Finite Group Databasebeta

← #17004264

Subgroups — #17004264

Order 4352.

766 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14538780C211yesmaximal, characteristic
2#14537496C211yesmaximal, characteristic, complemented
3C2×#11762874C211yesmaximal, characteristic, complemented
4#14527651C214yesmaximal, complemented
5#14527605C214yesmaximal, complemented
6#14527554C211yesmaximal, characteristic, complemented
7#14527210C211yesmaximal, characteristic, complemented
8#14527077C211yesmaximal, characteristic, complemented
9C17×#1480C211yesmaximal, characteristic, complemented
10C2×#10587634—21no—
11C22×C8×D17—21no—
12C2×C4×D68C2211yescharacteristic
13#11763120C412yes—
14#11763086C412yes—
15#11763001C412yes—
16#11762942C412yes—
17C4×#63228C2214yes—
18#11762887—21no—
19#11762887C2212yescomplemented
20D4×#3622—21no—
21D4×#3622C2212yescomplemented
22#11762874—21no—
23#11762874—21no—
24#11762874C2212yes—
25#11762874C2212yes—
26#11762862C2211yescharacteristic
27#11762833C2211yescharacteristic
28#11762776—21no—
29C2×#10587491—21no—
30C2×C4×#3622C2211yescharacteristic
31#11762559C2214yescomplemented
32#11762538C2214yes—
33#11762531C2212yescomplemented
34#11762530C2212yescomplemented
35C17×#532C2211yescharacteristic
36C17×#431C2211yescharacteristic
37C2×C17×C4⋊C8C2211yescharacteristic
38C17×D4⋊C8C2214yescomplemented
39C2×#41×C17—21no—
40C22×D68C2×C411yescharacteristic
41C22×C4×D17—21no—
42C22×C8×C17—21no—
43C17×C4⋊Q8C2×C411yescharacteristic
44C17×#127C2×C411yescharacteristic
45C17×#124—21no—
46C17×#121—21no—
47C17×#121C2×C412yes—
48C4×D4×C17—21no—
49C4×D4×C17C2312yes—
50C2×C42×C17C2311yescharacteristic
51#10587634—44no—
52C2×#63228C2×C414yes—
53C2×#63227—21no—
54C2×C4⋊Dic17C2×C411yescharacteristic
55C2×#63223—22no—
56C2×#63223—21no—
57C22×#3622—22no—
58C22×#3622D411yescharacteristic
59C22×#3622D411yescharacteristic
60C2×C8×D17—22no—
61C2×C8×D17—22no—
62C2×C8×D17—44no—
63C4×D68—21no—
64C4×D68—21no—
65C4×D68C2×C412yes—
66C4×D68C2312yes—
67C17×C4⋊C8—21no—
68C17×C4⋊C8—21no—
69C17×C4⋊C8C2312yes—
70C17×C22⋊C8—21no—
71C4×C8×C17C2312yes—
72#10587503C2×C414yes—
73C22⋊#3622—22no—
74C22⋊#3622—22no—
75C22⋊#3622C2×C412yes—
76#10587491—22no—
77#10587491—22no—
78#10587480C2×C414yes—
79#10587478C2×C414yes—
80C4⋊#3622—21no—
81C4⋊#3622C2×C412yes—
82C4⋊#3622C2312yes—
83C4×#3622D411yescharacteristic
84C4×#3622D411yescharacteristic
85C4×#3622C2312yes—
86C4×#3622C2311yescharacteristic
87C4×#3622C2311yescharacteristic
88C23×D17—21no—
89#41×C17—42no—
90#41×C17—42no—
91C2×Q8×C17—21no—
92C2×D4×C17—22no—
93C2×D4×C17—21no—
94C2×D4×C17—21no—
95C2×D4×C17C22×C412yes—
96C22×C4×C17—22no—
97C22×C4×C17C22×C411yescharacteristic
98C22×C4×C17C2×D411yescharacteristic
99C22×C4×C17#4111yescharacteristic
100C22×Dic17—21no—
101C2×D68—21no—
102C2×D68—24no—
103C2×D68C22×C412yes—
104C2×C4×D17—22no—
105C2×C4×D17—22no—
106C2×C4×D17—42no—
107C2×C4×D17—42no—
108C2×C8×C17—22no—
109C2×C8×C17—22no—
110C2×C8×C17—42no—
111C4⋊C4×C17—21no—
112C4⋊C4×C17C22×C412yes—
113C22⋊C4×C17—22no—
114C22⋊C4×C17—22no—
115C42×C17C22×C411yescharacteristic
116C42×C17C2×D411yescharacteristic
117C42×C17#4111yescharacteristic
118C42×C17C2411yescharacteristic
119#63228—28no—
120#63227—22no—
121#63227—22no—
122C4⋊Dic17—21no—
123C4⋊Dic17C22×C412yes—
124#63223—44no—
125#63223—42no—
126C2×#3622—22no—
127C2×#3622—22no—
128C2×#3622—22no—
129C2×#3622—44no—
130C2×#3622C22×C414yes—
131C2×#3622C2×D411yescharacteristic
132C2×#3622C2×D411yescharacteristic
133C2×#3622C2×D411yescharacteristic
134C2×#3622C2×D411yescharacteristic
135C8×D17—44no—
136C8×D17—44no—
137#17907—171nomaximal, Sylow
138C23×C17—22no—
139C23×C17C4×D411yescharacteristic
140C22×D17—24no—
141C22×D17—22no—
142C22×D17—42no—
143C22×D17—42no—
144Q8×C17—42no—
145D4×C17—24no—
146D4×C17—42no—
147D4×C17—44no—
148D4×C17—42no—
149C2×C4×C17—22no—
150C2×C4×C17—22no—
151C2×C4×C17—41no—
152C2×C4×C17—42no—
153C2×C4×C17—42no—
154C2×C4×C17C4×D411yescharacteristic
155C2×C4×C17C4×D411yescharacteristic
156C2×C4×C17C4×D411yescharacteristic
157C2×C4×C17AGL(1,ℤ/8)12yes—
158C2×C4×C17AGL(1,ℤ/8)12yes—
159C2×C4×C17C23×C411yescharacteristic
160C2×C4×C17C22×D411yescharacteristic
161C2×C4×C17C2×#4111yescharacteristic
162C2×Dic17—22no—
163C2×Dic17—42no—
164D68—24no—
165D68—22no—
166D68#13114yescomplemented
167C4×D17—42no—
168C4×D17—42no—
169C4×D17—44no—
170C8×C17—42no—
171C8×C17—42no—
172#3622—24no—
173#3622—21no—
174#3622—21no—
175#3622—21no—
176#3622—21no—
177#3622—42no—
178#3100—171no—
179#2931—171no—
180C2×C8×D4—171no—
181#1586—174no—
182#1580—174no—
183#1570—171no—
184#1486—171no—
185#1480D1711yescharacteristic, complemented
186#1452—171no—
187C22×C17—21no—
188C22×C17—21no—
189C22×C17—42no—
190C22×C17—42no—
191C22×C17—42no—
192C22×C17#44212yes—
193C22×C17C2×C4×D411yescharacteristic
194C2×D17—24no—
195C2×D17—48no—
196C2×D17—42no—
197C2×D17—42no—
198C4×C17—21no—
199C4×C17—21no—
200C4×C17—21no—
201C4×C17—42no—
202C4×C17#44212yes—
203C4×C17C2×#13112yes—
204C4×C17C2×AGL(1,ℤ/8)11yescharacteristic
205C4×C17C2×AGL(1,ℤ/8)11yescharacteristic
206Dic17—42no—
207C2×#131—341no—
208C23×C8—341no—
209#532C2×D1711yescharacteristic
210C2×C4×D4—171no—
211#480—172no—
212#474—172no—
213#459—172no—
214#447—172no—
215C4×D8—174no—
216#436—172no—
217#436—341no—
218C8×D4—172no—
219C8×D4—172no—
220C8×D4—172no—
221C8×D4—341no—
222C8×D4—341no—
223C8×D4—341no—
224#431—171no—
225#431C2×D1711yescharacteristic
226#423—171no—
227C2×C4⋊C8C2×D1711yescharacteristic
228#408—341no—
229C2×C22⋊C8—341no—
230C2×C4×C8—171no—
231#334—174no—
232D4⋊C8—174no—
233D4⋊C8C2×D1714yescomplemented
234#322—172no—
235#322—172no—
236C2×C17—22no—
237C2×C17—82no—
238C2×C17#292911yescharacteristic
239C2×C17C2×#44211yescharacteristic
240C2×C17#295511yescharacteristic
241D17—44no—
242C2×#41—21no—
243C22×D4—171no—
244C23×C4—341no—
245C2×D8—174no—
246#131—684no—
247C2×M16—342no—
248C2×M16—341no—
249C22×C8—21no—
250C22×C8—171no—
251C22×C8—171no—
252C22×C8—342no—
253C22×C8—342no—
254C22×C8—342no—
255C22×C8—684no—
256C4⋊Q8C4×D1711yescharacteristic
257#127C4×D1711yescharacteristic
258#124—21no—
259#121—21no—
260#121C4×D1712yes—
261C4×D4—21no—
262C4×D4—172no—
263C4×D4—172no—
264C4×D4—341no—
265C4×D4—341no—
266C4×D4C22×D1712yes—
267C2×C4⋊C4—171no—
268C2×C22⋊C4—341no—
269C2×C42C22×D1711yescharacteristic
270#107—174no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—172no—
274C4⋊C8—172no—
275C4⋊C8—341no—
276C4⋊C8C22×D1712yes—
277#102—174no—
278#102—174no—
279C22⋊C8—21no—
280C22⋊C8—172no—
281C22⋊C8—342no—
282C22⋊C8—342no—
283C22⋊C8—342no—
284C22⋊C8—342no—
285C4×C8—171no—
286C4×C8—171no—
287C4×C8—172no—
288C4×C8—171no—
289C4×C8—171no—
290C4×C8C22×D1712yes—
291C17#1790711yescharacteristic, complemented, Sylow
292C24—341no—
293#41—42no—
294#41—42no—
295C2×Q8—21no—
296C2×D4—22no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—172no—
300C2×D4—344no—
301C2×D4—341no—
302C2×D4C2×C4×D1712yes—
303C22×C4—22no—
304C22×C4—341no—
305C22×C4—342no—
306C22×C4—342no—
307C22×C4—682no—
308C22×C4—682no—
309C22×C4C2×C4×D1711yescharacteristic
310C22×C4D4×D1711yescharacteristic
311C22×C4#6325411yescharacteristic
312D8—348no—
313M16—684no—
314M16—682no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—42no—
318C2×C8—174no—
319C2×C8—171no—
320C2×C8—171no—
321C2×C8—171no—
322C2×C8—171no—
323C2×C8—342no—
324C2×C8—342no—
325C2×C8—342no—
326C2×C8—684no—
327C2×C8—684no—
328C2×C8—684no—
329C4⋊C4—21no—
330C4⋊C4—172no—
331C4⋊C4—341no—
332C4⋊C4C2×C4×D1712yes—
333C22⋊C4—22no—
334C22⋊C4—22no—
335C22⋊C4—342no—
336C22⋊C4—342no—
337C42C2×C4×D1711yescharacteristic
338C42D4×D1711yescharacteristic
339C42#6325411yescharacteristic
340C42C23×D1711yescharacteristic
341C23—22no—
342C23—344no—
343C23—342no—
344C23—682no—
345C23—682no—
346C23#1058756711yescharacteristic
347Q8—42no—
348D4—24no—
349D4—42no—
350D4—44no—
351D4—42no—
352D4—174no—
353D4—344no—
354D4—342no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—342no—
361C2×C4—682no—
362C2×C4—682no—
363C2×C4—682no—
364C2×C4—684no—
365C2×C4#1058756711yescharacteristic
366C2×C4#1058756711yescharacteristic
367C2×C4#1058756711yescharacteristic
368C2×C4#1058759712yes—
369C2×C4#1058760012yes—
370C2×C4C22×C4×D1711yescharacteristic
371C2×C4C2×D4×D1711yescharacteristic
372C2×C4C2×#6325411yescharacteristic
373C8—42no—
374C8—42no—
375C8—344no—
376C8—341no—
377C8—341no—
378C8—341no—
379C8—341no—
380C8—682no—
381C22—21no—
382C22—21no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—344no—
387C22—688no—
388C22—682no—
389C22—682no—
390C22#1176291812yes—
391C22C2×#1058756711yescharacteristic
392C4—21no—
393C4—21no—
394C4—21no—
395C4—42no—
396C4—682no—
397C4#1176291812yes—
398C4#131×D1712yes—
399C4C2×#1058759711yescharacteristic
400C4C2×#1058760011yescharacteristic
401C2—22no—
402C2—82no—
403C2—684no—
404C2#1453748111yescharacteristic
405C2C2×#1176291811yescharacteristic
406C2#1453768311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC42C23×D17
FittingC17×#1480C2
socleC22×C17C2×C4×D4
prime radical#14538780C2
cosocleC42×C17C24
hypercenter#1480D17
hypocenterC17#17907
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1480

Complements

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

NH ≅ G/Ntype
2. #14537496C2semidirect
3. C2×#11762874C2semidirect
4. #14527651C2semidirect
5. #14527605C2semidirect
6. #14527554C2semidirect
7. #14527210C2semidirect
8. #14527077C2semidirect
9. C17×#1480C2semidirect
19. #11762887C22semidirect
21. D4×#3622C22semidirect
31. #11762559C22semidirect
33. #11762531C22semidirect
34. #11762530C22semidirect
38. C17×D4⋊C8C22semidirect
166. D68#131semidirect
185. #1480D17semidirect
233. D4⋊C8C2×D17semidirect
291. C17#17907semidirect

Maximal subgroups

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

subgroupindex
#145387802
#145374962
C2×#117628742
#145276512
#145276052
#145275542
#145272102
#145270772
C17×#14802
#1790717