Finite Group Databasebeta

← #27297085

Subgroups — #27297085

Order 13568.

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

normal subgroups only

This group has 558 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#20276060C211yesmaximal, characteristic, complemented
2#20273654C212yesmaximal, complemented
3#20272718C212yesmaximal, complemented
4#20272661C211yesmaximal, characteristic, complemented
5C53×#2537C211yesmaximal, characteristic
6#16811254C2211yescharacteristic
7#16810774—22no—
8#16810663—21no—
9#16810392C2211yescharacteristic
10#16810383—21no—
11#16810359—22no—
12#16810349C2212yes—
13C2×#121×C53—22no—
14C2×C4×D4×C53—21no—
15C53×#394—21no—
16C53×#394C2212yes—
17C53×#384C2211yescharacteristic
18C53×#380—22no—
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×D4×C53—21no—
30C22×D4×C53—21no—
31C22×D4×C53D412yes—
32C23×C4×C53—22no—
33#121×C53—48no—
34C4×D4×C53—42no—
35C4×D4×C53—42no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53—22no—
38C2×C4⋊C4×C53D411yescharacteristic
39C2×C4⋊C4×C53C2311yescharacteristic
40C2×C22⋊C4×C53—22no—
41C2×C22⋊C4×C53—22no—
42C2×C22⋊C4×C53—22no—
43C2×C42×C53D411yescharacteristic
44#95×C53—21no—
45#95×C53D412yes—
46C24×C53—22no—
47C2×D4×C53—24no—
48C2×D4×C53—42no—
49C2×D4×C53—42no—
50C2×D4×C53—42no—
51C2×D4×C53—44no—
52C2×D4×C53—44no—
53C2×D4×C53—81no—
54C22×C4×C53—21no—
55C22×C4×C53—22no—
56C22×C4×C53—21no—
57C22×C4×C53—44no—
58C22×C4×C53—44no—
59C22×C4×C53—44no—
60C22×C4×C53C2×D411yescharacteristic
61C22×C4×C53C2×D412yes—
62C2×Dic106—22no—
63C4⋊C4×C53—41no—
64C4⋊C4×C53—42no—
65C4⋊C4×C53—44no—
66C4⋊C4×C53—41no—
67C22⋊C4×C53—44no—
68C22⋊C4×C53—44no—
69C22⋊C4×C53—42no—
70C22⋊C4×C53—42no—
71C42×C53—21no—
72C42×C53SD1612yescomplemented
73C22⋊Dic53—41no—
74#11719923—84no—
75C4⋊Dic53—22no—
76C4⋊Dic53—21no—
77C4⋊Dic53—21no—
78#11719919—42no—
79C4×Dic53—41no—
80#11719917—22no—
81#11719917—41no—
82C2×#88322—41no—
83C2×#88322—42no—
84C23×C53—24no—
85C23×C53—42no—
86C23×C53—42no—
87C23×C53—42no—
88C23×C53—42no—
89C23×C53—42no—
90C23×C53—42no—
91C23×C53#12011yescharacteristic
92D4×C53—48no—
93D4×C53—44no—
94D4×C53—82no—
95D4×C53—84no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—22no—
102C2×C4×C53—42no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×C4×C53—41no—
108C2×C4×C53—88no—
109C2×C4×C53C2×SD1611yescharacteristic
110C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
111C2×Dic53—41no—
112C2×Dic53—41no—
113Dic106—82no—
114#88322—42no—
115#88322—82no—
116#12246—531nomaximal, Sylow
117C22×C53—21no—
118C22×C53—21no—
119C22×C53—44no—
120C22×C53—44no—
121C22×C53—44no—
122C22×C53—82no—
123C22×C53—82no—
124C22×C53—82no—
125C22×C53—82no—
126C22×C53#45011yescharacteristic
127C22×C53#45611yescharacteristic
128C22×C53B(4,𝔽2)11yescharacteristic
129C4×C53—24no—
130C4×C53—21no—
131C4×C53—21no—
132C4×C53—42no—
133C4×C53—82no—
134C4×C53—81no—
135Dic53—81no—
136Dic53—81no—
137#2537D5311yescharacteristic
138#2060—531no—
139#1599—532no—
140#1345—532no—
141#1325—531no—
142C2×C53—21no—
143C2×C53—21no—
144C2×C53—84no—
145C2×C53#160411yescharacteristic
146C2×C53#219611yescharacteristic
147C2×C53#220411yescharacteristic
148#544—531no—
149C2×#121—22no—
150C2×C4×D4—21no—
151#450—1062no—
152#428—1061no—
153#394—21no—
154#394C2×D5312yes—
155#384C2×D5311yescharacteristic
156#380—22no—
157#343—531no—
158#339—1061no—
159#328—1062no—
160#324—532no—
161C53#1224611yescharacteristic, complemented, Sylow
162C22×D4—21no—
163C22×D4—21no—
164C22×D4C22⋊D5312yes—
165C23×C4—22no—
166C2×SD16—2122no—
167C2×M16—1061no—
168C4⋊Q8—2121no—
169#125—2121no—
170C22⋊Q8—1062no—
171#121—48no—
172C4×D4—42no—
173C4×D4—42no—
174#117—1061no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4C22⋊D5311yescharacteristic
178C2×C4⋊C4C22×D5311yescharacteristic
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C42C22⋊D5311yescharacteristic
183#107—2121no—
184#106—2121no—
185#102—2122no—
186C22⋊C8—1062no—
187#95—21no—
188#95C22⋊D5312yes—
189C24—22no—
190C2×Q8—1062no—
191C2×D4—24no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—44no—
196C2×D4—44no—
197C2×D4—81no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—44no—
204C22×C4C2×C22⋊D5311yescharacteristic
205C22×C4C2×C22⋊D5312yes—
206SD16—4244no—
207M16—1062no—
208M16—2121no—
209C2×C8—2122no—
210C2×C8—2121no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C4⋊C4—44no—
215C4⋊C4—1062no—
216C4⋊C4—1061no—
217C4⋊C4—1061no—
218C4⋊C4—2122no—
219C22⋊C4—44no—
220C22⋊C4—44no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—2121no—
224C42—21no—
225C42—2121no—
226C42#1171992312yescomplemented
227C23—24no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23#1313318611yescharacteristic
235Q8—4242no—
236D4—48no—
237D4—44no—
238D4—82no—
239D4—84no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—88no—
253C2×C4—2121no—
254C2×C4—2121no—
255C2×C4C2×#1171992311yescharacteristic
256C2×C4#1313324311yescharacteristic
257C8—4242no—
258C8—2122no—
259C22—21no—
260C22—21no—
261C22—44no—
262C22—44no—
263C22—44no—
264C22—82no—
265C22—82no—
266C22—82no—
267C22—82no—
268C22#1681077411yescharacteristic
269C22#1681080611yescharacteristic
270C22#1681081011yescharacteristic
271C4—24no—
272C4—21no—
273C4—21no—
274C4—42no—
275C4—82no—
276C4—81no—
277C4—4241no—
278C4—4241no—
279C2—21no—
280C2—21no—
281C2—84no—
282C2#2027368411yescharacteristic
283C2#2027669911yescharacteristic
284C2#2027674111yescharacteristic
subgroupG/H
centerC22#16810810
commutator, cosocleC2×C4⋊C4×C53C23
FrattiniC2×C4⋊C4C22×D53
Fitting, prime radicalC53×#2537C2
socleC22×C53B(4,𝔽2)
hypercenter#2537D53
hypocenterC53#12246
LCSG ▷ C2×C4⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2537

Complements

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

NH ≅ G/Ntype
1. #20276060C2semidirect
2. #20273654C2semidirect
3. #20272718C2semidirect
4. #20272661C2semidirect
72. C42×C53SD16semidirect
161. C53#12246semidirect
226. C42#11719923semidirect

Maximal subgroups

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

subgroupindex
#202760602
#202736542
#202727182
#202726612
C53×#25372
#1224653