Finite Group Databasebeta

← #17051572

Subgroups — #17051572

Order 4352.

982 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762810C211yesmaximal, characteristic
2#14531663C211yesmaximal, characteristic, complemented
3#14531625C211yesmaximal, characteristic
4#14529478C211yesmaximal, characteristic, complemented
5#14529464C211yesmaximal, characteristic, complemented
6#14529414C211yesmaximal, characteristic, complemented
7#14529407C211yesmaximal, characteristic, complemented
8#14529050C211yesmaximal, characteristic, complemented
9#14529036C211yesmaximal, characteristic, complemented
10#14528871C211yesmaximal, characteristic, complemented
11#14528815C211yesmaximal, characteristic, complemented
12#14528616C211yesmaximal, characteristic, complemented
13#103×Dic17C211yesmaximal, characteristic, complemented
14#14528570C211yesmaximal, characteristic, complemented
15C17×#1788C211yesmaximal, characteristic
16C22×#63223C2211yescharacteristic
17C2×Q8×Dic17C2211yescharacteristic
18C2×Q8×Dic17C2211yescharacteristic
19C22×C4⋊Dic17C2211yescharacteristic
20C2×#10587478C2211yescharacteristic
21C2×#10587479C2211yescharacteristic
22#11762810C418yescomplemented
23C2×#10587506C2211yescharacteristic
24C2×#10587506C2211yescharacteristic
25C2×#10587503C2211yescharacteristic
26C2×#10587503C2211yescharacteristic
27C2×#10587486C2211yescharacteristic
28C2×C8×Dic17C2211yescharacteristic
29#11762712—21no—
30#11762712—21no—
31#11762712C2211yescharacteristic
32#11762712C2211yescharacteristic
33#11762705C2211yescharacteristic
34#11762705C2211yescharacteristic
35#11762694—21no—
36#11762694—21no—
37#11762669—21no—
38#11762669C2211yescharacteristic
39#11762667—21no—
40C4⋊C4×Dic17C2211yescharacteristic
41C4⋊C4×Dic17C2211yescharacteristic
42C22⋊C4×Dic17—21no—
43C4×C22⋊Dic17—21no—
44C4×C22⋊Dic17C2211yescharacteristic
45#11762572C2211yescharacteristic
46#11762572C2211yescharacteristic
47#11762572C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762563C2211yescharacteristic
52#11762563C2211yescharacteristic
53C22×#41×C17C2211yescharacteristic
54C2×C17×#103C2211yescharacteristic
55C2×C17×#103C2211yescharacteristic
56C2×C17×#102C2211yescharacteristic
57C2×C17×#102C2211yescharacteristic
58C2×C17×C22⋊C8C2211yescharacteristic
59C17×#380C2211yescharacteristic
60C2×#41×C17—22no—
61C2×#41×C17—22no—
62C2×#41×C17—22no—
63C2×#41×C17—41no—
64C2×#41×C17—41no—
65C2×#41×C17C2×C414yes—
66C22×Q8×C17C2311yescharacteristic
67C22×D4×C17—21no—
68C22×D4×C17C2311yescharacteristic
69C23×C4×C17—21no—
70C23×C4×C17C2311yescharacteristic
71C23×Dic17—21no—
72C22×C8×C17C2311yescharacteristic
73C22×C8×C17C2311yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C22⋊C4×C17—21no—
77C2×C22⋊Dic17—21no—
78C2×C22⋊Dic17—21no—
79C2×C22⋊Dic17—21no—
80C2×C22⋊Dic17—21no—
81C2×C22⋊Dic17—21no—
82C2×C22⋊Dic17D411yescharacteristic
83C2×C22⋊Dic17D411yescharacteristic
84Q8×Dic17—22no—
85Q8×Dic17—22no—
86Q8×Dic17—41no—
87Q8×Dic17—41no—
88C2×C4⋊Dic17—21no—
89C2×C4⋊Dic17—22no—
90C2×C4⋊Dic17—21no—
91C2×C4⋊Dic17—21no—
92C2×C4⋊Dic17C2×C414yes—
93C2×C4⋊Dic17C2311yescharacteristic
94C2×C4⋊Dic17C2311yescharacteristic
95C2×C4⋊Dic17C2311yescharacteristic
96C2×C4⋊Dic17C2311yescharacteristic
97C2×#63225—21no—
98C2×#63225—21no—
99C2×C4×Dic17—21no—
100C2×C4×Dic17—21no—
101C2×C4×Dic17—21no—
102C2×C4×Dic17—21no—
103C2×C4×Dic17C2311yescharacteristic
104C2×C4×Dic17C2311yescharacteristic
105C2×#63223—41no—
106C2×#63223—41no—
107C2×#63223C2×C414yes—
108C22×#3622C2311yescharacteristic
109C22×#3622C2311yescharacteristic
110C17×#103—22no—
111C17×#103—22no—
112C17×#102—41no—
113C17×#102—41no—
114C17×C22⋊C8—22no—
115C17×#95—21no—
116#10587506C2×C414yes—
117#10587506C2×C414yes—
118#10587503C2×C414yes—
119#10587503C2×C414yes—
120#10587502—21no—
121#10587502—21no—
122#10587502—21no—
123#10587502—21no—
124#10587502—21no—
125#10587502D411yescharacteristic
126#10587502D411yescharacteristic
127#10587486—41no—
128C8×Dic17—41no—
129#10587479—41no—
130#10587478—41no—
131C24×C17—21no—
132C24×C17C2×D411yescharacteristic
133#41×C17—28no—
134#41×C17—44no—
135#41×C17—44no—
136#41×C17—44no—
137C2×Q8×C17—21no—
138C2×Q8×C17—21no—
139C2×Q8×C17—21no—
140C2×Q8×C17C22×C412yes—
141C2×Q8×C17#4112yes—
142C2×Q8×C17#4112yes—
143C2×D4×C17—24no—
144C2×D4×C17—21no—
145C2×D4×C17—21no—
146C2×D4×C17—21no—
147C2×D4×C17—21no—
148C2×D4×C17—21no—
149C2×D4×C17—42no—
150C2×D4×C17—42no—
151C2×D4×C17C22×C412yes—
152C22×C4×C17—21no—
153C22×C4×C17—21no—
154C22×C4×C17—21no—
155C22×C4×C17—21no—
156C22×C4×C17—21no—
157C22×C4×C17—24no—
158C22×C4×C17—21no—
159C22×C4×C17—21no—
160C22×C4×C17—42no—
161C22×C4×C17—42no—
162C22×C4×C17C22×C412yes—
163C22×C4×C17C2×D411yescharacteristic
164C22×C4×C17#4112yes—
165C22×C4×C17C2411yescharacteristic
166C22×Dic17—21no—
167C22×Dic17—21no—
168C22×Dic17—24no—
169C22×Dic17—21no—
170C22×Dic17—21no—
171C22×Dic17—81no—
172C22×Dic17—81no—
173C22×Dic17C2×D411yescharacteristic
174C22×Dic17C2×D411yescharacteristic
175C22×Dic17C2×D411yescharacteristic
176C22×Dic17C2×D411yescharacteristic
177C2×C8×C17—21no—
178C2×C8×C17—21no—
179C2×C8×C17—21no—
180C2×C8×C17—21no—
181C2×C8×C17—21no—
182C2×C8×C17—21no—
183C4⋊C4×C17—21no—
184C4⋊C4×C17—21no—
185C4⋊C4×C17—21no—
186C4⋊C4×C17—21no—
187C22⋊C4×C17—42no—
188C22⋊Dic17—22no—
189C22⋊Dic17—41no—
190C22⋊Dic17—42no—
191C22⋊Dic17—42no—
192C22⋊Dic17—81no—
193C22⋊Dic17—81no—
194C22⋊Dic17—81no—
195C4⋊Dic17—24no—
196C4⋊Dic17—21no—
197C4⋊Dic17—21no—
198C4⋊Dic17—21no—
199C4⋊Dic17—21no—
200C4⋊Dic17—21no—
201C4⋊Dic17—21no—
202C4⋊Dic17—42no—
203C4⋊Dic17—42no—
204C4⋊Dic17C22×C412yes—
205C4⋊Dic17C22×C412yes—
206#63225—81no—
207#63225—81no—
208C4×Dic17—21no—
209C4×Dic17—21no—
210C4×Dic17—21no—
211C4×Dic17—21no—
212C4×Dic17—42no—
213C4×Dic17—42no—
214C4×Dic17—81no—
215C4×Dic17—81no—
216#63223—44no—
217C2×#3622—21no—
218C2×#3622—21no—
219C2×#3622—21no—
220C2×#3622—21no—
221C2×#3622C22×C412yes—
222C2×#3622C22×C412yes—
223#22579—171nomaximal, Sylow
224C23×C17—21no—
225C23×C17—21no—
226C23×C17—21no—
227C23×C17—21no—
228C23×C17—21no—
229C23×C17—21no—
230C23×C17—22no—
231C23×C17—41no—
232C23×C17—41no—
233C23×C17—41no—
234C23×C17—41no—
235C23×C17—41no—
236C23×C17—41no—
237C23×C17C4×D412yes—
238C23×C17C22×D411yescharacteristic
239Q8×C17—22no—
240Q8×C17—22no—
241Q8×C17—22no—
242Q8×C17#11714yes—
243D4×C17—22no—
244D4×C17—22no—
245D4×C17—48no—
246D4×C17—41no—
247D4×C17—41no—
248C2×C4×C17—22no—
249C2×C4×C17—21no—
250C2×C4×C17—21no—
251C2×C4×C17—22no—
252C2×C4×C17—22no—
253C2×C4×C17—22no—
254C2×C4×C17—22no—
255C2×C4×C17—22no—
256C2×C4×C17—22no—
257C2×C4×C17—48no—
258C2×C4×C17—41no—
259C2×C4×C17—41no—
260C2×C4×C17—41no—
261C2×C4×C17—41no—
262C2×C4×C17—41no—
263C2×C4×C17—41no—
264C2×C4×C17C4×D412yes—
265C2×C4×C17C23×C411yescharacteristic
266C2×C4×C17C2×#4111yescharacteristic
267C2×C4×C17C2×#4111yescharacteristic
268C2×C4×C17C2×#4111yescharacteristic
269C2×C4×C17H(5,𝔽2)11yescharacteristic
270C2×C4×C17H(5,𝔽2)11yescharacteristic
271C2×Dic17—21no—
272C2×Dic17—21no—
273C2×Dic17—22no—
274C2×Dic17—22no—
275C2×Dic17—21no—
276C2×Dic17—21no—
277C2×Dic17—21no—
278C2×Dic17—21no—
279C2×Dic17—21no—
280C2×Dic17—21no—
281C2×Dic17—21no—
282C2×Dic17—21no—
283C2×Dic17—21no—
284C2×Dic17—41no—
285C2×Dic17—41no—
286C2×Dic17—41no—
287C2×Dic17—41no—
288C2×Dic17—41no—
289C2×Dic17—41no—
290C2×Dic17—41no—
291C2×Dic17—41no—
292C2×Dic17—41no—
293C2×Dic17—41no—
294C2×Dic17—84no—
295C2×Dic17#13712yes—
296C8×C17—41no—
297C8×C17—41no—
298#3622—22no—
299#3622—22no—
300C2×#417—171no—
301#2345—171no—
302#2341—171no—
303#1936—171no—
304#1934—171no—
305#1926—171no—
306#1925—171no—
307#1865—171no—
308#1863—171no—
309#1823—171no—
310#1809—171no—
311#1788D1711yescharacteristic
312#1763—171no—
313C4×#103—171no—
314#1753—171no—
315C22×C17—22no—
316C22×C17—22no—
317C22×C17—22no—
318C22×C17—41no—
319C22×C17—41no—
320C22×C17—41no—
321C22×C17—41no—
322C22×C17—42no—
323C22×C17—42no—
324C22×C17—42no—
325C22×C17—81no—
326C22×C17—81no—
327C22×C17—81no—
328C22×C17—81no—
329C22×C17C2×C4×D411yescharacteristic
330C22×C17#54611yescharacteristic
331C22×C17#54611yescharacteristic
332C22×C17C2×#13711yescharacteristic
333C22×C17#57511yescharacteristic
334C22×C17#57611yescharacteristic
335C22×C17#57711yescharacteristic
336C4×C17—22no—
337C4×C17—24no—
338C4×C17—81no—
339C4×C17—81no—
340C4×C17C2×#11711yescharacteristic
341C4×C17#51811yescharacteristic
342C4×C17#55111yescharacteristic
343C4×C17#55111yescharacteristic
344Dic17—22no—
345Dic17—41no—
346Dic17—42no—
347Dic17—42no—
348Dic17—41no—
349Dic17—41no—
350Dic17—81no—
351Dic17—81no—
352C22×#41C2×D1711yescharacteristic
353C22×M16—171no—
354C2×C4×Q8—171no—
355C2×C4×Q8—171no—
356C22×C4⋊C4—171no—
357C2×#107—171no—
358C2×#106—171no—
359#417—178no—
360C2×#103—171no—
361C2×#103—171no—
362C2×#103C2×D1711yescharacteristic
363C2×#103C2×D1711yescharacteristic
364C2×#102—171no—
365C2×#102—171no—
366C2×#102C2×D1711yescharacteristic
367C2×#102C2×D1711yescharacteristic
368C2×C22⋊C8C2×D1711yescharacteristic
369C2×#97—171no—
370C2×C4×C8—171no—
371#390—171no—
372#390—171no—
373#390—341no—
374#390—341no—
375#389—171no—
376#389—171no—
377#387—341no—
378#387—341no—
379#380—171no—
380#380—341no—
381#380—341no—
382#380C2×D1711yescharacteristic
383C4×C4⋊C4—171no—
384C4×C4⋊C4—171no—
385C4×C22⋊C4—171no—
386C4×C22⋊C4—341no—
387C4×C22⋊C4—341no—
388#340—171no—
389#340—171no—
390#340—171no—
391#340—171no—
392#340—171no—
393#340—171no—
394#336—171no—
395#336—171no—
396C2×C17—42no—
397C2×C17—82no—
398C2×C17#234211yescharacteristic
399C2×C17C4×#13711yescharacteristic
400C2×C17#294611yescharacteristic
401C2×C17#330911yescharacteristic
402C2×C17#331611yescharacteristic
403C2×C17#332611yescharacteristic
404C2×C17#333311yescharacteristic
405C2×#41—22no—
406C2×#41—22no—
407C2×#41—22no—
408C2×#41—41no—
409C2×#41—41no—
410C2×#41C4×D1714yes—
411C22×Q8C22×D1711yescharacteristic
412C22×D4—21no—
413C22×D4C22×D1711yescharacteristic
414C23×C4—21no—
415C23×C4—341no—
416C23×C4C22×D1711yescharacteristic
417C2×M16—174no—
418C2×M16—681no—
419C2×M16—681no—
420C22×C8—171no—
421C22×C8—171no—
422C22×C8C22×D1711yescharacteristic
423C22×C8C22×D1711yescharacteristic
424C4×Q8—342no—
425C4×Q8—342no—
426C4×Q8—681no—
427C4×Q8—681no—
428C2×C4⋊C4—174no—
429C2×C4⋊C4—171no—
430C2×C4⋊C4—171no—
431C2×C4⋊C4—171no—
432C2×C4⋊C4—171no—
433C2×C4⋊C4—341no—
434C2×C4⋊C4—342no—
435C2×C4⋊C4—341no—
436C2×C4⋊C4—341no—
437C2×C4⋊C4—341no—
438C2×C4⋊C4—341no—
439C2×C4⋊C4C22×D1711yescharacteristic
440C2×C4⋊C4C22×D1711yescharacteristic
441C2×C22⋊C4—21no—
442C2×C22⋊C4—171no—
443C2×C22⋊C4—171no—
444C2×C22⋊C4—341no—
445C2×C22⋊C4—341no—
446C2×C22⋊C4—341no—
447C2×C22⋊C4—341no—
448C2×C22⋊C4—341no—
449C2×C42—171no—
450C2×C42—171no—
451C2×C42—341no—
452C2×C42—341no—
453C2×C42—341no—
454C2×C42—341no—
455#107—681no—
456#106—681no—
457#103—22no—
458#103—22no—
459#103—174no—
460#103—174no—
461#102—41no—
462#102—41no—
463#102—174no—
464#102—174no—
465C22⋊C8—22no—
466#97—681no—
467C4×C8—681no—
468#95—21no—
469#95—171no—
470#95—171no—
471#95—341no—
472#95—341no—
473#95—341no—
474#95—341no—
475#95—341no—
476C17#2257911yescharacteristic, complemented, Sylow
477C24—21no—
478C24D4×D1711yescharacteristic
479#41—28no—
480#41—44no—
481#41—44no—
482#41—44no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8C2×C4×D1712yes—
487C2×Q8#6325412yes—
488C2×Q8#6325412yes—
489C2×D4—24no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—42no—
496C2×D4—42no—
497C2×D4C2×C4×D1712yes—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—24no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—42no—
507C22×C4—42no—
508C22×C4—171no—
509C22×C4—171no—
510C22×C4—171no—
511C22×C4—171no—
512C22×C4—341no—
513C22×C4—341no—
514C22×C4—344no—
515C22×C4—341no—
516C22×C4—341no—
517C22×C4—1361no—
518C22×C4—1361no—
519C22×C4C2×C4×D1712yes—
520C22×C4D4×D1711yescharacteristic
521C22×C4#6325412yes—
522C22×C4C23×D1711yescharacteristic
523M16—684no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—172no—
531C2×C8—172no—
532C2×C8—341no—
533C2×C8—341no—
534C2×C8—341no—
535C2×C8—341no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—172no—
541C4⋊C4—172no—
542C4⋊C4—341no—
543C4⋊C4—341no—
544C4⋊C4—344no—
545C4⋊C4—341no—
546C4⋊C4—341no—
547C4⋊C4—341no—
548C4⋊C4—341no—
549C4⋊C4—682no—
550C4⋊C4—682no—
551C4⋊C4—1361no—
552C4⋊C4—1361no—
553C22⋊C4—42no—
554C22⋊C4—342no—
555C22⋊C4—682no—
556C22⋊C4—681no—
557C22⋊C4—682no—
558C22⋊C4—1361no—
559C22⋊C4—1361no—
560C22⋊C4—1361no—
561C42—341no—
562C42—341no—
563C42—341no—
564C42—341no—
565C42—682no—
566C42—682no—
567C42—1361no—
568C42—1361no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—22no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23#1058756712yes—
583C23C2×D4×D1711yescharacteristic
584Q8—22no—
585Q8—22no—
586Q8—22no—
587Q8#1058756314yes—
588D4—22no—
589D4—22no—
590D4—48no—
591D4—41no—
592D4—41no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—48no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—172no—
610C2×C4—341no—
611C2×C4—341no—
612C2×C4—342no—
613C2×C4—342no—
614C2×C4—341no—
615C2×C4—341no—
616C2×C4—341no—
617C2×C4—341no—
618C2×C4—341no—
619C2×C4—341no—
620C2×C4—341no—
621C2×C4—341no—
622C2×C4—341no—
623C2×C4—681no—
624C2×C4—681no—
625C2×C4—681no—
626C2×C4—681no—
627C2×C4—681no—
628C2×C4—681no—
629C2×C4—681no—
630C2×C4—681no—
631C2×C4—681no—
632C2×C4—681no—
633C2×C4—1364no—
634C2×C4#1058756712yes—
635C2×C4C22×C4×D1711yescharacteristic
636C2×C4C2×#6325411yescharacteristic
637C2×C4C2×#6325411yescharacteristic
638C2×C4C2×#6325411yescharacteristic
639C2×C4#1058769111yescharacteristic
640C2×C4#1058769111yescharacteristic
641C8—41no—
642C8—41no—
643C8—342no—
644C8—342no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22C2×#1058756711yescharacteristic
660C22#1176344611yescharacteristic
661C22#1176344611yescharacteristic
662C22#137×D1711yescharacteristic
663C22#1176360511yescharacteristic
664C22#1176361111yescharacteristic
665C22#1176361611yescharacteristic
666C4—22no—
667C4—24no—
668C4—81no—
669C4—81no—
670C4—342no—
671C4—681no—
672C4—682no—
673C4—682no—
674C4—681no—
675C4—681no—
676C4—1361no—
677C4—1361no—
678C4C2×#1058756311yescharacteristic
679C4#1176328211yescharacteristic
680C4#1176349011yescharacteristic
681C4#1176349011yescharacteristic
682C2—42no—
683C2—82no—
684C2#1453163511yescharacteristic
685C2#1453760811yescharacteristic
686C2#1453761911yescharacteristic
687C2#1454078311yescharacteristic
688C2#1454088811yescharacteristic
689C2#1454103011yescharacteristic
690C2#1454111511yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1788C2
socleC23×C17C22×D4
prime radicalC22×#41×C17C22
cosocleC22×C4×C17C24
hypercenter#1788D17
hypocenterC17#22579
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1788

Complements

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

NH ≅ G/Ntype
2. #14531663C2semidirect
4. #14529478C2semidirect
5. #14529464C2semidirect
6. #14529414C2semidirect
7. #14529407C2semidirect
8. #14529050C2semidirect
9. #14529036C2semidirect
10. #14528871C2semidirect
11. #14528815C2semidirect
12. #14528616C2semidirect
13. #103×Dic17C2semidirect
14. #14528570C2semidirect
22. #11762810C4semidirect
476. C17#22579semidirect

Maximal subgroups

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

subgroupindex
C2×#117628102
#145316632
#145316252
#145294782
#145294642
#145294142
#145294072
#145290502
#145290362
#145288712
#145288152
#145286162
#103×Dic172
#145285702
C17×#17882
#2257917