Finite Group Databasebeta

← #17102456

Subgroups — #17102456

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#14531787C211yesmaximal, characteristic, complemented
2#14530101C211yesmaximal, characteristic, complemented
3#14529987C211yesmaximal, characteristic, complemented
4#14529932C211yesmaximal, characteristic, complemented
5#14529787C211yesmaximal, characteristic
6#14529742C211yesmaximal, characteristic, complemented
7#14529470C211yesmaximal, characteristic, complemented
8#14529401C211yesmaximal, characteristic, complemented
9#14529392C211yesmaximal, characteristic, complemented
10#14529376C211yesmaximal, characteristic, complemented
11#14529142C211yesmaximal, characteristic, complemented
12#14529102C211yesmaximal, characteristic
13#14528868C211yesmaximal, characteristic, complemented
14C4×#10587481C211yesmaximal, characteristic, complemented
15C17×#2490C211yesmaximal, characteristic
16C22×#63229C2211yescharacteristic
17C2×#10587610C2211yescharacteristic
18C2×C4×Dic34C2211yescharacteristic
19C22×C4⋊Dic17C2211yescharacteristic
20C2×#10587478C2211yescharacteristic
21C2×C4⋊#3622C2211yescharacteristic
22C2×#10587481C2211yescharacteristic
23C2×#10587481C2211yescharacteristic
24C2×#10587481C2211yescharacteristic
25C2×#10587503C2211yescharacteristic
26C2×#10587503C2211yescharacteristic
27C2×#10587503C2211yescharacteristic
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×#121C2211yescharacteristic
50C17×#397—21no—
51C17×#394—21no—
52C17×#390—21no—
53C17×#390C2211yescharacteristic
54C17×#390C2211yescharacteristic
55C17×#388—21no—
56C17×#380C2211yescharacteristic
57C4×C4⋊C4×C17C2211yescharacteristic
58C4×C22⋊C4×C17C2211yescharacteristic
59C22×D4×C17—21no—
60C22×D4×C17C2311yescharacteristic
61C23×C4×C17C2311yescharacteristic
62C23×Dic17—21no—
63C22×Dic34C2311yescharacteristic
64C17×#125—22no—
65C17×#125—41no—
66C17×#121—22no—
67C17×#121—41no—
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×C17—21no—
76C2×C22⋊C4×C17—21no—
77C2×C22⋊C4×C17—21no—
78C2×C22⋊C4×C17—21no—
79C2×C22⋊C4×C17D411yescharacteristic
80C2×C22⋊C4×C17D411yescharacteristic
81C2×C42×C17—21no—
82C2×C42×C17C2311yescharacteristic
83C2×C42×C17C2311yescharacteristic
84C2×C22⋊Dic17—21no—
85C2×#63229—22no—
86C2×#63229—41no—
87C2×#63229—41no—
88C2×C4⋊Dic17—21no—
89C2×C4⋊Dic17—22no—
90C2×C4⋊Dic17—41no—
91C2×C4⋊Dic17C2311yescharacteristic
92C2×C4⋊Dic17C2311yescharacteristic
93C2×C4⋊Dic17C2311yescharacteristic
94#10587610—22no—
95#10587610—22no—
96C2×#63225—21no—
97C2×#63225—21no—
98C2×#63225—21no—
99C2×C4×Dic17—21no—
100C2×C4×Dic17—21no—
101C22×#3622C2311yescharacteristic
102C22×#3622C2311yescharacteristic
103C22×#3622C2311yescharacteristic
104C22×#3622C2311yescharacteristic
105C4×Dic34—22no—
106C4×Dic34—41no—
107C17×#95—21no—
108C17×#95—21no—
109C17×#95D411yescharacteristic
110C17×#95D411yescharacteristic
111#10587503—22no—
112#10587503—22no—
113#10587503—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#10587478—22no—
124C4⋊#3622—41no—
125C4×#3622—41no—
126C24×C17—21no—
127C24×C17C2×D411yescharacteristic
128C2×D4×C17—21no—
129C2×D4×C17—21no—
130C2×D4×C17—21no—
131C2×D4×C17—21no—
132C2×D4×C17—21no—
133C2×D4×C17—21no—
134C2×D4×C17—42no—
135C2×D4×C17—42no—
136C2×D4×C17—81no—
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×C17—21no—
144C22×C4×C17—21no—
145C22×C4×C17—21no—
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2×D411yescharacteristic
151C22×C4×C17C2411yescharacteristic
152C22×Dic17—21no—
153C22×Dic17—21no—
154C22×Dic17—21no—
155C22×Dic17—21no—
156C22×Dic17—42no—
157C22×Dic17—42no—
158C22×Dic17—81no—
159C2×Dic34—21no—
160C2×Dic34—21no—
161C2×Dic34—21no—
162C2×Dic34—21no—
163C2×Dic34—21no—
164C2×Dic34#4112yes—
165C4⋊C4×C17—21no—
166C4⋊C4×C17—21no—
167C4⋊C4×C17—21no—
168C4⋊C4×C17—21no—
169C4⋊C4×C17—21no—
170C4⋊C4×C17—42no—
171C4⋊C4×C17—42no—
172C4⋊C4×C17#4112yes—
173C22⋊C4×C17—22no—
174C22⋊C4×C17—22no—
175C22⋊C4×C17—42no—
176C22⋊C4×C17—42no—
177C22⋊C4×C17—81no—
178C22⋊C4×C17—81no—
179C22⋊C4×C17—81no—
180C22⋊C4×C17—81no—
181C42×C17—21no—
182C42×C17—21no—
183C42×C17—21no—
184C42×C17—21no—
185C42×C17—81no—
186C22⋊Dic17—24no—
187#63229—44no—
188C4⋊Dic17—21no—
189C4⋊Dic17—21no—
190C4⋊Dic17—21no—
191C4⋊Dic17—21no—
192C4⋊Dic17—42no—
193C4⋊Dic17#4112yes—
194C4⋊Dic17#4112yes—
195#63225—24no—
196#63225—42no—
197#63225—81no—
198C4×Dic17—42no—
199C4×Dic17—81no—
200C2×#3622—21no—
201C2×#3622—21no—
202C2×#3622—21no—
203C2×#3622—21no—
204C2×#3622—21no—
205C2×#3622—21no—
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
228D4×C17—41no—
229D4×C17—41no—
230D4×C17—41no—
231D4×C17—41no—
232D4×C17—84no—
233C2×C4×C17—21no—
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—22no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—21no—
244C2×C4×C17—21no—
245C2×C4×C17—21no—
246C2×C4×C17—21no—
247C2×C4×C17—21no—
248C2×C4×C17—41no—
249C2×C4×C17—41no—
250C2×C4×C17—41no—
251C2×C4×C17—41no—
252C2×C4×C17—41no—
253C2×C4×C17—41no—
254C2×C4×C17—41no—
255C2×C4×C17—41no—
256C2×C4×C17—41no—
257C2×C4×C17—41no—
258C2×C4×C17—41no—
259C2×C4×C17—41no—
260C2×C4×C17#13712yes—
261C2×C4×C17C2×#4111yescharacteristic
262C2×C4×C17C2×#4111yescharacteristic
263C2×C4×C17C2×#4111yescharacteristic
264C2×C4×C17C2×#4111yescharacteristic
265C2×C4×C17H(5,𝔽2)11yescharacteristic
266C2×C4×C17H(5,𝔽2)11yescharacteristic
267C2×Dic17—22no—
268C2×Dic17—22no—
269C2×Dic17—22no—
270C2×Dic17—41no—
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—84no—
280Dic34—22no—
281Dic34—22no—
282Dic34—41no—
283Dic34—41no—
284#3622—41no—
285#3622—41no—
286#3622—41no—
287#3622—41no—
288#2490D1711yescharacteristic
289#2360—171no—
290#2055—171no—
291#2036—171no—
292#2026—171no—
293#2000—171no—
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—22no—
325C4×C17—41no—
326C4×C17—41no—
327C4×C17—41no—
328C4×C17—42no—
329C4×C17—41no—
330C4×C17—81no—
331C4×C17—81no—
332C4×C17#52911yescharacteristic
333C4×C17#53811yescharacteristic
334C4×C17#55511yescharacteristic
335C4×C17#55911yescharacteristic
336Dic17—42no—
337Dic17—42no—
338Dic17—42no—
339Dic17—81no—
340C22×SD16—171no—
341C2×#125C2×D1711yescharacteristic
342C2×C22⋊Q8—171no—
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×#103—171no—
351C2×#102—171no—
352C2×#102—171no—
353C2×#102—171no—
354C2×C22⋊C8—171no—
355C2×C22⋊C8—171no—
356C2×C4×C8—171no—
357#397—21no—
358#394—21no—
359#391—171no—
360#390—21no—
361#390C2×D1711yescharacteristic
362#390C2×D1711yescharacteristic
363#389—171no—
364#389—341no—
365#388—21no—
366#387—341no—
367#385—341no—
368#380C2×D1711yescharacteristic
369C4×C4⋊C4—171no—
370C4×C4⋊C4C2×D1711yescharacteristic
371C4×C22⋊C4—341no—
372C4×C22⋊C4C2×D1711yescharacteristic
373#340—171no—
374#340—171no—
375#340—171no—
376#340—171no—
377#340—171no—
378#340—171no—
379#340—171no—
380#340—171no—
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×Q8—171no—
393C22×D4—21no—
394C22×D4C22×D1711yescharacteristic
395C23×C4—341no—
396C23×C4C22×D1711yescharacteristic
397C2×SD16—342no—
398C2×SD16—681no—
399C2×SD16—681no—
400C22×C8—171no—
401C22×C8—171no—
402C22×C8—171no—
403C22×C8—171no—
404#125—22no—
405#125—41no—
406C22⋊Q8—342no—
407C22⋊Q8—342no—
408#121—22no—
409#121—41no—
410C4×Q8—342no—
411C4×Q8—681no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
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—681no—
423C2×C4⋊C4C22×D1711yescharacteristic
424C2×C4⋊C4C22×D1711yescharacteristic
425C2×C4⋊C4C22×D1711yescharacteristic
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—341no—
433C2×C22⋊C4C22⋊D1711yescharacteristic
434C2×C22⋊C4C22⋊D1711yescharacteristic
435C2×C42—21no—
436C2×C42—341no—
437C2×C42—341no—
438C2×C42C22×D1711yescharacteristic
439C2×C42C22×D1711yescharacteristic
440#107—342no—
441C4⋊C8—681no—
442#103—342no—
443#103—342no—
444#103—681no—
445#102—342no—
446#102—342no—
447#102—681no—
448C22⋊C8—681no—
449C22⋊C8—681no—
450C4×C8—681no—
451#95—21no—
452#95—21no—
453#95—341no—
454#95—341no—
455#95—341no—
456#95—341no—
457#95C22⋊D1711yescharacteristic
458#95C22⋊D1711yescharacteristic
459C17#2672711yescharacteristic, complemented, Sylow
460C24—21no—
461C24C2×C22⋊D1711yescharacteristic
462C2×Q8—172no—
463C2×Q8—341no—
464C2×Q8—341no—
465C2×Q8—341no—
466C2×Q8—341no—
467C2×Q8—341no—
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—341no—
487C22×C4—341no—
488C22×C4—341no—
489C22×C4—341no—
490C22×C4—682no—
491C22×C4—682no—
492C22×C4—1361no—
493C22×C4C2×C22⋊D1711yescharacteristic
494C22×C4C2×C22⋊D1711yescharacteristic
495C22×C4C2×C22⋊D1711yescharacteristic
496C22×C4C2×C22⋊D1711yescharacteristic
497C22×C4C2×C22⋊D1711yescharacteristic
498C22×C4C23×D1711yescharacteristic
499SD16—684no—
500C2×C8—341no—
501C2×C8—341no—
502C2×C8—341no—
503C2×C8—341no—
504C2×C8—341no—
505C2×C8—341no—
506C2×C8—341no—
507C2×C8—341no—
508C2×C8—341no—
509C2×C8—341no—
510C2×C8—341no—
511C2×C8—341no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—172no—
520C4⋊C4—172no—
521C4⋊C4—344no—
522C4⋊C4—341no—
523C4⋊C4—341no—
524C4⋊C4—341no—
525C4⋊C4—341no—
526C4⋊C4—682no—
527C4⋊C4—682no—
528C4⋊C4—1361no—
529C4⋊C4#6325412yes—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C22⋊C4—81no—
535C22⋊C4—81no—
536C22⋊C4—81no—
537C22⋊C4—81no—
538C22⋊C4—344no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—81no—
544C42—682no—
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#1058763612yes—
560C23C22×C22⋊D1711yescharacteristic
561Q8—342no—
562Q8—342no—
563Q8—681no—
564Q8—681no—
565D4—41no—
566D4—41no—
567D4—41no—
568D4—41no—
569D4—84no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—342no—
598C2×C4—342no—
599C2×C4—342no—
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—1364no—
610C2×C4#1058763712yes—
611C2×C4C2×#6325411yescharacteristic
612C2×C4#1058769111yescharacteristic
613C2×C4#41×D1711yescharacteristic
614C2×C4#41×D1711yescharacteristic
615C2×C4#41×D1711yescharacteristic
616C2×C4#1058769611yescharacteristic
617C8—681no—
618C8—681no—
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#1176331611yescharacteristic
636C22#1176344711yescharacteristic
637C22#1176345311yescharacteristic
638C22C2×#1058763611yescharacteristic
639C22C2×#1058763711yescharacteristic
640C22#1176361211yescharacteristic
641C22#1176362011yescharacteristic
642C4—22no—
643C4—41no—
644C4—41no—
645C4—42no—
646C4—41no—
647C4—41no—
648C4—81no—
649C4—81no—
650C4—682no—
651C4—682no—
652C4—682no—
653C4—1361no—
654C4#1176333811yescharacteristic
655C4#1176339311yescharacteristic
656C4#1176352111yescharacteristic
657C4#1176354111yescharacteristic
658C2—42no—
659C2—161no—
660C2#1453271711yescharacteristic
661C2#1453886011yescharacteristic
662C2#1453897711yescharacteristic
663C2#1454061911yescharacteristic
664C2#1454062811yescharacteristic
665C2#1454094911yescharacteristic
666C2#1454112211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2490C2
socleC23×C17C22×D4
prime radicalC2×C17×#121C22
hypercenter#2490D17
hypocenterC17#26727
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2490

Complements

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

NH ≅ G/Ntype
1. #14531787C2semidirect
2. #14530101C2semidirect
3. #14529987C2semidirect
4. #14529932C2semidirect
6. #14529742C2semidirect
7. #14529470C2semidirect
8. #14529401C2semidirect
9. #14529392C2semidirect
10. #14529376C2semidirect
11. #14529142C2semidirect
13. #14528868C2semidirect
14. C4×#10587481C2semidirect
459. C17#26727semidirect

Maximal subgroups

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

subgroupindex
#145317872
#145301012
#145299872
#145299322
#145297872
#145297422
#145294702
#145294012
#145293922
#145293762
#145291422
#145291022
#145288682
C4×#105874812
C17×#24902
#2672717