Finite Group Databasebeta

← #16925213

Subgroups — #16925213

Order 4352.

628 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14528546C211yesmaximal, characteristic
2C2×#11762546C212yesmaximal
3#14527011C211yesmaximal, characteristic
4#14526662C212yesmaximal
5C4×C17×C22⋊Q8C211yesmaximal, characteristic
6C2×C22⋊#3622C2212yes—
7C2×#10587474—21no—
8C2×#10587502—21no—
9C2×#10587502—21no—
10C2×#10587502C2212yes—
11#11762564—21no—
12#11762546C418yescomplemented
13C2×C17×C22⋊Q8C2211yescharacteristic
14C2×C4×Q8×C17C411yescharacteristic, complemented
15C22×C42×C17C2211yescharacteristic
16C17×#391C411yescharacteristic, complemented
17C17×#389C411yescharacteristic, complemented
18C17×#387—21no—
19C17×#385—21no—
20C17×#380C2211yescharacteristic
21C4×C4⋊C4×C17—21no—
22C4×C4⋊C4×C17C411yescharacteristic, complemented
23C4×C22⋊C4×C17—21no—
24C22×Q8×C17C2×C411yescharacteristic
25C23×C4×C17D411yescharacteristic
26C23×C4×C17D411yescharacteristic
27C23×C4×C17C2311yescharacteristic
28C23×Dic17—22no—
29C17×C22⋊Q8—22no—
30C17×C22⋊Q8C2×C414yes—
31C4×Q8×C17—42no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17—21no—
34C2×C4⋊C4×C17—21no—
35C2×C4⋊C4×C17C2×C411yescharacteristic
36C2×C4⋊C4×C17C2×C411yescharacteristic
37C2×C4⋊C4×C17C2×C411yescharacteristic
38C2×C22⋊C4×C17—21no—
39C2×C22⋊C4×C17—21no—
40C2×C42×C17—21no—
41C2×C42×C17—21no—
42C2×C42×C17—42no—
43C2×C42×C17C2×C411yescharacteristic
44C2×C42×C17C2×C411yescharacteristic
45C2×C42×C17D411yescharacteristic
46C2×C42×C17D411yescharacteristic
47C22×#3622—22no—
48C17×#95—21no—
49C17×#95—21no—
50C17×#95—21no—
51C17×#95—21no—
52C17×#95—21no—
53#10587502—22no—
54#10587502—42no—
55#10587502—42no—
56#10587502C2×C414yes—
57C22⋊#3622—22no—
58C22⋊#3622C2×C414yes—
59#10587474—42no—
60C24×C17C2×D411yescharacteristic
61C2×Q8×C17—21no—
62C2×Q8×C17—41no—
63C2×Q8×C17—41no—
64C2×Q8×C17C4212yes—
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17—21no—
68C22×C4×C17—22no—
69C22×C4×C17—21no—
70C22×C4×C17—22no—
71C22×C4×C17—21no—
72C22×C4×C17—21no—
73C22×C4×C17—41no—
74C22×C4×C17—41no—
75C22×C4×C17—41no—
76C22×C4×C17—41no—
77C22×C4×C17—41no—
78C22×C4×C17C22⋊C411yescharacteristic
79C22×C4×C17C22⋊C411yescharacteristic
80C22×C4×C17C22⋊C411yescharacteristic
81C22×C4×C17C22⋊C411yescharacteristic
82C22×C4×C17C22×C412yes—
83C22×C4×C17C22×C411yescharacteristic
84C22×C4×C17C2×D411yescharacteristic
85C22×C4×C17#4112yes—
86C22×Dic17—24no—
87C22×Dic17—42no—
88C22×Dic17—42no—
89C22×Dic17—42no—
90C22×Dic17—42no—
91C22×Dic17—42no—
92C22×Dic17—42no—
93C4⋊C4×C17—21no—
94C4⋊C4×C17—24no—
95C4⋊C4×C17—41no—
96C4⋊C4×C17—41no—
97C4⋊C4×C17—42no—
98C4⋊C4×C17—42no—
99C4⋊C4×C17C4212yes—
100C22⋊C4×C17—24no—
101C22⋊C4×C17—41no—
102C22⋊C4×C17—41no—
103C42×C17—22no—
104C42×C17—22no—
105C42×C17—41no—
106C42×C17—41no—
107C42×C17—42no—
108C42×C17—42no—
109C2×#3622—24no—
110C2×#3622—42no—
111C2×#3622—42no—
112#8217—171nomaximal, Sylow
113C23×C17—21no—
114C23×C17—21no—
115C23×C17—21no—
116C23×C17—21no—
117C23×C17C2×C22⋊C411yescharacteristic
118C23×C17C4×D412yes—
119C23×C17AGL(1,ℤ/8)11yescharacteristic
120C23×C17#13712yes—
121C23×C17#13711yescharacteristic
122Q8×C17—42no—
123Q8×C17—42no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—22no—
131C2×C4×C17—22no—
132C2×C4×C17—21no—
133C2×C4×C17—21no—
134C2×C4×C17—21no—
135C2×C4×C17—21no—
136C2×C4×C17—21no—
137C2×C4×C17—21no—
138C2×C4×C17—21no—
139C2×C4×C17—21no—
140C2×C4×C17—21no—
141C2×C4×C17—42no—
142C2×C4×C17—42no—
143C2×C4×C17—44no—
144C2×C4×C17—42no—
145C2×C4×C17—42no—
146C2×C4×C17—41no—
147C2×C4×C17—41no—
148C2×C4×C17—41no—
149C2×C4×C17—41no—
150C2×C4×C17—41no—
151C2×C4×C17—41no—
152C2×C4×C17—41no—
153C2×C4×C17—41no—
154C2×C4×C17—41no—
155C2×C4×C17—41no—
156C2×C4×C17#9912yes—
157C2×C4×C17C2×C4211yescharacteristic
158C2×C4×C17#11711yescharacteristic
159C2×C4×C17C4×D412yes—
160C2×Dic17—44no—
161C2×Dic17—44no—
162C2×Dic17—44no—
163C2×Dic17—82no—
164C2×Dic17—82no—
165C2×Dic17—82no—
166C2×Dic17—82no—
167#3622—44no—
168C4×C22⋊Q8D1711yescharacteristic
169#1749—171no—
170C2×#328—172no—
171#1434—171no—
172#1294—172no—
173C22×C17—22no—
174C22×C17—22no—
175C22×C17—22no—
176C22×C17—21no—
177C22×C17—21no—
178C22×C17—21no—
179C22×C17—21no—
180C22×C17—22no—
181C22×C17C4×C22⋊C411yescharacteristic
182C22×C17C2×#9911yescharacteristic
183C22×C17#41011yescharacteristic
184C22×C17#41711yescharacteristic
185C22×C17#41911yescharacteristic
186C22×C17#42111yescharacteristic
187C22×C17#42111yescharacteristic
188C22×C17#44012yes—
189C22×C17#44112yes—
190C4×C17—22no—
191C4×C17—21no—
192C4×C17—21no—
193C4×C17—22no—
194C4×C17—42no—
195C4×C17—42no—
196C4×C17—41no—
197C4×C17—41no—
198C4×C17—41no—
199C4×C17—81no—
200C4×C17—81no—
201Dic17—84no—
202C2×C22⋊Q8C2×D1711yescharacteristic
203C2×C4×Q8Dic1711yescharacteristic, complemented
204C22×C42C2×D1711yescharacteristic
205C2×C22⋊C8—172no—
206C2×#97—341no—
207#391Dic1711yescharacteristic, complemented
208#389Dic1711yescharacteristic, complemented
209#387—21no—
210#385—21no—
211#380C2×D1711yescharacteristic
212C4×C4⋊C4—21no—
213C4×C4⋊C4Dic1711yescharacteristic, complemented
214C4×C22⋊C4—21no—
215C2×#95—172no—
216C2×#95—341no—
217C2×#95—341no—
218#336—341no—
219#328—178no—
220C2×C17—22no—
221C2×C17—22no—
222C2×C17#151111yescharacteristic
223C2×C17#151211yescharacteristic
224C2×C17#151311yescharacteristic
225C2×C17#151511yescharacteristic
226C2×C17C4×#9911yescharacteristic
227C2×C17#175911yescharacteristic
228C2×C17#176311yescharacteristic
229C22×Q8C2×Dic1711yescharacteristic
230C23×C4—342no—
231C23×C4C22⋊D1711yescharacteristic
232C23×C4C22⋊D1711yescharacteristic
233C23×C4C22×D1711yescharacteristic
234C22×C8—342no—
235C22⋊Q8—22no—
236C22⋊Q8C4×D1714yes—
237C4×Q8—42no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4C2×Dic1711yescharacteristic
242C2×C4⋊C4C2×Dic1711yescharacteristic
243C2×C4⋊C4C2×Dic1711yescharacteristic
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—42no—
249C2×C42C2×Dic1711yescharacteristic
250C2×C42C2×Dic1711yescharacteristic
251C2×C42C22⋊D1711yescharacteristic
252C2×C42C22⋊D1711yescharacteristic
253C22⋊C8—174no—
254C22⋊C8—342no—
255#97—682no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—174no—
262#95—342no—
263#95—682no—
264#95—682no—
265C17#821711yescharacteristic, complemented, Sylow
266C24C2×C22⋊D1711yescharacteristic
267C2×Q8—21no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8C4×Dic1712yes—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—344no—
285C22×C4—682no—
286C22×C4—682no—
287C22×C4—682no—
288C22×C4—682no—
289C22×C4—682no—
290C22×C4—682no—
291C22×C4C22⋊Dic1711yescharacteristic
292C22×C4C22⋊Dic1711yescharacteristic
293C22×C4C22⋊Dic1711yescharacteristic
294C22×C4C22⋊Dic1711yescharacteristic
295C22×C4C2×C4×D1712yes—
296C22×C4#6325212yes—
297C22×C4C22×Dic1711yescharacteristic
298C22×C4C2×C22⋊D1711yescharacteristic
299C2×C8—344no—
300C2×C8—682no—
301C2×C8—682no—
302C4⋊C4—21no—
303C4⋊C4—24no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4C4×Dic1712yes—
309C22⋊C4—24no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C42—22no—
313C42—22no—
314C42—41no—
315C42—41no—
316C42—42no—
317C42—42no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23C4×C22⋊D1712yes—
323C23#1058762812yes—
324C23#1058763511yescharacteristic
325C23#1058763711yescharacteristic
326C23C2×C22⋊Dic1711yescharacteristic
327Q8—42no—
328Q8—42no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—684no—
362C2×C4—684no—
363C2×C4—684no—
364C2×C4—1362no—
365C2×C4—1362no—
366C2×C4—1362no—
367C2×C4—1362no—
368C2×C4#1058750512yes—
369C2×C4C2×C4×Dic1711yescharacteristic
370C2×C4#1058761211yescharacteristic
371C2×C4C4×C22⋊D1712yes—
372C8—684no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—22no—
381C22C4×C22⋊Dic1711yescharacteristic
382C22C2×#1058750511yescharacteristic
383C22#1176278311yescharacteristic
384C22#1176281011yescharacteristic
385C22#1176281811yescharacteristic
386C22#1176282611yescharacteristic
387C22#1176282611yescharacteristic
388C22#1176290712yes—
389C22#1176291412yes—
390C4—22no—
391C4—21no—
392C4—21no—
393C4—22no—
394C4—42no—
395C4—42no—
396C4—41no—
397C4—41no—
398C4—41no—
399C4—81no—
400C4—81no—
401C4—1364no—
402C2—22no—
403C2—22no—
404C2#1452731711yescharacteristic
405C2#1452732211yescharacteristic
406C2#1452732711yescharacteristic
407C2#1452733511yescharacteristic
408C2C4×#1058750511yescharacteristic
409C2#1452859411yescharacteristic
410C2#1452861411yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic17
commutatorC2×C4×C17C2×C42
FrattiniC23×C4C22×D17
FittingC4×C17×C22⋊Q8C2
socleC23×C17C2×C22⋊C4
prime radicalC24×C17C2×D4
cosocleC23×C4×C17C23
hypercenterC4×C22⋊Q8D17
hypocenterC17#8217
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
12. #11762546C4semidirect
14. C2×C4×Q8×C17C4semidirect
16. C17×#391C4semidirect
17. C17×#389C4semidirect
22. C4×C4⋊C4×C17C4semidirect
203. C2×C4×Q8Dic17semidirect
207. #391Dic17semidirect
208. #389Dic17semidirect
213. C4×C4⋊C4Dic17semidirect
265. C17#8217semidirect

Maximal subgroups

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

subgroupindex
#145285462
C2×#117625462
#145270112
#145266622
C4×C17×C22⋊Q82
#821717