Finite Group Databasebeta

← #16962781

Subgroups — #16962781

Order 4352.

1658 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×#11762702C212yesmaximal, complemented
2#14527033C212yesmaximal, complemented
3#14527029C212yesmaximal, complemented
4#14527025C212yesmaximal
5#14527021C212yesmaximal, complemented
6#14527016C212yesmaximal, complemented
7#14527014C211yesmaximal, characteristic, complemented
8#14527011C211yesmaximal, characteristic, complemented
9C2×C17×#390C211yesmaximal, characteristic
10C22×C4⋊Dic17C2212yes—
11C22×C22⋊Dic17C2212yes—
12C22×C22⋊Dic17C2212yes—
13C22×C22⋊Dic17C2212yes—
14C22×C22⋊Dic17C2212yes—
15C22×C4×Dic17C2212yes—
16#11762702—28no—
17C2×#10587502C2212yes—
18C2×#10587502C2211yescharacteristic
19C2×#10587502C2212yes—
20C2×#10587502C2212yes—
21C2×#10587502C2212yes—
22C2×#10587502C2211yescharacteristic
23C2×#10587502C2211yescharacteristic
24C2×#10587502C2212yes—
25C2×#10587502C2212yes—
26C2×#10587502C2211yescharacteristic
27C23×D4×C17C2211yescharacteristic
28C22×C4⋊C4×C17C2211yescharacteristic
29C22×C22⋊C4×C17C2212yes—
30C22×C22⋊C4×C17C2212yes—
31C22×C42×C17C2211yescharacteristic
32C17×#390C418yescomplemented
33C25×C17C2312yes—
34C22×D4×C17—22no—
35C22×D4×C17—22no—
36C22×D4×C17—42no—
37C22×D4×C17—42no—
38C22×D4×C17C2×C414yes—
39C23×C4×C17C2311yescharacteristic
40C23×C4×C17C2312yes—
41C23×C4×C17C2311yescharacteristic
42C23×C4×C17C2311yescharacteristic
43C23×Dic17C2312yes—
44C23×Dic17C2312yes—
45C23×Dic17C2312yes—
46C23×Dic17C2312yes—
47C2×C4⋊C4×C17—42no—
48C2×C4⋊C4×C17C2×C414yes—
49C2×C22⋊C4×C17—24no—
50C2×C22⋊C4×C17—24no—
51C2×C22⋊C4×C17—42no—
52C2×C22⋊C4×C17—42no—
53C2×C22⋊C4×C17C2×C418yes—
54C2×C22⋊C4×C17C2×C418yes—
55C2×C42×C17—42no—
56C2×C42×C17C2×C414yes—
57C2×C22⋊Dic17—24no—
58C2×C22⋊Dic17—24no—
59C2×C22⋊Dic17—24no—
60C2×C22⋊Dic17—24no—
61C2×C22⋊Dic17—42no—
62C2×C22⋊Dic17—42no—
63C2×C22⋊Dic17—42no—
64C2×C22⋊Dic17—42no—
65C2×C22⋊Dic17—42no—
66C2×C22⋊Dic17—42no—
67C2×C22⋊Dic17—42no—
68C2×C22⋊Dic17—42no—
69C2×C4⋊Dic17—24no—
70C2×C4⋊Dic17—24no—
71C2×C4⋊Dic17—42no—
72C2×C4×Dic17—24no—
73C2×C4×Dic17—42no—
74C2×C4×Dic17—42no—
75#10587502—24no—
76#10587502—24no—
77#10587502—22no—
78#10587502—24no—
79#10587502—42no—
80#10587502—41no—
81#10587502—42no—
82#10587502—41no—
83#10587502—42no—
84#10587502D414yescomplemented
85C24×C17—22no—
86C24×C17—22no—
87C24×C17—22no—
88C24×C17—22no—
89C24×C17—22no—
90C24×C17—22no—
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×C414yes—
100C24×C17C2411yescharacteristic
101C2×D4×C17—28no—
102C2×D4×C17—44no—
103C2×D4×C17—44no—
104C2×D4×C17—48no—
105C2×D4×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—21no—
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—21no—
120C22×C4×C17C22×C412yes—
121C22×C4×C17C22×C414yes—
122C22×C4×C17C22×C412yes—
123C22×C4×C17C22×C412yes—
124C22×C4×C17C2×D412yes—
125C22×C4×C17#4112yes—
126C22×C4×C17#4112yes—
127C22×C4×C17#4112yes—
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×Dic17—22no—
153C22×Dic17—22no—
154C22×Dic17C2×D414yes—
155C22×Dic17#4114yes—
156C4⋊C4×C17—44no—
157C22⋊C4×C17—216no—
158C22⋊C4×C17—48no—
159C42×C17—44no—
160C22⋊Dic17—48no—
161C22⋊Dic17—48no—
162C22⋊Dic17—48no—
163C22⋊Dic17—48no—
164C4⋊Dic17—48no—
165C4×Dic17—48no—
166#14444—171nomaximal, Sylow
167C23×C17—24no—
168C23×C17—24no—
169C23×C17—24no—
170C23×C17—24no—
171C23×C17—24no—
172C23×C17—24no—
173C23×C17—24no—
174C23×C17—42no—
175C23×C17—42no—
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—42no—
190C23×C17—42no—
191C23×C17—42no—
192C23×C17—42no—
193C23×C17—42no—
194C23×C17—42no—
195C23×C17—42no—
196C23×C17—42no—
197C23×C17—42no—
198C23×C17—42no—
199C23×C17—42no—
200C23×C17—42no—
201C23×C17—42no—
202C23×C17C23×C411yescharacteristic
203C23×C17C22×D411yescharacteristic
204C23×C17C2×#4111yescharacteristic
205C23×C17C2×#4111yescharacteristic
206C23×C17C2×#4111yescharacteristic
207C23×C17C2×#4112yes—
208C23×C17H(5,𝔽2)12yes—
209C23×C17H(5,𝔽2)12yes—
210C23×C17H(5,𝔽2)12yes—
211C23×C17H(5,𝔽2)11yescharacteristic
212C23×C172−1+411yescharacteristic
213D4×C17—416no—
214C2×C4×C17—22no—
215C2×C4×C17—22no—
216C2×C4×C17—22no—
217C2×C4×C17—22no—
218C2×C4×C17—24no—
219C2×C4×C17—24no—
220C2×C4×C17—24no—
221C2×C4×C17—24no—
222C2×C4×C17—24no—
223C2×C4×C17—22no—
224C2×C4×C17—22no—
225C2×C4×C17—24no—
226C2×C4×C17—24no—
227C2×C4×C17—22no—
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#11714yes—
235C2×C4×C17C4×D414yes—
236C2×Dic17—24no—
237C2×Dic17—24no—
238C2×Dic17—24no—
239C2×Dic17—24no—
240C2×Dic17—24no—
241C2×Dic17—24no—
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×#390D1711yescharacteristic
265C2×#388—172no—
266#1440—172no—
267#1439—172no—
268#1438—172no—
269#1437—172no—
270#1436—172no—
271C4⋊#95—171no—
272#1434—171no—
273C22×C17—44no—
274C22×C17—44no—
275C22×C17—44no—
276C22×C17—44no—
277C22×C17—44no—
278C22×C17—44no—
279C22×C17—44no—
280C22×C17—82no—
281C22×C17—82no—
282C22×C17—82no—
283C22×C17—82no—
284C22×C17—82no—
285C22×C17—82no—
286C22×C17—82no—
287C22×C17—82no—
288C22×C17C2×#11711yescharacteristic
289C22×C17C2×C4×D411yescharacteristic
290C22×C17C4×#4111yescharacteristic
291C22×C17#51811yescharacteristic
292C22×C17#51811yescharacteristic
293C22×C17#51811yescharacteristic
294C22×C17#52011yescharacteristic
295C22×C17#53512yes—
296C22×C17#53812yes—
297C22×C17#53812yes—
298C22×C17#53812yes—
299C22×C17#54012yes—
300C22×C17#54212yes—
301C22×C17#54612yes—
302C22×C17#54711yescharacteristic
303C22×C17#55011yescharacteristic
304C22×C17#55312yes—
305C22×C17#55312yes—
306C22×C17#55511yescharacteristic
307C22×C17#55611yescharacteristic
308C22×C17#55912yes—
309C22×C17#56012yes—
310C22×C17#56212yes—
311C4×C17—24no—
312C4×C17—24no—
313C4×C17—44no—
314C4×C17—42no—
315Dic17—44no—
316Dic17—44no—
317Dic17—82no—
318Dic17—82no—
319C23×D4C2×D1711yescharacteristic
320C22×C4⋊C4—172no—
321C22×C4⋊C4C2×D1711yescharacteristic
322C22×C22⋊C4—172no—
323C22×C22⋊C4—172no—
324C22×C22⋊C4—172no—
325C22×C22⋊C4—172no—
326C22×C22⋊C4C2×D1712yes—
327C22×C22⋊C4C2×D1712yes—
328C22×C42—172no—
329C22×C42C2×D1711yescharacteristic
330#390Dic1718yescomplemented
331#388—348no—
332C2×#95—171no—
333C2×#95—172no—
334C2×#95—171no—
335C2×#95—172no—
336C2×#95—172no—
337C2×#95—172no—
338C2×#95—171no—
339C2×#95—171no—
340C2×#95—172no—
341C2×#95—172no—
342C2×C17—84no—
343C2×C17#231211yescharacteristic
344C2×C17#232811yescharacteristic
345C2×C17#233711yescharacteristic
346C2×C17#235411yescharacteristic
347C2×C17#235711yescharacteristic
348C2×C17#236411yescharacteristic
349C2×C17#237711yescharacteristic
350C2×C17#271012yes—
351C2×C17#271412yes—
352C2×C17#280012yes—
353C2×C17#280312yes—
354C25C22×D1712yes—
355C22×D4—22no—
356C22×D4—22no—
357C22×D4—42no—
358C22×D4—42no—
359C22×D4C2×Dic1714yes—
360C23×C4—172no—
361C23×C4—172no—
362C23×C4—172no—
363C23×C4—172no—
364C23×C4C22×D1712yes—
365C23×C4C22×D1711yescharacteristic
366C23×C4C22×D1711yescharacteristic
367C23×C4C22×D1711yescharacteristic
368C2×C4⋊C4—42no—
369C2×C4⋊C4—344no—
370C2×C4⋊C4—344no—
371C2×C4⋊C4—682no—
372C2×C4⋊C4C2×Dic1714yes—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—42no—
376C2×C22⋊C4—42no—
377C2×C22⋊C4—344no—
378C2×C22⋊C4—344no—
379C2×C22⋊C4—344no—
380C2×C22⋊C4—344no—
381C2×C22⋊C4—682no—
382C2×C22⋊C4—682no—
383C2×C22⋊C4—682no—
384C2×C22⋊C4—682no—
385C2×C22⋊C4—682no—
386C2×C22⋊C4—682no—
387C2×C22⋊C4—682no—
388C2×C22⋊C4—682no—
389C2×C22⋊C4C2×Dic1718yes—
390C2×C22⋊C4C2×Dic1718yes—
391C2×C42—42no—
392C2×C42—344no—
393C2×C42—682no—
394C2×C42—682no—
395C2×C42C2×Dic1714yes—
396#95—174no—
397#95—344no—
398#95—344no—
399#95—344no—
400#95—342no—
401#95—682no—
402#95—682no—
403#95—682no—
404#95—681no—
405#95—681no—
406C17#1444411yescharacteristic, complemented, Sylow
407C24—22no—
408C24—22no—
409C24—22no—
410C24—22no—
411C24—22no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24—22no—
416C24—22no—
417C24—22no—
418C24—22no—
419C24—22no—
420C24—22no—
421C24C22×Dic1714yes—
422C24C23×D1711yescharacteristic
423C2×D4—28no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—48no—
427C2×D4—48no—
428C22×C4—22no—
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—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—174no—
443C22×C4—174no—
444C22×C4—342no—
445C22×C4—342no—
446C22×C4—342no—
447C22×C4—342no—
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#6325212yes—
471C22×C4#6325212yes—
472C22×C4D4×D1712yes—
473C22×C4#6325412yes—
474C22×C4C22×Dic1712yes—
475C22×C4C22×Dic1714yes—
476C22×C4C22×Dic1712yes—
477C22×C4C22×Dic1712yes—
478C4⋊C4—44no—
479C4⋊C4—688no—
480C22⋊C4—216no—
481C22⋊C4—48no—
482C22⋊C4—688no—
483C22⋊C4—688no—
484C22⋊C4—688no—
485C22⋊C4—688no—
486C42—44no—
487C42—688no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23C2×#6325211yescharacteristic
524C23C2×#6325211yescharacteristic
525C23C2×D4×D1711yescharacteristic
526C23C2×#6325411yescharacteristic
527C23#1058769112yes—
528C23#1058769112yes—
529C23#1058769112yes—
530C23#41×D1712yes—
531C23#1058769611yescharacteristic
532C23#1058769711yescharacteristic
533C23C23×Dic1711yescharacteristic
534D4—416no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—344no—
556C2×C4—344no—
557C2×C4—344no—
558C2×C4—344no—
559C2×C4—344no—
560C2×C4—344no—
561C2×C4—682no—
562C2×C4—682no—
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#1058761214yes—
584C2×C4D4×Dic1714yes—
585C22—44no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22C2×#1058761211yescharacteristic
601C22C2×D4×Dic1711yescharacteristic
602C22#41×Dic1711yescharacteristic
603C22#1176328411yescharacteristic
604C22#1176328411yescharacteristic
605C22#1176328411yescharacteristic
606C22#1176329211yescharacteristic
607C22#1176337212yes—
608C22#1176339312yes—
609C22#1176339712yes—
610C22#1176339712yes—
611C22#1176341312yes—
612C22#1176342312yes—
613C22#1176344812yes—
614C22#1176346411yescharacteristic
615C22#1176348011yescharacteristic
616C22#1176350212yes—
617C22#1176350912yes—
618C22#1176352411yescharacteristic
619C22#1176352911yescharacteristic
620C22#1176354212yes—
621C22#1176354712yes—
622C22#1176355412yes—
623C4—24no—
624C4—24no—
625C4—42no—
626C4—44no—
627C4—684no—
628C4—684no—
629C4—1362no—
630C4—1362no—
631C2—84no—
632C2#1453138211yescharacteristic
633C2#1453151011yescharacteristic
634C2#1453159911yescharacteristic
635C2#1453173311yescharacteristic
636C2#1453175711yescharacteristic
637C2#1453183611yescharacteristic
638C2#1453194111yescharacteristic
639C2#1453552312yes—
640C2#1453557712yes—
641C2#1453668112yes—
642C2#1453671812yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC2×C17×#390C2
socle, cosocleC24×C17C24
prime radicalC23×D4×C17C22
hypercenterC2×#390D17
hypocenterC17#14444
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
1. C2×#11762702C2semidirect
2. #14527033C2semidirect
3. #14527029C2semidirect
5. #14527021C2semidirect
6. #14527016C2semidirect
7. #14527014C2semidirect
8. #14527011C2semidirect
32. C17×#390C4semidirect
84. #10587502D4semidirect
330. #390Dic17semidirect
406. C17#14444semidirect

Maximal subgroups

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

subgroupindex
C2×#117627022
#145270332
#145270292
#145270252
#145270212
#145270162
#145270142
#145270112
C2×C17×#3902
#1444417