Finite Group Databasebeta

← #27284039

Subgroups — #27284039

Order 13568.

428 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 123 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#20285038C211yesmaximal, characteristic
2#20274853C211yesmaximal, characteristic, complemented
3#20273171C212yesmaximal, complemented
4#20273167C212yesmaximal
5C2×C82×C53C211yesmaximal, characteristic, complemented
6#16811180C2211yescharacteristic
7#16811173C2211yescharacteristic
8#16810978—22no—
9#16810974C412yes—
10#16810970C412yes—
11#16810647C2212yes—
12#16810620—22no—
13#16810388—22no—
14#16810355—24no—
15#16810352—24no—
16C2×C4×C8×C53C2212yes—
17C2×C4×C8×C53C2211yescharacteristic
18C82×C53—22no—
19C2×#11719945—21no—
20C2×#11719917—22no—
21#13133217C2×C411yescharacteristic
22#13133217C2×C412yes—
23#13133214C2×C411yescharacteristic
24#13133209—21no—
25#13133205—21no—
26#13133196—22no—
27#13133191—22no—
28C4×Dic106—22no—
29C4×D212—22no—
30#13133176—21no—
31C4⋊#88322—22no—
32C4⋊#88322—22no—
33#13133142—24no—
34#13133139—24no—
35C22×C8×C53—22no—
36C22×C8×C53D412yes—
37C2×C42×C53C2311yescharacteristic
38C4×C8×C53—22no—
39C4×C8×C53C2×C412yes—
40C4×C8×C53D414yes—
41C4×C8×C53D412yes—
42C22×C4×C53C2×D411yescharacteristic
43C22×C4×C53#4112yes—
44C2×C22⋊D53—41no—
45#11719945—44no—
46C2×D212—21no—
47C2×D212—22no—
48C2×C4×D53—41no—
49C2×Dic106—21no—
50C2×Dic106—22no—
51C2×C8×C53—28no—
52C2×C8×C53—24no—
53C2×C8×C53—24no—
54C2×C8×C53C22⋊C414yes—
55C42×C53C22×C411yescharacteristic
56C42×C53C2×D411yescharacteristic
57C42×C53C2×D412yes—
58C22⋊Dic53—41no—
59#11719921—42no—
60C4⋊Dic53—22no—
61C4⋊Dic53—22no—
62C4⋊Dic53—22no—
63#11719919—42no—
64C4×Dic53—41no—
65#11719917—44no—
66C2×#88322—42no—
67C23×C53#12411yescharacteristic
68C22×D53—41no—
69C2×C4×C53—24no—
70C2×C4×C53C2×C22⋊C411yescharacteristic
71C2×C4×C53C4×D412yes—
72C2×C4×C53#12112yes—
73C2×C4×C53#12711yescharacteristic
74C2×C4×C53#13514yes—
75C22⋊D53—82no—
76C2×Dic53—41no—
77C2×Dic53—42no—
78D212—42no—
79D212—44no—
80C4×D53—82no—
81Dic106—42no—
82Dic106—44no—
83C8×C53—28no—
84C8×C53—24no—
85#88322—82no—
86#9857—531nomaximal, Sylow
87C22×C53—22no—
88C22×C53#39011yescharacteristic
89C22×C53#48812yes—
90C2×D53—81no—
91C2×D53—82no—
92C4×C53—22no—
93C4×C53#41612yes—
94C4×C53#44314yes—
95C4×C53#49512yes—
96Dic53—81no—
97Dic53—82no—
98#3131—531no—
99#1837—531no—
100#1493—532no—
101#1492—532no—
102C2×C82D5311yescharacteristic, complemented
103C2×C53—22no—
104C2×C53#195811yescharacteristic
105C2×C53#197412yes—
106D53—161no—
107#533—531no—
108#532—531no—
109#488—1062no—
110#487—532no—
111#486—532no—
112#423—532no—
113#416—1062no—
114C2×C4×C8C2×D5312yes—
115C2×C4×C8C2×D5311yescharacteristic
116#341—1062no—
117Q8⋊C8—1064no—
118D4⋊C8—1064no—
119C82—22no—
120C53#985711yescharacteristic, complemented, Sylow
121C2×#41—1061no—
122C2×M16—1062no—
123C22×C8—22no—
124C22×C8D21212yes—
125C4⋊Q8—531no—
126C4⋊Q8—532no—
127#127—531no—
128#125—1061no—
129#124—1061no—
130C22⋊Q8—1062no—
131#121—1062no—
132C4×Q8—1062no—
133C4×D4—1062no—
134#117—1061no—
135C2×C42C22×D5311yescharacteristic
136C4⋊C8—1062no—
137C4⋊C8—1062no—
138#103—1064no—
139#102—1064no—
140C4×C8—22no—
141C4×C8C4×D5312yes—
142C4×C8D21214yes—
143C4×C8C22⋊D5312yes—
144#41—2124no—
145C2×Q8—1061no—
146C2×Q8—1062no—
147C2×D4—1061no—
148C2×D4—1062no—
149C2×D4—2121no—
150C22×C4—2121no—
151C22×C4C2×D21211yescharacteristic
152C22×C4#1171994512yes—
153M16—2124no—
154C2×C8—28no—
155C2×C8—24no—
156C2×C8—24no—
157C2×C8—2122no—
158C2×C8#1171992114yes—
159C4⋊C4—1062no—
160C4⋊C4—1062no—
161C4⋊C4—1062no—
162C4⋊C4—2122no—
163C22⋊C4—2121no—
164C22⋊C4—2122no—
165C42—2121no—
166C42C2×C4×D5311yescharacteristic
167C42C2×D21212yes—
168C42C2×C22⋊D5311yescharacteristic
169C23—2121no—
170C23#1313320511yescharacteristic
171Q8—2122no—
172Q8—2124no—
173D4—4242no—
174D4—2122no—
175D4—2124no—
176C2×C4—24no—
177C2×C4—4242no—
178C2×C4—2122no—
179C2×C4—2121no—
180C2×C4C2×#1171992111yescharacteristic
181C2×C4C4×D21212yes—
182C2×C4#1313319112yes—
183C2×C4#1313321411yescharacteristic
184C2×C4#1313323914yes—
185C8—28no—
186C8—24no—
187C8—4242no—
188C22—22no—
189C22—4241no—
190C22—4242no—
191C22#1681052511yescharacteristic
192C22#1681097812yes—
193C4—22no—
194C4—4241no—
195C4—4242no—
196C4#1681062012yes—
197C4#1681074114yes—
198C4#1681100712yes—
199C2—22no—
200C2—8481no—
201C2#2027552611yescharacteristic
202C2#2027560312yes—
subgroupG/H
centerC2×C4#13133214
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC2×C82×C53C2
socleC22×C53#390
prime radical#16811173C22
cosocleC2×C42×C53C23
hypercenterC2×C82D53
hypocenterC53#9857
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C82

Complements

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

NH ≅ G/Ntype
2. #20274853C2semidirect
3. #20273171C2semidirect
5. C2×C82×C53C2semidirect
102. C2×C82D53semidirect
120. C53#9857semidirect

Maximal subgroups

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

subgroupindex
#202850382
#202748532
#202731712
#202731672
C2×C82×C532
#985753