Finite Group Databasebeta

← #17038018

Subgroups — #17038018

Order 4352.

1686 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587561C211yesmaximal, characteristic
2C2×#11762819C212yesmaximal, complemented
3C2×#11762815C212yesmaximal
4#14531191C211yesmaximal, characteristic, complemented
5#14529600C214yesmaximal, complemented
6#14529595C214yesmaximal, complemented
7C2×C4×M16×C17C211yesmaximal, characteristic, complemented
8C23×Dic34C2211yescharacteristic
9C23×D68C2211yescharacteristic
10C2×#10587561—22no—
11C2×#10587561—24no—
12C2×#10587561C418yescomplemented
13C2×#10587561C2212yes—
14C2×#10587561C2212yes—
15C2×#10587612C2212yes—
16C22×#63227—22no—
17#11762819—28no—
18#11762815—28no—
19C2×#10587487C2218yescomplemented
20C2×#10587492C2218yes—
21#11762679C2212yescomplemented
22#11762675C2212yescomplemented
23#11762669—22no—
24C4×C4⋊Dic17—22no—
25C22×M16×C17C2212yes—
26C22×C42×C17C2211yescharacteristic
27C4×M16×C17—24no—
28C2×C17×#97C2212yescomplemented
29C2×C4×C8×C17C2212yescomplemented
30C24×D17—21no—
31C23×C4×C17D412yes—
32C23×C4×C17C2311yescharacteristic
33C23×Dic17—21no—
34C22×D68—24no—
35C22×D68—24no—
36C22×D68—21no—
37C22×D68—22no—
38C22×D68C2×C414yes—
39C22×D68C2312yes—
40C22×Dic34—24no—
41C22×Dic34—24no—
42C22×Dic34—21no—
43C22×Dic34—22no—
44C22×Dic34C2×C414yes—
45C22×Dic34C2312yes—
46C2×M16×C17—24no—
47C2×M16×C17—24no—
48C2×M16×C17D418yes—
49C22×C8×C17C2314yes—
50C2×C42×C17C2×C414yes—
51C2×C42×C17D412yes—
52C2×C42×C17D414yes—
53C2×C42×C17C2311yescharacteristic
54C2×C42×C17C2311yescharacteristic
55C2×C22⋊Dic17—22no—
56C2×#63227—216no—
57C2×#63227—24no—
58C2×#63227—24no—
59#10587612—24no—
60#10587612—24no—
61C2×C4⋊Dic17—24no—
62C2×C4⋊Dic17C2314yes—
63C2×C4×Dic17—22no—
64#10587561—216no—
65#10587561—24no—
66#10587561—24no—
67#10587561C2×C418yes—
68#10587561C2×C418yes—
69C17×#97—24no—
70C4×C8×C17—24no—
71#10587502—24no—
72#10587492—216no—
73#10587487—216no—
74C24×C17C2×D411yescharacteristic
75C23×D17—24no—
76C23×D17—22no—
77C23×D17—48no—
78C23×D17—42no—
79C23×D17—42no—
80C22×C4×C17—44no—
81C22×C4×C17C22⋊C418yes—
82C22×C4×C17C22×C412yes—
83C22×C4×C17C2×D412yes—
84C22×C4×C17C2×D412yes—
85C22×C4×C17C2×D411yescharacteristic
86C22×C4×C17C2×D414yes—
87C22×C4×C17C2×D412yes—
88C22×C4×C17#4114yes—
89C22×C4×C17C2411yescharacteristic
90C22×Dic17—24no—
91C22×Dic17—22no—
92C22×Dic17—24no—
93C22×Dic17—42no—
94C22×Dic17—42no—
95C2×D68—216no—
96C2×D68—24no—
97C2×D68—24no—
98C2×D68—24no—
99C2×D68—22no—
100C2×D68—48no—
101C2×D68—42no—
102C2×D68—42no—
103C2×D68C22×C414yes—
104C2×Dic34—216no—
105C2×Dic34—24no—
106C2×Dic34—24no—
107C2×Dic34—24no—
108C2×Dic34—22no—
109C2×Dic34—48no—
110C2×Dic34—42no—
111C2×Dic34—42no—
112C2×Dic34C22×C414yes—
113M16×C17—216no—
114C2×C8×C17—24no—
115C2×C8×C17—24no—
116C2×C8×C17C2×D418yes—
117C42×C17C22⋊C418yes—
118C42×C17C22×C412yes—
119C42×C17C22×C412yes—
120C42×C17C2×D412yes—
121C42×C17C2×D412yes—
122C22⋊Dic17—44no—
123#63227—216no—
124#63227—216no—
125C4⋊Dic17—24no—
126C4⋊Dic17—24no—
127C4⋊Dic17—48no—
128C4×Dic17—44no—
129#21204—171nomaximal, Sylow
130C23×C17—24no—
131C23×C17C2×C22⋊C412yes—
132C23×C17#12412yes—
133C23×C17#12712yes—
134C23×C17C22×D411yescharacteristic
135C22×D17—24no—
136C22×D17—416no—
137C22×D17—42no—
138C22×D17—42no—
139C22×D17—48no—
140C22×D17—48no—
141C22×D17—88no—
142C22×D17—88no—
143C2×C4×C17—24no—
144C2×C4×C17—24no—
145C2×C4×C17—48no—
146C2×C4×C17C2×C22⋊C414yes—
147C2×C4×C17C2×C22⋊C414yes—
148C2×C4×C17C2×C22⋊C412yes—
149C2×C4×C17C4×D418yes—
150C2×C4×C17#12118yes—
151C2×C4×C17#12412yes—
152C2×C4×C17#12712yes—
153C2×C4×C17C23×C411yescharacteristic
154C2×C4×C17C22×D412yes—
155C2×C4×C17C22×D411yescharacteristic
156C2×C4×C17C2×#4112yes—
157C2×Dic17—24no—
158C2×Dic17—42no—
159C2×Dic17—42no—
160C2×Dic17—48no—
161C2×Dic17—44no—
162C2×Dic17—48no—
163D68—28no—
164D68—44no—
165D68—48no—
166Dic34—28no—
167Dic34—44no—
168Dic34—48no—
169C8×C17—28no—
170C22×#124—171no—
171C2×#419—172no—
172C2×#418—172no—
173C2×C4×M16D1711yescharacteristic, complemented
174#2285—171no—
175#1959—174no—
176#1958—174no—
177C22×C17—24no—
178C22×C17—24no—
179C22×C17—24no—
180C22×C17#39014yes—
181C22×C17C22×C22⋊C411yescharacteristic
182C22×C17C2×#12411yescharacteristic
183C22×C17C2×#12711yescharacteristic
184C22×C17#57514yes—
185C2×D17—48no—
186C2×D17—44no—
187C2×D17—816no—
188C2×D17—84no—
189C2×D17—84no—
190C4×C17—24no—
191C4×C17—24no—
192C4×C17#39014yes—
193C4×C17C2×C4×D412yes—
194C4×C17C2×#12112yes—
195Dic17—44no—
196Dic17—84no—
197C23×Q8—171no—
198C23×D4—171no—
199C22×M16C2×D1712yes—
200C2×#124—178no—
201C2×#124—172no—
202C2×#124—172no—
203C2×#124—342no—
204C2×#124—344no—
205C2×#117—172no—
206C22×C22⋊C4—342no—
207C22×C42C2×D1711yescharacteristic
208#419—348no—
209#418—348no—
210C2×#103—178no—
211C2×#102—178no—
212C4×M16—24no—
213C2×#97C2×D1712yescomplemented
214C2×C4×C8C2×D1712yescomplemented
215#384—172no—
216#382—172no—
217#380—342no—
218C4×C4⋊C4—342no—
219C2×C17—24no—
220C2×C17C2×#39011yescharacteristic
221C2×C17#289612yes—
222C2×C17#314012yes—
223C2×C17#315312yes—
224D17—84no—
225C25—341no—
226C22×Q8—174no—
227C22×Q8—172no—
228C22×Q8—341no—
229C22×Q8—342no—
230C22×Q8—344no—
231C22×Q8—344no—
232C22×D4—174no—
233C22×D4—172no—
234C22×D4—344no—
235C22×D4—341no—
236C22×D4—344no—
237C22×D4—342no—
238C23×C4—341no—
239C23×C4D6812yes—
240C23×C4C22×D1711yescharacteristic
241C2×M16—24no—
242C2×M16—24no—
243C2×M16D6818yes—
244C22×C8C22×D1714yes—
245#124—178no—
246#124—178no—
247#124—344no—
248#124—344no—
249#124—3416no—
250#117—344no—
251#117—344no—
252C2×C4⋊C4—174no—
253C2×C4⋊C4—344no—
254C2×C22⋊C4—344no—
255C2×C22⋊C4—344no—
256C2×C22⋊C4—3416no—
257C2×C22⋊C4—342no—
258C2×C42—342no—
259C2×C42C4×D1714yes—
260C2×C42D6812yes—
261C2×C42C22⋊D1714yes—
262C2×C42C22×D1711yescharacteristic
263C2×C42C22×D1711yescharacteristic
264#103—3416no—
265#102—3416no—
266#97—24no—
267C4×C8—24no—
268#95—344no—
269C17#2120411yescharacteristic, complemented, Sylow
270C24—342no—
271C24—344no—
272C24—688no—
273C24—682no—
274C24—682no—
275C24C2×D6811yescharacteristic
276C2×Q8—174no—
277C2×Q8—342no—
278C2×Q8—344no—
279C2×Q8—344no—
280C2×Q8—344no—
281C2×Q8—3416no—
282C2×Q8—682no—
283C2×Q8—682no—
284C2×Q8—688no—
285C2×D4—174no—
286C2×D4—344no—
287C2×D4—344no—
288C2×D4—342no—
289C2×D4—3416no—
290C2×D4—344no—
291C2×D4—682no—
292C2×D4—682no—
293C2×D4—688no—
294C22×C4—44no—
295C22×C4—342no—
296C22×C4—344no—
297C22×C4—344no—
298C22×C4—682no—
299C22×C4—682no—
300C22×C4#6322718yes—
301C22×C4C2×C4×D1712yes—
302C22×C4C2×D6812yes—
303C22×C4C2×D6812yes—
304C22×C4C2×D6814yes—
305C22×C4C2×D6811yescharacteristic
306C22×C4#6325214yes—
307C22×C4C2×C22⋊D1712yes—
308C22×C4C23×D1711yescharacteristic
309M16—216no—
310C2×C8—24no—
311C2×C8—24no—
312C2×C8C2×D6818yes—
313C4⋊C4—344no—
314C4⋊C4—344no—
315C4⋊C4—688no—
316C22⋊C4—3416no—
317C22⋊C4—3416no—
318C22⋊C4—684no—
319C42—684no—
320C42#6322718yes—
321C42C2×C4×D1712yes—
322C42C2×C4×D1712yes—
323C42C2×C22⋊D1712yes—
324C42C2×C22⋊D1712yes—
325C23—24no—
326C23—344no—
327C23—6816no—
328C23—688no—
329C23—682no—
330C23—682no—
331C23—688no—
332C23—1368no—
333C23—1368no—
334C23#1058756012yes—
335C23#1058756112yes—
336C23C2×#6322712yes—
337C23C22×D6811yescharacteristic
338Q8—348no—
339Q8—688no—
340Q8—684no—
341D4—348no—
342D4—688no—
343D4—684no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—48no—
347C2×C4—344no—
348C2×C4—688no—
349C2×C4—682no—
350C2×C4—682no—
351C2×C4—684no—
352C2×C4—688no—
353C2×C4C4×D6818yes—
354C2×C4#1058756012yes—
355C2×C4#1058756112yes—
356C2×C4C2×#6322714yes—
357C2×C4C2×#6322714yes—
358C2×C4C2×#6322712yes—
359C2×C4#1058761618yes—
360C2×C4C22×C4×D1711yescharacteristic
361C2×C4C22×D6812yes—
362C2×C4C2×#6325212yes—
363C2×C4C22×C22⋊D1711yescharacteristic
364C8—28no—
365C22—24no—
366C22—24no—
367C22—24no—
368C22—688no—
369C22—684no—
370C22—13616no—
371C22—1364no—
372C22—1364no—
373C22#1176271114yes—
374C22C22×#6322711yescharacteristic
375C22C2×#1058756111yescharacteristic
376C22C2×#1058756011yescharacteristic
377C22#1176360914yes—
378C4—24no—
379C4—24no—
380C4—684no—
381C4—1364no—
382C4#1176271114yes—
383C4C2×C4×D6812yes—
384C4C2×#1058761612yes—
385C2—24no—
386C2—1364no—
387C2C2×#1176271111yescharacteristic
388C2#1453730412yes—
389C2#1453917112yes—
390C2#1453925012yes—
subgroupG/H
centerC23C22×D68
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC2×C4×M16×C17C2
socleC23×C17C22×D4
prime radicalC23×D68C22
cosocleC22×C4×C17C24
hypercenterC2×C4×M16D17
hypocenterC17#21204
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×C4×M16

Complements

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

NH ≅ G/Ntype
2. C2×#11762819C2semidirect
4. #14531191C2semidirect
5. #14529600C2semidirect
6. #14529595C2semidirect
7. C2×C4×M16×C17C2semidirect
12. C2×#10587561C4semidirect
19. C2×#10587487C22semidirect
21. #11762679C22semidirect
22. #11762675C22semidirect
28. C2×C17×#97C22semidirect
29. C2×C4×C8×C17C22semidirect
173. C2×C4×M16D17semidirect
213. C2×#97C2×D17semidirect
214. C2×C4×C8C2×D17semidirect
269. C17#21204semidirect

Maximal subgroups

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

subgroupindex
C22×#105875612
C2×#117628192
C2×#117628152
#145311912
#145296002
#145295952
C2×C4×M16×C172
#2120417