Finite Group Databasebeta

← #16961367

Subgroups — #16961367

Order 4352.

1650 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762726C211yesmaximal, characteristic
2C2×#11762711C211yesmaximal, characteristic
3#14527040C211yesmaximal, characteristic, complemented
4#14527037C211yesmaximal, characteristic, complemented
5#14527032C212yesmaximal, complemented
6#14527026C212yesmaximal, complemented
7#14527024C211yesmaximal, characteristic
8#14527020C212yesmaximal, complemented
9#14527020C212yesmaximal, complemented
10C4×#10587502C211yesmaximal, characteristic, complemented
11C2×C17×#387C211yesmaximal, characteristic, complemented
12C23×D68C2211yescharacteristic
13C22×C4⋊Dic17C2211yescharacteristic
14C22×#63227C2212yes—
15C22×#63227C2212yes—
16C22×#63227C2212yes—
17C22×#63227C2212yes—
18C22×#63227C2212yes—
19C22×#63227C2212yes—
20C22×C4×Dic17C2212yes—
21#11762726C418yescomplemented
22#11762711—24no—
23C2×#10587502C2211yescharacteristic
24C2×#10587502C2212yes—
25C2×#10587502C2212yes—
26C2×#10587502C2212yes—
27C2×#10587502C2212yes—
28C2×#10587502C2212yes—
29C2×#10587502C2211yescharacteristic
30C22×C4⋊C4×C17C2211yescharacteristic
31C22×C4⋊C4×C17C2211yescharacteristic
32C22×C42×C17C2211yescharacteristic
33C17×#387—24no—
34C2×C17×#95C2211yescharacteristic
35C2×C17×#95C2211yescharacteristic
36C2×C17×#95C2211yescharacteristic
37C2×C17×#95C2211yescharacteristic
38C24×D17C2312yes—
39C23×C4×C17C2311yescharacteristic
40C23×C4×C17C2311yescharacteristic
41C23×C4×C17C2311yescharacteristic
42C23×C4×C17C2311yescharacteristic
43C23×C4×C17C2311yescharacteristic
44C23×C4×C17C2311yescharacteristic
45C23×C4×C17C2311yescharacteristic
46C23×Dic17C2312yes—
47C23×Dic17C2312yes—
48C23×Dic17C2312yes—
49C22×D68—22no—
50C22×D68—22no—
51C22×D68—42no—
52C22×D68—42no—
53C22×D68C2×C414yes—
54C2×C4⋊C4×C17—22no—
55C2×C4⋊C4×C17—22no—
56C2×C4⋊C4×C17—22no—
57C2×C4⋊C4×C17—42no—
58C2×C4⋊C4×C17C2×C414yes—
59C2×C42×C17—42no—
60C2×C42×C17D414yes—
61C2×#63227—24no—
62C2×#63227—24no—
63C2×#63227—24no—
64C2×#63227—24no—
65C2×#63227—24no—
66C2×#63227—24no—
67C2×#63227—42no—
68C2×#63227—42no—
69C2×#63227—42no—
70C2×#63227—42no—
71C2×#63227—42no—
72C2×#63227—42no—
73C2×#63227—42no—
74C2×#63227—42no—
75C2×#63227—42no—
76C2×#63227—42no—
77C2×#63227C2×C418yes—
78C2×#63227C2×C418yes—
79C2×C4⋊Dic17—22no—
80C2×C4⋊Dic17—42no—
81C2×C4×Dic17—24no—
82C2×C4×Dic17—42no—
83C2×C4×Dic17—42no—
84C17×#95—22no—
85C17×#95—22no—
86C17×#95—22no—
87C17×#95—22no—
88#10587502—22no—
89#10587502—24no—
90#10587502—42no—
91#10587502—42no—
92#10587502—42no—
93#10587502—42no—
94#10587502C2×C414yes—
95C24×C17C2411yescharacteristic
96C23×D17—22no—
97C23×D17—22no—
98C23×D17—22no—
99C23×D17—22no—
100C23×D17—22no—
101C23×D17—22no—
102C23×D17—22no—
103C23×D17—22no—
104C23×D17—22no—
105C23×D17—22no—
106C23×D17—22no—
107C23×D17—22no—
108C23×D17—22no—
109C23×D17—22no—
110C23×D17C22×C414yes—
111C22×C4×C17—22no—
112C22×C4×C17—22no—
113C22×C4×C17—21no—
114C22×C4×C17—22no—
115C22×C4×C17—21no—
116C22×C4×C17—21no—
117C22×C4×C17—21no—
118C22×C4×C17—22no—
119C22×C4×C17—21no—
120C22×C4×C17—21no—
121C22×C4×C17—21no—
122C22×C4×C17—22no—
123C22×C4×C17—21no—
124C22×C4×C17—21no—
125C22×C4×C17—21no—
126C22×C4×C17—21no—
127C22×C4×C17—22no—
128C22×C4×C17—22no—
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17—21no—
132C22×C4×C17—22no—
133C22×C4×C17—22no—
134C22×C4×C17—21no—
135C22×C4×C17—22no—
136C22×C4×C17—21no—
137C22×C4×C17—21no—
138C22×C4×C17—22no—
139C22×C4×C17—21no—
140C22×C4×C17C22×C412yes—
141C22×C4×C17C22×C412yes—
142C22×C4×C17C22×C412yes—
143C22×C4×C17C2×D412yes—
144C22×C4×C17C2×D412yes—
145C22×C4×C17C2×D412yes—
146C22×C4×C17#4112yes—
147C22×C4×C17#4114yes—
148C22×Dic17—22no—
149C22×Dic17—22no—
150C22×Dic17—22no—
151C22×Dic17—22no—
152C22×Dic17—22no—
153C22×Dic17—22no—
154C22×Dic17—22no—
155C22×Dic17—22no—
156C22×Dic17—22no—
157C22×Dic17—22no—
158C22×Dic17—22no—
159C22×Dic17—22no—
160C22×Dic17—22no—
161C22×Dic17—22no—
162C22×Dic17—22no—
163C22×Dic17—22no—
164C22×Dic17—22no—
165C22×Dic17—22no—
166C22×Dic17—22no—
167C22×Dic17C22×C414yes—
168C22×Dic17#4114yes—
169C2×D68—28no—
170C2×D68—48no—
171C2×D68—44no—
172C2×D68—44no—
173C2×D68—48no—
174C4⋊C4×C17—28no—
175C4⋊C4×C17—44no—
176C42×C17—44no—
177#63227—48no—
178#63227—48no—
179#63227—48no—
180#63227—48no—
181#63227—48no—
182#63227—48no—
183C4⋊Dic17—44no—
184C4×Dic17—48no—
185#14325—171nomaximal, Sylow
186C23×C17C23×C411yescharacteristic
187C23×C17C22×D411yescharacteristic
188C23×C17C2×#4111yescharacteristic
189C23×C17C2×#4112yes—
190C23×C17C2×#4112yes—
191C23×C17H(5,𝔽2)12yes—
192C23×C17H(5,𝔽2)12yes—
193C23×C17H(5,𝔽2)11yescharacteristic
194C23×C17H(5,𝔽2)11yescharacteristic
195C23×C17H(5,𝔽2)11yescharacteristic
196C23×C172−1+411yescharacteristic
197C22×D17—24no—
198C22×D17—24no—
199C22×D17—24no—
200C22×D17—24no—
201C22×D17—24no—
202C22×D17—24no—
203C22×D17—24no—
204C22×D17—42no—
205C22×D17—42no—
206C22×D17—42no—
207C22×D17—42no—
208C22×D17—42no—
209C22×D17—42no—
210C22×D17—42no—
211C22×D17—42no—
212C22×D17—42no—
213C22×D17—42no—
214C22×D17—42no—
215C22×D17—42no—
216C22×D17—42no—
217C22×D17—42no—
218C22×D17—42no—
219C22×D17—42no—
220C22×D17—42no—
221C22×D17—42no—
222C22×D17—42no—
223C22×D17—42no—
224C22×D17—42no—
225C22×D17—42no—
226C22×D17—42no—
227C22×D17—42no—
228C22×D17—42no—
229C22×D17—42no—
230C22×D17—42no—
231C22×D17—42no—
232C2×C4×C17—24no—
233C2×C4×C17—22no—
234C2×C4×C17—22no—
235C2×C4×C17—22no—
236C2×C4×C17—22no—
237C2×C4×C17—24no—
238C2×C4×C17—24no—
239C2×C4×C17—24no—
240C2×C4×C17—22no—
241C2×C4×C17—22no—
242C2×C4×C17—24no—
243C2×C4×C17—22no—
244C2×C4×C17—22no—
245C2×C4×C17—22no—
246C2×C4×C17—22no—
247C2×C4×C17—22no—
248C2×C4×C17—41no—
249C2×C4×C17—41no—
250C2×C4×C17—41no—
251C2×C4×C17—41no—
252C2×C4×C17—42no—
253C2×C4×C17—42no—
254C2×C4×C17—42no—
255C2×C4×C17—41no—
256C2×C4×C17—41no—
257C2×C4×C17—41no—
258C2×C4×C17—41no—
259C2×C4×C17—41no—
260C2×C4×C17—41no—
261C2×C4×C17—41no—
262C2×C4×C17—41no—
263C2×C4×C17—42no—
264C2×C4×C17—42no—
265C2×C4×C17—42no—
266C2×C4×C17—42no—
267C2×C4×C17C4×D414yes—
268C2×C4×C17#12414yes—
269C2×Dic17—24no—
270C2×Dic17—24no—
271C2×Dic17—24no—
272C2×Dic17—24no—
273C2×Dic17—24no—
274C2×Dic17—24no—
275C2×Dic17—42no—
276C2×Dic17—42no—
277C2×Dic17—42no—
278C2×Dic17—42no—
279C2×Dic17—42no—
280C2×Dic17—42no—
281C2×Dic17—42no—
282C2×Dic17—42no—
283C2×Dic17—42no—
284C2×Dic17—42no—
285C2×Dic17—42no—
286C2×Dic17—42no—
287C2×Dic17—42no—
288C2×Dic17—42no—
289C2×Dic17—42no—
290D68—416no—
291C2×#394—171no—
292C2×#390—171no—
293C2×#387D1711yescharacteristic, complemented
294#1442—171no—
295#1441—171no—
296#1440—172no—
297#1439—172no—
298#1438—171no—
299#1437—172no—
300#1437—172no—
301C4×#95—171no—
302C22×C17C2×C4×D411yescharacteristic
303C22×C17C4×#4112yes—
304C22×C17#51812yes—
305C22×C17#51811yescharacteristic
306C22×C17#52011yescharacteristic
307C22×C17C2×#12411yescharacteristic
308C22×C17#52912yes—
309C22×C17#53211yescharacteristic
310C22×C17#53411yescharacteristic
311C22×C17#53511yescharacteristic
312C22×C17#53511yescharacteristic
313C22×C17#53711yescharacteristic
314C22×C17#53812yes—
315C22×C17#53812yes—
316C22×C17#53812yes—
317C22×C17#53812yes—
318C22×C17#53912yes—
319C22×C17#54012yes—
320C22×C17#54011yescharacteristic
321C22×C17#54012yes—
322C22×C17#54012yes—
323C22×C17#54212yes—
324C22×C17#54211yescharacteristic
325C2×D17—44no—
326C2×D17—44no—
327C2×D17—44no—
328C2×D17—44no—
329C2×D17—44no—
330C2×D17—44no—
331C2×D17—44no—
332C2×D17—82no—
333C2×D17—82no—
334C2×D17—82no—
335C2×D17—82no—
336C2×D17—82no—
337C2×D17—82no—
338C2×D17—82no—
339C2×D17—82no—
340C4×C17—24no—
341C4×C17—24no—
342C4×C17—42no—
343C4×C17—42no—
344C4×C17—42no—
345C4×C17—81no—
346C4×C17—81no—
347Dic17—44no—
348Dic17—44no—
349Dic17—82no—
350C23×D4—171no—
351C22×C4⋊C4—171no—
352C22×C4⋊C4C2×D1711yescharacteristic
353C22×C4⋊C4C2×D1711yescharacteristic
354C22×C22⋊C4—172no—
355C22×C22⋊C4—172no—
356C22×C22⋊C4—172no—
357C22×C22⋊C4—172no—
358C22×C22⋊C4—172no—
359C22×C22⋊C4—172no—
360C22×C42—172no—
361C22×C42C2×D1711yescharacteristic
362#394—178no—
363#390—344no—
364#387—24no—
365C2×#95—171no—
366C2×#95—171no—
367C2×#95—172no—
368C2×#95—172no—
369C2×#95—172no—
370C2×#95—172no—
371C2×#95—172no—
372C2×#95C2×D1711yescharacteristic
373C2×#95C2×D1711yescharacteristic
374C2×#95C2×D1711yescharacteristic
375C2×#95C2×D1711yescharacteristic
376C2×C17C4×#12411yescharacteristic
377C2×C17#232111yescharacteristic
378C2×C17#232411yescharacteristic
379C2×C17#233312yes—
380C2×C17#233512yes—
381C2×C17#263911yescharacteristic
382C2×C17#264111yescharacteristic
383C2×C17#264812yes—
384C2×C17#265112yes—
385C2×C17#266111yescharacteristic
386C2×C17#266211yescharacteristic
387D17—84no—
388C25—172no—
389C22×D4—174no—
390C22×D4—342no—
391C22×D4—342no—
392C22×D4—682no—
393C22×D4—682no—
394C23×C4—172no—
395C23×C4—172no—
396C23×C4—172no—
397C23×C4C22×D1711yescharacteristic
398C23×C4C22×D1711yescharacteristic
399C23×C4C22×D1711yescharacteristic
400C23×C4C22×D1711yescharacteristic
401C23×C4C22×D1711yescharacteristic
402C23×C4C22×D1711yescharacteristic
403C23×C4C22×D1711yescharacteristic
404C2×C4⋊C4—22no—
405C2×C4⋊C4—22no—
406C2×C4⋊C4—22no—
407C2×C4⋊C4—42no—
408C2×C4⋊C4—342no—
409C2×C4⋊C4—682no—
410C2×C4⋊C4C4×D1714yes—
411C2×C22⋊C4—178no—
412C2×C22⋊C4—178no—
413C2×C22⋊C4—344no—
414C2×C22⋊C4—344no—
415C2×C22⋊C4—344no—
416C2×C22⋊C4—344no—
417C2×C22⋊C4—344no—
418C2×C22⋊C4—344no—
419C2×C22⋊C4—682no—
420C2×C22⋊C4—682no—
421C2×C22⋊C4—682no—
422C2×C22⋊C4—682no—
423C2×C22⋊C4—682no—
424C2×C22⋊C4—682no—
425C2×C22⋊C4—682no—
426C2×C22⋊C4—682no—
427C2×C22⋊C4—682no—
428C2×C22⋊C4—682no—
429C2×C42—42no—
430C2×C42—344no—
431C2×C42—682no—
432C2×C42—682no—
433C2×C42C22⋊D1714yes—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—174no—
439#95—344no—
440#95—342no—
441#95—682no—
442#95—682no—
443#95—682no—
444#95—682no—
445C17#1432511yescharacteristic, complemented, Sylow
446C24—174no—
447C24—342no—
448C24—342no—
449C24—342no—
450C24—342no—
451C24—342no—
452C24—342no—
453C24—342no—
454C24—342no—
455C24—342no—
456C24—342no—
457C24—342no—
458C24—342no—
459C24—342no—
460C24—342no—
461C24C23×D1711yescharacteristic
462C2×D4—348no—
463C2×D4—688no—
464C2×D4—684no—
465C2×D4—684no—
466C2×D4—688no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—21no—
490C22×C4—22no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—174no—
497C22×C4—174no—
498C22×C4—342no—
499C22×C4—342no—
500C22×C4—342no—
501C22×C4—342no—
502C22×C4—342no—
503C22×C4—342no—
504C22×C4—342no—
505C22×C4—342no—
506C22×C4—342no—
507C22×C4—342no—
508C22×C4—342no—
509C22×C4—342no—
510C22×C4—342no—
511C22×C4—342no—
512C22×C4—342no—
513C22×C4—342no—
514C22×C4—342no—
515C22×C4—342no—
516C22×C4—342no—
517C22×C4C2×C4×D1712yes—
518C22×C4C2×C4×D1712yes—
519C22×C4C2×C4×D1712yes—
520C22×C4#6325212yes—
521C22×C4#6325614yes—
522C22×C4C2×C22⋊D1712yes—
523C22×C4C2×C22⋊D1712yes—
524C22×C4C2×C22⋊D1712yes—
525C4⋊C4—28no—
526C4⋊C4—44no—
527C4⋊C4—684no—
528C22⋊C4—688no—
529C22⋊C4—688no—
530C22⋊C4—688no—
531C22⋊C4—688no—
532C22⋊C4—688no—
533C22⋊C4—688no—
534C42—44no—
535C42—688no—
536C23—344no—
537C23—344no—
538C23—344no—
539C23—344no—
540C23—344no—
541C23—344no—
542C23—344no—
543C23—682no—
544C23—682no—
545C23—682no—
546C23—682no—
547C23—682no—
548C23—682no—
549C23—682no—
550C23—682no—
551C23—682no—
552C23—682no—
553C23—682no—
554C23—682no—
555C23—682no—
556C23—682no—
557C23—682no—
558C23—682no—
559C23—682no—
560C23—682no—
561C23—682no—
562C23—682no—
563C23—682no—
564C23—682no—
565C23—682no—
566C23—682no—
567C23—682no—
568C23—682no—
569C23—682no—
570C23—682no—
571C23C22×C4×D1711yescharacteristic
572C23C2×#6325211yescharacteristic
573C23#1058769112yes—
574C23C2×#6325612yes—
575C23#41×D1712yes—
576C23#1058769612yes—
577C23#1058769611yescharacteristic
578C23#1058769611yescharacteristic
579C23#1058769611yescharacteristic
580C23#1058769711yescharacteristic
581C23C22×C22⋊D1711yescharacteristic
582D4—6816no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—344no—
619C2×C4—344no—
620C2×C4—344no—
621C2×C4—344no—
622C2×C4—344no—
623C2×C4—344no—
624C2×C4—682no—
625C2×C4—682no—
626C2×C4—682no—
627C2×C4—682no—
628C2×C4—682no—
629C2×C4—682no—
630C2×C4—682no—
631C2×C4—682no—
632C2×C4—682no—
633C2×C4—682no—
634C2×C4—682no—
635C2×C4—682no—
636C2×C4—682no—
637C2×C4—682no—
638C2×C4—682no—
639C2×C4C4×C22⋊D1714yes—
640C2×C4#1058763314yes—
641C22—684no—
642C22—684no—
643C22—684no—
644C22—684no—
645C22—684no—
646C22—684no—
647C22—684no—
648C22—1362no—
649C22—1362no—
650C22—1362no—
651C22—1362no—
652C22—1362no—
653C22—1362no—
654C22—1362no—
655C22—1362no—
656C22C2×C4×C22⋊D1711yescharacteristic
657C22C4×#6325612yes—
658C22#1176328112yes—
659C22#1176328311yescharacteristic
660C22#1176328911yescharacteristic
661C22C2×#1058763311yescharacteristic
662C22#1176334612yes—
663C22#1176335511yescharacteristic
664C22#1176336811yescharacteristic
665C22#1176337311yescharacteristic
666C22#1176337311yescharacteristic
667C22#1176338711yescharacteristic
668C22#1176339212yes—
669C22#1176339512yes—
670C22#1176339812yes—
671C22#1176339912yes—
672C22#1176340812yes—
673C22#1176341112yes—
674C22#1176341411yescharacteristic
675C22#1176341512yes—
676C22#1176341512yes—
677C22#1176342512yes—
678C22#1176343011yescharacteristic
679C4—24no—
680C4—24no—
681C4—42no—
682C4—42no—
683C4—42no—
684C4—81no—
685C4—81no—
686C4—684no—
687C4—684no—
688C4—1362no—
689C2—1364no—
690C2C4×#1058763311yescharacteristic
691C2#1453143811yescharacteristic
692C2#1453147711yescharacteristic
693C2#1453154812yes—
694C2#1453157312yes—
695C2#1453471811yescharacteristic
696C2#1453473011yescharacteristic
697C2#1453477812yes—
698C2#1453482312yes—
699C2#1453491211yescharacteristic
700C2#1453492411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC2×C17×#387C2
socle, cosocleC24×C17C24
prime radicalC23×D68C22
hypercenterC2×#387D17
hypocenterC17#14325
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×#387

Complements

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

NH ≅ G/Ntype
3. #14527040C2semidirect
4. #14527037C2semidirect
5. #14527032C2semidirect
6. #14527026C2semidirect
8. #14527020C2semidirect
9. #14527020C2semidirect
10. C4×#10587502C2semidirect
11. C2×C17×#387C2semidirect
21. #11762726C4semidirect
293. C2×#387D17semidirect
445. C17#14325semidirect

Maximal subgroups

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

subgroupindex
C2×#117627262
C2×#117627112
#145270402
#145270372
#145270322
#145270262
#145270242
#145270202
C4×#105875022
C2×C17×#3872
#1432517