Finite Group Databasebeta

← #17045458

Subgroups — #17045458

Order 4352.

1308 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#121×Dic17C211yesmaximal, characteristic, complemented
2C2×#11762712C211yesmaximal, characteristic, complemented
3#14529651C212yesmaximal, complemented
4#14529469C212yesmaximal, complemented
5#14529406C212yesmaximal, complemented
6#14529134C212yesmaximal, complemented
7#14529102C211yesmaximal, characteristic, complemented
8#14528974C211yesmaximal, characteristic, complemented
9#14528715C211yesmaximal, characteristic, complemented
10C22⋊C8×Dic17C211yesmaximal, characteristic, complemented
11C2×C17×#447C211yesmaximal, characteristic
12C2×D4×Dic17—21no—
13C2×D4×Dic17C2211yescharacteristic, complemented
14C22×C4⋊Dic17C2211yescharacteristic
15C22×C22⋊Dic17—21no—
16C22×C22⋊Dic17—21no—
17C22×C4×Dic17C2211yescharacteristic
18C2×#10587489C2212yescomplemented
19C2×#10587503C2212yes—
20C2×#10587503C2212yes—
21C2×#10587503C2212yescomplemented
22C2×C22⋊#3622C2211yescharacteristic
23C2×C8×Dic17C2212yescomplemented
24#11762712—21no—
25#11762712—21no—
26#11762712C2212yes—
27#11762712C2212yes—
28#11762712C2211yescharacteristic, complemented
29#11762708C2211yescharacteristic, complemented
30#11762702—21no—
31#11762689—21no—
32#11762669C2211yescharacteristic
33C4⋊C4×Dic17C2211yescharacteristic, complemented
34C22⋊C4×Dic17—21no—
35#11762572C2212yescomplemented
36#11762571C2212yescomplemented
37#11762563C2212yescomplemented
38C23×D4×C17C2211yescharacteristic
39C22×D8×C17C2212yes—
40C2×C17×#121C2211yescharacteristic
41C17×#447C418yescomplemented
42C2×C17×#102C2212yes—
43C2×C17×C22⋊C8C2211yescharacteristic
44C25×C17—21no—
45C22×D4×C17—21no—
46C22×D4×C17—21no—
47C22×D4×C17—22no—
48C22×D4×C17—42no—
49C22×D4×C17—42no—
50C22×D4×C17C2×C414yescomplemented
51C22×D4×C17C2312yes—
52C22×D4×C17C2311yescharacteristic
53C23×C4×C17C2311yescharacteristic
54C23×Dic17—21no—
55C23×Dic17—21no—
56C23×Dic17D411yescharacteristic
57C23×Dic17D411yescharacteristic
58C2×D8×C17—42no—
59C2×D8×C17—42no—
60C2×D8×C17C2×C418yescomplemented
61C22×C8×C17C2312yes—
62C17×#121—22no—
63C17×#121C2×C414yes—
64C2×C4⋊C4×C17C2311yescharacteristic
65C2×C22⋊C4×C17—21no—
66C2×C22⋊Dic17—21no—
67C2×C22⋊Dic17—21no—
68C2×C22⋊Dic17—21no—
69C2×C22⋊Dic17—21no—
70C2×C22⋊Dic17—22no—
71C2×C22⋊Dic17—22no—
72C2×C22⋊Dic17—22no—
73C2×C22⋊Dic17—22no—
74C2×C22⋊Dic17—44no—
75C2×C22⋊Dic17—44no—
76D4×Dic17—42no—
77D4×Dic17—44no—
78C2×C4⋊Dic17—21no—
79C2×C4⋊Dic17—21no—
80C2×C4⋊Dic17—42no—
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4⋊Dic17C2312yes—
83C2×C4⋊Dic17C2311yescharacteristic
84C2×C4×Dic17—21no—
85C2×C4×Dic17—42no—
86C2×C4×Dic17D411yescharacteristic
87C2×C4×Dic17D411yescharacteristic
88C2×C4×Dic17C2311yescharacteristic
89C2×C4×Dic17C2311yescharacteristic
90C22×#3622C2312yes—
91C17×#102C2×C418yescomplemented
92C17×C22⋊C8C2×C414yescomplemented
93#10587503—42no—
94#10587503—42no—
95#10587503—42no—
96#10587502—21no—
97#10587502—21no—
98#10587502—21no—
99C22⋊#3622—22no—
100#10587489—42no—
101C8×Dic17—42no—
102C24×C17—24no—
103C24×C17—21no—
104C24×C17—22no—
105C24×C17—42no—
106C24×C17—42no—
107C24×C17—42no—
108C24×C17—42no—
109C24×C17—42no—
110C24×C17—42no—
111C24×C17C2×D411yescharacteristic
112C2×D4×C17—24no—
113C2×D4×C17—24no—
114C2×D4×C17—22no—
115C2×D4×C17—21no—
116C2×D4×C17—22no—
117C2×D4×C17—22no—
118C2×D4×C17—22no—
119C2×D4×C17—22no—
120C2×D4×C17—22no—
121C2×D4×C17—22no—
122C2×D4×C17—48no—
123C2×D4×C17—42no—
124C2×D4×C17—42no—
125C2×D4×C17—48no—
126C2×D4×C17—42no—
127C2×D4×C17—42no—
128C2×D4×C17C22×C412yes—
129C2×D4×C17C22×C414yes—
130C22×C4×C17—22no—
131C22×C4×C17—22no—
132C22×C4×C17—21no—
133C22×C4×C17C22×C412yes—
134C22×C4×C17C2×D411yescharacteristic
135C22×C4×C17#4112yes—
136C22×C4×C17C2411yescharacteristic
137C22×Dic17—22no—
138C22×Dic17—22no—
139C22×Dic17—21no—
140C22×Dic17—22no—
141C22×Dic17—21no—
142C22×Dic17—41no—
143C22×Dic17—41no—
144C22×Dic17—41no—
145C22×Dic17—42no—
146C22×Dic17—42no—
147C22×Dic17—42no—
148C22×Dic17—42no—
149C22×Dic17—42no—
150C22×Dic17—42no—
151C22×Dic17D814yescomplemented
152C22×Dic17C2×D411yescharacteristic
153C22×Dic17C2×D411yescharacteristic
154C22×Dic17C2×D411yescharacteristic
155C22×Dic17C2×D411yescharacteristic
156D8×C17—48no—
157C2×C8×C17—22no—
158C2×C8×C17—22no—
159C2×C8×C17C22×C414yes—
160C4⋊C4×C17—21no—
161C4⋊C4×C17C22×C412yes—
162C22⋊C4×C17—24no—
163C22⋊Dic17—42no—
164C22⋊Dic17—44no—
165C22⋊Dic17—42no—
166C22⋊Dic17—42no—
167C22⋊Dic17—44no—
168C22⋊Dic17—44no—
169C22⋊Dic17—42no—
170C22⋊Dic17—44no—
171C4⋊Dic17—22no—
172C4⋊Dic17—22no—
173C4⋊Dic17—22no—
174C4⋊Dic17—22no—
175C4⋊Dic17—42no—
176C4⋊Dic17—42no—
177C4×Dic17—22no—
178C4×Dic17—22no—
179C4×Dic17—41no—
180C4×Dic17—41no—
181C4×Dic17—42no—
182C2×#3622—22no—
183C2×#3622—22no—
184C2×#3622—22no—
185#22040—171nomaximal, Sylow
186C23×C17—21no—
187C23×C17—21no—
188C23×C17—21no—
189C23×C17—21no—
190C23×C17—22no—
191C23×C17—24no—
192C23×C17—48no—
193C23×C17—48no—
194C23×C17—41no—
195C23×C17—42no—
196C23×C17—42no—
197C23×C17—41no—
198C23×C17—42no—
199C23×C17—41no—
200C23×C17—41no—
201C23×C17—42no—
202C23×C17—42no—
203C23×C17—41no—
204C23×C17—41no—
205C23×C17—42no—
206C23×C17—48no—
207C23×C17—84no—
208C23×C17—84no—
209C23×C17—84no—
210C23×C17—84no—
211C23×C17C4×D412yes—
212C23×C17C2×D812yes—
213C23×C17#13512yes—
214C23×C17C22×D411yescharacteristic
215D4×C17—24no—
216D4×C17—24no—
217D4×C17—24no—
218D4×C17—48no—
219D4×C17—48no—
220D4×C17—42no—
221D4×C17—42no—
222D4×C17—42no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—24no—
226C2×C4×C17—24no—
227C2×C4×C17—22no—
228C2×C4×C17—41no—
229C2×C4×C17—41no—
230C2×C4×C17C4×D412yes—
231C2×C4×C17C23×C411yescharacteristic
232C2×C4×C17C2×#4111yescharacteristic
233C2×C4×C17H(5,𝔽2)12yes—
234C2×C4×C17H(5,𝔽2)12yes—
235C2×Dic17—21no—
236C2×Dic17—21no—
237C2×Dic17—21no—
238C2×Dic17—21no—
239C2×Dic17—21no—
240C2×Dic17—21no—
241C2×Dic17—21no—
242C2×Dic17—21no—
243C2×Dic17—21no—
244C2×Dic17—44no—
245C2×Dic17—44no—
246C2×Dic17—41no—
247C2×Dic17—42no—
248C2×Dic17—41no—
249C2×Dic17—42no—
250C2×Dic17—41no—
251C2×Dic17—41no—
252C2×Dic17—42no—
253C2×Dic17—41no—
254C2×Dic17—41no—
255C2×Dic17—84no—
256C2×Dic17—84no—
257C2×Dic17C2×D812yes—
258C2×Dic17C2×D812yes—
259C2×Dic17AGL(1,ℤ/8)12yes—
260C8×C17—24no—
261#3622—42no—
262C2×#447D1711yescharacteristic
263C4×#121—171no—
264C2×#390—171no—
265#1970—172no—
266#1934—172no—
267#1925—172no—
268#1879—172no—
269#1874—171no—
270#1852—171no—
271#1786—171no—
272C4×C22⋊C8—171no—
273C22×C17—22no—
274C22×C17—22no—
275C22×C17—22no—
276C22×C17—22no—
277C22×C17—22no—
278C22×C17—22no—
279C22×C17—44no—
280C22×C17—42no—
281C22×C17—42no—
282C22×C17—44no—
283C22×C17—42no—
284C22×C17—44no—
285C22×C17—816no—
286C22×C17—81no—
287C22×C17—82no—
288C22×C17—81no—
289C22×C17—82no—
290C22×C17—81no—
291C22×C17—82no—
292C22×C17—81no—
293C22×C17—82no—
294C22×C17C4×D814yes—
295C22×C17C2×C4×D411yescharacteristic
296C22×C17#53411yescharacteristic
297C22×C17#53511yescharacteristic
298C22×C17C22×D811yescharacteristic
299C22×C17C2×#13511yescharacteristic
300C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
301C22×C17#57511yescharacteristic
302C4×C17—22no—
303C4×C17—42no—
304C4×C17#51812yes—
305C4×C17#53812yes—
306Dic17—22no—
307Dic17—22no—
308Dic17—22no—
309Dic17—41no—
310Dic17—81no—
311Dic17—81no—
312Dic17—82no—
313C23×D4C2×D1711yescharacteristic
314C22×D8C2×D1712yes—
315C2×#121C2×D1711yescharacteristic
316C2×C4×D4—171no—
317C2×C4×D4—341no—
318C22×C4⋊C4—171no—
319C22×C22⋊C4—341no—
320C22×C22⋊C4—341no—
321C22×C42—171no—
322#447Dic1718yescomplemented
323C2×#107—172no—
324C2×#102—172no—
325C2×#102—172no—
326C2×#102—172no—
327C2×#102C2×D1712yes—
328C2×C22⋊C8—171no—
329C2×C22⋊C8C2×D1711yescharacteristic
330C2×C4×C8—172no—
331#390—172no—
332#390—172no—
333#390—171no—
334#390—341no—
335#390—341no—
336#389—171no—
337#388—341no—
338#386—341no—
339#380—171no—
340C4×C4⋊C4—171no—
341C4×C22⋊C4—341no—
342#340—172no—
343#340—172no—
344#336—172no—
345C2×C17—24no—
346C2×C17—82no—
347C2×C17—84no—
348C2×C17#232111yescharacteristic
349C2×C17C2×C4×D811yescharacteristic
350C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
351C2×C17#318611yescharacteristic
352C2×C17#318811yescharacteristic
353C2×C17#320511yescharacteristic
354C2×C17#321111yescharacteristic
355C25—21no—
356C22×D4—21no—
357C22×D4—21no—
358C22×D4—22no—
359C22×D4—42no—
360C22×D4—42no—
361C22×D4C2×Dic1714yescomplemented
362C22×D4C22×D1711yescharacteristic
363C22×D4C22×D1712yes—
364C23×C4—171no—
365C23×C4—171no—
366C23×C4—341no—
367C23×C4—341no—
368C23×C4C22×D1711yescharacteristic
369C2×D8—42no—
370C2×D8—42no—
371C2×D8C2×Dic1718yescomplemented
372C22×C8—172no—
373C22×C8C22×D1712yes—
374#121—22no—
375#121C2×Dic1714yes—
376C4×D4—682no—
377C4×D4—684no—
378C2×C4⋊C4—172no—
379C2×C4⋊C4—171no—
380C2×C4⋊C4—171no—
381C2×C4⋊C4—341no—
382C2×C4⋊C4—341no—
383C2×C4⋊C4—682no—
384C2×C4⋊C4C22×D1711yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—342no—
387C2×C22⋊C4—342no—
388C2×C22⋊C4—342no—
389C2×C22⋊C4—342no—
390C2×C22⋊C4—341no—
391C2×C22⋊C4—341no—
392C2×C22⋊C4—341no—
393C2×C22⋊C4—341no—
394C2×C22⋊C4—684no—
395C2×C22⋊C4—684no—
396C2×C42—171no—
397C2×C42—171no—
398C2×C42—171no—
399C2×C42—171no—
400C2×C42—341no—
401C2×C42—682no—
402#107—682no—
403#102—682no—
404#102—682no—
405#102—682no—
406#102C2×Dic1718yescomplemented
407C22⋊C8—342no—
408C22⋊C8C2×Dic1714yescomplemented
409C4×C8—682no—
410#95—341no—
411#95—341no—
412#95—341no—
413C17#2204011yescharacteristic, complemented, Sylow
414C24—21no—
415C24—22no—
416C24—24no—
417C24—42no—
418C24—42no—
419C24—42no—
420C24—42no—
421C24—42no—
422C24—42no—
423C24D4×D1711yescharacteristic
424C2×D4—24no—
425C2×D4—21no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—24no—
434C2×D4—48no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—48no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4C22×Dic1714yes—
441C2×D4C22×Dic1712yes—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—171no—
446C22×C4—171no—
447C22×C4—171no—
448C22×C4—174no—
449C22×C4—171no—
450C22×C4—342no—
451C22×C4—342no—
452C22×C4—341no—
453C22×C4—341no—
454C22×C4—342no—
455C22×C4—682no—
456C22×C4—681no—
457C22×C4—681no—
458C22×C4—681no—
459C22×C4—682no—
460C22×C4—682no—
461C22×C4—682no—
462C22×C4—682no—
463C22×C4—682no—
464C22×C4D4×D1711yescharacteristic
465C22×C4#6325412yes—
466C22×C4C22×Dic1712yes—
467C22×C4C23×D1711yescharacteristic
468D8—48no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—342no—
472C2×C8—342no—
473C2×C8—342no—
474C2×C8C22×Dic1714yes—
475C4⋊C4—21no—
476C4⋊C4—342no—
477C4⋊C4—342no—
478C4⋊C4—342no—
479C4⋊C4—342no—
480C4⋊C4—682no—
481C4⋊C4—682no—
482C4⋊C4C22×Dic1712yes—
483C22⋊C4—24no—
484C22⋊C4—682no—
485C22⋊C4—684no—
486C22⋊C4—684no—
487C22⋊C4—684no—
488C22⋊C4—682no—
489C22⋊C4—684no—
490C22⋊C4—682no—
491C22⋊C4—682no—
492C42—342no—
493C42—342no—
494C42—681no—
495C42—681no—
496C42—682no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—22no—
502C23—24no—
503C23—48no—
504C23—48no—
505C23—42no—
506C23—41no—
507C23—41no—
508C23—42no—
509C23—42no—
510C23—41no—
511C23—41no—
512C23—42no—
513C23—41no—
514C23—42no—
515C23—42no—
516C23—41no—
517C23—48no—
518C23—84no—
519C23—84no—
520C23—84no—
521C23—84no—
522C23D8×D1712yes—
523C23#1058759812yes—
524C23D4×Dic1712yes—
525C23C2×D4×D1711yescharacteristic
526D4—24no—
527D4—24no—
528D4—24no—
529D4—48no—
530D4—48no—
531D4—42no—
532D4—42no—
533D4—42no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—172no—
542C2×C4—172no—
543C2×C4—172no—
544C2×C4—341no—
545C2×C4—341no—
546C2×C4—341no—
547C2×C4—341no—
548C2×C4—341no—
549C2×C4—341no—
550C2×C4—341no—
551C2×C4—341no—
552C2×C4—341no—
553C2×C4—684no—
554C2×C4—684no—
555C2×C4—681no—
556C2×C4—682no—
557C2×C4—681no—
558C2×C4—682no—
559C2×C4—681no—
560C2×C4—681no—
561C2×C4—681no—
562C2×C4—682no—
563C2×C4—681no—
564C2×C4—1364no—
565C2×C4—1364no—
566C2×C4D4×Dic1712yes—
567C2×C4C2×#6325411yescharacteristic
568C2×C4#1058769112yes—
569C2×C4#1058769112yes—
570C2×C4C23×Dic1711yescharacteristic
571C8—24no—
572C8—682no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—42no—
580C22—44no—
581C22—44no—
582C22—42no—
583C22—44no—
584C22—42no—
585C22—816no—
586C22—81no—
587C22—82no—
588C22—82no—
589C22—81no—
590C22—82no—
591C22—81no—
592C22—82no—
593C22—81no—
594C22D8×Dic1714yes—
595C22C2×D4×Dic1711yescharacteristic
596C22#1176336911yescharacteristic
597C22#1176337411yescharacteristic
598C22C2×D8×D1711yescharacteristic
599C22C2×#1058759811yescharacteristic
600C22AGL(1,ℤ/8)×D1711yescharacteristic
601C22#1176360511yescharacteristic
602C4—22no—
603C4—42no—
604C4—342no—
605C4—342no—
606C4—342no—
607C4—681no—
608C4—1361no—
609C4—1362no—
610C4—1361no—
611C4#1176328412yes—
612C4#1176339412yes—
613C2—24no—
614C2—84no—
615C2—82no—
616C2#1453144511yescharacteristic
617C2C2×D8×Dic1711yescharacteristic
618C2AGL(1,ℤ/8)×Dic1711yescharacteristic
619C2#1453953311yescharacteristic
620C2#1453954611yescharacteristic
621C2#1453970011yescharacteristic
622C2#1453974611yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
Fitting, prime radicalC2×C17×#447C2
socleC23×C17C22×D4
cosocleC22×C4×C17C24
hypercenterC2×#447D17
hypocenterC17#22040
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#447

Complements

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

NH ≅ G/Ntype
1. #121×Dic17C2semidirect
2. C2×#11762712C2semidirect
3. #14529651C2semidirect
4. #14529469C2semidirect
5. #14529406C2semidirect
6. #14529134C2semidirect
7. #14529102C2semidirect
8. #14528974C2semidirect
9. #14528715C2semidirect
10. C22⋊C8×Dic17C2semidirect
13. C2×D4×Dic17C22semidirect
18. C2×#10587489C22semidirect
21. C2×#10587503C22semidirect
23. C2×C8×Dic17C22semidirect
28. #11762712C22semidirect
29. #11762708C22semidirect
33. C4⋊C4×Dic17C22semidirect
35. #11762572C22semidirect
36. #11762571C22semidirect
37. #11762563C22semidirect
41. C17×#447C4semidirect
50. C22×D4×C17C2×C4semidirect
60. C2×D8×C17C2×C4semidirect
91. C17×#102C2×C4semidirect
92. C17×C22⋊C8C2×C4semidirect
151. C22×Dic17D8semidirect
322. #447Dic17semidirect
361. C22×D4C2×Dic17semidirect
371. C2×D8C2×Dic17semidirect
406. #102C2×Dic17semidirect
408. C22⋊C8C2×Dic17semidirect
413. C17#22040semidirect

Maximal subgroups

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

subgroupindex
#121×Dic172
C2×#117627122
#145296512
#145294692
#145294062
#145291342
#145291022
#145289742
#145287152
C22⋊C8×Dic172
C2×C17×#4472
#2204017