Finite Group Databasebeta

← #17040723

Subgroups — #17040723

Order 4352.

1008 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763014C211yesmaximal, characteristic
2C2×#11762909C211yesmaximal, characteristic
3C2×C4×#63229C211yesmaximal, characteristic
4#14531503C211yesmaximal, characteristic, complemented
5#14529707C211yesmaximal, characteristic, complemented
6#14529552C211yesmaximal, characteristic, complemented
7#14529425C211yesmaximal, characteristic, complemented
8#14529418C211yesmaximal, characteristic, complemented
9#14529118C211yesmaximal, characteristic
10#14529118C211yesmaximal, characteristic
11#14528794C211yesmaximal, characteristic
12#14528641C211yesmaximal, characteristic, complemented
13#14528617C211yesmaximal, characteristic, complemented
14C4×#10587481C211yesmaximal, characteristic, complemented
15C17×#2275C211yesmaximal, characteristic
16C22×#63229C2211yescharacteristic
17C22×#63229C2211yescharacteristic
18C2×#10587555C2211yescharacteristic
19C2×C4×Dic34C2211yescharacteristic
20C2×C4×Dic34C2211yescharacteristic
21#11763014C418yescomplemented
22#11762909—24no—
23C4×#63229—24no—
24C2×#10587478C2211yescharacteristic
25C2×#10587479C2211yescharacteristic
26C2×C4⋊#3622C2211yescharacteristic
27C2×C4⋊#3622C2211yescharacteristic
28C2×#10587481C2211yescharacteristic
29C2×#10587481C2211yescharacteristic
30C2×#10587481C2211yescharacteristic
31C2×#10587481C2211yescharacteristic
32C2×#10587481C2211yescharacteristic
33C2×#10587481C2211yescharacteristic
34C2×#10587503C2211yescharacteristic
35C2×#10587503C2211yescharacteristic
36C2×#10587474C2211yescharacteristic
37C2×C4×#3622C2211yescharacteristic
38#11762715C2211yescharacteristic
39#11762715C2211yescharacteristic
40#11762694—21no—
41#11762694—21no—
42#11762694—21no—
43#11762694—21no—
44#11762675C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762564C2211yescharacteristic
50#11762564C2211yescharacteristic
51C2×C4×D4×C17C2211yescharacteristic
52C2×C4×D4×C17C2211yescharacteristic
53C2×C4×D4×C17C2211yescharacteristic
54C4×C4⋊C4×C17C2211yescharacteristic
55C4×C4⋊C4×C17C2211yescharacteristic
56C4×C4⋊C4×C17C2211yescharacteristic
57C4×C22⋊C4×C17—21no—
58C4×C22⋊C4×C17—21no—
59C4×C22⋊C4×C17—21no—
60C17×#376C2211yescharacteristic
61C2×C17×#95—21no—
62C22×D4×C17C2311yescharacteristic
63C23×C4×C17—21no—
64C23×C4×C17—21no—
65C23×C4×C17—21no—
66C22×Dic34C2311yescharacteristic
67C22×Dic34C2311yescharacteristic
68C4×D4×C17—22no—
69C4×D4×C17—22no—
70C4×D4×C17—22no—
71C4×D4×C17—22no—
72C4×D4×C17C2×C414yes—
73C4×D4×C17D414yes—
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C4⋊C4×C17C2311yescharacteristic
77C2×C22⋊C4×C17—21no—
78C2×C22⋊C4×C17—21no—
79C2×C22⋊C4×C17—21no—
80C2×C42×C17—21no—
81C2×C42×C17—21no—
82C2×C42×C17—21no—
83C2×C42×C17C2311yescharacteristic
84C2×C42×C17C2311yescharacteristic
85C2×C42×C17C2311yescharacteristic
86C2×#63229—22no—
87C2×#63229—22no—
88C2×#63229—22no—
89C2×#63229—22no—
90C2×#63229C2×C414yes—
91C2×#63229C2×C414yes—
92C2×C4⋊Dic17—21no—
93C2×C4⋊Dic17—21no—
94C2×C4⋊Dic17C2311yescharacteristic
95C2×C4⋊Dic17C2311yescharacteristic
96C2×#63225—21no—
97C2×#63225—21no—
98C2×C4×Dic17—21no—
99C2×C4×Dic17—21no—
100C22×#3622C2311yescharacteristic
101C22×#3622C2311yescharacteristic
102C22×#3622C2311yescharacteristic
103C22×#3622C2311yescharacteristic
104#10587555—22no—
105#10587555C2×C414yescomplemented
106C4×Dic34—22no—
107C4×Dic34—22no—
108C4×Dic34—22no—
109C4×Dic34—22no—
110C17×#95—21no—
111C17×#95—21no—
112C17×#95—21no—
113C17×#95D411yescharacteristic
114C17×#95D411yescharacteristic
115C17×#95D411yescharacteristic
116C17×#95D411yescharacteristic
117#10587503—22no—
118#10587503—22no—
119#10587502—21no—
120#10587502—21no—
121#10587502—21no—
122#10587502—21no—
123#10587502—21no—
124#10587502—21no—
125#10587481—22no—
126#10587481—22no—
127#10587481—22no—
128#10587481—22no—
129#10587481C2×C414yescomplemented
130#10587481C2×C414yescomplemented
131#10587479—22no—
132#10587478—22no—
133C4⋊#3622—22no—
134C4⋊#3622C2×C414yescomplemented
135#10587474—22no—
136C4×#3622—22no—
137C24×C17—21no—
138C2×D4×C17—21no—
139C2×D4×C17—21no—
140C2×D4×C17—21no—
141C2×D4×C17C22×C412yes—
142C2×D4×C17C2×D412yes—
143C2×D4×C17#4112yes—
144C22×C4×C17—21no—
145C22×C4×C17—21no—
146C22×C4×C17—21no—
147C22×C4×C17—24no—
148C22×C4×C17—24no—
149C22×C4×C17—24no—
150C22×C4×C17—81no—
151C22×C4×C17—81no—
152C22×C4×C17—81no—
153C22×C4×C17—81no—
154C22×C4×C17—81no—
155C22×C4×C17—81no—
156C22×C4×C17C2×D411yescharacteristic
157C22×C4×C17C2×D411yescharacteristic
158C22×C4×C17C2×D411yescharacteristic
159C22×C4×C17C2×D411yescharacteristic
160C22×C4×C17C2×D411yescharacteristic
161C22×C4×C17C2×D411yescharacteristic
162C22×C4×C17C2411yescharacteristic
163C22×Dic17—21no—
164C22×Dic17—21no—
165C22×Dic17—21no—
166C22×Dic17—21no—
167C2×Dic34—21no—
168C2×Dic34—21no—
169C2×Dic34—21no—
170C2×Dic34—21no—
171C2×Dic34—21no—
172C2×Dic34—21no—
173C2×Dic34—21no—
174C2×Dic34—21no—
175C2×Dic34—21no—
176C2×Dic34—21no—
177C2×Dic34C22×C412yes—
178C2×Dic34C22×C412yes—
179C4⋊C4×C17—21no—
180C4⋊C4×C17—21no—
181C4⋊C4×C17—21no—
182C4⋊C4×C17C22×C412yes—
183C4⋊C4×C17C2×D412yes—
184C4⋊C4×C17#4112yes—
185C22⋊C4×C17—24no—
186C22⋊C4×C17—24no—
187C22⋊C4×C17—24no—
188C42×C17—21no—
189C42×C17—21no—
190C42×C17—81no—
191C42×C17—81no—
192C42×C17—81no—
193C42×C17C22×C412yes—
194C42×C17C2×D412yes—
195C42×C17#4112yes—
196C42×C17#4112yes—
197#63229—28no—
198#63229—28no—
199C4⋊Dic17—24no—
200C4⋊Dic17—24no—
201C4⋊Dic17—21no—
202C4⋊Dic17—21no—
203C4⋊Dic17—21no—
204C4⋊Dic17—21no—
205#63225—42no—
206#63225—42no—
207C4×Dic17—42no—
208C4×Dic17—42no—
209C2×#3622—21no—
210C2×#3622—21no—
211C2×#3622—21no—
212C2×#3622—21no—
213C2×#3622—21no—
214C2×#3622—21no—
215C2×#3622—21no—
216C2×#3622—21no—
217C2×#3622—21no—
218C2×#3622—21no—
219C2×#3622C22×C412yes—
220C2×#3622C22×C412yes—
221#21514—171nomaximal, Sylow
222C23×C17—22no—
223C23×C17—22no—
224C23×C17—22no—
225C23×C17—41no—
226C23×C17—41no—
227C23×C17—41no—
228C23×C17—41no—
229C23×C17C22×D411yescharacteristic
230D4×C17—22no—
231D4×C17—22no—
232D4×C17—22no—
233D4×C17C4×D414yes—
234C2×C4×C17—21no—
235C2×C4×C17—21no—
236C2×C4×C17—21no—
237C2×C4×C17—21no—
238C2×C4×C17—21no—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—21no—
242C2×C4×C17—22no—
243C2×C4×C17—22no—
244C2×C4×C17—21no—
245C2×C4×C17—21no—
246C2×C4×C17—22no—
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—84no—
258C2×C4×C17—84no—
259C2×C4×C17—84no—
260C2×C4×C17C4×D412yes—
261C2×C4×C17C4×D412yes—
262C2×C4×C17#12112yes—
263C2×C4×C17#12112yes—
264C2×C4×C17C23×C411yescharacteristic
265C2×C4×C17C22×D411yescharacteristic
266C2×C4×C17C2×#4111yescharacteristic
267C2×C4×C17C2×#4111yescharacteristic
268C2×C4×C17C2×#4111yescharacteristic
269C2×C4×C17C2×#4111yescharacteristic
270C2×Dic17—22no—
271C2×Dic17—22no—
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—41no—
282Dic34—22no—
283Dic34—22no—
284Dic34—22no—
285Dic34—22no—
286Dic34—41no—
287Dic34—41no—
288Dic34—41no—
289Dic34—41no—
290#3622—22no—
291#3622—22no—
292#3622—41no—
293#3622—41no—
294C2×#461—171no—
295C2×#440—171no—
296C2×C4×SD16—171no—
297#2327—171no—
298#2275D1711yescharacteristic
299#1981—171no—
300#1949—171no—
301#1928—171no—
302#1927—171no—
303#1877—171no—
304#1877—171no—
305#1804—171no—
306#1768—171no—
307#1763—171no—
308C4×#103—171no—
309C22×C17—24no—
310C22×C17—42no—
311C22×C17—42no—
312C22×C17—42no—
313C22×C17—42no—
314C22×C17—42no—
315C22×C17—42no—
316C22×C17C2×C4×D411yescharacteristic
317C22×C17C2×#12111yescharacteristic
318C22×C17#52511yescharacteristic
319C22×C17#57711yescharacteristic
320C22×C17#57711yescharacteristic
321C22×C17#57711yescharacteristic
322C22×C17#57811yescharacteristic
323C4×C17—22no—
324C4×C17—22no—
325C4×C17—22no—
326C4×C17—22no—
327C4×C17—41no—
328C4×C17—41no—
329C4×C17—42no—
330C4×C17—42no—
331C4×C17—42no—
332C4×C17C2×C4×D411yescharacteristic
333C4×C17C4×#4111yescharacteristic
334C4×C17#52911yescharacteristic
335C4×C17#53211yescharacteristic
336Dic17—42no—
337Dic17—42no—
338Dic17—81no—
339Dic17—81no—
340C22×SD16—171no—
341C22×SD16—171no—
342C2×C4⋊Q8—171no—
343C2×C4×Q8—171no—
344C2×C4×Q8—171no—
345C2×C4×D4C2×D1711yescharacteristic
346C2×C4×D4C2×D1711yescharacteristic
347C2×C4×D4C2×D1711yescharacteristic
348#461—178no—
349#440—344no—
350C4×SD16—344no—
351C2×#107—171no—
352C2×#106—171no—
353C2×C4⋊C8—171no—
354C2×C4⋊C8—171no—
355C2×#103—171no—
356C2×#103—171no—
357C2×#103—171no—
358C2×#103—171no—
359C2×#103—171no—
360C2×#103—171no—
361C2×#102—171no—
362C2×#102—171no—
363C2×#97—171no—
364C2×C4×C8—171no—
365#391—171no—
366#391—171no—
367#387—341no—
368#387—341no—
369#387—341no—
370#387—341no—
371#382—171no—
372C4×C4⋊C4C2×D1711yescharacteristic
373C4×C4⋊C4C2×D1711yescharacteristic
374C4×C4⋊C4C2×D1711yescharacteristic
375C4×C22⋊C4—21no—
376C4×C22⋊C4—21no—
377C4×C22⋊C4—21no—
378#376C2×D1711yescharacteristic
379C2×#95—21no—
380#340—171no—
381#340—171no—
382#340—171no—
383#340—171no—
384#336—171no—
385#336—171no—
386C2×C17—44no—
387C2×C17C4×#12111yescharacteristic
388C2×C17#294711yescharacteristic
389C2×C17#294911yescharacteristic
390C2×C17#304211yescharacteristic
391C2×C17#304611yescharacteristic
392C2×C17#312011yescharacteristic
393C2×C17#312311yescharacteristic
394C22×Q8—171no—
395C22×Q8—171no—
396C22×D4C22×D1711yescharacteristic
397C23×C4—21no—
398C23×C4—21no—
399C23×C4—21no—
400C2×SD16—174no—
401C2×SD16—174no—
402C2×SD16—342no—
403C2×SD16—342no—
404C2×SD16—342no—
405C2×SD16—342no—
406C22×C8—171no—
407C22×C8—171no—
408C22×C8—171no—
409C22×C8—171no—
410C4⋊Q8—174no—
411C4⋊Q8—342no—
412C4×Q8—342no—
413C4×Q8—342no—
414C4×Q8—342no—
415C4×Q8—342no—
416C4×D4—22no—
417C4×D4—22no—
418C4×D4—22no—
419C4×D4—22no—
420C4×D4C4×D1714yes—
421C4×D4D6814yes—
422C2×C4⋊C4—171no—
423C2×C4⋊C4—171no—
424C2×C4⋊C4—341no—
425C2×C4⋊C4—341no—
426C2×C4⋊C4—341no—
427C2×C4⋊C4—341no—
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×C42—21no—
435C2×C42—21no—
436C2×C42—21no—
437C2×C42—341no—
438C2×C42—341no—
439C2×C42C22×D1711yescharacteristic
440C2×C42C22×D1711yescharacteristic
441C2×C42C22×D1711yescharacteristic
442#107—342no—
443#106—342no—
444C4⋊C8—174no—
445C4⋊C8—342no—
446#103—174no—
447#103—174no—
448#103—342no—
449#103—342no—
450#103—342no—
451#103—342no—
452#102—342no—
453#102—342no—
454#97—342no—
455C4×C8—342no—
456#95—21no—
457#95—21no—
458#95—21no—
459#95—341no—
460#95—341no—
461#95—341no—
462#95—341no—
463#95—341no—
464#95—341no—
465#95C22⋊D1711yescharacteristic
466#95C22⋊D1711yescharacteristic
467#95C22⋊D1711yescharacteristic
468#95C22⋊D1711yescharacteristic
469C17#2151411yescharacteristic, complemented, Sylow
470C24—21no—
471C2×Q8—172no—
472C2×Q8—172no—
473C2×Q8—341no—
474C2×Q8—341no—
475C2×Q8—341no—
476C2×Q8—341no—
477C2×Q8—341no—
478C2×Q8—341no—
479C2×Q8—341no—
480C2×Q8—341no—
481C2×Q8—341no—
482C2×Q8—341no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4C2×C4×D1712yes—
487C2×D4C2×D6812yes—
488C2×D4#6325212yes—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—24no—
495C22×C4—81no—
496C22×C4—81no—
497C22×C4—81no—
498C22×C4—81no—
499C22×C4—81no—
500C22×C4—81no—
501C22×C4—341no—
502C22×C4—341no—
503C22×C4—341no—
504C22×C4—341no—
505C22×C4C2×C22⋊D1711yescharacteristic
506C22×C4C2×C22⋊D1711yescharacteristic
507C22×C4C2×C22⋊D1711yescharacteristic
508C22×C4C2×C22⋊D1711yescharacteristic
509C22×C4C2×C22⋊D1711yescharacteristic
510C22×C4C2×C22⋊D1711yescharacteristic
511C22×C4C23×D1711yescharacteristic
512SD16—348no—
513SD16—348no—
514C2×C8—172no—
515C2×C8—172no—
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—
525C2×C8—341no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—344no—
530C4⋊C4—344no—
531C4⋊C4—341no—
532C4⋊C4—341no—
533C4⋊C4—341no—
534C4⋊C4—341no—
535C4⋊C4—682no—
536C4⋊C4—682no—
537C4⋊C4C2×C4×D1712yes—
538C4⋊C4C2×D6812yes—
539C4⋊C4#6325212yes—
540C22⋊C4—24no—
541C22⋊C4—24no—
542C22⋊C4—24no—
543C42—21no—
544C42—21no—
545C42—81no—
546C42—81no—
547C42—81no—
548C42—682no—
549C42—682no—
550C42C2×C4×D1712yes—
551C42C2×D6812yes—
552C42#6325212yes—
553C42#6325212yes—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23C22×C22⋊D1711yescharacteristic
562Q8—342no—
563Q8—342no—
564Q8—342no—
565Q8—342no—
566Q8—681no—
567Q8—681no—
568Q8—681no—
569Q8—681no—
570D4—22no—
571D4—22no—
572D4—22no—
573D4C4×D6814yes—
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—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4—84no—
600C2×C4—342no—
601C2×C4—342no—
602C2×C4—681no—
603C2×C4—681no—
604C2×C4—681no—
605C2×C4—681no—
606C2×C4—681no—
607C2×C4—681no—
608C2×C4—681no—
609C2×C4—681no—
610C2×C4—681no—
611C2×C4—681no—
612C2×C4C4×C22⋊D1712yes—
613C2×C4C4×C22⋊D1712yes—
614C2×C4#1058761612yes—
615C2×C4#1058761612yes—
616C2×C4C22×C4×D1711yescharacteristic
617C2×C4C22×D6811yescharacteristic
618C2×C4C2×#6325211yescharacteristic
619C2×C4C2×#6325211yescharacteristic
620C2×C4C2×#6325211yescharacteristic
621C2×C4C2×#6325211yescharacteristic
622C8—342no—
623C8—342no—
624C8—681no—
625C8—681no—
626C22—24no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22C2×C4×C22⋊D1711yescharacteristic
634C22C2×#1058761611yescharacteristic
635C22#1176332011yescharacteristic
636C22#1176362011yescharacteristic
637C22#1176362011yescharacteristic
638C22#1176362011yescharacteristic
639C22#1176362411yescharacteristic
640C4—22no—
641C4—22no—
642C4—22no—
643C4—22no—
644C4—41no—
645C4—42no—
646C4—42no—
647C4—42no—
648C4—41no—
649C4—682no—
650C4—682no—
651C4—1361no—
652C4—1361no—
653C4C2×C4×D6811yescharacteristic
654C4C4×#6325211yescharacteristic
655C4#1176334311yescharacteristic
656C4#1176335911yescharacteristic
657C2—44no—
658C2C4×#1058761611yescharacteristic
659C2#1453762311yescharacteristic
660C2#1453764111yescharacteristic
661C2#1453829611yescharacteristic
662C2#1453833411yescharacteristic
663C2#1453899811yescharacteristic
664C2#1453903611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#2275C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
cosocleC22×C4×C17C24
hypercenter#2275D17
hypocenterC17#21514
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ #376 ◁ #2275

Complements

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

NH ≅ G/Ntype
4. #14531503C2semidirect
5. #14529707C2semidirect
6. #14529552C2semidirect
7. #14529425C2semidirect
8. #14529418C2semidirect
12. #14528641C2semidirect
13. #14528617C2semidirect
14. C4×#10587481C2semidirect
21. #11763014C4semidirect
105. #10587555C2×C4semidirect
129. #10587481C2×C4semidirect
130. #10587481C2×C4semidirect
134. C4⋊#3622C2×C4semidirect
469. C17#21514semidirect

Maximal subgroups

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

subgroupindex
C2×#117630142
C2×#117629092
C2×C4×#632292
#145315032
#145297072
#145295522
#145294252
#145294182
#145291182
#145287942
#145286412
#145286172
C4×#105874812
C17×#22752
#2151417