Finite Group Databasebeta

← #17090467

Subgroups — #17090467

Order 4352.

1262 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14532429C212yesmaximal, complemented
2#14532149C211yesmaximal, characteristic
3#14530136C214yesmaximal, complemented
4#14529770C212yesmaximal, complemented
5#14529065C214yesmaximal
6C4×#10587505C211yesmaximal, characteristic
7C17×#3501C211yesmaximal, characteristic
8C23×Dic34C2211yescharacteristic
9C2×#10587630—22no—
10C2×#10587610—21no—
11C2×#10587505C2212yes—
12#11762729—21no—
13#11762729—24no—
14#11762721—21no—
15#11762715—21no—
16#11762714—22no—
17#11762689C2212yes—
18#11762681C2218yes—
19#11762681C2212yes—
20#11762669C2211yescharacteristic
21C4×C22⋊Dic17C2212yes—
22#11762576C2218yes—
23#11762576C2212yes—
24C17×#554—22no—
25C17×#554—24no—
26D4×Q8×C17—22no—
27C17×#544—21no—
28C17×#543—21no—
29C17×#543C2212yes—
30C17×#533—21no—
31C2×C17×C22⋊Q8—21no—
32C2×C17×C22⋊Q8C2212yes—
33C17×#518C2212yes—
34C2×C4×D4×C17C2211yescharacteristic
35C22×Q8×C17—21no—
36C22×Q8×C17D412yes—
37C22×D4×C17C2311yescharacteristic
38C23×C4×C17D411yescharacteristic
39C23×C4×C17C2311yescharacteristic
40C23×Dic17C2312yes—
41C22×Dic34—22no—
42C22×Dic34—21no—
43C22×Dic34—21no—
44C22×Dic34—22no—
45C22×Dic34—44no—
46C17×C4⋊Q8—21no—
47C17×C4⋊Q8—21no—
48C17×C4⋊Q8—24no—
49C17×#125—22no—
50C17×#125—24no—
51C17×C22⋊Q8—22no—
52C17×C22⋊Q8—22no—
53C17×C22⋊Q8—24no—
54C17×C22⋊Q8—24no—
55C17×C22⋊Q8—22no—
56C17×C22⋊Q8—22no—
57C17×C22⋊Q8—22no—
58C17×C22⋊Q8—22no—
59C17×C22⋊Q8—22no—
60C17×C22⋊Q8—22no—
61C17×C22⋊Q8—24no—
62C17×C22⋊Q8—24no—
63C4×Q8×C17—22no—
64C4×Q8×C17—22no—
65C4×D4×C17—22no—
66C4×D4×C17—22no—
67C4×D4×C17—24no—
68C4×D4×C17—24no—
69C17×#117—22no—
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17—22no—
72C2×C4⋊C4×C17D411yescharacteristic
73C2×C4⋊C4×C17D412yes—
74C2×C4⋊C4×C17D414yes—
75C2×C22⋊C4×C17—22no—
76C2×C22⋊C4×C17D411yescharacteristic
77C2×C22⋊C4×C17C2311yescharacteristic
78C2×C22⋊C4×C17C2314yes—
79C2×C42×C17D411yescharacteristic
80C2×C22⋊Dic17—22no—
81C2×C22⋊Dic17C2312yes—
82C2×C22⋊Dic17C2314yes—
83#10587630—48no—
84C2×C4⋊Dic17—21no—
85#10587610—44no—
86C2×#63225—22no—
87C2×#63225—28no—
88C2×C4×Dic17—22no—
89#10587505Q818yescomplemented
90#10587502—22no—
91#10587502—28no—
92#10587502—22no—
93#10587502—21no—
94#10587502—22no—
95C24×C17C2×D411yescharacteristic
96C24×C17C2411yescharacteristic
97C2×Q8×C17—22no—
98C2×Q8×C17—22no—
99C2×Q8×C17—22no—
100C2×Q8×C17—22no—
101C2×Q8×C17—44no—
102C2×Q8×C17—82no—
103C2×D4×C17—22no—
104C2×D4×C17—22no—
105C2×D4×C17C2×Q814yes—
106C22×C4×C17—22no—
107C22×C4×C17—22no—
108C22×C4×C17—21no—
109C22×C4×C17—21no—
110C22×C4×C17—42no—
111C22×C4×C17—42no—
112C22×C4×C17C2×D411yescharacteristic
113C22×C4×C17C2×D412yes—
114C22×C4×C17C2×D411yescharacteristic
115C22×C4×C17C2×D411yescharacteristic
116C22×C4×C17C2×D411yescharacteristic
117C22×C4×C17C2×D414yes—
118C22×C4×C17C2×D414yes—
119C22×C4×C17C2×D411yescharacteristic
120C22×C4×C17C2×D412yes—
121C22×Dic17—24no—
122C22×Dic17—22no—
123C22×Dic17—24no—
124C22×Dic17—22no—
125C22×Dic17—44no—
126C2×Dic34—24no—
127C2×Dic34—28no—
128C2×Dic34—44no—
129C2×Dic34—48no—
130C2×Dic34—42no—
131C2×Dic34—44no—
132C2×Dic34—41no—
133C2×Dic34—42no—
134C2×Dic34—41no—
135C4⋊C4×C17—24no—
136C4⋊C4×C17—22no—
137C4⋊C4×C17—22no—
138C4⋊C4×C17—24no—
139C4⋊C4×C17—22no—
140C4⋊C4×C17—22no—
141C4⋊C4×C17—22no—
142C4⋊C4×C17—24no—
143C4⋊C4×C17—24no—
144C4⋊C4×C17—22no—
145C4⋊C4×C17—22no—
146C22⋊C4×C17—24no—
147C22⋊C4×C17—24no—
148C22⋊C4×C17—22no—
149C22⋊C4×C17—24no—
150C22⋊C4×C17—22no—
151C42×C17—24no—
152C42×C17—21no—
153C42×C17—21no—
154C22⋊Dic17—24no—
155C22⋊Dic17—24no—
156C22⋊Dic17—44no—
157C22⋊Dic17C2×Q818yescomplemented
158C4⋊Dic17—42no—
159#63225—44no—
160#63225—416no—
161C4×Dic17—44no—
162#25852—171nomaximal, Sylow
163C23×C17—21no—
164C23×C17—21no—
165C23×C17—41no—
166C23×C17—41no—
167C23×C17—41no—
168C23×C17—41no—
169C23×C17C4⋊Q812yescomplemented
170C23×C17C22×D411yescharacteristic
171C23×C17C22×D412yes—
172C23×C17C22×Q812yes—
173C23×C172−1+412yes—
174Q8×C17—44no—
175Q8×C17—82no—
176Q8×C17—82no—
177Q8×C17—82no—
178D4×C17—28no—
179C2×C4×C17—21no—
180C2×C4×C17—21no—
181C2×C4×C17—22no—
182C2×C4×C17—24no—
183C2×C4×C17—21no—
184C2×C4×C17—24no—
185C2×C4×C17—24no—
186C2×C4×C17—21no—
187C2×C4×C17—42no—
188C2×C4×C17—41no—
189C2×C4×C17—42no—
190C2×C4×C17—41no—
191C2×C4×C17—44no—
192C2×C4×C17—41no—
193C2×C4×C17—44no—
194C2×C4×C17—81no—
195C2×C4×C17—82no—
196C2×C4×C17—82no—
197C2×C4×C17#12011yescharacteristic
198C2×C4×C17#12012yes—
199C2×C4×C17#12011yescharacteristic
200C2×C4×C17C4⋊Q812yescomplemented
201C2×Dic17—24no—
202C2×Dic17—28no—
203C2×Dic17—28no—
204C2×Dic17—44no—
205C2×Dic17—42no—
206C2×Dic17—44no—
207C2×Dic17—44no—
208C2×Dic17—42no—
209C2×Dic17—42no—
210C2×Dic17—42no—
211C2×Dic17—42no—
212Dic34—44no—
213Dic34—48no—
214Dic34—44no—
215#3501D1711yescharacteristic
216#2434—172no—
217#2409—171no—
218#2062—174no—
219#1996—172no—
220#1867—174no—
221C4×#99—171no—
222C22×C17—22no—
223C22×C17—21no—
224C22×C17—22no—
225C22×C17—21no—
226C22×C17—22no—
227C22×C17—42no—
228C22×C17—42no—
229C22×C17—42no—
230C22×C17—41no—
231C22×C17—41no—
232C22×C17—81no—
233C22×C17C2×#12011yescharacteristic
234C22×C17C2×C4⋊Q811yescharacteristic
235C22×C17#53611yescharacteristic
236C22×C17D4×Q814yes—
237C4×C17—22no—
238C4×C17—21no—
239C4×C17—22no—
240C4×C17—41no—
241C4×C17—44no—
242C4×C17—44no—
243C4×C17—41no—
244C4×C17B(4,𝔽2)12yes—
245Dic17—48no—
246Dic17—44no—
247Dic17—82no—
248C23×Q8—171no—
249#554—22no—
250#554—24no—
251D4×Q8—22no—
252#544—21no—
253#543—21no—
254#543C2×D1712yes—
255#533—21no—
256C2×C4⋊Q8—342no—
257C2×C22⋊Q8—21no—
258C2×C22⋊Q8—341no—
259C2×C22⋊Q8C2×D1712yes—
260#518C2×D1712yes—
261C2×C4×D4C2×D1711yescharacteristic
262C2×#99—172no—
263#395—341no—
264#395—344no—
265#393—341no—
266#391—342no—
267#391—341no—
268#386—172no—
269#385—172no—
270#385—178no—
271#380—171no—
272C4×C22⋊C4—172no—
273#342—172no—
274#342—178no—
275C2×C17—24no—
276C2×C17—21no—
277C2×C17—21no—
278C2×C17—42no—
279C2×C17#243411yescharacteristic
280C2×C17C2×B(4,𝔽2)11yescharacteristic
281C2×C17#302811yescharacteristic
282C22×Q8—21no—
283C22×Q8—341no—
284C22×Q8—342no—
285C22×Q8—341no—
286C22×Q8—342no—
287C22×Q8—684no—
288C22×Q8C22⋊D1712yes—
289C22×D4C22×D1711yescharacteristic
290C23×C4—172no—
291C23×C4C22⋊D1711yescharacteristic
292C23×C4C22×D1711yescharacteristic
293C4⋊Q8—21no—
294C4⋊Q8—21no—
295C4⋊Q8—24no—
296C4⋊Q8—688no—
297#125—22no—
298#125—24no—
299C22⋊Q8—22no—
300C22⋊Q8—22no—
301C22⋊Q8—24no—
302C22⋊Q8—24no—
303C22⋊Q8—22no—
304C22⋊Q8—22no—
305C22⋊Q8—22no—
306C22⋊Q8—22no—
307C22⋊Q8—22no—
308C22⋊Q8—22no—
309C22⋊Q8—24no—
310C22⋊Q8—24no—
311C22⋊Q8—684no—
312C4×Q8—22no—
313C4×Q8—22no—
314C4×D4—22no—
315C4×D4—22no—
316C4×D4—24no—
317C4×D4—24no—
318#117—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—341no—
322C2×C4⋊C4—342no—
323C2×C4⋊C4—348no—
324C2×C4⋊C4C22⋊D1711yescharacteristic
325C2×C4⋊C4C22⋊D1712yes—
326C2×C4⋊C4C22⋊D1714yes—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—174no—
329C2×C22⋊C4—172no—
330C2×C22⋊C4—342no—
331C2×C22⋊C4C22⋊D1711yescharacteristic
332C2×C22⋊C4C22×D1711yescharacteristic
333C2×C22⋊C4C22×D1714yes—
334C2×C42—342no—
335C2×C42C22⋊D1711yescharacteristic
336#99—178no—
337#95—342no—
338#95—341no—
339#95—342no—
340#95—348no—
341#95—342no—
342C17#2585211yescharacteristic, complemented, Sylow
343C24C2×C22⋊D1711yescharacteristic
344C24C23×D1711yescharacteristic
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—44no—
350C2×Q8—82no—
351C2×Q8—344no—
352C2×Q8—348no—
353C2×Q8—684no—
354C2×Q8—682no—
355C2×Q8—684no—
356C2×Q8—688no—
357C2×Q8—681no—
358C2×Q8—682no—
359C2×Q8—681no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4Q8×D1714yes—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—344no—
370C22×C4—342no—
371C22×C4—344no—
372C22×C4—342no—
373C22×C4—684no—
374C22×C4C2×C22⋊D1711yescharacteristic
375C22×C4C2×C22⋊D1712yes—
376C22×C4C2×C22⋊D1711yescharacteristic
377C22×C4C2×C22⋊D1711yescharacteristic
378C22×C4C2×C22⋊D1714yes—
379C22×C4C2×C22⋊D1714yes—
380C22×C4C2×C22⋊D1711yescharacteristic
381C22×C4C2×C22⋊D1711yescharacteristic
382C22×C4C2×C22⋊D1712yes—
383C4⋊C4—22no—
384C4⋊C4—24no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—24no—
388C4⋊C4—22no—
389C4⋊C4—24no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—24no—
394C4⋊C4—682no—
395C4⋊C4—684no—
396C4⋊C4—6816no—
397C22⋊C4—24no—
398C22⋊C4—24no—
399C22⋊C4—22no—
400C22⋊C4—24no—
401C22⋊C4—22no—
402C22⋊C4—178no—
403C22⋊C4—344no—
404C22⋊C4—344no—
405C22⋊C4—684no—
406C42—24no—
407C42—21no—
408C42—21no—
409C42—684no—
410C23—21no—
411C23—21no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23#1058763012yescomplemented
417C23C2×Q8×D1712yes—
418C23#1058769412yes—
419C23C22×C22⋊D1712yes—
420C23C22×C22⋊D1711yescharacteristic
421Q8—44no—
422Q8—82no—
423Q8—82no—
424Q8—82no—
425Q8—684no—
426Q8—688no—
427Q8—684no—
428D4—28no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—21no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—41no—
444C2×C4—81no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—344no—
448C2×C4—348no—
449C2×C4—348no—
450C2×C4—684no—
451C2×C4—684no—
452C2×C4—682no—
453C2×C4—684no—
454C2×C4—682no—
455C2×C4—682no—
456C2×C4—682no—
457C2×C4—682no—
458C2×C4#1058763012yescomplemented
459C2×C4#1058763911yescharacteristic
460C2×C4#1058763912yes—
461C2×C4#1058763911yescharacteristic
462C22—22no—
463C22—21no—
464C22—22no—
465C22—21no—
466C22—22no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—41no—
471C22—41no—
472C22—81no—
473C22C2×#1058763911yescharacteristic
474C22C2×#1058763011yescharacteristic
475C22#1176338311yescharacteristic
476C22Q8×C22⋊D1714yes—
477C4—22no—
478C4—21no—
479C4—22no—
480C4—41no—
481C4—44no—
482C4—44no—
483C4—41no—
484C4—688no—
485C4—684no—
486C4—1362no—
487C4#1176299612yes—
488C2—24no—
489C2—21no—
490C2—21no—
491C2—42no—
492C2#1453242411yescharacteristic
493C2C2×#1176299611yescharacteristic
494C2#1453819011yescharacteristic
subgroupG/H
centerC22C2×#10587639
commutator, cosocleC24×C17C24
FrattiniC24C23×D17
FittingC17×#3501C2
socleC22×C17C2×#120
prime radicalC22×D4×C17C23
hypercenter#3501D17
hypocenterC17#25852
LCSG ▷ C24×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. #14532429C2semidirect
3. #14530136C2semidirect
4. #14529770C2semidirect
89. #10587505Q8semidirect
157. C22⋊Dic17C2×Q8semidirect
169. C23×C17C4⋊Q8semidirect
200. C2×C4×C17C4⋊Q8semidirect
342. C17#25852semidirect
416. C23#10587630semidirect
458. C2×C4#10587630semidirect

Maximal subgroups

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

subgroupindex
#145324292
#145321492
#145301362
#145297702
#145290652
C4×#105875052
C17×#35012
#2585217