Finite Group Databasebeta

← #17102440

Subgroups — #17102440

Order 4352.

764 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14533160C211yesmaximal, characteristic
2#14531793C211yesmaximal, characteristic
3#14530104C211yesmaximal, characteristic, complemented
4#14529984C211yesmaximal, characteristic
5#14529923C211yesmaximal, characteristic, complemented
6#14529730C211yesmaximal, characteristic, complemented
7#14529462C211yesmaximal, characteristic, complemented
8#14529428C211yesmaximal, characteristic, complemented
9#14529387C211yesmaximal, characteristic
10#14529379C211yesmaximal, characteristic, complemented
11#14529127C211yesmaximal, characteristic, complemented
12#14529092C211yesmaximal, characteristic
13#14528875C211yesmaximal, characteristic
14#103×Dic17C211yesmaximal, characteristic, complemented
15C17×#2001C211yesmaximal, characteristic
16C2×#10587574C2211yescharacteristic
17C2×Q8×Dic17C2211yescharacteristic
18C22×C4⋊Dic17C2211yescharacteristic
19C2×#10587478C2211yescharacteristic
20C2×#10587485C2211yescharacteristic
21C2×#10587506C2211yescharacteristic
22C2×#10587506C2211yescharacteristic
23C2×#10587506C2211yescharacteristic
24C2×#10587506C2211yescharacteristic
25C2×C22⋊#3622C2211yescharacteristic
26C2×C8×Dic17C2211yescharacteristic
27#11762747—21no—
28#11762721—21no—
29#11762716C2211yescharacteristic
30#11762716C2211yescharacteristic
31#11762716C2211yescharacteristic
32#11762714—21no—
33#11762708—21no—
34#11762708C2211yescharacteristic
35#11762703—21no—
36#11762695—21no—
37#11762684—21no—
38#11762669C2211yescharacteristic
39C4⋊C4×Dic17C2211yescharacteristic
40C4⋊C4×Dic17C2211yescharacteristic
41C22⋊C4×Dic17—21no—
42C4×C22⋊Dic17C2211yescharacteristic
43#11762572C2211yescharacteristic
44#11762572C2211yescharacteristic
45#11762572C2211yescharacteristic
46#11762572C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52C22×Q16×C17C2211yescharacteristic
53C2×C17×C22⋊Q8C2211yescharacteristic
54C2×C17×C22⋊Q8C2211yescharacteristic
55C2×C17×#103C2211yescharacteristic
56C2×C17×#103C2211yescharacteristic
57C2×C17×C22⋊C8C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C22×Q8×C17C2311yescharacteristic
60C22×Q8×C17C2311yescharacteristic
61C23×C4×C17C2311yescharacteristic
62C23×Dic17—21no—
63C22×Dic34—21no—
64C2×Q16×C17—22no—
65C2×Q16×C17—41no—
66C2×Q16×C17—41no—
67C22×C8×C17C2311yescharacteristic
68C22×C8×C17C2311yescharacteristic
69C17×C22⋊Q8—22no—
70C17×C22⋊Q8—22no—
71C17×C22⋊Q8—22no—
72C17×C22⋊Q8—41no—
73C2×C4⋊C4×C17—21no—
74C2×C4⋊C4×C17—21no—
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C4⋊C4×C17C2311yescharacteristic
77C2×C22⋊C4×C17—21no—
78C2×C22⋊C4×C17—21no—
79C2×C22⋊Dic17—21no—
80C2×C22⋊Dic17D411yescharacteristic
81C2×C22⋊Dic17D411yescharacteristic
82Q8×Dic17—22no—
83Q8×Dic17—41no—
84C2×C4⋊Dic17—21no—
85C2×C4⋊Dic17—21no—
86C2×C4⋊Dic17—21no—
87C2×C4⋊Dic17—22no—
88C2×C4⋊Dic17—41no—
89C2×C4⋊Dic17C2311yescharacteristic
90C2×C4⋊Dic17C2311yescharacteristic
91C2×C4⋊Dic17C2311yescharacteristic
92C2×C4⋊Dic17C2311yescharacteristic
93C2×#63225—21no—
94C2×C4×Dic17—21no—
95C2×C4×Dic17—21no—
96C2×C4×Dic17—21no—
97C2×C4×Dic17C2311yescharacteristic
98C2×C4×Dic17C2311yescharacteristic
99C22×#3622C2311yescharacteristic
100C22×#3622C2311yescharacteristic
101#10587574—22no—
102#10587574—41no—
103C17×#103—22no—
104C17×#103—22no—
105C17×C22⋊C8—41no—
106#10587506—22no—
107#10587506—22no—
108#10587506—41no—
109#10587506—41no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587502—21no—
120#10587502D411yescharacteristic
121#10587502D411yescharacteristic
122C22⋊#3622—41no—
123#10587485—41no—
124C8×Dic17—41no—
125#10587478—22no—
126C24×C17C2×D411yescharacteristic
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—21no—
138C2×Q8×C17#4112yes—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17—21no—
142C22×C4×C17—21no—
143C22×C4×C17—21no—
144C22×C4×C17—21no—
145C22×C4×C17—21no—
146C22×C4×C17—21no—
147C22×C4×C17—21no—
148C22×C4×C17—21no—
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2411yescharacteristic
151C22×Dic17—21no—
152C22×Dic17—21no—
153C22×Dic17—21no—
154C22×Dic17—21no—
155C22×Dic17—42no—
156C22×Dic17—42no—
157C22×Dic17—81no—
158C22×Dic17C2×D411yescharacteristic
159C22×Dic17C2×D411yescharacteristic
160C22×Dic17C2×D411yescharacteristic
161C22×Dic17C2×D411yescharacteristic
162C2×Dic34—42no—
163C2×Dic34—42no—
164C2×Dic34—81no—
165Q16×C17—44no—
166C2×C8×C17—21no—
167C2×C8×C17—21no—
168C2×C8×C17—21no—
169C2×C8×C17—21no—
170C2×C8×C17—21no—
171C2×C8×C17—21no—
172C4⋊C4×C17—24no—
173C4⋊C4×C17—21no—
174C4⋊C4×C17—21no—
175C4⋊C4×C17—42no—
176C4⋊C4×C17#4112yes—
177C4⋊C4×C17#4112yes—
178C22⋊C4×C17—24no—
179C22⋊C4×C17—42no—
180C22⋊Dic17—41no—
181C22⋊Dic17—81no—
182C22⋊Dic17Q1614yescomplemented
183C4⋊Dic17—21no—
184C4⋊Dic17—21no—
185C4⋊Dic17—21no—
186C4⋊Dic17—21no—
187C4⋊Dic17—21no—
188C4⋊Dic17—21no—
189C4⋊Dic17—21no—
190C4⋊Dic17—42no—
191C4⋊Dic17—42no—
192C4⋊Dic17—81no—
193C4⋊Dic17#4112yes—
194#63225—42no—
195C4×Dic17—21no—
196C4×Dic17—21no—
197C4×Dic17—21no—
198C4×Dic17—21no—
199C4×Dic17—42no—
200C4×Dic17—81no—
201C4×Dic17—81no—
202C2×#3622—21no—
203C2×#3622—21no—
204C2×#3622—21no—
205C2×#3622—21no—
206C2×#3622—21no—
207C2×#3622—21no—
208#26726—171nomaximal, Sylow
209C23×C17—21no—
210C23×C17—21no—
211C23×C17—21no—
212C23×C17—21no—
213C23×C17—21no—
214C23×C17—21no—
215C23×C17C2×Q1612yes—
216C23×C17C22×D411yescharacteristic
217Q8×C17—22no—
218Q8×C17—22no—
219Q8×C17—41no—
220Q8×C17—41no—
221Q8×C17—41no—
222Q8×C17—41no—
223Q8×C17—41no—
224Q8×C17—41no—
225C2×C4×C17—22no—
226C2×C4×C17—22no—
227C2×C4×C17—21no—
228C2×C4×C17—22no—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17—41no—
232C2×C4×C17—41no—
233C2×C4×C17—41no—
234C2×C4×C17—41no—
235C2×C4×C17—41no—
236C2×C4×C17—41no—
237C2×C4×C17—41no—
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×C17C2×#4111yescharacteristic
245C2×C4×C17C2×#4111yescharacteristic
246C2×C4×C17C2×#4111yescharacteristic
247C2×C4×C17C2×#4111yescharacteristic
248C2×C4×C17H(5,𝔽2)11yescharacteristic
249C2×C4×C17H(5,𝔽2)11yescharacteristic
250C2×Dic17—21no—
251C2×Dic17—21no—
252C2×Dic17—21no—
253C2×Dic17—21no—
254C2×Dic17—21no—
255C2×Dic17—22no—
256C2×Dic17—21no—
257C2×Dic17—21no—
258C2×Dic17—21no—
259C2×Dic17—21no—
260C2×Dic17—21no—
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—
270C2×Dic17—41no—
271C2×Dic17—41no—
272C2×Dic17—84no—
273C2×Dic17C2×Q1612yes—
274C2×Dic17C2×Q1612yes—
275Dic34—84no—
276C8×C17—41no—
277C8×C17—41no—
278#3622—41no—
279#3622—41no—
280#2492—171no—
281#2360—171no—
282#2056—171no—
283#2036—171no—
284#2024—171no—
285#2001D1711yescharacteristic
286#1987—171no—
287#1933—171no—
288#1928—171no—
289#1923—171no—
290#1921—171no—
291#1878—171no—
292#1872—171no—
293#1824—171no—
294C4×#103—171no—
295C22×C17—22no—
296C22×C17—22no—
297C22×C17—22no—
298C22×C17—41no—
299C22×C17—41no—
300C22×C17—41no—
301C22×C17—41no—
302C22×C17#52511yescharacteristic
303C22×C17#54611yescharacteristic
304C22×C17#54611yescharacteristic
305C22×C17C22×Q1611yescharacteristic
306C22×C17#57511yescharacteristic
307C22×C17#57711yescharacteristic
308C22×C17#57711yescharacteristic
309C4×C17—42no—
310C4×C17—41no—
311C4×C17—42no—
312C4×C17—42no—
313C4×C17—81no—
314C4×C17#52911yescharacteristic
315C4×C17#53811yescharacteristic
316C4×C17#55511yescharacteristic
317C4×C17#55911yescharacteristic
318Dic17—22no—
319Dic17—22no—
320Dic17—41no—
321Dic17—42no—
322Dic17—41no—
323Dic17—81no—
324Dic17—81no—
325Dic17—81no—
326C22×Q16C2×D1711yescharacteristic
327C2×C4⋊Q8—171no—
328C2×C22⋊Q8C2×D1711yescharacteristic
329C2×C22⋊Q8C2×D1711yescharacteristic
330C2×C4×Q8—171no—
331C22×C4⋊C4—171no—
332C2×#107—171no—
333C2×C4⋊C8—171no—
334C2×#103—171no—
335C2×#103—171no—
336C2×#103—171no—
337C2×#103—171no—
338C2×#103C2×D1711yescharacteristic
339C2×#103C2×D1711yescharacteristic
340C2×C22⋊C8—171no—
341C2×C22⋊C8C2×D1711yescharacteristic
342C2×C4×C8—171no—
343#400—341no—
344#393—341no—
345#391—171no—
346#391—171no—
347#391—171no—
348#391—341no—
349#389—171no—
350#389—341no—
351#388—341no—
352#387—341no—
353#385—341no—
354#380—171no—
355C4×C4⋊C4—171no—
356C4×C4⋊C4—171no—
357C4×C22⋊C4—171no—
358C4×C22⋊C4—341no—
359#340—171no—
360#340—171no—
361#340—171no—
362#340—171no—
363#340—171no—
364#340—171no—
365#340—171no—
366#340C2×D1711yescharacteristic
367#336—171no—
368#336—171no—
369C2×C17—42no—
370C2×C17#245911yescharacteristic
371C2×C17#310411yescharacteristic
372C2×C17#312511yescharacteristic
373C2×C17#329911yescharacteristic
374C2×C17#329911yescharacteristic
375C2×C17#331111yescharacteristic
376C2×C17#332311yescharacteristic
377C22×Q8—341no—
378C22×Q8C22×D1711yescharacteristic
379C22×Q8C22×D1711yescharacteristic
380C23×C4—341no—
381C23×C4C22×D1711yescharacteristic
382C2×Q16—22no—
383C2×Q16—41no—
384C2×Q16—41no—
385C22×C8—171no—
386C22×C8—171no—
387C22×C8C22×D1711yescharacteristic
388C22×C8C22×D1711yescharacteristic
389C4⋊Q8—342no—
390C4⋊Q8—681no—
391C22⋊Q8—22no—
392C22⋊Q8—22no—
393C22⋊Q8—22no—
394C22⋊Q8—41no—
395C4×Q8—342no—
396C4×Q8—681no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—171no—
400C2×C4⋊C4—171no—
401C2×C4⋊C4—171no—
402C2×C4⋊C4—171no—
403C2×C4⋊C4—342no—
404C2×C4⋊C4—341no—
405C2×C4⋊C4—341no—
406C2×C4⋊C4—341no—
407C2×C4⋊C4—341no—
408C2×C4⋊C4—681no—
409C2×C4⋊C4C22×D1711yescharacteristic
410C2×C4⋊C4C22×D1711yescharacteristic
411C2×C22⋊C4—21no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—171no—
414C2×C22⋊C4—171no—
415C2×C22⋊C4—341no—
416C2×C42—171no—
417C2×C42—171no—
418C2×C42—341no—
419C2×C42—341no—
420C2×C42—341no—
421#107—342no—
422C4⋊C8—681no—
423#103—22no—
424#103—22no—
425#103—342no—
426#103—342no—
427#103—681no—
428#103—681no—
429C22⋊C8—41no—
430C22⋊C8—681no—
431C4×C8—681no—
432#95—171no—
433#95—171no—
434#95—341no—
435#95—341no—
436#95—341no—
437#95—341no—
438#95—341no—
439#95—341no—
440#95—341no—
441#95—341no—
442#95—341no—
443#95—341no—
444C17#2672611yescharacteristic, complemented, Sylow
445C24D4×D1711yescharacteristic
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—682no—
458C2×Q8—682no—
459C2×Q8—1361no—
460C2×Q8#6325412yes—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—171no—
472C22×C4—171no—
473C22×C4—171no—
474C22×C4—171no—
475C22×C4—341no—
476C22×C4—341no—
477C22×C4—341no—
478C22×C4—341no—
479C22×C4—682no—
480C22×C4—682no—
481C22×C4—1361no—
482C22×C4D4×D1711yescharacteristic
483C22×C4C23×D1711yescharacteristic
484Q16—44no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—341no—
492C2×C8—341no—
493C2×C8—341no—
494C2×C8—341no—
495C2×C8—341no—
496C2×C8—341no—
497C4⋊C4—24no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—42no—
501C4⋊C4—172no—
502C4⋊C4—341no—
503C4⋊C4—341no—
504C4⋊C4—341no—
505C4⋊C4—341no—
506C4⋊C4—341no—
507C4⋊C4—341no—
508C4⋊C4—341no—
509C4⋊C4—682no—
510C4⋊C4—682no—
511C4⋊C4—682no—
512C4⋊C4—1361no—
513C4⋊C4#6325412yes—
514C4⋊C4#6325412yes—
515C22⋊C4—24no—
516C22⋊C4—42no—
517C22⋊C4—174no—
518C22⋊C4—681no—
519C22⋊C4—1361no—
520C42—341no—
521C42—341no—
522C42—341no—
523C42—341no—
524C42—682no—
525C42—1361no—
526C42—1361no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23Q16×D1712yes—
534C23C2×D4×D1711yescharacteristic
535Q8—22no—
536Q8—22no—
537Q8—41no—
538Q8—41no—
539Q8—41no—
540Q8—41no—
541Q8—41no—
542Q8—41no—
543Q8—1364no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—172no—
564C2×C4—172no—
565C2×C4—341no—
566C2×C4—341no—
567C2×C4—341no—
568C2×C4—341no—
569C2×C4—341no—
570C2×C4—341no—
571C2×C4—341no—
572C2×C4—341no—
573C2×C4—341no—
574C2×C4—341no—
575C2×C4—342no—
576C2×C4—681no—
577C2×C4—681no—
578C2×C4—681no—
579C2×C4—681no—
580C2×C4—681no—
581C2×C4—681no—
582C2×C4—681no—
583C2×C4—681no—
584C2×C4—681no—
585C2×C4—681no—
586C2×C4—681no—
587C2×C4—1364no—
588C2×C4C2×#6325411yescharacteristic
589C2×C4C2×#6325411yescharacteristic
590C2×C4C2×#6325411yescharacteristic
591C2×C4#1058769111yescharacteristic
592C2×C4#1058769111yescharacteristic
593C2×C4#41×D1711yescharacteristic
594C8—41no—
595C8—41no—
596C8—681no—
597C8—681no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22#1176331511yescharacteristic
606C22#1176344611yescharacteristic
607C22#1176344811yescharacteristic
608C22C2×Q16×D1711yescharacteristic
609C22#1176360811yescharacteristic
610C22#1176361611yescharacteristic
611C22#1176361811yescharacteristic
612C4—42no—
613C4—41no—
614C4—42no—
615C4—42no—
616C4—81no—
617C4—342no—
618C4—342no—
619C4—681no—
620C4—682no—
621C4—681no—
622C4—1361no—
623C4—1361no—
624C4—1361no—
625C4#1176334011yescharacteristic
626C4#1176339411yescharacteristic
627C4#1176352311yescharacteristic
628C4#1176354511yescharacteristic
629C2—42no—
630C2#1453271611yescharacteristic
631C2#1453881811yescharacteristic
632C2#1453905411yescharacteristic
633C2#1454065311yescharacteristic
634C2#1454065511yescharacteristic
635C2#1454081511yescharacteristic
636C2#1454099311yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2001C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenter#2001D17
hypocenterC17#26726
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2001

Complements

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

NH ≅ G/Ntype
3. #14530104C2semidirect
5. #14529923C2semidirect
6. #14529730C2semidirect
7. #14529462C2semidirect
8. #14529428C2semidirect
10. #14529379C2semidirect
11. #14529127C2semidirect
14. #103×Dic17C2semidirect
182. C22⋊Dic17Q16semidirect
444. C17#26726semidirect

Maximal subgroups

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

subgroupindex
#145331602
#145317932
#145301042
#145299842
#145299232
#145297302
#145294622
#145294282
#145293872
#145293792
#145291272
#145290922
#145288752
#103×Dic172
C17×#20012
#2672617