Finite Group Databasebeta

← #17102439

Subgroups — #17102439

Order 4352.

764 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14533158C211yesmaximal, characteristic
2#14530106C211yesmaximal, characteristic, complemented
3#14529984C211yesmaximal, characteristic
4#14529925C211yesmaximal, characteristic, complemented
5#14529792C211yesmaximal, characteristic
6#14529732C211yesmaximal, characteristic, complemented
7#14529459C211yesmaximal, characteristic, complemented
8#14529425C211yesmaximal, characteristic, complemented
9#14529392C211yesmaximal, characteristic
10#14529373C211yesmaximal, characteristic, complemented
11#14529125C211yesmaximal, characteristic, complemented
12#14529091C211yesmaximal, characteristic
13#14528874C211yesmaximal, characteristic
14C4×#10587506C211yesmaximal, characteristic, complemented
15C17×#2360C211yesmaximal, characteristic
16C22×#63231C2211yescharacteristic
17C2×#10587555C2211yescharacteristic
18C2×#10587610C2211yescharacteristic
19C2×#10587478C2211yescharacteristic
20C2×C4⋊#3622C2211yescharacteristic
21C2×#10587506C2211yescharacteristic
22C2×#10587506C2211yescharacteristic
23C2×#10587506C2211yescharacteristic
24C2×#10587481C2211yescharacteristic
25C2×#10587481C2211yescharacteristic
26C2×#10587481C2211yescharacteristic
27C2×C22⋊#3622C2211yescharacteristic
28C2×C22⋊#3622C2211yescharacteristic
29C2×C4×#3622C2211yescharacteristic
30#11762751—21no—
31#11762721—21no—
32#11762715C2211yescharacteristic
33#11762715C2211yescharacteristic
34#11762713—21no—
35#11762702—21no—
36#11762669C2211yescharacteristic
37C4×C4⋊Dic17C2211yescharacteristic
38#11762571C2211yescharacteristic
39#11762571C2211yescharacteristic
40#11762571C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762564C2211yescharacteristic
47#11762564C2211yescharacteristic
48C2×C17×C22⋊Q8C2211yescharacteristic
49C2×C4×Q8×C17C2211yescharacteristic
50C22×C4⋊C4×C17C2211yescharacteristic
51C17×#391C2211yescharacteristic
52C17×#389—21no—
53C17×#389C2211yescharacteristic
54C17×#387—21no—
55C17×#385—21no—
56C4×C4⋊C4×C17C2211yescharacteristic
57C4×C22⋊C4×C17—21no—
58C4×C22⋊C4×C17C2211yescharacteristic
59C22×Q8×C17C2311yescharacteristic
60C23×C4×C17—21no—
61C23×C4×C17C2311yescharacteristic
62C22×Dic34—21no—
63C22×Dic34C2311yescharacteristic
64C17×C22⋊Q8—22no—
65C17×C22⋊Q8—22no—
66C4×Q8×C17—22no—
67C4×Q8×C17—41no—
68C2×C4⋊C4×C17—21no—
69C2×C4⋊C4×C17—22no—
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17—21no—
73C2×C4⋊C4×C17—41no—
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C4⋊C4×C17C2311yescharacteristic
77C2×C22⋊C4×C17—21no—
78C2×C22⋊C4×C17D411yescharacteristic
79C2×C22⋊C4×C17D411yescharacteristic
80C2×C42×C17—21no—
81C2×C42×C17—21no—
82C2×C42×C17C2311yescharacteristic
83C2×C42×C17C2311yescharacteristic
84C2×C22⋊Dic17—21no—
85C2×C22⋊Dic17—21no—
86C2×#63231—22no—
87C2×#63231—41no—
88C2×#63231—41no—
89C2×C4⋊Dic17—21no—
90C2×C4⋊Dic17C2311yescharacteristic
91C2×C4⋊Dic17C2311yescharacteristic
92C2×C4⋊Dic17C2311yescharacteristic
93#10587610—22no—
94#10587610—41no—
95C2×#63225—21no—
96C2×C4×Dic17—21no—
97C22×#3622C2311yescharacteristic
98C22×#3622C2311yescharacteristic
99C22×#3622C2311yescharacteristic
100C22×#3622C2311yescharacteristic
101#10587555—22no—
102#10587555—41no—
103C17×#95—21no—
104C17×#95—21no—
105C17×#95—21no—
106C17×#95—21no—
107C17×#95D411yescharacteristic
108C17×#95D411yescharacteristic
109#10587506—22no—
110#10587506—22no—
111#10587506—41no—
112#10587502—21no—
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118C22⋊#3622—41no—
119C22⋊#3622—41no—
120#10587481—22no—
121#10587481—22no—
122#10587481—41no—
123#10587478—22no—
124C4⋊#3622—41no—
125C4×#3622—41no—
126C24×C17C2×D411yescharacteristic
127C2×Q8×C17—21no—
128C2×Q8×C17—21no—
129C2×Q8×C17—21no—
130C2×Q8×C17—21no—
131C2×Q8×C17—21no—
132C2×Q8×C17#4112yes—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17—21no—
136C22×C4×C17—21no—
137C22×C4×C17—21no—
138C22×C4×C17—21no—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17—21no—
142C22×C4×C17—21no—
143C22×C4×C17—42no—
144C22×C4×C17—42no—
145C22×C4×C17—81no—
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2×D411yescharacteristic
151C22×C4×C17C2411yescharacteristic
152C22×Dic17—21no—
153C22×Dic17—21no—
154C22×Dic17—21no—
155C22×Dic17—21no—
156C2×Dic34—21no—
157C2×Dic34—21no—
158C2×Dic34—21no—
159C2×Dic34—21no—
160C2×Dic34—21no—
161C2×Dic34—21no—
162C2×Dic34—42no—
163C2×Dic34—42no—
164C2×Dic34—81no—
165C4⋊C4×C17—24no—
166C4⋊C4×C17—21no—
167C4⋊C4×C17—21no—
168C4⋊C4×C17—21no—
169C4⋊C4×C17—21no—
170C4⋊C4×C17—42no—
171C4⋊C4×C17—42no—
172C4⋊C4×C17—81no—
173C4⋊C4×C17#4112yes—
174C4⋊C4×C17#4112yes—
175C22⋊C4×C17—24no—
176C22⋊C4×C17—41no—
177C22⋊C4×C17Q1614yescomplemented
178C42×C17—21no—
179C42×C17—21no—
180C42×C17—21no—
181C42×C17—21no—
182C42×C17—42no—
183C42×C17—81no—
184C22⋊Dic17—42no—
185C22⋊Dic17—81no—
186#63231—44no—
187C4⋊Dic17—21no—
188C4⋊Dic17—21no—
189C4⋊Dic17—21no—
190C4⋊Dic17—21no—
191C4⋊Dic17—21no—
192C4⋊Dic17—42no—
193C4⋊Dic17#4112yes—
194#63225—42no—
195C4×Dic17—81no—
196C2×#3622—21no—
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—
208#26726—171nomaximal, Sylow
209C23×C17—21no—
210C23×C17—21no—
211C23×C17—21no—
212C23×C17—21no—
213C23×C17—21no—
214C23×C17—21no—
215C23×C17C2×Q1612yes—
216C23×C17C22×D411yescharacteristic
217Q8×C17—22no—
218Q8×C17—22no—
219Q8×C17—41no—
220Q8×C17—41no—
221C2×C4×C17—21no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—22no—
226C2×C4×C17—22no—
227C2×C4×C17—21no—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17—22no—
231C2×C4×C17—21no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—21no—
235C2×C4×C17—21no—
236C2×C4×C17—21no—
237C2×C4×C17—41no—
238C2×C4×C17—41no—
239C2×C4×C17—41no—
240C2×C4×C17—41no—
241C2×C4×C17—41no—
242C2×C4×C17—41no—
243C2×C4×C17—41no—
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—84no—
251C2×C4×C17C2×Q1612yes—
252C2×C4×C17C2×Q1612yes—
253C2×C4×C17C2×#4111yescharacteristic
254C2×C4×C17C2×#4111yescharacteristic
255C2×C4×C17C2×#4111yescharacteristic
256C2×C4×C17C2×#4111yescharacteristic
257C2×C4×C17H(5,𝔽2)11yescharacteristic
258C2×C4×C17H(5,𝔽2)11yescharacteristic
259C2×Dic17—22no—
260C2×Dic17—41no—
261C2×Dic17—41no—
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—
271Dic34—41no—
272Dic34—41no—
273Dic34—41no—
274Dic34—41no—
275Dic34—84no—
276#3622—41no—
277#3622—41no—
278#3622—41no—
279#3622—41no—
280#2492—171no—
281#2360D1711yescharacteristic
282#2056—171no—
283#2036—171no—
284#2024—171no—
285#2001—171no—
286#1987—171no—
287#1933—171no—
288#1928—171no—
289#1923—171no—
290#1921—171no—
291#1878—171no—
292#1872—171no—
293#1824—171no—
294C4×#103—171no—
295C22×C17—22no—
296C22×C17—22no—
297C22×C17—22no—
298C22×C17—41no—
299C22×C17—41no—
300C22×C17—41no—
301C22×C17—41no—
302C22×C17#52511yescharacteristic
303C22×C17#54611yescharacteristic
304C22×C17#54611yescharacteristic
305C22×C17C22×Q1611yescharacteristic
306C22×C17#57511yescharacteristic
307C22×C17#57711yescharacteristic
308C22×C17#57711yescharacteristic
309C4×C17—22no—
310C4×C17—22no—
311C4×C17—42no—
312C4×C17—41no—
313C4×C17—42no—
314C4×C17—41no—
315C4×C17—42no—
316C4×C17—41no—
317C4×C17—81no—
318C4×C17#52911yescharacteristic
319C4×C17#53811yescharacteristic
320C4×C17#55511yescharacteristic
321C4×C17#55911yescharacteristic
322Dic17—42no—
323Dic17—81no—
324Dic17—81no—
325Dic17—81no—
326C22×Q16—171no—
327C2×C4⋊Q8—171no—
328C2×C22⋊Q8—171no—
329C2×C22⋊Q8C2×D1711yescharacteristic
330C2×C4×Q8C2×D1711yescharacteristic
331C22×C4⋊C4C2×D1711yescharacteristic
332C2×#107—171no—
333C2×C4⋊C8—171no—
334C2×#103—171no—
335C2×#103—171no—
336C2×#103—171no—
337C2×#103—171no—
338C2×#103—171no—
339C2×#103—171no—
340C2×C22⋊C8—171no—
341C2×C22⋊C8—171no—
342C2×C4×C8—171no—
343#400—341no—
344#393—341no—
345#391—171no—
346#391—171no—
347#391—341no—
348#391C2×D1711yescharacteristic
349#389—21no—
350#389C2×D1711yescharacteristic
351#388—341no—
352#387—21no—
353#385—21no—
354#380—171no—
355C4×C4⋊C4—171no—
356C4×C4⋊C4C2×D1711yescharacteristic
357C4×C22⋊C4—21no—
358C4×C22⋊C4C2×D1711yescharacteristic
359#340—171no—
360#340—171no—
361#340—171no—
362#340—171no—
363#340—171no—
364#340—171no—
365#340—171no—
366#340—171no—
367#336—171no—
368#336—171no—
369C2×C17—42no—
370C2×C17#245911yescharacteristic
371C2×C17#310411yescharacteristic
372C2×C17#312511yescharacteristic
373C2×C17#329911yescharacteristic
374C2×C17#329911yescharacteristic
375C2×C17#331111yescharacteristic
376C2×C17#332311yescharacteristic
377C22×Q8—171no—
378C22×Q8—341no—
379C22×Q8C22×D1711yescharacteristic
380C23×C4—21no—
381C23×C4C22×D1711yescharacteristic
382C2×Q16—342no—
383C2×Q16—681no—
384C2×Q16—681no—
385C22×C8—171no—
386C22×C8—171no—
387C22×C8—171no—
388C22×C8—171no—
389C4⋊Q8—342no—
390C4⋊Q8—681no—
391C22⋊Q8—22no—
392C22⋊Q8—22no—
393C22⋊Q8—342no—
394C22⋊Q8—681no—
395C4×Q8—22no—
396C4×Q8—41no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—22no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—41no—
403C2×C4⋊C4—171no—
404C2×C4⋊C4—171no—
405C2×C4⋊C4—171no—
406C2×C4⋊C4—341no—
407C2×C4⋊C4—341no—
408C2×C4⋊C4C22×D1711yescharacteristic
409C2×C4⋊C4C22×D1711yescharacteristic
410C2×C4⋊C4C22×D1711yescharacteristic
411C2×C22⋊C4—21no—
412C2×C22⋊C4—341no—
413C2×C22⋊C4—341no—
414C2×C22⋊C4C22⋊D1711yescharacteristic
415C2×C22⋊C4C22⋊D1711yescharacteristic
416C2×C42—21no—
417C2×C42—21no—
418C2×C42—341no—
419C2×C42C22×D1711yescharacteristic
420C2×C42C22×D1711yescharacteristic
421#107—342no—
422C4⋊C8—681no—
423#103—342no—
424#103—342no—
425#103—342no—
426#103—342no—
427#103—681no—
428#103—681no—
429C22⋊C8—681no—
430C22⋊C8—681no—
431C4×C8—681no—
432#95—21no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95—341no—
437#95—341no—
438#95—341no—
439#95—341no—
440#95—341no—
441#95—341no—
442#95C22⋊D1711yescharacteristic
443#95C22⋊D1711yescharacteristic
444C17#2672611yescharacteristic, complemented, Sylow
445C24C2×C22⋊D1711yescharacteristic
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—341no—
452C2×Q8—341no—
453C2×Q8—341no—
454C2×Q8—341no—
455C2×Q8—341no—
456C2×Q8—341no—
457C2×Q8—682no—
458C2×Q8—682no—
459C2×Q8—1361no—
460C2×Q8#6325212yes—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—81no—
474C22×C4—341no—
475C22×C4—341no—
476C22×C4—341no—
477C22×C4—341no—
478C22×C4C2×C22⋊D1711yescharacteristic
479C22×C4C2×C22⋊D1711yescharacteristic
480C22×C4C2×C22⋊D1711yescharacteristic
481C22×C4C2×C22⋊D1711yescharacteristic
482C22×C4C2×C22⋊D1711yescharacteristic
483C22×C4C23×D1711yescharacteristic
484Q16—684no—
485C2×C8—341no—
486C2×C8—341no—
487C2×C8—341no—
488C2×C8—341no—
489C2×C8—341no—
490C2×C8—341no—
491C2×C8—341no—
492C2×C8—341no—
493C2×C8—341no—
494C2×C8—341no—
495C2×C8—341no—
496C2×C8—341no—
497C4⋊C4—24no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—81no—
505C4⋊C4—172no—
506C4⋊C4—341no—
507C4⋊C4—341no—
508C4⋊C4—341no—
509C4⋊C4—341no—
510C4⋊C4—341no—
511C4⋊C4—682no—
512C4⋊C4—682no—
513C4⋊C4#6325212yes—
514C4⋊C4#6325412yes—
515C22⋊C4—24no—
516C22⋊C4—41no—
517C22⋊C4—682no—
518C22⋊C4—1361no—
519C22⋊C4#6323114yescomplemented
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—42no—
525C42—81no—
526C42—1361no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23C2×#6323112yes—
534C23C22×C22⋊D1711yescharacteristic
535Q8—22no—
536Q8—22no—
537Q8—41no—
538Q8—41no—
539Q8—681no—
540Q8—681no—
541Q8—681no—
542Q8—681no—
543Q8—1364no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
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—84no—
574C2×C4—342no—
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×C4—681no—
584C2×C4—681no—
585C2×C4—681no—
586C2×C4C2×#6323112yes—
587C2×C4C2×#6323112yes—
588C2×C4C2×#6325211yescharacteristic
589C2×C4C2×#6325211yescharacteristic
590C2×C4C2×#6325411yescharacteristic
591C2×C4#1058769111yescharacteristic
592C2×C4#41×D1711yescharacteristic
593C2×C4#1058769611yescharacteristic
594C8—681no—
595C8—681no—
596C8—681no—
597C8—681no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22#1176332011yescharacteristic
606C22#1176344711yescharacteristic
607C22#1176345311yescharacteristic
608C22C22×#6323111yescharacteristic
609C22#1176360711yescharacteristic
610C22#1176361711yescharacteristic
611C22#1176362011yescharacteristic
612C4—22no—
613C4—22no—
614C4—41no—
615C4—42no—
616C4—42no—
617C4—41no—
618C4—42no—
619C4—41no—
620C4—81no—
621C4—682no—
622C4—1361no—
623C4—1361no—
624C4—1361no—
625C4#1176333911yescharacteristic
626C4#1176339211yescharacteristic
627C4#1176352411yescharacteristic
628C4#1176354211yescharacteristic
629C2—42no—
630C2#1453271511yescharacteristic
631C2#1453882311yescharacteristic
632C2#1453906111yescharacteristic
633C2#1454065411yescharacteristic
634C2#1454066011yescharacteristic
635C2#1454082011yescharacteristic
636C2#1454099411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2360C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenter#2360D17
hypocenterC17#26726
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2360

Complements

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

NH ≅ G/Ntype
2. #14530106C2semidirect
4. #14529925C2semidirect
6. #14529732C2semidirect
7. #14529459C2semidirect
8. #14529425C2semidirect
10. #14529373C2semidirect
11. #14529125C2semidirect
14. C4×#10587506C2semidirect
177. C22⋊C4×C17Q16semidirect
444. C17#26726semidirect
519. C22⋊C4#63231semidirect

Maximal subgroups

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

subgroupindex
#145331582
#145301062
#145299842
#145299252
#145297922
#145297322
#145294592
#145294252
#145293922
#145293732
#145291252
#145290912
#145288742
C4×#105875062
C17×#23602
#2672617