Finite Group Databasebeta

← #17045206

Subgroups — #17045206

Order 4352.

744 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#14531227C211yesmaximal, characteristic
2#14529580C211yesmaximal, characteristic
3#14529576C211yesmaximal, characteristic
4#14529506C211yesmaximal, characteristic, complemented
5#14529501C211yesmaximal, characteristic, complemented
6#14529400C211yesmaximal, characteristic, complemented
7#14529398C211yesmaximal, characteristic, complemented
8#14529393C211yesmaximal, characteristic, complemented
9#14529391C211yesmaximal, characteristic, complemented
10#14528854C211yesmaximal, characteristic, complemented
11#14528767C211yesmaximal, characteristic, complemented
12#14528764C211yesmaximal, characteristic, complemented
13#14528640C211yesmaximal, characteristic, complemented
14#14528534C211yesmaximal, characteristic, complemented
15C17×#1817C211yesmaximal, characteristic, complemented
16C2×C4×C22⋊D17C2211yescharacteristic
17C2×#10587558C2211yescharacteristic
18C2×C4⋊#3622C2211yescharacteristic
19C2×C4⋊#3622C2211yescharacteristic
20C2×#10587485C2211yescharacteristic
21C2×#10587485C2211yescharacteristic
22C2×#10587485C2211yescharacteristic
23C2×#10587485C2211yescharacteristic
24C2×C22⋊#3622C2211yescharacteristic
25C2×C22⋊#3622C2211yescharacteristic
26C2×#10587491C2211yescharacteristic
27C2×#10587491C2211yescharacteristic
28C2×#10587491C2211yescharacteristic
29C2×#10587491C2211yescharacteristic
30C2×#10587486C2211yescharacteristic
31C2×#10587486C2211yescharacteristic
32C2×C8×Dic17C2211yescharacteristic
33C2×C8×Dic17C2211yescharacteristic
34#11762700C411yescharacteristic
35#11762700C411yescharacteristic
36#11762693C411yescharacteristic
37#11762693C411yescharacteristic
38#11762688C411yescharacteristic
39#11762688C411yescharacteristic
40#11762682C411yescharacteristic
41#11762682C411yescharacteristic
42C4×#63225C2211yescharacteristic
43C4×#63227C2211yescharacteristic
44C4×C22⋊Dic17C2211yescharacteristic
45#11762655C2211yescharacteristic
46#11762564C2211yescharacteristic
47#11762564C2211yescharacteristic
48#11762563C2211yescharacteristic
49#11762563C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52C22×C42×C17C2211yescharacteristic
53C2×C17×C22⋊C8C2211yescharacteristic
54C2×C17×C22⋊C8C2211yescharacteristic
55C17×#336C2211yescharacteristic
56C17×#336C2211yescharacteristic
57C17×#336C2211yescharacteristic
58C17×#336C2211yescharacteristic
59C23×C4×C17C2×C411yescharacteristic
60C23×C4×C17C2×C411yescharacteristic
61C23×C4×C17C2311yescharacteristic
62C22×C22⋊D17C2×C411yescharacteristic
63C22×C4×D17C2311yescharacteristic
64C22×C8×C17C2311yescharacteristic
65C22×C8×C17C2311yescharacteristic
66C22×C8×C17C2311yescharacteristic
67C22×C8×C17C2311yescharacteristic
68C2×C42×C17—41no—
69C2×C42×C17—41no—
70C2×C42×C17C2×C411yescharacteristic
71C2×C42×C17C2×C411yescharacteristic
72C2×C42×C17C2311yescharacteristic
73C2×C42×C17C2311yescharacteristic
74C2×C22⋊Dic17C2×C411yescharacteristic
75C2×C22⋊Dic17C2×C411yescharacteristic
76C2×C22⋊Dic17C2×C411yescharacteristic
77C4×C22⋊D17—22no—
78C4×C22⋊D17—22no—
79C2×#63227C2×C411yescharacteristic
80C2×#63227C2×C411yescharacteristic
81C2×#63227C2×C411yescharacteristic
82C2×#63227C2×C411yescharacteristic
83C2×#63227C2×C411yescharacteristic
84C2×#63225C2×C411yescharacteristic
85C2×#63225C2×C411yescharacteristic
86C2×#63225C2×C411yescharacteristic
87C2×#63225C2×C411yescharacteristic
88C2×#63225C2×C411yescharacteristic
89C2×C4×Dic17C2311yescharacteristic
90C2×C4×Dic17C2311yescharacteristic
91C2×C4×Dic17C2311yescharacteristic
92C22×#3622C2311yescharacteristic
93C22×#3622C2311yescharacteristic
94C22×#3622C2311yescharacteristic
95C22×#3622C2311yescharacteristic
96#10587558—41no—
97#10587558—41no—
98C17×C22⋊C8—41no—
99C17×C22⋊C8—41no—
100#10587502C2×C411yescharacteristic
101#10587502C2×C411yescharacteristic
102#10587502C2×C411yescharacteristic
103#10587502C2×C411yescharacteristic
104#10587502C2×C411yescharacteristic
105#10587502C2×C411yescharacteristic
106#10587502C2×C411yescharacteristic
107#10587502C2×C411yescharacteristic
108#10587502C2×C411yescharacteristic
109#10587502C2×C411yescharacteristic
110C22⋊#3622—41no—
111C22⋊#3622—41no—
112#10587491—41no—
113#10587491—41no—
114#10587491—41no—
115#10587491—41no—
116#10587486—22no—
117#10587486—41no—
118#10587485—22no—
119#10587485—22no—
120#10587485—41no—
121#10587485—41no—
122C8×Dic17—22no—
123C8×Dic17—41no—
124C4⋊#3622—41no—
125C4⋊#3622—41no—
126C24×C17C22×C411yescharacteristic
127C23×D17C22×C411yescharacteristic
128C22×C4×C17—22no—
129C22×C4×C17—22no—
130C22×C4×C17—21no—
131C22×C4×C17—21no—
132C22×C4×C17—21no—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17—21no—
136C22×C4×C17—41no—
137C22×C4×C17—41no—
138C22×C4×C17—41no—
139C22×C4×C17—41no—
140C22×C4×C17C22×C411yescharacteristic
141C22×C4×C17C22×C411yescharacteristic
142C22×C4×C17C22×C411yescharacteristic
143C22×C4×C17C22×C411yescharacteristic
144C22×C4×C17C22×C411yescharacteristic
145C22×C4×C17C2411yescharacteristic
146C2×C22⋊D17—22no—
147C2×C22⋊D17—22no—
148C2×C22⋊D17—41no—
149C22×Dic17C22×C411yescharacteristic
150C22×Dic17C22×C411yescharacteristic
151C22×Dic17C22×C411yescharacteristic
152C22×Dic17C22×C411yescharacteristic
153C22×Dic17C22×C411yescharacteristic
154C22×Dic17C22×C411yescharacteristic
155C22×Dic17C22×C411yescharacteristic
156C2×C4×D17—21no—
157C2×C4×D17—21no—
158C2×C4×D17—21no—
159C2×C4×D17—21no—
160C2×C4×D17—21no—
161C2×C4×D17—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—21no—
169C2×C8×C17—21no—
170C2×C8×C17—21no—
171C2×C8×C17—21no—
172C2×C8×C17—21no—
173C2×C8×C17—21no—
174C42×C17—21no—
175C42×C17—21no—
176C42×C17—21no—
177C42×C17—21no—
178C42×C17—41no—
179C42×C17—41no—
180C22⋊Dic17—22no—
181C22⋊Dic17—22no—
182C22⋊Dic17—41no—
183#63227—22no—
184#63227—22no—
185#63227—41no—
186#63227—41no—
187#63227—41no—
188#63225—22no—
189#63225—22no—
190#63225—41no—
191#63225—41no—
192#63225—41no—
193C4×Dic17—21no—
194C4×Dic17—21no—
195C4×Dic17—21no—
196C4×Dic17—21no—
197C4×Dic17#4112yes—
198C4×Dic17#4112yes—
199C2×#3622—21no—
200C2×#3622—21no—
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—
211#22017—171nomaximal, Sylow
212C23×C17—21no—
213C23×C17—21no—
214C23×C17—21no—
215C23×C17—21no—
216C23×C17—21no—
217C23×C17—21no—
218C23×C17#13112yes—
219C23×C17C23×C411yescharacteristic
220C22×D17—21no—
221C22×D17—21no—
222C22×D17—21no—
223C22×D17—21no—
224C22×D17—21no—
225C22×D17—21no—
226C22×D17—21no—
227C2×C4×C17—21no—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
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—21no—
238C2×C4×C17—21no—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—44no—
242C2×C4×C17—44no—
243C2×C4×C17—41no—
244C2×C4×C17—41no—
245C2×C4×C17—41no—
246C2×C4×C17—41no—
247C2×C4×C17#13112yes—
248C2×C4×C17C2×#4111yescharacteristic
249C2×C4×C17C2×#4111yescharacteristic
250C2×C4×C17H(5,𝔽2)11yescharacteristic
251C2×C4×C17H(5,𝔽2)11yescharacteristic
252C2×C4×C17H(5,𝔽2)11yescharacteristic
253C2×C4×C172−1+411yescharacteristic
254C22⋊D17—44no—
255C2×Dic17—21no—
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#11712yes—
275C2×Dic17#11712yes—
276C4×D17—41no—
277C4×D17—41no—
278C4×D17—41no—
279C4×D17—41no—
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#2290—171no—
289#1955—171no—
290#1954—171no—
291#1940—171no—
292#1939—171no—
293#1924—171no—
294#1924—171no—
295#1923—171no—
296#1923—171no—
297#1818—171no—
298#1817D1711yescharacteristic, complemented
299#1798—171no—
300#1797—171no—
301#1768—171no—
302#1746—171no—
303C22×C17—22no—
304C22×C17—22no—
305C22×C17—22no—
306C22×C17—41no—
307C22×C17—41no—
308C22×C17—41no—
309C22×C17—41no—
310C22×C17C2×#11711yescharacteristic
311C22×C17#51811yescharacteristic
312C22×C17#52011yescharacteristic
313C22×C17C2×#13111yescharacteristic
314C22×C17C2×#13111yescharacteristic
315C22×C17#56811yescharacteristic
316C22×C17#56811yescharacteristic
317C2×D17—41no—
318C2×D17—41no—
319C2×D17—41no—
320C2×D17—41no—
321C2×D17—41no—
322C2×D17—41no—
323C2×D17—41no—
324C4×C17—22no—
325C4×C17—41no—
326C4×C17—41no—
327C4×C17—41no—
328C4×C17—41no—
329C4×C17#54011yescharacteristic
330C4×C17#54011yescharacteristic
331C4×C17#54211yescharacteristic
332C4×C17#54211yescharacteristic
333Dic17—22no—
334Dic17—22no—
335Dic17—41no—
336Dic17—41no—
337Dic17—41no—
338Dic17—41no—
339Dic17—41no—
340C2×C4×D4—171no—
341C2×#117—171no—
342C22×C42C2×D1711yescharacteristic
343C2×C4⋊C8—171no—
344C2×C4⋊C8—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⋊C8—171no—
353C2×C22⋊C8—171no—
354C2×C22⋊C8—171no—
355C2×C22⋊C8C2×D1711yescharacteristic
356C2×C22⋊C8C2×D1711yescharacteristic
357C2×#97—171no—
358C2×#97—171no—
359C2×C4×C8—171no—
360C2×C4×C8—171no—
361#388—171no—
362#388—171no—
363#387—171no—
364#387—171no—
365#386—171no—
366#386—171no—
367#385—171no—
368#385—171no—
369C4×C4⋊C4—171no—
370C4×C22⋊C4—171no—
371C4×C22⋊C4—171no—
372#376—171no—
373#336—171no—
374#336—171no—
375#336—171no—
376#336—171no—
377#336—171no—
378#336—171no—
379#336C2×D1711yescharacteristic
380#336C2×D1711yescharacteristic
381#336C2×D1711yescharacteristic
382#336C2×D1711yescharacteristic
383C2×C17—42no—
384C2×C17#231911yescharacteristic
385C2×C17#292411yescharacteristic
386C2×C17#292511yescharacteristic
387C2×C17#297711yescharacteristic
388C2×C17#297811yescharacteristic
389C2×C17#299211yescharacteristic
390C2×C17#299311yescharacteristic
391D17—81no—
392C22×D4—171no—
393C23×C4—171no—
394C23×C4C4×D1711yescharacteristic
395C23×C4C4×D1711yescharacteristic
396C23×C4C22×D1711yescharacteristic
397C22×C8—171no—
398C22×C8—171no—
399C22×C8—171no—
400C22×C8—171no—
401C22×C8C22×D1711yescharacteristic
402C22×C8C22×D1711yescharacteristic
403C22×C8C22×D1711yescharacteristic
404C22×C8C22×D1711yescharacteristic
405C4×D4—342no—
406C4×D4—342no—
407#117—681no—
408#117—681no—
409C2×C4⋊C4—171no—
410C2×C4⋊C4—171no—
411C2×C4⋊C4—171no—
412C2×C4⋊C4—171no—
413C2×C4⋊C4—171no—
414C2×C22⋊C4—171no—
415C2×C22⋊C4—171no—
416C2×C22⋊C4—171no—
417C2×C22⋊C4—171no—
418C2×C22⋊C4—171no—
419C2×C22⋊C4—171no—
420C2×C22⋊C4—171no—
421C2×C22⋊C4—171no—
422C2×C42—41no—
423C2×C42—41no—
424C2×C42—171no—
425C2×C42—171no—
426C2×C42—171no—
427C2×C42C4×D1711yescharacteristic
428C2×C42C4×D1711yescharacteristic
429C2×C42C22×D1711yescharacteristic
430C2×C42C22×D1711yescharacteristic
431C4⋊C8—342no—
432C4⋊C8—342no—
433C4⋊C8—681no—
434C4⋊C8—681no—
435C4⋊C8—681no—
436C4⋊C8—681no—
437C22⋊C8—41no—
438C22⋊C8—41no—
439C22⋊C8—681no—
440C22⋊C8—681no—
441C22⋊C8—681no—
442C22⋊C8—681no—
443C22⋊C8—681no—
444C22⋊C8—681no—
445#97—342no—
446#97—681no—
447C4×C8—342no—
448C4×C8—681no—
449#95—171no—
450#95—171no—
451#95—171no—
452#95—171no—
453#95—171no—
454#95—171no—
455#95—171no—
456#95—171no—
457#95—171no—
458#95—171no—
459C17#2201711yescharacteristic, complemented, Sylow
460C24—171no—
461C24C2×C4×D1711yescharacteristic
462C2×D4—342no—
463C2×D4—342no—
464C2×D4—681no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—171no—
478C22×C4—171no—
479C22×C4—171no—
480C22×C4—171no—
481C22×C4—171no—
482C22×C4—171no—
483C22×C4—171no—
484C22×C4—341no—
485C22×C4—341no—
486C22×C4—341no—
487C22×C4—341no—
488C22×C4—341no—
489C22×C4—341no—
490C22×C4C2×C4×D1711yescharacteristic
491C22×C4C2×C4×D1711yescharacteristic
492C22×C4C2×C4×D1711yescharacteristic
493C22×C4C2×C4×D1711yescharacteristic
494C22×C4C2×C4×D1711yescharacteristic
495C22×C4C23×D1711yescharacteristic
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
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—341no—
517C2×C8—341no—
518C2×C8—341no—
519C2×C8—341no—
520C4⋊C4—342no—
521C4⋊C4—342no—
522C4⋊C4—681no—
523C4⋊C4—681no—
524C4⋊C4—681no—
525C22⋊C4—342no—
526C22⋊C4—342no—
527C22⋊C4—342no—
528C22⋊C4—342no—
529C22⋊C4—681no—
530C22⋊C4—681no—
531C22⋊C4—681no—
532C22⋊C4—681no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—41no—
538C42—41no—
539C42—172no—
540C42—172no—
541C42—341no—
542C42—341no—
543C42—341no—
544C42—341no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—341no—
552C23—341no—
553C23—341no—
554C23—341no—
555C23—341no—
556C23—341no—
557C23—341no—
558C23#1058759312yes—
559C23C22×C4×D1711yescharacteristic
560D4—684no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—172no—
582C2×C4—172no—
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—681no—
603C2×C4—681no—
604C2×C4—681no—
605C2×C4—681no—
606C2×C4#1058758712yes—
607C2×C4#1058769111yescharacteristic
608C2×C4#1058769111yescharacteristic
609C2×C4#41×D1711yescharacteristic
610C2×C4#41×D1711yescharacteristic
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—681no—
629C22—681no—
630C22—681no—
631C22—681no—
632C22—681no—
633C22—681no—
634C22—681no—
635C22#117×D1711yescharacteristic
636C22#1176328211yescharacteristic
637C22#1176328911yescharacteristic
638C22C2×#1058758711yescharacteristic
639C22C2×#1058759311yescharacteristic
640C22#1176357511yescharacteristic
641C22#1176357511yescharacteristic
642C4—22no—
643C4—41no—
644C4—41no—
645C4—41no—
646C4—41no—
647C4—342no—
648C4—342no—
649C4—681no—
650C4—681no—
651C4—681no—
652C4—681no—
653C4—681no—
654C4#1176341111yescharacteristic
655C4#1176341111yescharacteristic
656C4#1176342711yescharacteristic
657C4#1176342711yescharacteristic
658C2—42no—
659C2—1361no—
660C2#1453142411yescharacteristic
661C2#1453744311yescharacteristic
662C2#1453744511yescharacteristic
663C2#1453783811yescharacteristic
664C2#1453784411yescharacteristic
665C2#1453792511yescharacteristic
666C2#1453793511yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D17
commutator, socleC23×C17C23×C4
FittingC17×#1817C2
prime radicalC22×C22⋊D17C2×C4
cosocleC22×C4×C17C24
hypercenter#1817D17
hypocenterC17#22017
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ #1817

Complements

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

NH ≅ G/Ntype
4. #14529506C2semidirect
5. #14529501C2semidirect
6. #14529400C2semidirect
7. #14529398C2semidirect
8. #14529393C2semidirect
9. #14529391C2semidirect
10. #14528854C2semidirect
11. #14528767C2semidirect
12. #14528764C2semidirect
13. #14528640C2semidirect
14. #14528534C2semidirect
15. C17×#1817C2semidirect
298. #1817D17semidirect
459. C17#22017semidirect

Maximal subgroups

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

subgroupindex
#145312272
#145295802
#145295762
#145295062
#145295012
#145294002
#145293982
#145293932
#145293912
#145288542
#145287672
#145287642
#145286402
#145285342
C17×#18172
#2201717