Finite Group Databasebeta

← #27286768

Subgroups — #27286768

Order 13568.

566 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 113 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#20285642C212yesmaximal
2#20283093C211yesmaximal, characteristic, complemented
3#20275543C211yesmaximal, characteristic
4#20273395C212yesmaximal, complemented
5C53×#1276C211yesmaximal, characteristic, complemented
6#16811173C2211yescharacteristic
7C2×#13133217C2212yes—
8#16810996—22no—
9#16810974—22no—
10#16810829—24no—
11#16810764—24no—
12#16810647C2211yescharacteristic
13C2×#13133145—22no—
14#16810529—24no—
15#16810489—22no—
16#16810371—24no—
17C53×#431C2212yes—
18C43×C53C2211yescharacteristic
19C2×#11719945—21no—
20C22×Dic106D414yes—
21C2×#11719913—28no—
22C2×#11719917—21no—
23#13133217—22no—
24#13133217—22no—
25#13133217—22no—
26#13133217—22no—
27#13133217D411yescharacteristic
28#13133214D411yescharacteristic
29#13133205—21no—
30#13133191—22no—
31C4×D212—22no—
32C2×C4⋊Dic53—22no—
33C2×C4⋊Dic53—22no—
34C4⋊#88322—22no—
35#13133145—44no—
36#13133142—28no—
37#13133124—28no—
38C2×C42×C53—22no—
39C2×C42×C53D412yes—
40C2×C42×C53C2311yescharacteristic
41C4⋊C8×C53D414yes—
42C22⋊C8×C53—22no—
43C4×C8×C53—22no—
44C22×C4×C53—22no—
45C22×C4×C53C2×D412yes—
46C22×C4×C53C2×D411yescharacteristic
47C2×C22⋊D53—41no—
48C22×Dic53—24no—
49#11719945—42no—
50#11719945—42no—
51C2×D212—22no—
52C2×D212—21no—
53C2×C4×D53—41no—
54C2×Dic106—24no—
55C2×Dic106—21no—
56C2×Dic106—24no—
57C2×Dic106—48no—
58C2×C8×C53—24no—
59C42×C53—22no—
60C42×C53—44no—
61C42×C53SD1614yescomplemented
62C42×C53C2×D411yescharacteristic
63C42×C53C2×D412yes—
64C42×C53#4111yescharacteristic
65#11719921—42no—
66C4⋊Dic53—22no—
67C4⋊Dic53—42no—
68C4⋊Dic53—42no—
69C4⋊Dic53—42no—
70C4⋊Dic53—42no—
71#11719917—42no—
72C2×#88322—41no—
73#11719913—88no—
74C23×C53#12011yescharacteristic
75C22×D53—41no—
76C2×C4×C53—24no—
77C2×C4×C53—24no—
78C2×C4×C53—44no—
79C2×C4×C53#12012yes—
80C2×C4×C53#12112yes—
81C2×C4×C53#12111yescharacteristic
82C2×C4×C53#12711yescharacteristic
83C2×C4×C53C2×SD1612yes—
84C2×C4×C53#13712yes—
85C22⋊D53—82no—
86C2×Dic53—48no—
87C2×Dic53—44no—
88C2×Dic53—41no—
89D212—44no—
90D212—42no—
91C4×D53—81no—
92C4×D53—81no—
93Dic106—48no—
94Dic106—42no—
95Dic106—84no—
96C8×C53—44no—
97#88322—81no—
98#10287—531nomaximal, Sylow
99C22×C53—22no—
100C22×C53#39211yescharacteristic
101C22×C53#44812yes—
102C2×D53—82no—
103C2×D53—81no—
104C4×C53—22no—
105C4×C53—44no—
106C4×C53#45514yes—
107C4×C53#46112yes—
108C4×C53#49211yescharacteristic
109C4×C53#49711yescharacteristic
110Dic53—84no—
111Dic53—81no—
112#3206—532no—
113#2862—531no—
114#1962—531no—
115#1548—532no—
116#1276D5311yescharacteristic, complemented
117C2×C53—22no—
118C2×C53#200412yes—
119C2×C53#201311yescharacteristic
120D53—161no—
121#532—531no—
122C2×C4⋊Q8—532no—
123#492—1062no—
124#487—1062no—
125#461—1064no—
126#448—1064no—
127#431C2×D5312yes—
128#423—531no—
129C2×#104—1062no—
130#391—1064no—
131#382—1062no—
132C43C2×D5311yescharacteristic
133#333—1064no—
134C53#1028711yescharacteristic, complemented, Sylow
135C2×#41—1061no—
136C22×Q8—534no—
137C2×SD16—1068no—
138C2×M16—1061no—
139C4⋊Q8—531no—
140C4⋊Q8—1062no—
141C4⋊Q8—1062no—
142C4⋊Q8—1062no—
143C4⋊Q8—1062no—
144#127—531no—
145#124—1061no—
146#121—1062no—
147C4×D4—1062no—
148C2×C4⋊C4—1062no—
149C2×C4⋊C4—1062no—
150C2×C42—22no—
151C2×C42D21212yes—
152C2×C42C22×D5311yescharacteristic
153C4⋊C8—1062no—
154C4⋊C8D21214yes—
155#104—2124no—
156#103—1068no—
157C22⋊C8—22no—
158C4×C8—22no—
159#95—1068no—
160#41—2122no—
161#41—2122no—
162C2×Q8—1064no—
163C2×Q8—1064no—
164C2×Q8—1061no—
165C2×Q8—2128no—
166C2×D4—1062no—
167C2×D4—1061no—
168C2×D4—2121no—
169C22×C4—22no—
170C22×C4—1064no—
171C22×C4—2121no—
172C22×C4C2×D21211yescharacteristic
173C22×C4D4×D5312yes—
174SD16—4248no—
175M16—2122no—
176C2×C8—24no—
177C2×C8—2121no—
178C4⋊C4—1062no—
179C4⋊C4—2122no—
180C4⋊C4—2122no—
181C4⋊C4—2122no—
182C4⋊C4—2122no—
183C22⋊C4—2122no—
184C42—22no—
185C42—44no—
186C42#1171991314yescomplemented
187C42C2×D21212yes—
188C42D4×D5311yescharacteristic
189C42#1171994911yescharacteristic
190C23—2121no—
191C23#1313318711yescharacteristic
192Q8—4244no—
193Q8—2128no—
194Q8—2122no—
195D4—4242no—
196D4—2124no—
197D4—2122no—
198C2×C4—24no—
199C2×C4—24no—
200C2×C4—44no—
201C2×C4—4241no—
202C2×C4—4241no—
203C2×C4—2128no—
204C2×C4—2121no—
205C2×C4—2124no—
206C2×C4#1313318712yes—
207C2×C4#1313319212yes—
208C2×C4#1313319211yescharacteristic
209C2×C4#1313321411yescharacteristic
210C2×C4C2×#1171991312yes—
211C2×C4#1313325012yes—
212C8—44no—
213C8—4241no—
214C22—22no—
215C22—4242no—
216C22—4241no—
217C22#1681053311yescharacteristic
218C22#1681076412yes—
219C4—22no—
220C4—44no—
221C4—4244no—
222C4—4241no—
223C4#1681080214yes—
224C4#1681082912yes—
225C4#1681099611yescharacteristic
226C4#1681101711yescharacteristic
227C2—22no—
228C2—8481no—
229C2#2027574712yes—
230C2#2027580011yescharacteristic
subgroupG/H
centerC22#16810533
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1276C2
socleC22×C53#392
prime radical#16811173C22
hypercenter#1276D53
hypocenterC53#10287
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
2. #20283093C2semidirect
4. #20273395C2semidirect
5. C53×#1276C2semidirect
61. C42×C53SD16semidirect
116. #1276D53semidirect
134. C53#10287semidirect
186. C42#11719913semidirect

Maximal subgroups

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

subgroupindex
#202856422
#202830932
#202755432
#202733952
C53×#12762
#1028753