Finite Group Databasebeta

← #17972194

Subgroups — #17972194

Order 4352.

1050 conjugacy classes of subgroups; only the 220 normal subgroups are shown.

normal subgroups only

This group has 1050 conjugacy classes of subgroups — too many to draw. Only its 220 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#14545231C212yesmaximal
2#14544975C218yesmaximal
3#14544682C214yesmaximal
4C2×#11763365C211yesmaximal, characteristic
5#11763754—22no—
6#11763753—22no—
7#11763752C2214yes—
8#11763751C2214yes—
9C4×C4⋊#595—22no—
10C4×C22⋊#595C2214yes—
11#11763745—22no—
12#11763687C22116yes—
13#11763365—22no—
14#11763365—22no—
15#11763365—22no—
16#11763365—22no—
17C2×#10587574C412yescomplemented
18C2×#10587556C411yescharacteristic, complemented
19C2×#10587582C2214yes—
20C2×#10587573C414yescomplemented
21C2×#10587578C2212yes—
22C2×C42×D17C2211yescharacteristic
23C2×C17×#125C411yescharacteristic, complemented
24C22×C4×D17C2311yescharacteristic
25C22×C4×D17C2312yes—
26C22×Dic34C2×C414yes—
27C2×C22⋊#595C2318yes—
28C2×C4⋊#595—24no—
29C2×C4×#595—24no—
30C17×#125—21no—
31C17×#125—22no—
32C17×#125—21no—
33C2×C4⋊C4×C17C2×C412yes—
34C2×C4⋊C4×C17C2×C414yes—
35C2×C42×C17C2×C411yescharacteristic
36C2×#63227C2314yes—
37C2×C4⋊Dic17C2×C412yes—
38C2×#63225C2×C414yes—
39C2×#63225C2×C414yes—
40C2×C4×Dic17C2×C414yes—
41C2×C4×Dic17C2×C412yes—
42C2×C4×Dic17C2×C411yescharacteristic
43#10587582—24no—
44#10587582—24no—
45#10587582—24no—
46#10587582—24no—
47#10587578—24no—
48#10587578—24no—
49#10587574—22no—
50#10587574—22no—
51#10587574—22no—
52#10587574—22no—
53#10587573—24no—
54#10587573—24no—
55#10587573—24no—
56#10587573—24no—
57C42×D17D412yes—
58C42×D17D412yes—
59C42×D17D412yes—
60C42×D17D412yes—
61#10587556—21no—
62#10587556—22no—
63#10587556—21no—
64(C2×#595)⋊C4—24no—
65(C2×#595)⋊C4—24no—
66(C2×#595)⋊C4—24no—
67(C2×#595)⋊C4—24no—
68C23×D17C2411yescharacteristic
69C22×#595—28no—
70C22×C4×C17C22×C411yescharacteristic
71C22×C4×C17C22×C414yes—
72C22×C4×C17C22×C412yes—
73C22×Dic17C22×C412yes—
74C22×Dic17C22×C411yescharacteristic
75C22×Dic17C22×C414yes—
76C2×C4×D17—24no—
77C2×C4×D17—44no—
78C2×C4×D17C2×D412yes—
79C2×C4×D17C2×D412yes—
80C2×C4×D17C2×D412yes—
81C2×C4×D17C2×D412yes—
82C2×C4×D17C2×D412yes—
83C2×C4×D17C2×D412yes—
84C2×Dic34—24no—
85C2×Dic34—24no—
86C2×Dic34—48no—
87C22⋊#595—216no—
88C4⋊#595—48no—
89C4×#595—48no—
90C4⋊C4×C17—24no—
91C4⋊C4×C17—24no—
92C4⋊C4×C17—22no—
93C4⋊C4×C17—22no—
94C42×C17C22⋊C411yescharacteristic
95C42×C17C22⋊C412yes—
96C42×C17C22⋊C411yescharacteristic
97#63227—28no—
98C4⋊Dic17—22no—
99C4⋊Dic17—22no—
100#63225—24no—
101#63225—24no—
102#63225—24no—
103#63225—24no—
104C4×Dic17—28no—
105C4×Dic17C22⋊C412yes—
106C4×Dic17C22⋊C412yes—
107C4×Dic17C22⋊C412yes—
108C4×Dic17C22⋊C412yes—
109C4×Dic17C22⋊C411yescharacteristic
110C4×Dic17C22⋊C412yes—
111C4×Dic17C22⋊C411yescharacteristic
112#21826—171nomaximal, Sylow
113C23×C17C23×C411yescharacteristic
114C22×D17—42no—
115C22×D17C22×D412yes—
116C22×D172−1+414yes—
117C2×#595—48no—
118C2×#595—48no—
119C2×#595—48no—
120C2×C4×C17—22no—
121C2×C4×C17—24no—
122C2×C4×C17—42no—
123C2×C4×C17—44no—
124C2×C4×C17C2×C22⋊C412yes—
125C2×C4×C17C2×C22⋊C412yes—
126C2×C4×C17C2×C22⋊C411yescharacteristic
127C2×C4×C17C2×C22⋊C411yescharacteristic
128C2×Dic17—22no—
129C2×Dic17—24no—
130C2×Dic17—24no—
131C2×Dic17—44no—
132C2×Dic17C2×C22⋊C412yes—
133C2×Dic17C2×C22⋊C412yes—
134C2×Dic17C2×C22⋊C412yes—
135C4×D17—24no—
136C4×D17—24no—
137C4×D17—24no—
138C4×D17—24no—
139C4×D17—42no—
140C4×D17—42no—
141Dic34—48no—
142Dic34—48no—
143C2×#533—171no—
144#2283—172no—
145#1996—178no—
146#1820—174no—
147C22×C17—22no—
148C22×C17C22×C22⋊C411yescharacteristic
149C22×C17#51912yes—
150C2×D17—22no—
151C2×D17—22no—
152C2×D17—44no—
153C2×D17#53614yes—
154#595—88no—
155C4×C17—21no—
156C4×C17—21no—
157C4×C17—22no—
158C4×C17—22no—
159C4×C17—44no—
160Dic17—22no—
161Dic17—22no—
162Dic17—44no—
163Dic17#41014yes—
164#533—342no—
165#533—342no—
166#533—342no—
167#533—342no—
168C2×C4⋊Q8—172no—
169C2×#125—171no—
170C2×#125#59511yescharacteristic, complemented
171C2×C22⋊Q8—174no—
172C2×C4×Q8—174no—
173C2×#117—172no—
174C22×C42—171no—
175#383—342no—
176#382—342no—
177#381—174no—
178#380—174no—
179C4×C4⋊C4—342no—
180C4×C22⋊C4—174no—
181#376—342no—
182#342—1716no—
183C2×C17—22no—
184C2×C17#231011yescharacteristic
185C2×C17C2×#41012yes—
186D17—24no—
187C22×Q8—174no—
188C23×C4—171no—
189C23×C4—172no—
190C4⋊Q8—342no—
191C4⋊Q8—342no—
192C4⋊Q8—342no—
193C4⋊Q8—342no—
194#125—21no—
195#125—22no—
196#125—21no—
197#125—341no—
198#125—342no—
199#125—341no—
200C22⋊Q8—344no—
201C22⋊Q8—344no—
202C22⋊Q8—344no—
203C22⋊Q8—344no—
204C4×Q8—344no—
205C4×Q8—344no—
206C4×Q8—344no—
207C4×Q8—344no—
208#117—344no—
209#117—344no—
210C2×C4⋊C4—172no—
211C2×C4⋊C4—174no—
212C2×C4⋊C4—174no—
213C2×C4⋊C4—344no—
214C2×C4⋊C4C2×#59512yes—
215C2×C4⋊C4C2×#59514yes—
216C2×C22⋊C4—174no—
217C2×C22⋊C4—178no—
218C2×C42—172no—
219C2×C42—171no—
220C2×C42—172no—
221C2×C42—172no—
222C2×C42—172no—
223C2×C42—172no—
224C2×C42—174no—
225C2×C42—344no—
226C2×C42C2×#59511yescharacteristic
227#95—344no—
228#95—344no—
229#95—344no—
230#95—344no—
231C17#2182611yescharacteristic, complemented, Sylow
232C24—171no—
233C2×Q8—344no—
234C2×Q8—344no—
235C2×Q8—688no—
236C22×C4—171no—
237C22×C4—172no—
238C22×C4—172no—
239C22×C4—172no—
240C22×C4—172no—
241C22×C4—172no—
242C22×C4—172no—
243C22×C4—172no—
244C22×C4—174no—
245C22×C4—344no—
246C22×C4—348no—
247C22×C4—684no—
248C22×C4C22×#59511yescharacteristic
249C22×C4C22×#59514yes—
250C22×C4C22×#59512yes—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—344no—
256C4⋊C4—344no—
257C4⋊C4—342no—
258C4⋊C4—342no—
259C4⋊C4—344no—
260C4⋊C4—344no—
261C4⋊C4—688no—
262C22⋊C4—348no—
263C22⋊C4—3416no—
264C42—172no—
265C42—172no—
266C42—172no—
267C42—172no—
268C42—171no—
269C42—172no—
270C42—171no—
271C42—348no—
272C42—688no—
273C42C22⋊#59511yescharacteristic
274C42C22⋊#59512yes—
275C42C22⋊#59511yescharacteristic
276C23—172no—
277C23—174no—
278C23—682no—
279C23C23×#59511yescharacteristic
280Q8—688no—
281Q8—688no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—172no—
287C2×C4—172no—
288C2×C4—172no—
289C2×C4—344no—
290C2×C4—344no—
291C2×C4—344no—
292C2×C4—344no—
293C2×C4—344no—
294C2×C4—342no—
295C2×C4—344no—
296C2×C4—682no—
297C2×C4—682no—
298C2×C4—684no—
299C2×C4—688no—
300C2×C4—688no—
301C2×C4—688no—
302C2×C4C2×C22⋊#59511yescharacteristic
303C2×C4C2×C22⋊#59511yescharacteristic
304C2×C4C2×C22⋊#59512yes—
305C2×C4C2×C22⋊#59512yes—
306C22—22no—
307C22—174no—
308C22—342no—
309C22—342no—
310C22—684no—
311C22C22×C22⋊#59511yescharacteristic
312C22#1176387712yes—
313C4—21no—
314C4—21no—
315C4—22no—
316C4—22no—
317C4—44no—
318C4—174no—
319C4—342no—
320C4—342no—
321C4—684no—
322C4—1368no—
323C2—22no—
324C2—344no—
325C2#1454525811yescharacteristic
326C2D17⋊#41012yes—
subgroupG/H
centerC22C22×C22⋊#595
commutatorC23×C17C23×C4
FrattiniC23C23×#595
FittingC2×C17×#125C4
socleC22×C17C22×C22⋊C4
prime radical, cosocleC23×D17C24
hypercenterC2×#125#595
hypocenterC17#21826
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
17. C2×#10587574C4semidirect
18. C2×#10587556C4semidirect
20. C2×#10587573C4semidirect
23. C2×C17×#125C4semidirect
170. C2×#125#595semidirect
231. C17#21826semidirect

Maximal subgroups

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

subgroupindex
#145452312
#145449752
#145446822
C2×#117633652
#2182617