Finite Group Databasebeta

← #27287608

Subgroups — #27287608

Order 13568.

630 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 127 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#20279631C211yesmaximal, characteristic
2#20274712C212yesmaximal, complemented
3C2×#16810352C212yesmaximal, complemented
4#20272598C211yesmaximal, characteristic, complemented
5C53×#1844C211yesmaximal, characteristic, complemented
6C2×#13133192—22no—
7C2×C4×D212C2212yes—
8C2×C4⋊#88322C2212yes—
9C2×#13133137—22no—
10#16810540—22no—
11#16810480—22no—
12#16810385—22no—
13#16810352—24no—
14C2×#106×C53—22no—
15C2×C4×C8×C53C2211yescharacteristic
16C53×#384C2211yescharacteristic
17C53×#384C2211yescharacteristic
18C22×D212—22no—
19C22×D212D412yes—
20C22×C4×D53—22no—
21C22×#88322—22no—
22#13133192—44no—
23#13133192—44no—
24C4×D212—22no—
25C4×D212—22no—
26C4×D212—22no—
27C4×D212—22no—
28C2×C4⋊Dic53D412yes—
29C2×#11719921—24no—
30C2×#11719921—22no—
31C4⋊#88322—22no—
32C4⋊#88322—22no—
33#13133137—44no—
34#13133124—22no—
35C22×C8×C53—21no—
36C2×C4⋊C4×C53—22no—
37C2×C4⋊C4×C53—22no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53D412yes—
40C2×C42×C53C2311yescharacteristic
41#106×C53—44no—
42C4×C8×C53—21no—
43C4×C8×C53D412yescomplemented
44C23×D53—22no—
45C22×C4×C53—22no—
46C22×C4×C53—41no—
47C22×C4×C53C2×D411yescharacteristic
48C22×C4×C53C2×D412yes—
49C22×Dic53—22no—
50C2×D212—22no—
51C2×D212—22no—
52C2×D212—22no—
53C2×D212—22no—
54C2×D212—22no—
55C2×D212—48no—
56C2×D212—82no—
57C2×D212SD1614yes—
58C2×C4×D53—42no—
59C2×C4×D53—42no—
60C2×C4×D53—42no—
61C2×C4×D53—42no—
62C2×C4×D53—42no—
63C2×C4×D53—42no—
64C2×C8×C53—22no—
65C2×C8×C53—22no—
66C2×C8×C53—22no—
67C4⋊C4×C53—42no—
68C4⋊C4×C53—42no—
69C4⋊C4×C53—42no—
70C4⋊C4×C53—42no—
71C4⋊C4×C53—42no—
72C4⋊C4×C53—42no—
73C42×C53C2×D411yescharacteristic
74C42×C53#4111yescharacteristic
75C42×C53#4112yes—
76#11719921—48no—
77#11719921—42no—
78#11719921—42no—
79C4⋊Dic53—22no—
80C4⋊Dic53—22no—
81C2×#88322—42no—
82C2×#88322—42no—
83C2×#88322—42no—
84C23×C53#12011yescharacteristic
85C22×D53—24no—
86C22×D53—42no—
87C22×D53—42no—
88C22×D53—42no—
89C22×D53—42no—
90C22×D53—42no—
91C22×D53—42no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—44no—
95C2×C4×C53—42no—
96C2×C4×C53—82no—
97C2×C4×C53—81no—
98C2×C4×C53#12111yescharacteristic
99C2×C4×C53#12112yes—
100C2×C4×C53#12312yes—
101C2×C4×C53#12411yescharacteristic
102C2×C4×C53C2×SD1612yes—
103C2×C4×C53#13512yes—
104C2×C4×C53AGL(1,ℤ/8)12yes—
105C2×C4×C53AGL(1,ℤ/8)12yes—
106C2×Dic53—42no—
107C2×Dic53—42no—
108C2×Dic53—42no—
109D212—24no—
110D212—24no—
111D212—42no—
112D212—42no—
113D212—88no—
114C4×D53—82no—
115C4×D53—82no—
116C4×D53—82no—
117C4×D53—82no—
118C8×C53—24no—
119#88322—82no—
120#10432—531nomaximal, Sylow
121C22×C53#39411yescharacteristic
122C22×C53#44912yes—
123C22×C53#45012yes—
124C22×C53#45311yescharacteristic
125C22×C53#45611yescharacteristic
126C2×D53—44no—
127C2×D53—44no—
128C2×D53—44no—
129C2×D53—82no—
130C2×D53—82no—
131C2×D53—82no—
132C2×D53—82no—
133C4×C53—21no—
134C4×C53—21no—
135C4×C53—82no—
136C4×C53—161no—
137C4×C53#46512yes—
138C4×C53#46912yes—
139C4×C53#48112yes—
140C4×C53#48212yes—
141Dic53—82no—
142#2537—531no—
143#1844D5311yescharacteristic, complemented
144#1802—532no—
145C2×D4⋊C8—532no—
146#1296—531no—
147C2×C53#162411yescharacteristic
148C2×C53#162911yescharacteristic
149C2×C53#165412yes—
150C2×C53#203412yes—
151C2×C53#203911yescharacteristic
152D53—84no—
153C2×#121—1062no—
154C2×C4×D4—532no—
155C2×#106—22no—
156C2×C4⋊C8—532no—
157C2×#102—1062no—
158C2×C4×C8C2×D5311yescharacteristic
159#394—1062no—
160#384C2×D5311yescharacteristic
161#384C2×D5311yescharacteristic
162#380—1062no—
163#340—1062no—
164D4⋊C8—1064no—
165C53#1043211yescharacteristic, complemented, Sylow
166C22×D4—532no—
167C22×D4—1062no—
168C23×C4—1062no—
169C22×C8—21no—
170C22×C8—1062no—
171#121—2124no—
172#121—2124no—
173C4×D4—1062no—
174C4×D4—1062no—
175C4×D4—1062no—
176C4×D4—1062no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—532no—
181C2×C4⋊C4C22⋊D5312yes—
182C2×C22⋊C4—1064no—
183C2×C22⋊C4—1062no—
184C2×C42C22×D5311yescharacteristic
185#106—44no—
186C4⋊C8—1062no—
187C4⋊C8—1062no—
188#102—2124no—
189C4×C8—21no—
190C4×C8D21212yescomplemented
191#95—1062no—
192C24—1062no—
193C2×D4—534no—
194C2×D4—1062no—
195C2×D4—1062no—
196C2×D4—1062no—
197C2×D4—1062no—
198C2×D4—1062no—
199C2×D4—4242no—
200C2×D4—2128no—
201C22×C4—22no—
202C22×C4—41no—
203C22×C4—1062no—
204C22×C4—2122no—
205C22×C4—2122no—
206C22×C4—2122no—
207C22×C4—2122no—
208C22×C4—2122no—
209C22×C4—2122no—
210C22×C4D4×D5312yes—
211C22×C4C2×C22⋊D5311yescharacteristic
212C2×C8—22no—
213C2×C8—22no—
214C2×C8—22no—
215C2×C8—2122no—
216C2×C8—2122no—
217C2×C8—2122no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—1062no—
225C4⋊C4—1062no—
226C22⋊C4—2128no—
227C22⋊C4—2122no—
228C22⋊C4—2122no—
229C42C2×D21211yescharacteristic
230C42#1171994511yescharacteristic
231C42#1171994912yes—
232C23—1064no—
233C23—2122no—
234C23—2122no—
235C23—2122no—
236C23—2122no—
237C23—2122no—
238C23—2122no—
239C23#1313318511yescharacteristic
240D4—1064no—
241D4—1064no—
242D4—4248no—
243D4—2122no—
244D4—2122no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—44no—
248C2×C4—42no—
249C2×C4—82no—
250C2×C4—81no—
251C2×C4—4242no—
252C2×C4—4242no—
253C2×C4—4242no—
254C2×C4—4242no—
255C2×C4—2122no—
256C2×C4—2122no—
257C2×C4—2122no—
258C2×C4#1313319111yescharacteristic
259C2×C4#1313319212yes—
260C2×C4#1313320212yes—
261C2×C4#1313320611yescharacteristic
262C2×C4SD16×D5312yes—
263C2×C4#1313323812yes—
264C2×C4#1313324412yes—
265C2×C4#1313324412yes—
266C8—24no—
267C8—4242no—
268C22—4242no—
269C22—4242no—
270C22—4242no—
271C22—4242no—
272C22—2124no—
273C22—2124no—
274C22—2124no—
275C22#1681054011yescharacteristic
276C22#1681076712yes—
277C22#1681077212yes—
278C22#1681079011yescharacteristic
279C22#1681080511yescharacteristic
280C4—21no—
281C4—21no—
282C4—82no—
283C4—161no—
284C4—4242no—
285C4#1681085312yes—
286C4#1681087712yes—
287C4#1681094312yes—
288C4#1681094912yes—
289C2—4244no—
290C2#2027379611yescharacteristic
291C2#2027382911yescharacteristic
292C2#2027399612yes—
293C2#2027590612yes—
294C2#2027593911yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1844C2
socleC23×C53#120
prime radical#20279631C2
hypercenter#1844D53
hypocenterC53#10432
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1844

Complements

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

NH ≅ G/Ntype
2. #20274712C2semidirect
3. C2×#16810352C2semidirect
4. #20272598C2semidirect
5. C53×#1844C2semidirect
43. C4×C8×C53D4semidirect
143. #1844D53semidirect
165. C53#10432semidirect
190. C4×C8D212semidirect

Maximal subgroups

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

subgroupindex
#202796312
#202747122
C2×#168103522
#202725982
C53×#18442
#1043253