Finite Group Databasebeta

← #17097365

Subgroups — #17097365

Order 4352.

884 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763109C211yesmaximal, characteristic
2C2×C4×#63229C211yesmaximal, characteristic
3#14532645C211yesmaximal, characteristic, complemented
4#14530323C211yesmaximal, characteristic, complemented
5#14530261C211yesmaximal, characteristic, complemented
6#14530176C211yesmaximal, characteristic, complemented
7#14530122C211yesmaximal, characteristic, complemented
8#14529514C211yesmaximal, characteristic, complemented
9#14529343C211yesmaximal, characteristic, complemented
10#14529118C211yesmaximal, characteristic
11#14529118C211yesmaximal, characteristic
12#14529051C211yesmaximal, characteristic, complemented
13#14529031C211yesmaximal, characteristic
14#14529031C211yesmaximal, characteristic
15D4×C4⋊C4×C17C211yesmaximal, characteristic
16C22×#63229C2211yescharacteristic
17C2×#10587556C2211yescharacteristic
18C2×C4×Dic34C2211yescharacteristic
19#11763109—24no—
20C4×#63229—24no—
21C2×#10587478C2211yescharacteristic
22C2×#10587478C2211yescharacteristic
23C2×#10587478C2211yescharacteristic
24C2×#10587479C2211yescharacteristic
25C2×#10587479C2211yescharacteristic
26C2×C4⋊#3622C2211yescharacteristic
27C2×#10587481C2211yescharacteristic
28C2×#10587481C2211yescharacteristic
29C2×#10587481C2211yescharacteristic
30C2×#10587503C2211yescharacteristic
31C2×#10587503C2211yescharacteristic
32C2×#10587503C2211yescharacteristic
33C2×C4×#3622C2211yescharacteristic
34#11762750—21no—
35#11762740—21no—
36#11762715C2211yescharacteristic
37#11762715C2211yescharacteristic
38#11762707C2211yescharacteristic
39#11762707C2211yescharacteristic
40#11762704—21no—
41#11762691—21no—
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762564C2211yescharacteristic
47#11762564C2211yescharacteristic
48#11762564C2211yescharacteristic
49#11762564C2211yescharacteristic
50C2×C4×D4×C17C2211yescharacteristic
51C2×C4×D4×C17C2211yescharacteristic
52C2×C4×D4×C17C2211yescharacteristic
53C22×C4⋊C4×C17—21no—
54C17×#389C2211yescharacteristic
55C17×#389C2211yescharacteristic
56C17×#385—21no—
57C17×#385—21no—
58C17×#384C2211yescharacteristic
59C17×#380—21no—
60C4×C4⋊C4×C17C2211yescharacteristic
61C22×D4×C17C2311yescharacteristic
62C23×C4×C17—21no—
63C23×C4×C17—21no—
64C23×C4×C17—21no—
65C22×Dic34C2311yescharacteristic
66C4×D4×C17—22no—
67C4×D4×C17—22no—
68C4×D4×C17—22no—
69C4×D4×C17—22no—
70C4×D4×C17—41no—
71C4×D4×C17—41no—
72C2×C4⋊C4×C17—21no—
73C2×C4⋊C4×C17—21no—
74C2×C4⋊C4×C17—21no—
75C2×C4⋊C4×C17—21no—
76C2×C4⋊C4×C17—44no—
77C2×C4⋊C4×C17D411yescharacteristic
78C2×C4⋊C4×C17D411yescharacteristic
79C2×C4⋊C4×C17C2311yescharacteristic
80C2×C4⋊C4×C17C2311yescharacteristic
81C2×C4⋊C4×C17C2311yescharacteristic
82C2×C22⋊C4×C17—21no—
83C2×C22⋊C4×C17—21no—
84C2×C22⋊C4×C17—21no—
85C2×C42×C17C2311yescharacteristic
86C2×C42×C17C2311yescharacteristic
87C2×C42×C17C2311yescharacteristic
88C2×#63229—22no—
89C2×#63229—22no—
90C2×#63229Q814yes—
91C2×C4⋊Dic17C2311yescharacteristic
92C2×C4⋊Dic17C2311yescharacteristic
93C2×C4⋊Dic17C2311yescharacteristic
94C2×#63225—21no—
95C2×#63225—21no—
96C2×#63225—21no—
97C2×#63225—21no—
98C2×#63225—21no—
99C2×C4×Dic17—21no—
100C22×#3622C2311yescharacteristic
101C22×#3622C2311yescharacteristic
102C22×#3622C2311yescharacteristic
103C22×#3622C2311yescharacteristic
104#10587556—22no—
105#10587556—41no—
106C4×Dic34—22no—
107C4×Dic34—41no—
108C17×#95—21no—
109C17×#95—21no—
110C17×#95D411yescharacteristic
111C17×#95D411yescharacteristic
112#10587503—22no—
113#10587503—22no—
114#10587503D414yes—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587502—21no—
120#10587502—21no—
121#10587481—22no—
122#10587481—22no—
123#10587481—41no—
124#10587479—22no—
125#10587479—22no—
126#10587478—22no—
127#10587478—22no—
128#10587478—41no—
129C4⋊#3622—22no—
130C4×#3622—22no—
131C24×C17—21no—
132C2×D4×C17—21no—
133C2×D4×C17—21no—
134C2×D4×C17—21no—
135C2×D4×C17C2×D412yes—
136C2×D4×C17C2×Q812yes—
137C2×D4×C17#4112yes—
138C22×C4×C17—24no—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17—21no—
142C22×C4×C17—42no—
143C22×C4×C17—42no—
144C22×C4×C17—42no—
145C22×C4×C17—42no—
146C22×C4×C17—42no—
147C22×C4×C17—42no—
148C22×C4×C17—81no—
149C22×C4×C17—81no—
150C22×C4×C17C2×D411yescharacteristic
151C22×C4×C17C2×D411yescharacteristic
152C22×C4×C17C2×D411yescharacteristic
153C22×C4×C17C2×D411yescharacteristic
154C22×C4×C17C2×D411yescharacteristic
155C22×C4×C17C2×D411yescharacteristic
156C22×C4×C17C2411yescharacteristic
157C22×Dic17—21no—
158C22×Dic17—21no—
159C22×Dic17—21no—
160C22×Dic17—21no—
161C2×Dic34—21no—
162C2×Dic34—21no—
163C2×Dic34—21no—
164C2×Dic34—21no—
165C2×Dic34—21no—
166C2×Dic34C2×Q812yes—
167C4⋊C4×C17—22no—
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—42no—
175C4⋊C4×C17—42no—
176C4⋊C4×C17—42no—
177C4⋊C4×C17—81no—
178C4⋊C4×C17#4112yes—
179C22⋊C4×C17—24no—
180C22⋊C4×C17—42no—
181C22⋊C4×C17—42no—
182C42×C17—21no—
183C42×C17—21no—
184C42×C17—21no—
185C42×C17—21no—
186C42×C17—21no—
187C42×C17—21no—
188#63229—28no—
189C4⋊Dic17—21no—
190C4⋊Dic17—21no—
191C4⋊Dic17—21no—
192C4⋊Dic17—21no—
193C4⋊Dic17—21no—
194C4⋊Dic17C2×D412yes—
195#63225—42no—
196#63225—42no—
197#63225—42no—
198#63225—42no—
199#63225—42no—
200C4×Dic17—42no—
201C2×#3622—21no—
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×#3622C2×D412yes—
212C2×#3622C2×Q812yes—
213#26374—171nomaximal, Sylow
214C23×C17—22no—
215C23×C17—22no—
216C23×C17—22no—
217C23×C17—41no—
218C23×C17—41no—
219C23×C17—41no—
220C23×C17—41no—
221C23×C17C22×D411yescharacteristic
222D4×C17—22no—
223D4×C17—22no—
224D4×C17—22no—
225D4×C17C22⋊Q814yes—
226C2×C4×C17—21no—
227C2×C4×C17—21no—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17—22no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—21no—
235C2×C4×C17—21no—
236C2×C4×C17—21no—
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—48no—
244C2×C4×C17—41no—
245C2×C4×C17—41no—
246C2×C4×C17—41no—
247C2×C4×C17—41no—
248C2×C4×C17—41no—
249C2×C4×C17—41no—
250C2×C4×C17—41no—
251C2×C4×C17—41no—
252C2×C4×C17—84no—
253C2×C4×C17—84no—
254C2×C4×C17#13512yes—
255C2×C4×C17AGL(1,ℤ/8)12yes—
256C2×C4×C17C22×D411yescharacteristic
257C2×C4×C17C22×Q811yescharacteristic
258C2×C4×C17C2×#4111yescharacteristic
259C2×C4×C17C2×#4111yescharacteristic
260C2×C4×C17H(5,𝔽2)11yescharacteristic
261C2×C4×C172−1+411yescharacteristic
262C2×Dic17—22no—
263C2×Dic17—22no—
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—
274Dic34—22no—
275Dic34—22no—
276Dic34—41no—
277Dic34—41no—
278#3622—22no—
279#3622—22no—
280#3622—41no—
281#3622—41no—
282C2×#478—171no—
283C2×C4×SD16—171no—
284#2453—171no—
285D4×C4⋊C4D1711yescharacteristic
286#2095—171no—
287#2084—171no—
288#2068—171no—
289#2059—171no—
290#1942—171no—
291#1916—171no—
292#1877—171no—
293#1877—171no—
294#1865—171no—
295#1862—171no—
296#1862—171no—
297C22×C17—24no—
298C22×C17—42no—
299C22×C17—42no—
300C22×C17—42no—
301C22×C17—42no—
302C22×C17—42no—
303C22×C17—42no—
304C22×C17#52511yescharacteristic
305C22×C17D4211yescharacteristic
306C22×C17D4×Q811yescharacteristic
307C22×C17C2×#13511yescharacteristic
308C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
309C22×C17#57711yescharacteristic
310C22×C17#57811yescharacteristic
311C4×C17—22no—
312C4×C17—22no—
313C4×C17—42no—
314C4×C17—41no—
315C4×C17—41no—
316C4×C17—41no—
317C4×C17—41no—
318C4×C17—81no—
319C4×C17—81no—
320C4×C17C2×C22⋊Q811yescharacteristic
321C4×C17#53911yescharacteristic
322C4×C17#54711yescharacteristic
323C4×C17#55411yescharacteristic
324Dic17—42no—
325Dic17—42no—
326Dic17—81no—
327Dic17—81no—
328C22×SD16—171no—
329C2×#125—171no—
330C2×C4×Q8—171no—
331C2×C4×D4C2×D1711yescharacteristic
332C2×C4×D4C2×D1711yescharacteristic
333C2×C4×D4C2×D1711yescharacteristic
334C22×C4⋊C4—21no—
335#478—344no—
336C4×SD16—344no—
337C2×#107—171no—
338C2×#107—171no—
339C2×#107—171no—
340C2×#106—171no—
341C2×#106—171no—
342C2×C4⋊C8—171no—
343C2×#103—171no—
344C2×#103—171no—
345C2×#103—171no—
346C2×#102—171no—
347C2×#102—171no—
348C2×#102—171no—
349C2×C4×C8—171no—
350#400—341no—
351#398—341no—
352#391—171no—
353#391—171no—
354#389—171no—
355#389—171no—
356#389—341no—
357#389C2×D1711yescharacteristic
358#389C2×D1711yescharacteristic
359#387—341no—
360#385—21no—
361#385—21no—
362#384C2×D1711yescharacteristic
363#380—21no—
364C4×C4⋊C4C2×D1711yescharacteristic
365#340—171no—
366#340—171no—
367#340—171no—
368#340—171no—
369#336—171no—
370#336—171no—
371#336—171no—
372#336—171no—
373C2×C17—44no—
374C2×C17#244911yescharacteristic
375C2×C17#310511yescharacteristic
376C2×C17#312011yescharacteristic
377C2×C17#326511yescharacteristic
378C2×C17#328311yescharacteristic
379C2×C17#338111yescharacteristic
380C2×C17#338611yescharacteristic
381C22×Q8—171no—
382C22×D4C22×D1711yescharacteristic
383C23×C4—21no—
384C23×C4—21no—
385C23×C4—21no—
386C2×SD16—174no—
387C2×SD16—342no—
388C2×SD16—342no—
389C22×C8—171no—
390C22×C8—171no—
391C22×C8—171no—
392C22×C8—171no—
393#125—342no—
394#125—681no—
395C4×Q8—342no—
396C4×Q8—681no—
397C4×D4—22no—
398C4×D4—22no—
399C4×D4—22no—
400C4×D4—22no—
401C4×D4—41no—
402C4×D4—41no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—44no—
408C2×C4⋊C4—171no—
409C2×C4⋊C4—171no—
410C2×C4⋊C4—171no—
411C2×C4⋊C4—341no—
412C2×C4⋊C4—341no—
413C2×C4⋊C4—341no—
414C2×C4⋊C4—341no—
415C2×C4⋊C4—341no—
416C2×C4⋊C4C22⋊D1711yescharacteristic
417C2×C4⋊C4C22⋊D1711yescharacteristic
418C2×C4⋊C4C22×D1711yescharacteristic
419C2×C4⋊C4C22×D1711yescharacteristic
420C2×C4⋊C4C22×D1711yescharacteristic
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C42—341no—
425C2×C42C22×D1711yescharacteristic
426C2×C42C22×D1711yescharacteristic
427C2×C42C22×D1711yescharacteristic
428#107—342no—
429#107—342no—
430#107—681no—
431#106—342no—
432#106—342no—
433C4⋊C8—342no—
434#103—342no—
435#103—342no—
436#103—681no—
437#102—174no—
438#102—342no—
439#102—342no—
440C4×C8—342no—
441#95—21no—
442#95—21no—
443#95—341no—
444#95—341no—
445#95—341no—
446#95—341no—
447#95—341no—
448#95—341no—
449#95C22⋊D1711yescharacteristic
450#95C22⋊D1711yescharacteristic
451C17#2637411yescharacteristic, complemented, Sylow
452C24—21no—
453C2×Q8—172no—
454C2×Q8—341no—
455C2×Q8—341no—
456C2×Q8—341no—
457C2×Q8—341no—
458C2×Q8—341no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4#6325212yes—
463C2×D4D4×D1712yes—
464C2×D4Q8×D1712yes—
465C22×C4—24no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—81no—
476C22×C4—81no—
477C22×C4—341no—
478C22×C4—341no—
479C22×C4—341no—
480C22×C4—341no—
481C22×C4C2×C22⋊D1711yescharacteristic
482C22×C4C2×C22⋊D1711yescharacteristic
483C22×C4C2×C22⋊D1711yescharacteristic
484C22×C4C2×C22⋊D1711yescharacteristic
485C22×C4C2×C22⋊D1711yescharacteristic
486C22×C4C2×C22⋊D1711yescharacteristic
487C22×C4C23×D1711yescharacteristic
488SD16—348no—
489C2×C8—172no—
490C2×C8—172no—
491C2×C8—341no—
492C2×C8—341no—
493C2×C8—341no—
494C2×C8—341no—
495C2×C8—341no—
496C2×C8—341no—
497C2×C8—341no—
498C2×C8—341no—
499C2×C8—341no—
500C2×C8—341no—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—81no—
512C4⋊C4—172no—
513C4⋊C4—341no—
514C4⋊C4—341no—
515C4⋊C4—341no—
516C4⋊C4—341no—
517C4⋊C4—341no—
518C4⋊C4—682no—
519C4⋊C4—682no—
520C4⋊C4—682no—
521C4⋊C4—682no—
522C4⋊C4—682no—
523C4⋊C4#6325212yes—
524C22⋊C4—24no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—682no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23C22×C22⋊D1711yescharacteristic
542Q8—342no—
543Q8—342no—
544Q8—681no—
545Q8—681no—
546D4—22no—
547D4—22no—
548D4—22no—
549D4#1058758214yes—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—48no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4—342no—
579C2×C4—342no—
580C2×C4—681no—
581C2×C4—681no—
582C2×C4—681no—
583C2×C4—681no—
584C2×C4—681no—
585C2×C4—681no—
586C2×C4—681no—
587C2×C4—681no—
588C2×C4—681no—
589C2×C4—681no—
590C2×C4#1058761812yes—
591C2×C4#1058763612yes—
592C2×C4C2×#6325211yescharacteristic
593C2×C4C2×#6325211yescharacteristic
594C2×C4C2×D4×D1711yescharacteristic
595C2×C4#1058769111yescharacteristic
596C2×C4C2×Q8×D1711yescharacteristic
597C2×C4#1058769411yescharacteristic
598C8—342no—
599C8—342no—
600C8—681no—
601C8—681no—
602C22—24no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22#1176332011yescharacteristic
610C22D4×C22⋊D1711yescharacteristic
611C22Q8×C22⋊D1711yescharacteristic
612C22C2×#1058763611yescharacteristic
613C22C2×#1058761811yescharacteristic
614C22#1176362011yescharacteristic
615C22#1176362411yescharacteristic
616C4—22no—
617C4—22no—
618C4—42no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—81no—
624C4—81no—
625C4—682no—
626C4—682no—
627C4—1361no—
628C4—1361no—
629C4C2×#1058758211yescharacteristic
630C4#1176340611yescharacteristic
631C4#1176346011yescharacteristic
632C4#1176351411yescharacteristic
633C2—44no—
634C2#1453259311yescharacteristic
635C2#1453883111yescharacteristic
636C2#1453899811yescharacteristic
637C2#1454027511yescharacteristic
638C2#1454050211yescharacteristic
639C2#1454165911yescharacteristic
640C2#1454170511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingD4×C4⋊C4×C17C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
hypercenterD4×C4⋊C4D17
hypocenterC17#26374
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ D4×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #14532645C2semidirect
4. #14530323C2semidirect
5. #14530261C2semidirect
6. #14530176C2semidirect
7. #14530122C2semidirect
8. #14529514C2semidirect
9. #14529343C2semidirect
12. #14529051C2semidirect
451. C17#26374semidirect

Maximal subgroups

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

subgroupindex
C2×#117631092
C2×C4×#632292
#145326452
#145303232
#145302612
#145301762
#145301222
#145295142
#145293432
#145291182
#145290512
#145290312
D4×C4⋊C4×C172
#2637417