Finite Group Databasebeta

← #16965005

Subgroups — #16965005

Order 4352.

1390 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762720C211yesmaximal, characteristic
2C2×#11762700C211yesmaximal, characteristic
3#14527055C211yesmaximal, characteristic, complemented
4#14527047C211yesmaximal, characteristic, complemented
5#14527042C212yesmaximal, complemented
6#14527039C212yesmaximal, complemented
7#14527032C212yesmaximal
8#14527028C212yesmaximal
9#14527026C211yesmaximal, characteristic
10#14527011C211yesmaximal, characteristic, complemented
11C2×C17×#391C211yesmaximal, characteristic, complemented
12C22×#63225C2211yescharacteristic
13C22×#63225C2212yes—
14C22×#63227C2211yescharacteristic
15C22×#63227C2211yescharacteristic
16C22×#63227C2211yescharacteristic
17C22×#63227C2212yes—
18C22×#63227C2212yes—
19#11762720C418yescomplemented
20#11762700—24no—
21C2×#10587502C2211yescharacteristic
22C2×#10587502C2211yescharacteristic
23C2×#10587502C2211yescharacteristic
24C2×#10587502C2212yes—
25C2×#10587502C2212yes—
26C2×#10587502C2212yes—
27C2×#10587502C2212yes—
28C2×#10587502C2212yes—
29C2×#10587502C2212yes—
30C2×#10587502C2211yescharacteristic
31C2×#10587502C2212yes—
32C23×Q8×C17C2211yescharacteristic
33C22×C4⋊C4×C17C2211yescharacteristic
34C22×C42×C17C2211yescharacteristic
35C17×#391—24no—
36C2×C17×#95C2212yes—
37C2×C17×#95C2212yes—
38C24×D17C2311yescharacteristic
39C22×Q8×C17—22no—
40C22×Q8×C17—22no—
41C22×Q8×C17—42no—
42C22×Q8×C17—42no—
43C22×Q8×C17C2×C414yes—
44C23×C4×C17C2312yes—
45C23×C4×C17C2312yes—
46C23×C4×C17C2311yescharacteristic
47C23×C4×C17C2311yescharacteristic
48C23×C4×C17C2311yescharacteristic
49C23×Dic17C2311yescharacteristic
50C23×Dic17C2311yescharacteristic
51C23×Dic17C2312yes—
52C23×Dic17C2311yescharacteristic
53C23×Dic17C2312yes—
54C2×C4⋊C4×C17—22no—
55C2×C4⋊C4×C17—42no—
56C2×C42×C17—42no—
57C2×C42×C17D414yes—
58C2×#63227—22no—
59C2×#63227—22no—
60C2×#63227—22no—
61C2×#63227—22no—
62C2×#63227—42no—
63C2×#63227—42no—
64C2×#63227—42no—
65C2×#63227—42no—
66C2×#63227—42no—
67C2×#63227—42no—
68C2×#63227—42no—
69C2×#63227C2×C414yes—
70C2×#63227C2×C418yes—
71C2×#63225—22no—
72C2×#63225—42no—
73C2×#63225—42no—
74C2×#63225—42no—
75C2×#63225—42no—
76C17×#95—24no—
77C17×#95—24no—
78#10587502—22no—
79#10587502—22no—
80#10587502—22no—
81#10587502—42no—
82#10587502—42no—
83#10587502—42no—
84#10587502—42no—
85#10587502—42no—
86#10587502—42no—
87#10587502C2×C418yes—
88#10587502C2×C414yes—
89C24×C17C2411yescharacteristic
90C23×D17—21no—
91C23×D17—21no—
92C23×D17—22no—
93C23×D17—22no—
94C23×D17—22no—
95C23×D17—21no—
96C23×D17—21no—
97C23×D17—21no—
98C23×D17—22no—
99C23×D17—21no—
100C23×D17C22×C412yes—
101C2×Q8×C17—28no—
102C2×Q8×C17—48no—
103C2×Q8×C17—44no—
104C2×Q8×C17—44no—
105C2×Q8×C17—48no—
106C22×C4×C17—22no—
107C22×C4×C17—22no—
108C22×C4×C17—22no—
109C22×C4×C17—22no—
110C22×C4×C17—22no—
111C22×C4×C17—22no—
112C22×C4×C17—22no—
113C22×C4×C17—22no—
114C22×C4×C17—21no—
115C22×C4×C17—22no—
116C22×C4×C17—22no—
117C22×C4×C17—22no—
118C22×C4×C17—21no—
119C22×C4×C17—21no—
120C22×C4×C17—22no—
121C22×C4×C17—21no—
122C22×C4×C17—22no—
123C22×C4×C17—22no—
124C22×C4×C17—22no—
125C22×C4×C17—22no—
126C22×C4×C17—22no—
127C22×C4×C17—22no—
128C22×C4×C17—22no—
129C22×C4×C17C22×C414yes—
130C22×C4×C17C22×C412yes—
131C22×C4×C17C2×D412yes—
132C22×C4×C17C2×D412yes—
133C22×C4×C17C2×D412yes—
134C22×C4×C17#4112yes—
135C22×Dic17—21no—
136C22×Dic17—21no—
137C22×Dic17—22no—
138C22×Dic17—22no—
139C22×Dic17—22no—
140C22×Dic17—22no—
141C22×Dic17—22no—
142C22×Dic17—22no—
143C22×Dic17—22no—
144C22×Dic17—22no—
145C22×Dic17—21no—
146C22×Dic17—21no—
147C22×Dic17—22no—
148C22×Dic17—22no—
149C22×Dic17—22no—
150C22×Dic17—22no—
151C22×Dic17—22no—
152C22×Dic17—22no—
153C22×Dic17—22no—
154C22×Dic17—22no—
155C22×Dic17—21no—
156C22×Dic17—22no—
157C22×Dic17—21no—
158C22×Dic17—21no—
159C22×Dic17—22no—
160C22×Dic17—22no—
161C22×Dic17—21no—
162C22×Dic17C22×C412yes—
163C22×Dic17C22×C414yes—
164C4⋊C4×C17—44no—
165C42×C17—44no—
166#63227—28no—
167#63227—48no—
168#63227—44no—
169#63227—44no—
170#63227—48no—
171#63225—44no—
172#63225—48no—
173#14625—171nomaximal, Sylow
174C23×C17C23×C411yescharacteristic
175C23×C17C22×D411yescharacteristic
176C23×C17C2×#4111yescharacteristic
177C23×C17H(5,𝔽2)12yes—
178C23×C17H(5,𝔽2)12yes—
179C23×C17H(5,𝔽2)12yes—
180C23×C17H(5,𝔽2)11yescharacteristic
181C23×C172−1+411yescharacteristic
182C23×C172−1+411yescharacteristic
183C23×C172−1+412yes—
184C23×C172−1+411yescharacteristic
185C22×D17—22no—
186C22×D17—24no—
187C22×D17—22no—
188C22×D17—22no—
189C22×D17—24no—
190C22×D17—41no—
191C22×D17—41no—
192C22×D17—41no—
193C22×D17—41no—
194C22×D17—41no—
195C22×D17—41no—
196C22×D17—42no—
197C22×D17—41no—
198C22×D17—41no—
199C22×D17—42no—
200C22×D17—42no—
201C22×D17—42no—
202C22×D17—42no—
203C22×D17—42no—
204C22×D17—42no—
205C22×D17—42no—
206C22×D17—42no—
207C22×D17—42no—
208Q8×C17—416no—
209C2×C4×C17—24no—
210C2×C4×C17—22no—
211C2×C4×C17—22no—
212C2×C4×C17—24no—
213C2×C4×C17—24no—
214C2×C4×C17—24no—
215C2×C4×C17—24no—
216C2×C4×C17—22no—
217C2×C4×C17—22no—
218C2×C4×C17—22no—
219C2×C4×C17—22no—
220C2×C4×C17—22no—
221C2×C4×C17—22no—
222C2×C4×C17—42no—
223C2×C4×C17—42no—
224C2×C4×C17—42no—
225C2×C4×C17—42no—
226C2×C4×C17—42no—
227C2×C4×C17—42no—
228C2×C4×C17—42no—
229C2×C4×C17—42no—
230C2×C4×C17—42no—
231C2×C4×C17—42no—
232C2×C4×C17—42no—
233C2×C4×C17—42no—
234C2×C4×C17—42no—
235C2×C4×C17—42no—
236C2×C4×C17—42no—
237C2×C4×C17C4×D414yes—
238C2×Dic17—22no—
239C2×Dic17—24no—
240C2×Dic17—24no—
241C2×Dic17—24no—
242C2×Dic17—24no—
243C2×Dic17—41no—
244C2×Dic17—41no—
245C2×Dic17—42no—
246C2×Dic17—42no—
247C2×Dic17—42no—
248C2×Dic17—42no—
249C2×Dic17—42no—
250C2×Dic17—41no—
251C2×Dic17—41no—
252C2×Dic17—41no—
253C2×Dic17—41no—
254C2×Dic17—41no—
255C2×Dic17—41no—
256C2×Dic17—42no—
257C2×Dic17—42no—
258C2×Dic17—42no—
259C2×Dic17—42no—
260C2×Dic17—42no—
261C2×Dic17—42no—
262C2×Dic17—42no—
263C2×Dic17—42no—
264C2×Dic17—42no—
265C2×Dic17—42no—
266C2×Dic17—42no—
267C2×#393—171no—
268C2×#391D1711yescharacteristic, complemented
269C2×#388—171no—
270#1446—171no—
271#1444—171no—
272#1442—172no—
273#1442—172no—
274#1440—172no—
275#1439—172no—
276#1439—171no—
277#1434—171no—
278C22×C17C2×C4×D411yescharacteristic
279C22×C17#51812yes—
280C22×C17#51911yescharacteristic
281C22×C17#52011yescharacteristic
282C22×C17#52012yes—
283C22×C17#53411yescharacteristic
284C22×C17#53511yescharacteristic
285C22×C17#53511yescharacteristic
286C22×C17#53611yescharacteristic
287C22×C17#53611yescharacteristic
288C22×C17#53711yescharacteristic
289C22×C17#53812yes—
290C22×C17#54111yescharacteristic
291C22×C17#54212yes—
292C22×C17#54211yescharacteristic
293C22×C17H(3,𝔽4)12yes—
294C22×C17#56212yes—
295C22×C17#56212yes—
296C22×C17#56212yes—
297C22×C17#56212yes—
298C22×C17#56312yes—
299C22×C17#56312yes—
300C22×C17#56312yes—
301C2×D17—42no—
302C2×D17—42no—
303C2×D17—42no—
304C2×D17—44no—
305C2×D17—44no—
306C2×D17—81no—
307C2×D17—81no—
308C2×D17—81no—
309C2×D17—82no—
310C2×D17—82no—
311C2×D17—81no—
312C4×C17—24no—
313C4×C17—42no—
314C4×C17—44no—
315C4×C17—42no—
316C4×C17—82no—
317Dic17—42no—
318Dic17—44no—
319Dic17—81no—
320Dic17—81no—
321Dic17—82no—
322C23×Q8C2×D1711yescharacteristic
323C22×C4⋊C4—171no—
324C22×C4⋊C4—172no—
325C22×C4⋊C4C2×D1711yescharacteristic
326C22×C22⋊C4—171no—
327C22×C22⋊C4—172no—
328C22×C22⋊C4—171no—
329C22×C22⋊C4—171no—
330C22×C22⋊C4—172no—
331C22×C42C2×D1711yescharacteristic
332#393—178no—
333#391—24no—
334#388—344no—
335C2×#95—171no—
336C2×#95—172no—
337C2×#95—171no—
338C2×#95—171no—
339C2×#95—171no—
340C2×#95—172no—
341C2×#95—172no—
342C2×#95—172no—
343C2×#95—172no—
344C2×#95—172no—
345C2×#95—172no—
346C2×#95C2×D1712yes—
347C2×#95C2×D1712yes—
348C2×C17#232111yescharacteristic
349C2×C17#232311yescharacteristic
350C2×C17#232611yescharacteristic
351C2×C17#237812yes—
352C2×C17#237912yes—
353C2×C17#281411yescharacteristic
354C2×C17#281611yescharacteristic
355C2×C17#281911yescharacteristic
356C2×C17#282111yescharacteristic
357C2×C17#283112yes—
358C2×C17#283212yes—
359D17—82no—
360C25—171no—
361C22×Q8—22no—
362C22×Q8—22no—
363C22×Q8—42no—
364C22×Q8—42no—
365C22×Q8C4×D1714yes—
366C23×C4—171no—
367C23×C4—172no—
368C23×C4—171no—
369C23×C4—171no—
370C23×C4—172no—
371C23×C4C22×D1712yes—
372C23×C4C22×D1712yes—
373C23×C4C22×D1711yescharacteristic
374C23×C4C22×D1711yescharacteristic
375C23×C4C22×D1711yescharacteristic
376C2×C4⋊C4—22no—
377C2×C4⋊C4—42no—
378C2×C4⋊C4—342no—
379C2×C4⋊C4—682no—
380C2×C4⋊C4—682no—
381C2×C4⋊C4—682no—
382C2×C4⋊C4—682no—
383C2×C22⋊C4—174no—
384C2×C22⋊C4—178no—
385C2×C22⋊C4—342no—
386C2×C22⋊C4—342no—
387C2×C22⋊C4—342no—
388C2×C22⋊C4—342no—
389C2×C22⋊C4—682no—
390C2×C22⋊C4—682no—
391C2×C22⋊C4—682no—
392C2×C22⋊C4—682no—
393C2×C22⋊C4—682no—
394C2×C22⋊C4—682no—
395C2×C22⋊C4—682no—
396C2×C42—42no—
397C2×C42C22⋊D1714yes—
398#95—24no—
399#95—24no—
400#95—178no—
401#95—174no—
402#95—342no—
403#95—342no—
404#95—342no—
405#95—682no—
406#95—682no—
407#95—682no—
408#95—682no—
409#95—682no—
410#95—682no—
411C17#1462511yescharacteristic, complemented, Sylow
412C24—172no—
413C24—341no—
414C24—342no—
415C24—342no—
416C24—341no—
417C24—341no—
418C24—342no—
419C24—341no—
420C24—342no—
421C24—341no—
422C24—341no—
423C24C23×D1711yescharacteristic
424C2×Q8—28no—
425C2×Q8—48no—
426C2×Q8—44no—
427C2×Q8—44no—
428C2×Q8—48no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—172no—
453C22×C4—174no—
454C22×C4—341no—
455C22×C4—342no—
456C22×C4—342no—
457C22×C4—342no—
458C22×C4—342no—
459C22×C4—342no—
460C22×C4—342no—
461C22×C4—342no—
462C22×C4—342no—
463C22×C4—341no—
464C22×C4—342no—
465C22×C4—342no—
466C22×C4—342no—
467C22×C4—342no—
468C22×C4—342no—
469C22×C4—341no—
470C22×C4—341no—
471C22×C4—342no—
472C22×C4—342no—
473C22×C4—342no—
474C22×C4—342no—
475C22×C4—342no—
476C22×C4—342no—
477C22×C4—341no—
478C22×C4—341no—
479C22×C4—341no—
480C22×C4—341no—
481C22×C4C2×C4×D1714yes—
482C22×C4C2×C4×D1712yes—
483C22×C4#6325212yes—
484C22×C4C2×C22⋊D1712yes—
485C22×C4C2×C22⋊D1712yes—
486C22×C4C2×C22⋊D1712yes—
487C4⋊C4—44no—
488C4⋊C4—684no—
489C4⋊C4—688no—
490C22⋊C4—348no—
491C22⋊C4—688no—
492C22⋊C4—684no—
493C22⋊C4—684no—
494C22⋊C4—688no—
495C42—44no—
496C23—342no—
497C23—344no—
498C23—342no—
499C23—344no—
500C23—342no—
501C23—682no—
502C23—682no—
503C23—682no—
504C23—682no—
505C23—682no—
506C23—682no—
507C23—682no—
508C23—682no—
509C23—681no—
510C23—681no—
511C23—681no—
512C23—681no—
513C23—681no—
514C23—681no—
515C23—682no—
516C23—681no—
517C23—681no—
518C23—682no—
519C23C22×C4×D1711yescharacteristic
520C23C2×#6325211yescharacteristic
521C23#1058769112yes—
522C23#1058769112yes—
523C23#1058769411yescharacteristic
524C23#1058769411yescharacteristic
525C23#1058769412yes—
526C23#1058769611yescharacteristic
527C23#1058769612yes—
528C23#1058769711yescharacteristic
529C23C22×C22⋊D1711yescharacteristic
530Q8—416no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—342no—
560C2×C4—344no—
561C2×C4—344no—
562C2×C4—344no—
563C2×C4—344no—
564C2×C4—682no—
565C2×C4—682no—
566C2×C4—682no—
567C2×C4—682no—
568C2×C4—681no—
569C2×C4—681no—
570C2×C4—682no—
571C2×C4—682no—
572C2×C4—682no—
573C2×C4—682no—
574C2×C4—682no—
575C2×C4—682no—
576C2×C4—682no—
577C2×C4—682no—
578C2×C4—682no—
579C2×C4—681no—
580C2×C4—681no—
581C2×C4—681no—
582C2×C4—681no—
583C2×C4—681no—
584C2×C4—681no—
585C2×C4—682no—
586C2×C4—682no—
587C2×C4—682no—
588C2×C4C4×C22⋊D1714yes—
589C22—684no—
590C22—682no—
591C22—684no—
592C22—682no—
593C22—682no—
594C22—1361no—
595C22—1361no—
596C22—1361no—
597C22—1361no—
598C22—1362no—
599C22—1362no—
600C22C2×C4×C22⋊D1711yescharacteristic
601C22#1176328212yes—
602C22#1176328511yescharacteristic
603C22#1176329112yes—
604C22#1176329111yescharacteristic
605C22#1176336811yescharacteristic
606C22#1176337311yescharacteristic
607C22#1176337611yescharacteristic
608C22#1176338311yescharacteristic
609C22#1176338311yescharacteristic
610C22#1176338711yescharacteristic
611C22#1176339712yes—
612C22#1176341711yescharacteristic
613C22#1176343111yescharacteristic
614C22#1176343112yes—
615C22#1176355112yes—
616C22#1176355312yes—
617C22#1176355512yes—
618C22#1176355512yes—
619C22#1176355512yes—
620C22#1176355812yes—
621C22#1176355812yes—
622C22#1176356012yes—
623C4—24no—
624C4—42no—
625C4—42no—
626C4—44no—
627C4—82no—
628C4—684no—
629C4—682no—
630C4—1361no—
631C4—1361no—
632C4—1362no—
633C2—1362no—
634C2#1453144211yescharacteristic
635C2#1453146011yescharacteristic
636C2#1453149111yescharacteristic
637C2#1453194512yes—
638C2#1453195612yes—
639C2#1453679511yescharacteristic
640C2#1453681111yescharacteristic
641C2#1453682911yescharacteristic
642C2#1453684311yescharacteristic
643C2#1453693612yes—
644C2#1453695112yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC2×C17×#391C2
socle, cosocleC24×C17C24
prime radicalC24×D17C23
hypercenterC2×#391D17
hypocenterC17#14625
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
3. #14527055C2semidirect
4. #14527047C2semidirect
5. #14527042C2semidirect
6. #14527039C2semidirect
10. #14527011C2semidirect
11. C2×C17×#391C2semidirect
19. #11762720C4semidirect
268. C2×#391D17semidirect
411. C17#14625semidirect

Maximal subgroups

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

subgroupindex
C2×#117627202
C2×#117627002
#145270552
#145270472
#145270422
#145270392
#145270322
#145270282
#145270262
#145270112
C2×C17×#3912
#1462517