Finite Group Databasebeta

← #17097873

Subgroups — #17097873

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#14532486C211yesmaximal, characteristic, complemented
2#14531798C211yesmaximal, characteristic
3#14530343C211yesmaximal, characteristic, complemented
4#14530272C211yesmaximal, characteristic, complemented
5#14530171C211yesmaximal, characteristic, complemented
6#14529921C211yesmaximal, characteristic, complemented
7#14529735C211yesmaximal, characteristic, complemented
8#14529356C211yesmaximal, characteristic, complemented
9#14529212C211yesmaximal, characteristic
10#14529099C211yesmaximal, characteristic
11#14529034C211yesmaximal, characteristic, complemented
12#14528974C211yesmaximal, characteristic, complemented
13#14528785C211yesmaximal, characteristic, complemented
14C2×#11762572C211yesmaximal, characteristic, complemented
15C17×#1850C211yesmaximal, characteristic, complemented
16C2×#10587555C2211yescharacteristic
17C2×#10587564—21no—
18C2×#10587616C2211yescharacteristic
19C2×C4×D68C2211yescharacteristic
20C2×#10587567—21no—
21C22×C4⋊Dic17C2211yescharacteristic
22C22×C4⋊Dic17C2211yescharacteristic
23C2×#10587489C2211yescharacteristic
24C2×#10587489C2211yescharacteristic
25C2×#10587492C2211yescharacteristic
26C2×#10587492C2211yescharacteristic
27C2×#10587492C2211yescharacteristic
28C2×#10587492C2211yescharacteristic
29#11762741—21no—
30#11762730—21no—
31#11762711C2211yescharacteristic
32#11762707C2211yescharacteristic
33#11762707C2211yescharacteristic
34#11762707C2211yescharacteristic
35#11762701—21no—
36#11762686—21no—
37#11762684—21no—
38#11762683—21no—
39#11762679C2211yescharacteristic
40#11762669C2211yescharacteristic
41C4×C4⋊Dic17C2211yescharacteristic
42#11762572C2211yescharacteristic
43#11762572C2211yescharacteristic
44#11762572C2211yescharacteristic
45#11762572C2211yescharacteristic
46#11762572C2211yescharacteristic
47#11762572C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762572C2211yescharacteristic
50#11762572C2211yescharacteristic
51#11762572C2211yescharacteristic
52C23×C8×C17C2211yescharacteristic
53C2×C17×C4⋊C8C2211yescharacteristic
54C2×C17×C4⋊C8C2211yescharacteristic
55C2×C17×C22⋊C8C2211yescharacteristic
56C4×C22⋊C4×C17C2211yescharacteristic
57C17×#336C2211yescharacteristic
58C17×#336C2211yescharacteristic
59C23×C4×C17C2311yescharacteristic
60C22×C22⋊D17—21no—
61C23×Dic17—21no—
62C23×Dic17—21no—
63C22×D68C2311yescharacteristic
64C22×C4×D17—21no—
65C22×Dic34—21no—
66C22×C8×C17—22no—
67C22×C8×C17—41no—
68C22×C8×C17—41no—
69C22×C8×C17C2311yescharacteristic
70C22×C8×C17C2311yescharacteristic
71C22×C8×C17C2311yescharacteristic
72C22×C8×C17C2311yescharacteristic
73C2×C22⋊C4×C17D411yescharacteristic
74C2×C22⋊C4×C17D411yescharacteristic
75C2×C42×C17C2311yescharacteristic
76C2×C42×C17C2311yescharacteristic
77C2×C22⋊Dic17—21no—
78#10587616—22no—
79#10587616Q814yescomplemented
80C2×#63227—21no—
81C2×#63227—21no—
82C2×#63227—21no—
83C2×#63227—21no—
84C2×C4⋊Dic17—21no—
85C2×C4⋊Dic17—21no—
86C2×C4⋊Dic17—21no—
87C2×C4⋊Dic17—21no—
88C2×C4⋊Dic17—21no—
89C2×C4⋊Dic17—22no—
90C2×C4⋊Dic17—22no—
91C2×C4⋊Dic17—41no—
92C2×C4⋊Dic17—41no—
93C2×C4⋊Dic17C2311yescharacteristic
94C2×C4⋊Dic17C2311yescharacteristic
95C2×C4⋊Dic17C2311yescharacteristic
96C2×C4⋊Dic17C2311yescharacteristic
97C2×C4⋊Dic17C2311yescharacteristic
98C2×C4⋊Dic17C2311yescharacteristic
99C2×C4⋊Dic17C2311yescharacteristic
100C2×#63225—21no—
101C2×#63225—21no—
102C2×#63225—21no—
103C2×C4×Dic17—21no—
104#10587567—44no—
105#10587564—44no—
106C4×D68—22no—
107C4×D68—41no—
108#10587555—22no—
109#10587555—41no—
110C17×C4⋊C8—41no—
111C17×C4⋊C8—41no—
112C17×C22⋊C8—22no—
113C17×#95D411yescharacteristic
114C17×#95D411yescharacteristic
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587502—21no—
120#10587492—22no—
121#10587492—41no—
122#10587492—41no—
123#10587492—41no—
124#10587489—22no—
125#10587489—41no—
126C24×C17C2×D411yescharacteristic
127C23×D17—21no—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17—21no—
132C22×C4×C17C2×D411yescharacteristic
133C22×C4×C17C2×D411yescharacteristic
134C22×C4×C17C2×D411yescharacteristic
135C22×C4×C17C2×D411yescharacteristic
136C22×C4×C17C2×D411yescharacteristic
137C22×C4×C17C2×Q812yes—
138C22×C4×C17#4112yes—
139C22×C4×C17C2411yescharacteristic
140C2×C22⋊D17—24no—
141C2×C22⋊D17—42no—
142C2×C22⋊D17—42no—
143C22×Dic17—21no—
144C22×Dic17—21no—
145C22×Dic17—21no—
146C22×Dic17—21no—
147C22×Dic17—21no—
148C22×Dic17—21no—
149C22×Dic17—21no—
150C22×Dic17—42no—
151C22×Dic17—42no—
152C22×Dic17—42no—
153C22×Dic17—42no—
154C22×Dic17—81no—
155C22×Dic17—81no—
156C2×D68—21no—
157C2×D68—21no—
158C2×D68—21no—
159C2×D68—21no—
160C2×D68—21no—
161C2×D68C2×Q812yes—
162C2×C4×D17—42no—
163C2×C4×D17—42no—
164C2×C4×D17—81no—
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—
174C2×C8×C17—21no—
175C2×C8×C17—21no—
176C2×C8×C17—21no—
177C2×C8×C17—21no—
178C2×C8×C17—21no—
179C2×C8×C17—21no—
180C2×C8×C17—44no—
181C22⋊C4×C17—22no—
182C22⋊C4×C17D814yes—
183C42×C17—21no—
184C42×C17—21no—
185C42×C17—21no—
186C42×C17—21no—
187C22⋊Dic17—42no—
188#63227—24no—
189#63227—42no—
190#63227—42no—
191#63227—81no—
192C4⋊Dic17—21no—
193C4⋊Dic17—21no—
194C4⋊Dic17—21no—
195C4⋊Dic17—21no—
196C4⋊Dic17—21no—
197C4⋊Dic17—21no—
198C4⋊Dic17—21no—
199C4⋊Dic17—21no—
200C4⋊Dic17—21no—
201C4⋊Dic17—21no—
202C4⋊Dic17—21no—
203C4⋊Dic17—42no—
204C4⋊Dic17—42no—
205C4⋊Dic17—42no—
206C4⋊Dic17—42no—
207C4⋊Dic17—81no—
208C4⋊Dic17C2×Q812yes—
209C4⋊Dic17#4112yes—
210C4⋊Dic17#4112yes—
211#63225—42no—
212#63225—42no—
213#63225—42no—
214C4×Dic17—42no—
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—
231C2×C4×C17—21no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—21no—
235C2×C4×C17—21no—
236C2×C4×C17—21no—
237C2×C4×C17—22no—
238C2×C4×C17—22no—
239C2×C4×C17—22no—
240C2×C4×C17—22no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—21no—
244C2×C4×C17—21no—
245C2×C4×C17—21no—
246C2×C4×C17—21no—
247C2×C4×C17C22⋊Q812yes—
248C2×C4×C17C2×D812yes—
249C2×C4×C17C2×D812yes—
250C2×C4×C17C22×Q811yescharacteristic
251C2×C4×C17C2×#4111yescharacteristic
252C2×C4×C17C2×#4111yescharacteristic
253C2×C4×C17C2×#4111yescharacteristic
254C2×C4×C17H(5,𝔽2)11yescharacteristic
255C2×C4×C17H(5,𝔽2)11yescharacteristic
256C22⋊D17—48no—
257C2×Dic17—22no—
258C2×Dic17—22no—
259C2×Dic17—22no—
260C2×Dic17—41no—
261C2×Dic17—41no—
262C2×Dic17—41no—
263C2×Dic17—41no—
264C2×Dic17—41no—
265C2×Dic17—41no—
266C2×Dic17—41no—
267C2×Dic17—41no—
268C2×Dic17—41no—
269C2×Dic17—41no—
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—84no—
279C2×Dic17—84no—
280D68—22no—
281D68—22no—
282D68—41no—
283D68—41no—
284C4×D17—84no—
285Dic34—84no—
286C8×C17—41no—
287C8×C17—41no—
288C8×C17—41no—
289C8×C17—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#1874—171no—
300#1862—171no—
301#1852—171no—
302#1850D1711yescharacteristic, complemented
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—41no—
330C4×C17—41no—
331C4×C17#52911yescharacteristic
332C4×C17#53811yescharacteristic
333C4×C17#55411yescharacteristic
334C4×C17#55411yescharacteristic
335Dic17—42no—
336Dic17—42no—
337Dic17—42no—
338Dic17—81no—
339Dic17—81no—
340Dic17—81no—
341Dic17—81no—
342C23×C8C2×D1711yescharacteristic
343C2×C4⋊Q8—171no—
344C2×C22⋊Q8—341no—
345C2×#121—171no—
346C2×C4×D4—171no—
347C2×C4×D4—341no—
348C22×C4⋊C4—171no—
349C22×C4⋊C4—171no—
350C2×#107—171no—
351C2×#107—171no—
352C2×C4⋊C8C2×D1711yescharacteristic
353C2×C4⋊C8C2×D1711yescharacteristic
354C2×#102—171no—
355C2×#102—171no—
356C2×#102—171no—
357C2×#102—171no—
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⋊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#336C2×D1711yescharacteristic
384#336C2×D1711yescharacteristic
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—341no—
395C22×D4—171no—
396C22×D4—341no—
397C23×C4—341no—
398C23×C4—341no—
399C23×C4—341no—
400C23×C4C22×D1711yescharacteristic
401C22×C8—22no—
402C22×C8—41no—
403C22×C8—41no—
404C22×C8C22×D1711yescharacteristic
405C22×C8C22×D1711yescharacteristic
406C22×C8C22×D1711yescharacteristic
407C22×C8C22×D1711yescharacteristic
408C4⋊Q8—342no—
409C4⋊Q8—681no—
410C22⋊Q8—684no—
411#121—174no—
412#121—342no—
413C4×D4—342no—
414C4×D4—681no—
415C4×D4—684no—
416C2×C4⋊C4—171no—
417C2×C4⋊C4—171no—
418C2×C4⋊C4—171no—
419C2×C4⋊C4—171no—
420C2×C4⋊C4—171no—
421C2×C4⋊C4—171no—
422C2×C4⋊C4—171no—
423C2×C4⋊C4—342no—
424C2×C4⋊C4—342no—
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—341no—
432C2×C4⋊C4—341no—
433C2×C4⋊C4—681no—
434C2×C4⋊C4—681no—
435C2×C22⋊C4—341no—
436C2×C22⋊C4—341no—
437C2×C22⋊C4—341no—
438C2×C22⋊C4—341no—
439C2×C22⋊C4—341no—
440C2×C22⋊C4D6811yescharacteristic
441C2×C22⋊C4D6811yescharacteristic
442C2×C42—341no—
443C2×C42C22×D1711yescharacteristic
444C2×C42C22×D1711yescharacteristic
445#107—342no—
446#107—681no—
447C4⋊C8—41no—
448C4⋊C8—41no—
449#102—342no—
450#102—681no—
451#102—681no—
452#102—681no—
453C22⋊C8—22no—
454#95—341no—
455#95—341no—
456#95—341no—
457#95—341no—
458#95—341no—
459#95D6811yescharacteristic
460#95D6811yescharacteristic
461C17#2640811yescharacteristic, complemented, Sylow
462C24—341no—
463C24C2×D6811yescharacteristic
464C2×Q8—682no—
465C2×Q8—682no—
466C2×Q8—1361no—
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—341no—
481C22×C4—341no—
482C22×C4—341no—
483C22×C4—341no—
484C22×C4—341no—
485C22×C4—341no—
486C22×C4—341no—
487C22×C4—682no—
488C22×C4—682no—
489C22×C4—682no—
490C22×C4—682no—
491C22×C4—682no—
492C22×C4—682no—
493C22×C4—1361no—
494C22×C4—1361no—
495C22×C4—1361no—
496C22×C4C2×D6811yescharacteristic
497C22×C4C2×D6811yescharacteristic
498C22×C4C2×D6811yescharacteristic
499C22×C4C2×D6811yescharacteristic
500C22×C4C2×D6811yescharacteristic
501C22×C4#6325412yes—
502C22×C4Q8×D1712yes—
503C22×C4C23×D1711yescharacteristic
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—44no—
517C4⋊C4—172no—
518C4⋊C4—172no—
519C4⋊C4—172no—
520C4⋊C4—341no—
521C4⋊C4—341no—
522C4⋊C4—341no—
523C4⋊C4—341no—
524C4⋊C4—341no—
525C4⋊C4—341no—
526C4⋊C4—341no—
527C4⋊C4—341no—
528C4⋊C4—341no—
529C4⋊C4—341no—
530C4⋊C4—341no—
531C4⋊C4—682no—
532C4⋊C4—682no—
533C4⋊C4—682no—
534C4⋊C4—682no—
535C4⋊C4—682no—
536C4⋊C4—682no—
537C4⋊C4—682no—
538C4⋊C4—1361no—
539C22⋊C4—22no—
540C22⋊C4—344no—
541C22⋊C4—682no—
542C22⋊C4—682no—
543C22⋊C4—682no—
544C22⋊C4—1361no—
545C22⋊C4D13614yes—
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—
562C23#1058758312yes—
563C23C2×D13612yes—
564C23#1058759512yes—
565C23C22×D6811yescharacteristic
566Q8—1364no—
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—342no—
589C2×C4—342no—
590C2×C4—342no—
591C2×C4—681no—
592C2×C4—681no—
593C2×C4—681no—
594C2×C4—681no—
595C2×C4—681no—
596C2×C4—681no—
597C2×C4—681no—
598C2×C4—681no—
599C2×C4—681no—
600C2×C4—681no—
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—1364no—
610C2×C4—1364no—
611C2×C4—1364no—
612C2×C4#1058758312yes—
613C2×C4C2×D13612yes—
614C2×C4C2×D13612yes—
615C2×C4C2×#6325411yescharacteristic
616C2×C4#1058769111yescharacteristic
617C2×C4#1058769111yescharacteristic
618C2×C4C2×Q8×D1711yescharacteristic
619C2×C4#41×D1711yescharacteristic
620C2×C4#41×D1711yescharacteristic
621C8—41no—
622C8—41no—
623C8—41no—
624C8—41no—
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#1176309014yes—
639C22C2×#1058758311yescharacteristic
640C22#1176345411yescharacteristic
641C22#1176345411yescharacteristic
642C22C22×D13611yescharacteristic
643C22C2×#1058759511yescharacteristic
644C22#1176362111yescharacteristic
645C22#1176362111yescharacteristic
646C4—22no—
647C4—22no—
648C4—22no—
649C4—41no—
650C4—41no—
651C4—682no—
652C4—682no—
653C4—682no—
654C4—1361no—
655C4—1361no—
656C4—1361no—
657C4—1361no—
658C4#1176333611yescharacteristic
659C4#1176339411yescharacteristic
660C4#1176351311yescharacteristic
661C4#1176351311yescharacteristic
662C2—24no—
663C2—1362no—
664C2#1453260611yescharacteristic
665C2C2×#1176309011yescharacteristic
666C2#1453864811yescharacteristic
667C2#1454059811yescharacteristic
668C2#1454059811yescharacteristic
669C2#1454089711yescharacteristic
670C2#1454095711yescharacteristic
subgroupG/H
centerC23C22×D68
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#1850C2
socleC23×C17C22×D4
prime radicalC2×#10587616C22
hypercenter#1850D17
hypocenterC17#26408
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1850

Complements

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

NH ≅ G/Ntype
1. #14532486C2semidirect
3. #14530343C2semidirect
4. #14530272C2semidirect
5. #14530171C2semidirect
6. #14529921C2semidirect
7. #14529735C2semidirect
8. #14529356C2semidirect
11. #14529034C2semidirect
12. #14528974C2semidirect
13. #14528785C2semidirect
14. C2×#11762572C2semidirect
15. C17×#1850C2semidirect
79. #10587616Q8semidirect
302. #1850D17semidirect
461. C17#26408semidirect

Maximal subgroups

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

subgroupindex
#145324862
#145317982
#145303432
#145302722
#145301712
#145299212
#145297352
#145293562
#145292122
#145290992
#145290342
#145289742
#145287852
C2×#117625722
C17×#18502
#2640817