Finite Group Databasebeta

← #17952429

Subgroups — #17952429

Order 4352.

1502 conjugacy classes of subgroups; only the 212 normal subgroups are shown.

normal subgroups only

This group has 1502 conjugacy classes of subgroups — too many to draw. Only its 212 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#14544606C212yesmaximal
2C2×#11763654C214yesmaximal, complemented
3C23×C4⋊Dic17C211yesmaximal, characteristic
4C2×C22⋊#3624—24no—
5C2×C22⋊#3624C2214yes—
6#11763654—28no—
7C24×Dic17C411yescharacteristic
8C24×Dic17C2211yescharacteristic
9C22×C4⋊Dic17—28no—
10C22×C4⋊Dic17—24no—
11C22×C4⋊Dic17C412yes—
12C22×C4⋊Dic17C2212yes—
13C24×C4×C17C411yescharacteristic
14C25×C17C2×C411yescharacteristic
15C23×C4×C17—22no—
16C23×C4×C17—22no—
17C23×C4×C17—24no—
18C23×C4×C17—24no—
19C23×C4×C17C814yescomplemented
20C23×C4×C17C2×C412yes—
21C23×Dic17—22no—
22C23×Dic17—24no—
23C23×Dic17—24no—
24C23×Dic17—44no—
25C23×Dic17—44no—
26C23×Dic17C2×C411yescharacteristic
27C23×Dic17C2×C412yes—
28C23×Dic17D414yes—
29C23×Dic17C2311yescharacteristic
30C22×#3624—24no—
31C2×C4⋊Dic17—232no—
32C2×C4⋊Dic17—24no—
33C2×C4⋊Dic17—24no—
34C2×C4⋊Dic17—24no—
35C2×C4⋊Dic17—24no—
36C2×C4⋊Dic17—24no—
37C2×C4⋊Dic17D418yes—
38C22⋊#3624—28no—
39C22⋊#3624—48no—
40C24×C17—24no—
41C24×C17—24no—
42C24×C17—22no—
43C24×C17—22no—
44C24×C17C22⋊C412yes—
45C24×C17C2×C812yes—
46C24×C17M1612yes—
47C24×C17C22×C411yescharacteristic
48C22×C4×C17—216no—
49C22×C4×C17—28no—
50C22×C4×C17—28no—
51C22×C4×C17—28no—
52C22×C4×C17—22no—
53C22×C4×C17—22no—
54C22×C4×C17—22no—
55C22×C4×C17—22no—
56C22×C4×C17—216no—
57C22×C4×C17C22⋊C414yes—
58C22×C4×C17C2×C814yes—
59C22×C4×C17M1614yes—
60C22×Dic17—28no—
61C22×Dic17—28no—
62C22×Dic17—22no—
63C22×Dic17—22no—
64C22×Dic17—22no—
65C22×Dic17—28no—
66C22×Dic17—42no—
67C22×Dic17—44no—
68C22×Dic17—42no—
69C22×Dic17—416no—
70C22×Dic17—416no—
71C22×Dic17—44no—
72C22×Dic17—44no—
73C22×Dic17—44no—
74C22×Dic17C22⋊C412yes—
75C22×Dic17C22⋊C414yes—
76C22×Dic17C2×D412yes—
77C22×Dic17C2×D414yes—
78C2×#3624—48no—
79C2×#3624—44no—
80C4⋊Dic17—216no—
81C4⋊Dic17—216no—
82#8299—171nomaximal, Sylow
83C23×C17—28no—
84C23×C17—216no—
85C23×C17—28no—
86C23×C17—28no—
87C23×C17—28no—
88C23×C17—22no—
89C23×C17—24no—
90C23×C17—22no—
91C23×C17—24no—
92C23×C17—22no—
93C23×C17—22no—
94C23×C17—22no—
95C23×C17—22no—
96C23×C17C22⋊C814yes—
97C23×C17#9914yes—
98C23×C17#10114yes—
99C23×C17C2×C22⋊C412yes—
100C23×C17C2×C22⋊C411yescharacteristic
101C23×C17C22×C811yescharacteristic
102C23×C17C2×M1611yescharacteristic
103C23×C17C2×M1612yes—
104C2×C4×C17—232no—
105C2×C4×C17—24no—
106C2×C4×C17—24no—
107C2×C4×C17—24no—
108C2×C4×C17—24no—
109C2×C4×C17—24no—
110C2×C4×C17—24no—
111C2×C4×C17C22⋊C818yescomplemented
112C2×Dic17—24no—
113C2×Dic17—22no—
114C2×Dic17—24no—
115C2×Dic17—24no—
116C2×Dic17—44no—
117C2×Dic17—416no—
118C2×Dic17—44no—
119C2×Dic17—416no—
120C2×Dic17—44no—
121C2×Dic17—44no—
122C2×Dic17—44no—
123C2×Dic17#12014yes—
124#3624—84no—
125C23×C4⋊C4—171no—
126#1779—172no—
127C2×#324—174no—
128C22×C17—216no—
129C22×C17—216no—
130C22×C17—24no—
131C22×C17—22no—
132C22×C17—22no—
133C22×C17—24no—
134C22×C17—24no—
135C22×C17—24no—
136C22×C17—24no—
137C22×C17—24no—
138C22×C17—22no—
139C22×C17—24no—
140C22×C17—22no—
141C22×C17#32418yescomplemented
142C22×C17#37911yescharacteristic
143C22×C17C2×C22⋊C812yes—
144C22×C17C2×C22⋊C811yescharacteristic
145C22×C17#40712yes—
146C22×C17#40711yescharacteristic
147C22×C17C2×#9912yes—
148C22×C17C2×#10112yes—
149C4×C17—28no—
150Dic17—24no—
151Dic17—42no—
152Dic17—44no—
153C24×C4—171no—
154C24×C4—171no—
155C24×C4#59511yescharacteristic
156C22×C4⋊C4—172no—
157C22×C4⋊C4—172no—
158C22×C4⋊C4—348no—
159C22×C4⋊C4—344no—
160C2×C22⋊C8—174no—
161C2×C22⋊C8—344no—
162#324—348no—
163C2×C17—28no—
164C2×C17—22no—
165C2×C17—22no—
166C2×C17C2×#32412yes—
167C2×C17#146312yes—
168C2×C17#177911yescharacteristic
169C2×C17#178111yescharacteristic
170C2×C17#178411yescharacteristic
171C25C2×#59511yescharacteristic
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—171no—
177C23×C4—174no—
178C23×C4—171no—
179C23×C4—172no—
180C23×C4—344no—
181C23×C4—344no—
182C23×C4—342no—
183C23×C4—684no—
184C23×C4—684no—
185C23×C4#362414yescomplemented
186C23×C4C2×#59512yes—
187C22×C8—344no—
188C2×C4⋊C4—178no—
189C2×C4⋊C4—344no—
190C2×C4⋊C4—344no—
191C2×C4⋊C4—344no—
192C2×C4⋊C4—344no—
193C2×C4⋊C4—3432no—
194C2×C4⋊C4—344no—
195C22⋊C8—348no—
196C22⋊C8—688no—
197C17#829911yescharacteristic, complemented, Sylow
198C24—22no—
199C24—22no—
200C24—24no—
201C24—24no—
202C24C2×#362412yes—
203C24#6324712yes—
204C24C22⋊#59512yes—
205C24C22×#59511yescharacteristic
206C22×C4—28no—
207C22×C4—28no—
208C22×C4—216no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—28no—
214C22×C4—216no—
215C22×C4—172no—
216C22×C4—174no—
217C22×C4—172no—
218C22×C4—174no—
219C22×C4—348no—
220C22×C4—348no—
221C22×C4—342no—
222C22×C4—342no—
223C22×C4—342no—
224C22×C4—348no—
225C22×C4—682no—
226C22×C4—684no—
227C22×C4—682no—
228C22×C4—6816no—
229C22×C4—6816no—
230C22×C4—684no—
231C22×C4—684no—
232C22×C4—684no—
233C22×C4C2×#362414yes—
234C22×C4#6324714yes—
235C22×C4C22⋊#59514yes—
236C2×C8—684no—
237C2×C8—688no—
238C4⋊C4—3416no—
239C4⋊C4—3416no—
240C23—28no—
241C23—216no—
242C23—28no—
243C23—22no—
244C23—24no—
245C23—24no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—28no—
252C23—28no—
253C23#1058753914yes—
254C23#1058754514yes—
255C23C22⋊#362414yes—
256C23C22×#362411yescharacteristic
257C23C2×#6324711yescharacteristic
258C23C2×#6324712yes—
259C23C2×C22⋊#59512yes—
260C23C2×C22⋊#59511yescharacteristic
261C2×C4—232no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—174no—
269C2×C4—344no—
270C2×C4—342no—
271C2×C4—344no—
272C2×C4—344no—
273C2×C4—684no—
274C2×C4—6816no—
275C2×C4—684no—
276C2×C4—684no—
277C2×C4—6816no—
278C2×C4—684no—
279C2×C4—684no—
280C2×C4C22⋊#362418yescomplemented
281C8—1364no—
282C22—216no—
283C22—216no—
284C22—22no—
285C22—22no—
286C22—24no—
287C22—24no—
288C22—24no—
289C22—24no—
290C22—24no—
291C22—22no—
292C22—24no—
293C22—22no—
294C22—24no—
295C22#1176365418yescomplemented
296C22#1176375011yescharacteristic
297C22C2×C22⋊#362412yes—
298C22C2×C22⋊#362411yescharacteristic
299C22#1176377912yes—
300C22#1176377911yescharacteristic
301C22C2×#1058753912yes—
302C22C2×#1058754512yes—
303C4—28no—
304C4—344no—
305C4—682no—
306C4—684no—
307C2—22no—
308C2—22no—
309C2—28no—
310C2C2×#1176365412yes—
311C2#1454403212yes—
312C2#1454460611yescharacteristic
313C2#1454461211yescharacteristic
314C2#1454461811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC23×C17C22×C8
FrattiniC24C22×#595
FittingC24×C4×C17C4
socleC23×C17C2×C22⋊C4
prime radicalC25×C17C2×C4
cosocleC23×Dic17C23
hypercenterC24×C4#595
hypocenterC17#8299
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C25 ◁ C24×C4

Complements

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

NH ≅ G/Ntype
2. C2×#11763654C2semidirect
19. C23×C4×C17C8semidirect
111. C2×C4×C17C22⋊C8semidirect
141. C22×C17#324semidirect
185. C23×C4#3624semidirect
197. C17#8299semidirect
280. C2×C4C22⋊#3624semidirect
295. C22#11763654semidirect

Maximal subgroups

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

subgroupindex
#145446062
C2×#117636542
C23×C4⋊Dic172
#829917