Finite Group Databasebeta

← #17090829

Subgroups — #17090829

Order 4352.

1292 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763177C211yesmaximal, characteristic, complemented
2C2×#11763109C211yesmaximal, characteristic, complemented
3#14533950C211yesmaximal, characteristic, complemented
4#14529984C212yesmaximal, complemented
5#14529977C212yesmaximal, complemented
6#14529708C211yesmaximal, characteristic, complemented
7#14529583C211yesmaximal, characteristic, complemented
8#14528724C212yesmaximal, complemented
9#14528724C212yesmaximal, complemented
10#14528609C211yesmaximal, characteristic, complemented
11C2×C17×#547C211yesmaximal, characteristic
12C2×#10587556C2211yescharacteristic
13#11763177—24no—
14#11763109—24no—
15C2×#10587478C2211yescharacteristic
16C2×#10587479C2211yescharacteristic
17C2×C4⋊#3622C2211yescharacteristic
18C2×#10587506C2212yes—
19C2×#10587506C2212yes—
20C2×#10587506C2212yes—
21C2×#10587506C2212yes—
22C2×#10587503C2211yescharacteristic
23C2×#10587503C2211yescharacteristic
24C2×#10587503C2212yes—
25C2×C22⋊#3622C2212yes—
26C2×C22⋊#3622C2212yes—
27C2×#10587474C2211yescharacteristic
28#11762747—21no—
29#11762723—21no—
30#11762703—22no—
31#11762669C2212yes—
32#11762669C2212yes—
33C4×C4⋊Dic17C2211yescharacteristic
34#11762571C2212yes—
35C22×#41×C17C2212yes—
36C17×#547—24no—
37C17×#547—24no—
38C2×C17×C4⋊Q8C2211yescharacteristic
39C2×C17×#123—22no—
40C2×C17×C22⋊Q8C2212yes—
41C2×C17×#121—21no—
42C2×C4×D4×C17C2211yescharacteristic
43C2×C4×D4×C17C2211yescharacteristic
44C22×C4⋊C4×C17—21no—
45C2×#41×C17—24no—
46C2×#41×C17—24no—
47C2×#41×C17—24no—
48C2×#41×C17—24no—
49C2×#41×C17—42no—
50C2×#41×C17—42no—
51C22×Q8×C17C2312yes—
52C22×D4×C17—22no—
53C22×D4×C17D411yescharacteristic
54C22×D4×C17C2311yescharacteristic
55C23×C4×C17—21no—
56C23×C4×C17—22no—
57C23×C4×C17C2312yes—
58C17×C4⋊Q8—22no—
59C17×C4⋊Q8D414yes—
60C17×#123—216no—
61C17×C22⋊Q8—24no—
62C17×C22⋊Q8—24no—
63C17×#121—28no—
64C4×D4×C17—22no—
65C4×D4×C17—22no—
66C4×D4×C17—22no—
67C4×D4×C17D414yes—
68C2×C4⋊C4×C17—28no—
69C2×C4⋊C4×C17—21no—
70C2×C4⋊C4×C17—22no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17D411yescharacteristic
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C22⋊C4×C17—22no—
75C2×C22⋊C4×C17—21no—
76C2×C22⋊C4×C17D412yes—
77C2×C42×C17C2311yescharacteristic
78C2×C22⋊Dic17—22no—
79C2×C22⋊Dic17—22no—
80C2×C4⋊Dic17C2311yescharacteristic
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4⋊Dic17C2312yes—
83C2×#63225—21no—
84C2×#63225—21no—
85C2×C4×Dic17—21no—
86C22×#3622C2312yes—
87C22×#3622C2311yescharacteristic
88C22×#3622C2311yescharacteristic
89#10587556—22no—
90#10587556—22no—
91#10587506—24no—
92#10587506—24no—
93#10587506—42no—
94#10587506—42no—
95#10587503—22no—
96#10587503—22no—
97#10587503—24no—
98#10587502—22no—
99#10587502—22no—
100#10587502—21no—
101C22⋊#3622—42no—
102C22⋊#3622—42no—
103#10587479—22no—
104#10587478—22no—
105C4⋊#3622—22no—
106#10587474—22no—
107C24×C17—21no—
108C24×C17C2×D412yes—
109#41×C17—48no—
110#41×C17—48no—
111#41×C17—48no—
112#41×C17—48no—
113C2×Q8×C17—22no—
114C2×Q8×C17—22no—
115C2×Q8×C17—22no—
116C2×Q8×C17—22no—
117C2×Q8×C17—22no—
118C2×Q8×C17C2×D414yes—
119C2×D4×C17—28no—
120C2×D4×C17—22no—
121C2×D4×C17—21no—
122C2×D4×C17—22no—
123C2×D4×C17—42no—
124C2×D4×C17—44no—
125C2×D4×C17—44no—
126C2×D4×C17C2×D412yes—
127C2×D4×C17#4114yes—
128C22×C4×C17—28no—
129C22×C4×C17—22no—
130C22×C4×C17—21no—
131C22×C4×C17—24no—
132C22×C4×C17—22no—
133C22×C4×C17—22no—
134C22×C4×C17—22no—
135C22×C4×C17—22no—
136C22×C4×C17—22no—
137C22×C4×C17—22no—
138C22×C4×C17—44no—
139C22×C4×C17—44no—
140C22×C4×C17—44no—
141C22×C4×C17C2×D412yes—
142C22×C4×C17C2×D411yescharacteristic
143C22×C4×C17C2×D411yescharacteristic
144C22×C4×C17C2411yescharacteristic
145C22×Dic17—22no—
146C22×Dic17—21no—
147C22×Dic17—21no—
148C4⋊C4×C17—28no—
149C4⋊C4×C17—24no—
150C4⋊C4×C17—24no—
151C4⋊C4×C17—22no—
152C4⋊C4×C17—21no—
153C4⋊C4×C17C2×D412yes—
154C22⋊C4×C17—28no—
155C22⋊C4×C17—24no—
156C22⋊C4×C17—24no—
157C42×C17C2×D412yes—
158C42×C17C2×D412yes—
159C22⋊Dic17—44no—
160C22⋊Dic17—82no—
161C4⋊Dic17—21no—
162C4⋊Dic17—21no—
163C4⋊Dic17—21no—
164C4⋊Dic17—21no—
165C4⋊Dic17—22no—
166C4⋊Dic17—22no—
167#63225—42no—
168#63225—42no—
169C4×Dic17—81no—
170C2×#3622—22no—
171C2×#3622—21no—
172C2×#3622—21no—
173C2×#3622—22no—
174C2×#3622—22no—
175C2×#3622—22no—
176C2×#3622—22no—
177#25892—171nomaximal, Sylow
178C23×C17—22no—
179C23×C17—24no—
180C23×C17—22no—
181C23×C17—22no—
182C23×C17—22no—
183C23×C17—22no—
184C23×C17—22no—
185C23×C17—22no—
186C23×C17—22no—
187C23×C17—42no—
188C23×C17—41no—
189C23×C17—41no—
190C23×C17C22×D411yescharacteristic
191Q8×C17—24no—
192Q8×C17—24no—
193Q8×C17—42no—
194Q8×C17—42no—
195D4×C17—22no—
196D4×C17—24no—
197D4×C17—44no—
198D4×C17—416no—
199D4×C17#12414yes—
200C2×C4×C17—22no—
201C2×C4×C17—21no—
202C2×C4×C17—21no—
203C2×C4×C17—22no—
204C2×C4×C17—22no—
205C2×C4×C17—24no—
206C2×C4×C17—22no—
207C2×C4×C17—22no—
208C2×C4×C17—48no—
209C2×C4×C17—416no—
210C2×C4×C17—42no—
211C2×C4×C17—42no—
212C2×C4×C17—42no—
213C2×C4×C17—42no—
214C2×C4×C17—42no—
215C2×C4×C17—42no—
216C2×C4×C17—42no—
217C2×C4×C17#12012yes—
218C2×C4×C17#12012yes—
219C2×C4×C17C22×D411yescharacteristic
220C2×C4×C17C22×D411yescharacteristic
221C2×C4×C17C2×#4112yes—
222C2×C4×C17H(5,𝔽2)12yes—
223C2×Dic17—41no—
224C2×Dic17—41no—
225C2×Dic17—42no—
226C2×Dic17—42no—
227C2×Dic17—42no—
228C2×Dic17—41no—
229C2×Dic17—42no—
230C2×Dic17—41no—
231#3622—41no—
232#3622—41no—
233#3622—42no—
234C2×#547D1711yescharacteristic
235C2×#491—171no—
236C2×#478—171no—
237#2562—171no—
238#2036—172no—
239#2035—172no—
240#1981—171no—
241#1955—171no—
242#1788—172no—
243#1788—172no—
244#1762—171no—
245C22×C17—24no—
246C22×C17—42no—
247C22×C17—42no—
248C22×C17—44no—
249C22×C17—44no—
250C22×C17—42no—
251C22×C17—42no—
252C22×C17—42no—
253C22×C17—42no—
254C22×C17—42no—
255C22×C17—42no—
256C22×C17—42no—
257C22×C17C2×#12011yescharacteristic
258C22×C17#54612yes—
259C22×C17#57512yes—
260C22×C17#57711yescharacteristic
261C22×C17#57811yescharacteristic
262C4×C17—22no—
263C4×C17—22no—
264C4×C17—42no—
265C4×C17—44no—
266C4×C17—42no—
267C4×C17C2×#12411yescharacteristic
268C4×C17#53511yescharacteristic
269C4×C17#54612yes—
270Dic17—81no—
271Dic17—81no—
272Dic17—82no—
273C22×#41C2×D1712yes—
274#547—24no—
275#547—24no—
276C2×C4⋊Q8C2×D1711yescharacteristic
277C2×#125—171no—
278C2×#123—22no—
279C2×C22⋊Q8C2×D1712yes—
280C2×#121—21no—
281C2×C4×D4C2×D1711yescharacteristic
282C2×C4×D4C2×D1711yescharacteristic
283C22×C4⋊C4—21no—
284#491—344no—
285#478—344no—
286C2×#107—171no—
287C2×#106—171no—
288C2×C4⋊C8—171no—
289C2×#103—172no—
290C2×#103—172no—
291C2×#103—172no—
292C2×#103—172no—
293C2×#102—171no—
294C2×#102—171no—
295C2×#102—172no—
296C2×C22⋊C8—172no—
297C2×C22⋊C8—172no—
298C2×#97—171no—
299#400—341no—
300#394—341no—
301#388—342no—
302#380—172no—
303#380—172no—
304C4×C4⋊C4—171no—
305#340—172no—
306C2×C17—44no—
307C2×C17—82no—
308C2×C17#243511yescharacteristic
309C2×C17#300511yescharacteristic
310C2×C17#301311yescharacteristic
311C2×C17#330712yes—
312C2×C17#331512yes—
313C2×#41—24no—
314C2×#41—24no—
315C2×#41—24no—
316C2×#41—24no—
317C2×#41—42no—
318C2×#41—42no—
319C22×Q8C22×D1712yes—
320C22×D4—22no—
321C22×D4C22⋊D1711yescharacteristic
322C22×D4C22×D1711yescharacteristic
323C23×C4—21no—
324C23×C4—22no—
325C23×C4C22×D1712yes—
326C22×C8—172no—
327C22×C8—171no—
328C22×C8—171no—
329C4⋊Q8—22no—
330C4⋊Q8C22⋊D1714yes—
331#125—342no—
332#125—342no—
333#123—216no—
334C22⋊Q8—24no—
335C22⋊Q8—24no—
336#121—28no—
337C4×D4—22no—
338C4×D4—22no—
339C4×D4—22no—
340C4×D4C22⋊D1714yes—
341C2×C4⋊C4—28no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—171no—
346C2×C4⋊C4—171no—
347C2×C4⋊C4—172no—
348C2×C4⋊C4—341no—
349C2×C4⋊C4—341no—
350C2×C4⋊C4C22⋊D1711yescharacteristic
351C2×C4⋊C4C22×D1711yescharacteristic
352C2×C22⋊C4—22no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—342no—
355C2×C22⋊C4—342no—
356C2×C22⋊C4C22⋊D1712yes—
357C2×C42—341no—
358C2×C42C22×D1711yescharacteristic
359#107—342no—
360#106—342no—
361C4⋊C8—342no—
362#103—344no—
363#103—344no—
364#103—682no—
365#103—682no—
366#102—344no—
367#102—342no—
368#102—342no—
369C22⋊C8—682no—
370C22⋊C8—682no—
371#97—342no—
372#95—341no—
373#95—342no—
374#95—342no—
375C17#2589211yescharacteristic, complemented, Sylow
376C24—21no—
377C24C2×C22⋊D1712yes—
378#41—48no—
379#41—48no—
380#41—48no—
381#41—48no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8C2×C22⋊D1714yes—
388C2×D4—28no—
389C2×D4—22no—
390C2×D4—21no—
391C2×D4—22no—
392C2×D4—42no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4#6325414yes—
396C2×D4C2×C22⋊D1712yes—
397C22×C4—28no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—342no—
411C22×C4—341no—
412C22×C4—341no—
413C22×C4C2×C22⋊D1712yes—
414C22×C4C2×C22⋊D1711yescharacteristic
415C22×C4C2×C22⋊D1711yescharacteristic
416C22×C4C23×D1711yescharacteristic
417C2×C8—342no—
418C2×C8—342no—
419C2×C8—342no—
420C2×C8—342no—
421C2×C8—342no—
422C2×C8—341no—
423C2×C8—341no—
424C4⋊C4—28no—
425C4⋊C4—21no—
426C4⋊C4—22no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—342no—
430C4⋊C4—342no—
431C4⋊C4—341no—
432C4⋊C4—341no—
433C4⋊C4—341no—
434C4⋊C4—341no—
435C4⋊C4—682no—
436C4⋊C4—682no—
437C4⋊C4C2×C22⋊D1712yes—
438C22⋊C4—28no—
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—684no—
442C22⋊C4—1362no—
443C42—1361no—
444C42C2×C22⋊D1712yes—
445C42C2×C22⋊D1712yes—
446C23—22no—
447C23—24no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—42no—
456C23—41no—
457C23—41no—
458C23C22×C22⋊D1711yescharacteristic
459Q8—24no—
460Q8—24no—
461Q8—42no—
462Q8—42no—
463D4—22no—
464D4—24no—
465D4—44no—
466D4—416no—
467D4#1058762214yes—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—48no—
477C2×C4—416no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—681no—
486C2×C4—681no—
487C2×C4—682no—
488C2×C4—682no—
489C2×C4—682no—
490C2×C4—681no—
491C2×C4—681no—
492C2×C4—682no—
493C2×C4#1058763912yes—
494C2×C4#1058763912yes—
495C2×C4C2×#6325412yes—
496C2×C4#1058769112yes—
497C2×C4C22×C22⋊D1711yescharacteristic
498C2×C4C22×C22⋊D1711yescharacteristic
499C8—681no—
500C8—681no—
501C8—682no—
502C22—24no—
503C22—42no—
504C22—44no—
505C22—42no—
506C22—44no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22C2×#1058763911yescharacteristic
515C22#1176345312yes—
516C22#1176360712yes—
517C22#1176361711yescharacteristic
518C22#1176362411yescharacteristic
519C4—22no—
520C4—22no—
521C4—42no—
522C4—42no—
523C4—44no—
524C4—1361no—
525C4—1361no—
526C4—1362no—
527C4C2×#1058762211yescharacteristic
528C4#1176337611yescharacteristic
529C4#1176345312yes—
530C2—44no—
531C2—82no—
532C2#1453243111yescharacteristic
533C2#1453802611yescharacteristic
534C2#1453809911yescharacteristic
535C2#1454076012yes—
536C2#1454088012yes—
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
Fitting, prime radicalC2×C17×#547C2
socleC23×C17C22×D4
hypercenterC2×#547D17
hypocenterC17#25892
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. C2×#11763177C2semidirect
2. C2×#11763109C2semidirect
3. #14533950C2semidirect
4. #14529984C2semidirect
5. #14529977C2semidirect
6. #14529708C2semidirect
7. #14529583C2semidirect
8. #14528724C2semidirect
9. #14528724C2semidirect
10. #14528609C2semidirect
375. C17#25892semidirect

Maximal subgroups

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

subgroupindex
C2×#117631772
C2×#117631092
#145339502
#145299842
#145299772
#145297082
#145295832
#145287242
#145286092
C2×C17×#5472
#2589217