Finite Group Databasebeta

← #17038114

Subgroups — #17038114

Order 4352.

1646 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14537348C211yesmaximal, characteristic, complemented
2C2×#11762793C212yesmaximal, complemented
3C22×#10587504C212yesmaximal, complemented
4#14537203C211yesmaximal, characteristic, complemented
5#14529634C214yesmaximal, complemented
6#14529625C214yesmaximal, complemented
7C2×C17×#535C211yesmaximal, characteristic
8C22×#63223C2212yes—
9C22×#63223C2212yes—
10#11762836—24no—
11#11762833C2214yescomplemented
12#11762793—28no—
13C2×#10587507C2214yescomplemented
14C2×#10587504—28no—
15C2×#10587504C2218yes—
16C2×#10587504C2214yescomplemented
17#11762765—24no—
18C4×#63223C2214yescomplemented
19C22×#41×C17C2211yescharacteristic
20C23×D4×C17C2211yescharacteristic
21C17×#535—24no—
22C17×#535—24no—
23C2×C17×#127C2212yes—
24C2×C17×#124C2212yes—
25C2×C17×#121C414yes—
26C2×C17×#120C414yes—
27C2×C17×#117C2211yescharacteristic
28C25×C17C2×C412yes—
29C2×#41×C17—22no—
30C2×#41×C17—22no—
31C2×#41×C17—24no—
32C2×#41×C17—24no—
33C22×Q8×C17C2311yescharacteristic
34C22×D4×C17—22no—
35C22×D4×C17—22no—
36C22×D4×C17—24no—
37C22×D4×C17—24no—
38C22×D4×C17C2×C412yes—
39C22×D4×C17C2×C412yes—
40C22×D4×C17C2×C418yes—
41C22×D4×C17C2312yes—
42C22×D4×C17C2311yescharacteristic
43C23×C4×C17C2×C412yes—
44C23×C4×C17C2311yescharacteristic
45C17×#127—24no—
46C17×#127—24no—
47C17×#124—24no—
48C17×#124—24no—
49C17×#121—216no—
50C17×#120—216no—
51C17×#117D414yes—
52C17×#117D414yes—
53C2×C4⋊C4×C17C2×C412yes—
54C2×C22⋊C4×C17C2×C418yes—
55C2×C22⋊C4×C17C2×C412yes—
56C2×C42×C17C2312yes—
57C2×#63223—22no—
58C2×#63223—22no—
59C2×#63223—24no—
60C2×#63223—24no—
61C2×#63223—24no—
62C2×#63223D418yescomplemented
63C2×#63223C2314yes—
64C2×#63223C2314yes—
65C22×#3622—22no—
66C22×#3622—22no—
67#10587507—28no—
68#10587504—216no—
69#10587504—28no—
70C4⋊#3622—28no—
71#10587474—24no—
72C4×#3622—24no—
73C24×C17—24no—
74C24×C17—28no—
75C24×C17—84no—
76C24×C17C22×C412yes—
77C24×C17C22×C414yes—
78C24×C17C22×C411yescharacteristic
79#41×C17—44no—
80#41×C17—44no—
81#41×C17—44no—
82#41×C17—44no—
83C2×Q8×C17—21no—
84C2×Q8×C17—21no—
85C2×Q8×C17—22no—
86C2×Q8×C17—22no—
87C2×D4×C17—24no—
88C2×D4×C17—28no—
89C2×D4×C17—24no—
90C2×D4×C17—216no—
91C2×D4×C17—28no—
92C2×D4×C17—22no—
93C2×D4×C17—22no—
94C2×D4×C17—21no—
95C2×D4×C17—21no—
96C2×D4×C17—24no—
97C2×D4×C17—22no—
98C2×D4×C17—24no—
99C2×D4×C17—22no—
100C2×D4×C17—44no—
101C2×D4×C17—44no—
102C2×D4×C17—44no—
103C2×D4×C17—44no—
104C2×D4×C17—88no—
105C22×C4×C17—24no—
106C22×C4×C17—28no—
107C22×C4×C17C22×C412yes—
108C22×C4×C17C22×C414yes—
109C22×C4×C17C22×C411yescharacteristic
110C22×C4×C17C2×D414yes—
111C22×C4×C17C2×D412yes—
112C22×C4×C17C2×D412yes—
113C22×C4×C17#4114yes—
114C22×C4×C17C2411yescharacteristic
115C4⋊C4×C17C22⋊C418yes—
116C22⋊C4×C17—216no—
117C22⋊C4×C17C22⋊C418yes—
118C42×C17C2×D414yes—
119C42×C17C2×D414yes—
120#63223—28no—
121#63223—24no—
122#63223—24no—
123#63223—24no—
124#63223—44no—
125#63223C2×D418yes—
126C2×#3622—28no—
127C2×#3622—24no—
128C2×#3622—24no—
129C2×#3622—44no—
130#21219—171nomaximal, Sylow
131C23×C17—24no—
132C23×C17—22no—
133C23×C17—24no—
134C23×C17—28no—
135C23×C17—42no—
136C23×C17—44no—
137C23×C17—48no—
138C23×C17—48no—
139C23×C17—88no—
140C23×C17—816no—
141C23×C17C2×C22⋊C412yes—
142C23×C17C4×D414yes—
143C23×C17C23×C411yescharacteristic
144Q8×C17—42no—
145Q8×C17—41no—
146Q8×C17—41no—
147D4×C17—48no—
148D4×C17—44no—
149D4×C17—42no—
150D4×C17—42no—
151D4×C17—41no—
152D4×C17—42no—
153D4×C17—41no—
154D4×C17—48no—
155D4×C17—816no—
156C2×C4×C17—22no—
157C2×C4×C17—24no—
158C2×C4×C17—44no—
159C2×C4×C17—48no—
160C2×C4×C17C2×C22⋊C418yes—
161C2×C4×C17C2×C22⋊C412yes—
162C2×C4×C17C4×D414yes—
163C2×C4×C17#12114yes—
164C2×C4×C17#12114yes—
165C2×C4×C17#12414yes—
166C2×C4×C17#12714yes—
167C2×C4×C17C22×D412yes—
168C2×C4×C17C22×D411yescharacteristic
169C2×C4×C17C22×D411yescharacteristic
170C2×C4×C17C2×#4112yes—
171#3622—28no—
172#3622—44no—
173C2×#535D1711yescharacteristic
174#2905—171no—
175C2×#413—172no—
176C22×#100—172no—
177#2873—171no—
178#1967—174no—
179#1965—174no—
180C22×C17—22no—
181C22×C17—24no—
182C22×C17—44no—
183C22×C17—44no—
184C22×C17—48no—
185C22×C17—42no—
186C22×C17—44no—
187C22×C17—84no—
188C22×C17—88no—
189C22×C17—816no—
190C22×C17#39014yes—
191C22×C17C22×C22⋊C411yescharacteristic
192C22×C17C2×C4×D412yes—
193C4×C17—28no—
194C4×C17—42no—
195C4×C17#39014yes—
196C4×C17C2×#12112yes—
197C4×C17C2×#12411yescharacteristic
198C4×C17C2×#12711yescharacteristic
199C22×#41C2×D1711yescharacteristic
200C23×D4C2×D1711yescharacteristic
201C22×M16—172no—
202C22×M16—172no—
203#535—24no—
204#535—24no—
205C2×#127C2×D1712yes—
206C2×#124C2×D1712yes—
207C2×#121Dic1714yes—
208C2×#120Dic1714yes—
209C2×#117C2×D1711yescharacteristic
210#424—344no—
211#423—174no—
212#413—348no—
213C2×#101—174no—
214C2×#100—178no—
215C2×#100—174no—
216C2×#100—348no—
217#405—344no—
218C4×M16—174no—
219C2×C17—22no—
220C2×C17—24no—
221C2×C17—82no—
222C2×C17—84no—
223C2×C17C2×#39011yescharacteristic
224C2×C17#288812yes—
225C25C2×Dic1712yes—
226C2×#41—24no—
227C2×#41—24no—
228C2×#41—22no—
229C2×#41—22no—
230C22×Q8C22×D1711yescharacteristic
231C22×D4—24no—
232C22×D4—24no—
233C22×D4—22no—
234C22×D4—22no—
235C22×D4C2×Dic1712yes—
236C22×D4C2×Dic1712yes—
237C22×D4C2×Dic1718yes—
238C22×D4C22×D1712yes—
239C22×D4C22×D1711yescharacteristic
240C23×C4C2×Dic1712yes—
241C23×C4C22×D1711yescharacteristic
242C2×M16—178no—
243C2×M16—174no—
244C2×M16—174no—
245C2×M16—344no—
246C2×M16—342no—
247C2×M16—342no—
248C2×M16—344no—
249C2×M16—344no—
250C22×C8—342no—
251C22×C8—342no—
252#127—24no—
253#127—24no—
254#124—24no—
255#124—24no—
256#121—216no—
257#120—216no—
258#117C22⋊D1714yes—
259#117C22⋊D1714yes—
260C2×C4⋊C4C2×Dic1712yes—
261C2×C22⋊C4C2×Dic1718yes—
262C2×C22⋊C4C2×Dic1712yes—
263C2×C42C22×D1712yes—
264C4⋊C8—348no—
265#101—348no—
266#100—3416no—
267#100—348no—
268#97—344no—
269C4×C8—344no—
270C17#2121911yescharacteristic, complemented, Sylow
271C24—28no—
272C24—24no—
273C24—84no—
274C24C22×Dic1714yes—
275C24C22×Dic1712yes—
276C24C22×Dic1711yescharacteristic
277#41—44no—
278#41—44no—
279#41—44no—
280#41—44no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×D4—28no—
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—22no—
289C2×D4—24no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—28no—
296C2×D4—24no—
297C2×D4—216no—
298C2×D4—44no—
299C2×D4—44no—
300C2×D4—44no—
301C2×D4—44no—
302C2×D4—88no—
303C22×C4—28no—
304C22×C4—24no—
305C22×C4D4×D1714yes—
306C22×C4#6325414yes—
307C22×C4C22×Dic1712yes—
308C22×C4C22×Dic1714yes—
309C22×C4C22×Dic1711yescharacteristic
310C22×C4C2×C22⋊D1712yes—
311C22×C4C2×C22⋊D1712yes—
312C22×C4C23×D1711yescharacteristic
313M16—178no—
314M16—344no—
315M16—348no—
316M16—344no—
317M16—344no—
318M16—684no—
319C2×C8—344no—
320C2×C8—348no—
321C2×C8—344no—
322C2×C8—684no—
323C4⋊C4C22⋊Dic1718yes—
324C22⋊C4—216no—
325C22⋊C4C22⋊Dic1718yes—
326C42C2×C22⋊D1714yes—
327C42C2×C22⋊D1714yes—
328C23—24no—
329C23—22no—
330C23—24no—
331C23—28no—
332C23—42no—
333C23—44no—
334C23—48no—
335C23—48no—
336C23—88no—
337C23—816no—
338C23D4×Dic1714yes—
339C23C2×C22⋊Dic1712yes—
340C23C23×Dic1711yescharacteristic
341Q8—42no—
342Q8—41no—
343Q8—41no—
344D4—48no—
345D4—48no—
346D4—44no—
347D4—42no—
348D4—42no—
349D4—41no—
350D4—42no—
351D4—41no—
352D4—816no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—44no—
356C2×C4—48no—
357C2×C4D4×Dic1714yes—
358C2×C4#1058762214yes—
359C2×C4#1058762414yes—
360C2×C4#1058762414yes—
361C2×C4#1058762614yes—
362C2×C4C2×C22⋊Dic1718yes—
363C2×C4C2×C22⋊Dic1712yes—
364C2×C4C2×D4×D1712yes—
365C2×C4C2×#6325412yes—
366C2×C4C22×C22⋊D1711yescharacteristic
367C2×C4C22×C22⋊D1711yescharacteristic
368C8—348no—
369C8—684no—
370C22—22no—
371C22—24no—
372C22—44no—
373C22—44no—
374C22—48no—
375C22—42no—
376C22—44no—
377C22—88no—
378C22—84no—
379C22—816no—
380C22#1176271214yes—
381C22C22×C22⋊Dic1711yescharacteristic
382C22C2×D4×Dic1712yes—
383C4—28no—
384C4—42no—
385C4#1176271214yes—
386C4C2×#1058762412yes—
387C4C2×#1058762211yescharacteristic
388C4C2×#1058762611yescharacteristic
389C2—22no—
390C2—24no—
391C2—84no—
392C2—82no—
393C2C2×#1176271211yescharacteristic
394C2#1453726312yes—
subgroupG/H
centerC22C22×C22⋊Dic17
commutatorC23×C17C23×C4
FrattiniC22×C4C23×D17
Fitting, prime radicalC2×C17×#535C2
socleC22×C17C22×C22⋊C4
cosocleC22×C4×C17C24
hypercenterC2×#535D17
hypocenterC17#21219
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×#117 ◁ C2×#535

Complements

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

NH ≅ G/Ntype
1. #14537348C2semidirect
2. C2×#11762793C2semidirect
3. C22×#10587504C2semidirect
4. #14537203C2semidirect
5. #14529634C2semidirect
6. #14529625C2semidirect
11. #11762833C22semidirect
13. C2×#10587507C22semidirect
16. C2×#10587504C22semidirect
18. C4×#63223C22semidirect
62. C2×#63223D4semidirect
270. C17#21219semidirect

Maximal subgroups

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

subgroupindex
#145373482
C2×#117627932
C22×#105875042
#145372032
#145296342
#145296252
C2×C17×#5352
#2121917