Finite Group Databasebeta

← #17060776

Subgroups — #17060776

Order 4352.

680 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#10587564C211yesmaximal, characteristic
2#14529724C211yesmaximal, characteristic, complemented
3#14529722C211yesmaximal, characteristic, complemented
4#14529506C211yesmaximal, characteristic, complemented
5#14529505C211yesmaximal, characteristic, complemented
6#14529400C211yesmaximal, characteristic
7#14529400C211yesmaximal, characteristic
8#14529399C211yesmaximal, characteristic
9#14529349C211yesmaximal, characteristic, complemented
10#14529348C211yesmaximal, characteristic, complemented
11#14529346C211yesmaximal, characteristic, complemented
12#14529345C211yesmaximal, characteristic, complemented
13#14528850C211yesmaximal, characteristic
14#14528762C211yesmaximal, characteristic
15C17×#1924C211yesmaximal, characteristic
16C2×#10587564C411yescharacteristic
17C2×C4×Dic34C2211yescharacteristic
18C22×C4×Dic17C2211yescharacteristic
19C2×C4⋊#3622C2211yescharacteristic
20C2×#10587485C2211yescharacteristic
21C2×#10587485C2211yescharacteristic
22C2×#10587485C2211yescharacteristic
23C2×#10587485C2211yescharacteristic
24C2×C22⋊#3622C2211yescharacteristic
25C2×C22⋊#3622C2211yescharacteristic
26C2×#10587474C2211yescharacteristic
27C2×#10587486C2211yescharacteristic
28C2×#10587486C2211yescharacteristic
29#11762713C411yescharacteristic
30#11762706C411yescharacteristic
31#11762696C411yescharacteristic
32#11762692C411yescharacteristic
33#11762684C411yescharacteristic
34#11762681C411yescharacteristic
35#11762667C411yescharacteristic
36C4×#63225C2211yescharacteristic
37C4×#63225C2211yescharacteristic
38C4×C4⋊Dic17C2211yescharacteristic
39C4×C22⋊Dic17C2211yescharacteristic
40#11762564C2211yescharacteristic
41#11762564C2211yescharacteristic
42#11762563C2211yescharacteristic
43#11762563C2211yescharacteristic
44#11762563C2211yescharacteristic
45#11762563C2211yescharacteristic
46#11762563C2211yescharacteristic
47#11762563C2211yescharacteristic
48#11762563C2211yescharacteristic
49#11762563C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52C2×C17×C4⋊C8C2211yescharacteristic
53C2×C17×C22⋊C8C2211yescharacteristic
54C2×C17×C22⋊C8C2211yescharacteristic
55C2×C17×#97C2211yescharacteristic
56C4×C22⋊C4×C17C2211yescharacteristic
57C17×#336C2211yescharacteristic
58C17×#336C2211yescharacteristic
59C23×C4×C17C2311yescharacteristic
60C23×Dic17C2×C411yescharacteristic
61C23×Dic17C2×C411yescharacteristic
62C22×Dic34C2×C411yescharacteristic
63C22×C8×C17C2311yescharacteristic
64C22×C8×C17C2311yescharacteristic
65C22×C8×C17C2311yescharacteristic
66C22×C8×C17C2311yescharacteristic
67C2×C22⋊C4×C17C2×C411yescharacteristic
68C2×C22⋊C4×C17C2×C411yescharacteristic
69C2×C42×C17C2311yescharacteristic
70C2×C42×C17C2311yescharacteristic
71C2×C22⋊Dic17C2×C411yescharacteristic
72C2×C22⋊Dic17C2×C411yescharacteristic
73C2×C4⋊Dic17C2×C411yescharacteristic
74C2×C4⋊Dic17C2×C411yescharacteristic
75C2×C4⋊Dic17C2×C411yescharacteristic
76C2×#63225C2×C411yescharacteristic
77C2×#63225C2×C411yescharacteristic
78C2×#63225C2×C411yescharacteristic
79C2×#63225C2×C411yescharacteristic
80C2×#63225C2×C411yescharacteristic
81C2×#63225C2×C411yescharacteristic
82C2×C4×Dic17—41no—
83C2×C4×Dic17—41no—
84C2×C4×Dic17C2×C411yescharacteristic
85C2×C4×Dic17C2×C411yescharacteristic
86C2×C4×Dic17C2311yescharacteristic
87C2×C4×Dic17C2311yescharacteristic
88C2×C4×Dic17C2311yescharacteristic
89C2×C4×Dic17C2311yescharacteristic
90C22×#3622C2311yescharacteristic
91C22×#3622C2311yescharacteristic
92C22×#3622C2311yescharacteristic
93C22×#3622C2311yescharacteristic
94#10587564—24no—
95C4×Dic34—41no—
96C4×Dic34—41no—
97C17×C4⋊C8—41no—
98C17×C22⋊C8—41no—
99C17×C22⋊C8—41no—
100C17×#97—41no—
101C17×#95C2×C411yescharacteristic
102C17×#95C2×C411yescharacteristic
103#10587502C2×C411yescharacteristic
104#10587502C2×C411yescharacteristic
105#10587502C2×C411yescharacteristic
106#10587502C2×C411yescharacteristic
107#10587502C2×C411yescharacteristic
108#10587502C2×C411yescharacteristic
109#10587502C2×C411yescharacteristic
110#10587502C2×C411yescharacteristic
111C22⋊#3622—41no—
112C22⋊#3622—41no—
113#10587486—41no—
114#10587486—41no—
115#10587485—41no—
116#10587485—41no—
117#10587485—41no—
118#10587485—41no—
119C4⋊#3622—41no—
120#10587474—41no—
121C24×C17C22×C411yescharacteristic
122C22×C4×C17—21no—
123C22×C4×C17—21no—
124C22×C4×C17—21no—
125C22×C4×C17—21no—
126C22×C4×C17—21no—
127C22×C4×C17—21no—
128C22×C4×C17C22×C411yescharacteristic
129C22×C4×C17C22×C411yescharacteristic
130C22×C4×C17C22×C411yescharacteristic
131C22×C4×C17C22×C411yescharacteristic
132C22×C4×C17C22×C411yescharacteristic
133C22×C4×C17C2411yescharacteristic
134C22×Dic17—41no—
135C22×Dic17—41no—
136C22×Dic17—41no—
137C22×Dic17—41no—
138C22×Dic17—41no—
139C22×Dic17M1614yes—
140C22×Dic17C22×C411yescharacteristic
141C22×Dic17C22×C411yescharacteristic
142C22×Dic17C22×C411yescharacteristic
143C22×Dic17C22×C411yescharacteristic
144C22×Dic17C22×C411yescharacteristic
145C22×Dic17C22×C411yescharacteristic
146C22×Dic17C22×C411yescharacteristic
147C22×Dic17C22×C411yescharacteristic
148C2×Dic34—22no—
149C2×Dic34—41no—
150C2×Dic34—41no—
151C2×C8×C17—21no—
152C2×C8×C17—21no—
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×C17—21no—
163C22⋊C4×C17—22no—
164C22⋊C4×C17—41no—
165C42×C17—21no—
166C42×C17—21no—
167C42×C17—21no—
168C42×C17—21no—
169C22⋊Dic17—22no—
170C22⋊Dic17—41no—
171C4⋊Dic17—22no—
172C4⋊Dic17—41no—
173C4⋊Dic17—41no—
174#63225—22no—
175#63225—22no—
176#63225—41no—
177#63225—41no—
178#63225—41no—
179#63225—41no—
180C4×Dic17—21no—
181C4×Dic17—21no—
182C4×Dic17—21no—
183C4×Dic17—21no—
184C4×Dic17—21no—
185C4×Dic17—21no—
186C4×Dic17—21no—
187C4×Dic17—21no—
188C4×Dic17—41no—
189C4×Dic17—41no—
190C2×#3622—21no—
191C2×#3622—21no—
192C2×#3622—21no—
193C2×#3622—21no—
194C2×#3622—21no—
195C2×#3622—21no—
196C2×#3622—21no—
197C2×#3622—21no—
198C2×#3622—21no—
199C2×#3622—21no—
200C2×#3622—21no—
201C2×#3622—21no—
202#23344—171nomaximal, Sylow
203C23×C17—21no—
204C23×C17—21no—
205C23×C17—21no—
206C23×C17—21no—
207C23×C17—21no—
208C23×C17—21no—
209C23×C17C2×M1612yes—
210C23×C17C23×C411yescharacteristic
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×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—21no—
226C2×C4×C17—41no—
227C2×C4×C17—41no—
228C2×C4×C17—41no—
229C2×C4×C17—41no—
230C2×C4×C17H(5,𝔽2)11yescharacteristic
231C2×C4×C17H(5,𝔽2)11yescharacteristic
232C2×C4×C17H(5,𝔽2)11yescharacteristic
233C2×C4×C172−1+411yescharacteristic
234C2×C4×C172−1+411yescharacteristic
235C2×C4×C172−1+411yescharacteristic
236C2×Dic17—21no—
237C2×Dic17—21no—
238C2×Dic17—21no—
239C2×Dic17—21no—
240C2×Dic17—21no—
241C2×Dic17—21no—
242C2×Dic17—21no—
243C2×Dic17—21no—
244C2×Dic17—21no—
245C2×Dic17—21no—
246C2×Dic17—21no—
247C2×Dic17—21no—
248C2×Dic17—21no—
249C2×Dic17—21no—
250C2×Dic17—21no—
251C2×Dic17—21no—
252C2×Dic17—21no—
253C2×Dic17—21no—
254C2×Dic17—21no—
255C2×Dic17—21no—
256C2×Dic17—21no—
257C2×Dic17—21no—
258C2×Dic17—44no—
259C2×Dic17—44no—
260C2×Dic17C2×M1612yes—
261C2×Dic17C2×M1612yes—
262Dic34—44no—
263C8×C17—41no—
264C8×C17—41no—
265C8×C17—41no—
266C8×C17—41no—
267#3622—41no—
268#3622—41no—
269#3622—41no—
270#3622—41no—
271C4×C22⋊Q8—171no—
272#1985—171no—
273#1985—171no—
274#1940—171no—
275#1940—171no—
276#1924—171no—
277#1924—171no—
278#1924—171no—
279#1924D1711yescharacteristic
280#1917—171no—
281#1917—171no—
282#1917—171no—
283#1917—171no—
284#1817—171no—
285#1797—171no—
286C22×C17—22no—
287C22×C17—22no—
288C22×C17—22no—
289C22×C17—41no—
290C22×C17—41no—
291C22×C17—41no—
292C22×C17—41no—
293C22×C17#51811yescharacteristic
294C22×C17#51911yescharacteristic
295C22×C17#52011yescharacteristic
296C22×C17C22×M1611yescharacteristic
297C22×C17#56811yescharacteristic
298C22×C17#56811yescharacteristic
299C22×C17#56811yescharacteristic
300C4×C17—41no—
301C4×C17—41no—
302C4×C17—41no—
303C4×C17—41no—
304C4×C17—41no—
305C4×C17#56211yescharacteristic
306C4×C17#56211yescharacteristic
307C4×C17#56311yescharacteristic
308C4×C17#56311yescharacteristic
309Dic17—22no—
310Dic17—22no—
311Dic17—41no—
312Dic17—41no—
313Dic17—41no—
314Dic17—41no—
315Dic17—41no—
316Dic17—41no—
317C2×C22⋊Q8—171no—
318C2×C4×Q8—171no—
319C22×C42—171no—
320C2×C4⋊C8—171no—
321C2×C4⋊C8—171no—
322C2×C4⋊C8—171no—
323C2×C4⋊C8—171no—
324C2×C4⋊C8—171no—
325C2×C4⋊C8C2×D1711yescharacteristic
326C2×C22⋊C8—171no—
327C2×C22⋊C8—171no—
328C2×C22⋊C8C2×D1711yescharacteristic
329C2×C22⋊C8C2×D1711yescharacteristic
330C2×#97—171no—
331C2×#97—171no—
332C2×#97—171no—
333C2×#97C2×D1711yescharacteristic
334#391—171no—
335#389—171no—
336#387—171no—
337#387—171no—
338#385—171no—
339#385—171no—
340#380—171no—
341C4×C4⋊C4—171no—
342C4×C4⋊C4—171no—
343C4×C4⋊C4—171no—
344C4×C22⋊C4—171no—
345C4×C22⋊C4C2×D1711yescharacteristic
346#336—171no—
347#336—171no—
348#336—171no—
349#336—171no—
350#336—171no—
351#336—171no—
352#336—171no—
353#336—171no—
354#336—171no—
355#336—171no—
356#336—171no—
357#336—171no—
358#336C2×D1711yescharacteristic
359#336C2×D1711yescharacteristic
360C2×C17—42no—
361C2×C17#237811yescharacteristic
362C2×C17#297311yescharacteristic
363C2×C17#297811yescharacteristic
364C2×C17#298311yescharacteristic
365C2×C17#298711yescharacteristic
366C2×C17#298911yescharacteristic
367C2×C17#299511yescharacteristic
368C22×Q8—171no—
369C23×C4—171no—
370C23×C4—171no—
371C23×C4C22×D1711yescharacteristic
372C22×C8—171no—
373C22×C8—171no—
374C22×C8—171no—
375C22×C8—171no—
376C22×C8C22×D1711yescharacteristic
377C22×C8C22×D1711yescharacteristic
378C22×C8C22×D1711yescharacteristic
379C22×C8C22×D1711yescharacteristic
380C22⋊Q8—344no—
381C4×Q8—681no—
382C4×Q8—681no—
383C2×C4⋊C4—171no—
384C2×C4⋊C4—171no—
385C2×C4⋊C4—171no—
386C2×C4⋊C4—171no—
387C2×C4⋊C4—171no—
388C2×C4⋊C4—171no—
389C2×C4⋊C4—171no—
390C2×C4⋊C4—171no—
391C2×C4⋊C4—171no—
392C2×C22⋊C4—171no—
393C2×C22⋊C4—171no—
394C2×C22⋊C4C4×D1711yescharacteristic
395C2×C22⋊C4C4×D1711yescharacteristic
396C2×C42—171no—
397C2×C42—171no—
398C2×C42—171no—
399C2×C42—171no—
400C2×C42—171no—
401C2×C42—171no—
402C2×C42—681no—
403C2×C42—681no—
404C2×C42C22×D1711yescharacteristic
405C2×C42C22×D1711yescharacteristic
406C4⋊C8—41no—
407C4⋊C8—681no—
408C4⋊C8—681no—
409C4⋊C8—681no—
410C4⋊C8—681no—
411C4⋊C8—681no—
412C22⋊C8—41no—
413C22⋊C8—41no—
414C22⋊C8—681no—
415C22⋊C8—681no—
416#97—41no—
417#97—681no—
418#97—681no—
419#97—681no—
420#95—171no—
421#95—171no—
422#95—171no—
423#95—171no—
424#95—171no—
425#95—171no—
426#95—171no—
427#95—171no—
428#95C4×D1711yescharacteristic
429#95C4×D1711yescharacteristic
430C17#2334411yescharacteristic, complemented, Sylow
431C24C2×C4×D1711yescharacteristic
432C2×Q8—342no—
433C2×Q8—681no—
434C2×Q8—681no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—171no—
442C22×C4—171no—
443C22×C4—174no—
444C22×C4—171no—
445C22×C4—171no—
446C22×C4—171no—
447C22×C4—171no—
448C22×C4—171no—
449C22×C4—171no—
450C22×C4—681no—
451C22×C4—681no—
452C22×C4—681no—
453C22×C4—681no—
454C22×C4—681no—
455C22×C4C2×C4×D1711yescharacteristic
456C22×C4C2×C4×D1711yescharacteristic
457C22×C4C2×C4×D1711yescharacteristic
458C22×C4C2×C4×D1711yescharacteristic
459C22×C4C2×C4×D1711yescharacteristic
460C22×C4C23×D1711yescharacteristic
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—341no—
474C2×C8—341no—
475C2×C8—341no—
476C2×C8—341no—
477C2×C8—341no—
478C2×C8—341no—
479C2×C8—341no—
480C2×C8—341no—
481C2×C8—341no—
482C2×C8—341no—
483C2×C8—341no—
484C2×C8—341no—
485C4⋊C4—342no—
486C4⋊C4—342no—
487C4⋊C4—342no—
488C4⋊C4—681no—
489C4⋊C4—681no—
490C4⋊C4—681no—
491C4⋊C4—681no—
492C4⋊C4—681no—
493C4⋊C4—681no—
494C22⋊C4—22no—
495C22⋊C4—41no—
496C22⋊C4—342no—
497C22⋊C4—681no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—341no—
503C42—341no—
504C42—341no—
505C42—341no—
506C42—341no—
507C42—341no—
508C42—341no—
509C42—341no—
510C42—681no—
511C42—681no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23M16×D1712yes—
519C23C22×C4×D1711yescharacteristic
520Q8—684no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—172no—
541C2×C4—172no—
542C2×C4—341no—
543C2×C4—341no—
544C2×C4—341no—
545C2×C4—341no—
546C2×C4—341no—
547C2×C4—341no—
548C2×C4—341no—
549C2×C4—341no—
550C2×C4—341no—
551C2×C4—341no—
552C2×C4—341no—
553C2×C4—341no—
554C2×C4—341no—
555C2×C4—341no—
556C2×C4—341no—
557C2×C4—341no—
558C2×C4—341no—
559C2×C4—341no—
560C2×C4—341no—
561C2×C4—341no—
562C2×C4—341no—
563C2×C4—341no—
564C2×C4—684no—
565C2×C4—684no—
566C2×C4#1058769111yescharacteristic
567C2×C4#1058769111yescharacteristic
568C2×C4#1058769611yescharacteristic
569C2×C4#1058769711yescharacteristic
570C2×C4#1058769711yescharacteristic
571C2×C4#1058769711yescharacteristic
572C8—41no—
573C8—41no—
574C8—41no—
575C8—41no—
576C8—681no—
577C8—681no—
578C8—681no—
579C8—681no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22#1176328111yescharacteristic
588C22#1176328611yescharacteristic
589C22#1176328811yescharacteristic
590C22C2×M16×D1711yescharacteristic
591C22#1176357411yescharacteristic
592C22#1176357411yescharacteristic
593C22#1176357511yescharacteristic
594C4—41no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—41no—
599C4—342no—
600C4—342no—
601C4—681no—
602C4—681no—
603C4—681no—
604C4—681no—
605C4—681no—
606C4—681no—
607C4#1176355311yescharacteristic
608C4#1176355411yescharacteristic
609C4#1176356111yescharacteristic
610C4#1176356211yescharacteristic
611C2—42no—
612C2#1453195111yescharacteristic
613C2#1453780911yescharacteristic
614C2#1453784111yescharacteristic
615C2#1453786811yescharacteristic
616C2#1453788811yescharacteristic
617C2#1453789611yescharacteristic
618C2#1453795511yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D17
commutator, socleC23×C17C23×C4
FittingC17×#1924C2
prime radicalC24×C17C22×C4
cosocleC22×C4×C17C24
hypercenter#1924D17
hypocenterC17#23344
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ #1924

Complements

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

NH ≅ G/Ntype
2. #14529724C2semidirect
3. #14529722C2semidirect
4. #14529506C2semidirect
5. #14529505C2semidirect
9. #14529349C2semidirect
10. #14529348C2semidirect
11. #14529346C2semidirect
12. #14529345C2semidirect
430. C17#23344semidirect

Maximal subgroups

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

subgroupindex
C4×#105875642
#145297242
#145297222
#145295062
#145295052
#145294002
#145293992
#145293492
#145293482
#145293462
#145293452
#145288502
#145287622
C17×#19242
#2334417