Finite Group Databasebeta

← #27287196

Subgroups — #27287196

Order 13568.

400 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 97 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#20283078C211yesmaximal, characteristic
2#20275692C211yesmaximal, characteristic
3#20273280C212yesmaximal
4#20272574C212yesmaximal
5C53×#1276C211yesmaximal, characteristic
6#16811180C2211yescharacteristic
7C2×#13133209—21no—
8#16810647C2211yescharacteristic
9#16810521—24no—
10#16810493—21no—
11#16810489C2212yes—
12#16810383—21no—
13#16810383—21no—
14#16810361—24no—
15C53×#431C2212yes—
16C43×C53C2211yescharacteristic
17C2×#11719917—21no—
18#13133217Q812yes—
19#13133209—21no—
20#13133209—42no—
21#13133209—42no—
22#13133196—22no—
23C4×Dic106—22no—
24#13133176—21no—
25C2×C4⋊Dic53—24no—
26C2×C4⋊Dic53—22no—
27C2×#11719919—24no—
28C4⋊#88322—22no—
29#13133124—22no—
30#13133124D414yes—
31C2×C42×C53—22no—
32C2×C42×C53D411yescharacteristic
33C2×C42×C53D411yescharacteristic
34C2×C42×C53C2311yescharacteristic
35C4⋊C8×C53—22no—
36C22⋊C8×C53—22no—
37C4×C8×C53—22no—
38C22×C4×C53—21no—
39C22×C4×C53—21no—
40C22×C4×C53C2×D412yes—
41C22×C4×C53C2×D411yescharacteristic
42C22×Dic53—24no—
43C2×Dic106—22no—
44C2×C8×C53—42no—
45C42×C53—21no—
46C42×C53—21no—
47C42×C53—44no—
48C42×C53SD1612yes—
49C42×C53Q1612yes—
50C42×C53C2×Q811yescharacteristic
51C42×C53#4112yes—
52C42×C53#4111yescharacteristic
53C22⋊Dic53—41no—
54C4⋊Dic53—22no—
55C4⋊Dic53—22no—
56C4⋊Dic53—48no—
57C4⋊Dic53—42no—
58C4⋊Dic53—42no—
59#11719919—48no—
60#11719919—42no—
61C4×Dic53—41no—
62#11719917—41no—
63#11719917—41no—
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×C53C22⋊Q812yes—
72C2×C4×C53C22⋊Q811yescharacteristic
73C2×C4×C53#12412yes—
74C2×C4×C53#12411yescharacteristic
75C2×C4×C53C2×SD1611yescharacteristic
76C2×C4×C53C2×Q1611yescharacteristic
77C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
78C2×C4×C53#13711yescharacteristic
79C2×Dic53—48no—
80C2×Dic53—42no—
81C2×Dic53—44no—
82Dic106—82no—
83C8×C53—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#47511yescharacteristic
95C4×C53#47711yescharacteristic
96C4×C53#48711yescharacteristic
97C4×C53#48911yescharacteristic
98Dic53—84no—
99Dic53—82no—
100#2859—531no—
101#1994—531no—
102#1518—532no—
103#1286—532no—
104#1276D5311yescharacteristic
105C2×C53—21no—
106C2×C53—21no—
107C2×C53#202411yescharacteristic
108C2×C53#203112yes—
109#533—531no—
110C2×#125—1061no—
111#431C2×D5312yes—
112#423—531no—
113#389—1064no—
114#384—1061no—
115#382—532no—
116C43C2×D5311yescharacteristic
117#339—1061no—
118#339—1061no—
119#328—1064no—
120C53#1036511yescharacteristic, complemented, Sylow
121C2×M16—1061no—
122C4⋊Q8—532no—
123#125—1061no—
124#125—2122no—
125#125—2122no—
126C22⋊Q8—1062no—
127C4×Q8—1062no—
128#117—1061no—
129C2×C4⋊C4—1064no—
130C2×C4⋊C4—1064no—
131C2×C4⋊C4—1062no—
132C2×C42—22no—
133C2×C42D21211yescharacteristic
134C2×C42D21211yescharacteristic
135C2×C42C22×D5311yescharacteristic
136C4⋊C8—22no—
137C4⋊C8—1062no—
138C22⋊C8—22no—
139C4×C8—22no—
140#95—534no—
141#95—1062no—
142C2×Q8—1062no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—1064no—
146C22×C4C2×D21211yescharacteristic
147C22×C4D4×D5312yes—
148M16—2121no—
149M16—2121no—
150C2×C8—42no—
151C2×C8—2121no—
152C4⋊C4—1062no—
153C4⋊C4—1062no—
154C4⋊C4—2128no—
155C4⋊C4—2128no—
156C4⋊C4—2122no—
157C4⋊C4—2122no—
158C4⋊C4—2122no—
159C22⋊C4—2121no—
160C42—21no—
161C42—21no—
162C42—44no—
163C42—2121no—
164C42#1171991312yes—
165C42Dic21212yes—
166C42#1171994512yes—
167C42#1171994711yescharacteristic
168C42Q8×D5311yescharacteristic
169C23#1313318711yescharacteristic
170Q8—4242no—
171C2×C4—24no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—2128no—
177C2×C4—2124no—
178C2×C4—2122no—
179C2×C4#1313319512yes—
180C2×C4#1313319711yescharacteristic
181C2×C4#1313320312yes—
182C2×C4#1313320511yescharacteristic
183C2×C4C2×#1171991311yescharacteristic
184C2×C4C2×Dic21211yescharacteristic
185C2×C4#1313324511yescharacteristic
186C2×C4#1313325011yescharacteristic
187C8—82no—
188C8—4241no—
189C22—22no—
190C22#1681053611yescharacteristic
191C22#1681077611yescharacteristic
192C22#1681078111yescharacteristic
193C4—24no—
194C4—22no—
195C4—22no—
196C4—42no—
197C4—4244no—
198C4—4242no—
199C4#1681090911yescharacteristic
200C4#1681091911yescharacteristic
201C4#1681097411yescharacteristic
202C4#1681098311yescharacteristic
203C2—21no—
204C2—21no—
205C2#2027585711yescharacteristic
206C2#2027588812yes—
subgroupG/H
centerC22#16810536
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1276C2
socleC22×C53#393
prime radicalC23×C53#120
hypercenter#1276D53
hypocenterC53#10365
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1276

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
#202830782
#202756922
#202732802
#202725742
C53×#12762
#1036553