Finite Group Databasebeta

← #28303378

Subgroups — #28303378

Order 13568.

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

normal subgroups only

This group has 522 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#20289946C212yesmaximal
2#20289543C212yesmaximal, complemented
3#20289539C212yesmaximal, complemented
4C4×#13133176C211yesmaximal, characteristic
5#16811639C2214yes—
6#16811468—22no—
7#16811466—22no—
8C2×#13133176C411yescharacteristic
9C2×#13133176C2211yescharacteristic
10C4×C4⋊Dic53—22no—
11C4×C22⋊Dic53—22no—
12#16810469C411yescharacteristic
13#16810469C2211yescharacteristic
14C42×Dic53C411yescharacteristic
15C42×Dic53C2211yescharacteristic
16C22×C42×C53C411yescharacteristic
17C4⋊#88324—24no—
18C22⋊#88324—24no—
19C4×#88324—24no—
20#13133176—22no—
21#13133176—22no—
22#13133176—22no—
23#13133176C814yescomplemented
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53D412yes—
26C2×C22⋊Dic53—21no—
27C2×C22⋊Dic53D412yes—
28C2×C4×Dic53—22no—
29C2×C4×Dic53—22no—
30C2×C4×Dic53C2×C411yescharacteristic
31C2×C4×Dic53C2×C411yescharacteristic
32C2×C4×Dic53C2×C411yescharacteristic
33C2×C4×Dic53C2311yescharacteristic
34#13133124—22no—
35#13133124—22no—
36C23×C4×C53—21no—
37C23×C4×C53C2×C411yescharacteristic
38C2×C42×C53—22no—
39C2×C42×C53—22no—
40C2×C42×C53—21no—
41C2×C42×C53C2×C411yescharacteristic
42C2×C42×C53C2×C411yescharacteristic
43C24×C53C22⋊C411yescharacteristic
44C22×C4×C53—21no—
45C22×C4×C53—21no—
46C22×C4×C53—21no—
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53—24no—
50C22×C4×C53—21no—
51C22×C4×C53—44no—
52C22×C4×C53C22⋊C411yescharacteristic
53C22×C4×C53C2×C812yes—
54C22×C4×C53M1612yes—
55C22×C4×C53C22×C411yescharacteristic
56C22×Dic53—21no—
57C22×Dic53—21no—
58C22×Dic53C22⋊C412yes—
59C22×Dic53C2×D412yes—
60C2×#88324—44no—
61C42×C53—24no—
62C42×C53—21no—
63C42×C53—21no—
64C42×C53M1612yes—
65C42×C53M1612yes—
66C22⋊Dic53—24no—
67C22⋊Dic53—24no—
68C4⋊Dic53—24no—
69C4⋊Dic53—24no—
70C4×Dic53—21no—
71C4×Dic53—21no—
72C4×Dic53—21no—
73C4×Dic53—21no—
74C4×Dic53—44no—
75C4×Dic53—44no—
76C4×Dic53C2×C812yes—
77C4×Dic53C2×C812yes—
78C4×Dic53#4112yes—
79C4×Dic53#4112yes—
80C23×C53—21no—
81C23×C53—21no—
82C23×C53—42no—
83C23×C53C22⋊C812yes—
84C23×C53C2×C22⋊C411yescharacteristic
85C2×C4×C53—24no—
86C2×C4×C53—21no—
87C2×C4×C53—22no—
88C2×C4×C53—21no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53—22no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—48no—
95C2×C4×C53—42no—
96C2×C4×C53C22⋊C812yes—
97C2×C4×C53#11711yescharacteristic
98C2×C4×C53#11711yescharacteristic
99C2×C4×C53C22×C811yescharacteristic
100C2×C4×C53C2×M1611yescharacteristic
101C2×C4×C53C2×M1611yescharacteristic
102C2×C4×C53C2×M1611yescharacteristic
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53—22no—
106C2×Dic53—42no—
107C2×Dic53—42no—
108C2×Dic53—42no—
109C2×Dic53—42no—
110C2×Dic53#12314yes—
111#88324—84no—
112#8731—531nomaximal, Sylow
113C22×C53—22no—
114C22×C53—22no—
115C22×C53—42no—
116C22×C53—42no—
117C22×C53—42no—
118C22×C53#38111yescharacteristic
119C22×C53C2×C22⋊C811yescharacteristic
120C22×C53#40711yescharacteristic
121C4×C53—24no—
122C4×C53—22no—
123C4×C53—42no—
124C4×C53#43111yescharacteristic
125C4×C53#43111yescharacteristic
126C4×C53#43211yescharacteristic
127C4×C53#43211yescharacteristic
128Dic53—24no—
129Dic53—42no—
130Dic53—42no—
131Dic53—42no—
132C4×#117—531no—
133#1458—532no—
134#1276—532no—
135#1274—532no—
136C2×C53—21no—
137C2×C53—21no—
138C2×C53—42no—
139C2×C53#181711yescharacteristic
140C2×C53#182712yes—
141C2×#117—531no—
142C2×#117—531no—
143C22×C42#615711yescharacteristic
144#431—534no—
145C4×C4⋊C4—1062no—
146C4×C22⋊C4—1062no—
147#376—531no—
148#376—531no—
149C43—531no—
150C43—531no—
151#324—1062no—
152#323—1062no—
153C53#873111yescharacteristic, complemented, Sylow
154C23×C4—21no—
155C23×C4C2×#615711yescharacteristic
156#117—534no—
157#117—1062no—
158#117—1062no—
159#117—1062no—
160C2×C4⋊C4—532no—
161C2×C4⋊C4—1061no—
162C2×C22⋊C4—532no—
163C2×C22⋊C4—1061no—
164C2×C42—21no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—531no—
168C2×C42—531no—
169C2×C42—531no—
170C2×C42—531no—
171C2×C42—1062no—
172C2×C42—1062no—
173C2×C42C2×#615711yescharacteristic
174C2×C42C2×#615711yescharacteristic
175C4⋊C8—1064no—
176C22⋊C8—1064no—
177C4×C8—1064no—
178#95—1062no—
179#95—1062no—
180C24C22⋊#615711yescharacteristic
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—24no—
187C22×C4—21no—
188C22×C4—44no—
189C22×C4—532no—
190C22×C4—532no—
191C22×C4—1061no—
192C22×C4—1061no—
193C22×C4D53⋊C812yes—
194C22×C4#1171993612yes—
195C22×C4C22⋊#615711yescharacteristic
196C22×C4C22×#615711yescharacteristic
197C2×C8—2124no—
198C4⋊C4—1064no—
199C4⋊C4—1064no—
200C22⋊C4—1064no—
201C22⋊C4—1064no—
202C42—24no—
203C42—21no—
204C42—21no—
205C42—532no—
206C42—532no—
207C42—532no—
208C42—532no—
209C42—1061no—
210C42—1061no—
211C42—1061no—
212C42—1061no—
213C42—2124no—
214C42—2124no—
215C42#1171994012yes—
216C42#1171994012yes—
217C23—21no—
218C23—21no—
219C23—42no—
220C23#1313327412yes—
221C23C2×C22⋊#615711yescharacteristic
222C2×C4—24no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—48no—
232C2×C4—42no—
233C2×C4—534no—
234C2×C4—1062no—
235C2×C4—1062no—
236C2×C4—1062no—
237C2×C4—2122no—
238C2×C4—2122no—
239C2×C4—2122no—
240C2×C4—2122no—
241C2×C4#1313327412yes—
242C2×C4#1313329611yescharacteristic
243C2×C4#1313329611yescharacteristic
244C2×C4C2×D53⋊C811yescharacteristic
245C2×C4C2×#1171994011yescharacteristic
246C2×C4C2×#1171994011yescharacteristic
247C2×C4C2×#1171993611yescharacteristic
248C8—4244no—
249C22—22no—
250C22—22no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22#1681156611yescharacteristic
255C22C2×#1313327411yescharacteristic
256C22#1681159311yescharacteristic
257C4—24no—
258C4—22no—
259C4—42no—
260C4—1064no—
261C4—2122no—
262C4—2122no—
263C4—2122no—
264C4#1681164011yescharacteristic
265C4#1681164011yescharacteristic
266C4#1681164411yescharacteristic
267C4#1681164411yescharacteristic
268C2—21no—
269C2—21no—
270C2—42no—
271C2#2029060611yescharacteristic
272C2#2029062412yes—
subgroupG/H
centerC2×C4#13133296
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC22×C42×C53C4
socleC22×C53#381
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC22×C42#6157
hypocenterC53#8731
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
2. #20289543C2semidirect
3. #20289539C2semidirect
23. #13133176C8semidirect
153. C53#8731semidirect

Maximal subgroups

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

subgroupindex
#202899462
#202895432
#202895392
C4×#131331762
#873153