Finite Group Databasebeta

← #17953106

Subgroups — #17953106

Order 4352.

1026 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 1026 conjugacy classes of subgroups — too many to draw. Only its 116 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#14544618C212yesmaximal
2#14544034C212yesmaximal
3#14544032C212yesmaximal
4C22×#10587574C211yesmaximal, characteristic
5C2×#10587545—24no—
6#11763779C2214yes—
7#11763655—24no—
8#11763654—24no—
9C23×Dic34C411yescharacteristic
10C23×Dic34C2211yescharacteristic
11C2×#10587574—24no—
12C2×#10587574—24no—
13C2×#10587574—24no—
14C22×C4⋊Dic17C2211yescharacteristic
15C22×#63225C411yescharacteristic
16C22×#63225C2211yescharacteristic
17C22×C4×Dic17C411yescharacteristic
18C22×C4⋊C4×C17C411yescharacteristic
19C23×C4×C17C2×C411yescharacteristic
20C23×C4×C17C2×C411yescharacteristic
21C23×C4×C17C2×C411yescharacteristic
22C23×Dic17C2×C411yescharacteristic
23C23×Dic17C2×C411yescharacteristic
24C23×Dic17C2×C411yescharacteristic
25C23×Dic17C2311yescharacteristic
26C22×Dic34—22no—
27C22×Dic34—22no—
28C22×Dic34—22no—
29C22×Dic34—22no—
30C22×Dic34—22no—
31C22×Dic34—44no—
32C22×Dic34—44no—
33C22×Dic34D414yes—
34C2×#63247—24no—
35C2×C4⋊C4×C17—22no—
36C2×C4⋊C4×C17—22no—
37C2×C4⋊C4×C17—22no—
38C2×C4⋊Dic17—22no—
39C2×C4⋊Dic17—22no—
40C2×C4⋊Dic17D414yes—
41C2×#63225—22no—
42C2×#63225—22no—
43C2×#63225—22no—
44C2×#63225—22no—
45C2×#63225—22no—
46C2×#63225D414yes—
47C2×C4×Dic17—22no—
48C2×C4×Dic17—22no—
49C2×C4×Dic17—22no—
50#10587574—232no—
51C22⋊#3624—24no—
52C22⋊#3624—24no—
53#10587545—48no—
54C24×C17C22×C411yescharacteristic
55C22×C4×C17—21no—
56C22×C4×C17—21no—
57C22×C4×C17—24no—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—21no—
61C22×C4×C17—21no—
62C22×C4×C17—42no—
63C22×C4×C17—42no—
64C22×C4×C17C22⋊C412yes—
65C22×C4×C17C22⋊C412yes—
66C22×C4×C17C22⋊C412yes—
67C22×Dic17—21no—
68C22×Dic17—21no—
69C22×Dic17—21no—
70C22×Dic17—21no—
71C22×Dic17—24no—
72C22×Dic17—24no—
73C22×Dic17—21no—
74C22×Dic17—21no—
75C22×Dic17—42no—
76C22×Dic17—42no—
77C22×Dic17C22⋊C412yes—
78C22×Dic17C22⋊C412yes—
79C22×Dic17C22⋊C412yes—
80C22×Dic17C2×D412yes—
81C22×Dic17C2×D412yes—
82C22×Dic17C2×D412yes—
83C2×Dic34—28no—
84C2×Dic34—28no—
85C2×Dic34—28no—
86C2×Dic34—44no—
87C2×Dic34—48no—
88C2×Dic34—48no—
89C2×Dic34—48no—
90C2×Dic34—48no—
91C2×Dic34—44no—
92C2×Dic34—44no—
93#63247—48no—
94C2×#3624—44no—
95C4⋊C4×C17—28no—
96C4⋊Dic17—28no—
97#63225—28no—
98#63225—28no—
99C4×Dic17—28no—
100#8551—171nomaximal, Sylow
101C23×C17—22no—
102C23×C17—22no—
103C23×C17—22no—
104C23×C17C2×C22⋊C411yescharacteristic
105C23×C17C2×C22⋊C411yescharacteristic
106C23×C17C2×C22⋊C411yescharacteristic
107C2×C4×C17—24no—
108C2×C4×C17—24no—
109C2×C4×C17—22no—
110C2×C4×C17—22no—
111C2×C4×C17—22no—
112C2×C4×C17—22no—
113C2×C4×C17—24no—
114C2×C4×C17—42no—
115C2×C4×C17—42no—
116C2×C4×C17—42no—
117C2×C4×C17—42no—
118C2×C4×C17—42no—
119C2×C4×C17#10114yescomplemented
120C2×Dic17—24no—
121C2×Dic17—24no—
122C2×Dic17—24no—
123C2×Dic17—24no—
124C2×Dic17—22no—
125C2×Dic17—22no—
126C2×Dic17—24no—
127C2×Dic17—22no—
128C2×Dic17—22no—
129C2×Dic17—22no—
130C2×Dic17—42no—
131C2×Dic17—42no—
132C2×Dic17—42no—
133C2×Dic17—42no—
134C2×Dic17—42no—
135C2×Dic17#9914yes—
136C2×Dic17#10114yescomplemented
137C2×Dic17#12014yes—
138Dic34—416no—
139Dic34—416no—
140#3624—84no—
141C22×C4⋊Q8—171no—
142#1784—172no—
143#1463—172no—
144#1463—172no—
145C22×C17—28no—
146C22×C17—21no—
147C22×C17—21no—
148C22×C17—21no—
149C22×C17—21no—
150C22×C17—21no—
151C22×C17—21no—
152C22×C17#37911yescharacteristic
153C22×C17C2×#9912yes—
154C22×C17C2×#10112yes—
155C22×C17C2×#10112yes—
156C4×C17—24no—
157C4×C17—42no—
158C4×C17—42no—
159Dic17—24no—
160Dic17—24no—
161Dic17—24no—
162Dic17—42no—
163C23×Q8—171no—
164C23×Q8—171no—
165C2×C4⋊Q8—344no—
166C2×C4⋊Q8—344no—
167C2×C4⋊Q8—344no—
168C22×C4⋊C4—171no—
169C22×C4⋊C4—171no—
170C22×C4⋊C4—171no—
171C22×C4⋊C4#59511yescharacteristic
172C22×C42—171no—
173C2×#101—344no—
174#407—174no—
175#324—344no—
176#324—344no—
177C2×C17—22no—
178C2×C17—22no—
179C2×C17—22no—
180C2×C17#178111yescharacteristic
181C2×C17#178411yescharacteristic
182C2×C17#178411yescharacteristic
183C22×Q8—174no—
184C22×Q8—342no—
185C22×Q8—342no—
186C22×Q8—342no—
187C22×Q8—342no—
188C22×Q8—342no—
189C22×Q8—684no—
190C22×Q8—684no—
191C23×C4—171no—
192C23×C4—171no—
193C23×C4—171no—
194C23×C4—171no—
195C23×C4C2×#59511yescharacteristic
196C23×C4C2×#59511yescharacteristic
197C23×C4C2×#59511yescharacteristic
198C2×M16—344no—
199C4⋊Q8—3432no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—174no—
204C2×C4⋊C4—174no—
205C2×C4⋊C4—342no—
206C2×C4⋊C4—342no—
207C2×C4⋊C4—342no—
208C2×C4⋊C4—342no—
209C2×C4⋊C4—342no—
210C2×C4⋊C4—342no—
211C2×C4⋊C4—342no—
212C2×C42—342no—
213C2×C42—342no—
214C2×C42—342no—
215#101—688no—
216C22⋊C8—344no—
217C22⋊C8—344no—
218C17#855111yescharacteristic, complemented, Sylow
219C24C22×#59511yescharacteristic
220C2×Q8—348no—
221C2×Q8—348no—
222C2×Q8—348no—
223C2×Q8—684no—
224C2×Q8—684no—
225C2×Q8—688no—
226C2×Q8—688no—
227C2×Q8—688no—
228C2×Q8—684no—
229C2×Q8—688no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—24no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—172no—
240C22×C4—172no—
241C22×C4—172no—
242C22×C4—172no—
243C22×C4—172no—
244C22×C4—172no—
245C22×C4—344no—
246C22×C4—344no—
247C22×C4—341no—
248C22×C4—341no—
249C22×C4—341no—
250C22×C4—341no—
251C22×C4—341no—
252C22×C4—341no—
253C22×C4—682no—
254C22×C4—682no—
255C22×C4C22⋊#59512yes—
256C22×C4C22⋊#59512yes—
257C22×C4C22⋊#59512yes—
258M16—688no—
259C2×C8—684no—
260C4⋊C4—28no—
261C4⋊C4—348no—
262C4⋊C4—348no—
263C4⋊C4—348no—
264C42—348no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23C2×C22⋊#59511yescharacteristic
269C23C2×C22⋊#59511yescharacteristic
270C23C2×C22⋊#59511yescharacteristic
271Q8—6816no—
272Q8—6816no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—24no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—174no—
286C2×C4—174no—
287C2×C4—174no—
288C2×C4—344no—
289C2×C4—344no—
290C2×C4—344no—
291C2×C4—344no—
292C2×C4—342no—
293C2×C4—342no—
294C2×C4—344no—
295C2×C4—342no—
296C2×C4—342no—
297C2×C4—342no—
298C2×C4—682no—
299C2×C4—682no—
300C2×C4—682no—
301C2×C4—682no—
302C2×C4—682no—
303C2×C4#1058754514yescomplemented
304C8—1364no—
305C22—28no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22#1176375011yescharacteristic
313C22D17⋊#9912yes—
314C22C2×#1058754512yes—
315C22D17⋊#10112yes—
316C4—24no—
317C4—42no—
318C4—42no—
319C4—344no—
320C4—344no—
321C4—344no—
322C4—682no—
323C2—22no—
324C2—22no—
325C2—22no—
326C2#1454461111yescharacteristic
327C2#1454461811yescharacteristic
328C2#1454461911yescharacteristic
subgroupG/H
centerC22#11763750
commutator, prime radicalC24×C17C22×C4
FrattiniC24C22×#595
FittingC22×C4⋊C4×C17C4
socleC22×C17#379
cosocleC23×Dic17C23
hypercenterC22×C4⋊C4#595
hypocenterC17#8551
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
119. C2×C4×C17#101semidirect
136. C2×Dic17#101semidirect
218. C17#8551semidirect
303. C2×C4#10587545semidirect

Maximal subgroups

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

subgroupindex
#145446182
#145440342
#145440322
C22×#105875742
#855117