Finite Group Databasebeta

← #28311336

Subgroups — #28311336

Order 13568.

638 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 88 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#20290549C212yesmaximal
2#20289812C214yesmaximal, complemented
3C2×#16810529C211yesmaximal, characteristic
4C2×#13133278—22no—
5C2×#13133278C2214yes—
6#16811593—22no—
7C23×Dic106C2211yescharacteristic
8C22×C4⋊Dic53C411yescharacteristic
9#16810529—22no—
10#16810529C412yes—
11#16810529C2212yes—
12C2×#13133124—22no—
13C22×C42×C53C411yescharacteristic
14C2×#11719940—24no—
15#13133278—28no—
16#13133278—44no—
17C22⋊#88324—42no—
18C22⋊#88324—42no—
19C22×Dic106—24no—
20C22×Dic106—42no—
21C22×Dic106—41no—
22C22×Dic106—41no—
23C22×Dic106C2×C411yescharacteristic
24C22×Dic106D412yes—
25C22×Dic106C2311yescharacteristic
26C23×Dic53—21no—
27C23×Dic53D412yes—
28C2×C4⋊Dic53—21no—
29C2×C4⋊Dic53—41no—
30C2×C4⋊Dic53—41no—
31C2×C4⋊Dic53C2×C412yes—
32#13133124—24no—
33#13133124—24no—
34C23×C4×C53—21no—
35C23×C4×C53C2×C411yescharacteristic
36C2×C42×C53—21no—
37C2×C42×C53—42no—
38C2×C42×C53C2×C412yes—
39C24×C53C22⋊C411yescharacteristic
40C22×C4×C53—21no—
41C22×C4×C53—21no—
42C22×C4×C53—22no—
43C22×C4×C53—22no—
44C22×C4×C53—24no—
45C22×C4×C53—22no—
46C22×C4×C53—42no—
47C22×C4×C53—42no—
48C22×C4×C53C22⋊C411yescharacteristic
49C22×C4×C53C22×C411yescharacteristic
50C22×Dic53—22no—
51C22×Dic53—22no—
52C22×Dic53—24no—
53C22×Dic53—22no—
54C22×Dic53—42no—
55C22×Dic53—44no—
56C22×Dic53—41no—
57C22×Dic53—41no—
58C22×Dic53C22⋊C412yes—
59C22×Dic53C2×D412yes—
60C2×Dic106—24no—
61C2×Dic106—22no—
62C2×Dic106—24no—
63C2×Dic106—41no—
64C2×Dic106—41no—
65C2×Dic106—42no—
66C2×Dic106—41no—
67C2×Dic106—41no—
68C2×Dic106—48no—
69C2×Dic106—48no—
70C2×Dic106#4114yes—
71#11719940—44no—
72#11719940—44no—
73C2×#88324—44no—
74C42×C53—42no—
75C42×C53—42no—
76C4⋊Dic53—42no—
77C4⋊Dic53—42no—
78C23×C53—22no—
79C23×C53—22no—
80C23×C53—21no—
81C23×C53—21no—
82C23×C53#9911yescharacteristic
83C23×C53#9911yescharacteristic
84C23×C53C2×C22⋊C411yescharacteristic
85C2×C4×C53—24no—
86C2×C4×C53—24no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—24no—
92C2×C4×C53—48no—
93C2×C4×C53—42no—
94C2×C4×C53—42no—
95C2×C4×C53#9912yes—
96C2×C4×C53#11712yes—
97C2×Dic53—22no—
98C2×Dic53—22no—
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53—22no—
102C2×Dic53—44no—
103C2×Dic53—44no—
104C2×Dic53—44no—
105C2×Dic53—42no—
106C2×Dic53—82no—
107C2×Dic53—82no—
108C2×Dic53#12314yes—
109Dic106—44no—
110Dic106—48no—
111Dic106—44no—
112#88324—84no—
113#12860—531nomaximal, Sylow
114C22×C53—24no—
115C22×C53—24no—
116C22×C53—21no—
117C22×C53—21no—
118C22×C53—21no—
119C22×C53—21no—
120C22×C53—21no—
121C22×C53—21no—
122C22×C53#35512yes—
123C22×C53#35612yes—
124C22×C53#38111yescharacteristic
125C22×C53C2×#9911yescharacteristic
126C22×C53C2×#9911yescharacteristic
127C4×C53—22no—
128C4×C53—22no—
129C4×C53—44no—
130Dic53—24no—
131Dic53—42no—
132Dic53—82no—
133C2×#391—531no—
134#1784—532no—
135#1394—534no—
136C2×C53—21no—
137C2×C53—22no—
138C2×C53—22no—
139C2×C53—21no—
140C2×C53#181911yescharacteristic
141C2×C53C2×#35511yescharacteristic
142C2×C53C2×#35611yescharacteristic
143C23×Q8—531no—
144C22×C4⋊C4—531no—
145C22×C42#615711yescharacteristic
146C2×#101—534no—
147C2×#101—1062no—
148#407—1062no—
149#391—532no—
150#391—532no—
151#391—1062no—
152C2×#95—1062no—
153C53#1286011yescharacteristic, complemented, Sylow
154C22×Q8—532no—
155C22×Q8—531no—
156C22×Q8—531no—
157C22×Q8—1064no—
158C22×Q8—2121no—
159C22×Q8—2121no—
160C22×Q8—2122no—
161C23×C4—21no—
162C23×C4—532no—
163C23×C4—1061no—
164C23×C4C2×#615711yescharacteristic
165C2×M16—1064no—
166C2×C4⋊C4—532no—
167C2×C4⋊C4—1061no—
168C2×C4⋊C4—2121no—
169C2×C4⋊C4—2121no—
170C2×C42—21no—
171C2×C42—42no—
172C2×C42C2×#615712yes—
173#101—1068no—
174#101—2124no—
175C22⋊C8—2122no—
176C22⋊C8—2122no—
177#95—1064no—
178#95—1064no—
179C24C22⋊#615711yescharacteristic
180C2×Q8—534no—
181C2×Q8—1064no—
182C2×Q8—1064no—
183C2×Q8—1062no—
184C2×Q8—2121no—
185C2×Q8—2121no—
186C2×Q8—2122no—
187C2×Q8—2121no—
188C2×Q8—2121no—
189C2×Q8—2128no—
190C2×Q8—2128no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—22no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—532no—
200C22×C4—532no—
201C22×C4—1064no—
202C22×C4—1062no—
203C22×C4—1062no—
204C22×C4—1062no—
205C22×C4—2122no—
206C22×C4—2124no—
207C22×C4—2121no—
208C22×C4—2121no—
209C22×C4C22⋊#615711yescharacteristic
210C22×C4C22×#615711yescharacteristic
211M16—2124no—
212M16—2124no—
213C2×C8—2124no—
214C4⋊C4—2122no—
215C4⋊C4—2122no—
216C42—42no—
217C42—42no—
218C23—22no—
219C23—22no—
220C23—21no—
221C23—21no—
222C23#1313327611yescharacteristic
223C23#1313327611yescharacteristic
224C23C2×C22⋊#615711yescharacteristic
225Q8—2124no—
226Q8—2128no—
227Q8—2124no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—24no—
235C2×C4—48no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—534no—
239C2×C4—1062no—
240C2×C4—1062no—
241C2×C4—1062no—
242C2×C4—1062no—
243C2×C4—1062no—
244C2×C4—4242no—
245C2×C4—4242no—
246C2×C4—2124no—
247C2×C4—2124no—
248C2×C4—2124no—
249C2×C4—2122no—
250C2×C4#1313327512yes—
251C2×C4#1313329612yes—
252C8—4244no—
253C22—24no—
254C22—24no—
255C22—21no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22#1681153212yes—
262C22#1681153412yes—
263C22#1681156611yescharacteristic
264C22C2×#1313327511yescharacteristic
265C22C2×#1313327611yescharacteristic
266C4—22no—
267C4—22no—
268C4—44no—
269C4—1064no—
270C4—4242no—
271C4—2122no—
272C2—21no—
273C2—22no—
274C2—22no—
275C2—21no—
276C2#2029061011yescharacteristic
277C2C2×#1681153211yescharacteristic
278C2C2×#1681153411yescharacteristic
subgroupG/H
centerC22C2×#13133276
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC22×C42×C53C4
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC22×Dic106C23
hypercenterC22×C42#6157
hypocenterC53#12860
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
2. #20289812C2semidirect
153. C53#12860semidirect

Maximal subgroups

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

subgroupindex
#202905492
#202898122
C2×#168105292
#1286053