Finite Group Databasebeta

← #16959592

Subgroups — #16959592

Order 4352.

2058 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587572C211yesmaximal, characteristic
2C2×C22⋊C4×Dic17C212yesmaximal, complemented
3#14527033C211yesmaximal, characteristic, complemented
4#14527023C212yesmaximal
5#14527020C214yesmaximal, complemented
6#14527018C214yesmaximal, complemented
7C17×#1440C211yesmaximal, characteristic, complemented
8C23×C22⋊D17C2211yescharacteristic
9C24×Dic17C2211yescharacteristic
10C2×#10587572—28no—
11C2×#10587572C418yescomplemented
12C22×C4⋊Dic17C2212yes—
13C22×C22⋊Dic17C2212yes—
14C22×#63227C2214yes—
15C22×#63227C2212yes—
16C22×C4×Dic17C2214yes—
17C22⋊C4×Dic17—28no—
18C2×#10587502C2214yes—
19C2×#10587502C2214yes—
20C2×#10587502C2214yes—
21C22×C22⋊C4×C17C2212yes—
22C22×C22⋊C4×C17C2211yescharacteristic
23C2×C17×#95C2214yes—
24C25×C17C2311yescharacteristic
25C24×D17C2311yescharacteristic
26C23×C4×C17C2312yes—
27C23×C4×C17C2314yes—
28C22×C22⋊D17—24no—
29C22×C22⋊D17—44no—
30C22×C22⋊D17C2×C414yes—
31C23×Dic17—44no—
32C23×Dic17C2×C414yes—
33C23×Dic17D418yes—
34C23×Dic17C2311yescharacteristic
35C23×Dic17C2314yes—
36C23×Dic17C2312yes—
37C2×C22⋊C4×C17—24no—
38C2×C22⋊C4×C17—24no—
39C2×C22⋊C4×C17—28no—
40C2×C22⋊C4×C17C2×C414yes—
41C2×C22⋊Dic17—24no—
42C2×C22⋊Dic17—28no—
43C2×#63227—28no—
44C2×#63227—24no—
45C2×#63227—24no—
46C2×#63227—28no—
47C2×#63227—44no—
48C2×#63227C2×C418yes—
49C2×C4⋊Dic17—24no—
50C2×C4⋊Dic17—24no—
51C2×C4⋊Dic17C2×C418yes—
52C2×C4×Dic17—28no—
53C2×C4×Dic17—44no—
54C2×C4×Dic17—44no—
55#10587572—232no—
56C17×#95—28no—
57#10587502—28no—
58#10587502—28no—
59#10587502—44no—
60C24×C17—24no—
61C24×C17—24no—
62C24×C17—24no—
63C24×C17C22×C412yes—
64C24×C17C2×D414yes—
65C24×C17C2411yescharacteristic
66C23×D17—22no—
67C23×D17—24no—
68C23×D17—24no—
69C23×D17—24no—
70C23×D17C22×C412yes—
71C22×C4×C17—24no—
72C22×C4×C17—22no—
73C22×C4×C17—22no—
74C22×C4×C17—24no—
75C22×C4×C17—24no—
76C22×C4×C17—24no—
77C22×C4×C17—24no—
78C22×C4×C17—24no—
79C22×C4×C17—24no—
80C22×C4×C17—24no—
81C22×C4×C17C22×C414yes—
82C22×C4×C17#4118yes—
83C2×C22⋊D17—28no—
84C2×C22⋊D17—48no—
85C2×C22⋊D17—416no—
86C22×Dic17—24no—
87C22×Dic17—24no—
88C22×Dic17—22no—
89C22×Dic17—24no—
90C22×Dic17—24no—
91C22×Dic17—24no—
92C22×Dic17—24no—
93C22×Dic17—24no—
94C22×Dic17—24no—
95C22×Dic17—24no—
96C22×Dic17—48no—
97C22×Dic17—416no—
98C22×Dic17C22⋊C4116yescomplemented
99C22×Dic17C22×C412yes—
100C22×Dic17C22×C414yes—
101C22×Dic17C2×D414yes—
102C22×Dic17C2×D414yes—
103C22×Dic17#4118yes—
104C22⋊C4×C17—216no—
105C22⋊C4×C17—28no—
106C22⋊Dic17—216no—
107#63227—216no—
108#63227—416no—
109C4⋊Dic17—216no—
110C4×Dic17—416no—
111#14167—171nomaximal, Sylow
112C23×C17—28no—
113C23×C17—24no—
114C23×C17—24no—
115C23×C17—24no—
116C23×C17—28no—
117C23×C17—28no—
118C23×C17—44no—
119C23×C17—44no—
120C23×C17—44no—
121C23×C17—44no—
122C23×C17C2×C22⋊C414yes—
123C23×C17C23×C411yescharacteristic
124C23×C17C22×D412yes—
125C23×C17C2×#4114yes—
126C23×C17C2×#4114yes—
127C23×C17H(5,𝔽2)14yes—
128C22×D17—22no—
129C22×D17—28no—
130C22×D17—24no—
131C22×D17—42no—
132C22×D17—42no—
133C22×D17—44no—
134C22×D17—44no—
135C22×D17—44no—
136C22×D17—44no—
137C22×D17—44no—
138C22×D17—44no—
139C2×C4×C17—24no—
140C2×C4×C17—28no—
141C2×C4×C17—28no—
142C2×C4×C17—28no—
143C2×C4×C17—28no—
144C2×C4×C17—42no—
145C2×C4×C17—42no—
146C2×C4×C17—44no—
147C2×C4×C17—44no—
148C2×C4×C17—44no—
149C2×C4×C17—44no—
150C2×C4×C17—44no—
151C22⋊D17—416no—
152C2×Dic17—24no—
153C2×Dic17—24no—
154C2×Dic17—28no—
155C2×Dic17—28no—
156C2×Dic17—22no—
157C2×Dic17—24no—
158C2×Dic17—28no—
159C2×Dic17—28no—
160C2×Dic17—28no—
161C2×Dic17—416no—
162C2×Dic17—44no—
163C2×Dic17—44no—
164C2×Dic17—44no—
165C2×Dic17—44no—
166C2×Dic17—44no—
167C2×Dic17C2×C22⋊C418yes—
168C22×#123—171no—
169C2×C4×C22⋊C4—172no—
170#1440—171no—
171#1440D1711yescharacteristic, complemented
172#1438—172no—
173#1437—174no—
174#1436—174no—
175C22×C17—24no—
176C22×C17—28no—
177C22×C17—48no—
178C22×C17—48no—
179C22×C17—48no—
180C22×C17C22×C22⋊C411yescharacteristic
181C22×C17C4×#4114yes—
182C22×C17#51812yes—
183C22×C17#52512yes—
184C22×C17#52512yes—
185C22×C17#53214yes—
186C22×C17#53414yes—
187C22×C17#55114yes—
188C22×C17#55314yes—
189C22×C17#55314yes—
190C22×C17#55914yes—
191C2×D17—42no—
192C2×D17—48no—
193C2×D17—44no—
194C2×D17—84no—
195C2×D17—84no—
196C4×C17—48no—
197C4×C17—44no—
198Dic17—28no—
199Dic17—42no—
200Dic17—48no—
201C23×D4—171no—
202C24×C4—171no—
203C2×#123—178no—
204C2×#123—348no—
205C22×C4⋊C4—172no—
206C22×C22⋊C4—172no—
207C22×C22⋊C4—172no—
208C22×C22⋊C4—174no—
209C22×C22⋊C4C2×D1712yes—
210C22×C22⋊C4C2×D1711yescharacteristic
211C22×C42—174no—
212C4×C22⋊C4—348no—
213C2×#95—174no—
214C2×#95—174no—
215C2×#95—174no—
216C2×#95C2×D1714yes—
217C2×C17—44no—
218C2×C17#229712yes—
219C2×C17#230811yescharacteristic
220C2×C17#237414yes—
221C2×C17#260214yes—
222C2×C17#260214yes—
223D17—82no—
224C25—171no—
225C25C22×D1711yescharacteristic
226C22×D4—174no—
227C22×D4—344no—
228C22×D4—684no—
229C23×C4—172no—
230C23×C4—174no—
231C23×C4—178no—
232C23×C4—174no—
233C23×C4—171no—
234C23×C4—684no—
235C23×C4C22×D1712yes—
236C23×C4C22×D1714yes—
237#123—3432no—
238C2×C4⋊C4—178no—
239C2×C4⋊C4—344no—
240C2×C4⋊C4—344no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—178no—
245C2×C22⋊C4—344no—
246C2×C22⋊C4—344no—
247C2×C22⋊C4—348no—
248C2×C22⋊C4—344no—
249C2×C22⋊C4—348no—
250C2×C22⋊C4—348no—
251C2×C22⋊C4—684no—
252C2×C22⋊C4C4×D1714yes—
253C2×C42—348no—
254C2×C42—684no—
255C2×C42—684no—
256#95—28no—
257#95—348no—
258#95—348no—
259#95—684no—
260C17#1416711yescharacteristic, complemented, Sylow
261C24—24no—
262C24—24no—
263C24—24no—
264C24—172no—
265C24—342no—
266C24—344no—
267C24—344no—
268C24—344no—
269C24C2×C4×D1712yes—
270C24D4×D1714yes—
271C24C23×D1711yescharacteristic
272C2×D4—348no—
273C2×D4—688no—
274C2×D4—6816no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—174no—
286C22×C4—174no—
287C22×C4—174no—
288C22×C4—1716no—
289C22×C4—172no—
290C22×C4—178no—
291C22×C4—344no—
292C22×C4—344no—
293C22×C4—344no—
294C22×C4—342no—
295C22×C4—344no—
296C22×C4—344no—
297C22×C4—344no—
298C22×C4—344no—
299C22×C4—344no—
300C22×C4—344no—
301C22×C4—688no—
302C22×C4—6816no—
303C22×C4C2×C4×D1714yes—
304C22×C4#6325418yes—
305C4⋊C4—3416no—
306C22⋊C4—216no—
307C22⋊C4—28no—
308C22⋊C4—3416no—
309C22⋊C4—3416no—
310C22⋊C4—6816no—
311C42—6816no—
312C23—28no—
313C23—24no—
314C23—24no—
315C23—24no—
316C23—28no—
317C23—28no—
318C23—44no—
319C23—44no—
320C23—44no—
321C23—44no—
322C23—342no—
323C23—348no—
324C23—344no—
325C23—684no—
326C23—684no—
327C23—684no—
328C23—684no—
329C23—682no—
330C23—682no—
331C23—684no—
332C23—684no—
333C23C22⋊C4×D1714yes—
334C23C22×C4×D1711yescharacteristic
335C23C2×D4×D1712yes—
336C23C2×#6325414yes—
337C23#41×D1714yes—
338C23#1058769614yes—
339D4—6816no—
340C2×C4—28no—
341C2×C4—24no—
342C2×C4—28no—
343C2×C4—28no—
344C2×C4—28no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—178no—
353C2×C4—344no—
354C2×C4—344no—
355C2×C4—348no—
356C2×C4—348no—
357C2×C4—348no—
358C2×C4—348no—
359C2×C4—342no—
360C2×C4—344no—
361C2×C4—348no—
362C2×C4—6816no—
363C2×C4—684no—
364C2×C4—684no—
365C2×C4—684no—
366C2×C4—684no—
367C2×C4—684no—
368C22—24no—
369C22—28no—
370C22—48no—
371C22—48no—
372C22—48no—
373C22—682no—
374C22—688no—
375C22—684no—
376C22—1364no—
377C22—1364no—
378C22C2×C22⋊C4×D1711yescharacteristic
379C22C4×#6325414yes—
380C22#1176328312yes—
381C22#1176331512yes—
382C22#1176331712yes—
383C22#1176335414yes—
384C22#1176337014yes—
385C22#1176348214yes—
386C22#1176351014yes—
387C22#1176351114yes—
388C22#1176354114yes—
389C4—44no—
390C4—48no—
391C4—348no—
392C4—682no—
393C4—688no—
394C2—44no—
395C2—1362no—
396C2#1453126912yes—
397C2#1453136111yescharacteristic
398C2#1453191414yes—
399C2#1453439114yes—
400C2#1453439714yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1440C2
socle, cosocleC24×C17C24
prime radicalC23×C22⋊D17C22
hypercenter#1440D17
hypocenterC17#14167
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1440

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊C4×Dic17C2semidirect
3. #14527033C2semidirect
5. #14527020C2semidirect
6. #14527018C2semidirect
7. C17×#1440C2semidirect
11. C2×#10587572C4semidirect
98. C22×Dic17C22⋊C4semidirect
171. #1440D17semidirect
260. C17#14167semidirect

Maximal subgroups

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

subgroupindex
C22×#105875722
C2×C22⋊C4×Dic172
#145270332
#145270232
#145270202
#145270182
C17×#14402
#1416717