Finite Group Databasebeta

← #17953531

Subgroups — #17953531

Order 4352.

786 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 108 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#14544607C212yesmaximal
2#14543622C214yesmaximal, complemented
3C2×#11762671C211yesmaximal, characteristic
4C2×#10587543—24no—
5C2×C22⋊#3624C2214yes—
6C23×C4×D17C2211yescharacteristic
7C22×#63227C412yes—
8#11762671—24no—
9C2×#10587502C411yescharacteristic
10C2×#10587502C2212yes—
11C2×C17×#95C411yescharacteristic
12C24×D17C2×C411yescharacteristic
13C23×C4×C17C2×C412yes—
14C23×C4×C17C2×C411yescharacteristic
15C23×Dic17C2×C412yes—
16C23×Dic17C2311yescharacteristic
17C22×C4×D17—24no—
18C22×C4×D17—44no—
19C22×C4×D17D414yes—
20C22×#3624—24no—
21C2×#63227—22no—
22C2×#63227—22no—
23C2×#63227—44no—
24C22⋊#3624—28no—
25#10587543—48no—
26C17×#95—22no—
27#10587502—22no—
28#10587502—24no—
29C24×C17C22×C411yescharacteristic
30C23×D17—22no—
31C23×D17—24no—
32C23×D17—22no—
33C23×D17—22no—
34C23×D17—42no—
35C23×D17C22⋊C412yes—
36C22×C4×C17—22no—
37C22×C4×C17—22no—
38C22×C4×C17—24no—
39C22×C4×C17—42no—
40C22×C4×C17—42no—
41C22×C4×C17—42no—
42C22×C4×C17C22⋊C412yes—
43C22×Dic17—22no—
44C22×Dic17—24no—
45C22×Dic17—42no—
46C22×Dic17—42no—
47C22×Dic17—42no—
48C22×Dic17C2×D412yes—
49C22×Dic17#4114yes—
50C2×C4×D17—28no—
51C2×C4×D17—28no—
52C2×C4×D17—44no—
53C2×C4×D17—416no—
54C2×#3624—48no—
55C2×#3624—44no—
56#63227—44no—
57#63227—44no—
58#8702—171nomaximal, Sylow
59C23×C17—22no—
60C23×C17—22no—
61C23×C17C2×C22⋊C411yescharacteristic
62C23×C17#11712yes—
63C23×C17#13112yes—
64C23×C17#13112yes—
65C22×D17—28no—
66C22×D17—22no—
67C22×D17—22no—
68C22×D17—22no—
69C22×D17—22no—
70C22×D17—22no—
71C22×D17—22no—
72C22×D17—48no—
73C22×D17—48no—
74C22×D17—44no—
75C22×D17—44no—
76C22×D17#9914yes—
77C2×C4×C17—22no—
78C2×C4×C17—24no—
79C2×C4×C17—44no—
80C2×C4×C17—42no—
81C2×C4×C17—42no—
82C2×C4×C17—42no—
83C2×C4×C17—84no—
84C2×Dic17—24no—
85C2×Dic17—44no—
86C2×Dic17—42no—
87C2×Dic17—42no—
88C2×Dic17—42no—
89C2×Dic17—84no—
90C2×Dic17#12314yes—
91C4×D17—416no—
92#3624—84no—
93C2×#381—171no—
94#1779—172no—
95#1280—174no—
96C22×C17—22no—
97C22×C17—22no—
98C22×C17—24no—
99C22×C17—24no—
100C22×C17#38111yescharacteristic
101C22×C17#40811yescharacteristic
102C22×C17#40811yescharacteristic
103C22×C17C2×#9912yes—
104C22×C17#41312yes—
105C22×C17#43314yes—
106C2×D17—24no—
107C2×D17—24no—
108C2×D17—24no—
109C2×D17—416no—
110C2×D17—42no—
111C2×D17—42no—
112C2×D17—42no—
113C2×D17—42no—
114C4×C17—42no—
115C4×C17—82no—
116Dic17—24no—
117Dic17—82no—
118C24×C4—171no—
119C22×C22⋊C4—172no—
120C2×C22⋊C8—174no—
121C2×C22⋊C8—344no—
122#381—344no—
123C2×#95—172no—
124C2×#95—171no—
125C2×#95#59511yescharacteristic
126C2×C17—24no—
127C2×C17#146812yes—
128C2×C17#146812yes—
129C2×C17#181811yescharacteristic
130C2×C17#181911yescharacteristic
131C2×C17#182211yescharacteristic
132D17—44no—
133C25—171no—
134C23×C4—171no—
135C23×C4—174no—
136C23×C4—172no—
137C23×C4—344no—
138C23×C4—684no—
139C23×C4C2×#59512yes—
140C23×C4C2×#59511yescharacteristic
141C22×C8—344no—
142C2×C22⋊C4—342no—
143C2×C22⋊C4—342no—
144C2×C22⋊C4—684no—
145C22⋊C8—348no—
146C22⋊C8—688no—
147#95—22no—
148#95—344no—
149#95—342no—
150C17#870211yescharacteristic, complemented, Sylow
151C24—172no—
152C24—344no—
153C24—342no—
154C24—342no—
155C24—342no—
156C24—682no—
157C24C22×#59511yescharacteristic
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—24no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—172no—
165C22×C4—174no—
166C22×C4—348no—
167C22×C4—348no—
168C22×C4—342no—
169C22×C4—344no—
170C22×C4—684no—
171C22×C4—6816no—
172C22×C4—682no—
173C22×C4—682no—
174C22×C4—682no—
175C22×C4C22⋊#59512yes—
176C2×C8—684no—
177C2×C8—688no—
178C22⋊C4—684no—
179C22⋊C4—684no—
180C23—22no—
181C23—22no—
182C23—174no—
183C23—348no—
184C23—342no—
185C23—342no—
186C23—342no—
187C23—342no—
188C23—342no—
189C23—342no—
190C23—688no—
191C23—688no—
192C23—684no—
193C23—684no—
194C23#1058767712yes—
195C23#1058767812yes—
196C23#1058768012yes—
197C23C2×C22⋊#59511yescharacteristic
198C2×C4—22no—
199C2×C4—24no—
200C2×C4—44no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—84no—
205C2×C4—174no—
206C2×C4—344no—
207C2×C4—6816no—
208C2×C4—684no—
209C2×C4—682no—
210C2×C4—682no—
211C2×C4—682no—
212C2×C4—1364no—
213C8—1364no—
214C22—22no—
215C22—22no—
216C22—24no—
217C22—24no—
218C22—344no—
219C22—344no—
220C22—344no—
221C22—6816no—
222C22—682no—
223C22—682no—
224C22—682no—
225C22—682no—
226C22#1176375211yescharacteristic
227C22#1176378311yescharacteristic
228C22#1176378411yescharacteristic
229C22D17⋊#9912yes—
230C22#1176379512yes—
231C22#1176383414yes—
232C4—42no—
233C4—82no—
234C4—344no—
235C4—1362no—
236C2—24no—
237C2—684no—
238C2#1454405012yes—
239C2#1454405212yes—
240C2#1454467811yescharacteristic
241C2#1454468011yescharacteristic
242C2#1454468511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC24×C17C22×C4
FrattiniC24C22×#595
FittingC2×C17×#95C4
socleC23×C17C2×C22⋊C4
prime radicalC24×D17C2×C4
cosocleC23×Dic17C23
hypercenterC2×#95#595
hypocenterC17#8702
LCSG ▷ C24×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#95

Complements

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

NH ≅ G/Ntype
2. #14543622C2semidirect
150. C17#8702semidirect

Maximal subgroups

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

subgroupindex
#145446072
#145436222
C2×#117626712
#870217