Finite Group Databasebeta

← #17107020

Subgroups — #17107020

Order 4352.

784 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14533184C211yesmaximal, characteristic, complemented
2#14530197C211yesmaximal, characteristic, complemented
3#14530172C211yesmaximal, characteristic, complemented
4#14529970C211yesmaximal, characteristic
5#14529963C211yesmaximal, characteristic
6#14529552C211yesmaximal, characteristic, complemented
7#14529514C211yesmaximal, characteristic, complemented
8#14529392C211yesmaximal, characteristic, complemented
9#14529376C211yesmaximal, characteristic, complemented
10#14529359C211yesmaximal, characteristic, complemented
11#14529031C211yesmaximal, characteristic, complemented
12#14528972C211yesmaximal, characteristic, complemented
13#14528874C211yesmaximal, characteristic, complemented
14C4×#10587503C211yesmaximal, characteristic, complemented
15C17×#2361C211yesmaximal, characteristic
16C2×#10587478C2211yescharacteristic
17C2×#10587478C2211yescharacteristic
18C2×#10587478C2211yescharacteristic
19C2×#10587479C2211yescharacteristic
20C2×C4⋊#3622C2211yescharacteristic
21C2×#10587503C2211yescharacteristic
22C2×#10587503C2211yescharacteristic
23C2×#10587503C2211yescharacteristic
24C2×#10587503C2211yescharacteristic
25C2×C22⋊#3622C2211yescharacteristic
26C2×C22⋊#3622C2211yescharacteristic
27C2×C4×#3622C2211yescharacteristic
28C17⋊#401—21no—
29#11762737—21no—
30#11762707C2211yescharacteristic
31#11762707C2211yescharacteristic
32#11762703—21no—
33#11762691—21no—
34#11762675C2211yescharacteristic
35#11762669C2211yescharacteristic
36#11762669C2211yescharacteristic
37C4×C4⋊Dic17C2211yescharacteristic
38#11762571C2211yescharacteristic
39#11762571C2211yescharacteristic
40#11762571C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762564C2211yescharacteristic
47#11762564C2211yescharacteristic
48C2×C17×#121C2211yescharacteristic
49C2×C4×D4×C17—21no—
50C2×C4×D4×C17C2211yescharacteristic
51C22×C4⋊C4×C17C2211yescharacteristic
52C17×#390C2211yescharacteristic
53C17×#389C2211yescharacteristic
54C17×#388—21no—
55C17×#386—21no—
56C17×#385—21no—
57C4×C4⋊C4×C17C2211yescharacteristic
58C4×C22⋊C4×C17C2211yescharacteristic
59C22×D4×C17—21no—
60C22×D4×C17C2311yescharacteristic
61C23×C4×C17—21no—
62C23×C4×C17—21no—
63C23×C4×C17C2311yescharacteristic
64C17×#121—22no—
65C17×#121—22no—
66C4×D4×C17—22no—
67C4×D4×C17—41no—
68C4×D4×C17—44no—
69C2×C4⋊C4×C17—22no—
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17—41no—
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C22⋊C4×C17—21no—
77C2×C22⋊C4×C17—21no—
78C2×C22⋊C4×C17—21no—
79C2×C22⋊C4×C17—21no—
80C2×C22⋊C4×C17D411yescharacteristic
81C2×C22⋊C4×C17D411yescharacteristic
82C2×C42×C17—21no—
83C2×C42×C17C2311yescharacteristic
84C2×C42×C17C2311yescharacteristic
85C2×C22⋊Dic17—21no—
86C2×C22⋊Dic17—21no—
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4⋊Dic17C2311yescharacteristic
89C2×C4⋊Dic17C2311yescharacteristic
90C2×C4⋊Dic17C2311yescharacteristic
91C2×#63225—21no—
92C2×#63225—21no—
93C2×C4×Dic17—21no—
94C22×#3622C2311yescharacteristic
95C22×#3622C2311yescharacteristic
96C22×#3622C2311yescharacteristic
97C22×#3622C2311yescharacteristic
98C17×#95—21no—
99C17×#95—21no—
100C17×#95D411yescharacteristic
101C17×#95D411yescharacteristic
102#10587503—22no—
103#10587503—22no—
104#10587503—22no—
105#10587503—41no—
106#10587502—21no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113C22⋊#3622—22no—
114C22⋊#3622—41no—
115#10587479—22no—
116#10587478—22no—
117#10587478—22no—
118#10587478—41no—
119C4⋊#3622—22no—
120C4×#3622—41no—
121C24×C17—21no—
122C24×C17C2×D411yescharacteristic
123C2×D4×C17—24no—
124C2×D4×C17—21no—
125C2×D4×C17—21no—
126C2×D4×C17—21no—
127C2×D4×C17—21no—
128C2×D4×C17—21no—
129C2×D4×C17—42no—
130C2×D4×C17—42no—
131C2×D4×C17#4112yes—
132C22×C4×C17—21no—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17—21no—
136C22×C4×C17—21no—
137C22×C4×C17—21no—
138C22×C4×C17—21no—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17—42no—
142C22×C4×C17—42no—
143C22×C4×C17—42no—
144C22×C4×C17—42no—
145C22×C4×C17—81no—
146C22×C4×C17—81no—
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2×D411yescharacteristic
151C22×C4×C17C2×D411yescharacteristic
152C22×C4×C17C2411yescharacteristic
153C22×Dic17—21no—
154C22×Dic17—21no—
155C22×Dic17—21no—
156C22×Dic17—21no—
157C4⋊C4×C17—21no—
158C4⋊C4×C17—21no—
159C4⋊C4×C17—21no—
160C4⋊C4×C17—21no—
161C4⋊C4×C17—42no—
162C4⋊C4×C17—42no—
163C4⋊C4×C17#4112yes—
164C4⋊C4×C17#4112yes—
165C22⋊C4×C17—24no—
166C22⋊C4×C17—22no—
167C22⋊C4×C17—22no—
168C22⋊C4×C17—42no—
169C22⋊C4×C17—42no—
170C22⋊C4×C17—81no—
171C42×C17—21no—
172C42×C17—21no—
173C42×C17—21no—
174C42×C17—21no—
175C42×C17—42no—
176C22⋊Dic17—42no—
177C22⋊Dic17—81no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184C4⋊Dic17—21no—
185C4⋊Dic17—21no—
186#63225—42no—
187#63225—81no—
188C4×Dic17—81no—
189C2×#3622—21no—
190C2×#3622—21no—
191C2×#3622—21no—
192C2×#3622—21no—
193C2×#3622—21no—
194C2×#3622—21no—
195C2×#3622—21no—
196C2×#3622—21no—
197C2×#3622—21no—
198C2×#3622—21no—
199C2×#3622—21no—
200C2×#3622#4112yes—
201#27033—171nomaximal, Sylow
202C23×C17—21no—
203C23×C17—21no—
204C23×C17—21no—
205C23×C17—21no—
206C23×C17—21no—
207C23×C17—21no—
208C23×C17—22no—
209C23×C17—41no—
210C23×C17—41no—
211C23×C17—41no—
212C23×C17—41no—
213C23×C17—41no—
214C23×C17—41no—
215C23×C17#13512yes—
216C23×C17C22×D411yescharacteristic
217D4×C17—22no—
218D4×C17—22no—
219D4×C17—48no—
220D4×C17—41no—
221D4×C17—41no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—22no—
226C2×C4×C17—21no—
227C2×C4×C17—22no—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17—21no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—21no—
235C2×C4×C17—21no—
236C2×C4×C17—21no—
237C2×C4×C17—21no—
238C2×C4×C17—41no—
239C2×C4×C17—41no—
240C2×C4×C17—41no—
241C2×C4×C17—41no—
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—84no—
250C2×C4×C17—84no—
251C2×C4×C17AGL(1,ℤ/8)12yes—
252C2×C4×C17C2×#4111yescharacteristic
253C2×C4×C17C2×#4111yescharacteristic
254C2×C4×C17C2×#4111yescharacteristic
255C2×C4×C17C2×#4111yescharacteristic
256C2×C4×C17H(5,𝔽2)11yescharacteristic
257C2×C4×C172−1+411yescharacteristic
258C2×Dic17—41no—
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—
268C2×Dic17—41no—
269C2×Dic17—41no—
270#3622—22no—
271#3622—41no—
272#3622—41no—
273#3622—41no—
274#2494—171no—
275#2361D1711yescharacteristic
276#2073—171no—
277#2067—171no—
278#2034—171no—
279#2033—171no—
280#1949—171no—
281#1942—171no—
282#1923—171no—
283#1921—171no—
284#1919—171no—
285#1862—171no—
286#1852—171no—
287#1824—171no—
288C4×#102—171no—
289C22×C17—22no—
290C22×C17—22no—
291C22×C17—22no—
292C22×C17—41no—
293C22×C17—41no—
294C22×C17—41no—
295C22×C17—41no—
296C22×C17—42no—
297C22×C17—42no—
298C22×C17—42no—
299C22×C17—81no—
300C22×C17—81no—
301C22×C17—81no—
302C22×C17—81no—
303C22×C17#52511yescharacteristic
304C22×C17#54611yescharacteristic
305C22×C17#54711yescharacteristic
306C22×C17C2×#13511yescharacteristic
307C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
308C22×C17#57611yescharacteristic
309C22×C17#57811yescharacteristic
310C4×C17—22no—
311C4×C17—41no—
312C4×C17—41no—
313C4×C17—42no—
314C4×C17—41no—
315C4×C17—41no—
316C4×C17—42no—
317C4×C17—81no—
318C4×C17#52911yescharacteristic
319C4×C17#53911yescharacteristic
320C4×C17#55311yescharacteristic
321C4×C17#55611yescharacteristic
322Dic17—81no—
323Dic17—81no—
324Dic17—81no—
325Dic17—81no—
326C2×#121C2×D1711yescharacteristic
327C2×C4×D4—21no—
328C2×C4×D4C2×D1711yescharacteristic
329C22×C4⋊C4C2×D1711yescharacteristic
330C2×#107—171no—
331C2×#107—171no—
332C2×#107—171no—
333C2×#106—171no—
334C2×C4⋊C8—171no—
335C2×#102—171no—
336C2×#102—171no—
337C2×#102—171no—
338C2×#102—171no—
339C2×C22⋊C8—171no—
340C2×C22⋊C8—171no—
341C2×C4×C8—171no—
342#401—341no—
343#397—341no—
344#390C2×D1711yescharacteristic
345#389—171no—
346#389—171no—
347#389C2×D1711yescharacteristic
348#388—21no—
349#388—341no—
350#387—341no—
351#386—21no—
352#385—21no—
353#382—171no—
354#380—171no—
355#380—171no—
356C4×C4⋊C4—171no—
357C4×C4⋊C4C2×D1711yescharacteristic
358C4×C22⋊C4C2×D1711yescharacteristic
359#340—171no—
360#340—171no—
361#340—171no—
362#340—171no—
363#340—171no—
364#340—171no—
365#340—171no—
366#340—171no—
367#336—171no—
368#336—171no—
369C2×C17—42no—
370C2×C17—82no—
371C2×C17#246611yescharacteristic
372C2×C17#310511yescharacteristic
373C2×C17#312411yescharacteristic
374C2×C17#330011yescharacteristic
375C2×C17#331811yescharacteristic
376C2×C17#333911yescharacteristic
377C2×C17#335311yescharacteristic
378C22×D4—21no—
379C22×D4C22×D1711yescharacteristic
380C23×C4—21no—
381C23×C4—21no—
382C23×C4C22×D1711yescharacteristic
383C22×C8—171no—
384C22×C8—171no—
385C22×C8—171no—
386C22×C8—171no—
387#121—22no—
388#121—22no—
389C4×D4—22no—
390C4×D4—41no—
391C4×D4—44no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—41no—
396C2×C4⋊C4—171no—
397C2×C4⋊C4—171no—
398C2×C4⋊C4—171no—
399C2×C4⋊C4—171no—
400C2×C4⋊C4—341no—
401C2×C4⋊C4—341no—
402C2×C4⋊C4C22×D1711yescharacteristic
403C2×C4⋊C4C22×D1711yescharacteristic
404C2×C4⋊C4C22×D1711yescharacteristic
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—341no—
410C2×C22⋊C4—341no—
411C2×C22⋊C4C22⋊D1711yescharacteristic
412C2×C22⋊C4C22⋊D1711yescharacteristic
413C2×C42—21no—
414C2×C42—341no—
415C2×C42C22×D1711yescharacteristic
416C2×C42C22×D1711yescharacteristic
417#107—342no—
418#107—342no—
419#107—681no—
420#106—342no—
421C4⋊C8—342no—
422#102—342no—
423#102—342no—
424#102—342no—
425#102—681no—
426C22⋊C8—342no—
427C22⋊C8—681no—
428C4×C8—681no—
429#95—21no—
430#95—21no—
431#95—341no—
432#95—341no—
433#95—341no—
434#95—341no—
435#95—341no—
436#95—341no—
437#95—341no—
438#95C22⋊D1711yescharacteristic
439#95C22⋊D1711yescharacteristic
440C17#2703311yescharacteristic, complemented, Sylow
441C24—21no—
442C24C2×C22⋊D1711yescharacteristic
443C2×D4—24no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4#6325212yes—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—81no—
466C22×C4—81no—
467C22×C4—341no—
468C22×C4—341no—
469C22×C4—341no—
470C22×C4—341no—
471C22×C4C2×C22⋊D1711yescharacteristic
472C22×C4C2×C22⋊D1711yescharacteristic
473C22×C4C2×C22⋊D1711yescharacteristic
474C22×C4C2×C22⋊D1711yescharacteristic
475C22×C4C2×C22⋊D1711yescharacteristic
476C22×C4C23×D1711yescharacteristic
477C2×C8—172no—
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—
488C2×C8—341no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—341no—
496C4⋊C4—341no—
497C4⋊C4—341no—
498C4⋊C4—341no—
499C4⋊C4—341no—
500C4⋊C4—341no—
501C4⋊C4—341no—
502C4⋊C4—341no—
503C4⋊C4—682no—
504C4⋊C4—1361no—
505C4⋊C4#6325212yes—
506C4⋊C4#6325412yes—
507C22⋊C4—24no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—81no—
513C22⋊C4—682no—
514C22⋊C4—1361no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—42no—
520C42—1361no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—22no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23#1058763612yes—
535C23C22×C22⋊D1711yescharacteristic
536D4—22no—
537D4—22no—
538D4—48no—
539D4—41no—
540D4—41no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
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—22no—
556C2×C4—21no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
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—84no—
569C2×C4—84no—
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—681no—
579C2×C4—681no—
580C2×C4—681no—
581C2×C4—681no—
582C2×C4#1058761812yes—
583C2×C4C2×#6325211yescharacteristic
584C2×C4C2×#6325211yescharacteristic
585C2×C4C2×#6325411yescharacteristic
586C2×C4#1058769111yescharacteristic
587C2×C4#41×D1711yescharacteristic
588C2×C4#1058769711yescharacteristic
589C8—342no—
590C8—681no—
591C8—681no—
592C8—681no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22#1176332011yescharacteristic
608C22#1176345311yescharacteristic
609C22#1176345811yescharacteristic
610C22C2×#1058763611yescharacteristic
611C22C2×#1058761811yescharacteristic
612C22#1176361211yescharacteristic
613C22#1176362411yescharacteristic
614C4—22no—
615C4—41no—
616C4—42no—
617C4—41no—
618C4—42no—
619C4—41no—
620C4—41no—
621C4—81no—
622C4—1361no—
623C4—1361no—
624C4—1361no—
625C4—1361no—
626C4#1176333911yescharacteristic
627C4#1176340111yescharacteristic
628C4#1176350911yescharacteristic
629C4#1176352911yescharacteristic
630C2—42no—
631C2—82no—
632C2#1453281411yescharacteristic
633C2#1453883111yescharacteristic
634C2#1453904611yescharacteristic
635C2#1454067111yescharacteristic
636C2#1454092511yescharacteristic
637C2#1454118011yescharacteristic
638C2#1454136811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2361C2
socleC23×C17C22×D4
prime radicalC2×C17×#121C22
hypercenter#2361D17
hypocenterC17#27033
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2361

Complements

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

NH ≅ G/Ntype
1. #14533184C2semidirect
2. #14530197C2semidirect
3. #14530172C2semidirect
6. #14529552C2semidirect
7. #14529514C2semidirect
8. #14529392C2semidirect
9. #14529376C2semidirect
10. #14529359C2semidirect
11. #14529031C2semidirect
12. #14528972C2semidirect
13. #14528874C2semidirect
14. C4×#10587503C2semidirect
440. C17#27033semidirect

Maximal subgroups

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

subgroupindex
#145331842
#145301972
#145301722
#145299702
#145299632
#145295522
#145295142
#145293922
#145293762
#145293592
#145290312
#145289722
#145288742
C4×#105875032
C17×#23612
#2703317