Finite Group Databasebeta

← #27297109

Subgroups — #27297109

Order 13568.

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

normal subgroups only

This group has 454 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#20276065C211yesmaximal, characteristic, complemented
2#20273654C212yesmaximal, complemented
3#20272718C212yesmaximal, complemented
4#20272659C211yesmaximal, characteristic, complemented
5C53×#2819C211yesmaximal, characteristic
6#16811254C2211yescharacteristic
7#16810774—22no—
8#16810663—21no—
9#16810392C2211yescharacteristic
10#16810383—21no—
11#16810359—22no—
12#16810349C2212yes—
13C2×#124×C53—21no—
14C53×#394—21no—
15C53×#394C2212yes—
16C53×#393—22no—
17C53×#392—22no—
18C53×#382C2211yescharacteristic
19C2×#11719923—42no—
20C2×#11719917—21no—
21#13133216—41no—
22#13133208—41no—
23#13133196—22no—
24#13133176—21no—
25#13133150—41no—
26#13133148—41no—
27#13133138—42no—
28C22⋊#88322—22no—
29C22×Q8×C53—21no—
30C22×D4×C53—21no—
31C22×D4×C53—21no—
32C22×D4×C53D412yes—
33#124×C53—42no—
34#124×C53—42no—
35C2×C4⋊C4×C53D411yescharacteristic
36C2×C4⋊C4×C53C2311yescharacteristic
37C2×C22⋊C4×C53—22no—
38C2×C22⋊C4×C53—22no—
39C2×C22⋊C4×C53—22no—
40C2×C22⋊C4×C53—22no—
41C2×C42×C53D411yescharacteristic
42#95×C53—21no—
43#95×C53—22no—
44#95×C53D412yes—
45C24×C53—22no—
46C2×Q8×C53—42no—
47C2×Q8×C53—81no—
48C2×Q8×C53—81no—
49C2×D4×C53—42no—
50C2×D4×C53—42no—
51C2×D4×C53—82no—
52C2×D4×C53—81no—
53C2×D4×C53—81no—
54C2×D4×C53—82no—
55C2×D4×C53—81no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—22no—
59C22×C4×C53C2×D411yescharacteristic
60C22×C4×C53C2×D412yes—
61C2×Dic106—22no—
62C4⋊C4×C53—41no—
63C4⋊C4×C53—41no—
64C22⋊C4×C53—44no—
65C22⋊C4×C53—42no—
66C22⋊C4×C53—42no—
67C22⋊C4×C53—82no—
68C22⋊C4×C53—82no—
69C42×C53—21no—
70C42×C53SD1612yescomplemented
71C22⋊Dic53—41no—
72#11719923—84no—
73C4⋊Dic53—22no—
74C4⋊Dic53—21no—
75C4⋊Dic53—21no—
76#11719919—42no—
77C4×Dic53—41no—
78#11719917—22no—
79#11719917—41no—
80C2×#88322—41no—
81C2×#88322—42no—
82C23×C53—42no—
83C23×C53—42no—
84C23×C53—42no—
85C23×C53—42no—
86C23×C53—42no—
87C23×C53—42no—
88C23×C53—42no—
89C23×C53#12011yescharacteristic
90Q8×C53—84no—
91D4×C53—44no—
92D4×C53—82no—
93D4×C53—84no—
94D4×C53—84no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—41no—
100C2×C4×C53—41no—
101C2×C4×C53—42no—
102C2×C4×C53—42no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53—41no—
107C2×C4×C53C2×SD1611yescharacteristic
108C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
109C2×Dic53—41no—
110C2×Dic53—41no—
111Dic106—82no—
112#88322—42no—
113#88322—82no—
114#12250—531nomaximal, Sylow
115C22×C53—21no—
116C22×C53—21no—
117C22×C53—82no—
118C22×C53—82no—
119C22×C53—82no—
120C22×C53—82no—
121C22×C53—82no—
122C22×C53—82no—
123C22×C53—82no—
124C22×C53#45011yescharacteristic
125C22×C53#45611yescharacteristic
126C22×C53B(4,𝔽2)11yescharacteristic
127C4×C53—21no—
128C4×C53—21no—
129C4×C53—42no—
130C4×C53—42no—
131C4×C53—82no—
132C4×C53—81no—
133Dic53—81no—
134Dic53—81no—
135#2819D5311yescharacteristic
136#2061—531no—
137#1599—532no—
138#1345—532no—
139#1324—531no—
140C2×C53—21no—
141C2×C53—21no—
142C2×C53—162no—
143C2×C53#160411yescharacteristic
144C2×C53#219911yescharacteristic
145C2×C53#220211yescharacteristic
146#544—531no—
147C2×#124—21no—
148#450—1062no—
149#428—1061no—
150#394—21no—
151#394C2×D5312yes—
152#393—22no—
153#392—22no—
154#382C2×D5311yescharacteristic
155#343—531no—
156#339—1061no—
157#328—1062no—
158#324—532no—
159C53#1225011yescharacteristic, complemented, Sylow
160C22×Q8—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4C22⋊D5312yes—
164C2×SD16—2122no—
165C2×M16—1061no—
166C4⋊Q8—2121no—
167#125—2121no—
168#124—42no—
169#124—42no—
170C22⋊Q8—1062no—
171#117—1061no—
172C2×C4⋊C4C22⋊D5311yescharacteristic
173C2×C4⋊C4C22×D5311yescharacteristic
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C42C22⋊D5311yescharacteristic
179#107—2121no—
180#106—2121no—
181#102—2122no—
182C22⋊C8—1062no—
183#95—21no—
184#95—22no—
185#95C22⋊D5312yes—
186C24—22no—
187C2×Q8—42no—
188C2×Q8—81no—
189C2×Q8—81no—
190C2×Q8—1062no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—82no—
194C2×D4—81no—
195C2×D4—81no—
196C2×D4—82no—
197C2×D4—81no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4C2×C22⋊D5311yescharacteristic
202C22×C4C2×C22⋊D5312yes—
203SD16—4244no—
204M16—1062no—
205M16—2121no—
206C2×C8—2122no—
207C2×C8—2121no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—1062no—
211C4⋊C4—1061no—
212C4⋊C4—1061no—
213C4⋊C4—2122no—
214C22⋊C4—44no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—82no—
218C22⋊C4—82no—
219C22⋊C4—2121no—
220C42—21no—
221C42—2121no—
222C42#1171992312yescomplemented
223C23—42no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23#1313318611yescharacteristic
231Q8—84no—
232Q8—4242no—
233D4—44no—
234D4—82no—
235D4—84no—
236D4—84no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—2121no—
250C2×C4—2121no—
251C2×C4C2×#1171992311yescharacteristic
252C2×C4#1313324311yescharacteristic
253C8—4242no—
254C8—2122no—
255C22—21no—
256C22—21no—
257C22—82no—
258C22—82no—
259C22—82no—
260C22—82no—
261C22—82no—
262C22—82no—
263C22—82no—
264C22#1681077411yescharacteristic
265C22#1681080611yescharacteristic
266C22#1681081011yescharacteristic
267C4—21no—
268C4—21no—
269C4—42no—
270C4—42no—
271C4—82no—
272C4—81no—
273C4—4241no—
274C4—4241no—
275C2—21no—
276C2—21no—
277C2—162no—
278C2#2027368411yescharacteristic
279C2#2027671611yescharacteristic
280C2#2027673211yescharacteristic
subgroupG/H
centerC22#16810810
commutator, cosocleC2×C4⋊C4×C53C23
FrattiniC2×C4⋊C4C22×D53
Fitting, prime radicalC53×#2819C2
socleC22×C53B(4,𝔽2)
hypercenter#2819D53
hypocenterC53#12250
LCSG ▷ C2×C4⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2819

Complements

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

NH ≅ G/Ntype
1. #20276065C2semidirect
2. #20273654C2semidirect
3. #20272718C2semidirect
4. #20272659C2semidirect
70. C42×C53SD16semidirect
159. C53#12250semidirect
222. C42#11719923semidirect

Maximal subgroups

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

subgroupindex
#202760652
#202736542
#202727182
#202726592
C53×#28192
#1225053