Finite Group Databasebeta

← #28302221

Subgroups — #28302221

Order 13568.

810 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 128 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#20290605C212yesmaximal
2C2×#16811469C212yesmaximal
3C2×#16811468C212yesmaximal
4C22×#13133207C211yesmaximal, characteristic
5C2×C22⋊#88324C2214yes—
6#16811489—24no—
7#16811469—24no—
8#16811468—24no—
9C2×#13133207—24no—
10C2×#13133207—24no—
11C2×#13133207—24no—
12C22×C4⋊Dic53C2211yescharacteristic
13C22×#11719919C411yescharacteristic
14C22×#11719919C411yescharacteristic
15C22×#11719919C411yescharacteristic
16C22×#11719919C2211yescharacteristic
17C22×C4×Dic53C2211yescharacteristic
18C22×C4⋊C4×C53C411yescharacteristic
19C22×#88324—24no—
20C22⋊#88324—24no—
21C22⋊#88324—24no—
22C23×Dic53C2×C411yescharacteristic
23C23×Dic53C2×C411yescharacteristic
24C23×Dic53C2×C411yescharacteristic
25C23×Dic53C2311yescharacteristic
26#13133207—232no—
27C2×C4⋊Dic53—22no—
28C2×C4⋊Dic53—22no—
29C2×C4⋊Dic53D414yes—
30C2×#11719919—22no—
31C2×#11719919—22no—
32C2×#11719919—22no—
33C2×#11719919—22no—
34C2×#11719919—22no—
35C2×#11719919—22no—
36C2×#11719919—22no—
37C2×#11719919—22no—
38C2×#11719919—22no—
39C2×#11719919—22no—
40C2×#11719919—22no—
41C2×#11719919D414yes—
42C2×C4×Dic53—22no—
43C2×C4×Dic53—22no—
44C2×C4×Dic53D414yes—
45C23×C4×C53C2×C411yescharacteristic
46C23×C4×C53C2×C411yescharacteristic
47C23×C4×C53C2×C411yescharacteristic
48C2×C4⋊C4×C53—22no—
49C2×C4⋊C4×C53—22no—
50C2×C4⋊C4×C53—22no—
51C24×C53C22×C411yescharacteristic
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—24no—
59C22×C4×C53—24no—
60C22×C4×C53—42no—
61C22×C4×C53C22⋊C412yes—
62C22×C4×C53C22⋊C412yes—
63C22×C4×C53C22⋊C412yes—
64C22×Dic53—21no—
65C22×Dic53—21no—
66C22×Dic53—24no—
67C22×Dic53—21no—
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C22×Dic53—24no—
72C22×Dic53—42no—
73C22×Dic53—42no—
74C22×Dic53C22⋊C412yes—
75C22×Dic53C22⋊C412yes—
76C22×Dic53C22⋊C412yes—
77C22×Dic53C2×D412yes—
78C22×Dic53C2×D412yes—
79C22×Dic53C2×D412yes—
80C2×#88324—48no—
81C2×#88324—44no—
82C4⋊C4×C53—28no—
83C4⋊Dic53—28no—
84#11719919—28no—
85#11719919—28no—
86#11719919—28no—
87#11719919—28no—
88C4×Dic53—28no—
89C23×C53—22no—
90C23×C53—22no—
91C23×C53C2×C22⋊C411yescharacteristic
92C23×C53C2×C22⋊C411yescharacteristic
93C23×C53C2×C22⋊C411yescharacteristic
94C23×C53#13114yes—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—24no—
99C2×C4×C53—24no—
100C2×C4×C53—42no—
101C2×C4×C53—42no—
102C2×C4×C53—42no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×Dic53—24no—
108C2×Dic53—24no—
109C2×Dic53—22no—
110C2×Dic53—24no—
111C2×Dic53—22no—
112C2×Dic53—22no—
113C2×Dic53—22no—
114C2×Dic53—24no—
115C2×Dic53—42no—
116C2×Dic53—42no—
117C2×Dic53—42no—
118C2×Dic53—42no—
119C2×Dic53—42no—
120C2×Dic53—42no—
121C2×Dic53—42no—
122C2×Dic53#10114yes—
123C2×Dic53#12014yes—
124#88324—84no—
125#8306—531nomaximal, Sylow
126C22×C53—28no—
127C22×C53—21no—
128C22×C53—21no—
129C22×C53—21no—
130C22×C53—21no—
131C22×C53#37911yescharacteristic
132C22×C53#40812yes—
133C22×C53#40812yes—
134C22×C53#40812yes—
135C22×C53#41012yes—
136C22×C53C2×#10112yes—
137C4×C53—42no—
138C4×C53—42no—
139C4×C53—42no—
140Dic53—24no—
141Dic53—24no—
142Dic53—42no—
143Dic53—42no—
144C22×#125—531no—
145#1817—532no—
146C2×#324—532no—
147C2×#324—532no—
148C2×C53—22no—
149C2×C53—22no—
150C2×C53#147014yes—
151C2×C53#178011yescharacteristic
152C2×C53#178211yescharacteristic
153C2×C53#178411yescharacteristic
154C2×#125—1064no—
155C2×#125—1064no—
156C2×#125—1064no—
157C22×C4⋊C4—531no—
158C22×C4⋊C4—531no—
159C22×C4⋊C4—531no—
160C22×C4⋊C4—531no—
161C22×C4⋊C4—531no—
162C22×C4⋊C4#615711yescharacteristic
163C22×C42—531no—
164C2×C22⋊C8—534no—
165#336—1064no—
166#324—1064no—
167#324—1064no—
168C53#830611yescharacteristic, complemented, Sylow
169C23×C4—531no—
170C23×C4—531no—
171C23×C4—531no—
172C23×C4—531no—
173C23×C4C2×#615711yescharacteristic
174C23×C4C2×#615711yescharacteristic
175C23×C4C2×#615711yescharacteristic
176C22×C8—1064no—
177#125—10632no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—534no—
182C2×C4⋊C4—534no—
183C2×C4⋊C4—1062no—
184C2×C4⋊C4—1062no—
185C2×C4⋊C4—1062no—
186C2×C4⋊C4—1062no—
187C2×C4⋊C4—1062no—
188C2×C4⋊C4—1062no—
189C2×C4⋊C4—1062no—
190C2×C4⋊C4—1062no—
191C2×C4⋊C4—1062no—
192C2×C4⋊C4—1062no—
193C2×C4⋊C4—1062no—
194C2×C4⋊C4—1062no—
195C2×C4⋊C4—1062no—
196C2×C42—534no—
197C2×C42—1062no—
198C2×C42—1062no—
199C22⋊C8—1064no—
200C22⋊C8—1064no—
201C24C22×#615711yescharacteristic
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—42no—
211C22×C4—532no—
212C22×C4—532no—
213C22×C4—532no—
214C22×C4—532no—
215C22×C4—532no—
216C22×C4—532no—
217C22×C4—1064no—
218C22×C4—1061no—
219C22×C4—1061no—
220C22×C4—1061no—
221C22×C4—1061no—
222C22×C4—1061no—
223C22×C4—1061no—
224C22×C4—1064no—
225C22×C4—2122no—
226C22×C4—2122no—
227C22×C4C22⋊#615712yes—
228C22×C4C22⋊#615712yes—
229C22×C4C22⋊#615712yes—
230C2×C8—2124no—
231C2×C8—2128no—
232C4⋊C4—28no—
233C4⋊C4—1068no—
234C4⋊C4—1068no—
235C4⋊C4—1068no—
236C4⋊C4—1068no—
237C4⋊C4—1068no—
238C42—1068no—
239C23—22no—
240C23—22no—
241C23C2×C22⋊#615711yescharacteristic
242C23C2×C22⋊#615711yescharacteristic
243C23C2×C22⋊#615711yescharacteristic
244C23#1313330414yes—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—534no—
258C2×C4—534no—
259C2×C4—1064no—
260C2×C4—1064no—
261C2×C4—1062no—
262C2×C4—1064no—
263C2×C4—1062no—
264C2×C4—1062no—
265C2×C4—1062no—
266C2×C4—1064no—
267C2×C4—2122no—
268C2×C4—2122no—
269C2×C4—2122no—
270C2×C4—2122no—
271C2×C4—2122no—
272C2×C4—2122no—
273C2×C4—2122no—
274C8—4244no—
275C22—28no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22#1681156411yescharacteristic
281C22#1681159612yes—
282C22#1681159612yes—
283C22#1681159712yes—
284C22#1681160212yes—
285C22D53⋊#10112yes—
286C4—42no—
287C4—42no—
288C4—42no—
289C4—1064no—
290C4—1064no—
291C4—2122no—
292C4—2122no—
293C2—22no—
294C2—22no—
295C2#2028998714yes—
296C2#2029053911yescharacteristic
297C2#2029054411yescharacteristic
298C2#2029055011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutator, prime radicalC24×C53C22×C4
FrattiniC24C22×#6157
FittingC22×C4⋊C4×C53C4
socleC23×C53C2×C22⋊C4
cosocleC23×Dic53C23
hypercenterC22×C4⋊C4#6157
hypocenterC53#8306
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
168. C53#8306semidirect

Maximal subgroups

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

subgroupindex
#202906052
C2×#168114692
C2×#168114682
C22×#131332072
#830653