Finite Group Databasebeta

← #17952525

Subgroups — #17952525

Order 4352.

1276 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1276 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14544607C211yesmaximal, characteristic
2C2×#11763662C214yesmaximal, complemented
3#14543958C211yesmaximal, characteristic
4C22×#10587583C211yesmaximal, characteristic
5C22×C22⋊#595—21no—
6C2×#10587543—21no—
7C2×#10587543C2212yes—
8C2×C22⋊#3624—21no—
9C2×(C2×#595)⋊C4—21no—
10C2×(C2×#595)⋊C4C2212yes—
11#11763662—28no—
12C23×Dic34C411yescharacteristic, complemented
13C23×C4×D17C2211yescharacteristic
14C2×#10587583—28no—
15C2×#10587583—21no—
16C2×#10587583—21no—
17C2×#10587583C412yescomplemented
18C2×#10587583C2212yes—
19C22×C4⋊Dic17—21no—
20C22×#63227—21no—
21C22×C4⋊C4×C17C411yescharacteristic, complemented
22C24×D17D411yescharacteristic
23C23×#595—22no—
24C23×C4×C17—21no—
25C23×C4×C17C2×C411yescharacteristic
26C23×Dic17—21no—
27C23×Dic17D411yescharacteristic
28C22×C4×D17—24no—
29C22×C4×D17—42no—
30C22×C4×D17—42no—
31C22×C4×D17C2×C411yescharacteristic
32C22×C4×D17D411yescharacteristic
33C22×C4×D17D411yescharacteristic
34C22×C4×D17C2311yescharacteristic
35C22×Dic34—21no—
36C22×Dic34—24no—
37C22×Dic34—42no—
38C22×Dic34—42no—
39C22×Dic34C2×C412yes—
40C2×C22⋊#595—24no—
41C2×C22⋊#595—42no—
42C2×C22⋊#595—42no—
43C22×#3624—22no—
44C2×C4⋊C4×C17—21no—
45C2×C4⋊C4×C17—24no—
46C2×C4⋊C4×C17C2×C412yes—
47C2×#63227—28no—
48C2×#63227—22no—
49C2×#63227—22no—
50C2×C4⋊Dic17—28no—
51C2×C4⋊Dic17—22no—
52C2×C4⋊Dic17—22no—
53#10587583—28no—
54#10587583—28no—
55#10587583—24no—
56#10587583—24no—
57#10587583—24no—
58#10587583D418yes—
59C22⋊#3624—42no—
60(C2×#595)⋊C4—24no—
61(C2×#595)⋊C4—42no—
62#10587543—24no—
63#10587543—42no—
64C24×C17C22⋊C411yescharacteristic
65C23×D17—22no—
66C23×D17—22no—
67C23×D17—22no—
68C23×D17—22no—
69C23×D17—41no—
70C23×D17—41no—
71C23×D17C22⋊C411yescharacteristic
72C23×D17SD1612yes—
73C23×D17Q1612yes—
74C23×D17C2×D411yescharacteristic
75C22×#595—24no—
76C22×#595—42no—
77C22×#595—42no—
78C22×#595—42no—
79C22×#595—42no—
80C22×#595—42no—
81C22×#595—42no—
82C22×C4×C17—22no—
83C22×C4×C17—22no—
84C22×C4×C17—22no—
85C22×C4×C17—24no—
86C22×C4×C17—22no—
87C22×C4×C17—42no—
88C22×C4×C17—42no—
89C22×C4×C17C22⋊C411yescharacteristic
90C22×C4×C17C22×C411yescharacteristic
91C22×Dic17—24no—
92C22×Dic17—22no—
93C22×Dic17—22no—
94C22×Dic17—22no—
95C22×Dic17—41no—
96C22×Dic17—42no—
97C22×Dic17—42no—
98C22×Dic17C22⋊C411yescharacteristic
99C22×Dic17C2×D411yescharacteristic
100C2×C4×D17—22no—
101C2×C4×D17—22no—
102C2×C4×D17—22no—
103C2×C4×D17—22no—
104C2×C4×D17—22no—
105C2×C4×D17—41no—
106C2×C4×D17—41no—
107C2×C4×D17—41no—
108C2×C4×D17—41no—
109C2×C4×D17—41no—
110C2×C4×D17—41no—
111C2×C4×D17—416no—
112C2×C4×D17C2×D414yes—
113C2×Dic34—24no—
114C2×Dic34—22no—
115C2×Dic34—42no—
116C2×Dic34—416no—
117C2×Dic34—42no—
118C2×Dic34—42no—
119C2×Dic34—42no—
120C2×Dic34C22⋊C414yescomplemented
121C22⋊#595—48no—
122C2×#3624—42no—
123C2×#3624—42no—
124C2×#3624—42no—
125C4⋊C4×C17—24no—
126C4⋊C4×C17—22no—
127C4⋊C4×C17C22⋊C414yescomplemented
128#63227—28no—
129#63227—28no—
130C4⋊Dic17—28no—
131C4⋊Dic17—28no—
132#8329—171nomaximal, Sylow
133C23×C17—21no—
134C23×C17—21no—
135C23×C17—21no—
136C23×C17—21no—
137C23×C17#9912yes—
138C23×C17#10312yes—
139C23×C17#10412yes—
140C23×C17C2×C22⋊C411yescharacteristic
141C22×D17—21no—
142C22×D17—21no—
143C22×D17—21no—
144C22×D17—21no—
145C22×D17—24no—
146C22×D17—24no—
147C22×D17—21no—
148C22×D17—21no—
149C22×D17—21no—
150C22×D17—21no—
151C22×D17—21no—
152C22×D17—21no—
153C22×D17—24no—
154C22×D17—24no—
155C22×D17—42no—
156C22×D17—42no—
157C22×D17—44no—
158C22×D17—44no—
159C22×D17—44no—
160C22×D17—44no—
161C22×D17#10312yes—
162C22×D17#12012yes—
163C22×D17C2×SD1611yescharacteristic
164C22×D17C2×Q1611yescharacteristic
165C22×D17#13712yes—
166C2×#595—44no—
167C2×#595—44no—
168C2×#595—44no—
169C2×#595—82no—
170C2×#595—82no—
171C2×#595—82no—
172C2×#595—82no—
173C2×C4×C17—28no—
174C2×C4×C17—21no—
175C2×C4×C17—21no—
176C2×C4×C17—21no—
177C2×C4×C17—21no—
178C2×C4×C17—24no—
179C2×C4×C17—48no—
180C2×C4×C17—42no—
181C2×C4×C17—42no—
182C2×C4×C17#9912yes—
183C2×C4×C17C2×C22⋊C412yes—
184C2×Dic17—21no—
185C2×Dic17—21no—
186C2×Dic17—24no—
187C2×Dic17—21no—
188C2×Dic17—21no—
189C2×Dic17—21no—
190C2×Dic17—44no—
191C2×Dic17—48no—
192C2×Dic17—42no—
193C2×Dic17—42no—
194C2×Dic17#12012yes—
195C4×D17—44no—
196C4×D17—44no—
197C4×D17—44no—
198C4×D17—44no—
199Dic34—48no—
200Dic34—44no—
201Dic34—44no—
202#3624—82no—
203C22×C22⋊Q8—171no—
204#1779—171no—
205C2×#328—174no—
206C22⋊#95—171no—
207C22×C17—22no—
208C22×C17—22no—
209C22×C17—22no—
210C22×C17—22no—
211C22×C17—22no—
212C22×C17—22no—
213C22×C17#32814yescomplemented
214C22×C17#37911yescharacteristic
215C22×C17C2×#9911yescharacteristic
216C22×C17C2×#9911yescharacteristic
217C22×C17C2×#10311yescharacteristic
218C22×C17#41911yescharacteristic
219C22×C17C2×#10411yescharacteristic
220C22×C17#42111yescharacteristic
221C2×D17—21no—
222C2×D17—21no—
223C2×D17—22no—
224C2×D17—22no—
225C2×D17—21no—
226C2×D17—21no—
227C2×D17—22no—
228C2×D17—22no—
229C2×D17—22no—
230C2×D17—22no—
231C2×D17—22no—
232C2×D17—48no—
233C2×D17—48no—
234C2×D17—41no—
235C2×D17—41no—
236C2×D17—41no—
237C2×D17—41no—
238C2×D17#44812yes—
239C2×D17#45112yes—
240#595—84no—
241C4×C17—22no—
242C4×C17—22no—
243C4×C17—44no—
244Dic17—22no—
245Dic17—44no—
246Dic17—41no—
247C23×Q8—171no—
248C24×C4—171no—
249C2×C22⋊Q8—172no—
250C2×C22⋊Q8—172no—
251C2×C22⋊Q8—348no—
252C2×C22⋊Q8—341no—
253C2×C22⋊Q8—341no—
254C22×C4⋊C4—341no—
255C22×C4⋊C4#59511yescharacteristic, complemented
256C22×C22⋊C4—341no—
257C22×C22⋊C4—341no—
258C2×C22⋊C8—172no—
259C2×C22⋊C8—341no—
260C2×C22⋊C8—341no—
261C2×#95—172no—
262C2×#95—341no—
263#328—348no—
264C2×C17—24no—
265C2×C17C2×#32811yescharacteristic
266C2×C17#150811yescharacteristic
267C2×C17#151111yescharacteristic
268C2×C17#152011yescharacteristic
269C2×C17#178111yescharacteristic
270C2×C17#178711yescharacteristic
271C2×C17#178911yescharacteristic
272D17—22no—
273D17—22no—
274D17—42no—
275C25—171no—
276C22×Q8—172no—
277C22×Q8—344no—
278C22×Q8—341no—
279C22×Q8—682no—
280C22×Q8—682no—
281C23×C4—21no—
282C23×C4—171no—
283C23×C4—171no—
284C23×C4—171no—
285C23×C4—171no—
286C23×C4—171no—
287C23×C4—344no—
288C23×C4—341no—
289C23×C4—342no—
290C23×C4—682no—
291C23×C4—682no—
292C23×C4C2×#59511yescharacteristic
293C22×C8—342no—
294C22⋊Q8—178no—
295C22⋊Q8—344no—
296C22⋊Q8—344no—
297C22⋊Q8—348no—
298C22⋊Q8—348no—
299C22⋊Q8—344no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—342no—
303C2×C4⋊C4—342no—
304C2×C4⋊C4—348no—
305C2×C4⋊C4C2×#59512yes—
306C2×C22⋊C4—342no—
307C2×C22⋊C4—342no—
308C2×C22⋊C4—348no—
309C2×C22⋊C4—344no—
310C2×C22⋊C4—682no—
311C2×C22⋊C4—682no—
312C22⋊C8—344no—
313C22⋊C8—682no—
314C22⋊C8—682no—
315#95—344no—
316#95—682no—
317C17#832911yescharacteristic, complemented, Sylow
318C24—171no—
319C24—171no—
320C24—172no—
321C24—172no—
322C24—342no—
323C24—342no—
324C24—342no—
325C24—342no—
326C24—681no—
327C24—681no—
328C24C22⋊#59511yescharacteristic
329C2×Q8—174no—
330C2×Q8—344no—
331C2×Q8—342no—
332C2×Q8—6816no—
333C2×Q8—682no—
334C2×Q8—682no—
335C2×Q8—682no—
336C2×Q8—682no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—171no—
345C22×C4—174no—
346C22×C4—171no—
347C22×C4—342no—
348C22×C4—342no—
349C22×C4—342no—
350C22×C4—344no—
351C22×C4—342no—
352C22×C4—342no—
353C22×C4—342no—
354C22×C4—342no—
355C22×C4—342no—
356C22×C4—344no—
357C22×C4—681no—
358C22×C4—681no—
359C22×C4—681no—
360C22×C4—681no—
361C22×C4—681no—
362C22×C4—681no—
363C22×C4—6816no—
364C22×C4—681no—
365C22×C4—682no—
366C22×C4—682no—
367C22×C4—682no—
368C22×C4—682no—
369C22×C4—682no—
370C22×C4—682no—
371C22×C4—682no—
372C22×C4—682no—
373C22×C4C22⋊#59511yescharacteristic
374C22×C4C22×#59511yescharacteristic
375C2×C8—682no—
376C2×C8—682no—
377C2×C8—682no—
378C4⋊C4—24no—
379C4⋊C4—22no—
380C4⋊C4—348no—
381C4⋊C4—348no—
382C4⋊C4C22⋊#59514yescomplemented
383C22⋊C4—348no—
384C22⋊C4—348no—
385C22⋊C4—688no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—172no—
391C23—171no—
392C23—171no—
393C23—172no—
394C23—172no—
395C23—344no—
396C23—344no—
397C23—341no—
398C23—341no—
399C23—341no—
400C23—341no—
401C23—341no—
402C23—341no—
403C23—341no—
404C23—341no—
405C23—341no—
406C23—341no—
407C23—344no—
408C23—344no—
409C23—682no—
410C23—682no—
411C23—684no—
412C23—684no—
413C23—684no—
414C23—684no—
415C23#1058753912yes—
416C23#1058754812yes—
417C23#1058754912yes—
418C23C2×C22⋊#59511yescharacteristic
419Q8—684no—
420Q8—684no—
421Q8—688no—
422C2×C4—28no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—24no—
428C2×C4—48no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—172no—
432C2×C4—341no—
433C2×C4—341no—
434C2×C4—344no—
435C2×C4—341no—
436C2×C4—341no—
437C2×C4—341no—
438C2×C4—684no—
439C2×C4—684no—
440C2×C4—684no—
441C2×C4—684no—
442C2×C4—684no—
443C2×C4—688no—
444C2×C4—682no—
445C2×C4—682no—
446C2×C4—684no—
447C2×C4—684no—
448C2×C4—684no—
449C2×C4—1362no—
450C2×C4—1362no—
451C2×C4—1362no—
452C2×C4—1362no—
453C2×C4#1058753912yes—
454C2×C4C2×C22⋊#59512yes—
455C8—1362no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—172no—
463C22—172no—
464C22—341no—
465C22—341no—
466C22—342no—
467C22—342no—
468C22—341no—
469C22—341no—
470C22—342no—
471C22—342no—
472C22—342no—
473C22—342no—
474C22—342no—
475C22—688no—
476C22—688no—
477C22—681no—
478C22—681no—
479C22—681no—
480C22—681no—
481C22#1176366214yescomplemented
482C22#1176375011yescharacteristic
483C22C2×#1058753911yescharacteristic
484C22C2×#1058753911yescharacteristic
485C22C2×#1058754911yescharacteristic
486C22#1176380211yescharacteristic
487C22C2×#1058754811yescharacteristic
488C22#1176380711yescharacteristic
489C4—22no—
490C4—22no—
491C4—44no—
492C4—342no—
493C4—681no—
494C4—684no—
495C4—1364no—
496C2—24no—
497C2—342no—
498C2—342no—
499C2—682no—
500C2C2×#1176366211yescharacteristic
501C2#1454418311yescharacteristic
502C2#1454419211yescharacteristic
503C2#1454422311yescharacteristic
504C2#1454461211yescharacteristic
505C2#1454462311yescharacteristic
506C2#1454462511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC22×C4⋊C4×C17C4
socleC23×C17C2×C22⋊C4
prime radicalC24×D17D4
cosocleC22×C4×D17C23
hypercenterC22×C4⋊C4#595
hypocenterC17#8329
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#11763662C2semidirect
12. C23×Dic34C4semidirect
17. C2×#10587583C4semidirect
21. C22×C4⋊C4×C17C4semidirect
120. C2×Dic34C22⋊C4semidirect
127. C4⋊C4×C17C22⋊C4semidirect
213. C22×C17#328semidirect
255. C22×C4⋊C4#595semidirect
317. C17#8329semidirect
382. C4⋊C4C22⋊#595semidirect
481. C22#11763662semidirect

Maximal subgroups

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

subgroupindex
#145446072
C2×#117636622
#145439582
C22×#105875832
#832917