Finite Group Databasebeta

← #17951220

Subgroups — #17951220

Order 4352.

726 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 176 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×#11763832C212yesmaximal
2#14543662C214yesmaximal, complemented
3D17×#384C211yesmaximal, characteristic
4#11763832—24no—
5#11763832C418yescomplemented
6C2×#10587544C2214yes—
7C2×#10587543—22no—
8C2×#10587542—22no—
9C2×C4⋊C4×D17—22no—
10C2×C4⋊C4×D17C412yes—
11C2×C42×D17C2211yescharacteristic
12#11762679C411yescharacteristic
13#11762678—22no—
14#11762678C2212yes—
15C17×#384C411yescharacteristic
16C22×C4×D17—22no—
17C22×C4×D17C2×C411yescharacteristic
18C22×C4×D17D412yes—
19C22×#3624—24no—
20C2×C4⋊C4×C17—22no—
21C2×C4⋊C4×C17C2×C412yes—
22C2×C42×C17C2×C411yescharacteristic
23C2×C4⋊Dic17—22no—
24C2×C4⋊Dic17C2×C412yes—
25C2×#63225—22no—
26C2×#63225—24no—
27C2×C4×Dic17D412yes—
28C2×C4×Dic17C2311yescharacteristic
29C4⋊C4×D17—48no—
30C4⋊C4×D17—44no—
31C42×D17C2×C414yes—
32C42×D17D412yes—
33C42×D17Q812yes—
34#10587544—24no—
35#10587544C2×C418yescomplemented
36#10587543—28no—
37#10587542—28no—
38C23×D17C22⋊C411yescharacteristic
39C22×C4×C17—22no—
40C22×C4×C17C22⋊C412yes—
41C22×C4×C17C22×C411yescharacteristic
42C22×Dic17—22no—
43C22×Dic17C22⋊C411yescharacteristic
44C22×Dic17C2×D412yes—
45C2×C4×D17—22no—
46C2×C4×D17—22no—
47C2×C4×D17—24no—
48C2×C4×D17—24no—
49C2×C4×D17—48no—
50C2×C4×D17—44no—
51C2×C4×D17C4212yes—
52C2×C4×D17C22⋊C414yes—
53C2×C4×D17C4⋊C412yes—
54C2×C4×D17C4⋊C414yes—
55C2×#3624—28no—
56C2×#3624—44no—
57C2×#3624—44no—
58C4⋊C4×C17—42no—
59C4⋊C4×C17—44no—
60C42×C17C4212yes—
61C42×C17C4⋊C412yes—
62C4⋊Dic17—44no—
63C4⋊Dic17—42no—
64#63225—44no—
65#63225—48no—
66C4×Dic17C22⋊C414yes—
67C4×Dic17C4⋊C414yes—
68C4×Dic17C22×C412yes—
69C4×Dic17C2×D411yescharacteristic
70C4×Dic17C2×Q811yescharacteristic
71#8040—171nomaximal, Sylow
72C23×C17C2×C22⋊C411yescharacteristic
73C22×D17—22no—
74C22×D17—22no—
75C22×D17#9512yes—
76C22×D17#10012yes—
77C22×D17#10112yes—
78C2×C4×C17—21no—
79C2×C4×C17—21no—
80C2×C4×C17—22no—
81C2×C4×C17—22no—
82C2×C4×C17—42no—
83C2×C4×C17—44no—
84C2×C4×C17#9514yes—
85C2×C4×C17C2×C4211yescharacteristic
86C2×C4×C17C2×C4⋊C411yescharacteristic
87C2×Dic17—21no—
88C2×Dic17—21no—
89C2×Dic17—42no—
90C2×Dic17—44no—
91C2×Dic17#9512yes—
92C2×Dic17C2×C22⋊C412yes—
93C2×Dic17C2×C4⋊C412yes—
94C2×Dic17AGL(1,ℤ/8)14yes—
95C2×Dic17#13714yes—
96C4×D17—24no—
97C4×D17—24no—
98C4×D17—28no—
99C4×D17—28no—
100C4×D17—88no—
101#3624—48no—
102C2×#432—172no—
103C2×#384—171no—
104#1296—174no—
105C22×C17C2×#9511yescharacteristic
106C22×C17C2×#10011yescharacteristic
107C22×C17C2×#10111yescharacteristic
108C22×C17#41812yes—
109C22×C17#41912yes—
110C2×D17—22no—
111C2×D17—24no—
112C2×D17—22no—
113C2×D17—24no—
114C2×D17#34314yes—
115C4×C17—24no—
116C4×C17—21no—
117C4×C17—21no—
118C4×C17—82no—
119Dic17—22no—
120Dic17—82no—
121Dic17#41714yes—
122Dic17#42814yes—
123C22×C4⋊C4—172no—
124C22×C4⋊C4—342no—
125C22×C42—171no—
126#432—178no—
127#432—344no—
128C2×C4⋊C8—174no—
129C2×C22⋊C8—342no—
130C2×#97—342no—
131#384—172no—
132#384—171no—
133#384—342no—
134#384#59511yescharacteristic
135C2×C17#155511yescharacteristic
136C2×C17#155612yes—
137C2×C17#155811yescharacteristic
138C2×C17#173612yes—
139C2×C17C2×#34311yescharacteristic
140D17—24no—
141C23×C4—172no—
142C23×C4—171no—
143C23×C4—342no—
144C22×C8—344no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—172no—
147C2×C4⋊C4—342no—
148C2×C4⋊C4—344no—
149C2×C4⋊C4—342no—
150C2×C4⋊C4—684no—
151C2×C4⋊C4—688no—
152C2×C4⋊C4C2×#59512yes—
153C2×C42—172no—
154C2×C42—171no—
155C2×C42—174no—
156C2×C42—172no—
157C2×C42—172no—
158C2×C42C2×#59511yescharacteristic
159C4⋊C8—178no—
160C4⋊C8—344no—
161C22⋊C8—348no—
162#97—348no—
163C17#804011yescharacteristic, complemented, Sylow
164C24—171no—
165C22×C4—22no—
166C22×C4—172no—
167C22×C4—171no—
168C22×C4—172no—
169C22×C4—172no—
170C22×C4—174no—
171C22×C4—174no—
172C22×C4—344no—
173C22×C4—344no—
174C22×C4—342no—
175C22×C4—342no—
176C22×C4—342no—
177C22×C4—684no—
178C22×C4—688no—
179C22×C4C22⋊#59512yes—
180C22×C4C22×#59511yescharacteristic
181C2×C8—348no—
182C2×C8—684no—
183C2×C8—684no—
184C4⋊C4—42no—
185C4⋊C4—44no—
186C4⋊C4—682no—
187C4⋊C4—684no—
188C4⋊C4—688no—
189C4⋊C4—684no—
190C42—172no—
191C42—171no—
192C42—171no—
193C42—174no—
194C42—174no—
195C42C4×#59512yes—
196C42C4⋊#59512yes—
197C23—172no—
198C23—172no—
199C23—172no—
200C23—342no—
201C23—342no—
202C23C2×C22⋊#59511yescharacteristic
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—42no—
208C2×C4—44no—
209C2×C4—172no—
210C2×C4—172no—
211C2×C4—174no—
212C2×C4—174no—
213C2×C4—172no—
214C2×C4—348no—
215C2×C4—348no—
216C2×C4—344no—
217C2×C4—344no—
218C2×C4—341no—
219C2×C4—341no—
220C2×C4—682no—
221C2×C4—684no—
222C2×C4—1368no—
223C2×C4(C2×#595)⋊C414yes—
224C2×C4C2×C4×#59511yescharacteristic
225C2×C4C2×C4⋊#59511yescharacteristic
226C8—688no—
227C22—174no—
228C22—344no—
229C22—342no—
230C22—344no—
231C22—342no—
232C22C2×(C2×#595)⋊C411yescharacteristic
233C22D17⋊#10011yescharacteristic
234C22D17⋊#10111yescharacteristic
235C22#1176380112yes—
236C22#1176380212yes—
237C4—24no—
238C4—21no—
239C4—21no—
240C4—82no—
241C4—174no—
242C4—174no—
243C4—342no—
244C4—1362no—
245C2—344no—
246C2#1454431211yescharacteristic
247C2#1454431412yes—
248C2#1454431811yescharacteristic
249C2#1454447112yes—
250C2#1454448711yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC2×C4×C17C2×C42
FrattiniC22×C4C22×#595
FittingC17×#384C4
socleC23×C17C2×C22⋊C4
prime radicalC23×D17C22⋊C4
cosocleC2×C4×Dic17C23
hypercenter#384#595
hypocenterC17#8040
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
2. #14543662C2semidirect
5. #11763832C4semidirect
35. #10587544C2×C4semidirect
163. C17#8040semidirect

Maximal subgroups

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

subgroupindex
C2×#117638322
#145436622
D17×#3842
#804017