Finite Group Databasebeta

← #17004442

Subgroups — #17004442

Order 4352.

678 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14538785C211yesmaximal, characteristic
2#14537862C211yesmaximal, characteristic, complemented
3#14537807C211yesmaximal, characteristic, complemented
4#14527647C212yesmaximal, complemented
5#14527612C212yesmaximal, complemented
6#14527607C212yesmaximal, complemented
7#14527578C212yesmaximal, complemented
8#14527226C211yesmaximal, characteristic, complemented
9#14527200C211yesmaximal, characteristic, complemented
10#14527079C211yesmaximal, characteristic, complemented
11C17×#1572C211yesmaximal, characteristic, complemented
12#11763359C2211yescharacteristic
13C2×C4×D68C2211yescharacteristic
14#11763123C412yes—
15#11763090C412yes—
16#11763005C412yes—
17#11762945C412yes—
18C4×D136C2212yes—
19C4×D136C2212yes—
20#11762886C2212yescomplemented
21#11762882—22no—
22C8×D68—22no—
23#11762874C2212yes—
24#11762874C2212yes—
25#11762874C2212yescomplemented
26#11762862C2211yescharacteristic
27#11762774—22no—
28C22⋊C8×D17—22no—
29C4×#63223C2211yescharacteristic
30#11762536C2212yes—
31#11762536C2212yes—
32#11762536C2212yescomplemented
33#11762536C2212yescomplemented
34#11762531C2211yescharacteristic, complemented
35#11762531C2211yescharacteristic, complemented
36#11762530C2211yescharacteristic, complemented
37C8×#3622C2211yescharacteristic, complemented
38C17×#432C2211yescharacteristic
39C17×#431C2211yescharacteristic
40C17×#431C2211yescharacteristic
41C17×#335C2212yescomplemented
42C17×#334C2212yescomplemented
43C2×#63252—21no—
44C22×D68C2×C411yescharacteristic
45C22×C4×D17—21no—
46C2×C42×C17C2311yescharacteristic
47#10587616—21no—
48#10587616C2×C412yes—
49C2×#63227—21no—
50C2×C4⋊Dic17C2×C411yescharacteristic
51C2×#63223—21no—
52C2×D136C2×C412yes—
53C2×D136C2×C412yes—
54C2×#63218—22no—
55C2×C8×D17—22no—
56C2×C8×D17—22no—
57C2×C8×D17—22no—
58#10587561—21no—
59#10587560C2×C411yescharacteristic
60C4×D68—21no—
61C4×D68—21no—
62C4×D68—21no—
63C4×D68C2×C412yes—
64C4×D68C2312yes—
65C4×D68C2312yes—
66#10587555C2×C411yescharacteristic
67C17×C4⋊C8C2×C412yes—
68C17×C4⋊C8C2312yes—
69C17×C4⋊C8C2312yes—
70C17×C22⋊C8—21no—
71C17×C22⋊C8—21no—
72C17×C22⋊C8C2×C412yes—
73C17×#97—21no—
74C4×C8×C17—21no—
75C4×C8×C17C2311yescharacteristic
76C4×C8×C17C2311yescharacteristic
77C22⋊#3622D412yes—
78#10587492C2×C412yes—
79#10587492C2×C412yes—
80#10587492C2×C412yes—
81#10587492C2×C412yes—
82#10587491—22no—
83#10587491—22no—
84#10587491—22no—
85#10587491—22no—
86#10587489C2×C412yes—
87#10587489C2×C412yes—
88C4⋊#3622D412yes—
89#10587474—21no—
90C4×#3622C2311yescharacteristic
91C4×#3622C2311yescharacteristic
92C4×#3622C2311yescharacteristic
93C4×#3622C2311yescharacteristic
94C23×D17—21no—
95C22×C4×C17C22×C411yescharacteristic
96C22×C4×C17C2×D411yescharacteristic
97C22×C4×C17#4111yescharacteristic
98C2×C22⋊D17—22no—
99C22×Dic17—21no—
100#63252—82no—
101C2×D68—21no—
102C2×D68—24no—
103C2×D68—21no—
104C2×D68—21no—
105C2×D68C22×C412yes—
106C2×D68C22×C412yes—
107C2×C4×D17—22no—
108C2×C4×D17—22no—
109C2×C4×D17—22no—
110C2×C4×D17—82no—
111C2×Dic34—21no—
112C2×C8×C17—22no—
113C2×C8×C17—22no—
114C2×C8×C17C22×C412yes—
115C2×C8×C17C22×C412yes—
116C42×C17C22×C411yescharacteristic
117C42×C17C2×D411yescharacteristic
118C42×C17#4111yescharacteristic
119C42×C17C2411yescharacteristic
120#63227—22no—
121#63227—22no—
122#63227—22no—
123#63227—22no—
124C4⋊Dic17—21no—
125C4⋊Dic17—21no—
126C4⋊Dic17C22×C412yes—
127C4⋊Dic17C22×C412yes—
128#63223—42no—
129C2×#3622—21no—
130C2×#3622—21no—
131C2×#3622C2×D412yes—
132C2×#3622C2×D412yes—
133D136—24no—
134D136—24no—
135#63218—82no—
136C8×D17—82no—
137C8×D17—82no—
138C8×D17—82no—
139#17922—171nomaximal, Sylow
140C23×C17C4×D411yescharacteristic
141C22×D17—22no—
142C22×D17—24no—
143C22×D17—22no—
144C22×D17—81no—
145C22×D17—81no—
146C2×C4×C17—22no—
147C2×C4×C17—22no—
148C2×C4×C17—41no—
149C2×C4×C17C4×D411yescharacteristic
150C2×C4×C17C4×D411yescharacteristic
151C2×C4×C17C4×D411yescharacteristic
152C2×C4×C17C23×C411yescharacteristic
153C2×C4×C17C22×D411yescharacteristic
154C2×C4×C17C2×#4111yescharacteristic
155C22⋊D17—82no—
156C2×Dic17—22no—
157C2×Dic17—22no—
158C2×Dic17—81no—
159D68—24no—
160D68—22no—
161D68—22no—
162D68—22no—
163D68—81no—
164D68—81no—
165D68#13114yes—
166C4×D17—42no—
167C4×D17—42no—
168C4×D17—42no—
169C4×D17—42no—
170C4×D17—82no—
171Dic34—81no—
172C8×C17—22no—
173C8×C17—22no—
174C8×C17—42no—
175C8×C17—42no—
176#3622—22no—
177#3622—22no—
178#3622—22no—
179#3622—41no—
180#3100—171no—
181#2981—171no—
182#2972—171no—
183#1586—172no—
184#1581—172no—
185#1580—172no—
186C8×D8—172no—
187#1572D1711yescharacteristic, complemented
188#1490—171no—
189#1484—171no—
190#1453—171no—
191C22×C17—21no—
192C22×C17—21no—
193C22×C17—21no—
194C22×C17C2×C4×D411yescharacteristic
195C2×D17—24no—
196C2×D17—42no—
197C2×D17—42no—
198C2×D17—42no—
199C2×D17—42no—
200C2×D17—42no—
201C2×D17—84no—
202C4×C17—21no—
203C4×C17—21no—
204C4×C17—21no—
205C4×C17—21no—
206C4×C17C2×#13112yes—
207C4×C17#57612yes—
208Dic17—42no—
209Dic17—42no—
210#532—171no—
211C2×C4×D4—171no—
212#480—172no—
213#474—172no—
214#459—172no—
215#447—172no—
216C4×D8—172no—
217C4×D8—172no—
218#436—172no—
219#435—342no—
220C8×D4—172no—
221C8×D4—172no—
222C8×D4—172no—
223C8×D4—342no—
224#432C2×D1711yescharacteristic
225#431—171no—
226#431C2×D1711yescharacteristic
227#431C2×D1711yescharacteristic
228#408—342no—
229C2×C22⋊C8—342no—
230C4×M16—171no—
231#335C2×D1712yescomplemented
232#334C2×D1712yescomplemented
233D4⋊C8—172no—
234D4⋊C8—172no—
235D4⋊C8—172no—
236D4⋊C8—172no—
237#322—171no—
238#322—171no—
239#322—171no—
240C82—171no—
241C2×C17—41no—
242C2×C17#292911yescharacteristic
243C2×C17#294811yescharacteristic
244C2×C17#295611yescharacteristic
245D17—44no—
246D17—82no—
247C2×#41—341no—
248C22×D4—171no—
249C23×C4—341no—
250C2×D8—172no—
251C2×D8—172no—
252C2×M16—342no—
253C2×M16—341no—
254C22×C8—342no—
255C22×C8—342no—
256C22×C8—342no—
257C4⋊Q8—171no—
258#127—171no—
259#124—341no—
260#121—172no—
261#121—341no—
262C4×D4—172no—
263C4×D4—172no—
264C4×D4—172no—
265C4×D4—341no—
266C4×D4—341no—
267C4×D4—341no—
268C2×C4⋊C4—171no—
269C2×C22⋊C4—341no—
270C2×C42C22×D1711yescharacteristic
271#107—172no—
272#107—172no—
273C4⋊C8—172no—
274C4⋊C8C4×D1712yes—
275C4⋊C8C22×D1712yes—
276C4⋊C8C22×D1712yes—
277#102—172no—
278#102—172no—
279#102—172no—
280#102—172no—
281C22⋊C8—21no—
282C22⋊C8—21no—
283C22⋊C8—172no—
284C22⋊C8—342no—
285C22⋊C8—342no—
286C22⋊C8—342no—
287C22⋊C8—342no—
288C22⋊C8C4×D1712yes—
289#97—21no—
290#97—341no—
291C4×C8—21no—
292C4×C8—171no—
293C4×C8—171no—
294C4×C8—171no—
295C4×C8—171no—
296C4×C8C22×D1711yescharacteristic
297C4×C8C22×D1711yescharacteristic
298C17#1792211yescharacteristic, complemented, Sylow
299C24—341no—
300#41—1362no—
301C2×Q8—341no—
302C2×D4—172no—
303C2×D4—172no—
304C2×D4—344no—
305C2×D4—342no—
306C2×D4—341no—
307C2×D4—341no—
308C2×D4—341no—
309C22×C4—341no—
310C22×C4—342no—
311C22×C4—342no—
312C22×C4—342no—
313C22×C4—1362no—
314C22×C4C2×C4×D1711yescharacteristic
315C22×C4D4×D1711yescharacteristic
316C22×C4#6325611yescharacteristic
317D8—344no—
318D8—344no—
319M16—682no—
320M16—1362no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—172no—
324C2×C8—172no—
325C2×C8—341no—
326C2×C8—341no—
327C2×C8—1362no—
328C2×C8—1362no—
329C2×C8—1362no—
330C2×C8C2×C4×D1712yes—
331C2×C8C2×C4×D1712yes—
332C4⋊C4—172no—
333C4⋊C4—172no—
334C4⋊C4—341no—
335C4⋊C4—341no—
336C22⋊C4—342no—
337C22⋊C4—342no—
338C22⋊C4—342no—
339C22⋊C4—342no—
340C42C2×C4×D1711yescharacteristic
341C42D4×D1711yescharacteristic
342C42#6325611yescharacteristic
343C42C23×D1711yescharacteristic
344C23—344no—
345C23—342no—
346C23—342no—
347C23—1361no—
348C23—1361no—
349C23#1058757911yescharacteristic
350Q8—1361no—
351D4—174no—
352D4—344no—
353D4—342no—
354D4—342no—
355D4—342no—
356D4—1361no—
357D4—1361no—
358D4—1362no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—342no—
363C2×C4—342no—
364C2×C4—682no—
365C2×C4—682no—
366C2×C4—682no—
367C2×C4—682no—
368C2×C4—1362no—
369C2×C4—1361no—
370C2×C4#1058757911yescharacteristic
371C2×C4#1058757911yescharacteristic
372C2×C4#1058757911yescharacteristic
373C2×C4C22×C4×D1711yescharacteristic
374C2×C4C2×D4×D1711yescharacteristic
375C2×C4C2×#6325611yescharacteristic
376C8—22no—
377C8—22no—
378C8—42no—
379C8—42no—
380C8—342no—
381C8—342no—
382C8—342no—
383C8—681no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—344no—
388C22—682no—
389C22—682no—
390C22—682no—
391C22—682no—
392C22—682no—
393C22—1364no—
394C22C2×#1058757911yescharacteristic
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—682no—
400C4—682no—
401C4#131×D1712yes—
402C4#1176361112yes—
403C2—41no—
404C2—684no—
405C2—1362no—
406C2#1453747711yescharacteristic
407C2#1453763511yescharacteristic
408C2#1453769311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC42C23×D17
FittingC17×#1572C2
socleC22×C17C2×C4×D4
prime radical#14538785C2
cosocleC42×C17C24
hypercenter#1572D17
hypocenterC17#17922
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1572

Complements

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

NH ≅ G/Ntype
2. #14537862C2semidirect
3. #14537807C2semidirect
4. #14527647C2semidirect
5. #14527612C2semidirect
6. #14527607C2semidirect
7. #14527578C2semidirect
8. #14527226C2semidirect
9. #14527200C2semidirect
10. #14527079C2semidirect
11. C17×#1572C2semidirect
20. #11762886C22semidirect
25. #11762874C22semidirect
32. #11762536C22semidirect
33. #11762536C22semidirect
34. #11762531C22semidirect
35. #11762531C22semidirect
36. #11762530C22semidirect
37. C8×#3622C22semidirect
41. C17×#335C22semidirect
42. C17×#334C22semidirect
187. #1572D17semidirect
231. #335C2×D17semidirect
232. #334C2×D17semidirect
298. C17#17922semidirect

Maximal subgroups

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

subgroupindex
#145387852
#145378622
#145378072
#145276472
#145276122
#145276072
#145275782
#145272262
#145272002
#145270792
C17×#15722
#1792217