Finite Group Databasebeta

← #17004438

Subgroups — #17004438

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#14538784C211yesmaximal, characteristic
2#14537862C211yesmaximal, characteristic, complemented
3#14537808C211yesmaximal, characteristic, complemented
4#14527645C212yesmaximal, complemented
5#14527610C212yesmaximal, complemented
6#14527605C212yesmaximal, complemented
7#14527576C212yesmaximal, complemented
8#14527565C211yesmaximal, characteristic, complemented
9#14527202C211yesmaximal, characteristic, complemented
10#14527081C211yesmaximal, characteristic, complemented
11C17×#1490C211yesmaximal, characteristic, complemented
12#11763359C2211yescharacteristic
13#11763122C412yes—
14#11763089C412yes—
15#11763004C412yes—
16#11762943C412yes—
17C4×#63228C2212yes—
18C4×#63228C2212yes—
19#11762886C2212yescomplemented
20#11762882—22no—
21D4×#3622—22no—
22D4×#3622C2212yes—
23D4×#3622C2212yes—
24#11762874C2212yescomplemented
25#11762862C2211yescharacteristic
26#11762862C2211yescharacteristic
27#11762862C2211yescharacteristic
28#11762774—22no—
29C2×C22⋊#3622—22no—
30#11762561C2212yescomplemented
31#11762559C2212yescomplemented
32#11762538C2212yescomplemented
33#11762538C2212yescomplemented
34#11762531C2211yescharacteristic, complemented
35#11762531C2211yescharacteristic, complemented
36#11762530C2211yescharacteristic, complemented
37C8×#3622C2211yescharacteristic, complemented
38C2×C4×D4×C17C2211yescharacteristic
39C17×#432C2211yescharacteristic
40C4×M16×C17C2211yescharacteristic
41C17×D4⋊C8C2212yes—
42C17×D4⋊C8C2212yes—
43C22×D4×C17C2×C411yescharacteristic
44C23×C4×C17—21no—
45C2×#63252—21no—
46C2×M16×C17—21no—
47C4×D4×C17—21no—
48C4×D4×C17—21no—
49C4×D4×C17C2×C412yes—
50C4×D4×C17C2312yes—
51C2×C4⋊C4×C17C2×C411yescharacteristic
52C2×C22⋊C4×C17—21no—
53C2×C42×C17C2311yescharacteristic
54C2×#63228C2×C412yes—
55C2×#63228C2×C412yes—
56#10587616—21no—
57#10587616C2×C412yes—
58C22×#3622—22no—
59C22×#3622—22no—
60C2×#63218—22no—
61C2×C8×D17—22no—
62#10587561—21no—
63#10587560C2×C411yescharacteristic
64C4×D68—21no—
65C4×D68C2312yes—
66#10587555C2×C411yescharacteristic
67C17×C4⋊C8C2312yes—
68C17×C22⋊C8—21no—
69C17×#97—21no—
70C17×#97—21no—
71C4×C8×C17C2311yescharacteristic
72C4×C8×C17C2311yescharacteristic
73#10587503C2×C412yes—
74#10587503C2×C412yes—
75C22⋊#3622—22no—
76C22⋊#3622—22no—
77C22⋊#3622—21no—
78C22⋊#3622C2×C412yes—
79C22⋊#3622D412yes—
80#10587491—22no—
81#10587491—22no—
82#10587480C2×C412yes—
83#10587480C2×C412yes—
84#10587478C2×C412yes—
85#10587478C2×C412yes—
86C4⋊#3622C2×C412yes—
87C4⋊#3622D412yes—
88C4⋊#3622C2312yes—
89C4×#3622—21no—
90C4×#3622C2311yescharacteristic
91C4×#3622C2311yescharacteristic
92C4×#3622C2311yescharacteristic
93C4×#3622C2311yescharacteristic
94C24×C17—21no—
95C2×D4×C17—24no—
96C2×D4×C17—21no—
97C2×D4×C17C22×C412yes—
98C22×C4×C17—21no—
99C22×C4×C17—22no—
100C22×C4×C17—22no—
101C22×C4×C17—82no—
102C22×C4×C17C22×C411yescharacteristic
103C22×C4×C17C2×D411yescharacteristic
104C22×C4×C17#4111yescharacteristic
105C2×C22⋊D17—22no—
106#63252—82no—
107C2×D68—21no—
108C2×D68—21no—
109C2×D68C22×C412yes—
110C2×C4×D17—22no—
111C2×Dic34—21no—
112M16×C17—42no—
113C2×C8×C17—21no—
114C2×C8×C17—22no—
115C2×C8×C17—21no—
116C4⋊C4×C17—21no—
117C4⋊C4×C17C22×C412yes—
118C22⋊C4×C17—22no—
119C22⋊C4×C17—22no—
120C42×C17C22×C411yescharacteristic
121C42×C17C2×D411yescharacteristic
122C42×C17#4111yescharacteristic
123C42×C17C2411yescharacteristic
124#63228—24no—
125#63228—24no—
126#63227—22no—
127#63227—22no—
128C4⋊Dic17—21no—
129C4⋊Dic17C22×C412yes—
130C2×#3622—22no—
131C2×#3622—82no—
132C2×#3622—82no—
133C2×#3622C22×C412yes—
134C2×#3622C22×C412yes—
135C2×#3622C2×D412yes—
136C2×#3622C2×D412yes—
137#63218—82no—
138C8×D17—82no—
139#17922—171nomaximal, Sylow
140C23×C17—24no—
141C23×C17—22no—
142C23×C17—81no—
143C23×C17—81no—
144C23×C17C4×D411yescharacteristic
145C22×D17—22no—
146D4×C17—24no—
147D4×C17—22no—
148D4×C17#13114yes—
149C2×C4×C17—22no—
150C2×C4×C17—22no—
151C2×C4×C17—22no—
152C2×C4×C17—41no—
153C2×C4×C17—42no—
154C2×C4×C17—42no—
155C2×C4×C17—82no—
156C2×C4×C17—81no—
157C2×C4×C17C4×D411yescharacteristic
158C2×C4×C17C4×D411yescharacteristic
159C2×C4×C17C4×D411yescharacteristic
160C2×C4×C17C23×C411yescharacteristic
161C2×C4×C17C22×D411yescharacteristic
162C2×C4×C17C2×#4111yescharacteristic
163C22⋊D17—82no—
164C2×Dic17—22no—
165D68—22no—
166D68—22no—
167D68—81no—
168D68—81no—
169C4×D17—42no—
170C4×D17—42no—
171Dic34—81no—
172C8×C17—22no—
173C8×C17—41no—
174C8×C17—42no—
175#3622—22no—
176#3622—22no—
177#3622—22no—
178#3622—22no—
179#3622—42no—
180#3100—171no—
181#2981—171no—
182#2972—171no—
183#1586—172no—
184#1581—172no—
185#1580—172no—
186C8×D8—172no—
187#1572—171no—
188#1490D1711yescharacteristic, complemented
189#1484—171no—
190#1453—171no—
191C22×C17—21no—
192C22×C17—24no—
193C22×C17—21no—
194C22×C17—21no—
195C22×C17—42no—
196C22×C17—42no—
197C22×C17—84no—
198C22×C17C2×C4×D411yescharacteristic
199C2×D17—42no—
200C2×D17—42no—
201C2×D17—42no—
202C4×C17—21no—
203C4×C17—21no—
204C4×C17—21no—
205C4×C17—21no—
206C4×C17—42no—
207C4×C17C2×#13112yes—
208C4×C17#57612yes—
209Dic17—42no—
210#532—171no—
211C2×C4×D4C2×D1711yescharacteristic
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#431—171no—
227#431—171no—
228#408—342no—
229C2×C22⋊C8—342no—
230C4×M16C2×D1711yescharacteristic
231#335—172no—
232#334—172no—
233D4⋊C8—172no—
234D4⋊C8—172no—
235D4⋊C8C2×D1712yes—
236D4⋊C8C2×D1712yes—
237#322—171no—
238#322—171no—
239#322—171no—
240C82—171no—
241C2×C17—41no—
242C2×C17—44no—
243C2×C17#292911yescharacteristic
244C2×C17#294811yescharacteristic
245C2×C17#295611yescharacteristic
246D17—82no—
247C2×#41—341no—
248C22×D4C4×D1711yescharacteristic
249C23×C4—21no—
250C2×D8—172no—
251C2×D8—172no—
252C2×M16—21no—
253C2×M16—342no—
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—21no—
263C4×D4—21no—
264C4×D4—172no—
265C4×D4—341no—
266C4×D4C4×D1712yes—
267C4×D4C22×D1712yes—
268C2×C4⋊C4C4×D1711yescharacteristic
269C2×C22⋊C4—21no—
270C2×C42C22×D1711yescharacteristic
271#107—172no—
272#107—172no—
273C4⋊C8—172no—
274C4⋊C8—172no—
275C4⋊C8—172no—
276C4⋊C8C22×D1712yes—
277#102—172no—
278#102—172no—
279#102—172no—
280#102—172no—
281C22⋊C8—21no—
282C22⋊C8—172no—
283C22⋊C8—172no—
284C22⋊C8—342no—
285C22⋊C8—342no—
286C22⋊C8—342no—
287C22⋊C8—342no—
288C22⋊C8—341no—
289#97—21no—
290#97—21no—
291C4×C8—171no—
292C4×C8—171no—
293C4×C8—171no—
294C4×C8—171no—
295C4×C8—341no—
296C4×C8C22×D1711yescharacteristic
297C4×C8C22×D1711yescharacteristic
298C17#1792211yescharacteristic, complemented, Sylow
299C24—21no—
300#41—1362no—
301C2×Q8—341no—
302C2×D4—24no—
303C2×D4—21no—
304C2×D4—172no—
305C2×D4—342no—
306C2×D4—341no—
307C2×D4—341no—
308C2×D4C2×C4×D1712yes—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—82no—
313C22×C4—342no—
314C22×C4C2×C4×D1711yescharacteristic
315C22×C4D4×D1711yescharacteristic
316C22×C4#6325411yescharacteristic
317D8—344no—
318D8—344no—
319M16—42no—
320M16—1362no—
321C2×C8—21no—
322C2×C8—22no—
323C2×C8—21no—
324C2×C8—172no—
325C2×C8—172no—
326C2×C8—172no—
327C2×C8—172no—
328C2×C8—342no—
329C2×C8—1362no—
330C2×C8—1362no—
331C2×C8—1362no—
332C4⋊C4—21no—
333C4⋊C4—172no—
334C4⋊C4—341no—
335C4⋊C4C2×C4×D1712yes—
336C22⋊C4—22no—
337C22⋊C4—22no—
338C22⋊C4—342no—
339C22⋊C4—342no—
340C42C2×C4×D1711yescharacteristic
341C42D4×D1711yescharacteristic
342C42#6325411yescharacteristic
343C42C23×D1711yescharacteristic
344C23—24no—
345C23—22no—
346C23—81no—
347C23—81no—
348C23—342no—
349C23#1058756711yescharacteristic
350Q8—1361no—
351D4—24no—
352D4—22no—
353D4—342no—
354D4—342no—
355D4—1361no—
356D4—1361no—
357D4—1362no—
358D4#1058759314yes—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4—342no—
368C2×C4—682no—
369C2×C4—682no—
370C2×C4#1058756711yescharacteristic
371C2×C4#1058756711yescharacteristic
372C2×C4#1058756711yescharacteristic
373C2×C4C22×C4×D1711yescharacteristic
374C2×C4C2×D4×D1711yescharacteristic
375C2×C4C2×#6325411yescharacteristic
376C8—22no—
377C8—41no—
378C8—42no—
379C8—342no—
380C8—342no—
381C8—342no—
382C8—342no—
383C8—682no—
384C22—21no—
385C22—24no—
386C22—21no—
387C22—21no—
388C22—42no—
389C22—42no—
390C22—84no—
391C22—682no—
392C22—682no—
393C22—682no—
394C22C2×#1058756711yescharacteristic
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—42no—
400C4—682no—
401C4C2×#1058759312yes—
402C4#1176361312yes—
403C2—41no—
404C2—44no—
405C2—1362no—
406C2#1453747411yescharacteristic
407C2#1453763211yescharacteristic
408C2#1453769011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC42C23×D17
FittingC17×#1490C2
socleC22×C17C2×C4×D4
prime radical#14538784C2
cosocleC42×C17C24
hypercenter#1490D17
hypocenterC17#17922
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1490

Complements

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

NH ≅ G/Ntype
2. #14537862C2semidirect
3. #14537808C2semidirect
4. #14527645C2semidirect
5. #14527610C2semidirect
6. #14527605C2semidirect
7. #14527576C2semidirect
8. #14527565C2semidirect
9. #14527202C2semidirect
10. #14527081C2semidirect
11. C17×#1490C2semidirect
19. #11762886C22semidirect
24. #11762874C22semidirect
30. #11762561C22semidirect
31. #11762559C22semidirect
32. #11762538C22semidirect
33. #11762538C22semidirect
34. #11762531C22semidirect
35. #11762531C22semidirect
36. #11762530C22semidirect
37. C8×#3622C22semidirect
188. #1490D17semidirect
298. C17#17922semidirect

Maximal subgroups

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

subgroupindex
#145387842
#145378622
#145378082
#145276452
#145276102
#145276052
#145275762
#145275652
#145272022
#145270812
C17×#14902
#1792217