Finite Group Databasebeta

← #17968913

Subgroups — #17968913

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×#11763779C212yesmaximal
2#14544526C218yesmaximal, complemented
3#14544514C214yesmaximal, complemented
4#14531162C211yesmaximal, characteristic
5C22×#63247C2214yes—
6#11763779C4116yescomplemented
7C2×C22⋊#3624C2214yes—
8C2×C22⋊#3624C2214yes—
9C2×#10587542C2214yescomplemented
10C2×C4×#3624C2214yescomplemented
11#11763675C2218yescomplemented
12C24×Dic17C2211yescharacteristic
13C2×#10587563C2212yes—
14#11762670C412yes—
15#11762667C414yes—
16#11762665C411yescharacteristic
17C22⋊C4×Dic17C2214yes—
18C17×#379C411yescharacteristic
19C25×C17C2×C411yescharacteristic
20C23×Dic17—44no—
21C23×Dic17C2×C412yes—
22C23×Dic17C2×C411yescharacteristic
23C23×Dic17C2×C418yes—
24C23×Dic17C2311yescharacteristic
25C23×Dic17C2312yes—
26C2×#63247—48no—
27C2×#63247C2×C4116yes—
28C22×#3624C2318yes—
29C2×C22⋊C4×C17C2×C412yes—
30C2×C22⋊C4×C17C2×C414yes—
31C2×C22⋊Dic17C2×C412yes—
32C2×C22⋊Dic17C2×C414yes—
33C2×#63225C2×C414yes—
34C2×C4×Dic17C2314yes—
35#10587563—44no—
36C22⋊#3624C2×C4116yes—
37C22⋊#3624C2×C4116yes—
38#10587542—44no—
39C4×#3624—44no—
40#10587502C2×C418yes—
41C24×C17—42no—
42C24×C17—44no—
43C24×C17C4214yes—
44C24×C17C22×C411yescharacteristic
45C24×C17C22×C412yes—
46C22×C4×C17C22×C414yes—
47C22×Dic17—28no—
48C22×Dic17—24no—
49C22×Dic17—416no—
50C22×Dic17—42no—
51C22×Dic17—44no—
52C22×Dic17C4218yes—
53C22×Dic17C4214yes—
54C22×Dic17C22×C414yes—
55C22×Dic17C22×C412yes—
56C22×Dic17C22×C411yescharacteristic
57C22×Dic17C22×C414yes—
58C22×Dic17C22×C414yes—
59C22×Dic17C22×C414yes—
60C22×Dic17C2411yescharacteristic
61#63247—416no—
62C2×#3624—28no—
63C2×#3624—28no—
64C2×#3624C22×C4116yes—
65C22⋊C4×C17—42no—
66C22⋊C4×C17—44no—
67C22⋊Dic17—42no—
68C22⋊Dic17—44no—
69#63225—44no—
70C4×Dic17—24no—
71C4×Dic17—24no—
72#20529—171nomaximal, Sylow
73C23×C17—22no—
74C23×C17—24no—
75C23×C17—24no—
76C23×C17—24no—
77C23×C17—24no—
78C23×C17—44no—
79C23×C17—48no—
80C23×C17—416no—
81C23×C17C2×C4214yes—
82C23×C17C2×C4212yes—
83C23×C17C23×C411yescharacteristic
84C2×C4×C17—24no—
85C2×C4×C17—28no—
86C2×Dic17—24no—
87C2×Dic17—22no—
88C2×Dic17—28no—
89C2×Dic17—28no—
90C2×Dic17—28no—
91C2×Dic17—24no—
92C2×Dic17—28no—
93C2×Dic17—44no—
94C2×Dic17—48no—
95C2×Dic17C2×C4214yes—
96C2×Dic17C2×C4212yes—
97C2×Dic17C23×C412yes—
98C2×Dic17H(5,𝔽2)14yes—
99#3624—216no—
100C2×#407—172no—
101#2280—171no—
102#1753—178no—
103#1750—174no—
104C22×C17—24no—
105C22×C17—24no—
106C22×C17—24no—
107C22×C17—24no—
108C22×C17—22no—
109C22×C17—416no—
110C22×C17—44no—
111C22×C17—48no—
112C22×C17C22×C4211yescharacteristic
113C22×C17#51812yes—
114C22×C17#56814yes—
115C4×C17—44no—
116Dic17—24no—
117Dic17—22no—
118Dic17—44no—
119Dic17#51814yes—
120C24×C4—171no—
121C22×M16—174no—
122C2×#117—172no—
123#407—1716no—
124C2×C22⋊C8—174no—
125C2×C22⋊C8—174no—
126C2×#97—174no—
127C2×C4×C8—174no—
128#381—172no—
129#380—174no—
130#379—171no—
131#379#59511yescharacteristic
132C4×C22⋊C4—174no—
133#336—178no—
134C2×C17—44no—
135C2×C17—42no—
136C2×C17#227311yescharacteristic
137C2×C17#287312yes—
138C2×C17#297814yes—
139C25C2×#59511yescharacteristic
140C23×C4—171no—
141C23×C4—172no—
142C23×C4—178no—
143C23×C4—172no—
144C23×C4—171no—
145C23×C4—684no—
146C2×M16—1716no—
147C2×M16—688no—
148C22×C8—178no—
149#117—684no—
150C2×C4⋊C4—174no—
151C2×C22⋊C4—172no—
152C2×C22⋊C4—174no—
153C2×C22⋊C4C2×#59512yes—
154C2×C22⋊C4C2×#59514yes—
155C2×C42—174no—
156C22⋊C8—1716no—
157C22⋊C8—1716no—
158#97—684no—
159C4×C8—684no—
160#95—178no—
161C17#2052911yescharacteristic, complemented, Sylow
162C24—42no—
163C24—44no—
164C24C4×#59514yes—
165C24C22×#59511yescharacteristic
166C24C22×#59512yes—
167C22×C4—171no—
168C22×C4—174no—
169C22×C4—174no—
170C22×C4—174no—
171C22×C4—172no—
172C22×C4—171no—
173C22×C4—178no—
174C22×C4—174no—
175C22×C4—174no—
176C22×C4—344no—
177C22×C4—348no—
178C22×C4—6816no—
179C22×C4—684no—
180C22×C4—682no—
181C22×C4C22×#59514yes—
182M16—6816no—
183C2×C8—1716no—
184C2×C8—348no—
185C2×C8—348no—
186C4⋊C4—684no—
187C22⋊C4—44no—
188C22⋊C4—42no—
189C22⋊C4—684no—
190C22⋊C4—682no—
191C42—344no—
192C42—344no—
193C23—22no—
194C23—24no—
195C23—24no—
196C23—24no—
197C23—24no—
198C23—44no—
199C23—48no—
200C23—416no—
201C23C2×C4×#59512yes—
202C23C2×C4×#59514yes—
203C23C23×#59511yescharacteristic
204C2×C4—24no—
205C2×C4—28no—
206C2×C4—172no—
207C2×C4—174no—
208C2×C4—172no—
209C2×C4—174no—
210C2×C4—348no—
211C2×C4—348no—
212C2×C4—348no—
213C2×C4—342no—
214C2×C4—344no—
215C2×C4—344no—
216C2×C4—348no—
217C2×C4—684no—
218C2×C4—688no—
219C8—3416no—
220C22—24no—
221C22—22no—
222C22—24no—
223C22—24no—
224C22—24no—
225C22—416no—
226C22—44no—
227C22—48no—
228C22C22×C4×#59511yescharacteristic
229C22#1176387612yes—
230C22#1176388714yes—
231C4—44no—
232C4—174no—
233C4—344no—
234C4—342no—
235C4—684no—
236C2—44no—
237C2—42no—
238C2#1454522011yescharacteristic
239C2#1454534812yes—
240C2#1454566014yes—
subgroupG/H
center, FrattiniC23C23×#595
commutatorC22×C17C22×C42
FittingC17×#379C4
socleC23×C17C23×C4
prime radicalC25×C17C2×C4
cosocleC22×Dic17C24
hypercenter#379#595
hypocenterC17#20529
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. #14544526C2semidirect
3. #14544514C2semidirect
6. #11763779C4semidirect
9. C2×#10587542C22semidirect
10. C2×C4×#3624C22semidirect
11. #11763675C22semidirect
161. C17#20529semidirect

Maximal subgroups

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

subgroupindex
C2×#117637792
#145445262
#145445142
#145311622
#2052917