Finite Group Databasebeta

← #28302229

Subgroups — #28302229

Order 13568.

1174 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 1174 conjugacy classes of subgroups — too many to draw. Only its 136 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#20290538C212yesmaximal
2C2×#16811469C214yesmaximal, complemented
3C22×#13133193C211yesmaximal, characteristic
4C2×#13133274—24no—
5C2×C22⋊#88324C2214yes—
6#16811469—28no—
7C23×C4×D53C2211yescharacteristic
8C2×#13133193—24no—
9C2×#13133193—28no—
10C22×C4⋊Dic53C411yescharacteristic
11C22×#11719919C2212yes—
12C22×#11719921C412yes—
13C23×Q8×C53C411yescharacteristic
14C22×#88324—24no—
15#13133274—48no—
16C22⋊#88324—28no—
17C24×D53C2×C411yescharacteristic
18C23×Dic53C2×C412yes—
19C23×Dic53C2311yescharacteristic
20C22×C4×D53—24no—
21C22×C4×D53—44no—
22C22×C4×D53D414yes—
23#13133193—232no—
24C2×C4⋊Dic53—22no—
25C2×C4⋊Dic53—24no—
26C2×#11719919—24no—
27C2×#11719919—24no—
28C2×#11719919D418yes—
29C2×#11719921—24no—
30C2×#11719921—24no—
31C2×#11719921—24no—
32C22×Q8×C53—24no—
33C22×Q8×C53—22no—
34C22×Q8×C53—44no—
35C23×C4×C53C2×C411yescharacteristic
36C23×C4×C53C2×C412yes—
37C24×C53C22×C411yescharacteristic
38C23×D53—22no—
39C23×D53—24no—
40C23×D53—24no—
41C23×D53—42no—
42C23×D53C22⋊C412yes—
43C2×Q8×C53—28no—
44C2×Q8×C53—416no—
45C2×Q8×C53—44no—
46C2×Q8×C53—44no—
47C22×C4×C53—22no—
48C22×C4×C53—24no—
49C22×C4×C53—22no—
50C22×C4×C53—22no—
51C22×C4×C53—44no—
52C22×C4×C53C22⋊C412yes—
53C22×C4×C53C22⋊C414yes—
54C22×Dic53—22no—
55C22×Dic53—22no—
56C22×Dic53—28no—
57C22×Dic53—42no—
58C22×Dic53C22⋊C414yes—
59C22×Dic53C2×D412yes—
60C22×Dic53C2×D414yes—
61C2×C4×D53—28no—
62C2×C4×D53—28no—
63C2×C4×D53—44no—
64C2×C4×D53—416no—
65C2×#88324—48no—
66C2×#88324—44no—
67#11719921—216no—
68C4⋊Dic53—28no—
69#11719919—216no—
70C23×C53—22no—
71C23×C53—22no—
72C23×C53C2×C22⋊C412yes—
73C23×C53C2×C22⋊C411yescharacteristic
74C23×C53#13114yes—
75C22×D53—28no—
76C22×D53—22no—
77C22×D53—24no—
78C22×D53—24no—
79C22×D53—22no—
80C22×D53—416no—
81C22×D53—44no—
82C22×D53—44no—
83C22×D53#10114yes—
84Q8×C53—416no—
85C2×C4×C53—28no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—24no—
89C2×C4×C53—44no—
90C2×C4×C53—44no—
91C2×C4×C53—44no—
92C2×C4×C53#9914yes—
93C2×Dic53—24no—
94C2×Dic53—24no—
95C2×Dic53—28no—
96C2×Dic53—44no—
97C2×Dic53—44no—
98C2×Dic53—44no—
99C2×Dic53#12014yes—
100C4×D53—416no—
101#88324—84no—
102#8308—531nomaximal, Sylow
103C22×C53—28no—
104C22×C53—22no—
105C22×C53—22no—
106C22×C53#37911yescharacteristic
107C22×C53#40814yes—
108C22×C53#40812yes—
109C22×C53C2×#9912yes—
110C22×C53C2×#10112yes—
111C2×D53—28no—
112C2×D53—24no—
113C2×D53—416no—
114C2×D53—44no—
115C2×D53—42no—
116C2×D53—42no—
117C4×C53—24no—
118C4×C53—44no—
119Dic53—24no—
120Dic53—44no—
121C22×C22⋊Q8—531no—
122#1779—532no—
123C2×#324—534no—
124C2×C53—24no—
125C2×C53#147214yes—
126C2×C53#178011yescharacteristic
127C2×C53#178111yescharacteristic
128C2×C53#178411yescharacteristic
129D53—44no—
130C23×Q8#615711yescharacteristic
131C24×C4—531no—
132C2×C22⋊Q8—1068no—
133C2×C22⋊Q8—1064no—
134C22×C4⋊C4—532no—
135C22×C4⋊C4—531no—
136C22×C22⋊C4—532no—
137C2×C22⋊C8—534no—
138C2×C22⋊C8—1064no—
139#324—1068no—
140C53#830811yescharacteristic, complemented, Sylow
141C25—531no—
142C22×Q8—24no—
143C22×Q8—22no—
144C22×Q8—44no—
145C23×C4—531no—
146C23×C4—532no—
147C23×C4—534no—
148C23×C4—1064no—
149C23×C4—2124no—
150C23×C4C2×#615712yes—
151C23×C4C2×#615711yescharacteristic
152C22×C8—1064no—
153C22⋊Q8—10632no—
154C2×C4⋊C4—538no—
155C2×C4⋊C4—1064no—
156C2×C4⋊C4—1064no—
157C2×C4⋊C4—1064no—
158C2×C4⋊C4—1062no—
159C2×C22⋊C4—1064no—
160C2×C22⋊C4—1064no—
161C2×C22⋊C4—1064no—
162C22⋊C8—1068no—
163C22⋊C8—2128no—
164C24—532no—
165C24—1064no—
166C24—1062no—
167C24—1064no—
168C24—2122no—
169C24C22×#615711yescharacteristic
170C2×Q8—28no—
171C2×Q8—416no—
172C2×Q8—44no—
173C2×Q8—44no—
174C22×C4—22no—
175C22×C4—24no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—44no—
179C22×C4—532no—
180C22×C4—534no—
181C22×C4—534no—
182C22×C4—1068no—
183C22×C4—1062no—
184C22×C4—1062no—
185C22×C4—1068no—
186C22×C4—1068no—
187C22×C4—2124no—
188C22×C4—21216no—
189C22×C4—2122no—
190C22×C4C22⋊#615712yes—
191C22×C4C22⋊#615714yes—
192C2×C8—2124no—
193C2×C8—2128no—
194C4⋊C4—10616no—
195C4⋊C4—1068no—
196C22⋊C4—10616no—
197C23—22no—
198C23—22no—
199C23—534no—
200C23—1068no—
201C23—1062no—
202C23—1064no—
203C23—1064no—
204C23—1062no—
205C23—21216no—
206C23—2124no—
207C23—2124no—
208C23C2×C22⋊#615712yes—
209C23C2×C22⋊#615711yescharacteristic
210C23#1313330514yes—
211Q8—416no—
212C2×C4—28no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—24no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—44no—
219C2×C4—534no—
220C2×C4—1064no—
221C2×C4—1064no—
222C2×C4—1068no—
223C2×C4—21216no—
224C2×C4—2124no—
225C2×C4—2124no—
226C2×C4—2124no—
227C2×C4#1313327514yes—
228C8—4244no—
229C22—28no—
230C22—22no—
231C22—22no—
232C22—1068no—
233C22—1064no—
234C22—21216no—
235C22—2124no—
236C22—2122no—
237C22—2122no—
238C22#1681156411yescharacteristic
239C22#1681159814yes—
240C22#1681159812yes—
241C22C2×#1313327512yes—
242C22D53⋊#10112yes—
243C4—24no—
244C4—44no—
245C4—1064no—
246C4—2124no—
247C2—24no—
248C2—2124no—
249C2#2028999514yes—
250C2#2029054011yescharacteristic
251C2#2029054111yescharacteristic
252C2#2029055011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC24×C53C22×C4
FrattiniC24C22×#6157
FittingC23×Q8×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×D53C2×C4
cosocleC23×Dic53C23
hypercenterC23×Q8#6157
hypocenterC53#8308
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#16811469C2semidirect
140. C53#8308semidirect

Maximal subgroups

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

subgroupindex
#202905382
C2×#168114692
C22×#131331932
#830853