4—1062no—
134#117—531no—
135M32—1062no—
136M32—1062no—
137M32—1062no—
138C2×C16—1062no—
139#107—1061no—
140#106—1061no—
141#102—1062no—
142#102—1062no—
143#97—531no—
144#97—531no—
145C4×C8—531no—
146C4×C8C2×#615711yescharacteristic
147C2×D4—22no—
148C2×D4—42no—
149C2×D4—2122no—
150C22×C4—531no—
151C22×C4—2122no—
152D8—42no—
153D8—41no—
154D8—41no—
155M16—2121no—
156C2×C8—531no—
157C2×C8—531no—
158C2×C8—1061no—
159C2×C8—1061no—
160C2×C8C22⋊#615711yescharacteristic
161C2×C8C22⋊#615711yescharacteristic
162C4⋊C4—1061no—
163C4⋊C4—1062no—
164C4⋊C4—2122no—
165C22⋊C4—1061no—
166C22⋊C4—2122no—
167C22⋊C4—2122no—
168C42—531no—
169C42C22×#615711yescharacteristic
170C16—1064no—
171C16—2122no—
172C23—42no—
173C23—531no—
174D4—42no—
175D4—42no—
176D4—162no—
177D4—8482no—
178C2×C4—21no—
179C2×C4—531no—
180C2×C4—8482no—
181C2×C4—1061no—
182C2×C4—1061no—
183C2×C4—1061no—
184C2×C4—2122no—
185C2×C4C2×C22⋊#615711yescharacteristic
186C8—21no—
187C8—1061no—
188C8—1061no—
189C8#1313328111yescharacteristic
190C8#1313328111yescharacteristic
191C22—82no—
192C22—82no—
193C22—82no—
194C22—1061no—
195C22—2121no—
196C22D53⋊#10011yescharacteristic
197C4—41no—
198C4—1061no—
199C4—4242no—
200C4—2121no—
201C4C2×#1313328111yescharacteristic
202C4#1681162111yescharacteristic
203C2—21no—
204C2—162no—
205C2—2121no—
206C2#2029018511yescharacteristic
subgroupG/H
centerC2#20290185
commutatorC42×C53C22×C4
FrattiniC42C22×#6157
FittingC53×#493C4
socleC2×C53#1530
prime radical#20285362C2
cosocle#13133128C23
hypercenter#493#6157
hypocenterC53#11913
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #493

Complements

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

NH ≅ G/Ntype
1. #20291090C2semidirect
2. #20290937C2semidirect
3. #20289717C2semidirect
123. C53#11913semidirect

Maximal subgroups

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

subgroupindex
#202910902
#202909372
#202897172
#202853622
#1191353