Finite Group Databasebeta

← #17033944

Subgroups — #17033944

Order 4352.

1438 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#63218C211yesmaximal, characteristic
2C2×#11762856C212yesmaximal, complemented
3D17×#407C212yesmaximal
4#14528825C212yesmaximal, complemented
5#14528823C212yesmaximal, complemented
6#14528822C211yesmaximal, characteristic, complemented
7#14528820C212yesmaximal, complemented
8#14528509C212yesmaximal, complemented
9C17×#1810C211yesmaximal, characteristic, complemented
10C23×C4×D17C2211yescharacteristic
11C22×#63218—28no—
12C22×#63218C418yes—
13C22×#63218C2211yescharacteristic
14C22×#63218C2211yescharacteristic
15C22×#63218C2211yescharacteristic
16C22×#63218C2211yescharacteristic
17C2×M16×D17C2214yes—
18C23×#3622C2211yescharacteristic
19#11762856—28no—
20C2×#10587490C2212yescomplemented
21C2×#10587493C2214yescomplemented
22#11762773C2212yes—
23#11762772C2214yes—
24#11762771—24no—
25C22⋊C8×D17—24no—
26#11762574C2212yescomplemented
27#11762573C2214yescomplemented
28C23×C8×C17C2211yescharacteristic
29C2×C17×#108C2212yescomplemented
30C17×#407C2212yes—
31C17×#341C2212yescomplemented
32C24×D17Q811yescharacteristic
33C23×C4×C17C2311yescharacteristic
34C23×Dic17Q811yescharacteristic
35C22×C4×D17—24no—
36C22×C4×D17—24no—
37C22×C4×D17C2×C414yes—
38C22×C4×D17D414yes—
39C22×C4×D17Q811yescharacteristic
40C22×C4×D17Q811yescharacteristic
41C22×C4×D17C2311yescharacteristic
42C22×C4×D17C2311yescharacteristic
43C2×M16×C17C2314yes—
44C22×C8×C17—24no—
45C22×C8×C17C2×C414yes—
46C22×C8×C17C2311yescharacteristic
47C22×C8×C17C2311yescharacteristic
48C2×#63223C2314yes—
49C22×#3622—24no—
50C22×#3622C2×C414yes—
51C22×#3622C2311yescharacteristic
52C22×#3622C2311yescharacteristic
53M16×D17—28no—
54M16×D17—28no—
55C2×#63218—24no—
56C2×#63218—232no—
57C2×#63218—24no—
58C2×#63218—24no—
59C2×#63218—24no—
60C2×#63218C2×C414yes—
61C2×#63218C2×C414yes—
62C2×#63218C2×C414yes—
63C2×#63218C2×C414yes—
64C2×#63218D418yes—
65C2×C8×D17—24no—
66C17×#108—24no—
67C17×C22⋊C8—24no—
68C22⋊#3622—24no—
69#10587493—28no—
70#10587491—28no—
71#10587490—24no—
72C24×C17C2×Q811yescharacteristic
73C23×D17—24no—
74C23×D17—44no—
75C23×D17C4⋊C414yes—
76C23×D17C2×Q811yescharacteristic
77C23×D17C2×Q811yescharacteristic
78C22×C4×C17—22no—
79C22×C4×C17—22no—
80C22×C4×C17C22×C412yes—
81C22×C4×C17C2×D412yes—
82C22×C4×C17C2×Q811yescharacteristic
83C22×C4×C17C2411yescharacteristic
84C22×Dic17—24no—
85C22×Dic17C4⋊C414yes—
86C22×Dic17C2×Q811yescharacteristic
87C22×Dic17C2×Q811yescharacteristic
88C2×C4×D17—24no—
89C2×C4×D17—22no—
90C2×C4×D17—22no—
91C2×C4×D17—24no—
92C2×C4×D17—28no—
93C2×C4×D17—28no—
94C2×C4×D17—216no—
95C2×C4×D17C4⋊C414yes—
96C2×C4×D17C4⋊C414yes—
97C2×C4×D17C22×C412yes—
98C2×C4×D17C22×C412yes—
99C2×C4×D17C2×D412yes—
100C2×C4×D17C2×D414yes—
101C2×C4×D17C2×D412yes—
102C2×C4×D17#4114yes—
103M16×C17—24no—
104M16×C17—24no—
105C2×C8×C17—22no—
106C2×C8×C17—28no—
107C2×C8×C17—24no—
108C2×C8×C17C22×C412yes—
109C2×C8×C17C22×C412yes—
110C2×C8×C17C2×D414yes—
111#63223—24no—
112#63223—24no—
113C2×#3622—22no—
114C2×#3622—28no—
115C2×#3622—24no—
116C2×#3622C22×C412yes—
117C2×#3622C22×C412yes—
118C2×#3622C2×D414yes—
119#63218—28no—
120#63218—28no—
121#63218—28no—
122#63218—48no—
123#63218C22⋊C4116yes—
124C8×D17—48no—
125#20667—171nomaximal, Sylow
126C23×C17—22no—
127C23×C17—24no—
128C23×C17C2×C4⋊C412yes—
129C23×C17C22×Q811yescharacteristic
130C22×D17—22no—
131C22×D17—28no—
132C22×D17—28no—
133C22×D17—44no—
134C22×D17—42no—
135C22×D17—42no—
136C22×D17—416no—
137C22×D17C2×C4⋊C412yes—
138C22×D17C2×C4⋊C412yes—
139C22×D17C22⋊Q814yes—
140C2×C4×C17—22no—
141C2×C4×C17—22no—
142C2×C4×C17—22no—
143C2×C4×C17—24no—
144C2×C4×C17C2×C4⋊C412yes—
145C2×C4×C17C23×C411yescharacteristic
146C2×C4×C17C22×D412yes—
147C2×C4×C17C22×D411yescharacteristic
148C2×C4×C17C2×#4112yes—
149C2×Dic17—22no—
150C2×Dic17—28no—
151C2×Dic17C2×C4⋊C412yes—
152C2×Dic17C2×C4⋊C412yes—
153C2×Dic17C22⋊Q814yes—
154C4×D17—28no—
155C4×D17—22no—
156C4×D17—22no—
157C4×D17—24no—
158C4×D17—28no—
159C4×D17C2×C22⋊C414yes—
160C4×D17#11714yes—
161C4×D17#12118yes—
162C8×C17—22no—
163C8×C17—44no—
164C8×C17C2×C22⋊C414yes—
165#3622—22no—
166#3622—44no—
167#3622C2×C22⋊C414yes—
168C23×M16—171no—
169C2×#430—172no—
170C2×#407—172no—
171#1811—172no—
172#1811—172no—
173#1810—172no—
174#1810—171no—
175#1810D1711yescharacteristic, complemented
176#1738—172no—
177C22×C17—28no—
178C22×C17—22no—
179C22×C17—22no—
180C22×C17—22no—
181C22×C17#43014yescomplemented
182C22×C17C22×C4⋊C411yescharacteristic
183C22×C17C2×C22⋊Q812yes—
184C2×D17—24no—
185C2×D17—24no—
186C2×D17—22no—
187C2×D17—44no—
188C2×D17—416no—
189C2×D17—44no—
190C2×D17#38014yes—
191C4×C17—22no—
192C4×C17C22×C22⋊C411yescharacteristic
193C4×C17C2×#11711yescharacteristic
194C4×C17C2×#12112yes—
195Dic17—22no—
196Dic17#38014yes—
197C24×C4—171no—
198C22×M16—178no—
199C22×M16—171no—
200C22×M16—171no—
201C22×M16—171no—
202C22×M16—171no—
203C22×M16—174no—
204C22×M16—348no—
205C23×C8—171no—
206C23×C8C2×D1711yescharacteristic
207#430—348no—
208C2×#108—174no—
209C2×#108—172no—
210C2×#108C2×D1712yescomplemented
211#407—172no—
212#407—174no—
213#407—344no—
214#407C2×D1712yes—
215C2×C22⋊C8—344no—
216#341—174no—
217#341—172no—
218#341C2×D1712yescomplemented
219C2×C17—22no—
220C2×C17—24no—
221C2×C17C2×#38011yescharacteristic
222C2×C17C2×#43012yes—
223D17—24no—
224D17—42no—
225C25—171no—
226C23×C4—171no—
227C23×C4—171no—
228C23×C4—174no—
229C23×C4—174no—
230C23×C4—171no—
231C23×C4—171no—
232C23×C4—171no—
233C23×C4—344no—
234C23×C4—344no—
235C23×C4C22×D1711yescharacteristic
236C2×M16—174no—
237C2×M16—174no—
238C2×M16—174no—
239C2×M16—174no—
240C2×M16—178no—
241C2×M16—174no—
242C2×M16—344no—
243C2×M16—3432no—
244C2×M16—344no—
245C2×M16—344no—
246C2×M16—344no—
247C2×M16—348no—
248C2×M16—348no—
249C2×M16C22×D1714yes—
250C22×C8—24no—
251C22×C8—171no—
252C22×C8—174no—
253C22×C8—171no—
254C22×C8—344no—
255C22×C8—344no—
256C22×C8C4×D1714yes—
257C22×C8C22×D1711yescharacteristic
258C22×C8C22×D1711yescharacteristic
259#108—24no—
260#108—348no—
261#108—344no—
262C22⋊C8—24no—
263C22⋊C8—348no—
264C22⋊C8—344no—
265C17#2066711yescharacteristic, complemented, Sylow
266C24—171no—
267C24—174no—
268C24—171no—
269C24—344no—
270C24—684no—
271C24Q8×D1711yescharacteristic
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—171no—
275C22×C4—172no—
276C22×C4—174no—
277C22×C4—172no—
278C22×C4—174no—
279C22×C4—171no—
280C22×C4—174no—
281C22×C4—172no—
282C22×C4—172no—
283C22×C4—174no—
284C22×C4—174no—
285C22×C4—348no—
286C22×C4—344no—
287C22×C4—348no—
288C22×C4—3416no—
289C22×C4—344no—
290C22×C4—344no—
291C22×C4—342no—
292C22×C4—342no—
293C22×C4C2×C4×D1712yes—
294C22×C4D4×D1712yes—
295C22×C4Q8×D1711yescharacteristic
296C22×C4C23×D1711yescharacteristic
297M16—24no—
298M16—24no—
299M16—1716no—
300M16—348no—
301M16—348no—
302M16—348no—
303M16—344no—
304M16—344no—
305M16—688no—
306C2×C8—22no—
307C2×C8—28no—
308C2×C8—24no—
309C2×C8—172no—
310C2×C8—174no—
311C2×C8—172no—
312C2×C8—348no—
313C2×C8—342no—
314C2×C8—344no—
315C2×C8—688no—
316C2×C8C2×C4×D1712yes—
317C2×C8C2×C4×D1712yes—
318C2×C8D4×D1714yes—
319C23—22no—
320C23—24no—
321C23—172no—
322C23—174no—
323C23—172no—
324C23—348no—
325C23—342no—
326C23—348no—
327C23—682no—
328C23—682no—
329C23—684no—
330C23—6816no—
331C23C4⋊C4×D1712yes—
332C23C2×Q8×D1711yescharacteristic
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—172no—
338C2×C4—174no—
339C2×C4—174no—
340C2×C4—174no—
341C2×C4—178no—
342C2×C4—172no—
343C2×C4—348no—
344C2×C4—348no—
345C2×C4—348no—
346C2×C4—342no—
347C2×C4—342no—
348C2×C4—342no—
349C2×C4—344no—
350C2×C4C4⋊C4×D1712yes—
351C2×C4C22×C4×D1711yescharacteristic
352C2×C4C2×D4×D1712yes—
353C2×C4C2×D4×D1711yescharacteristic
354C2×C4#41×D1712yes—
355C8—22no—
356C8—44no—
357C8—174no—
358C8—342no—
359C8—684no—
360C8C22⋊C4×D1714yes—
361C22—28no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—174no—
366C22—344no—
367C22—342no—
368C22—344no—
369C22—684no—
370C22—6816no—
371C22—684no—
372C22#1176285614yescomplemented
373C22C2×C4⋊C4×D1711yescharacteristic
374C22C22⋊Q8×D1712yes—
375C4—22no—
376C4—174no—
377C4—342no—
378C4C2×C22⋊C4×D1711yescharacteristic
379C4#117×D1711yescharacteristic
380C4#121×D1712yes—
381C2—22no—
382C2—24no—
383C2—344no—
384C2—682no—
385C2D17×#38011yescharacteristic
386C2C2×#1176285612yes—
subgroupG/H
centerC2×C4C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1810C2
socleC22×C17C22×C4⋊C4
prime radicalC24×D17Q8
cosocleC22×C4×C17C24
hypercenter#1810D17
hypocenterC17#20667
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1810

Complements

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

NH ≅ G/Ntype
2. C2×#11762856C2semidirect
4. #14528825C2semidirect
5. #14528823C2semidirect
6. #14528822C2semidirect
7. #14528820C2semidirect
8. #14528509C2semidirect
9. C17×#1810C2semidirect
20. C2×#10587490C22semidirect
21. C2×#10587493C22semidirect
26. #11762574C22semidirect
27. #11762573C22semidirect
29. C2×C17×#108C22semidirect
31. C17×#341C22semidirect
175. #1810D17semidirect
181. C22×C17#430semidirect
210. C2×#108C2×D17semidirect
218. #341C2×D17semidirect
265. C17#20667semidirect
372. C22#11762856semidirect

Maximal subgroups

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

subgroupindex
C23×#632182
C2×#117628562
D17×#4072
#145288252
#145288232
#145288222
#145288202
#145285092
C17×#18102
#2066717