Finite Group Databasebeta

← #17032604

Subgroups — #17032604

Order 4352.

1034 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762863C211yesmaximal, characteristic
2C22×#10587486C211yesmaximal, characteristic
3C4×#10587563C211yesmaximal, characteristic
4#14528660C212yesmaximal, complemented
5#14528657C212yesmaximal, complemented
6#14528562C211yesmaximal, characteristic
7#14528558C211yesmaximal, characteristic
8C8×C22⋊Dic17C211yesmaximal, characteristic
9#14528531C212yesmaximal, complemented
10#97×Dic17C212yesmaximal, complemented
11C17×#1752C211yesmaximal, characteristic
12C23×#3622C2211yescharacteristic
13C2×#10587563C412yes—
14C22×C4×Dic17C2211yescharacteristic
15#11762863C418yescomplemented
16C2×#10587485C2212yes—
17C2×C22⋊#3622C2211yescharacteristic
18C2×#10587474C2212yes—
19C2×#10587486C418yescomplemented
20C2×#10587486C2211yescharacteristic
21C2×#10587486C2212yes—
22C2×#10587486C2211yescharacteristic
23C2×#10587486C2211yescharacteristic
24C2×#10587486C2211yescharacteristic
25C2×C8×Dic17C2212yes—
26C2×C8×Dic17C2212yes—
27C4×#63225C2212yes—
28#11762660C412yes—
29C22⋊C4×Dic17C412yes—
30C4×C22⋊Dic17C2211yescharacteristic
31#11762654C412yes—
32C42×Dic17C2212yes—
33#11762564C2212yes—
34#11762563C2212yes—
35#11762563C2212yes—
36C23×C8×C17C2211yescharacteristic
37C2×C17×C22⋊C8C2211yescharacteristic
38C2×C17×#97C2212yes—
39C4×C22⋊C4×C17C2211yescharacteristic
40C17×#336C2212yes—
41C23×C4×C17C2311yescharacteristic
42C23×Dic17C2×C411yescharacteristic
43C23×Dic17C2×C411yescharacteristic
44C22×C8×C17—24no—
45C22×C8×C17C2×C414yes—
46C22×C8×C17C2312yes—
47C22×C8×C17C2311yescharacteristic
48C22×C8×C17C2311yescharacteristic
49C2×C22⋊C4×C17C2×C412yes—
50C2×C42×C17C2312yes—
51C2×C22⋊Dic17C2×C412yes—
52C2×#63225C2×C414yes—
53C2×C4×Dic17C2×C414yes—
54C2×C4×Dic17C2×C414yes—
55C2×C4×Dic17C2×C414yes—
56C2×C4×Dic17C2×C414yes—
57C2×C4×Dic17C2×C411yescharacteristic
58C2×C4×Dic17C2×C411yescharacteristic
59C2×C4×Dic17C2312yes—
60C2×C4×Dic17C2311yescharacteristic
61C2×C4×Dic17C2311yescharacteristic
62C22×#3622—24no—
63C22×#3622C2×C414yes—
64C22×#3622C2312yes—
65C22×#3622C2311yescharacteristic
66C22×#3622C2311yescharacteristic
67#10587563—28no—
68C17×C22⋊C8C2×C414yes—
69C17×#97—24no—
70C17×#95C2×C412yes—
71#10587502C2×C412yes—
72#10587502C2×C414yes—
73C22⋊#3622C2×C414yes—
74#10587486C2×C414yes—
75#10587486C2×C414yes—
76#10587486C2×C414yes—
77#10587486C2×C414yes—
78#10587486C2×C418yes—
79#10587485C2×C418yes—
80C8×Dic17—24no—
81C8×Dic17—24no—
82#10587474—24no—
83C24×C17C22×C411yescharacteristic
84C22×C4×C17—22no—
85C22×C4×C17—22no—
86C22×C4×C17C22×C414yes—
87C22×C4×C17C22×C411yescharacteristic
88C22×C4×C17C22×C412yes—
89C22×C4×C17C22×C412yes—
90C22×C4×C17C2411yescharacteristic
91C22×Dic17—24no—
92C22×Dic17—42no—
93C22×Dic17C4214yes—
94C22×Dic17C4214yes—
95C22×Dic17C22×C411yescharacteristic
96C22×Dic17C22×C411yescharacteristic
97C22×Dic17C22×C411yescharacteristic
98C22×Dic17C22×C411yescharacteristic
99C22×Dic17C22×C414yes—
100C2×C8×C17—22no—
101C2×C8×C17—22no—
102C2×C8×C17—22no—
103C2×C8×C17—28no—
104C2×C8×C17C22×C412yes—
105C2×C8×C17C22×C414yes—
106C2×C8×C17C22×C412yes—
107C2×C8×C17#4114yes—
108C22⋊C4×C17—24no—
109C42×C17—22no—
110C42×C17—22no—
111C22⋊Dic17M1618yes—
112#63225—28no—
113C4×Dic17—22no—
114C4×Dic17—22no—
115C4×Dic17—28no—
116C4×Dic17—44no—
117C4×Dic17C4214yes—
118C4×Dic17C4214yes—
119C4×Dic17C22×C412yes—
120C4×Dic17C22×C412yes—
121C4×Dic17C22×C412yes—
122C4×Dic17C22×C412yes—
123C2×#3622—22no—
124C2×#3622—22no—
125C2×#3622—22no—
126C2×#3622—28no—
127C2×#3622C22×C414yes—
128C2×#3622C22×C412yes—
129C2×#3622C22×C412yes—
130C2×#3622#4114yes—
131#20479—171nomaximal, Sylow
132C23×C17—22no—
133C23×C17—22no—
134C23×C17C2×C4212yes—
135C23×C17C2×M1614yes—
136C23×C17C23×C411yescharacteristic
137C2×C4×C17—22no—
138C2×C4×C17—22no—
139C2×C4×C17—22no—
140C2×C4×C17—24no—
141C2×C4×C17—24no—
142C2×C4×C17—24no—
143C2×C4×C17—24no—
144C2×C4×C17C2×C4212yes—
145C2×C4×C17C23×C411yescharacteristic
146C2×C4×C17C23×C411yescharacteristic
147C2×C4×C17C2×#4112yes—
148C2×C4×C17C2×#4112yes—
149C2×Dic17—24no—
150C2×Dic17—22no—
151C2×Dic17—22no—
152C2×Dic17—22no—
153C2×Dic17—22no—
154C2×Dic17—24no—
155C2×Dic17—28no—
156C2×Dic17—44no—
157C2×Dic17C2×C4212yes—
158C2×Dic17C2×C4212yes—
159C2×Dic17C2×C4212yes—
160C2×Dic17C2×C4212yes—
161C2×Dic17C2×M1618yes—
162C8×C17—22no—
163C8×C17—24no—
164C8×C17#11714yescomplemented
165#3622—22no—
166#3622—24no—
167#3622#11714yescomplemented
168C2×#432—171no—
169C22×#97—171no—
170C4×#117—171no—
171#1771—172no—
172#1771—172no—
173#1752—171no—
174#1752D1711yescharacteristic
175C8×C22⋊C4—171no—
176C8×C22⋊C4—171no—
177#1745—172no—
178C4×#97—172no—
179C22×C17—22no—
180C22×C17—28no—
181C22×C17—22no—
182C22×C17C4×M1614yes—
183C22×C17C22×C4211yescharacteristic
184C22×C17C4×#4112yes—
185C22×C17C22×M1612yes—
186C22×C17#56812yes—
187C4×C17—22no—
188C4×C17—44no—
189C4×C17C2×#11711yescharacteristic
190C4×C17C2×#11711yescharacteristic
191C4×C17C4×#4112yes—
192Dic17—22no—
193Dic17—22no—
194Dic17—22no—
195Dic17—22no—
196Dic17—28no—
197C23×C8—171no—
198C23×C8C2×D1711yescharacteristic
199C2×#117—172no—
200C22×C42—171no—
201#432—178no—
202C2×C4⋊C8—172no—
203C2×C22⋊C8—171no—
204C2×C22⋊C8C2×D1711yescharacteristic
205C2×#97—178no—
206C2×#97—171no—
207C2×#97—171no—
208C2×#97—171no—
209C2×#97—171no—
210C2×#97—172no—
211C2×#97—172no—
212C2×#97C2×D1712yes—
213C2×C4×C8—172no—
214C2×C4×C8—172no—
215C4×C4⋊C4—172no—
216C4×C4⋊C4—172no—
217C4×C22⋊C4—171no—
218C4×C22⋊C4—172no—
219C4×C22⋊C4C2×D1711yescharacteristic
220#376—172no—
221C43—172no—
222#336—172no—
223#336—172no—
224#336—172no—
225#336C2×D1712yes—
226C2×C17—24no—
227C2×C17C4×#11711yescharacteristic
228C2×C17C2×C4×M1611yescharacteristic
229C2×C17#287311yescharacteristic
230C2×C17#297014yes—
231C23×C4—171no—
232C23×C4—171no—
233C23×C4C22×D1711yescharacteristic
234C22×C8—24no—
235C22×C8—171no—
236C22×C8—174no—
237C22×C8—171no—
238C22×C8—172no—
239C22×C8—344no—
240C22×C8C4×D1714yes—
241C22×C8C22×D1711yescharacteristic
242C22×C8C22×D1712yes—
243C22×C8C22×D1711yescharacteristic
244#117—348no—
245C2×C4⋊C4—174no—
246C2×C22⋊C4—172no—
247C2×C22⋊C4C4×D1712yes—
248C2×C42—171no—
249C2×C42—171no—
250C2×C42—174no—
251C2×C42—174no—
252C2×C42—171no—
253C2×C42—171no—
254C2×C42—174no—
255C2×C42—172no—
256C2×C42—174no—
257C2×C42C22×D1712yes—
258C4⋊C8—178no—
259C22⋊C8—174no—
260C22⋊C8C4×D1714yes—
261#97—24no—
262#97—174no—
263#97—174no—
264#97—174no—
265#97—174no—
266#97—178no—
267#97—344no—
268C4×C8—344no—
269C4×C8—344no—
270#95—174no—
271#95—172no—
272#95C4×D1712yes—
273C17#2047911yescharacteristic, complemented, Sylow
274C24C2×C4×D1711yescharacteristic
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—171no—
278C22×C4—174no—
279C22×C4—171no—
280C22×C4—171no—
281C22×C4—171no—
282C22×C4—174no—
283C22×C4—174no—
284C22×C4—344no—
285C22×C4—682no—
286C22×C4C2×C4×D1714yes—
287C22×C4C2×C4×D1712yes—
288C22×C4C2×C4×D1712yes—
289C22×C4C2×C4×D1711yescharacteristic
290C22×C4C23×D1711yescharacteristic
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—28no—
295C2×C8—172no—
296C2×C8—174no—
297C2×C8—172no—
298C2×C8—174no—
299C2×C8—348no—
300C2×C8—342no—
301C2×C8—342no—
302C2×C8—342no—
303C2×C8C2×C4×D1712yes—
304C2×C8C2×C4×D1714yes—
305C2×C8C2×C4×D1712yes—
306C2×C8#6325414yes—
307C4⋊C4—348no—
308C22⋊C4—24no—
309C22⋊C4—178no—
310C42—22no—
311C42—22no—
312C42—172no—
313C42—172no—
314C42—172no—
315C42—172no—
316C42—174no—
317C42—174no—
318C42—348no—
319C42—342no—
320C42—342no—
321C42—684no—
322C23—22no—
323C23—22no—
324C23C42×D1712yes—
325C23M16×D1714yes—
326C23C22×C4×D1711yescharacteristic
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—172no—
335C2×C4—172no—
336C2×C4—172no—
337C2×C4—172no—
338C2×C4—178no—
339C2×C4—348no—
340C2×C4—344no—
341C2×C4—342no—
342C2×C4—342no—
343C2×C4—342no—
344C2×C4—344no—
345C2×C4—342no—
346C2×C4—684no—
347C2×C4C42×D1712yes—
348C2×C4C22×C4×D1711yescharacteristic
349C2×C4C22×C4×D1711yescharacteristic
350C2×C4C2×#6325412yes—
351C2×C4#41×D1712yes—
352C8—22no—
353C8—24no—
354C8—174no—
355C8—342no—
356C8—344no—
357C8#1058756314yescomplemented
358C22—22no—
359C22—28no—
360C22—22no—
361C22#1176276114yes—
362C22C2×C42×D1711yescharacteristic
363C22C4×#6325412yes—
364C22C2×M16×D1712yes—
365C22#1176357412yes—
366C4—22no—
367C4—44no—
368C4—348no—
369C4—342no—
370C4—342no—
371C4—342no—
372C4—342no—
373C4#117×D1711yescharacteristic
374C4C2×#1058756311yescharacteristic
375C4C4×#6325412yes—
376C2—24no—
377C2C4×#1058756311yescharacteristic
378C2C2×#1176276111yescharacteristic
379C2#1453720011yescharacteristic
380C2#1453779414yes—
subgroupG/H
center, FrattiniC22×C4C23×D17
commutatorC22×C17C22×C42
FittingC17×#1752C2
socleC23×C17C23×C4
prime radicalC24×C17C22×C4
cosocleC22×C4×C17C24
hypercenter#1752D17
hypocenterC17#20479
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ #1752

Complements

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

NH ≅ G/Ntype
4. #14528660C2semidirect
5. #14528657C2semidirect
9. #14528531C2semidirect
10. #97×Dic17C2semidirect
15. #11762863C4semidirect
19. C2×#10587486C4semidirect
164. C8×C17#117semidirect
167. #3622#117semidirect
273. C17#20479semidirect
357. C8#10587563semidirect

Maximal subgroups

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

subgroupindex
C2×#117628632
C22×#105874862
C4×#105875632
#145286602
#145286572
#145285622
#145285582
C8×C22⋊Dic172
#145285312
#97×Dic172
C17×#17522
#2047917