Finite Group Databasebeta

← #17060853

Subgroups — #17060853

Order 4352.

732 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
1C4×#10587571C211yesmaximal, characteristic
2#14529724C211yesmaximal, characteristic, complemented
3#14529722C211yesmaximal, characteristic, complemented
4#14529397C211yesmaximal, characteristic, complemented
5#14529396C211yesmaximal, characteristic, complemented
6#14529394C211yesmaximal, characteristic, complemented
7#14529079C211yesmaximal, characteristic
8#14529078C211yesmaximal, characteristic
9#14528969C211yesmaximal, characteristic, complemented
10#14528968C211yesmaximal, characteristic, complemented
11#14528968C211yesmaximal, characteristic, complemented
12#14528966C211yesmaximal, characteristic, complemented
13#14528900C211yesmaximal, characteristic, complemented
14#14528899C211yesmaximal, characteristic, complemented
15C17×#1924C211yesmaximal, characteristic, complemented
16C22×#63223C2211yescharacteristic
17C22×#63218C2211yescharacteristic
18C2×#10587571C411yescharacteristic
19C2×C4×Dic34C2211yescharacteristic
20C2×C4×C22⋊D17C2211yescharacteristic
21C2×C4⋊#3622C2211yescharacteristic
22C2×C4⋊#3622C2211yescharacteristic
23C2×#10587485C2211yescharacteristic
24C2×#10587485C2211yescharacteristic
25C2×#10587485C2211yescharacteristic
26C2×C22⋊#3622C2211yescharacteristic
27C2×#10587491C2211yescharacteristic
28C2×#10587491C2211yescharacteristic
29C2×#10587491C2211yescharacteristic
30C2×#10587486C2211yescharacteristic
31#11762713C411yescharacteristic
32#11762709C411yescharacteristic
33#11762702C411yescharacteristic
34#11762701C411yescharacteristic
35#11762698C411yescharacteristic
36#11762697C411yescharacteristic
37#11762673C411yescharacteristic
38C4×#63227C2211yescharacteristic
39C4×#63227C2211yescharacteristic
40C4×C22⋊Dic17C2211yescharacteristic
41C42×Dic17C2211yescharacteristic
42#11762564C2211yescharacteristic
43#11762564C2211yescharacteristic
44#11762563C2211yescharacteristic
45#11762563C2211yescharacteristic
46#11762563C2211yescharacteristic
47#11762563C2211yescharacteristic
48#11762563C2211yescharacteristic
49#11762563C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52C2×C17×C4⋊C8C2211yescharacteristic
53C2×C17×C22⋊C8C2211yescharacteristic
54C2×C17×C22⋊C8C2211yescharacteristic
55C2×C17×#97C2211yescharacteristic
56C4×C22⋊C4×C17C2211yescharacteristic
57C17×#336C2211yescharacteristic
58C17×#336C2211yescharacteristic
59C23×C4×C17C2311yescharacteristic
60C22×C22⋊D17C2×C411yescharacteristic
61C22×C4×D17C2311yescharacteristic
62C22×Dic34C2×C411yescharacteristic
63C22×C8×C17C2311yescharacteristic
64C22×C8×C17C2311yescharacteristic
65C22×C8×C17C2311yescharacteristic
66C22×C8×C17C2311yescharacteristic
67C2×C22⋊C4×C17C2×C411yescharacteristic
68C2×C22⋊C4×C17C2×C411yescharacteristic
69C2×C42×C17C2311yescharacteristic
70C2×C42×C17C2311yescharacteristic
71C2×C22⋊Dic17C2×C411yescharacteristic
72C2×C22⋊Dic17C2×C411yescharacteristic
73C2×C22⋊Dic17C2×C411yescharacteristic
74C4×C22⋊D17—41no—
75C4×C22⋊D17—41no—
76C2×#63227C2×C411yescharacteristic
77C2×#63227C2×C411yescharacteristic
78C2×#63227C2×C411yescharacteristic
79C2×#63227C2×C411yescharacteristic
80C2×#63227C2×C411yescharacteristic
81C2×C4⋊Dic17C2×C411yescharacteristic
82C2×#63225C2×C411yescharacteristic
83C2×#63225C2×C411yescharacteristic
84C2×#63225C2×C411yescharacteristic
85C2×C4×Dic17C2×C411yescharacteristic
86C2×C4×Dic17C2×C411yescharacteristic
87C2×C4×Dic17C2×C411yescharacteristic
88C2×C4×Dic17C2×C411yescharacteristic
89C2×C4×Dic17C2311yescharacteristic
90C2×C4×Dic17C2311yescharacteristic
91C2×C4×Dic17C2311yescharacteristic
92C2×#63223—41no—
93C2×#63223—41no—
94C2×#63223—41no—
95C22×#3622C2311yescharacteristic
96C22×#3622C2311yescharacteristic
97C22×#3622C2311yescharacteristic
98C22×#3622C2311yescharacteristic
99C2×#63218—41no—
100C2×#63218—41no—
101C2×#63218—41no—
102#10587571—24no—
103C4×Dic34—41no—
104C4×Dic34—41no—
105C17×C4⋊C8—41no—
106C17×C22⋊C8—41no—
107C17×C22⋊C8—41no—
108C17×#97—41no—
109C17×#95C2×C411yescharacteristic
110C17×#95C2×C411yescharacteristic
111#10587502C2×C411yescharacteristic
112#10587502C2×C411yescharacteristic
113#10587502C2×C411yescharacteristic
114#10587502C2×C411yescharacteristic
115#10587502C2×C411yescharacteristic
116#10587502C2×C411yescharacteristic
117C22⋊#3622—41no—
118#10587491—41no—
119#10587491—41no—
120#10587491—41no—
121#10587486—41no—
122#10587485—41no—
123#10587485—41no—
124#10587485—41no—
125C4⋊#3622—41no—
126C4⋊#3622—41no—
127C24×C17C22×C411yescharacteristic
128C23×D17C22×C411yescharacteristic
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17—21no—
132C22×C4×C17—21no—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17C22×C411yescharacteristic
136C22×C4×C17C22×C411yescharacteristic
137C22×C4×C17C22×C411yescharacteristic
138C22×C4×C17C22×C411yescharacteristic
139C22×C4×C17C22×C411yescharacteristic
140C22×C4×C17C2411yescharacteristic
141C2×C22⋊D17—22no—
142C2×C22⋊D17—41no—
143C2×C22⋊D17—41no—
144C22×Dic17C22×C411yescharacteristic
145C22×Dic17C22×C411yescharacteristic
146C22×Dic17C22×C411yescharacteristic
147C22×Dic17C22×C411yescharacteristic
148C22×Dic17C22×C411yescharacteristic
149C22×Dic17C22×C411yescharacteristic
150C22×Dic17C22×C411yescharacteristic
151C2×C4×D17—21no—
152C2×C4×D17—21no—
153C2×C4×D17—21no—
154C2×C4×D17—21no—
155C2×C4×D17—21no—
156C2×C4×D17—21no—
157C2×Dic34—22no—
158C2×Dic34—41no—
159C2×Dic34—41no—
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—21no—
169C2×C8×C17—21no—
170C2×C8×C17—21no—
171C2×C8×C17—21no—
172C22⋊C4×C17—22no—
173C22⋊C4×C17—41no—
174C42×C17—21no—
175C42×C17—21no—
176C42×C17—21no—
177C42×C17—21no—
178C22⋊Dic17—22no—
179C22⋊Dic17—41no—
180C22⋊Dic17—41no—
181#63227—22no—
182#63227—22no—
183#63227—41no—
184#63227—41no—
185#63227—41no—
186C4⋊Dic17—41no—
187#63225—41no—
188#63225—41no—
189#63225—41no—
190C4×Dic17—21no—
191C4×Dic17—21no—
192C4×Dic17—21no—
193C4×Dic17—21no—
194C4×Dic17—21no—
195C4×Dic17—21no—
196C4×Dic17—41no—
197C4×Dic17—41no—
198C4×Dic17—41no—
199C4×Dic17M1614yescomplemented
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#63218—44no—
214#23353—171nomaximal, Sylow
215C23×C17—21no—
216C23×C17—21no—
217C23×C17—21no—
218C23×C17—21no—
219C23×C17—21no—
220C23×C17—21no—
221C23×C17—21no—
222C23×C17C23×C411yescharacteristic
223C22×D17—21no—
224C22×D17—21no—
225C22×D17—21no—
226C22×D17—21no—
227C22×D17—21no—
228C22×D17—21no—
229C22×D17—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—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×C17C2×M1612yes—
249C2×C4×C17H(5,𝔽2)11yescharacteristic
250C2×C4×C17H(5,𝔽2)11yescharacteristic
251C2×C4×C17H(5,𝔽2)11yescharacteristic
252C2×C4×C17H(5,𝔽2)11yescharacteristic
253C2×C4×C172−1+411yescharacteristic
254C2×C4×C172−1+411yescharacteristic
255C22⋊D17—44no—
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×Dic17C2×M1612yes—
276C2×Dic17C2×M1612yes—
277C4×D17—41no—
278C4×D17—41no—
279C4×D17—41no—
280C4×D17—41no—
281Dic34—44no—
282C8×C17—41no—
283C8×C17—41no—
284C8×C17—41no—
285C8×C17—41no—
286#3622—41no—
287#3622—41no—
288#3622—41no—
289#3622—41no—
290C4×#124—171no—
291#1985—171no—
292#1985—171no—
293#1924—171no—
294#1924—171no—
295#1924—171no—
296#1924D1711yescharacteristic, complemented
297#1870—171no—
298#1870—171no—
299#1851—171no—
300#1851—171no—
301#1851—171no—
302#1851—171no—
303#1831—171no—
304#1831—171no—
305C22×C17—41no—
306C22×C17—41no—
307C22×C17—41no—
308C22×C17—41no—
309C22×C17—41no—
310C22×C17—41no—
311C22×C17—41no—
312C22×C17#51811yescharacteristic
313C22×C17#52011yescharacteristic
314C22×C17#52011yescharacteristic
315C22×C17C22×M1611yescharacteristic
316C22×C17#56811yescharacteristic
317C22×C17#56811yescharacteristic
318C22×C17#56811yescharacteristic
319C2×D17—41no—
320C2×D17—41no—
321C2×D17—41no—
322C2×D17—41no—
323C2×D17—41no—
324C2×D17—41no—
325C2×D17—41no—
326C4×C17—22no—
327C4×C17—41no—
328C4×C17—41no—
329C4×C17—41no—
330C4×C17—41no—
331C4×C17H(3,𝔽4)11yescharacteristic
332C4×C17#56211yescharacteristic
333C4×C17#56211yescharacteristic
334C4×C17#56311yescharacteristic
335Dic17—22no—
336Dic17—22no—
337Dic17—41no—
338Dic17—41no—
339Dic17—41no—
340Dic17—41no—
341Dic17—41no—
342C22×M16—171no—
343C22×M16—171no—
344C2×#124—171no—
345C2×C4×Q8—171no—
346C2×C4×D4—171no—
347C2×C4⋊C8—171no—
348C2×C4⋊C8—171no—
349C2×C4⋊C8—171no—
350C2×C4⋊C8—171no—
351C2×C4⋊C8—171no—
352C2×C4⋊C8C2×D1711yescharacteristic
353C2×C22⋊C8—171no—
354C2×C22⋊C8—171no—
355C2×C22⋊C8—171no—
356C2×C22⋊C8—171no—
357C2×C22⋊C8C2×D1711yescharacteristic
358C2×C22⋊C8C2×D1711yescharacteristic
359C2×#97—171no—
360C2×#97C2×D1711yescharacteristic
361#391—171no—
362#390—171no—
363#388—171no—
364#388—171no—
365#388—171no—
366#388—171no—
367#382—171no—
368C4×C22⋊C4—171no—
369C4×C22⋊C4—171no—
370C4×C22⋊C4—171no—
371C4×C22⋊C4C2×D1711yescharacteristic
372C43—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#336—171no—
382#336—171no—
383#336C2×D1711yescharacteristic
384#336C2×D1711yescharacteristic
385C2×C17—81no—
386C2×C17#237911yescharacteristic
387C2×C17#297511yescharacteristic
388C2×C17#298011yescharacteristic
389C2×C17#298911yescharacteristic
390C2×C17#298911yescharacteristic
391C2×C17#299511yescharacteristic
392C2×C17#299511yescharacteristic
393D17—81no—
394C22×Q8—171no—
395C22×D4—171no—
396C23×C4—171no—
397C23×C4C22×D1711yescharacteristic
398C2×M16—681no—
399C2×M16—681no—
400C2×M16—681no—
401C2×M16—681no—
402C2×M16—681no—
403C2×M16—681no—
404C22×C8—171no—
405C22×C8—171no—
406C22×C8—171no—
407C22×C8—171no—
408C22×C8C22×D1711yescharacteristic
409C22×C8C22×D1711yescharacteristic
410C22×C8C22×D1711yescharacteristic
411C22×C8C22×D1711yescharacteristic
412#124—344no—
413C4×Q8—681no—
414C4×Q8—681no—
415C4×D4—681no—
416C4×D4—681no—
417C2×C4⋊C4—171no—
418C2×C4⋊C4—171no—
419C2×C4⋊C4—171no—
420C2×C4⋊C4—171no—
421C2×C22⋊C4—171no—
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⋊C4C4×D1711yescharacteristic
430C2×C22⋊C4C4×D1711yescharacteristic
431C2×C42—171no—
432C2×C42—171no—
433C2×C42—171no—
434C2×C42—171no—
435C2×C42—171no—
436C2×C42—171no—
437C2×C42—171no—
438C2×C42C22×D1711yescharacteristic
439C2×C42C22×D1711yescharacteristic
440C4⋊C8—41no—
441C4⋊C8—681no—
442C4⋊C8—681no—
443C4⋊C8—681no—
444C4⋊C8—681no—
445C4⋊C8—681no—
446C22⋊C8—41no—
447C22⋊C8—41no—
448C22⋊C8—681no—
449C22⋊C8—681no—
450C22⋊C8—681no—
451C22⋊C8—681no—
452#97—41no—
453#97—681no—
454#95—171no—
455#95—171no—
456#95—171no—
457#95—171no—
458#95—171no—
459#95—171no—
460#95C4×D1711yescharacteristic
461#95C4×D1711yescharacteristic
462C17#2335311yescharacteristic, complemented, Sylow
463C24—171no—
464C24C2×C4×D1711yescharacteristic
465C2×Q8—342no—
466C2×Q8—681no—
467C2×Q8—681no—
468C2×D4—342no—
469C2×D4—681no—
470C2×D4—681no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
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
496M16—684no—
497M16—684no—
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—21no—
509C2×C8—21no—
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—
520C2×C8—341no—
521C2×C8—341no—
522C4⋊C4—681no—
523C4⋊C4—681no—
524C4⋊C4—681no—
525C4⋊C4—681no—
526C22⋊C4—22no—
527C22⋊C4—41no—
528C22⋊C4—342no—
529C22⋊C4—342no—
530C22⋊C4—342no—
531C22⋊C4—681no—
532C22⋊C4—681no—
533C22⋊C4—681no—
534C22⋊C4—681no—
535C22⋊C4—681no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—174no—
541C42—341no—
542C42—341no—
543C42—341no—
544C42—341no—
545C42—341no—
546C42—341no—
547C42—681no—
548C42—681no—
549C42—681no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—341no—
558C23—341no—
559C23—341no—
560C23—341no—
561C23—341no—
562C23—341no—
563C23—341no—
564C23C22×C4×D1711yescharacteristic
565Q8—684no—
566D4—684no—
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—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—172no—
586C2×C4—172no—
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—341no—
606C2×C4—681no—
607C2×C4—681no—
608C2×C4—681no—
609C2×C4—681no—
610C2×C4M16×D1712yes—
611C2×C4#1058769111yescharacteristic
612C2×C4#1058769111yescharacteristic
613C2×C4#1058769611yescharacteristic
614C2×C4#1058769611yescharacteristic
615C2×C4#1058769711yescharacteristic
616C2×C4#1058769711yescharacteristic
617C8—41no—
618C8—41no—
619C8—41no—
620C8—41no—
621C8—681no—
622C8—681no—
623C8—681no—
624C8—681no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—681no—
633C22—681no—
634C22—681no—
635C22—681no—
636C22—681no—
637C22—681no—
638C22—681no—
639C22#1176328111yescharacteristic
640C22#1176328811yescharacteristic
641C22#1176328911yescharacteristic
642C22C2×M16×D1711yescharacteristic
643C22#1176357411yescharacteristic
644C22#1176357411yescharacteristic
645C22#1176357511yescharacteristic
646C4—22no—
647C4—41no—
648C4—41no—
649C4—41no—
650C4—41no—
651C4—342no—
652C4—342no—
653C4—681no—
654C4—681no—
655C4—681no—
656C4—681no—
657C4—681no—
658C4#1176355111yescharacteristic
659C4#1176355411yescharacteristic
660C4#1176355611yescharacteristic
661C4#1176356111yescharacteristic
662C2—81no—
663C2—1361no—
664C2#1453195311yescharacteristic
665C2#1453782211yescharacteristic
666C2#1453785211yescharacteristic
667C2#1453789611yescharacteristic
668C2#1453790311yescharacteristic
669C2#1453794911yescharacteristic
670C2#1453795511yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D17
commutator, socleC23×C17C23×C4
FittingC17×#1924C2
prime radicalC22×C22⋊D17C2×C4
cosocleC22×C4×C17C24
hypercenter#1924D17
hypocenterC17#23353
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ #1924

Complements

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

NH ≅ G/Ntype
2. #14529724C2semidirect
3. #14529722C2semidirect
4. #14529397C2semidirect
5. #14529396C2semidirect
6. #14529394C2semidirect
9. #14528969C2semidirect
10. #14528968C2semidirect
11. #14528968C2semidirect
12. #14528966C2semidirect
13. #14528900C2semidirect
14. #14528899C2semidirect
15. C17×#1924C2semidirect
199. C4×Dic17M16semidirect
296. #1924D17semidirect
462. C17#23353semidirect

Maximal subgroups

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

subgroupindex
C4×#105875712
#145297242
#145297222
#145293972
#145293962
#145293942
#145290792
#145290782
#145289692
#145289682
#145289662
#145289002
#145288992
C17×#19242
#2335317