Finite Group Databasebeta

← #17953080

Subgroups — #17953080

Order 4352.

1266 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 1266 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#14544679C214yesmaximal, complemented
2#14544610C212yesmaximal
3C22×#10587615C211yesmaximal, characteristic
4C2×#10587551—24no—
5#11763750C2214yes—
6#11763687—28no—
7C23×C22⋊D17C2211yescharacteristic
8C2×#10587615—28no—
9C2×#10587615—24no—
10C22×#63225C412yescomplemented
11C22×C22⋊Dic17C411yescharacteristic, complemented
12C22×#63227C2212yes—
13C24×C4×C17C411yescharacteristic, complemented
14C25×C17C2×C411yescharacteristic
15C24×D17C2311yescharacteristic
16C23×C4×C17—22no—
17C23×C4×C17—24no—
18C23×C4×C17—44no—
19C23×C4×C17C2×C412yes—
20C22×C22⋊D17—24no—
21C22×C22⋊D17—44no—
22C22×C22⋊D17D414yes—
23C23×Dic17C2×C412yes—
24C23×Dic17C2×C411yescharacteristic
25C2×C22⋊#595—24no—
26C2×C22⋊#595—28no—
27C2×C22⋊Dic17—22no—
28C2×C22⋊Dic17—24no—
29#10587615—232no—
30C2×#63227—24no—
31C2×#63227—24no—
32C2×#63227D418yes—
33C2×#63225—24no—
34C2×#63225—24no—
35C2×#63225—24no—
36#10587551—48no—
37C24×C17—24no—
38C24×C17—22no—
39C24×C17—44no—
40C24×C17C22⋊C412yes—
41C24×C17C22×C411yescharacteristic
42C23×D17—42no—
43C23×D17—44no—
44C23×D17C2×D414yes—
45C23×D17C2×D412yes—
46C22×#595—44no—
47C22×C4×C17—28no—
48C22×C4×C17—22no—
49C22×C4×C17—22no—
50C22×C4×C17—416no—
51C22×C4×C17—44no—
52C22×C4×C17—44no—
53C22×C4×C17—44no—
54C22×C4×C17C22⋊C414yes—
55C2×C22⋊D17—28no—
56C2×C22⋊D17—28no—
57C2×C22⋊D17—44no—
58C2×C22⋊D17—416no—
59C22×Dic17—22no—
60C22×Dic17—22no—
61C22×Dic17—22no—
62C22×Dic17—42no—
63C22×Dic17—44no—
64C22×Dic17C22⋊C412yes—
65C22×Dic17C22⋊C414yes—
66C22⋊#595—416no—
67C22⋊#595—48no—
68C22⋊Dic17—28no—
69#63227—216no—
70#63225—216no—
71#8543—171nomaximal, Sylow
72C23×C17—22no—
73C23×C17—24no—
74C23×C17—28no—
75C23×C17—22no—
76C23×C17—22no—
77C23×C17—44no—
78C23×C17—416no—
79C23×C17—44no—
80C23×C17—44no—
81C23×C17#9914yes—
82C23×C17C2×C22⋊C411yescharacteristic
83C23×C17C2×C22⋊C412yes—
84C22×D17—24no—
85C22×D17—28no—
86C22×D17—44no—
87C22×D17—44no—
88C22×D17—44no—
89C22×D17—88no—
90C22×D17#12014yes—
91C2×#595—84no—
92C2×#595—88no—
93C2×C4×C17—24no—
94C2×C4×C17—44no—
95C2×C4×C17—416no—
96C2×C4×C17—44no—
97C2×C4×C17—44no—
98C22⋊D17—416no—
99C2×Dic17—24no—
100C2×Dic17—22no—
101C2×Dic17—24no—
102C2×Dic17—44no—
103C2×Dic17—44no—
104C2×Dic17—44no—
105C2×Dic17—44no—
106C22×#123—171no—
107#1819—174no—
108#1781—172no—
109C22×C17—28no—
110C22×C17—22no—
111C22×C17—22no—
112C22×C17—22no—
113C22×C17—24no—
114C22×C17—44no—
115C22×C17—44no—
116C22×C17—416no—
117C22×C17—44no—
118C22×C17#37911yescharacteristic
119C22×C17C2×#9912yes—
120C22×C17#41014yes—
121C2×D17—24no—
122C2×D17—28no—
123C2×D17—84no—
124C2×D17—88no—
125#595—164no—
126C4×C17—44no—
127Dic17—42no—
128Dic17—44no—
129C23×D4—171no—
130C24×C4#59511yescharacteristic, complemented
131C2×#123—348no—
132C2×#123—344no—
133C22×C4⋊C4—172no—
134C22×C22⋊C4—172no—
135C22×C22⋊C4—171no—
136C2×#99—344no—
137#379—174no—
138#342—348no—
139C2×C17—22no—
140C2×C17—24no—
141C2×C17—44no—
142C2×C17#178111yescharacteristic
143C2×C17#178212yes—
144D17—44no—
145C25—171no—
146C25C2×#59511yescharacteristic
147C22×D4—174no—
148C22×D4—344no—
149C22×D4—684no—
150C23×C4—22no—
151C23×C4—24no—
152C23×C4—44no—
153C23×C4—172no—
154C23×C4—171no—
155C23×C4C2×#59512yes—
156#123—3432no—
157C2×C4⋊C4—344no—
158C2×C4⋊C4—344no—
159C2×C4⋊C4—344no—
160C2×C22⋊C4—178no—
161C2×C22⋊C4—344no—
162C2×C22⋊C4—344no—
163C2×C22⋊C4—344no—
164C2×C22⋊C4—342no—
165C2×C22⋊C4—348no—
166C2×C22⋊C4—344no—
167#99—688no—
168C17#854311yescharacteristic, complemented, Sylow
169C24—22no—
170C24—24no—
171C24—44no—
172C24—174no—
173C24—172no—
174C24—682no—
175C24—684no—
176C24C22⋊#59512yes—
177C24C22×#59511yescharacteristic
178C2×D4—348no—
179C2×D4—348no—
180C2×D4—684no—
181C2×D4—6816no—
182C22×C4—28no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—416no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—44no—
189C22×C4—172no—
190C22×C4—174no—
191C22×C4—342no—
192C22×C4—342no—
193C22×C4—342no—
194C22×C4—682no—
195C22×C4—684no—
196C22×C4—684no—
197C22×C4C22⋊#59514yes—
198C4⋊C4—3416no—
199C22⋊C4—3416no—
200C22⋊C4—348no—
201C22⋊C4—688no—
202C22⋊C4—6816no—
203C23—22no—
204C23—24no—
205C23—28no—
206C23—22no—
207C23—22no—
208C23—44no—
209C23—416no—
210C23—44no—
211C23—44no—
212C23—174no—
213C23—344no—
214C23—348no—
215C23—684no—
216C23—684no—
217C23—684no—
218C23—1368no—
219C23#1058755114yes—
220C23C2×C22⋊#59511yescharacteristic
221C23C2×C22⋊#59512yes—
222D4—6816no—
223C2×C4—24no—
224C2×C4—44no—
225C2×C4—416no—
226C2×C4—44no—
227C2×C4—44no—
228C2×C4—344no—
229C2×C4—342no—
230C2×C4—344no—
231C2×C4—684no—
232C2×C4—684no—
233C2×C4—684no—
234C2×C4—684no—
235C2×C4—1364no—
236C2×C4—1368no—
237C22—28no—
238C22—22no—
239C22—22no—
240C22—22no—
241C22—24no—
242C22—44no—
243C22—44no—
244C22—416no—
245C22—44no—
246C22—344no—
247C22—348no—
248C22—1364no—
249C22—1368no—
250C22#1176375011yescharacteristic
251C22C2×#1058755112yes—
252C22#1176378814yes—
253C4—44no—
254C4—2724no—
255C4—682no—
256C4—684no—
257C2—22no—
258C2—24no—
259C2—44no—
260C2—684no—
261C2#1454461011yescharacteristic
262C2#1454461312yes—
subgroupG/H
centerC22#11763750
commutatorC24×C17C22×C4
FrattiniC24C22×#595
FittingC24×C4×C17C4
socleC22×C17#379
prime radicalC23×C22⋊D17C22
cosocleC24×D17C23
hypercenterC24×C4#595
hypocenterC17#8543
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C24×C4

Complements

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

NH ≅ G/Ntype
1. #14544679C2semidirect
10. C22×#63225C4semidirect
11. C22×C22⋊Dic17C4semidirect
13. C24×C4×C17C4semidirect
130. C24×C4#595semidirect
168. C17#8543semidirect

Maximal subgroups

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

subgroupindex
#145446792
#145446102
C22×#105876152
#854317