Finite Group Databasebeta

← #17097879

Subgroups — #17097879

Order 4352.

902 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14532480C211yesmaximal, characteristic, complemented
2#14531804C211yesmaximal, characteristic
3#14530345C211yesmaximal, characteristic, complemented
4#14530274C211yesmaximal, characteristic, complemented
5#14530170C211yesmaximal, characteristic, complemented
6#14529918C211yesmaximal, characteristic, complemented
7#14529734C211yesmaximal, characteristic, complemented
8#14529354C211yesmaximal, characteristic, complemented
9#14529213C211yesmaximal, characteristic
10#14529033C211yesmaximal, characteristic, complemented
11#14528974C211yesmaximal, characteristic, complemented
12#14528961C211yesmaximal, characteristic, complemented
13#14528782C211yesmaximal, characteristic, complemented
14C2×#11762571C211yesmaximal, characteristic, complemented
15C17×#1874C211yesmaximal, characteristic
16C23×#3622C2211yescharacteristic
17C2×#10587574C2211yescharacteristic
18C2×#10587610—21no—
19C2×D4×Dic17—21no—
20C2×D4×Dic17C2211yescharacteristic
21C22×C4⋊Dic17C2211yescharacteristic
22C2×#10587489C2211yescharacteristic
23C2×#10587489C2211yescharacteristic
24C2×#10587485C2211yescharacteristic
25C2×#10587485C2211yescharacteristic
26C2×#10587503C2211yescharacteristic
27C2×#10587503C2211yescharacteristic
28#11762742—21no—
29#11762729—21no—
30#11762712C2211yescharacteristic
31#11762708C2211yescharacteristic
32#11762708C2211yescharacteristic
33#11762705C2211yescharacteristic
34#11762702—21no—
35#11762689—21no—
36#11762684—21no—
37#11762682—21no—
38#11762678C2211yescharacteristic
39#11762669C2211yescharacteristic
40C4⋊C4×Dic17C2211yescharacteristic
41C4×C22⋊Dic17C2211yescharacteristic
42#11762572C2211yescharacteristic
43#11762572C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52C2×C17×#121C2211yescharacteristic
53C22×C4⋊C4×C17C2211yescharacteristic
54C2×C17×#102C2211yescharacteristic
55C2×C17×#102C2211yescharacteristic
56C2×C17×C22⋊C8C2211yescharacteristic
57C17×#340C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C22×D4×C17—21no—
60C22×D4×C17C2311yescharacteristic
61C23×C4×C17—21no—
62C23×C4×C17C2311yescharacteristic
63C23×Dic17—21no—
64C23×Dic17—21no—
65C22×Dic34—21no—
66C22×C8×C17C2311yescharacteristic
67C22×C8×C17C2311yescharacteristic
68C17×#121—22no—
69C17×#121Q814yescomplemented
70C2×C4⋊C4×C17—22no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17—41no—
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C22⋊C4×C17—21no—
77C2×C22⋊Dic17—21no—
78C2×C22⋊Dic17—21no—
79C2×C22⋊Dic17—21no—
80C2×C22⋊Dic17—21no—
81C2×C22⋊Dic17D411yescharacteristic
82C2×C22⋊Dic17D411yescharacteristic
83D4×Dic17—22no—
84D4×Dic17—41no—
85D4×Dic17—44no—
86C2×C4⋊Dic17—21no—
87C2×C4⋊Dic17—21no—
88C2×C4⋊Dic17—21no—
89C2×C4⋊Dic17—22no—
90C2×C4⋊Dic17—41no—
91C2×C4⋊Dic17C2311yescharacteristic
92C2×C4⋊Dic17C2311yescharacteristic
93C2×C4⋊Dic17C2311yescharacteristic
94C2×C4⋊Dic17C2311yescharacteristic
95#10587610—44no—
96C2×#63225—21no—
97C2×#63225—21no—
98C2×#63225—21no—
99C2×#63225—21no—
100C2×C4×Dic17—21no—
101C2×C4×Dic17C2311yescharacteristic
102C2×C4×Dic17C2311yescharacteristic
103C22×#3622—22no—
104C22×#3622—41no—
105C22×#3622—41no—
106C22×#3622C2311yescharacteristic
107C22×#3622C2311yescharacteristic
108#10587574—22no—
109#10587574—41no—
110C17×#102—22no—
111C17×#102—41no—
112C17×C22⋊C8—22no—
113#10587503—41no—
114#10587503—41no—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587502—21no—
120#10587502D411yescharacteristic
121#10587502D411yescharacteristic
122#10587489—22no—
123#10587489—41no—
124#10587485—41no—
125#10587485—41no—
126C24×C17—21no—
127C24×C17C2×D411yescharacteristic
128C2×D4×C17—24no—
129C2×D4×C17—21no—
130C2×D4×C17—21no—
131C2×D4×C17—21no—
132C2×D4×C17—21no—
133C2×D4×C17—21no—
134C2×D4×C17—42no—
135C2×D4×C17—42no—
136C2×D4×C17C2×Q812yes—
137C22×C4×C17—21no—
138C22×C4×C17—21no—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17—21no—
142C22×C4×C17—21no—
143C22×C4×C17—21no—
144C22×C4×C17—42no—
145C22×C4×C17—42no—
146C22×C4×C17—81no—
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17C2×Q812yes—
149C22×C4×C17#4112yes—
150C22×C4×C17C2411yescharacteristic
151C22×Dic17—21no—
152C22×Dic17—21no—
153C22×Dic17—21no—
154C22×Dic17—21no—
155C22×Dic17—42no—
156C22×Dic17—42no—
157C22×Dic17—42no—
158C22×Dic17—42no—
159C22×Dic17—81no—
160C22×Dic17—81no—
161C22×Dic17C2×D411yescharacteristic
162C22×Dic17C2×D411yescharacteristic
163C22×Dic17C2×D411yescharacteristic
164C22×Dic17C2×D411yescharacteristic
165C2×Dic34—42no—
166C2×Dic34—42no—
167C2×Dic34—81no—
168C2×C8×C17—21no—
169C2×C8×C17—21no—
170C2×C8×C17—21no—
171C2×C8×C17—21no—
172C2×C8×C17—21no—
173C2×C8×C17—21no—
174C4⋊C4×C17—21no—
175C4⋊C4×C17—21no—
176C4⋊C4×C17—21no—
177C4⋊C4×C17—21no—
178C4⋊C4×C17—42no—
179C4⋊C4×C17C2×Q812yes—
180C4⋊C4×C17#4112yes—
181C22⋊C4×C17—24no—
182C22⋊Dic17—22no—
183C22⋊Dic17—42no—
184C22⋊Dic17—42no—
185C22⋊Dic17—42no—
186C22⋊Dic17—81no—
187C22⋊Dic17D814yes—
188C4⋊Dic17—21no—
189C4⋊Dic17—21no—
190C4⋊Dic17—21no—
191C4⋊Dic17—21no—
192C4⋊Dic17—21no—
193C4⋊Dic17—21no—
194C4⋊Dic17—21no—
195C4⋊Dic17—42no—
196C4⋊Dic17—42no—
197C4⋊Dic17—42no—
198C4⋊Dic17#4112yes—
199#63225—42no—
200#63225—42no—
201#63225—42no—
202#63225—81no—
203C4×Dic17—21no—
204C4×Dic17—21no—
205C4×Dic17—21no—
206C4×Dic17—21no—
207C4×Dic17—42no—
208C2×#3622—21no—
209C2×#3622—21no—
210C2×#3622—21no—
211C2×#3622—21no—
212C2×#3622—21no—
213C2×#3622—21no—
214C2×#3622—44no—
215#26408—171nomaximal, Sylow
216C23×C17—21no—
217C23×C17—21no—
218C23×C17—21no—
219C23×C17—21no—
220C23×C17—22no—
221C23×C17—41no—
222C23×C17—41no—
223C23×C17—41no—
224C23×C17—41no—
225C23×C17—41no—
226C23×C17—41no—
227C23×C17C22⋊Q812yes—
228C23×C17C2×D812yes—
229C23×C17#13712yes—
230C23×C17C22×D411yescharacteristic
231D4×C17—22no—
232D4×C17—22no—
233D4×C17—48no—
234D4×C17—41no—
235D4×C17—41no—
236C2×C4×C17—21no—
237C2×C4×C17—21no—
238C2×C4×C17—22no—
239C2×C4×C17—22no—
240C2×C4×C17—22no—
241C2×C4×C17—22no—
242C2×C4×C17—22no—
243C2×C4×C17—22no—
244C2×C4×C17—41no—
245C2×C4×C17—41no—
246C2×C4×C17—41no—
247C2×C4×C17—41no—
248C2×C4×C17—41no—
249C2×C4×C17—41no—
250C2×C4×C17—41no—
251C2×C4×C17—84no—
252C2×C4×C17C22⋊Q812yes—
253C2×C4×C17C22×Q811yescharacteristic
254C2×C4×C17C2×#4111yescharacteristic
255C2×C4×C17C2×#4111yescharacteristic
256C2×C4×C17C2×#4111yescharacteristic
257C2×C4×C17H(5,𝔽2)11yescharacteristic
258C2×C4×C17H(5,𝔽2)11yescharacteristic
259C2×Dic17—21no—
260C2×Dic17—21no—
261C2×Dic17—21no—
262C2×Dic17—21no—
263C2×Dic17—21no—
264C2×Dic17—22no—
265C2×Dic17—21no—
266C2×Dic17—21no—
267C2×Dic17—21no—
268C2×Dic17—21no—
269C2×Dic17—21no—
270C2×Dic17—41no—
271C2×Dic17—41no—
272C2×Dic17—41no—
273C2×Dic17—41no—
274C2×Dic17—41no—
275C2×Dic17—41no—
276C2×Dic17—41no—
277C2×Dic17—41no—
278C2×Dic17—41no—
279C2×Dic17—41no—
280C2×Dic17—41no—
281C2×Dic17—84no—
282C2×Dic17—84no—
283C2×Dic17C2×D812yes—
284C2×Dic17C2×D812yes—
285Dic34—84no—
286C8×C17—41no—
287C8×C17—41no—
288#3622—41no—
289#3622—41no—
290#2440—171no—
291#2361—171no—
292#2099—171no—
293#2086—171no—
294#2067—171no—
295#2023—171no—
296#1988—171no—
297#1918—171no—
298#1893—171no—
299#1874D1711yescharacteristic
300#1862—171no—
301#1852—171no—
302#1850—171no—
303#1802—171no—
304C2×#340—171no—
305C22×C17—22no—
306C22×C17—22no—
307C22×C17—22no—
308C22×C17—22no—
309C22×C17—22no—
310C22×C17—22no—
311C22×C17—42no—
312C22×C17—42no—
313C22×C17—42no—
314C22×C17—81no—
315C22×C17—81no—
316C22×C17—81no—
317C22×C17—81no—
318C22×C17#47414yes—
319C22×C17C2×C22⋊Q811yescharacteristic
320C22×C17#54611yescharacteristic
321C22×C17#54611yescharacteristic
322C22×C17C22×D811yescharacteristic
323C22×C17C2×#13711yescharacteristic
324C22×C17#57711yescharacteristic
325C22×C17#57711yescharacteristic
326C4×C17—22no—
327C4×C17—42no—
328C4×C17—42no—
329C4×C17—81no—
330C4×C17#52911yescharacteristic
331C4×C17#53811yescharacteristic
332C4×C17#55411yescharacteristic
333C4×C17#55411yescharacteristic
334Dic17—22no—
335Dic17—22no—
336Dic17—42no—
337Dic17—41no—
338Dic17—41no—
339Dic17—81no—
340Dic17—81no—
341Dic17—81no—
342C23×C8—171no—
343C2×C4⋊Q8—171no—
344C2×C22⋊Q8—341no—
345C2×#121C2×D1711yescharacteristic
346C2×C4×D4—171no—
347C2×C4×D4—341no—
348C22×C4⋊C4—171no—
349C22×C4⋊C4C2×D1711yescharacteristic
350C2×#107—171no—
351C2×#107—171no—
352C2×C4⋊C8—171no—
353C2×C4⋊C8—171no—
354C2×#102—171no—
355C2×#102—171no—
356C2×#102C2×D1711yescharacteristic
357C2×#102C2×D1711yescharacteristic
358C2×C22⋊C8C2×D1711yescharacteristic
359#398—341no—
360#395—341no—
361#390—171no—
362#389—171no—
363#389—171no—
364#389—171no—
365#388—341no—
366#386—341no—
367#385—341no—
368#385—341no—
369#384—171no—
370#380—171no—
371C4×C4⋊C4—171no—
372C4×C22⋊C4—171no—
373#340—171no—
374#340—171no—
375#340—171no—
376#340—171no—
377#340—171no—
378#340—171no—
379#340—171no—
380#340—171no—
381#340C2×D1711yescharacteristic
382#340C2×D1711yescharacteristic
383#336—171no—
384#336—171no—
385C2×C17—24no—
386C2×C17—82no—
387C2×C17#245011yescharacteristic
388C2×C17C2×#47411yescharacteristic
389C2×C17#308411yescharacteristic
390C2×C17#329411yescharacteristic
391C2×C17#329411yescharacteristic
392C2×C17#331611yescharacteristic
393C2×C17#332011yescharacteristic
394C22×Q8—341no—
395C22×D4—21no—
396C22×D4C22×D1711yescharacteristic
397C23×C4—21no—
398C23×C4—341no—
399C23×C4—341no—
400C23×C4C22×D1711yescharacteristic
401C22×C8—171no—
402C22×C8—171no—
403C22×C8—342no—
404C22×C8—681no—
405C22×C8—681no—
406C22×C8C22×D1711yescharacteristic
407C22×C8C22×D1711yescharacteristic
408C4⋊Q8—342no—
409C4⋊Q8—681no—
410C22⋊Q8—684no—
411#121—22no—
412#121Dic3414yescomplemented
413C4×D4—342no—
414C4×D4—681no—
415C4×D4—684no—
416C2×C4⋊C4—22no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—41no—
419C2×C4⋊C4—171no—
420C2×C4⋊C4—171no—
421C2×C4⋊C4—171no—
422C2×C4⋊C4—171no—
423C2×C4⋊C4—342no—
424C2×C4⋊C4—341no—
425C2×C4⋊C4—341no—
426C2×C4⋊C4—341no—
427C2×C4⋊C4—341no—
428C2×C4⋊C4—341no—
429C2×C4⋊C4—341no—
430C2×C4⋊C4—341no—
431C2×C4⋊C4—681no—
432C2×C4⋊C4C22×D1711yescharacteristic
433C2×C4⋊C4C22×D1711yescharacteristic
434C2×C4⋊C4C22×D1711yescharacteristic
435C2×C22⋊C4—21no—
436C2×C22⋊C4—171no—
437C2×C22⋊C4—171no—
438C2×C22⋊C4—341no—
439C2×C22⋊C4—341no—
440C2×C22⋊C4—341no—
441C2×C22⋊C4—341no—
442C2×C42—171no—
443C2×C42—171no—
444C2×C42—341no—
445#107—342no—
446#107—681no—
447C4⋊C8—681no—
448C4⋊C8—681no—
449#102—22no—
450#102—41no—
451#102—681no—
452#102—681no—
453C22⋊C8—22no—
454#95—171no—
455#95—171no—
456#95—341no—
457#95—341no—
458#95—341no—
459#95—341no—
460#95—341no—
461C17#2640811yescharacteristic, complemented, Sylow
462C24—21no—
463C24D4×D1711yescharacteristic
464C2×Q8—682no—
465C2×Q8—682no—
466C2×Q8—1361no—
467C2×D4—24no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—42no—
474C2×D4—42no—
475C2×D4C2×Dic3412yes—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—81no—
486C22×C4—171no—
487C22×C4—171no—
488C22×C4—171no—
489C22×C4—171no—
490C22×C4—341no—
491C22×C4—341no—
492C22×C4—341no—
493C22×C4—341no—
494C22×C4—682no—
495C22×C4—682no—
496C22×C4—682no—
497C22×C4—682no—
498C22×C4—1361no—
499C22×C4—1361no—
500C22×C4C2×Dic3412yes—
501C22×C4D4×D1711yescharacteristic
502C22×C4#6325412yes—
503C22×C4C23×D1711yescharacteristic
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—341no—
511C2×C8—341no—
512C2×C8—341no—
513C2×C8—341no—
514C2×C8—341no—
515C2×C8—341no—
516C2×C8—684no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—42no—
522C4⋊C4—172no—
523C4⋊C4—341no—
524C4⋊C4—341no—
525C4⋊C4—341no—
526C4⋊C4—341no—
527C4⋊C4—341no—
528C4⋊C4—341no—
529C4⋊C4—341no—
530C4⋊C4—682no—
531C4⋊C4—682no—
532C4⋊C4—682no—
533C4⋊C4—682no—
534C4⋊C4—682no—
535C4⋊C4—682no—
536C4⋊C4—1361no—
537C4⋊C4C2×Dic3412yes—
538C4⋊C4#6325412yes—
539C22⋊C4—24no—
540C22⋊C4—174no—
541C22⋊C4—342no—
542C22⋊C4—682no—
543C22⋊C4—682no—
544C22⋊C4—682no—
545C22⋊C4—1361no—
546C42—341no—
547C42—341no—
548C42—341no—
549C42—341no—
550C42—682no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—22no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23#1058756412yes—
563C23D8×D1712yes—
564C23#1058760112yes—
565C23C2×D4×D1711yescharacteristic
566Q8—1364no—
567D4—22no—
568D4—22no—
569D4—48no—
570D4—41no—
571D4—41no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—84no—
588C2×C4—172no—
589C2×C4—172no—
590C2×C4—341no—
591C2×C4—341no—
592C2×C4—341no—
593C2×C4—341no—
594C2×C4—341no—
595C2×C4—341no—
596C2×C4—341no—
597C2×C4—341no—
598C2×C4—341no—
599C2×C4—341no—
600C2×C4—342no—
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×C4—681no—
610C2×C4—681no—
611C2×C4—681no—
612C2×C4—1364no—
613C2×C4—1364no—
614C2×C4#1058756412yes—
615C2×C4C22×Dic3411yescharacteristic
616C2×C4C2×#6325411yescharacteristic
617C2×C4C2×#6325411yescharacteristic
618C2×C4#1058769111yescharacteristic
619C2×C4#41×D1711yescharacteristic
620C2×C4#1058769611yescharacteristic
621C8—41no—
622C8—41no—
623C8—681no—
624C8—681no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22#1176308714yes—
639C22C2×#1058756411yescharacteristic
640C22#1176344611yescharacteristic
641C22#1176345111yescharacteristic
642C22C2×D8×D1711yescharacteristic
643C22C2×#1058760111yescharacteristic
644C22#1176361811yescharacteristic
645C22#1176361911yescharacteristic
646C4—22no—
647C4—42no—
648C4—42no—
649C4—81no—
650C4—342no—
651C4—342no—
652C4—682no—
653C4—681no—
654C4—681no—
655C4—1361no—
656C4—1361no—
657C4—1361no—
658C4#1176334011yescharacteristic
659C4#1176339611yescharacteristic
660C4#1176351611yescharacteristic
661C4#1176351811yescharacteristic
662C2—24no—
663C2—82no—
664C2#1453261211yescharacteristic
665C2C2×#1176308711yescharacteristic
666C2#1453865011yescharacteristic
667C2#1454059011yescharacteristic
668C2#1454059511yescharacteristic
669C2#1454089311yescharacteristic
670C2#1454095911yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#1874C2
socleC23×C17C22×D4
prime radicalC2×C17×#121C22
hypercenter#1874D17
hypocenterC17#26408
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1874

Complements

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

NH ≅ G/Ntype
1. #14532480C2semidirect
3. #14530345C2semidirect
4. #14530274C2semidirect
5. #14530170C2semidirect
6. #14529918C2semidirect
7. #14529734C2semidirect
8. #14529354C2semidirect
10. #14529033C2semidirect
11. #14528974C2semidirect
12. #14528961C2semidirect
13. #14528782C2semidirect
14. C2×#11762571C2semidirect
69. C17×#121Q8semidirect
412. #121Dic34semidirect
461. C17#26408semidirect

Maximal subgroups

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

subgroupindex
#145324802
#145318042
#145303452
#145302742
#145301702
#145299182
#145297342
#145293542
#145292132
#145290332
#145289742
#145289612
#145287822
C2×#117625712
C17×#18742
#2640817