Finite Group Databasebeta

← #16934404

Subgroups — #16934404

Order 4352.

598 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14528546C211yesmaximal, characteristic
2#14527014C211yesmaximal, characteristic
3#14526662C212yesmaximal
4#14526662C212yesmaximal
5C17×#2851C211yesmaximal, characteristic
6C22×C4⋊Dic17—21no—
7C2×C22⋊#3622C2212yes—
8C2×#10587474—21no—
9C2×#10587502—21no—
10C2×#10587502C2212yes—
11#11762564—21no—
12C2×C17×#125C411yescharacteristic, complemented
13C22×C42×C17C2211yescharacteristic
14C17×#389C411yescharacteristic, complemented
15C17×#389C411yescharacteristic, complemented
16C17×#387—21no—
17C17×#387—21no—
18C17×#385—21no—
19C17×#385—21no—
20C17×#382C411yescharacteristic, complemented
21C17×#380C2211yescharacteristic
22C17×#380C2211yescharacteristic
23C23×C4×C17D411yescharacteristic
24C23×C4×C17D411yescharacteristic
25C23×C4×C17C2311yescharacteristic
26C23×Dic17—22no—
27C17×#125—42no—
28C2×C4⋊C4×C17—21no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17C2×C411yescharacteristic
33C2×C4⋊C4×C17C2×C411yescharacteristic
34C2×C4⋊C4×C17C2×C411yescharacteristic
35C2×C4⋊C4×C17C2×C411yescharacteristic
36C2×C22⋊C4×C17—21no—
37C2×C22⋊C4×C17—21no—
38C2×C42×C17—42no—
39C2×C42×C17C2×C411yescharacteristic
40C2×C42×C17C2×C411yescharacteristic
41C2×C42×C17D411yescharacteristic
42C2×C42×C17D411yescharacteristic
43C2×C4⋊Dic17—24no—
44C2×C4⋊Dic17—42no—
45C2×C4⋊Dic17—42no—
46C22×#3622—22no—
47C17×#95—21no—
48C17×#95—21no—
49C17×#95—21no—
50C17×#95—21no—
51C17×#95—21no—
52C17×#95—21no—
53#10587502—24no—
54#10587502—42no—
55C22⋊#3622—22no—
56C22⋊#3622Q814yes—
57#10587474—42no—
58C24×C17C2×D411yescharacteristic
59C22×C4×C17—21no—
60C22×C4×C17—21no—
61C22×C4×C17—22no—
62C22×C4×C17—22no—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17—21no—
68C22×C4×C17—21no—
69C22×C4×C17—41no—
70C22×C4×C17—41no—
71C22×C4×C17—41no—
72C22×C4×C17—41no—
73C22×C4×C17C22⋊C411yescharacteristic
74C22×C4×C17C22⋊C411yescharacteristic
75C22×C4×C17C22⋊C411yescharacteristic
76C22×C4×C17C22⋊C411yescharacteristic
77C22×C4×C17C22×C411yescharacteristic
78C22×C4×C17C2×D411yescharacteristic
79C22×C4×C17C2×Q812yes—
80C22×C4×C17#4112yes—
81C22×Dic17—24no—
82C22×Dic17—42no—
83C22×Dic17—42no—
84C22×Dic17—42no—
85C22×Dic17—42no—
86C22×Dic17—42no—
87C22×Dic17—42no—
88C4⋊C4×C17—41no—
89C4⋊C4×C17—41no—
90C4⋊C4×C17—41no—
91C4⋊C4×C17—41no—
92C4⋊C4×C17—42no—
93C4⋊C4×C17—42no—
94C4⋊C4×C17—42no—
95C4⋊C4×C17—42no—
96C22⋊C4×C17—41no—
97C22⋊C4×C17—41no—
98C22⋊C4×C17—41no—
99C22⋊C4×C17—41no—
100C42×C17—22no—
101C42×C17—22no—
102C42×C17—41no—
103C42×C17—41no—
104C4⋊Dic17—48no—
105C2×#3622—24no—
106C2×#3622—42no—
107C2×#3622—42no—
108#9887—171nomaximal, Sylow
109C23×C17—21no—
110C23×C17—21no—
111C23×C17—21no—
112C23×C17—21no—
113C23×C17C2×C22⋊C411yescharacteristic
114C23×C17#12411yescharacteristic
115C23×C17C4⋊Q811yescharacteristic
116C23×C17AGL(1,ℤ/8)11yescharacteristic
117C23×C17AGL(1,ℤ/8)11yescharacteristic
118C23×C17#13711yescharacteristic
119C23×C17#13711yescharacteristic
120C2×C4×C17—24no—
121C2×C4×C17—24no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
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—21no—
131C2×C4×C17—21no—
132C2×C4×C17—21no—
133C2×C4×C17—21no—
134C2×C4×C17—21no—
135C2×C4×C17—21no—
136C2×C4×C17—42no—
137C2×C4×C17—42no—
138C2×C4×C17—42no—
139C2×C4×C17—42no—
140C2×C4×C17—41no—
141C2×C4×C17—41no—
142C2×C4×C17—41no—
143C2×C4×C17—41no—
144C2×C4×C17—41no—
145C2×C4×C17—41no—
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#9912yes—
153C2×C4×C17#10412yes—
154C2×C4×C17C4×Q812yes—
155C2×C4×C17C22⋊Q812yes—
156C2×Dic17—44no—
157C2×Dic17—44no—
158C2×Dic17—44no—
159C2×Dic17—82no—
160C2×Dic17—82no—
161C2×Dic17—82no—
162C2×Dic17—82no—
163#3622—44no—
164#2851D1711yescharacteristic
165#1749—171no—
166C4⋊#95—171no—
167#1294—172no—
168#1294—172no—
169C22×C17—21no—
170C22×C17—21no—
171C22×C17—21no—
172C22×C17—21no—
173C22×C17—21no—
174C22×C17—21no—
175C22×C17—21no—
176C22×C17—21no—
177C22×C17—21no—
178C22×C17—21no—
179C22×C17—21no—
180C22×C17—21no—
181C22×C17#39111yescharacteristic
182C22×C17C2×#9911yescharacteristic
183C22×C17#41011yescharacteristic
184C22×C17#41711yescharacteristic
185C22×C17#41711yescharacteristic
186C22×C17C2×#10411yescharacteristic
187C22×C17#42111yescharacteristic
188C22×C17#49011yescharacteristic
189C22×C17#49111yescharacteristic
190C22×C17#50111yescharacteristic
191C22×C17#50111yescharacteristic
192C4×C17—22no—
193C4×C17—22no—
194C4×C17—22no—
195C4×C17—22no—
196C4×C17—41no—
197C4×C17—41no—
198C4×C17—81no—
199C4×C17—81no—
200C4×C17—81no—
201C4×C17—81no—
202Dic17—84no—
203C2×#125Dic1711yescharacteristic, complemented
204C22×C4⋊C4—341no—
205C22×C42C2×D1711yescharacteristic
206C2×C22⋊C8—172no—
207C2×#97—341no—
208#389Dic1711yescharacteristic, complemented
209#389Dic1711yescharacteristic, complemented
210#387—21no—
211#387—21no—
212#385—21no—
213#385—21no—
214#382Dic1711yescharacteristic, complemented
215#380C2×D1711yescharacteristic
216#380C2×D1711yescharacteristic
217C2×#95—172no—
218C2×#95—341no—
219#336—341no—
220C2×C17—21no—
221C2×C17—21no—
222C2×C17—21no—
223C2×C17—21no—
224C2×C17#150411yescharacteristic
225C2×C17#150911yescharacteristic
226C2×C17#151211yescharacteristic
227C2×C17#151311yescharacteristic
228C2×C17#198611yescharacteristic
229C2×C17#199211yescharacteristic
230C2×C17#199611yescharacteristic
231C23×C4—342no—
232C23×C4C22⋊D1711yescharacteristic
233C23×C4C22⋊D1711yescharacteristic
234C23×C4C22×D1711yescharacteristic
235C22×C8—342no—
236#125—42no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—344no—
242C2×C4⋊C4—682no—
243C2×C4⋊C4—682no—
244C2×C4⋊C4C2×Dic1711yescharacteristic
245C2×C4⋊C4C2×Dic1711yescharacteristic
246C2×C4⋊C4C2×Dic1711yescharacteristic
247C2×C4⋊C4C2×Dic1711yescharacteristic
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C42—42no—
251C2×C42C2×Dic1711yescharacteristic
252C2×C42C2×Dic1711yescharacteristic
253C2×C42C22⋊D1711yescharacteristic
254C2×C42C22⋊D1711yescharacteristic
255C22⋊C8—174no—
256C22⋊C8—342no—
257#97—682no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—344no—
265#95—682no—
266C17#988711yescharacteristic, complemented, Sylow
267C24C2×C22⋊D1711yescharacteristic
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—344no—
283C22×C4—682no—
284C22×C4—682no—
285C22×C4—682no—
286C22×C4—682no—
287C22×C4—682no—
288C22×C4—682no—
289C22×C4C22⋊Dic1711yescharacteristic
290C22×C4C22⋊Dic1711yescharacteristic
291C22×C4C22⋊Dic1711yescharacteristic
292C22×C4C22⋊Dic1711yescharacteristic
293C22×C4Q8×D1712yes—
294C22×C4#6325612yes—
295C22×C4C22×Dic1711yescharacteristic
296C22×C4C2×C22⋊D1711yescharacteristic
297C2×C8—344no—
298C2×C8—682no—
299C2×C8—682no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—688no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C42—22no—
314C42—22no—
315C42—41no—
316C42—41no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23#1058763011yescharacteristic
322C23#1058763311yescharacteristic
323C23#1058763511yescharacteristic
324C23#1058763511yescharacteristic
325C23#1058763711yescharacteristic
326C23#1058763711yescharacteristic
327C23C2×C22⋊Dic1711yescharacteristic
328C2×C4—24no—
329C2×C4—24no—
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—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
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—684no—
361C2×C4—684no—
362C2×C4—684no—
363C2×C4—1362no—
364C2×C4—1362no—
365C2×C4—1362no—
366C2×C4—1362no—
367C2×C4#1058750512yes—
368C2×C4#1058750812yes—
369C2×C4Q8×Dic1712yes—
370C2×C4#1058763212yes—
371C8—684no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22#1176271611yescharacteristic
385C22C2×#1058750511yescharacteristic
386C22#1176278311yescharacteristic
387C22#1176281011yescharacteristic
388C22#1176281011yescharacteristic
389C22C2×#1058750811yescharacteristic
390C22#1176282611yescharacteristic
391C22#1176317511yescharacteristic
392C22#1176317911yescharacteristic
393C22#1176321911yescharacteristic
394C22#1176321911yescharacteristic
395C4—22no—
396C4—22no—
397C4—22no—
398C4—22no—
399C4—41no—
400C4—41no—
401C4—81no—
402C4—81no—
403C4—81no—
404C4—81no—
405C4—1364no—
406C2—21no—
407C2—21no—
408C2—21no—
409C2—21no—
410C2#1452728811yescharacteristic
411C2#1452730911yescharacteristic
412C2#1452732211yescharacteristic
413C2#1452732711yescharacteristic
414C2#1452972911yescharacteristic
415C2#1452975411yescharacteristic
416C2#1452977011yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic17
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#2851C2
socleC23×C17C2×C22⋊C4
prime radicalC24×C17C2×D4
cosocleC23×C4×C17C23
hypercenter#2851D17
hypocenterC17#9887
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2851

Complements

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

NH ≅ G/Ntype
12. C2×C17×#125C4semidirect
14. C17×#389C4semidirect
15. C17×#389C4semidirect
20. C17×#382C4semidirect
203. C2×#125Dic17semidirect
208. #389Dic17semidirect
209. #389Dic17semidirect
214. #382Dic17semidirect
266. C17#9887semidirect

Maximal subgroups

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

subgroupindex
#145285462
#145270142
#145266622
C17×#28512
#988717