Finite Group Databasebeta

← #17037992

Subgroups — #17037992

Order 4352.

1726 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763359C211yesmaximal, characteristic
2C2×#11762811C214yesmaximal, complemented
3C2×#11762679C211yesmaximal, characteristic, complemented
4#14529600C214yesmaximal, complemented
5#14529590C212yesmaximal, complemented
6#14529586C212yesmaximal, complemented
7C2×C4×M16×C17C211yesmaximal, characteristic, complemented
8C22×#63252C2212yes—
9#11763359—24no—
10#11763359C418yescomplemented
11C2×#10587555C2211yescharacteristic
12C2×#10587560C2211yescharacteristic
13C2×#10587561C2212yes—
14C2×#10587616—22no—
15C2×C4×D68—22no—
16C22×C4⋊Dic17—22no—
17C22×C4⋊Dic17C2212yes—
18#11762811—216no—
19C2×#10587487C2218yes—
20C2×#10587492C2218yes—
21#11762679—22no—
22#11762679C2212yes—
23#11762679C2212yes—
24C22×M16×C17C2212yes—
25C22×C42×C17C2211yescharacteristic
26C4×M16×C17—24no—
27C2×C17×#97C2212yes—
28C2×C4×C8×C17C2212yes—
29C23×C4×C17D412yes—
30C23×C4×C17C2311yescharacteristic
31C22×C22⋊D17—22no—
32C23×Dic17—22no—
33C2×#63252—24no—
34C2×#63252—28no—
35C2×#63252—28no—
36C2×#63252C2×C418yes—
37C22×D68—22no—
38C22×D68C2312yes—
39C22×C4×D17—22no—
40C22×Dic34C2312yes—
41C2×M16×C17—24no—
42C2×M16×C17—24no—
43C2×M16×C17D418yescomplemented
44C22×C8×C17C2314yes—
45C2×C42×C17C2×C414yes—
46C2×C42×C17D412yes—
47C2×C42×C17D414yes—
48C2×C42×C17C2311yescharacteristic
49C2×C42×C17C2311yescharacteristic
50#10587616—216no—
51C2×#63227—24no—
52C2×C4⋊Dic17—22no—
53C2×C4⋊Dic17—24no—
54C2×C4⋊Dic17—24no—
55C2×C4⋊Dic17—22no—
56C2×C4⋊Dic17—28no—
57C2×C4⋊Dic17—48no—
58C2×C4⋊Dic17C2314yes—
59#10587561—24no—
60#10587561C2×C418yescomplemented
61#10587560—22no—
62#10587560C2×C414yescomplemented
63C4×D68—216no—
64#10587555—22no—
65#10587555C2×C414yescomplemented
66C17×#97—24no—
67C4×C8×C17—24no—
68#10587492—216no—
69#10587487—216no—
70C24×C17C2×D411yescharacteristic
71C23×D17—22no—
72C22×C4×C17—24no—
73C22×C4×C17—24no—
74C22×C4×C17C22⋊C418yes—
75C22×C4×C17C22×C412yes—
76C22×C4×C17C2×D412yes—
77C22×C4×C17C2×D411yescharacteristic
78C22×C4×C17C2×D412yes—
79C22×C4×C17C2×D414yes—
80C22×C4×C17C2×D412yes—
81C22×C4×C17#4114yes—
82C22×C4×C17C2411yescharacteristic
83C2×C22⋊D17—28no—
84C2×C22⋊D17—44no—
85C2×C22⋊D17—44no—
86C22×Dic17—22no—
87C22×Dic17—24no—
88C22×Dic17—48no—
89C22×Dic17—44no—
90#63252—232no—
91#63252—416no—
92C2×D68—24no—
93C2×D68—24no—
94C2×D68—28no—
95C2×D68—22no—
96C2×D68—44no—
97C2×D68—44no—
98C2×D68C22×C414yes—
99C2×C4×D17—28no—
100C2×C4×D17—44no—
101C2×C4×D17—44no—
102C2×Dic34—24no—
103C2×Dic34—24no—
104C2×Dic34—22no—
105C2×Dic34C22×C414yes—
106M16×C17—216no—
107C2×C8×C17—24no—
108C2×C8×C17—24no—
109C2×C8×C17C2×D418yes—
110C42×C17C22⋊C418yes—
111C42×C17C22×C412yes—
112C42×C17C22×C412yes—
113C42×C17C2×D412yes—
114C42×C17C2×D412yes—
115#63227—216no—
116C4⋊Dic17—28no—
117C4⋊Dic17—28no—
118C4⋊Dic17—48no—
119C4⋊Dic17—44no—
120C4⋊Dic17—48no—
121#21200—171nomaximal, Sylow
122C23×C17—22no—
123C23×C17—22no—
124C23×C17C2×C22⋊C412yes—
125C23×C17#12412yes—
126C23×C17#12712yes—
127C23×C17C22×D411yescharacteristic
128C22×D17—24no—
129C22×D17—42no—
130C22×D17—42no—
131C22×D17—44no—
132C22×D17—44no—
133C2×C4×C17—22no—
134C2×C4×C17—22no—
135C2×C4×C17—216no—
136C2×C4×C17C2×C22⋊C414yes—
137C2×C4×C17C2×C22⋊C412yes—
138C2×C4×C17C2×C22⋊C414yes—
139C2×C4×C17C4×D418yes—
140C2×C4×C17#12118yes—
141C2×C4×C17#12412yes—
142C2×C4×C17#12712yes—
143C2×C4×C17AGL(1,ℤ/8)14yes—
144C2×C4×C17#13714yes—
145C2×C4×C17C23×C411yescharacteristic
146C2×C4×C17C22×D412yes—
147C2×C4×C17C22×D411yescharacteristic
148C2×C4×C17C2×#4112yes—
149C22⋊D17—416no—
150C2×Dic17—24no—
151C2×Dic17—42no—
152C2×Dic17—42no—
153C2×Dic17—44no—
154C2×Dic17—48no—
155C2×Dic17—88no—
156D68—28no—
157D68—44no—
158D68—416no—
159C4×D17—416no—
160Dic34—28no—
161Dic34—44no—
162C8×C17—28no—
163C2×#532—171no—
164C2×#417—174no—
165C2×C4×M16D1711yescharacteristic, complemented
166C2×#384—171no—
167#1959—174no—
168#1957—172no—
169#1956—172no—
170C22×C17—24no—
171C22×C17—24no—
172C22×C17—22no—
173C22×C17—22no—
174C22×C17#39014yes—
175C22×C17C22×C22⋊C411yescharacteristic
176C22×C17C2×#12411yescharacteristic
177C22×C17C2×#12711yescharacteristic
178C22×C17C2×AGL(1,ℤ/8)12yes—
179C22×C17C2×#13712yes—
180C2×D17—48no—
181C2×D17—44no—
182C2×D17—84no—
183C2×D17—82no—
184C2×D17—82no—
185C4×C17—24no—
186C4×C17#39014yes—
187C4×C17#41718yescomplemented
188C4×C17C2×C4×D412yes—
189C4×C17C2×#12112yes—
190Dic17—44no—
191Dic17—84no—
192C22×#41—172no—
193C22×M16C2×D1712yes—
194#532—178no—
195#532—344no—
196C2×C4⋊Q8—171no—
197C2×#127—171no—
198C2×#124—172no—
199C2×#121—342no—
200C2×C4×D4—342no—
201C22×C4⋊C4—172no—
202C22×C4⋊C4—342no—
203C22×C42C2×D1711yescharacteristic
204#417—3416no—
205C2×#103—178no—
206C2×#102—178no—
207C4×M16—24no—
208C2×#97C2×D1712yes—
209C2×C4×C8C2×D1712yes—
210#384—172no—
211#384—172no—
212#384—342no—
213C2×C17—24no—
214C2×C17C2×#39011yescharacteristic
215C2×C17C2×#41712yes—
216C2×C17#314112yes—
217C2×C17#315111yescharacteristic
218C2×C17#315211yescharacteristic
219D17—84no—
220C2×#41—178no—
221C2×#41—344no—
222C2×#41—348no—
223C2×#41—348no—
224C22×Q8—172no—
225C22×D4—172no—
226C22×D4—342no—
227C22×D4—342no—
228C23×C4—342no—
229C23×C4—342no—
230C23×C4D6812yes—
231C23×C4C22×D1711yescharacteristic
232C2×M16—24no—
233C2×M16—24no—
234C2×M16D6818yescomplemented
235C22×C8C22×D1714yes—
236C4⋊Q8—174no—
237C4⋊Q8—342no—
238#127—174no—
239#127—342no—
240#124—178no—
241#124—344no—
242#121—3416no—
243C4×D4—3416no—
244C2×C4⋊C4—174no—
245C2×C4⋊C4—342no—
246C2×C4⋊C4—348no—
247C2×C4⋊C4—342no—
248C2×C4⋊C4—344no—
249C2×C4⋊C4—344no—
250C2×C4⋊C4—688no—
251C2×C22⋊C4—344no—
252C2×C42C4×D1714yes—
253C2×C42D6812yes—
254C2×C42C22⋊D1714yes—
255C2×C42C22×D1711yescharacteristic
256C2×C42C22×D1711yescharacteristic
257#103—3416no—
258#102—3416no—
259#97—24no—
260C4×C8—24no—
261C17#2120011yescharacteristic, complemented, Sylow
262C24—342no—
263C24C2×D6811yescharacteristic
264#41—3432no—
265#41—6816no—
266C2×Q8—174no—
267C2×Q8—342no—
268C2×Q8—344no—
269C2×Q8—344no—
270C2×D4—174no—
271C2×D4—348no—
272C2×D4—344no—
273C2×D4—344no—
274C2×D4—342no—
275C2×D4—348no—
276C2×D4—684no—
277C2×D4—684no—
278C2×D4—684no—
279C2×D4—684no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—342no—
283C22×C4—348no—
284C22×C4—344no—
285C22×C4—684no—
286C22×C4—684no—
287C22×C4—688no—
288C22×C4—684no—
289C22×C4#6322718yes—
290C22×C4C2×C4×D1712yes—
291C22×C4C2×D6812yes—
292C22×C4C2×D6814yes—
293C22×C4C2×D6811yescharacteristic
294C22×C4C2×D6812yes—
295C22×C4#6325214yes—
296C22×C4C2×C22⋊D1712yes—
297C22×C4C23×D1711yescharacteristic
298M16—216no—
299C2×C8—24no—
300C2×C8—24no—
301C2×C8C2×D6818yes—
302C4⋊C4—348no—
303C4⋊C4—348no—
304C4⋊C4—688no—
305C4⋊C4—684no—
306C4⋊C4—688no—
307C22⋊C4—3416no—
308C42#6322718yes—
309C42C2×C4×D1712yes—
310C42C2×C4×D1712yes—
311C42C2×C22⋊D1712yes—
312C42C2×C22⋊D1712yes—
313C23—22no—
314C23—22no—
315C23—344no—
316C23—682no—
317C23—684no—
318C23—684no—
319C23—682no—
320C23#1058756012yes—
321C23#1058756112yes—
322C23C2×#6322712yes—
323C23C22×D6811yescharacteristic
324Q8—348no—
325Q8—684no—
326D4—348no—
327D4—6816no—
328D4—684no—
329D4—6816no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—216no—
333C2×C4—344no—
334C2×C4—682no—
335C2×C4—682no—
336C2×C4—6816no—
337C2×C4—688no—
338C2×C4—684no—
339C2×C4—1368no—
340C2×C4C4×D6818yes—
341C2×C4#1058756012yes—
342C2×C4#1058756112yes—
343C2×C4#1058759414yes—
344C2×C4#1058759514yes—
345C2×C4C2×#6322714yes—
346C2×C4C2×#6322712yes—
347C2×C4C2×#6322714yes—
348C2×C4#1058761618yes—
349C2×C4C22×C4×D1711yescharacteristic
350C2×C4C22×D6812yes—
351C2×C4C2×#6325212yes—
352C2×C4C22×C22⋊D1711yescharacteristic
353C8—28no—
354C22—24no—
355C22—24no—
356C22—22no—
357C22—22no—
358C22—688no—
359C22—684no—
360C22—1364no—
361C22—1362no—
362C22—1362no—
363C22#1176271114yes—
364C22C22×#6322711yescharacteristic
365C22C2×#1058756111yescharacteristic
366C22C2×#1058756011yescharacteristic
367C22C2×#1058759412yes—
368C22C2×#1058759512yes—
369C4—24no—
370C4—684no—
371C4—1364no—
372C4#1176271114yes—
373C4#1176281118yescomplemented
374C4C2×C4×D6812yes—
375C4C2×#1058761612yes—
376C2—24no—
377C2—1364no—
378C2C2×#1176271111yescharacteristic
379C2C2×#1176281112yes—
380C2#1453917812yes—
381C2#1453923611yescharacteristic
382C2#1453924311yescharacteristic
subgroupG/H
centerC23C22×D68
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC2×C4×M16×C17C2
socleC23×C17C22×D4
prime radicalC2×#11763359C2
cosocleC22×C4×C17C24
hypercenterC2×C4×M16D17
hypocenterC17#21200
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×#11762811C2semidirect
3. C2×#11762679C2semidirect
4. #14529600C2semidirect
5. #14529590C2semidirect
6. #14529586C2semidirect
7. C2×C4×M16×C17C2semidirect
10. #11763359C4semidirect
43. C2×M16×C17D4semidirect
60. #10587561C2×C4semidirect
62. #10587560C2×C4semidirect
65. #10587555C2×C4semidirect
165. C2×C4×M16D17semidirect
187. C4×C17#417semidirect
234. C2×M16D68semidirect
261. C17#21200semidirect
373. C4#11762811semidirect

Maximal subgroups

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

subgroupindex
C2×#117633592
C2×#117628112
C2×#117626792
#145296002
#145295902
#145295862
C2×C4×M16×C172
#2120017