Finite Group Databasebeta

← #27296848

Subgroups — #27296848

Order 13568.

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

normal subgroups only

This group has 404 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#20275871C211yesmaximal, characteristic
2#20273722C212yesmaximal, complemented
3#20272721C212yesmaximal, complemented
4#20272647C211yesmaximal, characteristic
5C53×#2724C211yesmaximal, characteristic
6#16811254C2211yescharacteristic
7#16810953—22no—
8#16810628—21no—
9#16810392C2211yescharacteristic
10#16810383—21no—
11#16810359—22no—
12#16810349C2212yes—
13C2×#123×C53—21no—
14C53×#398C2212yes—
15C53×#397—22no—
16C53×#390C2211yescharacteristic
17C53×#389—21no—
18C53×#385—22no—
19C2×#11719917—21no—
20#13133216—41no—
21#13133208—41no—
22#13133196—22no—
23#13133176—21no—
24#13133148—42no—
25#13133140—42no—
26#13133138—41no—
27#13133138—41no—
28C22⋊#88322—22no—
29C22×D4×C53D411yescharacteristic
30C23×C4×C53—21no—
31#123×C53—42no—
32#123×C53—42no—
33C2×C4⋊C4×C53—21no—
34C2×C4⋊C4×C53—21no—
35C2×C4⋊C4×C53—22no—
36C2×C4⋊C4×C53D412yes—
37C2×C4⋊C4×C53C2311yescharacteristic
38C2×C22⋊C4×C53—22no—
39C2×C22⋊C4×C53—21no—
40C2×C22⋊C4×C53—22no—
41C2×C42×C53D411yescharacteristic
42#95×C53—22no—
43#95×C53—22no—
44#95×C53D412yes—
45C24×C53—21no—
46C2×D4×C53—21no—
47C2×D4×C53—41no—
48C2×D4×C53—41no—
49C2×D4×C53SD1612yes—
50C22×C4×C53—21no—
51C22×C4×C53—22no—
52C22×C4×C53—22no—
53C22×C4×C53—42no—
54C22×C4×C53—81no—
55C22×C4×C53—81no—
56C22×C4×C53C2×D411yescharacteristic
57C22×C4×C53C2×D412yes—
58C2×Dic106—22no—
59C4⋊C4×C53—41no—
60C4⋊C4×C53—42no—
61C4⋊C4×C53—42no—
62C4⋊C4×C53—42no—
63C4⋊C4×C53—42no—
64C4⋊C4×C53—81no—
65C22⋊C4×C53—44no—
66C22⋊C4×C53—44no—
67C22⋊C4×C53—42no—
68C42×C53—41no—
69C22⋊Dic53—41no—
70C4⋊Dic53—21no—
71C4⋊Dic53—21no—
72C4⋊Dic53—22no—
73#11719919—42no—
74C4×Dic53—41no—
75#11719917—41no—
76#11719917—41no—
77C2×#88322—42no—
78C2×#88322—41no—
79C23×C53—22no—
80C23×C53—42no—
81C23×C53—42no—
82C23×C53—41no—
83C23×C53—41no—
84C23×C53#12011yescharacteristic
85D4×C53—42no—
86D4×C53—81no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—42no—
94C2×C4×C53—42no—
95C2×C4×C53—42no—
96C2×C4×C53—42no—
97C2×C4×C53—42no—
98C2×C4×C53—42no—
99C2×C4×C53—42no—
100C2×C4×C53—84no—
101C2×C4×C53C2×SD1611yescharacteristic
102C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
103C2×Dic53—41no—
104C2×Dic53—41no—
105Dic106—82no—
106#88322—81no—
107#88322—82no—
108#12209—531nomaximal, Sylow
109C22×C53—21no—
110C22×C53—21no—
111C22×C53—44no—
112C22×C53—42no—
113C22×C53—82no—
114C22×C53—81no—
115C22×C53—81no—
116C22×C53#45011yescharacteristic
117C22×C53#45511yescharacteristic
118C22×C53B(4,𝔽2)11yescharacteristic
119C4×C53—21no—
120C4×C53—21no—
121C4×C53—42no—
122C4×C53—42no—
123C4×C53—82no—
124C4×C53—82no—
125Dic53—81no—
126Dic53—81no—
127#2724D5311yescharacteristic
128#2027—531no—
129#1611—532no—
130#1346—532no—
131#1318—531no—
132C2×C53—21no—
133C2×C53—21no—
134C2×C53—82no—
135C2×C53#160111yescharacteristic
136C2×C53#219811yescharacteristic
137C2×C53#220311yescharacteristic
138#544—531no—
139C2×#123—21no—
140#483—1062no—
141#418—1061no—
142#398C2×D5312yes—
143#397—22no—
144#390C2×D5311yescharacteristic
145#389—21no—
146#385—22no—
147#343—531no—
148#339—1061no—
149#328—1062no—
150#324—532no—
151C53#1220911yescharacteristic, complemented, Sylow
152C22×D4C22⋊D5311yescharacteristic
153C23×C4—21no—
154C2×M16—1061no—
155C4⋊Q8—2121no—
156#125—2121no—
157#123—42no—
158#123—42no—
159C22⋊Q8—1062no—
160#117—1061no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4C22⋊D5312yes—
165C2×C4⋊C4C22×D5311yescharacteristic
166C2×C22⋊C4—22no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—22no—
169C2×C42C22⋊D5311yescharacteristic
170#106—2122no—
171#103—2122no—
172#102—2121no—
173#102—2121no—
174C22⋊C8—1062no—
175#95—22no—
176#95—22no—
177#95C22⋊D5312yes—
178C24—21no—
179C2×Q8—1062no—
180C2×D4—21no—
181C2×D4—41no—
182C2×D4—41no—
183C2×D4#1171992312yes—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—42no—
188C22×C4—81no—
189C22×C4—81no—
190C22×C4C2×C22⋊D5311yescharacteristic
191C22×C4C2×C22⋊D5312yes—
192M16—2121no—
193M16—2121no—
194C2×C8—2122no—
195C2×C8—2121no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—81no—
202C4⋊C4—1061no—
203C4⋊C4—1061no—
204C4⋊C4—1062no—
205C4⋊C4—2122no—
206C22⋊C4—44no—
207C22⋊C4—42no—
208C22⋊C4—44no—
209C22⋊C4—2121no—
210C42—41no—
211C42—2121no—
212C23—22no—
213C23—42no—
214C23—42no—
215C23—41no—
216C23—41no—
217C23#1313318611yescharacteristic
218Q8—4242no—
219D4—42no—
220D4—81no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—84no—
235C2×C4—2121no—
236C2×C4—2121no—
237C2×C4C2×#1171992311yescharacteristic
238C2×C4#1313324311yescharacteristic
239C8—4241no—
240C8—4242no—
241C22—21no—
242C22—21no—
243C22—44no—
244C22—42no—
245C22—82no—
246C22—81no—
247C22—81no—
248C22#1681077411yescharacteristic
249C22#1681080111yescharacteristic
250C22#1681081011yescharacteristic
251C4—21no—
252C4—21no—
253C4—42no—
254C4—42no—
255C4—82no—
256C4—82no—
257C4—4241no—
258C4—4241no—
259C2—21no—
260C2—21no—
261C2—82no—
262C2#2027366711yescharacteristic
263C2#2027671111yescharacteristic
264C2#2027673611yescharacteristic
subgroupG/H
centerC22#16810810
commutator, cosocleC2×C4⋊C4×C53C23
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2724C2
socleC22×C53B(4,𝔽2)
prime radicalC22×D4×C53D4
hypercenter#2724D53
hypocenterC53#12209
LCSG ▷ C2×C4⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2724

Complements

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

NH ≅ G/Ntype
2. #20273722C2semidirect
3. #20272721C2semidirect
151. C53#12209semidirect

Maximal subgroups

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

subgroupindex
#202758712
#202737222
#202727212
#202726472
C53×#27242
#1220953