Finite Group Databasebeta

← #16956328

Subgroups — #16956328

Order 4352.

2222 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊Dic17C211yesmaximal, characteristic
2C2×#11762692C212yesmaximal, complemented
3C2×#11762667C211yesmaximal, characteristic
4C2×C22⋊C4×Dic17C211yesmaximal, characteristic
5#14527053C212yesmaximal, complemented
6#14527036C212yesmaximal, complemented
7#14527018C212yesmaximal, complemented
8#14527007C211yesmaximal, characteristic
9#14527006C211yesmaximal, characteristic
10#14527006C211yesmaximal, characteristic
11C17×C22⋊#95C211yesmaximal, characteristic
12C24×Dic17C2211yescharacteristic
13C24×Dic17C2211yescharacteristic
14C22×C4⋊Dic17—28no—
15C22×C4⋊Dic17C418yescomplemented
16C22×C4⋊Dic17C2211yescharacteristic
17C22×C4⋊Dic17C2211yescharacteristic
18C22×C4⋊Dic17C2211yescharacteristic
19C22×C4⋊Dic17C2211yescharacteristic
20C22×#63225C2212yes—
21C22×C22⋊Dic17C2211yescharacteristic
22C22×C22⋊Dic17C2211yescharacteristic
23C22×C4×Dic17C2212yes—
24#11762692—28no—
25#11762667—24no—
26C22⋊C4×Dic17—24no—
27C2×#10587502C2212yes—
28C2×#10587502C2212yes—
29C2×#10587502C2212yes—
30C2×#10587502C2212yes—
31C2×#10587502C2212yes—
32C2×#10587502C2212yes—
33C2×#10587502C2212yes—
34C2×#10587502C2212yes—
35C24×C4×C17C2211yescharacteristic
36C22×C22⋊C4×C17C2211yescharacteristic
37C22×C22⋊C4×C17C2211yescharacteristic
38C2×C17×#95C2212yes—
39C2×C17×#95C2212yes—
40C25×C17C2311yescharacteristic
41C23×C4×C17—22no—
42C23×C4×C17—22no—
43C23×C4×C17—24no—
44C23×C4×C17—24no—
45C23×C4×C17C2×C414yes—
46C23×C4×C17C2312yes—
47C23×C4×C17C2312yes—
48C23×C4×C17C2311yescharacteristic
49C23×C4×C17C2311yescharacteristic
50C23×Dic17—22no—
51C23×Dic17—22no—
52C23×Dic17—22no—
53C23×Dic17—22no—
54C23×Dic17—24no—
55C23×Dic17—24no—
56C23×Dic17—42no—
57C23×Dic17—42no—
58C23×Dic17C2×C414yes—
59C23×Dic17C2×C414yes—
60C23×Dic17C2312yes—
61C23×Dic17C2311yescharacteristic
62C23×Dic17C2311yescharacteristic
63C23×Dic17C2312yes—
64C23×Dic17C2311yescharacteristic
65C23×Dic17C2311yescharacteristic
66C2×C22⋊C4×C17—22no—
67C2×C22⋊C4×C17—22no—
68C2×C22⋊C4×C17—22no—
69C2×C22⋊C4×C17—22no—
70C2×C22⋊C4×C17—22no—
71C2×C22⋊C4×C17Q814yes—
72C2×C22⋊Dic17—22no—
73C2×C22⋊Dic17—22no—
74C2×C22⋊Dic17—22no—
75C2×C22⋊Dic17—22no—
76C2×C22⋊Dic17—22no—
77C2×C22⋊Dic17D414yes—
78C2×C4⋊Dic17—232no—
79C2×C4⋊Dic17—24no—
80C2×C4⋊Dic17—24no—
81C2×C4⋊Dic17—24no—
82C2×C4⋊Dic17C2×C414yes—
83C2×C4⋊Dic17C2×C414yes—
84C2×C4⋊Dic17C2×C414yes—
85C2×C4⋊Dic17C2×C414yes—
86C2×C4⋊Dic17D418yes—
87C2×#63225—24no—
88C2×#63225—24no—
89C2×#63225—42no—
90C2×C4×Dic17—24no—
91C2×C4×Dic17—24no—
92C2×C4×Dic17—42no—
93C17×#95—24no—
94C17×#95—24no—
95#10587502—24no—
96#10587502—24no—
97#10587502—24no—
98#10587502—24no—
99#10587502—24no—
100#10587502—24no—
101#10587502—42no—
102#10587502—42no—
103C24×C17—22no—
104C24×C17—22no—
105C24×C17—22no—
106C24×C17—22no—
107C24×C17C22×C412yes—
108C24×C17C2×D412yes—
109C24×C17C2×Q812yes—
110C24×C17#4112yes—
111C24×C17#4112yes—
112C24×C17#4112yes—
113C24×C17#4112yes—
114C24×C17C2411yescharacteristic
115C22×C4×C17—216no—
116C22×C4×C17—216no—
117C22×C4×C17—28no—
118C22×C4×C17—22no—
119C22×C4×C17—21no—
120C22×C4×C17—21no—
121C22×C4×C17—22no—
122C22×C4×C17—21no—
123C22×C4×C17—21no—
124C22×C4×C17—28no—
125C22×C4×C17—28no—
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×C17—22no—
138C22×C4×C17—22no—
139C22×C4×C17C22×C412yes—
140C22×C4×C17C22×C412yes—
141C22×C4×C17C2×D414yes—
142C22×C4×C17C2×Q814yes—
143C22×C4×C17#4114yes—
144C22×Dic17—21no—
145C22×Dic17—21no—
146C22×Dic17—21no—
147C22×Dic17—21no—
148C22×Dic17—22no—
149C22×Dic17—22no—
150C22×Dic17—28no—
151C22×Dic17—28no—
152C22×Dic17—21no—
153C22×Dic17—21no—
154C22×Dic17—22no—
155C22×Dic17—22no—
156C22×Dic17—22no—
157C22×Dic17—21no—
158C22×Dic17—21no—
159C22×Dic17—22no—
160C22×Dic17—22no—
161C22×Dic17—22no—
162C22×Dic17—22no—
163C22×Dic17—22no—
164C22×Dic17—22no—
165C22×Dic17—22no—
166C22×Dic17—28no—
167C22×Dic17—28no—
168C22×Dic17—22no—
169C22×Dic17—22no—
170C22×Dic17—22no—
171C22×Dic17—22no—
172C22×Dic17—22no—
173C22×Dic17—22no—
174C22×Dic17—22no—
175C22×Dic17—44no—
176C22×Dic17—44no—
177C22×Dic17—48no—
178C22×Dic17—48no—
179C22×Dic17—48no—
180C22×Dic17—48no—
181C22×Dic17C22×C412yes—
182C22×Dic17C22×C412yes—
183C22×Dic17C22×C412yes—
184C22×Dic17C22×C412yes—
185C22×Dic17C2×D414yes—
186C22×Dic17C2×D414yes—
187C22×Dic17C2×D414yes—
188C22⋊C4×C17—28no—
189C22⋊C4×C17—44no—
190C22⋊Dic17—28no—
191C22⋊Dic17—44no—
192C4⋊Dic17—28no—
193C4⋊Dic17—28no—
194C4⋊Dic17—28no—
195C4⋊Dic17C22⋊C4116yescomplemented
196#63225—48no—
197C4×Dic17—48no—
198#13882—171nomaximal, Sylow
199C23×C17—22no—
200C23×C17—22no—
201C23×C17—22no—
202C23×C17—22no—
203C23×C17—24no—
204C23×C17—24no—
205C23×C17—24no—
206C23×C17—24no—
207C23×C17—24no—
208C23×C17—24no—
209C23×C17—24no—
210C23×C17—24no—
211C23×C17—24no—
212C23×C17—24no—
213C23×C17—24no—
214C23×C17—24no—
215C23×C17#11714yes—
216C23×C17C4×D414yes—
217C23×C17C4×Q814yes—
218C23×C17C22⋊Q814yes—
219C23×C17#12314yes—
220C23×C17#12514yes—
221C23×C17#12614yes—
222C23×C17C23×C411yescharacteristic
223C23×C17C22×D412yes—
224C23×C17C22×D411yescharacteristic
225C23×C17C22×Q811yescharacteristic
226C23×C17C2×#4111yescharacteristic
227C23×C17C2×#4111yescharacteristic
228C23×C17C2×#4111yescharacteristic
229C23×C17C2×#4112yes—
230C23×C17C2×#4111yescharacteristic
231C23×C17H(5,𝔽2)12yes—
232C23×C172−1+412yes—
233C2×C4×C17—232no—
234C2×C4×C17—22no—
235C2×C4×C17—22no—
236C2×C4×C17—22no—
237C2×C4×C17—22no—
238C2×C4×C17—24no—
239C2×C4×C17—24no—
240C2×C4×C17—24no—
241C2×C4×C17—24no—
242C2×C4×C17—24no—
243C2×C4×C17—24no—
244C2×C4×C17—24no—
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×C4×C17—42no—
252C2×C4×C17—42no—
253C2×C4×C17—42no—
254C2×C4×C17—42no—
255C2×C4×C17—42no—
256C2×C4×C17C2×C22⋊C414yes—
257C2×C4×C17#11714yes—
258C2×Dic17—22no—
259C2×Dic17—22no—
260C2×Dic17—22no—
261C2×Dic17—22no—
262C2×Dic17—24no—
263C2×Dic17—24no—
264C2×Dic17—24no—
265C2×Dic17—24no—
266C2×Dic17—22no—
267C2×Dic17—22no—
268C2×Dic17—22no—
269C2×Dic17—22no—
270C2×Dic17—22no—
271C2×Dic17—22no—
272C2×Dic17—24no—
273C2×Dic17—24no—
274C2×Dic17—24no—
275C2×Dic17—416no—
276C2×Dic17—416no—
277C2×Dic17—42no—
278C2×Dic17—42no—
279C2×Dic17—42no—
280C2×Dic17—42no—
281C2×Dic17—42no—
282C2×Dic17—42no—
283C2×Dic17—42no—
284C2×Dic17—42no—
285C2×Dic17—42no—
286C2×Dic17—42no—
287C2×Dic17—42no—
288C2×Dic17—42no—
289C2×Dic17—42no—
290C2×Dic17—42no—
291C2×Dic17—42no—
292C2×Dic17C2×C22⋊C414yes—
293C2×Dic17C2×C22⋊C414yes—
294C23×C4⋊C4—171no—
295C2×#387—172no—
296C2×#380—171no—
297C2×C4×C22⋊C4—171no—
298#1445—172no—
299#1441—172no—
300#1436—172no—
301C22⋊#95—171no—
302C22⋊#95—171no—
303C22⋊#95—171no—
304C22⋊#95D1711yescharacteristic
305C22×C17—24no—
306C22×C17—24no—
307C22×C17—24no—
308C22×C17—24no—
309C22×C17—24no—
310C22×C17—28no—
311C22×C17—28no—
312C22×C17—28no—
313C22×C17—28no—
314C22×C17—24no—
315C22×C17#38718yescomplemented
316C22×C17C22×C22⋊C411yescharacteristic
317C22×C17C2×#11711yescharacteristic
318C22×C17C2×#11711yescharacteristic
319C22×C17C2×C4×D411yescharacteristic
320C22×C17C2×C4×Q811yescharacteristic
321C22×C17#51811yescharacteristic
322C22×C17#51911yescharacteristic
323C22×C17C2×C22⋊Q811yescharacteristic
324C22×C17C2×#12311yescharacteristic
325C22×C17#52512yes—
326C22×C17#52512yes—
327C22×C17C2×#12511yescharacteristic
328C22×C17C2×#12611yescharacteristic
329C22×C17#53912yes—
330C22×C17#53912yes—
331C22×C17D4212yes—
332C22×C17#54612yes—
333C22×C17#54712yes—
334C22×C17#54812yes—
335C22×C17D4×Q812yes—
336C22×C17#55212yes—
337C22×C17#55312yes—
338C22×C17#55412yes—
339C4×C17—24no—
340C4×C17—24no—
341C4×C17—44no—
342C4×C17—82no—
343Dic17—24no—
344Dic17—24no—
345Dic17—42no—
346Dic17—42no—
347Dic17—44no—
348Dic17—82no—
349C24×C4—171no—
350C24×C4—171no—
351C24×C4C2×D1711yescharacteristic
352C22×C4⋊C4—171no—
353C22×C4⋊C4—178no—
354C22×C4⋊C4—171no—
355C22×C4⋊C4—171no—
356C22×C4⋊C4—171no—
357C22×C4⋊C4—172no—
358C22×C4⋊C4—348no—
359C22×C22⋊C4—171no—
360C22×C22⋊C4—171no—
361C22×C22⋊C4C2×D1711yescharacteristic
362C22×C22⋊C4C2×D1711yescharacteristic
363C22×C42—172no—
364#387—348no—
365#380—344no—
366C4×C22⋊C4—344no—
367C2×#95—172no—
368C2×#95—172no—
369C2×#95—172no—
370C2×#95—172no—
371C2×#95—172no—
372C2×#95—172no—
373C2×#95—172no—
374C2×#95—172no—
375C2×#95C2×D1712yes—
376C2×#95C2×D1712yes—
377C2×C17—28no—
378C2×C17#228011yescharacteristic
379C2×C17C2×#38711yescharacteristic
380C2×C17#231311yescharacteristic
381C2×C17D4×C22⋊C411yescharacteristic
382C2×C17Q8×C22⋊C411yescharacteristic
383C2×C17#234411yescharacteristic
384C2×C17#234511yescharacteristic
385C2×C17#244312yes—
386C2×C17#244912yes—
387C2×C17#247312yes—
388C2×C17#248112yes—
389C25C22×D1711yescharacteristic
390C23×C4—22no—
391C23×C4—22no—
392C23×C4—24no—
393C23×C4—24no—
394C23×C4—171no—
395C23×C4—171no—
396C23×C4—172no—
397C23×C4—171no—
398C23×C4—171no—
399C23×C4—174no—
400C23×C4—174no—
401C23×C4—172no—
402C23×C4—344no—
403C23×C4—344no—
404C23×C4—342no—
405C23×C4—342no—
406C23×C4—342no—
407C23×C4—342no—
408C23×C4—682no—
409C23×C4—682no—
410C23×C4C4×D1714yes—
411C23×C4C22×D1712yes—
412C23×C4C22×D1712yes—
413C23×C4C22×D1711yescharacteristic
414C23×C4C22×D1711yescharacteristic
415C2×C4⋊C4—174no—
416C2×C4⋊C4—178no—
417C2×C4⋊C4—174no—
418C2×C4⋊C4—174no—
419C2×C4⋊C4—174no—
420C2×C4⋊C4—3432no—
421C2×C4⋊C4—344no—
422C2×C4⋊C4—344no—
423C2×C4⋊C4—344no—
424C2×C4⋊C4—344no—
425C2×C4⋊C4—344no—
426C2×C4⋊C4—682no—
427C2×C22⋊C4—22no—
428C2×C22⋊C4—22no—
429C2×C22⋊C4—22no—
430C2×C22⋊C4—22no—
431C2×C22⋊C4—22no—
432C2×C22⋊C4—174no—
433C2×C22⋊C4—342no—
434C2×C22⋊C4—342no—
435C2×C22⋊C4—342no—
436C2×C22⋊C4—342no—
437C2×C22⋊C4—342no—
438C2×C22⋊C4Dic3414yes—
439C2×C42—344no—
440C2×C42—344no—
441C2×C42—682no—
442#95—24no—
443#95—24no—
444#95—344no—
445#95—344no—
446#95—344no—
447#95—344no—
448#95—344no—
449#95—344no—
450#95—682no—
451#95—682no—
452C17#1388211yescharacteristic, complemented, Sylow
453C24—22no—
454C24—22no—
455C24—22no—
456C24—22no—
457C24C2×Dic3412yes—
458C24C2×C4×D1712yes—
459C24#6325212yes—
460C24D4×D1712yes—
461C24#6325412yes—
462C24#6325412yes—
463C24#6325612yes—
464C24C23×D1711yescharacteristic
465C22×C4—216no—
466C22×C4—216no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—28no—
474C22×C4—28no—
475C22×C4—28no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—172no—
490C22×C4—172no—
491C22×C4—174no—
492C22×C4—174no—
493C22×C4—172no—
494C22×C4—172no—
495C22×C4—174no—
496C22×C4—348no—
497C22×C4—348no—
498C22×C4—341no—
499C22×C4—341no—
500C22×C4—341no—
501C22×C4—341no—
502C22×C4—342no—
503C22×C4—342no—
504C22×C4—342no—
505C22×C4—341no—
506C22×C4—341no—
507C22×C4—342no—
508C22×C4—342no—
509C22×C4—342no—
510C22×C4—342no—
511C22×C4—342no—
512C22×C4—341no—
513C22×C4—341no—
514C22×C4—342no—
515C22×C4—342no—
516C22×C4—342no—
517C22×C4—342no—
518C22×C4—348no—
519C22×C4—348no—
520C22×C4—342no—
521C22×C4—342no—
522C22×C4—342no—
523C22×C4—342no—
524C22×C4—342no—
525C22×C4—342no—
526C22×C4—342no—
527C22×C4—684no—
528C22×C4—684no—
529C22×C4—688no—
530C22×C4—688no—
531C22×C4—688no—
532C22×C4—688no—
533C22×C4C2×Dic3414yes—
534C22×C4C2×C4×D1712yes—
535C22×C4C2×C4×D1712yes—
536C22×C4D4×D1714yes—
537C22×C4#6325414yes—
538C4⋊C4—1716no—
539C4⋊C4—348no—
540C4⋊C4—348no—
541C4⋊C4—348no—
542C4⋊C4—688no—
543C22⋊C4—28no—
544C22⋊C4—44no—
545C22⋊C4—348no—
546C22⋊C4—684no—
547C42—688no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—22no—
554C23—24no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23C4×Dic3414yes—
565C23#1058756314yes—
566C23#1058756414yes—
567C23#1058756914yes—
568C23#1058757614yes—
569C23#1058757914yes—
570C23#1058758414yes—
571C23C22×Dic3411yescharacteristic
572C23C22×C4×D1711yescharacteristic
573C23C2×#6325211yescharacteristic
574C23C2×D4×D1711yescharacteristic
575C23C2×D4×D1712yes—
576C23C2×#6325412yes—
577C23C2×#6325411yescharacteristic
578C23C2×#6325411yescharacteristic
579C23C2×#6325611yescharacteristic
580C23#1058769612yes—
581C23#1058769712yes—
582C2×C4—232no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—174no—
606C2×C4—174no—
607C2×C4—342no—
608C2×C4—342no—
609C2×C4—342no—
610C2×C4—342no—
611C2×C4—344no—
612C2×C4—344no—
613C2×C4—344no—
614C2×C4—344no—
615C2×C4—344no—
616C2×C4—342no—
617C2×C4—342no—
618C2×C4—342no—
619C2×C4—342no—
620C2×C4—342no—
621C2×C4—342no—
622C2×C4—344no—
623C2×C4—344no—
624C2×C4—6816no—
625C2×C4—6816no—
626C2×C4—682no—
627C2×C4—682no—
628C2×C4—682no—
629C2×C4—682no—
630C2×C4—682no—
631C2×C4—682no—
632C2×C4—682no—
633C2×C4—682no—
634C2×C4—682no—
635C2×C4—682no—
636C2×C4—682no—
637C2×C4—682no—
638C2×C4—682no—
639C2×C4—682no—
640C2×C4—682no—
641C2×C4#1058756314yes—
642C2×C4C22⋊C4×D1714yes—
643C22—24no—
644C22—24no—
645C22—24no—
646C22—28no—
647C22—24no—
648C22—24no—
649C22—24no—
650C22—28no—
651C22—28no—
652C22—28no—
653C22#1176269218yescomplemented
654C22C2×C22⋊C4×D1711yescharacteristic
655C22C2×#1058756311yescharacteristic
656C22C2×#1058756311yescharacteristic
657C22C2×#1058757911yescharacteristic
658C22C2×C4×Dic3411yescharacteristic
659C22#1176328311yescharacteristic
660C22#1176328611yescharacteristic
661C22C2×#1058756411yescharacteristic
662C22C2×#1058756911yescharacteristic
663C22#1176331512yes—
664C22#1176331512yes—
665C22C2×#1058757611yescharacteristic
666C22C2×#1058758411yescharacteristic
667C22#1176340312yes—
668C22#1176340312yes—
669C22#1176344312yes—
670C22#1176345012yes—
671C22#1176345912yes—
672C22#1176346812yes—
673C22D4×Dic3412yes—
674C22#1176350112yes—
675C22#1176350512yes—
676C22#1176351612yes—
677C4—24no—
678C4—24no—
679C4—44no—
680C4—82no—
681C4—344no—
682C4—344no—
683C4—682no—
684C4—682no—
685C4—684no—
686C4—1362no—
687C2—28no—
688C2#1453116211yescharacteristic
689C2C2×#1176269211yescharacteristic
690C2#1453138511yescharacteristic
691C2#1453160411yescharacteristic
692C2C22⋊C4×Dic3411yescharacteristic
693C2#1453165511yescharacteristic
694C2#1453165911yescharacteristic
695C2#1453252512yes—
696C2#1453259912yes—
697C2#1453292412yes—
698C2#1453304212yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×C22⋊#95C2
socle, cosocleC24×C17C24
prime radicalC25×C17C23
hypercenterC22⋊#95D17
hypocenterC17#13882
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
2. C2×#11762692C2semidirect
5. #14527053C2semidirect
6. #14527036C2semidirect
7. #14527018C2semidirect
15. C22×C4⋊Dic17C4semidirect
195. C4⋊Dic17C22⋊C4semidirect
315. C22×C17#387semidirect
452. C17#13882semidirect
653. C22#11762692semidirect

Maximal subgroups

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

subgroupindex
C23×C4⋊Dic172
C2×#117626922
C2×#117626672
C2×C22⋊C4×Dic172
#145270532
#145270362
#145270182
#145270072
#145270062
C17×C22⋊#952
#1388217