Finite Group Databasebeta

← #17979379

Subgroups — #17979379

Order 4352.

2576 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14545323C213yesmaximal, complemented
2#14545320C213yesmaximal, complemented
3#14545315C216yesmaximal, complemented
4#14545306C213yesmaximal, complemented
5#14545285C216yesmaximal, complemented
6#14545274C213yesmaximal, complemented
7#14545269C213yesmaximal, complemented
8#120×#595C213yesmaximal, complemented
9D17×H(3,𝔽4)C211yesmaximal, characteristic
10C24×#595C2211yescharacteristic, complemented
11C2×Q8×#595C2213yescomplemented
12C2×D4×#595C2213yes—
13C2×D4×#595C2216yes—
14C22×C22⋊#595C2213yescomplemented
15(C2×#595)⋊Q8C2213yescomplemented
16(C2×#595)⋊Q8C2216yescomplemented
17#11763761C2216yes—
18#11763761C2213yes—
19#11763759C2216yescomplemented
20#11763759C2216yescomplemented
21#11763759C2216yescomplemented
22#11763759C2216yescomplemented
23#11763759C2216yescomplemented
24#11763759C2216yescomplemented
25#11763757C2216yes—
26#11763757C2213yes—
27#11763757C2213yes—
28#11763757C2216yes—
29#11763756C2216yescomplemented
30#11763756C2213yescomplemented
31#11763756C2213yescomplemented
32#11763756C2216yescomplemented
33#11763755C2211yescharacteristic, complemented
34#11763754C2212yescomplemented
35#11763750C2213yescomplemented
36#11763750C2211yescharacteristic, complemented
37C4×C4⋊#595C2216yescomplemented
38C22⋊C4×#595C2213yescomplemented
39C22⋊C4×#595C2216yescomplemented
40C22⋊C4×#595C2213yescomplemented
41C22⋊C4×#595C2216yescomplemented
42#11763745C2213yescomplemented
43#11763552C413yescomplemented
44#11763552C413yescomplemented
45#11763551C413yescomplemented
46#11763551C416yescomplemented
47#124×D17C2213yes—
48#124×D17C2216yes—
49#120×D17C2213yes—
50#120×D17C2213yes—
51C17×H(3,𝔽4)C411yescharacteristic, complemented
52C24×D17C2311yescharacteristic
53C24×D17C2311yescharacteristic
54C23×#595—46no—
55C23×#595C2313yes—
56C23×#595C2313yes—
57C23×#595C2313yes—
58C23×#595C2313yes—
59C2×Q8×D17C2313yes—
60C2×D4×D17C2316yes—
61C2×D4×D17C2313yes—
62C2×C22⋊#595—46no—
63C2×C22⋊#595C2316yes—
64C2×C22⋊#595C2313yes—
65C2×C22⋊#595C2316yes—
66C2×C22⋊#595C2313yes—
67C2×C22⋊#595C2313yes—
68C2×C22⋊#595C2316yes—
69C2×C22⋊#595C2313yes—
70C2×C22⋊#595C2316yes—
71Q8×#595—46no—
72D4×#595—46no—
73D4×#595—412no—
74C2×C4⋊#595C2316yes—
75C2×C4⋊#595C2316yes—
76C2×C4⋊#595C2313yes—
77C2×C4⋊#595C2313yes—
78C2×C4×#595C2313yes—
79C2×C4×#595C2313yes—
80C2×C4×#595C2316yes—
81C2×C4×#595C2316yes—
82C17×#124C2×C413yescomplemented
83C17×#124C2×C416yescomplemented
84C17×#120C2×C413yescomplemented
85C17×#120C2×C413yescomplemented
86#10587639C2×C413yescomplemented
87#10587639C2×C413yescomplemented
88#10587633C2×C413yescomplemented
89#10587633C2×C416yescomplemented
90#10587623C2×C413yescomplemented
91#10587623C2×C416yescomplemented
92#10587623C2×C413yescomplemented
93#10587623C2×C416yescomplemented
94#10587622C2×C413yescomplemented
95#10587622C2×C416yescomplemented
96#10587571C2×C416yescomplemented
97#10587571C2×C416yescomplemented
98#10587571C2×C416yescomplemented
99#10587571C2×C416yescomplemented
100#10587571C2×C416yescomplemented
101#10587571C2×C416yescomplemented
102#10587568C2×C416yescomplemented
103#10587568C2×C413yescomplemented
104#10587568C2×C416yescomplemented
105#10587568C2×C413yescomplemented
106C22⋊C4×D17C2313yes—
107C22⋊C4×D17C2316yes—
108C22⋊C4×D17C2313yes—
109C22⋊C4×D17C2316yes—
110#10587561C2×C413yescomplemented
111#10587561C2×C416yescomplemented
112C42×D17C2311yescharacteristic
113C42×D17C2312yes—
114(C2×#595)⋊C4C2316yes—
115(C2×#595)⋊C4C2316yes—
116(C2×#595)⋊C4C2316yes—
117(C2×#595)⋊C4C2316yes—
118(C2×#595)⋊C4C2316yes—
119(C2×#595)⋊C4C2316yes—
120C24×C17C22×C411yescharacteristic
121C24×C17C22×C411yescharacteristic, complemented
122C23×D17—43no—
123C23×D17—43no—
124C23×D17—43no—
125C23×D17—43no—
126C23×D17C22×C413yes—
127C23×D17C22×C413yescomplemented
128C23×D17C2413yes—
129C23×D17C2413yes—
130C22×#595—212no—
131C22×#595—26no—
132C22×#595—26no—
133C22×#595—43no—
134C22×#595—46no—
135C22×#595—46no—
136C22×#595—43no—
137C22×#595—46no—
138C22×#595—416no—
139C22×#595C2413yes—
140C22×#595C2413yes—
141C22×#595C2416yes—
142C22×#595C2411yescharacteristic
143C22×#595C2413yes—
144C2×Q8×C17C22×C413yescomplemented
145C2×D4×C17C22×C413yes—
146C2×D4×C17C22×C416yes—
147C2×C22⋊D17C22×C416yes—
148C2×C22⋊D17C22×C413yes—
149C2×C22⋊D17C22×C416yes—
150C2×C22⋊D17C22×C413yes—
151Q8×D17—43no—
152Q8×D17—43no—
153D4×D17—43no—
154D4×D17—46no—
155D4×D17—43no—
156D4×D17—46no—
157C2×D68C22×C416yes—
158C2×D68C22×C413yes—
159C2×C4×D17C2413yes—
160C2×C4×D17C2416yes—
161C2×Dic34C22×C413yescomplemented
162C2×Dic34C22×C416yescomplemented
163C22⋊#595—212no—
164C22⋊#595—43no—
165C22⋊#595—43no—
166C22⋊#595—46no—
167C22⋊#595—46no—
168C22⋊#595—46no—
169C22⋊#595—43no—
170C22⋊#595—43no—
171C4⋊#595—212no—
172C4⋊#595—46no—
173C4⋊#595—43no—
174C4⋊#595—43no—
175C4×#595—26no—
176C4×#595—43no—
177C4×#595—46no—
178C4×#595—46no—
179C22⋊C4×C17C22×C416yes—
180C22⋊C4×C17C22×C413yes—
181C22⋊C4×C17C22×C416yescomplemented
182C22⋊C4×C17C22×C413yescomplemented
183C42×C17C22×C411yescharacteristic, complemented
184C42×C17C22×C412yescomplemented
185C22⋊Dic17C22×C413yes—
186C22⋊Dic17C22×C416yes—
187C22⋊Dic17C22×C413yescomplemented
188C22⋊Dic17C22×C416yescomplemented
189#63227C22×C416yes—
190#63227C22×C416yes—
191#63227C22×C416yes—
192#63227C22×C416yescomplemented
193#63227C22×C416yescomplemented
194#63227C22×C416yescomplemented
195C4×Dic17C22×C413yescomplemented
196C4×Dic17C22×C416yescomplemented
197#34512—171nomaximal, Sylow
198C23×C17—43no—
199C23×C17—43no—
200C23×C17C23×C413yes—
201C23×C17C23×C413yes—
202C22×D17—26no—
203C22×D17—23no—
204C22×D17—26no—
205C22×D17—23no—
206C22×D17—26no—
207C22×D17—23no—
208C22×D17—23no—
209C22×D17—26no—
210C22×D17—46no—
211C22×D17—43no—
212C22×D17—43no—
213C22×D17—46no—
214C22×D17—44no—
215C22×D17—412no—
216C22×D17—82no—
217C22×D17—86no—
218C22×D17C23×C413yes—
219C22×D17C23×C413yes—
220C22×D17C2511yescharacteristic
221C2×#595—26no—
222C2×#595—23no—
223C2×#595—26no—
224C2×#595—26no—
225C2×#595—23no—
226C2×#595—23no—
227C2×#595—26no—
228C2×#595—26no—
229C2×#595—26no—
230C2×#595—412no—
231C2×#595—412no—
232C2×#595—86no—
233C2×#595—86no—
234C2×#595H(5,𝔽2)16yes—
235Q8×C17—43no—
236D4×C17—43no—
237D4×C17—46no—
238C2×C4×C17C23×C413yes—
239C2×C4×C17C23×C416yes—
240C22⋊D17—43no—
241C22⋊D17—43no—
242C22⋊D17—46no—
243C22⋊D17—46no—
244C2×Dic17C23×C413yes—
245C2×Dic17C23×C416yes—
246D68—43no—
247D68—46no—
248C4×D17—23no—
249C4×D17—26no—
250C4×D17—23no—
251C4×D17—26no—
252Dic34—46no—
253Dic34—43no—
254C2×H(3,𝔽4)—171no—
255#2375—173no—
256#2372—173no—
257#2367—176no—
258#2358—173no—
259#2337—176no—
260#2326—173no—
261#2321—173no—
262C4×#120—173no—
263C22×C17—23no—
264C22×C17—26no—
265C22×C17—26no—
266C22×C17—23no—
267C22×C17—44no—
268C22×C17—82no—
269C22×C17C24×C411yescharacteristic
270C2×D17—23no—
271C2×D17—26no—
272C2×D17—26no—
273C2×D17—23no—
274C2×D17—412no—
275C2×D17—43no—
276C2×D17—43no—
277C2×D17—43no—
278C2×D17—43no—
279C2×D17—46no—
280C2×D17—46no—
281C2×D17—86no—
282C2×D17C2×H(5,𝔽2)13yes—
283C2×D17#58513yes—
284#595—26no—
285#595—41no—
286#595—43no—
287#595—43no—
288#595—46no—
289C4×C17—23no—
290C4×C17—26no—
291Dic17—23no—
292Dic17—26no—
293C24×C4—171no—
294H(3,𝔽4)—176no—
295H(3,𝔽4)—173no—
296H(3,𝔽4)—173no—
297H(3,𝔽4)—173no—
298H(3,𝔽4)#59511yescharacteristic, complemented
299C2×#124—173no—
300C2×#124—176no—
301C2×#120—173no—
302C2×#120—173no—
303C2×C4×Q8—173no—
304C2×C4×D4—173no—
305C2×C4×D4—176no—
306C22×C22⋊C4—173no—
307#391—173no—
308#391—176no—
309#390—176no—
310#390—173no—
311#388—176no—
312#388—176no—
313#388—176no—
314#388—176no—
315#388—176no—
316#388—176no—
317#386—176no—
318#386—173no—
319#386—176no—
320#386—173no—
321#385—173no—
322#385—176no—
323#385—176no—
324#385—173no—
325#384—171no—
326#383—172no—
327#379—171no—
328#379—173no—
329C4×C4⋊C4—176no—
330C4×C22⋊C4—173no—
331C4×C22⋊C4—173no—
332C4×C22⋊C4—176no—
333C4×C22⋊C4—176no—
334#376—173no—
335C2×C17—43no—
336C2×C17—43no—
337C2×C17C4×H(5,𝔽2)13yes—
338D17—43no—
339D17—43no—
340D17#353211yescharacteristic
341D17#354013yes—
342C25—171no—
343C25—171no—
344C22×Q8—173no—
345C22×D4—173no—
346C22×D4—176no—
347C23×C4—173no—
348C23×C4—173no—
349C23×C4—173no—
350C23×C4—173no—
351C23×C4—686no—
352#124—173no—
353#124—176no—
354#124—173no—
355#124—176no—
356#124—176no—
357#124—176no—
358#124—176no—
359#124—176no—
360#124—176no—
361#124—176no—
362#124—173no—
363#124—176no—
364#124C2×#59513yescomplemented
365#124C2×#59516yescomplemented
366#120—173no—
367#120—173no—
368#120—176no—
369#120—173no—
370#120—176no—
371#120—173no—
372#120—173no—
373#120—176no—
374#120—173no—
375#120—176no—
376#120C2×#59513yescomplemented
377#120C2×#59513yescomplemented
378C4×Q8—686no—
379C4×D4—686no—
380C4×D4—6812no—
381C2×C4⋊C4—176no—
382C2×C4⋊C4—173no—
383C2×C4⋊C4—173no—
384C2×C4⋊C4—176no—
385C2×C22⋊C4—176no—
386C2×C22⋊C4—173no—
387C2×C22⋊C4—176no—
388C2×C22⋊C4—173no—
389C2×C22⋊C4—176no—
390C2×C22⋊C4—176no—
391C2×C22⋊C4—173no—
392C2×C22⋊C4—176no—
393C2×C22⋊C4—173no—
394C2×C22⋊C4—173no—
395C2×C22⋊C4—176no—
396C2×C22⋊C4—173no—
397C2×C22⋊C4—686no—
398C2×C42—171no—
399C2×C42—172no—
400C2×C42—173no—
401C2×C42—176no—
402C2×C42—176no—
403C2×C42—173no—
404#95—176no—
405#95—176no—
406#95—176no—
407#95—176no—
408#95—176no—
409#95—176no—
410C17#3451211yescharacteristic, complemented, Sylow
411C24—173no—
412C24—173no—
413C24—173no—
414C24—173no—
415C24—683no—
416C24—683no—
417C24—683no—
418C24—683no—
419C24C22×#59511yescharacteristic
420C24C22×#59511yescharacteristic, complemented
421C2×Q8—173no—
422C2×Q8—176no—
423C2×Q8—683no—
424C2×Q8—683no—
425C2×Q8C22×#59513yescomplemented
426C2×D4—173no—
427C2×D4—176no—
428C2×D4—173no—
429C2×D4—176no—
430C2×D4—173no—
431C2×D4—176no—
432C2×D4—683no—
433C2×D4—686no—
434C2×D4—683no—
435C2×D4—686no—
436C2×D4C22×#59513yes—
437C2×D4C22×#59516yes—
438C22×C4—173no—
439C22×C4—173no—
440C22×C4—176no—
441C22×C4—176no—
442C22×C4—171no—
443C22×C4—173no—
444C22×C4—173no—
445C22×C4—3412no—
446C22×C4—346no—
447C22×C4—346no—
448C22×C4—683no—
449C22×C4—686no—
450C22×C4—683no—
451C22×C4—686no—
452C22×C4—686no—
453C22×C4—6816no—
454C4⋊C4—3412no—
455C4⋊C4—686no—
456C4⋊C4—683no—
457C4⋊C4—683no—
458C22⋊C4—176no—
459C22⋊C4—173no—
460C22⋊C4—176no—
461C22⋊C4—173no—
462C22⋊C4—176no—
463C22⋊C4—176no—
464C22⋊C4—176no—
465C22⋊C4—176no—
466C22⋊C4—176no—
467C22⋊C4—176no—
468C22⋊C4—3412no—
469C22⋊C4—686no—
470C22⋊C4—683no—
471C22⋊C4—683no—
472C22⋊C4—686no—
473C22⋊C4—683no—
474C22⋊C4—683no—
475C22⋊C4—686no—
476C22⋊C4C22×#59516yes—
477C22⋊C4C22×#59513yes—
478C22⋊C4C22×#59516yescomplemented
479C22⋊C4C22×#59513yescomplemented
480C42—173no—
481C42—176no—
482C42—346no—
483C42—683no—
484C42—686no—
485C42—686no—
486C42C22×#59511yescharacteristic, complemented
487C42C22×#59512yescomplemented
488C23—43no—
489C23—43no—
490C23—171no—
491C23—173no—
492C23—173no—
493C23—346no—
494C23—343no—
495C23—346no—
496C23—343no—
497C23—346no—
498C23—343no—
499C23—343no—
500C23—346no—
501C23—683no—
502C23—686no—
503C23—686no—
504C23—683no—
505C23—684no—
506C23—6812no—
507C23—1362no—
508C23—1366no—
509C23C23×#59513yes—
510C23C23×#59513yes—
511Q8—43no—
512Q8—683no—
513Q8—686no—
514D4—43no—
515D4—46no—
516D4—683no—
517D4—686no—
518D4—683no—
519D4—683no—
520D4—686no—
521D4—686no—
522C2×C4—176no—
523C2×C4—173no—
524C2×C4—176no—
525C2×C4—343no—
526C2×C4—346no—
527C2×C4—343no—
528C2×C4—346no—
529C2×C4—346no—
530C2×C4—343no—
531C2×C4—346no—
532C2×C4—346no—
533C2×C4—346no—
534C2×C4—343no—
535C2×C4—346no—
536C2×C4—343no—
537C2×C4—346no—
538C2×C4—6812no—
539C2×C4—6812no—
540C2×C4—1366no—
541C2×C4—1366no—
542C2×C4C23×#59513yes—
543C2×C4C23×#59516yes—
544C22—23no—
545C22—26no—
546C22—26no—
547C22—23no—
548C22—44no—
549C22—82no—
550C22—173no—
551C22—173no—
552C22—343no—
553C22—346no—
554C22—346no—
555C22—343no—
556C22—683no—
557C22—683no—
558C22—683no—
559C22—683no—
560C22—686no—
561C22—686no—
562C22—6812no—
563C22—1366no—
564C22C24×#59511yescharacteristic
565C4—23no—
566C4—26no—
567C4—346no—
568C4—343no—
569C4—346no—
570C4—681no—
571C4—683no—
572C4—686no—
573C4—683no—
574C2—43no—
575C2—43no—
576C2—171no—
577C2—173no—
578C2—683no—
579C2—683no—
580C2H(5,𝔽2)×#59513yes—
subgroupG/H
center, FrattiniC22C24×#595
commutator, socleC22×C17C24×C4
FittingC17×H(3,𝔽4)C4
prime radicalD17×H(3,𝔽4)C2
cosocleC22×D17C25
hypercenterH(3,𝔽4)#595
hypocenterC17#34512
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #14545323C2semidirect
2. #14545320C2semidirect
3. #14545315C2semidirect
4. #14545306C2semidirect
5. #14545285C2semidirect
6. #14545274C2semidirect
7. #14545269C2semidirect
8. #120×#595C2semidirect
10. C24×#595C22semidirect
11. C2×Q8×#595C22semidirect
14. C22×C22⋊#595C22semidirect
15. (C2×#595)⋊Q8C22semidirect
16. (C2×#595)⋊Q8C22semidirect
19. #11763759C22semidirect
20. #11763759C22semidirect
21. #11763759C22semidirect
22. #11763759C22semidirect
23. #11763759C22semidirect
24. #11763759C22semidirect
29. #11763756C22semidirect
30. #11763756C22semidirect
31. #11763756C22semidirect
32. #11763756C22semidirect
33. #11763755C22semidirect
34. #11763754C22semidirect
35. #11763750C22semidirect
36. #11763750C22semidirect
37. C4×C4⋊#595C22semidirect
38. C22⋊C4×#595C22semidirect
39. C22⋊C4×#595C22semidirect
40. C22⋊C4×#595C22semidirect
41. C22⋊C4×#595C22semidirect
42. #11763745C22semidirect
43. #11763552C4semidirect
44. #11763552C4semidirect
45. #11763551C4semidirect
46. #11763551C4semidirect
51. C17×H(3,𝔽4)C4semidirect
82. C17×#124C2×C4semidirect
83. C17×#124C2×C4semidirect
84. C17×#120C2×C4semidirect
85. C17×#120C2×C4semidirect
86. #10587639C2×C4semidirect
87. #10587639C2×C4semidirect
88. #10587633C2×C4semidirect
89. #10587633C2×C4semidirect
90. #10587623C2×C4semidirect
91. #10587623C2×C4semidirect
92. #10587623C2×C4semidirect
93. #10587623C2×C4semidirect
94. #10587622C2×C4semidirect
95. #10587622C2×C4semidirect
96. #10587571C2×C4semidirect
97. #10587571C2×C4semidirect
98. #10587571C2×C4semidirect
99. #10587571C2×C4semidirect
100. #10587571C2×C4semidirect
101. #10587571C2×C4semidirect
102. #10587568C2×C4semidirect
103. #10587568C2×C4semidirect
104. #10587568C2×C4semidirect
105. #10587568C2×C4semidirect
110. #10587561C2×C4semidirect
111. #10587561C2×C4semidirect
121. C24×C17C22×C4semidirect
127. C23×D17C22×C4semidirect
144. C2×Q8×C17C22×C4semidirect
161. C2×Dic34C22×C4semidirect
162. C2×Dic34C22×C4semidirect
181. C22⋊C4×C17C22×C4semidirect
182. C22⋊C4×C17C22×C4semidirect
183. C42×C17C22×C4semidirect
184. C42×C17C22×C4semidirect
187. C22⋊Dic17C22×C4semidirect
188. C22⋊Dic17C22×C4semidirect
192. #63227C22×C4semidirect
193. #63227C22×C4semidirect
194. #63227C22×C4semidirect
195. C4×Dic17C22×C4semidirect
196. C4×Dic17C22×C4semidirect
298. H(3,𝔽4)#595semidirect
364. #124C2×#595semidirect
365. #124C2×#595semidirect
376. #120C2×#595semidirect
377. #120C2×#595semidirect
410. C17#34512semidirect
420. C24C22×#595semidirect
425. C2×Q8C22×#595semidirect
478. C22⋊C4C22×#595semidirect
479. C22⋊C4C22×#595semidirect
486. C42C22×#595semidirect
487. C42C22×#595semidirect

Maximal subgroups

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

subgroupindex
#145453232
#145453202
#145453152
#145453062
#145452852
#145452742
#145452692
#120×#5952
D17×H(3,𝔽4)2
#3451217