Finite Group Databasebeta

← #16940632

Subgroups — #16940632

Order 4352.

1118 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587508C211yesmaximal, characteristic
2C2×M16×Dic17C211yesmaximal, characteristic, complemented
3#14531275C211yesmaximal, characteristic
4C2×#11762570C211yesmaximal, characteristic, complemented
5C2×#11762569C211yesmaximal, characteristic, complemented
6C4×#10587502C211yesmaximal, characteristic, complemented
7C22×C17×#104C211yesmaximal, characteristic
8C22×#63223C2211yescharacteristic
9C2×#10587612C2211yescharacteristic
10C22×C4×Dic17C2211yescharacteristic
11C22×C4×Dic17C2211yescharacteristic
12C2×#10587508—22no—
13C2×#10587508—22no—
14C2×#10587508C414yescomplemented
15M16×Dic17—22no—
16M16×Dic17—22no—
17C2×#10587486—21no—
18C2×C8×Dic17—21no—
19#11762716C412yescomplemented
20#11762712C412yescomplemented
21#11762689—22no—
22C4×C22⋊Dic17—21no—
23C4×C22⋊Dic17—21no—
24C2×#10587502—22no—
25#11762570—22no—
26#11762569—22no—
27C22×#41×C17C2211yescharacteristic
28C22×M16×C17C2211yescharacteristic
29C22×C42×C17C2211yescharacteristic
30C2×C17×#104—22no—
31C2×C17×#104—22no—
32C2×C17×#104C414yescomplemented
33C2×#41×C17—21no—
34C2×#41×C17—21no—
35C2×#41×C17—21no—
36C2×#41×C17—21no—
37C2×#41×C17—21no—
38C2×#41×C17—21no—
39C2×#41×C17—21no—
40C2×#41×C17—21no—
41C2×#41×C17—21no—
42C2×#41×C17—21no—
43C2×#41×C17C2×C412yes—
44C2×#41×C17C2×C412yes—
45C22×Q8×C17C2×C411yescharacteristic
46C22×D4×C17—21no—
47C22×D4×C17C2×C411yescharacteristic
48C23×C4×C17—21no—
49C23×C4×C17—21no—
50C23×C4×C17C2311yescharacteristic
51C23×Dic17—21no—
52C23×Dic17D412yes—
53C2×M16×C17—21no—
54C2×M16×C17—21no—
55C2×M16×C17—21no—
56C2×M16×C17C2×C412yes—
57C2×M16×C17D412yes—
58C2×M16×C17D412yes—
59C22×C8×C17—21no—
60C2×C42×C17—21no—
61C2×C42×C17—21no—
62C2×C42×C17—21no—
63C2×C42×C17—21no—
64C2×C42×C17—21no—
65C2×C42×C17C2×C412yes—
66C2×C22⋊Dic17—21no—
67C2×C22⋊Dic17—22no—
68C2×C22⋊Dic17—22no—
69#10587612—21no—
70#10587612—21no—
71#10587612—21no—
72#10587612—21no—
73C2×C4⋊Dic17C2×C412yes—
74C2×C4×Dic17—21no—
75C2×C4×Dic17—21no—
76C2×C4×Dic17—21no—
77C2×C4×Dic17—21no—
78C2×C4×Dic17—21no—
79C2×C4×Dic17—21no—
80C2×C4×Dic17—21no—
81C2×C4×Dic17—21no—
82C2×C4×Dic17—21no—
83C2×C4×Dic17—21no—
84C2×C4×Dic17C2×C412yes—
85C2×C4×Dic17C2×C412yes—
86C2×C4×Dic17D411yescharacteristic
87C2×C4×Dic17D411yescharacteristic
88C2×#63223—21no—
89C2×#63223—21no—
90C2×#63223—21no—
91C2×#63223C2×C412yes—
92C2×#63223D412yes—
93C2×#63223D412yes—
94C22×#3622—21no—
95C17×#104—28no—
96#10587508—28no—
97#10587502—28no—
98#10587502—22no—
99#10587502—22no—
100#10587486—24no—
101C8×Dic17—24no—
102C24×C17—21no—
103C24×C17C2×D411yescharacteristic
104#41×C17—22no—
105#41×C17—22no—
106#41×C17—22no—
107#41×C17—22no—
108#41×C17—22no—
109#41×C17—22no—
110#41×C17—22no—
111#41×C17—22no—
112#41×C17—41no—
113#41×C17—41no—
114#41×C17—41no—
115#41×C17—41no—
116#41×C17—41no—
117#41×C17—41no—
118#41×C17—41no—
119#41×C17—41no—
120C2×Q8×C17—21no—
121C2×Q8×C17—21no—
122C2×Q8×C17—21no—
123C2×Q8×C17—21no—
124C2×Q8×C17—21no—
125C2×Q8×C17C4212yes—
126C2×D4×C17—22no—
127C2×D4×C17—22no—
128C2×D4×C17—21no—
129C2×D4×C17—21no—
130C2×D4×C17—21no—
131C2×D4×C17—21no—
132C2×D4×C17—21no—
133C2×D4×C17—41no—
134C2×D4×C17—41no—
135C2×D4×C17—41no—
136C2×D4×C17—41no—
137C2×D4×C17C4212yes—
138C22×C4×C17—24no—
139C22×C4×C17—21no—
140C22×C4×C17—22no—
141C22×C4×C17—22no—
142C22×C4×C17—22no—
143C22×C4×C17—42no—
144C22×C4×C17—41no—
145C22×C4×C17—41no—
146C22×C4×C17—41no—
147C22×C4×C17—41no—
148C22×C4×C17—41no—
149C22×C4×C17—41no—
150C22×C4×C17C22×C411yescharacteristic
151C22×C4×C17C22×C411yescharacteristic
152C22×C4×C17C22×C411yescharacteristic
153C22×C4×C17C2×D411yescharacteristic
154C22×C4×C17C2×D411yescharacteristic
155C22×C4×C17C2×D411yescharacteristic
156C22×C4×C17C2×D411yescharacteristic
157C22×C4×C17C2×D411yescharacteristic
158C22×C4×C17#4111yescharacteristic
159C22×C4×C17#4111yescharacteristic
160C22×C4×C17#4111yescharacteristic
161C22×C4×C17#4111yescharacteristic
162C22×C4×C17#4111yescharacteristic
163C22×C4×C17#4111yescharacteristic
164C22×Dic17—24no—
165C22×Dic17—22no—
166C22×Dic17—22no—
167C22×Dic17—22no—
168C22×Dic17—24no—
169C22×Dic17—22no—
170C22×Dic17—42no—
171C22×Dic17—42no—
172C22×Dic17—41no—
173C22×Dic17—41no—
174C22×Dic17C22⋊C412yes—
175C22×Dic17C22⋊C412yes—
176M16×C17—22no—
177M16×C17—22no—
178M16×C17—22no—
179M16×C17C22⋊C414yescomplemented
180C2×C8×C17—22no—
181C2×C8×C17—22no—
182C2×C8×C17—22no—
183C42×C17—22no—
184C42×C17—22no—
185C42×C17—41no—
186C42×C17—41no—
187C22⋊Dic17—44no—
188C22⋊Dic17—41no—
189C22⋊Dic17—41no—
190C22⋊Dic17—44no—
191C4⋊Dic17—22no—
192C4⋊Dic17—22no—
193C4×Dic17—22no—
194C4×Dic17—22no—
195C4×Dic17—21no—
196C4×Dic17—21no—
197C4×Dic17—21no—
198C4×Dic17—21no—
199C4×Dic17—22no—
200C4×Dic17—22no—
201C4×Dic17—41no—
202C4×Dic17—41no—
203C4×Dic17—41no—
204C4×Dic17—41no—
205#63223—22no—
206#63223—22no—
207#63223—22no—
208#63223C22⋊C414yescomplemented
209C2×#3622—22no—
210C2×#3622—22no—
211C2×#3622—22no—
212#10933—171nomaximal, Sylow
213C23×C17—22no—
214C23×C17—22no—
215C23×C17—22no—
216C23×C17—41no—
217C23×C17—41no—
218C23×C17—42no—
219C23×C17—42no—
220C23×C17C2×C22⋊C411yescharacteristic
221C23×C17C4×D411yescharacteristic
222C23×C17C4×D411yescharacteristic
223C23×C17#12111yescharacteristic
224C23×C17#12111yescharacteristic
225C23×C17#12411yescharacteristic
226C23×C17#12711yescharacteristic
227Q8×C17—22no—
228Q8×C17—22no—
229Q8×C17—41no—
230Q8×C17—41no—
231D4×C17—22no—
232D4×C17—22no—
233D4×C17—42no—
234D4×C17—42no—
235D4×C17—42no—
236D4×C17—42no—
237D4×C17—41no—
238D4×C17—41no—
239C2×C4×C17—24no—
240C2×C4×C17—22no—
241C2×C4×C17—44no—
242C2×C4×C17—44no—
243C2×C4×C17—42no—
244C2×C4×C17—41no—
245C2×C4×C17—42no—
246C2×C4×C17—41no—
247C2×C4×C17—42no—
248C2×C4×C17—42no—
249C2×C4×C17—42no—
250C2×C4×C17—42no—
251C2×C4×C17C2×C4211yescharacteristic
252C2×C4×C17C2×C22⋊C411yescharacteristic
253C2×C4×C17C2×C22⋊C411yescharacteristic
254C2×C4×C17#11711yescharacteristic
255C2×C4×C17#11711yescharacteristic
256C2×C4×C17#11711yescharacteristic
257C2×C4×C17C4×D411yescharacteristic
258C2×C4×C17C4×D411yescharacteristic
259C2×C4×C17C4×D411yescharacteristic
260C2×C4×C17C4×D411yescharacteristic
261C2×C4×C17C4×D411yescharacteristic
262C2×C4×C17C4×D411yescharacteristic
263C2×C4×C17C4×D411yescharacteristic
264C2×C4×C17C4×D411yescharacteristic
265C2×C4×C17C4×D411yescharacteristic
266C2×C4×C17C4×D411yescharacteristic
267C2×C4×C17#12011yescharacteristic
268C2×C4×C17#12111yescharacteristic
269C2×C4×C17#12111yescharacteristic
270C2×C4×C17#12111yescharacteristic
271C2×C4×C17#12111yescharacteristic
272C2×C4×C17#12311yescharacteristic
273C2×C4×C17#12311yescharacteristic
274C2×C4×C17#12311yescharacteristic
275C2×C4×C17#12411yescharacteristic
276C2×C4×C17#12411yescharacteristic
277C2×C4×C17#12611yescharacteristic
278C2×C4×C17#12611yescharacteristic
279C2×Dic17—22no—
280C2×Dic17—22no—
281C2×Dic17—24no—
282C2×Dic17—22no—
283C2×Dic17—22no—
284C2×Dic17—22no—
285C2×Dic17—44no—
286C2×Dic17—44no—
287C2×Dic17—44no—
288C2×Dic17—44no—
289C2×Dic17—42no—
290C2×Dic17—42no—
291C2×Dic17—42no—
292C2×Dic17—42no—
293C2×Dic17—42no—
294C2×Dic17#10414yescomplemented
295C8×C17—24no—
296#3622—24no—
297C22×#104—171no—
298C22×#104D1711yescharacteristic
299C2×C4×M16—171no—
300#2297—171no—
301C2×#339—171no—
302C2×#339—171no—
303C4×#95—171no—
304C22×C17—22no—
305C22×C17—22no—
306C22×C17—22no—
307C22×C17—22no—
308C22×C17—21no—
309C22×C17—21no—
310C22×C17—21no—
311C22×C17—21no—
312C22×C17—42no—
313C22×C17—42no—
314C22×C17—42no—
315C22×C17—82no—
316C22×C17—82no—
317C22×C17C4×C22⋊C411yescharacteristic
318C22×C17#38611yescharacteristic
319C22×C17#38811yescharacteristic
320C22×C17#38811yescharacteristic
321C22×C17#39011yescharacteristic
322C22×C17#39011yescharacteristic
323C22×C17#39011yescharacteristic
324C22×C17C2×#10412yes—
325C22×C17#42112yes—
326C4×C17—44no—
327C4×C17—42no—
328C4×C17C4×C22⋊C411yescharacteristic
329C4×C17C4×C22⋊C411yescharacteristic
330C4×C17#38611yescharacteristic
331C4×C17#38611yescharacteristic
332C4×C17#38811yescharacteristic
333C4×C17#38811yescharacteristic
334C4×C17#38811yescharacteristic
335C4×C17#38811yescharacteristic
336Dic17—24no—
337Dic17—44no—
338Dic17—42no—
339Dic17—82no—
340C22×#41C2×D1711yescharacteristic
341C22×M16—171no—
342C22×M16C2×D1711yescharacteristic
343C2×#117—171no—
344C22×C42—171no—
345C22×C42—171no—
346C22×C42C2×D1711yescharacteristic
347C2×#104—22no—
348C2×#104—22no—
349C2×#104—174no—
350C2×#104—342no—
351C2×#104—342no—
352C2×#104Dic1714yescomplemented
353C4×M16—342no—
354C4×M16—342no—
355C2×#97—341no—
356C2×C4×C8—341no—
357#391—172no—
358#390—172no—
359#386—342no—
360C4×C22⋊C4—341no—
361C4×C22⋊C4—341no—
362C2×#95—342no—
363#339—342no—
364#339—342no—
365C2×C17—21no—
366C2×C17—21no—
367C2×C17—21no—
368C2×C17—21no—
369C2×C17—82no—
370C2×C17#143711yescharacteristic
371C2×C17C4×#10411yescharacteristic
372C2×C17#175911yescharacteristic
373C2×C17#187112yes—
374C2×C17#198012yes—
375C2×#41—21no—
376C2×#41—21no—
377C2×#41—21no—
378C2×#41—21no—
379C2×#41—21no—
380C2×#41—21no—
381C2×#41—21no—
382C2×#41—21no—
383C2×#41—21no—
384C2×#41—21no—
385C2×#41C4×D1712yes—
386C2×#41C2×Dic1712yes—
387C22×Q8C4×D1711yescharacteristic
388C22×D4—21no—
389C22×D4C4×D1711yescharacteristic
390C23×C4—21no—
391C23×C4—21no—
392C23×C4—172no—
393C23×C4—341no—
394C23×C4C22×D1711yescharacteristic
395C2×M16—21no—
396C2×M16—21no—
397C2×M16—21no—
398C2×M16—172no—
399C2×M16—172no—
400C2×M16—172no—
401C2×M16—341no—
402C2×M16—341no—
403C2×M16—341no—
404C2×M16C2×Dic1712yes—
405C2×M16C22⋊D1712yes—
406C2×M16C22⋊D1712yes—
407C22×C8—21no—
408C22×C8—341no—
409#117—341no—
410#117—341no—
411#117—341no—
412#117—341no—
413C2×C4⋊C4—172no—
414C2×C22⋊C4—342no—
415C2×C22⋊C4—342no—
416C2×C22⋊C4—341no—
417C2×C42—21no—
418C2×C42—21no—
419C2×C42—21no—
420C2×C42—21no—
421C2×C42—21no—
422C2×C42—172no—
423C2×C42—171no—
424C2×C42—171no—
425C2×C42—172no—
426C2×C42—341no—
427C2×C42—341no—
428C2×C42—341no—
429C2×C42—341no—
430C2×C42—341no—
431C2×C42—341no—
432C2×C42—341no—
433C2×C42—341no—
434C2×C42—341no—
435C2×C42—341no—
436C2×C42C2×Dic1712yes—
437#104—28no—
438#104—348no—
439#97—344no—
440C4×C8—344no—
441#95—342no—
442#95—342no—
443#95—348no—
444C17#1093311yescharacteristic, complemented, Sylow
445C24—21no—
446C24D4×D1711yescharacteristic
447#41—22no—
448#41—22no—
449#41—22no—
450#41—22no—
451#41—22no—
452#41—22no—
453#41—22no—
454#41—22no—
455#41—41no—
456#41—41no—
457#41—41no—
458#41—41no—
459#41—41no—
460#41—41no—
461#41—41no—
462#41—41no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8C4×Dic1712yes—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—41no—
477C2×D4—41no—
478C2×D4—41no—
479C2×D4—41no—
480C2×D4C4×Dic1712yes—
481C22×C4—24no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—42no—
487C22×C4—41no—
488C22×C4—41no—
489C22×C4—41no—
490C22×C4—41no—
491C22×C4—41no—
492C22×C4—41no—
493C22×C4—172no—
494C22×C4—172no—
495C22×C4—344no—
496C22×C4—344no—
497C22×C4—342no—
498C22×C4—342no—
499C22×C4—342no—
500C22×C4—342no—
501C22×C4—682no—
502C22×C4—682no—
503C22×C4—681no—
504C22×C4—681no—
505C22×C4C2×C4×D1711yescharacteristic
506C22×C4C2×C4×D1711yescharacteristic
507C22×C4#6325211yescharacteristic
508C22×C4#6325211yescharacteristic
509C22×C4D4×D1711yescharacteristic
510C22×C4D4×D1711yescharacteristic
511C22×C4D4×D1711yescharacteristic
512C22×C4#6325411yescharacteristic
513C22×C4#6325411yescharacteristic
514C22×C4#6325411yescharacteristic
515C22×C4#6325411yescharacteristic
516C22×C4C22×Dic1711yescharacteristic
517C22×C4C2×C22⋊D1711yescharacteristic
518C22×C4C2×C22⋊D1711yescharacteristic
519M16—22no—
520M16—22no—
521M16—22no—
522M16—174no—
523M16—342no—
524M16—342no—
525M16—342no—
526M16C22⋊Dic1714yescomplemented
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—342no—
531C2×C8—342no—
532C2×C8—342no—
533C4⋊C4—342no—
534C4⋊C4—342no—
535C22⋊C4—684no—
536C22⋊C4—684no—
537C22⋊C4—681no—
538C22⋊C4—681no—
539C42—22no—
540C42—22no—
541C42—41no—
542C42—41no—
543C42—342no—
544C42—341no—
545C42—341no—
546C42—342no—
547C42—341no—
548C42—341no—
549C42—342no—
550C42—342no—
551C42—681no—
552C42—681no—
553C42—681no—
554C42—681no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—41no—
559C23—42no—
560C23—42no—
561C23—41no—
562C23C22⋊C4×D1711yescharacteristic
563C23#1058756711yescharacteristic
564C23#1058757011yescharacteristic
565C23#1058757111yescharacteristic
566C23D4×Dic1711yescharacteristic
567C23#1058762411yescharacteristic
568C23#1058762611yescharacteristic
569Q8—22no—
570Q8—22no—
571Q8—41no—
572Q8—41no—
573D4—22no—
574D4—22no—
575D4—42no—
576D4—42no—
577D4—42no—
578D4—42no—
579D4—41no—
580D4—41no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—174no—
594C2×C4—342no—
595C2×C4—342no—
596C2×C4—344no—
597C2×C4—342no—
598C2×C4—342no—
599C2×C4—342no—
600C2×C4—684no—
601C2×C4—684no—
602C2×C4—684no—
603C2×C4—682no—
604C2×C4—684no—
605C2×C4—682no—
606C2×C4—682no—
607C2×C4—682no—
608C2×C4—682no—
609C2×C4#1058756311yescharacteristic
610C2×C4#1058756311yescharacteristic
611C2×C4#1058756511yescharacteristic
612C2×C4#1058756511yescharacteristic
613C2×C4C22⋊C4×D1711yescharacteristic
614C2×C4#1058756711yescharacteristic
615C2×C4#1058756711yescharacteristic
616C2×C4#1058756711yescharacteristic
617C2×C4#1058756911yescharacteristic
618C2×C4#1058756911yescharacteristic
619C2×C4#1058757011yescharacteristic
620C2×C4#1058757111yescharacteristic
621C2×C4C2×C4×Dic1711yescharacteristic
622C2×C4#1058761211yescharacteristic
623C2×C4C4×C22⋊D1711yescharacteristic
624C2×C4C4×C22⋊D1711yescharacteristic
625C2×C4C4×C22⋊D1711yescharacteristic
626C2×C4C4×C22⋊D1711yescharacteristic
627C2×C4D4×Dic1711yescharacteristic
628C2×C4D4×Dic1711yescharacteristic
629C2×C4D4×Dic1711yescharacteristic
630C2×C4#1058762111yescharacteristic
631C2×C4#1058762211yescharacteristic
632C2×C4#1058762311yescharacteristic
633C2×C4#1058762411yescharacteristic
634C2×C4#1058762511yescharacteristic
635C2×C4#1058762511yescharacteristic
636C2×C4C2×C22⋊Dic1711yescharacteristic
637C8—24no—
638C8—344no—
639C22—22no—
640C22—22no—
641C22—21no—
642C22—21no—
643C22—21no—
644C22—21no—
645C22—22no—
646C22—22no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—82no—
651C22—82no—
652C22C22⋊C4×Dic1711yescharacteristic
653C22#1176268511yescharacteristic
654C22#1176269711yescharacteristic
655C22#1176270211yescharacteristic
656C22#1176270911yescharacteristic
657C22#1176270911yescharacteristic
658C22#1176271211yescharacteristic
659C22#104×D1712yes—
660C22#1176282412yes—
661C4—44no—
662C4—42no—
663C4—344no—
664C4—684no—
665C4—682no—
666C4—1362no—
667C4C4×C22⋊Dic1711yescharacteristic
668C4C22⋊C4×Dic1711yescharacteristic
669C4#1176268511yescharacteristic
670C4#1176268911yescharacteristic
671C4#1176269711yescharacteristic
672C4#1176270211yescharacteristic
673C4#1176270311yescharacteristic
674C4#1176270311yescharacteristic
675C2—21no—
676C2—21no—
677C2—21no—
678C2—21no—
679C2—82no—
680C2#1452702111yescharacteristic
681C2#104×Dic1711yescharacteristic
682C2#1452859611yescharacteristic
683C2#1452908312yes—
684C2#1452970112yes—
subgroupG/H
centerC22×C4D4×D17
commutatorC2×C4×C17C2×C42
FrattiniC23×C4C22×D17
FittingC22×C17×#104C2
socleC23×C17C2×C22⋊C4
prime radicalC22×#41×C17C22
cosocleC23×C4×C17C23
hypercenterC22×#104D17
hypocenterC17#10933
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C22×#104

Complements

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

NH ≅ G/Ntype
2. C2×M16×Dic17C2semidirect
4. C2×#11762570C2semidirect
5. C2×#11762569C2semidirect
6. C4×#10587502C2semidirect
14. C2×#10587508C4semidirect
19. #11762716C4semidirect
20. #11762712C4semidirect
32. C2×C17×#104C4semidirect
179. M16×C17C22⋊C4semidirect
208. #63223C22⋊C4semidirect
294. C2×Dic17#104semidirect
352. C2×#104Dic17semidirect
444. C17#10933semidirect
526. M16C22⋊Dic17semidirect

Maximal subgroups

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

subgroupindex
C22×#105875082
C2×M16×Dic172
#145312752
C2×#117625702
C2×#117625692
C4×#105875022
C22×C17×#1042
#1093317