Finite Group Databasebeta

← #17107515

Subgroups — #17107515

Order 4352.

762 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14533370C211yesmaximal, characteristic
2#14530327C211yesmaximal, characteristic, complemented
3#14530099C211yesmaximal, characteristic
4#14530027C211yesmaximal, characteristic, complemented
5#14529867C211yesmaximal, characteristic
6#14529641C211yesmaximal, characteristic
7#14529415C211yesmaximal, characteristic, complemented
8#14529411C211yesmaximal, characteristic
9#14529362C211yesmaximal, characteristic, complemented
10#14529343C211yesmaximal, characteristic, complemented
11#14529139C211yesmaximal, characteristic
12#14529139C211yesmaximal, characteristic
13#14528789C211yesmaximal, characteristic, complemented
14C4×#10587506C211yesmaximal, characteristic, complemented
15C17×#2365C211yesmaximal, characteristic, complemented
16C22×#63230C2211yescharacteristic
17C22×#63230C2211yescharacteristic
18C2×#10587584—21no—
19C2×#10587479C2211yescharacteristic
20C2×#10587479C2211yescharacteristic
21C2×C4⋊#3622C2211yescharacteristic
22C2×#10587506C2211yescharacteristic
23C2×#10587506C2211yescharacteristic
24C2×#10587506C2211yescharacteristic
25C2×#10587506C2211yescharacteristic
26C2×#10587506C2211yescharacteristic
27C2×#10587506C2211yescharacteristic
28C2×#10587480C2211yescharacteristic
29C2×#10587480C2211yescharacteristic
30C2×C4×#3622C2211yescharacteristic
31#11762736—21no—
32#11762731—21no—
33#11762711C2211yescharacteristic
34#11762711C2211yescharacteristic
35#11762711C2211yescharacteristic
36#11762707C2211yescharacteristic
37#11762701—21no—
38#11762679C2211yescharacteristic
39C4×C4⋊Dic17C2211yescharacteristic
40#11762571C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762564C2211yescharacteristic
45#11762564C2211yescharacteristic
46#11762564C2211yescharacteristic
47#11762564C2211yescharacteristic
48C2×C17×C4⋊Q8C2211yescharacteristic
49C2×C4×Q8×C17C2211yescharacteristic
50C2×C4×Q8×C17C2211yescharacteristic
51C17×#391C2211yescharacteristic
52C17×#391C2211yescharacteristic
53C17×#389—21no—
54C17×#387—21no—
55C17×#387—21no—
56C4×C4⋊C4×C17—21no—
57C4×C4⋊C4×C17C2211yescharacteristic
58C4×C4⋊C4×C17C2211yescharacteristic
59C22×Q8×C17C2311yescharacteristic
60C22×Q8×C17C2311yescharacteristic
61C22×D68C2311yescharacteristic
62C17×C4⋊Q8—22no—
63C17×C4⋊Q8—22no—
64C4×Q8×C17—22no—
65C4×Q8×C17—22no—
66C4×Q8×C17—41no—
67C4×Q8×C17—41no—
68C2×C4⋊C4×C17—21no—
69C2×C4⋊C4×C17—21no—
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17D411yescharacteristic
73C2×C4⋊C4×C17D411yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C42×C17—21no—
77C2×C42×C17—21no—
78C2×C42×C17—21no—
79C2×C42×C17C2311yescharacteristic
80C2×C42×C17C2311yescharacteristic
81C2×C42×C17C2311yescharacteristic
82C2×#63230—22no—
83C2×#63230—22no—
84C2×#63230—22no—
85C2×#63230—41no—
86C2×#63230—41no—
87C2×#63230—41no—
88C2×#63227—21no—
89C2×#63227—21no—
90C2×#63227—21no—
91C2×#63227—21no—
92C2×#63227—21no—
93C2×#63227—21no—
94C2×C4⋊Dic17—21no—
95C2×C4⋊Dic17—21no—
96C2×C4⋊Dic17C2311yescharacteristic
97C2×C4⋊Dic17C2311yescharacteristic
98C2×C4⋊Dic17C2311yescharacteristic
99C2×#63225—21no—
100C2×C4×Dic17—21no—
101C22×#3622C2311yescharacteristic
102C22×#3622C2311yescharacteristic
103C22×#3622C2311yescharacteristic
104C22×#3622C2311yescharacteristic
105#10587584—44no—
106C17×#95—21no—
107C17×#95—21no—
108C17×#95—21no—
109C17×#95—21no—
110C17×#95—21no—
111C17×#95D411yescharacteristic
112C17×#95D411yescharacteristic
113#10587506—22no—
114#10587506—22no—
115#10587506—22no—
116#10587506—41no—
117#10587506—41no—
118#10587506—41no—
119#10587502—21no—
120#10587502—21no—
121#10587480—22no—
122#10587480—41no—
123#10587479—22no—
124#10587479—41no—
125C4⋊#3622—41no—
126C4×#3622—41no—
127C23×D17—21no—
128C2×Q8×C17—21no—
129C2×Q8×C17—21no—
130C2×Q8×C17—21no—
131C2×Q8×C17—21no—
132C2×Q8×C17—21no—
133C2×Q8×C17—21no—
134C2×Q8×C17—21no—
135C2×Q8×C17—21no—
136C2×Q8×C17—21no—
137C2×Q8×C17—21no—
138C2×Q8×C17#4112yes—
139C2×Q8×C17#4112yes—
140C22×C4×C17—21no—
141C22×C4×C17—21no—
142C22×C4×C17—21no—
143C22×C4×C17—21no—
144C22×C4×C17C2×D411yescharacteristic
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2411yescharacteristic
151C22×Dic17—21no—
152C22×Dic17—21no—
153C22×Dic17—21no—
154C2×D68—21no—
155C2×D68—21no—
156C2×D68—21no—
157C2×D68—21no—
158C2×D68—21no—
159C2×D68—21no—
160C4⋊C4×C17—24no—
161C4⋊C4×C17—24no—
162C4⋊C4×C17—21no—
163C4⋊C4×C17—21no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—41no—
166C4⋊C4×C17—42no—
167C4⋊C4×C17—42no—
168C4⋊C4×C17SD1614yescomplemented
169C4⋊C4×C17#4112yes—
170C42×C17—21no—
171C42×C17—21no—
172C42×C17—21no—
173C42×C17—21no—
174C42×C17—21no—
175C42×C17—21no—
176C42×C17—42no—
177C42×C17—42no—
178C42×C17—42no—
179#63230—44no—
180#63230—44no—
181#63227—42no—
182#63227—42no—
183#63227—42no—
184#63227—42no—
185#63227—81no—
186#63227—81no—
187C4⋊Dic17—21no—
188C4⋊Dic17—21no—
189C4⋊Dic17—21no—
190C4⋊Dic17—21no—
191C4⋊Dic17—21no—
192C4⋊Dic17—21no—
193C4⋊Dic17—42no—
194C4⋊Dic17—81no—
195#63225—42no—
196C4×Dic17—42no—
197C2×#3622—21no—
198C2×#3622—21no—
199C2×#3622—21no—
200C2×#3622—21no—
201C2×#3622—21no—
202C2×#3622—21no—
203C2×#3622—21no—
204C2×#3622—21no—
205C2×#3622—21no—
206C2×#3622—21no—
207C2×#3622—21no—
208C2×#3622#4112yes—
209#27066—171nomaximal, Sylow
210C23×C17C22×D411yescharacteristic
211C22×D17—41no—
212C22×D17—41no—
213C22×D17—41no—
214C22×D17—41no—
215C22×D17—41no—
216C22×D17—41no—
217C22×D17—41no—
218Q8×C17—22no—
219Q8×C17—22no—
220Q8×C17—22no—
221Q8×C17—22no—
222Q8×C17—41no—
223Q8×C17—41no—
224Q8×C17—41no—
225Q8×C17—41no—
226C2×C4×C17—21no—
227C2×C4×C17—21no—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17—21no—
232C2×C4×C17—21no—
233C2×C4×C17—22no—
234C2×C4×C17—22no—
235C2×C4×C17—21no—
236C2×C4×C17—21no—
237C2×C4×C17—22no—
238C2×C4×C17—21no—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—21no—
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—41no—
253C2×C4×C17C2×SD1612yes—
254C2×C4×C17C2×SD1612yes—
255C2×C4×C17C2×SD1612yes—
256C2×C4×C17C2×#4111yescharacteristic
257C2×C4×C17C2×#4111yescharacteristic
258C2×C4×C17C2×#4111yescharacteristic
259C2×C4×C17C2×#4111yescharacteristic
260C2×C4×C17H(5,𝔽2)11yescharacteristic
261C2×C4×C17H(5,𝔽2)11yescharacteristic
262C2×Dic17—41no—
263C2×Dic17—41no—
264C2×Dic17—41no—
265C2×Dic17—41no—
266C2×Dic17—41no—
267C2×Dic17—41no—
268C2×Dic17—41no—
269C2×Dic17—41no—
270C2×Dic17—41no—
271D68—41no—
272D68—41no—
273D68—41no—
274D68—41no—
275#3622—22no—
276#3622—41no—
277#3622—41no—
278#3622—41no—
279#2512—171no—
280#2365D1711yescharacteristic, complemented
281#2096—171no—
282#2055—171no—
283#2043—171no—
284#2013—171no—
285#1968—171no—
286#1927—171no—
287#1926—171no—
288#1919—171no—
289#1916—171no—
290#1880—171no—
291#1880—171no—
292#1803—171no—
293C4×#103—171no—
294C22×C17#52511yescharacteristic
295C22×C17#54611yescharacteristic
296C22×C17#55011yescharacteristic
297C22×C17C22×SD1611yescharacteristic
298C22×C17#57511yescharacteristic
299C22×C17#57711yescharacteristic
300C22×C17#57811yescharacteristic
301C2×D17—81no—
302C2×D17—81no—
303C2×D17—81no—
304C2×D17—81no—
305C2×D17—81no—
306C2×D17—81no—
307C2×D17—81no—
308C4×C17—22no—
309C4×C17—22no—
310C4×C17—22no—
311C4×C17—41no—
312C4×C17—42no—
313C4×C17—42no—
314C4×C17—41no—
315C4×C17—41no—
316C4×C17—42no—
317C4×C17—81no—
318C4×C17#52911yescharacteristic
319C4×C17#53811yescharacteristic
320C4×C17#55111yescharacteristic
321C4×C17#55311yescharacteristic
322Dic17—81no—
323Dic17—81no—
324Dic17—81no—
325C22×SD16—171no—
326C22×SD16—171no—
327C2×C4⋊Q8C2×D1711yescharacteristic
328C2×#126—341no—
329C2×C4×Q8C2×D1711yescharacteristic
330C2×C4×Q8C2×D1711yescharacteristic
331C2×#106—171no—
332C2×#106—171no—
333C2×C4⋊C8—171no—
334C2×#103—171no—
335C2×#103—171no—
336C2×#103—171no—
337C2×#103—171no—
338C2×#103—171no—
339C2×#103—171no—
340C2×#102—171no—
341C2×#102—171no—
342C2×C4×C8—171no—
343#397—341no—
344#396—341no—
345#391C2×D1711yescharacteristic
346#391C2×D1711yescharacteristic
347#390—171no—
348#390—171no—
349#390—171no—
350#389—21no—
351#389—171no—
352#388—341no—
353#387—21no—
354#387—21no—
355#384—171no—
356C4×C4⋊C4—21no—
357C4×C4⋊C4—171no—
358C4×C4⋊C4C2×D1711yescharacteristic
359C4×C4⋊C4C2×D1711yescharacteristic
360#340—171no—
361#340—171no—
362#340—171no—
363#340—171no—
364#336—171no—
365#336—171no—
366#336—171no—
367#336—171no—
368C2×C17#246711yescharacteristic
369C2×C17#310311yescharacteristic
370C2×C17#312111yescharacteristic
371C2×C17#329811yescharacteristic
372C2×C17#330711yescharacteristic
373C2×C17#339211yescharacteristic
374C2×C17#340211yescharacteristic
375D17—161no—
376C22×Q8C22×D1711yescharacteristic
377C22×Q8C22×D1711yescharacteristic
378C22×D4—171no—
379C2×SD16—342no—
380C2×SD16—342no—
381C2×SD16—342no—
382C2×SD16—681no—
383C2×SD16—681no—
384C2×SD16—681no—
385C22×C8—171no—
386C22×C8—171no—
387C22×C8—171no—
388C22×C8—171no—
389C4⋊Q8—22no—
390C4⋊Q8—22no—
391#126—684no—
392C4×Q8—22no—
393C4×Q8—22no—
394C4×Q8—41no—
395C4×Q8—41no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—171no—
401C2×C4⋊C4—171no—
402C2×C4⋊C4—171no—
403C2×C4⋊C4—341no—
404C2×C4⋊C4—341no—
405C2×C4⋊C4—341no—
406C2×C4⋊C4C22⋊D1711yescharacteristic
407C2×C4⋊C4C22⋊D1711yescharacteristic
408C2×C4⋊C4C22×D1711yescharacteristic
409C2×C4⋊C4C22×D1711yescharacteristic
410C2×C22⋊C4—341no—
411C2×C22⋊C4—341no—
412C2×C22⋊C4—341no—
413C2×C22⋊C4—341no—
414C2×C22⋊C4—341no—
415C2×C22⋊C4—341no—
416C2×C42—21no—
417C2×C42—21no—
418C2×C42—21no—
419C2×C42—341no—
420C2×C42C22×D1711yescharacteristic
421C2×C42C22×D1711yescharacteristic
422C2×C42C22×D1711yescharacteristic
423#106—342no—
424#106—681no—
425C4⋊C8—681no—
426#103—342no—
427#103—342no—
428#103—342no—
429#103—681no—
430#103—681no—
431#103—681no—
432#102—342no—
433#102—681no—
434C4×C8—681no—
435#95—21no—
436#95—21no—
437#95—21no—
438#95—21no—
439#95—21no—
440#95—341no—
441#95—341no—
442#95C22⋊D1711yescharacteristic
443#95C22⋊D1711yescharacteristic
444C17#2706611yescharacteristic, complemented, Sylow
445C24—341no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8#6325212yes—
457C2×Q8#6325212yes—
458C2×D4—341no—
459C2×D4—341no—
460C2×D4—341no—
461C2×D4—341no—
462C2×D4—341no—
463C2×D4—341no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—341no—
469C22×C4—341no—
470C22×C4—341no—
471C22×C4C2×C22⋊D1711yescharacteristic
472C22×C4C2×C22⋊D1711yescharacteristic
473C22×C4C2×C22⋊D1711yescharacteristic
474C22×C4C2×C22⋊D1711yescharacteristic
475C22×C4C2×C22⋊D1711yescharacteristic
476C22×C4C2×C22⋊D1711yescharacteristic
477C22×C4C23×D1711yescharacteristic
478SD16—684no—
479SD16—684no—
480C2×C8—172no—
481C2×C8—341no—
482C2×C8—341no—
483C2×C8—341no—
484C2×C8—341no—
485C2×C8—341no—
486C2×C8—341no—
487C2×C8—341no—
488C2×C8—341no—
489C2×C8—341no—
490C2×C8—341no—
491C2×C8—341no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—41no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—341no—
501C4⋊C4—341no—
502C4⋊C4—341no—
503C4⋊C4—341no—
504C4⋊C4—341no—
505C4⋊C4—341no—
506C4⋊C4—682no—
507C4⋊C4—682no—
508C4⋊C4—1361no—
509C4⋊C4#6323014yescomplemented
510C4⋊C4#6325412yes—
511C22⋊C4—682no—
512C22⋊C4—682no—
513C22⋊C4—682no—
514C22⋊C4—682no—
515C22⋊C4—1361no—
516C22⋊C4—1361no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—42no—
524C42—42no—
525C42—42no—
526C42—682no—
527C23—681no—
528C23—681no—
529C23—681no—
530C23—681no—
531C23—681no—
532C23—681no—
533C23—681no—
534C23C22×C22⋊D1711yescharacteristic
535Q8—22no—
536Q8—22no—
537Q8—22no—
538Q8—22no—
539Q8—41no—
540Q8—41no—
541Q8—41no—
542Q8—41no—
543D4—681no—
544D4—681no—
545D4—681no—
546D4—681no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
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—22no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—681no—
575C2×C4—681no—
576C2×C4—681no—
577C2×C4—681no—
578C2×C4—681no—
579C2×C4—681no—
580C2×C4—681no—
581C2×C4—681no—
582C2×C4—681no—
583C2×C4C2×#6323012yes—
584C2×C4C2×#6323012yes—
585C2×C4C2×#6323012yes—
586C2×C4C2×#6325211yescharacteristic
587C2×C4C2×#6325211yescharacteristic
588C2×C4C2×#6325411yescharacteristic
589C2×C4#1058769111yescharacteristic
590C2×C4#41×D1711yescharacteristic
591C2×C4#1058769611yescharacteristic
592C8—342no—
593C8—681no—
594C8—681no—
595C8—681no—
596C22—1361no—
597C22—1361no—
598C22—1361no—
599C22—1361no—
600C22—1361no—
601C22—1361no—
602C22—1361no—
603C22#1176332011yescharacteristic
604C22#1176345311yescharacteristic
605C22#1176347811yescharacteristic
606C22C22×#6323011yescharacteristic
607C22#1176360711yescharacteristic
608C22#1176361711yescharacteristic
609C22#1176362411yescharacteristic
610C4—22no—
611C4—22no—
612C4—22no—
613C4—41no—
614C4—41no—
615C4—41no—
616C4—42no—
617C4—42no—
618C4—42no—
619C4—81no—
620C4—1361no—
621C4—1361no—
622C4—1361no—
623C4#1176333911yescharacteristic
624C4#1176339211yescharacteristic
625C4#1176348811yescharacteristic
626C4#1176350511yescharacteristic
627C2—2721no—
628C2#1453282911yescharacteristic
629C2#1453881511yescharacteristic
630C2#1453901311yescharacteristic
631C2#1454064911yescharacteristic
632C2#1454076011yescharacteristic
633C2#1454176111yescharacteristic
634C2#1454185911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2365C2
socleC23×C17C22×D4
prime radicalC22×D68C23
hypercenter#2365D17
hypocenterC17#27066
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2365

Complements

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

NH ≅ G/Ntype
2. #14530327C2semidirect
4. #14530027C2semidirect
7. #14529415C2semidirect
9. #14529362C2semidirect
10. #14529343C2semidirect
13. #14528789C2semidirect
14. C4×#10587506C2semidirect
15. C17×#2365C2semidirect
168. C4⋊C4×C17SD16semidirect
280. #2365D17semidirect
444. C17#27066semidirect
509. C4⋊C4#63230semidirect

Maximal subgroups

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

subgroupindex
#145333702
#145303272
#145300992
#145300272
#145298672
#145296412
#145294152
#145294112
#145293622
#145293432
#145291392
#145287892
C4×#105875062
C17×#23652
#2706617