Finite Group Databasebeta

← #324×#595 #17951763

Subgroups — #324×#595 #17951763

Order 4352.

730 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×#595C211yesmaximal, characteristic
2C22⋊C8×#595C212yesmaximal
3#14544013C211yesmaximal, characteristic
4#14543621C212yesmaximal
5D17×#324C211yesmaximal, characteristic
6C22×C4⋊#595C411yescharacteristic
7C22×C4×#595C411yescharacteristic
8C22×C4×#595C411yescharacteristic
9C22×C4×#595C2211yescharacteristic
10#11763777C2212yes—
11C2×C8×#595—22no—
12C4⋊C4×#595—22no—
13C4⋊C4×#595—22no—
14C2×(C2×#595)⋊C4C411yescharacteristic
15C2×(C2×#595)⋊C4C2211yescharacteristic
16#11763674—22no—
17#11763655C412yescomplemented
18#11763654C412yescomplemented
19C2×C4⋊C4×D17C2211yescharacteristic
20C22⋊C8×D17C2212yes—
21#11762535C411yescharacteristic, complemented
22#11762534C412yescomplemented
23C17×#324C411yescharacteristic, complemented
24C23×#595C2×C411yescharacteristic
25C23×#595C2×C411yescharacteristic
26C23×#595C2×C411yescharacteristic
27C23×#595C2×C411yescharacteristic
28C22×C4×D17C2×C411yescharacteristic
29C22×C4×D17C2×C411yescharacteristic
30C22×C4×D17C2311yescharacteristic
31C2×C4⋊#595—22no—
32C2×C4⋊#595—22no—
33C2×C4⋊#595C814yescomplemented
34C2×C4×#595—21no—
35C2×C4×#595—21no—
36C2×C4×#595—22no—
37C2×C4×#595—21no—
38C2×C4×#595—21no—
39C2×C4×#595—42no—
40C2×C4×#595—42no—
41C2×C4×#595C814yescomplemented
42C2×C4×#595D412yes—
43C2×C4×#595D412yes—
44C2×#63242—22no—
45C2×C4⋊C4×C17C2×C411yescharacteristic
46C2×C4⋊Dic17C2×C411yescharacteristic
47C2×#63225C2×C411yescharacteristic
48C2×#63225C2×C411yescharacteristic
49C2×C8×D17—22no—
50C4⋊C4×D17—22no—
51C4⋊C4×D17—22no—
52C22⋊#3624C2×C414yes—
53(C2×#595)⋊C4—22no—
54(C2×#595)⋊C4—22no—
55#10587543—22no—
56C8×#595—44no—
57C17×C22⋊C8C2×C412yes—
58C22⋊#3622C2×C412yes—
59#10587491—22no—
60C23×D17C22×C411yescharacteristic
61C22×#595—22no—
62C22×#595—22no—
63C22×#595—21no—
64C22×#595—21no—
65C22×#595—21no—
66C22×#595—21no—
67C22×#595—41no—
68C22×#595—41no—
69C22×#595—41no—
70C22×#595—41no—
71C22×#595—41no—
72C22×#595—41no—
73C22×#595C22⋊C411yescharacteristic
74C22×#595C22⋊C412yes—
75C22×#595C22⋊C411yescharacteristic
76C22×#595C2×C812yes—
77C22×#595C2×C812yes—
78C22×#595C2×C812yes—
79C22×#595C2×C812yes—
80C22×#595M1612yes—
81C22×#595M1612yes—
82C22×C4×C17C4211yescharacteristic
83C22×C4×C17C4211yescharacteristic
84C22×C4×C17C22×C411yescharacteristic
85C22×Dic17C4211yescharacteristic
86C22×Dic17C4211yescharacteristic
87C22×Dic17C22×C411yescharacteristic
88C2×C4×D17—21no—
89C2×C4×D17—21no—
90C2×C4×D17—21no—
91C2×C4×D17—21no—
92C2×C4×D17—21no—
93C2×C4×D17—21no—
94C2×C4×D17—41no—
95C2×C4×D17—41no—
96C2×C4×D17—41no—
97C2×C4×D17—41no—
98C2×C4×D17C2×C812yes—
99C2×C4×D17C2×C812yes—
100C2×C4×D17C2×D412yes—
101C2×C4×D17#4112yes—
102C2×#3624—24no—
103C4⋊#595—28no—
104C4×#595—24no—
105C4×#595—24no—
106C4×#595—24no—
107C4×#595—42no—
108C4×#595—42no—
109C4×#595—42no—
110#63242—42no—
111#63242—42no—
112C2×C8×C17—22no—
113C4⋊C4×C17—22no—
114C4⋊Dic17—22no—
115#63225—22no—
116#63225—22no—
117C2×#3622—22no—
118C8×D17—42no—
119C8×D17—42no—
120C4×#324—171nomaximal, Sylow
121C23×C17C2×C4211yescharacteristic
122C22×D17—21no—
123C22×D17—21no—
124C22×D17—21no—
125C22×D17—21no—
126C22×D17C2×C22⋊C411yescharacteristic
127C22×D17#11711yescharacteristic
128C22×D17C22×C811yescharacteristic
129C22×D17C22×C811yescharacteristic
130C22×D17C2×M1611yescharacteristic
131C22×D17C2×M1611yescharacteristic
132C2×#595—22no—
133C2×#595—21no—
134C2×#595—21no—
135C2×#595—22no—
136C2×#595—24no—
137C2×#595—24no—
138C2×#595—22no—
139C2×#595—22no—
140C2×#595—42no—
141C2×#595—42no—
142C2×#595—42no—
143C2×#595—42no—
144C2×#595—41no—
145C2×#595—41no—
146C2×#595—41no—
147C2×#595—41no—
148C2×#595—42no—
149C2×#595—42no—
150C2×#595C22⋊C812yes—
151C2×#595C22⋊C814yes—
152C2×#595C22⋊C812yes—
153C2×#595#9912yes—
154C2×#595#10112yes—
155C2×C4×C17—21no—
156C2×C4×C17—21no—
157C2×C4×C17—21no—
158C2×C4×C17—41no—
159C2×C4×C17—41no—
160C2×C4×C17C4×C812yescomplemented
161C2×C4×C17C4×D412yes—
162C2×Dic17—21no—
163C2×Dic17—21no—
164C2×Dic17—21no—
165C2×Dic17—41no—
166C2×Dic17—41no—
167C2×Dic17C4×C812yescomplemented
168C2×Dic17C4×D412yes—
169C4×D17—22no—
170C4×D17—22no—
171C4×D17—22no—
172C4×D17—22no—
173C4×D17—22no—
174C4×D17—22no—
175C4×D17—42no—
176C4×D17—42no—
177C4×D17—41no—
178C4×D17—41no—
179#3624—44no—
180C8×C17—42no—
181#3622—42no—
182C2×C4×C4⋊C4—171no—
183C4×C22⋊C8—172no—
184C2×#324—171no—
185C2×#324—171no—
186#1280—172no—
187C22×C17—21no—
188C22×C17—21no—
189C22×C17C4×C22⋊C411yescharacteristic
190C22×C17C2×C4×C811yescharacteristic
191C22×C17C4×M1611yescharacteristic
192C2×D17—21no—
193C2×D17—21no—
194C2×D17—21no—
195C2×D17—21no—
196C2×D17—21no—
197C2×D17—21no—
198C2×D17—21no—
199C2×D17—21no—
200C2×D17—21no—
201C2×D17—21no—
202C2×D17C2×C22⋊C811yescharacteristic
203C2×D17C2×C22⋊C811yescharacteristic
204C2×D17C2×#9911yescharacteristic
205C2×D17#41011yescharacteristic
206C2×D17C2×#10111yescharacteristic
207C2×D17#41311yescharacteristic
208C2×D17#43111yescharacteristic
209C2×D17#43111yescharacteristic
210#595—22no—
211#595—24no—
212#595—42no—
213#595—42no—
214#595#32414yesdirect product factor
215C4×C17—22no—
216C4×C17—22no—
217C4×C17—41no—
218Dic17—22no—
219Dic17—22no—
220Dic17—41no—
221C22×C4⋊C4—171no—
222C22×C4⋊C4—171no—
223C22×C42—171no—
224C22×C42—171no—
225C22×C42—171no—
226C2×C22⋊C8—172no—
227C2×C22⋊C8—172no—
228C2×C4×C8—342no—
229C4×C4⋊C4—342no—
230C4×C4⋊C4—342no—
231C2×#95—171no—
232C2×#95—171no—
233#336—342no—
234#324—172no—
235#324—172no—
236#324—171no—
237#324—172no—
238#324#59511yescharacteristic, direct product factor
239C2×C17—21no—
240C2×C17—21no—
241C2×C17C4×C22⋊C811yescharacteristic
242C2×C17C4×#9911yescharacteristic
243C2×C17C4×#10111yescharacteristic
244D17—21no—
245D17—21no—
246D17C2×#32411yescharacteristic
247D17#145811yescharacteristic
248D17#145911yescharacteristic
249D17#146111yescharacteristic
250C23×C4—171no—
251C23×C4—171no—
252C23×C4—171no—
253C23×C4—171no—
254C23×C4—171no—
255C23×C4—171no—
256C23×C4—171no—
257C22×C8—342no—
258C22×C8—342no—
259C2×C4⋊C4—174no—
260C2×C4⋊C4—171no—
261C2×C4⋊C4—171no—
262C2×C4⋊C4—171no—
263C2×C4⋊C4—342no—
264C2×C4⋊C4—342no—
265C2×C4⋊C4—342no—
266C2×C4⋊C4—342no—
267C2×C4⋊C4C2×#59511yescharacteristic
268C2×C42—174no—
269C2×C42—172no—
270C2×C42—172no—
271C2×C42—341no—
272C2×C42—341no—
273C2×C42—342no—
274C2×C42—341no—
275C2×C42—341no—
276C2×C42—682no—
277C2×C42—682no—
278C22⋊C8—174no—
279C22⋊C8—172no—
280C22⋊C8—342no—
281C22⋊C8—342no—
282C22⋊C8C2×#59512yes—
283C4×C8—684no—
284#95—342no—
285#95—342no—
286C17C4×#32411yescharacteristic, complemented, Sylow
287C24—171no—
288C22×C4—171no—
289C22×C4—172no—
290C22×C4—171no—
291C22×C4—172no—
292C22×C4—172no—
293C22×C4—171no—
294C22×C4—172no—
295C22×C4—172no—
296C22×C4—171no—
297C22×C4—172no—
298C22×C4—172no—
299C22×C4—172no—
300C22×C4—172no—
301C22×C4—172no—
302C22×C4—172no—
303C22×C4—171no—
304C22×C4—341no—
305C22×C4—341no—
306C22×C4—341no—
307C22×C4—341no—
308C22×C4—341no—
309C22×C4—341no—
310C22×C4—342no—
311C22×C4—342no—
312C22×C4—341no—
313C22×C4—341no—
314C22×C4—341no—
315C22×C4—341no—
316C22×C4—681no—
317C22×C4—681no—
318C22×C4—681no—
319C22×C4—681no—
320C22×C4—681no—
321C22×C4—681no—
322C22×C4—681no—
323C22×C4—681no—
324C22×C4—681no—
325C22×C4—681no—
326C22×C4C4×#59511yescharacteristic
327C22×C4C4×#59511yescharacteristic
328C22×C4C22×#59511yescharacteristic
329C2×C8—22no—
330C2×C8—344no—
331C2×C8—342no—
332C2×C8—682no—
333C2×C8—682no—
334C2×C8—682no—
335C2×C8—682no—
336C4⋊C4—22no—
337C4⋊C4—342no—
338C4⋊C4—342no—
339C4⋊C4—348no—
340C4⋊C4—342no—
341C42—344no—
342C42—344no—
343C42—344no—
344C42—682no—
345C42—682no—
346C42—682no—
347C23—171no—
348C23—171no—
349C23—171no—
350C23—171no—
351C23—171no—
352C23—171no—
353C23—341no—
354C23—341no—
355C23—341no—
356C23—341no—
357C23C2×C4×#59511yescharacteristic
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—172no—
364C2×C4—172no—
365C2×C4—172no—
366C2×C4—172no—
367C2×C4—174no—
368C2×C4—172no—
369C2×C4—172no—
370C2×C4—342no—
371C2×C4—342no—
372C2×C4—342no—
373C2×C4—342no—
374C2×C4—342no—
375C2×C4—342no—
376C2×C4—344no—
377C2×C4—344no—
378C2×C4—342no—
379C2×C4—341no—
380C2×C4—341no—
381C2×C4—342no—
382C2×C4—341no—
383C2×C4—341no—
384C2×C4—342no—
385C2×C4—342no—
386C2×C4—341no—
387C2×C4—682no—
388C2×C4—681no—
389C2×C4—681no—
390C2×C4—682no—
391C2×C4—681no—
392C2×C4—682no—
393C2×C4—682no—
394C2×C4—681no—
395C2×C4—681no—
396C2×C4—681no—
397C2×C4—681no—
398C2×C4—682no—
399C2×C4—682no—
400C2×C4—681no—
401C2×C4—682no—
402C2×C4—682no—
403C2×C4C8×#59512yescomplemented
404C2×C4D4×#59512yes—
405C8—42no—
406C8—682no—
407C8—684no—
408C22—21no—
409C22—21no—
410C22—171no—
411C22—171no—
412C22—171no—
413C22—171no—
414C22—171no—
415C22—171no—
416C22—171no—
417C22—171no—
418C22—341no—
419C22—341no—
420C22—341no—
421C22—341no—
422C22—341no—
423C22—341no—
424C22—341no—
425C22—341no—
426C22—341no—
427C22—341no—
428C22C22⋊C4×#59511yescharacteristic
429C22C2×C8×#59511yescharacteristic
430C22M16×#59511yescharacteristic
431C4—22no—
432C4—22no—
433C4—41no—
434C4—174no—
435C4—342no—
436C4—344no—
437C4—342no—
438C4—342no—
439C4—682no—
440C4—682no—
441C4—681no—
442C2—21no—
443C2—21no—
444C2—171no—
445C2—171no—
446C2—171no—
447C2—171no—
448C2—341no—
449C2—341no—
450C2C22⋊C8×#59511yescharacteristic
451C2#99×#59511yescharacteristic
452C2#101×#59511yescharacteristic
subgroupG/H
centerC22C22⋊C4×#595
commutatorC22×C17C2×C4×C8
FrattiniC22×C4C22×#595
FittingC17×#324C4
socleC22×C17C4×C22⋊C4
prime radicalC23×D17C22×C4
cosocleC22×C4×D17C23
hypercenter#324#595
hypocenterC17C4×#324
LCSG ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
17. #11763655C4semidirect
18. #11763654C4semidirect
21. #11762535C4semidirect
22. #11762534C4semidirect
23. C17×#324C4semidirect
33. C2×C4⋊#595C8semidirect
41. C2×C4×#595C8semidirect
160. C2×C4×C17C4×C8semidirect
167. C2×Dic17C4×C8semidirect
214. #595#324direct (N × H)
238. #324#595direct (N × H)
286. C17C4×#324semidirect
403. C2×C4C8×#595semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×#5952
C22⋊C8×#5952
#145440132
#145436212
D17×#3242
C4×#32417