Finite Group Databasebeta

← #17956698

Subgroups — #17956698

Order 4352.

1510 conjugacy classes of subgroups; only the 228 normal subgroups are shown.

normal subgroups only

This group has 1510 conjugacy classes of subgroups — too many to draw. Only its 228 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
2C2×#11763687C214yesmaximal
3C22×#10587578C211yesmaximal, characteristic
4C22×C22⋊#595C2214yes—
5#11763752—24no—
6#11763752C418yescomplemented
7C2×(C2×#595)⋊C4—24no—
8#11763687—28no—
9C23×C4×D17C2211yescharacteristic
10C2×#10587578—28no—
11C2×#10587578—24no—
12C22×C4⋊Dic17C411yescharacteristic, complemented
13C22×#63227C2212yes—
14C22×C4×Dic17C412yescomplemented
15C22×C4⋊C4×C17C411yescharacteristic, complemented
16C24×D17C2311yescharacteristic
17C23×#595—24no—
18C23×C4×C17C2×C411yescharacteristic
19C23×C4×C17C2×C412yes—
20C23×Dic17C2×C411yescharacteristic
21C23×Dic17C2×C412yes—
22C22×C4×D17—24no—
23C22×C4×D17—24no—
24C22×C4×D17C2×C414yes—
25C22×C4×D17D412yes—
26C22×C4×D17D414yes—
27C22×C4×D17Q812yes—
28C2×C22⋊#595—28no—
29C2×C22⋊#595—28no—
30C2×C22⋊#595—24no—
31C2×C22⋊#595C2×C418yes—
32C2×C4⋊C4×C17—24no—
33C2×C4⋊C4×C17—22no—
34C2×#63227—24no—
35C2×#63227—24no—
36C2×#63227D418yes—
37C2×C4⋊Dic17—22no—
38C2×C4⋊Dic17—24no—
39C2×C4×Dic17—24no—
40C2×C4×Dic17—24no—
41C2×C4×Dic17—24no—
42#10587578—232no—
43(C2×#595)⋊C4—216no—
44C24×C17C22×C411yescharacteristic
45C23×D17—42no—
46C23×D17—42no—
47C23×D17C22×C412yes—
48C23×D17C2×D412yes—
49C23×D17C2×D411yescharacteristic
50C23×D17C2×D414yes—
51C23×D17C2×Q811yescharacteristic
52C23×D17#4114yes—
53C22×#595—28no—
54C22×#595—48no—
55C22×#595—48no—
56C22×#595—44no—
57C22×#595—44no—
58C22×C4×C17—22no—
59C22×C4×C17—22no—
60C22×C4×C17—22no—
61C22×C4×C17—22no—
62C22×C4×C17—44no—
63C22×C4×C17C4212yes—
64C22×C4×C17C22⋊C412yes—
65C22×C4×C17C22⋊C414yes—
66C22×C4×C17C4⋊C412yes—
67C22×Dic17—22no—
68C22×Dic17—22no—
69C22×Dic17—22no—
70C22×Dic17—22no—
71C22×Dic17—44no—
72C22×Dic17C4212yes—
73C22×Dic17C22⋊C412yes—
74C22×Dic17C22⋊C414yes—
75C22×Dic17C4⋊C412yes—
76C2×C4×D17—216no—
77C2×C4×D17—28no—
78C2×C4×D17—28no—
79C2×C4×D17—28no—
80C2×C4×D17—28no—
81C2×C4×D17C22⋊C418yes—
82C2×C4×D17C4⋊C418yes—
83C22⋊#595—216no—
84C22⋊#595—48no—
85C4⋊C4×C17—28no—
86#63227—216no—
87C4⋊Dic17—28no—
88C4×Dic17—216no—
89#10726—171nomaximal, Sylow
90C23×C17—22no—
91C23×C17—22no—
92C23×C17C2×C4211yescharacteristic
93C23×C17C2×C22⋊C412yes—
94C23×C17C2×C22⋊C411yescharacteristic
95C23×C17C2×C4⋊C411yescharacteristic
96C23×C17#11712yes—
97C22×D17—24no—
98C22×D17—24no—
99C22×D17—44no—
100C22×D17—42no—
101C22×D17—44no—
102C22×D17—44no—
103C22×D17—42no—
104C22×D17—48no—
105C22×D17C2×C22⋊C412yes—
106C22×D17C2×C4⋊C412yes—
107C22×D17C4×D418yes—
108C22×D17#12014yes—
109C22×D17#12114yes—
110C22×D17C22⋊Q814yes—
111C22×D17#12314yes—
112C2×#595—416no—
113C2×#595—48no—
114C2×#595—88no—
115C2×#595—84no—
116C2×#595—84no—
117C2×C4×C17—24no—
118C2×C4×C17—22no—
119C2×C4×C17—24no—
120C2×C4×C17—22no—
121C2×C4×C17—24no—
122C2×C4×C17—44no—
123C2×C4×C17—44no—
124C2×C4×C17—44no—
125C2×C4×C17#9514yes—
126C2×Dic17—24no—
127C2×Dic17—22no—
128C2×Dic17—24no—
129C2×Dic17—22no—
130C2×Dic17—24no—
131C2×Dic17—44no—
132C2×Dic17—44no—
133C2×Dic17—44no—
134C2×Dic17#9514yes—
135C4×D17—216no—
136C4×D17—216no—
137C22×#117—171no—
138C2×#381—172no—
139C2×#342—174no—
140C22×C17—24no—
141C22×C17—22no—
142C22×C17—22no—
143C22×C17—22no—
144C22×C17—22no—
145C22×C17C2×#9511yescharacteristic
146C22×C17C4×C22⋊C412yes—
147C22×C17#37911yescharacteristic
148C22×C17#38012yes—
149C22×C17#38111yescharacteristic
150C22×C17#41012yes—
151C22×C17#41012yes—
152C2×D17—28no—
153C2×D17—24no—
154C2×D17—24no—
155C2×D17—24no—
156C2×D17—24no—
157C2×D17—48no—
158C2×D17—44no—
159C2×D17—44no—
160C2×D17#38514yes—
161C2×D17#38614yes—
162#595—88no—
163C4×C17—24no—
164C4×C17—44no—
165Dic17—24no—
166Dic17—44no—
167C24×C4—171no—
168C2×#117—348no—
169C2×#117—344no—
170C22×C4⋊C4—171no—
171C22×C4⋊C4#59511yescharacteristic, complemented
172C22×C22⋊C4—172no—
173C22×C22⋊C4—174no—
174C22×C42—172no—
175#381—178no—
176#381—344no—
177C2×#95—344no—
178#342—348no—
179C2×C17—22no—
180C2×C17—22no—
181C2×C17C22⋊#9511yescharacteristic
182C2×C17#174012yes—
183C2×C17#178211yescharacteristic
184C2×C17#178211yescharacteristic
185C2×C17#182012yes—
186D17—24no—
187D17—24no—
188C25—171no—
189C23×C4—171no—
190C23×C4—174no—
191C23×C4—172no—
192C23×C4—172no—
193C23×C4—174no—
194C23×C4—172no—
195C23×C4—344no—
196C23×C4—344no—
197C23×C4—344no—
198C23×C4C2×#59512yes—
199C23×C4C2×#59511yescharacteristic
200#117—3432no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—344no—
204C2×C4⋊C4—342no—
205C2×C22⋊C4—178no—
206C2×C22⋊C4—178no—
207C2×C22⋊C4—344no—
208C2×C22⋊C4—344no—
209C2×C22⋊C4—348no—
210C2×C22⋊C4—348no—
211C2×C22⋊C4—344no—
212C2×C42—344no—
213C2×C42—344no—
214C2×C42—344no—
215#95—3416no—
216C17#1072611yescharacteristic, complemented, Sylow
217C24—172no—
218C24—172no—
219C24—171no—
220C24—171no—
221C24—174no—
222C24—174no—
223C24—682no—
224C24—682no—
225C24C22×#59511yescharacteristic
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—44no—
231C22×C4—172no—
232C22×C4—172no—
233C22×C4—172no—
234C22×C4—178no—
235C22×C4—178no—
236C22×C4—174no—
237C22×C4—348no—
238C22×C4—3416no—
239C22×C4—348no—
240C22×C4—348no—
241C22×C4—348no—
242C22×C4—342no—
243C22×C4—342no—
244C22×C4—342no—
245C22×C4—342no—
246C22×C4—348no—
247C22×C4—684no—
248C22×C4—684no—
249C22×C4—684no—
250C22×C4—688no—
251C22×C4—688no—
252C22×C4C4×#59512yes—
253C22×C4C4⋊#59512yes—
254C22×C4C22⋊#59512yes—
255C22×C4C22⋊#59514yes—
256C4⋊C4—28no—
257C4⋊C4—348no—
258C22⋊C4—3416no—
259C22⋊C4—3416no—
260C22⋊C4—688no—
261C42—3416no—
262C23—22no—
263C23—22no—
264C23—172no—
265C23—172no—
266C23—174no—
267C23—174no—
268C23—178no—
269C23—174no—
270C23—174no—
271C23—344no—
272C23—344no—
273C23—688no—
274C23—684no—
275C23—682no—
276C23—682no—
277C23—684no—
278C23—684no—
279C23C2×C4×#59511yescharacteristic
280C23C2×C4⋊#59511yescharacteristic
281C23#1058767712yes—
282C23C2×C22⋊#59512yes—
283C23C2×C22⋊#59511yescharacteristic
284C2×C4—24no—
285C2×C4—22no—
286C2×C4—24no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—174no—
293C2×C4—3416no—
294C2×C4—3416no—
295C2×C4—344no—
296C2×C4—342no—
297C2×C4—344no—
298C2×C4—342no—
299C2×C4—344no—
300C2×C4—684no—
301C2×C4—684no—
302C2×C4—684no—
303C2×C4—688no—
304C2×C4—6816no—
305C2×C4—1368no—
306C2×C4—1364no—
307C2×C4—1364no—
308C2×C4(C2×#595)⋊C414yes—
309C22—24no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22—174no—
315C22—174no—
316C22—348no—
317C22—344no—
318C22—344no—
319C22—344no—
320C22—344no—
321C22—688no—
322C22—684no—
323C22—684no—
324C22C2×(C2×#595)⋊C411yescharacteristic
325C22C4×C22⋊#59512yes—
326C22#1176375011yescharacteristic
327C22#1176375112yes—
328C22#1176375211yescharacteristic
329C22#1176378912yes—
330C22#1176379012yes—
331C4—24no—
332C4—44no—
333C4—344no—
334C4—684no—
335C4—1368no—
336C2—22no—
337C2—22no—
338C2—344no—
339C2—344no—
340C2#1454395811yescharacteristic
341C2#1454448412yes—
342C2#1454461411yescharacteristic
343C2#1454461511yescharacteristic
344C2#1454468212yes—
subgroupG/H
centerC23C2×C22⋊#595
commutatorC23×C17C2×C42
FrattiniC24C22×#595
FittingC22×C4⋊C4×C17C4
socleC23×C17C2×C22⋊C4
prime radical, cosocleC24×D17C23
hypercenterC22×C4⋊C4#595
hypocenterC17#10726
LCSG ▷ C23×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
6. #11763752C4semidirect
12. C22×C4⋊Dic17C4semidirect
14. C22×C4×Dic17C4semidirect
15. C22×C4⋊C4×C17C4semidirect
171. C22×C4⋊C4#595semidirect
216. C17#10726semidirect

Maximal subgroups

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

subgroupindex
C2×#117637522
C2×#117636872
C22×#105875782
#1072617