Finite Group Databasebeta

← #28307903

Subgroups — #28307903

Order 13568.

686 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 112 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#20289946C212yesmaximal, complemented
2#20289639C214yesmaximal, complemented
3#20277297C211yesmaximal, characteristic
4#16811639—22no—
5#16811468—22no—
6#16811466C2214yes—
7C2×#13133176C411yescharacteristic
8C2×#13133176C2211yescharacteristic
9#16810526—22no—
10#16810526—22no—
11#16810479C412yes—
12#16810479C2212yes—
13C23×D4×C53C411yescharacteristic
14C4⋊#88324—42no—
15C22⋊#88324—24no—
16C4×#88324—42no—
17#13133176—42no—
18#13133176—42no—
19C2×C4⋊Dic53—21no—
20C2×C4⋊Dic53D412yes—
21C2×C22⋊Dic53—24no—
22C2×C22⋊Dic53—24no—
23C2×C22⋊Dic53C2×C412yes—
24C2×C22⋊Dic53C2×C411yescharacteristic
25C2×C22⋊Dic53C2311yescharacteristic
26C2×C4×Dic53—21no—
27C2×C4×Dic53D412yes—
28C25×C53C2×C412yes—
29C22×D4×C53—21no—
30C22×D4×C53—21no—
31C22×D4×C53—22no—
32C22×D4×C53—42no—
33C22×D4×C53—42no—
34C22×D4×C53C814yescomplemented
35C23×C4×C53C2×C411yescharacteristic
36C24×C53—22no—
37C24×C53—22no—
38C24×C53—42no—
39C24×C53—42no—
40C24×C53—84no—
41C24×C53C2×C814yes—
42C24×C53C22×C411yescharacteristic
43C2×D4×C53—24no—
44C2×D4×C53—24no—
45C2×D4×C53—42no—
46C2×D4×C53—42no—
47C2×D4×C53—48no—
48C2×D4×C53—48no—
49C2×D4×C53—42no—
50C2×D4×C53—42no—
51C22×C4×C53—22no—
52C22×C4×C53—22no—
53C22×C4×C53C22⋊C411yescharacteristic
54C22×C4×C53C22⋊C411yescharacteristic
55C22×C4×C53C2×C812yes—
56C22×C4×C53M1612yes—
57C22×Dic53—22no—
58C22×Dic53C22⋊C412yes—
59C22×Dic53C2×D412yes—
60C2×#88324—44no—
61C22⋊Dic53—41no—
62C22⋊Dic53—41no—
63C22⋊Dic53—42no—
64C22⋊Dic53—84no—
65C22⋊Dic53—84no—
66C4⋊Dic53—42no—
67C4⋊Dic53—42no—
68C4×Dic53—22no—
69C4×Dic53—22no—
70C4×Dic53—42no—
71C23×C53—21no—
72C23×C53—21no—
73C23×C53—24no—
74C23×C53—42no—
75C23×C53—44no—
76C23×C53—44no—
77C23×C53—44no—
78C23×C53—44no—
79C23×C53—42no—
80C23×C53—42no—
81C23×C53—84no—
82C23×C53—84no—
83C23×C53—88no—
84C23×C53—88no—
85C23×C53C2×C22⋊C411yescharacteristic
86C23×C53C22×C811yescharacteristic
87C23×C53C2×M1611yescharacteristic
88D4×C53—48no—
89D4×C53—48no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—24no—
93C2×C4×C53—24no—
94C2×C4×C53C22⋊C812yes—
95C2×C4×C53C22⋊C812yes—
96C2×C4×C53#9912yes—
97C2×C4×C53#10012yes—
98C2×Dic53—22no—
99C2×Dic53—22no—
100C2×Dic53—42no—
101C2×Dic53—42no—
102C2×Dic53—44no—
103C2×Dic53—42no—
104#88324—84no—
105#11427—531nomaximal, Sylow
106C22×C53—22no—
107C22×C53—21no—
108C22×C53—21no—
109C22×C53—41no—
110C22×C53—41no—
111C22×C53—42no—
112C22×C53—44no—
113C22×C53—44no—
114C22×C53—42no—
115C22×C53—44no—
116C22×C53—84no—
117C22×C53—88no—
118C22×C53—88no—
119C22×C53—84no—
120C22×C53C2×C22⋊C811yescharacteristic
121C22×C53C2×#9911yescharacteristic
122C22×C53C2×#10011yescharacteristic
123C4×C53—22no—
124C4×C53#32314yes—
125Dic53—42no—
126Dic53—42no—
127Dic53—82no—
128#2309—531no—
129#1458—532no—
130#1316—534no—
131C2×C53—21no—
132C2×C53—21no—
133C2×C53—41no—
134C2×C53—41no—
135C2×C53—84no—
136C2×C53C2×#32311yescharacteristic
137C2×C53#212212yes—
138C23×D4#615711yescharacteristic
139C2×#117—531no—
140C2×#117—531no—
141#431—1062no—
142#390—1062no—
143#390—1062no—
144#379—532no—
145#379—532no—
146#324—1062no—
147#323—534no—
148C53#1142711yescharacteristic, complemented, Sylow
149C25C2×#615712yes—
150C22×D4—22no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—42no—
154C22×D4—42no—
155C22×D4#8832414yescomplemented
156C23×C4C2×#615711yescharacteristic
157#117—2122no—
158#117—2122no—
159C2×C4⋊C4—532no—
160C2×C4⋊C4—1061no—
161C2×C22⋊C4—531no—
162C2×C22⋊C4—531no—
163C2×C22⋊C4—532no—
164C2×C22⋊C4—1064no—
165C2×C22⋊C4—1064no—
166C2×C42—532no—
167C2×C42—1061no—
168C4⋊C8—2122no—
169C22⋊C8—1064no—
170C4×C8—2122no—
171C24—22no—
172C24—22no—
173C24—42no—
174C24—42no—
175C24—84no—
176C24C2×#8832414yes—
177C24C22×#615711yescharacteristic
178C2×D4—24no—
179C2×D4—24no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—48no—
183C2×D4—48no—
184C2×D4—42no—
185C2×D4—42no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—532no—
189C22×C4—532no—
190C22×C4—1062no—
191C22×C4C2×#8832412yes—
192C22×C4#1171994012yes—
193C22×C4C22⋊#615711yescharacteristic
194C22×C4C22⋊#615711yescharacteristic
195C2×C8—2124no—
196C4⋊C4—2122no—
197C4⋊C4—2122no—
198C22⋊C4—4244no—
199C22⋊C4—4244no—
200C22⋊C4—2121no—
201C22⋊C4—2121no—
202C22⋊C4—2122no—
203C42—1062no—
204C42—1062no—
205C42—2122no—
206C23—21no—
207C23—21no—
208C23—24no—
209C23—42no—
210C23—44no—
211C23—44no—
212C23—44no—
213C23—44no—
214C23—42no—
215C23—42no—
216C23—84no—
217C23—84no—
218C23—88no—
219C23—88no—
220C23C2×C22⋊#615711yescharacteristic
221C23C22×#8832411yescharacteristic
222C23C2×#1171994011yescharacteristic
223D4—48no—
224D4—48no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—24no—
228C2×C4—24no—
229C2×C4—1062no—
230C2×C4—1062no—
231C2×C4—2124no—
232C2×C4—2122no—
233C2×C4—2122no—
234C2×C4—2122no—
235C2×C4C22⋊#8832412yes—
236C2×C4C22⋊#8832412yes—
237C2×C4#1313327512yes—
238C2×C4#1313327712yes—
239C8—4244no—
240C22—22no—
241C22—21no—
242C22—21no—
243C22—41no—
244C22—41no—
245C22—42no—
246C22—44no—
247C22—44no—
248C22—42no—
249C22—44no—
250C22—84no—
251C22—88no—
252C22—88no—
253C22—84no—
254C22C2×C22⋊#8832411yescharacteristic
255C22C2×#1313327511yescharacteristic
256C22C2×#1313327711yescharacteristic
257C4—22no—
258C4—4242no—
259C4—2122no—
260C4—2122no—
261C4#1681146614yes—
262C2—21no—
263C2—21no—
264C2—41no—
265C2—41no—
266C2—84no—
267C2C2×#1681146611yescharacteristic
268C2#2029097412yes—
subgroupG/H
centerC22C2×#13133275
commutatorC23×C53C22×C8
FrattiniC24C22×#6157
Fitting, prime radicalC23×D4×C53C4
socleC22×C53C2×#99
cosocleC2×C22⋊Dic53C23
hypercenterC23×D4#6157
hypocenterC53#11427
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. #20289946C2semidirect
2. #20289639C2semidirect
34. C22×D4×C53C8semidirect
148. C53#11427semidirect
155. C22×D4#88324semidirect

Maximal subgroups

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

subgroupindex
#202899462
#202896392
#202772972
#1142753