Finite Group Databasebeta

← #17107560

Subgroups — #17107560

Order 4352.

734 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14533381C211yesmaximal, characteristic
2#14530332C211yesmaximal, characteristic, complemented
3#14530099C211yesmaximal, characteristic
4#14530027C211yesmaximal, characteristic, complemented
5#14529867C211yesmaximal, characteristic
6#14529641C211yesmaximal, characteristic
7#14529422C211yesmaximal, characteristic, complemented
8#14529404C211yesmaximal, characteristic
9#14529369C211yesmaximal, characteristic, complemented
10#14529343C211yesmaximal, characteristic, complemented
11#14529139C211yesmaximal, characteristic
12#14529139C211yesmaximal, characteristic
13#14528799C211yesmaximal, characteristic, complemented
14C4×#10587506C211yesmaximal, characteristic, complemented
15C17×#2365C211yesmaximal, characteristic, complemented
16C22×#63230C2211yescharacteristic
17C22×#63230C2211yescharacteristic
18C2×#10587478C2211yescharacteristic
19C2×#10587479C2211yescharacteristic
20C2×C4⋊#3622C2211yescharacteristic
21C2×#10587506C2211yescharacteristic
22C2×#10587506C2211yescharacteristic
23C2×#10587506C2211yescharacteristic
24C2×#10587506C2211yescharacteristic
25C2×#10587506C2211yescharacteristic
26C2×#10587506C2211yescharacteristic
27C2×#10587480C2211yescharacteristic
28C2×#10587480C2211yescharacteristic
29C2×C4×#3622C2211yescharacteristic
30#11762738—21no—
31#11762735—21no—
32#11762711C2211yescharacteristic
33#11762711C2211yescharacteristic
34#11762711C2211yescharacteristic
35#11762707C2211yescharacteristic
36#11762699—21no—
37#11762698—21no—
38#11762675C2211yescharacteristic
39C4×C4⋊Dic17C2211yescharacteristic
40#11762571C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762564C2211yescharacteristic
45#11762564C2211yescharacteristic
46#11762564C2211yescharacteristic
47#11762564C2211yescharacteristic
48C2×C17×C4⋊Q8C2211yescharacteristic
49C2×C4×Q8×C17C2211yescharacteristic
50C2×C4×Q8×C17C2211yescharacteristic
51C17×#391C2211yescharacteristic
52C17×#391C2211yescharacteristic
53C17×#389—21no—
54C17×#387—21no—
55C17×#387—21no—
56C4×C4⋊C4×C17—21no—
57C4×C4⋊C4×C17C2211yescharacteristic
58C4×C4⋊C4×C17C2211yescharacteristic
59C22×Q8×C17C2311yescharacteristic
60C22×Q8×C17C2311yescharacteristic
61C22×D68C2311yescharacteristic
62C17×C4⋊Q8—22no—
63C17×C4⋊Q8—22no—
64C4×Q8×C17—22no—
65C4×Q8×C17—22no—
66C4×Q8×C17—41no—
67C4×Q8×C17—41no—
68C2×C4⋊C4×C17—21no—
69C2×C4⋊C4×C17—21no—
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17D411yescharacteristic
73C2×C4⋊C4×C17D411yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C42×C17—21no—
77C2×C42×C17—21no—
78C2×C42×C17—21no—
79C2×C42×C17C2311yescharacteristic
80C2×C42×C17C2311yescharacteristic
81C2×C42×C17C2311yescharacteristic
82C2×#63230—22no—
83C2×#63230—22no—
84C2×#63230—22no—
85C2×#63230—41no—
86C2×#63230—41no—
87C2×#63230—41no—
88C2×#63227—21no—
89C2×#63227—21no—
90C2×#63227—21no—
91C2×#63227—21no—
92C2×#63227—21no—
93C2×#63227—21no—
94C2×C4⋊Dic17C2311yescharacteristic
95C2×C4⋊Dic17C2311yescharacteristic
96C2×C4⋊Dic17C2311yescharacteristic
97C2×#63225—21no—
98C2×C4×Dic17—21no—
99C22×#3622C2311yescharacteristic
100C22×#3622C2311yescharacteristic
101C22×#3622C2311yescharacteristic
102C22×#3622C2311yescharacteristic
103C17×#95—21no—
104C17×#95—21no—
105C17×#95—21no—
106C17×#95—21no—
107C17×#95—21no—
108C17×#95D411yescharacteristic
109C17×#95D411yescharacteristic
110#10587506—22no—
111#10587506—22no—
112#10587506—22no—
113#10587506—41no—
114#10587506—41no—
115#10587506—41no—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587502—21no—
120#10587480—22no—
121#10587480—41no—
122#10587479—22no—
123#10587478—41no—
124C4⋊#3622—41no—
125C4×#3622—41no—
126C23×D17—21no—
127C2×Q8×C17—21no—
128C2×Q8×C17—21no—
129C2×Q8×C17—21no—
130C2×Q8×C17—21no—
131C2×Q8×C17—21no—
132C2×Q8×C17—21no—
133C2×Q8×C17—21no—
134C2×Q8×C17—21no—
135C2×Q8×C17—21no—
136C2×Q8×C17—21no—
137C2×Q8×C17#4112yes—
138C2×Q8×C17#4112yes—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17—21no—
142C22×C4×C17—21no—
143C22×C4×C17C2×D411yescharacteristic
144C22×C4×C17C2×D411yescharacteristic
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2411yescharacteristic
150C22×Dic17—21no—
151C22×Dic17—21no—
152C22×Dic17—21no—
153C2×D68—21no—
154C2×D68—21no—
155C2×D68—21no—
156C2×D68—21no—
157C2×D68—21no—
158C2×D68—21no—
159C4⋊C4×C17—24no—
160C4⋊C4×C17—24no—
161C4⋊C4×C17—22no—
162C4⋊C4×C17—22no—
163C4⋊C4×C17—21no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17—42no—
167C4⋊C4×C17—42no—
168C4⋊C4×C17#4112yes—
169C42×C17—21no—
170C42×C17—21no—
171C42×C17—21no—
172C42×C17—21no—
173C42×C17—21no—
174C42×C17—21no—
175C42×C17—42no—
176C42×C17—42no—
177C42×C17—42no—
178#63230—44no—
179#63230—44no—
180#63227—42no—
181#63227—42no—
182#63227—81no—
183#63227—81no—
184#63227—81no—
185#63227—81no—
186C4⋊Dic17—21no—
187C4⋊Dic17—21no—
188C4⋊Dic17—21no—
189C4⋊Dic17—21no—
190C4⋊Dic17—21no—
191C4⋊Dic17—21no—
192#63225—81no—
193C4×Dic17—81no—
194C2×#3622—21no—
195C2×#3622—21no—
196C2×#3622—21no—
197C2×#3622—21no—
198C2×#3622—21no—
199C2×#3622—21no—
200C2×#3622—21no—
201C2×#3622—21no—
202C2×#3622—21no—
203C2×#3622—21no—
204C2×#3622—21no—
205C2×#3622#4112yes—
206#27069—171nomaximal, Sylow
207C23×C17C22×D411yescharacteristic
208C22×D17—41no—
209C22×D17—41no—
210C22×D17—41no—
211C22×D17—41no—
212C22×D17—41no—
213C22×D17—41no—
214C22×D17—41no—
215Q8×C17—22no—
216Q8×C17—22no—
217Q8×C17—22no—
218Q8×C17—22no—
219Q8×C17—41no—
220Q8×C17—41no—
221Q8×C17—41no—
222Q8×C17—41no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—21no—
226C2×C4×C17—21no—
227C2×C4×C17—21no—
228C2×C4×C17—22no—
229C2×C4×C17—22no—
230C2×C4×C17—21no—
231C2×C4×C17—21no—
232C2×C4×C17—21no—
233C2×C4×C17—22no—
234C2×C4×C17—21no—
235C2×C4×C17—21no—
236C2×C4×C17—21no—
237C2×C4×C17—21no—
238C2×C4×C17—21no—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—21no—
242C2×C4×C17—41no—
243C2×C4×C17—41no—
244C2×C4×C17—41no—
245C2×C4×C17—41no—
246C2×C4×C17—41no—
247C2×C4×C17—41no—
248C2×C4×C17—41no—
249C2×C4×C17—41no—
250C2×C4×C17—41no—
251C2×C4×C17#13512yes—
252C2×C4×C17#13712yes—
253C2×C4×C17C2×#4111yescharacteristic
254C2×C4×C17C2×#4111yescharacteristic
255C2×C4×C17C2×#4111yescharacteristic
256C2×C4×C17C2×#4111yescharacteristic
257C2×C4×C17H(5,𝔽2)11yescharacteristic
258C2×C4×C17H(5,𝔽2)11yescharacteristic
259C2×Dic17—41no—
260C2×Dic17—41no—
261C2×Dic17—41no—
262C2×Dic17—41no—
263C2×Dic17—41no—
264C2×Dic17—41no—
265C2×Dic17—41no—
266C2×Dic17—41no—
267C2×Dic17—41no—
268D68—41no—
269D68—41no—
270D68—41no—
271D68—41no—
272#3622—22no—
273#3622—41no—
274#3622—41no—
275#3622—41no—
276#2513—171no—
277#2365D1711yescharacteristic, complemented
278#2097—171no—
279#2055—171no—
280#2043—171no—
281#2013—171no—
282#1968—171no—
283#1928—171no—
284#1925—171no—
285#1920—171no—
286#1916—171no—
287#1880—171no—
288#1880—171no—
289#1805—171no—
290C4×#103—171no—
291C22×C17#52511yescharacteristic
292C22×C17#54611yescharacteristic
293C22×C17#55011yescharacteristic
294C22×C17C2×#13511yescharacteristic
295C22×C17C2×#13711yescharacteristic
296C22×C17#57711yescharacteristic
297C22×C17#57711yescharacteristic
298C2×D17—81no—
299C2×D17—81no—
300C2×D17—81no—
301C2×D17—81no—
302C2×D17—81no—
303C2×D17—81no—
304C2×D17—81no—
305C4×C17—22no—
306C4×C17—22no—
307C4×C17—41no—
308C4×C17—42no—
309C4×C17—42no—
310C4×C17—41no—
311C4×C17—41no—
312C4×C17—42no—
313C4×C17—41no—
314C4×C17—81no—
315C4×C17#52911yescharacteristic
316C4×C17#53811yescharacteristic
317C4×C17#55111yescharacteristic
318C4×C17#55311yescharacteristic
319Dic17—81no—
320Dic17—81no—
321Dic17—81no—
322C22×SD16—171no—
323C22×SD16—171no—
324C2×C4⋊Q8C2×D1711yescharacteristic
325C2×C4×Q8C2×D1711yescharacteristic
326C2×C4×Q8C2×D1711yescharacteristic
327C2×#107—171no—
328C2×#106—171no—
329C2×C4⋊C8—171no—
330C2×#103—171no—
331C2×#103—171no—
332C2×#103—171no—
333C2×#103—171no—
334C2×#103—171no—
335C2×#103—171no—
336C2×#102—171no—
337C2×#102—171no—
338C2×C4×C8—171no—
339#397—341no—
340#397—341no—
341#391C2×D1711yescharacteristic
342#391C2×D1711yescharacteristic
343#390—171no—
344#390—171no—
345#390—171no—
346#389—21no—
347#389—171no—
348#388—341no—
349#388—341no—
350#387—21no—
351#387—21no—
352#382—171no—
353C4×C4⋊C4—21no—
354C4×C4⋊C4—171no—
355C4×C4⋊C4C2×D1711yescharacteristic
356C4×C4⋊C4C2×D1711yescharacteristic
357#340—171no—
358#340—171no—
359#340—171no—
360#340—171no—
361#336—171no—
362#336—171no—
363#336—171no—
364#336—171no—
365C2×C17#246711yescharacteristic
366C2×C17#310811yescharacteristic
367C2×C17#312611yescharacteristic
368C2×C17#330311yescharacteristic
369C2×C17#332011yescharacteristic
370C2×C17#339411yescharacteristic
371C2×C17#340011yescharacteristic
372D17—161no—
373C22×Q8C22×D1711yescharacteristic
374C22×Q8C22×D1711yescharacteristic
375C22×D4—171no—
376C2×SD16—342no—
377C2×SD16—342no—
378C2×SD16—342no—
379C2×SD16—681no—
380C2×SD16—681no—
381C2×SD16—681no—
382C22×C8—171no—
383C22×C8—171no—
384C22×C8—171no—
385C22×C8—171no—
386C4⋊Q8—22no—
387C4⋊Q8—22no—
388C4×Q8—22no—
389C4×Q8—22no—
390C4×Q8—41no—
391C4×Q8—41no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—171no—
397C2×C4⋊C4—171no—
398C2×C4⋊C4—171no—
399C2×C4⋊C4—341no—
400C2×C4⋊C4C22⋊D1711yescharacteristic
401C2×C4⋊C4C22⋊D1711yescharacteristic
402C2×C4⋊C4C22×D1711yescharacteristic
403C2×C4⋊C4C22×D1711yescharacteristic
404C2×C22⋊C4—341no—
405C2×C22⋊C4—341no—
406C2×C22⋊C4—341no—
407C2×C22⋊C4—341no—
408C2×C22⋊C4—341no—
409C2×C22⋊C4—341no—
410C2×C42—21no—
411C2×C42—21no—
412C2×C42—21no—
413C2×C42—341no—
414C2×C42C22×D1711yescharacteristic
415C2×C42C22×D1711yescharacteristic
416C2×C42C22×D1711yescharacteristic
417#107—681no—
418#106—342no—
419C4⋊C8—681no—
420#103—342no—
421#103—342no—
422#103—342no—
423#103—681no—
424#103—681no—
425#103—681no—
426#102—342no—
427#102—681no—
428C4×C8—681no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—341no—
435#95—341no—
436#95—341no—
437#95—341no—
438#95C22⋊D1711yescharacteristic
439#95C22⋊D1711yescharacteristic
440C17#2706911yescharacteristic, complemented, Sylow
441C24—341no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8#6325212yes—
453C2×Q8#6325212yes—
454C2×D4—341no—
455C2×D4—341no—
456C2×D4—341no—
457C2×D4—341no—
458C2×D4—341no—
459C2×D4—341no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—341no—
465C22×C4—341no—
466C22×C4—341no—
467C22×C4C2×C22⋊D1711yescharacteristic
468C22×C4C2×C22⋊D1711yescharacteristic
469C22×C4C2×C22⋊D1711yescharacteristic
470C22×C4C2×C22⋊D1711yescharacteristic
471C22×C4C2×C22⋊D1711yescharacteristic
472C22×C4C2×C22⋊D1711yescharacteristic
473C22×C4C23×D1711yescharacteristic
474SD16—684no—
475SD16—684no—
476C2×C8—172no—
477C2×C8—341no—
478C2×C8—341no—
479C2×C8—341no—
480C2×C8—341no—
481C2×C8—341no—
482C2×C8—341no—
483C2×C8—341no—
484C2×C8—341no—
485C2×C8—341no—
486C2×C8—341no—
487C2×C8—341no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—341no—
498C4⋊C4—341no—
499C4⋊C4—341no—
500C4⋊C4—341no—
501C4⋊C4—341no—
502C4⋊C4—341no—
503C4⋊C4—1361no—
504C4⋊C4#6325412yes—
505C22⋊C4—682no—
506C22⋊C4—682no—
507C22⋊C4—1361no—
508C22⋊C4—1361no—
509C22⋊C4—1361no—
510C22⋊C4—1361no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—42no—
518C42—42no—
519C42—42no—
520C42—1361no—
521C23—681no—
522C23—681no—
523C23—681no—
524C23—681no—
525C23—681no—
526C23—681no—
527C23—681no—
528C23C22×C22⋊D1711yescharacteristic
529Q8—22no—
530Q8—22no—
531Q8—22no—
532Q8—22no—
533Q8—41no—
534Q8—41no—
535Q8—41no—
536Q8—41no—
537D4—681no—
538D4—681no—
539D4—681no—
540D4—681no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—681no—
570C2×C4—681no—
571C2×C4—681no—
572C2×C4—681no—
573C2×C4—681no—
574C2×C4—681no—
575C2×C4—681no—
576C2×C4—681no—
577C2×C4—681no—
578C2×C4#1058762812yes—
579C2×C4#1058763612yes—
580C2×C4C2×#6325211yescharacteristic
581C2×C4C2×#6325211yescharacteristic
582C2×C4C2×#6325411yescharacteristic
583C2×C4#1058769111yescharacteristic
584C2×C4#41×D1711yescharacteristic
585C2×C4#1058769611yescharacteristic
586C8—342no—
587C8—681no—
588C8—681no—
589C8—681no—
590C22—1361no—
591C22—1361no—
592C22—1361no—
593C22—1361no—
594C22—1361no—
595C22—1361no—
596C22—1361no—
597C22#1176332011yescharacteristic
598C22#1176345311yescharacteristic
599C22#1176347811yescharacteristic
600C22C2×#1058763611yescharacteristic
601C22C2×#1058762811yescharacteristic
602C22#1176361711yescharacteristic
603C22#1176361711yescharacteristic
604C4—22no—
605C4—22no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—41no—
610C4—42no—
611C4—42no—
612C4—42no—
613C4—81no—
614C4—1361no—
615C4—1361no—
616C4—1361no—
617C4#1176333911yescharacteristic
618C4#1176339211yescharacteristic
619C4#1176348811yescharacteristic
620C4#1176350511yescharacteristic
621C2—2721no—
622C2#1453282911yescharacteristic
623C2#1453886411yescharacteristic
624C2#1453907511yescharacteristic
625C2#1454071211yescharacteristic
626C2#1454095511yescharacteristic
627C2#1454178011yescharacteristic
628C2#1454183311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2365C2
socleC23×C17C22×D4
prime radicalC22×D68C23
hypercenter#2365D17
hypocenterC17#27069
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2365

Complements

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

NH ≅ G/Ntype
2. #14530332C2semidirect
4. #14530027C2semidirect
7. #14529422C2semidirect
9. #14529369C2semidirect
10. #14529343C2semidirect
13. #14528799C2semidirect
14. C4×#10587506C2semidirect
15. C17×#2365C2semidirect
277. #2365D17semidirect
440. C17#27069semidirect

Maximal subgroups

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

subgroupindex
#145333812
#145303322
#145300992
#145300272
#145298672
#145296412
#145294222
#145294042
#145293692
#145293432
#145291392
#145287992
C4×#105875062
C17×#23652
#2706917