Finite Group Databasebeta

← #16929930

Subgroups — #16929930

Order 4352.

626 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587474C211yesmaximal, characteristic
2#14528546C211yesmaximal, characteristic
3C2×#11762551C212yesmaximal, complemented
4#14526665C212yesmaximal, complemented
5C17×#2851C211yesmaximal, characteristic
6C23×#3622—21no—
7C2×C22⋊#3622—21no—
8C2×#10587474—21no—
9C2×#10587474—24no—
10C2×#10587474C2212yes—
11C2×#10587474C2212yes—
12#11762564—21no—
13#11762551—24no—
14C2×C17×#125C2211yescharacteristic
15C22×C42×C17C2211yescharacteristic
16C17×#389C411yescharacteristic
17C17×#389C411yescharacteristic
18C17×#387—21no—
19C17×#387—21no—
20C17×#385—21no—
21C17×#385—21no—
22C17×#382C2211yescharacteristic
23C17×#380C411yescharacteristic
24C17×#380C411yescharacteristic
25C23×C4×C17C2×C411yescharacteristic
26C23×C4×C17D411yescharacteristic
27C23×C4×C17D411yescharacteristic
28C17×#125—22no—
29C17×#125—22no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17—21no—
34C2×C4⋊C4×C17C2×C411yescharacteristic
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—22no—
41C2×C42×C17—22no—
42C2×C42×C17C2×C411yescharacteristic
43C2×C42×C17D411yescharacteristic
44C2×C42×C17D411yescharacteristic
45C2×C42×C17C2311yescharacteristic
46C22×#3622—24no—
47C22×#3622—24no—
48C22×#3622—24no—
49C22×#3622—22no—
50C22×#3622—22no—
51C17×#95—21no—
52C17×#95—21no—
53C17×#95—21no—
54C17×#95—21no—
55C17×#95—21no—
56C17×#95—21no—
57C22⋊#3622—24no—
58#10587474—24no—
59#10587474—22no—
60#10587474—22no—
61#10587474—22no—
62#10587474D414yes—
63C24×C17C22⋊C411yescharacteristic
64C22×C4×C17—22no—
65C22×C4×C17—22no—
66C22×C4×C17—21no—
67C22×C4×C17—21no—
68C22×C4×C17—22no—
69C22×C4×C17—22no—
70C22×C4×C17—22no—
71C22×C4×C17—22no—
72C22×C4×C17—21no—
73C22×C4×C17—21no—
74C22×C4×C17—21no—
75C22×C4×C17—21no—
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×C17C2×D411yescharacteristic
83C22×C4×C17C2×D411yescharacteristic
84C4⋊C4×C17—22no—
85C4⋊C4×C17—22no—
86C4⋊C4×C17—41no—
87C4⋊C4×C17—41no—
88C4⋊C4×C17—42no—
89C4⋊C4×C17—42no—
90C4⋊C4×C17—41no—
91C4⋊C4×C17—42no—
92C4⋊C4×C17—41no—
93C22⋊C4×C17—42no—
94C22⋊C4×C17—42no—
95C42×C17—22no—
96C42×C17—21no—
97C42×C17—22no—
98C42×C17—21no—
99C42×C17C2×D412yes—
100C42×C17#4112yes—
101C2×#3622—216no—
102C2×#3622—24no—
103C2×#3622—24no—
104C2×#3622—24no—
105C2×#3622—24no—
106C2×#3622—42no—
107C2×#3622—42no—
108#9121—171nomaximal, Sylow
109C23×C17—21no—
110C23×C17—21no—
111C23×C17—21no—
112C23×C17—21no—
113C23×C17#10112yes—
114C23×C17#10412yes—
115C23×C17#10412yes—
116C23×C17C2×C22⋊C411yescharacteristic
117C2×C4×C17—24no—
118C2×C4×C17—24no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
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—44no—
134C2×C4×C17—44no—
135C2×C4×C17—41no—
136C2×C4×C17—41no—
137C2×C4×C17—41no—
138C2×C4×C17—41no—
139C2×C4×C17—41no—
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#10412yes—
148C2×C4×C17C4×D412yes—
149C2×C4×C17#12011yescharacteristic
150C2×C4×C17#12112yes—
151C2×C4×C17#12311yescharacteristic
152#3622—28no—
153#3622—44no—
154C22×#97—171no—
155#2851D1711yescharacteristic
156#1749—171no—
157C2×#330—172no—
158#1295—172no—
159C22×C17—22no—
160C22×C17—22no—
161C22×C17—22no—
162C22×C17—22no—
163C22×C17—22no—
164C22×C17—22no—
165C22×C17#33014yescomplemented
166C22×C17#38611yescharacteristic
167C22×C17C2×#10111yescharacteristic
168C22×C17#41311yescharacteristic
169C22×C17C2×#10411yescharacteristic
170C22×C17C2×#10411yescharacteristic
171C22×C17C2×#10411yescharacteristic
172C22×C17#42111yescharacteristic
173C4×C17—22no—
174C4×C17—21no—
175C4×C17—22no—
176C4×C17—21no—
177C4×C17—22no—
178C4×C17—41no—
179C4×C17—41no—
180C4×C17—81no—
181C4×C17—81no—
182C4×C17—81no—
183C4×C17—81no—
184C23×C8—341no—
185C2×#125C2×D1711yescharacteristic
186C22×C42C2×D1711yescharacteristic
187C2×C22⋊C8—341no—
188C2×#97—172no—
189C2×#97—172no—
190C2×#97—344no—
191C2×#97—341no—
192#389Dic1711yescharacteristic
193#389Dic1711yescharacteristic
194#387—21no—
195#387—21no—
196#385—21no—
197#385—21no—
198#382C2×D1711yescharacteristic
199#380Dic1711yescharacteristic
200#380Dic1711yescharacteristic
201#336—341no—
202#330—344no—
203C2×C17—24no—
204C2×C17C2×#33011yescharacteristic
205C2×C17#152411yescharacteristic
206C2×C17#153111yescharacteristic
207C2×C17#153211yescharacteristic
208C2×C17#187111yescharacteristic
209C2×C17#189611yescharacteristic
210C2×C17#190111yescharacteristic
211C23×C4C2×Dic1711yescharacteristic
212C23×C4C22⋊D1711yescharacteristic
213C23×C4C22⋊D1711yescharacteristic
214C22×C8—344no—
215C22×C8—344no—
216C22×C8—344no—
217C22×C8—342no—
218C22×C8—342no—
219#125—22no—
220#125—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4C2×Dic1711yescharacteristic
226C2×C4⋊C4C2×Dic1711yescharacteristic
227C2×C4⋊C4C2×Dic1711yescharacteristic
228C2×C4⋊C4C2×Dic1711yescharacteristic
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42C2×Dic1711yescharacteristic
234C2×C42C22⋊D1711yescharacteristic
235C2×C42C22⋊D1711yescharacteristic
236C2×C42C22×D1711yescharacteristic
237C22⋊C8—344no—
238#97—174no—
239#97—344no—
240#97—342no—
241#97—342no—
242#97—342no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95—21no—
249C17#912111yescharacteristic, complemented, Sylow
250C24C22⋊Dic1711yescharacteristic
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4C22⋊Dic1711yescharacteristic
266C22×C4C22⋊Dic1711yescharacteristic
267C22×C4C22⋊Dic1711yescharacteristic
268C22×C4C22×Dic1711yescharacteristic
269C22×C4C2×C22⋊D1711yescharacteristic
270C22×C4C2×C22⋊D1711yescharacteristic
271C2×C8—3416no—
272C2×C8—344no—
273C2×C8—344no—
274C2×C8—344no—
275C2×C8—344no—
276C2×C8—682no—
277C2×C8—682no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C42—22no—
290C42—21no—
291C42—21no—
292C42—22no—
293C42D4×D1712yes—
294C42#6325412yes—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23#1058750712yes—
300C23#1058750812yes—
301C23#1058750812yes—
302C23C2×C22⋊Dic1711yescharacteristic
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4#1058750812yes—
334C2×C4D4×Dic1712yes—
335C2×C4#1058762111yescharacteristic
336C2×C4#1058762311yescharacteristic
337C2×C4#1058762512yes—
338C8—348no—
339C8—684no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22#1176255114yescomplemented
347C22#1176268911yescharacteristic
348C22C2×#1058750711yescharacteristic
349C22#1176279311yescharacteristic
350C22C2×#1058750811yescharacteristic
351C22C2×#1058750811yescharacteristic
352C22C2×#1058750811yescharacteristic
353C22#1176282611yescharacteristic
354C4—22no—
355C4—21no—
356C4—22no—
357C4—21no—
358C4—22no—
359C4—41no—
360C4—41no—
361C4—81no—
362C4—81no—
363C4—81no—
364C4—81no—
365C2—24no—
366C2C2×#1176255111yescharacteristic
367C2#1452737211yescharacteristic
368C2#1452739911yescharacteristic
369C2#1452740411yescharacteristic
370C2#1452908711yescharacteristic
371C2#1452922811yescharacteristic
372C2#1452925311yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic17
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#2851C2
socleC23×C17C2×C22⋊C4
prime radicalC24×C17C22⋊C4
cosocleC2×C42×C17C23
hypercenter#2851D17
hypocenterC17#9121
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
3. C2×#11762551C2semidirect
4. #14526665C2semidirect
165. C22×C17#330semidirect
249. C17#9121semidirect
346. C22#11762551semidirect

Maximal subgroups

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

subgroupindex
C22×#105874742
#145285462
C2×#117625512
#145266652
C17×#28512
#912117