Finite Group Databasebeta

← #16962724

Subgroups — #16962724

Order 4352.

2018 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
1C2×#11762689C212yesmaximal
2C2×#11762684C212yesmaximal, complemented
3#14527037C212yesmaximal, complemented
4#14527029C212yesmaximal, complemented
5#14527029C212yesmaximal, complemented
6#14527014C211yesmaximal, characteristic, complemented
7#14527011C211yesmaximal, characteristic, complemented
8#14527006C212yesmaximal, complemented
9C2×C17×#390C211yesmaximal, characteristic
10C24×Dic17C2212yes—
11C22×C4⋊Dic17C2212yes—
12C22×C4⋊Dic17C2212yes—
13C22×C22⋊Dic17C2212yes—
14C22×C22⋊Dic17C2212yes—
15C22×C22⋊Dic17C2212yes—
16#11762689—28no—
17#11762684—28no—
18C2×#10587502C2212yes—
19C2×#10587502C2211yescharacteristic
20C2×#10587502C2212yes—
21C2×#10587502C2212yes—
22C2×#10587502C2211yescharacteristic
23C2×#10587502C2212yes—
24C2×#10587502C2212yes—
25C2×#10587502C2211yescharacteristic
26C2×#10587502C2212yes—
27C2×#10587502C2211yescharacteristic
28C23×D4×C17C2211yescharacteristic
29C22×C4⋊C4×C17C2211yescharacteristic
30C22×C22⋊C4×C17C2212yes—
31C22×C22⋊C4×C17C2212yes—
32C22×C42×C17C2211yescharacteristic
33C17×#390C418yescomplemented
34C25×C17C2312yes—
35C22×D4×C17—42no—
36C22×D4×C17—42no—
37C22×D4×C17C2×C414yes—
38C22×D4×C17D414yes—
39C22×D4×C17Q814yes—
40C23×C4×C17C2311yescharacteristic
41C23×C4×C17C2312yes—
42C23×C4×C17C2311yescharacteristic
43C23×C4×C17C2311yescharacteristic
44C23×Dic17—24no—
45C23×Dic17—24no—
46C23×Dic17—24no—
47C23×Dic17—42no—
48C23×Dic17—42no—
49C23×Dic17—42no—
50C23×Dic17—42no—
51C23×Dic17C2312yes—
52C23×Dic17C2312yes—
53C23×Dic17C2312yes—
54C23×Dic17C2312yes—
55C2×C4⋊C4×C17—42no—
56C2×C4⋊C4×C17C2×C414yescomplemented
57C2×C22⋊C4×C17—24no—
58C2×C22⋊C4×C17—24no—
59C2×C22⋊C4×C17—42no—
60C2×C22⋊C4×C17—42no—
61C2×C22⋊C4×C17C2×C418yescomplemented
62C2×C22⋊C4×C17C2×C418yescomplemented
63C2×C42×C17—42no—
64C2×C42×C17C2×C414yescomplemented
65C2×C22⋊Dic17—24no—
66C2×C22⋊Dic17—24no—
67C2×C22⋊Dic17—24no—
68C2×C22⋊Dic17—24no—
69C2×C22⋊Dic17—24no—
70C2×C22⋊Dic17—42no—
71C2×C22⋊Dic17—42no—
72C2×C22⋊Dic17—42no—
73C2×C22⋊Dic17—42no—
74C2×C4⋊Dic17—24no—
75C2×C4⋊Dic17—24no—
76C2×C4⋊Dic17—24no—
77C2×C4⋊Dic17—42no—
78C2×C4⋊Dic17—42no—
79C2×C4⋊Dic17—42no—
80#10587502—24no—
81#10587502—24no—
82#10587502—22no—
83#10587502—24no—
84#10587502—41no—
85#10587502—42no—
86#10587502—42no—
87#10587502—42no—
88#10587502—41no—
89#10587502D414yescomplemented
90C24×C17—22no—
91C24×C17—22no—
92C24×C17—22no—
93C24×C17—22no—
94C24×C17—22no—
95C24×C17—22no—
96C24×C17—22no—
97C24×C17—22no—
98C24×C17—22no—
99C24×C17—22no—
100C24×C17—22no—
101C24×C17—22no—
102C24×C17C22×C414yes—
103C24×C17C2×D414yes—
104C24×C17C2×Q814yes—
105C24×C17C2411yescharacteristic
106C2×D4×C17—44no—
107C2×D4×C17—44no—
108C2×D4×C17—48no—
109C2×D4×C17—48no—
110C2×D4×C17C4⋊C4116yescomplemented
111C22×C4×C17—22no—
112C22×C4×C17—22no—
113C22×C4×C17—22no—
114C22×C4×C17—22no—
115C22×C4×C17—22no—
116C22×C4×C17—22no—
117C22×C4×C17—21no—
118C22×C4×C17—22no—
119C22×C4×C17—22no—
120C22×C4×C17—22no—
121C22×C4×C17—22no—
122C22×C4×C17—22no—
123C22×C4×C17—22no—
124C22×C4×C17—21no—
125C22×C4×C17C22×C412yes—
126C22×C4×C17C22×C414yes—
127C22×C4×C17C22×C412yes—
128C22×C4×C17C22×C412yes—
129C22×C4×C17C2×D412yes—
130C22×C4×C17C2×D412yes—
131C22×C4×C17C2×Q812yes—
132C22×C4×C17#4112yes—
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—22no—
143C22×Dic17—22no—
144C22×Dic17—22no—
145C22×Dic17—22no—
146C22×Dic17—22no—
147C22×Dic17—22no—
148C22×Dic17—22no—
149C22×Dic17—22no—
150C22×Dic17—22no—
151C22×Dic17—22no—
152C22×Dic17—22no—
153C22×Dic17—22no—
154C22×Dic17—22no—
155C22×Dic17—22no—
156C22×Dic17—22no—
157C22×Dic17—22no—
158C22×Dic17—22no—
159C22×Dic17—48no—
160C22×Dic17—48no—
161C22×Dic17—48no—
162C22×Dic17—48no—
163C22×Dic17—48no—
164C22×Dic17—48no—
165C22×Dic17—48no—
166C22×Dic17C2×D414yes—
167C22×Dic17#4114yes—
168C4⋊C4×C17—44no—
169C22⋊C4×C17—216no—
170C22⋊C4×C17—48no—
171C42×C17—44no—
172C22⋊Dic17—48no—
173C22⋊Dic17—48no—
174C22⋊Dic17—48no—
175C4⋊Dic17—48no—
176C4⋊Dic17—48no—
177#14438—171nomaximal, Sylow
178C23×C17—24no—
179C23×C17—24no—
180C23×C17—24no—
181C23×C17—24no—
182C23×C17—24no—
183C23×C17—24no—
184C23×C17—24no—
185C23×C17—24no—
186C23×C17—24no—
187C23×C17—24no—
188C23×C17—24no—
189C23×C17—24no—
190C23×C17—24no—
191C23×C17—24no—
192C23×C17—24no—
193C23×C17—24no—
194C23×C17—24no—
195C23×C17—24no—
196C23×C17—42no—
197C23×C17—42no—
198C23×C17—42no—
199C23×C17—42no—
200C23×C17—42no—
201C23×C17—42no—
202C23×C17—42no—
203C23×C17—42no—
204C23×C17—42no—
205C23×C17—42no—
206C23×C17—42no—
207C23×C17—42no—
208C23×C17—42no—
209C23×C17—42no—
210C23×C17—42no—
211C23×C17—42no—
212C23×C17C2×C4⋊C418yes—
213C23×C17C23×C411yescharacteristic
214C23×C17C22×D411yescharacteristic
215C23×C17C22×D411yescharacteristic
216C23×C17C22×Q811yescharacteristic
217C23×C17C2×#4111yescharacteristic
218C23×C17C2×#4112yes—
219C23×C17H(5,𝔽2)12yes—
220C23×C17H(5,𝔽2)12yes—
221C23×C17H(5,𝔽2)12yes—
222C23×C17H(5,𝔽2)11yescharacteristic
223C23×C172−1+411yescharacteristic
224D4×C17—416no—
225C2×C4×C17—22no—
226C2×C4×C17—22no—
227C2×C4×C17—22no—
228C2×C4×C17—22no—
229C2×C4×C17—24no—
230C2×C4×C17—24no—
231C2×C4×C17—24no—
232C2×C4×C17—24no—
233C2×C4×C17—24no—
234C2×C4×C17—22no—
235C2×C4×C17—22no—
236C2×C4×C17—24no—
237C2×C4×C17—24no—
238C2×C4×C17—22no—
239C2×C4×C17—42no—
240C2×C4×C17—42no—
241C2×C4×C17—42no—
242C2×C4×C17—42no—
243C2×C4×C17—42no—
244C2×C4×C17—42no—
245C2×C4×C17C2×C4⋊C414yes—
246C2×C4×C17C4×D414yes—
247C2×Dic17—24no—
248C2×Dic17—24no—
249C2×Dic17—24no—
250C2×Dic17—24no—
251C2×Dic17—24no—
252C2×Dic17—24no—
253C2×Dic17—432no—
254C2×Dic17—42no—
255C2×Dic17—42no—
256C2×Dic17—42no—
257C2×Dic17—42no—
258C2×Dic17—42no—
259C2×Dic17—42no—
260C2×Dic17—42no—
261C2×Dic17—42no—
262C2×Dic17—42no—
263C2×Dic17—42no—
264C2×Dic17—42no—
265C2×Dic17—42no—
266C2×Dic17—42no—
267C2×Dic17—42no—
268C2×Dic17—42no—
269C2×Dic17—42no—
270C2×Dic17—42no—
271C2×Dic17—42no—
272C2×Dic17—42no—
273C2×Dic17—42no—
274C2×Dic17—42no—
275C2×Dic17—42no—
276C2×#390D1711yescharacteristic
277C2×#386—172no—
278C2×#385—172no—
279#1441—172no—
280#1439—172no—
281#1439—172no—
282C4⋊#95—171no—
283#1434—171no—
284C22⋊#95—172no—
285C22×C17—28no—
286C22×C17—28no—
287C22×C17—28no—
288C22×C17—44no—
289C22×C17—44no—
290C22×C17—44no—
291C22×C17—44no—
292C22×C17—44no—
293C22×C17—44no—
294C22×C17—44no—
295C22×C17—44no—
296C22×C17—44no—
297C22×C17—44no—
298C22×C17—44no—
299C22×C17—44no—
300C22×C17C22×C4⋊C411yescharacteristic
301C22×C17C2×C4×D411yescharacteristic
302C22×C17C4×#4111yescharacteristic
303C22×C17#51811yescharacteristic
304C22×C17#51811yescharacteristic
305C22×C17#51811yescharacteristic
306C22×C17#52011yescharacteristic
307C22×C17#53412yes—
308C22×C17#53512yes—
309C22×C17#53812yes—
310C22×C17#54012yes—
311C22×C17#54212yes—
312C22×C17#54312yes—
313C22×C17D4211yescharacteristic
314C22×C17#54612yes—
315C22×C17#54612yes—
316C22×C17#54711yescharacteristic
317C22×C17D4×Q811yescharacteristic
318C22×C17#55312yes—
319C22×C17#55411yescharacteristic
320C22×C17#55412yes—
321C22×C17#56012yes—
322C22×C17#56212yes—
323C4×C17—24no—
324C4×C17—24no—
325C4×C17—44no—
326C4×C17—42no—
327Dic17—44no—
328Dic17—44no—
329Dic17—82no—
330Dic17—82no—
331C23×D4C2×D1711yescharacteristic
332C24×C4—172no—
333C22×C4⋊C4—172no—
334C22×C4⋊C4—172no—
335C22×C4⋊C4C2×D1711yescharacteristic
336C22×C22⋊C4—172no—
337C22×C22⋊C4—172no—
338C22×C22⋊C4—172no—
339C22×C22⋊C4C2×D1712yes—
340C22×C22⋊C4C2×D1712yes—
341C22×C42C2×D1711yescharacteristic
342#390Dic1718yescomplemented
343#386—348no—
344#385—348no—
345C2×#95—171no—
346C2×#95—172no—
347C2×#95—171no—
348C2×#95—172no—
349C2×#95—172no—
350C2×#95—171no—
351C2×#95—172no—
352C2×#95—172no—
353C2×#95—171no—
354C2×#95—172no—
355C2×C17—48no—
356C2×C17#231511yescharacteristic
357C2×C17#232811yescharacteristic
358C2×C17#233711yescharacteristic
359C2×C17D4×C4⋊C411yescharacteristic
360C2×C17#234911yescharacteristic
361C2×C17#236411yescharacteristic
362C2×C17#237711yescharacteristic
363C2×C17#267812yes—
364C2×C17#269212yes—
365C2×C17#270112yes—
366C2×C17#278012yes—
367C25C22×D1712yes—
368C22×D4—42no—
369C22×D4—42no—
370C22×D4Dic3414yes—
371C22×D4D6814yes—
372C22×D4C2×Dic1714yes—
373C23×C4—172no—
374C23×C4—172no—
375C23×C4—172no—
376C23×C4—172no—
377C23×C4—344no—
378C23×C4—344no—
379C23×C4—344no—
380C23×C4—682no—
381C23×C4—682no—
382C23×C4—682no—
383C23×C4—682no—
384C23×C4C22×D1712yes—
385C23×C4C22×D1711yescharacteristic
386C23×C4C22×D1711yescharacteristic
387C23×C4C22×D1711yescharacteristic
388C2×C4⋊C4—42no—
389C2×C4⋊C4—344no—
390C2×C4⋊C4—344no—
391C2×C4⋊C4—344no—
392C2×C4⋊C4—682no—
393C2×C4⋊C4—682no—
394C2×C4⋊C4—682no—
395C2×C4⋊C4C2×Dic1714yescomplemented
396C2×C22⋊C4—24no—
397C2×C22⋊C4—24no—
398C2×C22⋊C4—42no—
399C2×C22⋊C4—42no—
400C2×C22⋊C4—344no—
401C2×C22⋊C4—344no—
402C2×C22⋊C4—344no—
403C2×C22⋊C4—344no—
404C2×C22⋊C4—344no—
405C2×C22⋊C4—682no—
406C2×C22⋊C4—682no—
407C2×C22⋊C4—682no—
408C2×C22⋊C4—682no—
409C2×C22⋊C4C2×Dic1718yescomplemented
410C2×C22⋊C4C2×Dic1718yescomplemented
411C2×C42—42no—
412C2×C42C2×Dic1714yescomplemented
413#95—174no—
414#95—344no—
415#95—344no—
416#95—344no—
417#95—342no—
418#95—682no—
419#95—682no—
420#95—682no—
421#95—681no—
422#95—681no—
423C17#1443811yescharacteristic, complemented, Sylow
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24—22no—
430C24—22no—
431C24—22no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24C2×Dic3414yes—
437C24C2×D6814yes—
438C24C22×Dic1714yes—
439C24C23×D1711yescharacteristic
440C2×D4—48no—
441C2×D4—48no—
442C2×D4—44no—
443C2×D4—44no—
444C2×D4C4⋊Dic17116yescomplemented
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—174no—
460C22×C4—174no—
461C22×C4—342no—
462C22×C4—342no—
463C22×C4—342no—
464C22×C4—342no—
465C22×C4—342no—
466C22×C4—342no—
467C22×C4—342no—
468C22×C4—342no—
469C22×C4—342no—
470C22×C4—342no—
471C22×C4—342no—
472C22×C4—342no—
473C22×C4—342no—
474C22×C4—342no—
475C22×C4—342no—
476C22×C4—342no—
477C22×C4—342no—
478C22×C4—342no—
479C22×C4—342no—
480C22×C4—342no—
481C22×C4—342no—
482C22×C4—342no—
483C22×C4—342no—
484C22×C4—342no—
485C22×C4—342no—
486C22×C4—342no—
487C22×C4—688no—
488C22×C4—688no—
489C22×C4—688no—
490C22×C4—688no—
491C22×C4—688no—
492C22×C4—688no—
493C22×C4—688no—
494C22×C4C2×Dic3412yes—
495C22×C4C2×D6812yes—
496C22×C4D4×D1712yes—
497C22×C4#6325412yes—
498C22×C4C22×Dic1712yes—
499C22×C4C22×Dic1714yes—
500C22×C4C22×Dic1712yes—
501C22×C4C22×Dic1712yes—
502C4⋊C4—44no—
503C4⋊C4—688no—
504C4⋊C4—688no—
505C22⋊C4—216no—
506C22⋊C4—48no—
507C22⋊C4—688no—
508C22⋊C4—688no—
509C22⋊C4—688no—
510C42—44no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23C2×C4⋊Dic1718yes—
546C23C22×Dic3411yescharacteristic
547C23C22×D6811yescharacteristic
548C23C2×D4×D1711yescharacteristic
549C23C2×#6325411yescharacteristic
550C23#1058769112yes—
551C23#1058769112yes—
552C23#1058769112yes—
553C23#41×D1712yes—
554C23#1058769611yescharacteristic
555C23#1058769711yescharacteristic
556C23C23×Dic1711yescharacteristic
557D4—416no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—344no—
579C2×C4—344no—
580C2×C4—344no—
581C2×C4—344no—
582C2×C4—344no—
583C2×C4—344no—
584C2×C4—6832no—
585C2×C4—682no—
586C2×C4—682no—
587C2×C4—682no—
588C2×C4—682no—
589C2×C4—682no—
590C2×C4—682no—
591C2×C4—682no—
592C2×C4—682no—
593C2×C4—682no—
594C2×C4—682no—
595C2×C4—682no—
596C2×C4—682no—
597C2×C4—682no—
598C2×C4—682no—
599C2×C4—682no—
600C2×C4—682no—
601C2×C4—682no—
602C2×C4—682no—
603C2×C4—682no—
604C2×C4—682no—
605C2×C4—682no—
606C2×C4—682no—
607C2×C4C2×C4⋊Dic1714yes—
608C2×C4D4×Dic1714yes—
609C22—28no—
610C22—28no—
611C22—28no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22C22×C4⋊Dic1711yescharacteristic
625C22C2×D4×Dic1711yescharacteristic
626C22#41×Dic1711yescharacteristic
627C22#1176328411yescharacteristic
628C22#1176328411yescharacteristic
629C22#1176328411yescharacteristic
630C22#1176329211yescharacteristic
631C22#1176337112yes—
632C22#1176337212yes—
633C22#1176339312yes—
634C22#1176341312yes—
635C22#1176342312yes—
636C22#1176343512yes—
637C22D4×D6811yescharacteristic
638C22#1176344812yes—
639C22#1176345412yes—
640C22#1176346311yescharacteristic
641C22D4×Dic3411yescharacteristic
642C22#1176350212yes—
643C22#1176351611yescharacteristic
644C22#1176351812yes—
645C22#1176354712yes—
646C22#1176355412yes—
647C4—24no—
648C4—24no—
649C4—42no—
650C4—44no—
651C4—684no—
652C4—684no—
653C4—1362no—
654C4—1362no—
655C2—48no—
656C2#1453139811yescharacteristic
657C2#1453151011yescharacteristic
658C2#1453159911yescharacteristic
659C2D4×C4⋊Dic1711yescharacteristic
660C2#1453169411yescharacteristic
661C2#1453183611yescharacteristic
662C2#1453194111yescharacteristic
663C2#1453510412yes—
664C2#1453527712yes—
665C2#1453539512yes—
666C2#1453643312yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC2×C17×#390C2
socle, cosocleC24×C17C24
prime radicalC23×D4×C17C22
hypercenterC2×#390D17
hypocenterC17#14438
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C2×#11762684C2semidirect
3. #14527037C2semidirect
4. #14527029C2semidirect
5. #14527029C2semidirect
6. #14527014C2semidirect
7. #14527011C2semidirect
8. #14527006C2semidirect
33. C17×#390C4semidirect
56. C2×C4⋊C4×C17C2×C4semidirect
61. C2×C22⋊C4×C17C2×C4semidirect
62. C2×C22⋊C4×C17C2×C4semidirect
64. C2×C42×C17C2×C4semidirect
89. #10587502D4semidirect
110. C2×D4×C17C4⋊C4semidirect
342. #390Dic17semidirect
395. C2×C4⋊C4C2×Dic17semidirect
409. C2×C22⋊C4C2×Dic17semidirect
410. C2×C22⋊C4C2×Dic17semidirect
412. C2×C42C2×Dic17semidirect
423. C17#14438semidirect
444. C2×D4C4⋊Dic17semidirect

Maximal subgroups

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

subgroupindex
C2×#117626892
C2×#117626842
#145270372
#145270292
#145270142
#145270112
#145270062
C2×C17×#3902
#1443817