Finite Group Databasebeta

← #17091821

Subgroups — #17091821

Order 4352.

928 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763105C211yesmaximal, characteristic, complemented
2C2×#11762819C211yesmaximal, characteristic
3#14532475C211yesmaximal, characteristic
4C22⋊C4×Dic34C211yesmaximal, characteristic
5#14530331C211yesmaximal, characteristic, complemented
6#14530178C211yesmaximal, characteristic, complemented
7#14530164C211yesmaximal, characteristic, complemented
8#14529926C211yesmaximal, characteristic, complemented
9#14529055C211yesmaximal, characteristic, complemented
10#14529055C211yesmaximal, characteristic, complemented
11#14528985C211yesmaximal, characteristic
12#14528817C211yesmaximal, characteristic
13#14528800C211yesmaximal, characteristic, complemented
14#14528720C211yesmaximal, characteristic
15C17×#1851C211yesmaximal, characteristic
16C23×Dic34C2211yescharacteristic
17C2×#10587555C2211yescharacteristic
18C2×C4×Dic34C2211yescharacteristic
19C2×C4×Dic34C2211yescharacteristic
20C2×#10587612C2211yescharacteristic
21#11763105—24no—
22C2×#10587489C2211yescharacteristic
23C2×#10587489C2211yescharacteristic
24C2×#10587488C2211yescharacteristic
25C2×#10587488C2211yescharacteristic
26#11762819—24no—
27C2×#10587487C2211yescharacteristic
28C2×#10587487C2211yescharacteristic
29C2×#10587487C2211yescharacteristic
30C2×#10587487C2211yescharacteristic
31C2×#10587487C2211yescharacteristic
32C2×#10587487C2211yescharacteristic
33C2×#10587487C2211yescharacteristic
34C2×#10587487C2211yescharacteristic
35#11762732—21no—
36#11762721—21no—
37#11762715C2211yescharacteristic
38#11762715C2211yescharacteristic
39#11762713—21no—
40#11762713—21no—
41#11762713—21no—
42#11762707C2211yescharacteristic
43#11762707C2211yescharacteristic
44#11762696—21no—
45#11762675C2211yescharacteristic
46#11762669C2211yescharacteristic
47#11762669C2211yescharacteristic
48#11762667—21no—
49C22⋊C4×Dic17—21no—
50#11762572C2211yescharacteristic
51#11762572C2211yescharacteristic
52#11762572C2211yescharacteristic
53#11762572C2211yescharacteristic
54C22×M16×C17C2211yescharacteristic
55C2×C17×C4⋊C8C2211yescharacteristic
56C2×C17×C4⋊C8C2211yescharacteristic
57C2×C17×C22⋊C8C2211yescharacteristic
58C4×C22⋊C4×C17C2211yescharacteristic
59C17×#336C2211yescharacteristic
60C17×#336C2211yescharacteristic
61C23×C4×C17C2311yescharacteristic
62C23×Dic17—21no—
63C22×Dic34—22no—
64C22×Dic34—22no—
65C22×Dic34—21no—
66C22×Dic34—22no—
67C22×Dic34—21no—
68C22×Dic34—41no—
69C22×Dic34—41no—
70C22×Dic34Q814yescomplemented
71C22×Dic34C2311yescharacteristic
72C22×Dic34C2311yescharacteristic
73C2×M16×C17—22no—
74C2×M16×C17—22no—
75C2×M16×C17—41no—
76C22×C8×C17C2311yescharacteristic
77C22×C8×C17C2311yescharacteristic
78C22×C8×C17C2311yescharacteristic
79C22×C8×C17C2311yescharacteristic
80C2×C22⋊C4×C17D411yescharacteristic
81C2×C22⋊C4×C17D411yescharacteristic
82C2×C42×C17C2311yescharacteristic
83C2×C42×C17C2311yescharacteristic
84C2×C22⋊Dic17—21no—
85C2×C22⋊Dic17—21no—
86C2×C22⋊Dic17—21no—
87#10587612—22no—
88#10587612—41no—
89C2×C4⋊Dic17—21no—
90C2×C4⋊Dic17C2311yescharacteristic
91C2×C4⋊Dic17C2311yescharacteristic
92C2×C4⋊Dic17C2311yescharacteristic
93C2×C4⋊Dic17C2311yescharacteristic
94C2×C4⋊Dic17C2311yescharacteristic
95C2×C4⋊Dic17C2311yescharacteristic
96C2×#63225—21no—
97C2×#63225—21no—
98C2×#63225—21no—
99C2×#63225—21no—
100C2×C4×Dic17—21no—
101C2×C4×Dic17—21no—
102C2×C4×Dic17—21no—
103#10587555—22no—
104#10587555—41no—
105C4×Dic34—22no—
106C4×Dic34—22no—
107C4×Dic34—41no—
108C4×Dic34—41no—
109C17×C4⋊C8—22no—
110C17×C4⋊C8—41no—
111C17×C22⋊C8—22no—
112C17×#95D411yescharacteristic
113C17×#95D411yescharacteristic
114#10587502—21no—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587502—21no—
120#10587502—21no—
121#10587502—21no—
122#10587502—21no—
123#10587502—21no—
124#10587489—22no—
125#10587489—22no—
126#10587488—41no—
127#10587488—41no—
128#10587487—22no—
129#10587487—22no—
130#10587487—22no—
131#10587487—22no—
132#10587487—22no—
133#10587487—41no—
134#10587487—41no—
135#10587487D414yes—
136C24×C17C2×D411yescharacteristic
137C22×C4×C17—21no—
138C22×C4×C17—21no—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17C2×D411yescharacteristic
142C22×C4×C17C2×D411yescharacteristic
143C22×C4×C17C2×D411yescharacteristic
144C22×C4×C17C2×D411yescharacteristic
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2×Q812yes—
147C22×C4×C17#4112yes—
148C22×C4×C17C2411yescharacteristic
149C22×Dic17—21no—
150C22×Dic17—24no—
151C22×Dic17—21no—
152C22×Dic17—21no—
153C22×Dic17—21no—
154C22×Dic17—21no—
155C22×Dic17—21no—
156C22×Dic17—21no—
157C22×Dic17—21no—
158C22×Dic17—42no—
159C22×Dic17—42no—
160C2×Dic34—21no—
161C2×Dic34—21no—
162C2×Dic34—21no—
163C2×Dic34—21no—
164C2×Dic34—21no—
165C2×Dic34—21no—
166C2×Dic34—21no—
167C2×Dic34—28no—
168C2×Dic34—24no—
169C2×Dic34—21no—
170C2×Dic34—44no—
171C2×Dic34—44no—
172C2×Dic34—44no—
173C2×Dic34—42no—
174C2×Dic34—42no—
175C2×Dic34—42no—
176C2×Dic34—42no—
177C2×Dic34—81no—
178C2×Dic34C2×D412yes—
179C2×Dic34C2×Q812yes—
180C2×Dic34C2×Q812yes—
181C2×Dic34#4112yes—
182M16×C17—44no—
183C2×C8×C17—21no—
184C2×C8×C17—21no—
185C2×C8×C17—21no—
186C2×C8×C17—21no—
187C2×C8×C17—21no—
188C2×C8×C17—21no—
189C2×C8×C17—21no—
190C2×C8×C17—21no—
191C2×C8×C17—21no—
192C2×C8×C17—21no—
193C2×C8×C17—21no—
194C2×C8×C17C2×D412yes—
195C22⋊C4×C17—22no—
196C22⋊C4×C17—41no—
197C42×C17—21no—
198C42×C17—21no—
199C42×C17—21no—
200C42×C17—21no—
201C22⋊Dic17—42no—
202C22⋊Dic17—42no—
203C22⋊Dic17—81no—
204C4⋊Dic17—21no—
205C4⋊Dic17—21no—
206C4⋊Dic17—21no—
207C4⋊Dic17—21no—
208C4⋊Dic17—21no—
209C4⋊Dic17—21no—
210C4⋊Dic17—21no—
211C4⋊Dic17—21no—
212C4⋊Dic17—21no—
213C4⋊Dic17—21no—
214C4⋊Dic17—21no—
215C4⋊Dic17—42no—
216C4⋊Dic17C2×D412yes—
217#63225—42no—
218#63225—42no—
219#63225—42no—
220#63225—81no—
221C4×Dic17—42no—
222C4×Dic17—42no—
223C4×Dic17—42no—
224#25979—171nomaximal, Sylow
225C23×C17—21no—
226C23×C17—21no—
227C23×C17—21no—
228C23×C17—21no—
229C23×C17—21no—
230C23×C17—21no—
231C23×C17C22⋊Q812yes—
232C23×C17C22×D411yescharacteristic
233C2×C4×C17—21no—
234C2×C4×C17—21no—
235C2×C4×C17—21no—
236C2×C4×C17—21no—
237C2×C4×C17—22no—
238C2×C4×C17—22no—
239C2×C4×C17—22no—
240C2×C4×C17—22no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—21no—
244C2×C4×C17—21no—
245C2×C4×C17—21no—
246C2×C4×C17—21no—
247C2×C4×C17—21no—
248C2×C4×C17—21no—
249C2×C4×C17—21no—
250C2×C4×C17C22⋊Q812yes—
251C2×C4×C17#13712yes—
252C2×C4×C17C22×D411yescharacteristic
253C2×C4×C17C22×Q811yescharacteristic
254C2×C4×C17C2×#4111yescharacteristic
255C2×C4×C17C2×#4111yescharacteristic
256C2×C4×C17H(5,𝔽2)11yescharacteristic
257C2×C4×C17H(5,𝔽2)11yescharacteristic
258C2×Dic17—22no—
259C2×Dic17—22no—
260C2×Dic17—22no—
261C2×Dic17—22no—
262C2×Dic17—22no—
263C2×Dic17—48no—
264C2×Dic17—41no—
265C2×Dic17—41no—
266C2×Dic17—41no—
267C2×Dic17—41no—
268C2×Dic17—41no—
269C2×Dic17—41no—
270C2×Dic17—41no—
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—41no—
282C2×Dic17—41no—
283Dic34—22no—
284Dic34—22no—
285Dic34—22no—
286Dic34—22no—
287Dic34—22no—
288Dic34—48no—
289Dic34—41no—
290Dic34—41no—
291Dic34—84no—
292Dic34C22⋊Q814yes—
293C8×C17—22no—
294C8×C17—41no—
295C8×C17—41no—
296C8×C17—41no—
297C2×#477—171no—
298C2×#419—171no—
299#2439—171no—
300Q8×C22⋊C4—171no—
301#2097—171no—
302#2068—171no—
303#2066—171no—
304#2024—171no—
305#1865—171no—
306#1865—171no—
307#1854—171no—
308#1851D1711yescharacteristic
309#1809—171no—
310#1805—171no—
311#1787—171no—
312C22×C17—22no—
313C22×C17—22no—
314C22×C17—22no—
315C22×C17—41no—
316C22×C17—41no—
317C22×C17—41no—
318C22×C17—41no—
319C22×C17C2×C22⋊Q811yescharacteristic
320C22×C17D4211yescharacteristic
321C22×C17#54611yescharacteristic
322C22×C17C2×#13711yescharacteristic
323C22×C17#57511yescharacteristic
324C22×C17#57711yescharacteristic
325C22×C17#57811yescharacteristic
326C4×C17—22no—
327C4×C17—22no—
328C4×C17—41no—
329C4×C17—41no—
330C4×C17—41no—
331C4×C17C2×C22⋊Q811yescharacteristic
332C4×C17#54311yescharacteristic
333C4×C17#54611yescharacteristic
334C4×C17#55111yescharacteristic
335Dic17—24no—
336Dic17—42no—
337Dic17—42no—
338Dic17—81no—
339Dic17—81no—
340Dic17—81no—
341Dic17—81no—
342Dic17—81no—
343C23×Q8—171no—
344C22×M16C2×D1711yescharacteristic
345C2×C4⋊Q8—171no—
346C2×C4×Q8—171no—
347C2×C4×Q8—171no—
348C2×#117—171no—
349#477—344no—
350C2×#107—171no—
351C2×#107—171no—
352C2×#106—171no—
353C2×#106—171no—
354C2×C4⋊C8C2×D1711yescharacteristic
355C2×C4⋊C8C2×D1711yescharacteristic
356#419—344no—
357C2×#103—171no—
358C2×#103—171no—
359C2×#103—171no—
360C2×#103—171no—
361C2×#103—171no—
362C2×#103—171no—
363C2×#103—171no—
364C2×#103—171no—
365C2×C22⋊C8C2×D1711yescharacteristic
366#396—341no—
367#393—341no—
368#391—171no—
369#391—171no—
370#391—341no—
371#391—341no—
372#391—341no—
373#389—171no—
374#389—171no—
375#387—341no—
376#382—171no—
377#380—171no—
378#380—171no—
379#380—341no—
380C4×C22⋊C4—341no—
381C4×C22⋊C4C2×D1711yescharacteristic
382#340—171no—
383#340—171no—
384#340—171no—
385#340—171no—
386#336C2×D1711yescharacteristic
387#336C2×D1711yescharacteristic
388C2×C17—42no—
389C2×C17#243711yescharacteristic
390C2×C17#307811yescharacteristic
391C2×C17#308211yescharacteristic
392C2×C17#327011yescharacteristic
393C2×C17#327511yescharacteristic
394C2×C17#331311yescharacteristic
395C2×C17#332911yescharacteristic
396C22×Q8—174no—
397C22×Q8—171no—
398C22×Q8—171no—
399C22×Q8—342no—
400C22×Q8—342no—
401C22×Q8—341no—
402C22×Q8—341no—
403C22×Q8—342no—
404C22×Q8—681no—
405C22×Q8—681no—
406C23×C4—341no—
407C23×C4C22×D1711yescharacteristic
408C2×M16—22no—
409C2×M16—22no—
410C2×M16—41no—
411C22×C8C22×D1711yescharacteristic
412C22×C8C22×D1711yescharacteristic
413C22×C8C22×D1711yescharacteristic
414C22×C8C22×D1711yescharacteristic
415C4⋊Q8—342no—
416C4⋊Q8—681no—
417C4×Q8—342no—
418C4×Q8—342no—
419C4×Q8—681no—
420C4×Q8—681no—
421#117—342no—
422#117—681no—
423C2×C4⋊C4—171no—
424C2×C4⋊C4—171no—
425C2×C4⋊C4—171no—
426C2×C4⋊C4—171no—
427C2×C4⋊C4—171no—
428C2×C4⋊C4—171no—
429C2×C4⋊C4—341no—
430C2×C4⋊C4—341no—
431C2×C4⋊C4—341no—
432C2×C4⋊C4—341no—
433C2×C4⋊C4—341no—
434C2×C22⋊C4—341no—
435C2×C22⋊C4—341no—
436C2×C22⋊C4—341no—
437C2×C22⋊C4D6811yescharacteristic
438C2×C22⋊C4D6811yescharacteristic
439C2×C42—341no—
440C2×C42—341no—
441C2×C42—341no—
442C2×C42C22×D1711yescharacteristic
443C2×C42C22×D1711yescharacteristic
444#107—342no—
445#107—342no—
446#106—681no—
447#106—681no—
448C4⋊C8—22no—
449C4⋊C8—41no—
450#103—174no—
451#103—342no—
452#103—342no—
453#103—342no—
454#103—342no—
455#103—342no—
456#103—681no—
457#103—681no—
458C22⋊C8—22no—
459#95—341no—
460#95—341no—
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#95D6811yescharacteristic
470#95D6811yescharacteristic
471C17#2597911yescharacteristic, complemented, Sylow
472C24C2×D6811yescharacteristic
473C2×Q8—172no—
474C2×Q8—172no—
475C2×Q8—172no—
476C2×Q8—172no—
477C2×Q8—348no—
478C2×Q8—341no—
479C2×Q8—341no—
480C2×Q8—341no—
481C2×Q8—341no—
482C2×Q8—344no—
483C2×Q8—341no—
484C2×Q8—341no—
485C2×Q8—341no—
486C2×Q8—341no—
487C2×Q8—682no—
488C2×Q8—682no—
489C2×Q8—682no—
490C2×Q8—684no—
491C2×Q8—684no—
492C2×Q8—684no—
493C2×Q8—682no—
494C2×Q8—1361no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—341no—
500C22×C4—341no—
501C22×C4—341no—
502C22×C4—344no—
503C22×C4—341no—
504C22×C4—341no—
505C22×C4—341no—
506C22×C4—341no—
507C22×C4—341no—
508C22×C4—682no—
509C22×C4—682no—
510C22×C4C2×D6811yescharacteristic
511C22×C4C2×D6811yescharacteristic
512C22×C4C2×D6811yescharacteristic
513C22×C4C2×D6811yescharacteristic
514C22×C4C2×D6811yescharacteristic
515C22×C4#6325412yes—
516C22×C4Q8×D1712yes—
517C22×C4C23×D1711yescharacteristic
518M16—44no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8D4×D1712yes—
531C4⋊C4—172no—
532C4⋊C4—341no—
533C4⋊C4—341no—
534C4⋊C4—341no—
535C4⋊C4—341no—
536C4⋊C4—341no—
537C4⋊C4—341no—
538C4⋊C4—341no—
539C4⋊C4—341no—
540C4⋊C4—341no—
541C4⋊C4—341no—
542C4⋊C4—341no—
543C4⋊C4—682no—
544C4⋊C4—682no—
545C4⋊C4—682no—
546C4⋊C4—682no—
547C4⋊C4—1361no—
548C22⋊C4—22no—
549C22⋊C4—41no—
550C22⋊C4—682no—
551C22⋊C4—682no—
552C22⋊C4—1361no—
553C42—21no—
554C42—21no—
555C42—21no—
556C42—21no—
557C42—682no—
558C42—682no—
559C42—682no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23#1058758312yes—
567C23C22×D6811yescharacteristic
568Q8—174no—
569Q8—342no—
570Q8—342no—
571Q8—342no—
572Q8—342no—
573Q8—342no—
574Q8—688no—
575Q8—681no—
576Q8—681no—
577Q8—1364no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—342no—
596C2×C4—342no—
597C2×C4—342no—
598C2×C4—342no—
599C2×C4—342no—
600C2×C4—688no—
601C2×C4—681no—
602C2×C4—681no—
603C2×C4—681no—
604C2×C4—681no—
605C2×C4—681no—
606C2×C4—681no—
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—681no—
618C2×C4—681no—
619C2×C4—681no—
620C2×C4#1058758312yes—
621C2×C4#1058759512yes—
622C2×C4C2×D4×D1711yescharacteristic
623C2×C4C2×#6325411yescharacteristic
624C2×C4#1058769111yescharacteristic
625C2×C4C2×Q8×D1711yescharacteristic
626C2×C4#41×D1711yescharacteristic
627C2×C4#1058769611yescharacteristic
628C8—22no—
629C8—41no—
630C8—41no—
631C8—41no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22C2×#1058758311yescharacteristic
640C22D4×D6811yescharacteristic
641C22#1176345411yescharacteristic
642C22C2×#1058759511yescharacteristic
643C22#1176360911yescharacteristic
644C22#1176362111yescharacteristic
645C22#1176362611yescharacteristic
646C4—22no—
647C4—22no—
648C4—41no—
649C4—41no—
650C4—41no—
651C4—344no—
652C4—682no—
653C4—682no—
654C4—1361no—
655C4—1361no—
656C4—1361no—
657C4—1361no—
658C4—1361no—
659C4C22⋊Q8×D1711yescharacteristic
660C4#1176343411yescharacteristic
661C4#1176344611yescharacteristic
662C4#1176348211yescharacteristic
663C2—42no—
664C2#1453244811yescharacteristic
665C2#1453859111yescharacteristic
666C2#1453862711yescharacteristic
667C2#1454035911yescharacteristic
668C2#1454042011yescharacteristic
669C2#1454085211yescharacteristic
670C2#1454107911yescharacteristic
subgroupG/H
centerC23C22×D68
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#1851C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenter#1851D17
hypocenterC17#25979
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1851

Complements

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

NH ≅ G/Ntype
1. C2×#11763105C2semidirect
5. #14530331C2semidirect
6. #14530178C2semidirect
7. #14530164C2semidirect
8. #14529926C2semidirect
9. #14529055C2semidirect
10. #14529055C2semidirect
13. #14528800C2semidirect
70. C22×Dic34Q8semidirect
471. C17#25979semidirect

Maximal subgroups

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

subgroupindex
C2×#117631052
C2×#117628192
#145324752
C22⋊C4×Dic342
#145303312
#145301782
#145301642
#145299262
#145290552
#145289852
#145288172
#145288002
#145287202
C17×#18512
#2597917