Finite Group Databasebeta

← #17091816

Subgroups — #17091816

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×#11763104C211yesmaximal, characteristic, complemented
2C2×#11762818C211yesmaximal, characteristic
3#14532475C211yesmaximal, characteristic
4#14530332C211yesmaximal, characteristic, complemented
5#14530179C211yesmaximal, characteristic, complemented
6#14530165C211yesmaximal, characteristic, complemented
7#14529925C211yesmaximal, characteristic, complemented
8#14529052C211yesmaximal, characteristic, complemented
9#14529052C211yesmaximal, characteristic, complemented
10#14528981C211yesmaximal, characteristic
11#14528969C211yesmaximal, characteristic
12#14528815C211yesmaximal, characteristic
13#14528799C211yesmaximal, characteristic, complemented
14#14528719C211yesmaximal, characteristic
15Q8×C22⋊C4×C17C211yesmaximal, characteristic
16C22×#63223C2211yescharacteristic
17C2×#10587555C2211yescharacteristic
18C2×#10587612C2211yescharacteristic
19#11763104—24no—
20C2×#10587478C2211yescharacteristic
21C2×#10587478C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×#10587479C2211yescharacteristic
24C2×C4⋊#3622C2211yescharacteristic
25C2×C4⋊#3622C2211yescharacteristic
26#11762818—24no—
27C2×#10587506C2211yescharacteristic
28C2×#10587506C2211yescharacteristic
29C2×#10587506C2211yescharacteristic
30C2×#10587506C2211yescharacteristic
31C2×#10587506C2211yescharacteristic
32C2×#10587506C2211yescharacteristic
33C2×#10587506C2211yescharacteristic
34C2×#10587506C2211yescharacteristic
35C2×C22⋊#3622C2211yescharacteristic
36#11762732—21no—
37#11762721—21no—
38#11762713—21no—
39#11762707C2211yescharacteristic
40#11762707C2211yescharacteristic
41#11762696—21no—
42#11762675C2211yescharacteristic
43#11762669C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762564C2211yescharacteristic
49#11762564C2211yescharacteristic
50C23×Q8×C17C2211yescharacteristic
51C2×C4×Q8×C17C2211yescharacteristic
52C2×C4×Q8×C17C2211yescharacteristic
53C17×#391—21no—
54C17×#391—21no—
55C17×#391C2211yescharacteristic
56C17×#391C2211yescharacteristic
57C17×#380—21no—
58C17×#380C2211yescharacteristic
59C4×C22⋊C4×C17—21no—
60C4×C22⋊C4×C17C2211yescharacteristic
61C22×Q8×C17—21no—
62C22×Q8×C17—22no—
63C22×Q8×C17—22no—
64C22×Q8×C17—22no—
65C22×Q8×C17—41no—
66C22×Q8×C17—41no—
67C22×Q8×C17Q814yescomplemented
68C22×Q8×C17C2311yescharacteristic
69C22×Q8×C17C2311yescharacteristic
70C23×C4×C17—21no—
71C23×C4×C17C2311yescharacteristic
72C22×Dic34—21no—
73C4×Q8×C17—22no—
74C4×Q8×C17—22no—
75C4×Q8×C17—41no—
76C4×Q8×C17—41no—
77C2×C4⋊C4×C17—21no—
78C2×C4⋊C4×C17—21no—
79C2×C4⋊C4×C17C2311yescharacteristic
80C2×C4⋊C4×C17C2311yescharacteristic
81C2×C22⋊C4×C17—21no—
82C2×C22⋊C4×C17D411yescharacteristic
83C2×C22⋊C4×C17D411yescharacteristic
84C2×C42×C17—21no—
85C2×C42×C17—21no—
86C2×C42×C17C2311yescharacteristic
87C2×C42×C17C2311yescharacteristic
88C2×C22⋊Dic17—21no—
89C2×C22⋊Dic17—21no—
90#10587612—22no—
91#10587612—41no—
92C2×C4⋊Dic17—21no—
93C2×C4⋊Dic17C2311yescharacteristic
94C2×C4⋊Dic17C2311yescharacteristic
95C2×C4⋊Dic17C2311yescharacteristic
96C2×C4⋊Dic17C2311yescharacteristic
97C2×#63225—21no—
98C2×#63225—21no—
99C2×C4×Dic17—21no—
100C2×#63223—22no—
101C2×#63223—22no—
102C2×#63223—41no—
103C22×#3622C2311yescharacteristic
104C22×#3622C2311yescharacteristic
105C22×#3622C2311yescharacteristic
106C22×#3622C2311yescharacteristic
107#10587555—22no—
108#10587555—41no—
109C17×#95—21no—
110C17×#95—21no—
111C17×#95—21no—
112C17×#95—21no—
113C17×#95—21no—
114C17×#95D411yescharacteristic
115C17×#95D411yescharacteristic
116#10587506—22no—
117#10587506—22no—
118#10587506—22no—
119#10587506—22no—
120#10587506—22no—
121#10587506—41no—
122#10587506—41no—
123#10587506D414yes—
124#10587502—21no—
125#10587502—21no—
126#10587502—21no—
127#10587502—21no—
128#10587502—21no—
129C22⋊#3622—22no—
130#10587479—41no—
131#10587479—41no—
132#10587478—22no—
133#10587478—22no—
134C4⋊#3622—22no—
135C4⋊#3622—41no—
136C24×C17C2×D411yescharacteristic
137C2×Q8×C17—24no—
138C2×Q8×C17—21no—
139C2×Q8×C17—21no—
140C2×Q8×C17—21no—
141C2×Q8×C17—21no—
142C2×Q8×C17—28no—
143C2×Q8×C17—21no—
144C2×Q8×C17—21no—
145C2×Q8×C17—21no—
146C2×Q8×C17—21no—
147C2×Q8×C17—42no—
148C2×Q8×C17—42no—
149C2×Q8×C17—44no—
150C2×Q8×C17—44no—
151C2×Q8×C17—44no—
152C2×Q8×C17C2×D412yes—
153C2×Q8×C17C2×Q812yes—
154C2×Q8×C17C2×Q812yes—
155C2×Q8×C17#4112yes—
156C22×C4×C17—21no—
157C22×C4×C17—21no—
158C22×C4×C17—21no—
159C22×C4×C17—21no—
160C22×C4×C17—21no—
161C22×C4×C17—24no—
162C22×C4×C17—21no—
163C22×C4×C17—21no—
164C22×C4×C17—21no—
165C22×C4×C17—42no—
166C22×C4×C17—42no—
167C22×C4×C17C2×D411yescharacteristic
168C22×C4×C17C2×D411yescharacteristic
169C22×C4×C17C2×D411yescharacteristic
170C22×C4×C17C2×D411yescharacteristic
171C22×C4×C17C2×D411yescharacteristic
172C22×C4×C17C2×Q812yes—
173C22×C4×C17#4112yes—
174C22×C4×C17C2411yescharacteristic
175C22×Dic17—21no—
176C22×Dic17—21no—
177C22×Dic17—21no—
178C22×Dic17—21no—
179C2×Dic34—42no—
180C2×Dic34—42no—
181C2×Dic34—81no—
182C4⋊C4×C17—21no—
183C4⋊C4×C17—21no—
184C4⋊C4×C17—21no—
185C4⋊C4×C17—21no—
186C4⋊C4×C17—42no—
187C4⋊C4×C17—42no—
188C22⋊C4×C17—22no—
189C22⋊C4×C17—41no—
190C22⋊C4×C17—42no—
191C42×C17—21no—
192C42×C17—21no—
193C42×C17—21no—
194C42×C17—21no—
195C42×C17—42no—
196C42×C17—42no—
197C22⋊Dic17—42no—
198C22⋊Dic17—81no—
199C4⋊Dic17—21no—
200C4⋊Dic17—21no—
201C4⋊Dic17—21no—
202C4⋊Dic17—21no—
203C4⋊Dic17—21no—
204C4⋊Dic17—21no—
205C4⋊Dic17—21no—
206C4⋊Dic17—42no—
207C4⋊Dic17C2×D412yes—
208#63225—42no—
209#63225—81no—
210C4×Dic17—42no—
211#63223—44no—
212C2×#3622—21no—
213C2×#3622—21no—
214C2×#3622—21no—
215C2×#3622—21no—
216C2×#3622—21no—
217C2×#3622—21no—
218C2×#3622—21no—
219C2×#3622—21no—
220C2×#3622—21no—
221C2×#3622—21no—
222C2×#3622—21no—
223C2×#3622C2×D412yes—
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
233Q8×C17—22no—
234Q8×C17—22no—
235Q8×C17—22no—
236Q8×C17—22no—
237Q8×C17—22no—
238Q8×C17—48no—
239Q8×C17—41no—
240Q8×C17—41no—
241Q8×C17C22⋊Q814yes—
242C2×C4×C17—22no—
243C2×C4×C17—21no—
244C2×C4×C17—22no—
245C2×C4×C17—21no—
246C2×C4×C17—22no—
247C2×C4×C17—21no—
248C2×C4×C17—21no—
249C2×C4×C17—22no—
250C2×C4×C17—22no—
251C2×C4×C17—22no—
252C2×C4×C17—22no—
253C2×C4×C17—22no—
254C2×C4×C17—21no—
255C2×C4×C17—21no—
256C2×C4×C17—21no—
257C2×C4×C17—21no—
258C2×C4×C17—21no—
259C2×C4×C17—21no—
260C2×C4×C17—21no—
261C2×C4×C17—21no—
262C2×C4×C17—21no—
263C2×C4×C17—48no—
264C2×C4×C17—41no—
265C2×C4×C17—41no—
266C2×C4×C17—41no—
267C2×C4×C17—41no—
268C2×C4×C17—41no—
269C2×C4×C17—41no—
270C2×C4×C17—41no—
271C2×C4×C17—41no—
272C2×C4×C17C22⋊Q812yes—
273C2×C4×C17#13712yes—
274C2×C4×C17C22×D411yescharacteristic
275C2×C4×C17C22×Q811yescharacteristic
276C2×C4×C17C2×#4111yescharacteristic
277C2×C4×C17C2×#4111yescharacteristic
278C2×C4×C17H(5,𝔽2)11yescharacteristic
279C2×C4×C17H(5,𝔽2)11yescharacteristic
280C2×Dic17—22no—
281C2×Dic17—41no—
282C2×Dic17—41no—
283C2×Dic17—41no—
284C2×Dic17—41no—
285C2×Dic17—41no—
286C2×Dic17—41no—
287C2×Dic17—41no—
288C2×Dic17—41no—
289C2×Dic17—41no—
290C2×Dic17—41no—
291C2×Dic17—41no—
292Dic34—84no—
293#3622—22no—
294#3622—41no—
295#3622—41no—
296#3622—41no—
297C2×#477—171no—
298C2×#419—171no—
299#2439—171no—
300Q8×C22⋊C4D1711yescharacteristic
301#2097—171no—
302#2068—171no—
303#2066—171no—
304#2024—171no—
305#1865—171no—
306#1865—171no—
307#1854—171no—
308#1851—171no—
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—24no—
328C4×C17—22no—
329C4×C17—41no—
330C4×C17—42no—
331C4×C17—41no—
332C4×C17—41no—
333C4×C17—81no—
334C4×C17—81no—
335C4×C17C2×C22⋊Q811yescharacteristic
336C4×C17#54311yescharacteristic
337C4×C17#54611yescharacteristic
338C4×C17#55111yescharacteristic
339Dic17—42no—
340Dic17—81no—
341Dic17—81no—
342Dic17—81no—
343C23×Q8C2×D1711yescharacteristic
344C22×M16—171no—
345C2×C4⋊Q8—171no—
346C2×C4×Q8C2×D1711yescharacteristic
347C2×C4×Q8C2×D1711yescharacteristic
348C2×#117—171no—
349#477—344no—
350C2×#107—171no—
351C2×#107—171no—
352C2×#106—171no—
353C2×#106—171no—
354C2×C4⋊C8—171no—
355C2×C4⋊C8—171no—
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⋊C8—171no—
366#396—341no—
367#393—341no—
368#391—21no—
369#391—21no—
370#391—341no—
371#391C2×D1711yescharacteristic
372#391C2×D1711yescharacteristic
373#389—171no—
374#389—171no—
375#387—341no—
376#382—171no—
377#380—21no—
378#380—171no—
379#380C2×D1711yescharacteristic
380C4×C22⋊C4—21no—
381C4×C22⋊C4C2×D1711yescharacteristic
382#340—171no—
383#340—171no—
384#340—171no—
385#340—171no—
386#336—171no—
387#336—171no—
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—22no—
397C22×Q8—21no—
398C22×Q8—22no—
399C22×Q8—22no—
400C22×Q8—41no—
401C22×Q8—41no—
402C22×Q8—341no—
403C22×Q8Dic3414yescomplemented
404C22×Q8C22×D1711yescharacteristic
405C22×Q8C22×D1711yescharacteristic
406C23×C4—21no—
407C23×C4C22×D1711yescharacteristic
408C2×M16—342no—
409C2×M16—342no—
410C2×M16—681no—
411C22×C8—171no—
412C22×C8—171no—
413C22×C8—171no—
414C22×C8—171no—
415C4⋊Q8—342no—
416C4⋊Q8—681no—
417C4×Q8—22no—
418C4×Q8—22no—
419C4×Q8—41no—
420C4×Q8—41no—
421#117—342no—
422#117—681no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—21no—
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⋊C4C22×D1711yescharacteristic
433C2×C4⋊C4C22×D1711yescharacteristic
434C2×C22⋊C4—21no—
435C2×C22⋊C4—341no—
436C2×C22⋊C4—341no—
437C2×C22⋊C4C22⋊D1711yescharacteristic
438C2×C22⋊C4C22⋊D1711yescharacteristic
439C2×C42—21no—
440C2×C42—21no—
441C2×C42—341no—
442C2×C42C22×D1711yescharacteristic
443C2×C42C22×D1711yescharacteristic
444#107—342no—
445#107—342no—
446#106—681no—
447#106—681no—
448C4⋊C8—342no—
449C4⋊C8—681no—
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—342no—
459#95—21no—
460#95—21no—
461#95—21no—
462#95—21no—
463#95—21no—
464#95—341no—
465#95—341no—
466#95—341no—
467#95—341no—
468#95—341no—
469#95C22⋊D1711yescharacteristic
470#95C22⋊D1711yescharacteristic
471C17#2597911yescharacteristic, complemented, Sylow
472C24C2×C22⋊D1711yescharacteristic
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—24no—
478C2×Q8—28no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—42no—
484C2×Q8—42no—
485C2×Q8—44no—
486C2×Q8—44no—
487C2×Q8—44no—
488C2×Q8—682no—
489C2×Q8—682no—
490C2×Q8—1361no—
491C2×Q8C2×Dic3412yes—
492C2×Q8C2×Dic3412yes—
493C2×Q8D4×D1712yes—
494C2×Q8#6325412yes—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—24no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—42no—
505C22×C4—42no—
506C22×C4—341no—
507C22×C4—341no—
508C22×C4—341no—
509C22×C4—341no—
510C22×C4C2×Dic3412yes—
511C22×C4#6325212yes—
512C22×C4C2×C22⋊D1711yescharacteristic
513C22×C4C2×C22⋊D1711yescharacteristic
514C22×C4C2×C22⋊D1711yescharacteristic
515C22×C4C2×C22⋊D1711yescharacteristic
516C22×C4C2×C22⋊D1711yescharacteristic
517C22×C4C23×D1711yescharacteristic
518M16—684no—
519C2×C8—172no—
520C2×C8—341no—
521C2×C8—341no—
522C2×C8—341no—
523C2×C8—341no—
524C2×C8—341no—
525C2×C8—341no—
526C2×C8—341no—
527C2×C8—341no—
528C2×C8—341no—
529C2×C8—341no—
530C2×C8—341no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—172no—
538C4⋊C4—341no—
539C4⋊C4—341no—
540C4⋊C4—341no—
541C4⋊C4—341no—
542C4⋊C4—341no—
543C4⋊C4—341no—
544C4⋊C4—341no—
545C4⋊C4—682no—
546C4⋊C4—682no—
547C4⋊C4—1361no—
548C22⋊C4—22no—
549C22⋊C4—41no—
550C22⋊C4—42no—
551C22⋊C4—682no—
552C22⋊C4—1361no—
553C42—21no—
554C42—21no—
555C42—21no—
556C42—21no—
557C42—42no—
558C42—42no—
559C42—682no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23#1058761012yes—
567C23C22×C22⋊D1711yescharacteristic
568Q8—22no—
569Q8—22no—
570Q8—22no—
571Q8—22no—
572Q8—22no—
573Q8—48no—
574Q8—41no—
575Q8—41no—
576Q8—1364no—
577Q8#1058756414yes—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—48no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
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×C4—681no—
620C2×C4#1058761012yes—
621C2×C4#1058762812yes—
622C2×C4C22×Dic3411yescharacteristic
623C2×C4C2×#6325211yescharacteristic
624C2×C4C2×D4×D1711yescharacteristic
625C2×C4C2×#6325411yescharacteristic
626C2×C4#1058769111yescharacteristic
627C2×C4#1058769111yescharacteristic
628C8—342no—
629C8—681no—
630C8—681no—
631C8—681no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22C2×#1058761011yescharacteristic
640C22D4×C22⋊D1711yescharacteristic
641C22#1176345311yescharacteristic
642C22C2×#1058762811yescharacteristic
643C22#1176360711yescharacteristic
644C22#1176361711yescharacteristic
645C22#1176362411yescharacteristic
646C4—22no—
647C4—22no—
648C4—24no—
649C4—41no—
650C4—41no—
651C4—42no—
652C4—41no—
653C4—81no—
654C4—81no—
655C4—682no—
656C4—1361no—
657C4—1361no—
658C4—1361no—
659C4C2×#1058756411yescharacteristic
660C4#1176343511yescharacteristic
661C4#1176344911yescharacteristic
662C4#1176348511yescharacteristic
663C2—42no—
664C2#1453244511yescharacteristic
665C2#1453858911yescharacteristic
666C2#1453862611yescharacteristic
667C2#1454035511yescharacteristic
668C2#1454041611yescharacteristic
669C2#1454085011yescharacteristic
670C2#1454107811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingQ8×C22⋊C4×C17C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenterQ8×C22⋊C4D17
hypocenterC17#25979
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ Q8×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#11763104C2semidirect
4. #14530332C2semidirect
5. #14530179C2semidirect
6. #14530165C2semidirect
7. #14529925C2semidirect
8. #14529052C2semidirect
9. #14529052C2semidirect
13. #14528799C2semidirect
67. C22×Q8×C17Q8semidirect
403. C22×Q8Dic34semidirect
471. C17#25979semidirect

Maximal subgroups

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

subgroupindex
C2×#117631042
C2×#117628182
#145324752
#145303322
#145301792
#145301652
#145299252
#145290522
#145289812
#145289692
#145288152
#145287992
#145287192
Q8×C22⋊C4×C172
#2597917