Finite Group Databasebeta

← #28302314

Subgroups — #28302314

Order 13568.

826 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 826 conjugacy classes of subgroups — too many to draw. Only its 120 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#20290442C212yesmaximal
2C2×#16811480C214yesmaximal, complemented
3C2×#125×D53C211yesmaximal, characteristic
4C2×#13133274—22no—
5C2×#13133272C2214yes—
6#16811489—22no—
7#16811480—28no—
8C2×#13133209C411yescharacteristic
9C2×#13133208C2212yes—
10C2×#13133207—22no—
11#125×D53—28no—
12C2×C4⋊C4×D53—22no—
13C2×C4⋊C4×D53C412yes—
14C2×C42×D53C2211yescharacteristic
15C2×#125×C53C411yescharacteristic
16C22×#88324—24no—
17#13133274—28no—
18#13133272—28no—
19C22×C4×D53—22no—
20C22×C4×D53C2×C411yescharacteristic
21C22×C4×D53D412yes—
22#13133209—24no—
23#13133208—24no—
24#13133208D418yes—
25#13133207—216no—
26C2×C4⋊Dic53—22no—
27C2×C4⋊Dic53C2×C412yes—
28C2×#11719919—22no—
29C2×#11719919—22no—
30C2×#11719919—22no—
31C4⋊C4×D53—216no—
32C4⋊C4×D53—24no—
33C4⋊C4×D53—24no—
34C2×C4×Dic53D412yes—
35C2×C4×Dic53C2311yescharacteristic
36C42×D53—24no—
37#125×C53—24no—
38C2×C4⋊C4×C53—22no—
39C2×C4⋊C4×C53C2×C412yes—
40C2×C42×C53C2×C411yescharacteristic
41C23×D53C22⋊C411yescharacteristic
42C22×C4×C53—22no—
43C22×C4×C53C22⋊C412yes—
44C22×C4×C53C22×C411yescharacteristic
45C22×Dic53—22no—
46C22×Dic53C22⋊C411yescharacteristic
47C22×Dic53C2×D412yes—
48C2×C4×D53—22no—
49C2×C4×D53—22no—
50C2×C4×D53—22no—
51C2×C4×D53—28no—
52C2×C4×D53—24no—
53C2×C4×D53—24no—
54C2×C4×D53—42no—
55C2×C4×D53—44no—
56C2×C4×D53—44no—
57C2×#88324—28no—
58C2×#88324—44no—
59C2×#88324—44no—
60C4⋊C4×C53—28no—
61C4⋊C4×C53—22no—
62C4⋊C4×C53C22⋊C414yes—
63C42×C53—22no—
64C4⋊Dic53—22no—
65C4⋊Dic53—28no—
66C4⋊Dic53C22⋊C414yes—
67#11719919—28no—
68#11719919—28no—
69#11719919—28no—
70C4×Dic53—24no—
71C4×Dic53C2×D414yes—
72C23×C53C2×C22⋊C411yescharacteristic
73C22×D53—21no—
74C22×D53—21no—
75C22×D53—22no—
76C22×D53#10112yes—
77C22×D53#10414yes—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—24no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—42no—
85C2×C4×C53—42no—
86C2×C4×C53C2×C22⋊C412yes—
87C2×Dic53—21no—
88C2×Dic53—21no—
89C2×Dic53—22no—
90C2×Dic53—22no—
91C2×Dic53—24no—
92C2×Dic53—42no—
93C2×Dic53—42no—
94C2×Dic53#10112yes—
95C2×Dic53#12014yes—
96C4×D53—28no—
97C4×D53—48no—
98C4×D53—416no—
99#88324—48no—
100#8335—531nomaximal, Sylow
101C22×C53#37911yescharacteristic
102C22×C53C2×#10111yescharacteristic
103C22×C53C2×#10111yescharacteristic
104C22×C53C2×#10412yes—
105C22×C53#42112yes—
106C2×D53—24no—
107C2×D53—22no—
108C2×D53—24no—
109C2×D53—22no—
110C2×D53#33014yescomplemented
111C4×C53—22no—
112C4×C53—42no—
113C4×C53—44no—
114Dic53—22no—
115Dic53—24no—
116Dic53—44no—
117C22×#125—531no—
118#1749—532no—
119C2×#330—534no—
120C2×C53C2×#33011yescharacteristic
121C2×C53#152812yes—
122C2×C53#153411yescharacteristic
123C2×C53#178411yescharacteristic
124C2×C53#178912yes—
125D53—24no—
126C2×#125—532no—
127C2×#125—531no—
128C2×#125—1062no—
129C2×#125—1068no—
130C2×#125#615711yescharacteristic
131C22×C4⋊C4—532no—
132C22×C4⋊C4—1062no—
133C22×C42—531no—
134C2×C22⋊C8—1062no—
135C2×#97—534no—
136#336—1062no—
137#330—1068no—
138C53#833511yescharacteristic, complemented, Sylow
139C23×C4—531no—
140C23×C4—532no—
141C23×C4—1062no—
142C22×C8—1064no—
143#125—24no—
144#125—538no—
145#125—1064no—
146#125—1064no—
147#125—10616no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—532no—
150C2×C4⋊C4—1064no—
151C2×C4⋊C4—10616no—
152C2×C4⋊C4—1062no—
153C2×C4⋊C4—1062no—
154C2×C4⋊C4—1062no—
155C2×C4⋊C4—1062no—
156C2×C4⋊C4—1064no—
157C2×C4⋊C4C2×#615712yes—
158C2×C42—531no—
159C2×C42—532no—
160C2×C42—1064no—
161C2×C42C2×#615711yescharacteristic
162C22⋊C8—1068no—
163#97—1068no—
164C24—531no—
165C22×C4—22no—
166C22×C4—531no—
167C22×C4—532no—
168C22×C4—1062no—
169C22×C4—1064no—
170C22×C4—1062no—
171C22×C4—1064no—
172C22×C4—1068no—
173C22×C4—1062no—
174C22×C4—1062no—
175C22×C4—2122no—
176C22×C4—2124no—
177C22×C4—2124no—
178C22×C4C22⋊#615712yes—
179C22×C4C22×#615711yescharacteristic
180C2×C8—1068no—
181C2×C8—2124no—
182C2×C8—2124no—
183C4⋊C4—28no—
184C4⋊C4—22no—
185C4⋊C4—534no—
186C4⋊C4—1068no—
187C4⋊C4—1068no—
188C4⋊C4—1068no—
189C4⋊C4—1068no—
190C4⋊C4—1062no—
191C4⋊C4C22⋊#615714yes—
192C42—22no—
193C42—534no—
194C42—1064no—
195C23—532no—
196C23—534no—
197C23—1061no—
198C23—1061no—
199C23—1062no—
200C23C2×C22⋊#615711yescharacteristic
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—24no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—534no—
210C2×C4—532no—
211C2×C4—1068no—
212C2×C4—1061no—
213C2×C4—1061no—
214C2×C4—1062no—
215C2×C4—1062no—
216C2×C4—1064no—
217C2×C4—2128no—
218C2×C4—21216no—
219C2×C4—2122no—
220C2×C4—2122no—
221C2×C4C2×C22⋊#615712yes—
222C8—2128no—
223C22—534no—
224C22—1064no—
225C22—1062no—
226C22—1062no—
227C22—1064no—
228C22#1681156411yescharacteristic
229C22D53⋊#10111yescharacteristic
230C22D53⋊#10111yescharacteristic
231C22D53⋊#10412yes—
232C22#1681162012yes—
233C4—22no—
234C4—42no—
235C4—44no—
236C4—1062no—
237C4—1064no—
238C4—2124no—
239C2—1064no—
240C2D53⋊#33011yescharacteristic
241C2#2029017812yes—
242C2#2029019811yescharacteristic
243C2#2029055011yescharacteristic
244C2#2029055712yes—
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×#125×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×#125#6157
hypocenterC53#8335
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
2. C2×#16811480C2semidirect
110. C2×D53#330semidirect
138. C53#8335semidirect

Maximal subgroups

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

subgroupindex
#202904422
C2×#168114802
C2×#125×D532
#833553