Finite Group Databasebeta

← #17898146

Subgroups — #17898146

Order 4352.

2038 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14541983C211yesmaximal, characteristic
2#14537452C213yesmaximal, complemented
3#14537449C213yesmaximal, complemented
4#14537441C213yesmaximal, complemented
5#14537441C213yesmaximal, complemented
6#14537434C213yesmaximal, complemented
7#14537427C213yesmaximal, complemented
8#14537421C213yesmaximal, complemented
9C2×#11762869C213yesmaximal, complemented
10#14537352C211yesmaximal, characteristic, complemented
11#14537231C211yesmaximal, characteristic
12#14537231C211yesmaximal, characteristic
13#14537210C211yesmaximal, characteristic, complemented
14#131×Dic17C211yesmaximal, characteristic, complemented
15C17×#2906C211yesmaximal, characteristic, complemented
16C2×#41×D17C2211yescharacteristic
17C2×#10587634C2211yescharacteristic
18C2×#10587634C2211yescharacteristic
19#11763291C411yescharacteristic
20#11763291C411yescharacteristic
21#11763288C413yes—
22#11763288C413yes—
23#11763285C411yescharacteristic
24#11763285C411yescharacteristic
25#11763281C413yes—
26#11763281C413yes—
27#41×Dic17C2211yescharacteristic
28C4×#63252C2213yes—
29#11763277C2213yes—
30C2×#10587558C2213yes—
31#117×D17C2213yes—
32#11762869C2211yescharacteristic
33#11762869C2213yes—
34#11762869C2213yes—
35#11762869C2213yes—
36#11762869C2216yes—
37#11762869C2213yescomplemented
38#11762869C2213yescomplemented
39#11762869C2213yescomplemented
40#11762869C2213yescomplemented
41#11762869C2216yescomplemented
42#11762869C2216yescomplemented
43#11762865C2216yescomplemented
44#11762865C2216yescomplemented
45#11762861C2216yescomplemented
46#11762861C2216yescomplemented
47#11762836C2213yescomplemented
48#11762836C2213yescomplemented
49#11762833C2213yescomplemented
50C2×C4⋊#3622C2213yescomplemented
51#11762777C2213yes—
52#11762777C2213yes—
53#11762775C2213yes—
54#11762775C2213yes—
55#11762772C2213yes—
56#11762772C2213yes—
57C2×#10587491C2213yes—
58C2×#10587491C2213yes—
59#11762767C2213yescomplemented
60#11762767C2213yescomplemented
61#11762764C2213yescomplemented
62#11762764C2213yescomplemented
63M16×Dic17C2213yescomplemented
64M16×Dic17C2213yescomplemented
65C2×#10587486C2213yescomplemented
66C2×C8×Dic17C2213yescomplemented
67C2×C17×#131C2211yescharacteristic
68C2×C17×#131C2211yescharacteristic
69C4×#41×C17C2211yescharacteristic
70C17×#424C2213yescomplemented
71C17×#424C2213yescomplemented
72C17×#423C2213yescomplemented
73C2×C17×C4⋊C8C2213yescomplemented
74C2×#41×C17C2311yescharacteristic
75#41×D17—24no—
76#41×D17—24no—
77C2×#63256C2×C411yescharacteristic
78C2×Q8×D17C2×C411yescharacteristic
79C2×#63254C2×C413yes—
80C2×D4×D17C2×C413yes—
81C2×#63252C2313yes—
82C22×C4×D17C2313yes—
83C17×#131—24no—
84C17×#131—24no—
85C2×M16×C17C2313yes—
86C2×M16×C17C2313yes—
87C22×C8×C17C2313yes—
88C22×C8×C17C2313yes—
89C4×Q8×C17C2×C411yescharacteristic
90C4×Q8×C17C2×C411yescharacteristic
91C4×D4×C17C2×C413yes—
92C4×D4×C17C2×C413yes—
93C17×#117C2313yes—
94C2×C42×C17C2313yes—
95#10587634—24no—
96#10587634—24no—
97Q8×Dic17C2×C411yescharacteristic
98Q8×Dic17C2×C411yescharacteristic
99D4×Dic17C2×C413yes—
100D4×Dic17C2×C413yes—
101C4×C22⋊D17C2×C416yes—
102C4×C22⋊D17C2×C416yes—
103C2×#63227C2×C413yes—
104C2×#63227C2×C413yes—
105#10587612C2313yes—
106C2×#63225C2×C413yes—
107C2×#63225C2×C413yes—
108C2×C4×Dic17C2313yes—
109C2×#63223C2313yes—
110C2×#63223C2313yes—
111C22×#3622C2313yes—
112C22×#3622C2313yes—
113#10587579C2×C413yes—
114#10587579C2×C413yes—
115#10587578C2×C413yes—
116#10587578C2×C413yes—
117C4⋊C4×D17C2×C413yes—
118C4⋊C4×D17C2×C413yes—
119#10587573C2×C413yes—
120#10587573C2×C413yes—
121#10587567C2×C416yes—
122#10587567C2×C416yes—
123C22⋊C4×D17C2×C416yes—
124#10587563C2×C416yes—
125C4×D68C2×C413yes—
126C4×D68C2×C413yes—
127#10587558C2×C413yes—
128#10587558C2×C413yes—
129#10587558C2×C413yes—
130#10587558C2×C413yes—
131#10587558C2313yes—
132#10587558C2311yescharacteristic
133#10587558C2313yes—
134#10587558C2313yes—
135C42×D17C2316yes—
136C4×Dic34C2×C413yes—
137C4×Dic34C2×C413yes—
138C17×C4⋊C8C2313yes—
139C17×C4⋊C8C2316yes—
140C17×C4⋊C8C2313yes—
141C17×C4⋊C8C2311yescharacteristic
142C17×C4⋊C8C2313yes—
143#10587491C2311yescharacteristic
144#10587491C2311yescharacteristic
145#10587491C2313yes—
146#10587491C2313yes—
147#10587491C2313yes—
148#10587491C2313yes—
149#10587491C2313yes—
150#10587491C2313yes—
151#10587491C2316yes—
152#10587491C2316yes—
153#10587486C2311yescharacteristic
154#10587486C2313yes—
155#10587486C2313yes—
156#10587486C2313yes—
157#10587486C2316yes—
158C8×Dic17C2311yescharacteristic
159C8×Dic17C2313yes—
160C8×Dic17C2313yes—
161C8×Dic17C2313yes—
162C8×Dic17C2316yes—
163C4⋊#3622C2311yescharacteristic
164C4⋊#3622C2313yes—
165C4⋊#3622C2313yes—
166C4⋊#3622C2313yes—
167C4⋊#3622C2316yes—
168C23×D17C22×C413yes—
169#41×C17#4114yes—
170#41×C17#4114yes—
171C2×Q8×C17C22×C411yescharacteristic
172C2×D4×C17C22×C413yes—
173C22×C4×C17C22×C413yes—
174C22×C4×C17C22×C413yes—
175C22×C4×C17C2413yes—
176C2×C22⋊D17C22×C416yes—
177C22×Dic17C22×C413yes—
178C22×Dic17C22×C413yes—
179C22×Dic17C22×C413yes—
180#63256—24no—
181Q8×D17—24no—
182#63254—212no—
183D4×D17—212no—
184#63252—26no—
185#63252—26no—
186C2×D68C22×C413yes—
187C2×C4×D17—26no—
188C2×C4×D17—26no—
189C2×C4×D17C22×C413yes—
190C2×C4×D17C22×C413yes—
191C2×C4×D17C22×C416yes—
192C2×C4×D17C22×C416yes—
193C2×C4×D17C2411yescharacteristic
194C2×C4×D17C2413yes—
195C2×Dic34C22×C413yes—
196M16×C17—26no—
197M16×C17—26no—
198C2×C8×C17—26no—
199C2×C8×C17—26no—
200C2×C8×C17C2411yescharacteristic
201C2×C8×C17C2411yescharacteristic
202C2×C8×C17C2413yes—
203C2×C8×C17C2413yes—
204C4⋊C4×C17C22×C413yes—
205C4⋊C4×C17C22×C413yes—
206C22⋊C4×C17C22×C416yes—
207C42×C17C22×C413yes—
208C42×C17C22×C413yes—
209C42×C17C2413yes—
210C42×C17C2411yescharacteristic
211C22⋊Dic17C22×C416yes—
212#63227C22×C411yescharacteristic
213#63227C22×C411yescharacteristic
214#63227C22×C413yes—
215#63227C22×C413yes—
216#63227C22×C413yes—
217#63227C22×C413yes—
218#63227C22×C413yes—
219#63227C22×C413yes—
220C4⋊Dic17C22×C413yes—
221C4⋊Dic17C22×C413yes—
222#63225C22×C413yes—
223#63225C22×C413yes—
224#63225C22×C411yescharacteristic
225#63225C22×C411yescharacteristic
226#63225C22×C413yes—
227#63225C22×C413yes—
228#63225C22×C413yes—
229#63225C22×C413yes—
230C4×Dic17C22×C416yes—
231C4×Dic17C22×C416yes—
232C4×Dic17C22×C413yes—
233C4×Dic17C22×C413yes—
234C4×Dic17C2413yes—
235C4×Dic17C2411yescharacteristic
236#63223—26no—
237#63223—26no—
238C2×#3622—26no—
239C2×#3622—26no—
240C2×#3622C2411yescharacteristic
241C2×#3622C2413yes—
242C2×#3622C2413yes—
243C2×#3622C2411yescharacteristic
244#60129—171nomaximal, Sylow
245C23×C17C23×C413yes—
246C22×D17—26no—
247C22×D17—43no—
248C22×D17—43no—
249C22×D17C23×C413yes—
250C22×D17C23×C411yescharacteristic
251Q8×C17#11714yes—
252D4×C17#117112yes—
253C2×C4×C17—43no—
254C2×C4×C17—43no—
255C2×C4×C17C23×C411yescharacteristic
256C2×C4×C17C23×C411yescharacteristic
257C2×C4×C17C23×C413yes—
258C2×C4×C17C23×C413yes—
259C2×C4×C17C23×C413yes—
260C2×C4×C17C2×#4116yes—
261C2×C4×C17C2×#4116yes—
262C2×C4×C17C2511yescharacteristic
263C22⋊D17—212no—
264C2×Dic17—26no—
265C2×Dic17—43no—
266C2×Dic17—43no—
267C2×Dic17C23×C413yes—
268C2×Dic17C23×C413yes—
269C2×Dic17C23×C413yes—
270C2×Dic17C23×C411yescharacteristic
271C2×Dic17C23×C411yescharacteristic
272C2×Dic17C23×C411yescharacteristic
273D68—26no—
274C4×D17—26no—
275C4×D17—26no—
276C4×D17—21no—
277C4×D17—21no—
278C4×D17—23no—
279C4×D17—23no—
280C4×D17—46no—
281C4×D17—46no—
282Dic34—26no—
283C8×C17—21no—
284C8×C17—21no—
285C8×C17—23no—
286C8×C17—23no—
287#3622—21no—
288#3622—21no—
289#3622—23no—
290#3622—23no—
291#3428—171no—
292#2925—173no—
293#2925—173no—
294#2924—173no—
295#2924—173no—
296#2923—173no—
297#2922—173no—
298#2921—173no—
299C2×#433—173no—
300#2906—171no—
301#2906D1711yescharacteristic, complemented
302#2880—171no—
303#2880—171no—
304#2875—171no—
305C4×#131—171no—
306C22×C17—23no—
307C22×C17—23no—
308C22×C17C2×#11716yes—
309C22×C17C24×C411yescharacteristic
310C2×D17—21no—
311C2×D17—23no—
312C2×D17—23no—
313C2×D17—23no—
314C2×D17—21no—
315C2×D17—21no—
316C2×D17—86no—
317C4×C17—23no—
318C4×C17—23no—
319C4×C17—21no—
320C4×C17—21no—
321C4×C17C2×#11716yes—
322C4×C17C22×#4111yescharacteristic
323C4×C17C22×#4111yescharacteristic
324Dic17—23no—
325Dic17—21no—
326Dic17—23no—
327Dic17—23no—
328Dic17—21no—
329Dic17—21no—
330C22×#41—171no—
331C2×#131—171no—
332C2×#131—171no—
333C2×#131C2×D1711yescharacteristic
334C2×#131C2×D1711yescharacteristic
335#520—171no—
336#520—171no—
337#520—173no—
338#520—173no—
339#519—171no—
340#519—171no—
341#518—173no—
342#518—173no—
343C4×#41—171no—
344C4×#41—173no—
345C4×#41—173no—
346C4×#41C2×D1711yescharacteristic
347C2×#117—173no—
348C2×#117—173no—
349#433—171no—
350#433—173no—
351#433—173no—
352#433—173no—
353#433—173no—
354#433—173no—
355#433—173no—
356#433—173no—
357#433—176no—
358#433—176no—
359#433—176no—
360#432—176no—
361#432—176no—
362#431—176no—
363#431—176no—
364#424—173no—
365#424—173no—
366#424C2×D1713yescomplemented
367#424C2×D1713yescomplemented
368#423—173no—
369#423C2×D1713yescomplemented
370C2×C4⋊C8—173no—
371C2×C4⋊C8C2×D1713yescomplemented
372#408—173no—
373#408—173no—
374#408—173no—
375#408—173no—
376#407—173no—
377#407—173no—
378C2×C22⋊C8—173no—
379C2×C22⋊C8—173no—
380#405—173no—
381#405—173no—
382#405—173no—
383#405—173no—
384C4×M16—173no—
385C4×M16—173no—
386C2×#97—173no—
387C2×C4×C8—173no—
388C2×C17—26no—
389C2×C17C22×#11711yescharacteristic
390C2×C17#357311yescharacteristic
391C2×C17#357311yescharacteristic
392D17—43no—
393D17—41no—
394C2×#41—171no—
395C2×#41—173no—
396C2×#41—173no—
397C2×#41—344no—
398C2×#41—344no—
399C2×#41C22×D1711yescharacteristic
400C22×Q8—171no—
401C22×D4—173no—
402C23×C4—173no—
403#131—24no—
404#131—24no—
405#131—344no—
406#131—344no—
407C2×M16—173no—
408C2×M16—173no—
409C2×M16C22×D1713yes—
410C2×M16C22×D1713yes—
411C22×C8—173no—
412C22×C8—173no—
413C22×C8C22×D1713yes—
414C22×C8C22×D1713yes—
415C4×Q8—171no—
416C4×Q8—171no—
417C4×Q8—173no—
418C4×Q8—173no—
419C4×Q8—173no—
420C4×Q8—173no—
421C4×Q8C4×D1711yescharacteristic
422C4×Q8C4×D1711yescharacteristic
423C4×D4—173no—
424C4×D4—173no—
425C4×D4—176no—
426C4×D4—176no—
427C4×D4—173no—
428C4×D4—173no—
429C4×D4—176no—
430C4×D4—176no—
431C4×D4—173no—
432C4×D4—173no—
433C4×D4C4×D1713yes—
434C4×D4C4×D1713yes—
435#117—173no—
436#117—173no—
437#117—173no—
438#117—176no—
439#117—173no—
440#117—173no—
441#117—173no—
442#117—173no—
443#117—173no—
444#117—171no—
445#117—173no—
446#117—173no—
447#117C22×D1713yes—
448C2×C4⋊C4—173no—
449C2×C4⋊C4—173no—
450C2×C4⋊C4—173no—
451C2×C4⋊C4—173no—
452C2×C22⋊C4—173no—
453C2×C22⋊C4—173no—
454C2×C22⋊C4—176no—
455C2×C42—173no—
456C2×C42—176no—
457C2×C42C22×D1713yes—
458C4⋊C8—171no—
459C4⋊C8—173no—
460C4⋊C8—173no—
461C4⋊C8—173no—
462C4⋊C8—176no—
463C4⋊C8C22×D1711yescharacteristic
464C4⋊C8C22×D1713yes—
465C4⋊C8C22×D1713yes—
466C4⋊C8C22×D1713yes—
467C4⋊C8C22×D1716yes—
468C22⋊C8—171no—
469C22⋊C8—171no—
470C22⋊C8—173no—
471C22⋊C8—173no—
472C22⋊C8—173no—
473C22⋊C8—173no—
474C22⋊C8—173no—
475C22⋊C8—173no—
476C22⋊C8—176no—
477C22⋊C8—176no—
478#97—171no—
479#97—173no—
480#97—173no—
481#97—173no—
482#97—176no—
483C4×C8—171no—
484C4×C8—173no—
485C4×C8—173no—
486C4×C8—173no—
487C4×C8—176no—
488C17#6012911yescharacteristic, complemented, Sylow
489C24—173no—
490#41—344no—
491#41—3412no—
492#41—346no—
493#41—346no—
494#41#6325414yes—
495#41#6325614yes—
496C2×Q8—173no—
497C2×Q8—344no—
498C2×Q8C2×C4×D1711yescharacteristic
499C2×D4—176no—
500C2×D4—173no—
501C2×D4—3412no—
502C2×D4C2×C4×D1713yes—
503C22×C4—173no—
504C22×C4—173no—
505C22×C4—173no—
506C22×C4—173no—
507C22×C4—173no—
508C22×C4—173no—
509C22×C4—171no—
510C22×C4—176no—
511C22×C4—176no—
512C22×C4—346no—
513C22×C4—346no—
514C22×C4C2×C4×D1713yes—
515C22×C4C2×C4×D1713yes—
516C22×C4C23×D1713yes—
517M16—26no—
518M16—26no—
519M16—346no—
520M16—346no—
521C2×C8—26no—
522C2×C8—26no—
523C2×C8—171no—
524C2×C8—173no—
525C2×C8—173no—
526C2×C8—171no—
527C2×C8—346no—
528C2×C8—346no—
529C2×C8C23×D1711yescharacteristic
530C2×C8C23×D1713yes—
531C2×C8C23×D1713yes—
532C2×C8C23×D1711yescharacteristic
533C4⋊C4—173no—
534C4⋊C4—173no—
535C4⋊C4—171no—
536C4⋊C4—171no—
537C4⋊C4—173no—
538C4⋊C4—173no—
539C4⋊C4—173no—
540C4⋊C4—173no—
541C4⋊C4—173no—
542C4⋊C4—173no—
543C4⋊C4C2×C4×D1713yes—
544C4⋊C4C2×C4×D1713yes—
545C22⋊C4—176no—
546C22⋊C4—173no—
547C22⋊C4—173no—
548C22⋊C4—171no—
549C22⋊C4—171no—
550C22⋊C4—173no—
551C22⋊C4—173no—
552C22⋊C4—173no—
553C22⋊C4—173no—
554C22⋊C4C2×C4×D1716yes—
555C42—173no—
556C42—173no—
557C42—176no—
558C42—176no—
559C42—171no—
560C42—173no—
561C42C2×C4×D1713yes—
562C42C2×C4×D1713yes—
563C42C23×D1713yes—
564C42C23×D1711yescharacteristic
565C23—171no—
566C23—173no—
567C23—346no—
568C23—683no—
569C23—683no—
570C23C22×C4×D1713yes—
571Q8—346no—
572Q8#1058757814yes—
573D4—3412no—
574D4—346no—
575D4#10587578112yes—
576C2×C4—43no—
577C2×C4—43no—
578C2×C4—173no—
579C2×C4—171no—
580C2×C4—173no—
581C2×C4—171no—
582C2×C4—173no—
583C2×C4—171no—
584C2×C4—346no—
585C2×C4—346no—
586C2×C4—346no—
587C2×C4—341no—
588C2×C4—343no—
589C2×C4—341no—
590C2×C4—343no—
591C2×C4—686no—
592C2×C4—686no—
593C2×C4—683no—
594C2×C4—683no—
595C2×C4C22×C4×D1713yes—
596C2×C4C22×C4×D1713yes—
597C2×C4C22×C4×D1711yescharacteristic
598C2×C4C22×C4×D1711yescharacteristic
599C2×C4C22×C4×D1713yes—
600C2×C4C2×#6325416yes—
601C2×C4C2×#6325616yes—
602C2×C4C24×D1711yescharacteristic
603C8—21no—
604C8—21no—
605C8—23no—
606C8—23no—
607C8—341no—
608C8—341no—
609C8—343no—
610C8—343no—
611C22—23no—
612C22—23no—
613C22—341no—
614C22—343no—
615C22—343no—
616C22—341no—
617C22—343no—
618C22—341no—
619C22—1366no—
620C22C2×#1058757816yes—
621C22C23×C4×D1711yescharacteristic
622C4—21no—
623C4—23no—
624C4—21no—
625C4—23no—
626C4—341no—
627C4—343no—
628C4—343no—
629C4—341no—
630C4—343no—
631C4—341no—
632C4C2×#1058757816yes—
633C4C22×#6325411yescharacteristic
634C4C22×#6325611yescharacteristic
635C2—26no—
636C2—681no—
637C2—683no—
638C2C22×#1058757811yescharacteristic
639C2#1454349511yescharacteristic
640C2#1454349711yescharacteristic
subgroupG/H
center, FrattiniC2×C4C24×D17
commutator, socleC22×C17C24×C4
FittingC17×#2906C2
prime radicalC2×#41×D17C22
cosocleC2×C4×C17C25
hypercenter#2906D17
hypocenterC17#60129
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ #2906

Complements

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

NH ≅ G/Ntype
2. #14537452C2semidirect
3. #14537449C2semidirect
4. #14537441C2semidirect
5. #14537441C2semidirect
6. #14537434C2semidirect
7. #14537427C2semidirect
8. #14537421C2semidirect
9. C2×#11762869C2semidirect
10. #14537352C2semidirect
13. #14537210C2semidirect
14. #131×Dic17C2semidirect
15. C17×#2906C2semidirect
37. #11762869C22semidirect
38. #11762869C22semidirect
39. #11762869C22semidirect
40. #11762869C22semidirect
41. #11762869C22semidirect
42. #11762869C22semidirect
43. #11762865C22semidirect
44. #11762865C22semidirect
45. #11762861C22semidirect
46. #11762861C22semidirect
47. #11762836C22semidirect
48. #11762836C22semidirect
49. #11762833C22semidirect
50. C2×C4⋊#3622C22semidirect
59. #11762767C22semidirect
60. #11762767C22semidirect
61. #11762764C22semidirect
62. #11762764C22semidirect
63. M16×Dic17C22semidirect
64. M16×Dic17C22semidirect
65. C2×#10587486C22semidirect
66. C2×C8×Dic17C22semidirect
70. C17×#424C22semidirect
71. C17×#424C22semidirect
72. C17×#423C22semidirect
73. C2×C17×C4⋊C8C22semidirect
301. #2906D17semidirect
366. #424C2×D17semidirect
367. #424C2×D17semidirect
369. #423C2×D17semidirect
371. C2×C4⋊C8C2×D17semidirect
488. C17#60129semidirect

Maximal subgroups

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

subgroupindex
#145419832
#145374522
#145374492
#145374412
#145374342
#145374272
#145374212
C2×#117628692
#145373522
#145372312
#145372102
#131×Dic172
C17×#29062
#6012917