Finite Group Databasebeta

← #28299565

Subgroups — #28299565

Order 13568.

608 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 110 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#20290642C212yesmaximal, complemented
2#20289532C214yesmaximal, complemented
3#20277277C211yesmaximal, characteristic
4#16811489—22no—
5#16811489—22no—
6#16811489C2214yes—
7C2×#13133214C411yescharacteristic
8C2×#13133179—22no—
9C2×#13133179C412yes—
10#16810524—22no—
11#16810524C2212yes—
12C42×Dic53C2211yescharacteristic
13C53×#384C411yescharacteristic
14C22×#88324—24no—
15C22×D212—22no—
16C22×D212C2×C412yes—
17C22×C4×D53—22no—
18#13133214—22no—
19#13133214C814yescomplemented
20#13133179—48no—
21#13133179—44no—
22C2×#11719921—22no—
23C2×#11719921—24no—
24C2×C4×Dic53C2×C411yescharacteristic
25C2×C4×Dic53D412yes—
26C2×C4×Dic53D412yes—
27C2×C4×Dic53C2311yescharacteristic
28C2×C4⋊C4×C53—22no—
29C2×C4⋊C4×C53C2×C412yes—
30C2×C42×C53C2×C411yescharacteristic
31C23×D53—22no—
32C22×C4×C53—22no—
33C22×C4×C53C22⋊C412yes—
34C22×C4×C53C22×C411yescharacteristic
35C22×Dic53C22⋊C411yescharacteristic
36C22×Dic53C22⋊C411yescharacteristic
37C22×Dic53C2×D412yes—
38C2×D212—28no—
39C2×D212—22no—
40C2×D212—44no—
41C2×D212—44no—
42C2×D212—42no—
43C2×D212—42no—
44C2×D212C2×C814yes—
45C2×C4×D53—48no—
46C2×C4×D53—44no—
47C2×#88324—44no—
48C2×#88324—44no—
49C2×#88324—44no—
50C4⋊C4×C53—44no—
51C4⋊C4×C53—42no—
52C42×C53C2×C812yes—
53C42×C53M1612yes—
54#11719921—44no—
55#11719921—48no—
56C4×Dic53—22no—
57C4×Dic53—24no—
58C4×Dic53—42no—
59C4×Dic53—41no—
60C23×C53C2×C22⋊C411yescharacteristic
61C22×D53—24no—
62C22×D53—42no—
63C22×D53—42no—
64C22×D53—44no—
65C22×D53—44no—
66C2×C4×C53—21no—
67C2×C4×C53—21no—
68C2×C4×C53—22no—
69C2×C4×C53—22no—
70C2×C4×C53—42no—
71C2×C4×C53—44no—
72C2×C4×C53C22⋊C814yes—
73C2×C4×C53C22×C811yescharacteristic
74C2×C4×C53C2×M1611yescharacteristic
75C2×Dic53—22no—
76C2×Dic53—22no—
77C2×Dic53—22no—
78C2×Dic53—21no—
79C2×Dic53—22no—
80C2×Dic53#10012yes—
81C2×Dic53AGL(1,ℤ/8)14yes—
82D212—44no—
83D212—44no—
84D212—416no—
85C4×D53—88no—
86#88324—84no—
87#7603—531nomaximal, Sylow
88C22×C53C2×C22⋊C811yescharacteristic
89C22×C53#41011yescharacteristic
90C22×C53C2×#10011yescharacteristic
91C22×C53#41812yes—
92C22×C53#42112yes—
93C2×D53—44no—
94C2×D53—48no—
95C2×D53—82no—
96C2×D53—82no—
97C2×D53—84no—
98C4×C53—21no—
99C4×C53—21no—
100C4×C53—24no—
101C4×C53—82no—
102Dic53—22no—
103Dic53—24no—
104Dic53—41no—
105C4×#127—531no—
106#1839—532no—
107#1270—534no—
108C2×C53#146011yescharacteristic
109C2×C53#147812yes—
110C2×C53#151412yes—
111C2×C53#153611yescharacteristic
112C2×C53#155511yescharacteristic
113D53—84no—
114C2×#127—531no—
115C2×C4×D4—532no—
116C2×C4×D4—1062no—
117#390—532no—
118#390—1062no—
119#384#615711yescharacteristic
120C43—531no—
121#336—534no—
122#336—1062no—
123#336—1062no—
124C53#760311yescharacteristic, complemented, Sylow
125C22×D4—532no—
126C22×D4—1062no—
127C23×C4—1062no—
128C22×C8—1064no—
129#127—534no—
130#127—1062no—
131C4×D4—2128no—
132C4×D4—2124no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4C2×#615712yes—
135C2×C22⋊C4—1062no—
136C2×C22⋊C4—1064no—
137C2×C42—532no—
138C2×C42—531no—
139C2×C42—532no—
140C2×C42—531no—
141C2×C42C2×#615711yescharacteristic
142C24—1062no—
143C2×D4—534no—
144C2×D4—1068no—
145C2×D4—1062no—
146C2×D4—2122no—
147C2×D4—2122no—
148C2×D4—2124no—
149C2×D4—2124no—
150C22×C4—22no—
151C22×C4—531no—
152C22×C4—532no—
153C22×C4—531no—
154C22×C4—2124no—
155C22×C4—2128no—
156C22×C4C22⋊#615712yes—
157C22×C4C22×#615711yescharacteristic
158C2×C8—2124no—
159C2×C8—2124no—
160C2×C8—2124no—
161C4⋊C4—42no—
162C4⋊C4—44no—
163C22⋊C4—2128no—
164C22⋊C4—2124no—
165C42—1062no—
166C42—1064no—
167C42—2121no—
168C42—2122no—
169C42D53⋊C812yes—
170C42#1171993612yes—
171C23—1064no—
172C23—2122no—
173C23—2122no—
174C23—2124no—
175C23—2124no—
176C23C2×C22⋊#615711yescharacteristic
177D4—2124no—
178D4—2124no—
179D4—21216no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—42no—
185C2×C4—44no—
186C2×C4—532no—
187C2×C4—534no—
188C2×C4—1062no—
189C2×C4—1062no—
190C2×C4—1062no—
191C2×C4—1061no—
192C2×C4—1062no—
193C2×C4—4248no—
194C2×C4#1313327414yes—
195C2×C4C2×D53⋊C811yescharacteristic
196C2×C4C2×#1171993611yescharacteristic
197C8—4244no—
198C22—4244no—
199C22—4242no—
200C22—4242no—
201C22—2124no—
202C22—2128no—
203C22C2×#1313327411yescharacteristic
204C22#1681160411yescharacteristic
205C22D53⋊#10011yescharacteristic
206C22#1681161512yes—
207C22#1681162212yes—
208C4—21no—
209C4—21no—
210C4—24no—
211C4—82no—
212C4—1064no—
213C4—1062no—
214C4—2121no—
215C2—4244no—
216C2#2028995611yescharacteristic
217C2#2029001612yes—
218C2#2029013612yes—
219C2#2029020511yescharacteristic
220C2#2029024311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC53×#384C4
socleC23×C53C2×C22⋊C4
prime radicalC2×#13133214C4
cosocleC2×C4×Dic53C23
hypercenter#384#6157
hypocenterC53#7603
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
1. #20290642C2semidirect
2. #20289532C2semidirect
19. #13133214C8semidirect
124. C53#7603semidirect

Maximal subgroups

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

subgroupindex
#202906422
#202895322
#202772772
#760353