Finite Group Databasebeta

← #17951219

Subgroups — #17951219

Order 4352.

726 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 184 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×#11763829C212yesmaximal
2#14543662C214yesmaximal, complemented
3C2×#11762678C211yesmaximal, characteristic
4#11763829—24no—
5#11763829C418yescomplemented
6C2×#10587544C2214yes—
7C2×C22⋊#3624—22no—
8C2×#10587542—22no—
9C22×C4⋊Dic17C411yescharacteristic
10C22×#63225—22no—
11C22×C4×Dic17C2211yescharacteristic
12#11762678—22no—
13#11762678C412yes—
14#11762678C2212yes—
15C22×C4⋊C4×C17C411yescharacteristic
16C23×C4×C17—21no—
17C23×C4×C17C2×C411yescharacteristic
18C23×Dic17—21no—
19C23×Dic17D412yes—
20C22×#3624—24no—
21C2×C4⋊C4×C17—21no—
22C2×C4⋊C4×C17—42no—
23C2×C4⋊C4×C17C2×C412yes—
24C2×C4⋊Dic17—21no—
25C2×C4⋊Dic17—42no—
26C2×C4⋊Dic17C2×C412yes—
27C2×#63225—24no—
28C2×#63225—24no—
29C2×#63225—48no—
30C2×C4×Dic17C2×C411yescharacteristic
31C2×C4×Dic17C2×C414yes—
32C2×C4×Dic17D412yes—
33C2×C4×Dic17D412yes—
34C2×C4×Dic17Q812yes—
35C2×C4×Dic17C2311yescharacteristic
36C22⋊#3624—28no—
37#10587544—24no—
38#10587544C2×C418yescomplemented
39#10587542—28no—
40C24×C17C22⋊C411yescharacteristic
41C22×C4×C17—22no—
42C22×C4×C17—22no—
43C22×C4×C17—22no—
44C22×C4×C17—44no—
45C22×C4×C17—42no—
46C22×C4×C17C4212yes—
47C22×C4×C17C22⋊C411yescharacteristic
48C22×C4×C17C4⋊C412yes—
49C22×C4×C17C22×C411yescharacteristic
50C22×Dic17—24no—
51C22×Dic17—24no—
52C22×Dic17—22no—
53C22×Dic17—44no—
54C22×Dic17—42no—
55C22×Dic17C22⋊C412yes—
56C22×Dic17C22⋊C414yes—
57C22×Dic17C4⋊C414yes—
58C22×Dic17C2×D412yes—
59C2×#3624—28no—
60C2×#3624—44no—
61C2×#3624—44no—
62C4⋊C4×C17—42no—
63C4⋊C4×C17—42no—
64C4⋊Dic17—42no—
65C4⋊Dic17—42no—
66#63225—48no—
67#63225—48no—
68C4×Dic17C4212yes—
69C4×Dic17C22⋊C414yes—
70C4×Dic17C4⋊C412yes—
71C4×Dic17C4⋊C414yes—
72C4×Dic17C22×C412yes—
73C4×Dic17C2×D411yescharacteristic
74C4×Dic17C2×Q811yescharacteristic
75#8040—171nomaximal, Sylow
76C23×C17—22no—
77C23×C17—22no—
78C23×C17#9512yes—
79C23×C17#10012yes—
80C23×C17#10112yes—
81C23×C17C2×C22⋊C411yescharacteristic
82C2×C4×C17—24no—
83C2×C4×C17—24no—
84C2×C4×C17—21no—
85C2×C4×C17—21no—
86C2×C4×C17—21no—
87C2×C4×C17—21no—
88C2×C4×C17—42no—
89C2×C4×C17—44no—
90C2×C4×C17—84no—
91C2×C4×C17#9512yes—
92C2×C4×C17C2×C4211yescharacteristic
93C2×C4×C17C2×C4⋊C411yescharacteristic
94C2×Dic17—22no—
95C2×Dic17—22no—
96C2×Dic17—28no—
97C2×Dic17—28no—
98C2×Dic17—42no—
99C2×Dic17—44no—
100C2×Dic17—84no—
101C2×Dic17#9514yes—
102C2×Dic17C2×C22⋊C412yes—
103C2×Dic17C2×C4⋊C412yes—
104C2×Dic17AGL(1,ℤ/8)14yes—
105C2×Dic17#13714yes—
106#3624—48no—
107C2×#432—172no—
108C2×#384—171no—
109#1296—174no—
110C22×C17—22no—
111C22×C17—24no—
112C22×C17—22no—
113C22×C17—24no—
114C22×C17#34314yes—
115C22×C17C2×#9511yescharacteristic
116C22×C17C2×#10011yescharacteristic
117C22×C17C2×#10111yescharacteristic
118C22×C17#41812yes—
119C22×C17#41912yes—
120C4×C17—22no—
121C4×C17—21no—
122C4×C17—21no—
123C4×C17—82no—
124Dic17—24no—
125Dic17—82no—
126Dic17#41714yes—
127Dic17#42814yes—
128C22×C4⋊C4—171no—
129C22×C4⋊C4—342no—
130C22×C4⋊C4#59511yescharacteristic
131C22×C42—171no—
132#432—178no—
133#432—344no—
134C2×C4⋊C8—174no—
135C2×C22⋊C8—342no—
136C2×#97—342no—
137#384—172no—
138#384—172no—
139#384—342no—
140C2×C17—24no—
141C2×C17#155511yescharacteristic
142C2×C17#155612yes—
143C2×C17#155811yescharacteristic
144C2×C17#173612yes—
145C2×C17C2×#34311yescharacteristic
146C23×C4—21no—
147C23×C4—172no—
148C23×C4—341no—
149C23×C4C2×#59511yescharacteristic
150C22×C8—344no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—172no—
154C2×C4⋊C4—344no—
155C2×C4⋊C4—344no—
156C2×C4⋊C4—341no—
157C2×C4⋊C4—682no—
158C2×C4⋊C4—688no—
159C2×C4⋊C4C2×#59512yes—
160C2×C42—174no—
161C2×C42—172no—
162C2×C42—172no—
163C2×C42—172no—
164C2×C42—171no—
165C2×C42—171no—
166C4⋊C8—178no—
167C4⋊C8—344no—
168C22⋊C8—348no—
169#97—348no—
170C17#804011yescharacteristic, complemented, Sylow
171C24C22⋊#59511yescharacteristic
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—44no—
176C22×C4—42no—
177C22×C4—172no—
178C22×C4—172no—
179C22×C4—174no—
180C22×C4—174no—
181C22×C4—344no—
182C22×C4—344no—
183C22×C4—342no—
184C22×C4—684no—
185C22×C4—682no—
186C22×C4C4×#59512yes—
187C22×C4C4⋊#59512yes—
188C22×C4C22⋊#59511yescharacteristic
189C22×C4C22×#59511yescharacteristic
190C2×C8—348no—
191C2×C8—684no—
192C2×C8—684no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—682no—
196C4⋊C4—682no—
197C4⋊C4—688no—
198C4⋊C4—688no—
199C42—172no—
200C42—172no—
201C42—171no—
202C42—172no—
203C42—171no—
204C42—174no—
205C42—174no—
206C23—22no—
207C23—22no—
208C23#1058754512yes—
209C23(C2×#595)⋊C412yes—
210C23#1058755312yes—
211C23C2×C22⋊#59511yescharacteristic
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—42no—
219C2×C4—44no—
220C2×C4—84no—
221C2×C4—172no—
222C2×C4—172no—
223C2×C4—174no—
224C2×C4—174no—
225C2×C4—174no—
226C2×C4—348no—
227C2×C4—348no—
228C2×C4—342no—
229C2×C4—342no—
230C2×C4—682no—
231C2×C4—684no—
232C2×C4—1364no—
233C2×C4(C2×#595)⋊C412yes—
234C2×C4C2×C4×#59511yescharacteristic
235C2×C4C2×C4⋊#59511yescharacteristic
236C8—688no—
237C22—24no—
238C22—22no—
239C22—24no—
240C22—22no—
241C22#1176368914yes—
242C22C2×(C2×#595)⋊C411yescharacteristic
243C22C2×#1058755311yescharacteristic
244C22C2×#1058754511yescharacteristic
245C22#1176380112yes—
246C22#1176380212yes—
247C4—21no—
248C4—21no—
249C4—22no—
250C4—82no—
251C4—174no—
252C4—174no—
253C4—344no—
254C4—1362no—
255C2—24no—
256C2#1454431111yescharacteristic
257C2#1454431312yes—
258C2#1454431711yescharacteristic
259C2#1454447112yes—
260C2C2×#1176368911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC2×C4×C17C2×C42
FrattiniC22×C4C22×#595
FittingC22×C4⋊C4×C17C4
socleC23×C17C2×C22⋊C4
prime radicalC24×C17C22⋊C4
cosocleC2×C4×Dic17C23
hypercenterC22×C4⋊C4#595
hypocenterC17#8040
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #14543662C2semidirect
5. #11763829C4semidirect
38. #10587544C2×C4semidirect
170. C17#8040semidirect

Maximal subgroups

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

subgroupindex
C2×#117638292
#145436622
C2×#117626782
#804017