Finite Group Databasebeta

← #27297774

Subgroups — #27297774

Order 13568.

298 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 69 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#20287604C211yesmaximal, characteristic, complemented
2#20276938C211yesmaximal, characteristic
3#20274072C211yesmaximal, characteristic
4#20272769C211yesmaximal, characteristic, complemented
5#20272693C211yesmaximal, characteristic, complemented
6#20272679C211yesmaximal, characteristic, complemented
7C53×#2207C211yesmaximal, characteristic
8#16811352—21no—
9#16811028C2211yescharacteristic
10#16811024—21no—
11#16810932—22no—
12#16810908—21no—
13#16810820—21no—
14C8×Dic106C2211yescharacteristic
15Q16×Dic53—21no—
16#16810424—22no—
17C4×#11719908C2211yescharacteristic
18#16810376C2211yescharacteristic
19#16810367—21no—
20C2×SD32×C53—21no—
21C53×#493C2211yescharacteristic
22C4×Q16×C53C2211yescharacteristic
23C53×C4⋊C16C2211yescharacteristic
24C53×#357—21no—
25C2×#11719923—41no—
26#13133217—21no—
27#13133217—41no—
28#13133208—22no—
29#13133208—42no—
30Q8×Dic53—22no—
31C4×Dic106—41no—
32C4×Dic106D411yescharacteristic
33C2×#11719908—21no—
34#13133151—21no—
35#13133151D412yes—
36#13133148—22no—
37C4⋊#88322—21no—
38C4⋊#88322D411yescharacteristic
39#13133146—21no—
40#13133141—22no—
41#13133138—41no—
42C8×Dic53—21no—
43C2×Q16×C53D411yescharacteristic
44C2×D8×C53—21no—
45C2×D8×C53—21no—
46#127×C53—21no—
47C4×Q8×C53—21no—
48SD32×C53—42no—
49C2×C16×C53—21no—
50#107×C53D411yescharacteristic
51#103×C53—21no—
52C4×C8×C53C2311yescharacteristic
53C2×Q8×C53—21no—
54C2×D4×C53—41no—
55C2×D4×C53—41no—
56C2×D4×C53—41no—
57C2×Dic106—41no—
58C2×Dic106D811yescharacteristic
59Q16×C53—21no—
60Q16×C53SD1612yescomplemented
61D8×C53—42no—
62D8×C53—41no—
63D8×C53—41no—
64C2×C8×C53C2×D411yescharacteristic
65C2×C8×C53C2×D411yescharacteristic
66C4⋊C4×C53—21no—
67C4⋊C4×C53—41no—
68C42×C53—41no—
69C42×C53C2×D411yescharacteristic
70#11719923—82no—
71C4⋊Dic53—22no—
72C4⋊Dic53—42no—
73C4⋊Dic53—42no—
74C4⋊Dic53—41no—
75C4⋊Dic53D811yescharacteristic
76#11719919—21no—
77#11719919—42no—
78C4×Dic53—21no—
79C4×Dic53—42no—
80C2×#88322—21no—
81C2×#88322—41no—
82C16×C53—41no—
83#11719908—41no—
84C23×C53—81no—
85Q8×C53—22no—
86Q8×C53—41no—
87D4×C53—81no—
88D4×C53—81no—
89D4×C53—81no—
90D4×C53—81no—
91D4×C53—161no—
92C2×C4×C53—41no—
93C2×C4×C53—41no—
94C2×C4×C53#12011yescharacteristic
95C2×C4×C53C2×D811yescharacteristic
96C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
97C2×Dic53—21no—
98C2×Dic53—21no—
99C2×Dic53—42no—
100Dic106—21no—
101Dic106—21no—
102Dic106—81no—
103Dic106—81no—
104C8×C53—21no—
105C8×C53C2×SD1611yescharacteristic
106C8×C53#13711yescharacteristic
107#88322—41no—
108#88322—81no—
109#12353—531nomaximal, Sylow
110C22×C53—161no—
111C22×C53—161no—
112C22×C53—161no—
113C22×C53#44711yescharacteristic
114C4×C53—21no—
115C4×C53—42no—
116C4×C53—81no—
117C4×C53#44811yescharacteristic
118C4×C53#45411yescharacteristic
119C4×C53#50811yescharacteristic
120C4×C53#50911yescharacteristic
121Dic53—41no—
122Dic53—41no—
123Dic53—82no—
124#3383—531no—
125#2240—531no—
126#2207D5311yescharacteristic
127#1668—531no—
128#1364—531no—
129Q8⋊C16—531no—
130#1332—531no—
131C2×C53—321no—
132C2×C53#162711yescharacteristic
133C2×C53#218611yescharacteristic
134C2×C53#219311yescharacteristic
135#557—1061no—
136C2×SD32—21no—
137#500—531no—
138#499—1061no—
139#493C2×D5311yescharacteristic
140#479—1062no—
141#475—1061no—
142#460—1061no—
143C8×Q8—531no—
144C4×Q16—1061no—
145C4×Q16C2×D5311yescharacteristic
146C4⋊C16C2×D5311yescharacteristic
147#357—21no—
148#357—1062no—
149C4×C16—531no—
150#335—531no—
151#331—1061no—
152C53#1235311yescharacteristic, complemented, Sylow
153C2×Q16C22⋊D5311yescharacteristic
154C2×SD16—2121no—
155C2×D8—21no—
156C2×D8—21no—
157C4⋊Q8—1061no—
158C4⋊Q8—2121no—
159#127—21no—
160#125—1062no—
161#125—2122no—
162C4×Q8—21no—
163C4×Q8—531no—
164C4×Q8—1062no—
165C4×Q8—2121no—
166SD32—42no—
167C2×C16—21no—
168C2×C16—1061no—
169#107—532no—
170#107—1061no—
171#107C22⋊D5311yescharacteristic
172#106—1062no—
173C4⋊C8—531no—
174C4⋊C8—1061no—
175C4⋊C8—1061no—
176#103—21no—
177#103—1062no—
178#102—2121no—
179C4×C8—1061no—
180C4×C8C22×D5311yescharacteristic
181C2×Q8—21no—
182C2×Q8—531no—
183C2×Q8—2121no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—41no—
187Q16—21no—
188Q16#1171992312yescomplemented
189SD16—4242no—
190D8—42no—
191D8—41no—
192D8—41no—
193C2×C8—1061no—
194C2×C8—2121no—
195C2×C8D4×D5311yescharacteristic
196C2×C8C2×C22⋊D5311yescharacteristic
197C4⋊C4—21no—
198C4⋊C4—41no—
199C4⋊C4—531no—
200C4⋊C4—1061no—
201C4⋊C4—1062no—
202C4⋊C4—2122no—
203C4⋊C4—2122no—
204C4⋊C4—2122no—
205C4⋊C4—2121no—
206C42—41no—
207C42—1061no—
208C42—2122no—
209C42D4×D5311yescharacteristic
210C16—41no—
211C16—2121no—
212C23—81no—
213Q8—22no—
214Q8—41no—
215Q8—1061no—
216Q8—1061no—
217Q8—4241no—
218Q8—4241no—
219D4—81no—
220D4—81no—
221D4—81no—
222D4—81no—
223D4—161no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—1061no—
227C2×C4—1061no—
228C2×C4—2122no—
229C2×C4#1313318511yescharacteristic
230C2×C4D8×D5311yescharacteristic
231C2×C4#1313324111yescharacteristic
232C8—21no—
233C8—4241no—
234C8—2121no—
235C8C2×#1171992311yescharacteristic
236C8#1313324711yescharacteristic
237C22—161no—
238C22—161no—
239C22—161no—
240C22#1681075511yescharacteristic
241C4—21no—
242C4—42no—
243C4—81no—
244C4—4242no—
245C4—2121no—
246C4—2121no—
247C4#1681076111yescharacteristic
248C4#1681079411yescharacteristic
249C4#1681105411yescharacteristic
250C4#1681106011yescharacteristic
251C2—321no—
252C2#2027381711yescharacteristic
253C2#2027663711yescharacteristic
254C2#2027667611yescharacteristic
subgroupG/H
centerC22#16810755
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
FittingC53×#2207C2
socleC22×C53#447
prime radicalC53×#493C22
hypercenter#2207D53
hypocenterC53#12353
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2207

Complements

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

NH ≅ G/Ntype
1. #20287604C2semidirect
4. #20272769C2semidirect
5. #20272693C2semidirect
6. #20272679C2semidirect
60. Q16×C53SD16semidirect
152. C53#12353semidirect
188. Q16#11719923semidirect

Maximal subgroups

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

subgroupindex
#202876042
#202769382
#202740722
#202727692
#202726932
#202726792
C53×#22072
#1235353