Finite Group Databasebeta

← #17040828

Subgroups — #17040828

Order 4352.

1054 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763048C211yesmaximal, characteristic
2C2×#11762901C211yesmaximal, characteristic, complemented
3C22×C8×Dic17C211yesmaximal, characteristic
4C22⋊Q8×Dic17C211yesmaximal, characteristic
5#14531312C211yesmaximal, characteristic
6#14529642C211yesmaximal, characteristic, complemented
7#14529516C211yesmaximal, characteristic, complemented
8#14529461C211yesmaximal, characteristic, complemented
9#14529457C211yesmaximal, characteristic, complemented
10#14529207C211yesmaximal, characteristic
11#14528811C211yesmaximal, characteristic
12#14528676C211yesmaximal, characteristic, complemented
13#103×Dic17C211yesmaximal, characteristic, complemented
14#14528603C211yesmaximal, characteristic, complemented
15C17×#1892C211yesmaximal, characteristic
16C22×#63231C2211yescharacteristic
17C23×#3622C2211yescharacteristic
18C2×#10587610C2211yescharacteristic
19C2×Q8×Dic17C2211yescharacteristic
20C2×#10587573C2211yescharacteristic
21C22×C4×Dic17C2211yescharacteristic
22#11763048C418yescomplemented
23#11762901—24no—
24C2×#10587478C2211yescharacteristic
25C2×#10587478C2211yescharacteristic
26C2×#10587478C2211yescharacteristic
27C2×#10587487C2211yescharacteristic
28C2×#10587487C2211yescharacteristic
29C2×#10587506C2211yescharacteristic
30C2×#10587481C2211yescharacteristic
31C2×C8×Dic17—24no—
32C2×C8×Dic17C2211yescharacteristic
33C2×C8×Dic17C2211yescharacteristic
34C2×C8×Dic17C2211yescharacteristic
35C2×C8×Dic17C2211yescharacteristic
36#11762716C2211yescharacteristic
37#11762714C2211yescharacteristic
38#11762708C2211yescharacteristic
39#11762705C2211yescharacteristic
40#11762695—21no—
41#11762691—21no—
42#11762684—21no—
43#11762681—21no—
44#11762669C2211yescharacteristic
45C4⋊C4×Dic17—21no—
46C4⋊C4×Dic17C2211yescharacteristic
47C4⋊C4×Dic17C2211yescharacteristic
48#11762660—21no—
49C22⋊C4×Dic17—21no—
50C22⋊C4×Dic17—21no—
51#11762572C2211yescharacteristic
52#11762572C2211yescharacteristic
53#11762571C2211yescharacteristic
54#11762571C2211yescharacteristic
55C23×C8×C17C2211yescharacteristic
56C2×C17×C22⋊Q8C2211yescharacteristic
57C2×C17×#103C2211yescharacteristic
58C2×C17×#103C2211yescharacteristic
59C17×#380C2211yescharacteristic
60C17×#340C2211yescharacteristic
61C17×#340C2211yescharacteristic
62C22×Q8×C17C2311yescharacteristic
63C23×C4×C17C2311yescharacteristic
64C23×Dic17D411yescharacteristic
65C23×Dic17D411yescharacteristic
66C22×Dic34C2311yescharacteristic
67C22×C8×C17—22no—
68C22×C8×C17—22no—
69C22×C8×C17—22no—
70C22×C8×C17C2311yescharacteristic
71C22×C8×C17C2311yescharacteristic
72C17×C22⋊Q8—22no—
73C17×C22⋊Q8C2×C414yes—
74C2×C4⋊C4×C17—21no—
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C4⋊C4×C17C2311yescharacteristic
77C2×C4⋊C4×C17C2311yescharacteristic
78C2×C22⋊C4×C17—21no—
79C2×C22⋊C4×C17—21no—
80C2×C22⋊Dic17—21no—
81C2×C22⋊Dic17—21no—
82Q8×Dic17—22no—
83Q8×Dic17—22no—
84C2×#63231—22no—
85C2×#63231—22no—
86C2×#63231C2×C414yescomplemented
87C2×C4⋊Dic17—21no—
88C2×C4⋊Dic17—21no—
89C2×C4⋊Dic17C2311yescharacteristic
90C2×C4⋊Dic17C2311yescharacteristic
91C2×C4⋊Dic17C2311yescharacteristic
92#10587610—22no—
93#10587610C2×C414yes—
94C2×#63225—21no—
95C2×#63225—21no—
96C2×#63225—21no—
97C2×C4×Dic17—22no—
98C2×C4×Dic17—22no—
99C2×C4×Dic17—21no—
100C2×C4×Dic17—21no—
101C2×C4×Dic17—21no—
102C2×C4×Dic17—21no—
103C2×C4×Dic17D411yescharacteristic
104C2×C4×Dic17D411yescharacteristic
105C2×C4×Dic17C2311yescharacteristic
106C2×C4×Dic17C2311yescharacteristic
107C22×#3622—22no—
108C22×#3622—22no—
109C22×#3622C2×C414yes—
110C22×#3622C2311yescharacteristic
111C22×#3622C2311yescharacteristic
112#10587573—22no—
113#10587573—22no—
114C17×#103—22no—
115C17×#103—22no—
116C17×#95—21no—
117#10587506C2×C414yescomplemented
118#10587502—21no—
119#10587502—21no—
120#10587502—21no—
121#10587502—21no—
122#10587502—21no—
123#10587502—21no—
124#10587502—21no—
125#10587502—21no—
126#10587502—21no—
127#10587487—22no—
128#10587487—22no—
129C8×Dic17—22no—
130C8×Dic17—22no—
131C8×Dic17—22no—
132C8×Dic17D414yes—
133#10587481C2×C414yescomplemented
134#10587478—22no—
135#10587478—22no—
136#10587478C2×C414yescomplemented
137C24×C17C2×D411yescharacteristic
138C2×Q8×C17—21no—
139C2×Q8×C17—21no—
140C2×Q8×C17—21no—
141C2×Q8×C17—21no—
142C2×Q8×C17—21no—
143C2×Q8×C17C22×C412yes—
144C22×C4×C17—21no—
145C22×C4×C17—21no—
146C22×C4×C17—21no—
147C22×C4×C17—21no—
148C22×C4×C17—21no—
149C22×C4×C17—21no—
150C22×C4×C17—21no—
151C22×C4×C17—21no—
152C22×C4×C17C22×C412yes—
153C22×C4×C17C2×D411yescharacteristic
154C22×C4×C17#4112yes—
155C22×C4×C17C2411yescharacteristic
156C22×Dic17—22no—
157C22×Dic17—22no—
158C22×Dic17—22no—
159C22×Dic17—22no—
160C22×Dic17—22no—
161C22×Dic17—21no—
162C22×Dic17—21no—
163C22×Dic17—21no—
164C22×Dic17—21no—
165C22×Dic17Q1614yescomplemented
166C22×Dic17C2×D411yescharacteristic
167C22×Dic17C2×D411yescharacteristic
168C22×Dic17C2×D411yescharacteristic
169C22×Dic17C2×D411yescharacteristic
170C2×Dic34—21no—
171C2×Dic34—21no—
172C2×Dic34—21no—
173C2×Dic34—21no—
174C2×Dic34—21no—
175C2×Dic34C22×C412yes—
176C2×C8×C17—28no—
177C2×C8×C17—21no—
178C2×C8×C17—21no—
179C2×C8×C17—21no—
180C2×C8×C17—21no—
181C2×C8×C17C2×D412yes—
182C2×C8×C17#4112yes—
183C4⋊C4×C17—24no—
184C4⋊C4×C17—21no—
185C4⋊C4×C17—21no—
186C4⋊C4×C17—21no—
187C4⋊C4×C17—21no—
188C4⋊C4×C17—21no—
189C4⋊C4×C17C22×C412yes—
190C22⋊C4×C17—24no—
191C22⋊C4×C17—42no—
192C22⋊Dic17—24no—
193C22⋊Dic17—42no—
194#63231—28no—
195C4⋊Dic17—21no—
196C4⋊Dic17—21no—
197C4⋊Dic17—21no—
198C4⋊Dic17—21no—
199C4⋊Dic17—21no—
200C4⋊Dic17—42no—
201C4⋊Dic17—42no—
202C4⋊Dic17C22×C412yes—
203#63225—24no—
204#63225—42no—
205#63225—42no—
206C4×Dic17—21no—
207C4×Dic17—21no—
208C4×Dic17—22no—
209C4×Dic17—22no—
210C4×Dic17—42no—
211C4×Dic17—42no—
212C4×Dic17—42no—
213C4×Dic17—42no—
214C4×Dic17C2×D412yes—
215C4×Dic17#4112yes—
216C2×#3622—21no—
217C2×#3622—21no—
218C2×#3622—28no—
219C2×#3622C22×C412yes—
220C2×#3622C22×C412yes—
221C2×#3622C2×D412yes—
222C2×#3622#4112yes—
223C4×#467—171nomaximal, Sylow
224C23×C17—21no—
225C23×C17—21no—
226C23×C17—21no—
227C23×C17—21no—
228C23×C17C4×D412yes—
229C23×C17C2×Q1612yes—
230C23×C17#13512yes—
231C23×C17C22×D411yescharacteristic
232Q8×C17—22no—
233Q8×C17—22no—
234Q8×C17—41no—
235Q8×C17—41no—
236C2×C4×C17—21no—
237C2×C4×C17—21no—
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—41no—
245C2×C4×C17—41no—
246C2×C4×C17—41no—
247C2×C4×C17—41no—
248C2×C4×C17—41no—
249C2×C4×C17—41no—
250C2×C4×C17—41no—
251C2×C4×C17—41no—
252C2×C4×C17—41no—
253C2×C4×C17—41no—
254C2×C4×C17C4×D412yes—
255C2×C4×C17C23×C411yescharacteristic
256C2×C4×C17C22×D411yescharacteristic
257C2×C4×C17C2×#4111yescharacteristic
258C2×C4×C17C2×#4111yescharacteristic
259C2×C4×C17C2×#4111yescharacteristic
260C2×C4×C17C2×#4111yescharacteristic
261C2×Dic17—28no—
262C2×Dic17—21no—
263C2×Dic17—21no—
264C2×Dic17—21no—
265C2×Dic17—21no—
266C2×Dic17—21no—
267C2×Dic17—21no—
268C2×Dic17—28no—
269C2×Dic17—22no—
270C2×Dic17—22no—
271C2×Dic17—41no—
272C2×Dic17—41no—
273C2×Dic17—41no—
274C2×Dic17—41no—
275C2×Dic17—41no—
276C2×Dic17—41no—
277C2×Dic17—41no—
278C2×Dic17—41no—
279C2×Dic17—41no—
280C2×Dic17—41no—
281C2×Dic17#12112yes—
282C2×Dic17#12112yes—
283C2×Dic17C2×Q1612yes—
284C2×Dic17C2×Q1612yes—
285C2×Dic17#13512yes—
286C2×Dic17#13512yes—
287Dic34—22no—
288Dic34—22no—
289Dic34—41no—
290Dic34—41no—
291C8×C17—22no—
292C8×C17—22no—
293#3622—22no—
294#3622C4×D414yes—
295C2×#467—171no—
296C2×C4×Q16—171no—
297C22×C4×C8—171no—
298C4×C22⋊Q8—171no—
299C4×C22⋊Q8—171no—
300#1969—171no—
301#1942—171no—
302#1933—171no—
303#1933—171no—
304#1892—171no—
305#1892D1711yescharacteristic
306#1808—171no—
307C4×#107—171no—
308C4×#103—171no—
309C4×#103—171no—
310C22×C17—22no—
311C22×C17—22no—
312C22×C17—22no—
313C22×C17—22no—
314C22×C17—22no—
315C22×C17—22no—
316C22×C17C4×Q1614yescomplemented
317C22×C17C2×C4×D411yescharacteristic
318C22×C17C2×#12111yescharacteristic
319C22×C17#53211yescharacteristic
320C22×C17C22×Q1611yescharacteristic
321C22×C17C2×#13511yescharacteristic
322C22×C17C2×#13511yescharacteristic
323C22×C17C2×#13511yescharacteristic
324C4×C17—22no—
325C4×C17—42no—
326C4×C17—42no—
327C4×C17—81no—
328C4×C17—81no—
329C4×C17C2×C4×D411yescharacteristic
330C4×C17C4×#4111yescharacteristic
331C4×C17#52511yescharacteristic
332C4×C17#52911yescharacteristic
333Dic17—22no—
334Dic17—22no—
335Dic17—22no—
336Dic17—42no—
337Dic17—42no—
338Dic17—81no—
339Dic17—81no—
340Dic17#46714yescomplemented
341C22×Q16—171no—
342C23×C8—171no—
343C23×C8C2×D1711yescharacteristic
344C2×C22⋊Q8—171no—
345C2×C22⋊Q8C2×D1711yescharacteristic
346C2×C4×Q8—171no—
347C2×C4×Q8—171no—
348C22×C42—171no—
349#467—178no—
350C4×Q16—344no—
351C2×#107—171no—
352C2×#107—171no—
353C2×#107—171no—
354C2×#103—171no—
355C2×#103—171no—
356C2×#103—171no—
357C2×#103—171no—
358C2×#103C2×D1711yescharacteristic
359C2×#103C2×D1711yescharacteristic
360C2×C4×C8—171no—
361C2×C4×C8—171no—
362C2×C4×C8—171no—
363C2×C4×C8—171no—
364C2×C4×C8—344no—
365#391—171no—
366#391—171no—
367#389—171no—
368#389—171no—
369#387—341no—
370#387—341no—
371#385—341no—
372#385—341no—
373#380—171no—
374#380C2×D1711yescharacteristic
375C4×C4⋊C4—171no—
376C4×C4⋊C4—171no—
377C4×C4⋊C4—341no—
378C4×C4⋊C4—341no—
379C4×C22⋊C4—341no—
380C4×C22⋊C4—341no—
381#340—171no—
382#340—171no—
383#340—171no—
384#340—171no—
385#340C2×D1711yescharacteristic
386#340C2×D1711yescharacteristic
387C2×C17—24no—
388C2×C17C4×#12111yescharacteristic
389C2×C17C2×C4×Q1611yescharacteristic
390C2×C17C4×#13511yescharacteristic
391C2×C17C2×#46711yescharacteristic
392C2×C17#305011yescharacteristic
393C2×C17#316811yescharacteristic
394C2×C17#317011yescharacteristic
395C22×Q8—171no—
396C22×Q8C22×D1711yescharacteristic
397C23×C4—171no—
398C23×C4—171no—
399C23×C4C22×D1711yescharacteristic
400C2×Q16—174no—
401C2×Q16—342no—
402C2×Q16—342no—
403C22×C8—22no—
404C22×C8—22no—
405C22×C8—22no—
406C22×C8—171no—
407C22×C8—174no—
408C22×C8—171no—
409C22×C8—342no—
410C22×C8—342no—
411C22×C8C22×D1711yescharacteristic
412C22×C8C22×D1711yescharacteristic
413C22⋊Q8—22no—
414C22⋊Q8—174no—
415C22⋊Q8—342no—
416C22⋊Q8C4×D1714yes—
417C4×Q8—342no—
418C4×Q8—342no—
419C4×Q8—342no—
420C4×Q8—342no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—171no—
423C2×C4⋊C4—171no—
424C2×C4⋊C4—171no—
425C2×C4⋊C4—341no—
426C2×C4⋊C4—341no—
427C2×C4⋊C4—341no—
428C2×C4⋊C4—341no—
429C2×C4⋊C4—341no—
430C2×C4⋊C4C22×D1711yescharacteristic
431C2×C4⋊C4C22×D1711yescharacteristic
432C2×C4⋊C4C22×D1711yescharacteristic
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—341no—
436C2×C22⋊C4—341no—
437C2×C42—171no—
438C2×C42—171no—
439C2×C42—171no—
440C2×C42—171no—
441C2×C42—342no—
442C2×C42—342no—
443C2×C42—341no—
444C2×C42—341no—
445C2×C42—341no—
446C2×C42—341no—
447#107—174no—
448#107—342no—
449#107—342no—
450#103—22no—
451#103—22no—
452#103—174no—
453#103—174no—
454#103—342no—
455#103—342no—
456C4×C8—174no—
457C4×C8—342no—
458C4×C8—342no—
459C4×C8—342no—
460#95—21no—
461#95—341no—
462#95—341no—
463#95—341no—
464#95—341no—
465#95—341no—
466#95—341no—
467#95—341no—
468#95—341no—
469#95—341no—
470C17C4×#46711yescharacteristic, complemented, Sylow
471C24D4×D1711yescharacteristic
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—172no—
478C2×Q8—341no—
479C2×Q8—341no—
480C2×Q8—341no—
481C2×Q8—341no—
482C2×Q8—341no—
483C2×Q8C2×C4×D1712yes—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—171no—
493C22×C4—171no—
494C22×C4—171no—
495C22×C4—171no—
496C22×C4—174no—
497C22×C4—342no—
498C22×C4—342no—
499C22×C4—342no—
500C22×C4—342no—
501C22×C4—342no—
502C22×C4—341no—
503C22×C4—341no—
504C22×C4—341no—
505C22×C4—341no—
506C22×C4C2×C4×D1712yes—
507C22×C4D4×D1711yescharacteristic
508C22×C4#6325412yes—
509C22×C4C23×D1711yescharacteristic
510Q16—348no—
511C2×C8—28no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—172no—
517C2×C8—172no—
518C2×C8—172no—
519C2×C8—172no—
520C2×C8—348no—
521C2×C8—341no—
522C2×C8—341no—
523C2×C8D4×D1712yes—
524C2×C8#6325412yes—
525C4⋊C4—21no—
526C4⋊C4—24no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—172no—
532C4⋊C4—341no—
533C4⋊C4—344no—
534C4⋊C4—341no—
535C4⋊C4—341no—
536C4⋊C4—341no—
537C4⋊C4—341no—
538C4⋊C4—682no—
539C4⋊C4—682no—
540C4⋊C4—682no—
541C4⋊C4—682no—
542C4⋊C4C2×C4×D1712yes—
543C22⋊C4—24no—
544C22⋊C4—42no—
545C22⋊C4—344no—
546C22⋊C4—682no—
547C42—172no—
548C42—172no—
549C42—342no—
550C42—342no—
551C42—341no—
552C42—341no—
553C42—682no—
554C42—682no—
555C42—682no—
556C42—682no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23#1058756712yes—
562C23#1058760212yes—
563C23Q16×D1712yes—
564C23C2×D4×D1711yescharacteristic
565Q8—22no—
566Q8—22no—
567Q8—41no—
568Q8—41no—
569Q8—342no—
570Q8—342no—
571Q8—681no—
572Q8—681no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—172no—
592C2×C4—172no—
593C2×C4—172no—
594C2×C4—172no—
595C2×C4—172no—
596C2×C4—172no—
597C2×C4—348no—
598C2×C4—348no—
599C2×C4—341no—
600C2×C4—341no—
601C2×C4—341no—
602C2×C4—341no—
603C2×C4—341no—
604C2×C4—341no—
605C2×C4—342no—
606C2×C4—342no—
607C2×C4—681no—
608C2×C4—681no—
609C2×C4—681no—
610C2×C4—681no—
611C2×C4—681no—
612C2×C4—681no—
613C2×C4—681no—
614C2×C4—681no—
615C2×C4—681no—
616C2×C4—681no—
617C2×C4#1058756712yes—
618C2×C4C22×C4×D1711yescharacteristic
619C2×C4C2×D4×D1711yescharacteristic
620C2×C4C2×#6325411yescharacteristic
621C2×C4C2×#6325411yescharacteristic
622C2×C4#41×D1711yescharacteristic
623C2×C4#41×D1711yescharacteristic
624C8—22no—
625C8—22no—
626C8—174no—
627C8—342no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22#1176290114yescomplemented
635C22C2×#1058756711yescharacteristic
636C22#121×D1711yescharacteristic
637C22#1176335411yescharacteristic
638C22C2×Q16×D1711yescharacteristic
639C22#135×D1711yescharacteristic
640C22#135×D1711yescharacteristic
641C22C2×#1058760211yescharacteristic
642C4—22no—
643C4—42no—
644C4—42no—
645C4—81no—
646C4—81no—
647C4—174no—
648C4—342no—
649C4—342no—
650C4—342no—
651C4—682no—
652C4—682no—
653C4—1361no—
654C4—1361no—
655C4C4×D4×D1711yescharacteristic
656C4C4×#6325411yescharacteristic
657C4#1176331511yescharacteristic
658C4#1176333611yescharacteristic
659C2—24no—
660C2#1453129411yescharacteristic
661C2C2×#1176290111yescharacteristic
662C2#1453755211yescharacteristic
663C2D17×#46711yescharacteristic
664C2#1453836011yescharacteristic
665C2#1453934711yescharacteristic
666C2#1453936511yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1892C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
cosocleC22×C4×C17C24
hypercenter#1892D17
hypocenterC17C4×#467
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1892

Complements

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

NH ≅ G/Ntype
2. C2×#11762901C2semidirect
6. #14529642C2semidirect
7. #14529516C2semidirect
8. #14529461C2semidirect
9. #14529457C2semidirect
12. #14528676C2semidirect
13. #103×Dic17C2semidirect
14. #14528603C2semidirect
22. #11763048C4semidirect
86. C2×#63231C2×C4semidirect
117. #10587506C2×C4semidirect
133. #10587481C2×C4semidirect
136. #10587478C2×C4semidirect
165. C22×Dic17Q16semidirect
316. C22×C17C4×Q16semidirect
340. Dic17#467semidirect
470. C17C4×#467semidirect
634. C22#11762901semidirect

Maximal subgroups

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

subgroupindex
C2×#117630482
C2×#117629012
C22×C8×Dic172
C22⋊Q8×Dic172
#145313122
#145296422
#145295162
#145294612
#145294572
#145292072
#145288112
#145286762
#103×Dic172
#145286032
C17×#18922
C4×#46717