Finite Group Databasebeta

← #16927821

Subgroups — #16927821

Order 4352.

814 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 99 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#14531362C211yesmaximal, characteristic
2#14528699C211yesmaximal, characteristic
3#14526688C214yesmaximal, complemented
4C17×#1781C211yesmaximal, characteristic
5C22×C22⋊Dic17C412yescomplemented
6C2×#10587505—22no—
7#11762689—22no—
8#11762689—22no—
9#11762672C412yescomplemented
10#11762665C2212yes—
11#11762665C2212yes—
12C23×D4×C17C2211yescharacteristic
13C2×C17×#99—22no—
14C17×#379C2212yes—
15C25×C17C2×C411yescharacteristic
16C25×C17C2311yescharacteristic
17C22×D4×C17—24no—
18C22×D4×C17—44no—
19C22×D4×C17D412yes—
20C22×D4×C17D412yes—
21C23×C4×C17C2×C411yescharacteristic
22C23×Dic17—22no—
23C2×C22⋊C4×C17—22no—
24C2×C22⋊C4×C17—24no—
25C2×C22⋊Dic17—22no—
26C2×C22⋊Dic17—22no—
27C2×C22⋊Dic17—22no—
28C2×C22⋊Dic17—22no—
29C2×C22⋊Dic17—24no—
30C2×C22⋊Dic17—44no—
31C2×C22⋊Dic17C2×C414yes—
32C17×#99—44no—
33#10587505—44no—
34#10587502—24no—
35C24×C17—24no—
36C24×C17—22no—
37C24×C17—42no—
38C24×C17—82no—
39C24×C17—82no—
40C24×C17C22⋊C412yes—
41C24×C17C22×C411yescharacteristic
42C24×C17C2×D411yescharacteristic
43C24×C17C2×D411yescharacteristic
44C24×C17#4114yes—
45C2×D4×C17—24no—
46C2×D4×C17—24no—
47C2×D4×C17—24no—
48C2×D4×C17—24no—
49C2×D4×C17—42no—
50C2×D4×C17—42no—
51C2×D4×C17—416no—
52C22×C4×C17—22no—
53C22×C4×C17—42no—
54C22×C4×C17—42no—
55C22×C4×C17C22⋊C412yes—
56C22×Dic17—24no—
57C22×Dic17—44no—
58C22×Dic17—42no—
59C22×Dic17—84no—
60C22⋊C4×C17—44no—
61C22⋊C4×C17—84no—
62C22⋊Dic17—44no—
63C22⋊Dic17—44no—
64C22⋊Dic17—44no—
65C22⋊Dic17—82no—
66C22⋊Dic17—82no—
67C22⋊Dic17—84no—
68#8746—171nomaximal, Sylow
69C23×C17—28no—
70C23×C17—22no—
71C23×C17—22no—
72C23×C17—22no—
73C23×C17—22no—
74C23×C17—22no—
75C23×C17—42no—
76C23×C17—42no—
77C23×C17—42no—
78C23×C17—44no—
79C23×C17—44no—
80C23×C17—44no—
81C23×C17—44no—
82C23×C17—82no—
83C23×C17—82no—
84C23×C17—88no—
85C23×C17—88no—
86C23×C17#9914yes—
87C23×C17C2×C22⋊C411yescharacteristic
88C23×C17#11712yes—
89C23×C17#12312yes—
90C23×C17#12312yes—
91D4×C17—48no—
92D4×C17—48no—
93C2×C4×C17—22no—
94C2×C4×C17—22no—
95C2×C4×C17—22no—
96C2×C4×C17—44no—
97C2×C4×C17—82no—
98C2×C4×C17—84no—
99C2×Dic17—44no—
100C2×Dic17—44no—
101C2×Dic17—44no—
102C2×Dic17—88no—
103C2×Dic17—82no—
104C2×Dic17—84no—
105#2308—171no—
106#1781—171no—
107#1781D1711yescharacteristic
108#1304—174no—
109C22×C17—22no—
110C22×C17—22no—
111C22×C17—22no—
112C22×C17—22no—
113C22×C17—24no—
114C22×C17—24no—
115C22×C17—24no—
116C22×C17—22no—
117C22×C17—22no—
118C22×C17—44no—
119C22×C17—44no—
120C22×C17—44no—
121C22×C17—44no—
122C22×C17—82no—
123C22×C17—82no—
124C22×C17—88no—
125C22×C17—88no—
126C22×C17#38111yescharacteristic
127C22×C17C2×#9912yes—
128C4×C17—42no—
129C4×C17—162no—
130Dic17—84no—
131Dic17—162no—
132C23×D4C2×D1711yescharacteristic
133C22×C22⋊C4—172no—
134C2×#99—22no—
135C2×#99—342no—
136#386—342no—
137#386—342no—
138#381—172no—
139#379—172no—
140#379—172no—
141#379C2×D1712yes—
142C2×C17—22no—
143C2×C17—42no—
144C2×C17—42no—
145C2×C17—42no—
146C2×C17—44no—
147C2×C17#181911yescharacteristic
148C2×C17#182812yes—
149C25C4×D1711yescharacteristic
150C25C22×D1711yescharacteristic
151C22×D4—24no—
152C22×D4—44no—
153C22×D4D6812yes—
154C22×D4C22⋊D1712yes—
155C23×C4—342no—
156C23×C4C4×D1711yescharacteristic
157C2×C22⋊C4—22no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—174no—
160C2×C22⋊C4—342no—
161C2×C22⋊C4—342no—
162C2×C22⋊C4—342no—
163C2×C22⋊C4—342no—
164C2×C22⋊C4—344no—
165C2×C22⋊C4—684no—
166#99—44no—
167#99—684no—
168#95—344no—
169C17#874611yescharacteristic, complemented, Sylow
170C24—24no—
171C24—22no—
172C24—42no—
173C24—82no—
174C24—82no—
175C24#6322712yes—
176C24C2×C4×D1711yescharacteristic
177C24C2×D6811yescharacteristic
178C24#6325414yes—
179C24C2×C22⋊D1711yescharacteristic
180C2×D4—24no—
181C2×D4—24no—
182C2×D4—24no—
183C2×D4—24no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—416no—
187C22×C4—22no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—344no—
191C22×C4—684no—
192C22×C4—682no—
193C22×C4—1364no—
194C22×C4#6322712yes—
195C22⋊C4—44no—
196C22⋊C4—84no—
197C22⋊C4—684no—
198C22⋊C4—684no—
199C22⋊C4—684no—
200C22⋊C4—1362no—
201C22⋊C4—1362no—
202C22⋊C4—1364no—
203C23—28no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—42no—
210C23—44no—
211C23—42no—
212C23—44no—
213C23—42no—
214C23—44no—
215C23—44no—
216C23—82no—
217C23—82no—
218C23—88no—
219C23—88no—
220C23#1058747714yes—
221C23#1058756312yes—
222C23#1058757212yes—
223C23C2×#6322711yescharacteristic
224C23#1058762112yes—
225D4—48no—
226D4—48no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—44no—
231C2×C4—82no—
232C2×C4—84no—
233C2×C4—684no—
234C2×C4—684no—
235C2×C4—684no—
236C2×C4—1362no—
237C2×C4—1364no—
238C2×C4—1368no—
239C22—22no—
240C22—22no—
241C22—22no—
242C22—22no—
243C22—24no—
244C22—24no—
245C22—24no—
246C22—22no—
247C22—22no—
248C22—44no—
249C22—44no—
250C22—44no—
251C22—44no—
252C22—82no—
253C22—82no—
254C22—88no—
255C22—88no—
256C22#1176267011yescharacteristic
257C22C2×#1058747712yes—
258C4—42no—
259C4—162no—
260C4—2722no—
261C4—1364no—
262C2—22no—
263C2—42no—
264C2—42no—
265C2—42no—
266C2—44no—
267C2#1452885611yescharacteristic
268C2#1452889012yes—
subgroupG/H
centerC22#11762670
commutatorC24×C17C22×C4
FrattiniC25C22×D17
FittingC17×#1781C2
socleC22×C17#381
prime radicalC23×D4×C17C22
cosocleC25×C17C23
hypercenter#1781D17
hypocenterC17#8746
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C25 ◁ #1781

Complements

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

NH ≅ G/Ntype
3. #14526688C2semidirect
5. C22×C22⋊Dic17C4semidirect
9. #11762672C4semidirect
169. C17#8746semidirect

Maximal subgroups

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

subgroupindex
#145313622
#145286992
#145266882
C17×#17812
#874617