Finite Group Databasebeta

← #17968910

Subgroups — #17968910

Order 4352.

1250 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587543C211yesmaximal, characteristic
2C2×#11763777C211yesmaximal, characteristic
3#14544522C212yesmaximal, complemented
4#14544521C212yesmaximal, complemented
5#14544518C212yesmaximal, complemented
6C22⋊C4×#3624C212yesmaximal, complemented
7#14544513C214yesmaximal, complemented
8#14531166C211yesmaximal, characteristic
9C22×#63242C2212yes—
10C23×#3624C2212yes—
11C2×#10587543C418yescomplemented
12C2×#10587543C2212yes—
13C2×#10587543C2212yes—
14C2×#10587543C2211yescharacteristic
15C2×#10587543C2211yescharacteristic
16#11763777C418yescomplemented
17C2×C22⋊#3624C2211yescharacteristic
18C2×C22⋊#3624C2211yescharacteristic
19C2×#10587542C2214yescomplemented
20C2×C4×#3624C2214yescomplemented
21#11763675C2214yescomplemented
22#11763675C2214yescomplemented
23C23×C4×D17C2211yescharacteristic
24C2×#10587578C2212yes—
25#11762671C412yes—
26#11762669C411yescharacteristic
27#11762666C412yes—
28#11762664C412yes—
29C22⋊C4×Dic17C2212yes—
30#11762656C2212yes—
31C17×#380C411yescharacteristic
32C24×D17C2×C411yescharacteristic
33C23×C4×C17C2×C411yescharacteristic
34C23×Dic17C2311yescharacteristic
35C22×C4×D17—44no—
36C22×C4×D17C2×C411yescharacteristic
37C22×C4×D17C2×C411yescharacteristic
38C22×C4×D17C2×C414yes—
39C22×C4×D17C2×C414yes—
40C22×C4×D17C2311yescharacteristic
41C22×C4×D17C2311yescharacteristic
42C22×#3624—44no—
43C22×#3624C2×C418yes—
44C22×#3624C2312yes—
45C22×#3624C2312yes—
46C22×#3624C2312yes—
47C22×#3624C2312yes—
48C2×#63242—44no—
49C2×#63242C2×C418yes—
50C2×C4⋊C4×C17C2×C412yes—
51C2×C22⋊C4×C17C2×C412yes—
52C2×C22⋊Dic17C2×C412yes—
53C2×#63227C2×C414yes—
54C2×#63227C2×C414yes—
55C2×C4⋊Dic17C2×C412yes—
56C2×C4×Dic17C2314yes—
57#10587578—44no—
58C22⋊#3624C2×C414yes—
59C22⋊#3624C2×C414yes—
60#10587543C2×C418yes—
61#10587543C2×C418yes—
62#10587543C2×C414yes—
63#10587543C2×C414yes—
64#10587542—44no—
65C4×#3624—44no—
66C17×#95C2×C412yes—
67#10587502C2×C412yes—
68#10587502C2×C414yes—
69C24×C17C22×C411yescharacteristic
70C23×D17—22no—
71C23×D17—22no—
72C23×D17—24no—
73C23×D17—42no—
74C23×D17C4214yes—
75C23×D17C22×C411yescharacteristic
76C23×D17C22×C411yescharacteristic
77C22×C4×C17—42no—
78C22×C4×C17C4214yes—
79C22×C4×C17C22×C414yes—
80C22×C4×C17C22×C411yescharacteristic
81C22×C4×C17C22×C411yescharacteristic
82C22×Dic17—24no—
83C22×Dic17C22×C414yes—
84C22×Dic17C22×C411yescharacteristic
85C22×Dic17C22×C412yes—
86C22×Dic17C22×C412yes—
87C22×Dic17C2411yescharacteristic
88C2×C4×D17—24no—
89C2×C4×D17—24no—
90C2×C4×D17—48no—
91C2×C4×D17—48no—
92C2×C4×D17—42no—
93C2×C4×D17—42no—
94C2×C4×D17C4214yes—
95C2×C4×D17C4214yes—
96C2×C4×D17C22×C412yes—
97C2×C4×D17C22×C412yes—
98C2×C4×D17C22×C412yes—
99C2×C4×D17C22×C412yes—
100C2×#3624—24no—
101C2×#3624—24no—
102C2×#3624—24no—
103C2×#3624—24no—
104C2×#3624—48no—
105C2×#3624C22×C414yes—
106C2×#3624C22×C414yes—
107C2×#3624C22×C414yes—
108C2×#3624C22×C414yes—
109#63242—48no—
110C4⋊C4×C17—42no—
111C22⋊C4×C17—24no—
112C22⋊Dic17—24no—
113#63227—28no—
114#63227—44no—
115C4⋊Dic17—42no—
116C4×Dic17—24no—
117C4×Dic17—24no—
118#20528—171nomaximal, Sylow
119C23×C17—22no—
120C23×C17—22no—
121C23×C17C2×C4212yes—
122C23×C17#13112yes—
123C23×C17#13112yes—
124C23×C17C23×C411yescharacteristic
125C22×D17—22no—
126C22×D17—21no—
127C22×D17—21no—
128C22×D17—22no—
129C22×D17—21no—
130C22×D17—22no—
131C22×D17—21no—
132C22×D17—28no—
133C22×D17—28no—
134C22×D17—44no—
135C22×D17—48no—
136C22×D17—48no—
137C22×D17C2×C4212yes—
138C22×D17C2×C4212yes—
139C22×D17#13114yes—
140C2×C4×C17—22no—
141C2×C4×C17—22no—
142C2×C4×C17—24no—
143C2×C4×C17—24no—
144C2×C4×C17—24no—
145C2×C4×C17—44no—
146C2×C4×C17C2×C4212yes—
147C2×C4×C17C2×C4212yes—
148C2×Dic17—22no—
149C2×Dic17—24no—
150C2×Dic17—24no—
151C2×Dic17—24no—
152C2×Dic17—24no—
153C2×Dic17—24no—
154C2×Dic17C2×C4212yes—
155C2×Dic17C23×C411yescharacteristic
156C2×Dic17C23×C411yescharacteristic
157C2×Dic17H(5,𝔽2)14yes—
158C4×D17—24no—
159C4×D17—24no—
160C4×D17—24no—
161C4×D17—24no—
162C4×D17—44no—
163C4×D17—44no—
164#3624—24no—
165#3624—24no—
166#3624—24no—
167#3624—24no—
168C22×C22⋊C8—171no—
169C22×C22⋊C8—171no—
170#2280—171no—
171#1752—172no—
172#1752—172no—
173C8×C22⋊C4—172no—
174C8×C22⋊C4—172no—
175#1750—174no—
176C22×C17—22no—
177C22×C17—24no—
178C22×C17—24no—
179C22×C17—22no—
180C22×C17#40514yes—
181C22×C17C22×C4211yescharacteristic
182C22×C17#51812yes—
183C22×C17C2×#13111yescharacteristic
184C22×C17C2×#13112yes—
185C22×C17C2×#13111yescharacteristic
186C2×D17—22no—
187C2×D17—24no—
188C2×D17—24no—
189C2×D17—22no—
190C2×D17—22no—
191C2×D17—22no—
192C2×D17—22no—
193C2×D17—416no—
194C2×D17—42no—
195C2×D17—42no—
196C2×D17#40514yes—
197C4×C17—22no—
198C4×C17—22no—
199C4×C17—44no—
200Dic17—22no—
201Dic17—44no—
202Dic17#51812yes—
203Dic17#51812yes—
204C24×C4—171no—
205C23×C8—172no—
206C23×C8—172no—
207C2×#117—172no—
208C2×C22⋊C8—171no—
209C2×C22⋊C8—171no—
210C2×C22⋊C8—178no—
211C2×C22⋊C8—178no—
212C2×C22⋊C8—172no—
213C2×C22⋊C8—172no—
214C2×C22⋊C8—171no—
215C2×C22⋊C8—171no—
216C2×#97—174no—
217C2×C4×C8—174no—
218#381—172no—
219#380—171no—
220#380—172no—
221#380#59511yescharacteristic
222#379—172no—
223C4×C22⋊C4—172no—
224C4×C22⋊C4—172no—
225#336—174no—
226#336—174no—
227C2×C17—24no—
228C2×C17#227311yescharacteristic
229C2×C17C2×#40511yescharacteristic
230C2×C17C2×#40511yescharacteristic
231C2×C17#297712yes—
232C2×C17#297712yes—
233D17—24no—
234D17—42no—
235C25—171no—
236C23×C4—171no—
237C23×C4—171no—
238C23×C4—171no—
239C23×C4—174no—
240C23×C4—174no—
241C23×C4—171no—
242C23×C4—171no—
243C23×C4—684no—
244C23×C4C2×#59511yescharacteristic
245C22×C8—172no—
246C22×C8—172no—
247C22×C8—172no—
248C22×C8—172no—
249C22×C8—178no—
250C22×C8—178no—
251C22×C8—684no—
252C22×C8—684no—
253#117—684no—
254C2×C4⋊C4—172no—
255C2×C4⋊C4C2×#59512yes—
256C2×C22⋊C4—172no—
257C2×C22⋊C4—174no—
258C2×C22⋊C4—174no—
259C2×C22⋊C4C2×#59512yes—
260C2×C42—174no—
261C22⋊C8—178no—
262C22⋊C8—178no—
263C22⋊C8—174no—
264C22⋊C8—174no—
265C22⋊C8—174no—
266C22⋊C8—174no—
267#97—684no—
268C4×C8—684no—
269#95—174no—
270#95—172no—
271#95C2×#59512yes—
272C17#2052811yescharacteristic, complemented, Sylow
273C24—171no—
274C24—171no—
275C24—174no—
276C24—344no—
277C24—342no—
278C24—342no—
279C24—682no—
280C24C22×#59511yescharacteristic
281C22×C4—42no—
282C22×C4—171no—
283C22×C4—172no—
284C22×C4—172no—
285C22×C4—171no—
286C22×C4—172no—
287C22×C4—172no—
288C22×C4—172no—
289C22×C4—172no—
290C22×C4—174no—
291C22×C4—174no—
292C22×C4—174no—
293C22×C4—344no—
294C22×C4—344no—
295C22×C4—344no—
296C22×C4—682no—
297C22×C4—682no—
298C22×C4—688no—
299C22×C4—688no—
300C22×C4C4×#59514yes—
301C22×C4C22×#59514yes—
302C22×C4C22×#59511yescharacteristic
303C22×C4C22×#59511yescharacteristic
304C2×C8—174no—
305C2×C8—174no—
306C2×C8—174no—
307C2×C8—174no—
308C2×C8—344no—
309C2×C8—344no—
310C2×C8—344no—
311C2×C8—344no—
312C2×C8—688no—
313C2×C8—688no—
314C4⋊C4—42no—
315C4⋊C4—682no—
316C22⋊C4—24no—
317C22⋊C4—348no—
318C22⋊C4—344no—
319C22⋊C4—684no—
320C42—344no—
321C42—344no—
322C23—22no—
323C23—22no—
324C23—172no—
325C23—174no—
326C23—172no—
327C23—348no—
328C23—342no—
329C23—341no—
330C23—341no—
331C23—348no—
332C23—342no—
333C23—341no—
334C23—342no—
335C23—341no—
336C23—684no—
337C23—688no—
338C23—688no—
339C23C2×C4×#59512yes—
340C23#1058767812yes—
341C23#1058768012yes—
342C23C23×#59511yescharacteristic
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—44no—
349C2×C4—171no—
350C2×C4—171no—
351C2×C4—174no—
352C2×C4—172no—
353C2×C4—344no—
354C2×C4—344no—
355C2×C4—344no—
356C2×C4—344no—
357C2×C4—344no—
358C2×C4—344no—
359C2×C4—342no—
360C2×C4—344no—
361C2×C4—344no—
362C2×C4—344no—
363C2×C4—684no—
364C2×C4—684no—
365C2×C4C2×C4×#59512yes—
366C2×C4C2×C4×#59512yes—
367C8—344no—
368C8—344no—
369C8—344no—
370C8—344no—
371C22—22no—
372C22—22no—
373C22—24no—
374C22—24no—
375C22—174no—
376C22—342no—
377C22—342no—
378C22—344no—
379C22—344no—
380C22—342no—
381C22—342no—
382C22—342no—
383C22—6816no—
384C22—682no—
385C22—682no—
386C22#1176377414yes—
387C22C22×C4×#59511yescharacteristic
388C22#1176387612yes—
389C22C2×#1058767811yescharacteristic
390C22D17⋊#13112yes—
391C22C2×#1058768011yescharacteristic
392C4—22no—
393C4—22no—
394C4—44no—
395C4—172no—
396C4—172no—
397C4—342no—
398C4—684no—
399C2—24no—
400C2—344no—
401C2—682no—
402C2#1454522011yescharacteristic
403C2#1454534411yescharacteristic
404C2C2×#1176377411yescharacteristic
405C2#1454565712yes—
406C2#1454565912yes—
subgroupG/H
center, FrattiniC23C23×#595
commutatorC22×C17C22×C42
FittingC17×#380C4
socleC23×C17C23×C4
prime radicalC24×D17C2×C4
cosocleC22×Dic17C24
hypercenter#380#595
hypocenterC17#20528
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #14544522C2semidirect
4. #14544521C2semidirect
5. #14544518C2semidirect
6. C22⋊C4×#3624C2semidirect
7. #14544513C2semidirect
11. C2×#10587543C4semidirect
16. #11763777C4semidirect
19. C2×#10587542C22semidirect
20. C2×C4×#3624C22semidirect
21. #11763675C22semidirect
22. #11763675C22semidirect
272. C17#20528semidirect

Maximal subgroups

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

subgroupindex
C22×#105875432
C2×#117637772
#145445222
#145445212
#145445182
C22⋊C4×#36242
#145445132
#145311662
#2052817