Finite Group Databasebeta

← #16956301

Subgroups — #16956301

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×#11762678C211yesmaximal, characteristic, complemented
3C2×#11762674C211yesmaximal, characteristic, complemented
4C2×C4×C22⋊Dic17C211yesmaximal, characteristic
5#14527027C214yesmaximal
6#14527015C214yesmaximal, complemented
7#14527013C211yesmaximal, characteristic, complemented
8#14527013C211yesmaximal, characteristic, complemented
9C2×C17×#380C211yesmaximal, characteristic
10C2×#10587612—24no—
11C2×#10587612—24no—
12C2×#10587612—24no—
13C22×C4⋊Dic17C2211yescharacteristic
14C22×C4⋊Dic17C2211yescharacteristic
15C22×#63225C2214yes—
16C22×C22⋊Dic17C2212yes—
17C22×C22⋊Dic17C2212yes—
18C22×C4×Dic17C2212yes—
19C22×C4×Dic17C2211yescharacteristic
20C22×C4×Dic17C2211yescharacteristic
21#11762678—24no—
22#11762674—24no—
23C4×C22⋊Dic17C418yescomplemented
24C2×#10587502C2214yes—
25C2×#10587502C2214yes—
26C2×#10587502C2214yes—
27C2×#10587502C2212yes—
28C24×C4×C17C2211yescharacteristic
29C22×C4⋊C4×C17C2211yescharacteristic
30C22×C4⋊C4×C17C2211yescharacteristic
31C22×C22⋊C4×C17C2212yes—
32C17×#380—24no—
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×C17C2312yes—
44C23×C4×C17C2311yescharacteristic
45C23×C4×C17C2311yescharacteristic
46C23×Dic17C2312yes—
47C23×Dic17C2312yes—
48C23×Dic17C2312yes—
49C23×Dic17C2312yes—
50C2×C4⋊C4×C17—22no—
51C2×C4⋊C4×C17—22no—
52C2×C4⋊C4×C17—22no—
53C2×C4⋊C4×C17—22no—
54C2×C4⋊C4×C17—22no—
55C2×C4⋊C4×C17—22no—
56C2×C22⋊C4×C17—24no—
57C2×C22⋊C4×C17—44no—
58C2×C22⋊Dic17—24no—
59C2×C22⋊Dic17—24no—
60C2×C22⋊Dic17—24no—
61C2×C22⋊Dic17—24no—
62C2×C22⋊Dic17—24no—
63C2×C22⋊Dic17C2×C418yes—
64#10587612—232no—
65C2×C4⋊Dic17—22no—
66C2×C4⋊Dic17—22no—
67C2×C4⋊Dic17—22no—
68C2×C4⋊Dic17—22no—
69C2×C4⋊Dic17D414yes—
70C2×C4⋊Dic17D414yes—
71C2×#63225—28no—
72C2×#63225—28no—
73C2×#63225—44no—
74C2×C4×Dic17—24no—
75C2×C4×Dic17—22no—
76C2×C4×Dic17—22no—
77C2×C4×Dic17—22no—
78C2×C4×Dic17—22no—
79C2×C4×Dic17—24no—
80C2×C4×Dic17—24no—
81C2×C4×Dic17C2×C414yes—
82C2×C4×Dic17C2×C414yes—
83C17×#95—24no—
84#10587502—28no—
85#10587502—28no—
86#10587502—44no—
87#10587502C2×C418yes—
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—21no—
103C22×C4×C17—21no—
104C22×C4×C17—28no—
105C22×C4×C17—28no—
106C22×C4×C17—28no—
107C22×C4×C17—28no—
108C22×C4×C17—28no—
109C22×C4×C17—28no—
110C22×C4×C17—22no—
111C22×C4×C17—22no—
112C22×C4×C17—22no—
113C22×C4×C17—22no—
114C22×C4×C17—24no—
115C22×C4×C17—24no—
116C22×C4×C17—22no—
117C22×C4×C17—22no—
118C22×C4×C17—24no—
119C22×C4×C17—24no—
120C22×C4×C17C4⋊C4116yescomplemented
121C22×C4×C17C22×C412yes—
122C22×C4×C17C22×C412yes—
123C22×C4×C17C2×D412yes—
124C22×C4×C17C2×D412yes—
125C22×C4×C17C2×D412yes—
126C22×C4×C17C2×D412yes—
127C22×C4×C17C2×Q812yes—
128C22×C4×C17C2×Q812yes—
129C22×C4×C17#4112yes—
130C22×C4×C17#4112yes—
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—24no—
141C22×Dic17—22no—
142C22×Dic17—24no—
143C22×Dic17—22no—
144C22×Dic17—22no—
145C22×Dic17—22no—
146C22×Dic17—24no—
147C22×Dic17—22no—
148C22×Dic17—24no—
149C22×Dic17C22×C414yes—
150C22×Dic17C22×C414yes—
151C22×Dic17C2×D414yes—
152C22×Dic17C2×D414yes—
153C22×Dic17#4114yes—
154C22×Dic17#4114yes—
155C4⋊C4×C17—44no—
156C4⋊C4×C17—44no—
157C22⋊C4×C17—48no—
158C22⋊Dic17—216no—
159C22⋊Dic17—216no—
160C4⋊Dic17—28no—
161C4⋊Dic17—28no—
162#63225—416no—
163C4×Dic17—216no—
164C4×Dic17—28no—
165C4×Dic17—28no—
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×#4111yescharacteristic
196C23×C17C2×#4111yescharacteristic
197C23×C17C2×#4112yes—
198C23×C17C2×#4112yes—
199C23×C17H(5,𝔽2)12yes—
200C23×C172−1+412yes—
201C2×C4×C17—22no—
202C2×C4×C17—22no—
203C2×C4×C17—22no—
204C2×C4×C17—22no—
205C2×C4×C17—22no—
206C2×C4×C17—22no—
207C2×C4×C17—232no—
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—44no—
215C2×C4×C17—44no—
216C2×C4×C17—44no—
217C2×C4×C17—44no—
218C2×C4×C17C2×C4⋊C414yes—
219C2×C4×C17C2×C4⋊C414yes—
220C2×C4×C17C4×D414yes—
221C2×C4×C17C4×D414yes—
222C2×C4×C17#12014yes—
223C2×C4×C17C22⋊Q814yes—
224C2×C4×C17C22⋊Q814yes—
225C2×C4×C17#12314yes—
226C2×Dic17—24no—
227C2×Dic17—24no—
228C2×Dic17—24no—
229C2×Dic17—24no—
230C2×Dic17—24no—
231C2×Dic17—24no—
232C2×Dic17—24no—
233C2×Dic17—24no—
234C2×Dic17—24no—
235C2×Dic17—24no—
236C2×Dic17—24no—
237C2×Dic17—28no—
238C2×Dic17—28no—
239C2×Dic17—42no—
240C2×Dic17—42no—
241C2×Dic17—42no—
242C2×Dic17—42no—
243C2×Dic17—42no—
244C2×Dic17—44no—
245C2×Dic17—42no—
246C2×Dic17—44no—
247C2×Dic17—42no—
248C2×Dic17—42no—
249C2×Dic17#11718yescomplemented
250C22×#117—171no—
251C2×#384—171no—
252C2×#382—171no—
253C2×#380D1711yescharacteristic
254C2×C4×C22⋊C4—171no—
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—24no—
294C4×C17—82no—
295C4×C17—82no—
296C4×C17#38518yes—
297Dic17—28no—
298Dic17—44no—
299Dic17—44no—
300Dic17—44no—
301C24×C4C2×D1711yescharacteristic
302C2×#117—344no—
303C2×#117—344no—
304C2×#117—344no—
305C22×C4⋊C4—171no—
306C22×C4⋊C4—171no—
307C22×C4⋊C4—174no—
308C22×C4⋊C4C2×D1711yescharacteristic
309C22×C4⋊C4C2×D1711yescharacteristic
310C22×C22⋊C4—172no—
311C22×C22⋊C4—172no—
312C22×C22⋊C4C2×D1712yes—
313C22×C42—171no—
314C22×C42—171no—
315C22×C42—172no—
316#384—344no—
317#382—344no—
318#380—24no—
319C4×C22⋊C4—178no—
320C2×#95—172no—
321C2×#95—174no—
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×C4C4×D1714yes—
348C23×C4C22⋊D1714yes—
349C23×C4C22×D1712yes—
350C23×C4C22×D1712yes—
351C23×C4C22×D1711yescharacteristic
352C23×C4C22×D1711yescharacteristic
353#117—3432no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—174no—
361C2×C4⋊C4—174no—
362C2×C4⋊C4—342no—
363C2×C4⋊C4—342no—
364C2×C4⋊C4—342no—
365C2×C4⋊C4—342no—
366C2×C4⋊C4—348no—
367C2×C4⋊C4—348no—
368C2×C4⋊C4—684no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—44no—
371C2×C22⋊C4—178no—
372C2×C22⋊C4—344no—
373C2×C22⋊C4—344no—
374C2×C22⋊C4—344no—
375C2×C22⋊C4—344no—
376C2×C22⋊C4—344no—
377C2×C42—174no—
378C2×C42—174no—
379C2×C42—344no—
380C2×C42—344no—
381C2×C42—342no—
382C2×C42—342no—
383C2×C42—342no—
384C2×C42—342no—
385C2×C42—344no—
386#95—24no—
387#95—178no—
388#95—348no—
389#95—348no—
390#95—684no—
391C17#1387911yescharacteristic, complemented, Sylow
392C24—21no—
393C24—21no—
394C24—21no—
395C24—21no—
396C24—22no—
397C24—22no—
398C24—22no—
399C24—22no—
400C24C2×Dic3412yes—
401C24C2×C4×D1712yes—
402C24C2×C22⋊D1712yes—
403C24C23×D1711yescharacteristic
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—174no—
425C22×C4—174no—
426C22×C4—174no—
427C22×C4—174no—
428C22×C4—174no—
429C22×C4—174no—
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—344no—
440C22×C4—344no—
441C22×C4—342no—
442C22×C4—344no—
443C22×C4—342no—
444C22×C4—342no—
445C22×C4—344no—
446C22×C4—342no—
447C22×C4—342no—
448C22×C4#63225116yescomplemented
449C22×C4C2×Dic3412yes—
450C22×C4C2×Dic3412yes—
451C22×C4C2×C4×D1712yes—
452C22×C4C2×C4×D1712yes—
453C22×C4D4×D1712yes—
454C22×C4D4×D1712yes—
455C22×C4#6325412yes—
456C22×C4#6325412yes—
457C22×C4C2×C22⋊D1712yes—
458C22×C4C2×C22⋊D1712yes—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—348no—
462C4⋊C4—348no—
463C4⋊C4—6816no—
464C22⋊C4—48no—
465C22⋊C4—3416no—
466C22⋊C4—3416no—
467C42—3416no—
468C42—348no—
469C42—348no—
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×#6322514yes—
493C23C22×Dic3411yescharacteristic
494C23C22×C4×D1711yescharacteristic
495C23C2×D4×D1711yescharacteristic
496C23C2×D4×D1711yescharacteristic
497C23C2×#6325411yescharacteristic
498C23C2×#6325411yescharacteristic
499C23#1058769112yes—
500C23#1058769412yes—
501C23#41×D1712yes—
502C23#41×D1712yes—
503C23C22×C22⋊D1711yescharacteristic
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—232no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—178no—
522C2×C4—344no—
523C2×C4—344no—
524C2×C4—344no—
525C2×C4—344no—
526C2×C4—344no—
527C2×C4—344no—
528C2×C4—348no—
529C2×C4—348no—
530C2×C4—344no—
531C2×C4—344no—
532C2×C4—344no—
533C2×C4—344no—
534C2×C4—344no—
535C2×C4—682no—
536C2×C4—684no—
537C2×C4—684no—
538C2×C4—682no—
539C2×C4—682no—
540C2×C4—682no—
541C2×C4—682no—
542C2×C4—682no—
543C2×C4—682no—
544C2×C4—682no—
545C2×C4#1058756414yes—
546C2×C4#1058756414yes—
547C2×C4#1058756714yes—
548C2×C4#1058756714yes—
549C2×C4C2×#6322514yes—
550C2×C4C2×#6322514yes—
551C2×C4#1058762114yes—
552C2×C4#1058762314yes—
553C22—24no—
554C22—24no—
555C22—24no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—44no—
560C22—44no—
561C22—42no—
562C22—44no—
563C22—44no—
564C22C22×#6322511yescharacteristic
565C22#117×D1712yes—
566C22C2×#1058756711yescharacteristic
567C22C2×#1058756711yescharacteristic
568C22#1176329012yes—
569C22C2×#1058762311yescharacteristic
570C22C2×#1058756411yescharacteristic
571C22C2×#1058756411yescharacteristic
572C22C2×#1058762111yescharacteristic
573C22#1176335511yescharacteristic
574C22#1176335511yescharacteristic
575C22#1176336212yes—
576C22#1176337611yescharacteristic
577C22#1176338311yescharacteristic
578C22#1176344012yes—
579C22#1176345612yes—
580C22#1176346112yes—
581C22#1176346912yes—
582C22#1176347712yes—
583C22#1176349312yes—
584C22#1176350612yes—
585C22#1176352312yes—
586C22#1176353012yes—
587C4—24no—
588C4—82no—
589C4—82no—
590C4—348no—
591C4—684no—
592C4—684no—
593C4—684no—
594C4#1176268118yes—
595C2—44no—
596C2#1453118711yescharacteristic
597C2C2×#1176268111yescharacteristic
598C2#1453140811yescharacteristic
599C2#1453170612yes—
600C2#1453172812yes—
601C2#1453240411yescharacteristic
602C2#1453241311yescharacteristic
603C2#1453247612yes—
604C2#1453249812yes—
605C2#1453323811yescharacteristic
606C2#1453324711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC2×C17×#380C2
socle, cosocleC24×C17C24
prime radicalC25×C17C23
hypercenterC2×#380D17
hypocenterC17#13879
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C2×#11762678C2semidirect
3. C2×#11762674C2semidirect
6. #14527015C2semidirect
7. #14527013C2semidirect
8. #14527013C2semidirect
23. C4×C22⋊Dic17C4semidirect
120. C22×C4×C17C4⋊C4semidirect
249. C2×Dic17#117semidirect
391. C17#13879semidirect
448. C22×C4#63225semidirect

Maximal subgroups

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

subgroupindex
C22×#105876122
C2×#117626782
C2×#117626742
C2×C4×C22⋊Dic172
#145270272
#145270152
#145270132
C2×C17×#3802
#1387917