Finite Group Databasebeta

← #16956079

Subgroups — #16956079

Order 4352.

1790 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587573C211yesmaximal, characteristic
2C2×C4⋊C4×Dic17C211yesmaximal, characteristic
3C2×#11762654C211yesmaximal, characteristic
4#14527054C212yesmaximal
5#14527050C212yesmaximal
6#14527038C211yesmaximal, characteristic
7#14527037C211yesmaximal, characteristic
8#14527034C211yesmaximal, characteristic
9#95×Dic17C212yesmaximal
10#95×Dic17C212yesmaximal
11C17×#1441C211yesmaximal, characteristic
12C23×Dic34C2211yescharacteristic
13C2×#10587573—24no—
14C2×#10587573—24no—
15C2×#10587573C418yescomplemented
16C22×C4⋊Dic17C2211yescharacteristic
17C22×#63225C2211yescharacteristic
18C22×#63225C2211yescharacteristic
19C22×C4×Dic17C2212yes—
20C22×C4×Dic17C2211yescharacteristic
21C22×C4×Dic17C2211yescharacteristic
22C22×C4×Dic17C2212yes—
23C22×C4×Dic17C2211yescharacteristic
24C4⋊C4×Dic17—24no—
25#11762654—24no—
26C2×#10587502C2212yes—
27C2×#10587502C2212yes—
28C2×#10587502C2212yes—
29C2×#10587502C2212yes—
30C2×#10587502C2212yes—
31C2×#10587502C2212yes—
32C2×#10587502C2211yescharacteristic
33C2×#10587502C2212yes—
34C2×#10587502C2212yes—
35C22×C4⋊C4×C17C2211yescharacteristic
36C22×C4⋊C4×C17C2211yescharacteristic
37C2×C17×#95C2212yes—
38C2×C17×#95C2212yes—
39C2×C17×#95C2211yescharacteristic
40C23×C4×C17C2311yescharacteristic
41C23×C4×C17C2312yes—
42C23×C4×C17C2312yes—
43C23×C4×C17C2311yescharacteristic
44C23×C4×C17C2311yescharacteristic
45C23×Dic17C2311yescharacteristic
46C23×Dic17C2312yes—
47C23×Dic17C2312yes—
48C23×Dic17C2311yescharacteristic
49C23×Dic17C2311yescharacteristic
50C23×Dic17C2311yescharacteristic
51C22×Dic34—22no—
52C22×Dic34—22no—
53C22×Dic34—22no—
54C22×Dic34—22no—
55C22×Dic34—22no—
56C22×Dic34—22no—
57C22×Dic34C2×C414yes—
58C2×C4⋊C4×C17—22no—
59C2×C4⋊C4×C17—22no—
60C2×C4⋊C4×C17—22no—
61C2×C4⋊C4×C17—24no—
62C2×C4⋊C4×C17C2×C414yes—
63C2×C4⋊Dic17—22no—
64C2×C4⋊Dic17—24no—
65C2×#63225—22no—
66C2×#63225—22no—
67C2×#63225—22no—
68C2×#63225—22no—
69C2×#63225C2×C414yes—
70C2×#63225C2×C414yes—
71C2×C4×Dic17—22no—
72C2×C4×Dic17—22no—
73C2×C4×Dic17—22no—
74C2×C4×Dic17—22no—
75C2×C4×Dic17—24no—
76C2×C4×Dic17—24no—
77C2×C4×Dic17—24no—
78C2×C4×Dic17—24no—
79C2×C4×Dic17—24no—
80C2×C4×Dic17C2×C414yes—
81C2×C4×Dic17C2×C414yes—
82C2×C4×Dic17C2×C414yes—
83C2×C4×Dic17D414yes—
84C2×C4×Dic17D414yes—
85C2×C4×Dic17Q818yescomplemented
86#10587573—232no—
87C17×#95—22no—
88C17×#95—24no—
89C17×#95—24no—
90#10587502—22no—
91#10587502—24no—
92#10587502—24no—
93#10587502—24no—
94#10587502—24no—
95#10587502—24no—
96#10587502—24no—
97#10587502—24no—
98#10587502—24no—
99C24×C17C2411yescharacteristic
100C22×C4×C17—21no—
101C22×C4×C17—21no—
102C22×C4×C17—21no—
103C22×C4×C17—21no—
104C22×C4×C17—22no—
105C22×C4×C17—22no—
106C22×C4×C17—22no—
107C22×C4×C17—22no—
108C22×C4×C17—21no—
109C22×C4×C17—21no—
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×C17C22×C412yes—
123C22×C4×C17C22×C412yes—
124C22×C4×C17C22×C412yes—
125C22×C4×C17C2×D412yes—
126C22×C4×C17C2×D412yes—
127C22×C4×C17C2×Q814yes—
128C22×C4×C17#4114yes—
129C22×C4×C17#4112yes—
130C22×C4×C17#4112yes—
131C22×Dic17—21no—
132C22×Dic17—21no—
133C22×Dic17—21no—
134C22×Dic17—21no—
135C22×Dic17—21no—
136C22×Dic17—21no—
137C22×Dic17—21no—
138C22×Dic17—21no—
139C22×Dic17—22no—
140C22×Dic17—22no—
141C22×Dic17—21no—
142C22×Dic17—21no—
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—22no—
153C22×Dic17—22no—
154C22×Dic17—22no—
155C22×Dic17C22×C412yes—
156C22×Dic17C22×C412yes—
157C22×Dic17C22×C412yes—
158C22×Dic17C22×C412yes—
159C22×Dic17C2×D412yes—
160C22×Dic17C2×D412yes—
161C22×Dic17C2×D412yes—
162C22×Dic17C2×D412yes—
163C22×Dic17C2×Q814yes—
164C22×Dic17C2×Q814yes—
165C22×Dic17#4114yes—
166C22×Dic17#4112yes—
167C22×Dic17#4112yes—
168C22×Dic17#4114yes—
169C2×Dic34—28no—
170C2×Dic34—28no—
171C2×Dic34—28no—
172C2×Dic34—28no—
173C2×Dic34—28no—
174C2×Dic34—28no—
175C2×Dic34—28no—
176C4⋊C4×C17—28no—
177C4⋊C4×C17—28no—
178C4⋊Dic17—28no—
179#63225—28no—
180#63225—28no—
181C4×Dic17—28no—
182C4×Dic17—28no—
183C4×Dic17—216no—
184C4×Dic17—48no—
185C4×Dic17C22⋊C4116yes—
186#13850—171nomaximal, Sylow
187C23×C17C23×C411yescharacteristic
188C23×C17C22×D411yescharacteristic
189C23×C17C22×D411yescharacteristic
190C23×C17C22×Q812yes—
191C23×C17C2×#4111yescharacteristic
192C23×C17C2×#4112yes—
193C23×C17C2×#4111yescharacteristic
194C23×C17C2×#4111yescharacteristic
195C23×C17C2×#4111yescharacteristic
196C23×C17C2×#4112yes—
197C23×C17C2×#4112yes—
198C2×C4×C17—22no—
199C2×C4×C17—22no—
200C2×C4×C17—24no—
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—24no—
208C2×C4×C17—22no—
209C2×C4×C17—24no—
210C2×C4×C17—24no—
211C2×C4×C17—24no—
212C2×C4×C17—24no—
213C2×C4×C17—24no—
214C2×C4×C17—24no—
215C2×C4×C17—24no—
216C2×C4×C17—41no—
217C2×C4×C17—41no—
218C2×C4×C17—42no—
219C2×C4×C17—42no—
220C2×C4×C17—42no—
221C2×C4×C17—42no—
222C2×C4×C17—42no—
223C2×C4×C17—42no—
224C2×C4×C17—42no—
225C2×C4×C17—42no—
226C2×C4×C17—42no—
227C2×C4×C17C2×C22⋊C414yes—
228C2×C4×C17#11714yes—
229C2×Dic17—22no—
230C2×Dic17—22no—
231C2×Dic17—22no—
232C2×Dic17—22no—
233C2×Dic17—22no—
234C2×Dic17—22no—
235C2×Dic17—22no—
236C2×Dic17—22no—
237C2×Dic17—22no—
238C2×Dic17—22no—
239C2×Dic17—22no—
240C2×Dic17—22no—
241C2×Dic17—22no—
242C2×Dic17—24no—
243C2×Dic17—24no—
244C2×Dic17—24no—
245C2×Dic17—24no—
246C2×Dic17—24no—
247C2×Dic17—24no—
248C2×Dic17—24no—
249C2×Dic17—41no—
250C2×Dic17—41no—
251C2×Dic17—42no—
252C2×Dic17—42no—
253C2×Dic17—42no—
254C2×Dic17—42no—
255C2×Dic17—42no—
256C2×Dic17—42no—
257C2×Dic17—42no—
258C2×Dic17—42no—
259C2×Dic17—42no—
260C2×Dic17C2×C22⋊C414yes—
261C2×Dic17C2×C22⋊C414yes—
262C2×Dic17#11714yes—
263C2×Dic17C4×Q818yes—
264C2×Dic17C22⋊Q818yes—
265C2×Dic17#12414yes—
266C2×Dic17C4⋊Q814yes—
267Dic34—232no—
268C22×C4×Q8—171no—
269C2×C4×C4⋊C4—171no—
270C2×#376—171no—
271#1446—172no—
272#1445—172no—
273#1441—171no—
274#1441—171no—
275#1441—171no—
276#1441D1711yescharacteristic
277C4×#95—172no—
278C4×#95—172no—
279C22×C17C22×C22⋊C411yescharacteristic
280C22×C17C2×#11711yescharacteristic
281C22×C17C2×#11711yescharacteristic
282C22×C17C2×C4×Q812yes—
283C22×C17C4×#4112yes—
284C22×C17C2×C22⋊Q812yes—
285C22×C17#52511yescharacteristic
286C22×C17#52511yescharacteristic
287C22×C17#52511yescharacteristic
288C22×C17#52511yescharacteristic
289C22×C17#52511yescharacteristic
290C22×C17#52511yescharacteristic
291C22×C17C2×#12411yescharacteristic
292C22×C17#52912yes—
293C22×C17#52912yes—
294C22×C17#52912yes—
295C22×C17#52912yes—
296C22×C17C2×C4⋊Q811yescharacteristic
297C22×C17#53212yes—
298C22×C17#53312yes—
299C22×C17#53312yes—
300C22×C17#53312yes—
301C22×C17#53312yes—
302C4×C17—24no—
303C4×C17—24no—
304C4×C17—44no—
305C4×C17—44no—
306C4×C17—42no—
307Dic17—24no—
308Dic17—24no—
309Dic17—44no—
310Dic17—44no—
311Dic17—42no—
312Dic17#39118yes—
313C23×Q8—171no—
314C2×C4×Q8—178no—
315C2×C4×Q8—344no—
316C2×C4×Q8—344no—
317C22×C4⋊C4—171no—
318C22×C4⋊C4—171no—
319C22×C4⋊C4—171no—
320C22×C4⋊C4C2×D1711yescharacteristic
321C22×C4⋊C4C2×D1711yescharacteristic
322C22×C42—171no—
323C22×C42—171no—
324C22×C42—171no—
325C22×C42—172no—
326C22×C42—172no—
327C4×C4⋊C4—344no—
328#376—344no—
329C2×#95—172no—
330C2×#95—172no—
331C2×#95—172no—
332C2×#95—172no—
333C2×#95—172no—
334C2×#95—172no—
335C2×#95—171no—
336C2×#95—172no—
337C2×#95—172no—
338C2×#95C2×D1712yes—
339C2×#95C2×D1712yes—
340C2×#95C2×D1711yescharacteristic
341C2×C17#228011yescharacteristic
342C2×C17C2×#39111yescharacteristic
343C2×C17#229512yes—
344C2×C17#229512yes—
345C2×C17#229711yescharacteristic
346C2×C17#239612yes—
347C2×C17#239612yes—
348C2×C17#239811yescharacteristic
349C2×C17#239811yescharacteristic
350C2×C17#239911yescharacteristic
351C2×C17#239911yescharacteristic
352C22×Q8—174no—
353C22×Q8—342no—
354C22×Q8—342no—
355C22×Q8—342no—
356C22×Q8—342no—
357C22×Q8—342no—
358C22×Q8—342no—
359C23×C4—171no—
360C23×C4—171no—
361C23×C4—171no—
362C23×C4—171no—
363C23×C4—172no—
364C23×C4—172no—
365C23×C4C22×D1711yescharacteristic
366C23×C4C22×D1712yes—
367C23×C4C22×D1712yes—
368C23×C4C22×D1711yescharacteristic
369C23×C4C22×D1711yescharacteristic
370C4×Q8—3432no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—24no—
375C2×C4⋊C4—174no—
376C2×C4⋊C4—174no—
377C2×C4⋊C4—342no—
378C2×C4⋊C4—342no—
379C2×C4⋊C4—344no—
380C2×C4⋊C4—342no—
381C2×C4⋊C4—342no—
382C2×C4⋊C4—342no—
383C2×C4⋊C4C4×D1714yes—
384C2×C42—174no—
385C2×C42—174no—
386C2×C42—174no—
387C2×C42—174no—
388C2×C42—174no—
389C2×C42—178no—
390C2×C42—342no—
391C2×C42—342no—
392C2×C42—342no—
393C2×C42—342no—
394C2×C42—344no—
395C2×C42—344no—
396C2×C42—344no—
397C2×C42—344no—
398C2×C42—344no—
399#95—22no—
400#95—24no—
401#95—24no—
402#95—344no—
403#95—344no—
404#95—344no—
405#95—344no—
406#95—344no—
407#95—344no—
408#95—342no—
409#95—344no—
410#95—344no—
411C17#1385011yescharacteristic, complemented, Sylow
412C24C23×D1711yescharacteristic
413C2×Q8—348no—
414C2×Q8—348no—
415C2×Q8—348no—
416C2×Q8—348no—
417C2×Q8—348no—
418C2×Q8—348no—
419C2×Q8—348no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—172no—
443C22×C4—172no—
444C22×C4—172no—
445C22×C4—174no—
446C22×C4—174no—
447C22×C4—172no—
448C22×C4—172no—
449C22×C4—172no—
450C22×C4—172no—
451C22×C4—172no—
452C22×C4—172no—
453C22×C4—172no—
454C22×C4—174no—
455C22×C4—174no—
456C22×C4—341no—
457C22×C4—341no—
458C22×C4—341no—
459C22×C4—341no—
460C22×C4—341no—
461C22×C4—341no—
462C22×C4—341no—
463C22×C4—341no—
464C22×C4—342no—
465C22×C4—342no—
466C22×C4—341no—
467C22×C4—341no—
468C22×C4—342no—
469C22×C4—342no—
470C22×C4—342no—
471C22×C4—342no—
472C22×C4—342no—
473C22×C4—342no—
474C22×C4—342no—
475C22×C4—342no—
476C22×C4—342no—
477C22×C4—342no—
478C22×C4—342no—
479C22×C4—342no—
480C22×C4C2×C4×D1712yes—
481C22×C4C2×C4×D1712yes—
482C22×C4C2×C4×D1712yes—
483C22×C4D4×D1712yes—
484C22×C4D4×D1712yes—
485C22×C4#6325414yes—
486C22×C4#6325412yes—
487C22×C4Q8×D1714yes—
488C22×C4#6325612yes—
489C4⋊C4—28no—
490C4⋊C4—28no—
491C4⋊C4—348no—
492C4⋊C4—348no—
493C4⋊C4—348no—
494C42—1716no—
495C42—348no—
496C42—348no—
497C42—3416no—
498C42—688no—
499C23C22×C4×D1711yescharacteristic
500C23C2×D4×D1711yescharacteristic
501C23C2×D4×D1711yescharacteristic
502C23C2×#6325411yescharacteristic
503C23C2×#6325412yes—
504C23C2×Q8×D1712yes—
505C23C2×#6325611yescharacteristic
506C23#41×D1711yescharacteristic
507C23#41×D1711yescharacteristic
508C23#41×D1712yes—
509C23#41×D1712yes—
510Q8—3432no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—174no—
541C2×C4—178no—
542C2×C4—178no—
543C2×C4—174no—
544C2×C4—174no—
545C2×C4—174no—
546C2×C4—174no—
547C2×C4—342no—
548C2×C4—342no—
549C2×C4—342no—
550C2×C4—342no—
551C2×C4—342no—
552C2×C4—342no—
553C2×C4—342no—
554C2×C4—342no—
555C2×C4—342no—
556C2×C4—342no—
557C2×C4—342no—
558C2×C4—342no—
559C2×C4—342no—
560C2×C4—344no—
561C2×C4—344no—
562C2×C4—344no—
563C2×C4—344no—
564C2×C4—344no—
565C2×C4—344no—
566C2×C4—344no—
567C2×C4—681no—
568C2×C4—681no—
569C2×C4—682no—
570C2×C4—682no—
571C2×C4—682no—
572C2×C4—682no—
573C2×C4—682no—
574C2×C4—682no—
575C2×C4—682no—
576C2×C4—682no—
577C2×C4—682no—
578C2×C4C22⋊C4×D1714yes—
579C2×C4#1058757814yes—
580C22C2×C22⋊C4×D1711yescharacteristic
581C22#117×D1711yescharacteristic
582C22C2×#1058757811yescharacteristic
583C22C4×Q8×D1712yes—
584C22C4×#6325412yes—
585C22C22⋊Q8×D1712yes—
586C22#1176331511yescharacteristic
587C22#1176331711yescharacteristic
588C22#1176331911yescharacteristic
589C22#1176331911yescharacteristic
590C22#1176332211yescharacteristic
591C22#1176332211yescharacteristic
592C22#124×D1711yescharacteristic
593C22#1176333612yes—
594C22#1176334012yes—
595C22#1176334212yes—
596C22#1176334412yes—
597C22C4⋊Q8×D1711yescharacteristic
598C22#1176335412yes—
599C22#1176336112yes—
600C22#1176336112yes—
601C22#1176336412yes—
602C22#1176336512yes—
603C4—24no—
604C4—24no—
605C4—42no—
606C4—44no—
607C4—44no—
608C4—178no—
609C4—344no—
610C4—344no—
611C4—682no—
612C4—684no—
613C4—684no—
614C2#1453116611yescharacteristic
615C2D17×#39111yescharacteristic
616C2#1453125512yes—
617C2#1453125712yes—
618C2#1453126911yescharacteristic
619C2#1453205912yes—
620C2#1453206112yes—
621C2#1453207211yescharacteristic
622C2#1453207311yescharacteristic
623C2#1453207411yescharacteristic
624C2#1453207611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1441C2
socle, prime radical, cosocleC24×C17C24
hypercenter#1441D17
hypocenterC17#13850
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
15. C2×#10587573C4semidirect
85. C2×C4×Dic17Q8semidirect
411. C17#13850semidirect

Maximal subgroups

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

subgroupindex
C22×#105875732
C2×C4⋊C4×Dic172
C2×#117626542
#145270542
#145270502
#145270382
#145270372
#145270342
#95×Dic172
C17×#14412
#1385017