Finite Group Databasebeta

← #17097194

Subgroups — #17097194

Order 4352.

860 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763109C211yesmaximal, characteristic, complemented
2C2×#11762852C211yesmaximal, characteristic, complemented
3#14532508C211yesmaximal, characteristic, complemented
4#14530305C211yesmaximal, characteristic, complemented
5#14530285C211yesmaximal, characteristic, complemented
6#14530158C211yesmaximal, characteristic, complemented
7#14530122C211yesmaximal, characteristic, complemented
8#14529552C211yesmaximal, characteristic, complemented
9#14529355C211yesmaximal, characteristic, complemented
10#14529223C211yesmaximal, characteristic
11#14529102C211yesmaximal, characteristic
12#14529031C211yesmaximal, characteristic, complemented
13#14528983C211yesmaximal, characteristic, complemented
14#14528969C211yesmaximal, characteristic, complemented
15C17×#2363C211yesmaximal, characteristic
16C22×#63223C2211yescharacteristic
17C2×#10587556C2211yescharacteristic
18C22×C4⋊Dic17C2211yescharacteristic
19#11763109—24no—
20#11762852—24no—
21C2×#10587478C2211yescharacteristic
22C2×#10587478C2211yescharacteristic
23C2×#10587478C2211yescharacteristic
24C2×#10587479C2211yescharacteristic
25C2×#10587479C2211yescharacteristic
26C2×#10587479C2211yescharacteristic
27C2×C4⋊#3622C2211yescharacteristic
28C2×C4⋊#3622C2211yescharacteristic
29C2×#10587503C2211yescharacteristic
30C2×#10587503C2211yescharacteristic
31C2×#10587503C2211yescharacteristic
32C2×#10587503C2211yescharacteristic
33C2×C22⋊#3622C2211yescharacteristic
34#11762751—21no—
35#11762743—21no—
36#11762707C2211yescharacteristic
37#11762707C2211yescharacteristic
38#11762681—21no—
39#11762675C2211yescharacteristic
40#11762669C2211yescharacteristic
41#11762667—21no—
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762564C2211yescharacteristic
49#11762564C2211yescharacteristic
50C2×C17×#121C2211yescharacteristic
51C2×C4×D4×C17C2211yescharacteristic
52C2×C17×#117C2211yescharacteristic
53C17×#390—21no—
54C17×#390C2211yescharacteristic
55C17×#389C2211yescharacteristic
56C17×#388—21no—
57C17×#386—21no—
58C4×C4⋊C4×C17C2211yescharacteristic
59C4×C22⋊C4×C17—21no—
60C4×C22⋊C4×C17C2211yescharacteristic
61C22×D4×C17—21no—
62C22×D4×C17C2311yescharacteristic
63C23×C4×C17—21no—
64C23×C4×C17C2311yescharacteristic
65C23×Dic17—21no—
66C17×#121—22no—
67C17×#121Q814yes—
68C4×D4×C17—22no—
69C4×D4×C17—41no—
70C17×#117—22no—
71C17×#117—41no—
72C2×C4⋊C4×C17—21no—
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C22⋊C4×C17—21no—
77C2×C22⋊C4×C17—21no—
78C2×C22⋊C4×C17—21no—
79C2×C22⋊C4×C17—21no—
80C2×C22⋊C4×C17—21no—
81C2×C22⋊C4×C17D411yescharacteristic
82C2×C22⋊C4×C17D411yescharacteristic
83C2×C42×C17—21no—
84C2×C42×C17—21no—
85C2×C42×C17C2311yescharacteristic
86C2×C42×C17C2311yescharacteristic
87C2×C22⋊Dic17—21no—
88C2×C4⋊Dic17—21no—
89C2×C4⋊Dic17—22no—
90C2×C4⋊Dic17—41no—
91C2×C4⋊Dic17C2311yescharacteristic
92C2×C4⋊Dic17C2311yescharacteristic
93C2×C4⋊Dic17C2311yescharacteristic
94C2×C4⋊Dic17C2311yescharacteristic
95C2×#63225—21no—
96C2×#63225—21no—
97C2×#63225—21no—
98C2×#63225—21no—
99C2×#63223—22no—
100C2×#63223—22no—
101C2×#63223—22no—
102C22×#3622C2311yescharacteristic
103C22×#3622C2311yescharacteristic
104C22×#3622C2311yescharacteristic
105C22×#3622C2311yescharacteristic
106#10587556—22no—
107#10587556—41no—
108C17×#95—21no—
109C17×#95—21no—
110C17×#95D411yescharacteristic
111C17×#95D411yescharacteristic
112#10587503—22no—
113#10587503—22no—
114#10587503—22no—
115#10587503—41no—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587502—21no—
120#10587502—21no—
121C22⋊#3622—22no—
122#10587479—22no—
123#10587479—22no—
124#10587479—22no—
125#10587478—22no—
126#10587478—41no—
127#10587478D414yescomplemented
128C4⋊#3622—22no—
129C4⋊#3622—22no—
130C24×C17—21no—
131C24×C17C2×D411yescharacteristic
132C2×D4×C17—24no—
133C2×D4×C17—21no—
134C2×D4×C17—21no—
135C2×D4×C17—21no—
136C2×D4×C17—21no—
137C2×D4×C17—21no—
138C2×D4×C17—81no—
139C2×D4×C17—81no—
140C2×D4×C17C2×Q812yes—
141C22×C4×C17—21no—
142C22×C4×C17—21no—
143C22×C4×C17—21no—
144C22×C4×C17—21no—
145C22×C4×C17—21no—
146C22×C4×C17—21no—
147C22×C4×C17—21no—
148C22×C4×C17—42no—
149C22×C4×C17—42no—
150C22×C4×C17—81no—
151C22×C4×C17C2×D411yescharacteristic
152C22×C4×C17C2×D411yescharacteristic
153C22×C4×C17C2×D411yescharacteristic
154C22×C4×C17C2×D411yescharacteristic
155C22×C4×C17C2×D411yescharacteristic
156C22×C4×C17C2×Q812yes—
157C22×C4×C17#4112yes—
158C22×C4×C17C2411yescharacteristic
159C22×Dic17—21no—
160C22×Dic17—21no—
161C22×Dic17—21no—
162C22×Dic17—21no—
163C22×Dic17—42no—
164C22×Dic17—42no—
165C22×Dic17—81no—
166C4⋊C4×C17—21no—
167C4⋊C4×C17—21no—
168C4⋊C4×C17—21no—
169C4⋊C4×C17—21no—
170C4⋊C4×C17—81no—
171C4⋊C4×C17C2×D412yes—
172C4⋊C4×C17C2×Q812yes—
173C22⋊C4×C17—24no—
174C22⋊C4×C17—41no—
175C22⋊C4×C17—41no—
176C22⋊C4×C17—42no—
177C22⋊C4×C17—42no—
178C22⋊C4×C17—42no—
179C22⋊C4×C17—81no—
180C42×C17—21no—
181C42×C17—21no—
182C42×C17—21no—
183C42×C17—21no—
184C42×C17—42no—
185C42×C17—81no—
186C22⋊Dic17—42no—
187C4⋊Dic17—21no—
188C4⋊Dic17—21no—
189C4⋊Dic17—21no—
190C4⋊Dic17—21no—
191C4⋊Dic17—21no—
192C4⋊Dic17—21no—
193C4⋊Dic17—21no—
194C4⋊Dic17—42no—
195C4⋊Dic17C2×D412yes—
196#63225—42no—
197#63225—42no—
198#63225—81no—
199#63225—81no—
200#63223—28no—
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×#3622#4112yes—
213#26363—171nomaximal, Sylow
214C23×C17—21no—
215C23×C17—21no—
216C23×C17—21no—
217C23×C17—21no—
218C23×C17—21no—
219C23×C17—21no—
220C23×C17—22no—
221C23×C17—41no—
222C23×C17—41no—
223C23×C17—41no—
224C23×C17—41no—
225C23×C17—41no—
226C23×C17—41no—
227C23×C17C22⋊Q812yes—
228C23×C17C22×D411yescharacteristic
229D4×C17—22no—
230D4×C17—22no—
231D4×C17—41no—
232D4×C17—41no—
233D4×C17—84no—
234C2×C4×C17—21no—
235C2×C4×C17—21no—
236C2×C4×C17—22no—
237C2×C4×C17—22no—
238C2×C4×C17—22no—
239C2×C4×C17—22no—
240C2×C4×C17—22no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—22no—
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—21no—
251C2×C4×C17—21no—
252C2×C4×C17—21no—
253C2×C4×C17—21no—
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—84no—
262C2×C4×C17C22⋊Q812yes—
263C2×C4×C17C22×D411yescharacteristic
264C2×C4×C17C22×Q811yescharacteristic
265C2×C4×C17C2×#4111yescharacteristic
266C2×C4×C17C2×#4111yescharacteristic
267C2×C4×C17H(5,𝔽2)11yescharacteristic
268C2×C4×C172−1+411yescharacteristic
269C2×Dic17—22no—
270C2×Dic17—41no—
271C2×Dic17—41no—
272C2×Dic17—41no—
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—84no—
282#3622—22no—
283#3622—22no—
284#3622—41no—
285#3622—41no—
286C2×#478—171no—
287C2×#428—171no—
288#2442—171no—
289#2363D1711yescharacteristic
290#2092—171no—
291#2088—171no—
292#2065—171no—
293#2059—171no—
294#1949—171no—
295#1918—171no—
296#1895—171no—
297#1874—171no—
298#1862—171no—
299#1854—171no—
300#1851—171no—
301C22×C17—22no—
302C22×C17—22no—
303C22×C17—22no—
304C22×C17—41no—
305C22×C17—41no—
306C22×C17—41no—
307C22×C17—41no—
308C22×C17—42no—
309C22×C17—42no—
310C22×C17—42no—
311C22×C17—81no—
312C22×C17—81no—
313C22×C17—81no—
314C22×C17—81no—
315C22×C17C2×C22⋊Q811yescharacteristic
316C22×C17D4211yescharacteristic
317C22×C17#54711yescharacteristic
318C22×C17#57511yescharacteristic
319C22×C17#57511yescharacteristic
320C22×C17#57611yescharacteristic
321C22×C17#57711yescharacteristic
322C4×C17—22no—
323C4×C17—42no—
324C4×C17—41no—
325C4×C17—41no—
326C4×C17—41no—
327C4×C17—41no—
328C4×C17—42no—
329C4×C17—81no—
330C4×C17#52511yescharacteristic
331C4×C17#53911yescharacteristic
332C4×C17D4×Q811yescharacteristic
333C4×C17#55411yescharacteristic
334Dic17—42no—
335Dic17—81no—
336Dic17—81no—
337Dic17—81no—
338C22×M16—171no—
339C2×#125—171no—
340C2×#121C2×D1711yescharacteristic
341C2×C4×D4C2×D1711yescharacteristic
342C2×#117C2×D1711yescharacteristic
343C22×C4⋊C4—171no—
344#478—344no—
345#428—344no—
346C2×#107—171no—
347C2×#107—171no—
348C2×#107—171no—
349C2×#106—171no—
350C2×#106—171no—
351C2×#106—171no—
352C2×C4⋊C8—171no—
353C2×C4⋊C8—171no—
354C2×#102—171no—
355C2×#102—171no—
356C2×#102—171no—
357C2×#102—171no—
358C2×C22⋊C8—171no—
359#400—341no—
360#398—341no—
361#390—21no—
362#390C2×D1711yescharacteristic
363#389—171no—
364#389—171no—
365#389C2×D1711yescharacteristic
366#388—21no—
367#386—21no—
368#385—341no—
369#382—171no—
370#380—171no—
371#380—341no—
372C4×C4⋊C4C2×D1711yescharacteristic
373C4×C22⋊C4—21no—
374C4×C22⋊C4C2×D1711yescharacteristic
375#340—171no—
376#340—171no—
377#340—171no—
378#340—171no—
379#340—171no—
380#340—171no—
381#336—171no—
382#336—171no—
383C2×C17—42no—
384C2×C17—82no—
385C2×C17#244911yescharacteristic
386C2×C17#308011yescharacteristic
387C2×C17#308111yescharacteristic
388C2×C17#327111yescharacteristic
389C2×C17#327311yescharacteristic
390C2×C17#334811yescharacteristic
391C2×C17#335511yescharacteristic
392C22×D4—21no—
393C22×D4C22×D1711yescharacteristic
394C23×C4—21no—
395C23×C4—341no—
396C23×C4C22×D1711yescharacteristic
397C2×M16—342no—
398C2×M16—342no—
399C2×M16—342no—
400C22×C8—171no—
401C22×C8—171no—
402C22×C8—171no—
403C22×C8—171no—
404#125—342no—
405#125—681no—
406#121—22no—
407#121Dic3414yes—
408C4×D4—22no—
409C4×D4—41no—
410#117—22no—
411#117—41no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—171no—
414C2×C4⋊C4—171no—
415C2×C4⋊C4—171no—
416C2×C4⋊C4—171no—
417C2×C4⋊C4—342no—
418C2×C4⋊C4—341no—
419C2×C4⋊C4—341no—
420C2×C4⋊C4—341no—
421C2×C4⋊C4—341no—
422C2×C4⋊C4—341no—
423C2×C4⋊C4—681no—
424C2×C4⋊C4C22×D1711yescharacteristic
425C2×C4⋊C4C22×D1711yescharacteristic
426C2×C4⋊C4C22×D1711yescharacteristic
427C2×C22⋊C4—21no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—341no—
433C2×C22⋊C4C22⋊D1711yescharacteristic
434C2×C22⋊C4C22⋊D1711yescharacteristic
435C2×C42—21no—
436C2×C42—21no—
437C2×C42C22×D1711yescharacteristic
438C2×C42C22×D1711yescharacteristic
439#107—174no—
440#107—342no—
441#107—681no—
442#106—342no—
443#106—342no—
444#106—342no—
445C4⋊C8—342no—
446C4⋊C8—342no—
447#102—342no—
448#102—342no—
449#102—342no—
450#102—681no—
451C22⋊C8—342no—
452#95—21no—
453#95—21no—
454#95—341no—
455#95—341no—
456#95—341no—
457#95—341no—
458#95—341no—
459#95C22⋊D1711yescharacteristic
460#95C22⋊D1711yescharacteristic
461C17#2636311yescharacteristic, complemented, Sylow
462C24—21no—
463C24C2×C22⋊D1711yescharacteristic
464C2×D4—24no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—81no—
471C2×D4—81no—
472C2×D4C2×Dic3412yes—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—81no—
483C22×C4—341no—
484C22×C4—341no—
485C22×C4—341no—
486C22×C4—341no—
487C22×C4—682no—
488C22×C4—682no—
489C22×C4—1361no—
490C22×C4C2×Dic3412yes—
491C22×C4#6325212yes—
492C22×C4C2×C22⋊D1711yescharacteristic
493C22×C4C2×C22⋊D1711yescharacteristic
494C22×C4C2×C22⋊D1711yescharacteristic
495C22×C4C2×C22⋊D1711yescharacteristic
496C22×C4C2×C22⋊D1711yescharacteristic
497C22×C4C23×D1711yescharacteristic
498M16—348no—
499C2×C8—172no—
500C2×C8—172no—
501C2×C8—341no—
502C2×C8—341no—
503C2×C8—341no—
504C2×C8—341no—
505C2×C8—341no—
506C2×C8—341no—
507C2×C8—341no—
508C2×C8—341no—
509C2×C8—341no—
510C2×C8—341no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—81no—
516C4⋊C4—172no—
517C4⋊C4—341no—
518C4⋊C4—341no—
519C4⋊C4—341no—
520C4⋊C4—341no—
521C4⋊C4—341no—
522C4⋊C4—341no—
523C4⋊C4—341no—
524C4⋊C4—682no—
525C4⋊C4—682no—
526C4⋊C4—682no—
527C4⋊C4—1361no—
528C4⋊C4—1361no—
529C4⋊C4C2×Dic3412yes—
530C4⋊C4D4×D1712yes—
531C22⋊C4—24no—
532C22⋊C4—41no—
533C22⋊C4—41no—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—42no—
537C22⋊C4—81no—
538C22⋊C4—682no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—42no—
544C42—81no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—22no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23#1058761012yes—
559C23C22×C22⋊D1711yescharacteristic
560D4—22no—
561D4—22no—
562D4—41no—
563D4—41no—
564D4—84no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—84no—
593C2×C4—342no—
594C2×C4—681no—
595C2×C4—681no—
596C2×C4—681no—
597C2×C4—681no—
598C2×C4—681no—
599C2×C4—681no—
600C2×C4—681no—
601C2×C4—681no—
602C2×C4—681no—
603C2×C4—681no—
604C2×C4—681no—
605C2×C4—1364no—
606C2×C4#1058761012yes—
607C2×C4C22×Dic3411yescharacteristic
608C2×C4C2×#6325211yescharacteristic
609C2×C4C2×D4×D1711yescharacteristic
610C2×C4#1058769111yescharacteristic
611C2×C4#41×D1711yescharacteristic
612C2×C4#1058769711yescharacteristic
613C8—342no—
614C8—342no—
615C8—681no—
616C8—681no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22C2×#1058761011yescharacteristic
632C22D4×C22⋊D1711yescharacteristic
633C22#1176345811yescharacteristic
634C22#1176360711yescharacteristic
635C22#1176360711yescharacteristic
636C22#1176361211yescharacteristic
637C22#1176362011yescharacteristic
638C4—22no—
639C4—41no—
640C4—41no—
641C4—41no—
642C4—42no—
643C4—42no—
644C4—41no—
645C4—81no—
646C4—682no—
647C4—1361no—
648C4—1361no—
649C4—1361no—
650C4#1176331811yescharacteristic
651C4#1176340111yescharacteristic
652C4D4×Dic3411yescharacteristic
653C4#1176351811yescharacteristic
654C2—42no—
655C2—82no—
656C2#1453258711yescharacteristic
657C2#1453860611yescharacteristic
658C2#1453861511yescharacteristic
659C2#1454036511yescharacteristic
660C2#1454039111yescharacteristic
661C2#1454129911yescharacteristic
662C2#1454138411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2363C2
socleC23×C17C22×D4
prime radicalC2×C17×#121C22
hypercenter#2363D17
hypocenterC17#26363
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2363

Complements

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

NH ≅ G/Ntype
1. C2×#11763109C2semidirect
2. C2×#11762852C2semidirect
3. #14532508C2semidirect
4. #14530305C2semidirect
5. #14530285C2semidirect
6. #14530158C2semidirect
7. #14530122C2semidirect
8. #14529552C2semidirect
9. #14529355C2semidirect
12. #14529031C2semidirect
13. #14528983C2semidirect
14. #14528969C2semidirect
127. #10587478D4semidirect
461. C17#26363semidirect

Maximal subgroups

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

subgroupindex
C2×#117631092
C2×#117628522
#145325082
#145303052
#145302852
#145301582
#145301222
#145295522
#145293552
#145292232
#145291022
#145290312
#145289832
#145289692
C17×#23632
#2636317