Finite Group Databasebeta

← #17037355

Subgroups — #17037355

Order 4352.

1042 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762919C211yesmaximal, characteristic
2#14531248C211yesmaximal, characteristic, complemented
3#14529411C211yesmaximal, characteristic, complemented
4#14529404C211yesmaximal, characteristic, complemented
5#14529369C211yesmaximal, characteristic, complemented
6#14529362C211yesmaximal, characteristic, complemented
7#14529132C211yesmaximal, characteristic
8#14529132C211yesmaximal, characteristic
9#14529045C211yesmaximal, characteristic, complemented
10#14529031C211yesmaximal, characteristic, complemented
11#14528682C211yesmaximal, characteristic, complemented
12#14528612C211yesmaximal, characteristic, complemented
13#14528609C211yesmaximal, characteristic, complemented
14#14528533C211yesmaximal, characteristic, complemented
15C42×D4×C17C211yesmaximal, characteristic, complemented
16C22×#63228C2211yescharacteristic
17C2×C4×D68C2211yescharacteristic
18C2×#10587558—21no—
19#11762919C418yescomplemented
20C2×#10587478C2211yescharacteristic
21C2×#10587479C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×#10587503C2211yescharacteristic
24C2×#10587503C2211yescharacteristic
25C2×#10587503C2211yescharacteristic
26C2×#10587480C2211yescharacteristic
27C2×#10587480C2211yescharacteristic
28C2×#10587480C2211yescharacteristic
29C2×#10587474C2211yescharacteristic
30C2×#10587474C2211yescharacteristic
31#11762711C2211yescharacteristic
32#11762711C2211yescharacteristic
33#11762707C2211yescharacteristic
34#11762707C2211yescharacteristic
35#11762700—21no—
36#11762700—21no—
37C4×C4⋊Dic17C2211yescharacteristic
38C4×#63227—21no—
39#11762571C2211yescharacteristic
40#11762571C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762564C2211yescharacteristic
46#11762564C2211yescharacteristic
47#11762564C2211yescharacteristic
48#11762564C2211yescharacteristic
49C2×C4×D4×C17C2211yescharacteristic
50C2×C4×D4×C17C2211yescharacteristic
51C2×C4×D4×C17C2211yescharacteristic
52C22×C42×C17—21no—
53C4×C4⋊C4×C17C2211yescharacteristic
54C4×C4⋊C4×C17C2211yescharacteristic
55C4×C4⋊C4×C17C2211yescharacteristic
56C4×C22⋊C4×C17—21no—
57C4×C22⋊C4×C17—21no—
58C4×C22⋊C4×C17—21no—
59C43×C17C2211yescharacteristic
60C22×D4×C17C2311yescharacteristic
61C23×C4×C17—21no—
62C23×C4×C17—21no—
63C23×C4×C17—21no—
64C22×D68C2311yescharacteristic
65C22×C4×D17—21no—
66C4×D4×C17—22no—
67C4×D4×C17—22no—
68C4×D4×C17—22no—
69C4×D4×C17—22no—
70C4×D4×C17—22no—
71C4×D4×C17C2×C414yes—
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C22⋊C4×C17—21no—
76C2×C22⋊C4×C17—21no—
77C2×C22⋊C4×C17—21no—
78C2×C42×C17—21no—
79C2×C42×C17—21no—
80C2×C42×C17—21no—
81C2×C42×C17—44no—
82C2×C42×C17D411yescharacteristic
83C2×C42×C17D411yescharacteristic
84C2×C42×C17D411yescharacteristic
85C2×C42×C17D411yescharacteristic
86C2×C42×C17C2311yescharacteristic
87C2×C42×C17C2311yescharacteristic
88C2×C42×C17C2311yescharacteristic
89C2×#63228—22no—
90C2×#63228—22no—
91C2×#63228C2×C414yes—
92C2×#63227—21no—
93C2×#63227—21no—
94C2×#63227—21no—
95C2×#63227—21no—
96C2×#63227—21no—
97C2×C4⋊Dic17C2311yescharacteristic
98C2×C4⋊Dic17C2311yescharacteristic
99C2×C4⋊Dic17C2311yescharacteristic
100C2×#63225—21no—
101C2×#63225—21no—
102C2×C4×Dic17—21no—
103C22×#3622C2311yescharacteristic
104C22×#3622C2311yescharacteristic
105C22×#3622C2311yescharacteristic
106C22×#3622C2311yescharacteristic
107C4×D68—22no—
108C4×D68C2×C414yescomplemented
109#10587558—44no—
110C17×#95—21no—
111C17×#95—21no—
112C17×#95—21no—
113#10587503—22no—
114#10587503—22no—
115#10587503C2×C414yes—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587480—22no—
120#10587480—22no—
121#10587480C2×C414yescomplemented
122#10587479—22no—
123#10587479C2×C414yescomplemented
124#10587478—22no—
125#10587474—22no—
126#10587474C2×C414yescomplemented
127C24×C17—21no—
128C23×D17—21no—
129C2×D4×C17—21no—
130C2×D4×C17—21no—
131C2×D4×C17—21no—
132C2×D4×C17C22×C412yes—
133C2×D4×C17#4112yes—
134C2×D4×C17#4112yes—
135C22×C4×C17—24no—
136C22×C4×C17—24no—
137C22×C4×C17—21no—
138C22×C4×C17—21no—
139C22×C4×C17—21no—
140C22×C4×C17—24no—
141C22×C4×C17—42no—
142C22×C4×C17—42no—
143C22×C4×C17—42no—
144C22×C4×C17—42no—
145C22×C4×C17—42no—
146C22×C4×C17—42no—
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2×D411yescharacteristic
151C22×C4×C17C2×D411yescharacteristic
152C22×C4×C17C2×D411yescharacteristic
153C22×C4×C17C2411yescharacteristic
154C22×Dic17—21no—
155C22×Dic17—21no—
156C22×Dic17—21no—
157C2×D68—21no—
158C2×D68—21no—
159C2×D68—21no—
160C2×D68—21no—
161C2×D68—21no—
162C2×D68C22×C412yes—
163C2×C4×D17—24no—
164C2×C4×D17—42no—
165C2×C4×D17—42no—
166C4⋊C4×C17—21no—
167C4⋊C4×C17—21no—
168C4⋊C4×C17—21no—
169C4⋊C4×C17C22×C412yes—
170C4⋊C4×C17#4112yes—
171C4⋊C4×C17#4112yes—
172C22⋊C4×C17—24no—
173C22⋊C4×C17—24no—
174C22⋊C4×C17—24no—
175C42×C17—22no—
176C42×C17—22no—
177C42×C17—21no—
178C42×C17—21no—
179C42×C17—21no—
180C42×C17—21no—
181C42×C17—41no—
182C42×C17—41no—
183C42×C17—42no—
184C42×C17—42no—
185C42×C17—42no—
186C42×C17C22×C412yes—
187C42×C17#4112yes—
188#63228—28no—
189#63227—24no—
190#63227—42no—
191#63227—42no—
192#63227—42no—
193#63227—42no—
194C4⋊Dic17—21no—
195C4⋊Dic17—21no—
196C4⋊Dic17—21no—
197C4⋊Dic17—21no—
198C4⋊Dic17—21no—
199C4⋊Dic17C22×C412yes—
200#63225—42no—
201#63225—42no—
202C4×Dic17—42no—
203C2×#3622—21no—
204C2×#3622—21no—
205C2×#3622—21no—
206C2×#3622—21no—
207C2×#3622—21no—
208C2×#3622—21no—
209C2×#3622—21no—
210C2×#3622—21no—
211C2×#3622—21no—
212C2×#3622—21no—
213C2×#3622C22×C412yes—
214C2×#3622C22×C412yes—
215#21122—171nomaximal, Sylow
216C23×C17—22no—
217C23×C17—22no—
218C23×C17—22no—
219C23×C17—41no—
220C23×C17—41no—
221C23×C17—41no—
222C23×C17—41no—
223C23×C17C22×D411yescharacteristic
224C22×D17—22no—
225C22×D17—41no—
226C22×D17—41no—
227C22×D17—41no—
228C22×D17—41no—
229C22×D17—41no—
230C22×D17—41no—
231D4×C17—22no—
232D4×C17—22no—
233D4×C17—22no—
234D4×C17#11714yescomplemented
235C2×C4×C17—21no—
236C2×C4×C17—21no—
237C2×C4×C17—21no—
238C2×C4×C17—21no—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—22no—
244C2×C4×C17—21no—
245C2×C4×C17—21no—
246C2×C4×C17—22no—
247C2×C4×C17—22no—
248C2×C4×C17—21no—
249C2×C4×C17—21no—
250C2×C4×C17—21no—
251C2×C4×C17—21no—
252C2×C4×C17—48no—
253C2×C4×C17—48no—
254C2×C4×C17—48no—
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×C17C4×D412yes—
262C2×C4×C17C4×D412yes—
263C2×C4×C17AGL(1,ℤ/8)12yes—
264C2×C4×C17AGL(1,ℤ/8)12yes—
265C2×C4×C17C23×C411yescharacteristic
266C2×C4×C17C2×#4111yescharacteristic
267C2×C4×C17C2×#4111yescharacteristic
268C2×C4×C17C2×#4111yescharacteristic
269C2×C4×C17C2×#4111yescharacteristic
270C2×C4×C17C2×#4111yescharacteristic
271C2×Dic17—22no—
272C2×Dic17—41no—
273C2×Dic17—41no—
274C2×Dic17—41no—
275C2×Dic17—41no—
276C2×Dic17—41no—
277C2×Dic17—41no—
278C2×Dic17—41no—
279C2×Dic17—41no—
280D68—22no—
281D68—22no—
282D68—41no—
283D68—41no—
284C4×D17—48no—
285#3622—22no—
286#3622—22no—
287#3622—41no—
288#3622—41no—
289C2×#442—171no—
290#2293—171no—
291C42×D4D1711yescharacteristic, complemented
292#1926—171no—
293#1925—171no—
294#1920—171no—
295#1919—171no—
296#1879—171no—
297#1879—171no—
298#1864—171no—
299#1862—171no—
300#1776—171no—
301#1762—171no—
302#1762—171no—
303#1745—171no—
304C22×C17—24no—
305C22×C17—42no—
306C22×C17—42no—
307C22×C17—42no—
308C22×C17—42no—
309C22×C17—42no—
310C22×C17—42no—
311C22×C17C2×C4×D411yescharacteristic
312C22×C17#52511yescharacteristic
313C22×C17#52511yescharacteristic
314C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
315C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
316C22×C17#57511yescharacteristic
317C22×C17#57511yescharacteristic
318C2×D17—42no—
319C2×D17—42no—
320C2×D17—42no—
321C2×D17—81no—
322C2×D17—81no—
323C2×D17—81no—
324C2×D17—81no—
325C4×C17—22no—
326C4×C17—22no—
327C4×C17—22no—
328C4×C17—22no—
329C4×C17—42no—
330C4×C17—41no—
331C4×C17—42no—
332C4×C17—41no—
333C4×C17—42no—
334C4×C17C2×#11711yescharacteristic
335C4×C17C4×#4111yescharacteristic
336C4×C17#52911yescharacteristic
337C4×C17#52911yescharacteristic
338Dic17—42no—
339Dic17—81no—
340Dic17—81no—
341C22×D8—171no—
342C2×C4×D4—171no—
343C2×C4×D4C2×D1711yescharacteristic
344C2×C4×D4C2×D1711yescharacteristic
345C2×C4×D4C2×D1711yescharacteristic
346C2×#117—341no—
347C22×C42—21no—
348#442—178no—
349C2×#107—171no—
350C2×#106—171no—
351C2×#106—171no—
352C2×#102—171no—
353C2×#102—171no—
354C2×#102—171no—
355C2×#102—171no—
356C2×#102—171no—
357C2×#102—171no—
358C2×#97—171no—
359C2×#97—171no—
360#390—171no—
361#390—171no—
362#389—171no—
363#389—171no—
364#388—341no—
365#388—341no—
366C4×C4⋊C4—171no—
367C4×C4⋊C4C2×D1711yescharacteristic
368C4×C4⋊C4C2×D1711yescharacteristic
369C4×C4⋊C4C2×D1711yescharacteristic
370C4×C22⋊C4—21no—
371C4×C22⋊C4—21no—
372C4×C22⋊C4—21no—
373C4×C22⋊C4—341no—
374C43C2×D1711yescharacteristic
375#340—171no—
376#340—171no—
377#340—171no—
378#340—171no—
379#340—171no—
380#340—171no—
381#336—171no—
382#336—171no—
383#336—171no—
384#336—171no—
385C2×C17—44no—
386C2×C17#229011yescharacteristic
387C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
388C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
389C2×C17#310911yescharacteristic
390C2×C17#311111yescharacteristic
391C2×C17#311411yescharacteristic
392C2×C17#311511yescharacteristic
393D17—82no—
394C22×D4—171no—
395C22×D4C22×D1711yescharacteristic
396C23×C4—21no—
397C23×C4—21no—
398C23×C4—21no—
399C23×C4—341no—
400C2×D8—174no—
401C2×D8—342no—
402C2×D8—342no—
403C22×C8—171no—
404C22×C8—171no—
405C22×C8—171no—
406C22×C8—171no—
407C4×D4—22no—
408C4×D4—22no—
409C4×D4—22no—
410C4×D4—22no—
411C4×D4—22no—
412C4×D4—174no—
413C4×D4—342no—
414C4×D4C4×D1714yes—
415#117—684no—
416C2×C4⋊C4—171no—
417C2×C4⋊C4—171no—
418C2×C4⋊C4—171no—
419C2×C4⋊C4—341no—
420C2×C4⋊C4—341no—
421C2×C4⋊C4C22×D1711yescharacteristic
422C2×C4⋊C4C22×D1711yescharacteristic
423C2×C4⋊C4C22×D1711yescharacteristic
424C2×C22⋊C4—21no—
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—341no—
428C2×C22⋊C4—341no—
429C2×C22⋊C4—341no—
430C2×C22⋊C4—341no—
431C2×C22⋊C4—341no—
432C2×C42—21no—
433C2×C42—21no—
434C2×C42—21no—
435C2×C42—44no—
436C2×C42—341no—
437C2×C42C22⋊D1711yescharacteristic
438C2×C42C22⋊D1711yescharacteristic
439C2×C42C22⋊D1711yescharacteristic
440C2×C42C22⋊D1711yescharacteristic
441C2×C42C22×D1711yescharacteristic
442C2×C42C22×D1711yescharacteristic
443C2×C42C22×D1711yescharacteristic
444#107—342no—
445#106—174no—
446#106—342no—
447#102—174no—
448#102—174no—
449#102—342no—
450#102—342no—
451#102—342no—
452#102—342no—
453#97—174no—
454#97—342no—
455#95—21no—
456#95—21no—
457#95—21no—
458#95—341no—
459#95—341no—
460#95—341no—
461C17#2112211yescharacteristic, complemented, Sylow
462C24—21no—
463C24—341no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—172no—
468C2×D4—341no—
469C2×D4—341no—
470C2×D4—341no—
471C2×D4—341no—
472C2×D4—341no—
473C2×D4C2×C4×D1712yes—
474C2×D4#6325212yes—
475C2×D4#6325212yes—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—24no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—341no—
489C22×C4—344no—
490C22×C4—341no—
491C22×C4—341no—
492C22×C4—682no—
493C22×C4—682no—
494C22×C4C2×C22⋊D1711yescharacteristic
495C22×C4C2×C22⋊D1711yescharacteristic
496C22×C4C2×C22⋊D1711yescharacteristic
497C22×C4C2×C22⋊D1711yescharacteristic
498C22×C4C2×C22⋊D1711yescharacteristic
499C22×C4C2×C22⋊D1711yescharacteristic
500C22×C4C23×D1711yescharacteristic
501D8—348no—
502C2×C8—172no—
503C2×C8—172no—
504C2×C8—341no—
505C2×C8—341no—
506C2×C8—341no—
507C2×C8—341no—
508C2×C8—341no—
509C2×C8—341no—
510C2×C8—341no—
511C2×C8—341no—
512C2×C8—341no—
513C2×C8—341no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—172no—
518C4⋊C4—341no—
519C4⋊C4—341no—
520C4⋊C4—341no—
521C4⋊C4—341no—
522C4⋊C4—341no—
523C4⋊C4—682no—
524C4⋊C4—682no—
525C4⋊C4C2×C4×D1712yes—
526C4⋊C4#6325212yes—
527C4⋊C4#6325212yes—
528C22⋊C4—24no—
529C22⋊C4—24no—
530C22⋊C4—24no—
531C22⋊C4—344no—
532C22⋊C4—682no—
533C22⋊C4—682no—
534C22⋊C4—682no—
535C22⋊C4—682no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—22no—
541C42—22no—
542C42—41no—
543C42—41no—
544C42—42no—
545C42—42no—
546C42—42no—
547C42—682no—
548C42C2×C4×D1712yes—
549C42#6325212yes—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—342no—
558C23—681no—
559C23—681no—
560C23—681no—
561C23—681no—
562C23—681no—
563C23—681no—
564C23C22×C22⋊D1711yescharacteristic
565D4—22no—
566D4—22no—
567D4—22no—
568D4—342no—
569D4—342no—
570D4—681no—
571D4—681no—
572D4#1058755814yescomplemented
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—48no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—342no—
600C2×C4—688no—
601C2×C4—681no—
602C2×C4—681no—
603C2×C4—681no—
604C2×C4—681no—
605C2×C4—681no—
606C2×C4—681no—
607C2×C4—681no—
608C2×C4—681no—
609C2×C4C4×C22⋊D1712yes—
610C2×C4C4×C22⋊D1712yes—
611C2×C4#1058761812yes—
612C2×C4#1058763512yes—
613C2×C4C22×C4×D1711yescharacteristic
614C2×C4C2×#6325211yescharacteristic
615C2×C4C2×#6325211yescharacteristic
616C2×C4C2×#6325211yescharacteristic
617C2×C4C2×#6325211yescharacteristic
618C2×C4C2×#6325211yescharacteristic
619C8—342no—
620C8—342no—
621C8—681no—
622C8—681no—
623C22—24no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—682no—
631C22—682no—
632C22—682no—
633C22—1361no—
634C22—1361no—
635C22—1361no—
636C22—1361no—
637C22C2×C4×C22⋊D1711yescharacteristic
638C22#1176332011yescharacteristic
639C22#1176332011yescharacteristic
640C22C2×#1058763511yescharacteristic
641C22C2×#1058761811yescharacteristic
642C22#1176360711yescharacteristic
643C22#1176360711yescharacteristic
644C4—22no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—42no—
649C4—42no—
650C4—41no—
651C4—42no—
652C4—41no—
653C4—682no—
654C4—1361no—
655C4—1361no—
656C4C2×#1058755811yescharacteristic
657C4C4×#6325211yescharacteristic
658C4#1176334311yescharacteristic
659C4#1176334311yescharacteristic
660C2—44no—
661C2—1362no—
662C2#1453122711yescharacteristic
663C2C4×#1058763511yescharacteristic
664C2C4×#1058761811yescharacteristic
665C2#1453887511yescharacteristic
666C2#1453889311yescharacteristic
667C2#1453892211yescharacteristic
668C2#1453893511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC42×D4×C17C2
socleC23×C17C22×D4
prime radicalC22×#63228C22
cosocleC22×C4×C17C24
hypercenterC42×D4D17
hypocenterC17#21122
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C43 ◁ C42×D4

Complements

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

NH ≅ G/Ntype
2. #14531248C2semidirect
3. #14529411C2semidirect
4. #14529404C2semidirect
5. #14529369C2semidirect
6. #14529362C2semidirect
9. #14529045C2semidirect
10. #14529031C2semidirect
11. #14528682C2semidirect
12. #14528612C2semidirect
13. #14528609C2semidirect
14. #14528533C2semidirect
15. C42×D4×C17C2semidirect
19. #11762919C4semidirect
108. C4×D68C2×C4semidirect
121. #10587480C2×C4semidirect
123. #10587479C2×C4semidirect
126. #10587474C2×C4semidirect
234. D4×C17#117semidirect
291. C42×D4D17semidirect
461. C17#21122semidirect
572. D4#10587558semidirect

Maximal subgroups

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

subgroupindex
C2×#117629192
#145312482
#145294112
#145294042
#145293692
#145293622
#145291322
#145290452
#145290312
#145286822
#145286122
#145286092
#145285332
C42×D4×C172
#2112217