Finite Group Databasebeta

← #16956300

Subgroups — #16956300

Order 4352.

2006 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
1C22×#10587612C211yesmaximal, characteristic
2C2×#11762679C211yesmaximal, characteristic, complemented
3C2×#11762675C211yesmaximal, characteristic, complemented
4C2×#11762669C211yesmaximal, characteristic
5#14527029C214yesmaximal
6#14527018C214yesmaximal, complemented
7#14527014C211yesmaximal, characteristic, complemented
8#14527014C211yesmaximal, characteristic, complemented
9C2×C4×C22⋊C4×C17C211yesmaximal, characteristic
10C2×#10587612—24no—
11C2×#10587612—24no—
12C2×#10587612—24no—
13C22×C4⋊Dic17C2211yescharacteristic
14C22×C4⋊Dic17C2211yescharacteristic
15C22×C4⋊Dic17C2214yes—
16C22×C4⋊Dic17C2211yescharacteristic
17C22×C4⋊Dic17C2211yescharacteristic
18C22×C22⋊Dic17C2212yes—
19C22×C22⋊Dic17C2212yes—
20C22×C4×Dic17C2212yes—
21#11762679—24no—
22#11762675—24no—
23#11762669—24no—
24C2×#10587502C2214yes—
25C2×#10587502C2212yes—
26C2×#10587502C2214yes—
27C2×#10587502C2214yes—
28C24×C4×C17C2211yescharacteristic
29C22×C22⋊C4×C17C2212yes—
30C22×C42×C17C2211yescharacteristic
31C22×C42×C17C2211yescharacteristic
32C4×C22⋊C4×C17C418yescomplemented
33C2×C17×#95C2212yes—
34C25×C17C2311yescharacteristic
35C23×C4×C17—22no—
36C23×C4×C17—22no—
37C23×C4×C17—22no—
38C23×C4×C17—22no—
39C23×C4×C17C2×C414yes—
40C23×C4×C17D414yes—
41C23×C4×C17Q814yes—
42C23×C4×C17C2312yes—
43C23×C4×C17C2311yescharacteristic
44C23×C4×C17C2312yes—
45C23×C4×C17C2311yescharacteristic
46C23×Dic17C2312yes—
47C23×Dic17C2312yes—
48C23×Dic17C2312yes—
49C23×Dic17C2312yes—
50C2×C22⋊C4×C17—24no—
51C2×C22⋊C4×C17—24no—
52C2×C22⋊C4×C17C2×C418yes—
53C2×C42×C17—22no—
54C2×C42×C17—22no—
55C2×C42×C17—22no—
56C2×C42×C17—22no—
57C2×C42×C17C2×C414yes—
58C2×C42×C17C2×C414yes—
59C2×C22⋊Dic17—24no—
60C2×C22⋊Dic17—24no—
61C2×C22⋊Dic17—24no—
62C2×C22⋊Dic17—24no—
63C2×C22⋊Dic17—44no—
64#10587612—232no—
65C2×C4⋊Dic17—22no—
66C2×C4⋊Dic17—22no—
67C2×C4⋊Dic17—22no—
68C2×C4⋊Dic17—22no—
69C2×C4⋊Dic17—22no—
70C2×C4⋊Dic17—22no—
71C2×C4⋊Dic17—22no—
72C2×C4⋊Dic17—22no—
73C2×C4⋊Dic17—28no—
74C2×C4⋊Dic17—22no—
75C2×C4⋊Dic17—22no—
76C2×C4⋊Dic17—28no—
77C2×C4⋊Dic17—44no—
78C2×C4⋊Dic17D414yes—
79C2×C4⋊Dic17D414yes—
80C2×C4×Dic17—24no—
81C2×C4×Dic17—24no—
82C2×C4×Dic17—24no—
83C17×#95C2×C418yes—
84#10587502—24no—
85#10587502—28no—
86#10587502—28no—
87#10587502—44no—
88C24×C17—21no—
89C24×C17—21no—
90C24×C17—21no—
91C24×C17—21no—
92C24×C17—22no—
93C24×C17—22no—
94C24×C17—22no—
95C24×C17—22no—
96C24×C17C22×C412yes—
97C24×C17C2×D412yes—
98C24×C17C2×Q812yes—
99C24×C17C2411yescharacteristic
100C22×C4×C17—21no—
101C22×C4×C17—21no—
102C22×C4×C17—22no—
103C22×C4×C17—21no—
104C22×C4×C17—21no—
105C22×C4×C17—22no—
106C22×C4×C17—22no—
107C22×C4×C17—22no—
108C22×C4×C17—28no—
109C22×C4×C17—28no—
110C22×C4×C17—28no—
111C22×C4×C17—28no—
112C22×C4×C17—28no—
113C22×C4×C17—28no—
114C22×C4×C17—22no—
115C22×C4×C17—22no—
116C22×C4×C17—22no—
117C22×C4×C17—22no—
118C22×C4×C17—22no—
119C22×C4×C17—22no—
120C22×C4×C17C4⋊C4116yescomplemented
121C22×C4×C17C22×C414yes—
122C22×C4×C17C22×C412yes—
123C22×C4×C17C22×C414yes—
124C22×C4×C17C22×C412yes—
125C22×C4×C17C2×D412yes—
126C22×C4×C17C2×D412yes—
127C22×C4×C17C2×D412yes—
128C22×C4×C17C2×D412yes—
129C22×C4×C17C2×Q812yes—
130C22×C4×C17C2×Q812yes—
131C22×C4×C17#4114yes—
132C22×C4×C17#4114yes—
133C22×C4×C17#4112yes—
134C22×C4×C17#4112yes—
135C22×Dic17—22no—
136C22×Dic17—22no—
137C22×Dic17—22no—
138C22×Dic17—22no—
139C22×Dic17—24no—
140C22×Dic17—24no—
141C22×Dic17—24no—
142C22×Dic17—24no—
143C22×Dic17—22no—
144C22×Dic17—22no—
145C22×Dic17—22no—
146C22×Dic17—22no—
147C22×Dic17—24no—
148C22×Dic17—24no—
149C22×Dic17—22no—
150C22×Dic17—22no—
151C22×Dic17—24no—
152C22×Dic17—24no—
153C22×Dic17C2×D414yes—
154C22×Dic17C2×D414yes—
155C22⋊C4×C17—216no—
156C42×C17—28no—
157C42×C17—28no—
158C22⋊Dic17—216no—
159C22⋊Dic17—48no—
160C4⋊Dic17—28no—
161C4⋊Dic17—28no—
162C4⋊Dic17—416no—
163C4⋊Dic17—44no—
164C4⋊Dic17—44no—
165C4×Dic17—216no—
166#13879—171nomaximal, Sylow
167C23×C17—24no—
168C23×C17—22no—
169C23×C17—22no—
170C23×C17—24no—
171C23×C17—22no—
172C23×C17—22no—
173C23×C17—22no—
174C23×C17—22no—
175C23×C17—22no—
176C23×C17—22no—
177C23×C17—24no—
178C23×C17—22no—
179C23×C17—22no—
180C23×C17—24no—
181C23×C17—42no—
182C23×C17—42no—
183C23×C17—42no—
184C23×C17—42no—
185C23×C17—42no—
186C23×C17—42no—
187C23×C17—42no—
188C23×C17—42no—
189C23×C17C2×C4⋊C414yes—
190C23×C17C23×C411yescharacteristic
191C23×C17C22×D411yescharacteristic
192C23×C17C22×D411yescharacteristic
193C23×C17C22×D411yescharacteristic
194C23×C17C22×Q811yescharacteristic
195C23×C17C2×#4112yes—
196C23×C17C2×#4112yes—
197C23×C17C2×#4111yescharacteristic
198C23×C17C2×#4111yescharacteristic
199C23×C17H(5,𝔽2)12yes—
200C23×C172−1+412yes—
201C2×C4×C17—24no—
202C2×C4×C17—22no—
203C2×C4×C17—22no—
204C2×C4×C17—24no—
205C2×C4×C17—22no—
206C2×C4×C17—22no—
207C2×C4×C17—22no—
208C2×C4×C17—22no—
209C2×C4×C17—24no—
210C2×C4×C17—24no—
211C2×C4×C17—24no—
212C2×C4×C17—24no—
213C2×C4×C17—232no—
214C2×C4×C17—24no—
215C2×C4×C17—24no—
216C2×C4×C17—24no—
217C2×C4×C17—42no—
218C2×C4×C17—42no—
219C2×C4×C17—42no—
220C2×C4×C17—42no—
221C2×C4×C17C2×C4⋊C414yes—
222C2×C4×C17C2×C4⋊C414yes—
223C2×C4×C17#11718yescomplemented
224C2×C4×C17C4×D414yes—
225C2×C4×C17C4×D414yes—
226C2×C4×C17#12014yes—
227C2×C4×C17C22⋊Q814yes—
228C2×C4×C17C22⋊Q814yes—
229C2×C4×C17#12314yes—
230C2×Dic17—24no—
231C2×Dic17—24no—
232C2×Dic17—28no—
233C2×Dic17—28no—
234C2×Dic17—42no—
235C2×Dic17—42no—
236C2×Dic17—42no—
237C2×Dic17—42no—
238C2×Dic17—44no—
239C2×Dic17—44no—
240C2×Dic17—42no—
241C2×Dic17—42no—
242C2×Dic17—42no—
243C2×Dic17—42no—
244C2×Dic17—42no—
245C2×Dic17—42no—
246C2×Dic17—44no—
247C2×Dic17—44no—
248C2×Dic17—44no—
249C2×Dic17—44no—
250C22×#117—171no—
251C2×#384—171no—
252C2×#382—171no—
253C2×#380—171no—
254C2×C4×C22⋊C4D1711yescharacteristic
255#1439—174no—
256#1436—174no—
257C4⋊#95—171no—
258C4⋊#95—171no—
259C22×C17—24no—
260C22×C17—24no—
261C22×C17—24no—
262C22×C17—44no—
263C22×C17—44no—
264C22×C17—42no—
265C22×C17—42no—
266C22×C17—44no—
267C22×C17—44no—
268C22×C17—42no—
269C22×C17—42no—
270C22×C17C22×C4⋊C411yescharacteristic
271C22×C17C2×#11712yes—
272C22×C17C2×C4×D411yescharacteristic
273C22×C17C2×C4×D411yescharacteristic
274C22×C17#52012yes—
275C22×C17C2×#12011yescharacteristic
276C22×C17C2×C22⋊Q811yescharacteristic
277C22×C17C2×C22⋊Q811yescharacteristic
278C22×C17C2×#12311yescharacteristic
279C22×C17#53211yescharacteristic
280C22×C17#53211yescharacteristic
281C22×C17#53312yes—
282C22×C17#53511yescharacteristic
283C22×C17#53611yescharacteristic
284C22×C17#54412yes—
285C22×C17#54612yes—
286C22×C17#54712yes—
287C22×C17#54812yes—
288C22×C17#55012yes—
289C22×C17#55212yes—
290C22×C17#55312yes—
291C22×C17#55512yes—
292C22×C17#55612yes—
293C4×C17—28no—
294C4×C17—24no—
295C4×C17—44no—
296C4×C17#38518yes—
297Dic17—44no—
298Dic17—44no—
299Dic17—82no—
300Dic17—82no—
301C24×C4C2×D1711yescharacteristic
302C2×#117—344no—
303C2×#117—344no—
304C2×#117—344no—
305C22×C4⋊C4—171no—
306C22×C4⋊C4—171no—
307C22×C4⋊C4—171no—
308C22×C4⋊C4—171no—
309C22×C4⋊C4—174no—
310C22×C22⋊C4—172no—
311C22×C22⋊C4—172no—
312C22×C22⋊C4C2×D1712yes—
313C22×C42—172no—
314C22×C42C2×D1711yescharacteristic
315C22×C42C2×D1711yescharacteristic
316#384—344no—
317#382—344no—
318#380—344no—
319C4×C22⋊C4Dic1718yescomplemented
320C2×#95—174no—
321C2×#95—172no—
322C2×#95—174no—
323C2×#95—174no—
324C2×#95C2×D1712yes—
325C2×C17—44no—
326C2×C17#228511yescharacteristic
327C2×C17C2×#38511yescharacteristic
328C2×C17#231711yescharacteristic
329C2×C17#235212yes—
330C2×C17#235412yes—
331C2×C17#243111yescharacteristic
332C2×C17#243211yescharacteristic
333C2×C17#243912yes—
334C2×C17#244112yes—
335C2×C17#249911yescharacteristic
336C2×C17#250011yescharacteristic
337C25C22×D1711yescharacteristic
338C23×C4—22no—
339C23×C4—22no—
340C23×C4—22no—
341C23×C4—22no—
342C23×C4—172no—
343C23×C4—172no—
344C23×C4—172no—
345C23×C4—172no—
346C23×C4Dic3414yes—
347C23×C4D6814yes—
348C23×C4C2×Dic1714yes—
349C23×C4C22×D1712yes—
350C23×C4C22×D1711yescharacteristic
351C23×C4C22×D1712yes—
352C23×C4C22×D1711yescharacteristic
353#117—3432no—
354C2×C4⋊C4—174no—
355C2×C4⋊C4—174no—
356C2×C4⋊C4—342no—
357C2×C4⋊C4—342no—
358C2×C4⋊C4—342no—
359C2×C4⋊C4—342no—
360C2×C4⋊C4—348no—
361C2×C4⋊C4—342no—
362C2×C4⋊C4—342no—
363C2×C4⋊C4—348no—
364C2×C4⋊C4—342no—
365C2×C4⋊C4—342no—
366C2×C4⋊C4—342no—
367C2×C4⋊C4—342no—
368C2×C4⋊C4—684no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—344no—
372C2×C22⋊C4—344no—
373C2×C22⋊C4—344no—
374C2×C22⋊C4—344no—
375C2×C22⋊C4—684no—
376C2×C22⋊C4C2×Dic1718yes—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—344no—
382C2×C42—344no—
383C2×C42—344no—
384C2×C42C2×Dic1714yes—
385C2×C42C2×Dic1714yes—
386#95—348no—
387#95—344no—
388#95—348no—
389#95—684no—
390#95C2×Dic1718yes—
391C17#1387911yescharacteristic, complemented, Sylow
392C24—22no—
393C24—22no—
394C24—21no—
395C24—21no—
396C24—21no—
397C24—21no—
398C24—22no—
399C24—22no—
400C24C2×Dic3412yes—
401C24C2×D6812yes—
402C24C22×Dic1712yes—
403C24C23×D1711yescharacteristic
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—28no—
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—342no—
427C22×C4—342no—
428C22×C4—344no—
429C22×C4—344no—
430C22×C4—342no—
431C22×C4—342no—
432C22×C4—344no—
433C22×C4—344no—
434C22×C4—342no—
435C22×C4—342no—
436C22×C4—344no—
437C22×C4—344no—
438C22×C4—342no—
439C22×C4—342no—
440C22×C4—342no—
441C22×C4—342no—
442C22×C4—344no—
443C22×C4—344no—
444C22×C4C4⋊Dic17116yescomplemented
445C22×C4C2×Dic3412yes—
446C22×C4C2×Dic3412yes—
447C22×C4C2×D6812yes—
448C22×C4C2×D6812yes—
449C22×C4#6325214yes—
450C22×C4#6325214yes—
451C22×C4D4×D1712yes—
452C22×C4D4×D1712yes—
453C22×C4#6325412yes—
454C22×C4#6325412yes—
455C22×C4C22×Dic1714yes—
456C22×C4C22×Dic1712yes—
457C22×C4C22×Dic1714yes—
458C22×C4C22×Dic1712yes—
459C4⋊C4—348no—
460C4⋊C4—348no—
461C4⋊C4—6816no—
462C4⋊C4—684no—
463C4⋊C4—684no—
464C22⋊C4—216no—
465C22⋊C4—3416no—
466C22⋊C4—688no—
467C42—28no—
468C42—28no—
469C42—3416no—
470C23—24no—
471C23—22no—
472C23—22no—
473C23—24no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—24no—
481C23—22no—
482C23—22no—
483C23—24no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23C2×C4⋊Dic1714yes—
493C23C22×Dic3411yescharacteristic
494C23C22×D6811yescharacteristic
495C23C2×#6325212yes—
496C23C2×#6325212yes—
497C23C2×D4×D1711yescharacteristic
498C23C2×D4×D1711yescharacteristic
499C23C2×#6325411yescharacteristic
500C23C2×#6325411yescharacteristic
501C23#1058769612yes—
502C23#1058769712yes—
503C23C23×Dic1711yescharacteristic
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—232no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—348no—
525C2×C4—344no—
526C2×C4—344no—
527C2×C4—348no—
528C2×C4—684no—
529C2×C4—684no—
530C2×C4—682no—
531C2×C4—682no—
532C2×C4—682no—
533C2×C4—682no—
534C2×C4—684no—
535C2×C4—684no—
536C2×C4—684no—
537C2×C4—682no—
538C2×C4—682no—
539C2×C4—682no—
540C2×C4—682no—
541C2×C4—682no—
542C2×C4—682no—
543C2×C4—684no—
544C2×C4#1058756414yes—
545C2×C4#1058756414yes—
546C2×C4#1058756814yes—
547C2×C4#1058757214yes—
548C2×C4C2×C4⋊Dic1714yes—
549C2×C4C2×C4⋊Dic1714yes—
550C2×C4#1058761218yescomplemented
551C2×C4D4×Dic1714yes—
552C2×C4D4×Dic1714yes—
553C22—24no—
554C22—24no—
555C22—24no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—44no—
560C22—42no—
561C22—44no—
562C22—44no—
563C22—44no—
564C22C22×C4⋊Dic1711yescharacteristic
565C22C2×#1058761212yes—
566C22C2×D4×Dic1711yescharacteristic
567C22C2×D4×Dic1711yescharacteristic
568C22#1176329212yes—
569C22C2×#1058756811yescharacteristic
570C22C2×#1058756411yescharacteristic
571C22C2×#1058756411yescharacteristic
572C22C2×#1058757211yescharacteristic
573C22#1176335911yescharacteristic
574C22#1176335911yescharacteristic
575C22#1176336612yes—
576C22#1176337811yescharacteristic
577C22#1176338511yescharacteristic
578C22#1176343712yes—
579C22#1176345512yes—
580C22#1176346412yes—
581C22#1176346812yes—
582C22#1176348012yes—
583C22#1176349912yes—
584C22#1176350512yes—
585C22#1176352412yes—
586C22#1176352912yes—
587C4—24no—
588C4—28no—
589C4—44no—
590C4—684no—
591C4—684no—
592C4—1362no—
593C4—1362no—
594C4#1176268418yes—
595C2—44no—
596C2#1453119111yescharacteristic
597C2C2×#1176268411yescharacteristic
598C2#1453140711yescharacteristic
599C2#1453171312yes—
600C2#1453173312yes—
601C2#1453240311yescharacteristic
602C2#1453241211yescharacteristic
603C2#1453247512yes—
604C2#1453249712yes—
605C2#1453323711yescharacteristic
606C2#1453324611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC2×C4×C22⋊C4×C17C2
socle, cosocleC24×C17C24
prime radicalC25×C17C23
hypercenterC2×C4×C22⋊C4D17
hypocenterC17#13879
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#11762679C2semidirect
3. C2×#11762675C2semidirect
6. #14527018C2semidirect
7. #14527014C2semidirect
8. #14527014C2semidirect
32. C4×C22⋊C4×C17C4semidirect
120. C22×C4×C17C4⋊C4semidirect
223. C2×C4×C17#117semidirect
319. C4×C22⋊C4Dic17semidirect
391. C17#13879semidirect
444. C22×C4C4⋊Dic17semidirect
550. C2×C4#10587612semidirect

Maximal subgroups

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

subgroupindex
C22×#105876122
C2×#117626792
C2×#117626752
C2×#117626692
#145270292
#145270182
#145270142
C2×C4×C22⋊C4×C172
#1387917