Finite Group Databasebeta

← #17102455

Subgroups — #17102455

Order 4352.

790 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14533138C211yesmaximal, characteristic
2#14531793C211yesmaximal, characteristic, complemented
3#14530097C211yesmaximal, characteristic, complemented
4#14529984C211yesmaximal, characteristic, complemented
5#14529933C211yesmaximal, characteristic, complemented
6#14529740C211yesmaximal, characteristic, complemented
7#14529469C211yesmaximal, characteristic, complemented
8#14529407C211yesmaximal, characteristic, complemented
9#14529387C211yesmaximal, characteristic, complemented
10#14529379C211yesmaximal, characteristic, complemented
11#14529141C211yesmaximal, characteristic, complemented
12#14529102C211yesmaximal, characteristic
13#14528871C211yesmaximal, characteristic, complemented
14#103×Dic17C211yesmaximal, characteristic, complemented
15C17×#2000C211yesmaximal, characteristic
16C2×#10587576C2211yescharacteristic
17C2×Q8×Dic17C2211yescharacteristic
18C22×C4⋊Dic17C2211yescharacteristic
19C2×#10587478C2211yescharacteristic
20C2×#10587485C2211yescharacteristic
21C2×#10587506C2211yescharacteristic
22C2×#10587506C2211yescharacteristic
23C2×#10587503C2211yescharacteristic
24C2×#10587503C2211yescharacteristic
25C2×C22⋊#3622C2211yescharacteristic
26C2×C8×Dic17C2211yescharacteristic
27#11762734—21no—
28#11762723—21no—
29#11762716C2211yescharacteristic
30#11762712—21no—
31#11762712C2211yescharacteristic
32#11762712C2211yescharacteristic
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×SD16×C17C2211yescharacteristic
53C2×C17×C22⋊Q8C2211yescharacteristic
54C2×C17×#121C2211yescharacteristic
55C2×C17×#103C2211yescharacteristic
56C2×C17×#102C2211yescharacteristic
57C2×C17×C22⋊C8C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C22×Q8×C17C2311yescharacteristic
60C22×D4×C17—21no—
61C22×D4×C17C2311yescharacteristic
62C23×C4×C17C2311yescharacteristic
63C23×Dic17—21no—
64C2×SD16×C17—22no—
65C2×SD16×C17—41no—
66C2×SD16×C17—41no—
67C22×C8×C17C2311yescharacteristic
68C22×C8×C17C2311yescharacteristic
69C17×C22⋊Q8—22no—
70C17×C22⋊Q8—22no—
71C17×#121—22no—
72C17×#121—41no—
73C2×C4⋊C4×C17—21no—
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C22⋊C4×C17—21no—
77C2×C22⋊C4×C17—21no—
78C2×C22⋊Dic17—21no—
79C2×C22⋊Dic17—21no—
80C2×C22⋊Dic17—21no—
81C2×C22⋊Dic17—21no—
82C2×C22⋊Dic17—21no—
83C2×C22⋊Dic17D411yescharacteristic
84C2×C22⋊Dic17D411yescharacteristic
85Q8×Dic17—22no—
86Q8×Dic17—41no—
87C2×C4⋊Dic17—21no—
88C2×C4⋊Dic17—21no—
89C2×C4⋊Dic17—21no—
90C2×C4⋊Dic17—21no—
91C2×C4⋊Dic17—22no—
92C2×C4⋊Dic17—41no—
93C2×C4⋊Dic17C2311yescharacteristic
94C2×C4⋊Dic17C2311yescharacteristic
95C2×C4⋊Dic17C2311yescharacteristic
96C2×C4⋊Dic17C2311yescharacteristic
97C2×#63225—21no—
98C2×C4×Dic17—21no—
99C2×C4×Dic17—21no—
100C2×C4×Dic17—21no—
101C2×C4×Dic17C2311yescharacteristic
102C2×C4×Dic17C2311yescharacteristic
103C22×#3622C2311yescharacteristic
104C22×#3622C2311yescharacteristic
105#10587576—22no—
106#10587576—41no—
107C17×#103—22no—
108C17×#102—22no—
109C17×C22⋊C8—41no—
110#10587506—22no—
111#10587506—41no—
112#10587503—22no—
113#10587503—41no—
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×C17—21no—
127C24×C17C2×D411yescharacteristic
128C2×Q8×C17—21no—
129C2×Q8×C17—21no—
130C2×Q8×C17—21no—
131C2×Q8×C17—21no—
132C2×Q8×C17—21no—
133C2×Q8×C17#4112yes—
134C2×D4×C17—21no—
135C2×D4×C17—21no—
136C2×D4×C17—21no—
137C2×D4×C17—21no—
138C2×D4×C17—21no—
139C2×D4×C17—21no—
140C2×D4×C17—42no—
141C2×D4×C17—42no—
142C2×D4×C17—81no—
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×C17—21no—
150C22×C4×C17—21no—
151C22×C4×C17—21no—
152C22×C4×C17C2×D411yescharacteristic
153C22×C4×C17C2411yescharacteristic
154C22×Dic17—21no—
155C22×Dic17—21no—
156C22×Dic17—21no—
157C22×Dic17—21no—
158C22×Dic17—42no—
159C22×Dic17—42no—
160C22×Dic17—81no—
161C22×Dic17C2×D411yescharacteristic
162C22×Dic17C2×D411yescharacteristic
163C22×Dic17C2×D411yescharacteristic
164C22×Dic17C2×D411yescharacteristic
165SD16×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#4112yes—
176C4⋊C4×C17#4112yes—
177C22⋊C4×C17—24no—
178C22⋊C4×C17—42no—
179C22⋊Dic17—22no—
180C22⋊Dic17—22no—
181C22⋊Dic17—42no—
182C22⋊Dic17—81no—
183C22⋊Dic17—81no—
184C22⋊Dic17—81no—
185C22⋊Dic17—81no—
186C4⋊Dic17—21no—
187C4⋊Dic17—21no—
188C4⋊Dic17—21no—
189C4⋊Dic17—21no—
190C4⋊Dic17—21no—
191C4⋊Dic17—21no—
192C4⋊Dic17—21no—
193C4⋊Dic17—42no—
194C4⋊Dic17—42no—
195C4⋊Dic17—42no—
196C4⋊Dic17—81no—
197C4⋊Dic17#4112yes—
198#63225—42no—
199C4×Dic17—21no—
200C4×Dic17—21no—
201C4×Dic17—21no—
202C4×Dic17—21no—
203C4×Dic17—42no—
204C4×Dic17—81no—
205C4×Dic17—81no—
206C2×#3622—21no—
207C2×#3622—21no—
208C2×#3622—21no—
209C2×#3622—21no—
210C2×#3622—21no—
211C2×#3622—21no—
212#26727—171nomaximal, Sylow
213C23×C17—21no—
214C23×C17—21no—
215C23×C17—21no—
216C23×C17—21no—
217C23×C17—21no—
218C23×C17—21no—
219C23×C17—41no—
220C23×C17—41no—
221C23×C17—41no—
222C23×C17—41no—
223C23×C17—41no—
224C23×C17—41no—
225C23×C17—41no—
226C23×C17#13512yes—
227C23×C17C22×D411yescharacteristic
228Q8×C17—22no—
229Q8×C17—22no—
230Q8×C17—41no—
231Q8×C17—41no—
232D4×C17—41no—
233D4×C17—41no—
234D4×C17—41no—
235D4×C17—41no—
236D4×C17—84no—
237C2×C4×C17—22no—
238C2×C4×C17—22no—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—22no—
242C2×C4×C17—21no—
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—41no—
252C2×C4×C17—41no—
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—21no—
260C2×Dic17—21no—
261C2×Dic17—22no—
262C2×Dic17—21no—
263C2×Dic17—21no—
264C2×Dic17—21no—
265C2×Dic17—21no—
266C2×Dic17—21no—
267C2×Dic17—21no—
268C2×Dic17—21no—
269C2×Dic17—21no—
270C2×Dic17—21no—
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—41no—
280C2×Dic17—41no—
281C2×Dic17—41no—
282C2×Dic17—84no—
283C2×Dic17#13712yes—
284C8×C17—41no—
285C8×C17—41no—
286#3622—41no—
287#3622—41no—
288#2490—171no—
289#2360—171no—
290#2055—171no—
291#2036—171no—
292#2026—171no—
293#2000D1711yescharacteristic
294#1989—171no—
295#1934—171no—
296#1925—171no—
297#1923—171no—
298#1921—171no—
299#1880—171no—
300#1874—171no—
301#1823—171no—
302C4×#103—171no—
303C22×C17—22no—
304C22×C17—22no—
305C22×C17—22no—
306C22×C17—41no—
307C22×C17—41no—
308C22×C17—41no—
309C22×C17—41no—
310C22×C17—81no—
311C22×C17—81no—
312C22×C17—81no—
313C22×C17—81no—
314C22×C17—81no—
315C22×C17—81no—
316C22×C17—81no—
317C22×C17#52511yescharacteristic
318C22×C17#54611yescharacteristic
319C22×C17#54611yescharacteristic
320C22×C17C2×#13511yescharacteristic
321C22×C17C2×#13711yescharacteristic
322C22×C17#57611yescharacteristic
323C22×C17#57711yescharacteristic
324C4×C17—42no—
325C4×C17—41no—
326C4×C17—42no—
327C4×C17—42no—
328C4×C17#52911yescharacteristic
329C4×C17#53811yescharacteristic
330C4×C17#55511yescharacteristic
331C4×C17#55911yescharacteristic
332Dic17—22no—
333Dic17—41no—
334Dic17—41no—
335Dic17—41no—
336Dic17—42no—
337Dic17—81no—
338Dic17—81no—
339Dic17—81no—
340C22×SD16C2×D1711yescharacteristic
341C2×#125—171no—
342C2×C22⋊Q8C2×D1711yescharacteristic
343C2×#121C2×D1711yescharacteristic
344C2×C4×Q8—171no—
345C22×C4⋊C4—171no—
346C2×#107—171no—
347C2×C4⋊C8—171no—
348C2×#103—171no—
349C2×#103—171no—
350C2×#103C2×D1711yescharacteristic
351C2×#102—171no—
352C2×#102—171no—
353C2×#102C2×D1711yescharacteristic
354C2×C22⋊C8—171no—
355C2×C22⋊C8C2×D1711yescharacteristic
356C2×C4×C8—171no—
357#397—341no—
358#394—341no—
359#391—171no—
360#390—171no—
361#390—171no—
362#390—341no—
363#389—171no—
364#389—341no—
365#388—341no—
366#387—341no—
367#385—341no—
368#380—171no—
369C4×C4⋊C4—171no—
370C4×C4⋊C4—171no—
371C4×C22⋊C4—171no—
372C4×C22⋊C4—341no—
373#340—171no—
374#340—171no—
375#340—171no—
376#340—171no—
377#340—171no—
378#340—171no—
379#340—171no—
380#340C2×D1711yescharacteristic
381#336—171no—
382#336—171no—
383C2×C17—42no—
384C2×C17—161no—
385C2×C17#245911yescharacteristic
386C2×C17#310811yescharacteristic
387C2×C17#311911yescharacteristic
388C2×C17#329611yescharacteristic
389C2×C17#329711yescharacteristic
390C2×C17#332011yescharacteristic
391C2×C17#333311yescharacteristic
392C22×Q8C22×D1711yescharacteristic
393C22×D4—21no—
394C22×D4C22×D1711yescharacteristic
395C23×C4—341no—
396C23×C4C22×D1711yescharacteristic
397C2×SD16—22no—
398C2×SD16—41no—
399C2×SD16—41no—
400C22×C8—171no—
401C22×C8—171no—
402C22×C8C22×D1711yescharacteristic
403C22×C8C22×D1711yescharacteristic
404#125—342no—
405#125—681no—
406C22⋊Q8—22no—
407C22⋊Q8—22no—
408#121—22no—
409#121—41no—
410C4×Q8—342no—
411C4×Q8—681no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—171no—
414C2×C4⋊C4—171no—
415C2×C4⋊C4—171no—
416C2×C4⋊C4—171no—
417C2×C4⋊C4—342no—
418C2×C4⋊C4—341no—
419C2×C4⋊C4—341no—
420C2×C4⋊C4—341no—
421C2×C4⋊C4—341no—
422C2×C4⋊C4—341no—
423C2×C4⋊C4—681no—
424C2×C4⋊C4C22×D1711yescharacteristic
425C2×C4⋊C4C22×D1711yescharacteristic
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—171no—
429C2×C22⋊C4—171no—
430C2×C22⋊C4—341no—
431C2×C22⋊C4—341no—
432C2×C22⋊C4—341no—
433C2×C22⋊C4—341no—
434C2×C22⋊C4—341no—
435C2×C42—171no—
436C2×C42—171no—
437C2×C42—341no—
438C2×C42—341no—
439C2×C42—341no—
440#107—342no—
441C4⋊C8—681no—
442#103—22no—
443#103—342no—
444#103—681no—
445#102—22no—
446#102—342no—
447#102—681no—
448C22⋊C8—41no—
449C22⋊C8—681no—
450C4×C8—681no—
451#95—171no—
452#95—171no—
453#95—341no—
454#95—341no—
455#95—341no—
456#95—341no—
457#95—341no—
458#95—341no—
459C17#2672711yescharacteristic, complemented, Sylow
460C24—21no—
461C24D4×D1711yescharacteristic
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8#6325412yes—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4—81no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—171no—
487C22×C4—171no—
488C22×C4—171no—
489C22×C4—171no—
490C22×C4—341no—
491C22×C4—341no—
492C22×C4—341no—
493C22×C4—341no—
494C22×C4—682no—
495C22×C4—682no—
496C22×C4—1361no—
497C22×C4D4×D1711yescharacteristic
498C22×C4C23×D1711yescharacteristic
499SD16—44no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—341no—
507C2×C8—341no—
508C2×C8—341no—
509C2×C8—341no—
510C2×C8—341no—
511C2×C8—341no—
512C4⋊C4—24no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—172no—
516C4⋊C4—341no—
517C4⋊C4—341no—
518C4⋊C4—341no—
519C4⋊C4—341no—
520C4⋊C4—341no—
521C4⋊C4—341no—
522C4⋊C4—341no—
523C4⋊C4—682no—
524C4⋊C4—682no—
525C4⋊C4—682no—
526C4⋊C4—682no—
527C4⋊C4—1361no—
528C4⋊C4#6325412yes—
529C4⋊C4#6325412yes—
530C22⋊C4—24no—
531C22⋊C4—42no—
532C22⋊C4—342no—
533C22⋊C4—342no—
534C22⋊C4—682no—
535C22⋊C4—1361no—
536C22⋊C4—1361no—
537C22⋊C4—1361no—
538C22⋊C4—1361no—
539C42—341no—
540C42—341no—
541C42—341no—
542C42—341no—
543C42—682no—
544C42—1361no—
545C42—1361no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23#1058760212yes—
560C23C2×D4×D1711yescharacteristic
561Q8—22no—
562Q8—22no—
563Q8—41no—
564Q8—41no—
565D4—41no—
566D4—41no—
567D4—41no—
568D4—41no—
569D4—84no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—172no—
587C2×C4—341no—
588C2×C4—341no—
589C2×C4—341no—
590C2×C4—341no—
591C2×C4—341no—
592C2×C4—341no—
593C2×C4—341no—
594C2×C4—341no—
595C2×C4—341no—
596C2×C4—341no—
597C2×C4—341no—
598C2×C4—342no—
599C2×C4—681no—
600C2×C4—681no—
601C2×C4—681no—
602C2×C4—681no—
603C2×C4—681no—
604C2×C4—681no—
605C2×C4—681no—
606C2×C4—681no—
607C2×C4—681no—
608C2×C4—681no—
609C2×C4—681no—
610C2×C4—1364no—
611C2×C4C2×#6325411yescharacteristic
612C2×C4C2×#6325411yescharacteristic
613C2×C4C2×#6325411yescharacteristic
614C2×C4#1058769111yescharacteristic
615C2×C4#1058769111yescharacteristic
616C2×C4#41×D1711yescharacteristic
617C8—41no—
618C8—41no—
619C8—681no—
620C8—681no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22#1176331511yescharacteristic
636C22#1176344611yescharacteristic
637C22#1176344811yescharacteristic
638C22C2×#1058760211yescharacteristic
639C22#137×D1711yescharacteristic
640C22#1176361111yescharacteristic
641C22#1176361911yescharacteristic
642C4—42no—
643C4—41no—
644C4—42no—
645C4—42no—
646C4—342no—
647C4—681no—
648C4—681no—
649C4—681no—
650C4—682no—
651C4—1361no—
652C4—1361no—
653C4—1361no—
654C4#1176334011yescharacteristic
655C4#1176339411yescharacteristic
656C4#1176352311yescharacteristic
657C4#1176354511yescharacteristic
658C2—42no—
659C2—161no—
660C2#1453271611yescharacteristic
661C2#1453885911yescharacteristic
662C2#1453897611yescharacteristic
663C2#1454062311yescharacteristic
664C2#1454064011yescharacteristic
665C2#1454095011yescharacteristic
666C2#1454111511yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2000C2
socleC23×C17C22×D4
prime radicalC2×C17×#121C22
hypercenter#2000D17
hypocenterC17#26727
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2000

Complements

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

NH ≅ G/Ntype
2. #14531793C2semidirect
3. #14530097C2semidirect
4. #14529984C2semidirect
5. #14529933C2semidirect
6. #14529740C2semidirect
7. #14529469C2semidirect
8. #14529407C2semidirect
9. #14529387C2semidirect
10. #14529379C2semidirect
11. #14529141C2semidirect
13. #14528871C2semidirect
14. #103×Dic17C2semidirect
459. C17#26727semidirect

Maximal subgroups

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

subgroupindex
#145331382
#145317932
#145300972
#145299842
#145299332
#145297402
#145294692
#145294072
#145293872
#145293792
#145291412
#145291022
#145288712
#103×Dic172
C17×#20002
#2672717