Finite Group Databasebeta

← #17970454

Subgroups — #17970454

Order 4352.

982 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587534C211yesmaximal, characteristic
2C2×#11763772C211yesmaximal, characteristic
3#14545265C212yesmaximal
4#14544890C214yesmaximal, complemented
5#14544888C214yesmaximal, complemented
6#14544473C212yesmaximal
7D17×#427C211yesmaximal, characteristic
8C2×D17⋊M16C2211yescharacteristic
9C2×#10587677C2211yescharacteristic
10C22×C4⋊#595C2212yes—
11C2×#10587534—24no—
12C2×#10587534C2212yes—
13C2×#10587534C2212yes—
14#11763772—24no—
15C2×#10587532C2212yes—
16C2×C8×#595C2212yes—
17#11763760C2218yes—
18#11763756—24no—
19#11763755—22no—
20#11763753—22no—
21#11763682C2218yes—
22C22×C8×D17C2211yescharacteristic
23#117×D17C2212yes—
24#11762847C411yescharacteristic, complemented
25#11762846C412yescomplemented
26#11762845C412yescomplemented
27#11762844C412yescomplemented
28#107×D17C2212yes—
29#106×D17C2212yes—
30C17×#427C411yescharacteristic, complemented
31C23×#595—22no—
32C22×C4×D17C2311yescharacteristic
33C2×C22⋊#595D412yes—
34C2×#63247—21no—
35#10587677—22no—
36#10587677—22no—
37C2×C4⋊#595—24no—
38C2×C4⋊#595—24no—
39C2×C4⋊#595—22no—
40C2×C4⋊#595—28no—
41C2×C4⋊#595D411yescharacteristic
42C2×C4⋊#595D411yescharacteristic
43C2×C4⋊#595C2314yes—
44C2×C4×#595C2312yes—
45D17⋊M16—22no—
46D17⋊M16Q814yes—
47C2×#63243—21no—
48C2×#63242C2312yes—
49C22×C8×C17C2×C411yescharacteristic
50C17×#117C2×C412yes—
51#10587612C2×C412yes—
52C22×#3622C2×C411yescharacteristic
53C2×C8×D17C2×C411yescharacteristic
54C2×C8×D17C2×C411yescharacteristic
55C2×C8×D17D414yes—
56C2×C8×D17Q814yes—
57C2×C8×D17C2311yescharacteristic
58C2×C8×D17C2311yescharacteristic
59#10587578C2×C414yes—
60C4⋊C4×D17C2314yes—
61C22⋊C4×D17—22no—
62#10587563—22no—
63#10587558—22no—
64C42×D17—22no—
65(C2×#595)⋊C4—28no—
66#10587534—24no—
67#10587534—24no—
68#10587532—24no—
69C8×#595—24no—
70C17×#107C2×C412yescomplemented
71C17×#106C2×C412yescomplemented
72#10587489C2×C412yescomplemented
73#10587488C2×C412yescomplemented
74#10587479C2×C414yescomplemented
75#10587478C2×C414yescomplemented
76C23×D17C2×D411yescharacteristic
77C22×#595—24no—
78C22×#595—44no—
79C22×#595—84no—
80C22×#595C2×D412yes—
81C22×#595C2×D412yes—
82C22×C4×C17C22×C411yescharacteristic
83C22×Dic17#4111yescharacteristic
84C2×C4×D17—24no—
85C2×C4×D17C22×C411yescharacteristic
86C2×C4×D17C2×D412yes—
87C2×C4×D17C2×D411yescharacteristic
88C2×C4×D17C2×Q812yes—
89C2×C4×D17C2×Q812yes—
90C2×C4×D17#4112yes—
91C2×C4×D17#4111yescharacteristic
92C2×C4×D17C2411yescharacteristic
93C22⋊#595—24no—
94#63247—24no—
95C2×#3624—22no—
96C4⋊#595—22no—
97C4⋊#595—22no—
98C4⋊#595—24no—
99C4⋊#595—24no—
100C4⋊#595—44no—
101C4⋊#595—88no—
102C4×#595—22no—
103C4×#595—22no—
104#63243—24no—
105#63242—22no—
106#63242C2×Q814yes—
107C2×C8×C17C4⋊C414yescomplemented
108C2×C8×C17C22×C411yescharacteristic
109C2×C8×C17C22×C411yescharacteristic
110C4⋊C4×C17C22×C414yes—
111C22⋊C4×C17—22no—
112C42×C17—22no—
113C22⋊Dic17—22no—
114#63227—24no—
115C4⋊Dic17C22×C414yes—
116#63225—24no—
117C4×Dic17—22no—
118C4×Dic17—24no—
119C2×#3622C4⋊C414yescomplemented
120C2×#3622C22×C411yescharacteristic
121C2×#3622C22×C411yescharacteristic
122C8×D17C4⋊C414yescomplemented
123C8×D17C4⋊C414yescomplemented
124C8×D17C2×D412yes—
125C8×D17C2×D412yes—
126C8×D17C2×Q812yes—
127C8×D17C2×Q812yes—
128#21156—171nomaximal, Sylow
129C23×C17C4×D411yescharacteristic
130C22×D17—22no—
131C22×D17—22no—
132C22×D17C4×D411yescharacteristic
133C22×D17#12112yes—
134C22×D17C4⋊Q812yes—
135C22×D17C22×D411yescharacteristic
136C2×#595—22no—
137C2×#595—22no—
138C2×#595—24no—
139C2×#595—24no—
140C2×#595—44no—
141C2×#595—48no—
142C2×#595—88no—
143C2×C4×C17—24no—
144C2×C4×C17C2×C4⋊C412yes—
145C2×C4×C17C4×D411yescharacteristic
146C2×C4×C17C4×Q812yes—
147C2×C4×C17C23×C411yescharacteristic
148C2×Dic17—24no—
149C2×Dic17C4×D411yescharacteristic
150C2×Dic17C22⋊Q812yes—
151C2×Dic17#12512yes—
152C2×Dic17C2×#4111yescharacteristic
153C4×D17—44no—
154C4×D17—44no—
155C4×D17C2×C4⋊C412yes—
156C4×D17C4×Q812yes—
157C4×D17#12112yes—
158C4×D17C22⋊Q812yes—
159C4×D17#12512yes—
160C4×D17C4⋊Q812yes—
161C4×D17C22×D411yescharacteristic
162C4×D17C22×Q811yescharacteristic
163C4×D17C22×Q811yescharacteristic
164C4×D17C2×#4111yescharacteristic
165#3624—24no—
166C8×C17C2×C4⋊C412yes—
167C8×C17C2×C4⋊C412yes—
168#3622C2×C4⋊C412yes—
169#3622C2×C4⋊C412yes—
170C2×#427—171no—
171C22×#107—171no—
172C2×#405—171no—
173#2317—172no—
174#1944—174no—
175#1942—174no—
176#1736—172no—
177C22×C17—22no—
178C22×C17#38912yes—
179C22×C17C2×C4×D411yescharacteristic
180C2×D17—22no—
181C2×D17—24no—
182C2×D17—22no—
183C2×D17—22no—
184C2×D17#38912yes—
185C2×D17C2×#12111yescharacteristic
186C2×D17C2×C4⋊Q811yescharacteristic
187C2×D17#57612yes—
188C2×D17#57812yes—
189#595—42no—
190#595—42no—
191#595—84no—
192C4×C17—44no—
193C4×C17#38912yes—
194C4×C17C22×C4⋊C411yescharacteristic
195C4×C17C2×C4×Q811yescharacteristic
196Dic17—44no—
197Dic17#38912yes—
198Dic17C2×C22⋊Q811yescharacteristic
199Dic17C2×#12511yescharacteristic
200C22×M16—171no—
201C23×C8—171no—
202C2×#117—171no—
203C2×#117—172no—
204C22×C4⋊C4—172no—
205#427—171no—
206#427—172no—
207#427—172no—
208#427—172no—
209#427#59511yescharacteristic, complemented
210C2×#107—172no—
211C2×#107—172no—
212C2×#107—172no—
213C2×#107—344no—
214C2×#106—172no—
215#405—344no—
216C2×#97—172no—
217C2×C4×C8—172no—
218#389—178no—
219#385—344no—
220#384—342no—
221#382—342no—
222#340—178no—
223C2×C17—22no—
224C2×C17C2×#38911yescharacteristic
225C2×C17#295112yes—
226D17—22no—
227D17#305811yescharacteristic
228D17#305911yescharacteristic
229D17#316412yes—
230C23×C4—171no—
231C23×C4—342no—
232C2×M16—174no—
233C2×M16—341no—
234C2×M16—342no—
235C2×M16—341no—
236C22×C8—171no—
237C22×C8—174no—
238C22×C8—174no—
239C22×C8—171no—
240C22×C8—171no—
241C22×C8—171no—
242C22×C8—171no—
243C22×C8—172no—
244C22×C8C2×#59511yescharacteristic
245#117—172no—
246#117—174no—
247#117—342no—
248#117—342no—
249#117—342no—
250#117—342no—
251#117C2×#59512yes—
252C2×C4⋊C4—174no—
253C2×C4⋊C4—171no—
254C2×C4⋊C4—171no—
255C2×C4⋊C4—174no—
256C2×C4⋊C4—344no—
257C2×C4⋊C4—344no—
258C2×C4⋊C4—342no—
259C2×C4⋊C4—348no—
260C2×C22⋊C4—172no—
261C2×C22⋊C4—342no—
262C2×C42—172no—
263C2×C42—342no—
264#107—172no—
265#107—174no—
266#107—344no—
267#107—344no—
268#107C2×#59512yescomplemented
269#106—172no—
270#106—174no—
271#106C2×#59512yescomplemented
272#97—344no—
273C4×C8—344no—
274#95—348no—
275C17#2115611yescharacteristic, complemented, Sylow
276C24—171no—
277C22×C4—171no—
278C22×C4—172no—
279C22×C4—172no—
280C22×C4—171no—
281C22×C4—171no—
282C22×C4—172no—
283C22×C4—172no—
284C22×C4—171no—
285C22×C4—171no—
286C22×C4—172no—
287C22×C4—172no—
288C22×C4—344no—
289C22×C4—344no—
290C22×C4—684no—
291C22×C4—1364no—
292C22×C4C22×#59511yescharacteristic
293M16—344no—
294M16—344no—
295C2×C8—174no—
296C2×C8—171no—
297C2×C8—171no—
298C2×C8—172no—
299C2×C8—172no—
300C2×C8—174no—
301C2×C8—174no—
302C2×C8—172no—
303C2×C8—172no—
304C2×C8—174no—
305C2×C8—342no—
306C2×C8—342no—
307C2×C8C4⋊#59514yescomplemented
308C2×C8C22×#59511yescharacteristic
309C2×C8C22×#59511yescharacteristic
310C4⋊C4—174no—
311C4⋊C4—344no—
312C4⋊C4—342no—
313C4⋊C4—342no—
314C4⋊C4—344no—
315C4⋊C4—344no—
316C4⋊C4—684no—
317C4⋊C4—1368no—
318C4⋊C4C22×#59514yes—
319C22⋊C4—22no—
320C22⋊C4—344no—
321C22⋊C4—344no—
322C22⋊C4—342no—
323C42—22no—
324C42—344no—
325C42—342no—
326C42—342no—
327C42—342no—
328C23—171no—
329C23—172no—
330C23—172no—
331C23—171no—
332C23—342no—
333C23—342no—
334C23D4×#59511yescharacteristic
335C2×C4—24no—
336C2×C4—172no—
337C2×C4—172no—
338C2×C4—171no—
339C2×C4—171no—
340C2×C4—172no—
341C2×C4—172no—
342C2×C4—172no—
343C2×C4—172no—
344C2×C4—171no—
345C2×C4—172no—
346C2×C4—171no—
347C2×C4—171no—
348C2×C4—171no—
349C2×C4—172no—
350C2×C4—342no—
351C2×C4—342no—
352C2×C4—344no—
353C2×C4—344no—
354C2×C4—344no—
355C2×C4—684no—
356C2×C4—684no—
357C2×C4—684no—
358C2×C4—688no—
359C2×C4—1368no—
360C2×C4C2×C4⋊#59512yes—
361C2×C4D4×#59511yescharacteristic
362C2×C4Q8×#59512yes—
363C2×C4C23×#59511yescharacteristic
364C8—172no—
365C8—172no—
366C8—344no—
367C8C2×C4⋊#59512yes—
368C8C2×C4⋊#59512yes—
369C22—22no—
370C22—171no—
371C22—171no—
372C22—172no—
373C22—172no—
374C22—172no—
375C22—342no—
376C22—342no—
377C22—344no—
378C22—342no—
379C22#1176376012yes—
380C22C2×D4×#59511yescharacteristic
381C4—44no—
382C4—171no—
383C4—171no—
384C4—172no—
385C4—682no—
386C4—682no—
387C4—684no—
388C4—1364no—
389C4#1176376012yes—
390C4C22×C4⋊#59511yescharacteristic
391C4C2×Q8×#59511yescharacteristic
392C2—22no—
393C2—171no—
394C2—171no—
395C2—172no—
396C2—342no—
397C2C2×#1176376011yescharacteristic
398C2#1454554612yes—
subgroupG/H
centerC22C2×D4×#595
commutatorC2×C4×C17C23×C4
FrattiniC2×C4C23×#595
FittingC17×#427C4
socleC22×C17C2×C4×D4
prime radicalC23×D17C2×D4
cosocleC2×C4×D17C24
hypercenter#427#595
hypocenterC17#21156
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
4. #14544890C2semidirect
5. #14544888C2semidirect
24. #11762847C4semidirect
25. #11762846C4semidirect
26. #11762845C4semidirect
27. #11762844C4semidirect
30. C17×#427C4semidirect
70. C17×#107C2×C4semidirect
71. C17×#106C2×C4semidirect
72. #10587489C2×C4semidirect
73. #10587488C2×C4semidirect
74. #10587479C2×C4semidirect
75. #10587478C2×C4semidirect
107. C2×C8×C17C4⋊C4semidirect
119. C2×#3622C4⋊C4semidirect
122. C8×D17C4⋊C4semidirect
123. C8×D17C4⋊C4semidirect
209. #427#595semidirect
268. #107C2×#595semidirect
271. #106C2×#595semidirect
275. C17#21156semidirect
307. C2×C8C4⋊#595semidirect

Maximal subgroups

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

subgroupindex
C22×#105875342
C2×#117637722
#145452652
#145448902
#145448882
#145444732
D17×#4272
#2115617