Finite Group Databasebeta

← #17039601

Subgroups — #17039601

Order 4352.

1070 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763419C211yesmaximal, characteristic
2C2×#11762764C211yesmaximal, characteristic
3#14531391C211yesmaximal, characteristic
4#14529751C211yesmaximal, characteristic, complemented
5#14529747C211yesmaximal, characteristic, complemented
6#14529602C211yesmaximal, characteristic, complemented
7#14529584C211yesmaximal, characteristic, complemented
8#14529218C211yesmaximal, characteristic
9#14529215C211yesmaximal, characteristic
10#14529207C211yesmaximal, characteristic
11#14528616C211yesmaximal, characteristic, complemented
12#14528613C211yesmaximal, characteristic, complemented
13#103×Dic17C211yesmaximal, characteristic, complemented
14#14528603C211yesmaximal, characteristic, complemented
15C17×#1892C211yesmaximal, characteristic
16C22×#63223C2211yescharacteristic
17#11763419—24no—
18#11763419C418yescomplemented
19C2×#10587630C2211yescharacteristic
20C2×#10587565—21no—
21C2×#10587565—21no—
22C2×#10587576—21no—
23C2×#10587576C2211yescharacteristic
24C2×#10587575—21no—
25C2×#10587610C2211yescharacteristic
26C2×Q8×Dic17C2211yescharacteristic
27C2×#10587573C2211yescharacteristic
28C2×#10587612C2211yescharacteristic
29C2×#10587487C2211yescharacteristic
30C2×#10587487C2211yescharacteristic
31C2×#10587506C2211yescharacteristic
32C2×#10587506C2211yescharacteristic
33C2×#10587481C2211yescharacteristic
34C2×#10587481C2211yescharacteristic
35#11762764—24no—
36C2×#10587486C2211yescharacteristic
37C2×#10587486C2211yescharacteristic
38C2×C8×Dic17C2211yescharacteristic
39C2×C8×Dic17C2211yescharacteristic
40#11762703—21no—
41#11762697—21no—
42#11762694—21no—
43#11762692—21no—
44#11762678C2211yescharacteristic
45#11762674C2211yescharacteristic
46#11762669C2211yescharacteristic
47C4⋊C4×Dic17C2211yescharacteristic
48C4⋊C4×Dic17C2211yescharacteristic
49#11762572C2211yescharacteristic
50#11762572C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762571C2211yescharacteristic
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
64C22×Dic34C2311yescharacteristic
65C22×C8×C17—22no—
66C22×C8×C17—22no—
67C22×C8×C17—22no—
68C22×C8×C17C2311yescharacteristic
69C22×C8×C17C2311yescharacteristic
70C17×C22⋊Q8—22no—
71C17×C22⋊Q8C2×C414yes—
72C2×C4⋊C4×C17—21no—
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C22⋊C4×C17—21no—
77C2×C22⋊C4×C17—21no—
78C2×C22⋊Dic17—21no—
79C2×C22⋊Dic17—21no—
80C2×C22⋊Dic17D411yescharacteristic
81C2×C22⋊Dic17D411yescharacteristic
82Q8×Dic17—22no—
83Q8×Dic17C2×C414yes—
84#10587630—22no—
85#10587630C2×C414yes—
86#10587612C2×C414yes—
87#10587612D414yes—
88C2×C4⋊Dic17—21no—
89C2×C4⋊Dic17—21no—
90C2×C4⋊Dic17D411yescharacteristic
91C2×C4⋊Dic17D411yescharacteristic
92C2×C4⋊Dic17C2311yescharacteristic
93C2×C4⋊Dic17C2311yescharacteristic
94C2×C4⋊Dic17C2311yescharacteristic
95#10587610—22no—
96#10587610C2×C414yes—
97C2×#63225—21no—
98C2×#63225—21no—
99C2×#63225—21no—
100C2×#63225—21no—
101C2×#63225—21no—
102C2×#63225—21no—
103C2×#63225—21no—
104C2×C4×Dic17—21no—
105C2×C4×Dic17—21no—
106C2×C4×Dic17—21no—
107C2×C4×Dic17—21no—
108C2×C4×Dic17C2311yescharacteristic
109C2×C4×Dic17C2311yescharacteristic
110C2×#63223—22no—
111C2×#63223—22no—
112C2×#63223—22no—
113C22×#3622C2311yescharacteristic
114C22×#3622C2311yescharacteristic
115#10587576—28no—
116#10587576—22no—
117#10587576C2×C414yes—
118#10587575—28no—
119#10587573—22no—
120#10587573C2×C414yes—
121#10587565—28no—
122#10587565—28no—
123C17×#103—22no—
124C17×#103—22no—
125C17×#95—21no—
126#10587506—22no—
127#10587506—22no—
128#10587502—21no—
129#10587502—21no—
130#10587502—21no—
131#10587502—21no—
132#10587502—21no—
133#10587487—22no—
134#10587487—22no—
135#10587486—22no—
136#10587486—22no—
137C8×Dic17—22no—
138C8×Dic17—22no—
139#10587481—22no—
140#10587481—22no—
141C24×C17C2×D411yescharacteristic
142C2×Q8×C17—21no—
143C2×Q8×C17—21no—
144C2×Q8×C17—21no—
145C2×Q8×C17—21no—
146C2×Q8×C17—21no—
147C2×Q8×C17C22×C412yes—
148C22×C4×C17—21no—
149C22×C4×C17—21no—
150C22×C4×C17—21no—
151C22×C4×C17—21no—
152C22×C4×C17—21no—
153C22×C4×C17—21no—
154C22×C4×C17—21no—
155C22×C4×C17—21no—
156C22×C4×C17C22×C412yes—
157C22×C4×C17C2×D411yescharacteristic
158C22×C4×C17C2×D412yes—
159C22×C4×C17C2411yescharacteristic
160C22×Dic17—21no—
161C22×Dic17—21no—
162C22×Dic17—21no—
163C22×Dic17—21no—
164C22×Dic17C2×D411yescharacteristic
165C22×Dic17C2×D411yescharacteristic
166C22×Dic17C2×D411yescharacteristic
167C22×Dic17C2×D411yescharacteristic
168C2×Dic34—21no—
169C2×Dic34—21no—
170C2×Dic34—21no—
171C2×Dic34—21no—
172C2×Dic34—21no—
173C2×Dic34C22×C412yes—
174C2×C8×C17—28no—
175C2×C8×C17—21no—
176C2×C8×C17—21no—
177C2×C8×C17—21no—
178C2×C8×C17—21no—
179C2×C8×C17#4112yes—
180C2×C8×C17#4112yes—
181C4⋊C4×C17—24no—
182C4⋊C4×C17—21no—
183C4⋊C4×C17—21no—
184C4⋊C4×C17—21no—
185C4⋊C4×C17—21no—
186C4⋊C4×C17—21no—
187C4⋊C4×C17C22×C412yes—
188C22⋊C4×C17—24no—
189C22⋊C4×C17—81no—
190C22⋊Dic17—24no—
191C22⋊Dic17—81no—
192C22⋊Dic17C22⋊C414yes—
193C22⋊Dic17C22⋊C414yes—
194C4⋊Dic17—21no—
195C4⋊Dic17—24no—
196C4⋊Dic17—24no—
197C4⋊Dic17—21no—
198C4⋊Dic17—21no—
199C4⋊Dic17—21no—
200C4⋊Dic17—21no—
201C4⋊Dic17C22⋊C414yes—
202C4⋊Dic17C22⋊C414yes—
203C4⋊Dic17C22×C412yes—
204#63225—24no—
205#63225—24no—
206#63225—24no—
207#63225—24no—
208#63225—24no—
209#63225—81no—
210#63225—81no—
211C4×Dic17—24no—
212C4×Dic17—24no—
213C4×Dic17—81no—
214C4×Dic17—81no—
215C4×Dic17C22×C412yes—
216C4×Dic17C22×C412yes—
217C4×Dic17C2×D412yes—
218C4×Dic17C2×D412yes—
219#63223—28no—
220C2×#3622—21no—
221C2×#3622—21no—
222C2×#3622—21no—
223C2×#3622—21no—
224C2×#3622#4112yes—
225C2×#3622#4112yes—
226#21413—171nomaximal, Sylow
227C23×C17—21no—
228C23×C17—21no—
229C23×C17—21no—
230C23×C17—21no—
231C23×C17—21no—
232C23×C17—21no—
233C23×C17C2×C22⋊C412yes—
234C23×C17C22×D411yescharacteristic
235Q8×C17—22no—
236Q8×C17—22no—
237Q8×C17—41no—
238Q8×C17—41no—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—22no—
242C2×C4×C17—22no—
243C2×C4×C17—22no—
244C2×C4×C17—22no—
245C2×C4×C17—22no—
246C2×C4×C17—22no—
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×C17—41no—
255C2×C4×C17—41no—
256C2×C4×C17—41no—
257C2×C4×C17C2×C22⋊C412yes—
258C2×C4×C17C23×C411yescharacteristic
259C2×C4×C17C22×D411yescharacteristic
260C2×C4×C17C2×#4111yescharacteristic
261C2×C4×C17C2×#4111yescharacteristic
262C2×C4×C17C2×#4111yescharacteristic
263C2×C4×C17C2×#4111yescharacteristic
264C2×Dic17—21no—
265C2×Dic17—21no—
266C2×Dic17—21no—
267C2×Dic17—21no—
268C2×Dic17—21no—
269C2×Dic17—21no—
270C2×Dic17—21no—
271C2×Dic17—21no—
272C2×Dic17—22no—
273C2×Dic17—22no—
274C2×Dic17—41no—
275C2×Dic17—41no—
276C2×Dic17—41no—
277C2×Dic17—41no—
278C2×Dic17—41no—
279C2×Dic17—41no—
280C2×Dic17—41no—
281C2×Dic17—41no—
282C2×Dic17—41no—
283C2×Dic17—41no—
284C2×Dic17C2×C22⋊C412yes—
285C2×Dic17C2×C22⋊C412yes—
286C2×Dic17C2×C22⋊C412yes—
287C2×Dic17C2×C22⋊C412yes—
288Dic34—22no—
289Dic34—22no—
290Dic34—41no—
291Dic34—41no—
292C8×C17—22no—
293C8×C17—22no—
294#3622—22no—
295#3622—22no—
296C2×#541—171no—
297C2×#405—171no—
298#2314—171no—
299#1992—171no—
300#1991—171no—
301#1960—171no—
302#1956—171no—
303#1894—171no—
304#1894—171no—
305#1892—171no—
306#1892D1711yescharacteristic
307#1763—171no—
308#1763—171no—
309C4×#103—171no—
310C4×#103—171no—
311C22×C17—22no—
312C22×C17—22no—
313C22×C17—22no—
314C22×C17—41no—
315C22×C17—41no—
316C22×C17—41no—
317C22×C17—41no—
318C22×C17C22×C22⋊C411yescharacteristic
319C22×C17#53211yescharacteristic
320C22×C17#53211yescharacteristic
321C22×C17#57711yescharacteristic
322C22×C17#57711yescharacteristic
323C22×C17#57811yescharacteristic
324C22×C17#57811yescharacteristic
325C4×C17—22no—
326C4×C17—42no—
327C4×C17—42no—
328C4×C17—81no—
329C4×C17—81no—
330C4×C17C4×#4111yescharacteristic
331C4×C17C4×#4111yescharacteristic
332C4×C17#52511yescharacteristic
333C4×C17#52511yescharacteristic
334Dic17—22no—
335Dic17—22no—
336Dic17—22no—
337Dic17—22no—
338Dic17—42no—
339Dic17—42no—
340Dic17—81no—
341Dic17—81no—
342C22×M16—171no—
343C23×C8C2×D1711yescharacteristic
344#541—178no—
345#541—344no—
346C2×C4⋊Q8—171no—
347C2×#126—341no—
348C2×#126—341no—
349C2×#125—171no—
350C2×#125—341no—
351C2×#125—341no—
352C2×C22⋊Q8—171no—
353C2×C22⋊Q8C2×D1711yescharacteristic
354C2×C4×Q8—171no—
355C2×C4×Q8—171no—
356C2×#117—171no—
357C2×#103—171no—
358C2×#103—171no—
359C2×#103—171no—
360C2×#103—171no—
361C2×#103—171no—
362C2×#103—171no—
363C2×#103C2×D1711yescharacteristic
364C2×#103C2×D1711yescharacteristic
365#405—344no—
366C2×#97—171no—
367C2×#97—171no—
368C2×C4×C8—171no—
369C2×C4×C8—171no—
370#388—341no—
371#388—341no—
372#387—341no—
373#387—341no—
374#384—171no—
375#382—171no—
376#380—171no—
377#380C2×D1711yescharacteristic
378C4×C4⋊C4—171no—
379C4×C4⋊C4—171no—
380#340—171no—
381#340—171no—
382#340—171no—
383#340—171no—
384#340—171no—
385#340—171no—
386#340C2×D1711yescharacteristic
387#340C2×D1711yescharacteristic
388C2×C17—42no—
389C2×C17#229711yescharacteristic
390C2×C17#289811yescharacteristic
391C2×C17#289811yescharacteristic
392C2×C17#317411yescharacteristic
393C2×C17#317711yescharacteristic
394C2×C17#317811yescharacteristic
395C2×C17#317811yescharacteristic
396C22×Q8—171no—
397C22×Q8C22×D1711yescharacteristic
398C23×C4C22×D1711yescharacteristic
399C2×M16—342no—
400C2×M16—342no—
401C2×M16—342no—
402C22×C8—22no—
403C22×C8—22no—
404C22×C8—22no—
405C22×C8—171no—
406C22×C8—171no—
407C22×C8C22×D1711yescharacteristic
408C22×C8C22×D1711yescharacteristic
409C4⋊Q8—174no—
410C4⋊Q8—342no—
411#126—348no—
412#126—348no—
413#125—174no—
414#125—342no—
415#125—348no—
416#125—348no—
417C22⋊Q8—22no—
418C22⋊Q8—174no—
419C22⋊Q8—342no—
420C22⋊Q8C4×D1714yes—
421C4×Q8—174no—
422C4×Q8—174no—
423C4×Q8—342no—
424C4×Q8—342no—
425#117—174no—
426#117—174no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—171no—
429C2×C4⋊C4—171no—
430C2×C4⋊C4—171no—
431C2×C4⋊C4—171no—
432C2×C4⋊C4—171no—
433C2×C4⋊C4—341no—
434C2×C4⋊C4—341no—
435C2×C4⋊C4—341no—
436C2×C4⋊C4—341no—
437C2×C4⋊C4—341no—
438C2×C4⋊C4—341no—
439C2×C4⋊C4—341no—
440C2×C4⋊C4—341no—
441C2×C4⋊C4—341no—
442C2×C4⋊C4C22×D1711yescharacteristic
443C2×C4⋊C4C22×D1711yescharacteristic
444C2×C4⋊C4C22×D1711yescharacteristic
445C2×C22⋊C4—21no—
446C2×C22⋊C4—21no—
447C2×C22⋊C4—171no—
448C2×C22⋊C4—171no—
449C2×C22⋊C4—341no—
450C2×C22⋊C4—341no—
451C2×C42—171no—
452C2×C42—171no—
453C2×C42—341no—
454C2×C42—341no—
455C2×C42—341no—
456C2×C42—341no—
457#103—22no—
458#103—22no—
459#103—342no—
460#103—342no—
461#103—342no—
462#103—342no—
463#103—342no—
464#103—342no—
465#97—342no—
466#97—342no—
467C4×C8—342no—
468C4×C8—342no—
469#95—21no—
470#95—341no—
471#95—341no—
472#95—341no—
473#95—341no—
474#95—341no—
475C17#2141311yescharacteristic, complemented, Sylow
476C24D4×D1711yescharacteristic
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—172no—
483C2×Q8—341no—
484C2×Q8—341no—
485C2×Q8—341no—
486C2×Q8—341no—
487C2×Q8—341no—
488C2×Q8C2×C4×D1712yes—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—171no—
498C22×C4—171no—
499C22×C4—171no—
500C22×C4—171no—
501C22×C4—341no—
502C22×C4—341no—
503C22×C4—341no—
504C22×C4—341no—
505C22×C4C2×C4×D1712yes—
506C22×C4D4×D1712yes—
507C22×C4D4×D1711yescharacteristic
508C22×C4C23×D1711yescharacteristic
509M16—348no—
510C2×C8—28no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—172no—
516C2×C8—172no—
517C2×C8—341no—
518C2×C8—341no—
519C2×C8—341no—
520C2×C8—341no—
521C2×C8#6325412yes—
522C2×C8#6325412yes—
523C4⋊C4—21no—
524C4⋊C4—24no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—174no—
530C4⋊C4—174no—
531C4⋊C4—172no—
532C4⋊C4—344no—
533C4⋊C4—344no—
534C4⋊C4—344no—
535C4⋊C4—344no—
536C4⋊C4—344no—
537C4⋊C4—344no—
538C4⋊C4—341no—
539C4⋊C4—344no—
540C4⋊C4—341no—
541C4⋊C4—341no—
542C4⋊C4—341no—
543C4⋊C4—341no—
544C4⋊C4—1361no—
545C4⋊C4—1361no—
546C4⋊C4C2×C4×D1712yes—
547C22⋊C4—24no—
548C22⋊C4—81no—
549C22⋊C4—174no—
550C22⋊C4—174no—
551C22⋊C4—344no—
552C22⋊C4—1361no—
553C42—172no—
554C42—172no—
555C42—172no—
556C42—172no—
557C42—344no—
558C42—344no—
559C42—1361no—
560C42—1361no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23C22⋊C4×D1712yes—
568C23C2×D4×D1711yescharacteristic
569Q8—22no—
570Q8—22no—
571Q8—41no—
572Q8—41no—
573Q8—342no—
574Q8—342no—
575Q8—681no—
576Q8—681no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—172no—
596C2×C4—172no—
597C2×C4—172no—
598C2×C4—172no—
599C2×C4—341no—
600C2×C4—341no—
601C2×C4—341no—
602C2×C4—341no—
603C2×C4—341no—
604C2×C4—341no—
605C2×C4—341no—
606C2×C4—341no—
607C2×C4—342no—
608C2×C4—342no—
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—681no—
618C2×C4—681no—
619C2×C4C22⋊C4×D1712yes—
620C2×C4C22×C4×D1711yescharacteristic
621C2×C4C2×D4×D1711yescharacteristic
622C2×C4C2×#6325411yescharacteristic
623C2×C4C2×#6325411yescharacteristic
624C2×C4#41×D1711yescharacteristic
625C2×C4#41×D1711yescharacteristic
626C8—22no—
627C8—22no—
628C8—342no—
629C8—342no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22C2×C22⋊C4×D1711yescharacteristic
638C22#1176335411yescharacteristic
639C22#1176335411yescharacteristic
640C22#1176361511yescharacteristic
641C22#1176361611yescharacteristic
642C22#1176362211yescharacteristic
643C22#1176362311yescharacteristic
644C4—22no—
645C4—42no—
646C4—42no—
647C4—81no—
648C4—81no—
649C4—342no—
650C4—342no—
651C4—342no—
652C4—342no—
653C4—682no—
654C4—682no—
655C4—1361no—
656C4—1361no—
657C4C4×#6325411yescharacteristic
658C4C4×#6325411yescharacteristic
659C4#1176331511yescharacteristic
660C4#1176331711yescharacteristic
661C2—42no—
662C2#1453126911yescharacteristic
663C2#1453731211yescharacteristic
664C2#1453731311yescharacteristic
665C2#1453940511yescharacteristic
666C2#1453943611yescharacteristic
667C2#1453944711yescharacteristic
668C2#1453945511yescharacteristic
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
hypocenterC17#21413
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
4. #14529751C2semidirect
5. #14529747C2semidirect
6. #14529602C2semidirect
7. #14529584C2semidirect
11. #14528616C2semidirect
12. #14528613C2semidirect
13. #103×Dic17C2semidirect
14. #14528603C2semidirect
18. #11763419C4semidirect
475. C17#21413semidirect

Maximal subgroups

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

subgroupindex
C2×#117634192
C2×#117627642
#145313912
#145297512
#145297472
#145296022
#145295842
#145292182
#145292152
#145292072
#145286162
#145286132
#103×Dic172
#145286032
C17×#18922
#2141317