Finite Group Databasebeta

← #17973669

Subgroups — #17973669

Order 4352.

722 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14544955C211yesmaximal, characteristic, complemented
2#14544954C211yesmaximal, characteristic, complemented
3#14544883C211yesmaximal, characteristic, complemented
4#14544882C211yesmaximal, characteristic, complemented
5#14544840C211yesmaximal, characteristic, complemented
6#14544838C211yesmaximal, characteristic, complemented
7#14544837C211yesmaximal, characteristic, complemented
8#14544837C211yesmaximal, characteristic, complemented
9#14544732C211yesmaximal, characteristic
10#14544730C211yesmaximal, characteristic
11#14544643C211yesmaximal, characteristic
12#14544518C211yesmaximal, characteristic
13#14544518C211yesmaximal, characteristic
14#14544451C211yesmaximal, characteristic
15#14532061C211yesmaximal, characteristic
16C22×#63243C2211yescharacteristic
17C22×#63242C2211yescharacteristic
18C2×#10587544C2211yescharacteristic
19C2×#10587544C2211yescharacteristic
20C2×#10587544C2211yescharacteristic
21C2×C4⋊#3624C2211yescharacteristic
22C2×C4⋊#3624C2211yescharacteristic
23C2×C4⋊#3624C2211yescharacteristic
24C2×#10587543C2211yescharacteristic
25C2×#10587543C2211yescharacteristic
26C2×C4×#3624C2211yescharacteristic
27C2×C4×#3624C2211yescharacteristic
28C2×C4×#3624C2211yescharacteristic
29C2×C4×#3624C2211yescharacteristic
30#11763675C2211yescharacteristic
31#11763675C2211yescharacteristic
32#11763675C2211yescharacteristic
33#11763675C2211yescharacteristic
34#11763675C2211yescharacteristic
35#11763675C2211yescharacteristic
36#11763675C2211yescharacteristic
37#11763675C2211yescharacteristic
38#11763675C2211yescharacteristic
39#11763675C2211yescharacteristic
40#11763675C2211yescharacteristic
41#11763675C2211yescharacteristic
42#11763675C2211yescharacteristic
43#11763675C2211yescharacteristic
44C2×#10587573C2211yescharacteristic
45C2×#10587578C2211yescharacteristic
46#11762751C411yescharacteristic
47#11762744C411yescharacteristic
48#11762742C411yescharacteristic
49#11762731C411yescharacteristic
50#11762731C411yescharacteristic
51#11762730C411yescharacteristic
52#11762722C411yescharacteristic
53C4⋊C4×Dic17C2211yescharacteristic
54C4⋊C4×Dic17C2211yescharacteristic
55#11762660C2211yescharacteristic
56#11762656C2211yescharacteristic
57#11762656C2211yescharacteristic
58C17×#398C411yescharacteristic
59C22×C4×D17C2311yescharacteristic
60C22×Dic34C2×C411yescharacteristic
61C22×#3624C2311yescharacteristic
62C22×#3624C2311yescharacteristic
63C22×#3624C2311yescharacteristic
64C22×#3624C2311yescharacteristic
65C22×#3624C2311yescharacteristic
66C22×#3624C2311yescharacteristic
67C22×#3624C2311yescharacteristic
68C22×#3624C2311yescharacteristic
69C2×#63243—22no—
70C2×#63243—41no—
71C2×#63243—41no—
72C2×#63242—41no—
73C2×#63242—41no—
74C2×#63242Q814yescomplemented
75C2×C4⋊C4×C17C2×C411yescharacteristic
76C2×C4⋊C4×C17C2×C411yescharacteristic
77C2×C4⋊C4×C17C2×C411yescharacteristic
78C2×C4⋊C4×C17C2×C411yescharacteristic
79C2×#63227C2×C411yescharacteristic
80C2×#63227C2×C411yescharacteristic
81C2×#63227C2×C411yescharacteristic
82C2×#63227C2×C411yescharacteristic
83C2×#63227C2×C411yescharacteristic
84C2×#63227C2×C411yescharacteristic
85C2×C4⋊Dic17C2×C411yescharacteristic
86C2×C4⋊Dic17C2×C411yescharacteristic
87C2×C4⋊Dic17C2×C411yescharacteristic
88C2×#63225C2×C411yescharacteristic
89C2×#63225C2×C411yescharacteristic
90C2×#63225C2×C411yescharacteristic
91C2×#63225C2×C411yescharacteristic
92C2×C4×Dic17C2311yescharacteristic
93C2×C4×Dic17C2311yescharacteristic
94C2×C4×Dic17C2311yescharacteristic
95C2×C4×Dic17C2311yescharacteristic
96C2×C4×Dic17C2311yescharacteristic
97C2×C4×Dic17C2311yescharacteristic
98#10587578—41no—
99#10587578—41no—
100#10587573—41no—
101#10587573—41no—
102#10587544—41no—
103#10587544—41no—
104#10587544—41no—
105#10587543—22no—
106#10587543—22no—
107C4⋊#3624—41no—
108C4⋊#3624—41no—
109C4⋊#3624—41no—
110C4×#3624—41no—
111C4×#3624—41no—
112C4×#3624—41no—
113C4×#3624—41no—
114C17×#95C2×C411yescharacteristic
115C17×#95C2×C411yescharacteristic
116C17×#95C2×C411yescharacteristic
117#10587502C2×C411yescharacteristic
118#10587502C2×C411yescharacteristic
119#10587502C2×C411yescharacteristic
120#10587502C2×C411yescharacteristic
121#10587502C2×C411yescharacteristic
122#10587502C2×C411yescharacteristic
123#10587502C2×C411yescharacteristic
124C23×D17C22×C411yescharacteristic
125C22×C4×C17C22×C411yescharacteristic
126C22×C4×C17C22×C411yescharacteristic
127C22×C4×C17C22×C411yescharacteristic
128C22×C4×C17C22×C411yescharacteristic
129C22×C4×C17C22×C411yescharacteristic
130C22×C4×C17C22×C411yescharacteristic
131C22×C4×C17C22×C411yescharacteristic
132C22×Dic17C22×C411yescharacteristic
133C22×Dic17C22×C411yescharacteristic
134C22×Dic17C22×C411yescharacteristic
135C22×Dic17C22×C411yescharacteristic
136C22×Dic17C22×C411yescharacteristic
137C22×Dic17C22×C411yescharacteristic
138C22×Dic17C2411yescharacteristic
139C2×C4×D17—21no—
140C2×C4×D17—21no—
141C2×C4×D17—21no—
142C2×C4×D17—21no—
143C2×C4×D17C2×Q812yes—
144C2×C4×D17#4112yes—
145C2×Dic34—22no—
146C2×Dic34—22no—
147C2×Dic34—41no—
148C2×#3624—21no—
149C2×#3624—21no—
150C2×#3624—21no—
151C2×#3624—21no—
152C2×#3624—21no—
153C2×#3624—21no—
154C2×#3624—21no—
155C2×#3624—21no—
156C2×#3624—21no—
157C2×#3624—21no—
158C2×#3624—21no—
159C2×#3624—21no—
160C2×#3624—21no—
161C2×#3624—21no—
162C2×#3624—21no—
163C2×#3624—21no—
164C2×#3624—21no—
165C2×#3624—21no—
166C2×#3624—21no—
167C2×#3624—21no—
168C2×#3624—21no—
169C2×#3624—21no—
170C2×#3624C2×Q812yes—
171C2×#3624C2×Q812yes—
172#63243—44no—
173#63242—44no—
174C4⋊C4×C17—22no—
175C4⋊C4×C17—22no—
176C4⋊C4×C17—41no—
177C4⋊C4×C17—41no—
178#63227—22no—
179#63227—22no—
180#63227—22no—
181#63227—22no—
182#63227—41no—
183#63227—41no—
184C4⋊Dic17—22no—
185C4⋊Dic17—22no—
186C4⋊Dic17—41no—
187#63225—22no—
188#63225—22no—
189#63225—41no—
190#63225—41no—
191C4×Dic17—21no—
192C4×Dic17—21no—
193C4×Dic17—21no—
194C4×Dic17—21no—
195C4×Dic17—21no—
196C4×Dic17—21no—
197C4×Dic17—21no—
198C4×Dic17—21no—
199C4×Dic17—21no—
200C4×Dic17—21no—
201C4×Dic17—21no—
202C4×Dic17—21no—
203#22288—171nomaximal, Sylow
204C23×C17C23×C411yescharacteristic
205C22×D17—21no—
206C22×D17—21no—
207C22×D17—21no—
208C22×D17—21no—
209C22×D17—21no—
210C22×D17—21no—
211C22×D17C4×Q812yes—
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—21no—
227C2×C4×C17—21no—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17C4×Q812yes—
231C2×C4×C17#13112yes—
232C2×C4×C17#13112yes—
233C2×Dic17—21no—
234C2×Dic17—21no—
235C2×Dic17—21no—
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#13112yes—
250C2×Dic17#13112yes—
251C2×Dic17C22×Q811yescharacteristic
252C2×Dic17C2×#4111yescharacteristic
253C2×Dic17H(5,𝔽2)11yescharacteristic
254C2×Dic17H(5,𝔽2)11yescharacteristic
255C2×Dic17H(5,𝔽2)11yescharacteristic
256C2×Dic172−1+411yescharacteristic
257C4×D17—22no—
258C4×D17—22no—
259C4×D17—22no—
260C4×D17—22no—
261Dic34—44no—
262#3624—22no—
263#3624—22no—
264#3624—41no—
265#3624—41no—
266#3624—41no—
267#3624—41no—
268#3624—41no—
269#3624—41no—
270#2396—171no—
271#1984—171no—
272#1984—171no—
273#1939—171no—
274#1939—171no—
275#1916—171no—
276#1916—171no—
277#1916—171no—
278#1916—171no—
279#1851—171no—
280#1851—171no—
281#1798—171no—
282C8×C22⋊C4—171no—
283C8×C22⋊C4—171no—
284#1729—171no—
285C22×C17C2×C4×Q811yescharacteristic
286C22×C17#51811yescharacteristic
287C22×C17#52011yescharacteristic
288C22×C17C2×#13111yescharacteristic
289C22×C17C2×#13111yescharacteristic
290C22×C17C2×#13111yescharacteristic
291C22×C17C2×#13111yescharacteristic
292C2×D17—22no—
293C2×D17—22no—
294C2×D17—22no—
295C2×D17—41no—
296C2×D17—41no—
297C2×D17—41no—
298C2×D17—41no—
299C4×C17—22no—
300C4×C17—22no—
301C4×C17—22no—
302C4×C17—41no—
303C4×C17—41no—
304C4×C17—41no—
305C4×C17—41no—
306Dic17—22no—
307Dic17—22no—
308Dic17—41no—
309Dic17—41no—
310Dic17—41no—
311Dic17—41no—
312Dic17#53811yescharacteristic
313Dic17#53911yescharacteristic
314Dic17#54311yescharacteristic
315Dic17#54411yescharacteristic
316C22×M16—171no—
317C23×C8—171no—
318C2×C4×Q8—171no—
319C2×#117—171no—
320C2×C4⋊C8—171no—
321C2×C4⋊C8—171no—
322C2×C4⋊C8—171no—
323C2×C4⋊C8—171no—
324C2×C4⋊C8—171no—
325C2×C4⋊C8—171no—
326C2×C22⋊C8—171no—
327C2×C22⋊C8—171no—
328C2×C4×C8—171no—
329C2×C4×C8—171no—
330C2×C4×C8—171no—
331C2×C4×C8—171no—
332#400—171no—
333#399—171no—
334#398—171no—
335#398#59511yescharacteristic
336#396—171no—
337#396—171no—
338#395—171no—
339#393—171no—
340C4×C4⋊C4—171no—
341C4×C4⋊C4—171no—
342C4×C4⋊C4—171no—
343C4×C22⋊C4—171no—
344C4×C22⋊C4—171no—
345#336—171no—
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#336—171no—
359C2×C17#232911yescharacteristic
360C2×C17#296011yescharacteristic
361C2×C17#296011yescharacteristic
362C2×C17#297711yescharacteristic
363C2×C17#297911yescharacteristic
364C2×C17#299411yescharacteristic
365C2×C17#299411yescharacteristic
366D17—42no—
367C22×Q8—171no—
368C23×C4—171no—
369C2×M16—342no—
370C2×M16—681no—
371C2×M16—681no—
372C22×C8—171no—
373C22×C8—171no—
374C22×C8—174no—
375C22×C8—171no—
376C22×C8—171no—
377C22×C8—171no—
378C22×C8—171no—
379C22×C8—171no—
380C22×C8—171no—
381C22×C8—681no—
382C22×C8—681no—
383C4×Q8—681no—
384C4×Q8—681no—
385#117—681no—
386#117—681no—
387C2×C4⋊C4—171no—
388C2×C4⋊C4—171no—
389C2×C4⋊C4—171no—
390C2×C4⋊C4—171no—
391C2×C4⋊C4—171no—
392C2×C4⋊C4—171no—
393C2×C4⋊C4—171no—
394C2×C4⋊C4C2×#59511yescharacteristic
395C2×C4⋊C4C2×#59511yescharacteristic
396C2×C4⋊C4C2×#59511yescharacteristic
397C2×C4⋊C4C2×#59511yescharacteristic
398C2×C22⋊C4—171no—
399C2×C22⋊C4—171no—
400C2×C22⋊C4—171no—
401C2×C22⋊C4—171no—
402C2×C22⋊C4—171no—
403C2×C22⋊C4—171no—
404C2×C42—171no—
405C2×C42—171no—
406C2×C42—171no—
407C2×C42—171no—
408C2×C42—171no—
409C2×C42—171no—
410C4⋊C8—681no—
411C4⋊C8—681no—
412C4⋊C8—681no—
413C4⋊C8—681no—
414C4⋊C8—681no—
415C4⋊C8—681no—
416C22⋊C8—342no—
417C22⋊C8—342no—
418C4×C8—681no—
419C4×C8—681no—
420C4×C8—681no—
421C4×C8—681no—
422#95—171no—
423#95—171no—
424#95—171no—
425#95—171no—
426#95—171no—
427#95—171no—
428#95—171no—
429#95C2×#59511yescharacteristic
430#95C2×#59511yescharacteristic
431#95C2×#59511yescharacteristic
432C17#2228811yescharacteristic, complemented, Sylow
433C24—171no—
434C2×Q8—342no—
435C2×Q8—342no—
436C2×Q8—681no—
437C22×C4—171no—
438C22×C4—171no—
439C22×C4—171no—
440C22×C4—172no—
441C22×C4—172no—
442C22×C4—171no—
443C22×C4—171no—
444C22×C4—171no—
445C22×C4—171no—
446C22×C4—341no—
447C22×C4—341no—
448C22×C4—341no—
449C22×C4—341no—
450C22×C4C22×#59511yescharacteristic
451C22×C4C22×#59511yescharacteristic
452C22×C4C22×#59511yescharacteristic
453C22×C4C22×#59511yescharacteristic
454C22×C4C22×#59511yescharacteristic
455C22×C4C22×#59511yescharacteristic
456C22×C4C22×#59511yescharacteristic
457M16—684no—
458C2×C8—172no—
459C2×C8—172no—
460C2×C8—341no—
461C2×C8—341no—
462C2×C8—341no—
463C2×C8—341no—
464C2×C8—341no—
465C2×C8—341no—
466C2×C8—341no—
467C2×C8—341no—
468C2×C8—341no—
469C2×C8—341no—
470C2×C8—341no—
471C2×C8—341no—
472C2×C8—341no—
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—684no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—342no—
488C4⋊C4—342no—
489C4⋊C4—342no—
490C4⋊C4—342no—
491C4⋊C4—681no—
492C4⋊C4—681no—
493C4⋊C4—681no—
494C22⋊C4—342no—
495C22⋊C4—342no—
496C22⋊C4—342no—
497C22⋊C4—342no—
498C22⋊C4—681no—
499C22⋊C4—681no—
500C42—341no—
501C42—341no—
502C42—341no—
503C42—341no—
504C42—341no—
505C42—341no—
506C42—341no—
507C42—341no—
508C42—341no—
509C42—341no—
510C42—341no—
511C42—341no—
512C23—172no—
513C23—341no—
514C23—341no—
515C23—341no—
516C23—341no—
517C23—341no—
518C23—341no—
519C23C23×#59511yescharacteristic
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—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—171no—
540C2×C4—171no—
541C2×C4—171no—
542C2×C4—171no—
543C2×C4—171no—
544C2×C4—171no—
545C2×C4—172no—
546C2×C4—172no—
547C2×C4—342no—
548C2×C4—342no—
549C2×C4—342no—
550C2×C4—342no—
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—341no—
565C2×C4—341no—
566C2×C4—341no—
567C2×C4#1058767812yes—
568C2×C4#1058768012yes—
569C2×C4Q8×#59512yes—
570C8—342no—
571C8—342no—
572C8—681no—
573C8—681no—
574C8—681no—
575C8—681no—
576C8—681no—
577C8—681no—
578C22—342no—
579C22—342no—
580C22—342no—
581C22—681no—
582C22—681no—
583C22—681no—
584C22—681no—
585C22C2×Q8×#59511yescharacteristic
586C22#1176387611yescharacteristic
587C22#1176387811yescharacteristic
588C22C2×#1058767811yescharacteristic
589C22D17⋊#13111yescharacteristic
590C22D17⋊#13111yescharacteristic
591C22C2×#1058768011yescharacteristic
592C4—22no—
593C4—22no—
594C4—22no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—41no—
599C4—171no—
600C4—171no—
601C4—171no—
602C4—171no—
603C4—342no—
604C4—342no—
605C4—681no—
606C4—681no—
607C4—681no—
608C4—681no—
609C2—682no—
610C2#1454527711yescharacteristic
611C2#1454557311yescharacteristic
612C2#1454557411yescharacteristic
613C2#1454565911yescharacteristic
614C2#1454566411yescharacteristic
615C2#1454573211yescharacteristic
616C2#1454573611yescharacteristic
subgroupG/H
center, FrattiniC23C23×#595
commutator, socleC23×C17C23×C4
FittingC17×#398C4
prime radicalC23×D17C22×C4
cosocleC22×Dic17C24
hypercenter#398#595
hypocenterC17#22288
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. #14544955C2semidirect
2. #14544954C2semidirect
3. #14544883C2semidirect
4. #14544882C2semidirect
5. #14544840C2semidirect
6. #14544838C2semidirect
7. #14544837C2semidirect
8. #14544837C2semidirect
74. C2×#63242Q8semidirect
432. C17#22288semidirect

Maximal subgroups

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

subgroupindex
#145449552
#145449542
#145448832
#145448822
#145448402
#145448382
#145448372
#145447322
#145447302
#145446432
#145445182
#145444512
#145320612
#2228817