Finite Group Databasebeta

← #17078199

Subgroups — #17078199

Order 4352.

932 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762965C211yesmaximal, characteristic
2C2×C4×#63231C211yesmaximal, characteristic, complemented
3#14532287C211yesmaximal, characteristic
4#14530332C211yesmaximal, characteristic, complemented
5#14530210C211yesmaximal, characteristic
6#14530106C211yesmaximal, characteristic, complemented
7#14529980C211yesmaximal, characteristic
8#14529463C211yesmaximal, characteristic, complemented
9#14529392C211yesmaximal, characteristic
10#14529128C211yesmaximal, characteristic, complemented
11#14529125C211yesmaximal, characteristic, complemented
12#14529052C211yesmaximal, characteristic, complemented
13#14528799C211yesmaximal, characteristic, complemented
14#14528719C211yesmaximal, characteristic
15Q8×C22⋊C4×C17C211yesmaximal, characteristic
16C22×#63231C2211yescharacteristic
17C22×#63231C2211yescharacteristic
18C2×#10587610C2211yescharacteristic
19C2×C4×Dic34C2211yescharacteristic
20#11762965—24no—
21C4×#63231—24no—
22C2×#10587478C2211yescharacteristic
23C2×#10587479C2211yescharacteristic
24C2×C4⋊#3622C2211yescharacteristic
25C2×#10587506C2211yescharacteristic
26C2×#10587506C2211yescharacteristic
27C2×#10587506C2211yescharacteristic
28C2×#10587506C2211yescharacteristic
29C2×#10587506C2211yescharacteristic
30C2×#10587506C2211yescharacteristic
31C2×#10587481C2211yescharacteristic
32C2×#10587481C2211yescharacteristic
33C2×C22⋊#3622C2211yescharacteristic
34C2×C22⋊#3622C2211yescharacteristic
35C2×C4×#3622C2211yescharacteristic
36#11762750—21no—
37#11762737—21no—
38#11762715C2211yescharacteristic
39#11762707C2211yescharacteristic
40#11762703—21no—
41#11762691—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—22no—
62C22×Q8×C17—22no—
63C22×Q8×C17—21no—
64C22×Q8×C17—22no—
65C22×Q8×C17—22no—
66C22×Q8×C17—41no—
67C22×Q8×C17—41no—
68C22×Q8×C17C2311yescharacteristic
69C22×Q8×C17C2311yescharacteristic
70C23×C4×C17—21no—
71C23×C4×C17C2311yescharacteristic
72C22×Dic34C2311yescharacteristic
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—
90C2×#63231—22no—
91C2×#63231—22no—
92C2×#63231—22no—
93C2×#63231—22no—
94C2×#63231—41no—
95C2×#63231D414yes—
96C2×C4⋊Dic17C2311yescharacteristic
97C2×C4⋊Dic17C2311yescharacteristic
98C2×C4⋊Dic17C2311yescharacteristic
99#10587610—22no—
100#10587610—41no—
101C2×#63225—21no—
102C2×#63225—21no—
103C2×#63225—21no—
104C2×C4×Dic17—21no—
105C22×#3622C2311yescharacteristic
106C22×#3622C2311yescharacteristic
107C22×#3622C2311yescharacteristic
108C22×#3622C2311yescharacteristic
109C4×Dic34—22no—
110C4×Dic34—41no—
111C17×#95—21no—
112C17×#95—21no—
113C17×#95—21no—
114C17×#95—21no—
115C17×#95—21no—
116C17×#95D411yescharacteristic
117C17×#95D411yescharacteristic
118#10587506—22no—
119#10587506—22no—
120#10587506—22no—
121#10587506—22no—
122#10587506—22no—
123#10587506—41no—
124#10587502—21no—
125#10587502—21no—
126#10587502—21no—
127#10587502—21no—
128#10587502—21no—
129#10587502—21no—
130C22⋊#3622—22no—
131C22⋊#3622—41no—
132#10587481—22no—
133#10587481—41no—
134#10587479—41no—
135#10587478—22no—
136C4⋊#3622—41no—
137C4×#3622—22no—
138C24×C17C2×D411yescharacteristic
139C2×Q8×C17—24no—
140C2×Q8×C17—21no—
141C2×Q8×C17—21no—
142C2×Q8×C17—21no—
143C2×Q8×C17—21no—
144C2×Q8×C17—21no—
145C2×Q8×C17—21no—
146C2×Q8×C17—21no—
147C2×Q8×C17—21no—
148C2×Q8×C17—42no—
149C2×Q8×C17—42no—
150C2×Q8×C17—44no—
151C2×Q8×C17—44no—
152C2×Q8×C17—44no—
153C2×Q8×C17—44no—
154C2×Q8×C17C2×D412yes—
155C2×Q8×C17#4112yes—
156C2×Q8×C17#4112yes—
157C2×Q8×C17#4112yes—
158C22×C4×C17—21no—
159C22×C4×C17—24no—
160C22×C4×C17—21no—
161C22×C4×C17—21no—
162C22×C4×C17—21no—
163C22×C4×C17—21no—
164C22×C4×C17—21no—
165C22×C4×C17—21no—
166C22×C4×C17—21no—
167C22×C4×C17—21no—
168C22×C4×C17—21no—
169C22×C4×C17—42no—
170C22×C4×C17—42no—
171C22×C4×C17C2×D411yescharacteristic
172C22×C4×C17C2×D411yescharacteristic
173C22×C4×C17C2×D411yescharacteristic
174C22×C4×C17C2×D411yescharacteristic
175C22×C4×C17C2×D411yescharacteristic
176C22×C4×C17C2411yescharacteristic
177C22×Dic17—21no—
178C22×Dic17—21no—
179C22×Dic17—21no—
180C22×Dic17—21no—
181C2×Dic34—21no—
182C2×Dic34—21no—
183C2×Dic34—21no—
184C2×Dic34—21no—
185C2×Dic34—21no—
186C2×Dic34C2×D412yes—
187C4⋊C4×C17—21no—
188C4⋊C4×C17—21no—
189C4⋊C4×C17—21no—
190C4⋊C4×C17—21no—
191C4⋊C4×C17—42no—
192C4⋊C4×C17—42no—
193C22⋊C4×C17—22no—
194C22⋊C4×C17—22no—
195C22⋊C4×C17—42no—
196C42×C17—21no—
197C42×C17—21no—
198C42×C17—21no—
199C42×C17—21no—
200C42×C17—42no—
201C42×C17—42no—
202C22⋊Dic17—42no—
203C22⋊Dic17—42no—
204#63231—28no—
205#63231—44no—
206C4⋊Dic17—21no—
207C4⋊Dic17—21no—
208C4⋊Dic17—21no—
209C4⋊Dic17—21no—
210C4⋊Dic17—21no—
211C4⋊Dic17—21no—
212#63225—42no—
213#63225—42no—
214#63225—42no—
215C4×Dic17—42no—
216C2×#3622—21no—
217C2×#3622—21no—
218C2×#3622—21no—
219C2×#3622—21no—
220C2×#3622—21no—
221C2×#3622—21no—
222C2×#3622—21no—
223C2×#3622—21no—
224C2×#3622—21no—
225C2×#3622—21no—
226C2×#3622—21no—
227C2×#3622C2×D412yes—
228#24902—171nomaximal, Sylow
229C23×C17—21no—
230C23×C17—21no—
231C23×C17—21no—
232C23×C17—21no—
233C23×C17—21no—
234C23×C17—21no—
235C23×C17#13512yes—
236C23×C17C22×D411yescharacteristic
237Q8×C17—22no—
238Q8×C17—22no—
239Q8×C17—22no—
240Q8×C17—22no—
241Q8×C17—22no—
242Q8×C17—48no—
243Q8×C17—41no—
244Q8×C17—41no—
245Q8×C17#12314yes—
246C2×C4×C17—22no—
247C2×C4×C17—21no—
248C2×C4×C17—22no—
249C2×C4×C17—22no—
250C2×C4×C17—21no—
251C2×C4×C17—22no—
252C2×C4×C17—21no—
253C2×C4×C17—21no—
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—21no—
264C2×C4×C17—48no—
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×C17—41no—
273C2×C4×C17—41no—
274C2×C4×C17—41no—
275C2×C4×C17—41no—
276C2×C4×C17—41no—
277C2×C4×C17#13712yes—
278C2×C4×C17C22×D411yescharacteristic
279C2×C4×C17C2×#4111yescharacteristic
280C2×C4×C17C2×#4111yescharacteristic
281C2×C4×C17C2×#4111yescharacteristic
282C2×C4×C17H(5,𝔽2)11yescharacteristic
283C2×C4×C17H(5,𝔽2)11yescharacteristic
284C2×Dic17—22no—
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—41no—
295C2×Dic17—41no—
296Dic34—22no—
297Dic34—22no—
298Dic34—41no—
299Dic34—41no—
300#3622—22no—
301#3622—41no—
302#3622—41no—
303#3622—41no—
304C2×#451—171no—
305C2×C4×Q16—171no—
306#2421—171no—
307Q8×C22⋊C4D1711yescharacteristic
308#2097—171no—
309#2075—171no—
310#2056—171no—
311#2035—171no—
312#1933—171no—
313#1923—171no—
314#1878—171no—
315#1878—171no—
316#1865—171no—
317#1805—171no—
318#1787—171no—
319C22×C17—22no—
320C22×C17—22no—
321C22×C17—22no—
322C22×C17—41no—
323C22×C17—41no—
324C22×C17—41no—
325C22×C17—41no—
326C22×C17#52511yescharacteristic
327C22×C17D4211yescharacteristic
328C22×C17#54611yescharacteristic
329C22×C17C2×#13511yescharacteristic
330C22×C17C2×#13711yescharacteristic
331C22×C17#57811yescharacteristic
332C22×C17#57811yescharacteristic
333C4×C17—24no—
334C4×C17—22no—
335C4×C17—41no—
336C4×C17—41no—
337C4×C17—42no—
338C4×C17—41no—
339C4×C17—41no—
340C4×C17—81no—
341C4×C17—81no—
342C4×C17C2×#12311yescharacteristic
343C4×C17#53811yescharacteristic
344C4×C17#54611yescharacteristic
345C4×C17#55111yescharacteristic
346Dic17—42no—
347Dic17—81no—
348Dic17—81no—
349Dic17—81no—
350C23×Q8C2×D1711yescharacteristic
351C22×Q16—171no—
352C22×Q16—171no—
353C2×C22⋊Q8—171no—
354C2×C4×Q8—171no—
355C2×C4×Q8C2×D1711yescharacteristic
356C2×C4×Q8C2×D1711yescharacteristic
357#451—344no—
358C4×Q16—344no—
359C2×#107—171no—
360C2×#106—171no—
361C2×C4⋊C8—171no—
362C2×#103—171no—
363C2×#103—171no—
364C2×#103—171no—
365C2×#103—171no—
366C2×#103—171no—
367C2×#103—171no—
368C2×#103—171no—
369C2×#103—171no—
370C2×C22⋊C8—171no—
371C2×C22⋊C8—171no—
372C2×C4×C8—171no—
373#400—341no—
374#397—341no—
375#391—21no—
376#391—21no—
377#391—171no—
378#391C2×D1711yescharacteristic
379#391C2×D1711yescharacteristic
380#389—171no—
381#388—341no—
382#387—341no—
383#382—171no—
384#380—21no—
385#380—171no—
386#380C2×D1711yescharacteristic
387C4×C22⋊C4—21no—
388C4×C22⋊C4C2×D1711yescharacteristic
389#340—171no—
390#340—171no—
391#340—171no—
392#340—171no—
393#336—171no—
394#336—171no—
395C2×C17—42no—
396C2×C17#241811yescharacteristic
397C2×C17#310811yescharacteristic
398C2×C17#312711yescharacteristic
399C2×C17#327011yescharacteristic
400C2×C17#328511yescharacteristic
401C2×C17#330011yescharacteristic
402C2×C17#332811yescharacteristic
403C22×Q8—22no—
404C22×Q8—22no—
405C22×Q8—21no—
406C22×Q8—22no—
407C22×Q8—22no—
408C22×Q8—41no—
409C22×Q8—41no—
410C22×Q8—171no—
411C22×Q8C22×D1711yescharacteristic
412C22×Q8C22×D1711yescharacteristic
413C23×C4—21no—
414C23×C4C22×D1711yescharacteristic
415C2×Q16—174no—
416C2×Q16—342no—
417C2×Q16—342no—
418C2×Q16—342no—
419C2×Q16—342no—
420C2×Q16—681no—
421C22×C8—171no—
422C22×C8—171no—
423C22×C8—171no—
424C22×C8—171no—
425C22⋊Q8—342no—
426C22⋊Q8—681no—
427C4×Q8—22no—
428C4×Q8—22no—
429C4×Q8—41no—
430C4×Q8—41no—
431C4×Q8—342no—
432C4×Q8—681no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—171no—
436C2×C4⋊C4—171no—
437C2×C4⋊C4—171no—
438C2×C4⋊C4—341no—
439C2×C4⋊C4—341no—
440C2×C4⋊C4—341no—
441C2×C4⋊C4C22×D1711yescharacteristic
442C2×C4⋊C4C22×D1711yescharacteristic
443C2×C22⋊C4—21no—
444C2×C22⋊C4—341no—
445C2×C22⋊C4—341no—
446C2×C22⋊C4C22⋊D1711yescharacteristic
447C2×C22⋊C4C22⋊D1711yescharacteristic
448C2×C42—21no—
449C2×C42—21no—
450C2×C42—341no—
451C2×C42C22×D1711yescharacteristic
452C2×C42C22×D1711yescharacteristic
453#107—342no—
454#106—681no—
455C4⋊C8—681no—
456#103—342no—
457#103—342no—
458#103—342no—
459#103—342no—
460#103—342no—
461#103—342no—
462#103—681no—
463#103—681no—
464C22⋊C8—342no—
465C22⋊C8—681no—
466C4×C8—342no—
467#95—21no—
468#95—21no—
469#95—21no—
470#95—21no—
471#95—21no—
472#95—341no—
473#95—341no—
474#95—341no—
475#95—341no—
476#95—341no—
477#95—341no—
478#95C22⋊D1711yescharacteristic
479#95C22⋊D1711yescharacteristic
480C17#2490211yescharacteristic, complemented, Sylow
481C24C2×C22⋊D1711yescharacteristic
482C2×Q8—24no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—42no—
492C2×Q8—42no—
493C2×Q8—44no—
494C2×Q8—44no—
495C2×Q8—44no—
496C2×Q8—44no—
497C2×Q8—172no—
498C2×Q8—341no—
499C2×Q8—341no—
500C2×Q8—341no—
501C2×Q8—341no—
502C2×Q8—341no—
503C2×Q8#6325212yes—
504C2×Q8#6325212yes—
505C2×Q8D4×D1712yes—
506C2×Q8#6325412yes—
507C22×C4—21no—
508C22×C4—24no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—42no—
519C22×C4—42no—
520C22×C4—341no—
521C22×C4—341no—
522C22×C4—341no—
523C22×C4—341no—
524C22×C4C2×C22⋊D1711yescharacteristic
525C22×C4C2×C22⋊D1711yescharacteristic
526C22×C4C2×C22⋊D1711yescharacteristic
527C22×C4C2×C22⋊D1711yescharacteristic
528C22×C4C2×C22⋊D1711yescharacteristic
529C22×C4C23×D1711yescharacteristic
530Q16—348no—
531Q16—684no—
532C2×C8—172no—
533C2×C8—341no—
534C2×C8—341no—
535C2×C8—341no—
536C2×C8—341no—
537C2×C8—341no—
538C2×C8—341no—
539C2×C8—341no—
540C2×C8—341no—
541C2×C8—341no—
542C2×C8—341no—
543C2×C8—341no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—341no—
551C4⋊C4—341no—
552C4⋊C4—341no—
553C4⋊C4—341no—
554C4⋊C4—341no—
555C4⋊C4—341no—
556C4⋊C4—682no—
557C4⋊C4—682no—
558C4⋊C4—682no—
559C22⋊C4—22no—
560C22⋊C4—22no—
561C22⋊C4—42no—
562C22⋊C4—682no—
563C22⋊C4—682no—
564C42—21no—
565C42—21no—
566C42—21no—
567C42—21no—
568C42—42no—
569C42—42no—
570C42—682no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23#1058763612yes—
578C23C22×C22⋊D1711yescharacteristic
579Q8—22no—
580Q8—22no—
581Q8—22no—
582Q8—22no—
583Q8—22no—
584Q8—48no—
585Q8—41no—
586Q8—41no—
587Q8—342no—
588Q8—342no—
589Q8—681no—
590Q8—681no—
591Q8#1058756914yes—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—48no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—342no—
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—681no—
634C2×C4—681no—
635C2×C4#1058762812yes—
636C2×C4C2×#6325211yescharacteristic
637C2×C4C2×#6325211yescharacteristic
638C2×C4C2×D4×D1711yescharacteristic
639C2×C4C2×#6325411yescharacteristic
640C2×C4#1058769111yescharacteristic
641C2×C4#1058769111yescharacteristic
642C8—342no—
643C8—681no—
644C8—681no—
645C8—681no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22#1176332011yescharacteristic
654C22D4×C22⋊D1711yescharacteristic
655C22#1176345311yescharacteristic
656C22C2×#1058763611yescharacteristic
657C22C2×#1058762811yescharacteristic
658C22#1176362411yescharacteristic
659C22#1176362411yescharacteristic
660C4—22no—
661C4—24no—
662C4—41no—
663C4—41no—
664C4—41no—
665C4—42no—
666C4—41no—
667C4—81no—
668C4—81no—
669C4—682no—
670C4—1361no—
671C4—1361no—
672C4—1361no—
673C4C2×#1058756911yescharacteristic
674C4#1176339711yescharacteristic
675C4#1176344911yescharacteristic
676C4#1176348511yescharacteristic
677C2—42no—
678C2#1453224611yescharacteristic
679C2#1453886411yescharacteristic
680C2#1453908411yescharacteristic
681C2#1454035511yescharacteristic
682C2#1454052111yescharacteristic
683C2#1454067111yescharacteristic
684C2#1454106711yescharacteristic
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#24902
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
2. C2×C4×#63231C2semidirect
4. #14530332C2semidirect
6. #14530106C2semidirect
8. #14529463C2semidirect
10. #14529128C2semidirect
11. #14529125C2semidirect
12. #14529052C2semidirect
13. #14528799C2semidirect
480. C17#24902semidirect

Maximal subgroups

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

subgroupindex
C2×#117629652
C2×C4×#632312
#145322872
#145303322
#145302102
#145301062
#145299802
#145294632
#145293922
#145291282
#145291252
#145290522
#145287992
#145287192
Q8×C22⋊C4×C172
#2490217