Finite Group Databasebeta

← #16963496

Subgroups — #16963496

Order 4352.

2070 conjugacy classes of subgroups.

The subgroup lattice has 752 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587621C211yesmaximal, characteristic
2C2×#11762712C211yesmaximal, characteristic
3#14527038C212yesmaximal, complemented
4#14527031C212yesmaximal, complemented
5#14527031C212yesmaximal, complemented
6#14527019C212yesmaximal, complemented
7#14527007C212yesmaximal, complemented
8#14527006C212yesmaximal, complemented
9C2×C17×#386C211yesmaximal, characteristic
10C24×Dic17C2211yescharacteristic
11C2×#10587621—24no—
12C2×#10587621—24no—
13C2×#10587621C418yescomplemented
14C22×C4⋊Dic17C2211yescharacteristic
15C22×#63225C2212yes—
16C22×C22⋊Dic17C2212yes—
17C22×C22⋊Dic17C2212yes—
18C22×C22⋊Dic17C2212yes—
19C22×C22⋊Dic17C2211yescharacteristic
20C22×C4×Dic17C2211yescharacteristic
21#11762712—24no—
22C2×#10587502C2212yes—
23C2×#10587502C2212yes—
24C2×#10587502C2212yes—
25C2×#10587502C2212yes—
26C2×#10587502C2212yes—
27C2×#10587502C2212yes—
28C2×#10587502C2212yes—
29C2×#10587502C2212yes—
30C23×D4×C17C2211yescharacteristic
31C24×C4×C17C2211yescharacteristic
32C22×C22⋊C4×C17C2212yes—
33C22×C22⋊C4×C17C2211yescharacteristic
34C17×#386—24no—
35C2×C17×#95C2212yes—
36C25×C17C2311yescharacteristic
37C25×C17C2311yescharacteristic
38C22×D4×C17—41no—
39C22×D4×C17—41no—
40C22×D4×C17—41no—
41C22×D4×C17—41no—
42C22×D4×C17C2×C414yes—
43C22×D4×C17D414yes—
44C22×D4×C17D414yes—
45C23×C4×C17—22no—
46C23×C4×C17—24no—
47C23×C4×C17—41no—
48C23×C4×C17—41no—
49C23×C4×C17—42no—
50C23×C4×C17C2312yes—
51C23×C4×C17C2312yes—
52C23×C4×C17C2311yescharacteristic
53C23×Dic17—22no—
54C23×Dic17—22no—
55C23×Dic17—41no—
56C23×Dic17—41no—
57C23×Dic17—41no—
58C23×Dic17—41no—
59C23×Dic17C2×C414yescomplemented
60C23×Dic17C2312yes—
61C23×Dic17C2312yes—
62C23×Dic17C2312yes—
63C23×Dic17C2312yes—
64C2×C22⋊C4×C17—24no—
65C2×C22⋊C4×C17—22no—
66C2×C22⋊C4×C17—24no—
67C2×C22⋊C4×C17—42no—
68C2×C22⋊C4×C17—41no—
69C2×C22⋊C4×C17—41no—
70C2×C22⋊Dic17—24no—
71C2×C22⋊Dic17—24no—
72C2×C22⋊Dic17—22no—
73C2×C22⋊Dic17—22no—
74C2×C22⋊Dic17—24no—
75C2×C22⋊Dic17—24no—
76C2×C22⋊Dic17—24no—
77C2×C22⋊Dic17—24no—
78C2×C22⋊Dic17—42no—
79C2×C22⋊Dic17—42no—
80C2×C22⋊Dic17C2×C418yes—
81C2×C22⋊Dic17C2×C414yescomplemented
82#10587621—232no—
83C2×C4⋊Dic17—22no—
84C2×C4⋊Dic17—41no—
85C2×C4⋊Dic17—41no—
86C2×#63225—24no—
87C2×#63225—24no—
88C2×#63225C2×C418yescomplemented
89C2×C4×Dic17—22no—
90C2×C4×Dic17—41no—
91C2×C4×Dic17—41no—
92C17×#95—24no—
93#10587502—24no—
94#10587502—24no—
95#10587502—42no—
96#10587502—42no—
97#10587502—42no—
98#10587502—42no—
99#10587502—42no—
100#10587502—42no—
101C24×C17—22no—
102C24×C17—22no—
103C24×C17—22no—
104C24×C17—22no—
105C24×C17—21no—
106C24×C17—21no—
107C24×C17—21no—
108C24×C17—21no—
109C24×C17—21no—
110C24×C17—21no—
111C24×C17—22no—
112C24×C17—21no—
113C24×C17—21no—
114C24×C17—22no—
115C24×C17—22no—
116C24×C17—22no—
117C24×C17C22×C412yes—
118C24×C17C22×C412yes—
119C24×C17C2×D412yes—
120C24×C17C2×D412yes—
121C24×C17C2×D412yes—
122C24×C17C2×D412yes—
123C24×C17C2411yescharacteristic
124C2×D4×C17—44no—
125C2×D4×C17—44no—
126C2×D4×C17—44no—
127C2×D4×C17—44no—
128C2×D4×C17—44no—
129C2×D4×C17—44no—
130C2×D4×C17C22⋊C4116yes—
131C22×C4×C17—21no—
132C22×C4×C17—21no—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17—22no—
136C22×C4×C17—22no—
137C22×C4×C17—22no—
138C22×C4×C17—22no—
139C22×C4×C17—22no—
140C22×C4×C17—22no—
141C22×C4×C17—22no—
142C22×C4×C17—22no—
143C22×C4×C17—22no—
144C22×C4×C17—22no—
145C22×C4×C17—22no—
146C22×C4×C17—22no—
147C22×C4×C17—22no—
148C22×C4×C17—44no—
149C22×C4×C17—48no—
150C22×C4×C17—48no—
151C22×C4×C17—44no—
152C22×C4×C17—44no—
153C22×C4×C17C22×C412yes—
154C22×C4×C17C2×D412yes—
155C22×C4×C17C2×D412yes—
156C22×C4×C17#4114yes—
157C22×Dic17—28no—
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×Dic17—22no—
168C22×Dic17—22no—
169C22×Dic17—22no—
170C22×Dic17—22no—
171C22×Dic17—22no—
172C22×Dic17—22no—
173C22×Dic17—22no—
174C22×Dic17—22no—
175C22×Dic17—22no—
176C22×Dic17—22no—
177C22×Dic17—22no—
178C22×Dic17—22no—
179C22×Dic17—22no—
180C22×Dic17—22no—
181C22×Dic17—22no—
182C22×Dic17—22no—
183C22×Dic17—44no—
184C22×Dic17—44no—
185C22×Dic17—44no—
186C22×Dic17—44no—
187C22×Dic17—44no—
188C22×Dic17—44no—
189C22×Dic17C22×C414yes—
190C22×Dic17C22×C414yes—
191C22×Dic17#4114yes—
192C22⋊C4×C17—48no—
193C22⋊C4×C17—44no—
194C22⋊Dic17—216no—
195C22⋊Dic17—28no—
196C22⋊Dic17—48no—
197C22⋊Dic17—48no—
198C4⋊Dic17—44no—
199#63225—216no—
200C4×Dic17—44no—
201#14505—171nomaximal, Sylow
202C23×C17—24no—
203C23×C17—24no—
204C23×C17—22no—
205C23×C17—22no—
206C23×C17—22no—
207C23×C17—22no—
208C23×C17—24no—
209C23×C17—24no—
210C23×C17—22no—
211C23×C17—22no—
212C23×C17—22no—
213C23×C17—22no—
214C23×C17—22no—
215C23×C17—22no—
216C23×C17—22no—
217C23×C17—22no—
218C23×C17—24no—
219C23×C17—24no—
220C23×C17—24no—
221C23×C17—24no—
222C23×C17—24no—
223C23×C17—24no—
224C23×C17—24no—
225C23×C17—24no—
226C23×C17—42no—
227C23×C17—42no—
228C23×C17—42no—
229C23×C17—42no—
230C23×C17—42no—
231C23×C17—42no—
232C23×C17—42no—
233C23×C17—42no—
234C23×C17—41no—
235C23×C17—41no—
236C23×C17—42no—
237C23×C17—42no—
238C23×C17—41no—
239C23×C17—41no—
240C23×C17—41no—
241C23×C17—41no—
242C23×C17—42no—
243C23×C17—42no—
244C23×C17—41no—
245C23×C17—41no—
246C23×C17C2×C22⋊C414yes—
247C23×C17C2×C22⋊C414yes—
248C23×C17C23×C411yescharacteristic
249C23×C17C22×D411yescharacteristic
250C23×C17C22×D411yescharacteristic
251C23×C17C2×#4112yes—
252C23×C17C2×#4112yes—
253C23×C17H(5,𝔽2)11yescharacteristic
254C23×C17H(5,𝔽2)11yescharacteristic
255C23×C17H(5,𝔽2)11yescharacteristic
256C23×C17H(5,𝔽2)11yescharacteristic
257C23×C17H(5,𝔽2)12yes—
258C23×C172−1+412yes—
259D4×C17—416no—
260C2×C4×C17—22no—
261C2×C4×C17—22no—
262C2×C4×C17—22no—
263C2×C4×C17—22no—
264C2×C4×C17—22no—
265C2×C4×C17—22no—
266C2×C4×C17—24no—
267C2×C4×C17—24no—
268C2×C4×C17—24no—
269C2×C4×C17—416no—
270C2×C4×C17—42no—
271C2×C4×C17—42no—
272C2×C4×C17—42no—
273C2×C4×C17—42no—
274C2×C4×C17—42no—
275C2×C4×C17—42no—
276C2×C4×C17—42no—
277C2×C4×C17—42no—
278C2×C4×C17—42no—
279C2×C4×C17—42no—
280C2×C4×C17—42no—
281C2×C4×C17C2×C22⋊C414yes—
282C2×Dic17—24no—
283C2×Dic17—24no—
284C2×Dic17—24no—
285C2×Dic17—24no—
286C2×Dic17—24no—
287C2×Dic17—24no—
288C2×Dic17—24no—
289C2×Dic17—24no—
290C2×Dic17—24no—
291C2×Dic17—416no—
292C2×Dic17—42no—
293C2×Dic17—42no—
294C2×Dic17—42no—
295C2×Dic17—42no—
296C2×Dic17—42no—
297C2×Dic17—42no—
298C2×Dic17—42no—
299C2×Dic17—42no—
300C2×Dic17—42no—
301C2×Dic17—42no—
302C2×Dic17—42no—
303C2×Dic17—42no—
304C2×Dic17—42no—
305C2×Dic17—42no—
306C2×Dic17—42no—
307C2×Dic17—42no—
308C2×Dic17—42no—
309C2×Dic17—42no—
310C2×Dic17—42no—
311C22×#123—171no—
312C2×#390—171no—
313C2×#386D1711yescharacteristic
314#1441—172no—
315#1440—172no—
316#1440—172no—
317#1436—172no—
318C22⋊#95—172no—
319C22⋊#95—172no—
320C22×C17—24no—
321C22×C17—24no—
322C22×C17—28no—
323C22×C17—28no—
324C22×C17—44no—
325C22×C17—44no—
326C22×C17—42no—
327C22×C17—42no—
328C22×C17—42no—
329C22×C17—42no—
330C22×C17—42no—
331C22×C17—42no—
332C22×C17—44no—
333C22×C17—44no—
334C22×C17—44no—
335C22×C17—44no—
336C22×C17—42no—
337C22×C17—42no—
338C22×C17—42no—
339C22×C17—42no—
340C22×C17—42no—
341C22×C17—42no—
342C22×C17C22×C22⋊C411yescharacteristic
343C22×C17C4×#4112yes—
344C22×C17#51811yescharacteristic
345C22×C17#51811yescharacteristic
346C22×C17#52012yes—
347C22×C17#53411yescharacteristic
348C22×C17#53411yescharacteristic
349C22×C17#53411yescharacteristic
350C22×C17#53411yescharacteristic
351C22×C17#53812yes—
352C22×C17#53912yes—
353C22×C17#54012yes—
354C22×C17#54212yes—
355C22×C17#54612yes—
356C22×C17#54612yes—
357C22×C17#54712yes—
358C22×C17#54712yes—
359C22×C17#55111yescharacteristic
360C22×C17#55312yes—
361C22×C17#55911yescharacteristic
362C22×C17#56011yescharacteristic
363C22×C17#56212yes—
364C22×C17#56311yescharacteristic
365C4×C17—24no—
366C4×C17—44no—
367C4×C17—82no—
368Dic17—44no—
369Dic17—44no—
370Dic17—44no—
371Dic17—82no—
372C23×D4C2×D1711yescharacteristic
373C24×C4—171no—
374C24×C4C2×D1711yescharacteristic
375C2×#123—178no—
376C2×#123—344no—
377C2×#123—344no—
378C22×C4⋊C4—172no—
379C22×C4⋊C4—171no—
380C22×C22⋊C4—172no—
381C22×C22⋊C4—171no—
382C22×C22⋊C4—172no—
383C22×C22⋊C4—172no—
384C22×C22⋊C4C2×D1712yes—
385C22×C22⋊C4C2×D1711yescharacteristic
386C22×C42—171no—
387#390—344no—
388#386—24no—
389C2×#95—172no—
390C2×#95—172no—
391C2×#95—172no—
392C2×#95—172no—
393C2×#95—172no—
394C2×#95—172no—
395C2×#95—172no—
396C2×#95—172no—
397C2×#95C2×D1712yes—
398C2×C17—44no—
399C2×C17—44no—
400C2×C17#230811yescharacteristic
401C2×C17#233512yes—
402C2×C17#233912yes—
403C2×C17#237411yescharacteristic
404C2×C17#237611yescharacteristic
405C2×C17#269711yescharacteristic
406C2×C17#270112yes—
407C2×C17#270812yes—
408C2×C17#270911yescharacteristic
409C2×C17#271711yescharacteristic
410C2×C17#272411yescharacteristic
411C25C22×D1711yescharacteristic
412C25C22×D1711yescharacteristic
413C22×D4—41no—
414C22×D4—41no—
415C22×D4—41no—
416C22×D4—41no—
417C22×D4C4×D1714yes—
418C22×D4D6814yes—
419C22×D4C22⋊D1714yes—
420C23×C4—22no—
421C23×C4—24no—
422C23×C4—41no—
423C23×C4—41no—
424C23×C4—42no—
425C23×C4—172no—
426C23×C4—172no—
427C23×C4—172no—
428C23×C4—174no—
429C23×C4—172no—
430C23×C4—342no—
431C23×C4—342no—
432C23×C4—681no—
433C23×C4—681no—
434C23×C4—681no—
435C23×C4—681no—
436C23×C4C22×D1712yes—
437C23×C4C22×D1712yes—
438C23×C4C22×D1711yescharacteristic
439#123—3432no—
440C2×C4⋊C4—178no—
441C2×C4⋊C4—344no—
442C2×C4⋊C4—344no—
443C2×C4⋊C4—342no—
444C2×C4⋊C4—681no—
445C2×C4⋊C4—681no—
446C2×C22⋊C4—24no—
447C2×C22⋊C4—22no—
448C2×C22⋊C4—24no—
449C2×C22⋊C4—42no—
450C2×C22⋊C4—41no—
451C2×C22⋊C4—41no—
452C2×C22⋊C4—178no—
453C2×C22⋊C4—174no—
454C2×C22⋊C4—342no—
455C2×C22⋊C4—344no—
456C2×C22⋊C4—342no—
457C2×C22⋊C4—344no—
458C2×C22⋊C4—344no—
459C2×C22⋊C4—344no—
460C2×C22⋊C4—344no—
461C2×C22⋊C4—344no—
462C2×C22⋊C4—682no—
463C2×C22⋊C4—682no—
464C2×C42—342no—
465C2×C42—681no—
466C2×C42—681no—
467#95—24no—
468#95—344no—
469#95—344no—
470#95—682no—
471#95—682no—
472#95—682no—
473#95—682no—
474#95—682no—
475#95—682no—
476C17#1450511yescharacteristic, complemented, Sylow
477C24—22no—
478C24—22no—
479C24—22no—
480C24—21no—
481C24—21no—
482C24—22no—
483C24—21no—
484C24—21no—
485C24—21no—
486C24—21no—
487C24—21no—
488C24—21no—
489C24—22no—
490C24—22no—
491C24—22no—
492C24—22no—
493C24C2×C4×D1712yes—
494C24C2×C4×D1712yes—
495C24C2×D6812yes—
496C24C2×D6812yes—
497C24C2×C22⋊D1712yes—
498C24C2×C22⋊D1712yes—
499C24C23×D1711yescharacteristic
500C2×D4—44no—
501C2×D4—44no—
502C2×D4—44no—
503C2×D4—44no—
504C2×D4—44no—
505C2×D4—44no—
506C2×D4#63227116yes—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—22no—
512C22×C4—22no—
513C22×C4—22no—
514C22×C4—22no—
515C22×C4—22no—
516C22×C4—22no—
517C22×C4—22no—
518C22×C4—22no—
519C22×C4—22no—
520C22×C4—22no—
521C22×C4—22no—
522C22×C4—22no—
523C22×C4—22no—
524C22×C4—44no—
525C22×C4—48no—
526C22×C4—48no—
527C22×C4—44no—
528C22×C4—44no—
529C22×C4—174no—
530C22×C4—174no—
531C22×C4—174no—
532C22×C4—342no—
533C22×C4—342no—
534C22×C4—342no—
535C22×C4—342no—
536C22×C4—342no—
537C22×C4—342no—
538C22×C4—342no—
539C22×C4—342no—
540C22×C4—342no—
541C22×C4—342no—
542C22×C4—342no—
543C22×C4—342no—
544C22×C4—342no—
545C22×C4—342no—
546C22×C4—342no—
547C22×C4—342no—
548C22×C4—342no—
549C22×C4—342no—
550C22×C4—348no—
551C22×C4—342no—
552C22×C4—342no—
553C22×C4—342no—
554C22×C4—342no—
555C22×C4—342no—
556C22×C4—342no—
557C22×C4—342no—
558C22×C4—684no—
559C22×C4—684no—
560C22×C4—684no—
561C22×C4—684no—
562C22×C4—684no—
563C22×C4—684no—
564C22×C4C2×C4×D1712yes—
565C22×C4C2×D6812yes—
566C22×C4#6325414yes—
567C22×C4C2×C22⋊D1712yes—
568C4⋊C4—3416no—
569C4⋊C4—684no—
570C22⋊C4—48no—
571C22⋊C4—44no—
572C22⋊C4—348no—
573C22⋊C4—3416no—
574C22⋊C4—688no—
575C22⋊C4—688no—
576C42—684no—
577C23—24no—
578C23—24no—
579C23—24no—
580C23—24no—
581C23—24no—
582C23—24no—
583C23—22no—
584C23—22no—
585C23—22no—
586C23—22no—
587C23—24no—
588C23—24no—
589C23—22no—
590C23—22no—
591C23—22no—
592C23—22no—
593C23—22no—
594C23—22no—
595C23—22no—
596C23—22no—
597C23—24no—
598C23—24no—
599C23—24no—
600C23—24no—
601C23—41no—
602C23—41no—
603C23—42no—
604C23—42no—
605C23—41no—
606C23—41no—
607C23—41no—
608C23—41no—
609C23—42no—
610C23—42no—
611C23—41no—
612C23—41no—
613C23—42no—
614C23—42no—
615C23—42no—
616C23—42no—
617C23—42no—
618C23—42no—
619C23—42no—
620C23—42no—
621C23C2×#6322714yes—
622C23C2×#6322714yes—
623C23C22×C4×D1711yescharacteristic
624C23C22×D6811yescharacteristic
625C23C2×#6325412yes—
626C23#1058769111yescharacteristic
627C23#1058769111yescharacteristic
628C23#1058769111yescharacteristic
629C23#1058769111yescharacteristic
630C23#1058769112yes—
631C23#41×D1712yes—
632C23#1058769712yes—
633C23C22×C22⋊D1711yescharacteristic
634D4—416no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—416no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—344no—
657C2×C4—344no—
658C2×C4—344no—
659C2×C4—344no—
660C2×C4—344no—
661C2×C4—344no—
662C2×C4—344no—
663C2×C4—344no—
664C2×C4—344no—
665C2×C4—6816no—
666C2×C4—682no—
667C2×C4—682no—
668C2×C4—682no—
669C2×C4—682no—
670C2×C4—682no—
671C2×C4—682no—
672C2×C4—682no—
673C2×C4—682no—
674C2×C4—682no—
675C2×C4—682no—
676C2×C4—682no—
677C2×C4—682no—
678C2×C4—682no—
679C2×C4—682no—
680C2×C4—682no—
681C2×C4—682no—
682C2×C4—682no—
683C2×C4—682no—
684C2×C4—682no—
685C2×C4C2×#6322714yes—
686C22—28no—
687C22—28no—
688C22—24no—
689C22—24no—
690C22—42no—
691C22—42no—
692C22—44no—
693C22—44no—
694C22—42no—
695C22—42no—
696C22—42no—
697C22—42no—
698C22—42no—
699C22—42no—
700C22—42no—
701C22—42no—
702C22—42no—
703C22—42no—
704C22—44no—
705C22—44no—
706C22—44no—
707C22—44no—
708C22C22×#6322711yescharacteristic
709C22C4×#6325412yes—
710C22#1176328211yescharacteristic
711C22#1176328211yescharacteristic
712C22#1176328812yes—
713C22#1176336811yescharacteristic
714C22#1176336811yescharacteristic
715C22#1176337111yescharacteristic
716C22#1176337111yescharacteristic
717C22#1176339412yes—
718C22#1176340212yes—
719C22#1176340912yes—
720C22#1176342612yes—
721C22#1176345312yes—
722C22#1176345412yes—
723C22#1176345812yes—
724C22#1176346312yes—
725C22#1176349011yescharacteristic
726C22#1176350612yes—
727C22#1176354311yescharacteristic
728C22#1176354811yescharacteristic
729C22#1176355412yes—
730C22#1176356211yescharacteristic
731C4—24no—
732C4—44no—
733C4—82no—
734C4—684no—
735C4—684no—
736C4—684no—
737C4—1362no—
738C2—44no—
739C2—44no—
740C2#1453135911yescharacteristic
741C2#1453157512yes—
742C2#1453160912yes—
743C2#1453191711yescharacteristic
744C2#1453192711yescharacteristic
745C2#1453534511yescharacteristic
746C2#1453539312yes—
747C2#1453550012yes—
748C2#1453551411yescharacteristic
749C2#1453562211yescharacteristic
750C2#1453571211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC2×C17×#386C2
socle, cosocleC24×C17C24
prime radicalC23×D4×C17C22
hypercenterC2×#386D17
hypocenterC17#14505
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×#386

Complements

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

NH ≅ G/Ntype
3. #14527038C2semidirect
4. #14527031C2semidirect
5. #14527031C2semidirect
6. #14527019C2semidirect
7. #14527007C2semidirect
8. #14527006C2semidirect
13. C2×#10587621C4semidirect
59. C23×Dic17C2×C4semidirect
81. C2×C22⋊Dic17C2×C4semidirect
88. C2×#63225C2×C4semidirect
476. C17#14505semidirect

Maximal subgroups

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

subgroupindex
C22×#105876212
C2×#117627122
#145270382
#145270312
#145270192
#145270072
#145270062
C2×C17×#3862
#1450517