Finite Group Databasebeta

← #27287192

Subgroups — #27287192

Order 13568.

400 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 101 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#20275692C211yesmaximal, characteristic
2#20273278C212yesmaximal
3#20272573C212yesmaximal
4#20272548C211yesmaximal, characteristic
5C53×#2859C211yesmaximal, characteristic
6#16811180C2211yescharacteristic
7#16810676C2212yes—
8#16810647C2211yescharacteristic
9#16810383—21no—
10#16810383—21no—
11#16810359—24no—
12C2×#125×C53—21no—
13C53×#389—24no—
14C53×#384—21no—
15C53×#382C2212yes—
16C43×C53C2211yescharacteristic
17C2×#11719917—21no—
18#13133217Q812yes—
19#13133209—21no—
20#13133196—22no—
21C4×Dic106—22no—
22#13133176—21no—
23C4⋊#88322—22no—
24C4⋊#88322—22no—
25C22⋊#88322—22no—
26C4×#88322—22no—
27#125×C53—42no—
28#125×C53—42no—
29C2×C4⋊C4×C53—24no—
30C2×C4⋊C4×C53—24no—
31C2×C4⋊C4×C53—22no—
32C2×C42×C53—22no—
33C2×C42×C53D411yescharacteristic
34C2×C42×C53D411yescharacteristic
35C2×C42×C53C2311yescharacteristic
36#95×C53—22no—
37#95×C53D414yes—
38C22×C4×C53—21no—
39C22×C4×C53—21no—
40C22×C4×C53—24no—
41C22×C4×C53C2×D412yes—
42C22×C4×C53C2×D411yescharacteristic
43C2×Dic106—22no—
44C4⋊C4×C53—48no—
45C4⋊C4×C53—48no—
46C4⋊C4×C53—42no—
47C4⋊C4×C53—42no—
48C42×C53—21no—
49C42×C53—21no—
50C42×C53—44no—
51C42×C53SD1612yes—
52C42×C53Q1612yes—
53C42×C53C2×Q811yescharacteristic
54C42×C53#4111yescharacteristic
55C42×C53#4112yes—
56C22⋊Dic53—41no—
57C4⋊Dic53—22no—
58C4⋊Dic53—22no—
59#11719919—42no—
60C4×Dic53—41no—
61#11719917—41no—
62#11719917—41no—
63C2×#88322—42no—
64C2×#88322—41no—
65C23×C53#12011yescharacteristic
66C2×C4×C53—24no—
67C2×C4×C53—22no—
68C2×C4×C53—22no—
69C2×C4×C53—42no—
70C2×C4×C53—42no—
71C2×C4×C53—48no—
72C2×C4×C53—44no—
73C2×C4×C53C22⋊Q812yes—
74C2×C4×C53C22⋊Q811yescharacteristic
75C2×C4×C53#12412yes—
76C2×C4×C53#12411yescharacteristic
77C2×C4×C53C2×SD1611yescharacteristic
78C2×C4×C53C2×Q1611yescharacteristic
79C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
80C2×C4×C53#13711yescharacteristic
81C2×Dic53—42no—
82Dic106—82no—
83#88322—82no—
84#88322—81no—
85#10365—531nomaximal, Sylow
86C22×C53—22no—
87C22×C53#39311yescharacteristic
88C22×C53#45011yescharacteristic
89C22×C53#45111yescharacteristic
90C4×C53—24no—
91C4×C53—22no—
92C4×C53—22no—
93C4×C53—42no—
94C4×C53—84no—
95C4×C53#47511yescharacteristic
96C4×C53#47711yescharacteristic
97C4×C53#48711yescharacteristic
98C4×C53#48911yescharacteristic
99Dic53—82no—
100#2859D5311yescharacteristic
101#1994—531no—
102#1518—532no—
103#1286—532no—
104#1276—531no—
105C2×C53—21no—
106C2×C53—21no—
107C2×C53#202411yescharacteristic
108C2×C53#203112yes—
109#533—531no—
110C2×#125—21no—
111#431—532no—
112#423—531no—
113#389—24no—
114#384—21no—
115#382C2×D5312yes—
116C43C2×D5311yescharacteristic
117#339—1061no—
118#339—1061no—
119#328—1064no—
120C53#1036511yescharacteristic, complemented, Sylow
121C2×M16—1061no—
122C4⋊Q8—532no—
123#125—42no—
124#125—42no—
125#125—1061no—
126C22⋊Q8—1062no—
127C4×Q8—1062no—
128#117—1061no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—22no—
132C2×C42—22no—
133C2×C42C22⋊D5311yescharacteristic
134C2×C42C22⋊D5311yescharacteristic
135C2×C42C22×D5311yescharacteristic
136C4⋊C8—1062no—
137C4⋊C8—1062no—
138C22⋊C8—1062no—
139C4×C8—1062no—
140#95—22no—
141#95C22⋊D5314yes—
142C2×Q8—1062no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—24no—
146C22×C4C2×C22⋊D5312yes—
147C22×C4C2×C22⋊D5311yescharacteristic
148M16—2121no—
149M16—2121no—
150C2×C8—2122no—
151C2×C8—2121no—
152C4⋊C4—48no—
153C4⋊C4—48no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—1062no—
157C4⋊C4—1062no—
158C4⋊C4—2122no—
159C22⋊C4—2121no—
160C42—21no—
161C42—21no—
162C42—44no—
163C42—2121no—
164C42#1171992312yes—
165C42#1171992512yes—
166C42Q8×D5311yescharacteristic
167C42#1171994911yescharacteristic
168C42#1171994912yes—
169C23#1313318611yescharacteristic
170Q8—4242no—
171C2×C4—24no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—48no—
177C2×C4—44no—
178C2×C4—2122no—
179C2×C4#1313319312yes—
180C2×C4#1313319311yescharacteristic
181C2×C4#1313320612yes—
182C2×C4#1313320611yescharacteristic
183C2×C4C2×#1171992311yescharacteristic
184C2×C4C2×#1171992511yescharacteristic
185C2×C4#1313324311yescharacteristic
186C2×C4#1313324811yescharacteristic
187C8—4242no—
188C8—4241no—
189C22—22no—
190C22#1681053411yescharacteristic
191C22#1681077411yescharacteristic
192C22#1681077911yescharacteristic
193C4—24no—
194C4—22no—
195C4—22no—
196C4—42no—
197C4—84no—
198C4—4242no—
199C4#1681090511yescharacteristic
200C4#1681091511yescharacteristic
201C4#1681097511yescharacteristic
202C4#1681098511yescharacteristic
203C2—21no—
204C2—21no—
205C2#2027585311yescharacteristic
206C2#2027588412yes—
subgroupG/H
centerC22#16810534
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2859C2
socleC22×C53#393
prime radicalC23×C53#120
hypercenter#2859D53
hypocenterC53#10365
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2859

Complements

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

NH ≅ G/Ntype
120. C53#10365semidirect

Maximal subgroups

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

subgroupindex
#202756922
#202732782
#202725732
#202725482
C53×#28592
#1036553