Finite Group Databasebeta

← #28302285

Subgroups — #28302285

Order 13568.

812 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 108 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#20290014C214yesmaximal, complemented
2#20289948C212yesmaximal, complemented
3C2×#16811227C211yesmaximal, characteristic
4#16811639C2214yes—
5D4⋊#88324—28no—
6#16811469—22no—
7#16811467—22no—
8#16811227—28no—
9C2×#13133209C411yescharacteristic
10C2×#13133198—22no—
11C2×#13133189—22no—
12C2×#13133188C412yes—
13C2×D4×Dic53C2212yes—
14C2×#13133174C2211yescharacteristic
15C2×#127×C53C411yescharacteristic
16C4⋊#88324—24no—
17C22⋊#88324—24no—
18C4×#88324—24no—
19C22×C22⋊D53—22no—
20C23×Dic53—22no—
21C22×C4×D53C2×C411yescharacteristic
22#13133209—24no—
23#13133198—216no—
24#13133189—216no—
25#13133188—24no—
26#13133188—24no—
27D4×Dic53—24no—
28D4×Dic53D418yes—
29#13133174—24no—
30C2×C4⋊Dic53C2×C412yes—
31C2×#11719919—22no—
32C2×#11719919D412yes—
33C2×C22⋊Dic53—22no—
34C2×C22⋊Dic53—22no—
35C2×#11719921D412yes—
36C2×C4×Dic53C2311yescharacteristic
37C22×D4×C53—22no—
38C22×D4×C53C2×C412yes—
39#127×C53—24no—
40C2×C42×C53C2×C411yescharacteristic
41C24×C53—22no—
42C23×D53C22⋊C411yescharacteristic
43C2×D4×C53—28no—
44C2×D4×C53—22no—
45C2×D4×C53—28no—
46C2×D4×C53—84no—
47C2×D4×C53C22⋊C414yes—
48C22×C4×C53C22⋊C412yes—
49C22×C4×C53C22×C411yescharacteristic
50C2×C22⋊D53—28no—
51C2×C22⋊D53—84no—
52C22×Dic53—22no—
53C22×Dic53—28no—
54C22×Dic53—84no—
55C22×Dic53C22⋊C411yescharacteristic
56C22×Dic53C2×D412yes—
57C2×C4×D53—22no—
58C2×C4×D53—24no—
59C2×#88324—44no—
60C42×C53—22no—
61C22⋊Dic53—28no—
62C22⋊Dic53—28no—
63#11719921—24no—
64C4⋊Dic53—22no—
65C4⋊Dic53C22⋊C414yes—
66#11719919—28no—
67#11719919—24no—
68C4×Dic53C2×D414yes—
69C23×C53—24no—
70C23×C53—44no—
71C23×C53—44no—
72C23×C53—44no—
73C23×C53C2×C22⋊C411yescharacteristic
74C22×D53—21no—
75C22×D53—22no—
76C22×D53—42no—
77D4×C53—28no—
78D4×C53—44no—
79D4×C53—88no—
80C2×C4×C53—22no—
81C2×C4×C53—42no—
82C2×C4×C53C2×C22⋊C412yes—
83C2×C4×C53#13114yes—
84C22⋊D53—88no—
85C2×Dic53—21no—
86C2×Dic53—24no—
87C2×Dic53—22no—
88C2×Dic53—42no—
89C2×Dic53—44no—
90C2×Dic53—88no—
91C2×Dic53#12014yes—
92C4×D53—44no—
93#88324—84no—
94#8326—531nomaximal, Sylow
95C22×C53—22no—
96C22×C53—44no—
97C22×C53—44no—
98C22×C53—44no—
99C22×C53—88no—
100C22×C53#37911yescharacteristic
101C22×C53#40812yes—
102C2×D53—42no—
103C2×D53—41no—
104C2×D53—82no—
105C4×C53—42no—
106C4×C53#40814yes—
107Dic53—24no—
108Dic53—44no—
109Dic53—41no—
110C2×#540—531no—
111#1478—534no—
112#1458—532no—
113C2×C53—22no—
114C2×C53—84no—
115C2×C53#178011yescharacteristic
116C2×C53#179512yes—
117D53—81no—
118#540—1068no—
119C2×#127#615711yescharacteristic
120C2×#125—531no—
121C2×#123—1062no—
122C2×#121—532no—
123C2×#121—1062no—
124C2×C4×D4—532no—
125C2×#117—531no—
126#431—534no—
127D4⋊C8—1068no—
128#324—1062no—
129#323—1062no—
130C53#832611yescharacteristic, complemented, Sylow
131C22×D4—22no—
132C22×D4—1062no—
133C22×D4C2×#615712yes—
134C23×C4—531no—
135C23×C4—1062no—
136#127—24no—
137#125—1064no—
138#123—10616no—
139#121—1064no—
140#121—10616no—
141#121—1064no—
142C4×D4—538no—
143C4×D4—1064no—
144#117—1064no—
145C2×C4⋊C4—532no—
146C2×C4⋊C4—532no—
147C2×C4⋊C4—1062no—
148C2×C22⋊C4—532no—
149C2×C22⋊C4—1062no—
150C2×C22⋊C4—1062no—
151C2×C42—531no—
152C2×C42C2×#615711yescharacteristic
153C4⋊C8—1064no—
154C22⋊C8—1064no—
155C4×C8—1064no—
156C24—22no—
157C24—531no—
158C2×D4—28no—
159C2×D4—22no—
160C2×D4—28no—
161C2×D4—84no—
162C2×D4—1068no—
163C2×D4—4244no—
164C2×D4C22⋊#615714yes—
165C22×C4—532no—
166C22×C4—531no—
167C22×C4—1068no—
168C22×C4—1062no—
169C22×C4—1062no—
170C22×C4—1064no—
171C22×C4—4244no—
172C22×C4C22⋊#615712yes—
173C22×C4C22×#615711yescharacteristic
174C2×C8—2124no—
175C4⋊C4—534no—
176C4⋊C4—1064no—
177C4⋊C4—1068no—
178C4⋊C4—1062no—
179C22⋊C4—1064no—
180C22⋊C4—1068no—
181C22⋊C4—1068no—
182C42—22no—
183C42—534no—
184C23—24no—
185C23—44no—
186C23—44no—
187C23—44no—
188C23—1062no—
189C23—1061no—
190C23—2122no—
191C23C2×C22⋊#615711yescharacteristic
192D4—28no—
193D4—44no—
194D4—88no—
195D4—4248no—
196C2×C4—22no—
197C2×C4—42no—
198C2×C4—534no—
199C2×C4—1061no—
200C2×C4—1064no—
201C2×C4—1062no—
202C2×C4—4248no—
203C2×C4—2122no—
204C2×C4—2124no—
205C2×C4—2124no—
206C2×C4C2×C22⋊#615712yes—
207C2×C4#1313330414yes—
208C8—4244no—
209C22—22no—
210C22—44no—
211C22—44no—
212C22—44no—
213C22—88no—
214C22—4242no—
215C22—2122no—
216C22—2121no—
217C22#1681156411yescharacteristic
218C22#1681159712yes—
219C4—42no—
220C4—1064no—
221C4—2124no—
222C4—2121no—
223C4#1681159614yes—
224C2—22no—
225C2—84no—
226C2—4241no—
227C2#2029053911yescharacteristic
228C2#2029056512yes—
subgroupG/H
centerC22#16811564
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×#127×C53C4
socleC22×C53#379
prime radicalC2×#16811227C2
cosocleC2×C4×Dic53C23
hypercenterC2×#127#6157
hypocenterC53#8326
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #20290014C2semidirect
2. #20289948C2semidirect
130. C53#8326semidirect

Maximal subgroups

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

subgroupindex
#202900142
#202899482
C2×#168112272
#832653