Finite Group Databasebeta

← #17056171

Subgroups — #17056171

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×#11763110C211yesmaximal, characteristic
2#14531878C211yesmaximal, characteristic, complemented
3#14531695C211yesmaximal, characteristic
4#14529749C211yesmaximal, characteristic, complemented
5#14529554C211yesmaximal, characteristic, complemented
6#14529517C211yesmaximal, characteristic, complemented
7#14529512C211yesmaximal, characteristic, complemented
8#14529377C211yesmaximal, characteristic, complemented
9#14529370C211yesmaximal, characteristic, complemented
10#14529048C211yesmaximal, characteristic
11#14529034C211yesmaximal, characteristic
12C4×#10587492C211yesmaximal, characteristic
13C4×#10587492C211yesmaximal, characteristic
14C4×#10587492C211yesmaximal, characteristic
15C8×C4⋊C4×C17C211yesmaximal, characteristic, complemented
16C2×#10587556C2211yescharacteristic
17C2×C4×D68C2211yescharacteristic
18C2×C4×D68C2211yescharacteristic
19C2×C4×D68C2211yescharacteristic
20#11763110C418yescomplemented
21C2×#10587489C2211yescharacteristic
22C2×#10587489C2211yescharacteristic
23C2×#10587488C2211yescharacteristic
24C2×#10587488C2211yescharacteristic
25C2×#10587492C2211yescharacteristic
26C2×#10587492C2211yescharacteristic
27C2×#10587492C2211yescharacteristic
28C2×#10587492C2211yescharacteristic
29#11762707C2211yescharacteristic
30#11762707C2211yescharacteristic
31#11762707C2211yescharacteristic
32#11762707C2211yescharacteristic
33#11762704—21no—
34#11762691—21no—
35#11762690—21no—
36#11762680—21no—
37#11762680—21no—
38#11762675C2211yescharacteristic
39#11762668—21no—
40C4×C4⋊Dic17C2211yescharacteristic
41C4×C4⋊Dic17C2211yescharacteristic
42C4×C4⋊Dic17C2211yescharacteristic
43#11762660—21no—
44#95×D17—21no—
45#11762572C2211yescharacteristic
46#11762572C2211yescharacteristic
47#11762572C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762572C2211yescharacteristic
50#11762572C2211yescharacteristic
51#11762572C2211yescharacteristic
52#11762572C2211yescharacteristic
53C2×C17×C4⋊C8C2211yescharacteristic
54C2×C4×C8×C17C2211yescharacteristic
55C2×C4×C8×C17C2211yescharacteristic
56C2×C4×C8×C17C2211yescharacteristic
57C4×C4⋊C4×C17C2211yescharacteristic
58C17×#336C2211yescharacteristic
59C17×#336C2211yescharacteristic
60C22×D68C2311yescharacteristic
61C22×C4×D17—21no—
62C22×C4×D17—21no—
63C22×C4×D17—21no—
64C22×C8×C17C2311yescharacteristic
65C22×C8×C17C2311yescharacteristic
66C22×C8×C17C2311yescharacteristic
67C22×C8×C17C2311yescharacteristic
68C2×C4⋊C4×C17D411yescharacteristic
69C2×C4⋊C4×C17D411yescharacteristic
70C2×C42×C17C2311yescharacteristic
71C2×C42×C17C2311yescharacteristic
72C2×C42×C17C2311yescharacteristic
73C2×#63227—21no—
74C2×#63227—21no—
75C2×#63227—21no—
76C2×C4⋊Dic17C2311yescharacteristic
77C2×C4⋊Dic17C2311yescharacteristic
78C2×C4⋊Dic17C2311yescharacteristic
79C2×C4⋊Dic17C2311yescharacteristic
80C2×C4⋊Dic17C2311yescharacteristic
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4⋊Dic17C2311yescharacteristic
83C2×#63225—21no—
84C2×#63225—21no—
85C2×#63225—21no—
86C2×#63225—21no—
87C2×#63225—21no—
88C2×#63225—21no—
89C2×C4×Dic17—21no—
90C2×C4×Dic17—21no—
91C2×C4×Dic17—21no—
92C4×D68—22no—
93C4×D68—22no—
94C4×D68—22no—
95C4×D68—41no—
96C4×D68—41no—
97C4×D68C2×C414yes—
98#10587556—22no—
99#10587556C2×C414yescomplemented
100C17×C4⋊C8C2×C414yescomplemented
101C4×C8×C17—22no—
102C4×C8×C17—41no—
103C4×C8×C17—41no—
104C17×#95D411yescharacteristic
105C17×#95D411yescharacteristic
106#10587502—21no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587502—21no—
115#10587492—22no—
116#10587492C2×C414yes—
117#10587492C2×C414yes—
118#10587492Q814yes—
119#10587489—41no—
120#10587489C2×C414yescomplemented
121#10587488—41no—
122#10587488C2×C414yescomplemented
123C23×D17—21no—
124C22×C4×C17C2×D411yescharacteristic
125C22×C4×C17C2×D411yescharacteristic
126C22×C4×C17C2×D411yescharacteristic
127C22×C4×C17C2×D411yescharacteristic
128C22×C4×C17C2×D411yescharacteristic
129C22×C4×C17C2×D411yescharacteristic
130C22×C4×C17C2411yescharacteristic
131C22×Dic17—21no—
132C22×Dic17—21no—
133C22×Dic17—21no—
134C22×Dic17—21no—
135C22×Dic17—21no—
136C22×Dic17—21no—
137C22×Dic17—21no—
138C2×D68—21no—
139C2×D68—21no—
140C2×D68—21no—
141C2×D68C22×C412yes—
142C2×D68C2×Q812yes—
143C2×D68#4112yes—
144C2×C4×D17—24no—
145C2×C4×D17—42no—
146C2×C4×D17—42no—
147C2×C4×D17—42no—
148C2×C4×D17—42no—
149C2×C4×D17—81no—
150C2×C4×D17—81no—
151C2×C4×D17—81no—
152C2×C4×D17—81no—
153C2×C8×C17—21no—
154C2×C8×C17—21no—
155C2×C8×C17—21no—
156C2×C8×C17—21no—
157C2×C8×C17—21no—
158C2×C8×C17—21no—
159C2×C8×C17—21no—
160C2×C8×C17—21no—
161C2×C8×C17—21no—
162C2×C8×C17C22×C412yes—
163C2×C8×C17C22×C412yes—
164C2×C8×C17C2×Q812yes—
165C4⋊C4×C17—22no—
166C4⋊C4×C17—22no—
167C42×C17—21no—
168C42×C17—21no—
169C42×C17—21no—
170C42×C17—21no—
171C42×C17C22×C412yes—
172C42×C17#4112yes—
173#63227—24no—
174#63227—42no—
175#63227—42no—
176C4⋊Dic17—21no—
177C4⋊Dic17—21no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184C4⋊Dic17—21no—
185C4⋊Dic17—21no—
186C4⋊Dic17C22×C412yes—
187C4⋊Dic17C22×C412yes—
188C4⋊Dic17C22×C412yes—
189C4⋊Dic17C2×Q812yes—
190#63225—24no—
191#63225—24no—
192#63225—42no—
193#63225—81no—
194#63225—81no—
195#63225—81no—
196C4×Dic17—42no—
197C4×Dic17—81no—
198C4×Dic17—81no—
199#22966—171nomaximal, Sylow
200C23×C17C22×D411yescharacteristic
201C22×D17—22no—
202C22×D17—22no—
203C22×D17—22no—
204C22×D17—41no—
205C22×D17—41no—
206C22×D17—41no—
207C22×D17—41no—
208C2×C4×C17—21no—
209C2×C4×C17—21no—
210C2×C4×C17—21no—
211C2×C4×C17—21no—
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—21no—
221C2×C4×C17—21no—
222C2×C4×C17C4×D412yes—
223C2×C4×C17C4×D412yes—
224C2×C4×C17#13512yes—
225C2×C4×C17#13512yes—
226C2×C4×C17C23×C411yescharacteristic
227C2×C4×C17C22×Q811yescharacteristic
228C2×C4×C17C2×#4111yescharacteristic
229C2×C4×C17C2×#4111yescharacteristic
230C2×C4×C17H(5,𝔽2)11yescharacteristic
231C2×C4×C172−1+411yescharacteristic
232C2×Dic17—22no—
233C2×Dic17—22no—
234C2×Dic17—22no—
235C2×Dic17—22no—
236C2×Dic17—41no—
237C2×Dic17—41no—
238C2×Dic17—41no—
239C2×Dic17—41no—
240C2×Dic17—41no—
241C2×Dic17—41no—
242C2×Dic17—41no—
243C2×Dic17—41no—
244C2×Dic17—41no—
245C2×Dic17—41no—
246C2×Dic17—41no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C2×Dic17—41no—
251C2×Dic17—41no—
252C2×Dic17—41no—
253D68—22no—
254D68—22no—
255D68—22no—
256D68C4×Q814yes—
257C4×D17—84no—
258C4×D17—84no—
259C4×D17—84no—
260C8×C17—22no—
261C8×C17—22no—
262C8×C17—22no—
263C8×C17—41no—
264C2×#478—171no—
265#2370—171no—
266#2349—171no—
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⋊C4D1711yescharacteristic, complemented
276C4×#102—171no—
277C4×#102—171no—
278C4×#102—171no—
279C22×C17C2×C4×D411yescharacteristic
280C22×C17#54611yescharacteristic
281C22×C17D4×Q811yescharacteristic
282C22×C17C2×#13511yescharacteristic
283C22×C17C2×#13511yescharacteristic
284C22×C17#57611yescharacteristic
285C22×C17#57711yescharacteristic
286C2×D17—24no—
287C2×D17—42no—
288C2×D17—42no—
289C2×D17—42no—
290C2×D17—42no—
291C2×D17—42no—
292C2×D17—42no—
293C4×C17—22no—
294C4×C17—22no—
295C4×C17—22no—
296C4×C17—22no—
297C4×C17—41no—
298C4×C17—41no—
299C4×C17C2×C4×Q811yescharacteristic
300C4×C17#52011yescharacteristic
301C4×C17#55211yescharacteristic
302C4×C17#55411yescharacteristic
303Dic17—42no—
304Dic17—42no—
305Dic17—42no—
306Dic17—42no—
307Dic17—81no—
308Dic17—81no—
309Dic17—81no—
310C2×#125—171no—
311C2×C4×D4—171no—
312C2×C4×D4—171no—
313C2×C4×D4—171no—
314#478—178no—
315C2×#107—171no—
316C2×#107—171no—
317C2×#106—171no—
318C2×#106—171no—
319C2×C4⋊C8C2×D1711yescharacteristic
320C2×#102—171no—
321C2×#102—171no—
322C2×#102—171no—
323C2×#102—171no—
324C2×C4×C8C2×D1711yescharacteristic
325C2×C4×C8C2×D1711yescharacteristic
326C2×C4×C8C2×D1711yescharacteristic
327#389—171no—
328#389—171no—
329#389—171no—
330#389—171no—
331#389—341no—
332#387—341no—
333#387—341no—
334#385—341no—
335#385—341no—
336#382—171no—
337#380—341no—
338C4×C4⋊C4—171no—
339C4×C4⋊C4—171no—
340C4×C4⋊C4—171no—
341C4×C4⋊C4—341no—
342C4×C4⋊C4C2×D1711yescharacteristic
343C2×#95—341no—
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#336C2×D1711yescharacteristic
353#336C2×D1711yescharacteristic
354C2×C17#236011yescharacteristic
355C2×C17C4×#13511yescharacteristic
356C2×C17#294611yescharacteristic
357C2×C17#329611yescharacteristic
358C2×C17#329711yescharacteristic
359C2×C17#338011yescharacteristic
360C2×C17#338111yescharacteristic
361D17—44no—
362C22×D4—171no—
363C23×C4—341no—
364C23×C4—341no—
365C23×C4—341no—
366C22×C8C22×D1711yescharacteristic
367C22×C8C22×D1711yescharacteristic
368C22×C8C22×D1711yescharacteristic
369C22×C8C22×D1711yescharacteristic
370#125—174no—
371#125—342no—
372C4×D4—174no—
373C4×D4—342no—
374C4×D4—342no—
375C4×D4—342no—
376C4×D4—681no—
377C4×D4—681no—
378C2×C4⋊C4—171no—
379C2×C4⋊C4—171no—
380C2×C4⋊C4—171no—
381C2×C4⋊C4—171no—
382C2×C4⋊C4—171no—
383C2×C4⋊C4—171no—
384C2×C4⋊C4—171no—
385C2×C4⋊C4—341no—
386C2×C4⋊C4—341no—
387C2×C4⋊C4—341no—
388C2×C4⋊C4—341no—
389C2×C4⋊C4—341no—
390C2×C4⋊C4—341no—
391C2×C4⋊C4D6811yescharacteristic
392C2×C4⋊C4D6811yescharacteristic
393C2×C22⋊C4—341no—
394C2×C22⋊C4—341no—
395C2×C22⋊C4—341no—
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⋊C8C4×D1714yescomplemented
407#102—174no—
408#102—174no—
409#102—174no—
410#102—342no—
411C4×C8—22no—
412C4×C8—41no—
413C4×C8—41no—
414#95—341no—
415#95—341no—
416#95—341no—
417#95—341no—
418#95—341no—
419#95—341no—
420#95—341no—
421#95—341no—
422#95—341no—
423#95D6811yescharacteristic
424#95D6811yescharacteristic
425C17#2296611yescharacteristic, complemented, Sylow
426C24—341no—
427C2×D4—172no—
428C2×D4—172no—
429C2×D4—172no—
430C2×D4—341no—
431C2×D4—341no—
432C2×D4—341no—
433C22×C4—341no—
434C22×C4—341no—
435C22×C4—341no—
436C22×C4—341no—
437C22×C4—344no—
438C22×C4—341no—
439C22×C4—341no—
440C22×C4—341no—
441C22×C4—682no—
442C22×C4—682no—
443C22×C4—682no—
444C22×C4—682no—
445C22×C4—1361no—
446C22×C4—1361no—
447C22×C4—1361no—
448C22×C4—1361no—
449C22×C4C2×D6811yescharacteristic
450C22×C4C2×D6811yescharacteristic
451C22×C4C2×D6811yescharacteristic
452C22×C4C2×D6811yescharacteristic
453C22×C4C2×D6811yescharacteristic
454C22×C4C2×D6811yescharacteristic
455C22×C4C23×D1711yescharacteristic
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8C2×C4×D1712yes—
466C2×C8C2×C4×D1712yes—
467C2×C8Q8×D1712yes—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—172no—
471C4⋊C4—172no—
472C4⋊C4—172no—
473C4⋊C4—172no—
474C4⋊C4—341no—
475C4⋊C4—341no—
476C4⋊C4—341no—
477C4⋊C4—341no—
478C4⋊C4—344no—
479C4⋊C4—344no—
480C4⋊C4—341no—
481C4⋊C4—341no—
482C4⋊C4—341no—
483C4⋊C4—341no—
484C4⋊C4—341no—
485C4⋊C4—341no—
486C4⋊C4—682no—
487C4⋊C4—1361no—
488C4⋊C4—1361no—
489C4⋊C4—1361no—
490C22⋊C4—344no—
491C22⋊C4—682no—
492C22⋊C4—682no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—682no—
498C42—1361no—
499C42—1361no—
500C42C2×C4×D1712yes—
501C42#6325212yes—
502C23—342no—
503C23—342no—
504C23—342no—
505C23—681no—
506C23—681no—
507C23—681no—
508C23—681no—
509C23C22×D6811yescharacteristic
510D4—174no—
511D4—342no—
512D4—342no—
513D4—342no—
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—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—342no—
529C2×C4—342no—
530C2×C4—342no—
531C2×C4—342no—
532C2×C4—681no—
533C2×C4—681no—
534C2×C4—681no—
535C2×C4—681no—
536C2×C4—681no—
537C2×C4—681no—
538C2×C4—681no—
539C2×C4—681no—
540C2×C4—681no—
541C2×C4—681no—
542C2×C4—681no—
543C2×C4—681no—
544C2×C4—681no—
545C2×C4—681no—
546C2×C4—681no—
547C2×C4—681no—
548C2×C4—681no—
549C2×C4—1364no—
550C2×C4—1364no—
551C2×C4—1364no—
552C2×C4C4×D6812yes—
553C2×C4C4×D6812yes—
554C2×C4#1058759012yes—
555C2×C4#1058759012yes—
556C2×C4C22×C4×D1711yescharacteristic
557C2×C4C2×#6325211yescharacteristic
558C2×C4#1058769111yescharacteristic
559C2×C4C2×Q8×D1711yescharacteristic
560C2×C4#41×D1711yescharacteristic
561C2×C4#1058769711yescharacteristic
562C8—22no—
563C8—22no—
564C8—22no—
565C8—41no—
566C22—344no—
567C22—682no—
568C22—682no—
569C22—682no—
570C22—682no—
571C22—682no—
572C22—682no—
573C22C2×C4×D6811yescharacteristic
574C22#1176345411yescharacteristic
575C22Q8×D6811yescharacteristic
576C22C2×#1058759011yescharacteristic
577C22C2×#1058759011yescharacteristic
578C22#1176361411yescharacteristic
579C22#1176362111yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—41no—
585C4—41no—
586C4—682no—
587C4—682no—
588C4—682no—
589C4—682no—
590C4—1361no—
591C4—1361no—
592C4—1361no—
593C4C4×Q8×D1711yescharacteristic
594C4#1176328811yescharacteristic
595C4#1176349811yescharacteristic
596C4#1176351411yescharacteristic
597C2—684no—
598C2#1453179111yescharacteristic
599C2C4×#1058759011yescharacteristic
600C2#1453761811yescharacteristic
601C2#1454062111yescharacteristic
602C2#1454063611yescharacteristic
603C2#1454165211yescharacteristic
604C2#1454166111yescharacteristic
subgroupG/H
centerC23C22×D68
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC8×C4⋊C4×C17C2
socleC23×C17C22×D4
prime radicalC22×D68C23
cosocleC22×C4×C17C24
hypercenterC8×C4⋊C4D17
hypocenterC17#22966
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #14531878C2semidirect
4. #14529749C2semidirect
5. #14529554C2semidirect
6. #14529517C2semidirect
7. #14529512C2semidirect
8. #14529377C2semidirect
9. #14529370C2semidirect
15. C8×C4⋊C4×C17C2semidirect
20. #11763110C4semidirect
99. #10587556C2×C4semidirect
100. C17×C4⋊C8C2×C4semidirect
120. #10587489C2×C4semidirect
122. #10587488C2×C4semidirect
275. C8×C4⋊C4D17semidirect
406. C4⋊C8C4×D17semidirect
425. C17#22966semidirect

Maximal subgroups

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

subgroupindex
C2×#117631102
#145318782
#145316952
#145297492
#145295542
#145295172
#145295122
#145293772
#145293702
#145290482
#145290342
C4×#105874922
C8×C4⋊C4×C172
#2296617