Finite Group Databasebeta

← #17097871

Subgroups — #17097871

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#14531798C211yesmaximal, characteristic
2#14530341C211yesmaximal, characteristic, complemented
3#14530275C211yesmaximal, characteristic, complemented
4#14530169C211yesmaximal, characteristic, complemented
5#14529917C211yesmaximal, characteristic, complemented
6#14529737C211yesmaximal, characteristic, complemented
7#14529352C211yesmaximal, characteristic, complemented
8#14529210C211yesmaximal, characteristic
9#14529101C211yesmaximal, characteristic
10#14529030C211yesmaximal, characteristic, complemented
11#14528971C211yesmaximal, characteristic, complemented
12#14528964C211yesmaximal, characteristic, complemented
13#14528781C211yesmaximal, characteristic, complemented
14C2×#11762571C211yesmaximal, characteristic, complemented
15C17×#2440C211yesmaximal, characteristic, complemented
16C23×#3622C2211yescharacteristic
17C2×#10587616C2211yescharacteristic
18C2×C4×D68C2211yescharacteristic
19C2×#10587567—21no—
20C22×C4⋊Dic17C2211yescharacteristic
21C2×#10587478C2211yescharacteristic
22C2×#10587478C2211yescharacteristic
23C2×C4⋊#3622C2211yescharacteristic
24C2×C4⋊#3622C2211yescharacteristic
25C2×#10587480C2211yescharacteristic
26C2×#10587480C2211yescharacteristic
27C2×#10587480C2211yescharacteristic
28C2×#10587480C2211yescharacteristic
29C2×C22⋊#3622C2211yescharacteristic
30#11762711C2211yescharacteristic
31#11762707C2211yescharacteristic
32#11762701—21no—
33#11762686—21no—
34#11762683—21no—
35C4×C4⋊Dic17C2211yescharacteristic
36#11762571C2211yescharacteristic
37#11762571C2211yescharacteristic
38#11762571C2211yescharacteristic
39#11762571C2211yescharacteristic
40#11762571C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762564C2211yescharacteristic
47#11762564C2211yescharacteristic
48C2×C17×C4⋊Q8C2211yescharacteristic
49C2×C17×C22⋊Q8—21no—
50C22×C4⋊C4×C17C2211yescharacteristic
51C17×#398—21no—
52C17×#395—21no—
53C17×#389C2211yescharacteristic
54C17×#389C2211yescharacteristic
55C17×#385—21no—
56C17×#384C2211yescharacteristic
57C17×#380C2211yescharacteristic
58C4×C22⋊C4×C17C2211yescharacteristic
59C22×Q8×C17—21no—
60C23×C4×C17—21no—
61C23×C4×C17C2311yescharacteristic
62C22×C22⋊D17—21no—
63C23×Dic17—21no—
64C22×D68C2311yescharacteristic
65C22×C4×D17—21no—
66C17×C4⋊Q8—22no—
67C17×C4⋊Q8—41no—
68C17×C22⋊Q8—44no—
69C2×C4⋊C4×C17—22no—
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17—21no—
73C2×C4⋊C4×C17—21no—
74C2×C4⋊C4×C17—21no—
75C2×C4⋊C4×C17—21no—
76C2×C4⋊C4×C17—41no—
77C2×C4⋊C4×C17C2311yescharacteristic
78C2×C4⋊C4×C17C2311yescharacteristic
79C2×C4⋊C4×C17C2311yescharacteristic
80C2×C4⋊C4×C17C2311yescharacteristic
81C2×C22⋊C4×C17—21no—
82C2×C22⋊C4×C17D411yescharacteristic
83C2×C22⋊C4×C17D411yescharacteristic
84C2×C42×C17C2311yescharacteristic
85C2×C42×C17C2311yescharacteristic
86#10587616—22no—
87#10587616Q814yescomplemented
88C2×#63227—21no—
89C2×#63227—21no—
90C2×#63227—21no—
91C2×#63227—21no—
92C2×C4⋊Dic17—21no—
93C2×C4⋊Dic17—22no—
94C2×C4⋊Dic17—41no—
95C2×C4⋊Dic17C2311yescharacteristic
96C2×C4⋊Dic17C2311yescharacteristic
97C2×C4⋊Dic17C2311yescharacteristic
98C2×#63225—21no—
99C2×C4×Dic17—21no—
100C22×#3622—22no—
101C22×#3622—41no—
102C22×#3622—41no—
103C22×#3622C2311yescharacteristic
104C22×#3622C2311yescharacteristic
105C22×#3622C2311yescharacteristic
106C22×#3622C2311yescharacteristic
107#10587567—44no—
108C4×D68—22no—
109C4×D68—41no—
110C17×#95—21no—
111C17×#95—21no—
112C17×#95—21no—
113C17×#95D411yescharacteristic
114C17×#95D411yescharacteristic
115#10587502—21no—
116#10587502—21no—
117C22⋊#3622—22no—
118#10587480—22no—
119#10587480—41no—
120#10587480—41no—
121#10587480—41no—
122#10587478—22no—
123#10587478—41no—
124C4⋊#3622—41no—
125C4⋊#3622—41no—
126C24×C17C2×D411yescharacteristic
127C23×D17—21no—
128C2×Q8×C17—42no—
129C2×Q8×C17—42no—
130C2×Q8×C17—81no—
131C22×C4×C17—21no—
132C22×C4×C17—21no—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17—21no—
136C22×C4×C17—21no—
137C22×C4×C17—21no—
138C22×C4×C17—21no—
139C22×C4×C17—42no—
140C22×C4×C17—42no—
141C22×C4×C17—81no—
142C22×C4×C17C2×D411yescharacteristic
143C22×C4×C17C2×D411yescharacteristic
144C22×C4×C17C2×D411yescharacteristic
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×Q812yes—
148C22×C4×C17#4112yes—
149C22×C4×C17C2411yescharacteristic
150C2×C22⋊D17—24no—
151C2×C22⋊D17—42no—
152C2×C22⋊D17—42no—
153C22×Dic17—21no—
154C22×Dic17—21no—
155C22×Dic17—21no—
156C22×Dic17—42no—
157C22×Dic17—42no—
158C22×Dic17—81no—
159C2×D68—21no—
160C2×D68—21no—
161C2×D68—21no—
162C2×D68—21no—
163C2×D68—21no—
164C2×D68C2×Q812yes—
165C2×C4×D17—42no—
166C2×C4×D17—42no—
167C2×C4×D17—81no—
168C4⋊C4×C17—21no—
169C4⋊C4×C17—21no—
170C4⋊C4×C17—21no—
171C4⋊C4×C17—21no—
172C4⋊C4×C17—21no—
173C4⋊C4×C17—21no—
174C4⋊C4×C17—21no—
175C4⋊C4×C17—42no—
176C4⋊C4×C17—42no—
177C4⋊C4×C17—42no—
178C4⋊C4×C17—42no—
179C4⋊C4×C17—42no—
180C4⋊C4×C17—81no—
181C4⋊C4×C17#4112yes—
182C22⋊C4×C17—22no—
183C22⋊C4×C17—42no—
184C22⋊C4×C17D814yes—
185C42×C17—21no—
186C42×C17—21no—
187C42×C17—21no—
188C42×C17—21no—
189#63227—24no—
190#63227—42no—
191#63227—42no—
192#63227—81no—
193C4⋊Dic17—21no—
194C4⋊Dic17—21no—
195C4⋊Dic17—21no—
196C4⋊Dic17—21no—
197C4⋊Dic17—42no—
198C4⋊Dic17C2×Q812yes—
199C4⋊Dic17#4112yes—
200#63225—42no—
201C4×Dic17—42no—
202C2×#3622—21no—
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×#3622—21no—
214C2×#3622—44no—
215#26408—171nomaximal, Sylow
216C23×C17—21no—
217C23×C17—21no—
218C23×C17—21no—
219C23×C17—21no—
220C23×C17C22⋊Q812yes—
221C23×C17C2×D812yes—
222C23×C17#13712yes—
223C23×C17C22×D411yescharacteristic
224C22×D17—22no—
225C22×D17—41no—
226C22×D17—41no—
227C22×D17—41no—
228C22×D17—41no—
229C22×D17—41no—
230C22×D17—41no—
231Q8×C17—84no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—21no—
235C2×C4×C17—21no—
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—21no—
243C2×C4×C17—21no—
244C2×C4×C17—22no—
245C2×C4×C17—21no—
246C2×C4×C17—21no—
247C2×C4×C17—21no—
248C2×C4×C17—21no—
249C2×C4×C17—41no—
250C2×C4×C17—41no—
251C2×C4×C17—41no—
252C2×C4×C17—41no—
253C2×C4×C17—41no—
254C2×C4×C17—41no—
255C2×C4×C17—41no—
256C2×C4×C17—41no—
257C2×C4×C17—41no—
258C2×C4×C17—41no—
259C2×C4×C17—41no—
260C2×C4×C17—84no—
261C2×C4×C17C22⋊Q812yes—
262C2×C4×C17C2×D812yes—
263C2×C4×C17C2×D812yes—
264C2×C4×C17C22×Q811yescharacteristic
265C2×C4×C17C2×#4111yescharacteristic
266C2×C4×C17C2×#4111yescharacteristic
267C2×C4×C17C2×#4111yescharacteristic
268C2×C4×C17H(5,𝔽2)11yescharacteristic
269C2×C4×C17H(5,𝔽2)11yescharacteristic
270C22⋊D17—48no—
271C2×Dic17—22no—
272C2×Dic17—22no—
273C2×Dic17—41no—
274C2×Dic17—41no—
275C2×Dic17—41no—
276C2×Dic17—41no—
277C2×Dic17—41no—
278C2×Dic17—41no—
279C2×Dic17—41no—
280C2×Dic17—84no—
281D68—22no—
282D68—22no—
283D68—41no—
284D68—41no—
285C4×D17—84no—
286#3622—41no—
287#3622—41no—
288#3622—41no—
289#3622—41no—
290#2440D1711yescharacteristic, complemented
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#1874—171no—
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#47414yes—
312C22×C17C2×C22⋊Q811yescharacteristic
313C22×C17#54611yescharacteristic
314C22×C17#54611yescharacteristic
315C22×C17C22×D811yescharacteristic
316C22×C17C2×#13711yescharacteristic
317C22×C17#57711yescharacteristic
318C22×C17#57711yescharacteristic
319C2×D17—42no—
320C2×D17—42no—
321C2×D17—42no—
322C2×D17—81no—
323C2×D17—81no—
324C2×D17—81no—
325C2×D17—81no—
326C4×C17—22no—
327C4×C17—22no—
328C4×C17—22no—
329C4×C17—42no—
330C4×C17—41no—
331C4×C17—41no—
332C4×C17—81no—
333C4×C17—81no—
334C4×C17—81no—
335C4×C17#52911yescharacteristic
336C4×C17#53811yescharacteristic
337C4×C17#55411yescharacteristic
338C4×C17#55411yescharacteristic
339Dic17—42no—
340Dic17—42no—
341Dic17—81no—
342C23×C8—171no—
343C2×C4⋊Q8C2×D1711yescharacteristic
344C2×C22⋊Q8—21no—
345C2×#121—171no—
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×#102—171no—
357C2×#102—171no—
358C2×C22⋊C8—171no—
359#398—21no—
360#395—21no—
361#390—171no—
362#389—171no—
363#389C2×D1711yescharacteristic
364#389C2×D1711yescharacteristic
365#388—341no—
366#386—341no—
367#385—21no—
368#385—341no—
369#384C2×D1711yescharacteristic
370#380C2×D1711yescharacteristic
371C4×C4⋊C4—171no—
372C4×C22⋊C4C2×D1711yescharacteristic
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#340—171no—
382#340—171no—
383#336—171no—
384#336—171no—
385C2×C17—24no—
386C2×C17#245011yescharacteristic
387C2×C17C2×#47411yescharacteristic
388C2×C17#308411yescharacteristic
389C2×C17#329411yescharacteristic
390C2×C17#329411yescharacteristic
391C2×C17#331611yescharacteristic
392C2×C17#332011yescharacteristic
393D17—82no—
394C22×Q8—21no—
395C22×D4—171no—
396C22×D4—341no—
397C23×C4—21no—
398C23×C4—341no—
399C23×C4—341no—
400C23×C4C22×D1711yescharacteristic
401C22×C8—171no—
402C22×C8—171no—
403C22×C8—171no—
404C22×C8—171no—
405C22×C8—342no—
406C22×C8—681no—
407C22×C8—681no—
408C4⋊Q8—22no—
409C4⋊Q8—41no—
410C22⋊Q8—44no—
411#121—174no—
412#121—342no—
413C4×D4—342no—
414C4×D4—681no—
415C4×D4—684no—
416C2×C4⋊C4—22no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—41no—
424C2×C4⋊C4—171no—
425C2×C4⋊C4—171no—
426C2×C4⋊C4—171no—
427C2×C4⋊C4—342no—
428C2×C4⋊C4—341no—
429C2×C4⋊C4—341no—
430C2×C4⋊C4—681no—
431C2×C4⋊C4C22×D1711yescharacteristic
432C2×C4⋊C4C22×D1711yescharacteristic
433C2×C4⋊C4C22×D1711yescharacteristic
434C2×C4⋊C4C22×D1711yescharacteristic
435C2×C22⋊C4—21no—
436C2×C22⋊C4—341no—
437C2×C22⋊C4—341no—
438C2×C22⋊C4—341no—
439C2×C22⋊C4—341no—
440C2×C22⋊C4C22⋊D1711yescharacteristic
441C2×C22⋊C4C22⋊D1711yescharacteristic
442C2×C42—341no—
443C2×C42C22×D1711yescharacteristic
444C2×C42C22×D1711yescharacteristic
445#107—342no—
446#107—681no—
447C4⋊C8—681no—
448C4⋊C8—681no—
449#102—342no—
450#102—681no—
451#102—681no—
452#102—681no—
453C22⋊C8—342no—
454#95—21no—
455#95—21no—
456#95—21no—
457#95—341no—
458#95—341no—
459#95C22⋊D1711yescharacteristic
460#95C22⋊D1711yescharacteristic
461C17#2640811yescharacteristic, complemented, Sylow
462C24—341no—
463C24C2×C22⋊D1711yescharacteristic
464C2×Q8—42no—
465C2×Q8—42no—
466C2×Q8—81no—
467C2×D4—172no—
468C2×D4—344no—
469C2×D4—341no—
470C2×D4—341no—
471C2×D4—341no—
472C2×D4—341no—
473C2×D4—341no—
474C2×D4—682no—
475C2×D4—682no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—81no—
487C22×C4—341no—
488C22×C4—341no—
489C22×C4—341no—
490C22×C4—682no—
491C22×C4—682no—
492C22×C4—682no—
493C22×C4—682no—
494C22×C4—1361no—
495C22×C4—1361no—
496C22×C4Q8×D1712yes—
497C22×C4#6325612yes—
498C22×C4C2×C22⋊D1711yescharacteristic
499C22×C4C2×C22⋊D1711yescharacteristic
500C22×C4C2×C22⋊D1711yescharacteristic
501C22×C4C2×C22⋊D1711yescharacteristic
502C22×C4C2×C22⋊D1711yescharacteristic
503C22×C4C23×D1711yescharacteristic
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—
514C2×C8—341no—
515C2×C8—341no—
516C2×C8—684no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—81no—
530C4⋊C4—172no—
531C4⋊C4—172no—
532C4⋊C4—341no—
533C4⋊C4—341no—
534C4⋊C4—341no—
535C4⋊C4—341no—
536C4⋊C4—682no—
537C4⋊C4—682no—
538C4⋊C4#6325412yes—
539C22⋊C4—22no—
540C22⋊C4—42no—
541C22⋊C4—344no—
542C22⋊C4—682no—
543C22⋊C4—682no—
544C22⋊C4—1361no—
545C22⋊C4#6322814yes—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—682no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—342no—
556C23—681no—
557C23—681no—
558C23—681no—
559C23—681no—
560C23—681no—
561C23—681no—
562C23C2×#6322812yes—
563C23#1058762812yes—
564C23#1058763212yes—
565C23C22×C22⋊D1711yescharacteristic
566Q8—84no—
567D4—342no—
568D4—342no—
569D4—688no—
570D4—681no—
571D4—681no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—84no—
601C2×C4—342no—
602C2×C4—342no—
603C2×C4—681no—
604C2×C4—681no—
605C2×C4—681no—
606C2×C4—681no—
607C2×C4—681no—
608C2×C4—681no—
609C2×C4—681no—
610C2×C4—1364no—
611C2×C4—1364no—
612C2×C4C2×#6322812yes—
613C2×C4C2×#6322812yes—
614C2×C4#1058763212yes—
615C2×C4C2×#6325411yescharacteristic
616C2×C4#1058769111yescharacteristic
617C2×C4C2×Q8×D1711yescharacteristic
618C2×C4C2×#6325611yescharacteristic
619C2×C4#41×D1711yescharacteristic
620C2×C4#1058769611yescharacteristic
621C8—681no—
622C8—681no—
623C8—681no—
624C8—681no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—682no—
632C22—682no—
633C22—682no—
634C22—1361no—
635C22—1361no—
636C22—1361no—
637C22—1361no—
638C22#1176308614yes—
639C22C2×#1058763211yescharacteristic
640C22#1176344711yescharacteristic
641C22#1176345311yescharacteristic
642C22C22×#6322811yescharacteristic
643C22C2×#1058762811yescharacteristic
644C22#1176361711yescharacteristic
645C22#1176362011yescharacteristic
646C4—22no—
647C4—22no—
648C4—22no—
649C4—42no—
650C4—41no—
651C4—41no—
652C4—81no—
653C4—81no—
654C4—81no—
655C4—682no—
656C4—682no—
657C4—1361no—
658C4#1176334511yescharacteristic
659C4#1176339911yescharacteristic
660C4#1176351311yescharacteristic
661C4#1176351711yescharacteristic
662C2—24no—
663C2—1362no—
664C2#1453260411yescharacteristic
665C2C2×#1176308611yescharacteristic
666C2#1453864211yescharacteristic
667C2#1454059111yescharacteristic
668C2#1454059711yescharacteristic
669C2#1454088911yescharacteristic
670C2#1454095511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2440C2
socleC23×C17C22×D4
prime radicalC2×#10587616C22
hypercenter#2440D17
hypocenterC17#26408
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2440

Complements

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

NH ≅ G/Ntype
2. #14530341C2semidirect
3. #14530275C2semidirect
4. #14530169C2semidirect
5. #14529917C2semidirect
6. #14529737C2semidirect
7. #14529352C2semidirect
10. #14529030C2semidirect
11. #14528971C2semidirect
12. #14528964C2semidirect
13. #14528781C2semidirect
14. C2×#11762571C2semidirect
15. C17×#2440C2semidirect
87. #10587616Q8semidirect
290. #2440D17semidirect
461. C17#26408semidirect

Maximal subgroups

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

subgroupindex
#145317982
#145303412
#145302752
#145301692
#145299172
#145297372
#145293522
#145292102
#145291012
#145290302
#145289712
#145289642
#145287812
C2×#117625712
C17×#24402
#2640817