Finite Group Databasebeta

← #16965395

Subgroups — #16965395

Order 4352.

1494 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762712C211yesmaximal, characteristic
2#14527036C212yesmaximal, complemented
3#14527031C212yesmaximal, complemented
4#14527031C212yesmaximal, complemented
5#14527027C212yesmaximal, complemented
6#14527027C212yesmaximal, complemented
7#14527025C211yesmaximal, characteristic
8#14527009C212yesmaximal, complemented
9C17×#1438C211yesmaximal, characteristic
10C22×C4⋊Dic17C2211yescharacteristic
11C22×#63225C2212yes—
12C22×C22⋊Dic17C2211yescharacteristic
13C22×C22⋊Dic17C2212yes—
14C22×C22⋊Dic17C2211yescharacteristic
15C22×C22⋊Dic17C2212yes—
16C22×C22⋊Dic17C2212yes—
17C22×C4×Dic17C2211yescharacteristic
18#11762712C418yescomplemented
19C2×#10587502C2212yes—
20C2×#10587502C2212yes—
21C2×#10587502C2212yes—
22C2×#10587502C2212yes—
23C2×#10587502C2212yes—
24C2×#10587502C2212yes—
25C2×#10587502C2212yes—
26C2×#10587502C2212yes—
27C23×D4×C17C2211yescharacteristic
28C22×C22⋊C4×C17C2211yescharacteristic
29C22×C22⋊C4×C17C2212yes—
30C22×C22⋊C4×C17C2211yescharacteristic
31C2×C17×#95C2212yes—
32C25×C17C2311yescharacteristic
33C25×C17C2311yescharacteristic
34C22×D4×C17—24no—
35C22×D4×C17—42no—
36C22×D4×C17—41no—
37C22×D4×C17—41no—
38C22×D4×C17C2×C414yes—
39C23×C4×C17C2312yes—
40C23×C4×C17C2312yes—
41C23×C4×C17C2311yescharacteristic
42C23×Dic17C2312yes—
43C23×Dic17C2312yes—
44C23×Dic17C2312yes—
45C23×Dic17C2312yes—
46C2×C22⋊C4×C17—42no—
47C2×C22⋊C4×C17—41no—
48C2×C22⋊C4×C17—42no—
49C2×C22⋊C4×C17—42no—
50C2×C22⋊C4×C17—42no—
51C2×C22⋊C4×C17—41no—
52C2×C22⋊C4×C17—42no—
53C2×C22⋊Dic17—42no—
54C2×C22⋊Dic17—42no—
55C2×C22⋊Dic17—42no—
56C2×C22⋊Dic17—42no—
57C2×C22⋊Dic17—42no—
58C2×C22⋊Dic17—41no—
59C2×C22⋊Dic17—42no—
60C2×C22⋊Dic17—42no—
61C2×C22⋊Dic17—42no—
62C2×C22⋊Dic17—41no—
63C2×C22⋊Dic17—42no—
64C2×C22⋊Dic17C2×C418yes—
65C2×C22⋊Dic17C2×C418yes—
66C2×C4⋊Dic17—41no—
67C2×C4⋊Dic17—41no—
68C2×C4⋊Dic17C2×C414yes—
69C2×#63225—42no—
70C2×#63225—42no—
71C2×#63225—42no—
72C2×C4×Dic17—42no—
73C2×C4×Dic17C2×C414yes—
74C17×#95—24no—
75#10587502—24no—
76#10587502—42no—
77#10587502—42no—
78#10587502—42no—
79#10587502—42no—
80#10587502—42no—
81#10587502—42no—
82#10587502—42no—
83C24×C17—22no—
84C24×C17—22no—
85C24×C17—22no—
86C24×C17—21no—
87C24×C17—21no—
88C24×C17—22no—
89C24×C17—21no—
90C24×C17—21no—
91C24×C17—22no—
92C24×C17—22no—
93C24×C17—22no—
94C24×C17—22no—
95C24×C17—22no—
96C24×C17—22no—
97C24×C17—22no—
98C24×C17—22no—
99C24×C17C22×C412yes—
100C24×C17C22×C412yes—
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—21no—
109C22×C4×C17—21no—
110C22×C4×C17—22no—
111C22×C4×C17—22no—
112C22×C4×C17—22no—
113C22×C4×C17—22no—
114C22×C4×C17—22no—
115C22×C4×C17—22no—
116C22×C4×C17—22no—
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—22no—
124C22×C4×C17C22×C412yes—
125C22×C4×C17#4114yes—
126C22×Dic17—22no—
127C22×Dic17—22no—
128C22×Dic17—22no—
129C22×Dic17—22no—
130C22×Dic17—22no—
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×Dic17C22×C414yes—
153C22×Dic17C22×C414yes—
154C22⋊C4×C17—48no—
155C22⋊C4×C17—44no—
156C22⋊C4×C17—44no—
157C22⋊Dic17—48no—
158C22⋊Dic17—48no—
159C22⋊Dic17—48no—
160C22⋊Dic17—44no—
161C22⋊Dic17—44no—
162C4⋊Dic17—44no—
163#63225—48no—
164C4×Dic17—44no—
165#14667—171nomaximal, Sylow
166C23×C17—22no—
167C23×C17—22no—
168C23×C17—24no—
169C23×C17—24no—
170C23×C17—22no—
171C23×C17—22no—
172C23×C17—22no—
173C23×C17—22no—
174C23×C17—24no—
175C23×C17—24no—
176C23×C17—42no—
177C23×C17—42no—
178C23×C17—42no—
179C23×C17—42no—
180C23×C17—42no—
181C23×C17—42no—
182C23×C17—42no—
183C23×C17—42no—
184C23×C17—42no—
185C23×C17—42no—
186C23×C17—42no—
187C23×C17—42no—
188C23×C17—42no—
189C23×C17—41no—
190C23×C17—42no—
191C23×C17—41no—
192C23×C17—42no—
193C23×C17—42no—
194C23×C17—42no—
195C23×C17—42no—
196C23×C17—41no—
197C23×C17—41no—
198C23×C17—41no—
199C23×C17—41no—
200C23×C17—42no—
201C23×C17—42no—
202C23×C17—42no—
203C23×C17—42no—
204C23×C17—42no—
205C23×C17—42no—
206C23×C17—41no—
207C23×C17—41no—
208C23×C17C23×C411yescharacteristic
209C23×C17C2×#4112yes—
210C23×C17H(5,𝔽2)11yescharacteristic
211C23×C17H(5,𝔽2)12yes—
212C23×C17H(5,𝔽2)11yescharacteristic
213C23×C17H(5,𝔽2)12yes—
214C23×C17H(5,𝔽2)12yes—
215C23×C17H(5,𝔽2)12yes—
216C23×C172−1+412yes—
217D4×C17—416no—
218C2×C4×C17—22no—
219C2×C4×C17—22no—
220C2×C4×C17—24no—
221C2×C4×C17—24no—
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#11714yes—
237C2×Dic17—24no—
238C2×Dic17—24no—
239C2×Dic17—24no—
240C2×Dic17—24no—
241C2×Dic17—24no—
242C2×Dic17—24no—
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×#390—171no—
266#1441—172no—
267#1440—172no—
268#1440—172no—
269#1439—172no—
270#1439—172no—
271#1438—171no—
272#1438D1711yescharacteristic
273#1434—172no—
274C22×C17—42no—
275C22×C17—42no—
276C22×C17—44no—
277C22×C17—44no—
278C22×C17—44no—
279C22×C17—44no—
280C22×C17—42no—
281C22×C17—42no—
282C22×C17—42no—
283C22×C17—42no—
284C22×C17—82no—
285C22×C17—82no—
286C22×C17—82no—
287C22×C17—82no—
288C22×C17—82no—
289C22×C17—82no—
290C22×C17—82no—
291C22×C17—82no—
292C22×C17C2×#11711yescharacteristic
293C22×C17#51812yes—
294C22×C17#51811yescharacteristic
295C22×C17#51811yescharacteristic
296C22×C17#52012yes—
297C22×C17#53812yes—
298C22×C17#53812yes—
299C22×C17#54012yes—
300C22×C17#54012yes—
301C22×C17#54212yes—
302C22×C17#54212yes—
303C22×C17#56012yes—
304C22×C17#56012yes—
305C22×C17#56012yes—
306C22×C17#56011yescharacteristic
307C22×C17#56011yescharacteristic
308C22×C17H(3,𝔽4)11yescharacteristic
309C22×C17#56212yes—
310C22×C17#56212yes—
311C22×C17#56212yes—
312C22×C17#56311yescharacteristic
313C4×C17—24no—
314C4×C17—82no—
315C4×C17—82no—
316Dic17—44no—
317Dic17—44no—
318Dic17—82no—
319Dic17—82no—
320C23×D4C2×D1711yescharacteristic
321C22×C4⋊C4—171no—
322C22×C4⋊C4—172no—
323C22×C22⋊C4—172no—
324C22×C22⋊C4—172no—
325C22×C22⋊C4—171no—
326C22×C22⋊C4—172no—
327C22×C22⋊C4—171no—
328C22×C22⋊C4C2×D1711yescharacteristic
329C22×C22⋊C4C2×D1712yes—
330C22×C22⋊C4C2×D1711yescharacteristic
331C22×C42—171no—
332#390—178no—
333C2×#95—172no—
334C2×#95—172no—
335C2×#95—172no—
336C2×#95—172no—
337C2×#95—172no—
338C2×#95—172no—
339C2×#95—172no—
340C2×#95—172no—
341C2×#95C2×D1712yes—
342C2×C17—82no—
343C2×C17—82no—
344C2×C17#231211yescharacteristic
345C2×C17#231912yes—
346C2×C17#237712yes—
347C2×C17#237911yescharacteristic
348C2×C17#238011yescharacteristic
349C2×C17#282812yes—
350C2×C17#282912yes—
351C2×C17#282912yes—
352C2×C17#283012yes—
353C25C22×D1711yescharacteristic
354C25C22×D1711yescharacteristic
355C22×D4—24no—
356C22×D4—42no—
357C22×D4—41no—
358C22×D4—41no—
359C22×D4C4×D1714yes—
360C23×C4—172no—
361C23×C4—172no—
362C23×C4—172no—
363C23×C4—172no—
364C23×C4C22×D1712yes—
365C23×C4C22×D1712yes—
366C23×C4C22×D1711yescharacteristic
367C2×C4⋊C4—174no—
368C2×C4⋊C4—682no—
369C2×C4⋊C4—682no—
370C2×C4⋊C4—682no—
371C2×C4⋊C4—681no—
372C2×C4⋊C4—681no—
373C2×C22⋊C4—42no—
374C2×C22⋊C4—41no—
375C2×C22⋊C4—42no—
376C2×C22⋊C4—42no—
377C2×C22⋊C4—42no—
378C2×C22⋊C4—42no—
379C2×C22⋊C4—41no—
380C2×C22⋊C4—178no—
381C2×C22⋊C4—178no—
382C2×C22⋊C4—682no—
383C2×C22⋊C4—682no—
384C2×C22⋊C4—682no—
385C2×C22⋊C4—682no—
386C2×C22⋊C4—682no—
387C2×C22⋊C4—681no—
388C2×C22⋊C4—682no—
389C2×C22⋊C4—682no—
390C2×C22⋊C4—681no—
391C2×C22⋊C4—682no—
392C2×C22⋊C4—682no—
393C2×C42—174no—
394C2×C42—682no—
395#95—24no—
396#95—344no—
397#95—682no—
398#95—682no—
399#95—682no—
400#95—682no—
401#95—682no—
402#95—682no—
403#95—682no—
404C17#1466711yescharacteristic, complemented, Sylow
405C24—22no—
406C24—22no—
407C24—22no—
408C24—22no—
409C24—22no—
410C24—22no—
411C24—21no—
412C24—22no—
413C24—21no—
414C24—22no—
415C24—21no—
416C24—21no—
417C24—22no—
418C24—22no—
419C24—22no—
420C24—22no—
421C24C2×C4×D1712yes—
422C24C2×C4×D1712yes—
423C24C23×D1711yescharacteristic
424C2×D4—28no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—48no—
428C2×D4—48no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
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—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—174no—
447C22×C4—174no—
448C22×C4—342no—
449C22×C4—342no—
450C22×C4—342no—
451C22×C4—342no—
452C22×C4—342no—
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×C4C2×C4×D1712yes—
475C22×C4#6325414yes—
476C4⋊C4—684no—
477C4⋊C4—688no—
478C22⋊C4—48no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—688no—
482C22⋊C4—688no—
483C22⋊C4—688no—
484C22⋊C4—684no—
485C22⋊C4—684no—
486C42—684no—
487C23—22no—
488C23—22no—
489C23—24no—
490C23—24no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—24no—
496C23—24no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—41no—
503C23—42no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—41no—
516C23—41no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23C22×C4×D1711yescharacteristic
530C23C2×#6325412yes—
531C23#1058769112yes—
532C23#1058769111yescharacteristic
533C23#1058769112yes—
534C23#1058769111yescharacteristic
535C23#1058769112yes—
536C23#1058769612yes—
537C23#1058769712yes—
538D4—416no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—42no—
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—344no—
558C2×C4—344no—
559C2×C4—344no—
560C2×C4—344no—
561C2×C4—344no—
562C2×C4—344no—
563C2×C4—682no—
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#1058756314yes—
586C22—42no—
587C22—42no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22C2×#1058756311yescharacteristic
605C22#1176328112yes—
606C22#1176328211yescharacteristic
607C22#1176328211yescharacteristic
608C22#1176328812yes—
609C22#1176339412yes—
610C22#1176339412yes—
611C22#1176341212yes—
612C22#1176341312yes—
613C22#1176342612yes—
614C22#1176342812yes—
615C22#1176354712yes—
616C22#1176354712yes—
617C22#1176354812yes—
618C22#1176354811yescharacteristic
619C22#1176354911yescharacteristic
620C22#1176355211yescharacteristic
621C22#1176355412yes—
622C22#1176355412yes—
623C22#1176355612yes—
624C22#1176356211yescharacteristic
625C4—24no—
626C4—82no—
627C4—82no—
628C4—684no—
629C4—684no—
630C4—1362no—
631C4—1362no—
632C2—82no—
633C2—82no—
634C2#1453138011yescharacteristic
635C2#1453142112yes—
636C2#1453193612yes—
637C2#1453195411yescharacteristic
638C2#1453195911yescharacteristic
639C2#1453690612yes—
640C2#1453691612yes—
641C2#1453692012yes—
642C2#1453693012yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1438C2
socle, cosocleC24×C17C24
prime radicalC23×D4×C17C22
hypercenter#1438D17
hypocenterC17#14667
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1438

Complements

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

NH ≅ G/Ntype
2. #14527036C2semidirect
3. #14527031C2semidirect
4. #14527031C2semidirect
5. #14527027C2semidirect
6. #14527027C2semidirect
8. #14527009C2semidirect
18. #11762712C4semidirect
404. C17#14667semidirect

Maximal subgroups

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

subgroupindex
C2×#117627122
#145270362
#145270312
#145270272
#145270252
#145270092
C17×#14382
#1466717