Finite Group Databasebeta

← #17056169

Subgroups — #17056169

Order 4352.

846 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763109C211yesmaximal, characteristic
2#14531878C211yesmaximal, characteristic, complemented
3#14529748C211yesmaximal, characteristic, complemented
4#14529550C211yesmaximal, characteristic, complemented
5#14529514C211yesmaximal, characteristic, complemented
6#14529509C211yesmaximal, characteristic, complemented
7#14529373C211yesmaximal, characteristic, complemented
8#14529366C211yesmaximal, characteristic, complemented
9#14529045C211yesmaximal, characteristic
10#14529031C211yesmaximal, characteristic
11C4⋊C4×#3622C211yesmaximal, characteristic, complemented
12C4×#10587503C211yesmaximal, characteristic
13C4×#10587503C211yesmaximal, characteristic
14C4×#10587503C211yesmaximal, characteristic
15C17×#2349C211yesmaximal, characteristic
16C2×#10587556C2211yescharacteristic
17#11763109C418yescomplemented
18C2×#10587478C2211yescharacteristic
19C2×#10587478C2211yescharacteristic
20C2×#10587479C2211yescharacteristic
21C2×#10587479C2211yescharacteristic
22C2×C4⋊#3622C2211yescharacteristic
23C2×#10587503C2211yescharacteristic
24C2×#10587503C2211yescharacteristic
25C2×#10587503C2211yescharacteristic
26C2×#10587503C2211yescharacteristic
27C2×C4×#3622C2211yescharacteristic
28C2×C4×#3622C2211yescharacteristic
29C2×C4×#3622C2211yescharacteristic
30#11762707C2211yescharacteristic
31#11762707C2211yescharacteristic
32#11762704—21no—
33#11762691—21no—
34#11762690—21no—
35C4×C4⋊Dic17C2211yescharacteristic
36C4×C4⋊Dic17C2211yescharacteristic
37C4×C4⋊Dic17C2211yescharacteristic
38#11762660—21no—
39#11762571C2211yescharacteristic
40#11762571C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762564C2211yescharacteristic
48#11762564C2211yescharacteristic
49C2×C4×D4×C17C2211yescharacteristic
50C2×C4×D4×C17C2211yescharacteristic
51C2×C4×D4×C17C2211yescharacteristic
52C17×#389C2211yescharacteristic
53C17×#389C2211yescharacteristic
54C17×#385—21no—
55C17×#385—21no—
56C17×#382C2211yescharacteristic
57C17×#380—21no—
58C4×C4⋊C4×C17C2211yescharacteristic
59C2×C17×#95—21no—
60C22×D4×C17C2311yescharacteristic
61C23×C4×C17—21no—
62C23×C4×C17—21no—
63C23×C4×C17—21no—
64C4×D4×C17—22no—
65C4×D4×C17—22no—
66C4×D4×C17—22no—
67C4×D4×C17—41no—
68C4×D4×C17—41no—
69C4×D4×C17C2×C414yes—
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17D411yescharacteristic
73C2×C4⋊C4×C17D411yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C4⋊C4×C17C2311yescharacteristic
77C2×C22⋊C4×C17—21no—
78C2×C22⋊C4×C17—21no—
79C2×C22⋊C4×C17—21no—
80C2×C42×C17C2311yescharacteristic
81C2×C42×C17C2311yescharacteristic
82C2×C42×C17C2311yescharacteristic
83C2×C4⋊Dic17C2311yescharacteristic
84C2×C4⋊Dic17C2311yescharacteristic
85C2×C4⋊Dic17C2311yescharacteristic
86C2×C4⋊Dic17C2311yescharacteristic
87C2×#63225—21no—
88C2×#63225—21no—
89C2×#63225—21no—
90C2×#63225—21no—
91C2×C4×Dic17—21no—
92C2×C4×Dic17—21no—
93C2×C4×Dic17—21no—
94C22×#3622C2311yescharacteristic
95C22×#3622C2311yescharacteristic
96C22×#3622C2311yescharacteristic
97C22×#3622C2311yescharacteristic
98#10587556—22no—
99#10587556C2×C414yescomplemented
100C17×#95—21no—
101C17×#95—21no—
102C17×#95—21no—
103C17×#95—21no—
104C17×#95D411yescharacteristic
105C17×#95D411yescharacteristic
106#10587503—22no—
107#10587503C2×C414yes—
108#10587503C2×C414yes—
109#10587503Q814yes—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587502—21no—
115#10587479—41no—
116#10587479C2×C414yescomplemented
117#10587478—41no—
118#10587478C2×C414yescomplemented
119C4⋊#3622C2×C414yescomplemented
120C4×#3622—22no—
121C4×#3622—41no—
122C4×#3622—41no—
123C24×C17—21no—
124C2×D4×C17—21no—
125C2×D4×C17—21no—
126C2×D4×C17—21no—
127C2×D4×C17C22×C412yes—
128C2×D4×C17C2×Q812yes—
129C2×D4×C17#4112yes—
130C22×C4×C17—21no—
131C22×C4×C17—24no—
132C22×C4×C17—21no—
133C22×C4×C17—21no—
134C22×C4×C17—42no—
135C22×C4×C17—42no—
136C22×C4×C17—42no—
137C22×C4×C17—42no—
138C22×C4×C17—81no—
139C22×C4×C17—81no—
140C22×C4×C17—81no—
141C22×C4×C17—81no—
142C22×C4×C17C2×D411yescharacteristic
143C22×C4×C17C2×D411yescharacteristic
144C22×C4×C17C2×D411yescharacteristic
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17C2411yescharacteristic
149C22×Dic17—21no—
150C22×Dic17—21no—
151C22×Dic17—21no—
152C22×Dic17—21no—
153C4⋊C4×C17—22no—
154C4⋊C4×C17—22no—
155C4⋊C4×C17—21no—
156C4⋊C4×C17—21no—
157C4⋊C4×C17—21no—
158C4⋊C4×C17—21no—
159C4⋊C4×C17—21no—
160C4⋊C4×C17—81no—
161C4⋊C4×C17—81no—
162C4⋊C4×C17C22×C412yes—
163C22⋊C4×C17—24no—
164C22⋊C4×C17—42no—
165C22⋊C4×C17—42no—
166C42×C17—21no—
167C42×C17—21no—
168C42×C17—21no—
169C42×C17—21no—
170C42×C17C22×C412yes—
171C42×C17#4112yes—
172C4⋊Dic17—21no—
173C4⋊Dic17—21no—
174C4⋊Dic17—21no—
175C4⋊Dic17—21no—
176C4⋊Dic17—21no—
177C4⋊Dic17C22×C412yes—
178C4⋊Dic17C22×C412yes—
179C4⋊Dic17C2×Q812yes—
180#63225—24no—
181#63225—24no—
182#63225—42no—
183#63225—81no—
184C4×Dic17—42no—
185C4×Dic17—81no—
186C4×Dic17—81no—
187C2×#3622—21no—
188C2×#3622—21no—
189C2×#3622—21no—
190C2×#3622—21no—
191C2×#3622—21no—
192C2×#3622—21no—
193C2×#3622—21no—
194C2×#3622—21no—
195C2×#3622—21no—
196C2×#3622C22×C412yes—
197C2×#3622C22×C412yes—
198C2×#3622C2×Q812yes—
199#22966—171nomaximal, Sylow
200C23×C17—22no—
201C23×C17—22no—
202C23×C17—22no—
203C23×C17—41no—
204C23×C17—41no—
205C23×C17—41no—
206C23×C17—41no—
207C23×C17C22×D411yescharacteristic
208D4×C17—22no—
209D4×C17—22no—
210D4×C17—22no—
211D4×C17C4×Q814yes—
212C2×C4×C17—21no—
213C2×C4×C17—21no—
214C2×C4×C17—21no—
215C2×C4×C17—21no—
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17—21no—
219C2×C4×C17—21no—
220C2×C4×C17—22no—
221C2×C4×C17—21no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—21no—
226C2×C4×C17—21no—
227C2×C4×C17—41no—
228C2×C4×C17—41no—
229C2×C4×C17—41no—
230C2×C4×C17—41no—
231C2×C4×C17—41no—
232C2×C4×C17—41no—
233C2×C4×C17—41no—
234C2×C4×C17—41no—
235C2×C4×C17—84no—
236C2×C4×C17—84no—
237C2×C4×C17—84no—
238C2×C4×C17C4×D412yes—
239C2×C4×C17C4×D412yes—
240C2×C4×C17#13512yes—
241C2×C4×C17#13512yes—
242C2×C4×C17C23×C411yescharacteristic
243C2×C4×C17C22×Q811yescharacteristic
244C2×C4×C17C2×#4111yescharacteristic
245C2×C4×C17C2×#4111yescharacteristic
246C2×C4×C17H(5,𝔽2)11yescharacteristic
247C2×C4×C172−1+411yescharacteristic
248C2×Dic17—22no—
249C2×Dic17—22no—
250C2×Dic17—22no—
251C2×Dic17—41no—
252C2×Dic17—41no—
253C2×Dic17—41no—
254C2×Dic17—41no—
255C2×Dic17—41no—
256C2×Dic17—41no—
257C2×Dic17—41no—
258C2×Dic17—41no—
259C2×Dic17—41no—
260#3622—22no—
261#3622—22no—
262#3622—22no—
263#3622—41no—
264C2×#478—171no—
265#2370—171no—
266#2349D1711yescharacteristic
267#1991—171no—
268#1949—171no—
269#1942—171no—
270#1941—171no—
271#1921—171no—
272#1920—171no—
273#1864—171no—
274#1862—171no—
275C8×C4⋊C4—171no—
276C4×#102—171no—
277C4×#102—171no—
278C4×#102—171no—
279C22×C17—24no—
280C22×C17—42no—
281C22×C17—42no—
282C22×C17—42no—
283C22×C17—42no—
284C22×C17—42no—
285C22×C17—42no—
286C22×C17C2×C4×D411yescharacteristic
287C22×C17#54611yescharacteristic
288C22×C17D4×Q811yescharacteristic
289C22×C17C2×#13511yescharacteristic
290C22×C17C2×#13511yescharacteristic
291C22×C17#57611yescharacteristic
292C22×C17#57711yescharacteristic
293C4×C17—22no—
294C4×C17—22no—
295C4×C17—22no—
296C4×C17—22no—
297C4×C17—42no—
298C4×C17—41no—
299C4×C17—41no—
300C4×C17—81no—
301C4×C17—81no—
302C4×C17C2×C4×Q811yescharacteristic
303C4×C17#52011yescharacteristic
304C4×C17#55211yescharacteristic
305C4×C17#55411yescharacteristic
306Dic17—42no—
307Dic17—42no—
308Dic17—42no—
309Dic17—81no—
310C2×#125—171no—
311C2×C4×D4C2×D1711yescharacteristic
312C2×C4×D4C2×D1711yescharacteristic
313C2×C4×D4C2×D1711yescharacteristic
314#478—178no—
315C2×#107—171no—
316C2×#107—171no—
317C2×#106—171no—
318C2×#106—171no—
319C2×C4⋊C8—171no—
320C2×#102—171no—
321C2×#102—171no—
322C2×#102—171no—
323C2×#102—171no—
324C2×C4×C8—171no—
325C2×C4×C8—171no—
326C2×C4×C8—171no—
327#389—171no—
328#389—171no—
329#389—341no—
330#389C2×D1711yescharacteristic
331#389C2×D1711yescharacteristic
332#387—341no—
333#387—341no—
334#385—21no—
335#385—21no—
336#382C2×D1711yescharacteristic
337#380—21no—
338C4×C4⋊C4—171no—
339C4×C4⋊C4—171no—
340C4×C4⋊C4—171no—
341C4×C4⋊C4—341no—
342C4×C4⋊C4C2×D1711yescharacteristic
343C2×#95—21no—
344#340—171no—
345#340—171no—
346#340—171no—
347#340—171no—
348#340—171no—
349#340—171no—
350#340—171no—
351#340—171no—
352#336—171no—
353#336—171no—
354C2×C17—44no—
355C2×C17#236011yescharacteristic
356C2×C17C4×#13511yescharacteristic
357C2×C17#294611yescharacteristic
358C2×C17#329611yescharacteristic
359C2×C17#329711yescharacteristic
360C2×C17#338011yescharacteristic
361C2×C17#338111yescharacteristic
362C22×D4C22×D1711yescharacteristic
363C23×C4—21no—
364C23×C4—21no—
365C23×C4—21no—
366C22×C8—171no—
367C22×C8—171no—
368C22×C8—171no—
369C22×C8—171no—
370#125—174no—
371#125—342no—
372C4×D4—22no—
373C4×D4—22no—
374C4×D4—22no—
375C4×D4—41no—
376C4×D4—41no—
377C4×D4C4×D1714yes—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—171no—
381C2×C4⋊C4—171no—
382C2×C4⋊C4—171no—
383C2×C4⋊C4—171no—
384C2×C4⋊C4—341no—
385C2×C4⋊C4—341no—
386C2×C4⋊C4—341no—
387C2×C4⋊C4—341no—
388C2×C4⋊C4C22⋊D1711yescharacteristic
389C2×C4⋊C4C22⋊D1711yescharacteristic
390C2×C4⋊C4C22×D1711yescharacteristic
391C2×C4⋊C4C22×D1711yescharacteristic
392C2×C4⋊C4C22×D1711yescharacteristic
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C42—341no—
397C2×C42—341no—
398C2×C42—341no—
399C2×C42C22×D1711yescharacteristic
400C2×C42C22×D1711yescharacteristic
401C2×C42C22×D1711yescharacteristic
402#107—174no—
403#107—681no—
404#106—174no—
405#106—681no—
406C4⋊C8—174no—
407#102—174no—
408#102—174no—
409#102—174no—
410#102—342no—
411C4×C8—342no—
412C4×C8—681no—
413C4×C8—681no—
414#95—21no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—341no—
419#95—341no—
420#95—341no—
421#95—341no—
422#95—341no—
423#95C22⋊D1711yescharacteristic
424#95C22⋊D1711yescharacteristic
425C17#2296611yescharacteristic, complemented, Sylow
426C24—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4C2×C4×D1712yes—
431C2×D4#6325412yes—
432C2×D4Q8×D1712yes—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—81no—
442C22×C4—81no—
443C22×C4—81no—
444C22×C4—81no—
445C22×C4—341no—
446C22×C4—341no—
447C22×C4—341no—
448C22×C4—341no—
449C22×C4C2×C22⋊D1711yescharacteristic
450C22×C4C2×C22⋊D1711yescharacteristic
451C22×C4C2×C22⋊D1711yescharacteristic
452C22×C4C2×C22⋊D1711yescharacteristic
453C22×C4C2×C22⋊D1711yescharacteristic
454C22×C4C2×C22⋊D1711yescharacteristic
455C22×C4C23×D1711yescharacteristic
456C2×C8—172no—
457C2×C8—172no—
458C2×C8—172no—
459C2×C8—341no—
460C2×C8—341no—
461C2×C8—341no—
462C2×C8—341no—
463C2×C8—341no—
464C2×C8—341no—
465C2×C8—341no—
466C2×C8—341no—
467C2×C8—341no—
468C4⋊C4—21no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—81no—
476C4⋊C4—81no—
477C4⋊C4—172no—
478C4⋊C4—172no—
479C4⋊C4—172no—
480C4⋊C4—341no—
481C4⋊C4—341no—
482C4⋊C4—341no—
483C4⋊C4—344no—
484C4⋊C4—344no—
485C4⋊C4—341no—
486C4⋊C4—341no—
487C4⋊C4—682no—
488C4⋊C4—1361no—
489C4⋊C4C2×C4×D1712yes—
490C22⋊C4—24no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—682no—
498C42—1361no—
499C42—1361no—
500C42C2×C4×D1712yes—
501C42#6325212yes—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23C22×C22⋊D1711yescharacteristic
510D4—22no—
511D4—22no—
512D4—22no—
513D4#1058757314yes—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—84no—
538C2×C4—84no—
539C2×C4—84no—
540C2×C4—342no—
541C2×C4—342no—
542C2×C4—342no—
543C2×C4—681no—
544C2×C4—681no—
545C2×C4—681no—
546C2×C4—681no—
547C2×C4—681no—
548C2×C4—681no—
549C2×C4—681no—
550C2×C4—681no—
551C2×C4—681no—
552C2×C4C4×C22⋊D1712yes—
553C2×C4C4×C22⋊D1712yes—
554C2×C4#1058763612yes—
555C2×C4#1058763612yes—
556C2×C4C22×C4×D1711yescharacteristic
557C2×C4C2×#6325211yescharacteristic
558C2×C4C2×#6325411yescharacteristic
559C2×C4#1058769111yescharacteristic
560C2×C4C2×Q8×D1711yescharacteristic
561C2×C4#1058769411yescharacteristic
562C8—342no—
563C8—342no—
564C8—342no—
565C8—681no—
566C22—24no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22C2×C4×C22⋊D1711yescharacteristic
574C22#1176345311yescharacteristic
575C22Q8×C22⋊D1711yescharacteristic
576C22C2×#1058763611yescharacteristic
577C22C2×#1058763611yescharacteristic
578C22#1176361211yescharacteristic
579C22#1176362011yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—41no—
585C4—41no—
586C4—42no—
587C4—81no—
588C4—81no—
589C4—682no—
590C4—682no—
591C4—682no—
592C4—1361no—
593C4C2×#1058757311yescharacteristic
594C4#1176329011yescharacteristic
595C4#1176349411yescharacteristic
596C4#1176351411yescharacteristic
597C2—44no—
598C2#1453178911yescharacteristic
599C2C4×#1058763611yescharacteristic
600C2#1453761611yescharacteristic
601C2#1454061911yescharacteristic
602C2#1454063411yescharacteristic
603C2#1454164711yescharacteristic
604C2#1454165911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#2349C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
cosocleC22×C4×C17C24
hypercenter#2349D17
hypocenterC17#22966
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
2. #14531878C2semidirect
3. #14529748C2semidirect
4. #14529550C2semidirect
5. #14529514C2semidirect
6. #14529509C2semidirect
7. #14529373C2semidirect
8. #14529366C2semidirect
11. C4⋊C4×#3622C2semidirect
17. #11763109C4semidirect
99. #10587556C2×C4semidirect
116. #10587479C2×C4semidirect
118. #10587478C2×C4semidirect
119. C4⋊#3622C2×C4semidirect
425. C17#22966semidirect

Maximal subgroups

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

subgroupindex
C2×#117631092
#145318782
#145297482
#145295502
#145295142
#145295092
#145293732
#145293662
#145290452
#145290312
C4⋊C4×#36222
C4×#105875032
C17×#23492
#2296617