Finite Group Databasebeta

← #28307906

Subgroups — #28307906

Order 13568.

526 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 112 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#20289946C212yesmaximal
2#20289641C214yesmaximal
3#20277301C211yesmaximal, characteristic
4#16811639—22no—
5#16811468—22no—
6#16811466C2214yes—
7C2×#13133176C411yescharacteristic
8C2×#13133176C2211yescharacteristic
9#16810530—22no—
10#16810530—22no—
11#16810486C412yes—
12#16810486C2212yes—
13C23×Q8×C53C411yescharacteristic
14C4⋊#88324—42no—
15C22⋊#88324—24no—
16C4×#88324—42no—
17#13133176—42no—
18#13133176—42no—
19C2×C4⋊Dic53—21no—
20C2×C4⋊Dic53D412yes—
21C2×C22⋊Dic53C2×C412yes—
22C2×C22⋊Dic53C2×C411yescharacteristic
23C2×C22⋊Dic53C2311yescharacteristic
24C2×C4×Dic53—21no—
25C2×C4×Dic53D412yes—
26#13133124—24no—
27#13133124—24no—
28C22×Q8×C53—21no—
29C22×Q8×C53—21no—
30C22×Q8×C53—22no—
31C22×Q8×C53—42no—
32C22×Q8×C53—42no—
33C22×Q8×C53C814yescomplemented
34C23×C4×C53C2×C411yescharacteristic
35C23×C4×C53C2×C412yes—
36C24×C53C22×C411yescharacteristic
37C2×Q8×C53—24no—
38C2×Q8×C53—24no—
39C2×Q8×C53—42no—
40C2×Q8×C53—42no—
41C2×Q8×C53—48no—
42C2×Q8×C53—48no—
43C2×Q8×C53—42no—
44C2×Q8×C53—42no—
45C22×C4×C53—22no—
46C22×C4×C53—22no—
47C22×C4×C53—22no—
48C22×C4×C53—22no—
49C22×C4×C53—42no—
50C22×C4×C53—42no—
51C22×C4×C53C22⋊C411yescharacteristic
52C22×C4×C53C22⋊C411yescharacteristic
53C22×C4×C53C2×C812yes—
54C22×C4×C53C2×C814yes—
55C22×C4×C53M1612yes—
56C22×Dic53—22no—
57C22×Dic53C22⋊C412yes—
58C22×Dic53C2×D412yes—
59C2×#88324—44no—
60C22⋊Dic53—41no—
61C22⋊Dic53—41no—
62C22⋊Dic53—42no—
63C4⋊Dic53—42no—
64C4⋊Dic53—42no—
65C4×Dic53—22no—
66C4×Dic53—22no—
67C4×Dic53—42no—
68C23×C53—21no—
69C23×C53—21no—
70C23×C53—42no—
71C23×C53C2×C22⋊C411yescharacteristic
72C23×C53C22×C811yescharacteristic
73C23×C53C2×M1611yescharacteristic
74Q8×C53—48no—
75Q8×C53—48no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53—24no—
79C2×C4×C53—24no—
80C2×C4×C53—24no—
81C2×C4×C53—44no—
82C2×C4×C53—42no—
83C2×C4×C53—42no—
84C2×C4×C53—44no—
85C2×C4×C53C22⋊C812yes—
86C2×C4×C53C22⋊C812yes—
87C2×C4×C53#9912yes—
88C2×C4×C53#10012yes—
89C2×Dic53—22no—
90C2×Dic53—22no—
91C2×Dic53—42no—
92C2×Dic53—42no—
93C2×Dic53—44no—
94C2×Dic53—42no—
95#88324—84no—
96#11428—531nomaximal, Sylow
97C22×C53—22no—
98C22×C53—21no—
99C22×C53—21no—
100C22×C53—41no—
101C22×C53—41no—
102C22×C53—42no—
103C22×C53—42no—
104C22×C53C2×C22⋊C811yescharacteristic
105C22×C53C2×#9911yescharacteristic
106C22×C53C2×#10011yescharacteristic
107C4×C53—22no—
108C4×C53—44no—
109C4×C53#32314yes—
110Dic53—42no—
111Dic53—42no—
112Dic53—82no—
113#2310—531no—
114#1458—532no—
115#1317—534no—
116C2×C53—21no—
117C2×C53—21no—
118C2×C53—41no—
119C2×C53—41no—
120C2×C53C2×#32311yescharacteristic
121C2×C53#212312yes—
122C23×Q8#615711yescharacteristic
123C2×#117—531no—
124C2×#117—531no—
125#431—1062no—
126#391—1062no—
127#391—1062no—
128#381—532no—
129#381—532no—
130#324—1062no—
131#323—534no—
132C53#1142811yescharacteristic, complemented, Sylow
133C22×Q8—22no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—42no—
137C22×Q8—42no—
138C22×Q8#8832414yescomplemented
139C23×C4C2×#615712yes—
140C23×C4C2×#615711yescharacteristic
141#117—2122no—
142#117—2122no—
143C2×C4⋊C4—532no—
144C2×C4⋊C4—1061no—
145C2×C22⋊C4—531no—
146C2×C22⋊C4—531no—
147C2×C22⋊C4—532no—
148C2×C42—532no—
149C2×C42—1061no—
150C4⋊C8—2122no—
151C22⋊C8—1064no—
152C4×C8—2122no—
153#95—1064no—
154#95—1064no—
155C24C22×#615711yescharacteristic
156C2×Q8—24no—
157C2×Q8—24no—
158C2×Q8—48no—
159C2×Q8—42no—
160C2×Q8—42no—
161C2×Q8—48no—
162C2×Q8—42no—
163C2×Q8—42no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—532no—
171C22×C4—532no—
172C22×C4—1062no—
173C22×C4C2×#8832412yes—
174C22×C4C2×#8832414yes—
175C22×C4#1171994012yes—
176C22×C4C22⋊#615711yescharacteristic
177C22×C4C22⋊#615711yescharacteristic
178C2×C8—2124no—
179C4⋊C4—2122no—
180C4⋊C4—2122no—
181C22⋊C4—2121no—
182C22⋊C4—2121no—
183C22⋊C4—2122no—
184C42—1062no—
185C42—1062no—
186C42—2122no—
187C23—21no—
188C23—21no—
189C23—42no—
190C23C2×C22⋊#615711yescharacteristic
191C23C22×#8832411yescharacteristic
192C23C2×#1171994011yescharacteristic
193Q8—48no—
194Q8—48no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—24no—
198C2×C4—24no—
199C2×C4—24no—
200C2×C4—44no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—44no—
204C2×C4—1062no—
205C2×C4—1062no—
206C2×C4—2124no—
207C2×C4—2122no—
208C2×C4—2122no—
209C2×C4—2122no—
210C2×C4C22⋊#8832412yes—
211C2×C4C22⋊#8832412yes—
212C2×C4#1313327512yes—
213C2×C4#1313327712yes—
214C8—4244no—
215C22—22no—
216C22—21no—
217C22—21no—
218C22—41no—
219C22—41no—
220C22—42no—
221C22—42no—
222C22C2×C22⋊#8832411yescharacteristic
223C22C2×#1313327511yescharacteristic
224C22C2×#1313327711yescharacteristic
225C4—22no—
226C4—44no—
227C4—4242no—
228C4—2122no—
229C4—2122no—
230C4#1681146614yes—
231C2—21no—
232C2—21no—
233C2—41no—
234C2—41no—
235C2C2×#1681146611yescharacteristic
236C2#2029097712yes—
subgroupG/H
centerC22C2×#13133275
commutatorC23×C53C22×C8
FrattiniC24C22×#6157
FittingC23×Q8×C53C4
socleC22×C53C2×#99
prime radicalC24×C53C22×C4
cosocleC2×C22⋊Dic53C23
hypercenterC23×Q8#6157
hypocenterC53#11428
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
33. C22×Q8×C53C8semidirect
132. C53#11428semidirect
138. C22×Q8#88324semidirect

Maximal subgroups

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

subgroupindex
#202899462
#202896412
#202773012
#1142853