Finite Group Databasebeta

← #16956247

Subgroups — #16956247

Order 4352.

1918 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762670C211yesmaximal, characteristic
2C2×#11762670C211yesmaximal, characteristic
3C2×#11762667C211yesmaximal, characteristic
4C22×#10587502C211yesmaximal, characteristic
5#14527044C212yesmaximal, complemented
6#14527039C212yesmaximal, complemented
7#14527017C212yesmaximal, complemented
8#14527011C212yesmaximal, complemented
9#14527007C211yesmaximal, characteristic
10#14527006C211yesmaximal, characteristic
11C2×C4×C22⋊C4×C17C211yesmaximal, characteristic
12C24×Dic17C2211yescharacteristic
13C24×Dic17C2211yescharacteristic
14C22×#63225C2212yes—
15C22×C22⋊Dic17C2211yescharacteristic
16C22×C22⋊Dic17C2211yescharacteristic
17#11762670—24no—
18#11762670—24no—
19#11762667—24no—
20C2×#10587502C418yescomplemented
21C2×#10587502C2212yes—
22C2×#10587502C2212yes—
23C2×#10587502C2212yes—
24C2×#10587502C2212yes—
25C2×#10587502C2212yes—
26C2×#10587502C2212yes—
27C2×#10587502C2212yes—
28C2×#10587502C2212yes—
29C2×#10587502C2212yes—
30C2×#10587502C2211yescharacteristic
31C2×#10587502C2211yescharacteristic
32C2×#10587502C2211yescharacteristic
33C2×#10587502C2211yescharacteristic
34C24×C4×C17C2211yescharacteristic
35C22×C22⋊C4×C17C2211yescharacteristic
36C22×C22⋊C4×C17C2211yescharacteristic
37C22×C42×C17C2212yes—
38C4×C22⋊C4×C17—24no—
39C2×C17×#95C2212yes—
40C25×C17C2311yescharacteristic
41C23×C4×C17—22no—
42C23×C4×C17—22no—
43C23×C4×C17—24no—
44C23×C4×C17—42no—
45C23×C4×C17C2×C414yes—
46C23×C4×C17C2311yescharacteristic
47C23×C4×C17C2312yes—
48C23×C4×C17C2312yes—
49C23×C4×C17C2311yescharacteristic
50C23×Dic17—22no—
51C23×Dic17—22no—
52C23×Dic17—22no—
53C23×Dic17—22no—
54C23×Dic17—22no—
55C23×Dic17—22no—
56C23×Dic17—22no—
57C23×Dic17—22no—
58C23×Dic17—24no—
59C23×Dic17—42no—
60C23×Dic17C2×C414yes—
61C23×Dic17C2×C414yes—
62C23×Dic17C2312yes—
63C23×Dic17C2312yes—
64C23×Dic17C2311yescharacteristic
65C23×Dic17C2311yescharacteristic
66C23×Dic17C2311yescharacteristic
67C23×Dic17C2311yescharacteristic
68C2×C22⋊C4×C17—22no—
69C2×C22⋊C4×C17—22no—
70C2×C22⋊C4×C17—22no—
71C2×C22⋊C4×C17—22no—
72C2×C22⋊C4×C17D414yes—
73C2×C22⋊C4×C17D414yes—
74C2×C42×C17—24no—
75C2×C42×C17—24no—
76C2×C42×C17—42no—
77C2×C22⋊Dic17—22no—
78C2×C22⋊Dic17—22no—
79C2×C22⋊Dic17—22no—
80C2×C22⋊Dic17—22no—
81C2×C22⋊Dic17—22no—
82C2×C22⋊Dic17—22no—
83C2×#63225—24no—
84C2×#63225—24no—
85C2×#63225—42no—
86C17×#95—24no—
87#10587502—24no—
88#10587502—24no—
89#10587502—24no—
90#10587502—24no—
91#10587502—24no—
92#10587502—24no—
93#10587502—24no—
94#10587502—42no—
95#10587502—42no—
96#10587502C2×C414yes—
97#10587502C2×C414yes—
98#10587502C2×C414yes—
99#10587502C2×C414yes—
100C24×C17—22no—
101C24×C17—22no—
102C24×C17—22no—
103C24×C17—22no—
104C24×C17C22×C412yes—
105C24×C17C2×D412yes—
106C24×C17C2×D412yes—
107C24×C17#4112yes—
108C24×C17#4112yes—
109C24×C17#4112yes—
110C24×C17#4112yes—
111C24×C17C2411yescharacteristic
112C22×C4×C17—21no—
113C22×C4×C17—21no—
114C22×C4×C17—22no—
115C22×C4×C17—28no—
116C22×C4×C17—21no—
117C22×C4×C17—22no—
118C22×C4×C17—21no—
119C22×C4×C17—22no—
120C22×C4×C17—22no—
121C22×C4×C17—22no—
122C22×C4×C17—22no—
123C22×C4×C17—22no—
124C22×C4×C17—22no—
125C22×C4×C17—22no—
126C22×C4×C17—22no—
127C22×C4×C17—22no—
128C22×C4×C17—22no—
129C22×C4×C17—22no—
130C22×C4×C17—22no—
131C22×C4×C17—22no—
132C22×C4×C17—28no—
133C22×C4×C17—44no—
134C22×C4×C17—48no—
135C22×C4×C17—48no—
136C22×C4×C17C22×C412yes—
137C22×C4×C17C22×C412yes—
138C22×C4×C17C2×D414yes—
139C22×C4×C17C2×D414yes—
140C22×C4×C17#4114yes—
141C22×Dic17—21no—
142C22×Dic17—21no—
143C22×Dic17—22no—
144C22×Dic17—28no—
145C22×Dic17—28no—
146C22×Dic17—28no—
147C22×Dic17—28no—
148C22×Dic17—21no—
149C22×Dic17—21no—
150C22×Dic17—21no—
151C22×Dic17—21no—
152C22×Dic17—21no—
153C22×Dic17—21no—
154C22×Dic17—21no—
155C22×Dic17—21no—
156C22×Dic17—21no—
157C22×Dic17—21no—
158C22×Dic17—21no—
159C22×Dic17—21no—
160C22×Dic17—21no—
161C22×Dic17—21no—
162C22×Dic17—22no—
163C22×Dic17—22no—
164C22×Dic17—28no—
165C22×Dic17—22no—
166C22×Dic17—22no—
167C22×Dic17—22no—
168C22×Dic17—22no—
169C22×Dic17—22no—
170C22×Dic17—22no—
171C22×Dic17—22no—
172C22×Dic17—22no—
173C22×Dic17—22no—
174C22×Dic17—22no—
175C22×Dic17—22no—
176C22×Dic17—22no—
177C22×Dic17—22no—
178C22×Dic17—22no—
179C22×Dic17—44no—
180C22×Dic17—48no—
181C22×Dic17—48no—
182C22×Dic17—44no—
183C22×Dic17—44no—
184C22×Dic17—44no—
185C22×Dic17—44no—
186C22×Dic17C22×C412yes—
187C22×Dic17C22×C412yes—
188C22×Dic17C22×C412yes—
189C22×Dic17C22×C412yes—
190C22×Dic17#4114yes—
191C22⋊C4×C17—28no—
192C22⋊C4×C17—28no—
193C42×C17—48no—
194C22⋊Dic17—44no—
195C22⋊Dic17—44no—
196#63225—48no—
197#13874—171nomaximal, Sylow
198C23×C17—22no—
199C23×C17—22no—
200C23×C17—22no—
201C23×C17—22no—
202C23×C17—24no—
203C23×C17—24no—
204C23×C17—24no—
205C23×C17—24no—
206C23×C17—24no—
207C23×C17—24no—
208C23×C17—24no—
209C23×C17—24no—
210C23×C17—24no—
211C23×C17—24no—
212C23×C17—24no—
213C23×C17—24no—
214C23×C17#11714yes—
215C23×C17C4×D414yes—
216C23×C17C4×D414yes—
217C23×C17#12114yes—
218C23×C17#12314yes—
219C23×C17#12414yes—
220C23×C17#12614yes—
221C23×C17C23×C411yescharacteristic
222C23×C17C22×D411yescharacteristic
223C23×C17C22×D411yescharacteristic
224C23×C17C2×#4111yescharacteristic
225C23×C17C2×#4111yescharacteristic
226C23×C17C2×#4111yescharacteristic
227C23×C17C2×#4111yescharacteristic
228C23×C17C2×#4112yes—
229C23×C17C2×#4112yes—
230C23×C17H(5,𝔽2)12yes—
231C23×C172−1+412yes—
232C2×C4×C17—22no—
233C2×C4×C17—22no—
234C2×C4×C17—24no—
235C2×C4×C17—24no—
236C2×C4×C17—22no—
237C2×C4×C17—22no—
238C2×C4×C17—22no—
239C2×C4×C17—24no—
240C2×C4×C17—24no—
241C2×C4×C17—24no—
242C2×C4×C17—24no—
243C2×C4×C17—24no—
244C2×C4×C17—24no—
245C2×C4×C17—416no—
246C2×C4×C17—42no—
247C2×C4×C17—42no—
248C2×C4×C17—42no—
249C2×C4×C17—42no—
250C2×C4×C17—42no—
251C2×C4×C17—42no—
252C2×C4×C17—42no—
253C2×C4×C17—42no—
254C2×C4×C17—42no—
255C2×C4×C17—42no—
256C2×C4×C17#11714yes—
257C2×Dic17—22no—
258C2×Dic17—22no—
259C2×Dic17—24no—
260C2×Dic17—24no—
261C2×Dic17—22no—
262C2×Dic17—22no—
263C2×Dic17—22no—
264C2×Dic17—22no—
265C2×Dic17—22no—
266C2×Dic17—22no—
267C2×Dic17—22no—
268C2×Dic17—22no—
269C2×Dic17—22no—
270C2×Dic17—416no—
271C2×Dic17—41no—
272C2×Dic17—41no—
273C2×Dic17—41no—
274C2×Dic17—41no—
275C2×Dic17—41no—
276C2×Dic17—41no—
277C2×Dic17—41no—
278C2×Dic17—41no—
279C2×Dic17—42no—
280C2×Dic17—42no—
281C2×Dic17—416no—
282C2×Dic17—42no—
283C2×Dic17—42no—
284C2×Dic17—42no—
285C2×Dic17—42no—
286C2×Dic17—42no—
287C2×Dic17—42no—
288C2×Dic17—42no—
289C2×Dic17—42no—
290C2×Dic17—42no—
291C2×Dic17—42no—
292C2×Dic17—42no—
293C2×Dic17—42no—
294C2×Dic17—42no—
295C2×Dic17—42no—
296C2×Dic17#11714yes—
297C2×#381—171no—
298C2×#381—171no—
299C2×#380—171no—
300C2×C4×C22⋊C4D1711yescharacteristic
301C22×#95—171no—
302#1443—172no—
303#1442—172no—
304#1436—172no—
305#1434—172no—
306C22⋊#95—171no—
307C22⋊#95—171no—
308C22×C17—24no—
309C22×C17—24no—
310C22×C17—24no—
311C22×C17—24no—
312C22×C17—24no—
313C22×C17—28no—
314C22×C17—28no—
315C22×C17—28no—
316C22×C17—28no—
317C22×C17—24no—
318C22×C17#38818yes—
319C22×C17C2×#11711yescharacteristic
320C22×C17C2×#11711yescharacteristic
321C22×C17C2×#11711yescharacteristic
322C22×C17C2×C4×D411yescharacteristic
323C22×C17C2×C4×D411yescharacteristic
324C22×C17#51811yescharacteristic
325C22×C17#51911yescharacteristic
326C22×C17C2×#12111yescharacteristic
327C22×C17C2×#12311yescharacteristic
328C22×C17C2×#12411yescharacteristic
329C22×C17C2×#12611yescharacteristic
330C22×C17#52912yes—
331C22×C17#52912yes—
332C22×C17#53912yes—
333C22×C17#53912yes—
334C22×C17#54612yes—
335C22×C17#54612yes—
336C22×C17#54712yes—
337C22×C17#54712yes—
338C22×C17#55112yes—
339C22×C17#55112yes—
340C22×C17#55212yes—
341C22×C17#55212yes—
342C4×C17—24no—
343C4×C17—42no—
344C4×C17—44no—
345C4×C17—44no—
346Dic17—24no—
347Dic17—42no—
348Dic17—42no—
349Dic17—42no—
350Dic17—82no—
351Dic17—82no—
352C24×C4—171no—
353C24×C4—171no—
354C24×C4C2×D1711yescharacteristic
355C22×C4⋊C4—172no—
356C22×C22⋊C4—171no—
357C22×C22⋊C4—171no—
358C22×C22⋊C4C2×D1711yescharacteristic
359C22×C22⋊C4C2×D1711yescharacteristic
360C22×C42C2×D1712yes—
361#381—344no—
362#381—344no—
363#380—344no—
364C4×C22⋊C4—24no—
365C2×#95—178no—
366C2×#95—171no—
367C2×#95—171no—
368C2×#95—171no—
369C2×#95—171no—
370C2×#95—172no—
371C2×#95—172no—
372C2×#95—172no—
373C2×#95—172no—
374C2×#95—172no—
375C2×#95—172no—
376C2×#95—172no—
377C2×#95—172no—
378C2×#95—172no—
379C2×#95C2×D1712yes—
380C2×C17—28no—
381C2×C17#228311yescharacteristic
382C2×C17C2×#38811yescharacteristic
383C2×C17#231311yescharacteristic
384C2×C17#234211yescharacteristic
385C2×C17#234211yescharacteristic
386C2×C17#234411yescharacteristic
387C2×C17#234411yescharacteristic
388C2×C17#241912yes—
389C2×C17#242512yes—
390C2×C17#248812yes—
391C2×C17#249312yes—
392C25C22×D1711yescharacteristic
393C23×C4—22no—
394C23×C4—22no—
395C23×C4—24no—
396C23×C4—42no—
397C23×C4—171no—
398C23×C4—174no—
399C23×C4—171no—
400C23×C4—171no—
401C23×C4—171no—
402C23×C4—174no—
403C23×C4—172no—
404C23×C4—172no—
405C23×C4—344no—
406C23×C4—342no—
407C23×C4—342no—
408C23×C4—342no—
409C23×C4—342no—
410C23×C4—342no—
411C23×C4—342no—
412C23×C4—342no—
413C23×C4—342no—
414C23×C4—682no—
415C23×C4C4×D1714yes—
416C23×C4C22×D1711yescharacteristic
417C23×C4C22×D1712yes—
418C23×C4C22×D1712yes—
419C23×C4C22×D1711yescharacteristic
420C2×C4⋊C4—344no—
421C2×C4⋊C4—344no—
422C2×C4⋊C4—682no—
423C2×C22⋊C4—22no—
424C2×C22⋊C4—22no—
425C2×C22⋊C4—22no—
426C2×C22⋊C4—22no—
427C2×C22⋊C4—342no—
428C2×C22⋊C4—342no—
429C2×C22⋊C4—342no—
430C2×C22⋊C4—342no—
431C2×C22⋊C4—342no—
432C2×C22⋊C4—342no—
433C2×C22⋊C4D6814yes—
434C2×C22⋊C4C22⋊D1714yes—
435C2×C42—24no—
436C2×C42—24no—
437C2×C42—42no—
438#95—24no—
439#95—174no—
440#95—174no—
441#95—174no—
442#95—174no—
443#95—344no—
444#95—344no—
445#95—344no—
446#95—344no—
447#95—344no—
448#95—344no—
449#95—344no—
450#95—682no—
451#95—682no—
452C17#1387411yescharacteristic, complemented, Sylow
453C24—22no—
454C24—22no—
455C24—22no—
456C24—22no—
457C24C2×C4×D1712yes—
458C24C2×D6812yes—
459C24#6325212yes—
460C24#6325212yes—
461C24#6325212yes—
462C24#6325212yes—
463C24C2×C22⋊D1712yes—
464C24C23×D1711yescharacteristic
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—28no—
485C22×C4—28no—
486C22×C4—44no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—172no—
490C22×C4—174no—
491C22×C4—172no—
492C22×C4—172no—
493C22×C4—172no—
494C22×C4—348no—
495C22×C4—341no—
496C22×C4—341no—
497C22×C4—342no—
498C22×C4—341no—
499C22×C4—341no—
500C22×C4—341no—
501C22×C4—341no—
502C22×C4—341no—
503C22×C4—341no—
504C22×C4—341no—
505C22×C4—341no—
506C22×C4—341no—
507C22×C4—341no—
508C22×C4—341no—
509C22×C4—341no—
510C22×C4—342no—
511C22×C4—342no—
512C22×C4—341no—
513C22×C4—341no—
514C22×C4—348no—
515C22×C4—348no—
516C22×C4—348no—
517C22×C4—348no—
518C22×C4—342no—
519C22×C4—342no—
520C22×C4—342no—
521C22×C4—342no—
522C22×C4—342no—
523C22×C4—342no—
524C22×C4—342no—
525C22×C4—342no—
526C22×C4—342no—
527C22×C4—342no—
528C22×C4—342no—
529C22×C4—342no—
530C22×C4—342no—
531C22×C4—342no—
532C22×C4—684no—
533C22×C4—688no—
534C22×C4—688no—
535C22×C4—684no—
536C22×C4—684no—
537C22×C4—684no—
538C22×C4—684no—
539C22×C4C2×C4×D1712yes—
540C22×C4C2×C4×D1712yes—
541C22×C4C2×D6814yes—
542C22×C4#6325414yes—
543C22×C4C2×C22⋊D1714yes—
544C4⋊C4—688no—
545C22⋊C4—28no—
546C22⋊C4—28no—
547C22⋊C4—684no—
548C22⋊C4—684no—
549C42—48no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—24no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23—24no—
565C23—24no—
566C23#1058755814yes—
567C23C4×D6814yes—
568C23#1058756114yes—
569C23#1058756214yes—
570C23C4×C22⋊D1714yes—
571C23#1058761514yes—
572C23#1058761614yes—
573C23C22×C4×D1711yescharacteristic
574C23C22×D6811yescharacteristic
575C23C2×#6325211yescharacteristic
576C23C2×#6325211yescharacteristic
577C23C2×#6325211yescharacteristic
578C23C2×#6325211yescharacteristic
579C23C2×#6325412yes—
580C23#1058769112yes—
581C23#41×D1712yes—
582C23#1058769712yes—
583C23C22×C22⋊D1711yescharacteristic
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—416no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—174no—
609C2×C4—344no—
610C2×C4—342no—
611C2×C4—342no—
612C2×C4—344no—
613C2×C4—342no—
614C2×C4—342no—
615C2×C4—342no—
616C2×C4—342no—
617C2×C4—342no—
618C2×C4—342no—
619C2×C4—342no—
620C2×C4—342no—
621C2×C4—342no—
622C2×C4—6816no—
623C2×C4—6816no—
624C2×C4—682no—
625C2×C4—681no—
626C2×C4—681no—
627C2×C4—681no—
628C2×C4—681no—
629C2×C4—681no—
630C2×C4—681no—
631C2×C4—681no—
632C2×C4—681no—
633C2×C4—682no—
634C2×C4—682no—
635C2×C4—682no—
636C2×C4—682no—
637C2×C4—682no—
638C2×C4—682no—
639C2×C4—682no—
640C2×C4—682no—
641C2×C4—682no—
642C2×C4—682no—
643C2×C4—682no—
644C2×C4—682no—
645C2×C4—682no—
646C2×C4—682no—
647C2×C4—682no—
648C2×C4#1058756314yes—
649C22—28no—
650C22—24no—
651C22—24no—
652C22—24no—
653C22—24no—
654C22—24no—
655C22—24no—
656C22—28no—
657C22—28no—
658C22—28no—
659C22#1176270018yes—
660C22#117×D1711yescharacteristic
661C22C2×#1058756311yescharacteristic
662C22C2×#1058755811yescharacteristic
663C22C2×C4×C22⋊D1711yescharacteristic
664C22C2×C4×D6811yescharacteristic
665C22#1176328211yescharacteristic
666C22#1176328611yescharacteristic
667C22C2×#1058761611yescharacteristic
668C22C2×#1058761511yescharacteristic
669C22C2×#1058756111yescharacteristic
670C22C2×#1058756211yescharacteristic
671C22#1176333912yes—
672C22#1176333912yes—
673C22#1176340112yes—
674C22#1176340112yes—
675C22#1176345312yes—
676C22#1176345412yes—
677C22#1176345812yes—
678C22#1176346312yes—
679C22#1176348512yes—
680C22#1176348812yes—
681C22#1176349212yes—
682C22#1176349912yes—
683C4—24no—
684C4—42no—
685C4—44no—
686C4—44no—
687C4—344no—
688C4—682no—
689C4—682no—
690C4—682no—
691C4—1362no—
692C4—1362no—
693C2—28no—
694C2#1453117811yescharacteristic
695C2C2×#1176270011yescharacteristic
696C2#1453138411yescharacteristic
697C2#1453162911yescharacteristic
698C2#1453163611yescharacteristic
699C2#1453165111yescharacteristic
700C2#1453165411yescharacteristic
701C2#1453225512yes—
702C2#1453232912yes—
703C2#1453310412yes—
704C2#1453316312yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC2×C4×C22⋊C4×C17C2
socle, cosocleC24×C17C24
prime radicalC25×C17C23
hypercenterC2×C4×C22⋊C4D17
hypocenterC17#13874
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
5. #14527044C2semidirect
6. #14527039C2semidirect
7. #14527017C2semidirect
8. #14527011C2semidirect
20. C2×#10587502C4semidirect
452. C17#13874semidirect

Maximal subgroups

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

subgroupindex
C2×#117626702
C2×#117626672
C22×#105875022
#145270442
#145270392
#145270172
#145270112
#145270072
#145270062
C2×C4×C22⋊C4×C172
#1387417