Finite Group Databasebeta

← #17076162

Subgroups — #17076162

Order 4352.

1076 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763036C212yesmaximal
2C2×#11763014C211yesmaximal, characteristic, complemented
3C22×C4⋊#3622C211yesmaximal, characteristic, complemented
4#14537136C211yesmaximal, characteristic, complemented
5#14530162C212yesmaximal, complemented
6#14530101C212yesmaximal, complemented
7#14529707C211yesmaximal, characteristic, complemented
8#14529547C211yesmaximal, characteristic, complemented
9#14529205C212yesmaximal, complemented
10#14528784C211yesmaximal, characteristic, complemented
11C4×C17×#121C211yesmaximal, characteristic
12C22×#63229C2212yes—
13C23×#3622C2212yes—
14C2×#10587555C2211yescharacteristic
15C2×#10587610C2212yes—
16#11763036—28no—
17#11763014—24no—
18C2×#10587479C2212yes—
19C2×C4⋊#3622—24no—
20C2×C4⋊#3622C2211yescharacteristic
21C2×C4⋊#3622C2212yes—
22C2×C4⋊#3622C2211yescharacteristic
23C2×#10587481C2212yes—
24C2×#10587481C2212yes—
25C2×#10587481C2212yes—
26C2×#10587503C2212yes—
27#11762715C2212yes—
28#11762707—21no—
29#11762707—21no—
30#11762679C2211yescharacteristic
31#11762669—21no—
32#11762669—21no—
33#11762571C2212yes—
34#11762571C2212yes—
35C2×C17×#121C2211yescharacteristic
36C2×C4×D4×C17—21no—
37C2×C4×D4×C17C2211yescharacteristic
38C22×C42×C17C2211yescharacteristic
39C17×#390C2211yescharacteristic
40C17×#389C2211yescharacteristic
41C17×#388—21no—
42C17×#386—21no—
43C17×#380C2211yescharacteristic
44C4×C4⋊C4×C17C2211yescharacteristic
45C4×C22⋊C4×C17—21no—
46C22×D4×C17—21no—
47C22×D4×C17C2311yescharacteristic
48C23×C4×C17—21no—
49C23×C4×C17D411yescharacteristic
50C23×C4×C17D411yescharacteristic
51C23×C4×C17C2311yescharacteristic
52C22×Dic34C2312yes—
53C17×#121—22no—
54C17×#121D414yes—
55C4×D4×C17—22no—
56C4×D4×C17—22no—
57C4×D4×C17—44no—
58C2×C4⋊C4×C17—21no—
59C2×C4⋊C4×C17C2311yescharacteristic
60C2×C4⋊C4×C17C2311yescharacteristic
61C2×C4⋊C4×C17C2311yescharacteristic
62C2×C22⋊C4×C17—21no—
63C2×C22⋊C4×C17—21no—
64C2×C22⋊C4×C17—21no—
65C2×C22⋊C4×C17—21no—
66C2×C22⋊C4×C17—21no—
67C2×C42×C17—22no—
68C2×C42×C17—22no—
69C2×C42×C17—21no—
70C2×C42×C17D411yescharacteristic
71C2×C42×C17D411yescharacteristic
72C2×C42×C17C2311yescharacteristic
73C2×C42×C17C2311yescharacteristic
74C2×C22⋊Dic17—22no—
75C2×#63229—24no—
76C2×#63229—24no—
77C2×#63229—42no—
78C2×C4⋊Dic17—21no—
79C2×C4⋊Dic17—21no—
80C2×C4⋊Dic17—21no—
81C2×C4⋊Dic17—21no—
82C2×C4⋊Dic17C2312yes—
83#10587610—24no—
84#10587610—24no—
85C2×#63225—22no—
86C22×#3622—24no—
87C22×#3622—24no—
88C22×#3622—42no—
89C22×#3622C2312yes—
90C22×#3622C2312yes—
91#10587555—22no—
92#10587555—22no—
93C17×#95—21no—
94C17×#95—21no—
95C17×#95—21no—
96#10587503—24no—
97#10587502—22no—
98#10587502—22no—
99#10587481—24no—
100#10587481—24no—
101#10587481—42no—
102#10587479—24no—
103C4⋊#3622—24no—
104C4⋊#3622—22no—
105C4⋊#3622D414yescomplemented
106C24×C17—21no—
107C24×C17C2×D411yescharacteristic
108C2×D4×C17—24no—
109C2×D4×C17—21no—
110C2×D4×C17—21no—
111C2×D4×C17—21no—
112C2×D4×C17—21no—
113C2×D4×C17—21no—
114C2×D4×C17—42no—
115C2×D4×C17—42no—
116C2×D4×C17C2×D412yes—
117C22×C4×C17—22no—
118C22×C4×C17—22no—
119C22×C4×C17—21no—
120C22×C4×C17—21no—
121C22×C4×C17—21no—
122C22×C4×C17—21no—
123C22×C4×C17—22no—
124C22×C4×C17—21no—
125C22×C4×C17—22no—
126C22×C4×C17—21no—
127C22×C4×C17—21no—
128C22×C4×C17—41no—
129C22×C4×C17—42no—
130C22×C4×C17—42no—
131C22×C4×C17—42no—
132C22×C4×C17SD1614yescomplemented
133C22×C4×C17C2×D412yes—
134C22×C4×C17C2×D411yescharacteristic
135C22×C4×C17C2×D411yescharacteristic
136C22×C4×C17C2×D411yescharacteristic
137C22×C4×C17C2×D411yescharacteristic
138C22×C4×C17C2×D411yescharacteristic
139C22×C4×C17#4112yes—
140C22×C4×C17C2411yescharacteristic
141C22×Dic17—22no—
142C22×Dic17—22no—
143C2×Dic34—22no—
144C2×Dic34—22no—
145C2×Dic34—22no—
146C2×Dic34—22no—
147C2×Dic34—22no—
148C2×Dic34—22no—
149C4⋊C4×C17—21no—
150C4⋊C4×C17—21no—
151C4⋊C4×C17—21no—
152C4⋊C4×C17—21no—
153C4⋊C4×C17—21no—
154C4⋊C4×C17—42no—
155C4⋊C4×C17C2×D412yes—
156C22⋊C4×C17—24no—
157C22⋊C4×C17—42no—
158C22⋊C4×C17—42no—
159C22⋊C4×C17—42no—
160C22⋊C4×C17—42no—
161C42×C17—22no—
162C42×C17—22no—
163C42×C17—21no—
164C42×C17—21no—
165C42×C17—42no—
166C42×C17C2×D412yes—
167C42×C17#4112yes—
168C22⋊Dic17—44no—
169#63229—48no—
170C4⋊Dic17—22no—
171C4⋊Dic17—22no—
172C4⋊Dic17—42no—
173C4⋊Dic17—42no—
174C4⋊Dic17—42no—
175C4⋊Dic17—42no—
176#63225—44no—
177C2×#3622—22no—
178C2×#3622—22no—
179C2×#3622—22no—
180C2×#3622—22no—
181C2×#3622—22no—
182C2×#3622—48no—
183C2×#3622C2×D414yes—
184#24751—171nomaximal, Sylow
185C23×C17—21no—
186C23×C17—21no—
187C23×C17—21no—
188C23×C17—21no—
189C23×C17—22no—
190C23×C17—41no—
191C23×C17—41no—
192C23×C17—41no—
193C23×C17—41no—
194C23×C17—41no—
195C23×C17—41no—
196C23×C17#12112yes—
197C23×C17C2×SD1612yes—
198C23×C17#13712yes—
199C23×C17C22×D411yescharacteristic
200D4×C17—22no—
201D4×C17—22no—
202D4×C17—48no—
203D4×C17—41no—
204D4×C17—41no—
205C2×C4×C17—21no—
206C2×C4×C17—21no—
207C2×C4×C17—21no—
208C2×C4×C17—21no—
209C2×C4×C17—21no—
210C2×C4×C17—21no—
211C2×C4×C17—22no—
212C2×C4×C17—22no—
213C2×C4×C17—22no—
214C2×C4×C17—22no—
215C2×C4×C17—28no—
216C2×C4×C17—21no—
217C2×C4×C17—22no—
218C2×C4×C17—21no—
219C2×C4×C17—21no—
220C2×C4×C17—44no—
221C2×C4×C17—41no—
222C2×C4×C17—41no—
223C2×C4×C17—41no—
224C2×C4×C17—41no—
225C2×C4×C17—41no—
226C2×C4×C17—41no—
227C2×C4×C17—41no—
228C2×C4×C17—41no—
229C2×C4×C17—84no—
230C2×C4×C17#12112yes—
231C2×C4×C17#12112yes—
232C2×C4×C17#12112yes—
233C2×C4×C17C2×SD1612yes—
234C2×C4×C17C2×SD1612yes—
235C2×C4×C17#13512yes—
236C2×C4×C17C22×D411yescharacteristic
237C2×C4×C17C22×D411yescharacteristic
238C2×C4×C17C2×#4111yescharacteristic
239C2×C4×C17C2×#4111yescharacteristic
240C2×C4×C17H(5,𝔽2)11yescharacteristic
241C2×C4×C172−1+411yescharacteristic
242C2×Dic17—42no—
243C2×Dic17—42no—
244C2×Dic17—42no—
245C2×Dic17—42no—
246C2×Dic17—42no—
247C2×Dic17—42no—
248Dic34—42no—
249Dic34—42no—
250Dic34—42no—
251Dic34—42no—
252#3622—24no—
253#3622—42no—
254C2×#465—172no—
255C2×#461—171no—
256C22×C4⋊C8—171no—
257#2857—171no—
258C4×#121D1711yescharacteristic
259#2066—172no—
260#2055—172no—
261#1981—171no—
262#1948—171no—
263#1892—172no—
264#1802—171no—
265C22×C17—22no—
266C22×C17—22no—
267C22×C17—22no—
268C22×C17—22no—
269C22×C17—22no—
270C22×C17—22no—
271C22×C17—42no—
272C22×C17—42no—
273C22×C17—42no—
274C22×C17—81no—
275C22×C17—81no—
276C22×C17—81no—
277C22×C17—81no—
278C22×C17#46114yescomplemented
279C22×C17C2×#12111yescharacteristic
280C22×C17C2×#12111yescharacteristic
281C22×C17#53711yescharacteristic
282C22×C17C22×SD1611yescharacteristic
283C22×C17C2×#13511yescharacteristic
284C22×C17C2×#13711yescharacteristic
285C22×C17#57511yescharacteristic
286C4×C17—22no—
287C4×C17—22no—
288C4×C17—22no—
289C4×C17—42no—
290C4×C17—41no—
291C4×C17—81no—
292C4×C17—81no—
293C4×C17#46514yescomplemented
294C4×C17D4211yescharacteristic
295C4×C17#54711yescharacteristic
296C4×C17#54811yescharacteristic
297C4×C17#55311yescharacteristic
298Dic17—82no—
299Dic17—82no—
300C22×SD16—172no—
301C23×C8—172no—
302C2×C4⋊Q8—171no—
303C2×C22⋊Q8—172no—
304C2×#121C2×D1711yescharacteristic
305C2×C4×D4—21no—
306C2×C4×D4C2×D1711yescharacteristic
307C22×C42C2×D1711yescharacteristic
308#465—348no—
309#461—344no—
310C2×#106—172no—
311C2×C4⋊C8—171no—
312C2×C4⋊C8—172no—
313C2×C4⋊C8—171no—
314C2×C4⋊C8—344no—
315C2×#103—172no—
316C2×#103—172no—
317C2×#103—172no—
318C2×#102—172no—
319#391—172no—
320#390C2×D1711yescharacteristic
321#389—341no—
322#389—341no—
323#389C2×D1711yescharacteristic
324#388—21no—
325#386—21no—
326#384—171no—
327#380—341no—
328#380—341no—
329#380C2×D1711yescharacteristic
330C4×C4⋊C4C2×D1711yescharacteristic
331C4×C22⋊C4—21no—
332#340—172no—
333#340—172no—
334C2×C17—24no—
335C2×C17—82no—
336C2×C17#241611yescharacteristic
337C2×C17C2×#46111yescharacteristic
338C2×C17#303511yescharacteristic
339C2×C17C2×#46511yescharacteristic
340C2×C17#305611yescharacteristic
341C2×C17#324611yescharacteristic
342C2×C17#324811yescharacteristic
343C22×Q8—172no—
344C22×D4—21no—
345C22×D4C22×D1711yescharacteristic
346C23×C4—21no—
347C23×C4C22⋊D1711yescharacteristic
348C23×C4C22⋊D1711yescharacteristic
349C23×C4C22×D1711yescharacteristic
350C2×SD16—344no—
351C2×SD16—344no—
352C2×SD16—682no—
353C22×C8—172no—
354C22×C8—172no—
355C22×C8—344no—
356C22×C8—344no—
357C22×C8—682no—
358C4⋊Q8—342no—
359C4⋊Q8—342no—
360C22⋊Q8—344no—
361C22⋊Q8—344no—
362#121—22no—
363#121D6814yes—
364C4×D4—22no—
365C4×D4—22no—
366C4×D4—44no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—172no—
369C2×C4⋊C4—342no—
370C2×C4⋊C4—341no—
371C2×C4⋊C4—341no—
372C2×C4⋊C4—341no—
373C2×C4⋊C4—341no—
374C2×C4⋊C4C22×D1711yescharacteristic
375C2×C4⋊C4C22×D1711yescharacteristic
376C2×C4⋊C4C22×D1711yescharacteristic
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—342no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—21no—
386C2×C42C22⋊D1711yescharacteristic
387C2×C42C22⋊D1711yescharacteristic
388C2×C42C22×D1711yescharacteristic
389C2×C42C22×D1711yescharacteristic
390#106—344no—
391C4⋊C8—174no—
392C4⋊C8—344no—
393C4⋊C8—342no—
394#103—344no—
395#103—344no—
396#103—682no—
397#102—344no—
398#95—21no—
399#95—21no—
400#95—21no—
401#95—342no—
402#95—342no—
403C17#2475111yescharacteristic, complemented, Sylow
404C24—21no—
405C24C2×C22⋊D1711yescharacteristic
406C2×Q8—342no—
407C2×Q8—342no—
408C2×Q8—342no—
409C2×Q8—342no—
410C2×Q8—342no—
411C2×Q8—342no—
412C2×D4—24no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4C2×D6812yes—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—41no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—342no—
437C22×C4—342no—
438C22×C4#6322914yescomplemented
439C22×C4C2×D6812yes—
440C22×C4#6325212yes—
441C22×C4C2×C22⋊D1711yescharacteristic
442C22×C4C2×C22⋊D1711yescharacteristic
443C22×C4C2×C22⋊D1711yescharacteristic
444C22×C4C2×C22⋊D1711yescharacteristic
445C22×C4C2×C22⋊D1711yescharacteristic
446C22×C4C23×D1711yescharacteristic
447SD16—688no—
448C2×C8—174no—
449C2×C8—342no—
450C2×C8—342no—
451C2×C8—342no—
452C2×C8—342no—
453C2×C8—342no—
454C2×C8—688no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—42no—
461C4⋊C4—342no—
462C4⋊C4—342no—
463C4⋊C4—682no—
464C4⋊C4—684no—
465C4⋊C4—682no—
466C4⋊C4—682no—
467C4⋊C4—682no—
468C4⋊C4C2×D6812yes—
469C22⋊C4—24no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—684no—
475C42—21no—
476C42—21no—
477C42—22no—
478C42—22no—
479C42—42no—
480C42D4×D1712yes—
481C42#6325412yes—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—22no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23#1058761612yes—
494C23C2×#6322912yes—
495C23#1058763712yes—
496C23C22×C22⋊D1711yescharacteristic
497Q8—682no—
498Q8—682no—
499Q8—682no—
500Q8—682no—
501D4—22no—
502D4—22no—
503D4—48no—
504D4—41no—
505D4—41no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—28no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—44no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—84no—
531C2×C4—682no—
532C2×C4—682no—
533C2×C4—682no—
534C2×C4—682no—
535C2×C4—682no—
536C2×C4—682no—
537C2×C4#1058761612yes—
538C2×C4C2×#6322912yes—
539C2×C4C2×#6322912yes—
540C2×C4#1058762512yes—
541C2×C4#1058762512yes—
542C2×C4#1058763612yes—
543C2×C4C22×D6811yescharacteristic
544C2×C4C2×#6325211yescharacteristic
545C2×C4C2×D4×D1711yescharacteristic
546C2×C4C2×#6325411yescharacteristic
547C2×C4#1058769611yescharacteristic
548C2×C4#1058769711yescharacteristic
549C8—344no—
550C8—682no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22#1176301414yescomplemented
565C22C2×#1058762511yescharacteristic
566C22C2×#1058761611yescharacteristic
567C22#1176338711yescharacteristic
568C22C22×#6322911yescharacteristic
569C22C2×#1058763611yescharacteristic
570C22C2×#1058763711yescharacteristic
571C22#1176360711yescharacteristic
572C4—22no—
573C4—22no—
574C4—22no—
575C4—41no—
576C4—42no—
577C4—81no—
578C4—81no—
579C4—1362no—
580C4—1362no—
581C4#1176303614yescomplemented
582C4D4×D6811yescharacteristic
583C4#1176346311yescharacteristic
584C4#1176346811yescharacteristic
585C4#1176350511yescharacteristic
586C2—24no—
587C2—82no—
588C2#1453222411yescharacteristic
589C2C2×#1176301411yescharacteristic
590C2#1453823911yescharacteristic
591C2C2×#1176303611yescharacteristic
592C2#1453841611yescharacteristic
593C2#1454010011yescharacteristic
594C2#1454012111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC4×C17×#121C2
socleC23×C17C22×D4
prime radicalC2×C17×#121C22
hypercenterC4×#121D17
hypocenterC17#24751
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
2. C2×#11763014C2semidirect
3. C22×C4⋊#3622C2semidirect
4. #14537136C2semidirect
5. #14530162C2semidirect
6. #14530101C2semidirect
7. #14529707C2semidirect
8. #14529547C2semidirect
9. #14529205C2semidirect
10. #14528784C2semidirect
105. C4⋊#3622D4semidirect
132. C22×C4×C17SD16semidirect
278. C22×C17#461semidirect
293. C4×C17#465semidirect
403. C17#24751semidirect
438. C22×C4#63229semidirect
564. C22#11763014semidirect
581. C4#11763036semidirect

Maximal subgroups

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

subgroupindex
C2×#117630362
C2×#117630142
C22×C4⋊#36222
#145371362
#145301622
#145301012
#145297072
#145295472
#145292052
#145287842
C4×C17×#1212
#2475117