Finite Group Databasebeta

← #17056118

Subgroups — #17056118

Order 4352.

926 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763087C211yesmaximal, characteristic
2#14531837C211yesmaximal, characteristic, complemented
3D4×#63225C211yesmaximal, characteristic
4#14529743C211yesmaximal, characteristic, complemented
5#14529539C211yesmaximal, characteristic, complemented
6#14529515C211yesmaximal, characteristic, complemented
7#14529514C211yesmaximal, characteristic, complemented
8#14529356C211yesmaximal, characteristic, complemented
9#14529353C211yesmaximal, characteristic, complemented
10#14529029C211yesmaximal, characteristic
11#14528645C211yesmaximal, characteristic, complemented
12#102×Dic17C211yesmaximal, characteristic
13#102×Dic17C211yesmaximal, characteristic
14C4×#10587503C211yesmaximal, characteristic
15C17×#1862C211yesmaximal, characteristic
16C2×#10587574C2211yescharacteristic
17C2×#10587573—21no—
18C2×D4×Dic17C2211yescharacteristic
19C2×D4×Dic17C2211yescharacteristic
20C22×#63225—21no—
21#11763087C418yescomplemented
22C2×#10587489C2211yescharacteristic
23C2×#10587478C2211yescharacteristic
24C2×#10587478C2211yescharacteristic
25C2×#10587485C2211yescharacteristic
26C2×#10587503C2211yescharacteristic
27C2×#10587503C2211yescharacteristic
28C2×C4×#3622C2211yescharacteristic
29C2×C8×Dic17C2211yescharacteristic
30C2×C8×Dic17C2211yescharacteristic
31#11762713—21no—
32#11762707C2211yescharacteristic
33#11762705C2211yescharacteristic
34#11762705C2211yescharacteristic
35#11762691—21no—
36#11762690—21no—
37#11762682—21no—
38#11762681—21no—
39#11762678C2211yescharacteristic
40#11762667—21no—
41C4⋊C4×Dic17C2211yescharacteristic
42C4⋊C4×Dic17C2211yescharacteristic
43C4×#63225C2211yescharacteristic
44C4×C4⋊Dic17C2211yescharacteristic
45#11762572C2211yescharacteristic
46#11762572C2211yescharacteristic
47#11762572C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762563C2211yescharacteristic
53C2×C4×D4×C17C2211yescharacteristic
54C2×C17×#107C2211yescharacteristic
55C2×C17×#102C2211yescharacteristic
56C2×C17×#102C2211yescharacteristic
57C17×#389C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C17×#336C2211yescharacteristic
60C22×D4×C17C2311yescharacteristic
61C23×C4×C17—21no—
62C23×Dic17—21no—
63C23×Dic17—21no—
64C22×Dic34—21no—
65C22×C8×C17C2311yescharacteristic
66C22×C8×C17C2311yescharacteristic
67C4×D4×C17—22no—
68C4×D4×C17—41no—
69C2×C4⋊C4×C17—21no—
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C4⋊C4×C17C2311yescharacteristic
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C22⋊C4×C17—21no—
74C2×C42×C17C2311yescharacteristic
75C2×C22⋊Dic17—21no—
76C2×C22⋊Dic17—21no—
77D4×Dic17—22no—
78D4×Dic17—22no—
79D4×Dic17—41no—
80D4×Dic17C2×C414yes—
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4⋊Dic17C2311yescharacteristic
83C2×C4⋊Dic17C2311yescharacteristic
84C2×C4⋊Dic17C2311yescharacteristic
85C2×#63225—21no—
86C2×#63225—21no—
87C2×#63225—21no—
88C2×#63225—21no—
89C2×#63225—21no—
90C2×#63225—21no—
91C2×#63225—44no—
92C2×#63225D411yescharacteristic
93C2×#63225D411yescharacteristic
94C2×C4×Dic17—21no—
95C2×C4×Dic17—21no—
96C2×C4×Dic17—21no—
97C2×C4×Dic17C2311yescharacteristic
98C2×C4×Dic17C2311yescharacteristic
99C22×#3622C2311yescharacteristic
100C22×#3622C2311yescharacteristic
101#10587574—22no—
102#10587574C2×C414yescomplemented
103#10587573—44no—
104C17×#107—41no—
105C17×#102—22no—
106C17×#102C2×C414yes—
107C17×#95—21no—
108#10587503C2×C414yes—
109#10587503Q814yescomplemented
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116#10587502D411yescharacteristic
117#10587502D411yescharacteristic
118#10587489C2×C414yescomplemented
119#10587485C2×C414yescomplemented
120C8×Dic17—22no—
121C8×Dic17—41no—
122#10587478—41no—
123#10587478C2×C414yescomplemented
124C4×#3622—41no—
125C24×C17—21no—
126C2×D4×C17—21no—
127C2×D4×C17—21no—
128C2×D4×C17—21no—
129C2×D4×C17C22×C412yes—
130C2×D4×C17C2×Q812yes—
131C2×D4×C17#4112yes—
132C22×C4×C17—21no—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17—42no—
136C22×C4×C17—42no—
137C22×C4×C17—81no—
138C22×C4×C17C2×D411yescharacteristic
139C22×C4×C17C2×D411yescharacteristic
140C22×C4×C17C2411yescharacteristic
141C22×Dic17—21no—
142C22×Dic17—21no—
143C22×Dic17—21no—
144C22×Dic17—24no—
145C22×Dic17—21no—
146C22×Dic17—42no—
147C22×Dic17—42no—
148C22×Dic17—42no—
149C22×Dic17—42no—
150C22×Dic17—81no—
151C22×Dic17C2×D411yescharacteristic
152C22×Dic17C2×D411yescharacteristic
153C22×Dic17C2×D411yescharacteristic
154C22×Dic17C2×D411yescharacteristic
155C2×Dic34—24no—
156C2×Dic34—42no—
157C2×Dic34—42no—
158C2×C8×C17—21no—
159C2×C8×C17—21no—
160C2×C8×C17—21no—
161C2×C8×C17—21no—
162C2×C8×C17—21no—
163C2×C8×C17C22×C412yes—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17—21no—
167C4⋊C4×C17—21no—
168C4⋊C4×C17—21no—
169C4⋊C4×C17—42no—
170C4⋊C4×C17C22×C412yes—
171C22⋊C4×C17—42no—
172C42×C17—21no—
173C42×C17—21no—
174C22⋊Dic17—24no—
175C22⋊Dic17—42no—
176C4⋊Dic17—21no—
177C4⋊Dic17—21no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17C22×C412yes—
182C4⋊Dic17C22×C412yes—
183C4⋊Dic17C2×Q812yes—
184#63225—24no—
185#63225—22no—
186#63225—42no—
187#63225—42no—
188#63225—42no—
189#63225—42no—
190#63225—81no—
191#63225D814yes—
192C4×Dic17—21no—
193C4×Dic17—21no—
194C4×Dic17—42no—
195C4×Dic17—42no—
196C4×Dic17—42no—
197C4×Dic17C22×C412yes—
198C4×Dic17#4112yes—
199C2×#3622—21no—
200C2×#3622—21no—
201C2×#3622—21no—
202C2×#3622—21no—
203C2×#3622C22×C412yes—
204C2×#3622C2×Q812yes—
205#22962—171nomaximal, Sylow
206C23×C17—22no—
207C23×C17—22no—
208C23×C17—22no—
209C23×C17—41no—
210C23×C17—41no—
211C23×C17—41no—
212C23×C17—41no—
213C23×C17C22×D411yescharacteristic
214D4×C17—22no—
215D4×C17—22no—
216D4×C17—22no—
217D4×C17C4×Q814yescomplemented
218C2×C4×C17—21no—
219C2×C4×C17—21no—
220C2×C4×C17—22no—
221C2×C4×C17—21no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17—41no—
225C2×C4×C17—41no—
226C2×C4×C17—41no—
227C2×C4×C17—41no—
228C2×C4×C17—41no—
229C2×C4×C17—41no—
230C2×C4×C17—41no—
231C2×C4×C17—41no—
232C2×C4×C17—84no—
233C2×C4×C17C2×D812yes—
234C2×C4×C17C23×C411yescharacteristic
235C2×C4×C17C22×Q811yescharacteristic
236C2×C4×C17C2×#4111yescharacteristic
237C2×C4×C17C2×#4111yescharacteristic
238C2×C4×C17H(5,𝔽2)11yescharacteristic
239C2×C4×C172−1+411yescharacteristic
240C2×Dic17—21no—
241C2×Dic17—21no—
242C2×Dic17—21no—
243C2×Dic17—21no—
244C2×Dic17—22no—
245C2×Dic17—22no—
246C2×Dic17—21no—
247C2×Dic17—22no—
248C2×Dic17—21no—
249C2×Dic17—21no—
250C2×Dic17—41no—
251C2×Dic17—41no—
252C2×Dic17—41no—
253C2×Dic17—41no—
254C2×Dic17—41no—
255C2×Dic17—41no—
256C2×Dic17—48no—
257C2×Dic17—41no—
258C2×Dic17—41no—
259C2×Dic17—41no—
260C2×Dic17—84no—
261C2×Dic17C4×D412yes—
262C2×Dic17C4×D412yes—
263C2×Dic17C2×D812yes—
264C2×Dic17C2×D812yes—
265C2×Dic17#13512yes—
266Dic34—48no—
267C8×C17—22no—
268C8×C17—41no—
269#3622—22no—
270#3622—22no—
271C2×#474—171no—
272#2365—171no—
273D4×C4⋊C4—171no—
274#1990—171no—
275#1947—171no—
276#1942—171no—
277#1942—171no—
278#1918—171no—
279#1918—171no—
280#1862—171no—
281#1862D1711yescharacteristic
282C8×C4⋊C4—171no—
283C4×#102—171no—
284C4×#102—171no—
285C4×#102—171no—
286C22×C17—24no—
287C22×C17—42no—
288C22×C17—42no—
289C22×C17—42no—
290C22×C17—42no—
291C22×C17—42no—
292C22×C17—42no—
293C22×C17C2×C4×D411yescharacteristic
294C22×C17#54611yescharacteristic
295C22×C17D4×Q811yescharacteristic
296C22×C17C22×D811yescharacteristic
297C22×C17C2×#13511yescharacteristic
298C22×C17#57711yescharacteristic
299C22×C17#57811yescharacteristic
300C4×C17—22no—
301C4×C17—42no—
302C4×C17—41no—
303C4×C17—81no—
304C4×C17—81no—
305C4×C17C2×C4×Q811yescharacteristic
306C4×C17#52011yescharacteristic
307C4×C17#55211yescharacteristic
308C4×C17#55411yescharacteristic
309Dic17—22no—
310Dic17—22no—
311Dic17—42no—
312Dic17—42no—
313Dic17—42no—
314Dic17—41no—
315Dic17—81no—
316Dic17C4×D814yes—
317C2×C4⋊Q8—171no—
318C2×C4×Q8—341no—
319C2×C4×D4—171no—
320C2×C4×D4—171no—
321C2×C4×D4C2×D1711yescharacteristic
322C22×C4⋊C4—341no—
323#474—178no—
324C2×#107—171no—
325C2×#107—171no—
326C2×#107—171no—
327C2×#107C2×D1711yescharacteristic
328C2×C4⋊C8—171no—
329C2×#102—171no—
330C2×#102—171no—
331C2×#102C2×D1711yescharacteristic
332C2×#102C2×D1711yescharacteristic
333C2×C4×C8—171no—
334C2×C4×C8—171no—
335C2×C4×C8—171no—
336#391—341no—
337#389—171no—
338#389—171no—
339#389—171no—
340#389C2×D1711yescharacteristic
341#387—341no—
342#387—341no—
343#385—341no—
344#385—341no—
345#384—171no—
346#380—341no—
347C4×C4⋊C4—171no—
348C4×C4⋊C4—171no—
349C4×C4⋊C4—171no—
350C4×C4⋊C4—171no—
351#340—171no—
352#340—171no—
353#340—171no—
354#340—171no—
355#340—171no—
356#340—171no—
357#340—171no—
358#340C2×D1711yescharacteristic
359#336—171no—
360#336C2×D1711yescharacteristic
361C2×C17—44no—
362C2×C17#236011yescharacteristic
363C2×C17C2×C4×D811yescharacteristic
364C2×C17#294711yescharacteristic
365C2×C17#329411yescharacteristic
366C2×C17#330211yescharacteristic
367C2×C17Q8×D811yescharacteristic
368C2×C17#338111yescharacteristic
369C22×Q8—341no—
370C22×D4C22×D1711yescharacteristic
371C23×C4—21no—
372C23×C4—341no—
373C23×C4—341no—
374C22×C8—171no—
375C22×C8—171no—
376C22×C8C22×D1711yescharacteristic
377C22×C8C22×D1711yescharacteristic
378C4⋊Q8—174no—
379C4⋊Q8—342no—
380C4×Q8—684no—
381C4×D4—22no—
382C4×D4—41no—
383C4×D4—174no—
384C4×D4—342no—
385C4×D4—342no—
386C4×D4—681no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—171no—
389C2×C4⋊C4—171no—
390C2×C4⋊C4—171no—
391C2×C4⋊C4—171no—
392C2×C4⋊C4—171no—
393C2×C4⋊C4—171no—
394C2×C4⋊C4—341no—
395C2×C4⋊C4—341no—
396C2×C4⋊C4—341no—
397C2×C4⋊C4—341no—
398C2×C4⋊C4—341no—
399C2×C4⋊C4—341no—
400C2×C4⋊C4—684no—
401C2×C4⋊C4C22×D1711yescharacteristic
402C2×C4⋊C4C22×D1711yescharacteristic
403C2×C4⋊C4C22×D1711yescharacteristic
404C2×C22⋊C4—21no—
405C2×C22⋊C4—341no—
406C2×C22⋊C4—341no—
407C2×C42—171no—
408C2×C42—171no—
409C2×C42—341no—
410C2×C42—341no—
411C2×C42—341no—
412C2×C42C22×D1711yescharacteristic
413#107—41no—
414#107—174no—
415#107—174no—
416#107—681no—
417C4⋊C8—174no—
418#102—22no—
419#102—174no—
420#102—174no—
421#102C4×D1714yes—
422C4×C8—342no—
423C4×C8—681no—
424C4×C8—681no—
425#95—21no—
426#95—171no—
427#95—171no—
428#95—341no—
429#95—341no—
430#95—341no—
431#95—341no—
432#95—341no—
433#95—341no—
434C17#2296211yescharacteristic, complemented, Sylow
435C24—21no—
436C2×Q8—344no—
437C2×Q8—682no—
438C2×Q8—682no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4C2×C4×D1712yes—
443C2×D4#6325412yes—
444C2×D4Q8×D1712yes—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—81no—
451C22×C4—171no—
452C22×C4—171no—
453C22×C4—171no—
454C22×C4—171no—
455C22×C4—344no—
456C22×C4—341no—
457C22×C4—341no—
458C22×C4—341no—
459C22×C4—341no—
460C22×C4—682no—
461C22×C4—682no—
462C22×C4—682no—
463C22×C4—682no—
464C22×C4—1361no—
465C22×C4D4×D1711yescharacteristic
466C22×C4D4×D1711yescharacteristic
467C22×C4C23×D1711yescharacteristic
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—172no—
474C2×C8—172no—
475C2×C8—341no—
476C2×C8—341no—
477C2×C8—341no—
478C2×C8—341no—
479C2×C8C2×C4×D1712yes—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—42no—
486C4⋊C4—174no—
487C4⋊C4—172no—
488C4⋊C4—172no—
489C4⋊C4—172no—
490C4⋊C4—342no—
491C4⋊C4—344no—
492C4⋊C4—341no—
493C4⋊C4—341no—
494C4⋊C4—341no—
495C4⋊C4—341no—
496C4⋊C4—341no—
497C4⋊C4—682no—
498C4⋊C4—682no—
499C4⋊C4—682no—
500C4⋊C4—682no—
501C4⋊C4—1361no—
502C4⋊C4C2×C4×D1712yes—
503C22⋊C4—42no—
504C22⋊C4—344no—
505C22⋊C4—682no—
506C42—21no—
507C42—21no—
508C42—172no—
509C42—172no—
510C42—341no—
511C42—341no—
512C42—682no—
513C42—682no—
514C42—682no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23C2×D4×D1711yescharacteristic
523Q8—688no—
524D4—22no—
525D4—22no—
526D4—22no—
527D4#1058757314yescomplemented
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—84no—
543C2×C4—172no—
544C2×C4—172no—
545C2×C4—172no—
546C2×C4—172no—
547C2×C4—172no—
548C2×C4—341no—
549C2×C4—341no—
550C2×C4—341no—
551C2×C4—341no—
552C2×C4—342no—
553C2×C4—342no—
554C2×C4—342no—
555C2×C4—341no—
556C2×C4—341no—
557C2×C4—341no—
558C2×C4—688no—
559C2×C4—681no—
560C2×C4—681no—
561C2×C4—681no—
562C2×C4—681no—
563C2×C4—681no—
564C2×C4—681no—
565C2×C4—681no—
566C2×C4—681no—
567C2×C4—681no—
568C2×C4—1364no—
569C2×C4D8×D1712yes—
570C2×C4C22×C4×D1711yescharacteristic
571C2×C4C2×#6325411yescharacteristic
572C2×C4#1058769111yescharacteristic
573C2×C4C2×Q8×D1711yescharacteristic
574C2×C4#41×D1711yescharacteristic
575C2×C4#1058769711yescharacteristic
576C8—22no—
577C8—41no—
578C8—342no—
579C8—342no—
580C22—24no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22C4×D4×D1711yescharacteristic
588C22#1176344611yescharacteristic
589C22#1176347211yescharacteristic
590C22C2×D8×D1711yescharacteristic
591C22#135×D1711yescharacteristic
592C22#1176361911yescharacteristic
593C22#1176362511yescharacteristic
594C4—22no—
595C4—42no—
596C4—41no—
597C4—81no—
598C4—81no—
599C4—174no—
600C4—342no—
601C4—342no—
602C4—682no—
603C4—682no—
604C4—682no—
605C4—681no—
606C4—1361no—
607C4C2×#1058757311yescharacteristic
608C4#1176328811yescharacteristic
609C4#1176349611yescharacteristic
610C4#1176351311yescharacteristic
611C2—44no—
612C2#1453178411yescharacteristic
613C2C4×D8×D1711yescharacteristic
614C2#1453762411yescharacteristic
615C2#1454059011yescharacteristic
616C2#1454069411yescharacteristic
617C2#1454163411yescharacteristic
618C2#1454165411yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1862C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
cosocleC22×C4×C17C24
hypercenter#1862D17
hypocenterC17#22962
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1862

Complements

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

NH ≅ G/Ntype
2. #14531837C2semidirect
4. #14529743C2semidirect
5. #14529539C2semidirect
6. #14529515C2semidirect
7. #14529514C2semidirect
8. #14529356C2semidirect
9. #14529353C2semidirect
11. #14528645C2semidirect
21. #11763087C4semidirect
102. #10587574C2×C4semidirect
109. #10587503Q8semidirect
118. #10587489C2×C4semidirect
119. #10587485C2×C4semidirect
123. #10587478C2×C4semidirect
217. D4×C17C4×Q8semidirect
434. C17#22962semidirect
527. D4#10587573semidirect

Maximal subgroups

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

subgroupindex
C2×#117630872
#145318372
D4×#632252
#145297432
#145295392
#145295152
#145295142
#145293562
#145293532
#145290292
#145286452
#102×Dic172
C4×#105875032
C17×#18622
#2296217