Finite Group Databasebeta

← #28298092

Subgroups — #28298092

Order 13568.

476 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 54 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#20289635C212yesmaximal, complemented
2#20274625C211yesmaximal, characteristic
3C22⋊#11719910—22no—
4C2×C22⋊#88322—22no—
5C2×C22⋊#88322C411yescharacteristic
6C2×C22⋊#88322C2211yescharacteristic
7C24×C4×C53C411yescharacteristic
8C2×#11719910—42no—
9C22×#88322—21no—
10C22×#88322D412yes—
11C22⋊#88322—41no—
12C22⋊#88322—41no—
13C22⋊#88322—44no—
14C25×C53C811yescharacteristic
15C23×C4×C53—21no—
16C23×C4×C53—21no—
17C23×C4×C53—44no—
18C23×C4×C53—41no—
19C23×C4×C53—41no—
20C23×C4×C53C811yescharacteristic
21C23×C4×C53C2×C411yescharacteristic
22C24×C53—21no—
23C24×C53—21no—
24C24×C53—42no—
25C24×C53—41no—
26C24×C53—42no—
27C24×C53—41no—
28C24×C53C1612yescomplemented
29C24×C53C2×C811yescharacteristic
30C22×C4×C53—24no—
31C22×C4×C53—21no—
32C22×C4×C53—21no—
33C22×C4×C53—21no—
34C22×C4×C53—42no—
35C22×C4×C53—42no—
36C22×C4×C53—42no—
37C22×C4×C53—42no—
38C22×C4×C53—42no—
39C22×C4×C53—42no—
40C22×C4×C53—44no—
41C22×C4×C53—44no—
42C22×C4×C53—44no—
43C22×C4×C53—41no—
44C22×C4×C53—41no—
45C22×C4×C53—41no—
46C22×C4×C53—44no—
47C22×C4×C53C22⋊C411yescharacteristic
48C22×C4×C53M1611yescharacteristic
49C2×#88322—22no—
50C2×#88322—42no—
51C2×#88322—42no—
52C2×#88322—41no—
53#11719910—82no—
54C23×C53—22no—
55C23×C53—22no—
56C23×C53—21no—
57C23×C53—22no—
58C23×C53—21no—
59C23×C53—41no—
60C23×C53—44no—
61C23×C53—42no—
62C23×C53—41no—
63C23×C53—41no—
64C23×C53—41no—
65C23×C53—44no—
66C23×C53—42no—
67C23×C53—42no—
68C23×C53—42no—
69C23×C53—42no—
70C23×C53—48no—
71C23×C53—42no—
72C23×C53—41no—
73C23×C53—41no—
74C23×C53C22⋊C811yescharacteristic
75C23×C53C2×C1611yescharacteristic
76C23×C53M3211yescharacteristic
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—21no—
80C2×C4×C53—44no—
81C2×C4×C53—42no—
82C2×C4×C53—41no—
83C2×C4×C53—48no—
84C2×C4×C53—42no—
85C2×C4×C53—41no—
86C2×C4×C53—44no—
87C2×C4×C53—42no—
88C2×C4×C53—41no—
89C2×C4×C53—42no—
90C2×C4×C53#9911yescharacteristic
91C2×C4×C53#10011yescharacteristic
92#88322—42no—
93#88322—81no—
94#7054—531nomaximal, Sylow
95C22×C53—21no—
96C22×C53—21no—
97C22×C53—24no—
98C22×C53—21no—
99C22×C53—21no—
100C22×C53—41no—
101C22×C53—41no—
102C22×C53—42no—
103C22×C53—42no—
104C22×C53—42no—
105C22×C53—42no—
106C22×C53—44no—
107C22×C53—44no—
108C22×C53—44no—
109C22×C53—44no—
110C22×C53—42no—
111C22×C53—42no—
112C22×C53—41no—
113C22×C53—42no—
114C22×C53—41no—
115C22×C53#32311yescharacteristic
116C22×C53C22⋊C1611yescharacteristic
117C22×C53#34911yescharacteristic
118C4×C53—21no—
119C4×C53—42no—
120C4×C53—41no—
121C4×C53AΣL(1,𝔽16)12yes—
122#1779—531no—
123#1314—532no—
124C2×C53—21no—
125C2×C53—21no—
126C2×C53—44no—
127C2×C53—41no—
128C2×C53—41no—
129C2×C53#131411yescharacteristic
130C2×C53#131611yescharacteristic
131C2×C53#132011yescharacteristic
132C24×C4#615711yescharacteristic
133C2×C22⋊C8—531no—
134C2×C22⋊C8—531no—
135C2×C22⋊C8—1062no—
136C22⋊C16—1062no—
137C53#705411yescharacteristic, complemented, Sylow
138C25#8832411yescharacteristic
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—44no—
142C23×C4—41no—
143C23×C4—41no—
144C23×C4#8832411yescharacteristic
145C23×C4C2×#615711yescharacteristic
146C22×C8—532no—
147C22×C8—1061no—
148C2×C16—2122no—
149C22⋊C8—2121no—
150C22⋊C8—2124no—
151C22⋊C8—2121no—
152C24—21no—
153C24—21no—
154C24—42no—
155C24—41no—
156C24—42no—
157C24—41no—
158C24#1171991012yescomplemented
159C24C2×#8832411yescharacteristic
160C22×C4—24no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—44no—
171C22×C4—44no—
172C22×C4—44no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—44no—
177C22×C4#1171994011yescharacteristic
178C22×C4C22⋊#615711yescharacteristic
179C2×C8—1062no—
180C2×C8—2122no—
181C2×C8—2121no—
182C2×C8—2122no—
183C16—4242no—
184C23—22no—
185C23—22no—
186C23—21no—
187C23—22no—
188C23—21no—
189C23—41no—
190C23—41no—
191C23—44no—
192C23—42no—
193C23—41no—
194C23—41no—
195C23—44no—
196C23—42no—
197C23—42no—
198C23—42no—
199C23—42no—
200C23—48no—
201C23—42no—
202C23—41no—
203C23—41no—
204C23C22⋊#8832411yescharacteristic
205C23C2×#1171991011yescharacteristic
206C23#1313329111yescharacteristic
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—44no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—48no—
214C2×C4—42no—
215C2×C4—41no—
216C2×C4—44no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—42no—
220C2×C4#1313327511yescharacteristic
221C2×C4#1313327711yescharacteristic
222C8—4241no—
223C8—2122no—
224C22—21no—
225C22—21no—
226C22—24no—
227C22—21no—
228C22—21no—
229C22—41no—
230C22—41no—
231C22—42no—
232C22—42no—
233C22—42no—
234C22—42no—
235C22—44no—
236C22—44no—
237C22—44no—
238C22—44no—
239C22—42no—
240C22—42no—
241C22—41no—
242C22—42no—
243C22—41no—
244C22#1681146611yescharacteristic
245C22C22⋊#1171991011yescharacteristic
246C22#1681151911yescharacteristic
247C4—21no—
248C4—42no—
249C4—41no—
250C4#1681152312yes—
251C2—21no—
252C2—21no—
253C2—44no—
254C2—41no—
255C2—41no—
256C2#2028963511yescharacteristic
257C2#2028963911yescharacteristic
258C2#2028964711yescharacteristic
subgroupG/H
centerC2×C4#13133275
commutatorC23×C53C2×C16
FrattiniC23×C4C2×#6157
FittingC24×C4×C53C4
socleC22×C53#323
prime radicalC25×C53C8
cosocleC2×C22⋊#88322C22
hypercenterC24×C4#6157
hypocenterC53#7054
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ C24×C4

Complements

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

NH ≅ G/Ntype
1. #20289635C2semidirect
28. C24×C53C16semidirect
137. C53#7054semidirect
158. C24#11719910semidirect

Maximal subgroups

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

subgroupindex
#202896352
#202746252
#705453