Finite Group Databasebeta

← #27287177

Subgroups — #27287177

Order 13568.

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

normal subgroups only

This group has 454 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#20275543C211yesmaximal, characteristic
2#20273274C212yesmaximal
3#20272573C212yesmaximal, complemented
4#20272548C211yesmaximal, characteristic, complemented
5C53×#2862C211yesmaximal, characteristic, complemented
6#16811173C2211yescharacteristic
7#16810676C2212yes—
8#16810647C2211yescharacteristic
9C2×#13133145—21no—
10C2×#13133145—21no—
11#16810356—24no—
12C2×C4⋊Q8×C53—21no—
13C53×#391—24no—
14C53×#382—21no—
15C53×#382C2212yes—
16C43×C53C2211yescharacteristic
17C2×#11719945—21no—
18C2×#11719917—21no—
19#13133217Q811yescharacteristic, complemented
20#13133214Q811yescharacteristic, complemented
21#13133205—21no—
22#13133191—22no—
23C4×D212—22no—
24C4⋊#88322—22no—
25C4⋊#88322—22no—
26#13133145—42no—
27#13133145—42no—
28C22⋊#88322—22no—
29C4×#88322—22no—
30C22×Q8×C53—22no—
31C4⋊Q8×C53—42no—
32C4⋊Q8×C53—42no—
33C2×C4⋊C4×C53—22no—
34C2×C4⋊C4×C53—22no—
35C2×C42×C53—22no—
36C2×C42×C53D411yescharacteristic
37C2×C42×C53D411yescharacteristic
38C2×C42×C53C2311yescharacteristic
39#95×C53—24no—
40#95×C53—22no—
41#95×C53D414yes—
42C2×Q8×C53—44no—
43C2×Q8×C53—48no—
44C22×C4×C53—21no—
45C22×C4×C53—21no—
46C22×C4×C53—24no—
47C22×C4×C53C2×D412yes—
48C22×C4×C53C2×D411yescharacteristic
49C2×C22⋊D53—41no—
50#11719945—41no—
51#11719945—41no—
52#11719945—41no—
53#11719945—41no—
54C2×D212—21no—
55C2×D212—22no—
56C2×C4×D53—41no—
57C2×Dic106—21no—
58C4⋊C4×C53—44no—
59C4⋊C4×C53—42no—
60C4⋊C4×C53—42no—
61C42×C53—21no—
62C42×C53—21no—
63C42×C53—44no—
64C42×C53D812yes—
65C42×C53SD1612yes—
66C42×C53C2×Q811yescharacteristic
67C42×C53#4111yescharacteristic
68C42×C53#4112yes—
69#11719921—42no—
70C4⋊Dic53—22no—
71#11719917—41no—
72#11719917—41no—
73C2×#88322—42no—
74C2×#88322—41no—
75C23×C53#12011yescharacteristic
76C22×D53—41no—
77Q8×C53—88no—
78C2×C4×C53—24no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53—42no—
82C2×C4×C53—42no—
83C2×C4×C53—48no—
84C2×C4×C53—44no—
85C2×C4×C53C22⋊Q812yes—
86C2×C4×C53C22⋊Q811yescharacteristic
87C2×C4×C53#12412yes—
88C2×C4×C53#12411yescharacteristic
89C2×C4×C53C2×D811yescharacteristic
90C2×C4×C53C2×SD1611yescharacteristic
91C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
92C2×C4×C53#13711yescharacteristic
93C22⋊D53—81no—
94C22⋊D53—81no—
95C2×Dic53—41no—
96D212—41no—
97D212—41no—
98D212—82no—
99C4×D53—81no—
100C4×D53—81no—
101Dic106—41no—
102Dic106—41no—
103#88322—82no—
104#88322—81no—
105#10362—531nomaximal, Sylow
106C22×C53—22no—
107C22×C53#39311yescharacteristic
108C22×C53#44711yescharacteristic
109C22×C53#44811yescharacteristic
110C2×D53—81no—
111C2×D53—82no—
112C4×C53—24no—
113C4×C53—22no—
114C4×C53—22no—
115C4×C53—42no—
116C4×C53—84no—
117C4×C53#47411yescharacteristic
118C4×C53#47611yescharacteristic
119C4×C53#48611yescharacteristic
120C4×C53#48911yescharacteristic
121Dic53—81no—
122#2862D5311yescharacteristic, complemented
123#1962—531no—
124#1517—532no—
125#1286—532no—
126#1276—531no—
127C2×C53—21no—
128C2×C53—21no—
129C2×C53#202311yescharacteristic
130C2×C53#203112yes—
131D53—161no—
132#532—531no—
133C2×C4⋊Q8—21no—
134#431—532no—
135#423—531no—
136C2×#104—1061no—
137C2×#104—1061no—
138#391—24no—
139#382—21no—
140#382C2×D5312yes—
141C43C2×D5311yescharacteristic
142#327—1064no—
143C53#1036211yescharacteristic, complemented, Sylow
144C2×#41—1061no—
145C22×Q8—22no—
146C2×M16—1061no—
147C4⋊Q8—42no—
148C4⋊Q8—42no—
149C4⋊Q8—531no—
150#127—531no—
151#124—1061no—
152#121—1062no—
153C4×D4—1062no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C42—22no—
157C2×C42C22⋊D5311yescharacteristic
158C2×C42C22⋊D5311yescharacteristic
159C2×C42C22×D5311yescharacteristic
160C4⋊C8—1062no—
161C4⋊C8—1062no—
162#104—2122no—
163#104—2122no—
164C22⋊C8—1062no—
165C4×C8—1062no—
166#95—24no—
167#95—22no—
168#95C22⋊D5314yes—
169#41—2121no—
170#41—2121no—
171#41—2121no—
172#41—2121no—
173C2×Q8—44no—
174C2×Q8—48no—
175C2×Q8—1061no—
176C2×D4—1062no—
177C2×D4—1061no—
178C2×D4—2121no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—24no—
182C22×C4—2121no—
183C22×C4C2×C22⋊D5312yes—
184C22×C4C2×C22⋊D5311yescharacteristic
185M16—2121no—
186M16—2121no—
187C2×C8—2122no—
188C2×C8—2121no—
189C4⋊C4—44no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—1062no—
193C22⋊C4—2122no—
194C42—21no—
195C42—21no—
196C42—44no—
197C42#1171992212yes—
198C42#1171992412yes—
199C42Q8×D5311yescharacteristic
200C42#1171994911yescharacteristic
201C42#1171994912yes—
202C23—2121no—
203C23#1313318611yescharacteristic
204Q8—88no—
205Q8—2121no—
206Q8—2121no—
207D4—4242no—
208D4—4241no—
209D4—4241no—
210D4—2121no—
211D4—2121no—
212C2×C4—24no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—48no—
218C2×C4—44no—
219C2×C4—4241no—
220C2×C4—4241no—
221C2×C4—2121no—
222C2×C4#1313319312yes—
223C2×C4#1313319311yescharacteristic
224C2×C4#1313320612yes—
225C2×C4#1313320611yescharacteristic
226C2×C4C2×#1171992211yescharacteristic
227C2×C4C2×#1171992411yescharacteristic
228C2×C4#1313324311yescharacteristic
229C2×C4#1313324811yescharacteristic
230C8—4242no—
231C8—4241no—
232C22—22no—
233C22—4241no—
234C22—4242no—
235C22#1681053411yescharacteristic
236C22#1681075711yescharacteristic
237C22#1681076211yescharacteristic
238C4—24no—
239C4—22no—
240C4—22no—
241C4—42no—
242C4—84no—
243C4—4241no—
244C4#1681090011yescharacteristic
245C4#1681091011yescharacteristic
246C4#1681097111yescharacteristic
247C4#1681098511yescharacteristic
248C2—21no—
249C2—21no—
250C2—8481no—
251C2#2027584811yescharacteristic
252C2#2027588412yes—
subgroupG/H
centerC22#16810534
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2862C2
socleC22×C53#393
prime radical#16811173C22
hypercenter#2862D53
hypocenterC53#10362
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2862

Complements

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

NH ≅ G/Ntype
3. #20272573C2semidirect
4. #20272548C2semidirect
5. C53×#2862C2semidirect
19. #13133217Q8semidirect
20. #13133214Q8semidirect
122. #2862D53semidirect
143. C53#10362semidirect

Maximal subgroups

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

subgroupindex
#202755432
#202732742
#202725732
#202725482
C53×#28622
#1036253