Finite Group Databasebeta

← #16929314

Subgroups — #16929314

Order 4352.

908 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14531398C211yesmaximal, characteristic
2#14528719C212yesmaximal, complemented
3C2×#11762533C211yesmaximal, characteristic
4#14526663C212yesmaximal, complemented
5C2×C17×#393C211yesmaximal, characteristic
6C22×C22⋊Dic17—21no—
7C2×#10587506—24no—
8C2×C22⋊#3622C2212yes—
9#11762684—24no—
10#11762669—21no—
11#11762669C2212yes—
12#11762533—22no—
13C23×Q8×C17C2211yescharacteristic
14C22×C22⋊C4×C17—21no—
15C22×C22⋊C4×C17C2211yescharacteristic
16C17×#393—22no—
17C17×#393C414yescomplemented
18C2×C17×#95—21no—
19C2×C17×#95C2211yescharacteristic
20C25×C17D411yescharacteristic
21C22×Q8×C17—21no—
22C22×Q8×C17—22no—
23C22×Q8×C17—22no—
24C22×Q8×C17—44no—
25C22×Q8×C17C2×C412yescomplemented
26C23×C4×C17—21no—
27C23×C4×C17—21no—
28C23×C4×C17D411yescharacteristic
29C23×C4×C17C2311yescharacteristic
30C23×Dic17—22no—
31C2×C22⋊C4×C17—22no—
32C2×C22⋊C4×C17—22no—
33C2×C22⋊C4×C17—24no—
34C2×C22⋊C4×C17—44no—
35C2×C22⋊C4×C17C2×C412yes—
36C2×C22⋊C4×C17D412yes—
37C2×C22⋊Dic17—22no—
38C2×C22⋊Dic17—22no—
39C2×C22⋊Dic17—42no—
40C2×C22⋊Dic17—42no—
41C2×C4⋊Dic17—22no—
42C2×C4⋊Dic17D414yes—
43C22×#3622—22no—
44C17×#95—24no—
45C17×#95—21no—
46C17×#95C2×C412yescomplemented
47#10587506—48no—
48#10587502—22no—
49#10587502—22no—
50C22⋊#3622—22no—
51C22⋊#3622—22no—
52C24×C17—21no—
53C24×C17—21no—
54C24×C17—21no—
55C24×C17—21no—
56C24×C17—42no—
57C24×C17—42no—
58C24×C17C22⋊C412yes—
59C24×C17SD1612yes—
60C24×C17Q1612yes—
61C24×C17C2×D411yescharacteristic
62C2×Q8×C17—28no—
63C2×Q8×C17—24no—
64C2×Q8×C17—44no—
65C2×Q8×C17—44no—
66C2×Q8×C17—44no—
67C2×Q8×C17—48no—
68C2×Q8×C17—42no—
69C22×C4×C17—24no—
70C22×C4×C17—22no—
71C22×C4×C17—21no—
72C22×C4×C17—21no—
73C22×C4×C17—22no—
74C22×C4×C17—42no—
75C22×C4×C17—42no—
76C22×C4×C17—41no—
77C22×C4×C17—41no—
78C22×C4×C17—42no—
79C22×C4×C17—42no—
80C22×C4×C17—42no—
81C22×C4×C17—41no—
82C22×C4×C17—41no—
83C22×C4×C17C22⋊C412yes—
84C22×C4×C17C22×C411yescharacteristic
85C22×C4×C17C2×D412yes—
86C22×C4×C17C2×D411yescharacteristic
87C22×C4×C17#4112yes—
88C22×Dic17—24no—
89C22×Dic17—44no—
90C22×Dic17—82no—
91C22×Dic17—82no—
92C22⋊C4×C17—24no—
93C22⋊C4×C17—24no—
94C22⋊C4×C17—48no—
95C22⋊Dic17—44no—
96C22⋊Dic17—42no—
97C22⋊Dic17—42no—
98C4⋊Dic17—44no—
99C4⋊Dic17—82no—
100C2×#3622—44no—
101C2×#3622—42no—
102#9013—171nomaximal, Sylow
103C23×C17—22no—
104C23×C17—22no—
105C23×C17—22no—
106C23×C17—22no—
107C23×C17—22no—
108C23×C17—22no—
109C23×C17—22no—
110C23×C17—22no—
111C23×C17—42no—
112C23×C17—42no—
113C23×C17—44no—
114C23×C17—44no—
115C23×C17—42no—
116C23×C17—42no—
117C23×C17—82no—
118C23×C17—82no—
119C23×C17—81no—
120C23×C17—81no—
121C23×C17#10314yescomplemented
122C23×C17C2×C22⋊C411yescharacteristic
123C23×C17#12011yescharacteristic
124C23×C17#12311yescharacteristic
125C23×C17C2×SD1611yescharacteristic
126C23×C17C2×Q1611yescharacteristic
127C23×C17AGL(1,ℤ/8)11yescharacteristic
128C23×C17#13711yescharacteristic
129Q8×C17—28no—
130Q8×C17—48no—
131Q8×C17—44no—
132C2×C4×C17—24no—
133C2×C4×C17—24no—
134C2×C4×C17—22no—
135C2×C4×C17—41no—
136C2×C4×C17—41no—
137C2×C4×C17—42no—
138C2×C4×C17—42no—
139C2×C4×C17—44no—
140C2×C4×C17—42no—
141C2×C4×C17—42no—
142C2×C4×C17—44no—
143C2×C4×C17—42no—
144C2×C4×C17—42no—
145C2×C4×C17—42no—
146C2×C4×C17—42no—
147C2×C4×C17—81no—
148C2×C4×C17—82no—
149C2×C4×C17—82no—
150C2×C4×C17C4×D412yes—
151C2×C4×C17#12112yes—
152C2×Dic17—44no—
153C2×Dic17—44no—
154C2×Dic17—44no—
155C2×Dic17—88no—
156#3622—82no—
157C2×#393D1711yescharacteristic
158#2315—171no—
159#1787—172no—
160C2×#323—171no—
161#1294—172no—
162C22×C17—22no—
163C22×C17—22no—
164C22×C17—22no—
165C22×C17—22no—
166C22×C17—21no—
167C22×C17—21no—
168C22×C17—21no—
169C22×C17—21no—
170C22×C17—24no—
171C22×C17—44no—
172C22×C17—44no—
173C22×C17—44no—
174C22×C17—42no—
175C22×C17—42no—
176C22×C17—82no—
177C22×C17—82no—
178C22×C17—82no—
179C22×C17#38611yescharacteristic
180C22×C17C2×#10311yescharacteristic
181C22×C17#41711yescharacteristic
182C22×C17#45011yescharacteristic
183C22×C17#45111yescharacteristic
184C22×C17#45512yes—
185C22×C17B(4,𝔽2)12yes—
186C22×C17#48311yescharacteristic
187C22×C17#48411yescharacteristic
188C4×C17—22no—
189C4×C17—22no—
190C4×C17—42no—
191C4×C17—44no—
192C4×C17—82no—
193Dic17—84no—
194C23×Q8C2×D1711yescharacteristic
195C22×C22⋊C4—21no—
196C22×C22⋊C4—341no—
197C22×C22⋊C4C2×D1711yescharacteristic
198C2×#103—344no—
199C2×C22⋊C8—172no—
200#393—22no—
201#393Dic1714yescomplemented
202#385—344no—
203#380—172no—
204#380—341no—
205C2×#95—21no—
206C2×#95C2×D1711yescharacteristic
207#323—342no—
208C2×C17—21no—
209C2×C17—21no—
210C2×C17—21no—
211C2×C17—21no—
212C2×C17—44no—
213C2×C17#160112yes—
214C2×C17#160212yes—
215C2×C17#187211yescharacteristic
216C2×C17#188511yescharacteristic
217C2×C17#188911yescharacteristic
218C25C22⋊D1711yescharacteristic
219C22×Q8—21no—
220C22×Q8—22no—
221C22×Q8—22no—
222C22×Q8—44no—
223C22×Q8C2×Dic1712yescomplemented
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—342no—
227C23×C4C22⋊D1711yescharacteristic
228C23×C4C22×D1711yescharacteristic
229C22×C8—342no—
230C2×C4⋊C4—174no—
231C2×C4⋊C4—342no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—342no—
237C2×C22⋊C4—342no—
238C2×C22⋊C4—682no—
239C2×C22⋊C4—682no—
240C2×C22⋊C4C2×Dic1712yes—
241C2×C22⋊C4C22⋊D1712yes—
242#103—688no—
243C22⋊C8—342no—
244C22⋊C8—342no—
245#95—24no—
246#95—21no—
247#95—342no—
248#95—342no—
249#95C2×Dic1712yescomplemented
250C17#901311yescharacteristic, complemented, Sylow
251C24—21no—
252C24—21no—
253C24—21no—
254C24—21no—
255C24—42no—
256C24—42no—
257C24#6323012yes—
258C24#6323112yes—
259C24C22⋊Dic1712yes—
260C24C2×C22⋊D1711yescharacteristic
261C2×Q8—28no—
262C2×Q8—24no—
263C2×Q8—44no—
264C2×Q8—44no—
265C2×Q8—44no—
266C2×Q8—48no—
267C2×Q8—42no—
268C22×C4—24no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—344no—
283C22×C4—684no—
284C22×C4—1362no—
285C22×C4—1362no—
286C22×C4C22⋊Dic1712yes—
287C22×C4D4×D1712yes—
288C22×C4#6325412yes—
289C22×C4C22×Dic1711yescharacteristic
290C22×C4C2×C22⋊D1711yescharacteristic
291C2×C8—684no—
292C2×C8—682no—
293C4⋊C4—684no—
294C4⋊C4—1362no—
295C22⋊C4—24no—
296C22⋊C4—24no—
297C22⋊C4—48no—
298C22⋊C4—684no—
299C22⋊C4—682no—
300C22⋊C4—682no—
301C23—22no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23—42no—
310C23—42no—
311C23—44no—
312C23—44no—
313C23—42no—
314C23—42no—
315C23—81no—
316C23—81no—
317C23—82no—
318C23—82no—
319C23#1058750614yescomplemented
320C23#1058762111yescharacteristic
321C23#1058762311yescharacteristic
322C23C2×#6323011yescharacteristic
323C23C2×#6323111yescharacteristic
324C23#1058763511yescharacteristic
325C23#1058763711yescharacteristic
326C23C2×C22⋊Dic1711yescharacteristic
327Q8—28no—
328Q8—48no—
329Q8—44no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—22no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—81no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—684no—
349C2×C4—684no—
350C2×C4—684no—
351C2×C4—1368no—
352C2×C4D4×Dic1712yes—
353C2×C4#1058762512yes—
354C8—1362no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—24no—
364C22—44no—
365C22—42no—
366C22—42no—
367C22—44no—
368C22—44no—
369C22—82no—
370C22—82no—
371C22—82no—
372C22#1176268911yescharacteristic
373C22C2×#1058750611yescharacteristic
374C22#1176281011yescharacteristic
375C22#1176295911yescharacteristic
376C22#1176296411yescharacteristic
377C22#1176298612yes—
378C22#1176299512yes—
379C22#1176313711yescharacteristic
380C22#1176314211yescharacteristic
381C4—22no—
382C4—22no—
383C4—42no—
384C4—44no—
385C4—82no—
386C4—1364no—
387C2—21no—
388C2—21no—
389C2—21no—
390C2—21no—
391C2—44no—
392C2#1452773512yes—
393C2#1452774012yes—
394C2#1452909211yescharacteristic
395C2#1452917311yescharacteristic
396C2#1452919311yescharacteristic
subgroupG/H
centerC23#10587623
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC2×C17×#393C2
socleC23×C17#120
prime radicalC25×C17D4
cosocleC23×C4×C17C23
hypercenterC2×#393D17
hypocenterC17#9013
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
2. #14528719C2semidirect
4. #14526663C2semidirect
17. C17×#393C4semidirect
25. C22×Q8×C17C2×C4semidirect
46. C17×#95C2×C4semidirect
121. C23×C17#103semidirect
201. #393Dic17semidirect
223. C22×Q8C2×Dic17semidirect
249. #95C2×Dic17semidirect
250. C17#9013semidirect
319. C23#10587506semidirect

Maximal subgroups

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

subgroupindex
#145313982
#145287192
C2×#117625332
#145266632
C2×C17×#3932
#901317