Finite Group Databasebeta

← #17019562

Subgroups — #17019562

Order 4352.

728 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#14539902C211yesmaximal, characteristic, complemented
2C2×#11763014C211yesmaximal, characteristic, complemented
3#14537808C211yesmaximal, characteristic, complemented
4#14528219C214yesmaximal, complemented
5#14528171C214yesmaximal, complemented
6#14527560C211yesmaximal, characteristic, complemented
7#14527457C211yesmaximal, characteristic, complemented
8#14527433C211yesmaximal, characteristic, complemented
9C17×#3036C211yesmaximal, characteristic
10C22×#63229—21no—
11C2×#10587555C2211yescharacteristic
12#11763212C2212yescomplemented
13#11763155C2212yescomplemented
14#11763120—22no—
15#11763089—22no—
16#11763014—22no—
17#11763014C2214yes—
18D4×#3622—22no—
19D4×#3622C2214yes—
20#11762862C2211yescharacteristic
21#11762862C2211yescharacteristic
22C2×C4⋊#3622C2211yescharacteristic
23C2×#10587481—21no—
24C2×C22⋊#3622—22no—
25#11762561C2214yescomplemented
26#11762556C2214yescomplemented
27#11762555C2212yescomplemented
28#11762553C2212yescomplemented
29C2×C17×AGL(1,ℤ/8)—21no—
30C17×#532C2211yescharacteristic
31C2×C4×D4×C17C2211yescharacteristic
32C17×#463—22no—
33C17×#461C2212yes—
34C17×#459C2212yes—
35C17×#423C2211yescharacteristic
36C17×#417—21no—
37C2×#41×C17—21no—
38C22×D4×C17D411yescharacteristic
39C23×C4×C17—21no—
40C22×Dic34—21no—
41C17×AGL(1,ℤ/8)—44no—
42C2×SD16×C17—22no—
43C2×D8×C17—22no—
44C2×M16×C17—21no—
45C17×C4⋊Q8C2311yescharacteristic
46C17×#127C2311yescharacteristic
47C17×#124—21no—
48C17×#121—21no—
49C17×#121—21no—
50C4×D4×C17—21no—
51C4×D4×C17—21no—
52C4×D4×C17—21no—
53C4×D4×C17—21no—
54C4×D4×C17D412yes—
55C4×D4×C17C2312yes—
56C2×C4⋊C4×C17D411yescharacteristic
57C2×C22⋊C4×C17—21no—
58C2×C42×C17C2311yescharacteristic
59C2×#63229—24no—
60C2×#63229—44no—
61C2×C4⋊Dic17—21no—
62C2×C4⋊Dic17—21no—
63C22×#3622—24no—
64C22×#3622—21no—
65#10587555—21no—
66#10587555—21no—
67#10587555—21no—
68#10587555C2312yes—
69C17×C4⋊C8—21no—
70C17×C4⋊C8C2312yes—
71C17×#103—22no—
72C17×#102—22no—
73#10587503—24no—
74C22⋊#3622—24no—
75C22⋊#3622—21no—
76C22⋊#3622D412yes—
77#10587481—24no—
78#10587478—24no—
79C4⋊#3622—21no—
80C4⋊#3622D412yes—
81C4⋊#3622C2312yes—
82C4⋊#3622C2312yes—
83C4×#3622—21no—
84C4×#3622C2312yes—
85C24×C17—21no—
86#41×C17—42no—
87#41×C17—42no—
88C2×Q8×C17—21no—
89C2×D4×C17—21no—
90C2×D4×C17—22no—
91C2×D4×C17—21no—
92C2×D4×C17—21no—
93C2×D4×C17—21no—
94C2×D4×C17—41no—
95C2×D4×C17D814yes—
96C2×D4×C17C2×D412yes—
97C22×C4×C17—21no—
98C22×C4×C17—22no—
99C22×C4×C17—22no—
100C22×C4×C17—41no—
101C22×C4×C17—82no—
102C22×C4×C17C2×D411yescharacteristic
103C22×C4×C17C2×D411yescharacteristic
104C22×C4×C17#4111yescharacteristic
105C22×Dic17—41no—
106C2×Dic34—22no—
107C2×Dic34—42no—
108C2×Dic34—42no—
109C2×Dic34—41no—
110SD16×C17—44no—
111D8×C17—44no—
112M16×C17—42no—
113C2×C8×C17—22no—
114C4⋊C4×C17—21no—
115C4⋊C4×C17—21no—
116C4⋊C4×C17—21no—
117C4⋊C4×C17C2×D412yes—
118C22⋊C4×C17—22no—
119C22⋊C4×C17—22no—
120C22⋊C4×C17—41no—
121C22⋊C4×C17—41no—
122C42×C17C2×D411yescharacteristic
123C42×C17C2×D411yescharacteristic
124C42×C17#4111yescharacteristic
125C42×C17C2411yescharacteristic
126#63229—48no—
127C4⋊Dic17—22no—
128C4⋊Dic17—22no—
129C4⋊Dic17—41no—
130C4⋊Dic17—41no—
131C2×#3622—22no—
132C2×#3622—22no—
133C2×#3622—42no—
134C2×#3622—84no—
135C2×#3622C2×D414yes—
136#19392—171nomaximal, Sylow
137C23×C17—22no—
138C23×C17—24no—
139C23×C17—41no—
140C23×C17—81no—
141C23×C17—81no—
142C23×C17#12111yescharacteristic
143Q8×C17—41no—
144Q8×C17—41no—
145D4×C17—22no—
146D4×C17—24no—
147D4×C17—41no—
148D4×C17—41no—
149D4×C17—42no—
150D4×C17—81no—
151D4×C17—82no—
152D4×C17C2×D814yes—
153C2×C4×C17—22no—
154C2×C4×C17—22no—
155C2×C4×C17—41no—
156C2×C4×C17—44no—
157C2×C4×C17—41no—
158C2×C4×C17—82no—
159C2×C4×C17—81no—
160C2×C4×C17—82no—
161C2×C4×C17#12111yescharacteristic
162C2×C4×C17#12111yescharacteristic
163C2×C4×C17#12111yescharacteristic
164C2×C4×C17C2×D812yes—
165C2×C4×C17#13512yes—
166C2×C4×C17C22×D411yescharacteristic
167C2×C4×C17C22×D411yescharacteristic
168C2×C4×C17C2×#4111yescharacteristic
169C2×Dic17—42no—
170C2×Dic17—82no—
171Dic34—42no—
172Dic34—42no—
173Dic34—82no—
174C8×C17—42no—
175#3622—24no—
176#3622—42no—
177#3622—42no—
178#3226—171no—
179#3036D1711yescharacteristic
180C2×#461—171no—
181#2972—171no—
182#1681—174no—
183#1673—174no—
184#1571—171no—
185#1546—171no—
186#1540—171no—
187C22×C17—21no—
188C22×C17—21no—
189C22×C17—24no—
190C22×C17—42no—
191C22×C17—42no—
192C22×C17—84no—
193C22×C17—81no—
194C22×C17—81no—
195C22×C17—81no—
196C22×C17#46612yes—
197C22×C17C2×#12111yescharacteristic
198C4×C17—21no—
199C4×C17—21no—
200C4×C17—21no—
201C4×C17—42no—
202C4×C17—81no—
203C4×C17#46612yes—
204C4×C17C22×D811yescharacteristic
205C4×C17C2×#13511yescharacteristic
206C4×C17#57712yes—
207Dic17—82no—
208C2×AGL(1,ℤ/8)—21no—
209C22×SD16—341no—
210#532C2×D1711yescharacteristic
211C2×C4⋊Q8—171no—
212C2×C4×D4C2×D1711yescharacteristic
213#500—172no—
214#486—172no—
215#480—342no—
216#474—342no—
217#463—22no—
218#461—174no—
219#461—342no—
220#461C2×D1712yes—
221#459C2×D1712yes—
222C8×D4—174no—
223C8×D4—342no—
224#431—171no—
225#431—171no—
226#423C2×D1711yescharacteristic
227C2×C4⋊C8—171no—
228#417—21no—
229C2×#103—341no—
230C2×C22⋊C8—342no—
231#335—174no—
232#332—172no—
233#332—174no—
234#331—172no—
235C2×C17—22no—
236C2×C17—44no—
237C2×C17—161no—
238C2×C17#304211yescharacteristic
239C2×C17C2×#46611yescharacteristic
240C2×C17#306911yescharacteristic
241C2×#41—21no—
242C22×Q8—341no—
243C22×D4C22⋊D1711yescharacteristic
244C23×C4—21no—
245AGL(1,ℤ/8)—44no—
246C2×SD16—22no—
247C2×SD16—344no—
248C2×SD16—684no—
249C2×D8—22no—
250C2×M16—21no—
251C22×C8—344no—
252C22×C8—341no—
253C4⋊Q8—172no—
254C4⋊Q8—341no—
255C4⋊Q8—341no—
256C4⋊Q8—341no—
257C4⋊Q8C22×D1711yescharacteristic
258#127C22×D1711yescharacteristic
259#124—21no—
260#121—21no—
261#121—21no—
262C4×D4—21no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—21no—
266C4×D4C22⋊D1712yes—
267C4×D4C22×D1712yes—
268C2×C4⋊C4—341no—
269C2×C4⋊C4—341no—
270C2×C4⋊C4C22⋊D1711yescharacteristic
271C2×C22⋊C4—21no—
272C2×C42C22×D1711yescharacteristic
273#107—344no—
274C4⋊C8—21no—
275C4⋊C8—172no—
276C4⋊C8—172no—
277C4⋊C8—172no—
278C4⋊C8—341no—
279C4⋊C8C22×D1712yes—
280#103—22no—
281#103—344no—
282#102—22no—
283#102—344no—
284C22⋊C8—172no—
285C22⋊C8—344no—
286C22⋊C8—341no—
287C4×C8—172no—
288C4×C8—341no—
289C17#1939211yescharacteristic, complemented, Sylow
290C24—21no—
291#41—42no—
292#41—42no—
293C2×Q8—21no—
294C2×Q8—342no—
295C2×Q8—681no—
296C2×Q8—682no—
297C2×Q8—682no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—22no—
303C2×D4—41no—
304C2×D4#6322814yes—
305C2×D4C2×C22⋊D1712yes—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—41no—
310C22×C4—82no—
311C22×C4—681no—
312C22×C4D4×D1711yescharacteristic
313C22×C4#6325411yescharacteristic
314C22×C4C2×C22⋊D1711yescharacteristic
315SD16—44no—
316SD16—688no—
317D8—44no—
318M16—42no—
319C2×C8—22no—
320C2×C8—174no—
321C2×C8—342no—
322C2×C8—342no—
323C2×C8—682no—
324C2×C8—1364no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—342no—
329C4⋊C4—342no—
330C4⋊C4—681no—
331C4⋊C4—681no—
332C4⋊C4C2×C22⋊D1712yes—
333C22⋊C4—22no—
334C22⋊C4—22no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C42D4×D1711yescharacteristic
338C42#6325411yescharacteristic
339C42C2×C22⋊D1711yescharacteristic
340C42C23×D1711yescharacteristic
341C23—22no—
342C23—24no—
343C23—41no—
344C23—81no—
345C23—81no—
346C23#1058762511yescharacteristic
347Q8—41no—
348Q8—41no—
349Q8—682no—
350Q8—682no—
351Q8—1362no—
352D4—24no—
353D4—22no—
354D4—41no—
355D4—41no—
356D4—42no—
357D4—81no—
358D4—82no—
359D4C2×#6322814yes—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—44no—
364C2×C4—41no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4—82no—
368C2×C4—682no—
369C2×C4—1362no—
370C2×C4C2×#6322812yes—
371C2×C4#1058762511yescharacteristic
372C2×C4#1058762511yescharacteristic
373C2×C4#1058762511yescharacteristic
374C2×C4#1058763612yes—
375C2×C4C2×D4×D1711yescharacteristic
376C2×C4C2×#6325411yescharacteristic
377C2×C4C22×C22⋊D1711yescharacteristic
378C8—42no—
379C8—344no—
380C8—682no—
381C8—682no—
382C22—21no—
383C22—21no—
384C22—24no—
385C22—42no—
386C22—42no—
387C22—84no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22#1176304312yes—
392C22C2×#1058762511yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—42no—
397C4—81no—
398C4—1362no—
399C4#1176304312yes—
400C4C22×#6322811yescharacteristic
401C4C2×#1058763611yescharacteristic
402C4#1176361912yes—
403C2—22no—
404C2—44no—
405C2—161no—
406C2#1453829811yescharacteristic
407C2C2×#1176304311yescharacteristic
408C2#1453852311yescharacteristic
subgroupG/H
centerC22C2×#10587625
commutator, cosocleC42×C17C24
FrattiniC42C23×D17
Fitting, prime radicalC17×#3036C2
socleC22×C17C2×#121
hypercenter#3036D17
hypocenterC17#19392
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3036

Complements

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

NH ≅ G/Ntype
1. #14539902C2semidirect
2. C2×#11763014C2semidirect
3. #14537808C2semidirect
4. #14528219C2semidirect
5. #14528171C2semidirect
6. #14527560C2semidirect
7. #14527457C2semidirect
8. #14527433C2semidirect
12. #11763212C22semidirect
13. #11763155C22semidirect
25. #11762561C22semidirect
26. #11762556C22semidirect
27. #11762555C22semidirect
28. #11762553C22semidirect
289. C17#19392semidirect

Maximal subgroups

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

subgroupindex
#145399022
C2×#117630142
#145378082
#145282192
#145281712
#145275602
#145274572
#145274332
C17×#30362
#1939217