Finite Group Databasebeta

← #27296668

Subgroups — #27296668

Order 13568.

372 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 63 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#20275939C211yesmaximal, characteristic
2#20273751C212yesmaximal
3#20272742C212yesmaximal, complemented
4#20272661C211yesmaximal, characteristic, complemented
5C53×#2571C211yesmaximal, characteristic, complemented
6#16811204C2211yescharacteristic
7#16810941—22no—
8C2×#13133145—21no—
9#16810622—21no—
10#16810392C2211yescharacteristic
11#16810356—22no—
12#16810349C2212yes—
13C53×#400—21no—
14C53×#400C2212yes—
15C53×#389—22no—
16C53×#384C2211yescharacteristic
17C53×#382—22no—
18C4×C4⋊C4×C53—21no—
19C2×#11719945—21no—
20C2×#11719917—21no—
21#13133197—41no—
22#13133192—41no—
23#13133191—22no—
24#13133148—42no—
25#13133145—42no—
26#13133140—41no—
27#13133137—41no—
28#13133137—42no—
29C22⋊#88322—22no—
30C2×C4⋊C4×C53—21no—
31C2×C4⋊C4×C53—21no—
32C2×C4⋊C4×C53—21no—
33C2×C4⋊C4×C53—22no—
34C2×C4⋊C4×C53D411yescharacteristic
35C2×C4⋊C4×C53D412yes—
36C2×C4⋊C4×C53C2311yescharacteristic
37C2×C42×C53—22no—
38C2×C42×C53D411yescharacteristic
39#95×C53—21no—
40#95×C53—22no—
41#95×C53—22no—
42#95×C53—22no—
43#95×C53D412yes—
44C22×C4×C53—21no—
45C22×C4×C53—22no—
46C22×C4×C53—22no—
47C22×C4×C53—21no—
48C22×C4×C53C2×D411yescharacteristic
49C22×C4×C53C2×D412yes—
50C2×C22⋊D53—41no—
51#11719945—41no—
52#11719945—41no—
53#11719945—41no—
54#11719945—41no—
55C2×D212—21no—
56C2×D212—22no—
57C2×C4×D53—41no—
58C2×Dic106—21no—
59C4⋊C4×C53—41no—
60C4⋊C4×C53—42no—
61C4⋊C4×C53—44no—
62C4⋊C4×C53—42no—
63C4⋊C4×C53—41no—
64C4⋊C4×C53—42no—
65C4⋊C4×C53—81no—
66C42×C53—21no—
67C42×C53—44no—
68C42×C53SD1612yes—
69#11719921—42no—
70C4⋊Dic53—22no—
71#11719917—41no—
72#11719917—41no—
73C2×#88322—42no—
74C2×#88322—41no—
75C23×C53#12011yescharacteristic
76C22×D53—41no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—22no—
83C2×C4×C53—42no—
84C2×C4×C53—42no—
85C2×C4×C53—42no—
86C2×C4×C53—42no—
87C2×C4×C53—42no—
88C2×C4×C53—42no—
89C2×C4×C53—42no—
90C2×C4×C53—41no—
91C2×C4×C53—42no—
92C2×C4×C53C2×SD1611yescharacteristic
93C2×C4×C53#13711yescharacteristic
94C22⋊D53—81no—
95C22⋊D53—81no—
96C2×Dic53—41no—
97D212—41no—
98D212—41no—
99D212—82no—
100C4×D53—81no—
101C4×D53—81no—
102Dic106—41no—
103Dic106—41no—
104#88322—81no—
105#88322—82no—
106#12181—531nomaximal, Sylow
107C22×C53—21no—
108C22×C53—21no—
109C22×C53#44811yescharacteristic
110C22×C53#45411yescharacteristic
111C22×C53B(4,𝔽2)11yescharacteristic
112C2×D53—81no—
113C2×D53—81no—
114C2×D53—81no—
115C4×C53—24no—
116C4×C53—21no—
117C4×C53—21no—
118C4×C53—42no—
119C4×C53—82no—
120C4×C53—81no—
121C4×C53—82no—
122Dic53—81no—
123#2571D5311yescharacteristic, complemented
124#2039—531no—
125#1616—532no—
126#1353—532no—
127#1325—531no—
128C2×C53—21no—
129C2×C53—21no—
130C2×C53#159911yescharacteristic
131C2×C53#219811yescharacteristic
132C2×C53#220411yescharacteristic
133D53—161no—
134#537—531no—
135#481—1062no—
136C2×#104—1061no—
137#417—1061no—
138#400—21no—
139#400C2×D5312yes—
140#389—22no—
141#384C2×D5311yescharacteristic
142#382—22no—
143C4×C4⋊C4—21no—
144#343—531no—
145#327—1062no—
146#324—532no—
147C53#1218111yescharacteristic, complemented, Sylow
148C2×#41—1061no—
149C2×M16—1061no—
150C22⋊Q8—2121no—
151#121—1062no—
152#121—2121no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4C22⋊D5311yescharacteristic
158C2×C4⋊C4C22⋊D5312yes—
159C2×C4⋊C4C22×D5311yescharacteristic
160C2×C42—22no—
161C2×C42C22⋊D5311yescharacteristic
162#106—2122no—
163#104—2122no—
164#103—2121no—
165#102—2121no—
166#102—2122no—
167C22⋊C8—1062no—
168#95—21no—
169#95—22no—
170#95—22no—
171#95—22no—
172#95C22⋊D5312yes—
173#41—2121no—
174#41—2121no—
175#41—2121no—
176#41—2121no—
177C2×Q8—1061no—
178C2×D4—1062no—
179C2×D4—1061no—
180C2×D4—2121no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—21no—
185C22×C4—2121no—
186C22×C4C2×C22⋊D5311yescharacteristic
187C22×C4C2×C22⋊D5312yes—
188M16—2121no—
189M16—2121no—
190C2×C8—2122no—
191C2×C8—2121no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—44no—
197C4⋊C4—42no—
198C4⋊C4—81no—
199C4⋊C4—1062no—
200C22⋊C4—2122no—
201C42—21no—
202C42—44no—
203C42#1171992412yes—
204C23—2121no—
205C23#1313318611yescharacteristic
206Q8—2121no—
207Q8—2121no—
208D4—4242no—
209D4—4241no—
210D4—4241no—
211D4—2121no—
212D4—2121no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—4241no—
229C2×C4—4241no—
230C2×C4—2121no—
231C2×C4C2×#1171992411yescharacteristic
232C2×C4#1313324811yescharacteristic
233C8—4241no—
234C8—4242no—
235C22—21no—
236C22—21no—
237C22—4241no—
238C22—4241no—
239C22—4241no—
240C22#1681076211yescharacteristic
241C22#1681079611yescharacteristic
242C22#1681081011yescharacteristic
243C4—24no—
244C4—21no—
245C4—21no—
246C4—42no—
247C4—82no—
248C4—82no—
249C4—81no—
250C4—4241no—
251C2—21no—
252C2—21no—
253C2—8481no—
254C2#2027365511yescharacteristic
255C2#2027671111yescharacteristic
256C2#2027674111yescharacteristic
subgroupG/H
centerC22#16810810
commutator, cosocleC2×C4⋊C4×C53C23
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2571C2
socleC22×C53B(4,𝔽2)
prime radical#16811204C22
hypercenter#2571D53
hypocenterC53#12181
LCSG ▷ C2×C4⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2571

Complements

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

NH ≅ G/Ntype
3. #20272742C2semidirect
4. #20272661C2semidirect
5. C53×#2571C2semidirect
123. #2571D53semidirect
147. C53#12181semidirect

Maximal subgroups

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

subgroupindex
#202759392
#202737512
#202727422
#202726612
C53×#25712
#1218153