Finite Group Databasebeta

← #17952554

Subgroups — #17952554

Order 4352.

814 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 120 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#14544510C212yesmaximal
2C2×#11763666C214yesmaximal, complemented
3C2×#11763362C211yesmaximal, characteristic
4C2×C22⋊#3624—22no—
5C2×#10587542C2214yes—
6#11763675—22no—
7#11763666—28no—
8#11763362—28no—
9C2×#10587574C412yes—
10C2×#10587576C2212yes—
11C2×#10587564—22no—
12C2×C4×Dic34—22no—
13C2×#10587612C411yescharacteristic
14C22×C4×Dic17C2211yescharacteristic
15C2×C17×#117C411yescharacteristic
16C23×C4×C17C2×C411yescharacteristic
17C23×Dic17D412yes—
18C22×Dic34—22no—
19C22×#3624—24no—
20C17×#117—24no—
21C2×C4⋊C4×C17C2×C412yes—
22C2×C22⋊C4×C17—21no—
23C2×C42×C17—21no—
24C2×C22⋊Dic17—21no—
25#10587612—24no—
26C2×C4⋊Dic17—22no—
27C2×C4⋊Dic17C2×C412yes—
28C2×#63225—22no—
29C2×#63225—22no—
30C2×C4×Dic17—24no—
31C2×C4×Dic17—21no—
32C2×C4×Dic17C2×C411yescharacteristic
33C2×C4×Dic17D412yes—
34C2×C4×Dic17C2311yescharacteristic
35#10587576—24no—
36#10587576D418yes—
37#10587574—24no—
38#10587574—24no—
39#10587564—216no—
40C4×Dic34—216no—
41C22⋊#3624—28no—
42#10587542—28no—
43C24×C17C22⋊C411yescharacteristic
44C22×C4×C17—22no—
45C22×C4×C17—22no—
46C22×C4×C17—22no—
47C22×C4×C17—22no—
48C22×C4×C17C22⋊C411yescharacteristic
49C22×C4×C17C22×C411yescharacteristic
50C22×Dic17—22no—
51C22×Dic17—24no—
52C22×Dic17—24no—
53C22×Dic17—42no—
54C22×Dic17C22⋊C412yes—
55C22×Dic17C2×D412yes—
56C2×Dic34—28no—
57C2×Dic34—44no—
58C2×Dic34—44no—
59C2×#3624—28no—
60C2×#3624—44no—
61C2×#3624—44no—
62C4⋊C4×C17—22no—
63C4⋊C4×C17C22⋊C414yes—
64C22⋊C4×C17—24no—
65C42×C17—24no—
66C22⋊Dic17—24no—
67C4⋊Dic17—22no—
68C4⋊Dic17—28no—
69C4⋊Dic17C22⋊C414yes—
70#63225—28no—
71#63225—28no—
72C4×Dic17—24no—
73C4×Dic17—22no—
74C4×Dic17—24no—
75C4×Dic17C2×D414yes—
76#8338—171nomaximal, Sylow
77C23×C17—21no—
78C23×C17—21no—
79C23×C17—22no—
80C23×C17—22no—
81C23×C17#10112yes—
82C23×C17C2×C22⋊C411yescharacteristic
83C2×C4×C17—28no—
84C2×C4×C17—21no—
85C2×C4×C17—21no—
86C2×C4×C17—21no—
87C2×C4×C17—21no—
88C2×C4×C17—24no—
89C2×C4×C17—42no—
90C2×C4×C17—42no—
91C2×C4×C17#10112yes—
92C2×C4×C17C2×C22⋊C412yes—
93C2×Dic17—22no—
94C2×Dic17—22no—
95C2×Dic17—22no—
96C2×Dic17—22no—
97C2×Dic17—24no—
98C2×Dic17—48no—
99C2×Dic17—42no—
100C2×Dic17—42no—
101C2×Dic17#10414yes—
102C2×Dic17#12014yes—
103Dic34—416no—
104#3624—48no—
105C2×#533—171no—
106#1749—172no—
107C2×#330—174no—
108C22×C17—22no—
109C22×C17—24no—
110C22×C17—41no—
111C22×C17—41no—
112C22×C17—42no—
113C22×C17#37911yescharacteristic
114C22×C17C2×#10111yescharacteristic
115C22×C17C2×#10111yescharacteristic
116C22×C17C2×#10412yes—
117C22×C17#42112yes—
118C4×C17—22no—
119C4×C17—22no—
120C4×C17—44no—
121Dic17—24no—
122Dic17—24no—
123Dic17—44no—
124#533—348no—
125C2×C4⋊Q8—172no—
126C2×#125—172no—
127C2×C22⋊Q8—342no—
128C2×C4×Q8—342no—
129C2×#117—171no—
130C2×#117#59511yescharacteristic
131C22×C42—171no—
132C2×C22⋊C8—342no—
133C2×#97—174no—
134#336—342no—
135#330—348no—
136C2×C17—42no—
137C2×C17#152611yescharacteristic
138C2×C17#153112yes—
139C2×C17#153711yescharacteristic
140C2×C17#178411yescharacteristic
141C2×C17#178912yes—
142C22×Q8—342no—
143C23×C4—172no—
144C23×C4C2×#59511yescharacteristic
145C22×C8—344no—
146C4⋊Q8—344no—
147C4⋊Q8—344no—
148#125—178no—
149#125—344no—
150C22⋊Q8—3416no—
151C4×Q8—3416no—
152#117—24no—
153#117—344no—
154C2×C4⋊C4—172no—
155C2×C4⋊C4—342no—
156C2×C4⋊C4—342no—
157C2×C4⋊C4—342no—
158C2×C4⋊C4C2×#59512yes—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—341no—
161C2×C42—21no—
162C2×C42—171no—
163C2×C42—172no—
164C2×C42—171no—
165C2×C42—344no—
166C2×C42—341no—
167C22⋊C8—348no—
168#97—348no—
169C17#833811yescharacteristic, complemented, Sylow
170C24C22⋊#59511yescharacteristic
171C2×Q8—348no—
172C2×Q8—684no—
173C2×Q8—684no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—172no—
179C22×C4—172no—
180C22×C4—344no—
181C22×C4—344no—
182C22×C4—342no—
183C22×C4—682no—
184C22×C4C22⋊#59511yescharacteristic
185C22×C4C22×#59511yescharacteristic
186C2×C8—348no—
187C2×C8—684no—
188C2×C8—684no—
189C4⋊C4—22no—
190C4⋊C4—174no—
191C4⋊C4—348no—
192C4⋊C4—348no—
193C4⋊C4—348no—
194C4⋊C4—342no—
195C4⋊C4C22⋊#59514yes—
196C22⋊C4—24no—
197C22⋊C4—344no—
198C42—24no—
199C42—174no—
200C42—344no—
201C42—344no—
202C42—342no—
203C23—21no—
204C23—21no—
205C23—22no—
206C23—22no—
207C23#1058754512yes—
208C23C2×C22⋊#59511yescharacteristic
209Q8—6816no—
210C2×C4—28no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—24no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—174no—
219C2×C4—174no—
220C2×C4—342no—
221C2×C4—342no—
222C2×C4—342no—
223C2×C4—342no—
224C2×C4—344no—
225C2×C4—688no—
226C2×C4—682no—
227C2×C4—682no—
228C2×C4#1058754512yes—
229C2×C4C2×C22⋊#59512yes—
230C8—688no—
231C22—24no—
232C22—22no—
233C22—41no—
234C22—41no—
235C22—42no—
236C22#1176375011yescharacteristic
237C22C2×#1058754511yescharacteristic
238C22C2×#1058754511yescharacteristic
239C22D17⋊#10412yes—
240C22#1176380612yes—
241C4—22no—
242C4—22no—
243C4—44no—
244C4—344no—
245C4—344no—
246C4—684no—
247C2—42no—
248C2#1454424111yescharacteristic
249C2#1454425612yes—
250C2#1454427511yescharacteristic
251C2#1454461811yescharacteristic
252C2#1454462612yes—
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C17×#117C4
socleC23×C17C2×C22⋊C4
prime radicalC24×C17C22⋊C4
cosocleC2×C4×Dic17C23
hypercenterC2×#117#595
hypocenterC17#8338
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. C2×#11763666C2semidirect
169. C17#8338semidirect

Maximal subgroups

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

subgroupindex
#145445102
C2×#117636662
C2×#117633622
#833817