Finite Group Databasebeta

← #16959815

Subgroups — #16959815

Order 4352.

1786 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic34C211yesmaximal, characteristic
2C2×#11762729C211yesmaximal, characteristic
3C2×#11762704C212yesmaximal
4#14527054C211yesmaximal, characteristic
5#14527048C211yesmaximal, characteristic
6#14527037C211yesmaximal, characteristic
7#14527019C212yesmaximal
8#14527015C212yesmaximal
9#14527015C212yesmaximal
10C4×#10587502C211yesmaximal, characteristic
11C2×C17×#387C211yesmaximal, characteristic
12C23×Dic34C2211yescharacteristic
13C2×C4×Dic34—24no—
14C2×C4×Dic34—28no—
15C22×C4⋊Dic17C2211yescharacteristic
16C22×#63225C2212yes—
17C22×#63225C2212yes—
18C22×#63225C2212yes—
19C22×C4×Dic17C2212yes—
20C22×C4×Dic17C2212yes—
21#11762729C418yescomplemented
22#11762704—28no—
23C2×#10587502C2211yescharacteristic
24C2×#10587502C2212yes—
25C2×#10587502C2212yes—
26C2×#10587502C2212yes—
27C2×#10587502C2212yes—
28C2×#10587502C2212yes—
29C2×#10587502C2211yescharacteristic
30C2×#10587502C2212yes—
31C2×#10587502C2212yes—
32C22×C4⋊C4×C17C2211yescharacteristic
33C22×C4⋊C4×C17C2211yescharacteristic
34C22×C42×C17C2211yescharacteristic
35C17×#387—24no—
36C2×C17×#95C2211yescharacteristic
37C2×C17×#95C2211yescharacteristic
38C2×C17×#95C2211yescharacteristic
39C2×C17×#95C2211yescharacteristic
40C23×C4×C17C2311yescharacteristic
41C23×C4×C17C2311yescharacteristic
42C23×C4×C17C2311yescharacteristic
43C23×C4×C17C2311yescharacteristic
44C23×C4×C17C2311yescharacteristic
45C23×C4×C17C2311yescharacteristic
46C23×C4×C17C2311yescharacteristic
47C23×Dic17C2312yes—
48C23×Dic17C2312yes—
49C23×Dic17C2312yes—
50C23×Dic17C2312yes—
51C22×Dic34—22no—
52C22×Dic34—22no—
53C22×Dic34—24no—
54C22×Dic34—24no—
55C22×Dic34C2×C414yes—
56C2×C4⋊C4×C17—22no—
57C2×C4⋊C4×C17—22no—
58C2×C4⋊C4×C17—24no—
59C2×C4⋊C4×C17—22no—
60C2×C4⋊C4×C17C2×C414yes—
61C2×C42×C17—24no—
62C2×C42×C17D414yes—
63C2×C4⋊Dic17—22no—
64C2×C4⋊Dic17—24no—
65C2×#63225—24no—
66C2×#63225—24no—
67C2×#63225—24no—
68C2×#63225—24no—
69C2×#63225—24no—
70C2×#63225—24no—
71C2×#63225—42no—
72C2×#63225C2×C418yes—
73C2×#63225D418yes—
74C2×C4×Dic17—24no—
75C2×C4×Dic17—24no—
76C2×C4×Dic17—24no—
77C2×C4×Dic17—24no—
78C2×C4×Dic17—24no—
79C2×C4×Dic17—42no—
80C4×Dic34—232no—
81C17×#95—22no—
82C17×#95—22no—
83C17×#95—22no—
84C17×#95—22no—
85#10587502—22no—
86#10587502—24no—
87#10587502—24no—
88#10587502—24no—
89#10587502—24no—
90#10587502—42no—
91#10587502C2×C418yes—
92#10587502C2×C414yes—
93#10587502Q818yescomplemented
94C24×C17C2411yescharacteristic
95C22×C4×C17—22no—
96C22×C4×C17—22no—
97C22×C4×C17—22no—
98C22×C4×C17—22no—
99C22×C4×C17—21no—
100C22×C4×C17—21no—
101C22×C4×C17—21no—
102C22×C4×C17—21no—
103C22×C4×C17—22no—
104C22×C4×C17—21no—
105C22×C4×C17—21no—
106C22×C4×C17—22no—
107C22×C4×C17—21no—
108C22×C4×C17—21no—
109C22×C4×C17—21no—
110C22×C4×C17—21no—
111C22×C4×C17—21no—
112C22×C4×C17—21no—
113C22×C4×C17—22no—
114C22×C4×C17—21no—
115C22×C4×C17—21no—
116C22×C4×C17—22no—
117C22×C4×C17—22no—
118C22×C4×C17—22no—
119C22×C4×C17—21no—
120C22×C4×C17—21no—
121C22×C4×C17—21no—
122C22×C4×C17—21no—
123C22×C4×C17C22×C412yes—
124C22×C4×C17C22×C412yes—
125C22×C4×C17C22×C412yes—
126C22×C4×C17C2×D414yes—
127C22×C4×C17C2×D412yes—
128C22×C4×C17C2×D412yes—
129C22×C4×C17C2×D412yes—
130C22×C4×C17C2×Q814yes—
131C22×C4×C17#4112yes—
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—22no—
143C22×Dic17—22no—
144C22×Dic17—22no—
145C22×Dic17—22no—
146C22×Dic17—22no—
147C22×Dic17—22no—
148C22×Dic17—22no—
149C22×Dic17—22no—
150C22×Dic17—22no—
151C22×Dic17—22no—
152C22×Dic17C22×C414yes—
153C22×Dic17C22×C414yes—
154C22×Dic17C2×D414yes—
155C22×Dic17C2×D414yes—
156C22×Dic17C2×Q814yes—
157C22×Dic17C2×Q814yes—
158C22×Dic17#4114yes—
159C22×Dic17#4114yes—
160C2×Dic34—216no—
161C2×Dic34—28no—
162C2×Dic34—28no—
163C2×Dic34—28no—
164C2×Dic34—216no—
165C4⋊C4×C17—28no—
166C4⋊C4×C17—44no—
167C42×C17—28no—
168C4⋊Dic17—28no—
169#63225—216no—
170#63225—216no—
171#63225—48no—
172C4×Dic17—216no—
173C4×Dic17—48no—
174#14186—171nomaximal, Sylow
175C23×C17C23×C411yescharacteristic
176C23×C17C22×D412yes—
177C23×C17C22×D411yescharacteristic
178C23×C17C22×Q812yes—
179C23×C17C2×#4111yescharacteristic
180C23×C17C2×#4112yes—
181C23×C17C2×#4112yes—
182C23×C17H(5,𝔽2)11yescharacteristic
183C23×C172−1+411yescharacteristic
184C23×C172−1+411yescharacteristic
185C23×C172−1+411yescharacteristic
186C2×C4×C17—24no—
187C2×C4×C17—22no—
188C2×C4×C17—22no—
189C2×C4×C17—22no—
190C2×C4×C17—24no—
191C2×C4×C17—22no—
192C2×C4×C17—22no—
193C2×C4×C17—22no—
194C2×C4×C17—24no—
195C2×C4×C17—24no—
196C2×C4×C17—24no—
197C2×C4×C17—24no—
198C2×C4×C17—22no—
199C2×C4×C17—22no—
200C2×C4×C17—24no—
201C2×C4×C17—22no—
202C2×C4×C17—22no—
203C2×C4×C17—41no—
204C2×C4×C17—41no—
205C2×C4×C17—42no—
206C2×C4×C17—41no—
207C2×C4×C17—41no—
208C2×C4×C17—41no—
209C2×C4×C17—41no—
210C2×C4×C17—41no—
211C2×C4×C17—41no—
212C2×C4×C17—41no—
213C2×C4×C17—41no—
214C2×C4×C17—41no—
215C2×C4×C17—41no—
216C2×C4×C17—42no—
217C2×C4×C17—42no—
218C2×C4×C17—42no—
219C2×C4×C17—42no—
220C2×C4×C17C4×D414yes—
221C2×C4×C17#12014yes—
222C2×C4×C17C4⋊Q814yes—
223C2×Dic17—24no—
224C2×Dic17—24no—
225C2×Dic17—24no—
226C2×Dic17—24no—
227C2×Dic17—24no—
228C2×Dic17—24no—
229C2×Dic17—24no—
230C2×Dic17—24no—
231C2×Dic17—24no—
232C2×Dic17—24no—
233C2×Dic17—24no—
234C2×Dic17—24no—
235C2×Dic17—24no—
236C2×Dic17—24no—
237C2×Dic17—24no—
238C2×Dic17—24no—
239C2×Dic17—24no—
240C2×Dic17—24no—
241C2×Dic17—42no—
242C2×Dic17—42no—
243C2×Dic17—42no—
244C2×Dic17—42no—
245C2×Dic17—42no—
246C2×Dic17—42no—
247C2×Dic17—42no—
248C2×Dic17—42no—
249C2×Dic17C4×D418yes—
250C2×Dic17C4×Q818yescomplemented
251Dic34—232no—
252C22×C4×Q8—171no—
253C2×#395—171no—
254C2×#389—172no—
255C2×#387D1711yescharacteristic
256#1446—171no—
257#1444—171no—
258#1441—171no—
259#1436—172no—
260#1436—172no—
261#1436—172no—
262C4×#95—171no—
263C22×C17C2×C4×D412yes—
264C22×C17C2×C4×D411yescharacteristic
265C22×C17C2×C4×Q812yes—
266C22×C17#51911yescharacteristic
267C22×C17#52011yescharacteristic
268C22×C17C2×#12011yescharacteristic
269C22×C17#52511yescharacteristic
270C22×C17#52512yes—
271C22×C17C2×C4⋊Q811yescharacteristic
272C22×C17#53211yescharacteristic
273C22×C17#53312yes—
274C22×C17#53611yescharacteristic
275C22×C17#53711yescharacteristic
276C22×C17#54111yescharacteristic
277C22×C17#54211yescharacteristic
278C22×C17#54612yes—
279C22×C17#54712yes—
280C22×C17D4×Q812yes—
281C22×C17D4×Q812yes—
282C22×C17#55212yes—
283C22×C17#55412yes—
284C22×C17#55612yes—
285C22×C17#55712yes—
286C4×C17—24no—
287C4×C17—24no—
288C4×C17—24no—
289C4×C17—42no—
290C4×C17—42no—
291C4×C17—81no—
292C4×C17—81no—
293Dic17—28no—
294Dic17—28no—
295Dic17—44no—
296Dic17—44no—
297C23×Q8—171no—
298C2×C4×Q8—348no—
299C2×C4×Q8—344no—
300C22×C4⋊C4—172no—
301C22×C4⋊C4—172no—
302C22×C4⋊C4—171no—
303C22×C4⋊C4—172no—
304C22×C4⋊C4C2×D1711yescharacteristic
305C22×C4⋊C4C2×D1711yescharacteristic
306C22×C42—172no—
307C22×C42—172no—
308C22×C42C2×D1711yescharacteristic
309#395—178no—
310#389—348no—
311#387—24no—
312C2×#95—171no—
313C2×#95—172no—
314C2×#95—172no—
315C2×#95—171no—
316C2×#95—172no—
317C2×#95—172no—
318C2×#95—172no—
319C2×#95—172no—
320C2×#95—172no—
321C2×#95C2×D1711yescharacteristic
322C2×#95C2×D1711yescharacteristic
323C2×#95C2×D1711yescharacteristic
324C2×#95C2×D1711yescharacteristic
325C2×C17C4×#12011yescharacteristic
326C2×C17C4×C4⋊Q811yescharacteristic
327C2×C17#232311yescharacteristic
328C2×C17#236012yes—
329C2×C17#236512yes—
330C2×C17#243011yescharacteristic
331C2×C17#243411yescharacteristic
332C2×C17#246212yes—
333C2×C17#246512yes—
334C2×C17#258411yescharacteristic
335C2×C17#260411yescharacteristic
336C22×Q8—174no—
337C22×Q8—344no—
338C22×Q8—344no—
339C22×Q8—342no—
340C22×Q8—342no—
341C23×C4—172no—
342C23×C4—172no—
343C23×C4—172no—
344C23×C4—172no—
345C23×C4C22×D1711yescharacteristic
346C23×C4C22×D1711yescharacteristic
347C23×C4C22×D1711yescharacteristic
348C23×C4C22×D1711yescharacteristic
349C23×C4C22×D1711yescharacteristic
350C23×C4C22×D1711yescharacteristic
351C23×C4C22×D1711yescharacteristic
352C4×Q8—3432no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—24no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—178no—
358C2×C4⋊C4—178no—
359C2×C4⋊C4—344no—
360C2×C4⋊C4—344no—
361C2×C4⋊C4—344no—
362C2×C4⋊C4—344no—
363C2×C4⋊C4—344no—
364C2×C4⋊C4—342no—
365C2×C4⋊C4—344no—
366C2×C4⋊C4—344no—
367C2×C4⋊C4—682no—
368C2×C4⋊C4C4×D1714yes—
369C2×C42—24no—
370C2×C42—344no—
371C2×C42—344no—
372C2×C42—344no—
373C2×C42—344no—
374C2×C42—344no—
375C2×C42—682no—
376C2×C42C22⋊D1714yes—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—174no—
382#95—178no—
383#95—178no—
384#95—344no—
385#95—344no—
386#95—344no—
387#95—344no—
388#95—342no—
389#95—682no—
390C17#1418611yescharacteristic, complemented, Sylow
391C24C23×D1711yescharacteristic
392C2×Q8—3416no—
393C2×Q8—348no—
394C2×Q8—348no—
395C2×Q8—3416no—
396C2×Q8—348no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—174no—
426C22×C4—174no—
427C22×C4—174no—
428C22×C4—174no—
429C22×C4—174no—
430C22×C4—174no—
431C22×C4—174no—
432C22×C4—174no—
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—342no—
442C22×C4—342no—
443C22×C4—342no—
444C22×C4—342no—
445C22×C4—342no—
446C22×C4—342no—
447C22×C4—342no—
448C22×C4—342no—
449C22×C4—342no—
450C22×C4—342no—
451C22×C4—342no—
452C22×C4—342no—
453C22×C4C2×C4×D1712yes—
454C22×C4C2×C4×D1712yes—
455C22×C4C2×C4×D1712yes—
456C22×C4#6325212yes—
457C22×C4D4×D1714yes—
458C22×C4Q8×D1714yes—
459C22×C4C2×C22⋊D1712yes—
460C22×C4C2×C22⋊D1712yes—
461C22×C4C2×C22⋊D1712yes—
462C4⋊C4—28no—
463C4⋊C4—44no—
464C4⋊C4—3416no—
465C4⋊C4—348no—
466C4⋊C4—3416no—
467C4⋊C4—688no—
468C42—28no—
469C42—3416no—
470C42—688no—
471C23C22×C4×D1711yescharacteristic
472C23C2×#6325211yescharacteristic
473C23C2×D4×D1712yes—
474C23C2×Q8×D1712yes—
475C23#41×D1712yes—
476C23#41×D1712yes—
477C23#1058769611yescharacteristic
478C23#1058769711yescharacteristic
479C23#1058769711yescharacteristic
480C23#1058769711yescharacteristic
481C23C22×C22⋊D1711yescharacteristic
482Q8—3432no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—178no—
518C2×C4—178no—
519C2×C4—344no—
520C2×C4—344no—
521C2×C4—344no—
522C2×C4—344no—
523C2×C4—344no—
524C2×C4—344no—
525C2×C4—344no—
526C2×C4—344no—
527C2×C4—344no—
528C2×C4—344no—
529C2×C4—344no—
530C2×C4—344no—
531C2×C4—344no—
532C2×C4—344no—
533C2×C4—344no—
534C2×C4—344no—
535C2×C4—344no—
536C2×C4—344no—
537C2×C4—682no—
538C2×C4—682no—
539C2×C4—682no—
540C2×C4—682no—
541C2×C4—682no—
542C2×C4—682no—
543C2×C4—682no—
544C2×C4—682no—
545C2×C4C4×C22⋊D1714yes—
546C2×C4#1058762314yes—
547C2×C4#1058763014yes—
548C22C4×D4×D1712yes—
549C22C2×C4×C22⋊D1711yescharacteristic
550C22C4×Q8×D1712yes—
551C22#1176328611yescharacteristic
552C22#1176328911yescharacteristic
553C22C2×#1058762311yescharacteristic
554C22#1176331611yescharacteristic
555C22#1176331812yes—
556C22C2×#1058763011yescharacteristic
557C22#1176335511yescharacteristic
558C22#1176336312yes—
559C22#1176338011yescharacteristic
560C22#1176338711yescharacteristic
561C22#1176342111yescharacteristic
562C22#1176343011yescharacteristic
563C22#1176345112yes—
564C22#1176345712yes—
565C22#1176347212yes—
566C22#1176347212yes—
567C22#1176349512yes—
568C22#1176351512yes—
569C22#1176352612yes—
570C22#1176353612yes—
571C4—24no—
572C4—24no—
573C4—24no—
574C4—42no—
575C4—42no—
576C4—81no—
577C4—81no—
578C4—348no—
579C4—348no—
580C4—684no—
581C4—684no—
582C2C4×#1058762311yescharacteristic
583C2C4×#1058763011yescharacteristic
584C2#1453146611yescharacteristic
585C2#1453179212yes—
586C2#1453183912yes—
587C2#1453239611yescharacteristic
588C2#1453242911yescharacteristic
589C2#1453276812yes—
590C2#1453280712yes—
591C2#1453417311yescharacteristic
592C2#1453442411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC2×C17×#387C2
socle, prime radical, cosocleC24×C17C24
hypercenterC2×#387D17
hypocenterC17#14186
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×#387

Complements

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

NH ≅ G/Ntype
21. #11762729C4semidirect
93. #10587502Q8semidirect
250. C2×Dic17C4×Q8semidirect
390. C17#14186semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic342
C2×#117627292
C2×#117627042
#145270542
#145270482
#145270372
#145270192
#145270152
C4×#105875022
C2×C17×#3872
#1418617