Finite Group Databasebeta

← #16962740

Subgroups — #16962740

Order 4352.

2118 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762710C211yesmaximal, characteristic
2C2×#11762687C212yesmaximal
3C2×#11762683C212yesmaximal, complemented
4C2×#11762680C212yesmaximal, complemented
5#14527037C211yesmaximal, characteristic, complemented
6#14527028C212yesmaximal, complemented
7#14527028C212yesmaximal, complemented
8#14527012C212yesmaximal, complemented
9C17×#1441C211yesmaximal, characteristic, complemented
10C23×D68C2211yescharacteristic
11C23×C4×D17C2212yes—
12C22×C4⋊Dic17C2211yescharacteristic
13C22×#63225C2212yes—
14C22×#63225C2212yes—
15C22×#63227C2212yes—
16C22×#63227C2212yes—
17C22×#63227C2212yes—
18C22×#63227C2212yes—
19C22×#63227C2212yes—
20C22×C4×Dic17C2211yescharacteristic
21#11762710C418yescomplemented
22#11762687—28no—
23#11762683—28no—
24#11762680—28no—
25C2×#10587502C2212yes—
26C2×#10587502C2212yes—
27C2×#10587502C2212yes—
28C2×#10587502C2212yes—
29C2×#10587502C2211yescharacteristic
30C22×C4⋊C4×C17C2211yescharacteristic
31C22×C4⋊C4×C17C2211yescharacteristic
32C2×C17×#95C2212yes—
33C2×C17×#95C2212yes—
34C2×C17×#95C2211yescharacteristic
35C24×D17C2312yes—
36C23×C4×C17C2312yes—
37C23×C4×C17C2311yescharacteristic
38C23×C4×C17C2311yescharacteristic
39C23×C4×C17C2312yes—
40C23×C4×C17C2311yescharacteristic
41C23×Dic17C2312yes—
42C23×Dic17C2312yes—
43C23×Dic17C2312yes—
44C22×D68—41no—
45C22×D68—41no—
46C22×D68—41no—
47C22×D68—41no—
48C22×D68C2×C414yes—
49C22×D68D414yes—
50C22×D68Q814yes—
51C22×C4×D17—24no—
52C22×C4×D17—24no—
53C22×C4×D17—24no—
54C22×C4×D17—42no—
55C22×C4×D17—42no—
56C22×C4×D17—42no—
57C22×C4×D17—42no—
58C2×C4⋊C4×C17—24no—
59C2×C4⋊C4×C17—41no—
60C2×C4⋊C4×C17—41no—
61C2×C4⋊C4×C17—41no—
62C2×C4⋊C4×C17C2×C414yescomplemented
63C2×#63227—24no—
64C2×#63227—24no—
65C2×#63227—24no—
66C2×#63227—24no—
67C2×#63227—24no—
68C2×#63227—24no—
69C2×#63227—24no—
70C2×#63227—24no—
71C2×#63227—24no—
72C2×#63227—42no—
73C2×#63227—42no—
74C2×#63227—42no—
75C2×#63227—42no—
76C2×#63227C2×C418yescomplemented
77C2×#63227C2×C418yescomplemented
78C2×C4⋊Dic17—24no—
79C2×C4⋊Dic17—41no—
80C2×#63225—24no—
81C2×#63225—42no—
82C2×#63225—42no—
83C2×#63225—42no—
84C2×#63225—42no—
85C2×#63225D418yescomplemented
86C2×C4×Dic17—41no—
87C2×C4×Dic17—41no—
88C2×C4×Dic17C2×C414yescomplemented
89C17×#95—24no—
90C17×#95—24no—
91C17×#95—41no—
92#10587502—24no—
93#10587502—24no—
94#10587502—24no—
95#10587502—24no—
96#10587502—41no—
97C24×C17C2411yescharacteristic
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—
111C23×D17C2×D414yes—
112C23×D17C2×Q814yes—
113C22×C4×C17—21no—
114C22×C4×C17—21no—
115C22×C4×C17—21no—
116C22×C4×C17—21no—
117C22×C4×C17—22no—
118C22×C4×C17—22no—
119C22×C4×C17—22no—
120C22×C4×C17—22no—
121C22×C4×C17—22no—
122C22×C4×C17—22no—
123C22×C4×C17—22no—
124C22×C4×C17—22no—
125C22×C4×C17—22no—
126C22×C4×C17—22no—
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17—22no—
132C22×C4×C17—22no—
133C22×C4×C17—22no—
134C22×C4×C17—22no—
135C22×C4×C17—22no—
136C22×C4×C17—22no—
137C22×C4×C17—22no—
138C22×C4×C17C22×C412yes—
139C22×C4×C17C22×C412yes—
140C22×C4×C17C22×C412yes—
141C22×C4×C17C2×D414yes—
142C22×C4×C17C2×D412yes—
143C22×C4×C17C2×Q812yes—
144C22×Dic17—22no—
145C22×Dic17—22no—
146C22×Dic17—22no—
147C22×Dic17—22no—
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×Dic17C22×C414yes—
162C22×Dic17C2×D414yes—
163C22×Dic17C2×D414yes—
164C22×Dic17#4114yes—
165C2×D68—44no—
166C2×D68—44no—
167C2×D68—44no—
168C2×D68—44no—
169C2×D68—44no—
170C2×D68—44no—
171C2×D68C4⋊C4116yescomplemented
172C2×C4×D17—48no—
173C2×C4×D17—48no—
174C2×C4×D17—48no—
175C2×C4×D17—48no—
176C2×C4×D17—48no—
177C2×C4×D17—48no—
178C2×C4×D17—48no—
179C4⋊C4×C17—44no—
180C4⋊C4×C17—44no—
181#63227—216no—
182#63227—216no—
183#63227—48no—
184#63227—48no—
185#63227—48no—
186C4⋊Dic17—44no—
187#63225—48no—
188#63225—48no—
189C4×Dic17—44no—
190#14440—171nomaximal, Sylow
191C23×C17C23×C411yescharacteristic
192C23×C17C22×D411yescharacteristic
193C23×C17C22×D412yes—
194C23×C17C22×Q811yescharacteristic
195C23×C17C2×#4112yes—
196C23×C17H(5,𝔽2)11yescharacteristic
197C23×C17H(5,𝔽2)11yescharacteristic
198C23×C17H(5,𝔽2)11yescharacteristic
199C23×C17H(5,𝔽2)11yescharacteristic
200C23×C17H(5,𝔽2)12yes—
201C23×C172−1+412yes—
202C22×D17—24no—
203C22×D17—24no—
204C22×D17—24no—
205C22×D17—24no—
206C22×D17—24no—
207C22×D17—24no—
208C22×D17—24no—
209C22×D17—24no—
210C22×D17—24no—
211C22×D17—24no—
212C22×D17—24no—
213C22×D17—24no—
214C22×D17—24no—
215C22×D17—24no—
216C22×D17—24no—
217C22×D17—24no—
218C22×D17—24no—
219C22×D17—24no—
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—
232C22×D17—42no—
233C22×D17—42no—
234C22×D17—42no—
235C22×D17—42no—
236C22×D17C2×C4⋊C418yes—
237C2×C4×C17—22no—
238C2×C4×C17—22no—
239C2×C4×C17—22no—
240C2×C4×C17—22no—
241C2×C4×C17—22no—
242C2×C4×C17—22no—
243C2×C4×C17—24no—
244C2×C4×C17—24no—
245C2×C4×C17—22no—
246C2×C4×C17—22no—
247C2×C4×C17—24no—
248C2×C4×C17—24no—
249C2×C4×C17—24no—
250C2×C4×C17—42no—
251C2×C4×C17—42no—
252C2×C4×C17—41no—
253C2×C4×C17—42no—
254C2×C4×C17—41no—
255C2×C4×C17—42no—
256C2×C4×C17—42no—
257C2×C4×C17—41no—
258C2×C4×C17—41no—
259C2×C4×C17—42no—
260C2×C4×C17—42no—
261C2×C4×C17—42no—
262C2×C4×C17—42no—
263C2×C4×C17—42no—
264C2×C4×C17—42no—
265C2×C4×C17—42no—
266C2×C4×C17—42no—
267C2×C4×C17C2×C4⋊C414yes—
268C2×Dic17—24no—
269C2×Dic17—24no—
270C2×Dic17—24no—
271C2×Dic17—24no—
272C2×Dic17—24no—
273C2×Dic17—24no—
274C2×Dic17—24no—
275C2×Dic17—24no—
276C2×Dic17—24no—
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×Dic17C4×D418yes—
289D68—416no—
290C4×D17—432no—
291C2×#390—171no—
292C2×#386—172no—
293C2×#385—172no—
294C2×#385—172no—
295#1441—171no—
296#1441D1711yescharacteristic, complemented
297#1439—172no—
298#1439—172no—
299C4⋊#95—172no—
300C22×C17C22×C4⋊C411yescharacteristic
301C22×C17C2×C4×D412yes—
302C22×C17#51811yescharacteristic
303C22×C17#51811yescharacteristic
304C22×C17#52012yes—
305C22×C17#53411yescharacteristic
306C22×C17#53411yescharacteristic
307C22×C17#53412yes—
308C22×C17#53712yes—
309C22×C17#53812yes—
310C22×C17#53912yes—
311C22×C17#54311yescharacteristic
312C22×C17#54311yescharacteristic
313C22×C17D4211yescharacteristic
314C22×C17D4212yes—
315C22×C17#54712yes—
316C22×C17D4×Q812yes—
317C22×C17#55012yes—
318C22×C17#55412yes—
319C22×C17#55911yescharacteristic
320C22×C17H(3,𝔽4)11yescharacteristic
321C22×C17#56212yes—
322C22×C17#56311yescharacteristic
323C2×D17—28no—
324C2×D17—28no—
325C2×D17—28no—
326C2×D17—44no—
327C2×D17—44no—
328C2×D17—44no—
329C2×D17—44no—
330C2×D17—44no—
331C2×D17—44no—
332C2×D17—44no—
333C2×D17—44no—
334C2×D17—44no—
335C2×D17—44no—
336C2×D17—44no—
337C2×D17—44no—
338C4×C17—24no—
339C4×C17—42no—
340C4×C17—42no—
341C4×C17—44no—
342C4×C17—82no—
343Dic17—28no—
344Dic17—44no—
345Dic17—82no—
346C23×D4—171no—
347C24×C4—172no—
348C22×C4⋊C4—172no—
349C22×C4⋊C4—171no—
350C22×C4⋊C4—172no—
351C22×C4⋊C4C2×D1711yescharacteristic
352C22×C4⋊C4C2×D1711yescharacteristic
353C22×C22⋊C4—172no—
354C22×C22⋊C4—172no—
355C22×C22⋊C4—172no—
356C22×C22⋊C4—172no—
357C22×C22⋊C4—172no—
358C22×C42—171no—
359#390—178no—
360#386—348no—
361#385—348no—
362#385—348no—
363C2×#95—172no—
364C2×#95—172no—
365C2×#95—172no—
366C2×#95—172no—
367C2×#95—171no—
368C2×#95C2×D1712yes—
369C2×#95C2×D1712yes—
370C2×#95C2×D1711yescharacteristic
371C2×C17#231511yescharacteristic
372C2×C17#232512yes—
373C2×C17D4×C4⋊C412yes—
374C2×C17#236611yescharacteristic
375C2×C17#237911yescharacteristic
376C2×C17#267911yescharacteristic
377C2×C17#268212yes—
378C2×C17#268311yescharacteristic
379C2×C17#272411yescharacteristic
380C2×C17#273212yes—
381C2×C17#280411yescharacteristic
382D17—48no—
383C25—172no—
384C22×D4—174no—
385C22×D4—174no—
386C22×D4—174no—
387C22×D4—681no—
388C22×D4—681no—
389C22×D4—681no—
390C22×D4—681no—
391C23×C4—172no—
392C23×C4—172no—
393C23×C4—172no—
394C23×C4—344no—
395C23×C4—344no—
396C23×C4—344no—
397C23×C4—682no—
398C23×C4—682no—
399C23×C4—682no—
400C23×C4—682no—
401C23×C4C22×D1712yes—
402C23×C4C22×D1711yescharacteristic
403C23×C4C22×D1711yescharacteristic
404C23×C4C22×D1712yes—
405C23×C4C22×D1711yescharacteristic
406C2×C4⋊C4—24no—
407C2×C4⋊C4—41no—
408C2×C4⋊C4—41no—
409C2×C4⋊C4—41no—
410C2×C4⋊C4—178no—
411C2×C4⋊C4—344no—
412C2×C4⋊C4—344no—
413C2×C4⋊C4—682no—
414C2×C4⋊C4—682no—
415C2×C4⋊C4—682no—
416C2×C4⋊C4—682no—
417C2×C4⋊C4—681no—
418C2×C4⋊C4C4×D1714yescomplemented
419C2×C22⋊C4—178no—
420C2×C22⋊C4—178no—
421C2×C22⋊C4—344no—
422C2×C22⋊C4—344no—
423C2×C22⋊C4—344no—
424C2×C22⋊C4—344no—
425C2×C22⋊C4—344no—
426C2×C22⋊C4—344no—
427C2×C22⋊C4—344no—
428C2×C22⋊C4—344no—
429C2×C22⋊C4—344no—
430C2×C22⋊C4—682no—
431C2×C22⋊C4—682no—
432C2×C22⋊C4—682no—
433C2×C22⋊C4—682no—
434C2×C42—174no—
435C2×C42—681no—
436C2×C42—681no—
437#95—24no—
438#95—24no—
439#95—41no—
440#95—344no—
441#95—344no—
442#95—344no—
443#95—344no—
444#95—681no—
445C17#1444011yescharacteristic, complemented, Sylow
446C24—174no—
447C24—174no—
448C24—174no—
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—1716no—
463C2×D4—684no—
464C2×D4—684no—
465C2×D4—684no—
466C2×D4—684no—
467C2×D4—684no—
468C2×D4—684no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—174no—
495C22×C4—174no—
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—688no—
516C22×C4—688no—
517C22×C4—688no—
518C22×C4—688no—
519C22×C4—688no—
520C22×C4—688no—
521C22×C4—688no—
522C22×C4C2×C4×D1712yes—
523C22×C4C2×C4×D1712yes—
524C22×C4C2×C4×D1712yes—
525C22×C4D4×D1714yes—
526C22×C4D4×D1712yes—
527C22×C4Q8×D1712yes—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—688no—
531C4⋊C4—684no—
532C4⋊C4—688no—
533C22⋊C4—3416no—
534C22⋊C4—3416no—
535C22⋊C4—688no—
536C22⋊C4—688no—
537C22⋊C4—688no—
538C42—684no—
539C23—178no—
540C23—344no—
541C23—344no—
542C23—344no—
543C23—344no—
544C23—344no—
545C23—344no—
546C23—344no—
547C23—344no—
548C23—344no—
549C23—344no—
550C23—344no—
551C23—344no—
552C23—344no—
553C23—344no—
554C23—344no—
555C23—344no—
556C23—344no—
557C23—344no—
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—
571C23—682no—
572C23—682no—
573C23—682no—
574C23C22×C4×D1711yescharacteristic
575C23C2×D4×D1711yescharacteristic
576C23C2×D4×D1712yes—
577C23#1058769111yescharacteristic
578C23#1058769111yescharacteristic
579C23C2×Q8×D1711yescharacteristic
580C23#1058769412yes—
581C23#41×D1712yes—
582C23#1058769612yes—
583C23#1058769611yescharacteristic
584C23#1058769611yescharacteristic
585D4—6816no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—178no—
617C2×C4—344no—
618C2×C4—344no—
619C2×C4—344no—
620C2×C4—344no—
621C2×C4—344no—
622C2×C4—344no—
623C2×C4—344no—
624C2×C4—344no—
625C2×C4—344no—
626C2×C4—6832no—
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×C4C4⋊C4×D1714yes—
639C22—348no—
640C22—348no—
641C22—348no—
642C22—684no—
643C22—684no—
644C22—684no—
645C22—684no—
646C22—684no—
647C22—684no—
648C22—684no—
649C22—684no—
650C22—684no—
651C22—684no—
652C22—684no—
653C22—684no—
654C22C2×C4⋊C4×D1711yescharacteristic
655C22C4×D4×D1712yes—
656C22#1176328211yescharacteristic
657C22#1176328311yescharacteristic
658C22#1176329112yes—
659C22#1176336711yescharacteristic
660C22#1176336711yescharacteristic
661C22#1176337012yes—
662C22#1176338912yes—
663C22#1176339812yes—
664C22#1176340712yes—
665C22#1176343411yescharacteristic
666C22#1176343411yescharacteristic
667C22#1176344111yescharacteristic
668C22#1176344312yes—
669C22#1176346112yes—
670C22#1176347412yes—
671C22#1176347712yes—
672C22#1176351512yes—
673C22#1176354311yescharacteristic
674C22#1176355111yescharacteristic
675C22#1176355512yes—
676C22#1176355811yescharacteristic
677C4—24no—
678C4—42no—
679C4—42no—
680C4—44no—
681C4—82no—
682C4—348no—
683C4—684no—
684C4—1362no—
685C2—688no—
686C2#1453139711yescharacteristic
687C2#1453148312yes—
688C2#1453168512yes—
689C2#1453184711yescharacteristic
690C2#1453195611yescharacteristic
691C2#1453511711yescharacteristic
692C2#1453514412yes—
693C2#1453515811yescharacteristic
694C2#1453571111yescharacteristic
695C2#1453580512yes—
696C2#1453672911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1441C2
socle, cosocleC24×C17C24
prime radicalC23×D68C22
hypercenter#1441D17
hypocenterC17#14440
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
3. C2×#11762683C2semidirect
4. C2×#11762680C2semidirect
5. #14527037C2semidirect
6. #14527028C2semidirect
7. #14527028C2semidirect
8. #14527012C2semidirect
9. C17×#1441C2semidirect
21. #11762710C4semidirect
62. C2×C4⋊C4×C17C2×C4semidirect
76. C2×#63227C2×C4semidirect
77. C2×#63227C2×C4semidirect
85. C2×#63225D4semidirect
88. C2×C4×Dic17C2×C4semidirect
171. C2×D68C4⋊C4semidirect
296. #1441D17semidirect
418. C2×C4⋊C4C4×D17semidirect
445. C17#14440semidirect

Maximal subgroups

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

subgroupindex
C2×#117627102
C2×#117626872
C2×#117626832
C2×#117626802
#145270372
#145270282
#145270122
C17×#14412
#1444017