Finite Group Databasebeta

← #17042178

Subgroups — #17042178

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#14538574C211yesmaximal, characteristic
2#14537724C211yesmaximal, characteristic, complemented
3#14537209C211yesmaximal, characteristic
4#14529727C211yesmaximal, characteristic, complemented
5#14529560C211yesmaximal, characteristic, complemented
6#14529384C211yesmaximal, characteristic, complemented
7#14529384C211yesmaximal, characteristic, complemented
8#14529062C211yesmaximal, characteristic
9#14529062C211yesmaximal, characteristic
10#14528805C211yesmaximal, characteristic
11#14528668C211yesmaximal, characteristic, complemented
12#14528663C211yesmaximal, characteristic, complemented
13#14528628C211yesmaximal, characteristic
14#14528621C211yesmaximal, characteristic
15C17×#1771C211yesmaximal, characteristic, complemented
16C2×#10587587C2211yescharacteristic
17C2×#10587587C2211yescharacteristic
18#11763366C2211yescharacteristic
19C4×#63252C2211yescharacteristic
20#11763117C411yescharacteristic, complemented
21#11763117C411yescharacteristic, complemented
22#11763110C411yescharacteristic, complemented
23#11763110C411yescharacteristic, complemented
24#11763105C411yescharacteristic, complemented
25#11763100C411yescharacteristic, complemented
26#11763095C411yescharacteristic, complemented
27#11763090C411yescharacteristic, complemented
28#11762937—21no—
29#11762937—21no—
30#11762937—21no—
31#11762937—21no—
32C2×#10587490C2211yescharacteristic
33C2×#10587490C2211yescharacteristic
34#11762847C2211yescharacteristic
35#11762847C2211yescharacteristic
36#11762835C2211yescharacteristic
37#11762835C2211yescharacteristic
38#11762833C2211yescharacteristic
39C2×#10587476C2211yescharacteristic
40C2×#10587476C2211yescharacteristic
41#11762806C2211yescharacteristic
42#11762806C2211yescharacteristic
43#11762766—21no—
44#11762766—21no—
45#11762764C2211yescharacteristic
46#11762764C2211yescharacteristic
47#11762764C2211yescharacteristic
48#11762764C2211yescharacteristic
49C4×#63223C2211yescharacteristic
50#11762761—21no—
51#97×D17—21no—
52#11762574C2211yescharacteristic
53#11762574C2211yescharacteristic
54#11762569C2211yescharacteristic
55#11762569C2211yescharacteristic
56#11762569C2211yescharacteristic
57#11762569C2211yescharacteristic
58#11762569C2211yescharacteristic
59#11762569C2211yescharacteristic
60C2×C17×C4⋊C8C2211yescharacteristic
61C2×C17×#97C2211yescharacteristic
62C2×C17×#97C2211yescharacteristic
63C2×C4×C8×C17C2211yescharacteristic
64C4×C4⋊C4×C17C2211yescharacteristic
65C17×#336C2211yescharacteristic
66C17×#336C2211yescharacteristic
67C2×#63252C2311yescharacteristic
68C22×C8×C17C2311yescharacteristic
69C22×C8×C17C2311yescharacteristic
70C22×C8×C17C2311yescharacteristic
71C22×C8×C17C2311yescharacteristic
72C2×C4⋊C4×C17—21no—
73C2×C42×C17C2311yescharacteristic
74C2×C42×C17C2311yescharacteristic
75C2×C42×C17C2311yescharacteristic
76C4×C22⋊D17—21no—
77C4×C22⋊D17—21no—
78#10587612C2311yescharacteristic
79#10587612C2311yescharacteristic
80#10587612C2311yescharacteristic
81#10587610—21no—
82#10587610—21no—
83C2×#63223C2311yescharacteristic
84C2×#63223C2311yescharacteristic
85C2×#63223C2311yescharacteristic
86C2×#63223C2311yescharacteristic
87#10587587—22no—
88#10587587—22no—
89#10587587—22no—
90#10587587Q814yescomplemented
91C2×#63218—21no—
92C2×#63218—21no—
93C2×C8×D17—21no—
94C2×C8×D17—21no—
95C4×D68C2×C411yescharacteristic
96C4×D68C2×C411yescharacteristic
97#10587558—21no—
98C42×D17—21no—
99#10587556C2×C411yescharacteristic
100#10587556C2×C411yescharacteristic
101#10587555C2×C411yescharacteristic
102#10587555C2×C411yescharacteristic
103C4×Dic34—21no—
104C4×Dic34—21no—
105C4×Dic34C2×C411yescharacteristic
106C4×Dic34C2×C411yescharacteristic
107C17×C4⋊C8C2×C411yescharacteristic
108C17×C4⋊C8C2×C411yescharacteristic
109C17×C4⋊C8C2×C411yescharacteristic
110C17×C4⋊C8C2×C411yescharacteristic
111C17×#97—22no—
112C17×#97D411yescharacteristic
113C17×#97D411yescharacteristic
114C17×#97D411yescharacteristic
115C17×#97D411yescharacteristic
116C4×C8×C17—22no—
117C17×#95—21no—
118#10587492C2×C411yescharacteristic
119#10587492C2×C411yescharacteristic
120#10587492C2×C411yescharacteristic
121#10587492C2×C411yescharacteristic
122#10587490—22no—
123#10587490—22no—
124#10587489C2×C411yescharacteristic
125#10587489C2×C411yescharacteristic
126#10587489C2×C411yescharacteristic
127#10587489C2×C411yescharacteristic
128#10587488C2×C411yescharacteristic
129#10587488C2×C411yescharacteristic
130#10587488C2×C411yescharacteristic
131#10587488C2×C411yescharacteristic
132#10587487C2×C411yescharacteristic
133#10587487C2×C411yescharacteristic
134#10587487C2×C411yescharacteristic
135#10587487C2×C411yescharacteristic
136#10587486—21no—
137#10587486—21no—
138#10587486—21no—
139#10587486—21no—
140#10587485—21no—
141#10587485—21no—
142#10587485—21no—
143#10587485—21no—
144C8×Dic17—21no—
145C8×Dic17—21no—
146C8×Dic17—21no—
147C8×Dic17—21no—
148#10587476—22no—
149#10587476—22no—
150C4⋊#3622—21no—
151C4⋊#3622—21no—
152#10587474—21no—
153C4×#3622—21no—
154C22×C4×C17—21no—
155C22×C4×C17—21no—
156C22×C4×C17#4111yescharacteristic
157C22×C4×C17#4111yescharacteristic
158C22×C4×C17C2411yescharacteristic
159C2×C22⋊D17—21no—
160#63252—21no—
161#63252—21no—
162#63252C2×Q812yes—
163#63252#4112yes—
164C2×D68C22×C411yescharacteristic
165C2×C4×D17—21no—
166C2×C4×D17—21no—
167C2×C4×D17—21no—
168C2×Dic34—21no—
169C2×Dic34—21no—
170C2×Dic34C22×C411yescharacteristic
171C2×C8×C17—21no—
172C2×C8×C17—21no—
173C2×C8×C17—21no—
174C2×C8×C17—21no—
175C2×C8×C17—21no—
176C2×C8×C17—21no—
177C2×C8×C17C22×C411yescharacteristic
178C2×C8×C17C22×C411yescharacteristic
179C2×C8×C17C22×C411yescharacteristic
180C2×C8×C17C22×C411yescharacteristic
181C2×C8×C17C2×D411yescharacteristic
182C2×C8×C17C2×D411yescharacteristic
183C2×C8×C17C2×D411yescharacteristic
184C2×C8×C17C2×D411yescharacteristic
185C2×C8×C17C2×Q812yes—
186C2×C8×C17#4112yes—
187C4⋊C4×C17—42no—
188C42×C17—21no—
189C42×C17—21no—
190C42×C17—21no—
191C42×C17—21no—
192C42×C17C22×C411yescharacteristic
193C42×C17C22×C411yescharacteristic
194C42×C17C2×D411yescharacteristic
195C42×C17C2×D411yescharacteristic
196C22⋊Dic17—21no—
197C22⋊Dic17—21no—
198C22⋊Dic17—21no—
199#63227—21no—
200#63227—21no—
201C4⋊Dic17—21no—
202C4⋊Dic17—21no—
203C4⋊Dic17C22×C411yescharacteristic
204C4⋊Dic17C22×C411yescharacteristic
205C4⋊Dic17C22×C411yescharacteristic
206C4⋊Dic17C22×C411yescharacteristic
207C4⋊Dic17C22×C411yescharacteristic
208C4⋊Dic17C22×C411yescharacteristic
209#63225—21no—
210#63225—21no—
211#63225—21no—
212#63225—21no—
213#63225—21no—
214#63225—21no—
215C4×Dic17—21no—
216C4×Dic17—21no—
217C4×Dic17—21no—
218C4×Dic17—21no—
219C4×Dic17—21no—
220#63223—21no—
221#63223—21no—
222#63223—21no—
223#63223—21no—
224#63223—21no—
225#63223—21no—
226#63223C2×Q812yes—
227#63223#4112yes—
228C2×#3622—21no—
229C2×#3622—21no—
230C2×#3622—21no—
231C2×#3622—21no—
232#63218—24no—
233#63218—24no—
234C8×D17—24no—
235C8×D17—24no—
236#21629—171nomaximal, Sylow
237C23×C17C2×#4111yescharacteristic
238C22×D17—21no—
239C2×C4×C17—22no—
240C2×C4×C17—41no—
241C2×C4×C17—41no—
242C2×C4×C17—41no—
243C2×C4×C17—41no—
244C2×C4×C17—41no—
245C2×C4×C17—41no—
246C2×C4×C17—41no—
247C2×C4×C17C4×D411yescharacteristic
248C2×C4×C17C4×D411yescharacteristic
249C2×C4×C17C4×D411yescharacteristic
250C2×C4×C17C4×D411yescharacteristic
251C2×C4×C17C23×C411yescharacteristic
252C2×C4×C17C22×D411yescharacteristic
253C2×C4×C17C22×Q811yescharacteristic
254C2×C4×C17C2×#4111yescharacteristic
255C2×C4×C17C2×#4111yescharacteristic
256C2×C4×C17C2×#4111yescharacteristic
257C22⋊D17—22no—
258C22⋊D17—22no—
259C2×Dic17—21no—
260C2×Dic17—21no—
261C2×Dic17—21no—
262C2×Dic17—21no—
263C2×Dic17—21no—
264C2×Dic17—21no—
265C2×Dic17—21no—
266D68—21no—
267D68C4×Q812yes—
268C4×D17—22no—
269C4×D17—22no—
270C4×D17—81no—
271C4×D17—81no—
272Dic34—21no—
273Dic34—81no—
274Dic34—81no—
275Dic34C4×Q812yes—
276C8×C17—21no—
277C8×C17—21no—
278C8×C17—21no—
279C8×C17—21no—
280C8×C17—21no—
281C8×C17—21no—
282C8×C17C22⋊Q812yes—
283C8×C17C22⋊Q812yes—
284#3622—22no—
285#3622—22no—
286#3622—41no—
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#1771D1711yescharacteristic, complemented
301#1765—171no—
302#1764—171no—
303C22×C17—21no—
304C22×C17—21no—
305C22×C17C2×C4×D411yescharacteristic
306C22×C17#52911yescharacteristic
307C22×C17#53311yescharacteristic
308C2×D17—22no—
309C2×D17—41no—
310C2×D17—41no—
311C4×C17—21no—
312C4×C17—21no—
313C4×C17—21no—
314C4×C17—21no—
315C4×C17—42no—
316C4×C17—81no—
317C4×C17C2×C4×Q811yescharacteristic
318C4×C17C4×#4111yescharacteristic
319C4×C17C2×C22⋊Q811yescharacteristic
320C4×C17#52511yescharacteristic
321Dic17—22no—
322Dic17—41no—
323Dic17—41no—
324Dic17—41no—
325Dic17—41no—
326Dic17—41no—
327Dic17—41no—
328C2×#131—171no—
329C2×#131—171no—
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⋊C8C2×D1711yescharacteristic
352C2×#104—171no—
353C2×#104—171no—
354#416—171no—
355#416—171no—
356#405—171no—
357#405—171no—
358#405—171no—
359#405—171no—
360#405—341no—
361#405—341no—
362C4×M16—171no—
363C4×M16—341no—
364C2×#97—341no—
365C2×#97C2×D1711yescharacteristic
366C2×#97C2×D1711yescharacteristic
367C2×C4×C8C2×D1711yescharacteristic
368C4×C4⋊C4C2×D1711yescharacteristic
369#341—171no—
370#341—171no—
371#339—171no—
372#339—171no—
373#339—171no—
374#339—171no—
375#339—171no—
376#339—171no—
377#336C2×D1711yescharacteristic
378#336C2×D1711yescharacteristic
379C2×C17—21no—
380C2×C17—21no—
381C2×C17C4×C22⋊Q811yescharacteristic
382C2×C17#295611yescharacteristic
383C2×C17#295711yescharacteristic
384D17—42no—
385C2×#41—171no—
386#131—174no—
387#131—342no—
388#131—342no—
389#131—342no—
390C2×M16—171no—
391C2×M16—171no—
392C2×M16—171no—
393C2×M16—171no—
394C2×M16—341no—
395C2×M16—341no—
396C22×C8—341no—
397C22×C8—341no—
398C22×C8C22×D1711yescharacteristic
399C22×C8C22×D1711yescharacteristic
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—171no—
419#117—341no—
420C2×C4⋊C4—21no—
421C2×C42—341no—
422C2×C42C22×D1711yescharacteristic
423C2×C42C22×D1711yescharacteristic
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—341no—
436C4⋊C8—341no—
437C4⋊C8—341no—
438C4⋊C8—341no—
439C4⋊C8—341no—
440C4⋊C8—341no—
441C4⋊C8C4×D1711yescharacteristic
442C4⋊C8C4×D1711yescharacteristic
443C4⋊C8C4×D1711yescharacteristic
444C4⋊C8C4×D1711yescharacteristic
445#104—342no—
446#104—342no—
447#103—171no—
448#103—171no—
449#103—171no—
450#103—171no—
451#102—171no—
452#102—171no—
453#102—171no—
454#102—171no—
455#97—22no—
456#97—341no—
457#97—341no—
458#97—341no—
459#97—341no—
460#97—341no—
461#97D6811yescharacteristic
462#97D6811yescharacteristic
463#97D6811yescharacteristic
464#97D6811yescharacteristic
465C4×C8—22no—
466C4×C8—341no—
467C4×C8—341no—
468C4×C8—341no—
469C4×C8—341no—
470C4×C8—341no—
471#95—21no—
472C17#2162911yescharacteristic, complemented, Sylow
473#41—172no—
474#41—172no—
475#41—341no—
476#41—341no—
477C2×Q8—171no—
478C2×Q8—341no—
479C2×Q8—341no—
480C2×D4—171no—
481C2×D4—341no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—341no—
485C22×C4—341no—
486C22×C4—341no—
487C22×C4#6325211yescharacteristic
488C22×C4#6325211yescharacteristic
489C22×C4C23×D1711yescharacteristic
490M16—172no—
491M16—172no—
492M16—344no—
493M16—341no—
494M16—341no—
495M16—344no—
496M16—341no—
497M16—341no—
498M16—341no—
499M16—341no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—344no—
507C2×C8—341no—
508C2×C8—341no—
509C2×C8—344no—
510C2×C8—341no—
511C2×C8—341no—
512C2×C8C2×C4×D1711yescharacteristic
513C2×C8C2×C4×D1711yescharacteristic
514C2×C8C2×C4×D1711yescharacteristic
515C2×C8C2×C4×D1711yescharacteristic
516C2×C8C2×D6811yescharacteristic
517C2×C8C2×D6811yescharacteristic
518C2×C8C2×D6811yescharacteristic
519C2×C8C2×D6811yescharacteristic
520C2×C8#6325412yes—
521C2×C8Q8×D1712yes—
522C4⋊C4—42no—
523C4⋊C4—171no—
524C4⋊C4—171no—
525C4⋊C4—171no—
526C4⋊C4—171no—
527C4⋊C4—171no—
528C4⋊C4—171no—
529C4⋊C4—341no—
530C4⋊C4—341no—
531C4⋊C4—341no—
532C4⋊C4—341no—
533C4⋊C4—341no—
534C4⋊C4—341no—
535C4⋊C4—341no—
536C4⋊C4—341no—
537C22⋊C4—341no—
538C22⋊C4—341no—
539C22⋊C4—341no—
540C22⋊C4—341no—
541C22⋊C4—341no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—341no—
547C42—341no—
548C42—341no—
549C42—341no—
550C42—341no—
551C42C2×C4×D1711yescharacteristic
552C42C2×C4×D1711yescharacteristic
553C42C2×D6811yescharacteristic
554C42C2×D6811yescharacteristic
555C23—341no—
556C23C2×#6325211yescharacteristic
557Q8—172no—
558Q8—341no—
559Q8—1361no—
560Q8—1361no—
561D4—172no—
562D4—342no—
563D4—342no—
564D4—341no—
565C2×C4—22no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—341no—
574C2×C4—342no—
575C2×C4—342no—
576C2×C4—341no—
577C2×C4—341no—
578C2×C4—341no—
579C2×C4—341no—
580C2×C4—341no—
581C2×C4—341no—
582C2×C4—1361no—
583C2×C4—1361no—
584C2×C4C4×D6811yescharacteristic
585C2×C4C4×D6811yescharacteristic
586C2×C4C4×D6811yescharacteristic
587C2×C4C4×D6811yescharacteristic
588C2×C4C22×C4×D1711yescharacteristic
589C2×C4C22×D6811yescharacteristic
590C2×C4C2×#6325411yescharacteristic
591C2×C4C2×Q8×D1711yescharacteristic
592C2×C4#41×D1711yescharacteristic
593C2×C4#41×D1711yescharacteristic
594C8—21no—
595C8—21no—
596C8—21no—
597C8—21no—
598C8—21no—
599C8—21no—
600C8—342no—
601C8—342no—
602C8—681no—
603C8—681no—
604C8#1058758312yes—
605C8#1058758312yes—
606C22—21no—
607C22—21no—
608C22—342no—
609C22—681no—
610C22—681no—
611C22C2×C4×D6811yescharacteristic
612C22#1176334111yescharacteristic
613C22#1176336311yescharacteristic
614C4—21no—
615C4—21no—
616C4—21no—
617C4—21no—
618C4—42no—
619C4—81no—
620C4—342no—
621C4—681no—
622C4—681no—
623C4—681no—
624C4—681no—
625C4—681no—
626C4—681no—
627C4C4×Q8×D1711yescharacteristic
628C4C4×#6325411yescharacteristic
629C4C2×#1058758311yescharacteristic
630C4#1176332111yescharacteristic
631C2—21no—
632C2—21no—
633C2—682no—
634C2C4×#1058758311yescharacteristic
635C2#1453769511yescharacteristic
636C2#1453770211yescharacteristic
subgroupG/H
centerC2×C4C22×D68
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1771C2
socleC22×C17C2×C4×D4
prime radicalC2×#63252C23
cosocleC22×C4×C17C24
hypercenter#1771D17
hypocenterC17#21629
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×#97 ◁ #1771

Complements

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

NH ≅ G/Ntype
2. #14537724C2semidirect
4. #14529727C2semidirect
5. #14529560C2semidirect
6. #14529384C2semidirect
7. #14529384C2semidirect
11. #14528668C2semidirect
12. #14528663C2semidirect
15. C17×#1771C2semidirect
20. #11763117C4semidirect
21. #11763117C4semidirect
22. #11763110C4semidirect
23. #11763110C4semidirect
24. #11763105C4semidirect
25. #11763100C4semidirect
26. #11763095C4semidirect
27. #11763090C4semidirect
90. #10587587Q8semidirect
300. #1771D17semidirect
472. C17#21629semidirect

Maximal subgroups

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

subgroupindex
#145385742
#145377242
#145372092
#145297272
#145295602
#145293842
#145290622
#145288052
#145286682
#145286632
#145286282
#145286212
C17×#17712
#2162917