Finite Group Databasebeta

← #17042180

Subgroups — #17042180

Order 4352.

730 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14538568C211yesmaximal, characteristic
2#14537722C211yesmaximal, characteristic, complemented
3#14537210C211yesmaximal, characteristic
4#14529726C211yesmaximal, characteristic, complemented
5#14529558C211yesmaximal, characteristic, complemented
6#14529386C211yesmaximal, characteristic, complemented
7#14529382C211yesmaximal, characteristic, complemented
8#14529061C211yesmaximal, characteristic
9#14529057C211yesmaximal, characteristic
10#14528802C211yesmaximal, characteristic
11#14528668C211yesmaximal, characteristic, complemented
12#14528664C211yesmaximal, characteristic, complemented
13#14528660C211yesmaximal, characteristic, complemented
14#14528619C211yesmaximal, characteristic
15C17×#1765C211yesmaximal, characteristic
16C2×#10587634C2211yescharacteristic
17#11763362C2211yescharacteristic
18#41×Dic17C2211yescharacteristic
19#11763118C411yescharacteristic, complemented
20#11763114C411yescharacteristic, complemented
21#11763111C411yescharacteristic, complemented
22#11763107C411yescharacteristic, complemented
23#11763102C411yescharacteristic, complemented
24#11763097C411yescharacteristic, complemented
25#11763092C411yescharacteristic, complemented
26#11763087C411yescharacteristic, complemented
27#11762939—21no—
28#11762939—21no—
29#11762937—21no—
30#11762937—21no—
31C2×#10587490C2211yescharacteristic
32C2×#10587493C2211yescharacteristic
33#11762847C2211yescharacteristic
34#11762845C2211yescharacteristic
35#11762837C2211yescharacteristic
36#11762836C2211yescharacteristic
37#11762832C2211yescharacteristic
38C2×#10587485C2211yescharacteristic
39C2×#10587508C2211yescharacteristic
40#11762805C2211yescharacteristic
41#11762767—21no—
42#11762767C2211yescharacteristic
43#11762764—21no—
44#11762764C2211yescharacteristic
45#11762764C2211yescharacteristic
46M16×Dic17—21no—
47M16×Dic17C2211yescharacteristic
48C2×#10587486—21no—
49C2×#10587486C2211yescharacteristic
50C2×C8×Dic17C2211yescharacteristic
51C4×#63225C2211yescharacteristic
52#11762573C2211yescharacteristic
53#11762570C2211yescharacteristic
54#11762570C2211yescharacteristic
55#11762569C2211yescharacteristic
56#11762569C2211yescharacteristic
57#11762568C2211yescharacteristic
58#11762564C2211yescharacteristic
59#11762563C2211yescharacteristic
60C2×C17×#131C2211yescharacteristic
61C2×C17×#104C2211yescharacteristic
62C17×#416C2211yescharacteristic
63C17×#405C2211yescharacteristic
64C2×C17×#97C2211yescharacteristic
65C17×#341C2211yescharacteristic
66C17×#339C2211yescharacteristic
67C2×#41×C17C2311yescharacteristic
68C17×#131—22no—
69C17×#131Q814yescomplemented
70C2×M16×C17—21no—
71C2×M16×C17C2311yescharacteristic
72C2×M16×C17C2311yescharacteristic
73C22×C8×C17—21no—
74C22×C8×C17C2311yescharacteristic
75C22×C8×C17C2311yescharacteristic
76C17×#117C2311yescharacteristic
77C2×C42×C17C2311yescharacteristic
78#10587634—22no—
79#10587634—22no—
80Q8×Dic17C2×C411yescharacteristic
81Q8×Dic17C2×C411yescharacteristic
82D4×Dic17—21no—
83D4×Dic17—21no—
84D4×Dic17C2×C411yescharacteristic
85D4×Dic17C2×C411yescharacteristic
86#10587612—21no—
87#10587612C2311yescharacteristic
88#10587612C2311yescharacteristic
89C2×#63225—21no—
90C2×C4×Dic17—21no—
91C2×C4×Dic17C2311yescharacteristic
92C2×C4×Dic17C2311yescharacteristic
93C2×#63223—21no—
94C2×#63223C2311yescharacteristic
95C2×#63223C2311yescharacteristic
96C22×#3622—21no—
97C22×#3622C2311yescharacteristic
98C22×#3622C2311yescharacteristic
99#10587576C2×C411yescharacteristic
100#10587576C2×C411yescharacteristic
101#10587574C2×C411yescharacteristic
102#10587574C2×C411yescharacteristic
103#10587564—21no—
104#10587564—21no—
105C4×Dic34—21no—
106C4×Dic34—21no—
107C17×#104—22no—
108C17×#103C2×C411yescharacteristic
109C17×#103C2×C411yescharacteristic
110C17×#102C2×C411yescharacteristic
111C17×#102C2×C411yescharacteristic
112C17×#97—21no—
113C17×#97—22no—
114C4×C8×C17—21no—
115#10587508—22no—
116#10587506C2×C411yescharacteristic
117#10587506C2×C411yescharacteristic
118#10587503C2×C411yescharacteristic
119#10587503C2×C411yescharacteristic
120#10587502—21no—
121#10587493—22no—
122#10587490—22no—
123#10587489C2×C411yescharacteristic
124#10587489C2×C411yescharacteristic
125#10587488C2×C411yescharacteristic
126#10587488C2×C411yescharacteristic
127#10587486—21no—
128#10587486—21no—
129#10587486—21no—
130#10587486—21no—
131#10587486D411yescharacteristic
132#10587486D411yescharacteristic
133#10587486D411yescharacteristic
134#10587486D411yescharacteristic
135#10587485—21no—
136#10587485—21no—
137#10587485—21no—
138#10587485—21no—
139#10587485C2×C411yescharacteristic
140#10587485C2×C411yescharacteristic
141#10587485C2×C411yescharacteristic
142#10587485C2×C411yescharacteristic
143C8×Dic17—21no—
144C8×Dic17—21no—
145C8×Dic17—21no—
146C8×Dic17—21no—
147C8×Dic17—22no—
148#10587479C2×C411yescharacteristic
149#10587479C2×C411yescharacteristic
150#10587478C2×C411yescharacteristic
151#10587478C2×C411yescharacteristic
152C4⋊#3622—21no—
153C4⋊#3622—21no—
154#41×C17—21no—
155#41×C17—21no—
156#41×C17C2×Q812yes—
157#41×C17#4112yes—
158C2×Q8×C17C22×C411yescharacteristic
159C2×D4×C17—21no—
160C2×D4×C17C22×C411yescharacteristic
161C22×C4×C17—21no—
162C22×C4×C17—21no—
163C22×C4×C17C2411yescharacteristic
164C22×Dic17—21no—
165C22×Dic17—21no—
166C22×Dic17—21no—
167C22×Dic17#4111yescharacteristic
168C22×Dic17#4111yescharacteristic
169C2×Dic34—21no—
170C2×Dic34—21no—
171M16×C17—21no—
172M16×C17—24no—
173M16×C17—21no—
174M16×C17—21no—
175M16×C17C2×Q812yes—
176C2×C8×C17—21no—
177C2×C8×C17—21no—
178C2×C8×C17—21no—
179C2×C8×C17—24no—
180C2×C8×C17—21no—
181C2×C8×C17C22×C411yescharacteristic
182C2×C8×C17C22×C411yescharacteristic
183C2×C8×C17C2×D411yescharacteristic
184C2×C8×C17C2×D411yescharacteristic
185C2×C8×C17C2×Q812yes—
186C2×C8×C17#4112yes—
187C4⋊C4×C17C22×C411yescharacteristic
188C4⋊C4×C17C22×C411yescharacteristic
189C22⋊C4×C17—21no—
190C42×C17—21no—
191C42×C17—21no—
192C42×C17—21no—
193C22⋊Dic17—21no—
194C22⋊Dic17—21no—
195C22⋊Dic17—21no—
196C22⋊Dic17—21no—
197C4⋊Dic17—21no—
198C4⋊Dic17—21no—
199C4⋊Dic17—21no—
200C4⋊Dic17—21no—
201C4⋊Dic17C22×C411yescharacteristic
202C4⋊Dic17C22×C411yescharacteristic
203C4⋊Dic17C22×C411yescharacteristic
204C4⋊Dic17C22×C411yescharacteristic
205#63225—21no—
206#63225—21no—
207#63225—21no—
208#63225—21no—
209#63225—42no—
210C4×Dic17—21no—
211C4×Dic17—21no—
212C4×Dic17—21no—
213C4×Dic17—21no—
214C4×Dic17—21no—
215C4×Dic17—21no—
216C4×Dic17C22×C411yescharacteristic
217C4×Dic17C22×C411yescharacteristic
218C4×Dic17C2×D411yescharacteristic
219C4×Dic17C2×D411yescharacteristic
220#63223—24no—
221#63223—21no—
222#63223—21no—
223#63223—21no—
224#63223#4112yes—
225C2×#3622—24no—
226C2×#3622—21no—
227C2×#3622—21no—
228C2×#3622—21no—
229C2×#3622—21no—
230C2×#3622—21no—
231C2×#3622—21no—
232C2×#3622C22×C411yescharacteristic
233C2×#3622C22×C411yescharacteristic
234C2×#3622C2×D411yescharacteristic
235C2×#3622C2×D411yescharacteristic
236#21629—171nomaximal, Sylow
237C23×C17—21no—
238C23×C17C2×#4111yescharacteristic
239Q8×C17—21no—
240Q8×C17C4×Q812yes—
241D4×C17—22no—
242D4×C17—22no—
243D4×C17—21no—
244D4×C17C4×Q812yes—
245C2×C4×C17—21no—
246C2×C4×C17—22no—
247C2×C4×C17—22no—
248C2×C4×C17—21no—
249C2×C4×C17—21no—
250C2×C4×C17—41no—
251C2×C4×C17—41no—
252C2×C4×C17—41no—
253C2×C4×C17C23×C411yescharacteristic
254C2×C4×C17C22×D411yescharacteristic
255C2×C4×C17C22×Q811yescharacteristic
256C2×C4×C17C2×#4111yescharacteristic
257C2×C4×C17C2×#4111yescharacteristic
258C2×C4×C17C2×#4111yescharacteristic
259C2×Dic17—22no—
260C2×Dic17—21no—
261C2×Dic17—21no—
262C2×Dic17—21no—
263C2×Dic17—21no—
264C2×Dic17—41no—
265C2×Dic17—41no—
266C2×Dic17—41no—
267C2×Dic17—41no—
268C2×Dic17—81no—
269C2×Dic17—81no—
270C2×Dic17C4×D411yescharacteristic
271C2×Dic17C4×D411yescharacteristic
272C2×Dic17C4×D411yescharacteristic
273C2×Dic17C4×D411yescharacteristic
274Dic34—81no—
275Dic34—81no—
276C8×C17—21no—
277C8×C17—21no—
278C8×C17—22no—
279C8×C17—41no—
280C8×C17C22⋊Q812yes—
281C8×C17C22⋊Q812yes—
282#3622—21no—
283#3622—21no—
284#3622—21no—
285#3622—22no—
286#3622—21no—
287#3622—41no—
288#3076—171no—
289#2961—171no—
290#2875—171no—
291#1986—171no—
292#1950—171no—
293#1922—171no—
294#1922—171no—
295#1866—171no—
296#1866—171no—
297#1806—171no—
298#1773—171no—
299#1772—171no—
300#1771—171no—
301#1765D1711yescharacteristic
302#1764—171no—
303C22×C17—21no—
304C22×C17—21no—
305C22×C17—22no—
306C22×C17—41no—
307C22×C17—41no—
308C22×C17C2×C4×D411yescharacteristic
309C22×C17#52911yescharacteristic
310C22×C17#53311yescharacteristic
311C4×C17—22no—
312C4×C17—41no—
313C4×C17—41no—
314C4×C17—81no—
315C4×C17C2×C4×Q811yescharacteristic
316C4×C17C4×#4111yescharacteristic
317C4×C17C2×C22⋊Q811yescharacteristic
318C4×C17#52511yescharacteristic
319Dic17—21no—
320Dic17—21no—
321Dic17—21no—
322Dic17—21no—
323Dic17—42no—
324Dic17—41no—
325Dic17—41no—
326Dic17—41no—
327Dic17—41no—
328C2×#131—171no—
329C2×#131C2×D1711yescharacteristic
330#533—171no—
331C4×#41—171no—
332#479—171no—
333#479—171no—
334#478—171no—
335#478—171no—
336#477—171no—
337#476—171no—
338#475—171no—
339#474—171no—
340#446—341no—
341#446—341no—
342#446—341no—
343#446—341no—
344C2×#108—171no—
345C2×#108—171no—
346#427—171no—
347#427—171no—
348#424—171no—
349#424—171no—
350#423—171no—
351C2×C4⋊C8—171no—
352C2×#104—171no—
353C2×#104C2×D1711yescharacteristic
354#416—171no—
355#416C2×D1711yescharacteristic
356#405—171no—
357#405—171no—
358#405—171no—
359#405—341no—
360#405—341no—
361#405C2×D1711yescharacteristic
362C4×M16—171no—
363C4×M16—341no—
364C2×#97—171no—
365C2×#97—341no—
366C2×#97C2×D1711yescharacteristic
367C2×C4×C8—171no—
368C4×C4⋊C4—171no—
369#341—171no—
370#341C2×D1711yescharacteristic
371#339—171no—
372#339—171no—
373#339—171no—
374#339—171no—
375#339—171no—
376#339C2×D1711yescharacteristic
377#336—171no—
378#336—171no—
379C2×C17—21no—
380C2×C17—21no—
381C2×C17—42no—
382C2×C17C4×C22⋊Q811yescharacteristic
383C2×C17#295611yescharacteristic
384C2×C17#295711yescharacteristic
385C2×#41C22×D1711yescharacteristic
386#131—22no—
387#131—342no—
388#131—342no—
389#131Dic3414yescomplemented
390C2×M16—21no—
391C2×M16—171no—
392C2×M16—171no—
393C2×M16—341no—
394C2×M16C22×D1711yescharacteristic
395C2×M16C22×D1711yescharacteristic
396C22×C8—21no—
397C22×C8—171no—
398C22×C8—171no—
399C22×C8—341no—
400C22×C8C22×D1711yescharacteristic
401C22×C8C22×D1711yescharacteristic
402C4⋊Q8—171no—
403C4⋊Q8—171no—
404#125—171no—
405#125—171no—
406C22⋊Q8—341no—
407C22⋊Q8—341no—
408C4×Q8—171no—
409C4×Q8—171no—
410C4×Q8—341no—
411C4×Q8—341no—
412C4×D4—171no—
413C4×D4—171no—
414C4×D4—341no—
415C4×D4—341no—
416#117—171no—
417#117—171no—
418#117—341no—
419#117C22×D1711yescharacteristic
420C2×C4⋊C4—341no—
421C2×C42—171no—
422C2×C42—171no—
423C2×C42—341no—
424C2×C42C22×D1711yescharacteristic
425#108—342no—
426#108—342no—
427#107—171no—
428#107—171no—
429#107—171no—
430#107—171no—
431#106—171no—
432#106—171no—
433#106—171no—
434#106—171no—
435C4⋊C8—171no—
436C4⋊C8—171no—
437C4⋊C8—171no—
438C4⋊C8—171no—
439C4⋊C8—341no—
440C4⋊C8—341no—
441C4⋊C8—341no—
442C4⋊C8—341no—
443C4⋊C8—341no—
444C4⋊C8—341no—
445#104—22no—
446#104—342no—
447#103—171no—
448#103—171no—
449#103C4×D1711yescharacteristic
450#103C4×D1711yescharacteristic
451#102—171no—
452#102—171no—
453#102C4×D1711yescharacteristic
454#102C4×D1711yescharacteristic
455#97—21no—
456#97—22no—
457#97—171no—
458#97—171no—
459#97—171no—
460#97—171no—
461#97—341no—
462#97—341no—
463#97—341no—
464#97—341no—
465C4×C8—21no—
466C4×C8—341no—
467C4×C8—341no—
468C4×C8—341no—
469C4×C8—341no—
470C4×C8—342no—
471#95—341no—
472C17#2162911yescharacteristic, complemented, Sylow
473#41—21no—
474#41—21no—
475#41C2×Dic3412yes—
476#41#6325212yes—
477C2×Q8—341no—
478C2×Q8—341no—
479C2×Q8C2×C4×D1711yescharacteristic
480C2×D4—21no—
481C2×D4C2×C4×D1711yescharacteristic
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—171no—
485C22×C4—171no—
486C22×C4—341no—
487C22×C4—341no—
488C22×C4—341no—
489C22×C4C23×D1711yescharacteristic
490M16—21no—
491M16—24no—
492M16—21no—
493M16—21no—
494M16—172no—
495M16—344no—
496M16—341no—
497M16—341no—
498M16—341no—
499M16C2×Dic3412yes—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—24no—
504C2×C8—21no—
505C2×C8—171no—
506C2×C8—171no—
507C2×C8—171no—
508C2×C8—171no—
509C2×C8—344no—
510C2×C8—341no—
511C2×C8—341no—
512C2×C8—341no—
513C2×C8—341no—
514C2×C8—341no—
515C2×C8—341no—
516C2×C8C2×Dic3412yes—
517C2×C8C2×C4×D1711yescharacteristic
518C2×C8C2×C4×D1711yescharacteristic
519C2×C8D4×D1711yescharacteristic
520C2×C8D4×D1711yescharacteristic
521C2×C8#6325412yes—
522C4⋊C4—171no—
523C4⋊C4—171no—
524C4⋊C4—171no—
525C4⋊C4—171no—
526C4⋊C4—341no—
527C4⋊C4—341no—
528C4⋊C4—341no—
529C4⋊C4—341no—
530C4⋊C4—341no—
531C4⋊C4—341no—
532C4⋊C4—341no—
533C4⋊C4—341no—
534C4⋊C4—682no—
535C4⋊C4C2×C4×D1711yescharacteristic
536C4⋊C4C2×C4×D1711yescharacteristic
537C22⋊C4—21no—
538C22⋊C4—341no—
539C22⋊C4—341no—
540C22⋊C4—341no—
541C22⋊C4—341no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—171no—
546C42—171no—
547C42—171no—
548C42—171no—
549C42—341no—
550C42—341no—
551C42—341no—
552C42—341no—
553C42—341no—
554C42—341no—
555C23—21no—
556C23#41×D1711yescharacteristic
557Q8—21no—
558Q8—1361no—
559Q8—1361no—
560Q8C4×Dic3412yes—
561D4—22no—
562D4—22no—
563D4—21no—
564D4C4×Dic3412yes—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—171no—
574C2×C4—171no—
575C2×C4—171no—
576C2×C4—171no—
577C2×C4—342no—
578C2×C4—341no—
579C2×C4—341no—
580C2×C4—341no—
581C2×C4—341no—
582C2×C4—681no—
583C2×C4—681no—
584C2×C4—681no—
585C2×C4—681no—
586C2×C4—1361no—
587C2×C4—1361no—
588C2×C4C22×Dic3411yescharacteristic
589C2×C4C22×C4×D1711yescharacteristic
590C2×C4C2×#6325211yescharacteristic
591C2×C4C2×D4×D1711yescharacteristic
592C2×C4C2×#6325411yescharacteristic
593C2×C4#41×D1711yescharacteristic
594C8—21no—
595C8—21no—
596C8—22no—
597C8—41no—
598C8—341no—
599C8—341no—
600C8—341no—
601C8—341no—
602C8—342no—
603C8—681no—
604C8#1058756412yes—
605C8#1058756412yes—
606C22—21no—
607C22—21no—
608C22—22no—
609C22—41no—
610C22—41no—
611C22C4×D4×D1711yescharacteristic
612C22#1176333911yescharacteristic
613C22#1176336211yescharacteristic
614C4—22no—
615C4—41no—
616C4—41no—
617C4—81no—
618C4—341no—
619C4—341no—
620C4—341no—
621C4—341no—
622C4—682no—
623C4—681no—
624C4—681no—
625C4—681no—
626C4—681no—
627C4C2×C4×Dic3411yescharacteristic
628C4C4×#6325411yescharacteristic
629C4C2×#1058756411yescharacteristic
630C4#1176331811yescharacteristic
631C2—21no—
632C2—21no—
633C2—42no—
634C2C4×#1058756411yescharacteristic
635C2#1453769211yescharacteristic
636C2#1453769811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1765C2
socleC22×C17C2×C4×D4
prime radicalC2×#41×C17C23
cosocleC22×C4×C17C24
hypercenter#1765D17
hypocenterC17#21629
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1765

Complements

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

NH ≅ G/Ntype
2. #14537722C2semidirect
4. #14529726C2semidirect
5. #14529558C2semidirect
6. #14529386C2semidirect
7. #14529382C2semidirect
11. #14528668C2semidirect
12. #14528664C2semidirect
13. #14528660C2semidirect
19. #11763118C4semidirect
20. #11763114C4semidirect
21. #11763111C4semidirect
22. #11763107C4semidirect
23. #11763102C4semidirect
24. #11763097C4semidirect
25. #11763092C4semidirect
26. #11763087C4semidirect
69. C17×#131Q8semidirect
389. #131Dic34semidirect
472. C17#21629semidirect

Maximal subgroups

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

subgroupindex
#145385682
#145377222
#145372102
#145297262
#145295582
#145293862
#145293822
#145290612
#145290572
#145288022
#145286682
#145286642
#145286602
#145286192
C17×#17652
#2162917