Finite Group Databasebeta

← #17102426

Subgroups — #17102426

Order 4352.

748 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14531787C211yesmaximal, characteristic
2#14530106C211yesmaximal, characteristic, complemented
3#14529980C211yesmaximal, characteristic
4#14529932C211yesmaximal, characteristic, complemented
5#14529792C211yesmaximal, characteristic
6#14529742C211yesmaximal, characteristic, complemented
7#14529463C211yesmaximal, characteristic, complemented
8#14529415C211yesmaximal, characteristic, complemented
9#14529392C211yesmaximal, characteristic
10#14529369C211yesmaximal, characteristic, complemented
11#14529128C211yesmaximal, characteristic, complemented
12#14529092C211yesmaximal, characteristic
13#14528868C211yesmaximal, characteristic
14C4×#10587481C211yesmaximal, characteristic, complemented
15C17×#2489C211yesmaximal, characteristic
16C22×#63231C2211yescharacteristic
17C2×#10587610C2211yescharacteristic
18C2×C4×Dic34C2211yescharacteristic
19C22×C4⋊Dic17C2211yescharacteristic
20C2×#10587479C2211yescharacteristic
21C2×C4⋊#3622C2211yescharacteristic
22C2×#10587506C2211yescharacteristic
23C2×#10587506C2211yescharacteristic
24C2×#10587506C2211yescharacteristic
25C2×#10587481C2211yescharacteristic
26C2×#10587481C2211yescharacteristic
27C2×#10587481C2211yescharacteristic
28C2×C22⋊#3622C2211yescharacteristic
29C2×C22⋊#3622C2211yescharacteristic
30C2×C4×#3622C2211yescharacteristic
31#11762715C2211yescharacteristic
32#11762707C2211yescharacteristic
33#11762706—21no—
34#11762696—21no—
35#11762681—21no—
36C4×C4⋊Dic17C2211yescharacteristic
37C22⋊C4×Dic17—21no—
38#11762571C2211yescharacteristic
39#11762571C2211yescharacteristic
40#11762571C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762564C2211yescharacteristic
47#11762564C2211yescharacteristic
48C2×C17×#125C2211yescharacteristic
49C2×C17×C22⋊Q8C2211yescharacteristic
50C17×#400—21no—
51C17×#397—21no—
52C17×#391C2211yescharacteristic
53C17×#391C2211yescharacteristic
54C17×#388—21no—
55C17×#382—21no—
56C17×#380C2211yescharacteristic
57C4×C4⋊C4×C17C2211yescharacteristic
58C4×C22⋊C4×C17C2211yescharacteristic
59C22×Q8×C17C2311yescharacteristic
60C23×C4×C17C2311yescharacteristic
61C23×Dic17—21no—
62C22×Dic34C2311yescharacteristic
63C17×#125—22no—
64C17×#125—41no—
65C17×C22⋊Q8—22no—
66C17×C22⋊Q8—41no—
67C2×C4⋊C4×C17—21no—
68C2×C4⋊C4×C17—21no—
69C2×C4⋊C4×C17—21no—
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C4⋊C4×C17C2311yescharacteristic
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C22⋊C4×C17—21no—
74C2×C22⋊C4×C17—21no—
75C2×C22⋊C4×C17D411yescharacteristic
76C2×C22⋊C4×C17D411yescharacteristic
77C2×C42×C17—21no—
78C2×C42×C17C2311yescharacteristic
79C2×C42×C17C2311yescharacteristic
80C2×C22⋊Dic17—21no—
81C2×#63231—22no—
82C2×#63231—41no—
83C2×#63231—41no—
84C2×C4⋊Dic17—21no—
85C2×C4⋊Dic17—22no—
86C2×C4⋊Dic17—41no—
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4⋊Dic17C2311yescharacteristic
89C2×C4⋊Dic17C2311yescharacteristic
90#10587610—22no—
91#10587610—22no—
92C2×#63225—21no—
93C2×#63225—21no—
94C2×#63225—21no—
95C2×C4×Dic17—21no—
96C2×C4×Dic17—21no—
97C22×#3622C2311yescharacteristic
98C22×#3622C2311yescharacteristic
99C22×#3622C2311yescharacteristic
100C22×#3622C2311yescharacteristic
101C4×Dic34—22no—
102C4×Dic34—41no—
103C17×#95—21no—
104C17×#95—21no—
105C17×#95—21no—
106C17×#95—21no—
107C17×#95—21no—
108C17×#95—21no—
109C17×#95D411yescharacteristic
110C17×#95D411yescharacteristic
111#10587506—22no—
112#10587506—22no—
113#10587506—41no—
114#10587502—21no—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118C22⋊#3622—41no—
119C22⋊#3622—41no—
120#10587481—22no—
121#10587481—22no—
122#10587481—41no—
123#10587479—22no—
124C4⋊#3622—41no—
125C4×#3622—41no—
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—
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—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×C17C2411yescharacteristic
149C22×Dic17—21no—
150C22×Dic17—21no—
151C22×Dic17—21no—
152C22×Dic17—21no—
153C22×Dic17—42no—
154C22×Dic17—42no—
155C22×Dic17—81no—
156C2×Dic34—21no—
157C2×Dic34—21no—
158C2×Dic34—21no—
159C2×Dic34—21no—
160C2×Dic34—21no—
161C2×Dic34#4112yes—
162C4⋊C4×C17—21no—
163C4⋊C4×C17—21no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17—21no—
167C4⋊C4×C17—42no—
168C4⋊C4×C17—42no—
169C4⋊C4×C17—42no—
170C4⋊C4×C17#4112yes—
171C22⋊C4×C17—22no—
172C22⋊C4×C17—22no—
173C22⋊C4×C17—42no—
174C22⋊C4×C17—42no—
175C42×C17—21no—
176C42×C17—21no—
177C42×C17—21no—
178C42×C17—21no—
179C42×C17—42no—
180C22⋊Dic17—24no—
181#63231—44no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184C4⋊Dic17—21no—
185C4⋊Dic17—21no—
186C4⋊Dic17—42no—
187C4⋊Dic17#4112yes—
188C4⋊Dic17#4112yes—
189#63225—24no—
190#63225—42no—
191#63225—81no—
192C4×Dic17—42no—
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—21no—
206#26725—171nomaximal, Sylow
207C23×C17—21no—
208C23×C17—21no—
209C23×C17—21no—
210C23×C17—21no—
211C23×C17—21no—
212C23×C17—21no—
213C23×C17#13512yes—
214C23×C17C22×D411yescharacteristic
215Q8×C17—41no—
216Q8×C17—41no—
217Q8×C17—41no—
218Q8×C17—41no—
219C2×C4×C17—21no—
220C2×C4×C17—21no—
221C2×C4×C17—21no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—22no—
226C2×C4×C17—21no—
227C2×C4×C17—21no—
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—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×C17—41no—
245C2×C4×C17—41no—
246C2×C4×C17—41no—
247C2×C4×C17—41no—
248C2×C4×C17—41no—
249C2×C4×C17#13712yes—
250C2×C4×C17C2×#4111yescharacteristic
251C2×C4×C17C2×#4111yescharacteristic
252C2×C4×C17C2×#4111yescharacteristic
253C2×C4×C17C2×#4111yescharacteristic
254C2×C4×C17H(5,𝔽2)11yescharacteristic
255C2×C4×C17H(5,𝔽2)11yescharacteristic
256C2×Dic17—22no—
257C2×Dic17—22no—
258C2×Dic17—22no—
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—84no—
269Dic34—22no—
270Dic34—22no—
271Dic34—41no—
272Dic34—41no—
273#3622—41no—
274#3622—41no—
275#3622—41no—
276#3622—41no—
277#2489D1711yescharacteristic
278#2360—171no—
279#2056—171no—
280#2035—171no—
281#2026—171no—
282#2001—171no—
283#1989—171no—
284#1933—171no—
285#1927—171no—
286#1923—171no—
287#1920—171no—
288#1878—171no—
289#1872—171no—
290#1823—171no—
291C4×#103—171no—
292C22×C17—22no—
293C22×C17—22no—
294C22×C17—22no—
295C22×C17—41no—
296C22×C17—41no—
297C22×C17—41no—
298C22×C17—41no—
299C22×C17#52511yescharacteristic
300C22×C17#54611yescharacteristic
301C22×C17#54611yescharacteristic
302C22×C17C2×#13511yescharacteristic
303C22×C17C2×#13711yescharacteristic
304C22×C17#57711yescharacteristic
305C22×C17#57811yescharacteristic
306C4×C17—22no—
307C4×C17—41no—
308C4×C17—42no—
309C4×C17—41no—
310C4×C17—41no—
311C4×C17—41no—
312C4×C17—81no—
313C4×C17—81no—
314C4×C17—81no—
315C4×C17#52911yescharacteristic
316C4×C17#53811yescharacteristic
317C4×C17#55511yescharacteristic
318C4×C17#55911yescharacteristic
319Dic17—42no—
320Dic17—42no—
321Dic17—42no—
322Dic17—81no—
323C22×Q16—171no—
324C2×#125C2×D1711yescharacteristic
325C2×C22⋊Q8—171no—
326C2×C22⋊Q8C2×D1711yescharacteristic
327C2×C4×Q8—171no—
328C22×C4⋊C4—171no—
329C2×#106—171no—
330C2×C4⋊C8—171no—
331C2×#103—171no—
332C2×#103—171no—
333C2×#103—171no—
334C2×#103—171no—
335C2×#103—171no—
336C2×#103—171no—
337C2×C22⋊C8—171no—
338C2×C22⋊C8—171no—
339C2×C4×C8—171no—
340#400—21no—
341#397—21no—
342#391—171no—
343#391C2×D1711yescharacteristic
344#391C2×D1711yescharacteristic
345#389—171no—
346#389—341no—
347#388—21no—
348#387—341no—
349#385—341no—
350#382—21no—
351#380C2×D1711yescharacteristic
352C4×C4⋊C4—171no—
353C4×C4⋊C4C2×D1711yescharacteristic
354C4×C22⋊C4—341no—
355C4×C22⋊C4C2×D1711yescharacteristic
356#340—171no—
357#340—171no—
358#340—171no—
359#340—171no—
360#340—171no—
361#340—171no—
362#340—171no—
363#340—171no—
364#336—171no—
365#336—171no—
366C2×C17—42no—
367C2×C17#245911yescharacteristic
368C2×C17#310811yescharacteristic
369C2×C17#312011yescharacteristic
370C2×C17#329611yescharacteristic
371C2×C17#330011yescharacteristic
372C2×C17#331211yescharacteristic
373C2×C17#333111yescharacteristic
374C22×Q8—171no—
375C22×Q8C22×D1711yescharacteristic
376C23×C4—341no—
377C23×C4C22×D1711yescharacteristic
378C2×Q16—342no—
379C2×Q16—681no—
380C2×Q16—681no—
381C22×C8—171no—
382C22×C8—171no—
383C22×C8—171no—
384C22×C8—171no—
385#125—22no—
386#125—41no—
387C22⋊Q8—22no—
388C22⋊Q8—41no—
389C22⋊Q8—342no—
390C22⋊Q8—342no—
391C4×Q8—342no—
392C4×Q8—681no—
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—342no—
400C2×C4⋊C4—341no—
401C2×C4⋊C4—341no—
402C2×C4⋊C4—341no—
403C2×C4⋊C4—341no—
404C2×C4⋊C4—681no—
405C2×C4⋊C4C22×D1711yescharacteristic
406C2×C4⋊C4C22×D1711yescharacteristic
407C2×C4⋊C4C22×D1711yescharacteristic
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—341no—
411C2×C22⋊C4C22⋊D1711yescharacteristic
412C2×C22⋊C4C22⋊D1711yescharacteristic
413C2×C42—21no—
414C2×C42—341no—
415C2×C42—341no—
416C2×C42C22×D1711yescharacteristic
417C2×C42C22×D1711yescharacteristic
418#106—342no—
419C4⋊C8—681no—
420#103—342no—
421#103—342no—
422#103—342no—
423#103—342no—
424#103—681no—
425#103—681no—
426C22⋊C8—681no—
427C22⋊C8—681no—
428C4×C8—681no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—21no—
435#95—341no—
436#95—341no—
437#95—341no—
438#95—341no—
439#95C22⋊D1711yescharacteristic
440#95C22⋊D1711yescharacteristic
441C17#2672511yescharacteristic, complemented, Sylow
442C24C2×C22⋊D1711yescharacteristic
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—172no—
450C2×Q8—341no—
451C2×Q8—341no—
452C2×Q8—341no—
453C2×Q8—341no—
454C2×Q8—341no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—341no—
466C22×C4—341no—
467C22×C4—341no—
468C22×C4—341no—
469C22×C4—682no—
470C22×C4—682no—
471C22×C4—1361no—
472C22×C4C2×C22⋊D1711yescharacteristic
473C22×C4C2×C22⋊D1711yescharacteristic
474C22×C4C2×C22⋊D1711yescharacteristic
475C22×C4C2×C22⋊D1711yescharacteristic
476C22×C4C2×C22⋊D1711yescharacteristic
477C22×C4C23×D1711yescharacteristic
478Q16—684no—
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—
489C2×C8—341no—
490C2×C8—341no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—172no—
500C4⋊C4—172no—
501C4⋊C4—344no—
502C4⋊C4—341no—
503C4⋊C4—341no—
504C4⋊C4—341no—
505C4⋊C4—341no—
506C4⋊C4—682no—
507C4⋊C4—682no—
508C4⋊C4—1361no—
509C4⋊C4#6325412yes—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—344no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—42no—
520C42—682no—
521C42—1361no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23#1058763612yes—
529C23C22×C22⋊D1711yescharacteristic
530Q8—41no—
531Q8—41no—
532Q8—41no—
533Q8—41no—
534Q8—342no—
535Q8—342no—
536Q8—681no—
537Q8—681no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—21no—
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—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—342no—
569C2×C4—342no—
570C2×C4—342no—
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—1364no—
581C2×C4#1058763712yes—
582C2×C4C2×#6325411yescharacteristic
583C2×C4#1058769111yescharacteristic
584C2×C4#41×D1711yescharacteristic
585C2×C4#41×D1711yescharacteristic
586C2×C4#41×D1711yescharacteristic
587C2×C4#1058769611yescharacteristic
588C8—681no—
589C8—681no—
590C8—681no—
591C8—681no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22#1176331611yescharacteristic
600C22#1176344711yescharacteristic
601C22#1176345311yescharacteristic
602C22C2×#1058763611yescharacteristic
603C22C2×#1058763711yescharacteristic
604C22#1176361711yescharacteristic
605C22#1176362411yescharacteristic
606C4—22no—
607C4—41no—
608C4—42no—
609C4—41no—
610C4—41no—
611C4—41no—
612C4—81no—
613C4—81no—
614C4—81no—
615C4—682no—
616C4—682no—
617C4—682no—
618C4—1361no—
619C4#1176333811yescharacteristic
620C4#1176339311yescharacteristic
621C4#1176352111yescharacteristic
622C4#1176354111yescharacteristic
623C2—42no—
624C2#1453271711yescharacteristic
625C2#1453886011yescharacteristic
626C2#1453899211yescharacteristic
627C2#1454061311yescharacteristic
628C2#1454067111yescharacteristic
629C2#1454082911yescharacteristic
630C2#1454110011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2489C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenter#2489D17
hypocenterC17#26725
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2489

Complements

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

NH ≅ G/Ntype
2. #14530106C2semidirect
4. #14529932C2semidirect
6. #14529742C2semidirect
7. #14529463C2semidirect
8. #14529415C2semidirect
10. #14529369C2semidirect
11. #14529128C2semidirect
14. C4×#10587481C2semidirect
441. C17#26725semidirect

Maximal subgroups

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

subgroupindex
#145317872
#145301062
#145299802
#145299322
#145297922
#145297422
#145294632
#145294152
#145293922
#145293692
#145291282
#145290922
#145288682
C4×#105874812
C17×#24892
#2672517