Finite Group Databasebeta

← #16965115

Subgroups — #16965115

Order 4352.

1442 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762721C211yesmaximal, characteristic
2#14527054C212yesmaximal, complemented
3#14527043C212yesmaximal, complemented
4#14527039C212yesmaximal, complemented
5#14527039C212yesmaximal, complemented
6#14527027C212yesmaximal
7#14527006C212yesmaximal
8#14527006C211yesmaximal, characteristic
9C17×#1439C211yesmaximal, characteristic
10C23×Dic34C2211yescharacteristic
11C24×Dic17C2211yescharacteristic
12C22×#63225C2212yes—
13C22×C22⋊Dic17C2211yescharacteristic
14C22×C22⋊Dic17C2212yes—
15#11762721C418yescomplemented
16C2×#10587502C2212yes—
17C2×#10587502C2212yes—
18C2×#10587502C2212yes—
19C2×#10587502C2212yes—
20C2×#10587502C2212yes—
21C2×#10587502C2212yes—
22C2×#10587502C2212yes—
23C2×#10587502C2212yes—
24C2×#10587502C2212yes—
25C2×#10587502C2211yescharacteristic
26C2×#10587502C2212yes—
27C22×C4⋊C4×C17C2211yescharacteristic
28C22×C22⋊C4×C17C2212yes—
29C22×C22⋊C4×C17C2211yescharacteristic
30C2×C17×#95C2211yescharacteristic
31C2×C17×#95C2212yes—
32C25×C17C2311yescharacteristic
33C23×C4×C17C2312yes—
34C23×C4×C17C2312yes—
35C23×C4×C17C2311yescharacteristic
36C23×C4×C17C2311yescharacteristic
37C23×Dic17—42no—
38C23×Dic17—41no—
39C23×Dic17—41no—
40C23×Dic17—42no—
41C23×Dic17Q814yescomplemented
42C23×Dic17C2312yes—
43C23×Dic17C2312yes—
44C23×Dic17C2312yes—
45C23×Dic17C2312yes—
46C22×Dic34—41no—
47C22×Dic34—41no—
48C22×Dic34—42no—
49C22×Dic34—42no—
50C22×Dic34C2×C414yescomplemented
51C2×C4⋊C4×C17—42no—
52C2×C4⋊C4×C17—41no—
53C2×C22⋊C4×C17—24no—
54C2×C22⋊C4×C17—41no—
55C2×C22⋊C4×C17—41no—
56C2×C22⋊C4×C17—42no—
57C2×C22⋊C4×C17—42no—
58C2×C22⋊C4×C17C2×C414yes—
59C2×C22⋊Dic17—22no—
60C2×C22⋊Dic17—42no—
61C2×C22⋊Dic17—42no—
62C2×C22⋊Dic17—41no—
63C2×C22⋊Dic17—41no—
64C2×C22⋊Dic17C2×C418yes—
65C2×#63225—42no—
66C2×#63225—42no—
67C2×#63225—42no—
68C17×#95—42no—
69C17×#95—41no—
70#10587502—42no—
71#10587502—42no—
72#10587502—42no—
73#10587502—42no—
74#10587502—42no—
75#10587502—42no—
76#10587502—42no—
77#10587502—42no—
78#10587502—42no—
79#10587502C2×C418yescomplemented
80#10587502C2×C414yescomplemented
81C24×C17—22no—
82C24×C17—21no—
83C24×C17—21no—
84C24×C17—22no—
85C24×C17—22no—
86C24×C17—21no—
87C24×C17—22no—
88C24×C17—21no—
89C24×C17C22×C412yes—
90C24×C17C2×Q812yes—
91C24×C17#4112yes—
92C24×C17C2411yescharacteristic
93C22×C4×C17—22no—
94C22×C4×C17—22no—
95C22×C4×C17—21no—
96C22×C4×C17—21no—
97C22×C4×C17—22no—
98C22×C4×C17—21no—
99C22×C4×C17—21no—
100C22×C4×C17—22no—
101C22×C4×C17—22no—
102C22×C4×C17—22no—
103C22×C4×C17—22no—
104C22×C4×C17—22no—
105C22×C4×C17—22no—
106C22×C4×C17—22no—
107C22×C4×C17—22no—
108C22×C4×C17—22no—
109C22×C4×C17—22no—
110C22×C4×C17—22no—
111C22×C4×C17—22no—
112C22×C4×C17—22no—
113C22×C4×C17—22no—
114C22×C4×C17—22no—
115C22×C4×C17C22×C412yes—
116C22×C4×C17C22×C412yes—
117C22×Dic17—22no—
118C22×Dic17—22no—
119C22×Dic17—22no—
120C22×Dic17—22no—
121C22×Dic17—22no—
122C22×Dic17—22no—
123C22×Dic17—22no—
124C22×Dic17—22no—
125C22×Dic17—22no—
126C22×Dic17—22no—
127C22×Dic17—22no—
128C22×Dic17—22no—
129C22×Dic17—22no—
130C22×Dic17—22no—
131C22×Dic17—22no—
132C22×Dic17—22no—
133C22×Dic17—22no—
134C22×Dic17—22no—
135C22×Dic17—22no—
136C22×Dic17—22no—
137C22×Dic17—22no—
138C22×Dic17—22no—
139C22×Dic17—22no—
140C22×Dic17—22no—
141C22×Dic17—22no—
142C22×Dic17—48no—
143C22×Dic17—44no—
144C22×Dic17—48no—
145C22×Dic17—44no—
146C22×Dic17—44no—
147C22×Dic17C22×C414yes—
148C22×Dic17C22×C414yes—
149C22×Dic17C2×Q814yes—
150C2×Dic34—48no—
151C2×Dic34—44no—
152C2×Dic34—44no—
153C2×Dic34—48no—
154C2×Dic34—44no—
155C4⋊C4×C17—44no—
156C22⋊C4×C17—44no—
157C22⋊C4×C17—48no—
158C22⋊Dic17—48no—
159C22⋊Dic17—44no—
160#63225—48no—
161#14635—171nomaximal, Sylow
162C23×C17—24no—
163C23×C17—22no—
164C23×C17—24no—
165C23×C17—22no—
166C23×C17—22no—
167C23×C17—24no—
168C23×C17—24no—
169C23×C17—22no—
170C23×C17—22no—
171C23×C17—24no—
172C23×C17—22no—
173C23×C17—24no—
174C23×C17—41no—
175C23×C17—42no—
176C23×C17—42no—
177C23×C17—41no—
178C23×C17—41no—
179C23×C17—42no—
180C23×C17—42no—
181C23×C17—42no—
182C23×C17—42no—
183C23×C17—41no—
184C23×C17C4×Q814yes—
185C23×C17C23×C411yescharacteristic
186C23×C17C22×Q811yescharacteristic
187C23×C17C2×#4111yescharacteristic
188C23×C17H(5,𝔽2)12yes—
189C23×C17H(5,𝔽2)11yescharacteristic
190C23×C17H(5,𝔽2)11yescharacteristic
191C23×C17H(5,𝔽2)12yes—
192C23×C17H(5,𝔽2)12yes—
193C23×C17H(5,𝔽2)11yescharacteristic
194C23×C172−1+412yes—
195C23×C172−1+411yescharacteristic
196C2×C4×C17—22no—
197C2×C4×C17—22no—
198C2×C4×C17—24no—
199C2×C4×C17—24no—
200C2×C4×C17—41no—
201C2×C4×C17—42no—
202C2×C4×C17—41no—
203C2×C4×C17—41no—
204C2×C4×C17—42no—
205C2×C4×C17—41no—
206C2×C4×C17—42no—
207C2×C4×C17—42no—
208C2×C4×C17—42no—
209C2×C4×C17—42no—
210C2×C4×C17—42no—
211C2×C4×C17—42no—
212C2×C4×C17—42no—
213C2×C4×C17—42no—
214C2×C4×C17—42no—
215C2×C4×C17—42no—
216C2×C4×C17—42no—
217C2×C4×C17—42no—
218C2×C4×C17—42no—
219C2×C4×C17—42no—
220C2×Dic17—24no—
221C2×Dic17—24no—
222C2×Dic17—24no—
223C2×Dic17—24no—
224C2×Dic17—24no—
225C2×Dic17—24no—
226C2×Dic17—24no—
227C2×Dic17—24no—
228C2×Dic17—24no—
229C2×Dic17—416no—
230C2×Dic17—42no—
231C2×Dic17—42no—
232C2×Dic17—42no—
233C2×Dic17—42no—
234C2×Dic17—42no—
235C2×Dic17—42no—
236C2×Dic17—42no—
237C2×Dic17—42no—
238C2×Dic17—42no—
239C2×Dic17—42no—
240C2×Dic17—42no—
241C2×Dic17—42no—
242C2×Dic17—42no—
243C2×Dic17—42no—
244C2×Dic17—42no—
245C2×Dic17—42no—
246C2×Dic17—42no—
247C2×Dic17—42no—
248C2×Dic17—42no—
249Dic34—416no—
250C2×#393—171no—
251#1446—172no—
252#1442—172no—
253#1442—172no—
254#1442—172no—
255#1439—172no—
256#1439D1711yescharacteristic
257C22⋊#95—172no—
258C22⋊#95—171no—
259C22×C17—24no—
260C22×C17—28no—
261C22×C17—42no—
262C22×C17—44no—
263C22×C17—42no—
264C22×C17—42no—
265C22×C17—44no—
266C22×C17—42no—
267C22×C17—42no—
268C22×C17—44no—
269C22×C17—42no—
270C22×C17C2×C4×Q811yescharacteristic
271C22×C17#51812yes—
272C22×C17#51811yescharacteristic
273C22×C17#52012yes—
274C22×C17#52011yescharacteristic
275C22×C17#53812yes—
276C22×C17#53811yescharacteristic
277C22×C17#53912yes—
278C22×C17#53911yescharacteristic
279C22×C17#54312yes—
280C22×C17#54311yescharacteristic
281C22×C17#54411yescharacteristic
282C22×C17#54412yes—
283C22×C17#56011yescharacteristic
284C22×C17#56012yes—
285C22×C17H(3,𝔽4)12yes—
286C22×C17H(3,𝔽4)11yescharacteristic
287C22×C17H(3,𝔽4)11yescharacteristic
288C22×C17#56212yes—
289C22×C17#56212yes—
290C22×C17#56212yes—
291C22×C17#56312yes—
292C22×C17#56311yescharacteristic
293C4×C17—42no—
294C4×C17—42no—
295C4×C17—82no—
296C4×C17—82no—
297Dic17—44no—
298Dic17—44no—
299Dic17—44no—
300Dic17—82no—
301C23×Q8—171no—
302C24×C4—171no—
303C22×C4⋊C4—172no—
304C22×C4⋊C4C2×D1711yescharacteristic
305C22×C22⋊C4—172no—
306C22×C22⋊C4—171no—
307C22×C22⋊C4C2×D1712yes—
308C22×C22⋊C4C2×D1711yescharacteristic
309#393—178no—
310C2×#95—171no—
311C2×#95—172no—
312C2×#95—172no—
313C2×#95—172no—
314C2×#95—172no—
315C2×#95—172no—
316C2×#95—172no—
317C2×#95—172no—
318C2×#95—172no—
319C2×#95—172no—
320C2×#95—172no—
321C2×#95C2×D1711yescharacteristic
322C2×#95C2×D1712yes—
323C2×C17—44no—
324C2×C17#232911yescharacteristic
325C2×C17#232912yes—
326C2×C17#237511yescharacteristic
327C2×C17#237912yes—
328C2×C17#237911yescharacteristic
329C2×C17#282411yescharacteristic
330C2×C17#282412yes—
331C2×C17#283511yescharacteristic
332C2×C17#284011yescharacteristic
333C2×C17#284112yes—
334C2×C17#284111yescharacteristic
335C25C22×D1711yescharacteristic
336C22×Q8—174no—
337C22×Q8—682no—
338C22×Q8—681no—
339C22×Q8—681no—
340C22×Q8—682no—
341C23×C4—172no—
342C23×C4—172no—
343C23×C4—172no—
344C23×C4—174no—
345C23×C4—172no—
346C23×C4—681no—
347C23×C4—682no—
348C23×C4—682no—
349C23×C4—681no—
350C23×C4C22×D1712yes—
351C23×C4C22×D1712yes—
352C23×C4C22×D1711yescharacteristic
353C23×C4C22×D1711yescharacteristic
354C2×C4⋊C4—42no—
355C2×C4⋊C4—41no—
356C2×C4⋊C4—682no—
357C2×C4⋊C4—682no—
358C2×C4⋊C4—682no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—41no—
361C2×C22⋊C4—41no—
362C2×C22⋊C4—42no—
363C2×C22⋊C4—42no—
364C2×C22⋊C4—178no—
365C2×C22⋊C4—342no—
366C2×C22⋊C4—682no—
367C2×C22⋊C4—682no—
368C2×C22⋊C4—681no—
369C2×C22⋊C4—681no—
370C2×C22⋊C4C4×D1714yes—
371#95—41no—
372#95—42no—
373#95—174no—
374#95—178no—
375#95—682no—
376#95—682no—
377#95—682no—
378#95—682no—
379#95—682no—
380#95—682no—
381#95—682no—
382#95—682no—
383#95—682no—
384C17#1463511yescharacteristic, complemented, Sylow
385C24—22no—
386C24—21no—
387C24—21no—
388C24—22no—
389C24—22no—
390C24—21no—
391C24—22no—
392C24—21no—
393C24C2×C4×D1712yes—
394C24#6325412yes—
395C24Q8×D1712yes—
396C24C23×D1711yescharacteristic
397C2×Q8—688no—
398C2×Q8—684no—
399C2×Q8—684no—
400C2×Q8—688no—
401C2×Q8—684no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—174no—
425C22×C4—174no—
426C22×C4—174no—
427C22×C4—342no—
428C22×C4—342no—
429C22×C4—342no—
430C22×C4—342no—
431C22×C4—342no—
432C22×C4—342no—
433C22×C4—342no—
434C22×C4—342no—
435C22×C4—342no—
436C22×C4—342no—
437C22×C4—342no—
438C22×C4—342no—
439C22×C4—342no—
440C22×C4—342no—
441C22×C4—342no—
442C22×C4—342no—
443C22×C4—342no—
444C22×C4—342no—
445C22×C4—342no—
446C22×C4—342no—
447C22×C4—342no—
448C22×C4—342no—
449C22×C4—342no—
450C22×C4—342no—
451C22×C4—342no—
452C22×C4—688no—
453C22×C4—684no—
454C22×C4—684no—
455C22×C4—688no—
456C22×C4—684no—
457C22×C4C2×C4×D1712yes—
458C22×C4C2×C4×D1712yes—
459C4⋊C4—44no—
460C4⋊C4—688no—
461C22⋊C4—44no—
462C22⋊C4—48no—
463C22⋊C4—688no—
464C22⋊C4—684no—
465C23—24no—
466C23—22no—
467C23—24no—
468C23—22no—
469C23—24no—
470C23—22no—
471C23—24no—
472C23—22no—
473C23—22no—
474C23—24no—
475C23—22no—
476C23—24no—
477C23—41no—
478C23—42no—
479C23—41no—
480C23—42no—
481C23—41no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—41no—
487C23#1058757314yes—
488C23C22×C4×D1711yescharacteristic
489C23C2×#6325411yescharacteristic
490C23#1058769112yes—
491C23#1058769111yescharacteristic
492C23#1058769111yescharacteristic
493C23#1058769112yes—
494C23C2×Q8×D1711yescharacteristic
495C23#1058769612yes—
496C23#1058769611yescharacteristic
497C23#1058769712yes—
498C23#1058769711yescharacteristic
499Q8—6816no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—344no—
525C2×C4—344no—
526C2×C4—344no—
527C2×C4—344no—
528C2×C4—344no—
529C2×C4—344no—
530C2×C4—344no—
531C2×C4—344no—
532C2×C4—344no—
533C2×C4—6816no—
534C2×C4—682no—
535C2×C4—682no—
536C2×C4—682no—
537C2×C4—682no—
538C2×C4—682no—
539C2×C4—682no—
540C2×C4—682no—
541C2×C4—682no—
542C2×C4—682no—
543C2×C4—682no—
544C2×C4—682no—
545C2×C4—682no—
546C2×C4—682no—
547C2×C4—682no—
548C2×C4—682no—
549C2×C4—682no—
550C2×C4—682no—
551C2×C4—682no—
552C2×C4—682no—
553C22—24no—
554C22—28no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—44no—
559C22—42no—
560C22—44no—
561C22—42no—
562C22—42no—
563C22—44no—
564C22C2×#1058757311yescharacteristic
565C22#1176328112yes—
566C22#1176328211yescharacteristic
567C22#1176328812yes—
568C22#1176328911yescharacteristic
569C22#1176339412yes—
570C22#1176339611yescharacteristic
571C22#1176340312yes—
572C22#1176340411yescharacteristic
573C22#1176343412yes—
574C22#1176343411yescharacteristic
575C22#1176343811yescharacteristic
576C22#1176343812yes—
577C22#1176354611yescharacteristic
578C22#1176354712yes—
579C22#1176355111yescharacteristic
580C22#1176355112yes—
581C22#1176355211yescharacteristic
582C22#1176355412yes—
583C22#1176355412yes—
584C22#1176355612yes—
585C22#1176356112yes—
586C22#1176356211yescharacteristic
587C4—42no—
588C4—42no—
589C4—82no—
590C4—82no—
591C4—684no—
592C4—684no—
593C4—684no—
594C4—1362no—
595C2—44no—
596C2#1453151411yescharacteristic
597C2#1453151512yes—
598C2#1453192111yescharacteristic
599C2#1453195312yes—
600C2#1453195411yescharacteristic
601C2#1453686611yescharacteristic
602C2#1453686712yes—
603C2#1453697411yescharacteristic
604C2#1453703611yescharacteristic
605C2#1453704512yes—
606C2#1453704611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1439C2
socle, cosocleC24×C17C24
prime radicalC25×C17C23
hypercenter#1439D17
hypocenterC17#14635
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
2. #14527054C2semidirect
3. #14527043C2semidirect
4. #14527039C2semidirect
5. #14527039C2semidirect
15. #11762721C4semidirect
41. C23×Dic17Q8semidirect
50. C22×Dic34C2×C4semidirect
79. #10587502C2×C4semidirect
80. #10587502C2×C4semidirect
384. C17#14635semidirect

Maximal subgroups

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

subgroupindex
C2×#117627212
#145270542
#145270432
#145270392
#145270272
#145270062
C17×#14392
#1463517