Finite Group Databasebeta

← #27300087

Subgroups — #27300087

Order 13568.

524 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 119 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
2#20274703C212yesmaximal
3#20272840C212yesmaximal
4C53×#2326C211yesmaximal, characteristic
5C2×#13133132C2214yes—
6#16810483—22no—
7#16810418—24no—
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×C4⋊Dic53—42no—
19C2×C22⋊Dic53—24no—
20#13133132—24no—
21#13133132—24no—
22#13133124—42no—
23C22×Q8×C53C2×C411yescharacteristic
24C22×D4×C53C2311yescharacteristic
25C23×C4×C53D412yes—
26#124×C53—22no—
27#124×C53D412yes—
28#124×C53Q812yes—
29C4×Q8×C53—42no—
30C4×D4×C53—42no—
31C2×C4⋊C4×C53—21no—
32C2×C4⋊C4×C53—22no—
33C2×C4⋊C4×C53C2×C411yescharacteristic
34C2×C4⋊C4×C53C2×C411yescharacteristic
35C2×C22⋊C4×C53—22no—
36C2×C22⋊C4×C53—22no—
37C2×C22⋊C4×C53D412yes—
38C2×C42×C53—21no—
39C2×C42×C53C2×C411yescharacteristic
40C2×C42×C53C2×C411yescharacteristic
41C2×C42×C53C2×C411yescharacteristic
42#95×C53—21no—
43#95×C53—21no—
44#95×C53—22no—
45C24×C53C2×D412yes—
46C2×Q8×C53—21no—
47C2×Q8×C53—41no—
48C2×Q8×C53—41no—
49C2×Q8×C53C4⋊C412yescomplemented
50C2×D4×C53—41no—
51C2×D4×C53—41no—
52C2×D4×C53C2×D411yescharacteristic
53C2×D4×C53C2×Q811yescharacteristic
54C2×D4×C53#4112yes—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—22no—
58C22×C4×C53—22no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—82no—
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22⋊C412yes—
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22×C411yescharacteristic
66C22×Dic53—44no—
67C4⋊C4×C53—41no—
68C4⋊C4×C53—41no—
69C4⋊C4×C53—42no—
70C4⋊C4×C53—82no—
71C22⋊C4×C53—24no—
72C22⋊C4×C53—24no—
73C22⋊C4×C53—42no—
74C22⋊C4×C53—82no—
75C42×C53—22no—
76C42×C53—21no—
77C42×C53—42no—
78C42×C53—41no—
79C42×C53C4⋊C412yescomplemented
80C22⋊Dic53—44no—
81C22⋊Dic53—44no—
82C4⋊Dic53—84no—
83C23×C53—22no—
84C23×C53—22no—
85C23×C53—42no—
86C23×C53—42no—
87C23×C53C2×C22⋊C411yescharacteristic
88C23×C53#12112yes—
89C23×C53C22⋊Q812yes—
90Q8×C53—42no—
91Q8×C53—42no—
92D4×C53—41no—
93D4×C53—41no—
94D4×C53—42no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—21no—
101C2×C4×C53—41no—
102C2×C4×C53—44no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—42no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—42no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—82no—
115C2×C4×C53#9912yes—
116C2×C4×C53C2×C4⋊C411yescharacteristic
117C2×C4×C53#11711yescharacteristic
118C2×Dic53—84no—
119C2×Dic53—84no—
120C2×Dic53—84no—
121#12800—531nomaximal, Sylow
122C22×C53—21no—
123C22×C53—21no—
124C22×C53—22no—
125C22×C53—22no—
126C22×C53—42no—
127C22×C53—42no—
128C22×C53—42no—
129C22×C53—42no—
130C22×C53—82no—
131C22×C53#38011yescharacteristic
132C22×C53C2×#9911yescharacteristic
133C22×C53#41011yescharacteristic
134C4×C53—21no—
135C4×C53—21no—
136C4×C53—22no—
137C4×C53—42no—
138C4×C53—42no—
139C4×C53—42no—
140C4×C53—41no—
141C4×C53—81no—
142C4×C53—81no—
143Dic53—164no—
144#2326D5311yescharacteristic
145C2×#354—532no—
146#1799—532no—
147#1393—532no—
148C2×C53—21no—
149C2×C53—21no—
150C2×C53—42no—
151C2×C53—42no—
152C2×C53#179911yescharacteristic
153C2×C53#212711yescharacteristic
154C2×C53#212911yescharacteristic
155C2×#124C2×D5311yescharacteristic
156C2×C4×Q8Dic5311yescharacteristic, complemented
157C2×C4×D4C2×D5311yescharacteristic
158C2×#99—534no—
159#391Dic5311yescharacteristic, complemented
160#390C2×D5311yescharacteristic
161#388—22no—
162#385—22no—
163#384Dic5311yescharacteristic, complemented
164#380—1062no—
165#376Dic5311yescharacteristic, complemented
166#354—1064no—
167#342—1062no—
168C53#1280011yescharacteristic, complemented, Sylow
169C22×Q8C2×Dic5311yescharacteristic
170C22×D4C22×D5311yescharacteristic
171C23×C4C22⋊D5312yes—
172#124—22no—
173#124Dic10612yes—
174#124D21212yes—
175C4×Q8—42no—
176C4×D4—42no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—2122no—
180C2×C4⋊C4C2×Dic5311yescharacteristic
181C2×C4⋊C4C2×Dic5311yescharacteristic
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—1064no—
185C2×C22⋊C4C22⋊D5312yes—
186C2×C42—21no—
187C2×C42C2×Dic5311yescharacteristic
188C2×C42C2×Dic5311yescharacteristic
189C2×C42C2×Dic5311yescharacteristic
190#99—1064no—
191#99—1064no—
192#95—21no—
193#95—22no—
194#95—21no—
195#95—2122no—
196C24C2×C22⋊D5312yes—
197C2×Q8—21no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8C4⋊Dic5312yescomplemented
201C2×D4—41no—
202C2×D4—41no—
203C2×D4C2×Dic10611yescharacteristic
204C2×D4C2×D21211yescharacteristic
205C2×D4#1171994512yes—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—82no—
213C22×C4—2124no—
214C22×C4C22⋊Dic5311yescharacteristic
215C22×C4C22⋊Dic5312yes—
216C22×C4C22⋊Dic5311yescharacteristic
217C22×C4C22×Dic5311yescharacteristic
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—42no—
221C4⋊C4—82no—
222C4⋊C4—4244no—
223C22⋊C4—24no—
224C22⋊C4—24no—
225C22⋊C4—42no—
226C22⋊C4—82no—
227C22⋊C4—2124no—
228C22⋊C4—2124no—
229C42—22no—
230C42—21no—
231C42—41no—
232C42—42no—
233C42C4⋊Dic5312yescomplemented
234C23—22no—
235C23—22no—
236C23—42no—
237C23—42no—
238C23C2×C22⋊Dic5311yescharacteristic
239C23#1313319112yes—
240C23#1313319612yes—
241Q8—42no—
242Q8—42no—
243D4—41no—
244D4—41no—
245D4—42no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—44no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—82no—
266C2×C4—4244no—
267C2×C4—4244no—
268C2×C4—4244no—
269C2×C4#1313313212yes—
270C2×C4C2×C4⋊Dic5311yescharacteristic
271C2×C4#1313317611yescharacteristic
272C22—21no—
273C22—21no—
274C22—22no—
275C22—22no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—82no—
281C22#1681048311yescharacteristic
282C22C2×#1313313211yescharacteristic
283C22#1681059711yescharacteristic
284C4—21no—
285C4—21no—
286C4—22no—
287C4—42no—
288C4—42no—
289C4—42no—
290C4—41no—
291C4—81no—
292C4—81no—
293C4—8484no—
294C2—21no—
295C2—21no—
296C2—42no—
297C2—42no—
298C2#2027470311yescharacteristic
299C2#2027638411yescharacteristic
300C2#2027639011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC22×D4C22×D53
FittingC53×#2326C2
socleC22×C53C2×#99
prime radical, cosocleC22×D4×C53C23
hypercenter#2326D53
hypocenterC53#12800
LCSG ▷ C22×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
10. C2×C4×Q8×C53C4semidirect
12. C53×#391C4semidirect
16. C53×#384C4semidirect
17. C53×#376C4semidirect
49. C2×Q8×C53C4⋊C4semidirect
79. C42×C53C4⋊C4semidirect
156. C2×C4×Q8Dic53semidirect
159. #391Dic53semidirect
163. #384Dic53semidirect
165. #376Dic53semidirect
168. C53#12800semidirect
200. C2×Q8C4⋊Dic53semidirect
233. C42C4⋊Dic53semidirect

Maximal subgroups

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

subgroupindex
C2×#168104182
#202747032
#202728402
C53×#23262
#1280053