Finite Group Databasebeta

← #16960698

Subgroups — #16960698

Order 4352.

2142 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587564C211yesmaximal, characteristic
2C22×#10587563C211yesmaximal, characteristic
3C2×#11762713C212yesmaximal, complemented
4C2×#11762667C211yesmaximal, characteristic
5#14527036C212yesmaximal, complemented
6#14527035C212yesmaximal, complemented
7#14527033C211yesmaximal, characteristic
8#14527031C211yesmaximal, characteristic
9#14527031C211yesmaximal, characteristic
10#14527012C212yesmaximal, complemented
11C17×#1440C211yesmaximal, characteristic
12C23×Dic34C2211yescharacteristic
13C24×Dic17C2211yescharacteristic
14C2×#10587564—24no—
15C2×#10587564—24no—
16C2×#10587564C418yescomplemented
17C2×#10587563—28no—
18C2×#10587563—24no—
19C22×C4⋊Dic17C2211yescharacteristic
20C22×#63225C2212yes—
21C22×#63225C2212yes—
22C22×#63225C2211yescharacteristic
23C22×#63225C2211yescharacteristic
24C22×C22⋊Dic17C2211yescharacteristic
25C22×C22⋊Dic17C2211yescharacteristic
26C22×C22⋊Dic17C2211yescharacteristic
27C22×C4×Dic17C2212yes—
28#11762713—28no—
29#11762667—24no—
30C2×#10587502C2212yes—
31C2×#10587502C2212yes—
32C2×#10587502C2212yes—
33C2×#10587502C2212yes—
34C2×#10587502C2212yes—
35C2×#10587502C2212yes—
36C2×#10587502C2212yes—
37C22×C22⋊C4×C17C2211yescharacteristic
38C22×C22⋊C4×C17C2211yescharacteristic
39C22×C22⋊C4×C17C2211yescharacteristic
40C2×C17×#95C2212yes—
41C2×C17×#95C2212yes—
42C25×C17C2311yescharacteristic
43C23×C4×C17C2312yes—
44C23×C4×C17C2311yescharacteristic
45C23×C4×C17C2311yescharacteristic
46C23×C4×C17C2312yes—
47C23×Dic17—24no—
48C23×Dic17—24no—
49C23×Dic17C2×C414yes—
50C23×Dic17D414yes—
51C23×Dic17D414yes—
52C23×Dic17C2312yes—
53C23×Dic17C2311yescharacteristic
54C23×Dic17C2311yescharacteristic
55C23×Dic17C2312yes—
56C23×Dic17C2311yescharacteristic
57C23×Dic17C2311yescharacteristic
58C22×Dic34—24no—
59C22×Dic34—22no—
60C22×Dic34—22no—
61C22×Dic34—24no—
62C22×Dic34C2×C414yes—
63C2×C22⋊C4×C17—22no—
64C2×C22⋊C4×C17—22no—
65C2×C22⋊C4×C17—22no—
66C2×C22⋊C4×C17—24no—
67C2×C22⋊C4×C17—22no—
68C2×C22⋊C4×C17—42no—
69C2×C22⋊C4×C17C2×C414yes—
70C2×C22⋊Dic17—22no—
71C2×C22⋊Dic17—22no—
72C2×C22⋊Dic17—22no—
73C2×C22⋊Dic17—24no—
74C2×C22⋊Dic17—22no—
75C2×C22⋊Dic17—42no—
76C2×C22⋊Dic17C2×C414yes—
77C2×C4⋊Dic17—22no—
78C2×C4⋊Dic17—22no—
79C2×C4⋊Dic17C2×C414yes—
80C2×#63225—24no—
81C2×#63225—24no—
82C2×#63225—24no—
83C2×#63225—22no—
84C2×#63225—22no—
85C2×#63225—22no—
86C2×#63225—22no—
87C2×#63225—24no—
88C2×#63225—24no—
89C2×#63225—24no—
90C2×#63225C2×C414yes—
91C2×#63225C2×C414yes—
92C2×C4×Dic17—24no—
93C2×C4×Dic17—24no—
94C2×C4×Dic17D418yes—
95#10587564—232no—
96#10587563—232no—
97C17×#95—24no—
98C17×#95—42no—
99#10587502—24no—
100#10587502—24no—
101#10587502—24no—
102#10587502—24no—
103#10587502—42no—
104#10587502—42no—
105#10587502—42no—
106C24×C17—22no—
107C24×C17—22no—
108C24×C17—22no—
109C24×C17—22no—
110C24×C17—22no—
111C24×C17—22no—
112C24×C17C22×C412yes—
113C24×C17C2×D412yes—
114C24×C17C2×D412yes—
115C24×C17C2411yescharacteristic
116C22×C4×C17—22no—
117C22×C4×C17—21no—
118C22×C4×C17—21no—
119C22×C4×C17—22no—
120C22×C4×C17—22no—
121C22×C4×C17—21no—
122C22×C4×C17—21no—
123C22×C4×C17—22no—
124C22×C4×C17—22no—
125C22×C4×C17—22no—
126C22×C4×C17—22no—
127C22×C4×C17—22no—
128C22×C4×C17—22no—
129C22×C4×C17—22no—
130C22×C4×C17—22no—
131C22×C4×C17—22no—
132C22×C4×C17—22no—
133C22×C4×C17—22no—
134C22×C4×C17—22no—
135C22×C4×C17—22no—
136C22×C4×C17—22no—
137C22×C4×C17C22×C412yes—
138C22×C4×C17C22×C412yes—
139C22×C4×C17C2×D414yes—
140C22×Dic17—22no—
141C22×Dic17—22no—
142C22×Dic17—216no—
143C22×Dic17—28no—
144C22×Dic17—216no—
145C22×Dic17—28no—
146C22×Dic17—22no—
147C22×Dic17—21no—
148C22×Dic17—21no—
149C22×Dic17—22no—
150C22×Dic17—22no—
151C22×Dic17—21no—
152C22×Dic17—21no—
153C22×Dic17—22no—
154C22×Dic17—22no—
155C22×Dic17—22no—
156C22×Dic17—22no—
157C22×Dic17—22no—
158C22×Dic17—22no—
159C22×Dic17—22no—
160C22×Dic17—22no—
161C22×Dic17—22no—
162C22×Dic17—22no—
163C22×Dic17—22no—
164C22×Dic17—22no—
165C22×Dic17—22no—
166C22×Dic17—22no—
167C22×Dic17C22⋊C4116yes—
168C22×Dic17C22×C412yes—
169C22×Dic17C22×C412yes—
170C22×Dic17C22×C412yes—
171C22×Dic17C22×C412yes—
172C22×Dic17C2×D414yes—
173C22×Dic17C2×D412yes—
174C22×Dic17C2×D412yes—
175C22×Dic17C2×D412yes—
176C22×Dic17C2×D412yes—
177C22×Dic17C2×D414yes—
178C22×Dic17#4114yes—
179C2×Dic34—28no—
180C2×Dic34—216no—
181C2×Dic34—44no—
182C2×Dic34—48no—
183C2×Dic34—44no—
184C22⋊C4×C17—28no—
185C22⋊C4×C17—28no—
186C22⋊C4×C17—44no—
187C22⋊Dic17—28no—
188C22⋊Dic17—28no—
189C22⋊Dic17—44no—
190C4⋊Dic17—28no—
191#63225—216no—
192#63225—28no—
193#63225—28no—
194#63225—48no—
195C4×Dic17—216no—
196#14258—171nomaximal, Sylow
197C23×C17—24no—
198C23×C17—22no—
199C23×C17—22no—
200C23×C17—24no—
201C23×C17—22no—
202C23×C17—24no—
203C23×C17—22no—
204C23×C17—22no—
205C23×C17—24no—
206C23×C17—24no—
207C23×C17—22no—
208C23×C17—24no—
209C23×C17—42no—
210C23×C17—42no—
211C23×C17—42no—
212C23×C17—42no—
213C23×C17—42no—
214C23×C17—42no—
215C23×C17—42no—
216C23×C17—42no—
217C23×C17C2×C22⋊C414yes—
218C23×C17C23×C411yescharacteristic
219C23×C17C22×D411yescharacteristic
220C23×C17C22×D411yescharacteristic
221C23×C17C22×D412yes—
222C23×C17C2×#4112yes—
223C23×C17H(5,𝔽2)12yes—
224C23×C172−1+412yes—
225C23×C172−1+412yes—
226C23×C172−1+412yes—
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—41no—
235C2×C4×C17—42no—
236C2×C4×C17—41no—
237C2×C4×C17—41no—
238C2×C4×C17—42no—
239C2×C4×C17—41no—
240C2×C4×C17—42no—
241C2×C4×C17—42no—
242C2×C4×C17—42no—
243C2×C4×C17—42no—
244C2×C4×C17—42no—
245C2×C4×C17—42no—
246C2×C4×C17—42no—
247C2×C4×C17—42no—
248C2×C4×C17—42no—
249C2×C4×C17—42no—
250C2×C4×C17—42no—
251C2×Dic17—24no—
252C2×Dic17—22no—
253C2×Dic17—24no—
254C2×Dic17—22no—
255C2×Dic17—232no—
256C2×Dic17—22no—
257C2×Dic17—22no—
258C2×Dic17—24no—
259C2×Dic17—24no—
260C2×Dic17—24no—
261C2×Dic17—24no—
262C2×Dic17—24no—
263C2×Dic17—41no—
264C2×Dic17—42no—
265C2×Dic17—41no—
266C2×Dic17—41no—
267C2×Dic17—41no—
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×Dic17—42no—
277C2×Dic17—42no—
278C2×Dic17—42no—
279C2×Dic17—42no—
280C2×Dic17C2×C22⋊C414yes—
281C2×Dic17C2×C22⋊C414yes—
282C2×Dic17C4×D418yes—
283C2×Dic17#12014yes—
284C2×Dic17#12118yes—
285C2×Dic17#12314yes—
286Dic34—416no—
287C22×C22⋊Q8—171no—
288C22×#117—171no—
289C2×#391—172no—
290C2×#380—171no—
291#1441—172no—
292#1441—172no—
293#1440—171no—
294#1440—171no—
295#1440—171no—
296#1440D1711yescharacteristic
297C4⋊#95—172no—
298C22×C17—24no—
299C22×C17—24no—
300C22×C17—24no—
301C22×C17—44no—
302C22×C17—44no—
303C22×C17—44no—
304C22×C17—44no—
305C22×C17—44no—
306C22×C17—44no—
307C22×C17C22×C22⋊C411yescharacteristic
308C22×C17C2×C4×D412yes—
309C22×C17#51911yescharacteristic
310C22×C17#51911yescharacteristic
311C22×C17#52012yes—
312C22×C17C2×#12011yescharacteristic
313C22×C17C2×#12112yes—
314C22×C17C2×#12311yescharacteristic
315C22×C17#53511yescharacteristic
316C22×C17#53612yes—
317C22×C17#53612yes—
318C22×C17#53612yes—
319C22×C17#53611yescharacteristic
320C22×C17#53612yes—
321C22×C17#53712yes—
322C22×C17#53712yes—
323C22×C17#53712yes—
324C22×C17#54112yes—
325C22×C17#54112yes—
326C22×C17#54212yes—
327C22×C17#54212yes—
328C4×C17—42no—
329C4×C17—44no—
330C4×C17—42no—
331C4×C17—82no—
332Dic17—24no—
333Dic17—42no—
334Dic17—44no—
335Dic17—42no—
336Dic17—82no—
337Dic17#38618yes—
338C23×Q8—171no—
339C24×C4—171no—
340C2×C22⋊Q8—178no—
341C2×C22⋊Q8—344no—
342C2×C22⋊Q8—344no—
343C2×#117—348no—
344C2×#117—344no—
345C22×C4⋊C4—171no—
346C22×C4⋊C4—171no—
347C22×C4⋊C4—171no—
348C22×C4⋊C4—172no—
349C22×C4⋊C4—172no—
350C22×C22⋊C4—171no—
351C22×C22⋊C4—171no—
352C22×C22⋊C4—171no—
353C22×C22⋊C4C2×D1711yescharacteristic
354C22×C22⋊C4C2×D1711yescharacteristic
355C22×C22⋊C4C2×D1711yescharacteristic
356C22×C42—172no—
357#391—348no—
358#380—344no—
359C2×#95—172no—
360C2×#95—172no—
361C2×#95—172no—
362C2×#95—172no—
363C2×#95—172no—
364C2×#95—172no—
365C2×#95—172no—
366C2×#95C2×D1712yes—
367C2×#95C2×D1712yes—
368C2×C17—44no—
369C2×C17C2×#38611yescharacteristic
370C2×C17#231011yescharacteristic
371C2×C17#231111yescharacteristic
372C2×C17#232312yes—
373C2×C17#232312yes—
374C2×C17#261612yes—
375C2×C17#262012yes—
376C2×C17#262112yes—
377C2×C17#262512yes—
378C25C22×D1711yescharacteristic
379C22×Q8—174no—
380C22×Q8—342no—
381C22×Q8—342no—
382C22×Q8—344no—
383C22×Q8—344no—
384C23×C4—171no—
385C23×C4—174no—
386C23×C4—174no—
387C23×C4—174no—
388C23×C4—171no—
389C23×C4—171no—
390C23×C4—171no—
391C23×C4—172no—
392C23×C4—172no—
393C23×C4—344no—
394C23×C4—344no—
395C23×C4C22×D1712yes—
396C23×C4C22×D1711yescharacteristic
397C23×C4C22×D1711yescharacteristic
398C23×C4C22×D1712yes—
399C22⋊Q8—3432no—
400#117—3432no—
401C2×C4⋊C4—174no—
402C2×C4⋊C4—174no—
403C2×C4⋊C4—174no—
404C2×C4⋊C4—344no—
405C2×C4⋊C4—344no—
406C2×C4⋊C4—342no—
407C2×C4⋊C4—342no—
408C2×C4⋊C4—342no—
409C2×C4⋊C4—342no—
410C2×C4⋊C4—342no—
411C2×C4⋊C4—342no—
412C2×C4⋊C4—344no—
413C2×C4⋊C4—344no—
414C2×C4⋊C4—344no—
415C2×C4⋊C4—344no—
416C2×C22⋊C4—22no—
417C2×C22⋊C4—22no—
418C2×C22⋊C4—22no—
419C2×C22⋊C4—24no—
420C2×C22⋊C4—22no—
421C2×C22⋊C4—42no—
422C2×C22⋊C4—174no—
423C2×C22⋊C4—344no—
424C2×C22⋊C4—342no—
425C2×C22⋊C4—342no—
426C2×C22⋊C4—342no—
427C2×C22⋊C4—342no—
428C2×C22⋊C4—682no—
429C2×C22⋊C4C4×D1714yes—
430C2×C42—178no—
431C2×C42—344no—
432C2×C42—344no—
433#95—24no—
434#95—42no—
435#95—344no—
436#95—344no—
437#95—344no—
438#95—344no—
439#95—682no—
440#95—682no—
441#95—682no—
442C17#1425811yescharacteristic, complemented, Sylow
443C24—22no—
444C24—22no—
445C24—22no—
446C24—22no—
447C24—22no—
448C24—22no—
449C24C2×C4×D1712yes—
450C24D4×D1712yes—
451C24D4×D1712yes—
452C24C23×D1711yescharacteristic
453C2×Q8—348no—
454C2×Q8—3416no—
455C2×Q8—684no—
456C2×Q8—688no—
457C2×Q8—684no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—172no—
480C22×C4—174no—
481C22×C4—172no—
482C22×C4—172no—
483C22×C4—174no—
484C22×C4—172no—
485C22×C4—172no—
486C22×C4—172no—
487C22×C4—1716no—
488C22×C4—172no—
489C22×C4—172no—
490C22×C4—174no—
491C22×C4—348no—
492C22×C4—3416no—
493C22×C4—3416no—
494C22×C4—348no—
495C22×C4—342no—
496C22×C4—342no—
497C22×C4—342no—
498C22×C4—342no—
499C22×C4—342no—
500C22×C4—341no—
501C22×C4—341no—
502C22×C4—341no—
503C22×C4—341no—
504C22×C4—342no—
505C22×C4—342no—
506C22×C4—342no—
507C22×C4—342no—
508C22×C4—342no—
509C22×C4—342no—
510C22×C4—342no—
511C22×C4—342no—
512C22×C4—342no—
513C22×C4—342no—
514C22×C4—342no—
515C22×C4—342no—
516C22×C4—342no—
517C22×C4—342no—
518C22×C4C2×C4×D1712yes—
519C22×C4C2×C4×D1712yes—
520C22×C4D4×D1714yes—
521C4⋊C4—348no—
522C4⋊C4—348no—
523C4⋊C4—348no—
524C4⋊C4—3416no—
525C4⋊C4—688no—
526C22⋊C4—28no—
527C22⋊C4—28no—
528C22⋊C4—44no—
529C22⋊C4—348no—
530C22⋊C4—348no—
531C22⋊C4—684no—
532C42—3416no—
533C23—24no—
534C23—22no—
535C23—22no—
536C23—24no—
537C23—22no—
538C23—24no—
539C23—22no—
540C23—22no—
541C23—24no—
542C23—24no—
543C23—22no—
544C23—24no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23C22⋊C4×D1714yes—
554C23C22×C4×D1711yescharacteristic
555C23C2×D4×D1711yescharacteristic
556C23C2×D4×D1711yescharacteristic
557C23C2×D4×D1712yes—
558C23#1058769112yes—
559C23#41×D1712yes—
560C23#1058769712yes—
561C23#1058769712yes—
562C23#1058769712yes—
563Q8—6816no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—174no—
589C2×C4—178no—
590C2×C4—174no—
591C2×C4—178no—
592C2×C4—174no—
593C2×C4—174no—
594C2×C4—3432no—
595C2×C4—344no—
596C2×C4—342no—
597C2×C4—344no—
598C2×C4—342no—
599C2×C4—342no—
600C2×C4—342no—
601C2×C4—344no—
602C2×C4—344no—
603C2×C4—344no—
604C2×C4—344no—
605C2×C4—344no—
606C2×C4—681no—
607C2×C4—682no—
608C2×C4—681no—
609C2×C4—681no—
610C2×C4—681no—
611C2×C4—682no—
612C2×C4—682no—
613C2×C4—682no—
614C2×C4—682no—
615C2×C4—682no—
616C2×C4—682no—
617C2×C4—682no—
618C2×C4—682no—
619C2×C4—682no—
620C2×C4—682no—
621C2×C4—682no—
622C2×C4—682no—
623C22—24no—
624C22—24no—
625C22—24no—
626C22—44no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22C2×C22⋊C4×D1711yescharacteristic
633C22C4×D4×D1712yes—
634C22#1176328611yescharacteristic
635C22#1176328611yescharacteristic
636C22#1176328812yes—
637C22#120×D1711yescharacteristic
638C22#121×D1712yes—
639C22#123×D1711yescharacteristic
640C22#1176337411yescharacteristic
641C22#1176337912yes—
642C22#1176338112yes—
643C22#1176338112yes—
644C22#1176338111yescharacteristic
645C22#1176338112yes—
646C22#1176338612yes—
647C22#1176338612yes—
648C22#1176338812yes—
649C22#1176341612yes—
650C22#1176341612yes—
651C22#1176342312yes—
652C22#1176342712yes—
653C4—42no—
654C4—44no—
655C4—42no—
656C4—82no—
657C4—178no—
658C4—344no—
659C4—682no—
660C4—684no—
661C4—682no—
662C4—1362no—
663C2—44no—
664C2D17×#38611yescharacteristic
665C2#1453136911yescharacteristic
666C2#1453137111yescharacteristic
667C2#1453145912yes—
668C2#1453146312yes—
669C2#1453451212yes—
670C2#1453454012yes—
671C2#1453454912yes—
672C2#1453458012yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1440C2
socle, cosocleC24×C17C24
prime radicalC25×C17C23
hypercenter#1440D17
hypocenterC17#14258
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1440

Complements

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

NH ≅ G/Ntype
3. C2×#11762713C2semidirect
5. #14527036C2semidirect
6. #14527035C2semidirect
10. #14527012C2semidirect
16. C2×#10587564C4semidirect
442. C17#14258semidirect

Maximal subgroups

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

subgroupindex
C22×#105875642
C22×#105875632
C2×#117627132
C2×#117626672
#145270362
#145270352
#145270332
#145270312
#145270122
C17×#14402
#1425817