Finite Group Databasebeta

← #16956391

Subgroups — #16956391

Order 4352.

1602 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762675C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊Dic17C211yesmaximal, characteristic
3#14527033C212yesmaximal
4#14527029C212yesmaximal
5#14527018C212yesmaximal, complemented
6#14527011C211yesmaximal, characteristic, complemented
7#14527011C211yesmaximal, characteristic, complemented
8#14527011C211yesmaximal, characteristic, complemented
9#14527009C212yesmaximal, complemented
10#14527007C211yesmaximal, characteristic
11C2×C4×C22⋊C4×C17C211yesmaximal, characteristic
12C22×C4⋊Dic17C2212yes—
13C22×C22⋊Dic17C2212yes—
14C22×C22⋊Dic17C2212yes—
15C22×C4×Dic17C2212yes—
16#11762675—24no—
17C4×C22⋊Dic17—24no—
18C2×#10587502C2212yes—
19C2×#10587502C2211yescharacteristic
20C2×#10587502C2211yescharacteristic
21C2×#10587502C2212yes—
22C2×#10587502C2212yes—
23C2×#10587502C2212yes—
24C2×#10587502C2211yescharacteristic
25C2×#10587502C2212yes—
26C2×#10587502C2212yes—
27C2×#10587502C2212yes—
28C2×#10587502C2212yes—
29C2×#10587502C2211yescharacteristic
30C24×C4×C17C2211yescharacteristic
31C22×C22⋊C4×C17C2211yescharacteristic
32C22×C22⋊C4×C17C2211yescharacteristic
33C22×C42×C17C2211yescharacteristic
34C22×C42×C17C2211yescharacteristic
35C4×C22⋊C4×C17C418yescomplemented
36C2×C17×#95C2211yescharacteristic
37C2×C17×#95C2211yescharacteristic
38C25×C17C2311yescharacteristic
39C23×C4×C17—22no—
40C23×C4×C17—24no—
41C23×C4×C17—22no—
42C23×C4×C17—42no—
43C23×C4×C17C2×C414yes—
44C23×C4×C17C2311yescharacteristic
45C23×C4×C17C2311yescharacteristic
46C23×C4×C17C2311yescharacteristic
47C23×C4×C17C2311yescharacteristic
48C23×C4×C17C2311yescharacteristic
49C23×C4×C17C2311yescharacteristic
50C23×Dic17C2312yes—
51C23×Dic17C2312yes—
52C23×Dic17C2312yes—
53C23×Dic17C2312yes—
54C2×C22⋊C4×C17—24no—
55C2×C22⋊C4×C17—22no—
56C2×C22⋊C4×C17—22no—
57C2×C22⋊C4×C17C2×C414yes—
58C2×C22⋊C4×C17C2×C414yes—
59C2×C42×C17—22no—
60C2×C42×C17—22no—
61C2×C42×C17—42no—
62C2×C42×C17C2×C414yes—
63C2×C42×C17C2×C414yes—
64C2×C22⋊Dic17—24no—
65C2×C22⋊Dic17—24no—
66C2×C22⋊Dic17—24no—
67C2×C22⋊Dic17—24no—
68C2×C22⋊Dic17—42no—
69C2×C22⋊Dic17—42no—
70C2×C4⋊Dic17—24no—
71C2×C4⋊Dic17—24no—
72C2×C4⋊Dic17—42no—
73C2×C4×Dic17—24no—
74C2×C4×Dic17—24no—
75C2×C4×Dic17—42no—
76C17×#95C2×C414yes—
77C17×#95C2×C414yes—
78#10587502—22no—
79#10587502—22no—
80#10587502—24no—
81#10587502—24no—
82#10587502—22no—
83#10587502—24no—
84#10587502—24no—
85#10587502—24no—
86#10587502—42no—
87#10587502—42no—
88#10587502—42no—
89#10587502D414yes—
90C24×C17—21no—
91C24×C17—22no—
92C24×C17—21no—
93C24×C17—22no—
94C24×C17—21no—
95C24×C17—21no—
96C24×C17—22no—
97C24×C17—21no—
98C24×C17—21no—
99C24×C17C22×C412yes—
100C24×C17#4114yes—
101C24×C17C2411yescharacteristic
102C22×C4×C17—21no—
103C22×C4×C17—21no—
104C22×C4×C17—22no—
105C22×C4×C17—21no—
106C22×C4×C17—21no—
107C22×C4×C17—22no—
108C22×C4×C17—22no—
109C22×C4×C17—21no—
110C22×C4×C17—21no—
111C22×C4×C17—21no—
112C22×C4×C17—22no—
113C22×C4×C17—22no—
114C22×C4×C17—21no—
115C22×C4×C17—22no—
116C22×C4×C17—22no—
117C22×C4×C17—21no—
118C22×C4×C17—21no—
119C22×C4×C17—21no—
120C22×C4×C17—22no—
121C22×C4×C17—22no—
122C22×C4×C17—21no—
123C22×C4×C17—28no—
124C22×C4×C17—28no—
125C22×C4×C17—44no—
126C22×C4×C17—48no—
127C22×C4×C17—48no—
128C22×C4×C17C22×C412yes—
129C22×C4×C17C22×C412yes—
130C22×C4×C17C22×C412yes—
131C22×C4×C17C22×C412yes—
132C22×C4×C17C22×C412yes—
133C22×C4×C17C22×C412yes—
134C22×C4×C17C2×D412yes—
135C22×C4×C17#4114yes—
136C22×C4×C17#4112yes—
137C22×C4×C17#4112yes—
138C22×C4×C17#4112yes—
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×Dic17—22no—
160C22×Dic17—22no—
161C22×Dic17—22no—
162C22×Dic17—22no—
163C22×Dic17—22no—
164C22×Dic17—22no—
165C22×Dic17C2×D414yes—
166C22×Dic17#4114yes—
167C22⋊C4×C17—28no—
168C22⋊C4×C17—28no—
169C42×C17—28no—
170C42×C17—44no—
171C22⋊Dic17—216no—
172C22⋊Dic17—48no—
173C4⋊Dic17—48no—
174C4×Dic17—48no—
175#13888—171nomaximal, Sylow
176C23×C17—22no—
177C23×C17—22no—
178C23×C17—22no—
179C23×C17—22no—
180C23×C17—22no—
181C23×C17—22no—
182C23×C17—24no—
183C23×C17—24no—
184C23×C17—24no—
185C23×C17—24no—
186C23×C17—24no—
187C23×C17—24no—
188C23×C17—42no—
189C23×C17—41no—
190C23×C17—41no—
191C23×C17—42no—
192C23×C17—42no—
193C23×C17—41no—
194C23×C17—41no—
195C23×C17—42no—
196C23×C17—42no—
197C23×C17—42no—
198C23×C17#11714yes—
199C23×C17C23×C411yescharacteristic
200C23×C17C22×D411yescharacteristic
201C23×C17C2×#4112yes—
202C23×C17C2×#4112yes—
203C23×C17C2×#4111yescharacteristic
204C23×C17C2×#4111yescharacteristic
205C23×C17C2×#4111yescharacteristic
206C23×C17C2×#4112yes—
207C23×C17H(5,𝔽2)12yes—
208C23×C17H(5,𝔽2)11yescharacteristic
209C23×C172−1+411yescharacteristic
210C2×C4×C17—22no—
211C2×C4×C17—22no—
212C2×C4×C17—22no—
213C2×C4×C17—22no—
214C2×C4×C17—22no—
215C2×C4×C17—22no—
216C2×C4×C17—24no—
217C2×C4×C17—24no—
218C2×C4×C17—22no—
219C2×C4×C17—22no—
220C2×C4×C17—24no—
221C2×C4×C17—24no—
222C2×C4×C17—22no—
223C2×C4×C17—22no—
224C2×C4×C17—22no—
225C2×C4×C17—22no—
226C2×C4×C17—22no—
227C2×C4×C17—22no—
228C2×C4×C17—22no—
229C2×C4×C17—22no—
230C2×C4×C17—22no—
231C2×C4×C17—22no—
232C2×C4×C17—22no—
233C2×C4×C17—416no—
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×C4×C17#11714yes—
241C2×C4×C17#11714yes—
242C2×C4×C17C4×D414yes—
243C2×Dic17—24no—
244C2×Dic17—24no—
245C2×Dic17—24no—
246C2×Dic17—24no—
247C2×Dic17—24no—
248C2×Dic17—24no—
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×Dic17—42no—
269C2×Dic17—42no—
270C2×Dic17—42no—
271C2×#382—171no—
272C2×C4×C22⋊C4—171no—
273C2×C4×C22⋊C4D1711yescharacteristic
274#1440—172no—
275#1439—172no—
276#1436—172no—
277#1434—171no—
278#1434—171no—
279#1434—171no—
280#1434—172no—
281C22⋊#95—171no—
282C22×C17—24no—
283C22×C17—24no—
284C22×C17—24no—
285C22×C17—42no—
286C22×C17—42no—
287C22×C17—42no—
288C22×C17—42no—
289C22×C17—44no—
290C22×C17—44no—
291C22×C17—44no—
292C22×C17—44no—
293C22×C17C2×#11711yescharacteristic
294C22×C17C2×#11711yescharacteristic
295C22×C17C2×#11711yescharacteristic
296C22×C17C2×C4×D411yescharacteristic
297C22×C17C4×#4111yescharacteristic
298C22×C17#51811yescharacteristic
299C22×C17#52011yescharacteristic
300C22×C17#52512yes—
301C22×C17#52912yes—
302C22×C17#52912yes—
303C22×C17#52912yes—
304C22×C17#53812yes—
305C22×C17#53912yes—
306C22×C17#54612yes—
307C22×C17#54611yescharacteristic
308C22×C17#54811yescharacteristic
309C22×C17#55112yes—
310C22×C17#55212yes—
311C22×C17#55211yescharacteristic
312C22×C17#55311yescharacteristic
313C22×C17#55312yes—
314C22×C17#55512yes—
315C22×C17#55912yes—
316C4×C17—24no—
317C4×C17—24no—
318C4×C17—24no—
319C4×C17—42no—
320C4×C17—42no—
321C4×C17—42no—
322Dic17—44no—
323Dic17—44no—
324Dic17—82no—
325Dic17—82no—
326C24×C4C2×D1711yescharacteristic
327C22×C4⋊C4—172no—
328C22×C22⋊C4—172no—
329C22×C22⋊C4—172no—
330C22×C22⋊C4C2×D1711yescharacteristic
331C22×C22⋊C4C2×D1711yescharacteristic
332C22×C42—172no—
333C22×C42C2×D1711yescharacteristic
334C22×C42C2×D1711yescharacteristic
335#382—344no—
336C4×C22⋊C4—344no—
337C4×C22⋊C4Dic1718yescomplemented
338C2×#95—171no—
339C2×#95—171no—
340C2×#95—172no—
341C2×#95—171no—
342C2×#95—172no—
343C2×#95—172no—
344C2×#95—171no—
345C2×#95—172no—
346C2×#95—172no—
347C2×#95—172no—
348C2×#95—172no—
349C2×#95—172no—
350C2×#95C2×D1711yescharacteristic
351C2×#95C2×D1711yescharacteristic
352C2×C17—44no—
353C2×C17#228311yescharacteristic
354C2×C17#229011yescharacteristic
355C2×C17#231811yescharacteristic
356C2×C17#234211yescharacteristic
357C2×C17#234611yescharacteristic
358C2×C17#235211yescharacteristic
359C2×C17#235711yescharacteristic
360C2×C17#245912yes—
361C2×C17#246012yes—
362C2×C17#251712yes—
363C2×C17#251812yes—
364C25C22×D1711yescharacteristic
365C23×C4—24no—
366C23×C4—22no—
367C23×C4—22no—
368C23×C4—42no—
369C23×C4—172no—
370C23×C4—172no—
371C23×C4—172no—
372C23×C4—172no—
373C23×C4C2×Dic1714yes—
374C23×C4C22×D1711yescharacteristic
375C23×C4C22×D1711yescharacteristic
376C23×C4C22×D1711yescharacteristic
377C23×C4C22×D1711yescharacteristic
378C23×C4C22×D1711yescharacteristic
379C23×C4C22×D1711yescharacteristic
380C2×C4⋊C4—344no—
381C2×C4⋊C4—344no—
382C2×C4⋊C4—682no—
383C2×C22⋊C4—24no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—344no—
387C2×C22⋊C4—344no—
388C2×C22⋊C4—344no—
389C2×C22⋊C4—344no—
390C2×C22⋊C4—682no—
391C2×C22⋊C4—682no—
392C2×C22⋊C4C2×Dic1714yes—
393C2×C22⋊C4C2×Dic1714yes—
394C2×C42—22no—
395C2×C42—22no—
396C2×C42—42no—
397C2×C42—344no—
398C2×C42—344no—
399C2×C42—682no—
400C2×C42C2×Dic1714yes—
401C2×C42C2×Dic1714yes—
402#95—174no—
403#95—344no—
404#95—344no—
405#95—344no—
406#95—344no—
407#95—344no—
408#95—342no—
409#95—342no—
410#95—342no—
411#95—682no—
412#95—682no—
413#95—682no—
414#95C2×Dic1714yes—
415#95C2×Dic1714yes—
416C17#1388811yescharacteristic, complemented, Sylow
417C24—21no—
418C24—21no—
419C24—22no—
420C24—21no—
421C24—21no—
422C24—22no—
423C24—22no—
424C24—21no—
425C24—21no—
426C24#6325214yes—
427C24C22×Dic1712yes—
428C24C23×D1711yescharacteristic
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—28no—
451C22×C4—28no—
452C22×C4—44no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—174no—
456C22×C4—174no—
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×C4—342no—
482C22×C4—342no—
483C22×C4#6325212yes—
484C22×C4#6325214yes—
485C22×C4#6325212yes—
486C22×C4D4×D1712yes—
487C22×C4#6325412yes—
488C22×C4C22×Dic1712yes—
489C22×C4C22×Dic1712yes—
490C22×C4C22×Dic1712yes—
491C22×C4C22×Dic1712yes—
492C22×C4C22×Dic1712yes—
493C22×C4C22×Dic1712yes—
494C4⋊C4—688no—
495C22⋊C4—28no—
496C22⋊C4—28no—
497C22⋊C4—3416no—
498C22⋊C4—688no—
499C42—28no—
500C42—44no—
501C42—688no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—41no—
515C23—42no—
516C23—42no—
517C23—41no—
518C23—41no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—41no—
524C23#1058761214yes—
525C23C2×#6325211yescharacteristic
526C23C2×#6325212yes—
527C23C2×#6325211yescharacteristic
528C23C2×#6325212yes—
529C23C2×D4×D1711yescharacteristic
530C23C2×#6325411yescharacteristic
531C23#1058769112yes—
532C23#41×D1712yes—
533C23#1058769611yescharacteristic
534C23#1058769711yescharacteristic
535C23C23×Dic1711yescharacteristic
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—416no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—344no—
567C2×C4—344no—
568C2×C4—344no—
569C2×C4—344no—
570C2×C4—344no—
571C2×C4—344no—
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—
592C2×C4—682no—
593C2×C4—682no—
594C2×C4#1058761214yes—
595C2×C4#1058761214yes—
596C2×C4D4×Dic1714yes—
597C22—24no—
598C22—24no—
599C22—24no—
600C22—42no—
601C22—44no—
602C22—44no—
603C22—42no—
604C22—44no—
605C22—44no—
606C22—42no—
607C22—42no—
608C22C2×#1058761211yescharacteristic
609C22C2×#1058761211yescharacteristic
610C22C2×#1058761211yescharacteristic
611C22C2×D4×Dic1711yescharacteristic
612C22#41×Dic1711yescharacteristic
613C22#1176328411yescharacteristic
614C22#1176329211yescharacteristic
615C22#1176331812yes—
616C22#1176333912yes—
617C22#1176334312yes—
618C22#1176334312yes—
619C22#1176339212yes—
620C22#1176340112yes—
621C22#1176344912yes—
622C22#1176345511yescharacteristic
623C22#1176346811yescharacteristic
624C22#1176348512yes—
625C22#1176349812yes—
626C22#1176349911yescharacteristic
627C22#1176350511yescharacteristic
628C22#1176350912yes—
629C22#1176352212yes—
630C22#1176354212yes—
631C4—24no—
632C4—24no—
633C4—24no—
634C4—42no—
635C4—42no—
636C4—42no—
637C4—684no—
638C4—684no—
639C4—1362no—
640C4—1362no—
641C2—44no—
642C2#1453118211yescharacteristic
643C2#1453122911yescharacteristic
644C2#1453141711yescharacteristic
645C2#1453163311yescharacteristic
646C2#1453166911yescharacteristic
647C2#1453171311yescharacteristic
648C2#1453175711yescharacteristic
649C2#1453272112yes—
650C2#1453273612yes—
651C2#1453342712yes—
652C2#1453344212yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC2×C4×C22⋊C4×C17C2
socle, cosocleC24×C17C24
prime radicalC25×C17C23
hypercenterC2×C4×C22⋊C4D17
hypocenterC17#13888
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#11762675C2semidirect
5. #14527018C2semidirect
6. #14527011C2semidirect
7. #14527011C2semidirect
8. #14527011C2semidirect
9. #14527009C2semidirect
35. C4×C22⋊C4×C17C4semidirect
337. C4×C22⋊C4Dic17semidirect
416. C17#13888semidirect

Maximal subgroups

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

subgroupindex
C2×#117626752
C2×C4×C22⋊Dic172
#145270332
#145270292
#145270182
#145270112
#145270092
#145270072
C2×C4×C22⋊C4×C172
#1388817