Finite Group Databasebeta

← #27299262

Subgroups — #27299262

Order 13568.

564 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C2×#16810418C212yesmaximal
2C4×#13133132C212yesmaximal
3#20272840C212yesmaximal
4C53×#2326C211yesmaximal, characteristic
5C2×#13133132C2214yes—
6C4×C22⋊Dic53—22no—
7#16810418C418yescomplemented
8#16810390—22no—
9C2×#124×C53C2211yescharacteristic
10C2×C4×Q8×C53C411yescharacteristic, complemented
11C2×C4×D4×C53C2211yescharacteristic
12C53×#391C411yescharacteristic, complemented
13C53×#390C2211yescharacteristic
14C53×#388—22no—
15C53×#385—22no—
16C53×#384C411yescharacteristic, complemented
17C53×#376C411yescharacteristic, complemented
18C2×C22⋊Dic53—24no—
19C2×C4×Dic53—42no—
20#13133132—24no—
21#13133132C2×C418yes—
22#13133124—42no—
23C22×Q8×C53C2×C411yescharacteristic
24C22×D4×C53C2311yescharacteristic
25C23×C4×C53D412yes—
26#124×C53—22no—
27#124×C53C2×C414yes—
28C4×Q8×C53—42no—
29C4×D4×C53—42no—
30C2×C4⋊C4×C53—21no—
31C2×C4⋊C4×C53—22no—
32C2×C4⋊C4×C53C2×C411yescharacteristic
33C2×C4⋊C4×C53C2×C411yescharacteristic
34C2×C22⋊C4×C53—22no—
35C2×C22⋊C4×C53—22no—
36C2×C22⋊C4×C53D412yes—
37C2×C42×C53—21no—
38C2×C42×C53C2×C411yescharacteristic
39C2×C42×C53C2×C411yescharacteristic
40C2×C42×C53C2×C411yescharacteristic
41#95×C53—21no—
42#95×C53—21no—
43#95×C53—22no—
44C24×C53C2×D412yes—
45C2×Q8×C53—21no—
46C2×Q8×C53—41no—
47C2×Q8×C53—41no—
48C2×Q8×C53C4212yescomplemented
49C2×D4×C53—41no—
50C2×D4×C53—41no—
51C2×D4×C53C22×C412yes—
52C2×D4×C53#4112yes—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—22no—
56C22×C4×C53—22no—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—82no—
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22⋊C412yes—
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22×C411yescharacteristic
64C22×Dic53—44no—
65C4⋊C4×C53—41no—
66C4⋊C4×C53—41no—
67C4⋊C4×C53—42no—
68C4⋊C4×C53—82no—
69C22⋊C4×C53—28no—
70C22⋊C4×C53—42no—
71C22⋊C4×C53—82no—
72C42×C53—22no—
73C42×C53—21no—
74C42×C53—42no—
75C42×C53—41no—
76C42×C53C4212yescomplemented
77C22⋊Dic53—28no—
78C22⋊Dic53—44no—
79C4×Dic53—84no—
80C23×C53—22no—
81C23×C53—22no—
82C23×C53—42no—
83C23×C53—42no—
84C23×C53C2×C22⋊C411yescharacteristic
85C23×C53C4×D414yes—
86Q8×C53—42no—
87Q8×C53—42no—
88D4×C53—42no—
89D4×C53—42no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—21no—
96C2×C4×C53—41no—
97C2×C4×C53—44no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53—42no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—42no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—82no—
110C2×C4×C53#9912yes—
111C2×C4×C53C2×C4211yescharacteristic
112C2×C4×C53#11711yescharacteristic
113C2×Dic53—48no—
114C2×Dic53—84no—
115C2×Dic53—84no—
116#12627—531nomaximal, Sylow
117C22×C53—21no—
118C22×C53—21no—
119C22×C53—24no—
120C22×C53—44no—
121C22×C53—42no—
122C22×C53—42no—
123C22×C53—82no—
124C22×C53C4×C22⋊C411yescharacteristic
125C22×C53C2×#9911yescharacteristic
126C22×C53#41011yescharacteristic
127C4×C53—22no—
128C4×C53—21no—
129C4×C53—21no—
130C4×C53—42no—
131C4×C53—42no—
132C4×C53—41no—
133C4×C53—42no—
134C4×C53—81no—
135C4×C53—81no—
136Dic53—88no—
137#2326D5311yescharacteristic
138C2×#354—532no—
139C4×#99—532no—
140#1393—532no—
141C2×C53—21no—
142C2×C53—21no—
143C2×C53—44no—
144C2×C53C4×#9911yescharacteristic
145C2×C53#212711yescharacteristic
146C2×C53#212911yescharacteristic
147C2×#124C2×D5311yescharacteristic
148C2×C4×Q8Dic5311yescharacteristic, complemented
149C2×C4×D4C2×D5311yescharacteristic
150C2×#99—534no—
151#391Dic5311yescharacteristic, complemented
152#390C2×D5311yescharacteristic
153#388—22no—
154#385—22no—
155#384Dic5311yescharacteristic, complemented
156C4×C22⋊C4—1062no—
157#376Dic5311yescharacteristic, complemented
158#354—538no—
159#342—1062no—
160C53#1262711yescharacteristic, complemented, Sylow
161C22×Q8C2×Dic5311yescharacteristic
162C22×D4C22×D5311yescharacteristic
163C23×C4C22⋊D5312yes—
164#124—22no—
165#124C4×D5314yes—
166C4×Q8—42no—
167C4×D4—42no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4C2×Dic5311yescharacteristic
171C2×C4⋊C4C2×Dic5311yescharacteristic
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—1064no—
175C2×C22⋊C4C22⋊D5312yes—
176C2×C42—21no—
177C2×C42—2122no—
178C2×C42C2×Dic5311yescharacteristic
179C2×C42C2×Dic5311yescharacteristic
180C2×C42C2×Dic5311yescharacteristic
181#99—538no—
182#99—1064no—
183#95—21no—
184#95—22no—
185#95—21no—
186#95—2122no—
187C24C2×C22⋊D5312yes—
188C2×Q8—21no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8C4×Dic5312yescomplemented
192C2×D4—41no—
193C2×D4—41no—
194C2×D4C2×C4×D5312yes—
195C2×D4#1171994512yes—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—82no—
203C22×C4—2124no—
204C22×C4C22⋊Dic5311yescharacteristic
205C22×C4C22⋊Dic5312yes—
206C22×C4C22⋊Dic5311yescharacteristic
207C22×C4C22×Dic5311yescharacteristic
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—42no—
211C4⋊C4—82no—
212C22⋊C4—28no—
213C22⋊C4—42no—
214C22⋊C4—82no—
215C22⋊C4—1068no—
216C22⋊C4—2124no—
217C42—22no—
218C42—21no—
219C42—41no—
220C42—42no—
221C42—4244no—
222C42C4×Dic5312yescomplemented
223C23—22no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23C2×C22⋊Dic5311yescharacteristic
228C23C4×C22⋊D5314yes—
229Q8—42no—
230Q8—42no—
231D4—42no—
232D4—42no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—41no—
240C2×C4—44no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—82no—
253C2×C4—4244no—
254C2×C4—4244no—
255C2×C4—2128no—
256C2×C4#1313313212yes—
257C2×C4C2×C4×Dic5311yescharacteristic
258C2×C4#1313317611yescharacteristic
259C22—21no—
260C22—21no—
261C22—24no—
262C22—44no—
263C22—42no—
264C22—42no—
265C22—82no—
266C22C4×C22⋊Dic5311yescharacteristic
267C22C2×#1313313211yescharacteristic
268C22#1681059711yescharacteristic
269C4—21no—
270C4—21no—
271C4—22no—
272C4—42no—
273C4—42no—
274C4—42no—
275C4—41no—
276C4—81no—
277C4—81no—
278C4—4248no—
279C2—21no—
280C2—21no—
281C2—44no—
282C2C4×#1313313211yescharacteristic
283C2#2027638411yescharacteristic
284C2#2027639011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC2×C4×C53C2×C42
FrattiniC22×D4C22×D53
FittingC53×#2326C2
socleC22×C53C2×#99
prime radical, cosocleC22×D4×C53C23
hypercenter#2326D53
hypocenterC53#12627
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2326

Complements

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

NH ≅ G/Ntype
7. #16810418C4semidirect
10. C2×C4×Q8×C53C4semidirect
12. C53×#391C4semidirect
16. C53×#384C4semidirect
17. C53×#376C4semidirect
48. C2×Q8×C53C42semidirect
76. C42×C53C42semidirect
148. C2×C4×Q8Dic53semidirect
151. #391Dic53semidirect
155. #384Dic53semidirect
157. #376Dic53semidirect
160. C53#12627semidirect
191. C2×Q8C4×Dic53semidirect
222. C42C4×Dic53semidirect

Maximal subgroups

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

subgroupindex
C2×#168104182
C4×#131331322
#202728402
C53×#23262
#1262753