Finite Group Databasebeta

← #28299685

Subgroups — #28299685

Order 13568.

636 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 98 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#20290432C212yesmaximal, complemented
2#20289532C214yesmaximal, complemented
3#20277277C211yesmaximal, characteristic
4C2×#13133272—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—
15#13133272—28no—
16C22×D212—22no—
17C22×D212C2×C412yes—
18C22×C4×D53—22no—
19#13133214—24no—
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—42no—
42C2×D212—42no—
43C2×D212—44no—
44C2×D212M1614yes—
45C2×C4×D53—48no—
46C2×C4×D53—44no—
47C2×#88324—28no—
48C2×#88324—44no—
49C2×#88324—44no—
50C4⋊C4×C53—44no—
51C4⋊C4×C53—42no—
52C42×C53—22no—
53#11719921—44no—
54#11719921—48no—
55C4×Dic53—24no—
56C4×Dic53—22no—
57C4×Dic53—24no—
58C4×Dic53—41no—
59C23×C53C2×C22⋊C411yescharacteristic
60C22×D53—24no—
61C22×D53—42no—
62C22×D53—42no—
63C22×D53—44no—
64C22×D53—44no—
65C2×C4×C53—22no—
66C2×C4×C53—21no—
67C2×C4×C53—21no—
68C2×C4×C53—22no—
69C2×C4×C53—42no—
70C2×C4×C53—44no—
71C2×C4×C53#10414yes—
72C2×C4×C53C2×M1612yes—
73C2×Dic53—22no—
74C2×Dic53—22no—
75C2×Dic53—22no—
76C2×Dic53—21no—
77C2×Dic53—22no—
78C2×Dic53#10012yes—
79C2×Dic53AGL(1,ℤ/8)14yes—
80D212—44no—
81D212—44no—
82D212—416no—
83C4×D53—88no—
84#88324—48no—
85#7637—531nomaximal, Sylow
86C22×C53#40711yescharacteristic
87C22×C53#41011yescharacteristic
88C22×C53C2×#10011yescharacteristic
89C22×C53#41812yes—
90C22×C53C2×#10412yes—
91C2×D53—44no—
92C2×D53—48no—
93C2×D53—82no—
94C2×D53—84no—
95C2×D53—82no—
96C4×C53—22no—
97C4×C53—24no—
98C4×C53—82no—
99Dic53—22no—
100Dic53—24no—
101Dic53—41no—
102C4×#127—531no—
103#1745—532no—
104#1270—534no—
105C2×C53#146611yescharacteristic
106C2×C53#149012yes—
107C2×C53#150512yes—
108C2×C53#152511yescharacteristic
109C2×C53#155511yescharacteristic
110D53—84no—
111C2×#127—531no—
112C2×C4×D4—532no—
113C2×C4×D4—1062no—
114C2×#97—1062no—
115#390—532no—
116#390—1062no—
117#384#615711yescharacteristic
118C43—531no—
119#336—534no—
120#336—1062no—
121C53#763711yescharacteristic, complemented, Sylow
122C22×D4—532no—
123C22×D4—1062no—
124C23×C4—1062no—
125C22×C8—1064no—
126#127—1064no—
127C4×D4—2128no—
128C4×D4—2124no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4C2×#615712yes—
131C2×C22⋊C4—1062no—
132C2×C22⋊C4—1064no—
133C2×C42—532no—
134C2×C42—532no—
135C2×C42—531no—
136C2×C42—531no—
137C2×C42C2×#615711yescharacteristic
138#97—1068no—
139C24—1062no—
140C2×D4—534no—
141C2×D4—1068no—
142C2×D4—1062no—
143C2×D4—2122no—
144C2×D4—2122no—
145C2×D4—2124no—
146C2×D4—2124no—
147C22×C4—22no—
148C22×C4—531no—
149C22×C4—532no—
150C22×C4—531no—
151C22×C4—2124no—
152C22×C4—2128no—
153C22×C4C22⋊#615712yes—
154C22×C4C22×#615711yescharacteristic
155C2×C8—1068no—
156C2×C8—2124no—
157C2×C8—2124no—
158C4⋊C4—42no—
159C4⋊C4—44no—
160C22⋊C4—2128no—
161C22⋊C4—2124no—
162C42—22no—
163C42—1062no—
164C42—1064no—
165C42—1064no—
166C42—2121no—
167C23—1064no—
168C23—2122no—
169C23—2122no—
170C23—2124no—
171C23—2124no—
172C23C2×C22⋊#615711yescharacteristic
173D4—2124no—
174D4—21216no—
175D4—2124no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—42no—
181C2×C4—44no—
182C2×C4—532no—
183C2×C4—534no—
184C2×C4—1062no—
185C2×C4—1062no—
186C2×C4—1062no—
187C2×C4—1061no—
188C2×C4—1062no—
189C2×C4—4248no—
190C2×C4#1313328214yes—
191C2×C4D53⋊M1612yes—
192C8—2128no—
193C22—4244no—
194C22—4242no—
195C22—4242no—
196C22—2128no—
197C22—2124no—
198C22#1681159511yescharacteristic
199C22#1681160411yescharacteristic
200C22D53⋊#10011yescharacteristic
201C22#1681161512yes—
202C22C2×#1313328212yes—
203C4—24no—
204C4—22no—
205C4—82no—
206C4—1064no—
207C4—1062no—
208C4—2121no—
209C2—4244no—
210C2#2028997611yescharacteristic
211C2#2029005612yes—
212C2#2029010612yes—
213C2#2029017111yescharacteristic
214C2#2029024311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#384C4
socleC23×C53C2×C22⋊C4
prime radicalC2×#13133214C4
cosocleC2×C4×Dic53C23
hypercenter#384#6157
hypocenterC53#7637
LCSG ▷ C22×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. #20290432C2semidirect
2. #20289532C2semidirect
121. C53#7637semidirect

Maximal subgroups

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

subgroupindex
#202904322
#202895322
#202772772
#763753