Finite Group Databasebeta

← #17084498

Subgroups — #17084498

Order 4352.

850 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763116C211yesmaximal, characteristic, complemented
2C2×D4×#3622C211yesmaximal, characteristic
3#14533950C211yesmaximal, characteristic
4#14530220C211yesmaximal, characteristic, complemented
5#14530210C211yesmaximal, characteristic, complemented
6#14529984C211yesmaximal, characteristic, complemented
7#14529977C211yesmaximal, characteristic, complemented
8#14529748C211yesmaximal, characteristic, complemented
9#14529550C211yesmaximal, characteristic, complemented
10#14529208C211yesmaximal, characteristic, complemented
11#14529208C211yesmaximal, characteristic, complemented
12#14528811C211yesmaximal, characteristic, complemented
13#14528807C211yesmaximal, characteristic, complemented
14C4×#10587506C211yesmaximal, characteristic, complemented
15C17×#2536C211yesmaximal, characteristic
16C23×#3622C2211yescharacteristic
17C23×#3622C2211yescharacteristic
18C2×#10587556C2211yescharacteristic
19#11763116—24no—
20D4×#3622—24no—
21C2×#10587478C2211yescharacteristic
22C2×#10587478C2211yescharacteristic
23C2×#10587479C2211yescharacteristic
24C2×#10587479C2211yescharacteristic
25C2×C4⋊#3622C2211yescharacteristic
26C2×#10587506C2211yescharacteristic
27C2×#10587506C2211yescharacteristic
28C2×#10587506C2211yescharacteristic
29C2×#10587506C2211yescharacteristic
30C2×C22⋊#3622C2211yescharacteristic
31C2×C22⋊#3622C2211yescharacteristic
32C2×C4×#3622C2211yescharacteristic
33#11762747—21no—
34#11762723—21no—
35#11762703—21no—
36#11762703—21no—
37#11762669C2211yescharacteristic
38#11762669C2211yescharacteristic
39#11762669C2211yescharacteristic
40#11762669C2211yescharacteristic
41C4×C4⋊Dic17C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50C2×C17×C22⋊Q8C2211yescharacteristic
51C2×C17×C22⋊Q8C2211yescharacteristic
52C2×C4×Q8×C17C2211yescharacteristic
53C2×C4×D4×C17C2211yescharacteristic
54C17×#398—21no—
55C17×#394—21no—
56C17×#388—21no—
57C17×#388—21no—
58C17×#382C2211yescharacteristic
59C17×#380C2211yescharacteristic
60C17×#380C2211yescharacteristic
61C22×Q8×C17C2311yescharacteristic
62C22×D4×C17D411yescharacteristic
63C23×C4×C17C2311yescharacteristic
64C23×C4×C17C2311yescharacteristic
65C17×C22⋊Q8—22no—
66C17×C22⋊Q8—22no—
67C17×C22⋊Q8—41no—
68C17×C22⋊Q8—41no—
69C4×Q8×C17—22no—
70C4×Q8×C17—22no—
71C4×D4×C17—22no—
72C4×D4×C17—22no—
73C2×C4⋊C4×C17—21no—
74C2×C4⋊C4×C17—21no—
75C2×C4⋊C4×C17—21no—
76C2×C4⋊C4×C17D411yescharacteristic
77C2×C4⋊C4×C17C2311yescharacteristic
78C2×C4⋊C4×C17C2311yescharacteristic
79C2×C4⋊C4×C17C2311yescharacteristic
80C2×C22⋊C4×C17—21no—
81C2×C22⋊C4×C17—21no—
82C2×C22⋊C4×C17—21no—
83C2×C22⋊C4×C17—21no—
84C2×C22⋊C4×C17D411yescharacteristic
85C2×C22⋊C4×C17D411yescharacteristic
86C2×C42×C17—21no—
87C2×C42×C17C2311yescharacteristic
88C2×C22⋊Dic17—21no—
89C2×C22⋊Dic17—21no—
90C2×C22⋊Dic17—21no—
91C2×C22⋊Dic17—21no—
92C2×C4⋊Dic17C2311yescharacteristic
93C2×C4⋊Dic17C2311yescharacteristic
94C2×C4⋊Dic17C2311yescharacteristic
95C2×C4⋊Dic17C2311yescharacteristic
96C2×#63225—21no—
97C2×#63225—21no—
98C2×C4×Dic17—21no—
99C22×#3622—22no—
100C22×#3622—22no—
101C22×#3622—22no—
102C22×#3622—22no—
103C22×#3622—41no—
104C22×#3622—41no—
105C22×#3622C2311yescharacteristic
106C22×#3622C2311yescharacteristic
107C22×#3622C2311yescharacteristic
108C22×#3622C2311yescharacteristic
109#10587556—22no—
110#10587556—22no—
111C17×#95—21no—
112C17×#95—21no—
113C17×#95—21no—
114C17×#95—21no—
115#10587506—22no—
116#10587506—22no—
117#10587506—22no—
118#10587506—22no—
119#10587502—21no—
120#10587502—21no—
121#10587502—21no—
122#10587502—21no—
123#10587502—21no—
124C22⋊#3622—22no—
125C22⋊#3622—22no—
126#10587479—22no—
127#10587479—41no—
128#10587478—22no—
129#10587478—41no—
130C4⋊#3622D414yes—
131C4×#3622—22no—
132C24×C17C2×D411yescharacteristic
133C24×C17C2×D411yescharacteristic
134C2×Q8×C17—21no—
135C2×Q8×C17—21no—
136C2×Q8×C17—21no—
137C2×Q8×C17—21no—
138C2×Q8×C17—21no—
139C2×Q8×C17#4112yes—
140C2×D4×C17—22no—
141C2×D4×C17—22no—
142C2×D4×C17—22no—
143C22×C4×C17—21no—
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×C17—21no—
153C22×C4×C17—21no—
154C22×C4×C17—21no—
155C22×C4×C17—21no—
156C22×C4×C17—21no—
157C22×C4×C17—21no—
158C22×C4×C17—21no—
159C22×C4×C17C2×D411yescharacteristic
160C22×C4×C17C2×D411yescharacteristic
161C22×C4×C17C2×D411yescharacteristic
162C22×C4×C17C2×D411yescharacteristic
163C22×C4×C17C2411yescharacteristic
164C22×Dic17—21no—
165C22×Dic17—21no—
166C22×Dic17—21no—
167C22×Dic17—21no—
168C4⋊C4×C17—22no—
169C4⋊C4×C17—21no—
170C4⋊C4×C17—21no—
171C4⋊C4×C17—21no—
172C4⋊C4×C17—21no—
173C4⋊C4×C17—21no—
174C4⋊C4×C17—21no—
175C4⋊C4×C17—42no—
176C4⋊C4×C17—42no—
177C4⋊C4×C17—42no—
178C22⋊C4×C17—22no—
179C22⋊C4×C17—22no—
180C22⋊C4×C17—42no—
181C22⋊C4×C17—42no—
182C22⋊C4×C17—42no—
183C22⋊C4×C17—42no—
184C42×C17—42no—
185C42×C17C2×D412yes—
186C42×C17#4112yes—
187C22⋊Dic17—42no—
188C22⋊Dic17—42no—
189C22⋊Dic17—81no—
190C22⋊Dic17—81no—
191C4⋊Dic17—21no—
192C4⋊Dic17—21no—
193C4⋊Dic17—21no—
194C4⋊Dic17—21no—
195C4⋊Dic17—21no—
196C4⋊Dic17—21no—
197C4⋊Dic17—21no—
198C4⋊Dic17—21no—
199#63225—42no—
200#63225—42no—
201C4×Dic17—81no—
202C2×#3622—21no—
203C2×#3622—21no—
204C2×#3622—21no—
205C2×#3622—21no—
206C2×#3622—21no—
207C2×#3622—21no—
208C2×#3622—21no—
209C2×#3622—21no—
210C2×#3622—21no—
211C2×#3622—44no—
212C2×#3622—44no—
213C2×#3622C2×D412yes—
214C2×#3622C2×D412yes—
215C2×#3622#4112yes—
216#25343—171nomaximal, Sylow
217C23×C17—21no—
218C23×C17—21no—
219C23×C17—21no—
220C23×C17—21no—
221C23×C17—21no—
222C23×C17—21no—
223C23×C17—21no—
224C23×C17—21no—
225C23×C17—21no—
226C23×C17—21no—
227C23×C17—21no—
228C23×C17—21no—
229C23×C17#13512yes—
230C23×C17#13512yes—
231C23×C17C22×D411yescharacteristic
232Q8×C17—22no—
233Q8×C17—22no—
234Q8×C17—41no—
235Q8×C17—41no—
236D4×C17—28no—
237C2×C4×C17—21no—
238C2×C4×C17—21no—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—22no—
244C2×C4×C17—21no—
245C2×C4×C17—21no—
246C2×C4×C17—21no—
247C2×C4×C17—21no—
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×C17—41no—
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×C17—41no—
265C2×C4×C17—41no—
266C2×C4×C17—41no—
267C2×C4×C17#12112yes—
268C2×C4×C17#12112yes—
269C2×C4×C17C22×D411yescharacteristic
270C2×C4×C17C2×#4111yescharacteristic
271C2×C4×C17C2×#4111yescharacteristic
272C2×C4×C17C2×#4111yescharacteristic
273C2×C4×C17H(5,𝔽2)11yescharacteristic
274C2×C4×C172−1+411yescharacteristic
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×Dic17—41no—
285C2×Dic17—41no—
286C2×Dic17—41no—
287#3622—22no—
288#3622—22no—
289#3622—22no—
290#3622—41no—
291C2×#479—171no—
292C2×C8×D4—171no—
293#2562—171no—
294#2536D1711yescharacteristic
295#2077—171no—
296#2075—171no—
297#2036—171no—
298#2035—171no—
299#1991—171no—
300#1949—171no—
301#1892—171no—
302#1892—171no—
303#1808—171no—
304#1807—171no—
305C4×#103—171no—
306C22×C17—22no—
307C22×C17—22no—
308C22×C17—22no—
309C22×C17—22no—
310C22×C17—22no—
311C22×C17—22no—
312C22×C17—41no—
313C22×C17—41no—
314C22×C17—41no—
315C22×C17—41no—
316C22×C17—41no—
317C22×C17—41no—
318C22×C17—41no—
319C22×C17—41no—
320C22×C17C2×#12111yescharacteristic
321C22×C17#54611yescharacteristic
322C22×C17#54711yescharacteristic
323C22×C17C2×#13511yescharacteristic
324C22×C17C2×#13511yescharacteristic
325C22×C17#57711yescharacteristic
326C22×C17#57811yescharacteristic
327C4×C17—22no—
328C4×C17—22no—
329C4×C17—41no—
330C4×C17—41no—
331C4×C17—42no—
332C4×C17—81no—
333C4×C17—81no—
334C4×C17—81no—
335C4×C17#52911yescharacteristic
336C4×C17#54211yescharacteristic
337C4×C17#54611yescharacteristic
338C4×C17#54811yescharacteristic
339Dic17—81no—
340Dic17—81no—
341Dic17—81no—
342Dic17—81no—
343C23×C8—171no—
344C23×C8—171no—
345C2×#125—171no—
346C2×C22⋊Q8C2×D1711yescharacteristic
347C2×C22⋊Q8C2×D1711yescharacteristic
348C2×C4×Q8C2×D1711yescharacteristic
349C2×C4×D4C2×D1711yescharacteristic
350#479—344no—
351C8×D4—344no—
352C2×#107—171no—
353C2×#107—171no—
354C2×#106—171no—
355C2×#106—171no—
356C2×C4⋊C8—171no—
357C2×#103—171no—
358C2×#103—171no—
359C2×#103—171no—
360C2×#103—171no—
361C2×C22⋊C8—171no—
362C2×C22⋊C8—171no—
363C2×C4×C8—171no—
364#400—341no—
365#398—21no—
366#394—21no—
367#394—341no—
368#388—21no—
369#388—21no—
370#388—341no—
371#388—341no—
372#382C2×D1711yescharacteristic
373#380—171no—
374#380—171no—
375#380—171no—
376#380—171no—
377#380C2×D1711yescharacteristic
378#380C2×D1711yescharacteristic
379C4×C4⋊C4—171no—
380#340—171no—
381#340—171no—
382#340—171no—
383#340—171no—
384#340—171no—
385#340—171no—
386#340—171no—
387#340—171no—
388C2×C17—42no—
389C2×C17—42no—
390C2×C17#242711yescharacteristic
391C2×C17#305011yescharacteristic
392C2×C17#306311yescharacteristic
393C2×C17#330011yescharacteristic
394C2×C17#330111yescharacteristic
395C2×C17#333611yescharacteristic
396C2×C17#333711yescharacteristic
397C22×Q8C22×D1711yescharacteristic
398C22×D4C22⋊D1711yescharacteristic
399C23×C4C22×D1711yescharacteristic
400C23×C4C22×D1711yescharacteristic
401C22×C8—171no—
402C22×C8—171no—
403C22×C8—171no—
404C22×C8—171no—
405C22×C8—342no—
406C22×C8—342no—
407C22×C8—342no—
408C22×C8—342no—
409C22×C8—681no—
410C22×C8—681no—
411#125—342no—
412#125—342no—
413C22⋊Q8—22no—
414C22⋊Q8—22no—
415C22⋊Q8—41no—
416C22⋊Q8—41no—
417C4×Q8—22no—
418C4×Q8—22no—
419C4×D4—22no—
420C4×D4—22no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—171no—
425C2×C4⋊C4—171no—
426C2×C4⋊C4—171no—
427C2×C4⋊C4—171no—
428C2×C4⋊C4—341no—
429C2×C4⋊C4—341no—
430C2×C4⋊C4C22⋊D1711yescharacteristic
431C2×C4⋊C4C22×D1711yescharacteristic
432C2×C4⋊C4C22×D1711yescharacteristic
433C2×C4⋊C4C22×D1711yescharacteristic
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—21no—
438C2×C22⋊C4—341no—
439C2×C22⋊C4—341no—
440C2×C22⋊C4—341no—
441C2×C22⋊C4—341no—
442C2×C22⋊C4C22⋊D1711yescharacteristic
443C2×C22⋊C4C22⋊D1711yescharacteristic
444C2×C42—21no—
445C2×C42—341no—
446C2×C42C22×D1711yescharacteristic
447#107—342no—
448#107—681no—
449#106—342no—
450#106—681no—
451C4⋊C8—174no—
452#103—342no—
453#103—342no—
454#103—342no—
455#103—342no—
456C22⋊C8—342no—
457C22⋊C8—342no—
458C4×C8—342no—
459#95—21no—
460#95—21no—
461#95—21no—
462#95—21no—
463#95—341no—
464#95—341no—
465#95—341no—
466#95—341no—
467#95—341no—
468C17#2534311yescharacteristic, complemented, Sylow
469C24C2×C22⋊D1711yescharacteristic
470C24C2×C22⋊D1711yescharacteristic
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8#6325412yes—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—22no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
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—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—341no—
497C22×C4—341no—
498C22×C4—341no—
499C22×C4—341no—
500C22×C4C2×C22⋊D1711yescharacteristic
501C22×C4C2×C22⋊D1711yescharacteristic
502C22×C4C2×C22⋊D1711yescharacteristic
503C22×C4C2×C22⋊D1711yescharacteristic
504C22×C4C23×D1711yescharacteristic
505C2×C8—172no—
506C2×C8—172no—
507C2×C8—172no—
508C2×C8—341no—
509C2×C8—341no—
510C2×C8—341no—
511C2×C8—341no—
512C2×C8—341no—
513C2×C8—341no—
514C2×C8—341no—
515C2×C8—341no—
516C2×C8—341no—
517C2×C8—684no—
518C2×C8—684no—
519C4⋊C4—22no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—341no—
530C4⋊C4—341no—
531C4⋊C4—341no—
532C4⋊C4—341no—
533C4⋊C4—341no—
534C4⋊C4—341no—
535C4⋊C4—341no—
536C4⋊C4—341no—
537C4⋊C4—682no—
538C4⋊C4—682no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—42no—
545C22⋊C4—682no—
546C22⋊C4—682no—
547C22⋊C4—1361no—
548C22⋊C4—1361no—
549C42—42no—
550C42—1361no—
551C42D4×D1712yes—
552C42#6325412yes—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23#1058763612yes—
566C23#1058763612yes—
567C23C22×C22⋊D1711yescharacteristic
568Q8—22no—
569Q8—22no—
570Q8—41no—
571Q8—41no—
572D4—28no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—41no—
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—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
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#1058762512yes—
616C2×C4#1058762512yes—
617C2×C4C2×D4×D1711yescharacteristic
618C2×C4C2×#6325411yescharacteristic
619C2×C4C2×#6325411yescharacteristic
620C2×C4#1058769111yescharacteristic
621C2×C4#41×D1711yescharacteristic
622C2×C4#1058769711yescharacteristic
623C8—342no—
624C8—342no—
625C8—342no—
626C8—681no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22C2×#1058762511yescharacteristic
642C22#1176345311yescharacteristic
643C22#1176345811yescharacteristic
644C22C2×#1058763611yescharacteristic
645C22C2×#1058763611yescharacteristic
646C22#1176361711yescharacteristic
647C22#1176362411yescharacteristic
648C4—22no—
649C4—22no—
650C4—41no—
651C4—41no—
652C4—42no—
653C4—81no—
654C4—81no—
655C4—81no—
656C4—1361no—
657C4—1361no—
658C4—1361no—
659C4—1361no—
660C4#1176334011yescharacteristic
661C4#1176342611yescharacteristic
662C4#1176345611yescharacteristic
663C4#1176346511yescharacteristic
664C2—42no—
665C2—42no—
666C2#1453236311yescharacteristic
667C2#1453836211yescharacteristic
668C2#1453847111yescharacteristic
669C2#1454067111yescharacteristic
670C2#1454068611yescharacteristic
671C2#1454114611yescharacteristic
672C2#1454116311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2536C2
socleC23×C17C22×D4
prime radicalC22×D4×C17D4
hypercenter#2536D17
hypocenterC17#25343
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2536

Complements

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

NH ≅ G/Ntype
1. C2×#11763116C2semidirect
4. #14530220C2semidirect
5. #14530210C2semidirect
6. #14529984C2semidirect
7. #14529977C2semidirect
8. #14529748C2semidirect
9. #14529550C2semidirect
10. #14529208C2semidirect
11. #14529208C2semidirect
12. #14528811C2semidirect
13. #14528807C2semidirect
14. C4×#10587506C2semidirect
468. C17#25343semidirect

Maximal subgroups

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

subgroupindex
C2×#117631162
C2×D4×#36222
#145339502
#145302202
#145302102
#145299842
#145299772
#145297482
#145295502
#145292082
#145288112
#145288072
C4×#105875062
C17×#25362
#2534317