Finite Group Databasebeta

← #28307905

Subgroups — #28307905

Order 13568.

686 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 104 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#20289640C214yesmaximal, complemented
3#20277295C211yesmaximal, characteristic
4#16811639—22no—
5#16811468—22no—
6#16811466C2214yes—
7C23×D212C411yescharacteristic
8C2×#13133176C2211yescharacteristic
9#16810525—22no—
10#16810524—22no—
11#16810478C412yes—
12#16810478C2212yes—
13C2×#117×C53C411yescharacteristic
14C4⋊#88324—42no—
15C22⋊#88324—24no—
16C4×#88324—42no—
17C24×D53C2×C412yes—
18C22×D212—21no—
19C22×D212—21no—
20C22×D212—22no—
21C22×D212—42no—
22C22×D212—42no—
23C22×D212C814yescomplemented
24#13133176—42no—
25C2×C4⋊Dic53D412yes—
26C2×C22⋊Dic53C2×C411yescharacteristic
27C2×C22⋊Dic53C2311yescharacteristic
28C2×#11719921—24no—
29C2×#11719921—24no—
30C2×C4×Dic53D412yes—
31C23×C4×C53C2×C411yescharacteristic
32#117×C53—42no—
33C2×C4⋊C4×C53—21no—
34C2×C22⋊C4×C53C2×C412yes—
35C2×C42×C53—21no—
36C24×C53C22×C411yescharacteristic
37C23×D53—22no—
38C23×D53—22no—
39C23×D53—42no—
40C23×D53—42no—
41C23×D53—84no—
42C23×D53C2×C814yes—
43C22×C4×C53—22no—
44C22×C4×C53—22no—
45C22×C4×C53—22no—
46C22×C4×C53C22⋊C411yescharacteristic
47C22×C4×C53C22⋊C411yescharacteristic
48C22×C4×C53C2×C812yes—
49C22×C4×C53M1612yes—
50C22×Dic53C22⋊C412yes—
51C22×Dic53C2×D412yes—
52C2×D212—24no—
53C2×D212—24no—
54C2×D212—42no—
55C2×D212—42no—
56C2×D212—48no—
57C2×D212—42no—
58C2×D212—42no—
59C2×D212—48no—
60C2×#88324—44no—
61C4⋊C4×C53—42no—
62C22⋊C4×C53—42no—
63C42×C53—42no—
64C22⋊Dic53—41no—
65C22⋊Dic53—41no—
66#11719921—84no—
67#11719921—84no—
68C4⋊Dic53—42no—
69C4×Dic53—22no—
70C4×Dic53—22no—
71C23×C53—21no—
72C23×C53—21no—
73C23×C53—42no—
74C23×C53C2×C22⋊C411yescharacteristic
75C23×C53C22×C811yescharacteristic
76C23×C53C2×M1611yescharacteristic
77C22×D53—24no—
78C22×D53—44no—
79C22×D53—44no—
80C22×D53—44no—
81C22×D53—42no—
82C22×D53—42no—
83C22×D53—44no—
84C22×D53—84no—
85C22×D53—84no—
86C22×D53—88no—
87C22×D53—88no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—24no—
91C2×C4×C53—24no—
92C2×C4×C53—44no—
93C2×C4×C53—42no—
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—
102D212—48no—
103D212—48no—
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—42no—
113C22×C53C2×C22⋊C811yescharacteristic
114C22×C53C2×#9911yescharacteristic
115C22×C53C2×#10011yescharacteristic
116C2×D53—44no—
117C2×D53—44no—
118C2×D53—44no—
119C2×D53—84no—
120C2×D53—88no—
121C2×D53—84no—
122C2×D53—88no—
123C4×C53—22no—
124C4×C53—82no—
125C4×C53#32314yes—
126Dic53—42no—
127Dic53—42no—
128#2309—531no—
129#1458—532no—
130#1316—534no—
131C2×C53—21no—
132C2×C53—21no—
133C2×C53—41no—
134C2×C53—41no—
135C2×C53C2×#32311yescharacteristic
136C2×C53#212212yes—
137D53—84no—
138C23×D4—531no—
139C2×#117—531no—
140C2×#117#615711yescharacteristic
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
149C25—532no—
150C22×D4—534no—
151C22×D4—1062no—
152C22×D4—1061no—
153C22×D4—1061no—
154C22×D4—2122no—
155C22×D4—2122no—
156C23×C4C2×#615711yescharacteristic
157#117—42no—
158#117—2122no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—532no—
161C2×C22⋊C4—531no—
162C2×C22⋊C4—531no—
163C2×C22⋊C4—1064no—
164C2×C22⋊C4—1064no—
165C2×C22⋊C4C2×#615712yes—
166C2×C42—21no—
167C2×C42—532no—
168C4⋊C8—2122no—
169C22⋊C8—1064no—
170C4×C8—2122no—
171C24—534no—
172C24—1062no—
173C24—1062no—
174C24—4244no—
175C24—2122no—
176C24—2122no—
177C24C22×#615711yescharacteristic
178C2×D4—1064no—
179C2×D4—1064no—
180C2×D4—2122no—
181C2×D4—2122no—
182C2×D4—2128no—
183C2×D4—2128no—
184C2×D4—2122no—
185C2×D4—2122no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—532no—
190C22×C4—532no—
191C22×C4D53⋊C812yes—
192C22×C4#1171993612yes—
193C22×C4C22⋊#615711yescharacteristic
194C22×C4C22⋊#615711yescharacteristic
195C2×C8—2124no—
196C4⋊C4—42no—
197C4⋊C4—2122no—
198C22⋊C4—42no—
199C22⋊C4—4244no—
200C22⋊C4—4244no—
201C22⋊C4—2121no—
202C22⋊C4—2121no—
203C42—42no—
204C42—1062no—
205C42—1062no—
206C23—21no—
207C23—21no—
208C23—42no—
209C23—1064no—
210C23—4248no—
211C23—4248no—
212C23—4244no—
213C23—4244no—
214C23—2124no—
215C23—2124no—
216C23—2124no—
217C23—2124no—
218C23—2122no—
219C23—2122no—
220C23C2×C22⋊#615711yescharacteristic
221C23C2×D53⋊C811yescharacteristic
222C23C2×#1171993611yescharacteristic
223D4—2128no—
224D4—2128no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—24no—
228C2×C4—24no—
229C2×C4—44no—
230C2×C4—42no—
231C2×C4—1062no—
232C2×C4—1062no—
233C2×C4—2122no—
234C2×C4—2122no—
235C2×C4#1313327412yes—
236C2×C4#1313327412yes—
237C2×C4#1313327612yes—
238C2×C4#1313327712yes—
239C8—4244no—
240C22—22no—
241C22—21no—
242C22—21no—
243C22—41no—
244C22—41no—
245C22—42no—
246C22—42no—
247C22—4244no—
248C22—4248no—
249C22—4248no—
250C22—4244no—
251C22—2124no—
252C22—2124no—
253C22—2124no—
254C22C2×#1313327411yescharacteristic
255C22C2×#1313327611yescharacteristic
256C22C2×#1313327711yescharacteristic
257C4—22no—
258C4—82no—
259C4—2122no—
260C4—2122no—
261C4#1681146714yes—
262C2—21no—
263C2—21no—
264C2—41no—
265C2—41no—
266C2—4244no—
267C2C2×#1681146711yescharacteristic
268C2#2029097612yes—
subgroupG/H
centerC22C2×#13133276
commutatorC23×C53C22×C8
FrattiniC24C22×#6157
FittingC2×#117×C53C4
socleC22×C53C2×#99
prime radicalC23×D212C4
cosocleC2×C22⋊Dic53C23
hypercenterC2×#117#6157
hypocenterC53#11427
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. #20289946C2semidirect
2. #20289640C2semidirect
23. C22×D212C8semidirect
148. C53#11427semidirect

Maximal subgroups

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

subgroupindex
#202899462
#202896402
#202772952
#1142753