Finite Group Databasebeta

← #27287191

Subgroups — #27287191

Order 13568.

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

normal subgroups only

This group has 438 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, complemented
2#20275543C211yesmaximal, characteristic
3#20273276C212yesmaximal
4#20272574C212yesmaximal, complemented
5C53×#1276C211yesmaximal, characteristic, complemented
6#16811173C2211yescharacteristic
7C2×#13133209—21no—
8#16810647C2211yescharacteristic
9C2×#13133145—21no—
10C2×#13133145—21no—
11#16810521—24no—
12#16810493—21no—
13#16810489C2212yes—
14#16810358—24no—
15C53×#431C2212yes—
16C43×C53C2211yescharacteristic
17C2×#11719945—21no—
18C2×#11719917—21no—
19#13133217Q811yescharacteristic
20#13133214Q811yescharacteristic
21#13133209—42no—
22#13133209—42no—
23#13133205—21no—
24#13133191—22no—
25C4×D212—22no—
26C2×C4⋊Dic53—24no—
27C2×C4⋊Dic53—22no—
28C2×#11719919—24no—
29C4⋊#88322—22no—
30#13133145—42no—
31#13133145—42no—
32#13133124—22no—
33#13133124D414yes—
34C2×C42×C53—22no—
35C2×C42×C53D411yescharacteristic
36C2×C42×C53D411yescharacteristic
37C2×C42×C53C2311yescharacteristic
38C4⋊C8×C53—22no—
39C22⋊C8×C53—22no—
40C4×C8×C53—22no—
41C22×C4×C53—21no—
42C22×C4×C53—21no—
43C22×C4×C53C2×D412yes—
44C22×C4×C53C2×D411yescharacteristic
45C2×C22⋊D53—41no—
46C22×Dic53—24no—
47#11719945—41no—
48#11719945—41no—
49#11719945—41no—
50#11719945—41no—
51C2×D212—21no—
52C2×D212—22no—
53C2×C4×D53—41no—
54C2×Dic106—21no—
55C2×C8×C53—42no—
56C42×C53—21no—
57C42×C53—21no—
58C42×C53—44no—
59C42×C53D812yes—
60C42×C53SD1612yes—
61C42×C53C2×Q811yescharacteristic
62C42×C53#4112yes—
63C42×C53#4111yescharacteristic
64#11719921—42no—
65C4⋊Dic53—22no—
66C4⋊Dic53—48no—
67C4⋊Dic53—42no—
68C4⋊Dic53—42no—
69#11719919—48no—
70#11719917—41no—
71#11719917—41no—
72C2×#88322—41no—
73C23×C53#12011yescharacteristic
74C22×D53—41no—
75C2×C4×C53—24no—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53—42no—
79C2×C4×C53—42no—
80C2×C4×C53C22⋊Q812yes—
81C2×C4×C53C22⋊Q811yescharacteristic
82C2×C4×C53#12412yes—
83C2×C4×C53#12411yescharacteristic
84C2×C4×C53C2×D811yescharacteristic
85C2×C4×C53C2×SD1611yescharacteristic
86C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
87C2×C4×C53#13711yescharacteristic
88C22⋊D53—81no—
89C22⋊D53—81no—
90C2×Dic53—48no—
91C2×Dic53—41no—
92C2×Dic53—44no—
93D212—41no—
94D212—41no—
95D212—82no—
96C4×D53—81no—
97C4×D53—81no—
98Dic106—41no—
99Dic106—41no—
100C8×C53—82no—
101#88322—81no—
102#10364—531nomaximal, Sylow
103C22×C53—22no—
104C22×C53#39311yescharacteristic
105C22×C53#44711yescharacteristic
106C22×C53#44811yescharacteristic
107C2×D53—81no—
108C2×D53—82no—
109C4×C53—24no—
110C4×C53—22no—
111C4×C53—22no—
112C4×C53—42no—
113C4×C53#47411yescharacteristic
114C4×C53#47611yescharacteristic
115C4×C53#48611yescharacteristic
116C4×C53#48911yescharacteristic
117Dic53—84no—
118Dic53—81no—
119#2859—531no—
120#1962—531no—
121#1517—532no—
122#1286—532no—
123#1276D5311yescharacteristic, complemented
124C2×C53—21no—
125C2×C53—21no—
126C2×C53#202311yescharacteristic
127C2×C53#203112yes—
128D53—161no—
129#532—531no—
130C2×#125—1061no—
131#431C2×D5312yes—
132#423—531no—
133C2×#104—1061no—
134C2×#104—1061no—
135#389—1064no—
136#384—1061no—
137#382—532no—
138C43C2×D5311yescharacteristic
139#327—1064no—
140C53#1036411yescharacteristic, complemented, Sylow
141C2×#41—1061no—
142C2×M16—1061no—
143C4⋊Q8—531no—
144#127—531no—
145#125—2122no—
146#125—2122no—
147#124—1061no—
148#121—1062no—
149C4×D4—1062no—
150C2×C4⋊C4—1064no—
151C2×C4⋊C4—1064no—
152C2×C4⋊C4—1062no—
153C2×C42—22no—
154C2×C42D21211yescharacteristic
155C2×C42D21211yescharacteristic
156C2×C42C22×D5311yescharacteristic
157C4⋊C8—22no—
158C4⋊C8—1062no—
159#104—2122no—
160#104—2122no—
161C22⋊C8—22no—
162C4×C8—22no—
163#95—534no—
164#95—1062no—
165#41—2121no—
166#41—2121no—
167#41—2121no—
168#41—2121no—
169C2×Q8—1061no—
170C2×D4—1062no—
171C2×D4—1061no—
172C2×D4—2121no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—1064no—
176C22×C4—2121no—
177C22×C4C2×D21211yescharacteristic
178C22×C4D4×D5312yes—
179M16—2121no—
180M16—2121no—
181C2×C8—42no—
182C2×C8—2121no—
183C4⋊C4—1062no—
184C4⋊C4—2128no—
185C4⋊C4—2128no—
186C4⋊C4—2122no—
187C4⋊C4—2122no—
188C22⋊C4—2122no—
189C42—21no—
190C42—21no—
191C42—44no—
192C42#1171991312yes—
193C42D42412yes—
194C42#1171994512yes—
195C42#1171994711yescharacteristic
196C42Q8×D5311yescharacteristic
197C23—2121no—
198C23#1313318711yescharacteristic
199Q8—2121no—
200Q8—2121no—
201D4—4242no—
202D4—4241no—
203D4—4241no—
204D4—2121no—
205D4—2121no—
206C2×C4—24no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—4241no—
212C2×C4—4241no—
213C2×C4—2128no—
214C2×C4—2124no—
215C2×C4—2121no—
216C2×C4#1313319512yes—
217C2×C4#1313319711yescharacteristic
218C2×C4#1313320312yes—
219C2×C4#1313320511yescharacteristic
220C2×C4C2×D42411yescharacteristic
221C2×C4C2×#1171991311yescharacteristic
222C2×C4#1313324511yescharacteristic
223C2×C4#1313325011yescharacteristic
224C8—82no—
225C8—4241no—
226C22—22no—
227C22—4241no—
228C22—4242no—
229C22#1681053611yescharacteristic
230C22#1681075911yescharacteristic
231C22#1681076411yescharacteristic
232C4—24no—
233C4—22no—
234C4—22no—
235C4—42no—
236C4—4244no—
237C4—4241no—
238C4#1681090411yescharacteristic
239C4#1681091411yescharacteristic
240C4#1681097011yescharacteristic
241C4#1681098311yescharacteristic
242C2—21no—
243C2—21no—
244C2—8481no—
245C2#2027585211yescharacteristic
246C2#2027588812yes—
subgroupG/H
centerC22#16810536
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1276C2
socleC22×C53#393
prime radical#16811173C22
hypercenter#1276D53
hypocenterC53#10364
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
1. #20283078C2semidirect
4. #20272574C2semidirect
5. C53×#1276C2semidirect
123. #1276D53semidirect
140. C53#10364semidirect

Maximal subgroups

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

subgroupindex
#202830782
#202755432
#202732762
#202725742
C53×#12762
#1036453