Finite Group Databasebeta

← #27273279

Subgroups — #27273279

Order 13568.

478 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 83 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#20281741C211yesmaximal, characteristic
2#20276258C211yesmaximal, characteristic, complemented
3#20276120C212yesmaximal, complemented
4#20272535C212yesmaximal, complemented
5C53×#1272C211yesmaximal, characteristic, complemented
6C2×#13133206—21no—
7C2×#13133148—21no—
8C2×#13133141—21no—
9C2×#13133139—22no—
10#16810540—21no—
11#16810533—22no—
12#16810525C2212yes—
13#16810515—22no—
14#16810493C2211yescharacteristic
15#16810386—22no—
16#16810378C2211yescharacteristic
17C53×#391C2211yescharacteristic
18C53×#336C2212yes—
19C22×D212—21no—
20C22×D212—21no—
21C22×D212D412yes—
22C22×#88322—21no—
23#13133206—42no—
24#13133206—42no—
25C2×C4⋊Dic53—22no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53D412yes—
28C2×#11719921—22no—
29C2×#11719921—22no—
30C2×#11719921—22no—
31C2×#11719921—22no—
32C2×C4×Dic53—21no—
33#13133148—42no—
34#13133141—42no—
35#13133139—44no—
36#13133124—21no—
37C22×Q8×C53D411yescharacteristic
38C22×C8×C53—22no—
39C2×C4⋊C4×C53D411yescharacteristic
40C2×C42×C53C2311yescharacteristic
41#95×C53—22no—
42C23×D53—22no—
43C2×Q8×C53—21no—
44C2×Q8×C53—41no—
45C2×Q8×C53—41no—
46C2×Q8×C53SD1612yes—
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53C2×D411yescharacteristic
50C22×C4×C53C2×D412yes—
51C22×Dic53—22no—
52C22×Dic53—41no—
53C2×D212—22no—
54C2×D212—22no—
55C2×D212—42no—
56C2×D212—42no—
57C2×D212—42no—
58C2×D212—81no—
59C2×D212—81no—
60C2×D212—81no—
61C2×D212—82no—
62C2×C8×C53—42no—
63C2×C8×C53—42no—
64C2×C8×C53—42no—
65C4⋊C4×C53—41no—
66C42×C53—41no—
67#11719921—44no—
68#11719921—42no—
69#11719921—42no—
70#11719921—42no—
71#11719921—42no—
72#11719921—82no—
73C4⋊Dic53—42no—
74C4⋊Dic53—41no—
75C4⋊Dic53—41no—
76C4⋊Dic53—42no—
77C4⋊Dic53—42no—
78C4×Dic53—42no—
79C2×#88322—22no—
80C2×#88322—41no—
81C2×#88322—41no—
82C23×C53#12011yescharacteristic
83C22×D53—42no—
84C22×D53—42no—
85C22×D53—42no—
86C22×D53—42no—
87C22×D53—42no—
88C22×D53—42no—
89C22×D53—42no—
90Q8×C53—42no—
91Q8×C53—81no—
92C2×C4×C53—21no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—21no—
96C2×C4×C53—22no—
97C2×C4×C53—41no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53—42no—
101C2×C4×C53C2×SD1611yescharacteristic
102C2×C4×C53#13512yes—
103C2×C4×C53AGL(1,ℤ/8)12yes—
104C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
105C2×Dic53—42no—
106C2×Dic53—42no—
107C2×Dic53—42no—
108C2×Dic53—82no—
109C2×Dic53—81no—
110D212—44no—
111D212—82no—
112D212—84no—
113D212—84no—
114C8×C53—82no—
115#88322—42no—
116#7810—531nomaximal, Sylow
117C22×C53#44912yes—
118C22×C53#45011yescharacteristic
119C22×C53#45311yescharacteristic
120C22×C53#45512yes—
121C22×C53#45811yescharacteristic
122C2×D53—82no—
123C2×D53—82no—
124C2×D53—82no—
125C2×D53—82no—
126C2×D53—82no—
127C2×D53—82no—
128C2×D53—82no—
129C4×C53—22no—
130C4×C53—22no—
131C4×C53—21no—
132C4×C53—21no—
133C4×C53—42no—
134C4×C53—81no—
135Dic53—82no—
136Dic53—161no—
137#2733—531no—
138#2096—531no—
139#2071—532no—
140#1272D5311yescharacteristic, complemented
141#1270—532no—
142C2×C53#161212yes—
143C2×C53#161311yescharacteristic
144C2×C53#162411yescharacteristic
145C2×C53#164212yes—
146C2×C53#165911yescharacteristic
147D53—162no—
148C2×#124—1061no—
149C2×#106—1061no—
150C2×#103—1061no—
151C2×#102—1062no—
152#394—1061no—
153#392—1062no—
154#391C2×D5311yescharacteristic
155#390—532no—
156#388—1062no—
157#384—531no—
158#340—1062no—
159#336—531no—
160#336C2×D5312yes—
161C53#781011yescharacteristic, complemented, Sylow
162C22×Q8D21211yescharacteristic
163C22×D4—532no—
164C22×D4—1061no—
165C22×D4—1061no—
166C22×C8—22no—
167C22×C8—1061no—
168#124—2122no—
169#124—2122no—
170C2×C4⋊C4—532no—
171C2×C4⋊C4—1061no—
172C2×C4⋊C4—1062no—
173C2×C4⋊C4D21211yescharacteristic
174C2×C22⋊C4—1062no—
175C2×C22⋊C4—1062no—
176C2×C22⋊C4—1062no—
177C2×C22⋊C4—1062no—
178C2×C42—1061no—
179C2×C42C22×D5311yescharacteristic
180#106—2122no—
181#103—2122no—
182#102—2124no—
183#95—22no—
184#95—1061no—
185C24—1062no—
186C2×Q8—21no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8#1171991312yes—
190C2×D4—1062no—
191C2×D4—1062no—
192C2×D4—4241no—
193C2×D4—4241no—
194C2×D4—4241no—
195C2×D4—4242no—
196C2×D4—2122no—
197C2×D4—2122no—
198C2×D4—2122no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—1062no—
202C22×C4—2121no—
203C22×C4C2×D21211yescharacteristic
204C22×C4D4×D5312yes—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—1062no—
209C2×C8—2121no—
210C2×C8—2121no—
211C4⋊C4—41no—
212C4⋊C4—2121no—
213C4⋊C4—2121no—
214C4⋊C4—2122no—
215C4⋊C4—2122no—
216C4⋊C4—2122no—
217C22⋊C4—4242no—
218C22⋊C4—2124no—
219C22⋊C4—2122no—
220C22⋊C4—2122no—
221C22⋊C4—2122no—
222C22⋊C4—2122no—
223C42—41no—
224C42—2122no—
225C23—2122no—
226C23—2122no—
227C23—2122no—
228C23—2122no—
229C23—2122no—
230C23—2122no—
231C23—2122no—
232C23#1313318711yescharacteristic
233Q8—42no—
234Q8—81no—
235D4—4242no—
236D4—4244no—
237D4—4244no—
238D4—2124no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—4242no—
249C2×C4—4241no—
250C2×C4—2122no—
251C2×C4—2122no—
252C2×C4—2122no—
253C2×C4C2×#1171991311yescharacteristic
254C2×C4#1313324012yes—
255C2×C4#1313324412yes—
256C2×C4#1313324511yescharacteristic
257C8—82no—
258C8—2122no—
259C22—4242no—
260C22—4242no—
261C22—4242no—
262C22—4242no—
263C22—4242no—
264C22—4242no—
265C22—4242no—
266C22#1681077112yes—
267C22#1681077611yescharacteristic
268C22#1681079311yescharacteristic
269C22#1681080212yes—
270C22#1681081411yescharacteristic
271C4—22no—
272C4—22no—
273C4—21no—
274C4—21no—
275C4—42no—
276C4—81no—
277C4—8481no—
278C4—4242no—
279C2—8482no—
280C2#2027373212yes—
281C2#2027373711yescharacteristic
282C2#2027379911yescharacteristic
283C2#2027391812yes—
284C2#2027402211yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1272C2
socleC23×C53#120
prime radical#20281741C2
hypercenter#1272D53
hypocenterC53#7810
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
2. #20276258C2semidirect
3. #20276120C2semidirect
4. #20272535C2semidirect
5. C53×#1272C2semidirect
140. #1272D53semidirect
161. C53#7810semidirect

Maximal subgroups

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

subgroupindex
#202817412
#202762582
#202761202
#202725352
C53×#12722
#781053