Finite Group Databasebeta

← #17056408

Subgroups — #17056408

Order 4352.

692 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14538568C211yesmaximal, characteristic
2#14537730C211yesmaximal, characteristic, complemented
3#14537350C211yesmaximal, characteristic
4#14529763C211yesmaximal, characteristic, complemented
5#14529556C211yesmaximal, characteristic, complemented
6#14529535C211yesmaximal, characteristic, complemented
7#14529529C211yesmaximal, characteristic, complemented
8#14529384C211yesmaximal, characteristic, complemented
9#14529381C211yesmaximal, characteristic, complemented
10#14529057C211yesmaximal, characteristic
11#14528640C211yesmaximal, characteristic, complemented
12#14528631C211yesmaximal, characteristic
13#14528619C211yesmaximal, characteristic
14#14528596C211yesmaximal, characteristic
15C17×#1866C211yesmaximal, 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#11762938—21no—
28#11762938—21no—
29#11762936—21no—
30#11762933—21no—
31C2×#10587493C2211yescharacteristic
32#11762847C2211yescharacteristic
33#11762845C2211yescharacteristic
34#11762837—21no—
35#11762837C2211yescharacteristic
36#11762837C2211yescharacteristic
37#11762835—21no—
38#11762835C2211yescharacteristic
39#11762832—21no—
40#11762832C2211yescharacteristic
41C2×#10587485—21no—
42C2×#10587485C2211yescharacteristic
43C2×#10587485C2211yescharacteristic
44C2×#10587508C2211yescharacteristic
45#11762805C2211yescharacteristic
46#11762765C2211yescharacteristic
47#11762764C2211yescharacteristic
48M16×Dic17C2211yescharacteristic
49C2×#10587486C2211yescharacteristic
50C2×C8×Dic17C2211yescharacteristic
51#11762655C2211yescharacteristic
52#11762574C2211yescharacteristic
53#11762573C2211yescharacteristic
54#11762570C2211yescharacteristic
55#11762570C2211yescharacteristic
56#11762569C2211yescharacteristic
57#11762569C2211yescharacteristic
58#11762568C2211yescharacteristic
59#11762564C2211yescharacteristic
60C2×C17×#131C2211yescharacteristic
61C2×C17×#108C2211yescharacteristic
62C17×#424C2211yescharacteristic
63C2×C17×#104C2211yescharacteristic
64C17×#416C2211yescharacteristic
65C17×#339C2211yescharacteristic
66C17×#336C2211yescharacteristic
67C2×#41×C17C2311yescharacteristic
68C17×#131—22no—
69C17×#131—41no—
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—41no—
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×C4×Dic17—21no—
90C2×C4×Dic17C2311yescharacteristic
91C2×C4×Dic17C2311yescharacteristic
92C2×#63223—21no—
93C2×#63223C2311yescharacteristic
94C2×#63223C2311yescharacteristic
95C22×#3622—21no—
96C22×#3622C2311yescharacteristic
97C22×#3622C2311yescharacteristic
98#10587576C2×C411yescharacteristic
99#10587576C2×C411yescharacteristic
100#10587574C2×C411yescharacteristic
101#10587574C2×C411yescharacteristic
102#10587564—21no—
103#10587564—21no—
104C4×Dic34—21no—
105C4×Dic34—21no—
106C17×#108—41no—
107C17×C4⋊C8—21no—
108C17×C4⋊C8—21no—
109C17×#104—22no—
110C17×#103C2×C411yescharacteristic
111C17×#103C2×C411yescharacteristic
112C17×#102C2×C411yescharacteristic
113C17×#102C2×C411yescharacteristic
114#10587508Q814yescomplemented
115#10587506C2×C411yescharacteristic
116#10587506C2×C411yescharacteristic
117#10587503C2×C411yescharacteristic
118#10587503C2×C411yescharacteristic
119#10587502—21no—
120#10587502—21no—
121#10587493—41no—
122#10587489C2×C411yescharacteristic
123#10587489C2×C411yescharacteristic
124#10587488C2×C411yescharacteristic
125#10587488C2×C411yescharacteristic
126#10587486—21no—
127#10587486—21no—
128#10587486—41no—
129#10587485—21no—
130#10587485—21no—
131#10587485—21no—
132#10587485—21no—
133#10587485—21no—
134#10587485—21no—
135#10587485—21no—
136#10587485—21no—
137#10587485C2×C411yescharacteristic
138#10587485C2×C411yescharacteristic
139#10587485C2×C411yescharacteristic
140#10587485C2×C411yescharacteristic
141#10587485D411yescharacteristic
142#10587485D411yescharacteristic
143#10587485D411yescharacteristic
144#10587485D411yescharacteristic
145C8×Dic17—21no—
146C8×Dic17—21no—
147C8×Dic17—41no—
148#10587479C2×C411yescharacteristic
149#10587479C2×C411yescharacteristic
150#10587478C2×C411yescharacteristic
151#10587478C2×C411yescharacteristic
152#10587474—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—21no—
173M16×C17—21no—
174M16×C17—21no—
175M16×C17—42no—
176C2×C8×C17—21no—
177C2×C8×C17—21no—
178C2×C8×C17—21no—
179C2×C8×C17—21no—
180C2×C8×C17—21no—
181C2×C8×C17—21no—
182C2×C8×C17—42no—
183C2×C8×C17C22×C411yescharacteristic
184C2×C8×C17C22×C411yescharacteristic
185C2×C8×C17C2×D411yescharacteristic
186C2×C8×C17C2×D411yescharacteristic
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—
209C4×Dic17—21no—
210C4×Dic17—21no—
211C4×Dic17—21no—
212C4×Dic17—21no—
213C4×Dic17—21no—
214C4×Dic17C22×C411yescharacteristic
215C4×Dic17C22×C411yescharacteristic
216C4×Dic17C2×D411yescharacteristic
217C4×Dic17C2×D411yescharacteristic
218C4×Dic17C2×Q812yes—
219#63223—21no—
220#63223—21no—
221#63223—21no—
222#63223—42no—
223#63223C2×Q812yes—
224C2×#3622—21no—
225C2×#3622—21no—
226C2×#3622—21no—
227C2×#3622—21no—
228C2×#3622—21no—
229C2×#3622—21no—
230C2×#3622—42no—
231C2×#3622C22×C411yescharacteristic
232C2×#3622C22×C411yescharacteristic
233C2×#3622C2×D411yescharacteristic
234C2×#3622C2×D411yescharacteristic
235#22984—171nomaximal, Sylow
236C23×C17—21no—
237C23×C17C2×#4111yescharacteristic
238Q8×C17—21no—
239Q8×C17C4×Q812yes—
240D4×C17—22no—
241D4×C17—22no—
242D4×C17—21no—
243D4×C17C4×Q812yes—
244C2×C4×C17—21no—
245C2×C4×C17—22no—
246C2×C4×C17—22no—
247C2×C4×C17—21no—
248C2×C4×C17—21no—
249C2×C4×C17—41no—
250C2×C4×C17—41no—
251C2×C4×C17—41no—
252C2×C4×C17C23×C411yescharacteristic
253C2×C4×C17C22×D411yescharacteristic
254C2×C4×C17C22×Q811yescharacteristic
255C2×C4×C17C2×#4111yescharacteristic
256C2×C4×C17H(5,𝔽2)11yescharacteristic
257C2×C4×C172−1+411yescharacteristic
258C2×Dic17—21no—
259C2×Dic17—21no—
260C2×Dic17—21no—
261C2×Dic17—21no—
262C2×Dic17—22no—
263C2×Dic17—41no—
264C2×Dic17—41no—
265C2×Dic17—42no—
266C2×Dic17—41no—
267C2×Dic17—41no—
268C2×Dic17—81no—
269C2×Dic17C4×D411yescharacteristic
270C2×Dic17C4×D411yescharacteristic
271C2×Dic17C4×D411yescharacteristic
272C2×Dic17C4×D411yescharacteristic
273Dic34—42no—
274Dic34—81no—
275C8×C17—21no—
276C8×C17—21no—
277C8×C17—21no—
278C8×C17—21no—
279C8×C17—41no—
280C8×C17—41no—
281#3622—21no—
282#3622—21no—
283#3622—21no—
284#3622—21no—
285#3622—22no—
286#3622—41no—
287#3076—171no—
288#2962—171no—
289#2906—171no—
290#1995—171no—
291#1950—171no—
292#1946—171no—
293#1945—171no—
294#1922—171no—
295#1922—171no—
296#1866—171no—
297#1866D1711yescharacteristic
298#1768—171no—
299#1765—171no—
300#1764—171no—
301#1759—171no—
302C22×C17—21no—
303C22×C17—21no—
304C22×C17—22no—
305C22×C17—41no—
306C22×C17—41no—
307C22×C17C2×C4×D411yescharacteristic
308C22×C17#55211yescharacteristic
309C22×C17#55411yescharacteristic
310C4×C17—22no—
311C4×C17—41no—
312C4×C17—41no—
313C4×C17—81no—
314C4×C17C2×C4×Q811yescharacteristic
315C4×C17#52011yescharacteristic
316C4×C17#54611yescharacteristic
317C4×C17D4×Q811yescharacteristic
318Dic17—21no—
319Dic17—21no—
320Dic17—21no—
321Dic17—21no—
322Dic17—42no—
323Dic17—41no—
324Dic17—41no—
325Dic17—41no—
326Dic17—41no—
327C2×#131—171no—
328C2×#131C2×D1711yescharacteristic
329#533—171no—
330C4×#41—171no—
331#479—171no—
332#479—171no—
333#478—171no—
334#478—171no—
335#477—171no—
336#476—171no—
337#475—171no—
338#474—171no—
339#446—341no—
340#446—341no—
341#446—341no—
342C8×Q8—341no—
343C2×#108—171no—
344C2×#108C2×D1711yescharacteristic
345#427—171no—
346#427—171no—
347#424—171no—
348#424—171no—
349#424—171no—
350#424—341no—
351#424—341no—
352#424C2×D1711yescharacteristic
353#423—171no—
354#423—341no—
355C2×C4⋊C8—171no—
356C2×C4⋊C8—171no—
357C2×C4⋊C8—341no—
358C2×#104—171no—
359C2×#104C2×D1711yescharacteristic
360#416—171no—
361#416C2×D1711yescharacteristic
362#405—171no—
363#405—171no—
364C4×M16—171no—
365C2×#97—171no—
366C2×C4×C8—171no—
367#376—171no—
368#341—171no—
369#341—171no—
370#339—171no—
371#339—171no—
372#339—171no—
373#339—171no—
374#339—171no—
375#339C2×D1711yescharacteristic
376#336—171no—
377#336C2×D1711yescharacteristic
378C2×C17—21no—
379C2×C17—21no—
380C2×C17—42no—
381C2×C17#236011yescharacteristic
382C2×C17#295511yescharacteristic
383C2×C17#295711yescharacteristic
384C2×#41C22×D1711yescharacteristic
385#131—22no—
386#131—41no—
387#131—342no—
388#131—681no—
389C2×M16—21no—
390C2×M16—171no—
391C2×M16—171no—
392C2×M16—341no—
393C2×M16C22×D1711yescharacteristic
394C2×M16C22×D1711yescharacteristic
395C22×C8—21no—
396C22×C8—171no—
397C22×C8—171no—
398C22×C8—341no—
399C22×C8C22×D1711yescharacteristic
400C22×C8C22×D1711yescharacteristic
401C4⋊Q8—171no—
402C4⋊Q8—171no—
403#125—171no—
404#125—171no—
405C22⋊Q8—341no—
406C22⋊Q8—341no—
407C4×Q8—171no—
408C4×Q8—171no—
409C4×Q8—341no—
410C4×Q8—341no—
411C4×D4—171no—
412C4×D4—171no—
413C4×D4—341no—
414C4×D4—341no—
415#117—171no—
416#117—171no—
417#117—341no—
418#117C22×D1711yescharacteristic
419C2×C42—171no—
420C2×C42—171no—
421C2×C42—341no—
422C2×C42C22×D1711yescharacteristic
423#108—41no—
424#108—681no—
425#107—171no—
426#107—171no—
427#107—171no—
428#107—171no—
429#106—171no—
430#106—171no—
431#106—171no—
432#106—171no—
433C4⋊C8—21no—
434C4⋊C8—21no—
435C4⋊C8—171no—
436C4⋊C8—171no—
437C4⋊C8—171no—
438C4⋊C8—171no—
439C4⋊C8—171no—
440C4⋊C8—171no—
441C4⋊C8—171no—
442C4⋊C8—171no—
443C4⋊C8—341no—
444C4⋊C8—341no—
445C4⋊C8—341no—
446C4⋊C8—341no—
447C4⋊C8—341no—
448C4⋊C8—341no—
449C4⋊C8—341no—
450C4⋊C8—341no—
451#104—22no—
452#104—174no—
453#103—171no—
454#103—171no—
455#103C4×D1711yescharacteristic
456#103C4×D1711yescharacteristic
457#102—171no—
458#102—171no—
459#102C4×D1711yescharacteristic
460#102C4×D1711yescharacteristic
461#97—341no—
462#97—341no—
463#97—341no—
464#97—681no—
465C4×C8—341no—
466C4×C8—341no—
467C4×C8—341no—
468C4×C8—681no—
469#95—341no—
470#95—341no—
471C17#2298411yescharacteristic, complemented, Sylow
472#41—21no—
473#41—21no—
474#41#6325412yes—
475#41Q8×D1712yes—
476C2×Q8—341no—
477C2×Q8—341no—
478C2×Q8C2×C4×D1711yescharacteristic
479C2×D4—21no—
480C2×D4C2×C4×D1711yescharacteristic
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—171no—
484C22×C4—171no—
485C22×C4—341no—
486C22×C4—341no—
487C22×C4—341no—
488C22×C4C23×D1711yescharacteristic
489M16—21no—
490M16—21no—
491M16—21no—
492M16—21no—
493M16—42no—
494M16—172no—
495M16—341no—
496M16—341no—
497M16—341no—
498M16—682no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—42no—
506C2×C8—171no—
507C2×C8—171no—
508C2×C8—171no—
509C2×C8—171no—
510C2×C8—341no—
511C2×C8—341no—
512C2×C8—341no—
513C2×C8—341no—
514C2×C8—341no—
515C2×C8—341no—
516C2×C8—682no—
517C2×C8C2×C4×D1711yescharacteristic
518C2×C8C2×C4×D1711yescharacteristic
519C2×C8D4×D1711yescharacteristic
520C2×C8D4×D1711yescharacteristic
521C4⋊C4—171no—
522C4⋊C4—171no—
523C4⋊C4—171no—
524C4⋊C4—171no—
525C4⋊C4—341no—
526C4⋊C4—341no—
527C4⋊C4—341no—
528C4⋊C4—341no—
529C4⋊C4—341no—
530C4⋊C4—341no—
531C4⋊C4—341no—
532C4⋊C4—341no—
533C4⋊C4C2×C4×D1711yescharacteristic
534C4⋊C4C2×C4×D1711yescharacteristic
535C22⋊C4—21no—
536C22⋊C4—341no—
537C22⋊C4—341no—
538C22⋊C4—341no—
539C22⋊C4—341no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—171no—
544C42—171no—
545C42—171no—
546C42—171no—
547C42—172no—
548C42—341no—
549C42—341no—
550C42—341no—
551C42—341no—
552C42—341no—
553C23—21no—
554C23#41×D1711yescharacteristic
555Q8—21no—
556Q8—682no—
557Q8—1361no—
558Q8#1058757312yes—
559D4—22no—
560D4—22no—
561D4—21no—
562D4#1058757312yes—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—171no—
572C2×C4—171no—
573C2×C4—171no—
574C2×C4—171no—
575C2×C4—342no—
576C2×C4—341no—
577C2×C4—341no—
578C2×C4—341no—
579C2×C4—341no—
580C2×C4—681no—
581C2×C4—681no—
582C2×C4—682no—
583C2×C4—681no—
584C2×C4—681no—
585C2×C4—1361no—
586C2×C4C22×C4×D1711yescharacteristic
587C2×C4C2×D4×D1711yescharacteristic
588C2×C4C2×#6325411yescharacteristic
589C2×C4#1058769111yescharacteristic
590C2×C4C2×Q8×D1711yescharacteristic
591C2×C4#1058769711yescharacteristic
592C8—21no—
593C8—21no—
594C8—21no—
595C8—21no—
596C8—41no—
597C8—41no—
598C8—342no—
599C8—341no—
600C8—341no—
601C8—341no—
602C8—341no—
603C8—681no—
604C22—21no—
605C22—21no—
606C22—22no—
607C22—41no—
608C22—41no—
609C22C4×D4×D1711yescharacteristic
610C22#1176349611yescharacteristic
611C22#1176351311yescharacteristic
612C4—22no—
613C4—41no—
614C4—41no—
615C4—81no—
616C4—341no—
617C4—341no—
618C4—341no—
619C4—341no—
620C4—682no—
621C4—681no—
622C4—681no—
623C4—681no—
624C4—681no—
625C4C2×#1058757311yescharacteristic
626C4#1176328811yescharacteristic
627C4#1176344611yescharacteristic
628C4#1176347211yescharacteristic
629C2—21no—
630C2—21no—
631C2—42no—
632C2#1453178411yescharacteristic
633C2#1453768211yescharacteristic
634C2#1453769811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1866C2
socleC22×C17C2×C4×D4
prime radicalC2×#41×C17C23
cosocleC22×C4×C17C24
hypercenter#1866D17
hypocenterC17#22984
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1866

Complements

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

NH ≅ G/Ntype
2. #14537730C2semidirect
4. #14529763C2semidirect
5. #14529556C2semidirect
6. #14529535C2semidirect
7. #14529529C2semidirect
8. #14529384C2semidirect
9. #14529381C2semidirect
11. #14528640C2semidirect
19. #11763118C4semidirect
20. #11763114C4semidirect
21. #11763111C4semidirect
22. #11763107C4semidirect
23. #11763102C4semidirect
24. #11763097C4semidirect
25. #11763092C4semidirect
26. #11763087C4semidirect
114. #10587508Q8semidirect
471. C17#22984semidirect

Maximal subgroups

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

subgroupindex
#145385682
#145377302
#145373502
#145297632
#145295562
#145295352
#145295292
#145293842
#145293812
#145290572
#145286402
#145286312
#145286192
#145285962
C17×#18662
#2298417