Finite Group Databasebeta

← #16964639

Subgroups — #16964639

Order 4352.

1442 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14527046C212yesmaximal, complemented
2#14527037C212yesmaximal, complemented
3#14527029C211yesmaximal, characteristic
4#14527021C212yesmaximal
5#14527016C212yesmaximal, complemented
6#14527016C212yesmaximal, complemented
7#14527006C212yesmaximal
8#14527006C211yesmaximal, characteristic
9C2×C17×#399C211yesmaximal, characteristic
10C24×Dic17C2211yescharacteristic
11C22×C4⋊Dic17C2211yescharacteristic
12C22×C22⋊Dic17C2212yes—
13C22×C22⋊Dic17C2211yescharacteristic
14C22×C4×Dic17C2212yes—
15C2×#10587502C2212yes—
16C2×#10587502C2212yes—
17C2×#10587502C2212yes—
18C2×#10587502C2212yes—
19C2×#10587502C2212yes—
20C2×#10587502C2212yes—
21C2×#10587502C2212yes—
22C2×#10587502C2212yes—
23C2×#10587502C2211yescharacteristic
24C2×#10587502C2212yes—
25C2×#10587502C2212yes—
26C22×C4⋊C4×C17C2211yescharacteristic
27C22×C4⋊C4×C17C2212yes—
28C22×C22⋊C4×C17C2212yes—
29C22×C22⋊C4×C17C2211yescharacteristic
30C17×#399C418yescomplemented
31C2×C17×#95C2211yescharacteristic
32C25×C17C2311yescharacteristic
33C23×C4×C17C2312yes—
34C23×C4×C17C2311yescharacteristic
35C23×C4×C17C2311yescharacteristic
36C23×C4×C17C2312yes—
37C23×Dic17—22no—
38C23×Dic17—24no—
39C23×Dic17—41no—
40C23×Dic17—42no—
41C23×Dic17—41no—
42C23×Dic17C2312yes—
43C23×Dic17C2312yes—
44C23×Dic17C2312yes—
45C23×Dic17C2312yes—
46C2×C4⋊C4×C17—41no—
47C2×C4⋊C4×C17—42no—
48C2×C4⋊C4×C17—41no—
49C2×C4⋊C4×C17—42no—
50C2×C4⋊C4×C17C2×C414yescomplemented
51C2×C4⋊C4×C17C2×C418yescomplemented
52C2×C22⋊C4×C17—42no—
53C2×C22⋊C4×C17—41no—
54C2×C22⋊C4×C17—42no—
55C2×C22⋊C4×C17—41no—
56C2×C22⋊C4×C17C2×C418yes—
57C2×C22⋊C4×C17C2×C414yes—
58C2×C22⋊Dic17—24no—
59C2×C22⋊Dic17—24no—
60C2×C22⋊Dic17—41no—
61C2×C22⋊Dic17—41no—
62C2×C22⋊Dic17—42no—
63C2×C22⋊Dic17Q814yes—
64C2×C4⋊Dic17—41no—
65C2×C4⋊Dic17—42no—
66C2×C4×Dic17—42no—
67C2×C4×Dic17—42no—
68C2×C4×Dic17—42no—
69C17×#95C2×C414yescomplemented
70#10587502—24no—
71#10587502—24no—
72#10587502—24no—
73#10587502—24no—
74#10587502—42no—
75#10587502—42no—
76#10587502—42no—
77#10587502—42no—
78#10587502—41no—
79#10587502—42no—
80#10587502—42no—
81C24×C17—22no—
82C24×C17—22no—
83C24×C17—21no—
84C24×C17—21no—
85C24×C17—22no—
86C24×C17—21no—
87C24×C17—22no—
88C24×C17—21no—
89C24×C17C22×C412yes—
90C24×C17C2×Q812yes—
91C24×C17#4112yes—
92C24×C17C2411yescharacteristic
93C22×C4×C17—22no—
94C22×C4×C17—22no—
95C22×C4×C17—22no—
96C22×C4×C17—22no—
97C22×C4×C17—22no—
98C22×C4×C17—21no—
99C22×C4×C17—22no—
100C22×C4×C17—22no—
101C22×C4×C17—22no—
102C22×C4×C17—22no—
103C22×C4×C17—22no—
104C22×C4×C17—22no—
105C22×C4×C17—22no—
106C22×C4×C17—21no—
107C22×C4×C17—22no—
108C22×C4×C17—22no—
109C22×C4×C17—22no—
110C22×C4×C17—21no—
111C22×C4×C17—21no—
112C22×C4×C17—22no—
113C22×C4×C17C22×C412yes—
114C22×C4×C17C22×C414yes—
115C22×C4×C17C22×C414yes—
116C22×C4×C17C22×C412yes—
117C22×Dic17—22no—
118C22×Dic17—22no—
119C22×Dic17—22no—
120C22×Dic17—22no—
121C22×Dic17—22no—
122C22×Dic17—22no—
123C22×Dic17—22no—
124C22×Dic17—22no—
125C22×Dic17—22no—
126C22×Dic17—22no—
127C22×Dic17—22no—
128C22×Dic17—22no—
129C22×Dic17—22no—
130C22×Dic17—22no—
131C22×Dic17—22no—
132C22×Dic17—22no—
133C22×Dic17—22no—
134C22×Dic17—22no—
135C22×Dic17—22no—
136C22×Dic17—22no—
137C22×Dic17—22no—
138C22×Dic17—22no—
139C22×Dic17—22no—
140C22×Dic17—22no—
141C22×Dic17—22no—
142C22×Dic17—44no—
143C22×Dic17—48no—
144C22×Dic17—48no—
145C22×Dic17—44no—
146C22×Dic17—44no—
147C22×Dic17C2×Q814yes—
148C22×Dic17#4114yes—
149C22×Dic17#4114yes—
150C4⋊C4×C17—48no—
151C4⋊C4×C17—44no—
152C22⋊C4×C17—48no—
153C22⋊C4×C17—44no—
154C22⋊Dic17—44no—
155C22⋊Dic17—48no—
156C4⋊Dic17—44no—
157C4×Dic17—48no—
158#14592—171nomaximal, Sylow
159C23×C17—24no—
160C23×C17—22no—
161C23×C17—24no—
162C23×C17—22no—
163C23×C17—22no—
164C23×C17—22no—
165C23×C17—22no—
166C23×C17—22no—
167C23×C17—24no—
168C23×C17—24no—
169C23×C17—24no—
170C23×C17—24no—
171C23×C17—42no—
172C23×C17—42no—
173C23×C17—42no—
174C23×C17—42no—
175C23×C17—41no—
176C23×C17—42no—
177C23×C17—41no—
178C23×C17—41no—
179C23×C17—41no—
180C23×C17—42no—
181C23×C17C4×Q814yes—
182C23×C17C23×C411yescharacteristic
183C23×C17C22×Q811yescharacteristic
184C23×C17C2×#4111yescharacteristic
185C23×C17C2×#4112yes—
186C23×C17C2×#4112yes—
187C23×C17H(5,𝔽2)12yes—
188C23×C17H(5,𝔽2)11yescharacteristic
189C23×C17H(5,𝔽2)11yescharacteristic
190C23×C17H(5,𝔽2)12yes—
191C23×C17H(5,𝔽2)11yescharacteristic
192C23×C172−1+411yescharacteristic
193C2×C4×C17—24no—
194C2×C4×C17—24no—
195C2×C4×C17—24no—
196C2×C4×C17—22no—
197C2×C4×C17—22no—
198C2×C4×C17—24no—
199C2×C4×C17—24no—
200C2×C4×C17—24no—
201C2×C4×C17—24no—
202C2×C4×C17—24no—
203C2×C4×C17—42no—
204C2×C4×C17—42no—
205C2×C4×C17—42no—
206C2×C4×C17—42no—
207C2×C4×C17—41no—
208C2×C4×C17—42no—
209C2×C4×C17—41no—
210C2×C4×C17—42no—
211C2×C4×C17—41no—
212C2×C4×C17—42no—
213C2×C4×C17—42no—
214C2×C4×C17—42no—
215C2×C4×C17—42no—
216C2×C4×C17—41no—
217C2×Dic17—24no—
218C2×Dic17—24no—
219C2×Dic17—24no—
220C2×Dic17—24no—
221C2×Dic17—24no—
222C2×Dic17—24no—
223C2×Dic17—24no—
224C2×Dic17—24no—
225C2×Dic17—24no—
226C2×Dic17—416no—
227C2×Dic17—42no—
228C2×Dic17—42no—
229C2×Dic17—42no—
230C2×Dic17—42no—
231C2×Dic17—42no—
232C2×Dic17—42no—
233C2×Dic17—42no—
234C2×Dic17—42no—
235C2×Dic17—42no—
236C2×Dic17—42no—
237C2×Dic17—42no—
238C2×Dic17—42no—
239C2×Dic17—42no—
240C2×Dic17—42no—
241C2×Dic17—42no—
242C2×Dic17—42no—
243C2×Dic17—42no—
244C2×Dic17—42no—
245C2×Dic17—42no—
246C2×#399D1711yescharacteristic
247#1443—172no—
248#1441—172no—
249#1439—171no—
250#1437—172no—
251#1436—172no—
252#1436—172no—
253C22⋊#95—171no—
254C22⋊#95—172no—
255C22×C17—24no—
256C22×C17—28no—
257C22×C17—44no—
258C22×C17—42no—
259C22×C17—42no—
260C22×C17—42no—
261C22×C17—42no—
262C22×C17—44no—
263C22×C17—42no—
264C22×C17—44no—
265C22×C17—42no—
266C22×C17C2×C4×Q811yescharacteristic
267C22×C17C4×#4112yes—
268C22×C17#51811yescharacteristic
269C22×C17#51812yes—
270C22×C17#52011yescharacteristic
271C22×C17#53812yes—
272C22×C17#53811yescharacteristic
273C22×C17#53912yes—
274C22×C17#53911yescharacteristic
275C22×C17#54012yes—
276C22×C17#54212yes—
277C22×C17#54311yescharacteristic
278C22×C17#54411yescharacteristic
279C22×C17#55112yes—
280C22×C17#55112yes—
281C22×C17#55111yescharacteristic
282C22×C17#55312yes—
283C22×C17#55412yes—
284C22×C17#55412yes—
285C22×C17#55911yescharacteristic
286C22×C17#56011yescharacteristic
287C22×C17#56011yescharacteristic
288C22×C17#56012yes—
289C4×C17—44no—
290C4×C17—42no—
291C4×C17—44no—
292C4×C17—42no—
293Dic17—44no—
294Dic17—44no—
295Dic17—44no—
296Dic17—82no—
297C24×C4—171no—
298C22×C4⋊C4—171no—
299C22×C4⋊C4C2×D1711yescharacteristic
300C22×C4⋊C4C2×D1712yes—
301C22×C22⋊C4—171no—
302C22×C22⋊C4—172no—
303C22×C22⋊C4C2×D1712yes—
304C22×C22⋊C4C2×D1711yescharacteristic
305C22×C42—172no—
306#399Dic1718yescomplemented
307C2×#95—172no—
308C2×#95—172no—
309C2×#95—172no—
310C2×#95—172no—
311C2×#95—172no—
312C2×#95—172no—
313C2×#95—172no—
314C2×#95—172no—
315C2×#95—171no—
316C2×#95—172no—
317C2×#95—172no—
318C2×#95C2×D1711yescharacteristic
319C2×C17—44no—
320C2×C17#232911yescharacteristic
321C2×C17#233512yes—
322C2×C17#236812yes—
323C2×C17#237411yescharacteristic
324C2×C17#238011yescharacteristic
325C2×C17#273511yescharacteristic
326C2×C17#274211yescharacteristic
327C2×C17#274612yes—
328C2×C17#275211yescharacteristic
329C2×C17#278012yes—
330C2×C17#280711yescharacteristic
331C25C22×D1711yescharacteristic
332C23×C4—172no—
333C23×C4—172no—
334C23×C4—172no—
335C23×C4—172no—
336C23×C4—344no—
337C23×C4—342no—
338C23×C4—681no—
339C23×C4—682no—
340C23×C4—681no—
341C23×C4C22×D1711yescharacteristic
342C23×C4C22×D1712yes—
343C23×C4C22×D1711yescharacteristic
344C23×C4C22×D1712yes—
345C2×C4⋊C4—41no—
346C2×C4⋊C4—42no—
347C2×C4⋊C4—41no—
348C2×C4⋊C4—42no—
349C2×C4⋊C4—681no—
350C2×C4⋊C4—682no—
351C2×C4⋊C4C2×Dic1714yescomplemented
352C2×C4⋊C4C2×Dic1718yescomplemented
353C2×C22⋊C4—42no—
354C2×C22⋊C4—41no—
355C2×C22⋊C4—42no—
356C2×C22⋊C4—41no—
357C2×C22⋊C4—174no—
358C2×C22⋊C4—344no—
359C2×C22⋊C4—344no—
360C2×C22⋊C4—681no—
361C2×C22⋊C4—681no—
362C2×C22⋊C4—682no—
363C2×C22⋊C4C2×Dic1718yes—
364C2×C22⋊C4C2×Dic1714yes—
365C2×C42—682no—
366C2×C42—682no—
367C2×C42—682no—
368#95—344no—
369#95—344no—
370#95—344no—
371#95—344no—
372#95—681no—
373#95—682no—
374#95—682no—
375#95—682no—
376#95—682no—
377#95—682no—
378#95—682no—
379#95C2×Dic1714yescomplemented
380C17#1459211yescharacteristic, complemented, Sylow
381C24—22no—
382C24—21no—
383C24—22no—
384C24—21no—
385C24—22no—
386C24—22no—
387C24—21no—
388C24—21no—
389C24Q8×D1712yes—
390C24#6325612yes—
391C24C22×Dic1712yes—
392C24C23×D1711yescharacteristic
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—174no—
414C22×C4—174no—
415C22×C4—174no—
416C22×C4—342no—
417C22×C4—342no—
418C22×C4—342no—
419C22×C4—342no—
420C22×C4—342no—
421C22×C4—342no—
422C22×C4—342no—
423C22×C4—342no—
424C22×C4—342no—
425C22×C4—342no—
426C22×C4—342no—
427C22×C4—342no—
428C22×C4—342no—
429C22×C4—342no—
430C22×C4—342no—
431C22×C4—342no—
432C22×C4—342no—
433C22×C4—342no—
434C22×C4—342no—
435C22×C4—342no—
436C22×C4—342no—
437C22×C4—342no—
438C22×C4—342no—
439C22×C4—342no—
440C22×C4—342no—
441C22×C4—684no—
442C22×C4—688no—
443C22×C4—688no—
444C22×C4—684no—
445C22×C4—684no—
446C22×C4C22×Dic1712yes—
447C22×C4C22×Dic1714yes—
448C22×C4C22×Dic1712yes—
449C22×C4C22×Dic1714yes—
450C4⋊C4—44no—
451C4⋊C4—48no—
452C4⋊C4—684no—
453C22⋊C4—44no—
454C22⋊C4—48no—
455C22⋊C4—684no—
456C22⋊C4—688no—
457C42—688no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—41no—
471C23—42no—
472C23—41no—
473C23—41no—
474C23—42no—
475C23—41no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23Q8×Dic1714yes—
481C23#1058769112yes—
482C23#1058769112yes—
483C23#1058769111yescharacteristic
484C23#1058769111yescharacteristic
485C23C2×Q8×D1711yescharacteristic
486C23C2×#6325611yescharacteristic
487C23#41×D1712yes—
488C23#41×D1712yes—
489C23#1058769611yescharacteristic
490C23#1058769711yescharacteristic
491C23C23×Dic1711yescharacteristic
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—344no—
517C2×C4—344no—
518C2×C4—344no—
519C2×C4—344no—
520C2×C4—344no—
521C2×C4—344no—
522C2×C4—344no—
523C2×C4—344no—
524C2×C4—344no—
525C2×C4—6816no—
526C2×C4—682no—
527C2×C4—682no—
528C2×C4—682no—
529C2×C4—682no—
530C2×C4—682no—
531C2×C4—682no—
532C2×C4—682no—
533C2×C4—682no—
534C2×C4—682no—
535C2×C4—682no—
536C2×C4—682no—
537C2×C4—682no—
538C2×C4—682no—
539C2×C4—682no—
540C2×C4—682no—
541C2×C4—682no—
542C2×C4—682no—
543C2×C4—682no—
544C2×C4—682no—
545C22—28no—
546C22—24no—
547C22—44no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—44no—
555C22—44no—
556C22C2×Q8×Dic1711yescharacteristic
557C22#41×Dic1712yes—
558C22#1176328412yes—
559C22#1176328411yescharacteristic
560C22#1176329211yescharacteristic
561C22#1176339512yes—
562C22#1176339911yescharacteristic
563C22#1176340212yes—
564C22#1176340811yescharacteristic
565C22#1176341112yes—
566C22#1176342712yes—
567C22#1176343411yescharacteristic
568C22#1176343811yescharacteristic
569C22#1176348612yes—
570C22#1176348612yes—
571C22#1176348711yescharacteristic
572C22#1176350412yes—
573C22#1176351312yes—
574C22#1176351312yes—
575C22#1176354311yescharacteristic
576C22#1176354612yes—
577C22#1176354611yescharacteristic
578C22#1176354611yescharacteristic
579C4—42no—
580C4—44no—
581C4—42no—
582C4—44no—
583C4—684no—
584C4—684no—
585C4—684no—
586C4—1362no—
587C2—44no—
588C2#1453151911yescharacteristic
589C2#1453157812yes—
590C2#1453186912yes—
591C2#1453192011yescharacteristic
592C2#1453196111yescharacteristic
593C2#1453582611yescharacteristic
594C2#1453591411yescharacteristic
595C2#1453597412yes—
596C2#1453605311yescharacteristic
597C2#1453643712yes—
598C2#1453674911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC2×C17×#399C2
socle, cosocleC24×C17C24
prime radicalC25×C17C23
hypercenterC2×#399D17
hypocenterC17#14592
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
1. #14527046C2semidirect
2. #14527037C2semidirect
5. #14527016C2semidirect
6. #14527016C2semidirect
30. C17×#399C4semidirect
50. C2×C4⋊C4×C17C2×C4semidirect
51. C2×C4⋊C4×C17C2×C4semidirect
69. C17×#95C2×C4semidirect
306. #399Dic17semidirect
351. C2×C4⋊C4C2×Dic17semidirect
352. C2×C4⋊C4C2×Dic17semidirect
379. #95C2×Dic17semidirect
380. C17#14592semidirect

Maximal subgroups

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

subgroupindex
#145270462
#145270372
#145270292
#145270212
#145270162
#145270062
C2×C17×#3992
#1459217