Finite Group Databasebeta

← #17963593

Subgroups — #17963593

Order 4352.

1998 conjugacy classes of subgroups; only the 320 normal subgroups are shown.

normal subgroups only

This group has 1998 conjugacy classes of subgroups — too many to draw. Only its 320 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×#11763752C212yesmaximal
2#14543973C218yesmaximal, complemented
3#14543958C214yesmaximal
4C22×C4⋊C4×D17C211yesmaximal, characteristic
5C22×C22⋊#595C2214yes—
6#11763752—28no—
7C2×(C2×#595)⋊C4C2218yes—
8C2×(C2×#595)⋊C4C22116yes—
9C23×C4×D17C2211yescharacteristic
10C23×C4×D17C2212yes—
11C22×C4⋊Dic17C411yescharacteristic, complemented
12C22×#63225C412yescomplemented
13C2×C4⋊C4×D17—28no—
14C2×C4⋊C4×D17C414yescomplemented
15C2×C4⋊C4×D17C2214yes—
16C22×C4⋊C4×C17C411yescharacteristic, complemented
17C24×D17C2311yescharacteristic
18C23×#595C2318yes—
19C23×C4×C17C2×C411yescharacteristic
20C23×C4×C17C2×C412yes—
21C23×Dic17C2×C411yescharacteristic
22C23×Dic17C2×C412yes—
23C22×C4×D17—28no—
24C22×C4×D17—28no—
25C22×C4×D17—48no—
26C22×C4×D17C2×C412yes—
27C22×C4×D17C2×C414yes—
28C22×C4×D17D418yes—
29C22×C4×D17C2314yes—
30C22×C4×D17C2312yes—
31C2×C22⋊#595—28no—
32C2×C22⋊#595—216no—
33C2×C4⋊C4×C17—24no—
34C2×C4⋊C4×C17C2×C414yes—
35C2×C4⋊Dic17—24no—
36C2×C4⋊Dic17C2×C414yes—
37C2×#63225—28no—
38C2×#63225C2×C418yes—
39C4⋊C4×D17—216no—
40C4⋊C4×D17—216no—
41(C2×#595)⋊C4—216no—
42(C2×#595)⋊C4—416no—
43C24×C17C22×C411yescharacteristic
44C23×D17—24no—
45C23×D17—24no—
46C23×D17C22×C411yescharacteristic
47C23×D17C2×D414yes—
48C23×D17#4118yes—
49C23×D17C2411yescharacteristic
50C22×#595—28no—
51C22×#595—216no—
52C22×#595—216no—
53C22×#595—216no—
54C22×C4×C17—24no—
55C22×C4×C17—24no—
56C22×C4×C17—44no—
57C22×C4×C17C22⋊C414yes—
58C22×C4×C17C22×C414yes—
59C22×C4×C17C22×C412yes—
60C22×Dic17—24no—
61C22×Dic17—24no—
62C22×Dic17—44no—
63C22×Dic17C22⋊C414yes—
64C22×Dic17C22×C414yes—
65C22×Dic17C22×C412yes—
66C2×C4×D17—28no—
67C2×C4×D17—232no—
68C2×C4×D17—28no—
69C2×C4×D17—28no—
70C2×C4×D17—28no—
71C2×C4×D17—216no—
72C2×C4×D17—48no—
73C2×C4×D17—432no—
74C2×C4×D17C22⋊C418yes—
75C2×C4×D17C2×D418yes—
76C22⋊#595—416no—
77C4⋊C4×C17—28no—
78C4⋊Dic17—28no—
79#63225—216no—
80#14342—171nomaximal, Sylow
81C23×C17—22no—
82C23×C17—22no—
83C23×C17C2×C22⋊C412yes—
84C23×C17#11714yes—
85C23×C17C23×C411yescharacteristic
86C22×D17—22no—
87C22×D17—22no—
88C22×D17—28no—
89C22×D17—28no—
90C22×D17—232no—
91C22×D17C2×C22⋊C412yes—
92C22×D17#11714yes—
93C22×D17#123116yes—
94C22×D17C22×D412yes—
95C22×D17C2×#4114yes—
96C22×D17H(5,𝔽2)14yes—
97C22×D172−1+414yes—
98C2×#595—48no—
99C2×#595—48no—
100C2×#595—48no—
101C2×#595—432no—
102C2×C4×C17—24no—
103C2×C4×C17—28no—
104C2×C4×C17—24no—
105C2×C4×C17—28no—
106C2×C4×C17—48no—
107C2×C4×C17C2×C22⋊C414yes—
108C2×Dic17—24no—
109C2×Dic17—28no—
110C2×Dic17—24no—
111C2×Dic17—28no—
112C2×Dic17—48no—
113C2×Dic17C2×C22⋊C414yes—
114C4×D17—216no—
115C4×D17—216no—
116C4×D17—416no—
117C4×D17—416no—
118C23×C4⋊C4—171no—
119C2×#381—172no—
120#1445—178no—
121C22⋊#95—174no—
122C22×C17—22no—
123C22×C17—22no—
124C22×C17—28no—
125C22×C17#38114yes—
126C22×C17C22×C22⋊C411yescharacteristic
127C22×C17C2×#11712yes—
128C22×C17#51812yes—
129C22×C17#51912yes—
130C2×D17—22no—
131C2×D17—22no—
132C2×D17—28no—
133C2×D17—232no—
134C2×D17#38114yes—
135C2×D17C2×#12314yes—
136C2×D17#53514yes—
137C2×D17#53614yes—
138C2×D17#539116yes—
139#595—88no—
140C4×C17—24no—
141C4×C17—44no—
142Dic17—24no—
143Dic17—44no—
144C24×C4—171no—
145C24×C4—172no—
146C22×C4⋊C4—172no—
147C22×C4⋊C4—171no—
148C22×C4⋊C4—174no—
149C22×C4⋊C4—174no—
150C22×C4⋊C4—348no—
151C22×C4⋊C4#59511yescharacteristic, complemented
152C22×C22⋊C4—174no—
153#381—348no—
154C2×#95—178no—
155C2×#95—1716no—
156C2×C17—24no—
157C2×C17C2×#38111yescharacteristic
158C2×C17#230911yescharacteristic
159C2×C17#231011yescharacteristic
160C2×C17#231314yes—
161D17—24no—
162D17#262318yes—
163C25—171no—
164C23×C4—171no—
165C23×C4—178no—
166C23×C4—172no—
167C23×C4—172no—
168C23×C4—172no—
169C23×C4—174no—
170C23×C4—174no—
171C23×C4—178no—
172C23×C4—348no—
173C23×C4—348no—
174C23×C4—688no—
175C23×C4C2×#59511yescharacteristic
176C23×C4C2×#59512yes—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—178no—
179C2×C4⋊C4—174no—
180C2×C4⋊C4—3416no—
181C2×C4⋊C4—3416no—
182C2×C4⋊C4—348no—
183C2×C4⋊C4—344no—
184C2×C4⋊C4C2×#59514yes—
185C2×C22⋊C4—3416no—
186C2×C22⋊C4—348no—
187#95—3416no—
188#95—6816no—
189C17#1434211yescharacteristic, complemented, Sylow
190C24—171no—
191C24—178no—
192C24—174no—
193C24—171no—
194C24—344no—
195C24—344no—
196C24C22×#59511yescharacteristic
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—44no—
200C22×C4—174no—
201C22×C4—172no—
202C22×C4—178no—
203C22×C4—178no—
204C22×C4—174no—
205C22×C4—3432no—
206C22×C4—348no—
207C22×C4—348no—
208C22×C4—3416no—
209C22×C4—348no—
210C22×C4—348no—
211C22×C4—344no—
212C22×C4—344no—
213C22×C4—348no—
214C22×C4—3416no—
215C22×C4—3416no—
216C22×C4—3416no—
217C22×C4—688no—
218C22×C4—6832no—
219C22×C4—684no—
220C22×C4C22⋊#59514yes—
221C22×C4C22×#59514yes—
222C22×C4C22×#59512yes—
223C4⋊C4—28no—
224C4⋊C4—3416no—
225C4⋊C4—348no—
226C22⋊C4—6816no—
227C23—22no—
228C23—22no—
229C23—172no—
230C23—174no—
231C23—174no—
232C23—174no—
233C23—1716no—
234C23—172no—
235C23—174no—
236C23—348no—
237C23—348no—
238C23—3432no—
239C23—342no—
240C23—342no—
241C23#1058767714yes—
242C23C2×C22⋊#59512yes—
243C23C23×#59511yescharacteristic
244C2×C4—28no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—28no—
248C2×C4—48no—
249C2×C4—174no—
250C2×C4—3416no—
251C2×C4—3416no—
252C2×C4—348no—
253C2×C4—344no—
254C2×C4—344no—
255C2×C4—348no—
256C2×C4—6816no—
257C2×C4—6816no—
258C2×C4—688no—
259C2×C4—688no—
260C2×C4—688no—
261C2×C4—688no—
262C2×C4—6832no—
263C2×C4C2×C22⋊#59514yes—
264C22—22no—
265C22—22no—
266C22—28no—
267C22—174no—
268C22—174no—
269C22—174no—
270C22—1716no—
271C22—174no—
272C22—3432no—
273C22—342no—
274C22—342no—
275C22—348no—
276C22#1176375214yes—
277C22C22×C22⋊#59511yescharacteristic
278C22C2×#1058767712yes—
279C22#1176387612yes—
280C22#1176387712yes—
281C4—24no—
282C4—44no—
283C4—344no—
284C4—684no—
285C4—1368no—
286C2—24no—
287C2—178no—
288C2—344no—
289C2C2×#1176375211yescharacteristic
290C2#1454525711yescharacteristic
291C2#1454525811yescharacteristic
292C2#1454526114yes—
subgroupG/H
center, FrattiniC23C23×#595
commutator, socleC23×C17C23×C4
FittingC22×C4⋊C4×C17C4
prime radicalC24×D17C23
cosocleC23×D17C24
hypercenterC22×C4⋊C4#595
hypocenterC17#14342
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #14543973C2semidirect
11. C22×C4⋊Dic17C4semidirect
12. C22×#63225C4semidirect
14. C2×C4⋊C4×D17C4semidirect
16. C22×C4⋊C4×C17C4semidirect
151. C22×C4⋊C4#595semidirect
189. C17#14342semidirect

Maximal subgroups

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

subgroupindex
C2×#117637522
#145439732
#145439582
C22×C4⋊C4×D172
#1434217