Finite Group Databasebeta

← #27297778

Subgroups — #27297778

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#20276939C211yesmaximal, characteristic
3#20276755C211yesmaximal, characteristic
4#20274076C211yesmaximal, characteristic
5#20272771C211yesmaximal, characteristic, complemented
6#20272695C211yesmaximal, characteristic, complemented
7C53×#1332C211yesmaximal, characteristic, complemented
8#16811352—21no—
9C2×#13133160—21no—
10#16811028C2211yescharacteristic
11#16811024—21no—
12#16811000C2211yescharacteristic
13#16810932—22no—
14#16810908—21no—
15#16810824—21no—
16C8×Dic106C2211yescharacteristic
17Q16×Dic53—21no—
18C4⋊#11719908C2211yescharacteristic
19#16810425—22no—
20#16810423—21no—
21#16810366—21no—
22C4×Q16×C53C2211yescharacteristic
23C4×C16×C53C2211yescharacteristic
24C53×#335C2211yescharacteristic
25C2×#11719913—41no—
26C2×D424—21no—
27C2×D424—21no—
28#13133217—21no—
29#13133217—41no—
30#13133214—21no—
31#13133208—22no—
32#13133208—42no—
33Q8×Dic53—22no—
34C4×Dic106—41no—
35C4×Dic106D411yescharacteristic
36#13133160—42no—
37C2×#11719908—21no—
38#13133151—21no—
39#13133151D412yes—
40#13133148—22no—
41C4⋊#88322D411yescharacteristic
42#13133146—21no—
43#13133141—22no—
44#13133139—41no—
45C8×Dic53—21no—
46C2×Q16×C53D411yescharacteristic
47C4×Q8×C53—21no—
48C2×C16×C53—21no—
49#107×C53D411yescharacteristic
50C4⋊C8×C53—21no—
51#103×C53—21no—
52C4×C8×C53C2311yescharacteristic
53C2×Q8×C53—21no—
54C2×D212—41no—
55C2×D212—41no—
56C2×D212—41no—
57C2×Dic106—41no—
58C2×Dic106D811yescharacteristic
59Q16×C53—21no—
60Q16×C53SD1612yescomplemented
61C2×C8×C53—41no—
62C2×C8×C53C2×D411yescharacteristic
63C2×C8×C53C2×D411yescharacteristic
64C4⋊C4×C53—21no—
65C4⋊C4×C53—41no—
66C42×C53—41no—
67C42×C53C2×D411yescharacteristic
68C4⋊Dic53—22no—
69C4⋊Dic53—42no—
70C4⋊Dic53—42no—
71C4⋊Dic53—41no—
72C4⋊Dic53D811yescharacteristic
73#11719919—21no—
74#11719919—42no—
75C4×Dic53—21no—
76C4×Dic53—42no—
77C2×#88322—21no—
78D424—42no—
79D424—41no—
80D424—41no—
81#11719913—82no—
82C16×C53—41no—
83#11719908—41no—
84C22×D53—81no—
85Q8×C53—22no—
86Q8×C53—41no—
87C2×C4×C53—41no—
88C2×C4×C53—41no—
89C2×C4×C53#12011yescharacteristic
90C2×C4×C53C2×D811yescharacteristic
91C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
92C2×Dic53—21no—
93C2×Dic53—21no—
94C2×Dic53—42no—
95D212—81no—
96D212—81no—
97D212—81no—
98D212—81no—
99D212—161no—
100Dic106—21no—
101Dic106—21no—
102Dic106—81no—
103Dic106—81no—
104C8×C53—21no—
105C8×C53—81no—
106C8×C53C2×SD1611yescharacteristic
107C8×C53#13711yescharacteristic
108#88322—41no—
109#12353—531nomaximal, Sylow
110C22×C53#44711yescharacteristic
111C2×D53—161no—
112C2×D53—161no—
113C2×D53—161no—
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#2207—531no—
127#1668—531no—
128#1364—531no—
129Q8⋊C16—531no—
130#1332D5311yescharacteristic, complemented
131C2×C53#162711yescharacteristic
132C2×C53#218611yescharacteristic
133C2×C53#219311yescharacteristic
134D53—321no—
135#557—1061no—
136C2×SD32—1061no—
137#500—531no—
138#499—1061no—
139#493—531no—
140#479—1062no—
141#475—1061no—
142#460—1061no—
143C8×Q8—531no—
144C4×Q16—1061no—
145C4×Q16C2×D5311yescharacteristic
146C4⋊C16—531no—
147#357—1062no—
148#357—1061no—
149C4×C16C2×D5311yescharacteristic
150#335C2×D5311yescharacteristic
151#331—1061no—
152C53#1235311yescharacteristic, complemented, Sylow
153C2×Q16D21211yescharacteristic
154C2×SD16—2121no—
155C2×D8—1061no—
156C2×D8—1061no—
157C4⋊Q8—1061no—
158C4⋊Q8—2121no—
159#127—1061no—
160#125—1062no—
161#125—2122no—
162C4×Q8—21no—
163C4×Q8—531no—
164C4×Q8—1062no—
165C4×Q8—2121no—
166SD32—2122no—
167C2×C16—21no—
168C2×C16—1061no—
169#107—532no—
170#107—1061no—
171#107D21211yescharacteristic
172#106—1062no—
173C4⋊C8—21no—
174C4⋊C8—531no—
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—2121no—
185C2×D4—2121no—
186C2×D4—2121no—
187Q16—21no—
188Q16#1171991312yescomplemented
189SD16—4242no—
190D8—2122no—
191D8—2121no—
192D8—2121no—
193C2×C8—41no—
194C2×C8—1061no—
195C2×C8C2×D21211yescharacteristic
196C2×C8D4×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—4241no—
213Q8—22no—
214Q8—41no—
215Q8—1061no—
216Q8—1061no—
217Q8—4241no—
218Q8—4241no—
219D4—8481no—
220D4—4241no—
221D4—4241no—
222D4—4241no—
223D4—4241no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—1061no—
227C2×C4—1061no—
228C2×C4—2122no—
229C2×C4#1313318711yescharacteristic
230C2×C4D8×D5311yescharacteristic
231C2×C4#1313324411yescharacteristic
232C8—21no—
233C8—81no—
234C8—2121no—
235C8C2×#1171991311yescharacteristic
236C8#1313325011yescharacteristic
237C22—8481no—
238C22—8481no—
239C22—8481no—
240C22#1681075811yescharacteristic
241C4—21no—
242C4—42no—
243C4—81no—
244C4—4242no—
245C4—2121no—
246C4—2121no—
247C4#1681076411yescharacteristic
248C4#1681079711yescharacteristic
249C4#1681105611yescharacteristic
250C4#1681106011yescharacteristic
251C2—16961no—
252C2#2027382111yescharacteristic
253C2#2027664111yescharacteristic
254C2#2027668011yescharacteristic
subgroupG/H
centerC22#16810758
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
FittingC53×#1332C2
socleC22×C53#447
prime radical#16811000C22
hypercenter#1332D53
hypocenterC53#12353
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1332

Complements

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

NH ≅ G/Ntype
1. #20287604C2semidirect
5. #20272771C2semidirect
6. #20272695C2semidirect
7. C53×#1332C2semidirect
60. Q16×C53SD16semidirect
130. #1332D53semidirect
152. C53#12353semidirect
188. Q16#11719913semidirect

Maximal subgroups

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

subgroupindex
#202876042
#202769392
#202767552
#202740762
#202727712
#202726952
C53×#13322
#1235353