Finite Group Databasebeta

← #17068211

Subgroups — #17068211

Order 4352.

1434 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763086C211yesmaximal, characteristic, complemented
2C2×C4×#63228C211yesmaximal, characteristic, complemented
3#14533697C211yesmaximal, characteristic, complemented
4#14529963C212yesmaximal, complemented
5#14529780C212yesmaximal, complemented
6#14529746C211yesmaximal, characteristic, complemented
7#14529579C211yesmaximal, characteristic, complemented
8#14529101C212yesmaximal, complemented
9#14528972C212yesmaximal, complemented
10#14528784C211yesmaximal, characteristic, complemented
11C2×C17×#547C211yesmaximal, characteristic, complemented
12C22×#63228C2211yescharacteristic
13C2×#10587616C2212yes—
14C2×#10587624—21no—
15C2×C4×D68C2211yescharacteristic
16#11763086—24no—
17C4×#63228—24no—
18C2×#10587478C2212yes—
19C2×#10587478C2211yescharacteristic
20C2×C4⋊#3622C2211yescharacteristic
21C2×#10587503C2211yescharacteristic
22C2×#10587503C2212yes—
23C2×#10587480C2212yes—
24C2×#10587480C2211yescharacteristic
25C2×C22⋊#3622C2212yes—
26C2×C22⋊#3622C2212yes—
27C2×C4×#3622C2211yescharacteristic
28#11762725—22no—
29#11762679C2211yescharacteristic
30#11762669C2212yes—
31#11762666—21no—
32#11762571C2212yes—
33#11762571C2212yes—
34#11762571C2212yes—
35C22×#41×C17—21no—
36C17×#547—24no—
37C17×#547—24no—
38C2×C17×C4⋊Q8C2211yescharacteristic
39C2×C17×#123—22no—
40C2×C17×C22⋊Q8—21no—
41C2×C17×#121C2212yes—
42C2×C4×D4×C17C2211yescharacteristic
43C2×C4×D4×C17C2211yescharacteristic
44C22×C4⋊C4×C17C2212yes—
45C2×#41×C17—28no—
46C2×#41×C17—44no—
47C2×#41×C17—44no—
48C22×Q8×C17—21no—
49C22×D4×C17—22no—
50C22×D4×C17D411yescharacteristic
51C22×D4×C17C2311yescharacteristic
52C23×C4×C17—22no—
53C23×C4×C17—21no—
54C23×C4×C17C2312yes—
55C22×C22⋊D17—22no—
56C22×D68C2311yescharacteristic
57C22×C4×D17—21no—
58C17×C4⋊Q8—22no—
59C17×C4⋊Q8D414yescomplemented
60C17×#123—216no—
61C17×C22⋊Q8—28no—
62C17×#121—24no—
63C17×#121—24no—
64C4×D4×C17—22no—
65C4×D4×C17—22no—
66C4×D4×C17—22no—
67C4×D4×C17D414yescomplemented
68C2×C4⋊C4×C17—22no—
69C2×C4⋊C4×C17—21no—
70C2×C4⋊C4×C17—24no—
71C2×C4⋊C4×C17—24no—
72C2×C4⋊C4×C17D411yescharacteristic
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C4⋊C4×C17C2312yes—
75C2×C22⋊C4×C17—21no—
76C2×C22⋊C4×C17—22no—
77C2×C22⋊C4×C17D412yes—
78C2×C42×C17C2311yescharacteristic
79C2×C22⋊Dic17—22no—
80#10587624—44no—
81C2×#63228—22no—
82C2×#63228—22no—
83C2×#63228—22no—
84#10587616—24no—
85#10587616—42no—
86C2×#63227—21no—
87C2×#63227—22no—
88C2×C4⋊Dic17—21no—
89C2×C4⋊Dic17—21no—
90C2×C4⋊Dic17C2311yescharacteristic
91C2×C4⋊Dic17C2312yes—
92C22×#3622C2311yescharacteristic
93C22×#3622C2312yes—
94C22×#3622C2311yescharacteristic
95C4×D68—22no—
96C4×D68—22no—
97#10587503—22no—
98#10587503—42no—
99#10587502—22no—
100C22⋊#3622—24no—
101C22⋊#3622—42no—
102#10587480—24no—
103#10587480—22no—
104#10587478—24no—
105#10587478—22no—
106C4⋊#3622—22no—
107C4×#3622—22no—
108C24×C17—21no—
109C24×C17C2×D412yes—
110C23×D17—21no—
111#41×C17—432no—
112C2×Q8×C17—24no—
113C2×Q8×C17—42no—
114C2×Q8×C17—42no—
115C2×D4×C17—28no—
116C2×D4×C17—22no—
117C2×D4×C17—22no—
118C2×D4×C17—21no—
119C2×D4×C17—21no—
120C2×D4×C17—21no—
121C2×D4×C17—21no—
122C2×D4×C17—21no—
123C2×D4×C17—44no—
124C2×D4×C17—44no—
125C2×D4×C17D814yescomplemented
126C2×D4×C17C2×D412yes—
127C22×C4×C17—28no—
128C22×C4×C17—21no—
129C22×C4×C17—22no—
130C22×C4×C17—24no—
131C22×C4×C17—22no—
132C22×C4×C17—22no—
133C22×C4×C17—22no—
134C22×C4×C17—22no—
135C22×C4×C17—22no—
136C22×C4×C17—22no—
137C22×C4×C17—44no—
138C22×C4×C17—44no—
139C22×C4×C17—42no—
140C22×C4×C17—42no—
141C22×C4×C17C2×D412yes—
142C22×C4×C17C2×D411yescharacteristic
143C22×C4×C17C2×D411yescharacteristic
144C22×C4×C17C2411yescharacteristic
145C2×C22⋊D17—44no—
146C2×C22⋊D17—44no—
147C2×C22⋊D17—82no—
148C22×Dic17—21no—
149C22×Dic17—22no—
150C2×D68—21no—
151C2×D68—21no—
152C2×D68—21no—
153C2×D68—21no—
154C2×D68—21no—
155C2×D68#4112yescomplemented
156C2×C4×D17—42no—
157C2×C4×D17—42no—
158C2×C4×D17—42no—
159C4⋊C4×C17—28no—
160C4⋊C4×C17—22no—
161C4⋊C4×C17—24no—
162C4⋊C4×C17—22no—
163C4⋊C4×C17—21no—
164C4⋊C4×C17C2×D414yes—
165C4⋊C4×C17C2×D412yes—
166C22⋊C4×C17—28no—
167C22⋊C4×C17—24no—
168C22⋊C4×C17—24no—
169C42×C17C2×D412yes—
170C42×C17C2×D412yes—
171C22⋊Dic17—44no—
172#63228—28no—
173#63227—44no—
174#63227—42no—
175C4⋊Dic17—21no—
176C4⋊Dic17—21no—
177C4⋊Dic17—22no—
178C4⋊Dic17—22no—
179C4⋊Dic17—42no—
180C4⋊Dic17—81no—
181C2×#3622—21no—
182C2×#3622—21no—
183C2×#3622—21no—
184C2×#3622—22no—
185C2×#3622—22no—
186C2×#3622—21no—
187C2×#3622—22no—
188C2×#3622—21no—
189C2×#3622#4112yescomplemented
190#24116—171nomaximal, Sylow
191C23×C17—22no—
192C23×C17—22no—
193C23×C17—22no—
194C23×C17—22no—
195C23×C17—22no—
196C23×C17—22no—
197C23×C17—22no—
198C23×C17—41no—
199C23×C17—41no—
200C23×C17—41no—
201C23×C17—41no—
202C23×C17—41no—
203C23×C17—41no—
204C23×C17C2×D814yes—
205C23×C17C22×D411yescharacteristic
206C22×D17—22no—
207C22×D17—41no—
208C22×D17—41no—
209C22×D17—41no—
210C22×D17—41no—
211C22×D17—41no—
212C22×D17—41no—
213Q8×C17—48no—
214D4×C17—22no—
215D4×C17—22no—
216D4×C17—28no—
217D4×C17—416no—
218D4×C17—41no—
219D4×C17—41no—
220C2×C4×C17—21no—
221C2×C4×C17—21no—
222C2×C4×C17—22no—
223C2×C4×C17—22no—
224C2×C4×C17—24no—
225C2×C4×C17—22no—
226C2×C4×C17—22no—
227C2×C4×C17—416no—
228C2×C4×C17—41no—
229C2×C4×C17—42no—
230C2×C4×C17—41no—
231C2×C4×C17—42no—
232C2×C4×C17—42no—
233C2×C4×C17—42no—
234C2×C4×C17—42no—
235C2×C4×C17—42no—
236C2×C4×C17—48no—
237C2×C4×C17#12012yes—
238C2×C4×C17#12012yes—
239C2×C4×C17C2×D812yes—
240C2×C4×C17AGL(1,ℤ/8)12yes—
241C2×C4×C17C22×D411yescharacteristic
242C2×C4×C17C22×D411yescharacteristic
243C2×C4×C17C2×#4111yescharacteristic
244C2×C4×C17C2×#4111yescharacteristic
245C2×C4×C17H(5,𝔽2)11yescharacteristic
246C2×C4×C172−1+411yescharacteristic
247C22⋊D17—88no—
248C2×Dic17—41no—
249C2×Dic17—42no—
250C2×Dic17—41no—
251C2×Dic17—42no—
252C2×Dic17—41no—
253C2×Dic17—42no—
254D68—22no—
255D68—22no—
256D68—41no—
257D68—41no—
258C4×D17—84no—
259#3622—22no—
260#3622—41no—
261#3622—42no—
262C2×#547D1711yescharacteristic, complemented
263C2×#474—171no—
264C2×C4×D8—171no—
265#2537—171no—
266#2033—172no—
267#1999—172no—
268#1990—171no—
269#1954—171no—
270#1874—172no—
271#1852—172no—
272#1802—171no—
273C22×C17—24no—
274C22×C17—24no—
275C22×C17—24no—
276C22×C17—42no—
277C22×C17—42no—
278C22×C17—42no—
279C22×C17—42no—
280C22×C17—42no—
281C22×C17—42no—
282C22×C17—42no—
283C22×C17—81no—
284C22×C17—81no—
285C22×C17—81no—
286C22×C17—81no—
287C22×C17C2×#12011yescharacteristic
288C22×C17#54611yescharacteristic
289C22×C17#54711yescharacteristic
290C22×C17C22×D811yescharacteristic
291C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
292C22×C17#57711yescharacteristic
293C22×C17#57811yescharacteristic
294C2×D17—42no—
295C2×D17—42no—
296C2×D17—42no—
297C2×D17—81no—
298C2×D17—81no—
299C2×D17—81no—
300C2×D17—81no—
301C4×C17—22no—
302C4×C17—42no—
303C4×C17—42no—
304C4×C17—44no—
305C4×C17#44714yes—
306C4×C17#52511yescharacteristic
307C4×C17#53711yescharacteristic
308C4×C17#54811yescharacteristic
309C4×C17#55011yescharacteristic
310Dic17—81no—
311Dic17—82no—
312C22×#41—21no—
313C22×D8—171no—
314#547—24no—
315#547—24no—
316C2×C4⋊Q8C2×D1711yescharacteristic
317C2×#123—22no—
318C2×C22⋊Q8—21no—
319C2×#121—172no—
320C2×#121—341no—
321C2×#121C2×D1712yes—
322C2×C4×D4—171no—
323C2×C4×D4C2×D1711yescharacteristic
324C2×C4×D4C2×D1711yescharacteristic
325C22×C4⋊C4C2×D1712yes—
326#474—344no—
327C4×D8—344no—
328C2×#107—171no—
329C2×#107—172no—
330C2×C4⋊C8—171no—
331C2×#102—171no—
332C2×#102—171no—
333C2×#102—172no—
334C2×#102—172no—
335C2×C22⋊C8—172no—
336C2×C22⋊C8—172no—
337C2×C4×C8—171no—
338#394—342no—
339#384—171no—
340#380—172no—
341#380—341no—
342#340—172no—
343#340—172no—
344#340—172no—
345C2×C17—44no—
346C2×C17—82no—
347C2×C17#240711yescharacteristic
348C2×C17C2×#44711yescharacteristic
349C2×C17#301311yescharacteristic
350C2×C17#329411yescharacteristic
351C2×C17#333011yescharacteristic
352C2×C17#333411yescharacteristic
353C2×C17#335411yescharacteristic
354D17—82no—
355C2×#41—28no—
356C2×#41—44no—
357C2×#41—44no—
358C22×Q8—21no—
359C22×D4—22no—
360C22×D4—171no—
361C22×D4—342no—
362C22×D4C22⋊D1711yescharacteristic
363C22×D4C22×D1711yescharacteristic
364C23×C4—22no—
365C23×C4—21no—
366C23×C4—341no—
367C23×C4C22×D1712yes—
368C2×D8—342no—
369C2×D8—342no—
370C2×D8—342no—
371C22×C8—171no—
372C22×C8—171no—
373C22×C8—172no—
374C4⋊Q8—22no—
375C4⋊Q8C22⋊D1714yescomplemented
376#123—216no—
377C22⋊Q8—28no—
378#121—24no—
379#121—24no—
380#121—344no—
381#121—682no—
382#121—684no—
383C4×D4—22no—
384C4×D4—22no—
385C4×D4—22no—
386C4×D4—342no—
387C4×D4—342no—
388C4×D4C22⋊D1714yescomplemented
389C2×C4⋊C4—22no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—24no—
392C2×C4⋊C4—24no—
393C2×C4⋊C4—171no—
394C2×C4⋊C4—172no—
395C2×C4⋊C4—341no—
396C2×C4⋊C4—341no—
397C2×C4⋊C4C22⋊D1711yescharacteristic
398C2×C4⋊C4C22×D1711yescharacteristic
399C2×C4⋊C4C22×D1712yes—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—22no—
402C2×C22⋊C4—342no—
403C2×C22⋊C4—341no—
404C2×C22⋊C4—342no—
405C2×C22⋊C4C22⋊D1712yes—
406C2×C42C22×D1711yescharacteristic
407#107—342no—
408#107—344no—
409C4⋊C8—342no—
410#102—342no—
411#102—342no—
412#102—344no—
413#102—682no—
414C22⋊C8—344no—
415C22⋊C8—682no—
416C4×C8—342no—
417#95—342no—
418C17#2411611yescharacteristic, complemented, Sylow
419C24—21no—
420C24—341no—
421C24C2×C22⋊D1712yes—
422#41—432no—
423C2×Q8—24no—
424C2×Q8—42no—
425C2×Q8—42no—
426C2×D4—28no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—172no—
437C2×D4—341no—
438C2×D4—341no—
439C2×D4—341no—
440C2×D4—341no—
441C2×D4—341no—
442C2×D4—684no—
443C2×D4—684no—
444C2×D4—1362no—
445C2×D4#6322814yescomplemented
446C2×D4C2×C22⋊D1712yes—
447C22×C4—28no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—341no—
462C22×C4—342no—
463C22×C4—682no—
464C22×C4—682no—
465C22×C4—682no—
466C22×C4C2×C22⋊D1712yes—
467C22×C4C2×C22⋊D1711yescharacteristic
468C22×C4C2×C22⋊D1711yescharacteristic
469C22×C4C23×D1711yescharacteristic
470D8—348no—
471C2×C8—172no—
472C2×C8—341no—
473C2×C8—341no—
474C2×C8—342no—
475C2×C8—341no—
476C2×C8—342no—
477C2×C8—341no—
478C2×C8—341no—
479C2×C8—342no—
480C4⋊C4—28no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—342no—
486C4⋊C4—342no—
487C4⋊C4—341no—
488C4⋊C4—341no—
489C4⋊C4—682no—
490C4⋊C4—1361no—
491C4⋊C4C2×C22⋊D1712yes—
492C4⋊C4C2×C22⋊D1714yes—
493C22⋊C4—28no—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—684no—
497C22⋊C4—684no—
498C22⋊C4—682no—
499C42C2×C22⋊D1712yes—
500C42C2×C22⋊D1712yes—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—342no—
515C23—681no—
516C23—681no—
517C23—681no—
518C23—681no—
519C23—681no—
520C23—681no—
521C23C2×#6322814yes—
522C23C22×C22⋊D1711yescharacteristic
523Q8—48no—
524D4—22no—
525D4—22no—
526D4—28no—
527D4—416no—
528D4—41no—
529D4—41no—
530D4—342no—
531D4—342no—
532D4—681no—
533D4—681no—
534D4—1368no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—416no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—48no—
552C2×C4—681no—
553C2×C4—681no—
554C2×C4—682no—
555C2×C4—681no—
556C2×C4—682no—
557C2×C4—682no—
558C2×C4—1364no—
559C2×C4C2×#6322812yes—
560C2×C4#1058761812yes—
561C2×C4#1058763912yes—
562C2×C4#1058763912yes—
563C2×C4#41×D1711yescharacteristic
564C2×C4#41×D1711yescharacteristic
565C2×C4#1058769611yescharacteristic
566C2×C4#1058769711yescharacteristic
567C2×C4C22×C22⋊D1711yescharacteristic
568C2×C4C22×C22⋊D1711yescharacteristic
569C8—342no—
570C8—682no—
571C8—681no—
572C22—24no—
573C22—24no—
574C22—24no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—682no—
587C22—682no—
588C22—682no—
589C22—1361no—
590C22—1361no—
591C22—1361no—
592C22—1361no—
593C22C2×#1058763911yescharacteristic
594C22#1176344711yescharacteristic
595C22#1176345811yescharacteristic
596C22C22×#6322811yescharacteristic
597C22C2×#1058761811yescharacteristic
598C22#1176361711yescharacteristic
599C22#1176362411yescharacteristic
600C4—22no—
601C4—42no—
602C4—42no—
603C4—44no—
604C4—1362no—
605C4—1361no—
606C4#1176294314yes—
607C4#1176331611yescharacteristic
608C4#1176338711yescharacteristic
609C4#1176346611yescharacteristic
610C4#1176347811yescharacteristic
611C2—44no—
612C2—82no—
613C2—1362no—
614C2#1453212311yescharacteristic
615C2C2×#1176294311yescharacteristic
616C2#1453809911yescharacteristic
617C2#1454059111yescharacteristic
618C2#1454108311yescharacteristic
619C2#1454112711yescharacteristic
620C2#1454137911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#547C2
socleC23×C17C22×D4
hypercenterC2×#547D17
hypocenterC17#24116
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. C2×#11763086C2semidirect
2. C2×C4×#63228C2semidirect
3. #14533697C2semidirect
4. #14529963C2semidirect
5. #14529780C2semidirect
6. #14529746C2semidirect
7. #14529579C2semidirect
8. #14529101C2semidirect
9. #14528972C2semidirect
10. #14528784C2semidirect
11. C2×C17×#547C2semidirect
59. C17×C4⋊Q8D4semidirect
67. C4×D4×C17D4semidirect
125. C2×D4×C17D8semidirect
155. C2×D68#41semidirect
189. C2×#3622#41semidirect
262. C2×#547D17semidirect
375. C4⋊Q8C22⋊D17semidirect
388. C4×D4C22⋊D17semidirect
418. C17#24116semidirect
445. C2×D4#63228semidirect

Maximal subgroups

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

subgroupindex
C2×#117630862
C2×C4×#632282
#145336972
#145299632
#145297802
#145297462
#145295792
#145291012
#145289722
#145287842
C2×C17×#5472
#2411617