Finite Group Databasebeta

← #17039204

Subgroups — #17039204

Order 4352.

894 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762864C211yesmaximal, characteristic
2#14531188C211yesmaximal, characteristic
3C4×#10587578C211yesmaximal, characteristic
4#14529546C211yesmaximal, characteristic, complemented
5#14529544C211yesmaximal, characteristic, complemented
6#14529541C211yesmaximal, characteristic, complemented
7#14529539C211yesmaximal, characteristic, complemented
8#14528986C211yesmaximal, characteristic
9#14528984C211yesmaximal, characteristic
10#14528982C211yesmaximal, characteristic
11#14528980C211yesmaximal, characteristic
12#14528682C211yesmaximal, characteristic, complemented
13#14528682C211yesmaximal, characteristic, complemented
14#14528681C211yesmaximal, characteristic, complemented
15C17×#1776C211yesmaximal, characteristic, complemented
16C2×#10587578C2211yescharacteristic
17C2×#10587578C2211yescharacteristic
18C2×#10587558—21no—
19C2×C42×D17C2211yescharacteristic
20#11762864C418yescomplemented
21C2×#10587489C2211yescharacteristic
22C2×#10587478C2211yescharacteristic
23C2×#10587478C2211yescharacteristic
24C2×#10587488C2211yescharacteristic
25C2×#10587479C2211yescharacteristic
26C2×#10587479C2211yescharacteristic
27C2×#10587485C2211yescharacteristic
28C2×#10587485C2211yescharacteristic
29C2×#10587491C2211yescharacteristic
30C2×#10587491C2211yescharacteristic
31C2×#10587474C2211yescharacteristic
32#11762678C2211yescharacteristic
33#11762678C2211yescharacteristic
34#11762673—21no—
35#11762668—21no—
36#11762666C2211yescharacteristic
37#11762666C2211yescharacteristic
38C4⋊C4×Dic17C2211yescharacteristic
39C4⋊C4×Dic17C2211yescharacteristic
40C4×#63225—21no—
41C4×C4⋊Dic17C2211yescharacteristic
42C4×C4⋊Dic17C2211yescharacteristic
43C4×#63227—21no—
44#11762656—21no—
45#11762654—21no—
46C42×Dic17—21no—
47#11762572C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762571C2211yescharacteristic
53C2×C17×#107C2211yescharacteristic
54C2×C17×#106C2211yescharacteristic
55C2×C17×#97C2211yescharacteristic
56C4×C4⋊C4×C17C2211yescharacteristic
57C4×C4⋊C4×C17C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C17×#340C2211yescharacteristic
60C22×C4×D17D411yescharacteristic
61C22×C4×D17D411yescharacteristic
62C22×C4×D17C2311yescharacteristic
63C22×C8×C17C2311yescharacteristic
64C22×C8×C17C2311yescharacteristic
65C2×C4⋊C4×C17C2311yescharacteristic
66C2×C4⋊C4×C17C2311yescharacteristic
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C42×C17—21no—
70C2×C42×C17—21no—
71C2×C42×C17C2311yescharacteristic
72C2×#63227—21no—
73C2×#63227—21no—
74C2×#63227—21no—
75C2×#63227—21no—
76C2×C4⋊Dic17C2311yescharacteristic
77C2×C4⋊Dic17C2311yescharacteristic
78C2×C4⋊Dic17C2311yescharacteristic
79C2×C4⋊Dic17C2311yescharacteristic
80C2×#63225—21no—
81C2×#63225—21no—
82C2×#63225—21no—
83C2×#63225—21no—
84C2×C4×Dic17—21no—
85C2×C4×Dic17—21no—
86C2×C4×Dic17—21no—
87C2×C4×Dic17—21no—
88C2×C4×Dic17—21no—
89C2×C4×Dic17—21no—
90C2×C4×Dic17D411yescharacteristic
91C2×C4×Dic17D411yescharacteristic
92C2×C4×Dic17C2311yescharacteristic
93C22×#3622C2311yescharacteristic
94C22×#3622C2311yescharacteristic
95#10587578—22no—
96#10587578—22no—
97#10587578—22no—
98#10587578—22no—
99#10587558—44no—
100C42×D17C2×C414yes—
101C42×D17Q814yes—
102C17×#107—22no—
103C17×#106—22no—
104C17×#97C2×C414yes—
105C17×#95—21no—
106C17×#95—21no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587491C2×C414yes—
114#10587491C2×C414yes—
115#10587489—22no—
116#10587488—22no—
117#10587485C2×C414yes—
118#10587485C2×C414yes—
119#10587479—22no—
120#10587479—22no—
121#10587478—22no—
122#10587478—22no—
123#10587474C2×C414yes—
124C23×D17C2×D411yescharacteristic
125C22×C4×C17—21no—
126C22×C4×C17—21no—
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17C2×D411yescharacteristic
130C22×C4×C17C2×D411yescharacteristic
131C22×C4×C17C2411yescharacteristic
132C22×Dic17—21no—
133C22×Dic17—21no—
134C22×Dic17—21no—
135C22×Dic17—21no—
136C22×Dic17C2×D411yescharacteristic
137C22×Dic17C2×D411yescharacteristic
138C22×Dic17C2×D411yescharacteristic
139C2×C4×D17—21no—
140C2×C4×D17—21no—
141C2×C4×D17—21no—
142C2×C4×D17—21no—
143C2×C4×D17—22no—
144C2×C4×D17—22no—
145C2×C4×D17—41no—
146C2×C4×D17—41no—
147C2×C4×D17C4⋊C414yes—
148C2×C4×D17C4⋊C414yes—
149C2×C4×D17C22×C412yes—
150C2×C4×D17C2×Q812yes—
151C2×C8×C17—21no—
152C2×C8×C17—21no—
153C2×C8×C17—21no—
154C2×C8×C17—21no—
155C2×C8×C17C22×C412yes—
156C2×C8×C17C22×C412yes—
157C4⋊C4×C17—21no—
158C4⋊C4×C17—21no—
159C4⋊C4×C17—21no—
160C4⋊C4×C17—21no—
161C4⋊C4×C17—21no—
162C4⋊C4×C17—21no—
163C4⋊C4×C17#4112yes—
164C4⋊C4×C17#4112yes—
165C42×C17—42no—
166C42×C17—42no—
167C42×C17C22×C412yes—
168C42×C17C2×Q812yes—
169#63227—24no—
170#63227—24no—
171#63227—42no—
172#63227—42no—
173C4⋊Dic17—21no—
174C4⋊Dic17—21no—
175C4⋊Dic17—21no—
176C4⋊Dic17—21no—
177C4⋊Dic17—21no—
178C4⋊Dic17—21no—
179C4⋊Dic17#4112yes—
180C4⋊Dic17#4112yes—
181#63225—42no—
182#63225—42no—
183#63225—42no—
184#63225—42no—
185C4×Dic17—24no—
186C4×Dic17—24no—
187C4×Dic17—42no—
188C4×Dic17—42no—
189C4×Dic17—42no—
190C4×Dic17—42no—
191C4×Dic17C4⋊C414yes—
192C4×Dic17C4⋊C414yes—
193C4×Dic17C22×C412yes—
194C4×Dic17C2×Q812yes—
195C2×#3622—21no—
196C2×#3622—21no—
197C2×#3622—21no—
198C2×#3622—21no—
199C2×#3622C22×C412yes—
200C2×#3622C22×C412yes—
201#21364—171nomaximal, Sylow
202C23×C17C22×D411yescharacteristic
203C22×D17—21no—
204C22×D17—21no—
205C22×D17—21no—
206C22×D17—21no—
207C22×D17—21no—
208C22×D17—21no—
209C22×D17C2×C4⋊C412yes—
210C2×C4×C17—21no—
211C2×C4×C17—21no—
212C2×C4×C17—21no—
213C2×C4×C17—21no—
214C2×C4×C17—22no—
215C2×C4×C17—22no—
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×C17—41no—
226C2×C4×C17C2×C4⋊C412yes—
227C2×C4×C17C2×C4⋊C412yes—
228C2×C4×C17C23×C411yescharacteristic
229C2×C4×C17C22×Q811yescharacteristic
230C2×C4×C17C2×#4111yescharacteristic
231C2×C4×C17C2×#4111yescharacteristic
232C2×C4×C17C2×#4111yescharacteristic
233C2×C4×C17C2×#4111yescharacteristic
234C2×Dic17—21no—
235C2×Dic17—21no—
236C2×Dic17—21no—
237C2×Dic17—21no—
238C2×Dic17—22no—
239C2×Dic17—22no—
240C2×Dic17—41no—
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×Dic17—41no—
250C2×Dic17C2×C4⋊C412yes—
251C2×Dic17C2×C4⋊C412yes—
252C2×Dic17C2×C4⋊C412yes—
253C2×Dic17AGL(1,ℤ/8)12yes—
254C2×Dic17#13712yes—
255C4×D17—22no—
256C4×D17—22no—
257C4×D17—22no—
258C4×D17—22no—
259C4×D17—28no—
260C4×D17—44no—
261C8×C17—22no—
262C8×C17—22no—
263#3622—22no—
264#3622—22no—
265C2×#432—171no—
266#2285—171no—
267C4×#117—171no—
268#1948—171no—
269#1948—171no—
270#1947—171no—
271#1947—171no—
272#1854—171no—
273#1854—171no—
274#1854—171no—
275#1854—171no—
276#1776—171no—
277#1776—171no—
278#1776—171no—
279#1776D1711yescharacteristic, complemented
280C22×C17C22×C4⋊C411yescharacteristic
281C22×C17#52511yescharacteristic
282C22×C17#52511yescharacteristic
283C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
284C22×C17C2×#13711yescharacteristic
285C22×C17#57511yescharacteristic
286C22×C17#57511yescharacteristic
287C2×D17—22no—
288C2×D17—22no—
289C2×D17—22no—
290C2×D17—41no—
291C2×D17—41no—
292C2×D17—41no—
293C2×D17—41no—
294C4×C17—22no—
295C4×C17—22no—
296C4×C17—42no—
297C4×C17—42no—
298C4×C17—81no—
299C4×C17—81no—
300C4×C17C4×#4111yescharacteristic
301C4×C17C4×#4111yescharacteristic
302C4×C17#53311yescharacteristic
303C4×C17#53311yescharacteristic
304Dic17—22no—
305Dic17—22no—
306Dic17—42no—
307Dic17—42no—
308Dic17—81no—
309Dic17—81no—
310Dic17#42814yes—
311C2×#117—171no—
312C2×#117—171no—
313C2×#117—341no—
314C22×C42—171no—
315#432—178no—
316C2×#107—171no—
317C2×#107—171no—
318C2×#107—171no—
319C2×#107C2×D1711yescharacteristic
320C2×#106—171no—
321C2×#106—171no—
322C2×#106—171no—
323C2×#106C2×D1711yescharacteristic
324C2×C4⋊C8—171no—
325C2×C4⋊C8—171no—
326C2×C22⋊C8—171no—
327C2×C22⋊C8—171no—
328C2×#97—171no—
329C2×#97C2×D1711yescharacteristic
330#384—171no—
331#384—171no—
332#382—341no—
333#380—171no—
334#380—171no—
335#380—341no—
336C4×C4⋊C4—171no—
337C4×C4⋊C4—171no—
338C4×C4⋊C4—171no—
339C4×C4⋊C4—171no—
340C4×C4⋊C4—341no—
341C4×C4⋊C4C2×D1711yescharacteristic
342C4×C4⋊C4C2×D1711yescharacteristic
343C4×C22⋊C4—341no—
344C4×C22⋊C4—341no—
345#376—341no—
346C43—341no—
347#340—171no—
348#340—171no—
349#340—171no—
350#340—171no—
351#340—171no—
352#340—171no—
353#340C2×D1711yescharacteristic
354#340C2×D1711yescharacteristic
355C2×C17#229611yescharacteristic
356C2×C17C2×#42811yescharacteristic
357C2×C17#291111yescharacteristic
358C2×C17#311511yescharacteristic
359C2×C17#311511yescharacteristic
360C2×C17#311611yescharacteristic
361C2×C17#311611yescharacteristic
362D17—42no—
363C23×C4—171no—
364C23×C4—171no—
365C23×C4—171no—
366C22×C8—171no—
367C22×C8—171no—
368C22×C8C22×D1711yescharacteristic
369C22×C8C22×D1711yescharacteristic
370#117—342no—
371#117—342no—
372#117—342no—
373#117—342no—
374#117—684no—
375C2×C4⋊C4—171no—
376C2×C4⋊C4—171no—
377C2×C4⋊C4—171no—
378C2×C4⋊C4—171no—
379C2×C4⋊C4—341no—
380C2×C4⋊C4—341no—
381C2×C4⋊C4—341no—
382C2×C4⋊C4—341no—
383C2×C4⋊C4C22×D1711yescharacteristic
384C2×C4⋊C4C22×D1711yescharacteristic
385C2×C4⋊C4C22×D1711yescharacteristic
386C2×C4⋊C4C22×D1711yescharacteristic
387C2×C22⋊C4—341no—
388C2×C22⋊C4—341no—
389C2×C22⋊C4—341no—
390C2×C22⋊C4—341no—
391C2×C42—21no—
392C2×C42—21no—
393C2×C42—171no—
394C2×C42—171no—
395C2×C42—171no—
396C2×C42—174no—
397C2×C42—174no—
398C2×C42—341no—
399C2×C42—341no—
400C2×C42—341no—
401C2×C42—341no—
402C2×C42—341no—
403C2×C42—341no—
404C2×C42C22×D1711yescharacteristic
405#107—22no—
406#107—342no—
407#107—342no—
408#107—342no—
409#106—22no—
410#106—342no—
411#106—342no—
412#106—342no—
413C4⋊C8—174no—
414C4⋊C8—174no—
415C22⋊C8—174no—
416C22⋊C8—174no—
417#97—174no—
418#97C4×D1714yes—
419#95—21no—
420#95—21no—
421#95—341no—
422#95—341no—
423#95—341no—
424#95—341no—
425#95—341no—
426#95—341no—
427C17#2136411yescharacteristic, complemented, Sylow
428C24—171no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—171no—
434C22×C4—171no—
435C22×C4—171no—
436C22×C4—172no—
437C22×C4—172no—
438C22×C4—174no—
439C22×C4—174no—
440C22×C4—342no—
441C22×C4—342no—
442C22×C4—341no—
443C22×C4—341no—
444C22×C4—341no—
445C22×C4—341no—
446C22×C4—341no—
447C22×C4—341no—
448C22×C4—341no—
449C22×C4—341no—
450C22×C4—681no—
451C22×C4—681no—
452C22×C4D4×D1711yescharacteristic
453C22×C4D4×D1711yescharacteristic
454C22×C4C23×D1711yescharacteristic
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—172no—
460C2×C8—172no—
461C2×C8—341no—
462C2×C8—341no—
463C2×C8—341no—
464C2×C8—341no—
465C2×C8C2×C4×D1712yes—
466C2×C8C2×C4×D1712yes—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—172no—
474C4⋊C4—172no—
475C4⋊C4—341no—
476C4⋊C4—341no—
477C4⋊C4—341no—
478C4⋊C4—341no—
479C4⋊C4—341no—
480C4⋊C4—341no—
481C4⋊C4—682no—
482C4⋊C4—682no—
483C4⋊C4—682no—
484C4⋊C4—682no—
485C4⋊C4#6325212yes—
486C4⋊C4#6325212yes—
487C22⋊C4—344no—
488C22⋊C4—344no—
489C22⋊C4—682no—
490C22⋊C4—682no—
491C42—42no—
492C42—42no—
493C42—172no—
494C42—172no—
495C42—174no—
496C42—174no—
497C42—344no—
498C42—344no—
499C42—682no—
500C42—682no—
501C42—682no—
502C42—682no—
503C42C2×C4×D1712yes—
504C42Q8×D1712yes—
505C23—172no—
506C23—341no—
507C23—341no—
508C23—341no—
509C23—341no—
510C23—341no—
511C23—341no—
512C23C2×D4×D1711yescharacteristic
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—41no—
520C2×C4—41no—
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—172no—
530C2×C4—172no—
531C2×C4—172no—
532C2×C4—172no—
533C2×C4—172no—
534C2×C4—348no—
535C2×C4—341no—
536C2×C4—341no—
537C2×C4—341no—
538C2×C4—341no—
539C2×C4—342no—
540C2×C4—342no—
541C2×C4—342no—
542C2×C4—342no—
543C2×C4—342no—
544C2×C4—342no—
545C2×C4—684no—
546C2×C4—681no—
547C2×C4—681no—
548C2×C4—681no—
549C2×C4—681no—
550C2×C4—681no—
551C2×C4—681no—
552C2×C4—681no—
553C2×C4—681no—
554C2×C4—681no—
555C2×C4—681no—
556C2×C4C4⋊C4×D1712yes—
557C2×C4C4⋊C4×D1712yes—
558C2×C4C22×C4×D1711yescharacteristic
559C2×C4C2×#6325211yescharacteristic
560C2×C4C2×#6325211yescharacteristic
561C2×C4C2×Q8×D1711yescharacteristic
562C2×C4#41×D1711yescharacteristic
563C2×C4#41×D1711yescharacteristic
564C8—22no—
565C8—22no—
566C8—342no—
567C8—342no—
568C22—342no—
569C22—342no—
570C22—342no—
571C22—681no—
572C22—681no—
573C22—681no—
574C22—681no—
575C22C2×C4⋊C4×D1711yescharacteristic
576C22#1176331811yescharacteristic
577C22#1176331811yescharacteristic
578C22AGL(1,ℤ/8)×D1711yescharacteristic
579C22#137×D1711yescharacteristic
580C22#1176360511yescharacteristic
581C22#1176360811yescharacteristic
582C4—22no—
583C4—22no—
584C4—42no—
585C4—42no—
586C4—81no—
587C4—81no—
588C4—174no—
589C4—342no—
590C4—342no—
591C4—682no—
592C4—682no—
593C4—1361no—
594C4—1361no—
595C4#1176327711yescharacteristic
596C4C4×#6325211yescharacteristic
597C4#1176336311yescharacteristic
598C4#1176336311yescharacteristic
599C2—682no—
600C2#1453126411yescharacteristic
601C2D17×#42811yescharacteristic
602C2#1453737111yescharacteristic
603C2#1453893111yescharacteristic
604C2#1453893311yescharacteristic
605C2#1453894211yescharacteristic
606C2#1453894411yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1776C2
socleC23×C17C22×D4
prime radicalC23×D17C2×D4
cosocleC22×C4×C17C24
hypercenter#1776D17
hypocenterC17#21364
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
4. #14529546C2semidirect
5. #14529544C2semidirect
6. #14529541C2semidirect
7. #14529539C2semidirect
12. #14528682C2semidirect
13. #14528682C2semidirect
14. #14528681C2semidirect
15. C17×#1776C2semidirect
20. #11762864C4semidirect
279. #1776D17semidirect
427. C17#21364semidirect

Maximal subgroups

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

subgroupindex
C2×#117628642
#145311882
C4×#105875782
#145295462
#145295442
#145295412
#145295392
#145289862
#145289842
#145289822
#145289802
#145286822
#145286812
C17×#17762
#2136417