Finite Group Databasebeta

← #17969080

Subgroups — #17969080

Order 4352.

954 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763841C211yesmaximal, characteristic
2C2×#11763838C211yesmaximal, characteristic
3#14544583C211yesmaximal, characteristic, complemented
4#14544583C211yesmaximal, characteristic, complemented
5#14544570C211yesmaximal, characteristic, complemented
6#14544526C211yesmaximal, characteristic, complemented
7#14544525C211yesmaximal, characteristic, complemented
8#14544522C211yesmaximal, characteristic, complemented
9#14544521C211yesmaximal, characteristic, complemented
10#14544514C211yesmaximal, characteristic, complemented
11#14544513C211yesmaximal, characteristic, complemented
12#14544501C211yesmaximal, characteristic, complemented
13#14544451C211yesmaximal, characteristic, complemented
14#14544450C211yesmaximal, characteristic, complemented
15#14531244C211yesmaximal, characteristic
16C22×#63243C2211yescharacteristic
17C22×#63247C2211yescharacteristic
18C22×#63242C2211yescharacteristic
19C23×#3624C2211yescharacteristic
20#11763841C418yescomplemented
21#11763838C418yescomplemented
22C2×C4⋊#3624C2211yescharacteristic
23C2×C4⋊#3624C2211yescharacteristic
24C2×#10587543C2211yescharacteristic
25C2×#10587543C2211yescharacteristic
26C2×C22⋊#3624C2211yescharacteristic
27C2×C22⋊#3624C2211yescharacteristic
28C2×#10587542C2211yescharacteristic
29C2×#10587542C2211yescharacteristic
30C2×#10587542C2211yescharacteristic
31C2×#10587542C2211yescharacteristic
32C2×#10587542C2211yescharacteristic
33C2×#10587542C2211yescharacteristic
34C2×C4×#3624C2211yescharacteristic
35C2×C4×#3624C2211yescharacteristic
36#11763675C2211yescharacteristic
37#11763675C2211yescharacteristic
38#11763675C2211yescharacteristic
39#11763675C2211yescharacteristic
40#11763675C2211yescharacteristic
41#11763675C2211yescharacteristic
42#11763675C2211yescharacteristic
43#11763675C2211yescharacteristic
44#11763675C2211yescharacteristic
45#11763675C2211yescharacteristic
46C2×#10587567C2211yescharacteristic
47C2×#10587578C2211yescharacteristic
48C2×#10587563C2211yescharacteristic
49#11762709C411yescharacteristic
50#11762709C411yescharacteristic
51#11762706C411yescharacteristic
52#11762704C411yescharacteristic
53#11762702C411yescharacteristic
54#11762700C411yescharacteristic
55#11762698C411yescharacteristic
56#11762660C2211yescharacteristic
57C22⋊C4×Dic17C2211yescharacteristic
58#11762656C2211yescharacteristic
59C42×Dic17C2211yescharacteristic
60C17×#388C411yescharacteristic
61C22×C22⋊D17C2×C411yescharacteristic
62C23×Dic17C2311yescharacteristic
63C22×C4×D17C2311yescharacteristic
64C2×#63247—41no—
65C2×#63247—41no—
66C2×#63247C2×C414yes—
67C22×#3624—41no—
68C22×#3624—41no—
69C22×#3624C2×C414yes—
70C22×#3624C2311yescharacteristic
71C22×#3624C2311yescharacteristic
72C22×#3624C2311yescharacteristic
73C22×#3624C2311yescharacteristic
74C22×#3624C2311yescharacteristic
75C22×#3624C2311yescharacteristic
76C22×#3624C2311yescharacteristic
77C22×#3624C2311yescharacteristic
78C2×#63243—41no—
79C2×#63243—41no—
80C2×#63243C2×C414yes—
81C2×#63242—41no—
82C2×#63242—41no—
83C2×#63242C2×C414yes—
84C2×C4⋊C4×C17C2×C411yescharacteristic
85C2×C22⋊C4×C17C2×C411yescharacteristic
86C2×C22⋊C4×C17C2×C411yescharacteristic
87C2×C22⋊C4×C17C2×C411yescharacteristic
88C2×C42×C17C2×C411yescharacteristic
89C2×C22⋊Dic17C2×C411yescharacteristic
90C2×C22⋊Dic17C2×C411yescharacteristic
91C2×#63227C2×C411yescharacteristic
92C2×#63227C2×C411yescharacteristic
93C2×#63227C2×C411yescharacteristic
94C2×#63227C2×C411yescharacteristic
95C2×#63227C2×C411yescharacteristic
96C2×C4⋊Dic17C2×C411yescharacteristic
97C2×#63225C2×C411yescharacteristic
98C2×#63225C2×C411yescharacteristic
99C2×#63225C2×C411yescharacteristic
100C2×#63225C2×C411yescharacteristic
101C2×#63225C2×C411yescharacteristic
102C2×C4×Dic17C2×C411yescharacteristic
103C2×C4×Dic17C2×C411yescharacteristic
104C2×C4×Dic17C2×C411yescharacteristic
105C2×C4×Dic17C2311yescharacteristic
106C2×C4×Dic17C2311yescharacteristic
107C2×C4×Dic17C2311yescharacteristic
108C2×C4×Dic17C2311yescharacteristic
109C2×C4×Dic17C2311yescharacteristic
110#10587578—41no—
111#10587578—41no—
112#10587567C2×C414yes—
113#10587567C2×C414yes—
114#10587563—41no—
115#10587563—41no—
116C22⋊#3624C2×C414yes—
117C22⋊#3624C2×C414yes—
118#10587543C2×C414yes—
119#10587543C2×C414yes—
120#10587542—41no—
121#10587542—41no—
122#10587542—41no—
123#10587542—41no—
124#10587542C2×C414yes—
125#10587542C2×C414yes—
126C4⋊#3624C2×C414yes—
127C4⋊#3624C2×C414yes—
128C4×#3624—41no—
129C4×#3624—41no—
130C17×#95C2×C411yescharacteristic
131C17×#95C2×C411yescharacteristic
132#10587502C2×C411yescharacteristic
133#10587502C2×C411yescharacteristic
134#10587502C2×C411yescharacteristic
135#10587502C2×C411yescharacteristic
136C24×C17C22×C411yescharacteristic
137C23×D17C22×C411yescharacteristic
138C22×C4×C17C22×C411yescharacteristic
139C22×C4×C17C22×C411yescharacteristic
140C22×C4×C17C22×C411yescharacteristic
141C22×C4×C17C22×C411yescharacteristic
142C22×C4×C17C22×C411yescharacteristic
143C22×C4×C17C22×C411yescharacteristic
144C2×C22⋊D17—41no—
145C2×C22⋊D17—41no—
146C2×C22⋊D17C4214yes—
147C22×Dic17—21no—
148C22×Dic17—21no—
149C22×Dic17—21no—
150C22×Dic17—21no—
151C22×Dic17C22×C411yescharacteristic
152C22×Dic17C22×C411yescharacteristic
153C22×Dic17C22×C411yescharacteristic
154C22×Dic17C22×C411yescharacteristic
155C22×Dic17C22×C411yescharacteristic
156C22×Dic17C22×C411yescharacteristic
157C22×Dic17C22×C412yes—
158C22×Dic17C22×C412yes—
159C22×Dic17C2411yescharacteristic
160C2×C4×D17—21no—
161C2×C4×D17—21no—
162C2×C4×D17—21no—
163C2×C4×D17—21no—
164C2×C4×D17C22×C412yes—
165C2×C4×D17C22×C412yes—
166#63247—44no—
167C2×#3624—21no—
168C2×#3624—21no—
169C2×#3624—21no—
170C2×#3624—21no—
171C2×#3624—21no—
172C2×#3624—21no—
173C2×#3624—21no—
174C2×#3624—21no—
175C2×#3624—21no—
176C2×#3624—21no—
177C2×#3624—21no—
178C2×#3624—21no—
179C2×#3624—21no—
180C2×#3624—21no—
181C2×#3624—21no—
182C2×#3624—21no—
183C2×#3624—44no—
184C2×#3624C22×C412yes—
185C2×#3624C22×C412yes—
186C2×#3624C22×C412yes—
187C2×#3624C22×C412yes—
188C2×#3624C22×C412yes—
189C2×#3624C22×C412yes—
190C2×#3624C22×C412yes—
191C2×#3624C22×C412yes—
192#63243—44no—
193#63242—44no—
194C4⋊C4×C17—41no—
195C22⋊C4×C17—41no—
196C22⋊C4×C17—41no—
197C22⋊C4×C17C4214yes—
198C42×C17—41no—
199C22⋊Dic17—41no—
200C22⋊Dic17—41no—
201#63227—41no—
202#63227—41no—
203#63227—41no—
204#63227—41no—
205#63227C4214yes—
206C4⋊Dic17—41no—
207#63225—41no—
208#63225—41no—
209#63225—41no—
210#63225—41no—
211#63225C4214yes—
212C4×Dic17—21no—
213C4×Dic17—21no—
214C4×Dic17—21no—
215C4×Dic17—21no—
216C4×Dic17—21no—
217C4×Dic17—21no—
218C4×Dic17—21no—
219C4×Dic17—21no—
220C4×Dic17—41no—
221C4×Dic17—41no—
222C4×Dic17—41no—
223C4×Dic17C22×C412yes—
224C4×Dic17C22×C412yes—
225#20563—171nomaximal, Sylow
226C23×C17—21no—
227C23×C17—21no—
228C23×C17—21no—
229C23×C17—21no—
230C23×C17—21no—
231C23×C17—21no—
232C23×C17C2×C4212yes—
233C23×C17C23×C411yescharacteristic
234C22×D17—21no—
235C22×D17—21no—
236C22×D17—21no—
237C22×D17—21no—
238C22×D17—21no—
239C22×D17—21no—
240C22×D17C2×C4212yes—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—21no—
244C2×C4×C17—21no—
245C2×C4×C17—21no—
246C2×C4×C17—21no—
247C2×C4×C17—21no—
248C2×C4×C17—21no—
249C2×C4×C17—21no—
250C2×C4×C17—21no—
251C2×C4×C17—21no—
252C2×C4×C17—21no—
253C2×C4×C17—21no—
254C2×C4×C17—21no—
255C2×C4×C17—21no—
256C2×C4×C17—21no—
257C2×C4×C17C2×C4212yes—
258C2×C4×C17C2×C4212yes—
259C22⋊D17—44no—
260C2×Dic17—21no—
261C2×Dic17—21no—
262C2×Dic17—21no—
263C2×Dic17—21no—
264C2×Dic17—22no—
265C2×Dic17—22no—
266C2×Dic17—22no—
267C2×Dic17—22no—
268C2×Dic17—21no—
269C2×Dic17—21no—
270C2×Dic17—21no—
271C2×Dic17—21no—
272C2×Dic17—21no—
273C2×Dic17—21no—
274C2×Dic17—21no—
275C2×Dic17—21no—
276C2×Dic17—21no—
277C2×Dic17—21no—
278C2×Dic17—21no—
279C2×Dic17—21no—
280C2×Dic17C2×C4212yes—
281C2×Dic17C2×C4212yes—
282C2×Dic17C23×C411yescharacteristic
283C2×Dic17C23×C411yescharacteristic
284C2×Dic17H(5,𝔽2)11yescharacteristic
285C2×Dic17H(5,𝔽2)11yescharacteristic
286C2×Dic17H(5,𝔽2)11yescharacteristic
287C2×Dic172−1+411yescharacteristic
288C4×D17—22no—
289C4×D17—22no—
290C4×D17—22no—
291C4×D17—22no—
292#3624—22no—
293#3624—22no—
294#3624—22no—
295#3624—22no—
296#3624—22no—
297#3624—22no—
298#3624—22no—
299#3624—22no—
300C2×#435—171no—
301C2×#435—171no—
302#2293—171no—
303#1771—171no—
304#1771—171no—
305#1767—171no—
306#1753—171no—
307#1753—171no—
308#1752—171no—
309#1752—171no—
310#1750—171no—
311#1750—171no—
312#1745—171no—
313#1729—171no—
314#1729—171no—
315C22×C17—22no—
316C22×C17—22no—
317C22×C17—22no—
318C22×C17—41no—
319C22×C17—41no—
320C22×C17—41no—
321C22×C17—41no—
322C22×C17C22×C4211yescharacteristic
323C22×C17#51811yescharacteristic
324C22×C17#52011yescharacteristic
325C22×C17#56811yescharacteristic
326C22×C17#56811yescharacteristic
327C22×C17#56811yescharacteristic
328C22×C17#56811yescharacteristic
329C2×D17—22no—
330C2×D17—22no—
331C2×D17—22no—
332C2×D17—41no—
333C2×D17—41no—
334C2×D17—41no—
335C2×D17—41no—
336C4×C17—22no—
337C4×C17—22no—
338C4×C17—41no—
339C4×C17—41no—
340C4×C17—41no—
341C4×C17—41no—
342Dic17—22no—
343Dic17—22no—
344Dic17—41no—
345Dic17—41no—
346Dic17—41no—
347Dic17—41no—
348Dic17#51811yescharacteristic
349Dic17#51811yescharacteristic
350Dic17#52011yescharacteristic
351Dic17#52011yescharacteristic
352C22×M16—171no—
353C22×M16—171no—
354C23×C8—171no—
355C23×C8—171no—
356C2×C4×D4—171no—
357C2×#117—171no—
358C2×#117—171no—
359#435—178no—
360#435—178no—
361C2×C4⋊C8—171no—
362C2×C4⋊C8—171no—
363C2×C22⋊C8—171no—
364C2×C22⋊C8—171no—
365C2×C22⋊C8—171no—
366C2×C22⋊C8—171no—
367C2×#97—171no—
368C2×#97—171no—
369C2×#97—171no—
370C2×#97—171no—
371C2×#97—171no—
372C2×#97—171no—
373C2×C4×C8—171no—
374C2×C4×C8—171no—
375#390—171no—
376#390—171no—
377#389—171no—
378#389—171no—
379#388—171no—
380#388—171no—
381#388—171no—
382#388#59511yescharacteristic
383C4×C4⋊C4—171no—
384C4×C22⋊C4—171no—
385C4×C22⋊C4—171no—
386C43—171no—
387#336—171no—
388#336—171no—
389#336—171no—
390#336—171no—
391#336—171no—
392#336—171no—
393#336—171no—
394#336—171no—
395#336—171no—
396#336—171no—
397C2×C17—42no—
398C2×C17#227511yescharacteristic
399C2×C17#287311yescharacteristic
400C2×C17#287511yescharacteristic
401C2×C17#298011yescharacteristic
402C2×C17#298011yescharacteristic
403C2×C17#299511yescharacteristic
404C2×C17#299511yescharacteristic
405D17—42no—
406C22×D4—171no—
407C23×C4—171no—
408C23×C4—171no—
409C2×M16—174no—
410C2×M16—174no—
411C2×M16—681no—
412C2×M16—681no—
413C2×M16—681no—
414C2×M16—681no—
415C22×C8—171no—
416C22×C8—171no—
417C22×C8—171no—
418C22×C8—171no—
419C22×C8—171no—
420C22×C8—171no—
421C22×C8—171no—
422C22×C8—171no—
423C22×C8—174no—
424C22×C8—174no—
425C22×C8—681no—
426C22×C8—681no—
427C22×C8—681no—
428C22×C8—681no—
429C4×D4—174no—
430C4×D4—174no—
431#117—681no—
432#117—681no—
433#117—681no—
434#117—681no—
435C2×C4⋊C4—171no—
436C2×C4⋊C4—171no—
437C2×C4⋊C4—171no—
438C2×C4⋊C4—171no—
439C2×C4⋊C4—171no—
440C2×C4⋊C4—171no—
441C2×C4⋊C4C2×#59511yescharacteristic
442C2×C22⋊C4—171no—
443C2×C22⋊C4—171no—
444C2×C22⋊C4—171no—
445C2×C22⋊C4—171no—
446C2×C22⋊C4—171no—
447C2×C22⋊C4—171no—
448C2×C22⋊C4—171no—
449C2×C22⋊C4C2×#59511yescharacteristic
450C2×C22⋊C4C2×#59511yescharacteristic
451C2×C22⋊C4C2×#59511yescharacteristic
452C2×C42—171no—
453C2×C42—171no—
454C2×C42—171no—
455C2×C42—171no—
456C2×C42—171no—
457C2×C42—171no—
458C2×C42—171no—
459C2×C42—171no—
460C2×C42C2×#59511yescharacteristic
461C4⋊C8—174no—
462C4⋊C8—174no—
463C22⋊C8—174no—
464C22⋊C8—174no—
465C22⋊C8—174no—
466C22⋊C8—174no—
467#97—174no—
468#97—174no—
469#97—681no—
470#97—681no—
471#97—681no—
472#97—681no—
473C4×C8—681no—
474C4×C8—681no—
475#95—171no—
476#95—171no—
477#95—171no—
478#95—171no—
479#95C2×#59511yescharacteristic
480#95C2×#59511yescharacteristic
481C17#2056311yescharacteristic, complemented, Sylow
482C24—171no—
483C24C22×#59511yescharacteristic
484C2×D4—174no—
485C2×D4—681no—
486C2×D4—681no—
487C22×C4—171no—
488C22×C4—172no—
489C22×C4—172no—
490C22×C4—171no—
491C22×C4—171no—
492C22×C4—172no—
493C22×C4—172no—
494C22×C4—171no—
495C22×C4—171no—
496C22×C4—171no—
497C22×C4—171no—
498C22×C4—341no—
499C22×C4—341no—
500C22×C4—341no—
501C22×C4—341no—
502C22×C4—341no—
503C22×C4—341no—
504C22×C4—341no—
505C22×C4—341no—
506C22×C4C22×#59511yescharacteristic
507C22×C4C22×#59511yescharacteristic
508C22×C4C22×#59511yescharacteristic
509C22×C4C22×#59511yescharacteristic
510C22×C4C22×#59511yescharacteristic
511C22×C4C22×#59511yescharacteristic
512M16—684no—
513M16—684no—
514C2×C8—172no—
515C2×C8—172no—
516C2×C8—172no—
517C2×C8—172no—
518C2×C8—172no—
519C2×C8—172no—
520C2×C8—172no—
521C2×C8—172no—
522C2×C8—341no—
523C2×C8—341no—
524C2×C8—341no—
525C2×C8—341no—
526C2×C8—341no—
527C2×C8—341no—
528C2×C8—341no—
529C2×C8—341no—
530C2×C8—341no—
531C2×C8—341no—
532C2×C8—341no—
533C2×C8—341no—
534C2×C8—341no—
535C2×C8—341no—
536C2×C8—341no—
537C2×C8—341no—
538C2×C8—684no—
539C2×C8—684no—
540C4⋊C4—41no—
541C4⋊C4—174no—
542C4⋊C4—681no—
543C4⋊C4—681no—
544C4⋊C4—681no—
545C4⋊C4—681no—
546C4⋊C4—681no—
547C22⋊C4—41no—
548C22⋊C4—41no—
549C22⋊C4—174no—
550C22⋊C4—681no—
551C22⋊C4—681no—
552C22⋊C4—681no—
553C22⋊C4—681no—
554C22⋊C4—681no—
555C22⋊C4—681no—
556C22⋊C4C4×#59514yes—
557C42—41no—
558C42—172no—
559C42—172no—
560C42—341no—
561C42—341no—
562C42—341no—
563C42—341no—
564C42—341no—
565C42—341no—
566C42—341no—
567C42—341no—
568C42—681no—
569C42—681no—
570C42—681no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—172no—
578C23—341no—
579C23—341no—
580C23—341no—
581C23—341no—
582C23—341no—
583C23—341no—
584C23C2×C4×#59512yes—
585C23C23×#59511yescharacteristic
586D4—684no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—171no—
604C2×C4—171no—
605C2×C4—171no—
606C2×C4—171no—
607C2×C4—171no—
608C2×C4—171no—
609C2×C4—172no—
610C2×C4—172no—
611C2×C4—342no—
612C2×C4—342no—
613C2×C4—342no—
614C2×C4—342no—
615C2×C4—342no—
616C2×C4—342no—
617C2×C4—342no—
618C2×C4—342no—
619C2×C4—341no—
620C2×C4—341no—
621C2×C4—341no—
622C2×C4—341no—
623C2×C4—341no—
624C2×C4—341no—
625C2×C4—341no—
626C2×C4—341no—
627C2×C4—341no—
628C2×C4—341no—
629C2×C4—341no—
630C2×C4—341no—
631C2×C4—341no—
632C2×C4—341no—
633C2×C4—341no—
634C2×C4—341no—
635C2×C4C2×C4×#59512yes—
636C2×C4C2×C4×#59512yes—
637C8—342no—
638C8—342no—
639C8—342no—
640C8—342no—
641C8—342no—
642C8—342no—
643C8—342no—
644C8—342no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—342no—
653C22—342no—
654C22—342no—
655C22—681no—
656C22—681no—
657C22—681no—
658C22—681no—
659C22C22×C4×#59511yescharacteristic
660C22#1176387611yescharacteristic
661C22#1176387811yescharacteristic
662C22#1176388711yescharacteristic
663C22#1176388811yescharacteristic
664C22#1176388811yescharacteristic
665C22#1176388811yescharacteristic
666C4—22no—
667C4—22no—
668C4—41no—
669C4—41no—
670C4—41no—
671C4—41no—
672C4—171no—
673C4—171no—
674C4—171no—
675C4—171no—
676C4—342no—
677C4—342no—
678C4—681no—
679C4—681no—
680C4—681no—
681C4—681no—
682C2—42no—
683C2—682no—
684C2#1454522311yescharacteristic
685C2#1454534911yescharacteristic
686C2#1454535511yescharacteristic
687C2#1454566811yescharacteristic
688C2#1454567011yescharacteristic
689C2#1454573811yescharacteristic
690C2#1454573911yescharacteristic
subgroupG/H
center, FrattiniC23C23×#595
commutatorC22×C17C22×C42
FittingC17×#388C4
socleC23×C17C23×C4
prime radicalC22×C22⋊D17C2×C4
cosocleC22×Dic17C24
hypercenter#388#595
hypocenterC17#20563
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
3. #14544583C2semidirect
4. #14544583C2semidirect
5. #14544570C2semidirect
6. #14544526C2semidirect
7. #14544525C2semidirect
8. #14544522C2semidirect
9. #14544521C2semidirect
10. #14544514C2semidirect
11. #14544513C2semidirect
12. #14544501C2semidirect
13. #14544451C2semidirect
14. #14544450C2semidirect
20. #11763841C4semidirect
21. #11763838C4semidirect
481. C17#20563semidirect

Maximal subgroups

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

subgroupindex
C2×#117638412
C2×#117638382
#145445832
#145445702
#145445262
#145445252
#145445222
#145445212
#145445142
#145445132
#145445012
#145444512
#145444502
#145312442
#2056317