Finite Group Databasebeta

← #17102409

Subgroups — #17102409

Order 4352.

826 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14533147C211yesmaximal, characteristic
2#14530099C211yesmaximal, characteristic, complemented
3#14529977C211yesmaximal, characteristic, complemented
4#14529925C211yesmaximal, characteristic, complemented
5#14529785C211yesmaximal, characteristic
6#14529736C211yesmaximal, characteristic, complemented
7#14529466C211yesmaximal, characteristic, complemented
8#14529411C211yesmaximal, characteristic, complemented
9#14529392C211yesmaximal, characteristic, complemented
10#14529366C211yesmaximal, characteristic, complemented
11#14529139C211yesmaximal, characteristic, complemented
12#14529101C211yesmaximal, characteristic
13#14528874C211yesmaximal, characteristic, complemented
14C4×#10587506C211yesmaximal, characteristic, complemented
15C17×#2360C211yesmaximal, characteristic, complemented
16C22×#63230C2211yescharacteristic
17C2×#10587555C2211yescharacteristic
18C2×#10587571—21no—
19C2×#10587616C2211yescharacteristic
20C2×#10587479C2211yescharacteristic
21C2×C4⋊#3622C2211yescharacteristic
22C2×#10587506C2211yescharacteristic
23C2×#10587506C2211yescharacteristic
24C2×#10587506C2211yescharacteristic
25C2×#10587480C2211yescharacteristic
26C2×#10587480C2211yescharacteristic
27C2×#10587480C2211yescharacteristic
28C2×C22⋊#3622C2211yescharacteristic
29C2×C22⋊#3622C2211yescharacteristic
30C2×C4×#3622C2211yescharacteristic
31#11762725—21no—
32#11762722—21no—
33#11762711C2211yescharacteristic
34#11762711C2211yescharacteristic
35#11762701—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×C22⋊D17—21no—
63C22×D68C2311yescharacteristic
64C22×Dic34—21no—
65C17×C22⋊Q8—22no—
66C17×C22⋊Q8—22no—
67C4×Q8×C17—22no—
68C4×Q8×C17—41no—
69C2×C4⋊C4×C17—21no—
70C2×C4⋊C4×C17—22no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17—21no—
73C2×C4⋊C4×C17—21no—
74C2×C4⋊C4×C17—41no—
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C4⋊C4×C17C2311yescharacteristic
77C2×C4⋊C4×C17C2311yescharacteristic
78C2×C22⋊C4×C17—21no—
79C2×C22⋊C4×C17D411yescharacteristic
80C2×C22⋊C4×C17D411yescharacteristic
81C2×C42×C17—21no—
82C2×C42×C17—21no—
83C2×C42×C17C2311yescharacteristic
84C2×C42×C17C2311yescharacteristic
85C2×C22⋊Dic17—21no—
86C2×#63230—22no—
87C2×#63230—41no—
88C2×#63230—41no—
89#10587616—22no—
90#10587616—41no—
91C2×#63227—21no—
92C2×#63227—21no—
93C2×#63227—21no—
94C2×#63227—21no—
95C2×#63227—21no—
96C2×C4⋊Dic17—21no—
97C2×C4⋊Dic17C2311yescharacteristic
98C2×C4⋊Dic17C2311yescharacteristic
99C2×C4⋊Dic17C2311yescharacteristic
100C2×C4×Dic17—21no—
101C22×#3622C2311yescharacteristic
102C22×#3622C2311yescharacteristic
103C22×#3622C2311yescharacteristic
104C22×#3622C2311yescharacteristic
105#10587571—44no—
106#10587555—22no—
107#10587555—41no—
108C17×#95—21no—
109C17×#95—21no—
110C17×#95—21no—
111C17×#95—21no—
112C17×#95D411yescharacteristic
113C17×#95D411yescharacteristic
114#10587506—22no—
115#10587506—22no—
116#10587506—41no—
117#10587502—21no—
118#10587502—21no—
119C22⋊#3622—41no—
120C22⋊#3622—41no—
121#10587480—22no—
122#10587480—22no—
123#10587480—41no—
124#10587479—22no—
125C4⋊#3622—41no—
126C4×#3622—41no—
127C24×C17C2×D411yescharacteristic
128C23×D17—21no—
129C2×Q8×C17—21no—
130C2×Q8×C17—21no—
131C2×Q8×C17—21no—
132C2×Q8×C17—21no—
133C2×Q8×C17—21no—
134C2×Q8×C17#4112yes—
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—21no—
144C22×C4×C17—21no—
145C22×C4×C17—42no—
146C22×C4×C17—42no—
147C22×C4×C17—81no—
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2×D411yescharacteristic
151C22×C4×C17C2×D411yescharacteristic
152C22×C4×C17C2×D411yescharacteristic
153C22×C4×C17C2411yescharacteristic
154C2×C22⋊D17—42no—
155C2×C22⋊D17—42no—
156C2×C22⋊D17—81no—
157C22×Dic17—21no—
158C22×Dic17—21no—
159C22×Dic17—21no—
160C2×D68—21no—
161C2×D68—21no—
162C2×D68—21no—
163C2×D68—21no—
164C2×D68—21no—
165C2×D68—21no—
166C2×Dic34—42no—
167C2×Dic34—42no—
168C2×Dic34—81no—
169C4⋊C4×C17—24no—
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—81no—
177C4⋊C4×C17#4112yes—
178C4⋊C4×C17#4112yes—
179C22⋊C4×C17—24no—
180C22⋊C4×C17—41no—
181C22⋊C4×C17SD1614yescomplemented
182C42×C17—21no—
183C42×C17—21no—
184C42×C17—21no—
185C42×C17—21no—
186C42×C17—42no—
187C42×C17—81no—
188C22⋊Dic17—42no—
189#63230—44no—
190#63227—42no—
191#63227—42no—
192#63227—42no—
193#63227—42no—
194#63227—81no—
195C4⋊Dic17—21no—
196C4⋊Dic17—21no—
197C4⋊Dic17—21no—
198C4⋊Dic17—21no—
199C4⋊Dic17—21no—
200C4⋊Dic17—42no—
201C4⋊Dic17#4112yes—
202C4×Dic17—42no—
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—21no—
215#26724—171nomaximal, Sylow
216C23×C17—21no—
217C23×C17—21no—
218C23×C17—21no—
219C23×C17—21no—
220C23×C17—21no—
221C23×C17—21no—
222C23×C17C2×SD1612yes—
223C23×C17C22×D411yescharacteristic
224C22×D17—41no—
225C22×D17—41no—
226C22×D17—41no—
227C22×D17—41no—
228C22×D17—41no—
229C22×D17—41no—
230C22×D17—41no—
231Q8×C17—22no—
232Q8×C17—22no—
233Q8×C17—41no—
234Q8×C17—41no—
235C2×C4×C17—21no—
236C2×C4×C17—21no—
237C2×C4×C17—21no—
238C2×C4×C17—21no—
239C2×C4×C17—22no—
240C2×C4×C17—22no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—21no—
244C2×C4×C17—22no—
245C2×C4×C17—21no—
246C2×C4×C17—21no—
247C2×C4×C17—21no—
248C2×C4×C17—21no—
249C2×C4×C17—21no—
250C2×C4×C17—21no—
251C2×C4×C17—41no—
252C2×C4×C17—41no—
253C2×C4×C17—41no—
254C2×C4×C17—41no—
255C2×C4×C17—41no—
256C2×C4×C17—41no—
257C2×C4×C17—41no—
258C2×C4×C17—41no—
259C2×C4×C17—41no—
260C2×C4×C17—41no—
261C2×C4×C17—41no—
262C2×C4×C17—41no—
263C2×C4×C17—41no—
264C2×C4×C17—84no—
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
273C22⋊D17—84no—
274C2×Dic17—22no—
275C2×Dic17—41no—
276C2×Dic17—41no—
277C2×Dic17—41no—
278C2×Dic17—41no—
279C2×Dic17—41no—
280C2×Dic17—41no—
281C2×Dic17—41no—
282C2×Dic17—41no—
283D68—41no—
284D68—41no—
285D68—41no—
286D68—41no—
287Dic34—84no—
288#3622—41no—
289#3622—41no—
290#3622—41no—
291#3622—41no—
292#2491—171no—
293#2360D1711yescharacteristic, complemented
294#2055—171no—
295#2035—171no—
296#2024—171no—
297#2000—171no—
298#1988—171no—
299#1934—171no—
300#1926—171no—
301#1923—171no—
302#1920—171no—
303#1880—171no—
304#1874—171no—
305#1824—171no—
306C4×#103—171no—
307C22×C17—22no—
308C22×C17—22no—
309C22×C17—22no—
310C22×C17—41no—
311C22×C17—41no—
312C22×C17—41no—
313C22×C17—41no—
314C22×C17#52511yescharacteristic
315C22×C17#54611yescharacteristic
316C22×C17#54611yescharacteristic
317C22×C17C22×SD1611yescharacteristic
318C22×C17#57511yescharacteristic
319C22×C17#57611yescharacteristic
320C22×C17#57811yescharacteristic
321C2×D17—81no—
322C2×D17—81no—
323C2×D17—81no—
324C2×D17—81no—
325C2×D17—81no—
326C2×D17—81no—
327C2×D17—81no—
328C4×C17—22no—
329C4×C17—22no—
330C4×C17—42no—
331C4×C17—41no—
332C4×C17—42no—
333C4×C17—41no—
334C4×C17—42no—
335C4×C17—41no—
336C4×C17—81no—
337C4×C17#52911yescharacteristic
338C4×C17#53811yescharacteristic
339C4×C17#55511yescharacteristic
340C4×C17#55911yescharacteristic
341Dic17—42no—
342Dic17—81no—
343Dic17—81no—
344C22×SD16—171no—
345C2×C4⋊Q8—171no—
346C2×#124—341no—
347C2×C22⋊Q8C2×D1711yescharacteristic
348C2×#121—171no—
349C2×C4×Q8C2×D1711yescharacteristic
350C22×C4⋊C4C2×D1711yescharacteristic
351C2×#106—171no—
352C2×C4⋊C8—171no—
353C2×#103—171no—
354C2×#103—171no—
355C2×#103—171no—
356C2×#102—171no—
357C2×#102—171no—
358C2×#102—171no—
359C2×C22⋊C8—171no—
360C2×C22⋊C8—171no—
361C2×C4×C8—171no—
362#394—341no—
363#393—341no—
364#391C2×D1711yescharacteristic
365#390—171no—
366#390—171no—
367#389—21no—
368#389C2×D1711yescharacteristic
369#388—341no—
370#387—21no—
371#385—21no—
372#380—171no—
373C4×C4⋊C4—171no—
374C4×C4⋊C4C2×D1711yescharacteristic
375C4×C22⋊C4—21no—
376C4×C22⋊C4C2×D1711yescharacteristic
377#340—171no—
378#340—171no—
379#340—171no—
380#340—171no—
381#340—171no—
382#340—171no—
383#340—171no—
384#340—171no—
385#336—171no—
386#336—171no—
387C2×C17—42no—
388C2×C17#245911yescharacteristic
389C2×C17#310311yescharacteristic
390C2×C17#312411yescharacteristic
391C2×C17#329511yescharacteristic
392C2×C17#329811yescharacteristic
393C2×C17#331911yescharacteristic
394C2×C17#332711yescharacteristic
395D17—161no—
396C22×Q8—341no—
397C22×Q8C22×D1711yescharacteristic
398C22×D4—171no—
399C22×D4—341no—
400C23×C4—21no—
401C23×C4C22×D1711yescharacteristic
402C2×SD16—342no—
403C2×SD16—681no—
404C2×SD16—681no—
405C22×C8—171no—
406C22×C8—171no—
407C22×C8—171no—
408C22×C8—171no—
409C4⋊Q8—342no—
410C4⋊Q8—681no—
411#124—684no—
412C22⋊Q8—22no—
413C22⋊Q8—22no—
414#121—342no—
415#121—681no—
416C4×Q8—22no—
417C4×Q8—41no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—22no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—41no—
424C2×C4⋊C4—171no—
425C2×C4⋊C4—171no—
426C2×C4⋊C4—171no—
427C2×C4⋊C4—341no—
428C2×C4⋊C4C22×D1711yescharacteristic
429C2×C4⋊C4C22×D1711yescharacteristic
430C2×C4⋊C4C22×D1711yescharacteristic
431C2×C22⋊C4—21no—
432C2×C22⋊C4—341no—
433C2×C22⋊C4—341no—
434C2×C22⋊C4—341no—
435C2×C22⋊C4—341no—
436C2×C22⋊C4—341no—
437C2×C22⋊C4—341no—
438C2×C22⋊C4C22⋊D1711yescharacteristic
439C2×C22⋊C4C22⋊D1711yescharacteristic
440C2×C42—21no—
441C2×C42—21no—
442C2×C42—341no—
443C2×C42C22×D1711yescharacteristic
444C2×C42C22×D1711yescharacteristic
445#106—342no—
446C4⋊C8—681no—
447#103—342no—
448#103—342no—
449#103—681no—
450#102—342no—
451#102—342no—
452#102—681no—
453C22⋊C8—681no—
454C22⋊C8—681no—
455C4×C8—681no—
456#95—21no—
457#95—21no—
458#95—21no—
459#95—21no—
460#95—341no—
461#95—341no—
462#95C22⋊D1711yescharacteristic
463#95C22⋊D1711yescharacteristic
464C17#2672411yescharacteristic, complemented, Sylow
465C24—341no—
466C24C2×C22⋊D1711yescharacteristic
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—682no—
473C2×Q8—682no—
474C2×Q8—1361no—
475C2×Q8#6325212yes—
476C2×D4—341no—
477C2×D4—341no—
478C2×D4—341no—
479C2×D4—341no—
480C2×D4—341no—
481C2×D4—341no—
482C2×D4—682no—
483C2×D4—682no—
484C2×D4—1361no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—81no—
498C22×C4—341no—
499C22×C4—341no—
500C22×C4—341no—
501C22×C4C2×C22⋊D1711yescharacteristic
502C22×C4C2×C22⋊D1711yescharacteristic
503C22×C4C2×C22⋊D1711yescharacteristic
504C22×C4C2×C22⋊D1711yescharacteristic
505C22×C4C2×C22⋊D1711yescharacteristic
506C22×C4C23×D1711yescharacteristic
507SD16—684no—
508C2×C8—341no—
509C2×C8—341no—
510C2×C8—341no—
511C2×C8—341no—
512C2×C8—341no—
513C2×C8—341no—
514C2×C8—341no—
515C2×C8—341no—
516C2×C8—341no—
517C2×C8—341no—
518C2×C8—341no—
519C2×C8—341no—
520C4⋊C4—24no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—81no—
528C4⋊C4—172no—
529C4⋊C4—341no—
530C4⋊C4—341no—
531C4⋊C4—341no—
532C4⋊C4—341no—
533C4⋊C4—341no—
534C4⋊C4—682no—
535C4⋊C4#6325212yes—
536C4⋊C4#6325412yes—
537C22⋊C4—24no—
538C22⋊C4—41no—
539C22⋊C4—682no—
540C22⋊C4—682no—
541C22⋊C4—682no—
542C22⋊C4—682no—
543C22⋊C4—682no—
544C22⋊C4—1361no—
545C22⋊C4#6323014yescomplemented
546C42—21no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—42no—
551C42—81no—
552C42—682no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—681no—
560C23—681no—
561C23—681no—
562C23—681no—
563C23—681no—
564C23—681no—
565C23—681no—
566C23C2×#6323012yes—
567C23C22×C22⋊D1711yescharacteristic
568Q8—22no—
569Q8—22no—
570Q8—41no—
571Q8—41no—
572Q8—1364no—
573D4—681no—
574D4—681no—
575D4—681no—
576D4—681no—
577D4—1364no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
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—22no—
593C2×C4—21no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—84no—
608C2×C4—342no—
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×C4C2×#6323012yes—
618C2×C4C2×#6323012yes—
619C2×C4C2×#6325211yescharacteristic
620C2×C4C2×#6325211yescharacteristic
621C2×C4C2×#6325411yescharacteristic
622C2×C4#1058769111yescharacteristic
623C2×C4#41×D1711yescharacteristic
624C2×C4#1058769611yescharacteristic
625C8—681no—
626C8—681no—
627C8—681no—
628C8—681no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—1361no—
637C22—1361no—
638C22—1361no—
639C22—1361no—
640C22—1361no—
641C22—1361no—
642C22—1361no—
643C22#1176332011yescharacteristic
644C22#1176344711yescharacteristic
645C22#1176345311yescharacteristic
646C22C22×#6323011yescharacteristic
647C22#1176360711yescharacteristic
648C22#1176361211yescharacteristic
649C22#1176362411yescharacteristic
650C4—22no—
651C4—22no—
652C4—42no—
653C4—41no—
654C4—42no—
655C4—41no—
656C4—42no—
657C4—41no—
658C4—81no—
659C4—682no—
660C4—1361no—
661C4—1361no—
662C4#1176333911yescharacteristic
663C4#1176339211yescharacteristic
664C4#1176352411yescharacteristic
665C4#1176354211yescharacteristic
666C2—42no—
667C2—2721no—
668C2#1453271511yescharacteristic
669C2#1453881511yescharacteristic
670C2#1453904611yescharacteristic
671C2#1454060211yescharacteristic
672C2#1454064911yescharacteristic
673C2#1454094011yescharacteristic
674C2#1454104611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2360C2
socleC23×C17C22×D4
prime radicalC2×#10587616C22
hypercenter#2360D17
hypocenterC17#26724
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. #14530099C2semidirect
3. #14529977C2semidirect
4. #14529925C2semidirect
6. #14529736C2semidirect
7. #14529466C2semidirect
8. #14529411C2semidirect
9. #14529392C2semidirect
10. #14529366C2semidirect
11. #14529139C2semidirect
13. #14528874C2semidirect
14. C4×#10587506C2semidirect
15. C17×#2360C2semidirect
181. C22⋊C4×C17SD16semidirect
293. #2360D17semidirect
464. C17#26724semidirect
545. C22⋊C4#63230semidirect

Maximal subgroups

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

subgroupindex
#145331472
#145300992
#145299772
#145299252
#145297852
#145297362
#145294662
#145294112
#145293922
#145293662
#145291392
#145291012
#145288742
C4×#105875062
C17×#23602
#2672417