Finite Group Databasebeta

← #27276724

Subgroups — #27276724

Order 13568.

1026 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 1026 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C22×#13133215C211yesmaximal, characteristic
2#20274639C211yesmaximal, characteristic
3#20273044C214yesmaximal
4C53×#1784C211yesmaximal, characteristic
5C23×Dic106C411yescharacteristic
6C2×#13133215—28no—
7C2×#13133215—22no—
8C2×#13133215—22no—
9C22×#11719919C412yes—
10C22×#11719919C2212yes—
11C22×C4×Dic53C411yescharacteristic
12C2×#13133136—22no—
13#16810587C2212yes—
14#16810348—28no—
15C23×Q8×C53C2211yescharacteristic
16C2×#101×C53—22no—
17C53×#407C2212yes—
18C22×Dic106—22no—
19C22×Dic106—24no—
20C22×Dic106—44no—
21C23×Dic53C2×C412yes—
22C23×Dic53C2×C412yes—
23C2×#11719917—22no—
24#13133215—216no—
25#13133215—216no—
26C2×#11719919—24no—
27C2×#11719919—22no—
28C2×#11719919—22no—
29C2×#11719919—24no—
30C2×#11719919—24no—
31C2×#11719919—24no—
32C2×#11719919D418yes—
33C2×C4×Dic53—22no—
34C2×C4×Dic53—24no—
35#13133136—44no—
36C22⋊#88322—24no—
37C22×Q8×C53—24no—
38C22×Q8×C53—44no—
39C22×Q8×C53D412yes—
40C22×Q8×C53D412yes—
41C23×C4×C53C2×C411yescharacteristic
42C23×C4×C53C2×C411yescharacteristic
43C23×C4×C53C2311yescharacteristic
44C2×M16×C53—22no—
45#101×C53—44no—
46C22⋊C8×C53—24no—
47C24×C53C22×C411yescharacteristic
48C2×Q8×C53—24no—
49C2×Q8×C53—24no—
50C2×Q8×C53—24no—
51C2×Q8×C53—24no—
52C2×Q8×C53—42no—
53C2×Q8×C53—42no—
54C2×Q8×C53—416no—
55C22×C4×C53—22no—
56C22×C4×C53—24no—
57C22×C4×C53—22no—
58C22×C4×C53—42no—
59C22×C4×C53—42no—
60C22×C4×C53C22⋊C412yes—
61C22×C4×C53C22⋊C412yes—
62C22×C4×C53C2×D411yescharacteristic
63C22×C4×C53C2×D411yescharacteristic
64C22×C4×C53C2×D414yes—
65C22×Dic53—22no—
66C22×Dic53—22no—
67C22×Dic53—28no—
68C22×Dic53—22no—
69C22×Dic53—22no—
70C22×Dic53—44no—
71C22×Dic53C22⋊C414yes—
72C22×Dic53C22⋊C414yes—
73C2×Dic106—28no—
74C2×Dic106—416no—
75C2×Dic106—44no—
76C2×Dic106—44no—
77M16×C53—44no—
78C2×C8×C53—42no—
79#11719919—216no—
80#11719919—28no—
81#11719919—28no—
82C4×Dic53—28no—
83#11719917—44no—
84C2×#88322—42no—
85C23×C53—22no—
86C23×C53—24no—
87C23×C53C2×C22⋊C412yes—
88C23×C53C2×C22⋊C411yescharacteristic
89Q8×C53—48no—
90Q8×C53—48no—
91C2×C4×C53—28no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—44no—
100C2×C4×C53—44no—
101C2×C4×C53#9914yes—
102C2×C4×C53#12012yes—
103C2×C4×C53#12012yes—
104C2×Dic53—28no—
105C2×Dic53—28no—
106C2×Dic53—24no—
107C2×Dic53—24no—
108C2×Dic53—24no—
109C2×Dic53—44no—
110C2×Dic53—44no—
111C2×Dic53—44no—
112C2×Dic53#10118yescomplemented
113Dic106—416no—
114C8×C53—82no—
115#88322—82no—
116#8551—531nomaximal, Sylow
117C22×C53—28no—
118C22×C53—22no—
119C22×C53—22no—
120C22×C53—22no—
121C22×C53#37911yescharacteristic
122C22×C53C2×#9912yes—
123C22×C53C2×#10114yes—
124C4×C53—22no—
125C4×C53—22no—
126C4×C53—24no—
127C4×C53—42no—
128Dic53—28no—
129Dic53—44no—
130C22×C4⋊Q8—531no—
131#1784—531no—
132#1784D5311yescharacteristic
133#1463—534no—
134C2×C53—22no—
135C2×C53—24no—
136C2×C53#178111yescharacteristic
137C2×C53#178412yes—
138C23×Q8—531no—
139C23×Q8C2×D5311yescharacteristic
140C2×C4⋊Q8—1068no—
141C2×C4⋊Q8—1062no—
142C2×C4⋊Q8—1062no—
143C22×C4⋊C4—532no—
144C22×C4⋊C4—532no—
145C22×C42—531no—
146C2×#101—22no—
147C2×#101—1062no—
148#407—532no—
149#407C2×D5312yes—
150#324—1068no—
151C53#855111yescharacteristic, complemented, Sylow
152C22×Q8—24no—
153C22×Q8—44no—
154C22×Q8—1064no—
155C22×Q8—1062no—
156C22×Q8—2124no—
157C22×Q8D21212yes—
158C22×Q8C22⋊D5312yes—
159C23×C4—532no—
160C23×C4—532no—
161C23×C4C4×D5311yescharacteristic
162C23×C4C4×D5311yescharacteristic
163C23×C4C22×D5311yescharacteristic
164C2×M16—22no—
165C2×M16—1062no—
166C4⋊Q8—10616no—
167C4⋊Q8—10616no—
168C2×C4⋊C4—538no—
169C2×C4⋊C4—1064no—
170C2×C4⋊C4—1064no—
171C2×C4⋊C4—1062no—
172C2×C4⋊C4—1064no—
173C2×C4⋊C4—1062no—
174C2×C4⋊C4—1064no—
175C2×C42—1064no—
176C2×C42—1062no—
177#101—44no—
178#101—2124no—
179C22⋊C8—24no—
180C22⋊C8—1064no—
181C24C2×C4×D5311yescharacteristic
182C2×Q8—24no—
183C2×Q8—24no—
184C2×Q8—24no—
185C2×Q8—24no—
186C2×Q8—42no—
187C2×Q8—42no—
188C2×Q8—416no—
189C2×Q8—1068no—
190C2×Q8—21216no—
191C2×Q8—2124no—
192C2×Q8—2124no—
193C22×C4—22no—
194C22×C4—24no—
195C22×C4—22no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—534no—
199C22×C4—534no—
200C22×C4—1068no—
201C22×C4—1062no—
202C22×C4—1062no—
203C22×C4—1062no—
204C22×C4—1062no—
205C22×C4—2124no—
206C22×C4#1171992112yes—
207C22×C4#1171992112yes—
208C22×C4C2×D21211yescharacteristic
209C22×C4D4×D5314yes—
210C22×C4C2×C22⋊D5311yescharacteristic
211M16—44no—
212M16—2124no—
213C2×C8—42no—
214C2×C8—2122no—
215C4⋊C4—10616no—
216C4⋊C4—1068no—
217C4⋊C4—1068no—
218C42—1068no—
219C23—22no—
220C23—24no—
221C23C22⋊C4×D5312yes—
222C23C2×#1171992111yescharacteristic
223Q8—48no—
224Q8—48no—
225Q8—21216no—
226C2×C4—28no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—44no—
235C2×C4—44no—
236C2×C4—538no—
237C2×C4—1068no—
238C2×C4—1068no—
239C2×C4—1064no—
240C2×C4—1064no—
241C2×C4—1064no—
242C2×C4—2124no—
243C2×C4—2124no—
244C2×C4—2124no—
245C2×C4#1313313114yes—
246C2×C4#1313318512yes—
247C2×C4#1313318712yes—
248C8—82no—
249C8—4242no—
250C22—28no—
251C22—22no—
252C22—22no—
253C22—22no—
254C22#1681047811yescharacteristic
255C22C2×#1313313112yes—
256C22#101×D5314yes—
257C4—22no—
258C4—22no—
259C4—24no—
260C4—42no—
261C4—1068no—
262C4—2124no—
263C2—22no—
264C2—24no—
265C2#2027462911yescharacteristic
266C2#2027463712yes—
subgroupG/H
centerC22#16810478
commutator, prime radicalC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1784C2
socleC22×C53#379
cosocleC23×C4×C53C23
hypercenter#1784D53
hypocenterC53#8551
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1784

Complements

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

NH ≅ G/Ntype
112. C2×Dic53#101semidirect
151. C53#8551semidirect

Maximal subgroups

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

subgroupindex
C22×#131332152
#202746392
#202730442
C53×#17842
#855153