Finite Group Databasebeta

← #27287226

Subgroups — #27287226

Order 13568.

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

normal subgroups only

This group has 416 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#20278008C211yesmaximal, characteristic
2#20275696C211yesmaximal, characteristic
3#20273215C212yesmaximal
4#20272617C212yesmaximal
5C53×#1274C211yesmaximal, characteristic
6#16811180C2211yescharacteristic
7C2×C4×Dic106—21no—
8#16810647C2211yescharacteristic
9#16810555—22no—
10#16810544—22no—
11C4×C4⋊Dic53—21no—
12C4×C4⋊Dic53C2212yes—
13#16810383—21no—
14#16810383—21no—
15#16810361—22no—
16#16810361—22no—
17C53×#431C2212yes—
18C53×#376C2211yescharacteristic
19C22×Dic106—21no—
20C2×#11719917—21no—
21#13133217Q812yescomplemented
22#13133209—21no—
23#13133196—22no—
24C4×Dic106—22no—
25C4×Dic106—42no—
26C4×Dic106—42no—
27#13133176—21no—
28C2×C4⋊Dic53—22no—
29C2×C4⋊Dic53—21no—
30C2×C4⋊Dic53—22no—
31C2×#11719919—22no—
32C2×C4×Dic53—22no—
33C4⋊#88322—22no—
34#13133124—21no—
35#13133124—21no—
36#13133124D412yes—
37#13133124D412yes—
38C2×C42×C53D411yescharacteristic
39C2×C42×C53D411yescharacteristic
40C2×C42×C53C2311yescharacteristic
41C4⋊C8×C53—22no—
42C22⋊C8×C53—22no—
43C4×C8×C53—22no—
44#95×C53—22no—
45C22×C4×C53—21no—
46C22×C4×C53—21no—
47C22×C4×C53C2×D412yes—
48C22×C4×C53C2×D411yescharacteristic
49C22×Dic53—22no—
50C22×Dic53—22no—
51C2×Dic106—22no—
52C2×Dic106—44no—
53C2×Dic106—42no—
54C2×C8×C53—42no—
55C42×C53—21no—
56C42×C53—21no—
57C42×C53SD1612yes—
58C42×C53Q1612yes—
59C42×C53C2×Q811yescharacteristic
60C42×C53#4112yes—
61C42×C53#4111yescharacteristic
62C22⋊Dic53—41no—
63C4⋊Dic53—22no—
64C4⋊Dic53—22no—
65C4⋊Dic53—44no—
66C4⋊Dic53—44no—
67C4⋊Dic53—42no—
68#11719919—44no—
69#11719919—42no—
70C4×Dic53—41no—
71C4×Dic53—42no—
72C4×Dic53—42no—
73#11719917—41no—
74#11719917—41no—
75C2×#88322—41no—
76C23×C53#12011yescharacteristic
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—42no—
80C2×C4×C53—42no—
81C2×C4×C53—42no—
82C2×C4×C53C22⋊Q812yes—
83C2×C4×C53C22⋊Q811yescharacteristic
84C2×C4×C53#12412yes—
85C2×C4×C53#12411yescharacteristic
86C2×C4×C53C2×SD1611yescharacteristic
87C2×C4×C53C2×Q1611yescharacteristic
88C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
89C2×C4×C53#13711yescharacteristic
90C2×Dic53—24no—
91C2×Dic53—42no—
92C2×Dic53—42no—
93C2×Dic53—42no—
94C2×Dic53—42no—
95C2×Dic53—42no—
96C2×Dic53—42no—
97Dic106—48no—
98Dic106—82no—
99C8×C53—82no—
100#88322—81no—
101#10371—531nomaximal, Sylow
102C22×C53—22no—
103C22×C53#39311yescharacteristic
104C22×C53#45011yescharacteristic
105C22×C53#45111yescharacteristic
106C4×C53—22no—
107C4×C53—22no—
108C4×C53—42no—
109C4×C53—42no—
110C4×C53#47511yescharacteristic
111C4×C53#47711yescharacteristic
112C4×C53#48711yescharacteristic
113C4×C53#48911yescharacteristic
114Dic53—44no—
115Dic53—82no—
116Dic53—82no—
117#2399—531no—
118#1995—531no—
119#1503—532no—
120#1303—532no—
121#1274D5311yescharacteristic
122C2×C53—21no—
123C2×C53—21no—
124C2×C53#202411yescharacteristic
125C2×C53#203112yes—
126#533—531no—
127C2×C4×Q8—1061no—
128#431C2×D5312yes—
129#423—531no—
130#398—1062no—
131#395—1062no—
132C4×C4⋊C4—532no—
133C4×C4⋊C4—1061no—
134#376C2×D5311yescharacteristic
135#339—1061no—
136#339—1061no—
137#328—1062no—
138#328—1062no—
139C53#1037111yescharacteristic, complemented, Sylow
140C22×Q8—1061no—
141C2×M16—1061no—
142C4⋊Q8—532no—
143#125—1061no—
144C22⋊Q8—1062no—
145C4×Q8—1062no—
146C4×Q8—2122no—
147C4×Q8—2122no—
148#117—1061no—
149C2×C4⋊C4—1062no—
150C2×C4⋊C4—1061no—
151C2×C4⋊C4—1062no—
152C2×C4⋊C4—1062no—
153C2×C42—1062no—
154C2×C42D21211yescharacteristic
155C2×C42D21211yescharacteristic
156C2×C42C22×D5311yescharacteristic
157C4⋊C8—22no—
158C4⋊C8—1062no—
159C22⋊C8—22no—
160C4×C8—22no—
161#95—22no—
162#95—532no—
163#95—532no—
164#95—1061no—
165#95—1061no—
166C2×Q8—1062no—
167C2×Q8—2122no—
168C2×Q8—2124no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—1062no—
172C22×C4—1062no—
173C22×C4C2×D21211yescharacteristic
174C22×C4D4×D5312yes—
175M16—2121no—
176M16—2121no—
177C2×C8—42no—
178C2×C8—2121no—
179C4⋊C4—1062no—
180C4⋊C4—1062no—
181C4⋊C4—2124no—
182C4⋊C4—2122no—
183C4⋊C4—2124no—
184C4⋊C4—2124no—
185C4⋊C4—2122no—
186C22⋊C4—2121no—
187C42—21no—
188C42—21no—
189C42—2121no—
190C42—2122no—
191C42—2122no—
192C42#1171991312yes—
193C42Dic21212yes—
194C42#1171994512yes—
195C42#1171994711yescharacteristic
196C42Q8×D5311yescharacteristic
197C23#1313318711yescharacteristic
198Q8—4242no—
199Q8—2128no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—1064no—
206C2×C4—2122no—
207C2×C4—2122no—
208C2×C4—2122no—
209C2×C4—2122no—
210C2×C4—2122no—
211C2×C4—2122no—
212C2×C4#1313319512yes—
213C2×C4#1313319711yescharacteristic
214C2×C4#1313320312yes—
215C2×C4#1313320511yescharacteristic
216C2×C4C2×#1171991311yescharacteristic
217C2×C4C2×Dic21211yescharacteristic
218C2×C4#1313324511yescharacteristic
219C2×C4#1313325011yescharacteristic
220C8—82no—
221C8—4241no—
222C22—22no—
223C22#1681053611yescharacteristic
224C22#1681077611yescharacteristic
225C22#1681078111yescharacteristic
226C4—22no—
227C4—22no—
228C4—42no—
229C4—42no—
230C4—4242no—
231C4—4242no—
232C4—2124no—
233C4#1681090911yescharacteristic
234C4#1681091911yescharacteristic
235C4#1681097411yescharacteristic
236C4#1681098311yescharacteristic
237C2—21no—
238C2—21no—
239C2#2027585711yescharacteristic
240C2#2027588612yes—
subgroupG/H
centerC2×C4#13133205
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1274C2
socleC22×C53#393
prime radicalC23×C53#120
hypercenter#1274D53
hypocenterC53#10371
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1274

Complements

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

NH ≅ G/Ntype
21. #13133217Q8semidirect
139. C53#10371semidirect

Maximal subgroups

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

subgroupindex
#202780082
#202756962
#202732152
#202726172
C53×#12742
#1037153