Finite Group Databasebeta

← #16961095

Subgroups — #16961095

Order 4352.

1830 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic34C211yesmaximal, characteristic, complemented
2C2×#11762722C211yesmaximal, characteristic
3C2×#11762701C212yesmaximal
4C2×#11762700C211yesmaximal, characteristic
5#14527054C211yesmaximal, characteristic, complemented
6#14527049C211yesmaximal, characteristic, complemented
7#14527048C212yesmaximal, complemented
8#14527030C211yesmaximal, characteristic
9#14527028C212yesmaximal
10#14527008C212yesmaximal, complemented
11C17×#1434C211yesmaximal, characteristic, complemented
12C23×Dic34C2211yescharacteristic
13C2×C4×Dic34—24no—
14C2×C4×Dic34—28no—
15C22×C4⋊Dic17C2211yescharacteristic
16C22×C4⋊Dic17C2212yes—
17C22×#63225C2212yes—
18C22×#63225C2211yescharacteristic
19C22×#63227C2211yescharacteristic
20C22×#63227C2212yes—
21C22×#63227C2211yescharacteristic
22C22×#63227C2211yescharacteristic
23C22×#63227C2212yes—
24C22×C4×Dic17C2212yes—
25#11762722C418yescomplemented
26#11762701—28no—
27#11762700—24no—
28C2×#10587502C2211yescharacteristic
29C2×#10587502C2212yes—
30C2×#10587502C2212yes—
31C2×#10587502C2211yescharacteristic
32C2×#10587502C2212yes—
33C2×#10587502C2212yes—
34C2×#10587502C2212yes—
35C22×C42×C17C2211yescharacteristic
36C2×C17×#95C2212yes—
37C2×C17×#95C2211yescharacteristic
38C2×C17×#95C2212yes—
39C2×C17×#95C2211yescharacteristic
40C24×D17C2311yescharacteristic
41C23×C4×C17C2312yes—
42C23×C4×C17C2312yes—
43C23×C4×C17C2311yescharacteristic
44C23×C4×C17C2311yescharacteristic
45C23×C4×C17C2311yescharacteristic
46C23×Dic17C2311yescharacteristic
47C23×Dic17C2311yescharacteristic
48C23×Dic17C2311yescharacteristic
49C23×Dic17C2312yes—
50C23×Dic17C2312yes—
51C22×Dic34—22no—
52C22×Dic34—22no—
53C22×Dic34—24no—
54C22×Dic34—24no—
55C22×Dic34C2×C414yes—
56C2×C42×C17—24no—
57C2×C42×C17D414yes—
58C2×#63227—22no—
59C2×#63227—22no—
60C2×#63227—22no—
61C2×#63227—24no—
62C2×#63227—22no—
63C2×#63227—24no—
64C2×#63227—24no—
65C2×#63227—24no—
66C2×#63227—42no—
67C2×#63227—42no—
68C2×#63227—42no—
69C2×#63227C2×C414yes—
70C2×#63227C2×C418yes—
71C2×C4⋊Dic17—22no—
72C2×C4⋊Dic17—24no—
73C2×C4⋊Dic17—24no—
74C2×C4⋊Dic17—42no—
75C2×C4⋊Dic17—42no—
76C2×#63225—24no—
77C2×#63225—22no—
78C2×#63225—24no—
79C2×#63225—24no—
80C2×#63225—42no—
81C2×C4×Dic17—24no—
82C2×C4×Dic17—24no—
83C2×C4×Dic17D418yes—
84C4×Dic34—232no—
85C17×#95—24no—
86C17×#95—22no—
87C17×#95—42no—
88C17×#95C2×C414yes—
89#10587502—22no—
90#10587502—22no—
91#10587502—24no—
92#10587502—24no—
93#10587502—42no—
94#10587502—42no—
95#10587502C2×C418yes—
96C24×C17C2411yescharacteristic
97C23×D17—21no—
98C23×D17—22no—
99C23×D17—21no—
100C23×D17—22no—
101C23×D17—21no—
102C23×D17—22no—
103C23×D17—21no—
104C23×D17—21no—
105C23×D17—21no—
106C23×D17—22no—
107C23×D17C22×C412yes—
108C22×C4×C17—22no—
109C22×C4×C17—22no—
110C22×C4×C17—21no—
111C22×C4×C17—21no—
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—21no—
121C22×C4×C17—22no—
122C22×C4×C17—22no—
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×C17C22×C414yes—
131C22×C4×C17C22×C412yes—
132C22×C4×C17C2×D412yes—
133C22×C4×C17C2×D412yes—
134C22×C4×C17C2×D412yes—
135C22×C4×C17C2×D414yes—
136C22×C4×C17#4112yes—
137C22×Dic17—22no—
138C22×Dic17—22no—
139C22×Dic17—22no—
140C22×Dic17—22no—
141C22×Dic17—21no—
142C22×Dic17—22no—
143C22×Dic17—22no—
144C22×Dic17—21no—
145C22×Dic17—22no—
146C22×Dic17—22no—
147C22×Dic17—22no—
148C22×Dic17—22no—
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—21no—
158C22×Dic17—21no—
159C22×Dic17—21no—
160C22×Dic17—21no—
161C22×Dic17C22×C412yes—
162C22×Dic17C22×C414yes—
163C22×Dic17C2×D414yes—
164C22×Dic17C2×D414yes—
165C22×Dic17#4114yes—
166C2×Dic34—216no—
167C2×Dic34—28no—
168C2×Dic34—28no—
169C2×Dic34—28no—
170C2×Dic34—216no—
171C42×C17—28no—
172#63227—28no—
173#63227—216no—
174#63227—48no—
175#63227—44no—
176#63227—44no—
177C4⋊Dic17—28no—
178C4⋊Dic17—48no—
179#63225—216no—
180#63225—44no—
181C4×Dic17—216no—
182#14301—171nomaximal, Sylow
183C23×C17C23×C411yescharacteristic
184C23×C17C22×D411yescharacteristic
185C23×C17C22×D412yes—
186C23×C17C2×#4111yescharacteristic
187C23×C17C2×#4112yes—
188C23×C17H(5,𝔽2)11yescharacteristic
189C23×C17H(5,𝔽2)12yes—
190C23×C172−1+411yescharacteristic
191C23×C172−1+411yescharacteristic
192C23×C172−1+411yescharacteristic
193C23×C172−1+412yes—
194C22×D17—24no—
195C22×D17—22no—
196C22×D17—22no—
197C22×D17—22no—
198C22×D17—24no—
199C22×D17—41no—
200C22×D17—42no—
201C22×D17—41no—
202C22×D17—42no—
203C22×D17—41no—
204C22×D17—42no—
205C22×D17—42no—
206C22×D17—41no—
207C22×D17—41no—
208C22×D17—42no—
209C22×D17—41no—
210C22×D17—41no—
211C22×D17—42no—
212C22×D17—42no—
213C22×D17—41no—
214C22×D17—42no—
215C22×D17—42no—
216C22×D17—42no—
217C2×C4×C17—24no—
218C2×C4×C17—22no—
219C2×C4×C17—22no—
220C2×C4×C17—24no—
221C2×C4×C17—24no—
222C2×C4×C17—24no—
223C2×C4×C17—22no—
224C2×C4×C17—22no—
225C2×C4×C17—24no—
226C2×C4×C17—24no—
227C2×C4×C17—24no—
228C2×C4×C17—22no—
229C2×C4×C17—22no—
230C2×C4×C17—22no—
231C2×C4×C17—42no—
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—42no—
241C2×C4×C17—42no—
242C2×C4×C17—42no—
243C2×C4×C17—42no—
244C2×C4×C17C4×D414yes—
245C2×C4×C17#12014yes—
246C2×Dic17—24no—
247C2×Dic17—24no—
248C2×Dic17—24no—
249C2×Dic17—24no—
250C2×Dic17—24no—
251C2×Dic17—24no—
252C2×Dic17—22no—
253C2×Dic17—24no—
254C2×Dic17—24no—
255C2×Dic17—24no—
256C2×Dic17—24no—
257C2×Dic17—42no—
258C2×Dic17—42no—
259C2×Dic17—41no—
260C2×Dic17—41no—
261C2×Dic17—42no—
262C2×Dic17—42no—
263C2×Dic17—42no—
264C2×Dic17—41no—
265C2×Dic17—41no—
266C2×Dic17—41no—
267C2×Dic17—42no—
268C2×Dic17—41no—
269C2×Dic17—41no—
270C2×Dic17—42no—
271C2×Dic17—41no—
272C2×Dic17—42no—
273C2×Dic17—42no—
274C2×Dic17C4×D418yes—
275Dic34—232no—
276C22×C4×Q8—171no—
277C2×#393—171no—
278C2×#388—171no—
279C2×#388—172no—
280#1446—171no—
281#1444—172no—
282#1444—171no—
283#1439—172no—
284#1439—171no—
285#1434—172no—
286#1434D1711yescharacteristic, complemented
287C22×C17C2×C4×D412yes—
288C22×C17C2×C4×D411yescharacteristic
289C22×C17#51911yescharacteristic
290C22×C17#52011yescharacteristic
291C22×C17#52012yes—
292C22×C17C2×#12011yescharacteristic
293C22×C17#52512yes—
294C22×C17#52511yescharacteristic
295C22×C17#53512yes—
296C22×C17#53511yescharacteristic
297C22×C17#53612yes—
298C22×C17#53612yes—
299C22×C17#53611yescharacteristic
300C22×C17#53611yescharacteristic
301C22×C17#53712yes—
302C22×C17#53711yescharacteristic
303C22×C17#54111yescharacteristic
304C22×C17#54112yes—
305C22×C17#54212yes—
306C22×C17#54211yescharacteristic
307C22×C17#54212yes—
308C22×C17#54212yes—
309C22×C17#54212yes—
310C2×D17—44no—
311C2×D17—42no—
312C2×D17—44no—
313C2×D17—42no—
314C2×D17—42no—
315C2×D17—81no—
316C2×D17—81no—
317C2×D17—81no—
318C2×D17—82no—
319C2×D17—81no—
320C2×D17—82no—
321C4×C17—24no—
322C4×C17—24no—
323C4×C17—44no—
324C4×C17—42no—
325C4×C17—82no—
326Dic17—28no—
327Dic17—42no—
328Dic17—44no—
329Dic17—81no—
330Dic17—81no—
331C23×Q8—171no—
332C2×C4×Q8—348no—
333C2×C4×Q8—344no—
334C22×C4⋊C4—172no—
335C22×C4⋊C4—171no—
336C22×C4⋊C4—171no—
337C22×C4⋊C4—172no—
338C22×C22⋊C4—172no—
339C22×C22⋊C4—171no—
340C22×C22⋊C4—171no—
341C22×C22⋊C4—172no—
342C22×C22⋊C4—171no—
343C22×C42—172no—
344C22×C42C2×D1711yescharacteristic
345#393—178no—
346#388—348no—
347#388—344no—
348C2×#95—172no—
349C2×#95—171no—
350C2×#95—172no—
351C2×#95—172no—
352C2×#95—171no—
353C2×#95—172no—
354C2×#95—172no—
355C2×#95C2×D1712yes—
356C2×#95C2×D1711yescharacteristic
357C2×#95C2×D1712yes—
358C2×#95C2×D1711yescharacteristic
359C2×C17C4×#12011yescharacteristic
360C2×C17#232312yes—
361C2×C17#232311yescharacteristic
362C2×C17#232612yes—
363C2×C17#232611yescharacteristic
364C2×C17#261411yescharacteristic
365C2×C17#261511yescharacteristic
366C2×C17#263211yescharacteristic
367C2×C17#263212yes—
368C2×C17#263711yescharacteristic
369C2×C17#263712yes—
370D17—82no—
371C25—171no—
372C22×Q8—174no—
373C22×Q8—344no—
374C22×Q8—344no—
375C22×Q8—342no—
376C22×Q8—342no—
377C23×C4—172no—
378C23×C4—172no—
379C23×C4—171no—
380C23×C4—171no—
381C23×C4—171no—
382C23×C4C22×D1712yes—
383C23×C4C22×D1712yes—
384C23×C4C22×D1711yescharacteristic
385C23×C4C22×D1711yescharacteristic
386C23×C4C22×D1711yescharacteristic
387C4×Q8—3432no—
388C2×C4⋊C4—344no—
389C2×C4⋊C4—344no—
390C2×C4⋊C4—344no—
391C2×C4⋊C4—344no—
392C2×C4⋊C4—342no—
393C2×C4⋊C4—344no—
394C2×C4⋊C4—342no—
395C2×C4⋊C4—682no—
396C2×C4⋊C4—682no—
397C2×C4⋊C4—682no—
398C2×C22⋊C4—178no—
399C2×C22⋊C4—174no—
400C2×C22⋊C4—344no—
401C2×C22⋊C4—344no—
402C2×C22⋊C4—342no—
403C2×C22⋊C4—342no—
404C2×C22⋊C4—344no—
405C2×C22⋊C4—342no—
406C2×C22⋊C4—342no—
407C2×C22⋊C4—344no—
408C2×C22⋊C4—682no—
409C2×C22⋊C4—682no—
410C2×C22⋊C4—682no—
411C2×C42—24no—
412C2×C42—178no—
413C2×C42—344no—
414C2×C42—344no—
415C2×C42D6814yes—
416#95—22no—
417#95—24no—
418#95—42no—
419#95—178no—
420#95—344no—
421#95—344no—
422#95—342no—
423#95—342no—
424#95—682no—
425#95—682no—
426#95C4×D1714yes—
427C17#1430111yescharacteristic, complemented, Sylow
428C24—172no—
429C24—341no—
430C24—341no—
431C24—342no—
432C24—341no—
433C24—341no—
434C24—342no—
435C24—342no—
436C24—341no—
437C24—341no—
438C24—342no—
439C24C23×D1711yescharacteristic
440C2×Q8—3416no—
441C2×Q8—348no—
442C2×Q8—3416no—
443C2×Q8—348no—
444C2×Q8—348no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—174no—
468C22×C4—174no—
469C22×C4—174no—
470C22×C4—172no—
471C22×C4—174no—
472C22×C4—342no—
473C22×C4—342no—
474C22×C4—342no—
475C22×C4—342no—
476C22×C4—341no—
477C22×C4—342no—
478C22×C4—341no—
479C22×C4—342no—
480C22×C4—342no—
481C22×C4—342no—
482C22×C4—342no—
483C22×C4—342no—
484C22×C4—342no—
485C22×C4—342no—
486C22×C4—341no—
487C22×C4—341no—
488C22×C4—342no—
489C22×C4—341no—
490C22×C4—341no—
491C22×C4—342no—
492C22×C4—342no—
493C22×C4—342no—
494C22×C4—341no—
495C22×C4—341no—
496C22×C4C2×C4×D1714yes—
497C22×C4C2×C4×D1712yes—
498C22×C4C2×D6812yes—
499C22×C4C2×D6812yes—
500C22×C4C2×D6812yes—
501C22×C4#6325212yes—
502C22×C4D4×D1714yes—
503C4⋊C4—3416no—
504C4⋊C4—348no—
505C4⋊C4—684no—
506C4⋊C4—688no—
507C22⋊C4—3416no—
508C22⋊C4—348no—
509C22⋊C4—684no—
510C22⋊C4—688no—
511C22⋊C4—684no—
512C42—28no—
513C42—3416no—
514C23—344no—
515C23—342no—
516C23—342no—
517C23—342no—
518C23—344no—
519C23—681no—
520C23—682no—
521C23—681no—
522C23—682no—
523C23—681no—
524C23—682no—
525C23—682no—
526C23—681no—
527C23—681no—
528C23—682no—
529C23—681no—
530C23—681no—
531C23—682no—
532C23—682no—
533C23—681no—
534C23—682no—
535C23—682no—
536C23—682no—
537C23C22×C4×D1711yescharacteristic
538C23C22×D6811yescharacteristic
539C23C2×#6325211yescharacteristic
540C23C2×D4×D1712yes—
541C23#1058769111yescharacteristic
542C23#1058769411yescharacteristic
543C23#41×D1712yes—
544C23#1058769612yes—
545C23#1058769711yescharacteristic
546C23#1058769711yescharacteristic
547C23#1058769712yes—
548Q8—3432no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—178no—
577C2×C4—344no—
578C2×C4—344no—
579C2×C4—344no—
580C2×C4—344no—
581C2×C4—344no—
582C2×C4—344no—
583C2×C4—342no—
584C2×C4—344no—
585C2×C4—344no—
586C2×C4—344no—
587C2×C4—344no—
588C2×C4—682no—
589C2×C4—681no—
590C2×C4—682no—
591C2×C4—681no—
592C2×C4—682no—
593C2×C4—682no—
594C2×C4—682no—
595C2×C4—681no—
596C2×C4—681no—
597C2×C4—682no—
598C2×C4—681no—
599C2×C4—681no—
600C2×C4—681no—
601C2×C4—682no—
602C2×C4—681no—
603C2×C4—682no—
604C2×C4—682no—
605C2×C4C4×D6814yes—
606C2×C4#1058756814yes—
607C22—684no—
608C22—682no—
609C22—684no—
610C22—682no—
611C22—682no—
612C22—1361no—
613C22—1362no—
614C22—1361no—
615C22—1361no—
616C22—1361no—
617C22—1362no—
618C22C4×D4×D1712yes—
619C22C2×C4×D6811yescharacteristic
620C22#1176328511yescharacteristic
621C22#1176328811yescharacteristic
622C22#1176328912yes—
623C22C2×#1058756811yescharacteristic
624C22#1176331812yes—
625C22#1176332111yescharacteristic
626C22#1176337512yes—
627C22#1176337811yescharacteristic
628C22#1176338112yes—
629C22#1176338112yes—
630C22#1176338511yescharacteristic
631C22#1176338511yescharacteristic
632C22#1176338912yes—
633C22#1176339011yescharacteristic
634C22#1176341711yescharacteristic
635C22#1176341812yes—
636C22#1176342312yes—
637C22#1176342411yescharacteristic
638C22#1176342512yes—
639C22#1176342512yes—
640C22#1176343012yes—
641C4—24no—
642C4—24no—
643C4—42no—
644C4—44no—
645C4—82no—
646C4—348no—
647C4—684no—
648C4—682no—
649C4—1361no—
650C4—1361no—
651C2—1362no—
652C2C4×#1058756811yescharacteristic
653C2#1453146512yes—
654C2#1453146711yescharacteristic
655C2#1453149312yes—
656C2#1453149511yescharacteristic
657C2#1453450611yescharacteristic
658C2#1453451011yescharacteristic
659C2#1453464711yescharacteristic
660C2#1453464812yes—
661C2#1453469411yescharacteristic
662C2#1453469812yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1434C2
socle, cosocleC24×C17C24
prime radicalC24×D17C23
hypercenter#1434D17
hypocenterC17#14301
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
1. C22×C4×Dic34C2semidirect
5. #14527054C2semidirect
6. #14527049C2semidirect
7. #14527048C2semidirect
10. #14527008C2semidirect
11. C17×#1434C2semidirect
25. #11762722C4semidirect
286. #1434D17semidirect
427. C17#14301semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic342
C2×#117627222
C2×#117627012
C2×#117627002
#145270542
#145270492
#145270482
#145270302
#145270282
#145270082
C17×#14342
#1430117