Finite Group Databasebeta

← #17956675

Subgroups — #17956675

Order 4352.

1374 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 1374 conjugacy classes of subgroups — too many to draw. Only its 216 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#14545228C212yesmaximal
2C2×#11763677C214yesmaximal, complemented
3C22×M16×D17C211yesmaximal, characteristic
4C2×#10587677C2214yes—
5#11763752—22no—
6#11763751—24no—
7#11763750—22no—
8C4×C22⋊#595C418yescomplemented
9#11763677—28no—
10C23×C4×D17C2211yescharacteristic
11C22×#63223C411yescharacteristic, complemented
12C22×#63218C2212yes—
13C2×M16×D17—28no—
14C2×M16×D17—24no—
15C22×C8×D17C412yescomplemented
16C22×M16×C17C411yescharacteristic, complemented
17C24×D17D411yescharacteristic
18C23×C4×C17C2×C411yescharacteristic
19C23×Dic17Q811yescharacteristic
20C22×C4×D17—24no—
21C22×C4×D17—24no—
22C22×C4×D17C2×C411yescharacteristic
23C22×C4×D17C2×C414yes—
24C22×C4×D17D411yescharacteristic
25C22×C4×D17D414yes—
26C22×C4×D17Q811yescharacteristic
27C22×C4×D17C2311yescharacteristic
28C2×C22⋊#595—28no—
29C2×C22⋊#595—24no—
30#10587677—28no—
31#10587677—28no—
32C2×C4⋊#595—24no—
33C2×C4×#595C2×C418yes—
34C2×M16×C17—22no—
35C2×M16×C17—24no—
36C22×C8×C17C2×C412yes—
37C2×#63223—24no—
38C2×#63223—22no—
39C22×#3622C2×C412yes—
40M16×D17—232no—
41C2×#63218—24no—
42C2×#63218—24no—
43C2×#63218D418yes—
44C2×C8×D17—24no—
45C2×C8×D17—24no—
46C2×C8×D17—24no—
47(C2×#595)⋊C4—28no—
48C24×C17C4⋊C411yescharacteristic
49C23×D17—24no—
50C23×D17—44no—
51C23×D17C22⋊C414yes—
52C23×D17C4⋊C411yescharacteristic
53C23×D17C2×D411yescharacteristic
54C22×#595—28no—
55C22×C4×C17—22no—
56C22×C4×C17—22no—
57C22×C4×C17C4212yes—
58C22×C4×C17C22⋊C412yes—
59C22×C4×C17C4⋊C411yescharacteristic
60C22×C4×C17C22×C411yescharacteristic
61C22×Dic17—24no—
62C22×Dic17C4⋊C411yescharacteristic
63C22×Dic17C4⋊C414yes—
64C22×Dic17C2×Q811yescharacteristic
65C2×C4×D17—24no—
66C2×C4×D17—22no—
67C2×C4×D17—22no—
68C2×C4×D17—24no—
69C2×C4×D17—216no—
70C2×C4×D17—28no—
71C2×C4×D17—28no—
72C2×C4×D17C4212yes—
73C2×C4×D17C22⋊C414yes—
74C2×C4×D17C22⋊C412yes—
75C2×C4×D17C4⋊C414yes—
76C2×C4×D17C22×C412yes—
77C2×C4×D17C2×D414yes—
78C2×C4×D17C2×D412yes—
79C2×C4×D17#4114yes—
80C22⋊#595—416no—
81C22⋊#595—48no—
82C4⋊#595—44no—
83C4⋊#595—44no—
84C4×#595—216no—
85M16×C17—28no—
86C2×C8×C17—22no—
87C2×C8×C17—22no—
88C2×C8×C17C22⋊C414yescomplemented
89#63223—28no—
90C2×#3622—22no—
91C2×#3622—22no—
92C2×#3622C22⋊C414yescomplemented
93#63218—216no—
94C8×D17—216no—
95#10717—171nomaximal, Sylow
96C23×C17—22no—
97C23×C17—42no—
98C23×C17#9512yes—
99C23×C17C2×C4⋊C411yescharacteristic
100C22×D17—22no—
101C22×D17—24no—
102C22×D17—28no—
103C22×D17—28no—
104C22×D17—44no—
105C22×D17—42no—
106C22×D17—416no—
107C22×D17#9512yes—
108C22×D17C2×C22⋊C412yes—
109C22×D17#12114yes—
110C2×#595—416no—
111C2×#595—48no—
112C2×C4×C17—22no—
113C2×C4×C17—24no—
114C2×C4×C17—22no—
115C2×C4×C17—22no—
116C2×C4×C17#9512yes—
117C2×C4×C17C2×C4211yescharacteristic
118C2×C4×C17C2×C22⋊C412yes—
119C2×C4×C17C2×C22⋊C411yescharacteristic
120C2×C4×C17#11712yes—
121C2×Dic17—22no—
122C2×Dic17—28no—
123C2×Dic17#9512yes—
124C2×Dic17C2×C4⋊C412yes—
125C2×Dic17C22⋊Q814yes—
126C4×D17—28no—
127C4×D17—24no—
128C4×D17—22no—
129C4×D17—22no—
130C4×D17—28no—
131C4×D17C4×D418yes—
132C4×D17#12014yes—
133C4×D17#12314yes—
134C8×C17—24no—
135#3622—24no—
136C23×M16—171no—
137#2280—172no—
138C2×H(3,ℤ/4)—174no—
139C22×C17—22no—
140C22×C17—22no—
141C22×C17—24no—
142C22×C17—44no—
143C22×C17C2×#9511yescharacteristic
144C22×C17#38012yes—
145C2×D17—28no—
146C2×D17—22no—
147C2×D17—24no—
148C2×D17—22no—
149C2×D17—44no—
150C2×D17—416no—
151C2×D17H(3,ℤ/4)14yescomplemented
152C2×D17#38614yes—
153#595—88no—
154C4×C17—22no—
155C4×C17C4×C22⋊C412yes—
156C4×C17#37911yescharacteristic
157C4×C17#38111yescharacteristic
158Dic17—22no—
159Dic17#38514yes—
160C24×C4—171no—
161C22×M16—171no—
162C22×M16—172no—
163C22×M16—348no—
164C22×M16—344no—
165C22×M16#59511yescharacteristic, complemented
166C23×C8—172no—
167C2×#117—174no—
168#381—342no—
169#380—344no—
170#379—342no—
171C4×C22⋊C4—178no—
172H(3,ℤ/4)—348no—
173C2×C17—22no—
174C2×C17—42no—
175C2×C17C22⋊#9511yescharacteristic
176C2×C17C2×H(3,ℤ/4)12yes—
177D17—24no—
178D17—24no—
179C25—171no—
180C23×C4—171no—
181C23×C4—171no—
182C23×C4—174no—
183C23×C4—171no—
184C23×C4—171no—
185C23×C4—174no—
186C23×C4—171no—
187C23×C4—344no—
188C23×C4—344no—
189C23×C4C2×#59511yescharacteristic
190C2×M16—22no—
191C2×M16—24no—
192C2×M16—178no—
193C2×M16—344no—
194C2×M16—344no—
195C2×M16—344no—
196C2×M16—3432no—
197C2×M16—342no—
198C22×C8—172no—
199C22×C8—344no—
200C22×C8—344no—
201C22×C8—344no—
202C22×C8C2×#59512yes—
203#117—348no—
204#117—348no—
205C2×C4⋊C4—344no—
206C2×C22⋊C4—344no—
207C2×C22⋊C4—348no—
208C2×C42—178no—
209#95—348no—
210C17#1071711yescharacteristic, complemented, Sylow
211C24—171no—
212C24—171no—
213C24—174no—
214C24—344no—
215C24—684no—
216C24C4⋊#59511yescharacteristic
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—171no—
220C22×C4—172no—
221C22×C4—174no—
222C22×C4—172no—
223C22×C4—174no—
224C22×C4—171no—
225C22×C4—174no—
226C22×C4—172no—
227C22×C4—172no—
228C22×C4—174no—
229C22×C4—174no—
230C22×C4—348no—
231C22×C4—344no—
232C22×C4—3416no—
233C22×C4—348no—
234C22×C4—344no—
235C22×C4—344no—
236C22×C4—342no—
237C22×C4—342no—
238C22×C4—348no—
239C22×C4C4×#59512yes—
240C22×C4C4⋊#59511yescharacteristic
241C22×C4C22⋊#59512yes—
242C22×C4C22×#59511yescharacteristic
243M16—28no—
244M16—3416no—
245M16—348no—
246C2×C8—22no—
247C2×C8—22no—
248C2×C8—174no—
249C2×C8—3416no—
250C2×C8—342no—
251C2×C8—342no—
252C2×C8C22⋊#59514yescomplemented
253C4⋊C4—684no—
254C4⋊C4—684no—
255C22⋊C4—688no—
256C22⋊C4—6816no—
257C42—3416no—
258C23—22no—
259C23—42no—
260C23—172no—
261C23—172no—
262C23—174no—
263C23—344no—
264C23—348no—
265C23—348no—
266C23—342no—
267C23—682no—
268C23—684no—
269C23—6816no—
270C23(C2×#595)⋊C412yes—
271C23C2×C4⋊#59511yescharacteristic
272C2×C4—22no—
273C2×C4—24no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—172no—
277C2×C4—174no—
278C2×C4—178no—
279C2×C4—174no—
280C2×C4—174no—
281C2×C4—172no—
282C2×C4—348no—
283C2×C4—348no—
284C2×C4—348no—
285C2×C4—342no—
286C2×C4—344no—
287C2×C4—342no—
288C2×C4—342no—
289C2×C4—688no—
290C2×C4—6816no—
291C2×C4(C2×#595)⋊C412yes—
292C2×C4C2×C4×#59511yescharacteristic
293C2×C4#1058767712yes—
294C2×C4C2×C22⋊#59512yes—
295C2×C4C2×C22⋊#59511yescharacteristic
296C8—24no—
297C8—344no—
298C22—22no—
299C22—24no—
300C22—22no—
301C22—44no—
302C22—174no—
303C22—174no—
304C22—348no—
305C22—344no—
306C22—342no—
307C22—342no—
308C22—684no—
309C22—6816no—
310C22C2×(C2×#595)⋊C411yescharacteristic
311C22#1176375112yes—
312C4—22no—
313C4—174no—
314C4—342no—
315C4—1368no—
316C4C4×C22⋊#59512yes—
317C4#1176375011yescharacteristic
318C4#1176375211yescharacteristic
319C2—22no—
320C2—42no—
321C2—344no—
322C2—344no—
323C2#1454395811yescharacteristic
324C2#1454445612yes—
subgroupG/H
centerC2×C4C2×C4×#595
commutatorC2×C4×C17C2×C42
FrattiniC22×C4C22×#595
FittingC22×M16×C17C4
socleC22×C17C2×#95
prime radicalC24×D17D4
cosocleC22×C4×D17C23
hypercenterC22×M16#595
hypocenterC17#10717
LCSG ▷ C2×C4×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
2. C2×#11763677C2semidirect
8. C4×C22⋊#595C4semidirect
11. C22×#63223C4semidirect
15. C22×C8×D17C4semidirect
16. C22×M16×C17C4semidirect
88. C2×C8×C17C22⋊C4semidirect
92. C2×#3622C22⋊C4semidirect
151. C2×D17H(3,ℤ/4)semidirect
165. C22×M16#595semidirect
210. C17#10717semidirect
252. C2×C8C22⋊#595semidirect

Maximal subgroups

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

subgroupindex
#145452282
C2×#117636772
C22×M16×D172
#1071717