Finite Group Databasebeta

← #16965385

Subgroups — #16965385

Order 4352.

1602 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14527046C212yesmaximal, complemented
2#14527041C212yesmaximal, complemented
3#14527033C212yesmaximal, complemented
4#14527029C212yesmaximal, complemented
5#14527029C212yesmaximal, complemented
6#14527025C211yesmaximal, characteristic
7#14527025C211yesmaximal, characteristic
8#14527007C212yesmaximal, complemented
9C2×C17×#386C211yesmaximal, characteristic
10C22×C4⋊Dic17C2212yes—
11C22×C22⋊Dic17C2212yes—
12C22×C22⋊Dic17C2212yes—
13C22×C22⋊Dic17C2211yescharacteristic
14C22×C22⋊Dic17C2212yes—
15C22×C22⋊Dic17C2211yescharacteristic
16C2×#10587502C2212yes—
17C2×#10587502C2212yes—
18C2×#10587502C2212yes—
19C2×#10587502C2212yes—
20C2×#10587502C2211yescharacteristic
21C2×#10587502C2212yes—
22C2×#10587502C2211yescharacteristic
23C2×#10587502C2212yes—
24C2×#10587502C2212yes—
25C2×#10587502C2212yes—
26C23×D4×C17C2211yescharacteristic
27C24×C4×C17C2211yescharacteristic
28C22×C22⋊C4×C17C2211yescharacteristic
29C22×C22⋊C4×C17C2212yes—
30C17×#386C418yescomplemented
31C2×C17×#95C2212yes—
32C25×C17C2311yescharacteristic
33C25×C17C2311yescharacteristic
34C22×D4×C17—24no—
35C22×D4×C17—41no—
36C22×D4×C17—41no—
37C22×D4×C17—42no—
38C22×D4×C17C2×C414yes—
39C23×C4×C17—41no—
40C23×C4×C17—41no—
41C23×C4×C17—42no—
42C23×C4×C17—42no—
43C23×C4×C17C2×C414yes—
44C23×C4×C17C2312yes—
45C23×C4×C17C2312yes—
46C23×C4×C17C2311yescharacteristic
47C23×Dic17C2312yes—
48C23×Dic17C2312yes—
49C23×Dic17C2312yes—
50C23×Dic17C2312yes—
51C2×C22⋊C4×C17—42no—
52C2×C22⋊C4×C17—42no—
53C2×C22⋊C4×C17—42no—
54C2×C22⋊C4×C17C2×C418yes—
55C2×C22⋊C4×C17C2×C414yes—
56C2×C22⋊Dic17—24no—
57C2×C22⋊Dic17—24no—
58C2×C22⋊Dic17—42no—
59C2×C22⋊Dic17—42no—
60C2×C22⋊Dic17—42no—
61C2×C22⋊Dic17—42no—
62C2×C22⋊Dic17—41no—
63C2×C22⋊Dic17—41no—
64C2×C22⋊Dic17—42no—
65C2×C22⋊Dic17—42no—
66C2×C22⋊Dic17—42no—
67C2×C22⋊Dic17—42no—
68C2×C22⋊Dic17—42no—
69C2×C4⋊Dic17—42no—
70C2×C4⋊Dic17—42no—
71C2×C4⋊Dic17—42no—
72C17×#95C2×C418yes—
73#10587502—41no—
74#10587502—41no—
75#10587502—42no—
76#10587502—42no—
77#10587502—42no—
78#10587502—42no—
79#10587502—42no—
80#10587502—42no—
81#10587502—42no—
82#10587502—42no—
83C24×C17—21no—
84C24×C17—21no—
85C24×C17—22no—
86C24×C17—22no—
87C24×C17—22no—
88C24×C17—22no—
89C24×C17—22no—
90C24×C17—22no—
91C24×C17—21no—
92C24×C17—22no—
93C24×C17—21no—
94C24×C17—22no—
95C24×C17—22no—
96C24×C17—22no—
97C24×C17—22no—
98C24×C17C22×C412yes—
99C24×C17C22×C412yes—
100C24×C17#4114yes—
101C24×C17C2411yescharacteristic
102C2×D4×C17—28no—
103C2×D4×C17—44no—
104C2×D4×C17—44no—
105C2×D4×C17—48no—
106C2×D4×C17—48no—
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—21no—
117C22×C4×C17—22no—
118C22×C4×C17—22no—
119C22×C4×C17—22no—
120C22×C4×C17—22no—
121C22×C4×C17—22no—
122C22×C4×C17—22no—
123C22×C4×C17—48no—
124C22×C4×C17—44no—
125C22×C4×C17—48no—
126C22×C4×C17—44no—
127C22×C4×C17—44no—
128C22×C4×C17C22×C414yes—
129C22×C4×C17C22×C412yes—
130C22×C4×C17C22×C414yes—
131C22×Dic17—22no—
132C22×Dic17—22no—
133C22×Dic17—22no—
134C22×Dic17—22no—
135C22×Dic17—22no—
136C22×Dic17—22no—
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—22no—
146C22×Dic17—22no—
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—22no—
156C22×Dic17—22no—
157C22×Dic17—22no—
158C22×Dic17—22no—
159C22⋊C4×C17—48no—
160C22⋊C4×C17—44no—
161C22⋊Dic17—48no—
162C22⋊Dic17—48no—
163C22⋊Dic17—48no—
164C22⋊Dic17—44no—
165C22⋊Dic17—44no—
166C4⋊Dic17—48no—
167#14666—171nomaximal, Sylow
168C23×C17—22no—
169C23×C17—22no—
170C23×C17—24no—
171C23×C17—24no—
172C23×C17—24no—
173C23×C17—24no—
174C23×C17—24no—
175C23×C17—24no—
176C23×C17—24no—
177C23×C17—24no—
178C23×C17—22no—
179C23×C17—24no—
180C23×C17—22no—
181C23×C17—22no—
182C23×C17—24no—
183C23×C17—42no—
184C23×C17—41no—
185C23×C17—42no—
186C23×C17—42no—
187C23×C17—41no—
188C23×C17—41no—
189C23×C17—42no—
190C23×C17—42no—
191C23×C17—42no—
192C23×C17—41no—
193C23×C17—42no—
194C23×C17—42no—
195C23×C17—42no—
196C23×C17—42no—
197C23×C17—42no—
198C23×C17—42no—
199C23×C17—42no—
200C23×C17—41no—
201C23×C17—42no—
202C23×C17—42no—
203C23×C17—41no—
204C23×C17—41no—
205C23×C17—42no—
206C23×C17—42no—
207C23×C17—42no—
208C23×C17—41no—
209C23×C17#11714yes—
210C23×C17C23×C411yescharacteristic
211C23×C17C2×#4112yes—
212C23×C17H(5,𝔽2)11yescharacteristic
213C23×C17H(5,𝔽2)12yes—
214C23×C17H(5,𝔽2)11yescharacteristic
215C23×C17H(5,𝔽2)12yes—
216C23×C17H(5,𝔽2)12yes—
217C23×C17H(5,𝔽2)12yes—
218C23×C172−1+412yes—
219D4×C17—416no—
220C2×C4×C17—22no—
221C2×C4×C17—24no—
222C2×C4×C17—24no—
223C2×C4×C17—24no—
224C2×C4×C17—24no—
225C2×C4×C17—24no—
226C2×C4×C17—22no—
227C2×C4×C17—22no—
228C2×C4×C17—24no—
229C2×C4×C17—416no—
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×C17—42no—
238C2×C4×C17—42no—
239C2×C4×C17—42no—
240C2×Dic17—42no—
241C2×Dic17—42no—
242C2×Dic17—42no—
243C2×Dic17—42no—
244C2×Dic17—42no—
245C2×Dic17—42no—
246C2×Dic17—42no—
247C2×Dic17—42no—
248C2×Dic17—42no—
249C2×Dic17—42no—
250C2×Dic17—42no—
251C2×Dic17—42no—
252C2×Dic17—42no—
253C2×Dic17—42no—
254C2×Dic17—42no—
255C2×Dic17—42no—
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×Dic17—42no—
268C2×#386D1711yescharacteristic
269#1443—172no—
270#1442—172no—
271#1440—172no—
272#1439—172no—
273#1439—172no—
274#1438—171no—
275#1438—171no—
276C22⋊#95—172no—
277C22×C17—24no—
278C22×C17—28no—
279C22×C17—44no—
280C22×C17—44no—
281C22×C17—44no—
282C22×C17—44no—
283C22×C17—42no—
284C22×C17—42no—
285C22×C17—44no—
286C22×C17—42no—
287C22×C17—44no—
288C22×C17—42no—
289C22×C17—42no—
290C22×C17—44no—
291C22×C17—82no—
292C22×C17—82no—
293C22×C17—82no—
294C22×C17—82no—
295C22×C17C2×#11711yescharacteristic
296C22×C17#51811yescharacteristic
297C22×C17#51811yescharacteristic
298C22×C17#51812yes—
299C22×C17#52012yes—
300C22×C17#53812yes—
301C22×C17#53812yes—
302C22×C17#53812yes—
303C22×C17#53812yes—
304C22×C17#53912yes—
305C22×C17#53912yes—
306C22×C17#56012yes—
307C22×C17#56012yes—
308C22×C17#56012yes—
309C22×C17#56011yescharacteristic
310C22×C17#56011yescharacteristic
311C22×C17H(3,𝔽4)11yescharacteristic
312C22×C17#56212yes—
313C22×C17#56212yes—
314C22×C17#56212yes—
315C22×C17#56311yescharacteristic
316C4×C17—42no—
317C4×C17—44no—
318C4×C17—44no—
319Dic17—82no—
320Dic17—82no—
321Dic17—82no—
322Dic17—82no—
323C23×D4C2×D1711yescharacteristic
324C24×C4C2×D1711yescharacteristic
325C22×C4⋊C4—172no—
326C22×C22⋊C4—172no—
327C22×C22⋊C4—172no—
328C22×C22⋊C4—171no—
329C22×C22⋊C4—172no—
330C22×C22⋊C4—171no—
331C22×C22⋊C4C2×D1711yescharacteristic
332C22×C22⋊C4C2×D1712yes—
333#386Dic1718yescomplemented
334C2×#95—172no—
335C2×#95—172no—
336C2×#95—172no—
337C2×#95—172no—
338C2×#95—171no—
339C2×#95—172no—
340C2×#95—171no—
341C2×#95—172no—
342C2×#95—172no—
343C2×#95—172no—
344C2×#95C2×D1712yes—
345C2×C17—44no—
346C2×C17—82no—
347C2×C17#231211yescharacteristic
348C2×C17#231812yes—
349C2×C17#237511yescharacteristic
350C2×C17#237611yescharacteristic
351C2×C17#237712yes—
352C2×C17#282412yes—
353C2×C17#282512yes—
354C2×C17#282612yes—
355C2×C17#282712yes—
356C25C22×D1711yescharacteristic
357C25C22×D1711yescharacteristic
358C22×D4—24no—
359C22×D4—42no—
360C22×D4—41no—
361C22×D4—41no—
362C22×D4C2×Dic1714yes—
363C23×C4—41no—
364C23×C4—41no—
365C23×C4—42no—
366C23×C4—42no—
367C23×C4—172no—
368C23×C4—172no—
369C23×C4—172no—
370C23×C4—172no—
371C23×C4C2×Dic1714yes—
372C23×C4C22×D1712yes—
373C23×C4C22×D1712yes—
374C23×C4C22×D1711yescharacteristic
375C2×C4⋊C4—682no—
376C2×C4⋊C4—682no—
377C2×C4⋊C4—682no—
378C2×C22⋊C4—42no—
379C2×C22⋊C4—42no—
380C2×C22⋊C4—42no—
381C2×C22⋊C4—344no—
382C2×C22⋊C4—344no—
383C2×C22⋊C4—682no—
384C2×C22⋊C4—682no—
385C2×C22⋊C4—682no—
386C2×C22⋊C4—682no—
387C2×C22⋊C4—681no—
388C2×C22⋊C4—681no—
389C2×C22⋊C4—682no—
390C2×C22⋊C4—682no—
391C2×C22⋊C4—682no—
392C2×C22⋊C4—682no—
393C2×C22⋊C4—682no—
394C2×C22⋊C4C2×Dic1714yes—
395C2×C22⋊C4C2×Dic1718yes—
396#95—682no—
397#95—682no—
398#95—682no—
399#95—682no—
400#95—682no—
401#95—682no—
402#95—681no—
403#95—681no—
404#95—682no—
405#95—682no—
406#95C2×Dic1718yes—
407C17#1466611yescharacteristic, complemented, Sylow
408C24—22no—
409C24—22no—
410C24—22no—
411C24—21no—
412C24—21no—
413C24—22no—
414C24—22no—
415C24—22no—
416C24—22no—
417C24—22no—
418C24—22no—
419C24—21no—
420C24—21no—
421C24—22no—
422C24—22no—
423C24#6325214yes—
424C24C22×Dic1712yes—
425C24C22×Dic1712yes—
426C24C23×D1711yescharacteristic
427C2×D4—28no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—48no—
431C2×D4—48no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—48no—
449C22×C4—44no—
450C22×C4—48no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—342no—
454C22×C4—342no—
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—342no—
464C22×C4—342no—
465C22×C4—342no—
466C22×C4—342no—
467C22×C4—342no—
468C22×C4—342no—
469C22×C4—342no—
470C22×C4—342no—
471C22×C4—342no—
472C22×C4—342no—
473C22×C4—342no—
474C22×C4—342no—
475C22×C4—342no—
476C22×C4—342no—
477C22×C4—342no—
478C22×C4—342no—
479C22×C4—342no—
480C22×C4—342no—
481C22×C4C22×Dic1712yes—
482C22×C4C22×Dic1714yes—
483C22×C4C22×Dic1714yes—
484C4⋊C4—688no—
485C22⋊C4—48no—
486C22⋊C4—44no—
487C22⋊C4—688no—
488C22⋊C4—688no—
489C22⋊C4—688no—
490C22⋊C4—684no—
491C22⋊C4—684no—
492C23—22no—
493C23—22no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—22no—
503C23—24no—
504C23—22no—
505C23—22no—
506C23—24no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—41no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—41no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23#1058761214yes—
534C23C2×#6325212yes—
535C23#1058769112yes—
536C23#1058769112yes—
537C23#1058769112yes—
538C23#1058769111yescharacteristic
539C23#1058769111yescharacteristic
540C23#1058769612yes—
541C23#1058769712yes—
542C23C23×Dic1711yescharacteristic
543D4—416no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—416no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—682no—
565C2×C4—682no—
566C2×C4—682no—
567C2×C4—682no—
568C2×C4—682no—
569C2×C4—682no—
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—682no—
580C2×C4—682no—
581C2×C4—682no—
582C2×C4—682no—
583C2×C4—682no—
584C2×C4—682no—
585C2×C4—682no—
586C2×C4—682no—
587C2×C4—682no—
588C2×C4—682no—
589C2×C4—682no—
590C2×C4—682no—
591C2×C4—682no—
592C22—24no—
593C22—28no—
594C22—42no—
595C22—42no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—42no—
602C22—42no—
603C22—44no—
604C22—42no—
605C22—44no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22C2×#1058761211yescharacteristic
611C22#1176328411yescharacteristic
612C22#1176328411yescharacteristic
613C22#1176328412yes—
614C22#1176329212yes—
615C22#1176339212yes—
616C22#1176339212yes—
617C22#1176339712yes—
618C22#1176339712yes—
619C22#1176340112yes—
620C22#1176340112yes—
621C22#1176354712yes—
622C22#1176354712yes—
623C22#1176354812yes—
624C22#1176354811yescharacteristic
625C22#1176354911yescharacteristic
626C22#1176355211yescharacteristic
627C22#1176355412yes—
628C22#1176355412yes—
629C22#1176355612yes—
630C22#1176356211yescharacteristic
631C4—42no—
632C4—44no—
633C4—44no—
634C4—1362no—
635C4—1362no—
636C4—1362no—
637C4—1362no—
638C2—44no—
639C2—82no—
640C2#1453138211yescharacteristic
641C2#1453141712yes—
642C2#1453192511yescharacteristic
643C2#1453193211yescharacteristic
644C2#1453194112yes—
645C2#1453686112yes—
646C2#1453687012yes—
647C2#1453687912yes—
648C2#1453688812yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC2×C17×#386C2
socle, cosocleC24×C17C24
prime radicalC23×D4×C17C22
hypercenterC2×#386D17
hypocenterC17#14666
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×#386

Complements

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

NH ≅ G/Ntype
1. #14527046C2semidirect
2. #14527041C2semidirect
3. #14527033C2semidirect
4. #14527029C2semidirect
5. #14527029C2semidirect
8. #14527007C2semidirect
30. C17×#386C4semidirect
333. #386Dic17semidirect
407. C17#14666semidirect

Maximal subgroups

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

subgroupindex
#145270462
#145270412
#145270332
#145270292
#145270252
#145270072
C2×C17×#3862
#1466617