Finite Group Databasebeta

← #17045107

Subgroups — #17045107

Order 4352.

752 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14531249C211yesmaximal, characteristic
2#14529394C211yesmaximal, characteristic, complemented
3#14529393C211yesmaximal, characteristic, complemented
4#14529347C211yesmaximal, characteristic, complemented
5#14529342C211yesmaximal, characteristic, complemented
6#14529082C211yesmaximal, characteristic
7#14528966C211yesmaximal, characteristic, complemented
8#14528962C211yesmaximal, characteristic, complemented
9#14528926C211yesmaximal, characteristic, complemented
10#14528899C211yesmaximal, characteristic, complemented
11#14528853C211yesmaximal, characteristic, complemented
12#14528766C211yesmaximal, characteristic, complemented
13#14528552C211yesmaximal, characteristic, complemented
14#14528480C211yesmaximal, characteristic, complemented
15C17×#1869C211yesmaximal, characteristic
16C22×#63223C2211yescharacteristic
17C2×#10587612C2211yescharacteristic
18C2×#10587612C2211yescharacteristic
19C2×#10587485C2211yescharacteristic
20C2×#10587485C2211yescharacteristic
21C2×#10587485C2211yescharacteristic
22C2×#10587485C2211yescharacteristic
23C2×C22⋊#3622C2211yescharacteristic
24C2×C22⋊#3622C2211yescharacteristic
25C2×C22⋊#3622C2211yescharacteristic
26C2×#10587486C2211yescharacteristic
27C2×C8×Dic17C2211yescharacteristic
28#11762712C411yescharacteristic
29#11762712C411yescharacteristic
30#11762708C411yescharacteristic
31#11762708C411yescharacteristic
32#11762702C411yescharacteristic
33#11762702C411yescharacteristic
34#11762702C411yescharacteristic
35#11762702C411yescharacteristic
36C4×C4⋊Dic17C2211yescharacteristic
37C4×C22⋊Dic17C2211yescharacteristic
38C4×C22⋊Dic17C2211yescharacteristic
39C42×Dic17C2211yescharacteristic
40#11762564C2211yescharacteristic
41#11762564C2211yescharacteristic
42#11762563C2211yescharacteristic
43#11762563C2211yescharacteristic
44#11762563C2211yescharacteristic
45#11762563C2211yescharacteristic
46#11762563C2211yescharacteristic
47#11762563C2211yescharacteristic
48#11762563C2211yescharacteristic
49#11762563C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52C23×C8×C17C2211yescharacteristic
53C2×C4×D4×C17C2211yescharacteristic
54C2×C17×C22⋊C8C2211yescharacteristic
55C2×C17×C22⋊C8C2211yescharacteristic
56C2×C17×C22⋊C8C2211yescharacteristic
57C17×#336C2211yescharacteristic
58C17×#336C2211yescharacteristic
59C22×D4×C17C2×C411yescharacteristic
60C23×C4×C17C2311yescharacteristic
61C23×C4×C17C2311yescharacteristic
62C22×C8×C17—22no—
63C22×C8×C17—41no—
64C22×C8×C17—41no—
65C22×C8×C17C2311yescharacteristic
66C22×C8×C17C2311yescharacteristic
67C22×C8×C17C2311yescharacteristic
68C22×C8×C17C2311yescharacteristic
69C4×D4×C17—22no—
70C4×D4×C17—22no—
71C2×C4⋊C4×C17C2×C411yescharacteristic
72C2×C22⋊C4×C17C2×C411yescharacteristic
73C2×C22⋊C4×C17C2×C411yescharacteristic
74C2×C42×C17C2311yescharacteristic
75C2×C22⋊Dic17C2×C411yescharacteristic
76C2×C22⋊Dic17C2×C411yescharacteristic
77C2×C22⋊Dic17C2×C411yescharacteristic
78C2×C22⋊Dic17C2×C411yescharacteristic
79C2×C22⋊Dic17C2×C411yescharacteristic
80C2×C22⋊Dic17C2×C411yescharacteristic
81C2×C22⋊Dic17C2×C411yescharacteristic
82C2×C22⋊Dic17C2×C411yescharacteristic
83#10587612—41no—
84#10587612—41no—
85#10587612—41no—
86#10587612—41no—
87C2×C4⋊Dic17C2×C411yescharacteristic
88C2×C4⋊Dic17C2×C411yescharacteristic
89C2×C4⋊Dic17C2×C411yescharacteristic
90C2×C4⋊Dic17C2×C411yescharacteristic
91C2×C4⋊Dic17C2×C411yescharacteristic
92C2×C4⋊Dic17C2×C411yescharacteristic
93C2×C4×Dic17C2×C411yescharacteristic
94C2×C4×Dic17C2×C411yescharacteristic
95C2×C4×Dic17C2×C411yescharacteristic
96C2×C4×Dic17C2×C411yescharacteristic
97C2×C4×Dic17C2311yescharacteristic
98C2×C4×Dic17C2311yescharacteristic
99C2×C4×Dic17C2311yescharacteristic
100C2×C4×Dic17C2311yescharacteristic
101C2×#63223—22no—
102C2×#63223—41no—
103C2×#63223—41no—
104C22×#3622C2311yescharacteristic
105C22×#3622C2311yescharacteristic
106C22×#3622C2311yescharacteristic
107C22×#3622C2311yescharacteristic
108C17×C22⋊C8—22no—
109C17×C22⋊C8—41no—
110C17×C22⋊C8—41no—
111#10587502C2×C411yescharacteristic
112#10587502C2×C411yescharacteristic
113#10587502C2×C411yescharacteristic
114#10587502C2×C411yescharacteristic
115#10587502C2×C411yescharacteristic
116#10587502C2×C411yescharacteristic
117C22⋊#3622—22no—
118C22⋊#3622—41no—
119C22⋊#3622—41no—
120#10587486—41no—
121#10587485—41no—
122#10587485—41no—
123#10587485—41no—
124#10587485—41no—
125C8×Dic17—41no—
126C24×C17C22×C411yescharacteristic
127C24×C17C22×C411yescharacteristic
128C2×D4×C17—22no—
129C2×D4×C17—22no—
130C2×D4×C17—41no—
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—21no—
140C22×C4×C17—21no—
141C22×C4×C17C22×C411yescharacteristic
142C22×C4×C17C22×C411yescharacteristic
143C22×C4×C17C22×C411yescharacteristic
144C22×C4×C17C22×C411yescharacteristic
145C22×C4×C17#4112yes—
146C22×C4×C17#4112yes—
147C22×C4×C17C2411yescharacteristic
148C22×Dic17C22×C411yescharacteristic
149C22×Dic17C22×C411yescharacteristic
150C22×Dic17C22×C411yescharacteristic
151C22×Dic17C22×C411yescharacteristic
152C22×Dic17C22×C411yescharacteristic
153C22×Dic17C22×C411yescharacteristic
154C22×Dic17C22×C411yescharacteristic
155C22×Dic17C22×C411yescharacteristic
156C2×C8×C17—21no—
157C2×C8×C17—21no—
158C2×C8×C17—21no—
159C2×C8×C17—21no—
160C2×C8×C17—21no—
161C2×C8×C17—21no—
162C2×C8×C17—21no—
163C2×C8×C17—21no—
164C2×C8×C17—21no—
165C2×C8×C17—21no—
166C2×C8×C17—21no—
167C2×C8×C17—21no—
168C2×C8×C17—44no—
169C4⋊C4×C17—22no—
170C22⋊C4×C17—22no—
171C22⋊C4×C17—22no—
172C42×C17—21no—
173C42×C17—21no—
174C22⋊Dic17—22no—
175C22⋊Dic17—22no—
176C22⋊Dic17—41no—
177C22⋊Dic17—41no—
178C22⋊Dic17—41no—
179C22⋊Dic17—41no—
180C22⋊Dic17—41no—
181C22⋊Dic17—41no—
182C4⋊Dic17—22no—
183C4⋊Dic17—41no—
184C4⋊Dic17—41no—
185C4⋊Dic17—41no—
186C4⋊Dic17—41no—
187C4⋊Dic17—41no—
188C4×Dic17—22no—
189C4×Dic17—21no—
190C4×Dic17—21no—
191C4×Dic17—21no—
192C4×Dic17—21no—
193C4×Dic17—21no—
194C4×Dic17—21no—
195C4×Dic17—21no—
196C4×Dic17—21no—
197C4×Dic17—22no—
198C4×Dic17—41no—
199C4×Dic17—41no—
200#63223—44no—
201C2×#3622—21no—
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—
213#22007—171nomaximal, Sylow
214C23×C17—21no—
215C23×C17—21no—
216C23×C17—21no—
217C23×C17—21no—
218C23×C17—21no—
219C23×C17—21no—
220C23×C17—21no—
221C23×C17—21no—
222C23×C17—21no—
223C23×C17—21no—
224C23×C17—21no—
225C23×C17—21no—
226C23×C17—21no—
227C23×C17#11712yes—
228C23×C17C23×C411yescharacteristic
229D4×C17—44no—
230C2×C4×C17—21no—
231C2×C4×C17—21no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—22no—
235C2×C4×C17—22no—
236C2×C4×C17—22no—
237C2×C4×C17—22no—
238C2×C4×C17—21no—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—21no—
244C2×C4×C17—41no—
245C2×C4×C17—41no—
246C2×C4×C17—41no—
247C2×C4×C17—41no—
248C2×C4×C17#11712yes—
249C2×C4×C17#13112yes—
250C2×C4×C17C2×#4111yescharacteristic
251C2×C4×C17C2×#4111yescharacteristic
252C2×C4×C17H(5,𝔽2)11yescharacteristic
253C2×C4×C17H(5,𝔽2)11yescharacteristic
254C2×C4×C17H(5,𝔽2)11yescharacteristic
255C2×C4×C172−1+411yescharacteristic
256C2×Dic17—21no—
257C2×Dic17—21no—
258C2×Dic17—21no—
259C2×Dic17—21no—
260C2×Dic17—21no—
261C2×Dic17—21no—
262C2×Dic17—21no—
263C2×Dic17—21no—
264C2×Dic17—21no—
265C2×Dic17—21no—
266C2×Dic17—21no—
267C2×Dic17—21no—
268C2×Dic17—21no—
269C2×Dic17—21no—
270C2×Dic17—21no—
271C2×Dic17—21no—
272C2×Dic17—21no—
273C2×Dic17—21no—
274C2×Dic17—21no—
275C2×Dic17—21no—
276C2×Dic17—21no—
277C2×Dic17—21no—
278C2×Dic17—21no—
279C2×Dic17#13112yes—
280C8×C17—41no—
281C8×C17—41no—
282C8×C17—41no—
283C8×C17—41no—
284#3622—41no—
285#3622—41no—
286#3622—41no—
287#3622—41no—
288#2293—171no—
289#1924—171no—
290#1923—171no—
291#1917—171no—
292#1916—171no—
293#1870—171no—
294#1869D1711yescharacteristic
295#1851—171no—
296#1850—171no—
297#1839—171no—
298#1831—171no—
299#1818—171no—
300#1798—171no—
301#1750—171no—
302#1729—171no—
303C22×C17—22no—
304C22×C17—22no—
305C22×C17—22no—
306C22×C17—41no—
307C22×C17—41no—
308C22×C17—41no—
309C22×C17—41no—
310C22×C17—41no—
311C22×C17—41no—
312C22×C17—41no—
313C22×C17—41no—
314C22×C17—41no—
315C22×C17—41no—
316C22×C17—41no—
317C22×C17C2×#11711yescharacteristic
318C22×C17#51811yescharacteristic
319C22×C17#52011yescharacteristic
320C22×C17C2×#13111yescharacteristic
321C22×C17C2×#13111yescharacteristic
322C22×C17#56811yescharacteristic
323C22×C17#56811yescharacteristic
324C4×C17—22no—
325C4×C17—22no—
326C4×C17—41no—
327C4×C17—41no—
328C4×C17#53811yescharacteristic
329C4×C17#53811yescharacteristic
330C4×C17#53911yescharacteristic
331C4×C17#53911yescharacteristic
332Dic17—22no—
333Dic17—41no—
334Dic17—41no—
335Dic17—41no—
336Dic17—41no—
337Dic17—41no—
338Dic17—41no—
339Dic17—41no—
340C22×M16—171no—
341C23×C8C2×D1711yescharacteristic
342C2×C4×D4C2×D1711yescharacteristic
343C2×#117—171no—
344C2×#117—171no—
345C2×C4⋊C8—171no—
346C2×C4⋊C8—171no—
347C2×C4⋊C8—171no—
348C2×C4⋊C8—171no—
349C2×C22⋊C8—171no—
350C2×C22⋊C8—171no—
351C2×C22⋊C8—171no—
352C2×C22⋊C8C2×D1711yescharacteristic
353C2×C22⋊C8C2×D1711yescharacteristic
354C2×C22⋊C8C2×D1711yescharacteristic
355C2×#97—171no—
356C2×C4×C8—171no—
357#390—171no—
358#390—171no—
359#389—171no—
360#389—171no—
361#388—171no—
362#388—171no—
363#388—171no—
364#388—171no—
365C4×C4⋊C4—171no—
366C4×C22⋊C4—171no—
367C4×C22⋊C4—171no—
368C43—171no—
369#336—171no—
370#336—171no—
371#336—171no—
372#336—171no—
373#336—171no—
374#336—171no—
375#336—171no—
376#336—171no—
377#336—171no—
378#336—171no—
379#336—171no—
380#336—171no—
381#336C2×D1711yescharacteristic
382#336C2×D1711yescharacteristic
383C2×C17—42no—
384C2×C17—81no—
385C2×C17#231811yescharacteristic
386C2×C17#292411yescharacteristic
387C2×C17#292511yescharacteristic
388C2×C17#298111yescharacteristic
389C2×C17#298111yescharacteristic
390C2×C17#299411yescharacteristic
391C2×C17#299511yescharacteristic
392C22×D4C4×D1711yescharacteristic
393C23×C4C22×D1711yescharacteristic
394C23×C4C22×D1711yescharacteristic
395C2×M16—342no—
396C2×M16—681no—
397C2×M16—681no—
398C22×C8—22no—
399C22×C8—41no—
400C22×C8—41no—
401C22×C8—171no—
402C22×C8—171no—
403C22×C8—171no—
404C22×C8—171no—
405C22×C8C22×D1711yescharacteristic
406C22×C8C22×D1711yescharacteristic
407C22×C8C22×D1711yescharacteristic
408C22×C8C22×D1711yescharacteristic
409C4×D4—22no—
410C4×D4—22no—
411#117—681no—
412#117—681no—
413#117—681no—
414#117—681no—
415C2×C4⋊C4—171no—
416C2×C4⋊C4—171no—
417C2×C4⋊C4—171no—
418C2×C4⋊C4—171no—
419C2×C4⋊C4—171no—
420C2×C4⋊C4—171no—
421C2×C4⋊C4C4×D1711yescharacteristic
422C2×C22⋊C4—171no—
423C2×C22⋊C4—171no—
424C2×C22⋊C4—171no—
425C2×C22⋊C4—171no—
426C2×C22⋊C4—171no—
427C2×C22⋊C4—171no—
428C2×C22⋊C4—171no—
429C2×C22⋊C4—171no—
430C2×C22⋊C4C4×D1711yescharacteristic
431C2×C22⋊C4C4×D1711yescharacteristic
432C2×C42—171no—
433C2×C42—171no—
434C2×C42—171no—
435C2×C42—171no—
436C2×C42—171no—
437C2×C42—171no—
438C2×C42—171no—
439C2×C42—171no—
440C2×C42C22×D1711yescharacteristic
441C4⋊C8—681no—
442C4⋊C8—681no—
443C4⋊C8—681no—
444C4⋊C8—681no—
445C22⋊C8—22no—
446C22⋊C8—41no—
447C22⋊C8—41no—
448C22⋊C8—342no—
449C22⋊C8—681no—
450C22⋊C8—681no—
451#97—681no—
452C4×C8—681no—
453#95—171no—
454#95—171no—
455#95—171no—
456#95—171no—
457#95—171no—
458#95—171no—
459C17#2200711yescharacteristic, complemented, Sylow
460C24C2×C4×D1711yescharacteristic
461C24C2×C4×D1711yescharacteristic
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—41no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—171no—
476C22×C4—171no—
477C22×C4—171no—
478C22×C4—171no—
479C22×C4—171no—
480C22×C4—171no—
481C22×C4—171no—
482C22×C4—171no—
483C22×C4C2×C4×D1711yescharacteristic
484C22×C4C2×C4×D1711yescharacteristic
485C22×C4C2×C4×D1711yescharacteristic
486C22×C4C2×C4×D1711yescharacteristic
487C22×C4#6325412yes—
488C22×C4#6325412yes—
489C22×C4C23×D1711yescharacteristic
490M16—684no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—44no—
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—
516C4⋊C4—22no—
517C4⋊C4—342no—
518C4⋊C4—681no—
519C4⋊C4—681no—
520C4⋊C4—681no—
521C4⋊C4—681no—
522C4⋊C4—681no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—342no—
526C22⋊C4—342no—
527C22⋊C4—681no—
528C22⋊C4—681no—
529C22⋊C4—681no—
530C22⋊C4—681no—
531C22⋊C4—681no—
532C22⋊C4—681no—
533C42—21no—
534C42—21no—
535C42—342no—
536C42—341no—
537C42—341no—
538C42—341no—
539C42—341no—
540C42—341no—
541C42—341no—
542C42—341no—
543C42—341no—
544C42—342no—
545C42—681no—
546C42—681no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23#1058756312yes—
561C23C22×C4×D1711yescharacteristic
562D4—44no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—172no—
582C2×C4—341no—
583C2×C4—341no—
584C2×C4—341no—
585C2×C4—341no—
586C2×C4—341no—
587C2×C4—341no—
588C2×C4—341no—
589C2×C4—341no—
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—341no—
601C2×C4—341no—
602C2×C4—341no—
603C2×C4—341no—
604C2×C4—341no—
605C2×C4#1058756312yes—
606C2×C4#1058758712yes—
607C2×C4C2×#6325411yescharacteristic
608C2×C4C2×#6325411yescharacteristic
609C2×C4#1058769111yescharacteristic
610C2×C4#1058769111yescharacteristic
611C2×C4#1058769611yescharacteristic
612C2×C4#1058769711yescharacteristic
613C8—41no—
614C8—41no—
615C8—41no—
616C8—41no—
617C8—681no—
618C8—681no—
619C8—681no—
620C8—681no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22C2×#1058756311yescharacteristic
636C22#1176328111yescharacteristic
637C22#1176328811yescharacteristic
638C22#131×D1711yescharacteristic
639C22C2×#1058758711yescharacteristic
640C22#1176357411yescharacteristic
641C22#1176357511yescharacteristic
642C4—22no—
643C4—22no—
644C4—41no—
645C4—41no—
646C4—342no—
647C4—681no—
648C4—681no—
649C4—681no—
650C4—681no—
651C4—681no—
652C4—681no—
653C4—681no—
654C4#1176339411yescharacteristic
655C4#1176339611yescharacteristic
656C4#1176340311yescharacteristic
657C4#1176340411yescharacteristic
658C2—42no—
659C2—81no—
660C2#1453141011yescharacteristic
661C2#1453743811yescharacteristic
662C2#1453744611yescharacteristic
663C2#1453785611yescharacteristic
664C2#1453785811yescharacteristic
665C2#1453793911yescharacteristic
666C2#1453795011yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D17
commutator, socleC23×C17C23×C4
FittingC17×#1869C2
prime radicalC22×D4×C17C2×C4
cosocleC22×C4×C17C24
hypercenter#1869D17
hypocenterC17#22007
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ #1869

Complements

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

NH ≅ G/Ntype
2. #14529394C2semidirect
3. #14529393C2semidirect
4. #14529347C2semidirect
5. #14529342C2semidirect
7. #14528966C2semidirect
8. #14528962C2semidirect
9. #14528926C2semidirect
10. #14528899C2semidirect
11. #14528853C2semidirect
12. #14528766C2semidirect
13. #14528552C2semidirect
14. #14528480C2semidirect
459. C17#22007semidirect

Maximal subgroups

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

subgroupindex
#145312492
#145293942
#145293932
#145293472
#145293422
#145290822
#145289662
#145289622
#145289262
#145288992
#145288532
#145287662
#145285522
#145284802
C17×#18692
#2200717