Finite Group Databasebeta

← #27276705

Subgroups — #27276705

Order 13568.

1222 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 1222 conjugacy classes of subgroups — too many to draw. Only its 147 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#20274789C214yesmaximal, complemented
2#20274630C212yesmaximal
3C22×#124×C53C211yesmaximal, characteristic
4C2×#13133132—24no—
5#16810479C2214yes—
6#16810390—28no—
7C23×Q8×C53C411yescharacteristic, complemented
8C23×D4×C53C2211yescharacteristic
9C2×#124×C53—24no—
10C2×#124×C53—28no—
11C22×C22⋊C4×C53C412yescomplemented
12C22×C22⋊C4×C53C2212yes—
13C22×C42×C53C411yescharacteristic, complemented
14C2×C22⋊Dic53—24no—
15C2×C22⋊Dic53—28no—
16#13133132—48no—
17C25×C53C2×C411yescharacteristic
18C25×C53C2311yescharacteristic
19C22×Q8×C53—24no—
20C22×Q8×C53—22no—
21C22×Q8×C53—44no—
22C22×D4×C53—24no—
23C22×D4×C53—44no—
24C22×D4×C53D414yes—
25C23×C4×C53C2×C411yescharacteristic
26C23×C4×C53C2×C412yes—
27C23×C4×C53C2×C412yes—
28#124×C53—232no—
29C2×C22⋊C4×C53—24no—
30C2×C22⋊C4×C53—24no—
31C2×C22⋊C4×C53—24no—
32C2×C22⋊C4×C53—24no—
33C2×C22⋊C4×C53—24no—
34C2×C22⋊C4×C53D418yes—
35C2×C42×C53—24no—
36C2×C42×C53—22no—
37C24×C53—22no—
38C24×C53—42no—
39C24×C53—42no—
40C24×C53—44no—
41C24×C53—44no—
42C24×C53C22⋊C412yes—
43C24×C53C22×C411yescharacteristic
44C24×C53C2×D412yes—
45C24×C53C2×D414yes—
46C2×Q8×C53—28no—
47C2×Q8×C53—416no—
48C2×Q8×C53—44no—
49C2×Q8×C53—44no—
50C2×D4×C53—28no—
51C2×D4×C53—28no—
52C2×D4×C53—44no—
53C2×D4×C53—416no—
54C22×C4×C53—22no—
55C22×C4×C53—24no—
56C22×C4×C53—22no—
57C22×C4×C53—22no—
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53—44no—
61C22×C4×C53—44no—
62C22×C4×C53C22⋊C412yes—
63C22×C4×C53C22⋊C414yes—
64C22×C4×C53C22⋊C414yes—
65C22×Dic53—44no—
66C22⋊C4×C53—216no—
67C22⋊C4×C53—216no—
68C42×C53—28no—
69C22⋊Dic53—48no—
70C22⋊Dic53—416no—
71C23×C53—22no—
72C23×C53—24no—
73C23×C53—24no—
74C23×C53—28no—
75C23×C53—22no—
76C23×C53—24no—
77C23×C53—44no—
78C23×C53—44no—
79C23×C53—44no—
80C23×C53—44no—
81C23×C53—44no—
82C23×C53—44no—
83C23×C53—44no—
84C23×C53—88no—
85C23×C53—88no—
86C23×C53C2×C22⋊C411yescharacteristic
87C23×C53C2×C22⋊C412yes—
88C23×C53#12014yes—
89Q8×C53—416no—
90D4×C53—416no—
91C2×C4×C53—28no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—24no—
95C2×C4×C53—24no—
96C2×C4×C53—44no—
97C2×C4×C53—44no—
98C2×C4×C53—44no—
99C2×C4×C53—44no—
100C2×C4×C53—44no—
101C2×C4×C53—44no—
102C2×C4×C53#9914yes—
103C2×Dic53—84no—
104C2×Dic53—88no—
105#8544—531nomaximal, Sylow
106C22×C53—28no—
107C22×C53—22no—
108C22×C53—22no—
109C22×C53—22no—
110C22×C53—24no—
111C22×C53—28no—
112C22×C53—42no—
113C22×C53—44no—
114C22×C53—84no—
115C22×C53—88no—
116C22×C53—84no—
117C22×C53—88no—
118C22×C53#37911yescharacteristic
119C22×C53C2×#9912yes—
120C22×C53#41014yes—
121C4×C53—24no—
122C4×C53—44no—
123C4×C53—44no—
124Dic53—164no—
125C22×#124D5311yescharacteristic
126#1819—534no—
127#1781—532no—
128C2×C53—22no—
129C2×C53—24no—
130C2×C53—44no—
131C2×C53—82no—
132C2×C53#178111yescharacteristic
133C2×C53#178212yes—
134C23×Q8Dic5311yescharacteristic, complemented
135C23×D4C2×D5311yescharacteristic
136C2×#124—24no—
137C2×#124—28no—
138C22×C22⋊C4Dic5312yescomplemented
139C22×C22⋊C4C2×D5312yes—
140C22×C42Dic5311yescharacteristic, complemented
141C2×#99—1064no—
142#379—534no—
143#342—1068no—
144C53#854411yescharacteristic, complemented, Sylow
145C25C2×Dic5311yescharacteristic
146C25C22×D5311yescharacteristic
147C22×Q8—24no—
148C22×Q8—22no—
149C22×Q8—44no—
150C22×D4—24no—
151C22×D4—44no—
152C22×D4C22⋊D5314yes—
153C23×C4C2×Dic5311yescharacteristic
154C23×C4C2×Dic5312yes—
155C23×C4C2×Dic5312yes—
156#124—232no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—1068no—
163C2×C22⋊C4—1064no—
164C2×C22⋊C4C22⋊D5318yes—
165C2×C42—24no—
166C2×C42—22no—
167#99—2128no—
168C24—22no—
169C24—42no—
170C24—42no—
171C24—44no—
172C24—44no—
173C24C22⋊Dic5312yes—
174C24C22×Dic5311yescharacteristic
175C24C2×C22⋊D5312yes—
176C24C2×C22⋊D5314yes—
177C2×Q8—28no—
178C2×Q8—416no—
179C2×Q8—44no—
180C2×Q8—44no—
181C2×D4—28no—
182C2×D4—28no—
183C2×D4—44no—
184C2×D4—416no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4—2124no—
194C22×C4C22⋊Dic5312yes—
195C22×C4C22⋊Dic5314yes—
196C22×C4C22⋊Dic5314yes—
197C22⋊C4—216no—
198C22⋊C4—216no—
199C22⋊C4—2128no—
200C22⋊C4—21216no—
201C42—28no—
202C23—22no—
203C23—24no—
204C23—24no—
205C23—28no—
206C23—22no—
207C23—24no—
208C23—44no—
209C23—44no—
210C23—44no—
211C23—44no—
212C23—44no—
213C23—44no—
214C23—44no—
215C23—88no—
216C23—88no—
217C23C2×C22⋊Dic5311yescharacteristic
218C23C2×C22⋊Dic5312yes—
219C23#1313318614yes—
220Q8—416no—
221D4—416no—
222C2×C4—28no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—24no—
226C2×C4—24no—
227C2×C4—44no—
228C2×C4—44no—
229C2×C4—44no—
230C2×C4—44no—
231C2×C4—44no—
232C2×C4—44no—
233C2×C4—4244no—
234C2×C4—4248no—
235C2×C4#1313313214yes—
236C22—28no—
237C22—22no—
238C22—22no—
239C22—22no—
240C22—24no—
241C22—28no—
242C22—42no—
243C22—44no—
244C22—84no—
245C22—88no—
246C22—84no—
247C22—88no—
248C22#1681047911yescharacteristic
249C22C2×#1313313212yes—
250C22#1681059714yes—
251C4—24no—
252C4—44no—
253C4—44no—
254C4—8484no—
255C2—22no—
256C2—24no—
257C2—44no—
258C2—82no—
259C2#2027463011yescharacteristic
260C2#2027463312yes—
subgroupG/H
centerC22#16810479
commutatorC24×C53C22×C4
FrattiniC25C22×D53
FittingC22×#124×C53C2
socleC22×C53#379
prime radicalC23×D4×C53C22
cosocleC25×C53C23
hypercenterC22×#124D53
hypocenterC53#8544
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C25 ◁ C22×#124

Complements

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

NH ≅ G/Ntype
1. #20274789C2semidirect
7. C23×Q8×C53C4semidirect
11. C22×C22⋊C4×C53C4semidirect
13. C22×C42×C53C4semidirect
134. C23×Q8Dic53semidirect
138. C22×C22⋊C4Dic53semidirect
140. C22×C42Dic53semidirect
144. C53#8544semidirect

Maximal subgroups

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

subgroupindex
#202747892
#202746302
C22×#124×C532
#854453