Finite Group Databasebeta

← #16956088

Subgroups — #16956088

Order 4352.

2118 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587578C211yesmaximal, characteristic
2C2×C4⋊C4×Dic17C211yesmaximal, characteristic, complemented
3C2×#11762660C211yesmaximal, characteristic, complemented
4C2×#11762656C212yesmaximal
5#14527037C211yesmaximal, characteristic, complemented
6#14527036C212yesmaximal, complemented
7#14527030C211yesmaximal, characteristic
8#14527028C212yesmaximal
9#14527026C211yesmaximal, characteristic
10#95×Dic17C212yesmaximal, complemented
11C17×#1441C211yesmaximal, characteristic, complemented
12C23×C4×D17C2211yescharacteristic
13C2×#10587578—24no—
14C2×#10587578—24no—
15C2×#10587578C418yescomplemented
16C22×C4⋊Dic17C2211yescharacteristic
17C22×C4⋊Dic17C2211yescharacteristic
18C22×#63225C2212yes—
19C22×#63227C2212yes—
20C22×#63227C2212yes—
21C22×#63227C2211yescharacteristic
22C22×#63227C2211yescharacteristic
23C22×C4×Dic17C2212yes—
24C22×C4×Dic17C2212yes—
25C22×C4×Dic17C2211yescharacteristic
26C22×C4×Dic17C2211yescharacteristic
27C4⋊C4×Dic17—24no—
28#11762660—24no—
29#11762656—28no—
30C2×#10587502C2212yes—
31C2×#10587502C2212yes—
32C2×#10587502C2212yes—
33C2×#10587502C2212yes—
34C2×#10587502C2211yescharacteristic
35C2×#10587502C2212yes—
36C22×C4⋊C4×C17C2211yescharacteristic
37C22×C4⋊C4×C17C2211yescharacteristic
38C2×C17×#95C2212yes—
39C2×C17×#95C2212yes—
40C2×C17×#95C2211yescharacteristic
41C24×D17C2311yescharacteristic
42C23×C4×C17C2311yescharacteristic
43C23×C4×C17C2312yes—
44C23×C4×C17C2312yes—
45C23×C4×C17C2311yescharacteristic
46C23×C4×C17C2311yescharacteristic
47C23×Dic17C2311yescharacteristic
48C23×Dic17C2312yes—
49C23×Dic17C2312yes—
50C23×Dic17C2311yescharacteristic
51C23×Dic17C2311yescharacteristic
52C22×C4×D17—22no—
53C22×C4×D17—22no—
54C22×C4×D17—22no—
55C22×C4×D17—22no—
56C22×C4×D17C2×C414yes—
57C22×C4×D17D414yes—
58C22×C4×D17Q814yes—
59C2×C4⋊C4×C17—22no—
60C2×C4⋊C4×C17—22no—
61C2×C4⋊C4×C17—22no—
62C2×C4⋊C4×C17—24no—
63C2×C4⋊C4×C17C2×C414yes—
64C2×#63227—24no—
65C2×#63227—24no—
66C2×#63227—22no—
67C2×#63227—22no—
68C2×#63227—24no—
69C2×#63227—24no—
70C2×#63227—22no—
71C2×#63227—22no—
72C2×#63227—24no—
73C2×#63227—24no—
74C2×#63227C2×C414yes—
75C2×#63227C2×C414yes—
76C2×C4⋊Dic17—22no—
77C2×C4⋊Dic17—22no—
78C2×C4⋊Dic17—22no—
79C2×C4⋊Dic17—24no—
80C2×C4⋊Dic17C2×C414yes—
81C2×#63225—24no—
82C2×#63225—24no—
83C2×#63225—24no—
84C2×C4×Dic17—22no—
85C2×C4×Dic17—22no—
86C2×C4×Dic17—24no—
87C2×C4×Dic17—24no—
88C2×C4×Dic17—24no—
89C2×C4×Dic17—24no—
90C2×C4×Dic17—24no—
91C2×C4×Dic17—24no—
92C2×C4×Dic17C2×C414yes—
93C2×C4×Dic17C2×C414yes—
94C2×C4×Dic17D414yes—
95C2×C4×Dic17D414yes—
96#10587578—232no—
97C17×#95—24no—
98C17×#95—22no—
99C17×#95—24no—
100#10587502—24no—
101#10587502—22no—
102#10587502—24no—
103#10587502—24no—
104#10587502—24no—
105#10587502—24no—
106C24×C17C2411yescharacteristic
107C23×D17—21no—
108C23×D17—21no—
109C23×D17—21no—
110C23×D17—21no—
111C23×D17—22no—
112C23×D17—22no—
113C23×D17—22no—
114C23×D17—22no—
115C23×D17C22×C412yes—
116C23×D17C2×D412yes—
117C23×D17C2×Q812yes—
118C22×C4×C17—21no—
119C22×C4×C17—21no—
120C22×C4×C17—21no—
121C22×C4×C17—21no—
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—21no—
129C22×C4×C17—21no—
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×C17—22no—
140C22×C4×C17C22×C412yes—
141C22×C4×C17C22×C412yes—
142C22×C4×C17C22×C412yes—
143C22×C4×C17C2×D412yes—
144C22×C4×C17C2×D412yes—
145C22×C4×C17C2×D412yes—
146C22×C4×C17C2×Q812yes—
147C22×C4×C17#4114yes—
148C22×C4×C17#4114yes—
149C22×Dic17—22no—
150C22×Dic17—22no—
151C22×Dic17—22no—
152C22×Dic17—22no—
153C22×Dic17—22no—
154C22×Dic17—22no—
155C22×Dic17—21no—
156C22×Dic17—21no—
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×C412yes—
168C22×Dic17C22×C412yes—
169C22×Dic17C22×C412yes—
170C22×Dic17C2×D412yes—
171C22×Dic17C2×D412yes—
172C22×Dic17C2×D412yes—
173C22×Dic17C2×D412yes—
174C22×Dic17C2×D412yes—
175C22×Dic17C2×Q812yes—
176C22×Dic17#4114yes—
177C22×Dic17#4114yes—
178C22×Dic17#4112yes—
179C22×Dic17#4112yes—
180C2×C4×D17—28no—
181C2×C4×D17—28no—
182C2×C4×D17—28no—
183C2×C4×D17—28no—
184C2×C4×D17—28no—
185C2×C4×D17—28no—
186C2×C4×D17C4⋊C4116yescomplemented
187C4⋊C4×C17—28no—
188C4⋊C4×C17—28no—
189#63227—216no—
190#63227—216no—
191#63227—28no—
192#63227—28no—
193C4⋊Dic17—28no—
194C4⋊Dic17—28no—
195#63225—216no—
196C4×Dic17—28no—
197C4×Dic17—48no—
198C4×Dic17—48no—
199C4×Dic17C22⋊C4116yescomplemented
200#13852—171nomaximal, Sylow
201C23×C17C23×C411yescharacteristic
202C23×C17C22×D411yescharacteristic
203C23×C17C22×D411yescharacteristic
204C23×C17C22×D411yescharacteristic
205C23×C17C22×Q811yescharacteristic
206C23×C17C2×#4112yes—
207C23×C17C2×#4112yes—
208C23×C17C2×#4111yescharacteristic
209C23×C17C2×#4111yescharacteristic
210C23×C17C2×#4112yes—
211C23×C17C2×#4112yes—
212C22×D17—22no—
213C22×D17—22no—
214C22×D17—24no—
215C22×D17—24no—
216C22×D17—22no—
217C22×D17—22no—
218C22×D17—22no—
219C22×D17—22no—
220C22×D17—24no—
221C22×D17—24no—
222C22×D17—24no—
223C22×D17—24no—
224C22×D17—41no—
225C22×D17—41no—
226C22×D17—42no—
227C22×D17—42no—
228C22×D17—42no—
229C22×D17—42no—
230C22×D17—42no—
231C22×D17—42no—
232C22×D17—41no—
233C22×D17—41no—
234C22×D17C2×C4⋊C414yes—
235C2×C4×C17—22no—
236C2×C4×C17—22no—
237C2×C4×C17—24no—
238C2×C4×C17—22no—
239C2×C4×C17—22no—
240C2×C4×C17—22no—
241C2×C4×C17—22no—
242C2×C4×C17—22no—
243C2×C4×C17—22no—
244C2×C4×C17—24no—
245C2×C4×C17—22no—
246C2×C4×C17—24no—
247C2×C4×C17—24no—
248C2×C4×C17—24no—
249C2×C4×C17—24no—
250C2×C4×C17—24no—
251C2×C4×C17—24no—
252C2×C4×C17—24no—
253C2×C4×C17—42no—
254C2×C4×C17—42no—
255C2×C4×C17—41no—
256C2×C4×C17—41no—
257C2×C4×C17—42no—
258C2×C4×C17—42no—
259C2×C4×C17—42no—
260C2×C4×C17—42no—
261C2×C4×C17—42no—
262C2×C4×C17—42no—
263C2×C4×C17—42no—
264C2×C4×C17C2×C22⋊C414yes—
265C2×C4×C17C2×C4⋊C414yes—
266C2×Dic17—24no—
267C2×Dic17—24no—
268C2×Dic17—22no—
269C2×Dic17—22no—
270C2×Dic17—22no—
271C2×Dic17—24no—
272C2×Dic17—24no—
273C2×Dic17—24no—
274C2×Dic17—24no—
275C2×Dic17—24no—
276C2×Dic17—24no—
277C2×Dic17—24no—
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—41no—
287C2×Dic17—42no—
288C2×Dic17—41no—
289C2×Dic17C2×C22⋊C414yes—
290C2×Dic17C2×C22⋊C414yes—
291C2×Dic17C2×C4⋊C414yes—
292C2×Dic17#11714yes—
293C2×Dic17#12114yes—
294C2×Dic17#12114yes—
295C2×Dic17C22⋊Q814yes—
296C2×Dic17C22⋊Q814yes—
297C4×D17—232no—
298C22×#117—171no—
299C2×C4×C4⋊C4—171no—
300C2×C4×C4⋊C4—171no—
301C2×C4×C22⋊C4—172no—
302#1441—172no—
303#1441—171no—
304#1441D1711yescharacteristic, complemented
305#1439—172no—
306#1439—171no—
307#1439—171no—
308C4×#95—172no—
309C22×C17C22×C22⋊C411yescharacteristic
310C22×C17C22×C4⋊C411yescharacteristic
311C22×C17C2×#11711yescharacteristic
312C22×C17C4×#4112yes—
313C22×C17C4×#4112yes—
314C22×C17C2×#12111yescharacteristic
315C22×C17C2×#12111yescharacteristic
316C22×C17C2×C22⋊Q811yescharacteristic
317C22×C17C2×C22⋊Q811yescharacteristic
318C22×C17#52511yescharacteristic
319C22×C17#52511yescharacteristic
320C22×C17#52512yes—
321C22×C17#52512yes—
322C22×C17#52512yes—
323C22×C17#52912yes—
324C22×C17#52912yes—
325C22×C17#52912yes—
326C22×C17#52912yes—
327C22×C17#53212yes—
328C22×C17#53211yescharacteristic
329C22×C17#53211yescharacteristic
330C22×C17#53312yes—
331C22×C17#53312yes—
332C2×D17—24no—
333C2×D17—28no—
334C2×D17—42no—
335C2×D17—42no—
336C2×D17—44no—
337C2×D17—42no—
338C2×D17—42no—
339C2×D17—44no—
340C2×D17—42no—
341C2×D17—42no—
342C2×D17—44no—
343C4×C17—24no—
344C4×C17—24no—
345C4×C17—44no—
346C4×C17—44no—
347C4×C17—42no—
348Dic17—24no—
349Dic17—44no—
350Dic17—44no—
351Dic17—42no—
352Dic17#38018yes—
353C24×C4—171no—
354C2×#117—178no—
355C2×#117—344no—
356C2×#117—344no—
357C22×C4⋊C4—171no—
358C22×C4⋊C4—172no—
359C22×C4⋊C4—171no—
360C22×C4⋊C4C2×D1711yescharacteristic
361C22×C4⋊C4C2×D1711yescharacteristic
362C22×C22⋊C4—171no—
363C22×C22⋊C4—171no—
364C22×C22⋊C4—172no—
365C22×C22⋊C4—172no—
366C22×C42—171no—
367C22×C42—171no—
368C22×C42—172no—
369C22×C42—172no—
370C4×C4⋊C4—344no—
371C4×C4⋊C4—344no—
372C4×C22⋊C4—348no—
373C2×#95—172no—
374C2×#95—172no—
375C2×#95—172no—
376C2×#95—172no—
377C2×#95—171no—
378C2×#95—172no—
379C2×#95C2×D1712yes—
380C2×#95C2×D1711yescharacteristic
381C2×#95C2×D1712yes—
382C2×C17C2×#38011yescharacteristic
383C2×C17#229612yes—
384C2×C17#229711yescharacteristic
385C2×C17#229711yescharacteristic
386C2×C17#229812yes—
387C2×C17#239211yescharacteristic
388C2×C17#239211yescharacteristic
389C2×C17#239212yes—
390C2×C17#239611yescharacteristic
391C2×C17#239611yescharacteristic
392C2×C17#239612yes—
393D17—44no—
394C25—171no—
395C23×C4—171no—
396C23×C4—171no—
397C23×C4—174no—
398C23×C4—174no—
399C23×C4—174no—
400C23×C4—171no—
401C23×C4—172no—
402C23×C4—172no—
403C23×C4—342no—
404C23×C4—342no—
405C23×C4—342no—
406C23×C4—342no—
407C23×C4C22×D1711yescharacteristic
408C23×C4C22×D1712yes—
409C23×C4C22×D1712yes—
410C23×C4C22×D1711yescharacteristic
411C23×C4C22×D1711yescharacteristic
412#117—3432no—
413C2×C4⋊C4—22no—
414C2×C4⋊C4—22no—
415C2×C4⋊C4—22no—
416C2×C4⋊C4—24no—
417C2×C4⋊C4—174no—
418C2×C4⋊C4—342no—
419C2×C4⋊C4—342no—
420C2×C4⋊C4—344no—
421C2×C4⋊C4—344no—
422C2×C4⋊C4—344no—
423C2×C4⋊C4—342no—
424C2×C4⋊C4—344no—
425C2×C4⋊C4C4×D1714yes—
426C2×C22⋊C4—174no—
427C2×C22⋊C4—174no—
428C2×C22⋊C4—342no—
429C2×C22⋊C4—342no—
430C2×C22⋊C4—344no—
431C2×C22⋊C4—344no—
432C2×C22⋊C4—344no—
433C2×C22⋊C4—344no—
434C2×C22⋊C4—342no—
435C2×C22⋊C4—342no—
436C2×C22⋊C4—344no—
437C2×C22⋊C4—344no—
438C2×C42—174no—
439C2×C42—174no—
440C2×C42—174no—
441C2×C42—174no—
442C2×C42—342no—
443C2×C42—342no—
444C2×C42—344no—
445C2×C42—344no—
446C2×C42—344no—
447C2×C42—344no—
448C2×C42—344no—
449C2×C42—344no—
450#95—24no—
451#95—22no—
452#95—24no—
453#95—344no—
454#95—344no—
455#95—344no—
456#95—344no—
457#95—342no—
458#95—344no—
459C17#1385211yescharacteristic, complemented, Sylow
460C24—172no—
461C24—172no—
462C24—172no—
463C24—341no—
464C24—341no—
465C24—342no—
466C24—342no—
467C24—342no—
468C24—342no—
469C24—341no—
470C24—341no—
471C24C23×D1711yescharacteristic
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—172no—
495C22×C4—172no—
496C22×C4—172no—
497C22×C4—172no—
498C22×C4—172no—
499C22×C4—172no—
500C22×C4—172no—
501C22×C4—172no—
502C22×C4—172no—
503C22×C4—172no—
504C22×C4—1716no—
505C22×C4—172no—
506C22×C4—174no—
507C22×C4—174no—
508C22×C4—348no—
509C22×C4—348no—
510C22×C4—348no—
511C22×C4—348no—
512C22×C4—348no—
513C22×C4—348no—
514C22×C4—342no—
515C22×C4—342no—
516C22×C4—342no—
517C22×C4—342no—
518C22×C4—342no—
519C22×C4—342no—
520C22×C4—341no—
521C22×C4—341no—
522C22×C4—342no—
523C22×C4—342no—
524C22×C4—342no—
525C22×C4—342no—
526C22×C4—342no—
527C22×C4—342no—
528C22×C4—342no—
529C22×C4—342no—
530C22×C4—342no—
531C22×C4—342no—
532C22×C4C2×C4×D1712yes—
533C22×C4C2×C4×D1712yes—
534C22×C4C2×C4×D1712yes—
535C22×C4D4×D1712yes—
536C22×C4D4×D1712yes—
537C22×C4D4×D1712yes—
538C22×C4#6325414yes—
539C22×C4Q8×D1712yes—
540C22×C4#6325614yes—
541C4⋊C4—28no—
542C4⋊C4—28no—
543C4⋊C4—348no—
544C4⋊C4—3416no—
545C4⋊C4—348no—
546C22⋊C4—348no—
547C22⋊C4—3416no—
548C22⋊C4—3416no—
549C22⋊C4—348no—
550C42—1716no—
551C42—348no—
552C42—688no—
553C42—688no—
554C23—174no—
555C23—342no—
556C23—342no—
557C23—344no—
558C23—344no—
559C23—342no—
560C23—342no—
561C23—342no—
562C23—342no—
563C23—344no—
564C23—344no—
565C23—344no—
566C23—344no—
567C23—681no—
568C23—681no—
569C23—682no—
570C23—682no—
571C23—682no—
572C23—682no—
573C23—682no—
574C23—682no—
575C23—681no—
576C23—681no—
577C23C22×C4×D1711yescharacteristic
578C23C2×D4×D1711yescharacteristic
579C23C2×D4×D1711yescharacteristic
580C23C2×D4×D1711yescharacteristic
581C23C2×#6325412yes—
582C23C2×Q8×D1711yescharacteristic
583C23C2×#6325612yes—
584C23#41×D1711yescharacteristic
585C23#41×D1711yescharacteristic
586C23#41×D1712yes—
587C23#41×D1712yes—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—174no—
618C2×C4—174no—
619C2×C4—174no—
620C2×C4—174no—
621C2×C4—174no—
622C2×C4—174no—
623C2×C4—174no—
624C2×C4—174no—
625C2×C4—3432no—
626C2×C4—344no—
627C2×C4—344no—
628C2×C4—342no—
629C2×C4—342no—
630C2×C4—342no—
631C2×C4—344no—
632C2×C4—344no—
633C2×C4—344no—
634C2×C4—344no—
635C2×C4—344no—
636C2×C4—344no—
637C2×C4—344no—
638C2×C4—682no—
639C2×C4—682no—
640C2×C4—682no—
641C2×C4—682no—
642C2×C4—682no—
643C2×C4—682no—
644C2×C4—682no—
645C2×C4—682no—
646C2×C4—681no—
647C2×C4—682no—
648C2×C4—681no—
649C2×C4C22⋊C4×D1714yes—
650C2×C4C4⋊C4×D1714yes—
651C22—344no—
652C22—348no—
653C22—682no—
654C22—682no—
655C22—684no—
656C22—682no—
657C22—682no—
658C22—684no—
659C22—682no—
660C22—682no—
661C22—684no—
662C22C2×C22⋊C4×D1711yescharacteristic
663C22C2×C4⋊C4×D1711yescharacteristic
664C22#117×D1711yescharacteristic
665C22C4×#6325412yes—
666C22C4×#6325612yes—
667C22#121×D1711yescharacteristic
668C22#121×D1711yescharacteristic
669C22C22⋊Q8×D1711yescharacteristic
670C22C22⋊Q8×D1711yescharacteristic
671C22#1176331511yescharacteristic
672C22#1176331711yescharacteristic
673C22#1176331912yes—
674C22#1176332212yes—
675C22#1176332212yes—
676C22#1176333612yes—
677C22#1176334212yes—
678C22#1176334412yes—
679C22#1176334512yes—
680C22#1176335412yes—
681C22#1176335811yescharacteristic
682C22#1176336011yescharacteristic
683C22#1176336412yes—
684C22#1176336512yes—
685C4—24no—
686C4—24no—
687C4—44no—
688C4—44no—
689C4—42no—
690C4—178no—
691C4—344no—
692C4—684no—
693C4—684no—
694C4—682no—
695C2—684no—
696C2D17×#38011yescharacteristic
697C2#1453126512yes—
698C2#1453126911yescharacteristic
699C2#1453127311yescharacteristic
700C2#1453127612yes—
701C2#1453202311yescharacteristic
702C2#1453202511yescharacteristic
703C2#1453202912yes—
704C2#1453205511yescharacteristic
705C2#1453205711yescharacteristic
706C2#1453206112yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1441C2
socle, cosocleC24×C17C24
prime radicalC24×D17C23
hypercenter#1441D17
hypocenterC17#13852
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×Dic17C2semidirect
3. C2×#11762660C2semidirect
5. #14527037C2semidirect
6. #14527036C2semidirect
10. #95×Dic17C2semidirect
11. C17×#1441C2semidirect
15. C2×#10587578C4semidirect
186. C2×C4×D17C4⋊C4semidirect
199. C4×Dic17C22⋊C4semidirect
304. #1441D17semidirect
459. C17#13852semidirect

Maximal subgroups

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

subgroupindex
C22×#105875782
C2×C4⋊C4×Dic172
C2×#117626602
C2×#117626562
#145270372
#145270362
#145270302
#145270282
#145270262
#95×Dic172
C17×#14412
#1385217