Finite Group Databasebeta

← #27276702

Subgroups — #27276702

Order 13568.

1266 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 1266 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×#123×C53C211yesmaximal, characteristic
4C2×#13133132—24no—
5#16810479C2214yes—
6#16810390—28no—
7C23×D4×C53C2211yescharacteristic
8C24×C4×C53C411yescharacteristic, complemented
9C2×#123×C53—24no—
10C2×#123×C53—28no—
11C22×C4⋊C4×C53C412yescomplemented
12C22×C22⋊C4×C53C411yescharacteristic, complemented
13C22×C22⋊C4×C53C2212yes—
14C2×C22⋊Dic53—24no—
15C2×C22⋊Dic53—28no—
16#13133132—48no—
17C25×C53C2×C411yescharacteristic
18C25×C53C2311yescharacteristic
19C22×D4×C53—24no—
20C22×D4×C53—44no—
21C22×D4×C53D414yes—
22C23×C4×C53—22no—
23C23×C4×C53—24no—
24C23×C4×C53—44no—
25C23×C4×C53C2×C412yes—
26C23×C4×C53C2×C412yes—
27C23×C4×C53C2×C411yescharacteristic
28#123×C53—232no—
29C2×C4⋊C4×C53—24no—
30C2×C4⋊C4×C53—24no—
31C2×C4⋊C4×C53—24no—
32C2×C22⋊C4×C53—22no—
33C2×C22⋊C4×C53—24no—
34C2×C22⋊C4×C53—24no—
35C2×C22⋊C4×C53—24no—
36C2×C22⋊C4×C53D418yes—
37C24×C53—24no—
38C24×C53—22no—
39C24×C53—42no—
40C24×C53—44no—
41C24×C53—44no—
42C24×C53C22⋊C412yes—
43C24×C53C22×C411yescharacteristic
44C24×C53C2×D412yes—
45C24×C53C2×D414yes—
46C2×D4×C53—28no—
47C2×D4×C53—28no—
48C2×D4×C53—44no—
49C2×D4×C53—416no—
50C22×C4×C53—28no—
51C22×C4×C53—22no—
52C22×C4×C53—22no—
53C22×C4×C53—22no—
54C22×C4×C53—22no—
55C22×C4×C53—22no—
56C22×C4×C53—42no—
57C22×C4×C53—416no—
58C22×C4×C53—44no—
59C22×C4×C53—44no—
60C22×C4×C53—44no—
61C22×C4×C53—44no—
62C22×C4×C53C22⋊C412yes—
63C22×C4×C53C22⋊C414yes—
64C22×C4×C53C22⋊C414yes—
65C22×Dic53—44no—
66C4⋊C4×C53—216no—
67C22⋊C4×C53—28no—
68C22⋊C4×C53—216no—
69C22⋊Dic53—48no—
70C22⋊Dic53—416no—
71C23×C53—22no—
72C23×C53—24no—
73C23×C53—28no—
74C23×C53—22no—
75C23×C53—22no—
76C23×C53—24no—
77C23×C53—28no—
78C23×C53—44no—
79C23×C53—44no—
80C23×C53—44no—
81C23×C53—44no—
82C23×C53—416no—
83C23×C53—44no—
84C23×C53—44no—
85C23×C53—88no—
86C23×C53#9914yes—
87C23×C53C2×C22⋊C411yescharacteristic
88C23×C53C2×C22⋊C412yes—
89C23×C53#12014yes—
90D4×C53—416no—
91C2×C4×C53—24no—
92C2×C4×C53—22no—
93C2×C4×C53—24no—
94C2×C4×C53—24no—
95C2×C4×C53—44no—
96C2×C4×C53—44no—
97C2×C4×C53—416no—
98C2×C4×C53—44no—
99C2×C4×C53—44no—
100C2×C4×C53—44no—
101C2×C4×C53—44no—
102C2×C4×C53—44no—
103C2×Dic53—84no—
104C2×Dic53—88no—
105#8543—531nomaximal, Sylow
106C22×C53—28no—
107C22×C53—22no—
108C22×C53—22no—
109C22×C53—22no—
110C22×C53—24no—
111C22×C53—24no—
112C22×C53—28no—
113C22×C53—44no—
114C22×C53—44no—
115C22×C53—416no—
116C22×C53—44no—
117C22×C53—84no—
118C22×C53—88no—
119C22×C53#37911yescharacteristic
120C22×C53C2×#9912yes—
121C22×C53#41014yes—
122C4×C53—42no—
123C4×C53—44no—
124C4×C53—44no—
125Dic53—164no—
126C22×#123D5311yescharacteristic
127#1819—534no—
128#1781—532no—
129C2×C53—22no—
130C2×C53—24no—
131C2×C53—44no—
132C2×C53—44no—
133C2×C53#178111yescharacteristic
134C2×C53#178212yes—
135C23×D4C2×D5311yescharacteristic
136C24×C4Dic5311yescharacteristic, complemented
137C2×#123—24no—
138C2×#123—28no—
139C22×C4⋊C4Dic5312yescomplemented
140C22×C22⋊C4Dic5311yescharacteristic, complemented
141C22×C22⋊C4C2×D5312yes—
142C2×#99—1064no—
143#379—534no—
144#342—1068no—
145C53#854311yescharacteristic, complemented, Sylow
146C25C2×Dic5311yescharacteristic
147C25C22×D5311yescharacteristic
148C22×D4—24no—
149C22×D4—44no—
150C22×D4C22⋊D5314yes—
151C23×C4—22no—
152C23×C4—24no—
153C23×C4—44no—
154C23×C4C2×Dic5312yes—
155C23×C4C2×Dic5312yes—
156C23×C4C2×Dic5311yescharacteristic
157#123—232no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—1068no—
166C2×C22⋊C4—1064no—
167C2×C22⋊C4C22⋊D5318yes—
168#99—2128no—
169C24—24no—
170C24—22no—
171C24—42no—
172C24—44no—
173C24—44no—
174C24C22⋊Dic5312yes—
175C24C22×Dic5311yescharacteristic
176C24C2×C22⋊D5312yes—
177C24C2×C22⋊D5314yes—
178C2×D4—28no—
179C2×D4—28no—
180C2×D4—44no—
181C2×D4—416no—
182C22×C4—28no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—42no—
189C22×C4—416no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4—44no—
194C22×C4—2124no—
195C22×C4C22⋊Dic5312yes—
196C22×C4C22⋊Dic5314yes—
197C22×C4C22⋊Dic5314yes—
198C4⋊C4—216no—
199C22⋊C4—28no—
200C22⋊C4—216no—
201C22⋊C4—2128no—
202C22⋊C4—21216no—
203C23—22no—
204C23—24no—
205C23—28no—
206C23—22no—
207C23—22no—
208C23—24no—
209C23—28no—
210C23—44no—
211C23—44no—
212C23—44no—
213C23—44no—
214C23—416no—
215C23—44no—
216C23—44no—
217C23—88no—
218C23#1313313214yes—
219C23C2×C22⋊Dic5311yescharacteristic
220C23C2×C22⋊Dic5312yes—
221C23#1313318614yes—
222D4—416no—
223C2×C4—24no—
224C2×C4—24no—
225C2×C4—24no—
226C2×C4—22no—
227C2×C4—44no—
228C2×C4—44no—
229C2×C4—416no—
230C2×C4—44no—
231C2×C4—44no—
232C2×C4—44no—
233C2×C4—44no—
234C2×C4—44no—
235C2×C4—4244no—
236C2×C4—4248no—
237C22—28no—
238C22—22no—
239C22—22no—
240C22—22no—
241C22—24no—
242C22—24no—
243C22—28no—
244C22—44no—
245C22—44no—
246C22—416no—
247C22—44no—
248C22—84no—
249C22—88no—
250C22#1681047911yescharacteristic
251C22C2×#1313313212yes—
252C22#1681059714yes—
253C4—42no—
254C4—44no—
255C4—44no—
256C4—8484no—
257C2—22no—
258C2—24no—
259C2—44no—
260C2—44no—
261C2#2027463011yescharacteristic
262C2#2027463312yes—
subgroupG/H
centerC22#16810479
commutatorC24×C53C22×C4
FrattiniC25C22×D53
FittingC22×#123×C53C2
socleC22×C53#379
prime radicalC23×D4×C53C22
cosocleC25×C53C23
hypercenterC22×#123D53
hypocenterC53#8543
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C25 ◁ C22×#123

Complements

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

NH ≅ G/Ntype
1. #20274789C2semidirect
8. C24×C4×C53C4semidirect
11. C22×C4⋊C4×C53C4semidirect
12. C22×C22⋊C4×C53C4semidirect
136. C24×C4Dic53semidirect
139. C22×C4⋊C4Dic53semidirect
140. C22×C22⋊C4Dic53semidirect
145. C53#8543semidirect

Maximal subgroups

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

subgroupindex
#202747892
#202746302
C22×#123×C532
#854353