Finite Group Databasebeta

← #16960740

Subgroups — #16960740

Order 4352.

2142 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D17C211yesmaximal, characteristic
2C2×#11762680C212yesmaximal
3C2×#11762671C211yesmaximal, characteristic
4C2×#11762671C211yesmaximal, characteristic
5#14527048C211yesmaximal, characteristic, complemented
6#14527047C212yesmaximal, complemented
7#14527043C211yesmaximal, characteristic, complemented
8#14527041C211yesmaximal, characteristic, complemented
9#14527039C212yesmaximal, complemented
10#14527005C212yesmaximal
11C17×#1444C211yesmaximal, characteristic, complemented
12C23×C4×D17C2211yescharacteristic
13C23×C4×D17C2212yes—
14C22×C4⋊Dic17C2211yescharacteristic
15C22×#63225C2212yes—
16C22×#63225C2212yes—
17C2×C4⋊C4×D17—28no—
18C2×C4⋊C4×D17—24no—
19C22×#63227C2211yescharacteristic
20C22×#63227C2211yescharacteristic
21C22×#63227C2212yes—
22#11762680—28no—
23#11762671—24no—
24#11762671C418yescomplemented
25C2×#10587502C2211yescharacteristic
26C2×#10587502C2212yes—
27C2×#10587502C2212yes—
28C2×#10587502C2211yescharacteristic
29C2×#10587502C2211yescharacteristic
30C2×#10587502C2212yes—
31C2×#10587502C2212yes—
32C2×#10587502C2212yes—
33C2×#10587502C2212yes—
34C2×#10587502C2211yescharacteristic
35C22×C4⋊C4×C17C2211yescharacteristic
36C22×C4⋊C4×C17C2212yes—
37C2×C17×#95C2211yescharacteristic
38C2×C17×#95C2212yes—
39C2×C17×#95C2211yescharacteristic
40C24×D17C2311yescharacteristic
41C23×C4×C17C2311yescharacteristic
42C23×C4×C17C2311yescharacteristic
43C23×C4×C17C2312yes—
44C23×C4×C17C2312yes—
45C23×C4×C17C2311yescharacteristic
46C23×Dic17C2311yescharacteristic
47C23×Dic17C2311yescharacteristic
48C23×Dic17C2312yes—
49C23×Dic17C2312yes—
50C23×Dic17C2311yescharacteristic
51C22×C4×D17—24no—
52C22×C4×D17—24no—
53C22×C4×D17—24no—
54C22×C4×D17—24no—
55C22×C4×D17—42no—
56C22×C4×D17—42no—
57C22×C4×D17—42no—
58C22×C4×D17—42no—
59C22×C4×D17C2×C414yes—
60C22×C4×D17D414yes—
61C22×C4×D17D418yes—
62C22×C4×D17Q814yes—
63C2×C4⋊C4×C17—24no—
64C2×C4⋊C4×C17—22no—
65C2×C4⋊C4×C17—24no—
66C2×C4⋊C4×C17—42no—
67C2×C4⋊C4×C17—42no—
68C2×#63227—24no—
69C2×#63227—22no—
70C2×#63227—22no—
71C2×#63227—24no—
72C2×#63227—24no—
73C2×#63227—24no—
74C2×#63227C2×C418yes—
75C2×C4⋊Dic17—22no—
76C2×C4⋊Dic17—24no—
77C2×#63225—24no—
78C2×#63225—24no—
79C2×#63225—24no—
80C2×#63225—24no—
81C2×#63225—42no—
82C2×#63225—42no—
83C4⋊C4×D17—232no—
84C17×#95—22no—
85C17×#95—42no—
86C17×#95C2×C414yescomplemented
87#10587502—22no—
88#10587502—24no—
89#10587502—22no—
90#10587502—22no—
91#10587502—24no—
92#10587502—42no—
93#10587502—42no—
94#10587502—42no—
95#10587502C2×C414yescomplemented
96#10587502C2×C418yescomplemented
97C24×C17C2411yescharacteristic
98C23×D17—21no—
99C23×D17—22no—
100C23×D17—21no—
101C23×D17—22no—
102C23×D17—21no—
103C23×D17—21no—
104C23×D17C22×C412yes—
105C23×D17C2×D412yes—
106C23×D17C2×D414yes—
107C23×D17C2×Q812yes—
108C23×D17#4114yes—
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×C17—22no—
116C22×C4×C17—22no—
117C22×C4×C17—22no—
118C22×C4×C17—22no—
119C22×C4×C17—22no—
120C22×C4×C17—22no—
121C22×C4×C17—22no—
122C22×C4×C17—22no—
123C22×C4×C17—21no—
124C22×C4×C17—21no—
125C22×C4×C17—22no—
126C22×C4×C17—22no—
127C22×C4×C17—22no—
128C22×C4×C17—21no—
129C22×C4×C17—22no—
130C22×C4×C17—21no—
131C22×C4×C17—21no—
132C22×C4×C17—21no—
133C22×C4×C17C22×C414yes—
134C22×C4×C17C22×C412yes—
135C22×C4×C17C2×D414yes—
136C22×C4×C17C2×D412yes—
137C22×C4×C17C2×Q812yes—
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—21no—
153C22×Dic17—22no—
154C22×Dic17—21no—
155C22×Dic17—22no—
156C22×Dic17—21no—
157C22×Dic17—21no—
158C22×Dic17—22no—
159C22×Dic17—21no—
160C22×Dic17—22no—
161C22×Dic17—21no—
162C22×Dic17C22×C414yes—
163C22×Dic17C22×C412yes—
164C22×Dic17C2×D414yes—
165C22×Dic17C2×D412yes—
166C22×Dic17C2×Q812yes—
167C2×C4×D17—216no—
168C2×C4×D17—28no—
169C2×C4×D17—216no—
170C2×C4×D17—28no—
171C2×C4×D17—216no—
172C2×C4×D17—48no—
173C2×C4×D17—48no—
174C2×C4×D17—48no—
175C2×C4×D17—48no—
176C2×C4×D17—48no—
177C2×C4×D17—48no—
178C2×C4×D17C4⋊C4116yescomplemented
179C4⋊C4×C17—28no—
180C4⋊C4×C17—48no—
181#63227—216no—
182#63227—44no—
183#63227—44no—
184C4⋊Dic17—28no—
185#63225—216no—
186#63225—48no—
187#14263—171nomaximal, Sylow
188C23×C17C23×C411yescharacteristic
189C23×C17C22×D411yescharacteristic
190C23×C17C22×D412yes—
191C23×C17C22×Q811yescharacteristic
192C23×C17C2×#4112yes—
193C23×C17H(5,𝔽2)11yescharacteristic
194C23×C17H(5,𝔽2)12yes—
195C23×C17H(5,𝔽2)11yescharacteristic
196C23×C172−1+412yes—
197C23×C172−1+411yescharacteristic
198C23×C172−1+411yescharacteristic
199C22×D17—22no—
200C22×D17—24no—
201C22×D17—22no—
202C22×D17—24no—
203C22×D17—24no—
204C22×D17—22no—
205C22×D17—22no—
206C22×D17—22no—
207C22×D17—24no—
208C22×D17—22no—
209C22×D17—24no—
210C22×D17—24no—
211C22×D17—24no—
212C22×D17—24no—
213C22×D17—24no—
214C22×D17—24no—
215C22×D17—22no—
216C22×D17—22no—
217C22×D17C2×C4⋊C414yes—
218C22×D17C4×D418yes—
219C22×D17#12014yes—
220C22×D17C22⋊Q818yes—
221C22×D17#12314yes—
222C2×C4×C17—24no—
223C2×C4×C17—22no—
224C2×C4×C17—24no—
225C2×C4×C17—22no—
226C2×C4×C17—24no—
227C2×C4×C17—24no—
228C2×C4×C17—24no—
229C2×C4×C17—24no—
230C2×C4×C17—24no—
231C2×C4×C17—24no—
232C2×C4×C17—42no—
233C2×C4×C17—42no—
234C2×C4×C17—42no—
235C2×C4×C17—42no—
236C2×C4×C17—42no—
237C2×C4×C17—42no—
238C2×C4×C17—42no—
239C2×C4×C17—42no—
240C2×C4×C17—41no—
241C2×C4×C17—41no—
242C2×C4×C17—42no—
243C2×C4×C17—42no—
244C2×C4×C17—41no—
245C2×C4×C17—41no—
246C2×C4×C17—41no—
247C2×C4×C17—41no—
248C2×C4×C17—42no—
249C2×C4×C17—42no—
250C2×C4×C17C2×C4⋊C414yes—
251C2×Dic17—24no—
252C2×Dic17—22no—
253C2×Dic17—24no—
254C2×Dic17—22no—
255C2×Dic17—24no—
256C2×Dic17—24no—
257C2×Dic17—24no—
258C2×Dic17—24no—
259C2×Dic17—24no—
260C2×Dic17—24no—
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—41no—
270C2×Dic17—41no—
271C2×Dic17—41no—
272C2×Dic17—41no—
273C2×Dic17—42no—
274C2×Dic17—41no—
275C2×Dic17—42no—
276C2×Dic17—41no—
277C2×Dic17—42no—
278C2×Dic17—42no—
279C2×Dic17C2×C4⋊C414yes—
280C4×D17—232no—
281C4×D17—432no—
282C23×C4⋊C4—171no—
283C2×#385—172no—
284C2×#381—171no—
285C2×#381—171no—
286#1444—171no—
287#1444—172no—
288#1444D1711yescharacteristic, complemented
289#1442—172no—
290#1442—171no—
291#1442—171no—
292C22⋊#95—172no—
293C22×C17C22×C4⋊C411yescharacteristic
294C22×C17C2×C4×D412yes—
295C22×C17#52011yescharacteristic
296C22×C17#52012yes—
297C22×C17#52011yescharacteristic
298C22×C17C2×#12011yescharacteristic
299C22×C17C2×C22⋊Q812yes—
300C22×C17C2×#12311yescharacteristic
301C22×C17#53412yes—
302C22×C17#53412yes—
303C22×C17#53511yescharacteristic
304C22×C17#53511yescharacteristic
305C22×C17#53611yescharacteristic
306C22×C17#53612yes—
307C22×C17#53611yescharacteristic
308C22×C17#53612yes—
309C22×C17#53912yes—
310C22×C17#53912yes—
311C22×C17#53912yes—
312C22×C17#53912yes—
313C22×C17#54412yes—
314C22×C17#54411yescharacteristic
315C22×C17#54411yescharacteristic
316C2×D17—24no—
317C2×D17—24no—
318C2×D17—28no—
319C2×D17—28no—
320C2×D17—24no—
321C2×D17—24no—
322C2×D17—24no—
323C2×D17—24no—
324C2×D17—28no—
325C2×D17—28no—
326C2×D17#38518yes—
327C4×C17—24no—
328C4×C17—44no—
329C4×C17—44no—
330C4×C17—81no—
331C4×C17—81no—
332Dic17—24no—
333Dic17—44no—
334Dic17—44no—
335Dic17—81no—
336Dic17—81no—
337C24×C4—171no—
338C24×C4—172no—
339C22×C4⋊C4—172no—
340C22×C4⋊C4—171no—
341C22×C4⋊C4—172no—
342C22×C4⋊C4—344no—
343C22×C4⋊C4—348no—
344C22×C4⋊C4C2×D1711yescharacteristic
345C22×C4⋊C4C2×D1712yes—
346C22×C22⋊C4—172no—
347C22×C22⋊C4—171no—
348C22×C22⋊C4—171no—
349#385—348no—
350#381—178no—
351#381—344no—
352C2×#95—172no—
353C2×#95—171no—
354C2×#95—171no—
355C2×#95—171no—
356C2×#95—172no—
357C2×#95—172no—
358C2×#95—171no—
359C2×#95—172no—
360C2×#95—172no—
361C2×#95—172no—
362C2×#95C2×D1711yescharacteristic
363C2×#95C2×D1712yes—
364C2×#95C2×D1711yescharacteristic
365C2×C17C2×#38511yescharacteristic
366C2×C17#231711yescharacteristic
367C2×C17#231711yescharacteristic
368C2×C17#232212yes—
369C2×C17#232212yes—
370C2×C17#261311yescharacteristic
371C2×C17#261411yescharacteristic
372C2×C17#262311yescharacteristic
373C2×C17#262311yescharacteristic
374C2×C17#262712yes—
375C2×C17#262712yes—
376D17—28no—
377C25—171no—
378C23×C4—171no—
379C23×C4—174no—
380C23×C4—174no—
381C23×C4—174no—
382C23×C4—172no—
383C23×C4—172no—
384C23×C4—178no—
385C23×C4—171no—
386C23×C4—171no—
387C23×C4—344no—
388C23×C4—344no—
389C23×C4—344no—
390C23×C4—344no—
391C23×C4—682no—
392C23×C4—682no—
393C23×C4—682no—
394C23×C4—682no—
395C23×C4C22×D1711yescharacteristic
396C23×C4C22×D1711yescharacteristic
397C23×C4C22×D1712yes—
398C23×C4C22×D1712yes—
399C23×C4C22×D1711yescharacteristic
400C2×C4⋊C4—24no—
401C2×C4⋊C4—22no—
402C2×C4⋊C4—24no—
403C2×C4⋊C4—42no—
404C2×C4⋊C4—42no—
405C2×C4⋊C4—3432no—
406C2×C4⋊C4—344no—
407C2×C4⋊C4—344no—
408C2×C4⋊C4—344no—
409C2×C4⋊C4—344no—
410C2×C4⋊C4—342no—
411C2×C4⋊C4—344no—
412C2×C4⋊C4—682no—
413C2×C4⋊C4—682no—
414C2×C22⋊C4—178no—
415C2×C22⋊C4—344no—
416C2×C22⋊C4—344no—
417C2×C22⋊C4—342no—
418C2×C22⋊C4—342no—
419C2×C22⋊C4—344no—
420C2×C22⋊C4—344no—
421#95—22no—
422#95—42no—
423#95—178no—
424#95—174no—
425#95—342no—
426#95—342no—
427#95—344no—
428#95—342no—
429#95—344no—
430#95—682no—
431#95—682no—
432#95—682no—
433#95C4×D1714yescomplemented
434C17#1426311yescharacteristic, complemented, Sylow
435C24—172no—
436C24—172no—
437C24—174no—
438C24—172no—
439C24—174no—
440C24—341no—
441C24—342no—
442C24—342no—
443C24—341no—
444C24—341no—
445C24—341no—
446C24C23×D1711yescharacteristic
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—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—172no—
472C22×C4—172no—
473C22×C4—172no—
474C22×C4—1716no—
475C22×C4—174no—
476C22×C4—174no—
477C22×C4—348no—
478C22×C4—3416no—
479C22×C4—3416no—
480C22×C4—348no—
481C22×C4—3416no—
482C22×C4—342no—
483C22×C4—342no—
484C22×C4—342no—
485C22×C4—342no—
486C22×C4—342no—
487C22×C4—342no—
488C22×C4—342no—
489C22×C4—342no—
490C22×C4—342no—
491C22×C4—342no—
492C22×C4—342no—
493C22×C4—342no—
494C22×C4—342no—
495C22×C4—342no—
496C22×C4—341no—
497C22×C4—342no—
498C22×C4—341no—
499C22×C4—341no—
500C22×C4—342no—
501C22×C4—341no—
502C22×C4—342no—
503C22×C4—342no—
504C22×C4—341no—
505C22×C4—341no—
506C22×C4—688no—
507C22×C4—688no—
508C22×C4—688no—
509C22×C4—688no—
510C22×C4—688no—
511C22×C4—688no—
512C22×C4C2×C4×D1714yes—
513C22×C4C2×C4×D1712yes—
514C22×C4D4×D1714yes—
515C22×C4D4×D1712yes—
516C22×C4Q8×D1712yes—
517C4⋊C4—28no—
518C4⋊C4—48no—
519C4⋊C4—3416no—
520C4⋊C4—348no—
521C4⋊C4—688no—
522C22⋊C4—3416no—
523C22⋊C4—684no—
524C22⋊C4—684no—
525C23—174no—
526C23—178no—
527C23—174no—
528C23—178no—
529C23—174no—
530C23—342no—
531C23—344no—
532C23—342no—
533C23—344no—
534C23—344no—
535C23—342no—
536C23—342no—
537C23—342no—
538C23—344no—
539C23—342no—
540C23—344no—
541C23—344no—
542C23—344no—
543C23—344no—
544C23—344no—
545C23—344no—
546C23—342no—
547C23—342no—
548C23C22×C4×D1711yescharacteristic
549C23C2×D4×D1711yescharacteristic
550C23C2×D4×D1712yes—
551C23#1058769111yescharacteristic
552C23#1058769112yes—
553C23C2×Q8×D1711yescharacteristic
554C23#1058769412yes—
555C23#1058769411yescharacteristic
556C23#41×D1712yes—
557C23#1058769611yescharacteristic
558C23#1058769711yescharacteristic
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—174no—
588C2×C4—3432no—
589C2×C4—344no—
590C2×C4—342no—
591C2×C4—344no—
592C2×C4—342no—
593C2×C4—344no—
594C2×C4—344no—
595C2×C4—344no—
596C2×C4—344no—
597C2×C4—344no—
598C2×C4—344no—
599C2×C4—6832no—
600C2×C4—682no—
601C2×C4—682no—
602C2×C4—682no—
603C2×C4—682no—
604C2×C4—682no—
605C2×C4—682no—
606C2×C4—682no—
607C2×C4—682no—
608C2×C4—681no—
609C2×C4—681no—
610C2×C4—681no—
611C2×C4—681no—
612C2×C4—682no—
613C2×C4—681no—
614C2×C4—682no—
615C2×C4—681no—
616C2×C4—682no—
617C2×C4—682no—
618C2×C4C4⋊C4×D1714yes—
619C22—178no—
620C22—344no—
621C22—344no—
622C22—344no—
623C22—344no—
624C22—348no—
625C22—348no—
626C22—344no—
627C22—344no—
628C22—348no—
629C22—348no—
630C22C2×C4⋊C4×D1711yescharacteristic
631C22C4×D4×D1712yes—
632C22#1176328811yescharacteristic
633C22#1176329012yes—
634C22#1176329111yescharacteristic
635C22#120×D1711yescharacteristic
636C22C22⋊Q8×D1712yes—
637C22#123×D1711yescharacteristic
638C22#1176336712yes—
639C22#1176336912yes—
640C22#1176337211yescharacteristic
641C22#1176337511yescharacteristic
642C22#1176338211yescharacteristic
643C22#1176338212yes—
644C22#1176338411yescharacteristic
645C22#1176338412yes—
646C22#1176340212yes—
647C22#1176340512yes—
648C22#1176340512yes—
649C22#1176340712yes—
650C22#1176343612yes—
651C22#1176343611yescharacteristic
652C22#1176343911yescharacteristic
653C4—24no—
654C4—44no—
655C4—44no—
656C4—81no—
657C4—81no—
658C4—344no—
659C4—684no—
660C4—684no—
661C4—1361no—
662C4—1361no—
663C2—348no—
664C2D17×#38511yescharacteristic
665C2#1453140511yescharacteristic
666C2#1453140611yescharacteristic
667C2#1453144812yes—
668C2#1453145112yes—
669C2#1453449511yescharacteristic
670C2#1453450411yescharacteristic
671C2#1453456511yescharacteristic
672C2#1453456911yescharacteristic
673C2#1453459612yes—
674C2#1453460212yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1444C2
socle, cosocleC24×C17C24
prime radicalC24×D17C23
hypercenter#1444D17
hypocenterC17#14263
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
5. #14527048C2semidirect
6. #14527047C2semidirect
7. #14527043C2semidirect
8. #14527041C2semidirect
9. #14527039C2semidirect
11. C17×#1444C2semidirect
24. #11762671C4semidirect
86. C17×#95C2×C4semidirect
95. #10587502C2×C4semidirect
96. #10587502C2×C4semidirect
178. C2×C4×D17C4⋊C4semidirect
288. #1444D17semidirect
433. #95C4×D17semidirect
434. C17#14263semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D172
C2×#117626802
C2×#117626712
#145270482
#145270472
#145270432
#145270412
#145270392
#145270052
C17×#14442
#1426317