Finite Group Databasebeta

← #17953117

Subgroups — #17953117

Order 4352.

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

normal subgroups only

This group has 778 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#14544683C212yesmaximal
2#14544032C212yesmaximal
3#14544032C212yesmaximal
4C22×#10587576C211yesmaximal, characteristic
5#11763779C2214yes—
6#11763689—24no—
7#11763654—24no—
8#11763654—24no—
9C2×#10587576—24no—
10C2×#10587576—24no—
11C2×#10587576—24no—
12C22×C4⋊Dic17C411yescharacteristic
13C22×C4⋊Dic17C2211yescharacteristic
14C22×C4⋊Dic17C2211yescharacteristic
15C22×#63225C411yescharacteristic
16C22×#63225C2211yescharacteristic
17C22×C4×Dic17C411yescharacteristic
18C22×C4⋊C4×C17C411yescharacteristic
19C23×C4×C17C2×C411yescharacteristic
20C23×C4×C17C2×C411yescharacteristic
21C23×C4×C17C2×C411yescharacteristic
22C23×Dic17C2×C411yescharacteristic
23C23×Dic17C2×C411yescharacteristic
24C23×Dic17C2×C411yescharacteristic
25C23×Dic17C2311yescharacteristic
26C2×#63247—24no—
27C2×C4⋊C4×C17—22no—
28C2×C4⋊C4×C17—22no—
29C2×C4⋊C4×C17—22no—
30C2×C4⋊Dic17—22no—
31C2×C4⋊Dic17—22no—
32C2×C4⋊Dic17—22no—
33C2×C4⋊Dic17—22no—
34C2×C4⋊Dic17—22no—
35C2×C4⋊Dic17—22no—
36C2×C4⋊Dic17—22no—
37C2×C4⋊Dic17D414yes—
38C2×C4⋊Dic17D414yes—
39C2×#63225—22no—
40C2×#63225—22no—
41C2×#63225—22no—
42C2×#63225—22no—
43C2×#63225—22no—
44C2×#63225D414yes—
45C2×C4×Dic17—22no—
46C2×C4×Dic17—22no—
47C2×C4×Dic17—22no—
48#10587576—232no—
49C22⋊#3624—24no—
50C22⋊#3624—24no—
51C24×C17C22×C411yescharacteristic
52C22×C4×C17—21no—
53C22×C4×C17—21no—
54C22×C4×C17—24no—
55C22×C4×C17—21no—
56C22×C4×C17—21no—
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17—42no—
60C22×C4×C17—42no—
61C22×C4×C17C22⋊C412yes—
62C22×C4×C17C22⋊C412yes—
63C22×C4×C17C22⋊C412yes—
64C22×Dic17—21no—
65C22×Dic17—21no—
66C22×Dic17—21no—
67C22×Dic17—21no—
68C22×Dic17—21no—
69C22×Dic17—21no—
70C22×Dic17—42no—
71C22×Dic17—42no—
72C22×Dic17—42no—
73C22×Dic17—42no—
74C22×Dic17C22⋊C412yes—
75C22×Dic17C22⋊C412yes—
76C22×Dic17C22⋊C412yes—
77C22×Dic17C2×D412yes—
78C22×Dic17C2×D412yes—
79C22×Dic17C2×D412yes—
80#63247—48no—
81C2×#3624—44no—
82C4⋊C4×C17—28no—
83C4⋊Dic17—28no—
84C4⋊Dic17—28no—
85C4⋊Dic17—28no—
86#63225—28no—
87#63225—28no—
88C4×Dic17—28no—
89#8554—171nomaximal, Sylow
90C23×C17—22no—
91C23×C17—22no—
92C23×C17—22no—
93C23×C17C2×C22⋊C411yescharacteristic
94C23×C17C2×C22⋊C411yescharacteristic
95C23×C17C2×C22⋊C411yescharacteristic
96C2×C4×C17—24no—
97C2×C4×C17—24no—
98C2×C4×C17—22no—
99C2×C4×C17—22no—
100C2×C4×C17—22no—
101C2×C4×C17—22no—
102C2×C4×C17—24no—
103C2×C4×C17—42no—
104C2×C4×C17—42no—
105C2×C4×C17—42no—
106C2×C4×C17—42no—
107C2×C4×C17—42no—
108C2×C4×C17#10114yes—
109C2×Dic17—24no—
110C2×Dic17—22no—
111C2×Dic17—22no—
112C2×Dic17—22no—
113C2×Dic17—42no—
114C2×Dic17—42no—
115C2×Dic17—42no—
116C2×Dic17—42no—
117C2×Dic17—42no—
118C2×Dic17—42no—
119C2×Dic17—42no—
120C2×Dic17—42no—
121C2×Dic17—42no—
122C2×Dic17—42no—
123C2×Dic17—42no—
124C2×Dic17#12014yes—
125#3624—84no—
126C22×#125—171no—
127#1821—172no—
128#1463—172no—
129#1463—172no—
130C22×C17—28no—
131C22×C17—21no—
132C22×C17—21no—
133C22×C17—21no—
134C22×C17—21no—
135C22×C17—21no—
136C22×C17—21no—
137C22×C17#37911yescharacteristic
138C22×C17#41012yes—
139C22×C17C2×#10112yes—
140C22×C17#41312yes—
141C4×C17—24no—
142C4×C17—42no—
143C4×C17—42no—
144Dic17—24no—
145Dic17—42no—
146Dic17—42no—
147Dic17—42no—
148C2×#125—344no—
149C2×#125—344no—
150C2×#125—344no—
151C22×C4⋊C4—171no—
152C22×C4⋊C4—171no—
153C22×C4⋊C4—171no—
154C22×C4⋊C4—171no—
155C22×C4⋊C4—171no—
156C22×C4⋊C4#59511yescharacteristic
157C22×C42—171no—
158#407—174no—
159#343—344no—
160#324—344no—
161#324—344no—
162C2×C17—22no—
163C2×C17—22no—
164C2×C17—22no—
165C2×C17#178211yescharacteristic
166C2×C17#178411yescharacteristic
167C2×C17#178511yescharacteristic
168C23×C4—171no—
169C23×C4—171no—
170C23×C4—171no—
171C23×C4—171no—
172C23×C4C2×#59511yescharacteristic
173C23×C4C2×#59511yescharacteristic
174C23×C4C2×#59511yescharacteristic
175C2×M16—344no—
176#125—3432no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—174no—
181C2×C4⋊C4—174no—
182C2×C4⋊C4—174no—
183C2×C4⋊C4—342no—
184C2×C4⋊C4—342no—
185C2×C4⋊C4—342no—
186C2×C4⋊C4—342no—
187C2×C4⋊C4—342no—
188C2×C4⋊C4—342no—
189C2×C4⋊C4—342no—
190C2×C4⋊C4—342no—
191C2×C4⋊C4—342no—
192C2×C4⋊C4—342no—
193C2×C4⋊C4—342no—
194C2×C4⋊C4—342no—
195C2×C42—342no—
196C2×C42—342no—
197C2×C42—342no—
198C22⋊C8—344no—
199C22⋊C8—344no—
200C17#855411yescharacteristic, complemented, Sylow
201C24C22×#59511yescharacteristic
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—24no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—172no—
212C22×C4—172no—
213C22×C4—172no—
214C22×C4—172no—
215C22×C4—172no—
216C22×C4—172no—
217C22×C4—341no—
218C22×C4—341no—
219C22×C4—341no—
220C22×C4—341no—
221C22×C4—341no—
222C22×C4—341no—
223C22×C4—682no—
224C22×C4—682no—
225C22×C4—682no—
226C22×C4—682no—
227C22×C4C22⋊#59512yes—
228C22×C4C22⋊#59512yes—
229C22×C4C22⋊#59512yes—
230M16—688no—
231C2×C8—684no—
232C4⋊C4—28no—
233C4⋊C4—348no—
234C4⋊C4—348no—
235C4⋊C4—348no—
236C4⋊C4—348no—
237C4⋊C4—348no—
238C42—348no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23C2×C22⋊#59511yescharacteristic
243C23C2×C22⋊#59511yescharacteristic
244C23C2×C22⋊#59511yescharacteristic
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—24no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—174no—
258C2×C4—344no—
259C2×C4—342no—
260C2×C4—342no—
261C2×C4—342no—
262C2×C4—682no—
263C2×C4—682no—
264C2×C4—682no—
265C2×C4—682no—
266C2×C4—682no—
267C2×C4—682no—
268C2×C4—682no—
269C2×C4—682no—
270C2×C4—682no—
271C2×C4—682no—
272C2×C4—682no—
273C2×C4#1058754514yes—
274C8—1364no—
275C22—28no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22#1176375011yescharacteristic
283C22#1176379012yes—
284C22C2×#1058754512yes—
285C22#1176379612yes—
286C4—24no—
287C4—42no—
288C4—42no—
289C4—344no—
290C4—682no—
291C4—682no—
292C4—682no—
293C2—22no—
294C2—22no—
295C2—22no—
296C2#1454461511yescharacteristic
297C2#1454461811yescharacteristic
298C2#1454462111yescharacteristic
subgroupG/H
centerC22#11763750
commutator, prime radicalC24×C17C22×C4
FrattiniC24C22×#595
FittingC22×C4⋊C4×C17C4
socleC22×C17#379
cosocleC23×Dic17C23
hypercenterC22×C4⋊C4#595
hypocenterC17#8554
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
200. C17#8554semidirect

Maximal subgroups

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

subgroupindex
#145446832
#145440322
C22×#105875762
#855417