Finite Group Databasebeta

← #17957957

Subgroups — #17957957

Order 4352.

1426 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 1426 conjugacy classes of subgroups — too many to draw. Only its 180 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×#11763687C214yesmaximal
2#14543958C212yesmaximal
3C22×#10587583C211yesmaximal, characteristic
4C22×C22⋊#595C2214yes—
5C2×(C2×#595)⋊C4—24no—
6#11763687—28no—
7C23×Dic34C411yescharacteristic, complemented
8C23×C4×D17C2211yescharacteristic
9C2×#10587583—28no—
10C2×#10587583—24no—
11C22×C4⋊Dic17C412yescomplemented
12C22×#63227C2212yes—
13C22×C4⋊C4×C17C411yescharacteristic, complemented
14C24×D17C2311yescharacteristic
15C23×#595—24no—
16C23×C4×C17C2×C411yescharacteristic
17C23×C4×C17C2×C412yes—
18C23×Dic17C2×C412yes—
19C23×Dic17C2×C411yescharacteristic
20C22×C4×D17—24no—
21C22×C4×D17—44no—
22C22×C4×D17D414yes—
23C22×Dic34—22no—
24C22×Dic34—24no—
25C22×Dic34—44no—
26C2×C22⋊#595—28no—
27C2×C22⋊#595—28no—
28C2×C22⋊#595—24no—
29C2×C22⋊#595Q818yescomplemented
30C2×C4⋊C4×C17—24no—
31C2×C4⋊C4×C17—22no—
32C2×#63227—24no—
33C2×#63227—24no—
34C2×#63227D418yes—
35C2×C4⋊Dic17—24no—
36C2×C4⋊Dic17—24no—
37C2×C4⋊Dic17—24no—
38#10587583—232no—
39(C2×#595)⋊C4—48no—
40C24×C17C22×C411yescharacteristic
41C23×D17—42no—
42C23×D17—42no—
43C23×D17C2×D412yes—
44C23×D17C2×D414yes—
45C23×D17C2×Q814yes—
46C23×D17#4114yes—
47C22×#595—28no—
48C22×#595—48no—
49C22×#595—48no—
50C22×#595—44no—
51C22×#595—44no—
52C22×C4×C17—22no—
53C22×C4×C17—24no—
54C22×C4×C17—22no—
55C22×C4×C17—22no—
56C22×C4×C17—44no—
57C22×C4×C17C22⋊C412yes—
58C22×C4×C17C22⋊C414yes—
59C22×Dic17—22no—
60C22×Dic17—22no—
61C22×Dic17—22no—
62C22×Dic17—42no—
63C22×Dic17—44no—
64C22×Dic17C22⋊C412yes—
65C22×Dic17C22⋊C414yes—
66C2×C4×D17—28no—
67C2×C4×D17—28no—
68C2×C4×D17—44no—
69C2×C4×D17—416no—
70C2×Dic34—28no—
71C2×Dic34—416no—
72C2×Dic34—44no—
73C2×Dic34—44no—
74C22⋊#595—216no—
75C22⋊#595—48no—
76C4⋊C4×C17—28no—
77#63227—216no—
78C4⋊Dic17—216no—
79#11381—171nomaximal, Sylow
80C23×C17—22no—
81C23×C17—22no—
82C23×C17C2×C22⋊C412yes—
83C23×C17C2×C22⋊C411yescharacteristic
84C23×C17C4×Q814yes—
85C22×D17—24no—
86C22×D17—24no—
87C22×D17—44no—
88C22×D17—44no—
89C22×D17—48no—
90C22×D17—48no—
91C22×D17#12014yes—
92C22×D17C22⋊Q814yes—
93C22×D17C22⋊Q818yes—
94C22×D17#12412yes—
95C22×D17#12414yes—
96C22×D17C4⋊Q812yes—
97C22×D17C4⋊Q814yes—
98C2×#595—416no—
99C2×#595—48no—
100C2×#595—88no—
101C2×#595—84no—
102C2×#595—84no—
103C2×C4×C17—28no—
104C2×C4×C17—22no—
105C2×C4×C17—22no—
106C2×C4×C17—24no—
107C2×C4×C17—44no—
108C2×C4×C17—44no—
109C2×C4×C17—44no—
110C2×C4×C17#9914yes—
111C2×Dic17—24no—
112C2×Dic17—22no—
113C2×Dic17—24no—
114C2×Dic17—44no—
115C2×Dic17—44no—
116C2×Dic17—44no—
117C2×Dic17—44no—
118C4×D17—416no—
119Dic34—416no—
120C22×C22⋊Q8—171no—
121C2×#342—174no—
122C22⋊#95—172no—
123C22×C17—28no—
124C22×C17—22no—
125C22×C17—22no—
126C22×C17#37911yescharacteristic
127C22×C17#39114yes—
128C22×C17#39112yes—
129C22×C17C2×#9912yes—
130C22×C17#41012yes—
131C2×D17—28no—
132C2×D17—24no—
133C2×D17—24no—
134C2×D17—24no—
135C2×D17—24no—
136C2×D17—48no—
137C2×D17—44no—
138C2×D17—44no—
139C2×D17#39314yes—
140C2×D17#39514yes—
141#595—88no—
142C4×C17—24no—
143C4×C17—44no—
144Dic17—44no—
145Dic17—42no—
146C23×Q8—171no—
147C24×C4—171no—
148C2×C22⋊Q8—348no—
149C2×C22⋊Q8—344no—
150C22×C4⋊C4—172no—
151C22×C4⋊C4#59511yescharacteristic, complemented
152C22×C22⋊C4—172no—
153C22×C22⋊C4—174no—
154C2×#95—344no—
155#342—348no—
156C2×C17—24no—
157C2×C17#144611yescharacteristic
158C2×C17#178111yescharacteristic
159C2×C17#178211yescharacteristic
160C2×C17#199614yes—
161D17—24no—
162D17—24no—
163C25—171no—
164C22×Q8—344no—
165C22×Q8—342no—
166C22×Q8—684no—
167C23×C4—172no—
168C23×C4—171no—
169C23×C4—174no—
170C23×C4—344no—
171C23×C4—344no—
172C23×C4—684no—
173C23×C4C2×#59512yes—
174C23×C4C2×#59511yescharacteristic
175C22⋊Q8—3432no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—344no—
179C2×C4⋊C4—344no—
180C2×C4⋊C4—344no—
181C2×C22⋊C4—178no—
182C2×C22⋊C4—178no—
183C2×C22⋊C4—344no—
184C2×C22⋊C4—344no—
185C2×C22⋊C4—348no—
186C2×C22⋊C4—348no—
187C2×C22⋊C4—344no—
188#95—688no—
189C17#1138111yescharacteristic, complemented, Sylow
190C24—172no—
191C24—174no—
192C24—174no—
193C24—174no—
194C24—682no—
195C24—682no—
196C24C22×#59511yescharacteristic
197C2×Q8—348no—
198C2×Q8—6816no—
199C2×Q8—684no—
200C2×Q8—684no—
201C22×C4—22no—
202C22×C4—24no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—44no—
206C22×C4—172no—
207C22×C4—174no—
208C22×C4—348no—
209C22×C4—342no—
210C22×C4—342no—
211C22×C4—342no—
212C22×C4—348no—
213C22×C4—348no—
214C22×C4—684no—
215C22×C4—6816no—
216C22×C4—682no—
217C22×C4—684no—
218C22×C4—684no—
219C22×C4—684no—
220C22×C4—688no—
221C22×C4—688no—
222C22×C4C22⋊#59512yes—
223C22×C4C22⋊#59514yes—
224C4⋊C4—28no—
225C4⋊C4—3416no—
226C22⋊C4—3416no—
227C22⋊C4—3416no—
228C22⋊C4—688no—
229C23—22no—
230C23—22no—
231C23—174no—
232C23—174no—
233C23—172no—
234C23—172no—
235C23—178no—
236C23—174no—
237C23—174no—
238C23—344no—
239C23—344no—
240C23—684no—
241C23—688no—
242C23—688no—
243C23—684no—
244C23Q8×#59514yes—
245C23C2×C22⋊#59512yes—
246C23C2×C22⋊#59511yescharacteristic
247Q8—6816no—
248C2×C4—28no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—24no—
252C2×C4—44no—
253C2×C4—44no—
254C2×C4—44no—
255C2×C4—344no—
256C2×C4—342no—
257C2×C4—344no—
258C2×C4—6816no—
259C2×C4—684no—
260C2×C4—684no—
261C2×C4—684no—
262C2×C4—684no—
263C2×C4—688no—
264C2×C4—6816no—
265C2×C4—1368no—
266C2×C4—1364no—
267C2×C4—1364no—
268C2×C4#1058753914yes—
269C22—28no—
270C22—22no—
271C22—22no—
272C22—174no—
273C22—174no—
274C22—348no—
275C22—344no—
276C22—344no—
277C22—344no—
278C22—344no—
279C22—684no—
280C22—688no—
281C22—684no—
282C22#1176375011yescharacteristic
283C22(C2×#595)⋊Q814yes—
284C22(C2×#595)⋊Q812yes—
285C22C2×#1058753912yes—
286C22#1176378912yes—
287C4—24no—
288C4—44no—
289C4—682no—
290C4—684no—
291C4—1368no—
292C2—24no—
293C2—344no—
294C2—344no—
295C2#1454397411yescharacteristic
296C2#1454461211yescharacteristic
297C2#1454461411yescharacteristic
298C2#1454497414yes—
subgroupG/H
centerC23C2×C22⋊#595
commutatorC24×C17C22×C4
FrattiniC24C22×#595
FittingC22×C4⋊C4×C17C4
socleC23×C17C2×C22⋊C4
prime radical, cosocleC24×D17C23
hypercenterC22×C4⋊C4#595
hypocenterC17#11381
LCSG ▷ C24×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
7. C23×Dic34C4semidirect
11. C22×C4⋊Dic17C4semidirect
13. C22×C4⋊C4×C17C4semidirect
29. C2×C22⋊#595Q8semidirect
151. C22×C4⋊C4#595semidirect
189. C17#11381semidirect

Maximal subgroups

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

subgroupindex
C2×#117636872
#145439582
C22×#105875832
#1138117