Finite Group Databasebeta

← #17962983

Subgroups — #17962983

Order 4352.

1406 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763753C211yesmaximal, characteristic
2C2×C4×C4⋊#595C211yesmaximal, characteristic
3#14543964C214yesmaximal
4#14543964C214yesmaximal
5#14543959C212yesmaximal
6#95×#595C212yesmaximal
7D17×#383C211yesmaximal, characteristic
8C22×C4⋊#595C2212yes—
9C22×C4⋊#595C2212yes—
10C22×C4×#595C2214yes—
11C22×C4×#595C2211yescharacteristic
12C22×C4×#595C2211yescharacteristic
13#11763753C418yescomplemented
14C4×C4⋊#595C418yescomplemented
15C2×(C2×#595)⋊C4C2214yes—
16C2×(C2×#595)⋊C4C2214yes—
17C2×(C2×#595)⋊C4C2214yes—
18C2×(C2×#595)⋊C4C2212yes—
19C2×(C2×#595)⋊C4C2212yes—
20C2×(C2×#595)⋊C4C2212yes—
21C2×C42×D17C2211yescharacteristic
22#11762677C411yescharacteristic, complemented
23#11762676C412yescomplemented
24#11762676C412yescomplemented
25#11762676C412yescomplemented
26#95×D17C2212yes—
27#95×D17C2212yes—
28#95×D17C2212yes—
29C17×#383C411yescharacteristic, complemented
30C23×#595C2314yes—
31C23×#595C2312yes—
32C23×#595C2312yes—
33C22×C4×D17C2314yes—
34C22×C4×D17C2311yescharacteristic
35C22×C4×D17C2311yescharacteristic
36C22×C4×D17C2311yescharacteristic
37C2×C4⋊#595—28no—
38C2×C4⋊#595—44no—
39C2×C4⋊#595C2×C418yes—
40C2×C4⋊#595C2×C418yes—
41C2×C4×#595—28no—
42C2×C4×#595—42no—
43C2×C4×#595—42no—
44C2×C4×#595—44no—
45C2×C4×#595—44no—
46C2×C4×#595C2×C414yes—
47C2×C4×#595C2×C414yes—
48C2×C42×C17C2×C411yescharacteristic
49C2×C4×Dic17C2×C411yescharacteristic
50C2×C4×Dic17C2×C411yescharacteristic
51C2×C4×Dic17C2×C411yescharacteristic
52C42×D17C2×C414yes—
53C42×D17C2×C414yes—
54(C2×#595)⋊C4—28no—
55(C2×#595)⋊C4—44no—
56(C2×#595)⋊C4—44no—
57(C2×#595)⋊C4C2×C418yes—
58(C2×#595)⋊C4C2×C418yes—
59(C2×#595)⋊C4C2×C418yes—
60C17×#95C2×C412yes—
61C17×#95C2×C412yes—
62C17×#95C2×C412yes—
63#10587502C2×C414yes—
64#10587502C2×C414yes—
65#10587502C2×C414yes—
66#10587502C2×C412yes—
67#10587502C2×C412yes—
68#10587502C2×C412yes—
69C23×D17C2411yescharacteristic
70C22×#595—24no—
71C22×#595—22no—
72C22×#595—22no—
73C22×#595—24no—
74C22×#595—24no—
75C22×#595—24no—
76C22×#595—24no—
77C22×#595—24no—
78C22×#595—24no—
79C22×#595—24no—
80C22×#595—22no—
81C22×#595—22no—
82C22×#595—24no—
83C22×#595C22×C414yes—
84C22×#595C22×C418yes—
85C22×#595C22×C414yes—
86C22×#595#4118yes—
87C22×C4×C17C22×C414yes—
88C22×C4×C17C22×C411yescharacteristic
89C22×C4×C17C22×C411yescharacteristic
90C22×C4×C17C22×C411yescharacteristic
91C22×Dic17C22×C414yes—
92C22×Dic17C22×C411yescharacteristic
93C22×Dic17C22×C411yescharacteristic
94C22×Dic17C22×C411yescharacteristic
95C2×C4×D17—24no—
96C2×C4×D17—22no—
97C2×C4×D17—22no—
98C2×C4×D17—22no—
99C2×C4×D17—22no—
100C2×C4×D17—24no—
101C2×C4×D17—24no—
102C2×C4×D17—24no—
103C2×C4×D17—24no—
104C2×C4×D17—24no—
105C2×C4×D17—24no—
106C2×C4×D17C22×C412yes—
107C2×C4×D17C22×C412yes—
108C2×C4×D17C22×C412yes—
109C2×C4×D17C22×C412yes—
110C2×C4×D17C22×C412yes—
111C2×C4×D17C22×C412yes—
112C4⋊#595—216no—
113C4⋊#595—48no—
114C4×#595—44no—
115C4×#595—44no—
116C4×#595—416no—
117C42×C17C4214yescomplemented
118C4×Dic17C4214yescomplemented
119C4×Dic17C4214yescomplemented
120C4×Dic17C4214yescomplemented
121#13781—171nomaximal, Sylow
122C23×C17C23×C411yescharacteristic
123C22×D17C23×C411yescharacteristic
124C22×D17C23×C411yescharacteristic
125C22×D17C2×#4114yes—
126C22×D17H(5,𝔽2)12yes—
127C22×D17H(5,𝔽2)12yes—
128C22×D172−1+412yes—
129C22×D172−1+412yes—
130C2×#595—24no—
131C2×#595—24no—
132C2×#595—28no—
133C2×#595—28no—
134C2×#595—24no—
135C2×#595—28no—
136C2×#595—24no—
137C2×#595—28no—
138C2×#595—24no—
139C2×#595—28no—
140C2×#595—44no—
141C2×#595—44no—
142C2×#595—44no—
143C2×#595—44no—
144C2×#595—44no—
145C2×#595—44no—
146C2×#595#11718yes—
147C2×C4×C17—22no—
148C2×C4×C17—22no—
149C2×C4×C17—22no—
150C2×C4×C17—24no—
151C2×C4×C17—24no—
152C2×C4×C17—24no—
153C2×C4×C17C2×C4212yes—
154C2×C4×C17C2×C4212yes—
155C2×C4×C17C2×C4212yes—
156C2×Dic17—22no—
157C2×Dic17—22no—
158C2×Dic17—22no—
159C2×Dic17—24no—
160C2×Dic17—24no—
161C2×Dic17—24no—
162C2×Dic17C2×C4212yes—
163C2×Dic17C2×C4212yes—
164C2×Dic17C2×C4212yes—
165C4×D17—24no—
166C4×D17—24no—
167C4×D17—22no—
168C4×D17—22no—
169C4×D17—22no—
170C4×D17—22no—
171C4×D17—24no—
172C4×D17—24no—
173C4×D17—44no—
174C4×D17—44no—
175C4×D17—44no—
176C4×D17—44no—
177C2×#383—171no—
178C2×#382—171no—
179C2×C4×C4⋊C4—171no—
180#1436—174no—
181#1436—174no—
182#1434—172no—
183C4×#95—172no—
184C22×C17C22×C4211yescharacteristic
185C22×C17C4×#4112yes—
186C22×C17#52012yes—
187C22×C17#52012yes—
188C2×D17C2×#11712yes—
189C2×D17C4×#4112yes—
190C2×D17#51811yescharacteristic
191C2×D17#51811yescharacteristic
192C2×D17#51911yescharacteristic
193C2×D17#51911yescharacteristic
194C2×D17#52012yes—
195C2×D17#52012yes—
196C2×D17#53814yes—
197C2×D17#53914yes—
198C2×D17#54114yes—
199C2×D17#54214yes—
200#595—28no—
201#595—44no—
202#595—48no—
203C4×C17—22no—
204C4×C17—22no—
205C4×C17—22no—
206C4×C17—44no—
207Dic17—22no—
208Dic17—22no—
209Dic17—22no—
210Dic17—44no—
211C22×C4⋊C4—172no—
212C22×C4⋊C4—172no—
213C22×C42—171no—
214C22×C42—171no—
215C22×C42—171no—
216C22×C42—174no—
217#383—171no—
218#383—172no—
219#383—172no—
220#383—172no—
221#383#59511yescharacteristic, complemented
222#382—178no—
223C4×C4⋊C4—178no—
224C2×#95—172no—
225C2×#95—172no—
226C2×#95—172no—
227C2×#95—172no—
228C2×#95—172no—
229C2×#95—172no—
230C2×#95—174no—
231C2×#95—174no—
232C2×#95—174no—
233C2×C17C4×#11711yescharacteristic
234C2×C17#227411yescharacteristic
235C2×C17#227411yescharacteristic
236C2×C17#233414yes—
237D17#231812yes—
238D17#232012yes—
239D17#233612yes—
240D17#233712yes—
241C23×C4—172no—
242C23×C4—171no—
243C23×C4—171no—
244C23×C4—171no—
245C23×C4—174no—
246C23×C4—172no—
247C23×C4—174no—
248C2×C4⋊C4—178no—
249C2×C4⋊C4—178no—
250C2×C4⋊C4—348no—
251C2×C4⋊C4—684no—
252C2×C42—171no—
253C2×C42—171no—
254C2×C42—171no—
255C2×C42—174no—
256C2×C42—174no—
257C2×C42—174no—
258C2×C42—174no—
259C2×C42—348no—
260C2×C42—682no—
261C2×C42—682no—
262C2×C42—684no—
263C2×C42—684no—
264C2×C42C2×#59511yescharacteristic
265#95—178no—
266#95—178no—
267#95—178no—
268#95—174no—
269#95—174no—
270#95—174no—
271#95—172no—
272#95—172no—
273#95—172no—
274#95—348no—
275#95—684no—
276#95—684no—
277#95C2×#59512yes—
278#95C2×#59512yes—
279#95C2×#59512yes—
280C17#1378111yescharacteristic, complemented, Sylow
281C24—171no—
282C22×C4—174no—
283C22×C4—178no—
284C22×C4—171no—
285C22×C4—171no—
286C22×C4—171no—
287C22×C4—172no—
288C22×C4—172no—
289C22×C4—172no—
290C22×C4—172no—
291C22×C4—172no—
292C22×C4—172no—
293C22×C4—178no—
294C22×C4—174no—
295C22×C4—174no—
296C22×C4—342no—
297C22×C4—344no—
298C22×C4—342no—
299C22×C4—342no—
300C22×C4—342no—
301C22×C4—344no—
302C22×C4—344no—
303C22×C4—344no—
304C22×C4—344no—
305C22×C4—344no—
306C22×C4—344no—
307C22×C4—344no—
308C22×C4—342no—
309C22×C4—342no—
310C22×C4—344no—
311C22×C4—342no—
312C22×C4—344no—
313C22×C4—344no—
314C22×C4—344no—
315C22×C4—344no—
316C22×C4—344no—
317C22×C4—344no—
318C22×C4—344no—
319C22×C4—342no—
320C22×C4C22×#59514yes—
321C22×C4C22×#59511yescharacteristic
322C22×C4C22×#59511yescharacteristic
323C22×C4C22×#59511yescharacteristic
324C4⋊C4—3416no—
325C4⋊C4—688no—
326C42—174no—
327C42—174no—
328C42—174no—
329C42—684no—
330C42—684no—
331C42—6816no—
332C42C4×#59514yescomplemented
333C23—171no—
334C23—171no—
335C23—174no—
336C23—172no—
337C23—172no—
338C23—172no—
339C23—172no—
340C23C23×#59511yescharacteristic
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—178no—
348C2×C4—172no—
349C2×C4—172no—
350C2×C4—172no—
351C2×C4—342no—
352C2×C4—344no—
353C2×C4—342no—
354C2×C4—344no—
355C2×C4—344no—
356C2×C4—344no—
357C2×C4—342no—
358C2×C4—342no—
359C2×C4—344no—
360C2×C4—348no—
361C2×C4—344no—
362C2×C4—348no—
363C2×C4—342no—
364C2×C4—342no—
365C2×C4—342no—
366C2×C4—344no—
367C2×C4—344no—
368C2×C4—348no—
369C2×C4—348no—
370C2×C4—348no—
371C2×C4—344no—
372C2×C4—344no—
373C2×C4—344no—
374C2×C4—344no—
375C2×C4—684no—
376C2×C4—684no—
377C2×C4—684no—
378C2×C4—684no—
379C2×C4—684no—
380C2×C4—684no—
381C2×C4—684no—
382C2×C4—684no—
383C2×C4—684no—
384C2×C4—684no—
385C2×C4C2×C4×#59512yes—
386C2×C4C2×C4×#59512yes—
387C2×C4C2×C4×#59512yes—
388C22—172no—
389C22—172no—
390C22—171no—
391C22—171no—
392C22—172no—
393C22—172no—
394C22—171no—
395C22—171no—
396C22—174no—
397C22—174no—
398C22—174no—
399C22—174no—
400C22C22×C4×#59511yescharacteristic
401C22#41×#59512yes—
402C22#1176387812yes—
403C22#1176387812yes—
404C4—22no—
405C4—22no—
406C4—22no—
407C4—44no—
408C4—348no—
409C4—342no—
410C4—342no—
411C4—342no—
412C4—684no—
413C4—688no—
414C4—684no—
415C2—172no—
416C2—172no—
417C2—172no—
418C2—172no—
419C2#117×#59511yescharacteristic
420C2#1454522111yescharacteristic
421C2#1454522111yescharacteristic
422C2#1454528214yes—
subgroupG/H
center, FrattiniC23C23×#595
commutatorC22×C17C22×C42
FittingC17×#383C4
socleC23×C17C23×C4
prime radical, cosocleC23×D17C24
hypercenter#383#595
hypocenterC17#13781
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ #383

Complements

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

NH ≅ G/Ntype
13. #11763753C4semidirect
14. C4×C4⋊#595C4semidirect
22. #11762677C4semidirect
23. #11762676C4semidirect
24. #11762676C4semidirect
25. #11762676C4semidirect
29. C17×#383C4semidirect
117. C42×C17C42semidirect
118. C4×Dic17C42semidirect
119. C4×Dic17C42semidirect
120. C4×Dic17C42semidirect
221. #383#595semidirect
280. C17#13781semidirect
332. C42C4×#595semidirect

Maximal subgroups

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

subgroupindex
C2×#117637532
C2×C4×C4⋊#5952
#145439642
#145439592
#95×#5952
D17×#3832
#1378117