Finite Group Databasebeta

← #17968915

Subgroups — #17968915

Order 4352.

1170 conjugacy classes of subgroups; only the 312 normal subgroups are shown.

normal subgroups only

This group has 1170 conjugacy classes of subgroups — too many to draw. Only its 312 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
1C2×#11763780C212yesmaximal
2#14544525C218yesmaximal, complemented
3#14544514C214yesmaximal, complemented
4#14531164C211yesmaximal, characteristic
5C22×#63243C2214yes—
6#11763780C4116yescomplemented
7C2×#10587543C2214yes—
8C2×C22⋊#3624C2214yes—
9C2×#10587542C2214yescomplemented
10C2×C4×#3624C2214yescomplemented
11#11763675C2218yescomplemented
12C23×C4×D17C2211yescharacteristic
13C2×#10587563C2212yes—
14#11762672C411yescharacteristic
15#11762668C414yes—
16#11762664C412yes—
17#11762656C2214yes—
18C17×#381C411yescharacteristic
19C24×D17C2×C411yescharacteristic
20C23×C4×C17C2×C411yescharacteristic
21C23×Dic17C2311yescharacteristic
22C22×C4×D17—44no—
23C22×C4×D17C2×C412yes—
24C22×C4×D17C2×C418yes—
25C22×C4×D17C2312yes—
26C22×#3624C2318yes—
27C2×#63243—48no—
28C2×#63243C2×C4116yes—
29C2×C22⋊C4×C17C2×C412yes—
30C2×C22⋊Dic17C2×C412yes—
31C2×#63227C2×C418yes—
32C2×#63225C2×C414yes—
33C2×C4×Dic17C2314yes—
34#10587563—44no—
35C22⋊#3624C2×C4116yes—
36#10587543C2×C4116yes—
37#10587542—44no—
38C4×#3624—44no—
39C17×#95C2×C414yes—
40#10587502C2×C414yes—
41C24×C17C22×C411yescharacteristic
42C23×D17—42no—
43C23×D17—44no—
44C23×D17C4214yes—
45C23×D17C22×C412yes—
46C22×C4×C17—42no—
47C22×C4×C17C4214yes—
48C22×C4×C17C22×C414yes—
49C22×C4×C17C22×C412yes—
50C22×Dic17—24no—
51C22×Dic17C22×C414yes—
52C22×Dic17C22×C411yescharacteristic
53C22×Dic17C22×C414yes—
54C22×Dic17C2411yescharacteristic
55C2×C4×D17—28no—
56C2×C4×D17—416no—
57C2×C4×D17—44no—
58C2×C4×D17C4218yes—
59C2×C4×D17C22×C414yes—
60C2×C4×D17C22×C414yes—
61C2×#3624—28no—
62C2×#3624—28no—
63C2×#3624C22×C4116yes—
64#63243—416no—
65C22⋊C4×C17—42no—
66C22⋊Dic17—42no—
67#63227—48no—
68#63225—44no—
69C4×Dic17—24no—
70C4×Dic17—24no—
71#20529—171nomaximal, Sylow
72C23×C17—22no—
73C23×C17—24no—
74C23×C17C2×C4212yes—
75C23×C17C23×C411yescharacteristic
76C22×D17—24no—
77C22×D17—24no—
78C22×D17—24no—
79C22×D17—44no—
80C22×D17—48no—
81C22×D17—416no—
82C22×D17C2×C4214yes—
83C2×C4×C17—24no—
84C2×C4×C17—24no—
85C2×C4×C17—28no—
86C2×C4×C17—44no—
87C2×C4×C17C2×C4214yes—
88C2×Dic17—22no—
89C2×Dic17—28no—
90C2×Dic17—24no—
91C2×Dic17—28no—
92C2×Dic17C2×C4212yes—
93C2×Dic17C23×C412yes—
94C2×Dic17H(5,𝔽2)14yes—
95C4×D17—28no—
96C4×D17—28no—
97C4×D17—48no—
98#3624—216no—
99C2×#407—172no—
100#2280—171no—
101#1753—178no—
102#1750—174no—
103C22×C17—24no—
104C22×C17—22no—
105C22×C17—44no—
106C22×C17C22×C4211yescharacteristic
107C22×C17#51812yes—
108C22×C17#56814yes—
109C2×D17—24no—
110C2×D17—24no—
111C2×D17—24no—
112C2×D17—416no—
113C2×D17—48no—
114C4×C17—24no—
115C4×C17—44no—
116Dic17—22no—
117Dic17—44no—
118Dic17#51814yes—
119C24×C4—171no—
120C22×M16—174no—
121C2×#117—172no—
122#407—1716no—
123C2×C22⋊C8—174no—
124C2×C22⋊C8—174no—
125C2×#97—174no—
126C2×C4×C8—174no—
127#381—171no—
128#381#59511yescharacteristic
129#380—174no—
130#379—172no—
131C4×C22⋊C4—174no—
132#336—178no—
133C2×C17—42no—
134C2×C17#227311yescharacteristic
135C2×C17#287312yes—
136C2×C17#297814yes—
137D17—44no—
138C25—171no—
139C23×C4—171no—
140C23×C4—172no—
141C23×C4—178no—
142C23×C4—172no—
143C23×C4—684no—
144C23×C4C2×#59511yescharacteristic
145C2×M16—1716no—
146C2×M16—688no—
147C22×C8—178no—
148#117—684no—
149C2×C4⋊C4—174no—
150C2×C22⋊C4—172no—
151C2×C22⋊C4—178no—
152C2×C22⋊C4C2×#59512yes—
153C2×C42—174no—
154C22⋊C8—1716no—
155C22⋊C8—1716no—
156#97—684no—
157C4×C8—684no—
158#95—174no—
159#95C2×#59514yes—
160C17#2052911yescharacteristic, complemented, Sylow
161C24—172no—
162C24—174no—
163C24—682no—
164C24—684no—
165C24C22×#59511yescharacteristic
166C22×C4—42no—
167C22×C4—171no—
168C22×C4—174no—
169C22×C4—171no—
170C22×C4—174no—
171C22×C4—174no—
172C22×C4—178no—
173C22×C4—174no—
174C22×C4—348no—
175C22×C4—344no—
176C22×C4—684no—
177C22×C4—6816no—
178C22×C4C4×#59514yes—
179C22×C4C22×#59514yes—
180C22×C4C22×#59512yes—
181M16—6816no—
182C2×C8—1716no—
183C2×C8—348no—
184C2×C8—348no—
185C4⋊C4—684no—
186C22⋊C4—42no—
187C22⋊C4—688no—
188C22⋊C4—682no—
189C42—344no—
190C42—344no—
191C23—22no—
192C23—24no—
193C23—174no—
194C23—344no—
195C23—344no—
196C23—344no—
197C23—684no—
198C23—688no—
199C23—6816no—
200C23C2×C4×#59512yes—
201C23C23×#59511yescharacteristic
202C2×C4—24no—
203C2×C4—24no—
204C2×C4—28no—
205C2×C4—44no—
206C2×C4—172no—
207C2×C4—174no—
208C2×C4—172no—
209C2×C4—348no—
210C2×C4—348no—
211C2×C4—348no—
212C2×C4—342no—
213C2×C4—344no—
214C2×C4—348no—
215C2×C4—688no—
216C2×C4C2×C4×#59514yes—
217C8—3416no—
218C22—22no—
219C22—24no—
220C22—44no—
221C22—344no—
222C22—344no—
223C22—344no—
224C22—6816no—
225C22—688no—
226C22C22×C4×#59511yescharacteristic
227C22#1176387612yes—
228C22#1176388814yes—
229C4—24no—
230C4—44no—
231C4—174no—
232C4—342no—
233C4—684no—
234C2—42no—
235C2—684no—
236C2#1454522011yescharacteristic
237C2#1454534912yes—
238C2#1454566214yes—
subgroupG/H
center, FrattiniC23C23×#595
commutatorC22×C17C22×C42
FittingC17×#381C4
socleC23×C17C23×C4
prime radicalC24×D17C2×C4
cosocleC22×Dic17C24
hypercenter#381#595
hypocenterC17#20529
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
2. #14544525C2semidirect
3. #14544514C2semidirect
6. #11763780C4semidirect
9. C2×#10587542C22semidirect
10. C2×C4×#3624C22semidirect
11. #11763675C22semidirect
160. C17#20529semidirect

Maximal subgroups

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

subgroupindex
C2×#117637802
#145445252
#145445142
#145311642
#2052917