Finite Group Databasebeta

← #17094421

Subgroups — #17094421

Order 4352.

904 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763014C211yesmaximal, characteristic
2C2×C4×#63229C211yesmaximal, characteristic
3#14532560C211yesmaximal, characteristic, complemented
4#14530328C211yesmaximal, characteristic, complemented
5#14530285C211yesmaximal, characteristic, complemented
6#14530193C211yesmaximal, characteristic, complemented
7#14530030C211yesmaximal, characteristic, complemented
8#14529459C211yesmaximal, characteristic, complemented
9#14529343C211yesmaximal, characteristic, complemented
10#14529118C211yesmaximal, characteristic
11#14529118C211yesmaximal, characteristic
12#14529118C211yesmaximal, characteristic
13#14529051C211yesmaximal, characteristic, complemented
14#14529045C211yesmaximal, characteristic
15D4×C4⋊C4×C17C211yesmaximal, characteristic
16C22×#63229C2211yescharacteristic
17C22×#63229C2211yescharacteristic
18C2×#10587555C2211yescharacteristic
19C2×C4×Dic34C2211yescharacteristic
20#11763014—24no—
21C4×#63229—24no—
22C2×#10587479C2211yescharacteristic
23C2×#10587479C2211yescharacteristic
24C2×C4⋊#3622C2211yescharacteristic
25C2×#10587481C2211yescharacteristic
26C2×#10587481C2211yescharacteristic
27C2×#10587481C2211yescharacteristic
28C2×#10587481C2211yescharacteristic
29C2×#10587481C2211yescharacteristic
30C2×#10587481C2211yescharacteristic
31C2×#10587503C2211yescharacteristic
32C2×#10587503C2211yescharacteristic
33C2×C4×#3622C2211yescharacteristic
34#11762751—21no—
35#11762742—21no—
36#11762715C2211yescharacteristic
37#11762715C2211yescharacteristic
38#11762715C2211yescharacteristic
39#11762707C2211yescharacteristic
40#11762696—21no—
41#11762695—21no—
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762564C2211yescharacteristic
47#11762564C2211yescharacteristic
48#11762564C2211yescharacteristic
49#11762564C2211yescharacteristic
50C2×C4×D4×C17C2211yescharacteristic
51C2×C4×D4×C17C2211yescharacteristic
52C2×C4×D4×C17C2211yescharacteristic
53C22×C4⋊C4×C17—21no—
54C17×#389C2211yescharacteristic
55C17×#389C2211yescharacteristic
56C17×#385—21no—
57C17×#385—21no—
58C17×#384C2211yescharacteristic
59C17×#380—21no—
60C4×C4⋊C4×C17C2211yescharacteristic
61C22×D4×C17C2311yescharacteristic
62C23×C4×C17—21no—
63C23×C4×C17—21no—
64C23×C4×C17—21no—
65C22×Dic34C2311yescharacteristic
66C22×Dic34C2311yescharacteristic
67C4×D4×C17—22no—
68C4×D4×C17—22no—
69C4×D4×C17—22no—
70C4×D4×C17—22no—
71C4×D4×C17—41no—
72C4×D4×C17—41no—
73C2×C4⋊C4×C17—21no—
74C2×C4⋊C4×C17—21no—
75C2×C4⋊C4×C17—21no—
76C2×C4⋊C4×C17—21no—
77C2×C4⋊C4×C17—44no—
78C2×C4⋊C4×C17D411yescharacteristic
79C2×C4⋊C4×C17D411yescharacteristic
80C2×C4⋊C4×C17C2311yescharacteristic
81C2×C4⋊C4×C17C2311yescharacteristic
82C2×C4⋊C4×C17C2311yescharacteristic
83C2×C22⋊C4×C17—21no—
84C2×C22⋊C4×C17—21no—
85C2×C22⋊C4×C17—21no—
86C2×C42×C17C2311yescharacteristic
87C2×C42×C17C2311yescharacteristic
88C2×C42×C17C2311yescharacteristic
89C2×#63229—22no—
90C2×#63229—22no—
91C2×#63229—22no—
92C2×#63229—22no—
93C2×#63229—22no—
94C2×#63229D414yes—
95C2×C4⋊Dic17—21no—
96C2×C4⋊Dic17—21no—
97C2×C4⋊Dic17C2311yescharacteristic
98C2×C4⋊Dic17C2311yescharacteristic
99C2×#63225—21no—
100C2×#63225—21no—
101C2×C4×Dic17—21no—
102C22×#3622C2311yescharacteristic
103C22×#3622C2311yescharacteristic
104C22×#3622C2311yescharacteristic
105C22×#3622C2311yescharacteristic
106#10587555—22no—
107#10587555—41no—
108C4×Dic34—22no—
109C4×Dic34—41no—
110C17×#95—21no—
111C17×#95—21no—
112C17×#95D411yescharacteristic
113C17×#95D411yescharacteristic
114#10587503—22no—
115#10587503—22no—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587502—21no—
120#10587502—21no—
121#10587502—21no—
122#10587502—21no—
123#10587481—22no—
124#10587481—22no—
125#10587481—22no—
126#10587481—22no—
127#10587481—22no—
128#10587481—41no—
129#10587479—22no—
130#10587479—41no—
131C4⋊#3622—22no—
132C4×#3622—22no—
133C24×C17—21no—
134C2×D4×C17—21no—
135C2×D4×C17—21no—
136C2×D4×C17—21no—
137C2×D4×C17C2×D412yes—
138C2×D4×C17#4112yes—
139C2×D4×C17#4112yes—
140C22×C4×C17—24no—
141C22×C4×C17—21no—
142C22×C4×C17—21no—
143C22×C4×C17—21no—
144C22×C4×C17—42no—
145C22×C4×C17—42no—
146C22×C4×C17—42no—
147C22×C4×C17—42no—
148C22×C4×C17—42no—
149C22×C4×C17—42no—
150C22×C4×C17—81no—
151C22×C4×C17—81no—
152C22×C4×C17C2×D411yescharacteristic
153C22×C4×C17C2×D411yescharacteristic
154C22×C4×C17C2×D411yescharacteristic
155C22×C4×C17C2×D411yescharacteristic
156C22×C4×C17C2×D411yescharacteristic
157C22×C4×C17C2×D411yescharacteristic
158C22×C4×C17C2411yescharacteristic
159C22×Dic17—21no—
160C22×Dic17—21no—
161C22×Dic17—21no—
162C22×Dic17—21no—
163C2×Dic34—21no—
164C2×Dic34—21no—
165C2×Dic34—21no—
166C2×Dic34—21no—
167C2×Dic34—21no—
168C2×Dic34—21no—
169C2×Dic34—21no—
170C2×Dic34—21no—
171C2×Dic34—21no—
172C2×Dic34—21no—
173C2×Dic34—21no—
174C2×Dic34C2×D412yes—
175C4⋊C4×C17—21no—
176C4⋊C4×C17—21no—
177C4⋊C4×C17—21no—
178C4⋊C4×C17—21no—
179C4⋊C4×C17—21no—
180C4⋊C4×C17—41no—
181C4⋊C4×C17—42no—
182C4⋊C4×C17—42no—
183C4⋊C4×C17—42no—
184C4⋊C4×C17—81no—
185C4⋊C4×C17SD1614yescomplemented
186C4⋊C4×C17#4112yes—
187C22⋊C4×C17—24no—
188C22⋊C4×C17—42no—
189C22⋊C4×C17—42no—
190C42×C17—21no—
191C42×C17—21no—
192C42×C17—21no—
193C42×C17—21no—
194C42×C17—21no—
195C42×C17—21no—
196#63229—28no—
197#63229—44no—
198C4⋊Dic17—21no—
199C4⋊Dic17—21no—
200C4⋊Dic17—21no—
201C4⋊Dic17—21no—
202C4⋊Dic17—42no—
203C4⋊Dic17—42no—
204#63225—42no—
205#63225—42no—
206C4×Dic17—42no—
207C2×#3622—21no—
208C2×#3622—21no—
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×#3622C2×D412yes—
219#26166—171nomaximal, Sylow
220C23×C17—22no—
221C23×C17—22no—
222C23×C17—22no—
223C23×C17—41no—
224C23×C17—41no—
225C23×C17—41no—
226C23×C17—41no—
227C23×C17C22×D411yescharacteristic
228D4×C17—22no—
229D4×C17—22no—
230D4×C17—22no—
231D4×C17#12314yes—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—21no—
235C2×C4×C17—21no—
236C2×C4×C17—21no—
237C2×C4×C17—21no—
238C2×C4×C17—21no—
239C2×C4×C17—22no—
240C2×C4×C17—21no—
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—48no—
249C2×C4×C17—41no—
250C2×C4×C17—41no—
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×C17C2×SD1612yes—
260C2×C4×C17C2×SD1612yes—
261C2×C4×C17C2×SD1612yes—
262C2×C4×C17C22×D411yescharacteristic
263C2×C4×C17C2×#4111yescharacteristic
264C2×C4×C17C2×#4111yescharacteristic
265C2×C4×C17C2×#4111yescharacteristic
266C2×C4×C17H(5,𝔽2)11yescharacteristic
267C2×C4×C172−1+411yescharacteristic
268C2×Dic17—22no—
269C2×Dic17—41no—
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—
280Dic34—22no—
281Dic34—22no—
282Dic34—41no—
283Dic34—41no—
284Dic34—41no—
285Dic34—41no—
286Dic34—41no—
287Dic34—41no—
288#3622—22no—
289#3622—41no—
290#3622—41no—
291#3622—41no—
292C2×#461—171no—
293C2×C4×SD16—171no—
294#2446—171no—
295D4×C4⋊C4D1711yescharacteristic
296#2096—171no—
297#2088—171no—
298#2072—171no—
299#2043—171no—
300#1933—171no—
301#1916—171no—
302#1877—171no—
303#1877—171no—
304#1877—171no—
305#1865—171no—
306#1864—171no—
307C22×C17—24no—
308C22×C17—42no—
309C22×C17—42no—
310C22×C17—42no—
311C22×C17—42no—
312C22×C17—42no—
313C22×C17—42no—
314C22×C17#52511yescharacteristic
315C22×C17D4211yescharacteristic
316C22×C17#54811yescharacteristic
317C22×C17C22×SD1611yescharacteristic
318C22×C17#57511yescharacteristic
319C22×C17#57711yescharacteristic
320C22×C17#57811yescharacteristic
321C4×C17—22no—
322C4×C17—22no—
323C4×C17—22no—
324C4×C17—42no—
325C4×C17—41no—
326C4×C17—41no—
327C4×C17—41no—
328C4×C17—81no—
329C4×C17—81no—
330C4×C17C2×#12311yescharacteristic
331C4×C17#53911yescharacteristic
332C4×C17#54711yescharacteristic
333C4×C17#55311yescharacteristic
334Dic17—42no—
335Dic17—81no—
336Dic17—81no—
337Dic17—81no—
338C22×SD16—171no—
339C22×SD16—171no—
340C2×C4⋊Q8—171no—
341C2×C4×Q8—171no—
342C2×C4×D4C2×D1711yescharacteristic
343C2×C4×D4C2×D1711yescharacteristic
344C2×C4×D4C2×D1711yescharacteristic
345C22×C4⋊C4—21no—
346#461—344no—
347C4×SD16—344no—
348C2×#106—171no—
349C2×#106—171no—
350C2×C4⋊C8—171no—
351C2×#103—171no—
352C2×#103—171no—
353C2×#103—171no—
354C2×#103—171no—
355C2×#103—171no—
356C2×#103—171no—
357C2×#102—171no—
358C2×#102—171no—
359C2×C4×C8—171no—
360#400—341no—
361#398—341no—
362#391—171no—
363#391—171no—
364#391—171no—
365#389—171no—
366#389C2×D1711yescharacteristic
367#389C2×D1711yescharacteristic
368#387—341no—
369#387—341no—
370#385—21no—
371#385—21no—
372#384C2×D1711yescharacteristic
373#380—21no—
374C4×C4⋊C4C2×D1711yescharacteristic
375#340—171no—
376#340—171no—
377#340—171no—
378#340—171no—
379#336—171no—
380#336—171no—
381#336—171no—
382#336—171no—
383C2×C17—44no—
384C2×C17#244311yescharacteristic
385C2×C17#310111yescharacteristic
386C2×C17#312011yescharacteristic
387C2×C17#326911yescharacteristic
388C2×C17D4×SD1611yescharacteristic
389C2×C17#335911yescharacteristic
390C2×C17#337611yescharacteristic
391C22×Q8—171no—
392C22×Q8—171no—
393C22×D4C22×D1711yescharacteristic
394C23×C4—21no—
395C23×C4—21no—
396C23×C4—21no—
397C2×SD16—174no—
398C2×SD16—342no—
399C2×SD16—342no—
400C2×SD16—342no—
401C2×SD16—342no—
402C2×SD16—342no—
403C22×C8—171no—
404C22×C8—171no—
405C22×C8—171no—
406C22×C8—171no—
407C4⋊Q8—342no—
408C4⋊Q8—681no—
409C4×Q8—342no—
410C4×Q8—681no—
411C4×D4—22no—
412C4×D4—22no—
413C4×D4—22no—
414C4×D4—22no—
415C4×D4—41no—
416C4×D4—41no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—44no—
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×C4⋊C4C22×D1711yescharacteristic
432C2×C4⋊C4C22×D1711yescharacteristic
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C42—341no—
437C2×C42C22×D1711yescharacteristic
438C2×C42C22×D1711yescharacteristic
439C2×C42C22×D1711yescharacteristic
440#106—342no—
441#106—681no—
442C4⋊C8—342no—
443#103—342no—
444#103—342no—
445#103—342no—
446#103—342no—
447#103—342no—
448#103—681no—
449#102—342no—
450#102—342no—
451C4×C8—342no—
452#95—21no—
453#95—21no—
454#95—341no—
455#95—341no—
456#95—341no—
457#95—341no—
458#95—341no—
459#95—341no—
460#95—341no—
461#95C22⋊D1711yescharacteristic
462#95C22⋊D1711yescharacteristic
463C17#2616611yescharacteristic, complemented, Sylow
464C24—21no—
465C2×Q8—172no—
466C2×Q8—341no—
467C2×Q8—341no—
468C2×Q8—341no—
469C2×Q8—341no—
470C2×Q8—341no—
471C2×Q8—341no—
472C2×Q8—341no—
473C2×Q8—341no—
474C2×Q8—341no—
475C2×Q8—341no—
476C2×Q8—341no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4#6325212yes—
481C2×D4D4×D1712yes—
482C2×D4#6325612yes—
483C22×C4—24no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—81no—
494C22×C4—81no—
495C22×C4—341no—
496C22×C4—341no—
497C22×C4—341no—
498C22×C4—341no—
499C22×C4C2×C22⋊D1711yescharacteristic
500C22×C4C2×C22⋊D1711yescharacteristic
501C22×C4C2×C22⋊D1711yescharacteristic
502C22×C4C2×C22⋊D1711yescharacteristic
503C22×C4C2×C22⋊D1711yescharacteristic
504C22×C4C2×C22⋊D1711yescharacteristic
505C22×C4C23×D1711yescharacteristic
506SD16—348no—
507SD16—684no—
508C2×C8—172no—
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—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—41no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—81no—
530C4⋊C4—341no—
531C4⋊C4—341no—
532C4⋊C4—341no—
533C4⋊C4—341no—
534C4⋊C4—682no—
535C4⋊C4—682no—
536C4⋊C4—682no—
537C4⋊C4—682no—
538C4⋊C4#6322914yescomplemented
539C4⋊C4#6325212yes—
540C22⋊C4—24no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—682no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23C22×C22⋊D1711yescharacteristic
558Q8—342no—
559Q8—342no—
560Q8—681no—
561Q8—681no—
562Q8—681no—
563Q8—681no—
564Q8—681no—
565Q8—681no—
566D4—22no—
567D4—22no—
568D4—22no—
569D4#1058758014yes—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—48no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4—342no—
598C2×C4—681no—
599C2×C4—681no—
600C2×C4—681no—
601C2×C4—681no—
602C2×C4—681no—
603C2×C4—681no—
604C2×C4—681no—
605C2×C4—681no—
606C2×C4—681no—
607C2×C4—681no—
608C2×C4—681no—
609C2×C4C2×#6322912yes—
610C2×C4C2×#6322912yes—
611C2×C4C2×#6322912yes—
612C2×C4C2×#6325211yescharacteristic
613C2×C4C2×#6325211yescharacteristic
614C2×C4C2×D4×D1711yescharacteristic
615C2×C4#1058769111yescharacteristic
616C2×C4C2×#6325611yescharacteristic
617C2×C4#1058769411yescharacteristic
618C8—342no—
619C8—681no—
620C8—681no—
621C8—681no—
622C22—24no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22#1176332011yescharacteristic
630C22D4×C22⋊D1711yescharacteristic
631C22#1176347011yescharacteristic
632C22C22×#6322911yescharacteristic
633C22#1176360711yescharacteristic
634C22#1176362011yescharacteristic
635C22#1176362411yescharacteristic
636C4—22no—
637C4—22no—
638C4—22no—
639C4—42no—
640C4—41no—
641C4—41no—
642C4—41no—
643C4—81no—
644C4—81no—
645C4—682no—
646C4—1361no—
647C4—1361no—
648C4—1361no—
649C4C2×#1058758011yescharacteristic
650C4#1176340611yescharacteristic
651C4#1176346011yescharacteristic
652C4#1176350711yescharacteristic
653C2—44no—
654C2#1453251911yescharacteristic
655C2#1453879211yescharacteristic
656C2#1453899811yescharacteristic
657C2#1454033511yescharacteristic
658C2D4×#6322911yescharacteristic
659C2#1454142611yescharacteristic
660C2#1454161111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingD4×C4⋊C4×C17C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
hypercenterD4×C4⋊C4D17
hypocenterC17#26166
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ D4×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #14532560C2semidirect
4. #14530328C2semidirect
5. #14530285C2semidirect
6. #14530193C2semidirect
7. #14530030C2semidirect
8. #14529459C2semidirect
9. #14529343C2semidirect
13. #14529051C2semidirect
185. C4⋊C4×C17SD16semidirect
463. C17#26166semidirect
538. C4⋊C4#63229semidirect

Maximal subgroups

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

subgroupindex
C2×#117630142
C2×C4×#632292
#145325602
#145303282
#145302852
#145301932
#145300302
#145294592
#145293432
#145291182
#145290512
#145290452
D4×C4⋊C4×C172
#2616617