Finite Group Databasebeta

← #16942156

Subgroups — #16942156

Order 4352.

866 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×#11762808C211yesmaximal, characteristic
2C2×#11762808C211yesmaximal, characteristic
3C2×#11762775C211yesmaximal, characteristic
4C2×#11762578C211yesmaximal, characteristic, complemented
5C2×#11762578C211yesmaximal, characteristic, complemented
6C2×#11762574C211yesmaximal, characteristic, complemented
7C2×C17×#340C211yesmaximal, characteristic, complemented
8C22×#63252C2211yescharacteristic
9C22×#63223C2211yescharacteristic
10C22×#63223C2211yescharacteristic
11C22×#63223C2211yescharacteristic
12#11762808—22no—
13#11762808—22no—
14#11762808—22no—
15#11762808C414yescomplemented
16C2×#10587481—21no—
17C2×#10587481—21no—
18C2×#10587480—21no—
19C2×#10587480—21no—
20#11762775—22no—
21#11762775—22no—
22C2×#10587491—22no—
23#11762578—22no—
24#11762578—22no—
25#11762574—22no—
26C23×C8×C17C2211yescharacteristic
27C22×C4⋊C4×C17C2211yescharacteristic
28C22×C4⋊C4×C17C2211yescharacteristic
29C17×#340—22no—
30C23×C4×C17—21no—
31C23×C4×C17—21no—
32C23×C4×C17C2311yescharacteristic
33C22×C22⋊D17—21no—
34C2×#63252—21no—
35C2×#63252—21no—
36C2×#63252—21no—
37C2×#63252—21no—
38C2×#63252—21no—
39C2×#63252—21no—
40C2×#63252—21no—
41C2×#63252—21no—
42C2×#63252—21no—
43C2×#63252—21no—
44C2×#63252C2×C412yes—
45C2×#63252D412yes—
46C22×D68D411yescharacteristic
47C22×C4×D17—21no—
48C22×Dic34D411yescharacteristic
49C22×C8×C17—21no—
50C22×C8×C17—21no—
51C22×C8×C17—21no—
52C22×C8×C17—21no—
53C22×C8×C17—21no—
54C22×C8×C17—21no—
55C22×C8×C17D412yes—
56C2×C4⋊C4×C17—21no—
57C2×C4⋊C4×C17—21no—
58C2×C4⋊C4×C17—21no—
59C2×C4⋊C4×C17—21no—
60C2×C4⋊C4×C17—21no—
61C2×C4⋊C4×C17—21no—
62C2×C4⋊C4×C17—21no—
63C2×C4⋊C4×C17—21no—
64C2×C4⋊C4×C17—21no—
65C2×C4⋊C4×C17—21no—
66C2×C4⋊C4×C17C2×C412yes—
67C2×C4⋊C4×C17D412yes—
68C2×#63223—21no—
69C2×#63223—21no—
70C2×#63223—21no—
71C2×#63223—21no—
72C2×#63223—21no—
73C2×#63223—21no—
74C2×#63223—21no—
75C2×#63223—21no—
76C2×#63223—21no—
77C2×#63223—21no—
78C2×#63223—21no—
79C2×#63223—21no—
80C2×#63223—21no—
81C2×#63223—21no—
82C2×#63223—21no—
83C2×#63223C2×C412yes—
84C2×#63223D412yes—
85C2×#63223D412yes—
86C22×#3622—21no—
87C22×#3622—21no—
88C22×#3622—21no—
89#10587491—28no—
90#10587481—24no—
91#10587481—24no—
92#10587480—24no—
93#10587480—24no—
94C24×C17#4111yescharacteristic
95C23×D17—21no—
96C22×C4×C17—22no—
97C22×C4×C17—22no—
98C22×C4×C17—41no—
99C22×C4×C17—41no—
100C22×C4×C17—41no—
101C22×C4×C17—41no—
102C22×C4×C17—81no—
103C22×C4×C17—81no—
104C22×C4×C17—81no—
105C22×C4×C17—81no—
106C22×C4×C17C22×C411yescharacteristic
107C22×C4×C17C2×D411yescharacteristic
108C22×C4×C17C2×D411yescharacteristic
109C22×C4×C17C2×D411yescharacteristic
110C22×C4×C17C2×D411yescharacteristic
111C22×C4×C17C2×D411yescharacteristic
112C22×C4×C17C2×D411yescharacteristic
113C22×C4×C17#4111yescharacteristic
114C22×C4×C17#4111yescharacteristic
115C22×C4×C17#4111yescharacteristic
116C22×C4×C17#4111yescharacteristic
117C22×C4×C17#4111yescharacteristic
118C22×C4×C17#4111yescharacteristic
119C22×C4×C17#4111yescharacteristic
120C2×C22⋊D17—22no—
121C2×C22⋊D17—22no—
122C2×C22⋊D17—41no—
123C2×C22⋊D17—41no—
124C2×C22⋊D17—41no—
125C2×C22⋊D17—41no—
126C22×Dic17—21no—
127#63252—22no—
128#63252—22no—
129#63252—22no—
130#63252—22no—
131#63252—22no—
132#63252—22no—
133#63252—22no—
134#63252—22no—
135#63252—41no—
136#63252—41no—
137#63252—41no—
138#63252—41no—
139#63252—41no—
140#63252—41no—
141#63252—41no—
142#63252—41no—
143C2×D68—21no—
144C2×D68—21no—
145C2×D68—21no—
146C2×D68—21no—
147C2×D68—21no—
148C2×D68C22⋊C412yes—
149C2×C4×D17—22no—
150C2×C4×D17—22no—
151C2×C4×D17—41no—
152C2×C4×D17—41no—
153C2×C4×D17—41no—
154C2×C4×D17—41no—
155C2×Dic34—21no—
156C2×Dic34—21no—
157C2×Dic34—21no—
158C2×Dic34—21no—
159C2×Dic34—21no—
160C2×Dic34C22⋊C412yes—
161C2×C8×C17—22no—
162C2×C8×C17—22no—
163C2×C8×C17—22no—
164C2×C8×C17—41no—
165C2×C8×C17—41no—
166C2×C8×C17—41no—
167C2×C8×C17—41no—
168C4⋊C4×C17—22no—
169C4⋊C4×C17—22no—
170C4⋊C4×C17—22no—
171C4⋊C4×C17—22no—
172C4⋊C4×C17—41no—
173C4⋊C4×C17—41no—
174C4⋊C4×C17—41no—
175C4⋊C4×C17—41no—
176#63223—22no—
177#63223—22no—
178#63223—22no—
179#63223—22no—
180#63223—22no—
181#63223—22no—
182#63223—41no—
183#63223—41no—
184#63223—41no—
185#63223—41no—
186#63223—41no—
187#63223—41no—
188C2×#3622—22no—
189C2×#3622—22no—
190C2×#3622—22no—
191C2×#3622—41no—
192C2×#3622—41no—
193C2×#3622—41no—
194C2×#3622—41no—
195C2×#3622—41no—
196C2×#3622—41no—
197#11192—171nomaximal, Sylow
198C23×C17—21no—
199C23×C17—21no—
200C23×C17—21no—
201C23×C17—21no—
202C23×C17C4×D411yescharacteristic
203C23×C17#12111yescharacteristic
204C23×C17#12311yescharacteristic
205C23×C17#12311yescharacteristic
206C23×C17#12411yescharacteristic
207C23×C17#12611yescharacteristic
208C23×C17#12611yescharacteristic
209C22×D17—22no—
210C22×D17—41no—
211C22×D17—41no—
212C22×D17—41no—
213C22×D17—41no—
214C22×D17—41no—
215C22×D17—41no—
216C2×C4×C17—42no—
217C2×C4×C17—42no—
218C2×C4×C17—82no—
219C2×C4×C17—82no—
220C2×C4×C17—81no—
221C2×C4×C17—81no—
222C2×C4×C17—81no—
223C2×C4×C17—81no—
224C2×C4×C17—82no—
225C2×C4×C17—82no—
226C2×C4×C17C2×C22⋊C411yescharacteristic
227C2×C4×C17#11711yescharacteristic
228C2×C4×C17#11711yescharacteristic
229C2×C4×C17C4×D411yescharacteristic
230C2×C4×C17C4×D411yescharacteristic
231C2×C4×C17C4×D411yescharacteristic
232C2×C4×C17#12011yescharacteristic
233C2×C4×C17#12011yescharacteristic
234C2×C4×C17#12111yescharacteristic
235C2×C4×C17#12111yescharacteristic
236C2×C4×C17#12111yescharacteristic
237C2×C4×C17#12111yescharacteristic
238C2×C4×C17#12111yescharacteristic
239C2×C4×C17#12111yescharacteristic
240C2×C4×C17#12111yescharacteristic
241C2×C4×C17#12311yescharacteristic
242C2×C4×C17#12311yescharacteristic
243C2×C4×C17#12311yescharacteristic
244C2×C4×C17#12311yescharacteristic
245C2×C4×C17#12311yescharacteristic
246C2×C4×C17#12311yescharacteristic
247C2×C4×C17#12311yescharacteristic
248C2×C4×C17#12311yescharacteristic
249C2×C4×C17#12411yescharacteristic
250C2×C4×C17#12411yescharacteristic
251C2×C4×C17#12411yescharacteristic
252C2×C4×C17#12611yescharacteristic
253C2×C4×C17#12611yescharacteristic
254C22⋊D17—42no—
255C22⋊D17—42no—
256C22⋊D17—42no—
257C22⋊D17—42no—
258C2×Dic17—22no—
259C2×Dic17—41no—
260C2×Dic17—41no—
261D68—22no—
262D68—22no—
263D68—41no—
264D68—41no—
265C4×D17—42no—
266C4×D17—42no—
267C4×D17—42no—
268C4×D17—42no—
269Dic34—22no—
270Dic34—22no—
271Dic34—41no—
272Dic34—41no—
273C8×C17—42no—
274#3622—42no—
275#3622—42no—
276#3622—42no—
277C2×#417—171no—
278C2×#417—171no—
279C2×#408—171no—
280C2×#343—171no—
281C2×#343—171no—
282C2×#341—171no—
283C2×#340D1711yescharacteristic, complemented
284C22×C17—21no—
285C22×C17—21no—
286C22×C17—21no—
287C22×C17—21no—
288C22×C17—21no—
289C22×C17—21no—
290C22×C17—21no—
291C22×C17—21no—
292C22×C17—21no—
293C22×C17—21no—
294C22×C17—21no—
295C22×C17—21no—
296C22×C17#38611yescharacteristic
297C22×C17#38811yescharacteristic
298C22×C17#38811yescharacteristic
299C22×C17#39411yescharacteristic
300C22×C17#39711yescharacteristic
301C22×C17#39711yescharacteristic
302C22×C17#39711yescharacteristic
303C22×C17#47111yescharacteristic
304C22×C17#47111yescharacteristic
305C22×C17#47211yescharacteristic
306C22×C17#47311yescharacteristic
307C2×D17—42no—
308C2×D17—42no—
309C2×D17—42no—
310C2×D17—81no—
311C2×D17—81no—
312C2×D17—81no—
313C2×D17—81no—
314C4×C17—82no—
315C4×C17—82no—
316C4×C17#38111yescharacteristic
317C4×C17#38611yescharacteristic
318C4×C17#38811yescharacteristic
319C4×C17#38811yescharacteristic
320C4×C17#39411yescharacteristic
321C4×C17#39411yescharacteristic
322C4×C17#39411yescharacteristic
323C4×C17#39711yescharacteristic
324Dic17—42no—
325C22×#41—171no—
326C22×M16—171no—
327C22×M16—171no—
328C22×M16—171no—
329C23×C8C2×D1711yescharacteristic
330C22×C4⋊C4C2×D1711yescharacteristic
331C22×C4⋊C4C2×D1711yescharacteristic
332#417—174no—
333#417—342no—
334#417—342no—
335#417—342no—
336C2×#103—341no—
337C2×#103—341no—
338C2×#102—341no—
339C2×#102—341no—
340#408—342no—
341#408—342no—
342C2×C22⋊C8—342no—
343#343—342no—
344#343—342no—
345#341—342no—
346#340—22no—
347C2×C17—21no—
348C2×C17—21no—
349C2×C17—21no—
350C2×C17—21no—
351C2×C17#144011yescharacteristic
352C2×C17#190711yescharacteristic
353C2×C17#190811yescharacteristic
354C2×C17#205111yescharacteristic
355C2×C17#205211yescharacteristic
356C2×C17#207811yescharacteristic
357C2×C17#207911yescharacteristic
358D17—82no—
359C2×#41—172no—
360C2×#41—172no—
361C2×#41—341no—
362C2×#41—341no—
363C2×#41—341no—
364C2×#41—341no—
365C2×#41—341no—
366C2×#41—341no—
367C2×#41—341no—
368C2×#41—341no—
369C2×#41—341no—
370C2×#41—341no—
371C22×Q8—171no—
372C22×D4—171no—
373C22×D4—341no—
374C23×C4—21no—
375C23×C4—21no—
376C23×C4—341no—
377C23×C4C22×D1711yescharacteristic
378C2×M16—172no—
379C2×M16—172no—
380C2×M16—172no—
381C2×M16—341no—
382C2×M16—341no—
383C2×M16—341no—
384C2×M16—341no—
385C2×M16—341no—
386C2×M16—341no—
387C2×M16—341no—
388C2×M16—341no—
389C2×M16—341no—
390C2×M16—341no—
391C2×M16—341no—
392C2×M16—341no—
393C2×M16—341no—
394C2×M16—341no—
395C2×M16—341no—
396C22×C8—21no—
397C22×C8—21no—
398C22×C8—21no—
399C22×C8—21no—
400C22×C8—21no—
401C22×C8—21no—
402C22×C8—341no—
403C22×C8—341no—
404C22×C8—341no—
405C22×C8C22⋊D1712yes—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4C4×D1712yes—
417C2×C4⋊C4D6812yes—
418#103—344no—
419#103—344no—
420#102—344no—
421#102—344no—
422C22⋊C8—348no—
423C17#1119211yescharacteristic, complemented, Sylow
424C24—341no—
425C24#6325411yescharacteristic
426#41—342no—
427#41—342no—
428#41—342no—
429#41—342no—
430#41—342no—
431#41—342no—
432#41—342no—
433#41—342no—
434#41—681no—
435#41—681no—
436#41—681no—
437#41—681no—
438#41—681no—
439#41—681no—
440#41—681no—
441#41—681no—
442C2×Q8—172no—
443C2×Q8—341no—
444C2×Q8—341no—
445C2×Q8—341no—
446C2×Q8—341no—
447C2×Q8—341no—
448C2×D4—172no—
449C2×D4—342no—
450C2×D4—342no—
451C2×D4—341no—
452C2×D4—341no—
453C2×D4—341no—
454C2×D4—341no—
455C2×D4—341no—
456C2×D4—681no—
457C2×D4—681no—
458C2×D4—681no—
459C2×D4—681no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—81no—
467C22×C4—81no—
468C22×C4—81no—
469C22×C4—81no—
470C22×C4—341no—
471C22×C4—342no—
472C22×C4—342no—
473C22×C4—681no—
474C22×C4—681no—
475C22×C4—681no—
476C22×C4—681no—
477C22×C4C2×C4×D1711yescharacteristic
478C22×C4C2×D6811yescharacteristic
479C22×C4#6325211yescharacteristic
480C22×C4#6325211yescharacteristic
481C22×C4#6325211yescharacteristic
482C22×C4#6325211yescharacteristic
483C22×C4D4×D1711yescharacteristic
484C22×C4D4×D1711yescharacteristic
485C22×C4D4×D1711yescharacteristic
486C22×C4D4×D1711yescharacteristic
487C22×C4#6325411yescharacteristic
488C22×C4#6325611yescharacteristic
489C22×C4#6325611yescharacteristic
490C22×C4C2×C22⋊D1711yescharacteristic
491M16—342no—
492M16—342no—
493M16—342no—
494M16—342no—
495M16—342no—
496M16—342no—
497M16—681no—
498M16—681no—
499M16—681no—
500M16—681no—
501M16—681no—
502M16—681no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—41no—
507C2×C8—41no—
508C2×C8—41no—
509C2×C8—41no—
510C2×C8—342no—
511C2×C8—342no—
512C2×C8—342no—
513C2×C8—681no—
514C2×C8—681no—
515C2×C8—681no—
516C2×C8—681no—
517C2×C8—681no—
518C2×C8—681no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—41no—
524C4⋊C4—41no—
525C4⋊C4—41no—
526C4⋊C4—41no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—342no—
532C23—681no—
533C23—681no—
534C23—681no—
535C23—681no—
536C23—681no—
537C23—681no—
538C23#1058756711yescharacteristic
539C23#1058756911yescharacteristic
540C23#1058757111yescharacteristic
541C23#1058757211yescharacteristic
542C23#1058758411yescharacteristic
543C23#1058758411yescharacteristic
544C23#1058762411yescharacteristic
545Q8—342no—
546Q8—342no—
547Q8—681no—
548Q8—681no—
549D4—342no—
550D4—342no—
551D4—682no—
552D4—682no—
553D4—682no—
554D4—682no—
555D4—681no—
556D4—681no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—342no—
568C2×C4—681no—
569C2×C4—681no—
570C2×C4—682no—
571C2×C4—682no—
572C2×C4—682no—
573C2×C4—682no—
574C2×C4#1058755811yescharacteristic
575C2×C4C4×D6811yescharacteristic
576C2×C4#1058756111yescharacteristic
577C2×C4#1058756211yescharacteristic
578C2×C4C22⋊C4×D1711yescharacteristic
579C2×C4#1058756811yescharacteristic
580C2×C4#1058756911yescharacteristic
581C2×C4#1058756911yescharacteristic
582C2×C4#1058757011yescharacteristic
583C2×C4#1058757011yescharacteristic
584C2×C4#1058757011yescharacteristic
585C2×C4#1058757111yescharacteristic
586C2×C4#1058757811yescharacteristic
587C2×C4#1058757911yescharacteristic
588C2×C4#1058758011yescharacteristic
589C2×C4#1058758011yescharacteristic
590C2×C4#1058758011yescharacteristic
591C2×C4#1058758011yescharacteristic
592C2×C4#1058758011yescharacteristic
593C2×C4#1058758111yescharacteristic
594C2×C4#1058758111yescharacteristic
595C2×C4#1058758411yescharacteristic
596C2×C4C4×C22⋊D1711yescharacteristic
597C2×C4#1058761511yescharacteristic
598C2×C4#1058761611yescharacteristic
599C2×C4#1058762311yescharacteristic
600C2×C4#1058762511yescharacteristic
601C2×C4#1058763311yescharacteristic
602C8—42no—
603C8—682no—
604C8—682no—
605C8—682no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—682no—
619C22—682no—
620C22—682no—
621C22—1361no—
622C22—1361no—
623C22—1361no—
624C22—1361no—
625C22#1176268511yescharacteristic
626C22#1176269811yescharacteristic
627C22#1176270111yescharacteristic
628C22#1176272511yescharacteristic
629C22#1176273511yescharacteristic
630C22#1176273611yescharacteristic
631C22#1176273811yescharacteristic
632C22#1176306911yescharacteristic
633C22#1176306911yescharacteristic
634C22#1176307611yescharacteristic
635C22#1176308111yescharacteristic
636C4—82no—
637C4—82no—
638C4—682no—
639C4#1176267111yescharacteristic
640C4#1176268711yescharacteristic
641C4#1176269911yescharacteristic
642C4#1176270011yescharacteristic
643C4#1176272411yescharacteristic
644C4#1176272611yescharacteristic
645C4#1176272711yescharacteristic
646C4#1176273811yescharacteristic
647C2—21no—
648C2—21no—
649C2—21no—
650C2—21no—
651C2—1362no—
652C2#1452703211yescharacteristic
653C2#1452929511yescharacteristic
654C2#1452930211yescharacteristic
655C2#1453007711yescharacteristic
656C2#1453008411yescharacteristic
657C2#1453022511yescharacteristic
658C2#1453023011yescharacteristic
subgroupG/H
centerC23#10587624
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC2×C17×#340C2
socleC23×C17#121
prime radicalC22×#63252C22
cosocleC23×C4×C17C23
hypercenterC2×#340D17
hypocenterC17#11192
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
4. C2×#11762578C2semidirect
5. C2×#11762578C2semidirect
6. C2×#11762574C2semidirect
7. C2×C17×#340C2semidirect
15. #11762808C4semidirect
283. C2×#340D17semidirect
423. C17#11192semidirect

Maximal subgroups

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

subgroupindex
C2×#117628082
C2×#117627752
C2×#117625782
C2×#117625742
C2×C17×#3402
#1119217