Finite Group Databasebeta

← #27281921

Subgroups — #27281921

Order 13568.

614 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 103 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#20278989C211yesmaximal, characteristic
2#20274815C212yesmaximal, complemented
3#20273042C211yesmaximal, characteristic
4#20272568C212yesmaximal, complemented
5C53×#2312C211yesmaximal, characteristic
6C2×#13133176C2211yescharacteristic
7#16810587C2212yes—
8#16810559C412yescomplemented
9#16810548C412yescomplemented
10#16810511—22no—
11#16810496—21no—
12#16810486C2211yescharacteristic
13#16810479—21no—
14#16810383—24no—
15#16810347—22no—
16C22×C22⋊C4×C53C2211yescharacteristic
17C53×#388—24no—
18C53×#380—21no—
19C53×#379—21no—
20C4×C22⋊C4×C53C2212yes—
21C2×#11719917—22no—
22#13133176—22no—
23#13133176—22no—
24C2×C4⋊Dic53C2×C411yescharacteristic
25C2×C4⋊Dic53C2×C411yescharacteristic
26C2×#11719919—22no—
27C2×C22⋊Dic53—22no—
28C2×C22⋊Dic53—22no—
29C2×C22⋊Dic53—21no—
30C2×C22⋊Dic53—21no—
31C2×C4×Dic53—21no—
32C22⋊#88322—22no—
33C22⋊#88322—22no—
34#13133124—22no—
35#13133124—21no—
36#13133124C2×C412yes—
37C25×C53D411yescharacteristic
38C23×C4×C53D411yescharacteristic
39C23×C4×C53C2311yescharacteristic
40C2×C4⋊C4×C53—22no—
41C2×C22⋊C4×C53—22no—
42C2×C22⋊C4×C53—22no—
43C2×C22⋊C4×C53—22no—
44C2×C22⋊C4×C53—22no—
45C2×C22⋊C4×C53—22no—
46C2×C22⋊C4×C53C2×C412yes—
47C2×C22⋊C4×C53D412yes—
48C2×C42×C53D414yes—
49#95×C53—22no—
50#95×C53—22no—
51C24×C53—21no—
52C24×C53—21no—
53C24×C53—42no—
54C24×C53—42no—
55C24×C53—81no—
56C24×C53C22⋊C412yes—
57C24×C53C2×D411yescharacteristic
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—24no—
61C22×C4×C53—42no—
62C22×C4×C53—81no—
63C22×C4×C53C22⋊C412yes—
64C22×C4×C53C22×C411yescharacteristic
65C22×C4×C53C2×D412yes—
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53#4112yes—
68C22×Dic53—22no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C4⋊C4×C53—82no—
72C22⋊C4×C53—24no—
73C22⋊C4×C53—24no—
74C22⋊C4×C53—44no—
75C22⋊C4×C53—82no—
76C22⋊C4×C53—82no—
77C42×C53—44no—
78C22⋊Dic53—42no—
79C22⋊Dic53—44no—
80C22⋊Dic53—41no—
81C22⋊Dic53—41no—
82C22⋊Dic53—82no—
83C4⋊Dic53—22no—
84C4⋊Dic53—22no—
85#11719919—82no—
86C4×Dic53—42no—
87#11719917—44no—
88C2×#88322—42no—
89C23×C53—22no—
90C23×C53—22no—
91C23×C53—22no—
92C23×C53—22no—
93C23×C53—22no—
94C23×C53—41no—
95C23×C53—42no—
96C23×C53—42no—
97C23×C53—42no—
98C23×C53—42no—
99C23×C53—42no—
100C23×C53—42no—
101C23×C53—81no—
102C23×C53—81no—
103C23×C53—88no—
104C23×C53C2×C22⋊C411yescharacteristic
105C23×C53#12011yescharacteristic
106C23×C53#12311yescharacteristic
107C2×C4×C53—24no—
108C2×C4×C53—22no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—42no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—44no—
115C2×C4×C53—44no—
116C2×C4×C53—44no—
117C2×C4×C53—81no—
118C2×C4×C53C4×D412yes—
119C2×C4×C53#12112yes—
120C2×Dic53—42no—
121C2×Dic53—42no—
122C2×Dic53—42no—
123C2×Dic53—41no—
124C2×Dic53—41no—
125C2×Dic53—42no—
126C2×Dic53—42no—
127#88322—82no—
128#9481—531nomaximal, Sylow
129C22×C53—21no—
130C22×C53—21no—
131C22×C53—21no—
132C22×C53—21no—
133C22×C53—21no—
134C22×C53—21no—
135C22×C53—21no—
136C22×C53—44no—
137C22×C53—44no—
138C22×C53—42no—
139C22×C53—42no—
140C22×C53—42no—
141C22×C53—82no—
142C22×C53—84no—
143C22×C53—84no—
144C22×C53#38611yescharacteristic
145C22×C53#42112yes—
146C22×C53#45812yes—
147C4×C53—22no—
148C4×C53—22no—
149C4×C53—42no—
150C4×C53—84no—
151Dic53—82no—
152Dic53—81no—
153Dic53—81no—
154#2482—531no—
155#2312D5311yescharacteristic
156#1826—532no—
157#1462—531no—
158#1284—532no—
159C2×C53—22no—
160C2×C53—22no—
161C2×C53—41no—
162C2×C53—82no—
163C2×C53#189011yescharacteristic
164C2×C53#189711yescharacteristic
165C2×C53#191111yescharacteristic
166C2×#117—531no—
167C22×C22⋊C4C2×D5311yescharacteristic
168#407—532no—
169#399—532no—
170#397—532no—
171#388—24no—
172#388—1062no—
173#385—1061no—
174#381—531no—
175#380—21no—
176#379—21no—
177#379—1061no—
178C4×C22⋊C4C2×D5312yes—
179#339—1064no—
180#323—1062no—
181C53#948111yescharacteristic, complemented, Sylow
182C25C22⋊D5311yescharacteristic
183C23×C4C22⋊D5311yescharacteristic
184C23×C4C22×D5311yescharacteristic
185C2×M16—1062no—
186#117—1062no—
187#117—1062no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—531no—
190C2×C4⋊C4—531no—
191C2×C4⋊C4—1062no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—1062no—
198C2×C22⋊C4—1061no—
199C2×C22⋊C4—1061no—
200C2×C22⋊C4—1062no—
201C2×C22⋊C4C4×D5312yes—
202C2×C22⋊C4D21212yes—
203C2×C42—1061no—
204C2×C42C22⋊D5314yes—
205C22⋊C8—1062no—
206C22⋊C8—1062no—
207#95—22no—
208#95—22no—
209#95—532no—
210#95—1061no—
211#95—1062no—
212C24—21no—
213C24—21no—
214C24—42no—
215C24—42no—
216C24—81no—
217C24#1171992112yes—
218C24C2×C22⋊D5311yescharacteristic
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—24no—
222C22×C4—42no—
223C22×C4—81no—
224C22×C4—1062no—
225C22×C4—1061no—
226C22×C4—1061no—
227C22×C4#1171992112yes—
228C22×C4C2×C4×D5311yescharacteristic
229C22×C4C2×D21211yescharacteristic
230C22×C4#1171994512yes—
231C22×C4C2×C22⋊D5312yes—
232M16—2124no—
233C2×C8—2122no—
234C4⋊C4—82no—
235C4⋊C4—1062no—
236C4⋊C4—1062no—
237C4⋊C4—4242no—
238C22⋊C4—24no—
239C22⋊C4—24no—
240C22⋊C4—44no—
241C22⋊C4—82no—
242C22⋊C4—82no—
243C22⋊C4—4242no—
244C22⋊C4—2122no—
245C22⋊C4—2124no—
246C22⋊C4—2121no—
247C22⋊C4—2121no—
248C42—44no—
249C42—2122no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—41no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—81no—
263C23—81no—
264C23—88no—
265C23C2×#1171992111yescharacteristic
266C23#1313318611yescharacteristic
267C23#1313320011yescharacteristic
268C2×C4—24no—
269C2×C4—22no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—81no—
279C2×C4—2122no—
280C2×C4—2122no—
281C2×C4—2122no—
282C2×C4—2121no—
283C2×C4—2121no—
284C2×C4—2122no—
285C2×C4—2122no—
286C2×C4C4×C22⋊D5312yes—
287C2×C4#1313319112yes—
288C8—4242no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—44no—
297C22—44no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—82no—
302C22—84no—
303C22—84no—
304C22#1681050211yescharacteristic
305C22#1681064112yes—
306C22#1681081312yes—
307C4—22no—
308C4—22no—
309C4—42no—
310C4—84no—
311C4—4242no—
312C4—4241no—
313C4—4241no—
314C2—22no—
315C2—22no—
316C2—41no—
317C2—82no—
318C2#2027512811yescharacteristic
319C2#2027516311yescharacteristic
320C2#2027524711yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2312C2
socleC22×C53#386
prime radicalC25×C53D4
cosocleC23×C4×C53C23
hypercenter#2312D53
hypocenterC53#9481
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2312

Complements

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

NH ≅ G/Ntype
2. #20274815C2semidirect
4. #20272568C2semidirect
8. #16810559C4semidirect
9. #16810548C4semidirect
181. C53#9481semidirect

Maximal subgroups

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

subgroupindex
#202789892
#202748152
#202730422
#202725682
C53×#23122
#948153