Finite Group Databasebeta

← #27296297

Subgroups — #27296297

Order 13568.

470 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 59 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#20280380C211yesmaximal, characteristic, complemented
2#20273667C212yesmaximal
3#20272735C212yesmaximal, complemented
4#20272645C211yesmaximal, characteristic, complemented
5C53×#2022C211yesmaximal, characteristic
6#16810761—22no—
7#16810559—21no—
8#16810535C2212yes—
9#16810527—22no—
10#16810510—22no—
11C4×C22⋊Dic53—21no—
12#16810357—22no—
13#16810347C2212yes—
14C53×#534C2211yescharacteristic
15C2×#123×C53—21no—
16C2×#99×C53—21no—
17C53×#381C2211yescharacteristic
18C53×#342C2211yescharacteristic
19C22×Dic106D412yes—
20C2×#11719923—42no—
21C2×C4⋊Dic53—21no—
22C2×#11719919—22no—
23C2×C22⋊Dic53—22no—
24C2×C4×Dic53—21no—
25C2×C4×Dic53—21no—
26#13133140—42no—
27C22⋊#88322—22no—
28#13133124—22no—
29#13133124—21no—
30#13133124—22no—
31#13133124D412yes—
32C22×D4×C53—21no—
33C23×C4×C53D411yescharacteristic
34#123×C53—41no—
35#123×C53—42no—
36#123×C53—42no—
37#121×C53—22no—
38#120×C53—41no—
39C2×C4⋊C4×C53—41no—
40C2×C22⋊C4×C53—21no—
41C2×C22⋊C4×C53—41no—
42C2×C22⋊C4×C53D411yescharacteristic
43C2×C22⋊C4×C53C2311yescharacteristic
44#99×C53—42no—
45#95×C53—22no—
46C24×C53—21no—
47C24×C53C2×D411yescharacteristic
48C2×D4×C53—22no—
49C2×D4×C53—41no—
50C2×D4×C53—41no—
51C2×D4×C53—41no—
52C2×D4×C53—41no—
53C2×D4×C53—41no—
54C2×D4×C53—41no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—41no—
60C22×C4×C53—41no—
61C22×C4×C53—41no—
62C22×C4×C53SD1612yescomplemented
63C22×C4×C53C2×D412yes—
64C22×Dic53—22no—
65C22×Dic53—22no—
66C2×Dic106—42no—
67C2×Dic106—44no—
68C4⋊C4×C53—22no—
69C4⋊C4×C53—44no—
70C22⋊C4×C53—22no—
71C22⋊C4×C53—41no—
72C22⋊C4×C53—42no—
73C22⋊C4×C53—44no—
74C22⋊C4×C53—41no—
75C22⋊C4×C53—41no—
76C22⋊Dic53—44no—
77#11719923—84no—
78C4⋊Dic53—81no—
79#11719919—44no—
80C4×Dic53—81no—
81C4×Dic53—81no—
82C2×#88322—42no—
83C23×C53—21no—
84C23×C53—21no—
85C23×C53—42no—
86C23×C53—41no—
87C23×C53—41no—
88C23×C53—41no—
89C23×C53—82no—
90C23×C53#12011yescharacteristic
91C23×C53C2×SD1611yescharacteristic
92C23×C53#13711yescharacteristic
93D4×C53—82no—
94D4×C53—81no—
95D4×C53—81no—
96D4×C53—81no—
97D4×C53—81no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53—42no—
103C2×C4×C53—42no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—81no—
111C2×C4×C53—82no—
112C2×Dic53—24no—
113C2×Dic53—42no—
114C2×Dic53—42no—
115C2×Dic53—42no—
116C2×Dic53—42no—
117C2×Dic53—42no—
118Dic106—44no—
119Dic106—82no—
120#88322—82no—
121#12121—531nomaximal, Sylow
122C22×C53—22no—
123C22×C53—21no—
124C22×C53—21no—
125C22×C53—41no—
126C22×C53—41no—
127C22×C53—42no—
128C22×C53—81no—
129C22×C53—82no—
130C22×C53—81no—
131C22×C53—82no—
132C22×C53—161no—
133C22×C53#44811yescharacteristic
134C22×C53#45311yescharacteristic
135C22×C53B(4,𝔽2)11yescharacteristic
136C4×C53—42no—
137C4×C53—42no—
138C4×C53—81no—
139C4×C53—82no—
140C4×C53—81no—
141Dic53—44no—
142Dic53—82no—
143#2607—531no—
144#2022D5311yescharacteristic
145#1601—532no—
146#1351—532no—
147#1317—531no—
148C2×C53—21no—
149C2×C53—21no—
150C2×C53—41no—
151C2×C53—41no—
152C2×C53—161no—
153C2×C53#159611yescharacteristic
154C2×C53#219712yes—
155#534C2×D5311yescharacteristic
156C2×#123—21no—
157#448—1062no—
158C2×#99—21no—
159#399—1061no—
160#393—532no—
161#391—1062no—
162#387—1062no—
163#381C2×D5311yescharacteristic
164C4×C22⋊C4—1061no—
165#342C2×D5311yescharacteristic
166#327—1062no—
167#323—532no—
168C53#1212111yescharacteristic, complemented, Sylow
169C22×Q8—532no—
170C22×D4—21no—
171C23×C4C22⋊D5311yescharacteristic
172C2×SD16—2122no—
173#123—42no—
174#123—41no—
175#123—42no—
176#121—22no—
177#120—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—1061no—
180C2×C4⋊C4—1062no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—1062no—
184C2×C22⋊C4C22⋊D5311yescharacteristic
185C2×C22⋊C4C22×D5311yescharacteristic
186C2×C42—1061no—
187C2×C42—1061no—
188#103—2122no—
189#99—42no—
190C22⋊C8—1062no—
191#95—22no—
192#95—532no—
193#95—1062no—
194#95—1061no—
195#95—1062no—
196C24—21no—
197C24C2×C22⋊D5311yescharacteristic
198C2×Q8—2122no—
199C2×Q8—2124no—
200C2×D4—22no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4—41no—
205C2×D4—41no—
206C2×D4—41no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—1062no—
215C22×C4—1062no—
216C22×C4#1171992312yescomplemented
217C22×C4D4×D5312yes—
218SD16—4244no—
219C2×C8—2122no—
220C4⋊C4—22no—
221C4⋊C4—44no—
222C4⋊C4—4241no—
223C4⋊C4—2124no—
224C22⋊C4—22no—
225C22⋊C4—41no—
226C22⋊C4—42no—
227C22⋊C4—44no—
228C22⋊C4—41no—
229C22⋊C4—41no—
230C22⋊C4—2124no—
231C42—4241no—
232C42—4241no—
233C23—21no—
234C23—21no—
235C23—42no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—82no—
240C23#1313318511yescharacteristic
241C23C2×#1171992311yescharacteristic
242C23#1313324711yescharacteristic
243Q8—4242no—
244Q8—2124no—
245D4—82no—
246D4—81no—
247D4—81no—
248D4—81no—
249D4—81no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—81no—
263C2×C4—82no—
264C2×C4—1064no—
265C2×C4—2122no—
266C2×C4—2122no—
267C2×C4—2122no—
268C2×C4—2122no—
269C2×C4—2122no—
270C8—4242no—
271C22—22no—
272C22—21no—
273C22—21no—
274C22—41no—
275C22—41no—
276C22—42no—
277C22—81no—
278C22—82no—
279C22—82no—
280C22—81no—
281C22—161no—
282C22#1681076111yescharacteristic
283C22#1681079011yescharacteristic
284C22#1681080911yescharacteristic
285C4—42no—
286C4—42no—
287C4—81no—
288C4—82no—
289C4—81no—
290C4—4242no—
291C4—2124no—
292C2—21no—
293C2—21no—
294C2—41no—
295C2—41no—
296C2—161no—
297C2#2027363711yescharacteristic
298C2#2027670212yes—
subgroupG/H
centerC22#16810809
commutator, cosocleC2×C22⋊C4×C53C23
FrattiniC2×C22⋊C4C22×D53
FittingC53×#2022C2
socleC22×C53B(4,𝔽2)
prime radicalC53×#534C22
hypercenter#2022D53
hypocenterC53#12121
LCSG ▷ C2×C22⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2022

Complements

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

NH ≅ G/Ntype
1. #20280380C2semidirect
3. #20272735C2semidirect
4. #20272645C2semidirect
62. C22×C4×C53SD16semidirect
168. C53#12121semidirect
216. C22×C4#11719923semidirect

Maximal subgroups

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

subgroupindex
#202803802
#202736672
#202727352
#202726452
C53×#20222
#1212153