Finite Group Databasebeta

← #17046447

Subgroups — #17046447

Order 4352.

836 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763098C211yesmaximal, characteristic
2C4×#10587574C211yesmaximal, characteristic, complemented
3#14531242C211yesmaximal, characteristic
4#14529730C211yesmaximal, characteristic, complemented
5#14529541C211yesmaximal, characteristic, complemented
6#14529528C211yesmaximal, characteristic, complemented
7#14529526C211yesmaximal, characteristic, complemented
8#14529482C211yesmaximal, characteristic
9#14529479C211yesmaximal, characteristic
10#14529405C211yesmaximal, characteristic
11#14529404C211yesmaximal, characteristic
12#14529366C211yesmaximal, characteristic, complemented
13#14528783C211yesmaximal, characteristic
14#14528636C211yesmaximal, characteristic, complemented
15C17×#1920C211yesmaximal, characteristic, complemented
16C2×#10587574C2211yescharacteristic
17C2×C4×Dic34—21no—
18C2×#10587579C2211yescharacteristic
19C2×#10587578—21no—
20#11763098C418yescomplemented
21C2×#10587489C2211yescharacteristic
22C2×#10587478C2211yescharacteristic
23C2×#10587479C2211yescharacteristic
24C2×#10587485C2211yescharacteristic
25C2×#10587485C2211yescharacteristic
26C2×#10587492C2211yescharacteristic
27C2×#10587492C2211yescharacteristic
28C2×#10587480C2211yescharacteristic
29C2×#10587480C2211yescharacteristic
30C2×#10587486C2211yescharacteristic
31C2×#10587486C2211yescharacteristic
32#11762713—21no—
33#11762711C2211yescharacteristic
34#11762710C2211yescharacteristic
35#11762708C2211yescharacteristic
36#11762708C2211yescharacteristic
37#11762706—21no—
38#11762705C2211yescharacteristic
39#11762701—21no—
40#11762699—21no—
41#11762678C2211yescharacteristic
42C4⋊C4×Dic17C2211yescharacteristic
43C4×#63225—21no—
44C4×C4⋊Dic17C2211yescharacteristic
45#11762656—21no—
46C42×Dic17C2211yescharacteristic
47#11762572C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762564C2211yescharacteristic
53C2×C17×#106C2211yescharacteristic
54C17×#389C2211yescharacteristic
55C4×C4⋊C4×C17C2211yescharacteristic
56C17×#340C2211yescharacteristic
57C17×#340C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C17×#336C2211yescharacteristic
60C22×D68C2311yescharacteristic
61C22×C4×D17—21no—
62C22×Dic34—21no—
63C22×C8×C17C2311yescharacteristic
64C22×C8×C17C2311yescharacteristic
65C2×C4⋊C4×C17—21no—
66C2×C4⋊C4×C17C2311yescharacteristic
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C42×C17—21no—
71C2×C42×C17C2311yescharacteristic
72C2×#63227—21no—
73C2×#63227—21no—
74C2×#63227—21no—
75C2×#63227—21no—
76C2×#63227—21no—
77C2×C4⋊Dic17—21no—
78C2×C4⋊Dic17—21no—
79C2×C4⋊Dic17C2311yescharacteristic
80C2×C4⋊Dic17C2311yescharacteristic
81C2×C4⋊Dic17C2311yescharacteristic
82C2×#63225—21no—
83C2×#63225—21no—
84C2×#63225—21no—
85C2×#63225—21no—
86C2×C4×Dic17—21no—
87C2×C4×Dic17—21no—
88C2×C4×Dic17D411yescharacteristic
89C2×C4×Dic17D411yescharacteristic
90C2×C4×Dic17D411yescharacteristic
91C2×C4×Dic17D411yescharacteristic
92C2×C4×Dic17C2311yescharacteristic
93C2×C4×Dic17C2311yescharacteristic
94C22×#3622C2311yescharacteristic
95C22×#3622C2311yescharacteristic
96#10587579—22no—
97#10587579C2×C414yes—
98#10587578—44no—
99#10587574—22no—
100#10587574C2×C414yes—
101C4×Dic34—44no—
102C17×#106C2×C414yes—
103C17×#95—21no—
104C17×#95—21no—
105#10587502—21no—
106#10587502—21no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587492—41no—
111#10587492C2×C414yes—
112#10587489—41no—
113#10587486—41no—
114#10587486—41no—
115#10587485—22no—
116#10587485C2×C414yes—
117#10587480—41no—
118#10587480C2×C414yes—
119#10587479C2×C414yes—
120#10587478—41no—
121C23×D17—21no—
122C22×C4×C17—21no—
123C22×C4×C17—21no—
124C22×C4×C17—21no—
125C22×C4×C17—21no—
126C22×C4×C17C2×D411yescharacteristic
127C22×C4×C17C2×D411yescharacteristic
128C22×C4×C17C2411yescharacteristic
129C22×Dic17—21no—
130C22×Dic17—21no—
131C22×Dic17—21no—
132C22×Dic17C2×D411yescharacteristic
133C22×Dic17C2×D411yescharacteristic
134C22×Dic17C2×D411yescharacteristic
135C22×Dic17C2×D411yescharacteristic
136C2×D68—21no—
137C2×D68—21no—
138C2×D68—21no—
139C2×D68—21no—
140C2×D68—21no—
141C2×D68C22×C412yes—
142C2×C4×D17—24no—
143C2×C4×D17—42no—
144C2×C4×D17—42no—
145C2×Dic34—24no—
146C2×Dic34—42no—
147C2×Dic34—42no—
148C2×C8×C17—21no—
149C2×C8×C17—21no—
150C2×C8×C17—21no—
151C2×C8×C17—21no—
152C2×C8×C17—21no—
153C2×C8×C17C22×C412yes—
154C4⋊C4×C17—21no—
155C4⋊C4×C17—21no—
156C4⋊C4×C17—21no—
157C4⋊C4×C17—21no—
158C4⋊C4×C17—21no—
159C4⋊C4×C17—21no—
160C4⋊C4×C17—42no—
161C4⋊C4×C17C22×C412yes—
162C4⋊C4×C17C22×C412yes—
163C42×C17—21no—
164C42×C17—21no—
165C42×C17—42no—
166#63227—24no—
167#63227—42no—
168#63227—42no—
169#63227—42no—
170#63227—42no—
171C4⋊Dic17—21no—
172C4⋊Dic17—21no—
173C4⋊Dic17—21no—
174C4⋊Dic17—21no—
175C4⋊Dic17—21no—
176C4⋊Dic17—42no—
177C4⋊Dic17—42no—
178C4⋊Dic17C22×C412yes—
179#63225—24no—
180#63225—42no—
181#63225—42no—
182#63225—42no—
183C4×Dic17—22no—
184C4×Dic17—21no—
185C4×Dic17—21no—
186C4×Dic17—22no—
187C4×Dic17—22no—
188C4×Dic17—41no—
189C4×Dic17—42no—
190C4×Dic17—42no—
191C4×Dic17C22×C412yes—
192C4×Dic17#4112yes—
193C2×#3622—21no—
194C2×#3622—21no—
195C2×#3622—21no—
196C2×#3622—21no—
197C2×#3622—21no—
198C2×#3622C22×C412yes—
199#22117—171nomaximal, Sylow
200C23×C17C22×D411yescharacteristic
201C22×D17—22no—
202C22×D17—41no—
203C22×D17—41no—
204C22×D17—41no—
205C22×D17—41no—
206C22×D17—41no—
207C22×D17—41no—
208C2×C4×C17—21no—
209C2×C4×C17—21no—
210C2×C4×C17—22no—
211C2×C4×C17—22no—
212C2×C4×C17—21no—
213C2×C4×C17—21no—
214C2×C4×C17—21no—
215C2×C4×C17—41no—
216C2×C4×C17—41no—
217C2×C4×C17—41no—
218C2×C4×C17—41no—
219C2×C4×C17—41no—
220C2×C4×C17—41no—
221C2×C4×C17—41no—
222C2×C4×C17—41no—
223C2×C4×C17—41no—
224C2×C4×C17—41no—
225C2×C4×C17AGL(1,ℤ/8)12yes—
226C2×C4×C17C23×C411yescharacteristic
227C2×C4×C17C2×#4111yescharacteristic
228C2×C4×C17H(5,𝔽2)11yescharacteristic
229C2×C4×C172−1+411yescharacteristic
230C2×C4×C172−1+411yescharacteristic
231C2×C4×C172−1+411yescharacteristic
232C2×Dic17—21no—
233C2×Dic17—21no—
234C2×Dic17—22no—
235C2×Dic17—21no—
236C2×Dic17—21no—
237C2×Dic17—21no—
238C2×Dic17—21no—
239C2×Dic17—21no—
240C2×Dic17—21no—
241C2×Dic17—41no—
242C2×Dic17—41no—
243C2×Dic17—41no—
244C2×Dic17—41no—
245C2×Dic17—41no—
246C2×Dic17—41no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17C4×D412yes—
250C2×Dic17C4×D412yes—
251C2×Dic17AGL(1,ℤ/8)12yes—
252C2×Dic17#13712yes—
253D68—22no—
254D68—22no—
255D68—41no—
256D68—41no—
257C4×D17—48no—
258Dic34—48no—
259C8×C17—22no—
260C8×C17—41no—
261#3622—22no—
262#3622—41no—
263C2×#476—171no—
264C4×C4⋊Q8—171no—
265#2293—171no—
266#1987—171no—
267#1947—171no—
268#1944—171no—
269#1944—171no—
270#1936—171no—
271#1936—171no—
272#1925—171no—
273#1925—171no—
274#1920—171no—
275#1920D1711yescharacteristic, complemented
276#1802—171no—
277#1767—171no—
278C22×C17C2×C4×D411yescharacteristic
279C22×C17#53611yescharacteristic
280C22×C17#53711yescharacteristic
281C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
283C22×C17C2×#13711yescharacteristic
284C22×C17#57511yescharacteristic
285C2×D17—42no—
286C2×D17—42no—
287C2×D17—42no—
288C2×D17—81no—
289C2×D17—81no—
290C2×D17—81no—
291C2×D17—81no—
292C4×C17—22no—
293C4×C17—42no—
294C4×C17—42no—
295C4×C17—41no—
296C4×C17—81no—
297C4×C17—81no—
298C4×C17#51911yescharacteristic
299C4×C17#52011yescharacteristic
300C4×C17#54111yescharacteristic
301C4×C17#54211yescharacteristic
302Dic17—22no—
303Dic17—42no—
304Dic17—41no—
305Dic17—41no—
306Dic17—81no—
307Dic17—81no—
308Dic17#44014yes—
309C2×C4⋊Q8—171no—
310C2×C4×Q8—341no—
311C2×C4×D4—171no—
312C2×#117—341no—
313#476—178no—
314C2×#107—171no—
315C2×#107—171no—
316C2×#106—171no—
317C2×#106C2×D1711yescharacteristic
318C2×C4⋊C8—171no—
319C2×C4⋊C8—171no—
320C2×#102—171no—
321C2×#102—171no—
322C2×#102—171no—
323C2×#102—171no—
324C2×#97—171no—
325C2×#97—171no—
326#391—341no—
327#390—171no—
328#390—171no—
329#389—171no—
330#389—171no—
331#389—171no—
332#389—341no—
333#389C2×D1711yescharacteristic
334#388—341no—
335#388—341no—
336#384—171no—
337C4×C4⋊C4—171no—
338C4×C4⋊C4—171no—
339C4×C4⋊C4—341no—
340C4×C4⋊C4C2×D1711yescharacteristic
341C4×C22⋊C4—341no—
342C43—171no—
343#340—171no—
344#340—171no—
345#340—171no—
346#340—171no—
347#340—171no—
348#340C2×D1711yescharacteristic
349#340C2×D1711yescharacteristic
350#340C2×D1711yescharacteristic
351#336—171no—
352#336C2×D1711yescharacteristic
353C2×C17#232311yescharacteristic
354C2×C17C2×#44011yescharacteristic
355C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
356C2×C17#323511yescharacteristic
357C2×C17#323811yescharacteristic
358C2×C17#325211yescharacteristic
359C2×C17#325311yescharacteristic
360D17—82no—
361C22×Q8—341no—
362C22×D4—171no—
363C23×C4—341no—
364C22×C8—171no—
365C22×C8—171no—
366C22×C8C22×D1711yescharacteristic
367C22×C8C22×D1711yescharacteristic
368C4⋊Q8—174no—
369C4⋊Q8—342no—
370C4×Q8—684no—
371C4×D4—174no—
372C4×D4—342no—
373#117—684no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—171no—
376C2×C4⋊C4—171no—
377C2×C4⋊C4—171no—
378C2×C4⋊C4—341no—
379C2×C4⋊C4—341no—
380C2×C4⋊C4—341no—
381C2×C4⋊C4—341no—
382C2×C4⋊C4—341no—
383C2×C4⋊C4—341no—
384C2×C4⋊C4C22×D1711yescharacteristic
385C2×C4⋊C4C22×D1711yescharacteristic
386C2×C4⋊C4C22×D1711yescharacteristic
387C2×C4⋊C4C22×D1711yescharacteristic
388C2×C22⋊C4—341no—
389C2×C22⋊C4—341no—
390C2×C22⋊C4—341no—
391C2×C22⋊C4—341no—
392C2×C22⋊C4—341no—
393C2×C42—21no—
394C2×C42—171no—
395C2×C42—171no—
396C2×C42—171no—
397C2×C42—171no—
398C2×C42—171no—
399C2×C42—171no—
400C2×C42—341no—
401C2×C42—341no—
402C2×C42C22×D1711yescharacteristic
403#107—681no—
404#107—681no—
405#106—174no—
406#106C4×D1714yes—
407C4⋊C8—174no—
408C4⋊C8—342no—
409#102—174no—
410#102—174no—
411#102—681no—
412#102—681no—
413#97—681no—
414#97—681no—
415#95—21no—
416#95—21no—
417#95—341no—
418#95—341no—
419#95—341no—
420#95—341no—
421#95—341no—
422C17#2211711yescharacteristic, complemented, Sylow
423C24—341no—
424C2×Q8—344no—
425C2×Q8—682no—
426C2×Q8—682no—
427C2×D4—172no—
428C2×D4—341no—
429C2×D4—341no—
430C2×D4—341no—
431C2×D4—341no—
432C2×D4—341no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—171no—
438C22×C4—171no—
439C22×C4—171no—
440C22×C4—171no—
441C22×C4—344no—
442C22×C4—341no—
443C22×C4—341no—
444C22×C4—341no—
445C22×C4—682no—
446C22×C4—682no—
447C22×C4D4×D1711yescharacteristic
448C22×C4D4×D1711yescharacteristic
449C22×C4C23×D1711yescharacteristic
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—172no—
456C2×C8—341no—
457C2×C8—341no—
458C2×C8—341no—
459C2×C8—341no—
460C2×C8—341no—
461C2×C8C2×C4×D1712yes—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—42no—
469C4⋊C4—172no—
470C4⋊C4—344no—
471C4⋊C4—341no—
472C4⋊C4—341no—
473C4⋊C4—341no—
474C4⋊C4—341no—
475C4⋊C4—341no—
476C4⋊C4—682no—
477C4⋊C4—682no—
478C4⋊C4—682no—
479C4⋊C4—682no—
480C4⋊C4—682no—
481C4⋊C4C2×C4×D1712yes—
482C4⋊C4C2×C4×D1712yes—
483C22⋊C4—344no—
484C22⋊C4—682no—
485C22⋊C4—682no—
486C22⋊C4—682no—
487C22⋊C4—682no—
488C42—21no—
489C42—21no—
490C42—42no—
491C42—172no—
492C42—172no—
493C42—342no—
494C42—342no—
495C42—341no—
496C42—341no—
497C42—342no—
498C42—681no—
499C42—682no—
500C42—682no—
501C23—342no—
502C23—681no—
503C23—681no—
504C23—681no—
505C23—681no—
506C23—681no—
507C23—681no—
508C23C2×D4×D1711yescharacteristic
509Q8—688no—
510D4—342no—
511D4—342no—
512D4—681no—
513D4—681no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—41no—
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—41no—
531C2×C4—172no—
532C2×C4—172no—
533C2×C4—172no—
534C2×C4—172no—
535C2×C4—341no—
536C2×C4—341no—
537C2×C4—342no—
538C2×C4—341no—
539C2×C4—341no—
540C2×C4—341no—
541C2×C4—341no—
542C2×C4—341no—
543C2×C4—341no—
544C2×C4—688no—
545C2×C4—681no—
546C2×C4—681no—
547C2×C4—681no—
548C2×C4—681no—
549C2×C4—681no—
550C2×C4—681no—
551C2×C4—681no—
552C2×C4—681no—
553C2×C4#1058760012yes—
554C2×C4C22×C4×D1711yescharacteristic
555C2×C4#1058769411yescharacteristic
556C2×C4#41×D1711yescharacteristic
557C2×C4#1058769611yescharacteristic
558C2×C4#1058769711yescharacteristic
559C2×C4#1058769711yescharacteristic
560C8—22no—
561C8—41no—
562C8—342no—
563C8—681no—
564C22—682no—
565C22—682no—
566C22—682no—
567C22—1361no—
568C22—1361no—
569C22—1361no—
570C22—1361no—
571C22C4×D4×D1711yescharacteristic
572C22#1176338111yescharacteristic
573C22#1176338911yescharacteristic
574C22AGL(1,ℤ/8)×D1711yescharacteristic
575C22C2×#1058760011yescharacteristic
576C22#137×D1711yescharacteristic
577C22#1176360611yescharacteristic
578C4—22no—
579C4—42no—
580C4—42no—
581C4—41no—
582C4—81no—
583C4—81no—
584C4—174no—
585C4—342no—
586C4—682no—
587C4—681no—
588C4—681no—
589C4—1361no—
590C4—1361no—
591C4#1176328511yescharacteristic
592C4#1176328911yescharacteristic
593C4#1176341811yescharacteristic
594C4#1176342511yescharacteristic
595C2—1362no—
596C2#1453146511yescharacteristic
597C2D17×#44011yescharacteristic
598C2C4×#1058760011yescharacteristic
599C2#1453997911yescharacteristic
600C2#1454000811yescharacteristic
601C2#1454015611yescharacteristic
602C2#1454016411yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1920C2
socleC23×C17C22×D4
prime radicalC22×D68C23
cosocleC22×C4×C17C24
hypercenter#1920D17
hypocenterC17#22117
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1920

Complements

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

NH ≅ G/Ntype
2. C4×#10587574C2semidirect
4. #14529730C2semidirect
5. #14529541C2semidirect
6. #14529528C2semidirect
7. #14529526C2semidirect
12. #14529366C2semidirect
14. #14528636C2semidirect
15. C17×#1920C2semidirect
20. #11763098C4semidirect
275. #1920D17semidirect
422. C17#22117semidirect

Maximal subgroups

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

subgroupindex
C2×#117630982
C4×#105875742
#145312422
#145297302
#145295412
#145295282
#145295262
#145294822
#145294792
#145294052
#145294042
#145293662
#145287832
#145286362
C17×#19202
#2211717