Finite Group Databasebeta

← #17107485

Subgroups — #17107485

Order 4352.

848 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14533359C211yesmaximal, characteristic, complemented
2#14530317C211yesmaximal, characteristic, complemented
3#14530101C211yesmaximal, characteristic, complemented
4#14530015C211yesmaximal, characteristic
5#14529872C211yesmaximal, characteristic
6#14529637C211yesmaximal, characteristic, complemented
7#14529418C211yesmaximal, characteristic, complemented
8#14529408C211yesmaximal, characteristic, complemented
9#14529369C211yesmaximal, characteristic, complemented
10#14529343C211yesmaximal, characteristic, complemented
11#14529118C211yesmaximal, characteristic, complemented
12#14529118C211yesmaximal, characteristic, complemented
13#14528784C211yesmaximal, characteristic, complemented
14C4×#10587503C211yesmaximal, characteristic, complemented
15C17×#2366C211yesmaximal, characteristic
16C22×#63229C2211yescharacteristic
17C22×#63229C2211yescharacteristic
18C2×#10587479C2211yescharacteristic
19C2×#10587479C2211yescharacteristic
20C2×C4⋊#3622C2211yescharacteristic
21C2×#10587481C2211yescharacteristic
22C2×#10587481C2211yescharacteristic
23C2×#10587503C2211yescharacteristic
24C2×#10587503C2211yescharacteristic
25C2×#10587503C2211yescharacteristic
26C2×#10587503C2211yescharacteristic
27C2×#10587503C2211yescharacteristic
28C2×#10587503C2211yescharacteristic
29C2×C4×#3622C2211yescharacteristic
30#11762751—21no—
31#11762748—21no—
32#11762715C2211yescharacteristic
33#11762715C2211yescharacteristic
34#11762715C2211yescharacteristic
35#11762707C2211yescharacteristic
36#11762695—21no—
37#11762692—21no—
38#11762679C2211yescharacteristic
39C4×C4⋊Dic17C2211yescharacteristic
40#11762571C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762564C2211yescharacteristic
45#11762564C2211yescharacteristic
46#11762564C2211yescharacteristic
47#11762564C2211yescharacteristic
48C2×C17×#127C2211yescharacteristic
49C2×C4×D4×C17—21no—
50C2×C4×D4×C17C2211yescharacteristic
51C2×C4×D4×C17C2211yescharacteristic
52C17×#390—21no—
53C17×#390C2211yescharacteristic
54C17×#390C2211yescharacteristic
55C17×#386—21no—
56C17×#386—21no—
57C4×C4⋊C4×C17C2211yescharacteristic
58C4×C4⋊C4×C17C2211yescharacteristic
59C22×D4×C17—21no—
60C22×D4×C17—21no—
61C22×D4×C17C2311yescharacteristic
62C22×D4×C17C2311yescharacteristic
63C23×C4×C17—21no—
64C23×C4×C17—21no—
65C22×Dic34C2311yescharacteristic
66C17×#127—22no—
67C17×#127—22no—
68C4×D4×C17—22no—
69C4×D4×C17—22no—
70C4×D4×C17—41no—
71C4×D4×C17—41no—
72C4×D4×C17—44no—
73C2×C4⋊C4×C17D411yescharacteristic
74C2×C4⋊C4×C17D411yescharacteristic
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C4⋊C4×C17C2311yescharacteristic
77C2×C22⋊C4×C17—21no—
78C2×C22⋊C4×C17—21no—
79C2×C22⋊C4×C17—21no—
80C2×C22⋊C4×C17—21no—
81C2×C22⋊C4×C17—21no—
82C2×C22⋊C4×C17—21no—
83C2×C42×C17—21no—
84C2×C42×C17C2311yescharacteristic
85C2×C42×C17C2311yescharacteristic
86C2×C42×C17C2311yescharacteristic
87C2×#63229—22no—
88C2×#63229—22no—
89C2×#63229—22no—
90C2×#63229—41no—
91C2×#63229—41no—
92C2×#63229—41no—
93C2×C4⋊Dic17—21no—
94C2×C4⋊Dic17—21no—
95C2×C4⋊Dic17C2311yescharacteristic
96C2×C4⋊Dic17C2311yescharacteristic
97C2×C4⋊Dic17C2311yescharacteristic
98C2×#63225—21no—
99C2×C4×Dic17—21no—
100C22×#3622C2311yescharacteristic
101C22×#3622C2311yescharacteristic
102C22×#3622C2311yescharacteristic
103C22×#3622C2311yescharacteristic
104C17×#95—21no—
105C17×#95D411yescharacteristic
106C17×#95D411yescharacteristic
107#10587503—22no—
108#10587503—22no—
109#10587503—22no—
110#10587503—41no—
111#10587503—41no—
112#10587503—41no—
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587502—21no—
120#10587502—21no—
121#10587481—22no—
122#10587481—41no—
123#10587479—22no—
124#10587479—41no—
125C4⋊#3622—41no—
126C4×#3622—41no—
127C24×C17—21no—
128C24×C17—21no—
129C2×D4×C17—24no—
130C2×D4×C17—24no—
131C2×D4×C17—21no—
132C2×D4×C17—21no—
133C2×D4×C17—21no—
134C2×D4×C17—21no—
135C2×D4×C17—21no—
136C2×D4×C17—21no—
137C2×D4×C17—21no—
138C2×D4×C17—21no—
139C2×D4×C17—21no—
140C2×D4×C17—21no—
141C2×D4×C17—42no—
142C2×D4×C17—42no—
143C2×D4×C17—81no—
144C2×D4×C17—81no—
145C2×D4×C17#4112yes—
146C2×D4×C17#4112yes—
147C22×C4×C17—21no—
148C22×C4×C17—21no—
149C22×C4×C17—42no—
150C22×C4×C17—42no—
151C22×C4×C17—42no—
152C22×C4×C17—42no—
153C22×C4×C17—81no—
154C22×C4×C17—81no—
155C22×C4×C17C2×D411yescharacteristic
156C22×C4×C17C2×D411yescharacteristic
157C22×C4×C17C2×D411yescharacteristic
158C22×C4×C17C2×D411yescharacteristic
159C22×C4×C17C2×D411yescharacteristic
160C22×C4×C17C2×D411yescharacteristic
161C22×C4×C17C2411yescharacteristic
162C22×Dic17—21no—
163C22×Dic17—21no—
164C22×Dic17—21no—
165C22×Dic17—21no—
166C2×Dic34—21no—
167C2×Dic34—21no—
168C2×Dic34—21no—
169C2×Dic34—21no—
170C2×Dic34—21no—
171C2×Dic34—21no—
172C4⋊C4×C17—21no—
173C4⋊C4×C17—21no—
174C4⋊C4×C17—21no—
175C4⋊C4×C17—41no—
176C4⋊C4×C17SD1614yescomplemented
177C4⋊C4×C17#4112yes—
178C22⋊C4×C17—42no—
179C22⋊C4×C17—42no—
180C22⋊C4×C17—42no—
181C22⋊C4×C17—42no—
182C22⋊C4×C17—81no—
183C22⋊C4×C17—81no—
184C42×C17—21no—
185C42×C17—21no—
186C42×C17—21no—
187C42×C17—21no—
188C42×C17—21no—
189C42×C17—21no—
190C42×C17—42no—
191#63229—44no—
192#63229—44no—
193C4⋊Dic17—21no—
194C4⋊Dic17—21no—
195C4⋊Dic17—21no—
196C4⋊Dic17—21no—
197C4⋊Dic17—21no—
198C4⋊Dic17—21no—
199C4⋊Dic17—42no—
200C4⋊Dic17—81no—
201#63225—81no—
202C4×Dic17—81no—
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×#3622—21no—
212C2×#3622—21no—
213C2×#3622—21no—
214C2×#3622#4112yes—
215#27064—171nomaximal, Sylow
216C23×C17—22no—
217C23×C17—22no—
218C23×C17—41no—
219C23×C17—41no—
220C23×C17—41no—
221C23×C17—41no—
222C23×C17—41no—
223C23×C17—41no—
224C23×C17—41no—
225C23×C17—41no—
226C23×C17—41no—
227C23×C17—41no—
228C23×C17—41no—
229C23×C17—41no—
230C23×C17C22×D411yescharacteristic
231D4×C17—22no—
232D4×C17—22no—
233D4×C17—22no—
234D4×C17—22no—
235D4×C17—48no—
236D4×C17—41no—
237D4×C17—41no—
238D4×C17—41no—
239D4×C17—41no—
240D4×C17—84no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—21no—
244C2×C4×C17—21no—
245C2×C4×C17—21no—
246C2×C4×C17—21no—
247C2×C4×C17—21no—
248C2×C4×C17—21no—
249C2×C4×C17—21no—
250C2×C4×C17—22no—
251C2×C4×C17—21no—
252C2×C4×C17—21no—
253C2×C4×C17—21no—
254C2×C4×C17—21no—
255C2×C4×C17—21no—
256C2×C4×C17—21no—
257C2×C4×C17—41no—
258C2×C4×C17—41no—
259C2×C4×C17—41no—
260C2×C4×C17—41no—
261C2×C4×C17—41no—
262C2×C4×C17—84no—
263C2×C4×C17—84no—
264C2×C4×C17C2×SD1612yes—
265C2×C4×C17C2×SD1612yes—
266C2×C4×C17C2×SD1612yes—
267C2×C4×C17C2×#4111yescharacteristic
268C2×C4×C17C2×#4111yescharacteristic
269C2×C4×C17C2×#4111yescharacteristic
270C2×C4×C17C2×#4111yescharacteristic
271C2×C4×C17H(5,𝔽2)11yescharacteristic
272C2×C4×C17H(5,𝔽2)11yescharacteristic
273C2×Dic17—41no—
274C2×Dic17—41no—
275C2×Dic17—41no—
276C2×Dic17—41no—
277C2×Dic17—41no—
278C2×Dic17—41no—
279C2×Dic17—41no—
280C2×Dic17—41no—
281C2×Dic17—41no—
282C2×Dic17—41no—
283C2×Dic17—41no—
284C2×Dic17—41no—
285Dic34—41no—
286Dic34—41no—
287Dic34—41no—
288Dic34—41no—
289#3622—22no—
290#3622—41no—
291#3622—41no—
292#3622—41no—
293#2511—171no—
294#2366D1711yescharacteristic
295#2094—171no—
296#2055—171no—
297#2041—171no—
298#2014—171no—
299#1968—171no—
300#1927—171no—
301#1926—171no—
302#1920—171no—
303#1916—171no—
304#1877—171no—
305#1877—171no—
306#1802—171no—
307C4×#102—171no—
308C22×C17—42no—
309C22×C17—42no—
310C22×C17—42no—
311C22×C17—42no—
312C22×C17—42no—
313C22×C17—42no—
314C22×C17—81no—
315C22×C17—81no—
316C22×C17—81no—
317C22×C17—81no—
318C22×C17—81no—
319C22×C17—81no—
320C22×C17—81no—
321C22×C17—81no—
322C22×C17#52511yescharacteristic
323C22×C17#54611yescharacteristic
324C22×C17#55011yescharacteristic
325C22×C17C22×SD1611yescharacteristic
326C22×C17#57511yescharacteristic
327C22×C17#57611yescharacteristic
328C22×C17#57711yescharacteristic
329C4×C17—22no—
330C4×C17—22no—
331C4×C17—22no—
332C4×C17—41no—
333C4×C17—42no—
334C4×C17—41no—
335C4×C17—41no—
336C4×C17—81no—
337C4×C17#52911yescharacteristic
338C4×C17#53811yescharacteristic
339C4×C17#55111yescharacteristic
340C4×C17#55311yescharacteristic
341Dic17—81no—
342Dic17—81no—
343Dic17—81no—
344Dic17—81no—
345C22×SD16—171no—
346C22×SD16—171no—
347C2×#127C2×D1711yescharacteristic
348C2×C4×D4—21no—
349C2×C4×D4C2×D1711yescharacteristic
350C2×C4×D4C2×D1711yescharacteristic
351C2×#106—171no—
352C2×#106—171no—
353C2×C4⋊C8—171no—
354C2×#103—171no—
355C2×#103—171no—
356C2×#102—171no—
357C2×#102—171no—
358C2×#102—171no—
359C2×#102—171no—
360C2×#102—171no—
361C2×#102—171no—
362C2×C4×C8—171no—
363#400—341no—
364#400—341no—
365#391—171no—
366#391—171no—
367#391—171no—
368#390—21no—
369#390C2×D1711yescharacteristic
370#390C2×D1711yescharacteristic
371#389—171no—
372#387—341no—
373#387—341no—
374#386—21no—
375#386—21no—
376#384—171no—
377C4×C4⋊C4—171no—
378C4×C4⋊C4C2×D1711yescharacteristic
379C4×C4⋊C4C2×D1711yescharacteristic
380#340—171no—
381#340—171no—
382#340—171no—
383#340—171no—
384#336—171no—
385#336—171no—
386#336—171no—
387#336—171no—
388C2×C17—82no—
389C2×C17—82no—
390C2×C17#246711yescharacteristic
391C2×C17#310111yescharacteristic
392C2×C17#311811yescharacteristic
393C2×C17#329511yescharacteristic
394C2×C17#331711yescharacteristic
395C2×C17#339211yescharacteristic
396C2×C17#340311yescharacteristic
397C22×Q8—171no—
398C22×D4—21no—
399C22×D4—21no—
400C22×D4C22×D1711yescharacteristic
401C22×D4C22×D1711yescharacteristic
402C23×C4—21no—
403C23×C4—21no—
404C2×SD16—342no—
405C2×SD16—342no—
406C2×SD16—342no—
407C2×SD16—681no—
408C2×SD16—681no—
409C2×SD16—681no—
410C22×C8—171no—
411C22×C8—171no—
412C22×C8—171no—
413C22×C8—171no—
414#127—22no—
415#127—22no—
416C4×D4—22no—
417C4×D4—22no—
418C4×D4—41no—
419C4×D4—41no—
420C4×D4—44no—
421C2×C4⋊C4—171no—
422C2×C4⋊C4—171no—
423C2×C4⋊C4—171no—
424C2×C4⋊C4—341no—
425C2×C4⋊C4—341no—
426C2×C4⋊C4—341no—
427C2×C4⋊C4C22⋊D1711yescharacteristic
428C2×C4⋊C4C22⋊D1711yescharacteristic
429C2×C4⋊C4C22×D1711yescharacteristic
430C2×C4⋊C4C22×D1711yescharacteristic
431C2×C22⋊C4—21no—
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C42—21no—
438C2×C42—341no—
439C2×C42C22×D1711yescharacteristic
440C2×C42C22×D1711yescharacteristic
441C2×C42C22×D1711yescharacteristic
442#106—342no—
443#106—681no—
444C4⋊C8—681no—
445#103—342no—
446#103—681no—
447#102—342no—
448#102—342no—
449#102—342no—
450#102—681no—
451#102—681no—
452#102—681no—
453C4×C8—681no—
454#95—21no—
455#95—341no—
456#95—341no—
457#95—341no—
458#95—341no—
459#95—341no—
460#95—341no—
461#95—341no—
462#95—341no—
463#95C22⋊D1711yescharacteristic
464#95C22⋊D1711yescharacteristic
465C17#2706411yescharacteristic, complemented, Sylow
466C24—21no—
467C24—21no—
468C2×Q8—341no—
469C2×Q8—341no—
470C2×Q8—341no—
471C2×Q8—341no—
472C2×Q8—341no—
473C2×Q8—341no—
474C2×D4—24no—
475C2×D4—24no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—42no—
487C2×D4—42no—
488C2×D4—81no—
489C2×D4—81no—
490C2×D4#6325212yes—
491C2×D4#6325212yes—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—81no—
499C22×C4—81no—
500C22×C4—341no—
501C22×C4—341no—
502C22×C4—341no—
503C22×C4—341no—
504C22×C4C2×C22⋊D1711yescharacteristic
505C22×C4C2×C22⋊D1711yescharacteristic
506C22×C4C2×C22⋊D1711yescharacteristic
507C22×C4C2×C22⋊D1711yescharacteristic
508C22×C4C2×C22⋊D1711yescharacteristic
509C22×C4C2×C22⋊D1711yescharacteristic
510C22×C4C23×D1711yescharacteristic
511SD16—684no—
512SD16—684no—
513C2×C8—172no—
514C2×C8—341no—
515C2×C8—341no—
516C2×C8—341no—
517C2×C8—341no—
518C2×C8—341no—
519C2×C8—341no—
520C2×C8—341no—
521C2×C8—341no—
522C2×C8—341no—
523C2×C8—341no—
524C2×C8—341no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—41no—
529C4⋊C4—341no—
530C4⋊C4—341no—
531C4⋊C4—341no—
532C4⋊C4—341no—
533C4⋊C4—341no—
534C4⋊C4—341no—
535C4⋊C4—682no—
536C4⋊C4—1361no—
537C4⋊C4—1361no—
538C4⋊C4#6322914yescomplemented
539C4⋊C4#6325412yes—
540C22⋊C4—42no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—81no—
545C22⋊C4—81no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42—21no—
552C42—42no—
553C42—1361no—
554C23—22no—
555C23—22no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23C22×C22⋊D1711yescharacteristic
569Q8—681no—
570Q8—681no—
571Q8—681no—
572Q8—681no—
573D4—22no—
574D4—22no—
575D4—22no—
576D4—22no—
577D4—48no—
578D4—41no—
579D4—41no—
580D4—41no—
581D4—41no—
582D4—84no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
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—22no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—84no—
605C2×C4—84no—
606C2×C4—681no—
607C2×C4—681no—
608C2×C4—681no—
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×C4C2×#6322912yes—
619C2×C4C2×#6322912yes—
620C2×C4C2×#6322912yes—
621C2×C4C2×#6325211yescharacteristic
622C2×C4C2×#6325211yescharacteristic
623C2×C4C2×#6325411yescharacteristic
624C2×C4#1058769111yescharacteristic
625C2×C4#41×D1711yescharacteristic
626C2×C4#1058769611yescharacteristic
627C8—342no—
628C8—681no—
629C8—681no—
630C8—681no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22#1176332011yescharacteristic
646C22#1176345311yescharacteristic
647C22#1176347811yescharacteristic
648C22C22×#6322911yescharacteristic
649C22#1176360711yescharacteristic
650C22#1176361211yescharacteristic
651C22#1176362011yescharacteristic
652C4—22no—
653C4—22no—
654C4—22no—
655C4—42no—
656C4—41no—
657C4—41no—
658C4—41no—
659C4—81no—
660C4—1361no—
661C4—1361no—
662C4—1361no—
663C4—1361no—
664C4#1176333911yescharacteristic
665C4#1176339211yescharacteristic
666C4#1176348811yescharacteristic
667C4#1176350511yescharacteristic
668C2—82no—
669C2—82no—
670C2#1453282911yescharacteristic
671C2#1453879211yescharacteristic
672C2#1453896811yescharacteristic
673C2#1454060811yescharacteristic
674C2#1454091011yescharacteristic
675C2#1454176511yescharacteristic
676C2#1454186811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2366C2
socleC23×C17C22×D4
prime radicalC2×C17×#127C22
hypercenter#2366D17
hypocenterC17#27064
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2366

Complements

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

NH ≅ G/Ntype
1. #14533359C2semidirect
2. #14530317C2semidirect
3. #14530101C2semidirect
6. #14529637C2semidirect
7. #14529418C2semidirect
8. #14529408C2semidirect
9. #14529369C2semidirect
10. #14529343C2semidirect
11. #14529118C2semidirect
12. #14529118C2semidirect
13. #14528784C2semidirect
14. C4×#10587503C2semidirect
176. C4⋊C4×C17SD16semidirect
465. C17#27064semidirect
538. C4⋊C4#63229semidirect

Maximal subgroups

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

subgroupindex
#145333592
#145303172
#145301012
#145300152
#145298722
#145296372
#145294182
#145294082
#145293692
#145293432
#145291182
#145287842
C4×#105875032
C17×#23662
#2706417