Finite Group Databasebeta

← #16964606

Subgroups — #16964606

Order 4352.

1650 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762702C212yesmaximal, complemented
2#14527037C211yesmaximal, characteristic, complemented
3#14527029C212yesmaximal, complemented
4#14527025C212yesmaximal
5#14527021C212yesmaximal, complemented
6#14527021C212yesmaximal, complemented
7#14527013C211yesmaximal, characteristic, complemented
8#14527009C212yesmaximal, complemented
9C2×C17×#394C211yesmaximal, characteristic
10C22×C4⋊Dic17C2211yescharacteristic
11C22×C4⋊Dic17C2211yescharacteristic
12C22×C22⋊Dic17C2212yes—
13C22×C22⋊Dic17C2212yes—
14C22×C22⋊Dic17C2212yes—
15C22×C22⋊Dic17C2212yes—
16C22×C4×Dic17C2212yes—
17C22×C4×Dic17C2211yescharacteristic
18#11762702—28no—
19C2×#10587502C2212yes—
20C2×#10587502C2212yes—
21C2×#10587502C2212yes—
22C2×#10587502C2212yes—
23C2×#10587502C2212yes—
24C2×#10587502C2212yes—
25C2×#10587502C2212yes—
26C2×#10587502C2211yescharacteristic
27C23×D4×C17C2211yescharacteristic
28C22×C4⋊C4×C17C2211yescharacteristic
29C22×C22⋊C4×C17C2212yes—
30C22×C22⋊C4×C17C2212yes—
31C17×#394C418yescomplemented
32C2×C17×#95C2211yescharacteristic
33C25×C17C2312yes—
34C22×D4×C17—42no—
35C22×D4×C17—42no—
36C22×D4×C17—41no—
37C22×D4×C17—41no—
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—41no—
48C2×C4⋊C4×C17—41no—
49C2×C4⋊C4×C17C2×C414yes—
50C2×C22⋊C4×C17—24no—
51C2×C22⋊C4×C17—24no—
52C2×C22⋊C4×C17—42no—
53C2×C22⋊C4×C17—42no—
54C2×C22⋊C4×C17C2×C418yes—
55C2×C22⋊C4×C17C2×C418yes—
56C2×C22⋊Dic17—24no—
57C2×C22⋊Dic17—24no—
58C2×C22⋊Dic17—24no—
59C2×C22⋊Dic17—24no—
60C2×C22⋊Dic17—24no—
61C2×C22⋊Dic17—24no—
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×C4⋊Dic17—41no—
69C2×C4⋊Dic17—41no—
70C2×C4⋊Dic17—42no—
71C2×C4⋊Dic17—41no—
72C2×C4⋊Dic17D414yescomplemented
73C2×C4×Dic17—22no—
74C2×C4×Dic17—24no—
75C2×C4×Dic17—24no—
76C2×C4×Dic17—42no—
77C2×C4×Dic17—41no—
78C2×C4×Dic17—41no—
79C17×#95C2×C414yes—
80#10587502—24no—
81#10587502—24no—
82#10587502—24no—
83#10587502—42no—
84#10587502—42no—
85#10587502—42no—
86#10587502—42no—
87#10587502—41no—
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×C17—22no—
100C24×C17—22no—
101C24×C17—22no—
102C24×C17C22×C414yes—
103C24×C17C2411yescharacteristic
104C2×D4×C17—44no—
105C2×D4×C17—48no—
106C2×D4×C17—44no—
107C2×D4×C17—48no—
108C2×D4×C17—44no—
109C22×C4×C17—21no—
110C22×C4×C17—21no—
111C22×C4×C17—21no—
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×C17—21no—
121C22×C4×C17—22no—
122C22×C4×C17—22no—
123C22×C4×C17—21no—
124C22×C4×C17—22no—
125C22×C4×C17—22no—
126C22×C4×C17—21no—
127C22×C4×C17C22×C412yes—
128C22×C4×C17C22×C414yes—
129C22×C4×C17C22×C412yes—
130C22×C4×C17C22×C412yes—
131C22×C4×C17C2×D412yes—
132C22×C4×C17#4112yes—
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×Dic17C2×D414yes—
159C22×Dic17#4114yes—
160C22×Dic17#4114yes—
161C4⋊C4×C17—44no—
162C22⋊C4×C17—216no—
163C22⋊C4×C17—48no—
164C22⋊Dic17—48no—
165C22⋊Dic17—48no—
166C22⋊Dic17—48no—
167C22⋊Dic17—48no—
168C4⋊Dic17—44no—
169C4⋊Dic17—44no—
170C4×Dic17—48no—
171C4×Dic17—44no—
172#14589—171nomaximal, Sylow
173C23×C17—24no—
174C23×C17—24no—
175C23×C17—24no—
176C23×C17—24no—
177C23×C17—24no—
178C23×C17—24no—
179C23×C17—24no—
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×C17—42no—
203C23×C17—42no—
204C23×C17—42no—
205C23×C17—42no—
206C23×C17—42no—
207C23×C17—42no—
208C23×C17C23×C411yescharacteristic
209C23×C17C22×D411yescharacteristic
210C23×C17C2×#4111yescharacteristic
211C23×C17C2×#4112yes—
212C23×C17C2×#4112yes—
213C23×C17H(5,𝔽2)11yescharacteristic
214C23×C17H(5,𝔽2)11yescharacteristic
215C23×C17H(5,𝔽2)12yes—
216C23×C17H(5,𝔽2)12yes—
217C23×C17H(5,𝔽2)11yescharacteristic
218C23×C172−1+411yescharacteristic
219D4×C17—416no—
220C2×C4×C17—22no—
221C2×C4×C17—22no—
222C2×C4×C17—22no—
223C2×C4×C17—22no—
224C2×C4×C17—22no—
225C2×C4×C17—22no—
226C2×C4×C17—24no—
227C2×C4×C17—24no—
228C2×C4×C17—24no—
229C2×C4×C17—24no—
230C2×C4×C17—22no—
231C2×C4×C17—22no—
232C2×C4×C17—24no—
233C2×C4×C17—42no—
234C2×C4×C17—42no—
235C2×C4×C17—42no—
236C2×C4×C17—42no—
237C2×C4×C17—41no—
238C2×C4×C17—42no—
239C2×C4×C17—41no—
240C2×C4×C17—42no—
241C2×C4×C17—41no—
242C2×C4×C17—41no—
243C2×C4×C17C4×D414yes—
244C2×Dic17—24no—
245C2×Dic17—24no—
246C2×Dic17—24no—
247C2×Dic17—24no—
248C2×Dic17—24no—
249C2×Dic17—24no—
250C2×Dic17—24no—
251C2×Dic17—24no—
252C2×Dic17—24no—
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×Dic17—42no—
269C2×Dic17—42no—
270C2×Dic17—42no—
271C2×Dic17—42no—
272C2×#394D1711yescharacteristic
273C2×#388—172no—
274#1441—171no—
275#1439—172no—
276#1438—172no—
277#1437—172no—
278#1437—172no—
279C4⋊#95—171no—
280#1434—172no—
281C22×C17—44no—
282C22×C17—44no—
283C22×C17—44no—
284C22×C17—44no—
285C22×C17—44no—
286C22×C17—44no—
287C22×C17—44no—
288C22×C17—82no—
289C22×C17—82no—
290C22×C17—82no—
291C22×C17—82no—
292C22×C17—82no—
293C22×C17—82no—
294C22×C17—82no—
295C22×C17—82no—
296C22×C17C2×C4×D411yescharacteristic
297C22×C17C4×#4112yes—
298C22×C17#51811yescharacteristic
299C22×C17#51812yes—
300C22×C17#52011yescharacteristic
301C22×C17#53411yescharacteristic
302C22×C17#53711yescharacteristic
303C22×C17#53812yes—
304C22×C17#53811yescharacteristic
305C22×C17#53912yes—
306C22×C17#53911yescharacteristic
307C22×C17#54012yes—
308C22×C17#54212yes—
309C22×C17#54612yes—
310C22×C17#55012yes—
311C22×C17#55112yes—
312C22×C17#55312yes—
313C22×C17#55511yescharacteristic
314C22×C17#55511yescharacteristic
315C22×C17#55512yes—
316C22×C17#56011yescharacteristic
317C22×C17#56012yes—
318C22×C17H(3,𝔽4)11yescharacteristic
319C4×C17—24no—
320C4×C17—44no—
321C4×C17—42no—
322C4×C17—42no—
323Dic17—44no—
324Dic17—44no—
325Dic17—44no—
326Dic17—82no—
327C23×D4C2×D1711yescharacteristic
328C22×C4⋊C4—171no—
329C22×C4⋊C4—171no—
330C22×C4⋊C4C2×D1711yescharacteristic
331C22×C22⋊C4—172no—
332C22×C22⋊C4—172no—
333C22×C22⋊C4—172no—
334C22×C22⋊C4—172no—
335C22×C22⋊C4C2×D1712yes—
336C22×C22⋊C4C2×D1712yes—
337C22×C42—172no—
338C22×C42—171no—
339#394Dic1718yescomplemented
340#388—348no—
341C2×#95—172no—
342C2×#95—172no—
343C2×#95—172no—
344C2×#95—172no—
345C2×#95—172no—
346C2×#95—172no—
347C2×#95—172no—
348C2×#95—171no—
349C2×#95C2×D1711yescharacteristic
350C2×C17—84no—
351C2×C17#232511yescharacteristic
352C2×C17#233512yes—
353C2×C17#236312yes—
354C2×C17#237211yescharacteristic
355C2×C17#237511yescharacteristic
356C2×C17#269911yescharacteristic
357C2×C17#270512yes—
358C2×C17#273311yescharacteristic
359C2×C17#275612yes—
360C2×C17#278511yescharacteristic
361C2×C17#279511yescharacteristic
362C25C22×D1712yes—
363C22×D4—42no—
364C22×D4—42no—
365C22×D4—41no—
366C22×D4—41no—
367C22×D4C2×Dic1714yes—
368C23×C4—172no—
369C23×C4—172no—
370C23×C4—172no—
371C23×C4—172no—
372C23×C4C22×D1711yescharacteristic
373C23×C4C22×D1712yes—
374C23×C4C22×D1711yescharacteristic
375C23×C4C22×D1711yescharacteristic
376C2×C4⋊C4—41no—
377C2×C4⋊C4—41no—
378C2×C4⋊C4—174no—
379C2×C4⋊C4—681no—
380C2×C4⋊C4—681no—
381C2×C4⋊C4—682no—
382C2×C4⋊C4—681no—
383C2×C4⋊C4C2×Dic1714yes—
384C2×C22⋊C4—24no—
385C2×C22⋊C4—24no—
386C2×C22⋊C4—42no—
387C2×C22⋊C4—42no—
388C2×C22⋊C4—344no—
389C2×C22⋊C4—344no—
390C2×C22⋊C4—344no—
391C2×C22⋊C4—344no—
392C2×C22⋊C4—344no—
393C2×C22⋊C4—344no—
394C2×C22⋊C4—682no—
395C2×C22⋊C4—682no—
396C2×C22⋊C4—682no—
397C2×C22⋊C4—682no—
398C2×C22⋊C4—682no—
399C2×C22⋊C4—682no—
400C2×C22⋊C4C2×Dic1718yes—
401C2×C22⋊C4C2×Dic1718yes—
402C2×C42—344no—
403C2×C42—344no—
404C2×C42—342no—
405C2×C42—681no—
406C2×C42—681no—
407C2×C42—682no—
408#95—344no—
409#95—344no—
410#95—344no—
411#95—682no—
412#95—682no—
413#95—681no—
414#95—682no—
415#95—682no—
416#95C2×Dic1714yes—
417C17#1458911yescharacteristic, complemented, Sylow
418C24—22no—
419C24—22no—
420C24—22no—
421C24—22no—
422C24—22no—
423C24—22no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24—22no—
430C24—22no—
431C24—22no—
432C24C22×Dic1714yes—
433C24C23×D1711yescharacteristic
434C2×D4—44no—
435C2×D4—48no—
436C2×D4—48no—
437C2×D4—44no—
438C2×D4—44no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—174no—
458C22×C4—174no—
459C22×C4—174no—
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×C4—342no—
482C22×C4—342no—
483C22×C4—342no—
484C22×C4—342no—
485C22×C4D4×D1712yes—
486C22×C4#6325412yes—
487C22×C4C22×Dic1714yes—
488C22×C4C22×Dic1712yes—
489C22×C4C22×Dic1712yes—
490C22×C4C22×Dic1712yes—
491C4⋊C4—44no—
492C4⋊C4—684no—
493C4⋊C4—684no—
494C22⋊C4—216no—
495C22⋊C4—48no—
496C22⋊C4—688no—
497C22⋊C4—688no—
498C22⋊C4—688no—
499C22⋊C4—688no—
500C42—684no—
501C42—688no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
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—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23C2×D4×D1711yescharacteristic
538C23C2×#6325411yescharacteristic
539C23#1058769112yes—
540C23#1058769112yes—
541C23#1058769111yescharacteristic
542C23#1058769111yescharacteristic
543C23#41×D1712yes—
544C23#41×D1712yes—
545C23#1058769611yescharacteristic
546C23#1058769711yescharacteristic
547C23C23×Dic1711yescharacteristic
548D4—416no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—344no—
573C2×C4—344no—
574C2×C4—344no—
575C2×C4—344no—
576C2×C4—344no—
577C2×C4—344no—
578C2×C4—344no—
579C2×C4—344no—
580C2×C4—344no—
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—
592C2×C4—682no—
593C2×C4—682no—
594C2×C4—682no—
595C2×C4—682no—
596C2×C4—682no—
597C2×C4—682no—
598C2×C4—682no—
599C2×C4—682no—
600C2×C4D4×Dic1714yes—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22C2×D4×Dic1711yescharacteristic
617C22#41×Dic1712yes—
618C22#1176328411yescharacteristic
619C22#1176328412yes—
620C22#1176329211yescharacteristic
621C22#1176336711yescharacteristic
622C22#1176338611yescharacteristic
623C22#1176339512yes—
624C22#1176339611yescharacteristic
625C22#1176340212yes—
626C22#1176340411yescharacteristic
627C22#1176341112yes—
628C22#1176342712yes—
629C22#1176345612yes—
630C22#1176347712yes—
631C22#1176348912yes—
632C22#1176350612yes—
633C22#1176352111yescharacteristic
634C22#1176352111yescharacteristic
635C22#1176352312yes—
636C22#1176354812yes—
637C22#1176354811yescharacteristic
638C22#1176355211yescharacteristic
639C4—24no—
640C4—42no—
641C4—42no—
642C4—44no—
643C4—684no—
644C4—684no—
645C4—684no—
646C4—1362no—
647C2—84no—
648C2#1453148911yescharacteristic
649C2#1453157812yes—
650C2#1453182612yes—
651C2#1453189611yescharacteristic
652C2#1453192511yescharacteristic
653C2#1453536611yescharacteristic
654C2#1453545012yes—
655C2#1453580711yescharacteristic
656C2#1453610512yes—
657C2#1453649111yescharacteristic
658C2#1453661711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC2×C17×#394C2
socle, cosocleC24×C17C24
prime radicalC23×D4×C17C22
hypercenterC2×#394D17
hypocenterC17#14589
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
1. C2×#11762702C2semidirect
2. #14527037C2semidirect
3. #14527029C2semidirect
5. #14527021C2semidirect
6. #14527021C2semidirect
7. #14527013C2semidirect
8. #14527009C2semidirect
31. C17×#394C4semidirect
72. C2×C4⋊Dic17D4semidirect
339. #394Dic17semidirect
417. C17#14589semidirect

Maximal subgroups

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

subgroupindex
C2×#117627022
#145270372
#145270292
#145270252
#145270212
#145270132
#145270092
C2×C17×#3942
#1458917