Finite Group Databasebeta

← #17044260

Subgroups — #17044260

Order 4352.

1366 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763359C211yesmaximal, characteristic
2#14531182C211yesmaximal, characteristic, complemented
3#14529735C212yesmaximal, complemented
4#14529710C212yesmaximal, complemented
5#14529222C214yesmaximal, complemented
6#14528795C214yesmaximal, complemented
7C2×C17×#423C211yesmaximal, characteristic, complemented
8C22×#63252—21no—
9#11763359—24no—
10#11763359C418yescomplemented
11C2×#10587555C2211yescharacteristic
12C2×#10587560C2211yescharacteristic
13C2×#10587561—21no—
14C2×#10587616—21no—
15C2×#10587616C2212yes—
16C2×C4×D68—21no—
17C2×C4×D68C2212yes—
18C2×#10587492C2218yes—
19#11762677—21no—
20#11762675C2212yes—
21#11762672—21no—
22#11762669C2212yes—
23C4×C4⋊Dic17C2212yes—
24C4×C22⋊Dic17—21no—
25#11762655—21no—
26#11762572C2218yes—
27C22×M16×C17C2212yes—
28C22×C42×C17C2211yescharacteristic
29C17×#423—24no—
30C2×C17×C4⋊C8C2212yes—
31C2×C17×C4⋊C8C2212yes—
32C23×C4×C17D411yescharacteristic
33C23×C4×C17D411yescharacteristic
34C23×C4×C17C2311yescharacteristic
35C22×C22⋊D17—22no—
36C2×#63252—28no—
37C2×#63252—48no—
38C22×D68—21no—
39C22×D68—21no—
40C22×D68C2312yes—
41C22×C4×D17—22no—
42C22×Dic34—21no—
43C2×M16×C17—24no—
44C2×M16×C17—44no—
45C22×C8×C17C2314yes—
46C2×C42×C17C2×C414yes—
47C2×C42×C17D411yescharacteristic
48C2×C42×C17D411yescharacteristic
49C2×C42×C17D414yes—
50C2×C42×C17C2311yescharacteristic
51C2×C42×C17C2311yescharacteristic
52C2×C22⋊Dic17—22no—
53#10587616—28no—
54#10587616—24no—
55#10587616C2×C418yes—
56C2×#63227—22no—
57C2×#63227—22no—
58C2×C4⋊Dic17—21no—
59C2×C4⋊Dic17C2314yes—
60C2×C4⋊Dic17C2312yes—
61C2×C4×Dic17—22no—
62#10587561—28no—
63#10587560—22no—
64#10587560C2×C414yes—
65C4×D68—28no—
66C4×D68—24no—
67C4×D68C2×C418yes—
68#10587555—22no—
69#10587555C2×C414yes—
70C17×C4⋊C8—24no—
71C17×C4⋊C8—24no—
72#10587502—22no—
73#10587502—22no—
74#10587502—22no—
75#10587502—22no—
76#10587492—48no—
77C24×C17C2×D411yescharacteristic
78C23×D17—22no—
79C22×C4×C17—24no—
80C22×C4×C17—22no—
81C22×C4×C17—22no—
82C22×C4×C17—42no—
83C22×C4×C17C22⋊C414yes—
84C22×C4×C17C22⋊C414yes—
85C22×C4×C17C22×C412yes—
86C22×C4×C17C2×D411yescharacteristic
87C22×C4×C17C2×D411yescharacteristic
88C22×C4×C17C2×D411yescharacteristic
89C22×C4×C17C2×D411yescharacteristic
90C22×C4×C17C2×D411yescharacteristic
91C22×C4×C17C2×D412yes—
92C22×C4×C17C2411yescharacteristic
93C2×C22⋊D17—28no—
94C2×C22⋊D17—48no—
95C22×Dic17—22no—
96C22×Dic17—24no—
97#63252—432no—
98C2×D68—24no—
99C2×D68—24no—
100C2×D68—24no—
101C2×D68—24no—
102C2×D68—22no—
103C2×D68—44no—
104C2×D68—44no—
105C2×D68C22×C414yes—
106C2×C4×D17—28no—
107C2×C4×D17—48no—
108C2×Dic34—24no—
109C2×Dic34—44no—
110M16×C17—48no—
111C2×C8×C17—24no—
112C2×C8×C17—28no—
113C42×C17C22⋊C414yes—
114C42×C17C22⋊C414yes—
115C42×C17C22×C412yes—
116C42×C17C22×C412yes—
117C42×C17C2×D412yes—
118C42×C17C2×D412yes—
119C22⋊Dic17—44no—
120#63227—28no—
121#63227—28no—
122C4⋊Dic17—24no—
123C4⋊Dic17—22no—
124C4⋊Dic17—24no—
125C4⋊Dic17—24no—
126C4⋊Dic17C22×C414yes—
127C4×Dic17—44no—
128#21891—171nomaximal, Sylow
129C23×C17—22no—
130C23×C17—22no—
131C23×C17—22no—
132C23×C17C2×C22⋊C412yes—
133C23×C17C22×D411yescharacteristic
134C22×D17—24no—
135C22×D17—44no—
136C22×D17—44no—
137C22×D17—44no—
138C2×C4×C17—22no—
139C2×C4×C17—22no—
140C2×C4×C17—22no—
141C2×C4×C17—22no—
142C2×C4×C17—24no—
143C2×C4×C17—22no—
144C2×C4×C17—22no—
145C2×C4×C17—28no—
146C2×C4×C17—44no—
147C2×C4×C17C2×C22⋊C412yes—
148C2×C4×C17C2×C22⋊C412yes—
149C2×C4×C17C2×C22⋊C412yes—
150C2×C4×C17C2×C22⋊C412yes—
151C2×C4×C17C2×C22⋊C412yes—
152C2×C4×C17AGL(1,ℤ/8)14yes—
153C2×C4×C17C23×C411yescharacteristic
154C2×C4×C17C22×D411yescharacteristic
155C2×C4×C17H(5,𝔽2)12yes—
156C2×C4×C172−1+412yes—
157C22⋊D17—416no—
158C2×Dic17—24no—
159C2×Dic17—44no—
160C2×Dic17—44no—
161C2×Dic17—44no—
162C2×Dic17—44no—
163D68—28no—
164D68—48no—
165D68—42no—
166D68—42no—
167D68—48no—
168C4×D17—416no—
169Dic34—48no—
170C8×C17—44no—
171C2×#532—171no—
172C2×#423D1711yescharacteristic, complemented
173#2283—171no—
174#1988—172no—
175#1982—172no—
176#1895—174no—
177#1804—174no—
178C22×C17—22no—
179C22×C17—22no—
180C22×C17—22no—
181C22×C17—44no—
182C22×C17C22×C22⋊C411yescharacteristic
183C22×C17#53712yes—
184C22×C17C2×AGL(1,ℤ/8)12yes—
185C22×C17#57512yes—
186C2×D17—44no—
187C2×D17—44no—
188C2×D17—44no—
189C2×D17—88no—
190C4×C17—22no—
191C4×C17—22no—
192C4×C17—22no—
193C4×C17—22no—
194C4×C17#41814yes—
195C4×C17#52012yes—
196C4×C17#53511yescharacteristic
197C4×C17#53611yescharacteristic
198Dic17—44no—
199Dic17—84no—
200C22×#41—341no—
201C22×M16C2×D1712yes—
202#532—178no—
203#532—344no—
204C2×C4⋊Q8—171no—
205C2×#127—171no—
206C2×#124—341no—
207C2×#121—172no—
208C2×#121—341no—
209C2×C4×D4—172no—
210C2×C4×D4—341no—
211C22×C42C2×D1711yescharacteristic
212#423—24no—
213C2×C4⋊C8C2×D1712yes—
214C2×C4⋊C8C2×D1712yes—
215C2×#102—178no—
216#383—341no—
217#382—172no—
218#381—341no—
219#380—172no—
220C4×C4⋊C4—172no—
221C4×C22⋊C4—341no—
222#376—341no—
223#340—178no—
224C2×C17—42no—
225C2×C17#231111yescharacteristic
226C2×C17C2×#41811yescharacteristic
227C2×C17#289611yescharacteristic
228C2×C17#325114yes—
229D17—84no—
230C2×#41—348no—
231C2×#41—688no—
232C22×Q8—341no—
233C22×D4—172no—
234C22×D4—342no—
235C22×D4—341no—
236C22×D4—341no—
237C23×C4—342no—
238C23×C4D6811yescharacteristic
239C23×C4D6811yescharacteristic
240C23×C4C22×D1711yescharacteristic
241C2×M16—24no—
242C2×M16—44no—
243C22×C8C22×D1714yes—
244C4⋊Q8—174no—
245C4⋊Q8—342no—
246#127—174no—
247#127—342no—
248#124—348no—
249#121—178no—
250#121—344no—
251#121—348no—
252C4×D4—178no—
253C4×D4—344no—
254C4×D4—348no—
255C2×C4⋊C4—172no—
256C2×C4⋊C4—174no—
257C2×C4⋊C4—341no—
258C2×C22⋊C4—342no—
259C2×C22⋊C4—342no—
260C2×C22⋊C4—342no—
261C2×C42—342no—
262C2×C42C4×D1714yes—
263C2×C42D6811yescharacteristic
264C2×C42D6811yescharacteristic
265C2×C42C22⋊D1714yes—
266C2×C42C22×D1711yescharacteristic
267C2×C42C22×D1711yescharacteristic
268C4⋊C8—24no—
269C4⋊C8—24no—
270#102—688no—
271#95—342no—
272#95—342no—
273#95—342no—
274#95—342no—
275C17#2189111yescharacteristic, complemented, Sylow
276C24—342no—
277C24C2×D6811yescharacteristic
278#41—6832no—
279C2×Q8—344no—
280C2×Q8—684no—
281C2×D4—174no—
282C2×D4—348no—
283C2×D4—344no—
284C2×D4—344no—
285C2×D4—342no—
286C2×D4—344no—
287C2×D4—344no—
288C2×D4—684no—
289C2×D4—688no—
290C2×D4—684no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—42no—
295C22×C4—342no—
296C22×C4—348no—
297C22×C4—344no—
298C22×C4—688no—
299C22×C4#6322714yes—
300C22×C4#6322714yes—
301C22×C4C2×C4×D1712yes—
302C22×C4C2×D6811yescharacteristic
303C22×C4C2×D6811yescharacteristic
304C22×C4C2×D6811yescharacteristic
305C22×C4C2×D6811yescharacteristic
306C22×C4C2×D6811yescharacteristic
307C22×C4C2×C22⋊D1712yes—
308C22×C4C23×D1711yescharacteristic
309M16—48no—
310C2×C8—28no—
311C2×C8—24no—
312C4⋊C4—174no—
313C4⋊C4—342no—
314C4⋊C4—344no—
315C4⋊C4—344no—
316C4⋊C4—344no—
317C22⋊C4—348no—
318C22⋊C4—348no—
319C22⋊C4—684no—
320C42—684no—
321C42#6322714yes—
322C42#6322714yes—
323C42C2×C4×D1712yes—
324C42C2×C4×D1712yes—
325C42C2×C22⋊D1712yes—
326C42C2×C22⋊D1712yes—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—344no—
331C23—684no—
332C23—684no—
333C23—684no—
334C23C2×#6322712yes—
335C23C22×D6811yescharacteristic
336Q8—688no—
337D4—348no—
338D4—688no—
339D4—6816no—
340D4—688no—
341D4—682no—
342D4—682no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—24no—
348C2×C4—28no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—44no—
352C2×C4—344no—
353C2×C4—6816no—
354C2×C4—684no—
355C2×C4—684no—
356C2×C4—684no—
357C2×C4—684no—
358C2×C4#1058759414yes—
359C2×C4C2×#6322712yes—
360C2×C4C2×#6322712yes—
361C2×C4C2×#6322712yes—
362C2×C4C2×#6322712yes—
363C2×C4C2×#6322712yes—
364C2×C4C22×C4×D1711yescharacteristic
365C2×C4#1058769112yes—
366C2×C4#1058769412yes—
367C2×C4C22×C22⋊D1711yescharacteristic
368C8—44no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—44no—
373C22—684no—
374C22—684no—
375C22—684no—
376C22—1368no—
377C22C22×#6322711yescharacteristic
378C22#1176339012yes—
379C22C2×#1058759412yes—
380C22#1176360912yes—
381C4—22no—
382C4—22no—
383C4—22no—
384C4—22no—
385C4—684no—
386C4—1364no—
387C4#1176281514yes—
388C4#1176329012yes—
389C4#1176337611yescharacteristic
390C4#1176338311yescharacteristic
391C2—42no—
392C2—1364no—
393C2#1453137711yescharacteristic
394C2C2×#1176281511yescharacteristic
395C2#1453730411yescharacteristic
396C2#1454014714yes—
subgroupG/H
centerC23C22×D68
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC2×C17×#423C2
socleC23×C17C22×D4
prime radicalC2×#11763359C2
cosocleC22×C4×C17C24
hypercenterC2×#423D17
hypocenterC17#21891
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#423

Complements

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

NH ≅ G/Ntype
2. #14531182C2semidirect
3. #14529735C2semidirect
4. #14529710C2semidirect
5. #14529222C2semidirect
6. #14528795C2semidirect
7. C2×C17×#423C2semidirect
10. #11763359C4semidirect
172. C2×#423D17semidirect
275. C17#21891semidirect

Maximal subgroups

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

subgroupindex
C2×#117633592
#145311822
#145297352
#145297102
#145292222
#145287952
C2×C17×#4232
#2189117