Finite Group Databasebeta

← #17034209

Subgroups — #17034209

Order 4352.

1418 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587612C211yesmaximal, characteristic
2C2×#11762867C212yesmaximal, complemented
3#14528853C212yesmaximal, complemented
4#14528766C212yesmaximal, complemented
5#14528762C211yesmaximal, characteristic, complemented
6#14528762C211yesmaximal, characteristic, complemented
7#14528694C211yesmaximal, characteristic
8#14528552C212yesmaximal, complemented
9#14528548C211yesmaximal, characteristic, complemented
10C22⋊C8×Dic17C211yesmaximal, characteristic, complemented
11C17×#1779C211yesmaximal, characteristic
12C2×#10587612—28no—
13C2×#10587612C411yescharacteristic
14C2×#10587612C411yescharacteristic
15C2×#10587612C411yescharacteristic
16C2×#10587612C411yescharacteristic
17C2×#10587612C2211yescharacteristic
18C2×#10587612C2211yescharacteristic
19C2×#10587612C2211yescharacteristic
20C2×#10587612C2211yescharacteristic
21C22×C4⋊Dic17C411yescharacteristic
22C22×C4⋊Dic17C411yescharacteristic
23C22×C22⋊Dic17C411yescharacteristic
24C22×C22⋊Dic17C411yescharacteristic
25C22×C4×Dic17C2211yescharacteristic
26C22×C4×Dic17C2211yescharacteristic
27#11762867—28no—
28C2×#10587485C2212yescomplemented
29C2×#10587485C2212yescomplemented
30C2×C22⋊#3622C2211yescharacteristic
31C2×C22⋊#3622C2211yescharacteristic
32C2×C22⋊#3622C2212yes—
33C2×C22⋊#3622C2212yes—
34C2×#10587486C2212yescomplemented
35C2×C8×Dic17C2212yescomplemented
36#11762563C2212yescomplemented
37#11762563C2212yescomplemented
38#11762563C2212yescomplemented
39#11762563C2212yescomplemented
40C24×C4×C17C2211yescharacteristic
41C2×C17×C22⋊C8C2211yescharacteristic
42C2×C17×C22⋊C8C2211yescharacteristic
43C2×C17×C22⋊C8C2212yes—
44C2×C17×C22⋊C8C2212yes—
45C25×C17C2×C411yescharacteristic
46C23×C4×C17—24no—
47C23×C4×C17—24no—
48C23×C4×C17—22no—
49C23×C4×C17—22no—
50C23×C4×C17C2×C411yescharacteristic
51C23×C4×C17C2×C411yescharacteristic
52C23×C4×C17C2×C411yescharacteristic
53C23×C4×C17C2311yescharacteristic
54C23×C4×C17C2311yescharacteristic
55C23×C4×C17C2311yescharacteristic
56C23×Dic17C2×C411yescharacteristic
57C23×Dic17C2×C411yescharacteristic
58C23×Dic17C2×C411yescharacteristic
59C23×Dic17C2×C411yescharacteristic
60C22×C8×C17C2312yes—
61C22×C8×C17C2312yes—
62C2×C22⋊Dic17—24no—
63C2×C22⋊Dic17—24no—
64C2×C22⋊Dic17C2×C411yescharacteristic
65C2×C22⋊Dic17C2×C411yescharacteristic
66C2×C22⋊Dic17C2×C411yescharacteristic
67C2×C22⋊Dic17C2×C411yescharacteristic
68C2×C22⋊Dic17C2×C411yescharacteristic
69C2×C22⋊Dic17C2×C411yescharacteristic
70C2×C22⋊Dic17C2×C411yescharacteristic
71C2×C22⋊Dic17C2×C411yescharacteristic
72#10587612—24no—
73#10587612—24no—
74#10587612—24no—
75#10587612—24no—
76#10587612—24no—
77#10587612—24no—
78#10587612—24no—
79#10587612D418yes—
80C2×C4⋊Dic17—24no—
81C2×C4⋊Dic17—24no—
82C2×C4⋊Dic17C2×C411yescharacteristic
83C2×C4⋊Dic17C2×C411yescharacteristic
84C2×C4⋊Dic17C2×C411yescharacteristic
85C2×C4⋊Dic17C2×C411yescharacteristic
86C2×C4⋊Dic17C2×C411yescharacteristic
87C2×C4⋊Dic17C2×C411yescharacteristic
88C2×C4⋊Dic17C2×C411yescharacteristic
89C2×C4⋊Dic17C2×C411yescharacteristic
90C2×C4×Dic17—24no—
91C2×C4×Dic17—24no—
92C2×C4×Dic17C2×C411yescharacteristic
93C2×C4×Dic17C2×C411yescharacteristic
94C2×C4×Dic17C2×C411yescharacteristic
95C2×C4×Dic17C2×C411yescharacteristic
96C2×C4×Dic17C2311yescharacteristic
97C2×C4×Dic17C2311yescharacteristic
98C2×C4×Dic17C2311yescharacteristic
99C2×C4×Dic17C2311yescharacteristic
100C22×#3622C2312yes—
101C22×#3622C2312yes—
102C17×C22⋊C8—24no—
103C17×C22⋊C8—22no—
104C17×C22⋊C8—22no—
105C17×C22⋊C8—24no—
106C22⋊#3622—24no—
107C22⋊#3622—22no—
108C22⋊#3622—22no—
109C22⋊#3622—24no—
110#10587486—24no—
111#10587485—24no—
112#10587485—24no—
113C8×Dic17—24no—
114C24×C17—22no—
115C24×C17—22no—
116C24×C17—22no—
117C24×C17—22no—
118C24×C17—22no—
119C24×C17—41no—
120C24×C17—41no—
121C24×C17C22×C411yescharacteristic
122C24×C17C22×C411yescharacteristic
123C24×C17C22×C411yescharacteristic
124C22×C4×C17—216no—
125C22×C4×C17—216no—
126C22×C4×C17—22no—
127C22×C4×C17—22no—
128C22×C4×C17—22no—
129C22×C4×C17—22no—
130C22×C4×C17—22no—
131C22×C4×C17—22no—
132C22×C4×C17—22no—
133C22×C4×C17—22no—
134C22×C4×C17—22no—
135C22×C4×C17—22no—
136C22×C4×C17—22no—
137C22×C4×C17—21no—
138C22×C4×C17—21no—
139C22×C4×C17—22no—
140C22×C4×C17—22no—
141C22×C4×C17—21no—
142C22×C4×C17—21no—
143C22×C4×C17C22×C411yescharacteristic
144C22×C4×C17C22×C411yescharacteristic
145C22×C4×C17C22×C411yescharacteristic
146C22×C4×C17C2×D414yes—
147C22×C4×C17#4114yes—
148C22×C4×C17#4112yes—
149C22×C4×C17#4112yes—
150C22×C4×C17C2411yescharacteristic
151C22×Dic17—22no—
152C22×Dic17—22no—
153C22×Dic17—22no—
154C22×Dic17—22no—
155C22×Dic17—22no—
156C22×Dic17—22no—
157C22×Dic17—22no—
158C22×Dic17—22no—
159C22×Dic17—41no—
160C22×Dic17—41no—
161C22×Dic17—41no—
162C22×Dic17—41no—
163C22×Dic17C22×C411yescharacteristic
164C22×Dic17C22×C411yescharacteristic
165C22×Dic17C22×C411yescharacteristic
166C22×Dic17C22×C411yescharacteristic
167C22×Dic17C22×C411yescharacteristic
168C22×Dic17C22×C411yescharacteristic
169C22×Dic17C22×C411yescharacteristic
170C22×Dic17C22×C411yescharacteristic
171C2×C8×C17—22no—
172C2×C8×C17—22no—
173C2×C8×C17—22no—
174C2×C8×C17—22no—
175C2×C8×C17—22no—
176C2×C8×C17—22no—
177C22⋊Dic17—22no—
178C22⋊Dic17—22no—
179C22⋊Dic17—22no—
180C22⋊Dic17—22no—
181C22⋊Dic17—22no—
182C22⋊Dic17—22no—
183C22⋊Dic17C22⋊C414yes—
184C22⋊Dic17C22⋊C414yes—
185C4⋊Dic17—22no—
186C4⋊Dic17—22no—
187C4⋊Dic17—22no—
188C4⋊Dic17—22no—
189C4⋊Dic17—22no—
190C4⋊Dic17—22no—
191C4⋊Dic17C22⋊C414yes—
192C4⋊Dic17C22⋊C414yes—
193C4×Dic17—22no—
194C4×Dic17—22no—
195C4×Dic17—22no—
196C4×Dic17—22no—
197C4×Dic17—22no—
198C4×Dic17—22no—
199C4×Dic17C2×D414yes—
200C4×Dic17C2×D414yes—
201C2×#3622—22no—
202C2×#3622—22no—
203C2×#3622—22no—
204C2×#3622—22no—
205C2×#3622—22no—
206C2×#3622—22no—
207#20716—171nomaximal, Sylow
208C23×C17—21no—
209C23×C17—21no—
210C23×C17—21no—
211C23×C17—21no—
212C23×C17—28no—
213C23×C17—21no—
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—44no—
226C23×C17—44no—
227C23×C17—44no—
228C23×C17—44no—
229C23×C17—44no—
230C23×C17—44no—
231C23×C17C2×C22⋊C412yes—
232C23×C17#11712yes—
233C23×C17#11712yes—
234C23×C17#13112yes—
235C23×C17#13112yes—
236C23×C17C23×C411yescharacteristic
237C2×C4×C17—28no—
238C2×C4×C17—28no—
239C2×C4×C17—28no—
240C2×C4×C17—21no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—21no—
244C2×C4×C17—21no—
245C2×C4×C17—21no—
246C2×C4×C17—24no—
247C2×C4×C17—24no—
248C2×C4×C17—24no—
249C2×C4×C17—24no—
250C2×C4×C17—24no—
251C2×C4×C17—24no—
252C2×C4×C17C2×C22⋊C412yes—
253C2×C4×C17#11712yes—
254C2×C4×C17#11712yes—
255C2×C4×C17C22×D412yes—
256C2×C4×C17C2×#4111yescharacteristic
257C2×C4×C17C2×#4111yescharacteristic
258C2×C4×C17C2×#4112yes—
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—44no—
280C2×Dic17—44no—
281C2×Dic17—44no—
282C2×Dic17—44no—
283C2×Dic17C2×C22⋊C412yes—
284C2×Dic17C2×C22⋊C412yes—
285C2×Dic17C2×C22⋊C412yes—
286C2×Dic17C2×C22⋊C412yes—
287C8×C17—42no—
288C8×C17—42no—
289#3622—42no—
290#3622—42no—
291C22×#117—171no—
292C2×#433—172no—
293#1818—172no—
294#1798—172no—
295#1797—171no—
296#1797—171no—
297#1779—171no—
298#1779D1711yescharacteristic
299#1750—172no—
300#1749—171no—
301C4×C22⋊C8—171no—
302C22×C17—22no—
303C22×C17—22no—
304C22×C17—22no—
305C22×C17—22no—
306C22×C17—22no—
307C22×C17—22no—
308C22×C17—22no—
309C22×C17—22no—
310C22×C17—22no—
311C22×C17—22no—
312C22×C17—22no—
313C22×C17—22no—
314C22×C17—416no—
315C22×C17—41no—
316C22×C17—41no—
317C22×C17—41no—
318C22×C17—41no—
319C22×C17—41no—
320C22×C17—41no—
321C22×C17—41no—
322C22×C17—41no—
323C22×C17#43314yescomplemented
324C22×C17C22×C22⋊C411yescharacteristic
325C22×C17C2×#11711yescharacteristic
326C22×C17C2×#11711yescharacteristic
327C22×C17C2×#13111yescharacteristic
328C22×C17C2×#13111yescharacteristic
329C22×C17#56811yescharacteristic
330C22×C17#56811yescharacteristic
331C4×C17—22no—
332C4×C17—22no—
333C4×C17—22no—
334C4×C17#52512yes—
335C4×C17#52512yes—
336Dic17—22no—
337Dic17—22no—
338Dic17—22no—
339Dic17—22no—
340Dic17—41no—
341Dic17—41no—
342Dic17—41no—
343Dic17—41no—
344C24×C4C2×D1711yescharacteristic
345C2×#117—171no—
346C2×#117—171no—
347C2×#117—171no—
348C2×#117—171no—
349C2×#117—171no—
350C2×#117—171no—
351C2×#117—171no—
352C2×#117—171no—
353C2×#117—348no—
354C22×C4⋊C4—171no—
355C22×C4⋊C4—171no—
356C22×C22⋊C4—171no—
357C22×C22⋊C4—171no—
358C22×C42—171no—
359C22×C42—171no—
360#433—348no—
361C2×C4⋊C8—172no—
362C2×C4⋊C8—172no—
363C2×C22⋊C8—171no—
364C2×C22⋊C8—171no—
365C2×C22⋊C8—172no—
366C2×C22⋊C8—172no—
367C2×C22⋊C8C2×D1711yescharacteristic
368C2×C22⋊C8C2×D1711yescharacteristic
369C2×C22⋊C8C2×D1712yes—
370C2×C22⋊C8C2×D1712yes—
371C2×#97—172no—
372C2×C4×C8—172no—
373#336—172no—
374#336—172no—
375#336—172no—
376#336—172no—
377C2×C17—24no—
378C2×C17—42no—
379C2×C17—42no—
380C2×C17#228011yescharacteristic
381C2×C17#288011yescharacteristic
382C2×C17#288011yescharacteristic
383C2×C17C2×#43311yescharacteristic
384C2×C17#292111yescharacteristic
385C2×C17#292411yescharacteristic
386C2×C17#292411yescharacteristic
387C25C4×D1711yescharacteristic
388C23×C4—24no—
389C23×C4—24no—
390C23×C4—22no—
391C23×C4—22no—
392C23×C4—171no—
393C23×C4—171no—
394C23×C4—171no—
395C23×C4—171no—
396C23×C4C4×D1711yescharacteristic
397C23×C4C4×D1711yescharacteristic
398C23×C4C4×D1711yescharacteristic
399C23×C4C22×D1711yescharacteristic
400C23×C4C22×D1711yescharacteristic
401C23×C4C22×D1711yescharacteristic
402C22×C8—172no—
403C22×C8—172no—
404C22×C8C22×D1712yes—
405C22×C8C22×D1712yes—
406#117—178no—
407#117—344no—
408#117—344no—
409#117—344no—
410#117—344no—
411#117—344no—
412#117—344no—
413#117—344no—
414C2×C4⋊C4—171no—
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⋊C4—171no—
422C2×C4⋊C4—344no—
423C2×C4⋊C4—344no—
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⋊C4—171no—
431C2×C22⋊C4—171no—
432C2×C22⋊C4—344no—
433C2×C22⋊C4—344no—
434C2×C42—171no—
435C2×C42—171no—
436C2×C42—171no—
437C2×C42—171no—
438C2×C42—171no—
439C2×C42—171no—
440C2×C42—171no—
441C2×C42—171no—
442C2×C42—344no—
443C2×C42—344no—
444C4⋊C8—344no—
445C4⋊C8—344no—
446C22⋊C8—24no—
447C22⋊C8—22no—
448C22⋊C8—22no—
449C22⋊C8—24no—
450C22⋊C8—344no—
451C22⋊C8—342no—
452C22⋊C8—342no—
453C22⋊C8—344no—
454#97—344no—
455C4×C8—344no—
456C17#2071611yescharacteristic, complemented, Sylow
457C24—22no—
458C24—22no—
459C24—22no—
460C24—22no—
461C24—22no—
462C24—41no—
463C24—41no—
464C24C2×C4×D1711yescharacteristic
465C24C2×C4×D1711yescharacteristic
466C24C2×C4×D1711yescharacteristic
467C22×C4—216no—
468C22×C4—216no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—171no—
487C22×C4—171no—
488C22×C4—171no—
489C22×C4—171no—
490C22×C4—171no—
491C22×C4—171no—
492C22×C4—171no—
493C22×C4—171no—
494C22×C4—342no—
495C22×C4—342no—
496C22×C4—342no—
497C22×C4—342no—
498C22×C4—342no—
499C22×C4—342no—
500C22×C4—342no—
501C22×C4—342no—
502C22×C4—681no—
503C22×C4—681no—
504C22×C4—681no—
505C22×C4—681no—
506C22×C4C2×C4×D1711yescharacteristic
507C22×C4C2×C4×D1711yescharacteristic
508C22×C4C2×C4×D1711yescharacteristic
509C22×C4D4×D1714yes—
510C22×C4#6325414yes—
511C22×C4#6325412yes—
512C22×C4#6325412yes—
513C22×C4C23×D1711yescharacteristic
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—342no—
521C2×C8—342no—
522C2×C8—342no—
523C2×C8—342no—
524C2×C8—342no—
525C2×C8—342no—
526C4⋊C4—174no—
527C4⋊C4—174no—
528C4⋊C4—342no—
529C4⋊C4—342no—
530C4⋊C4—342no—
531C4⋊C4—342no—
532C4⋊C4—342no—
533C4⋊C4—342no—
534C22⋊C4—174no—
535C22⋊C4—174no—
536C22⋊C4—342no—
537C22⋊C4—342no—
538C22⋊C4—342no—
539C22⋊C4—342no—
540C22⋊C4—342no—
541C22⋊C4—342no—
542C42—174no—
543C42—174no—
544C42—342no—
545C42—342no—
546C42—342no—
547C42—342no—
548C42—342no—
549C42—342no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—28no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23#1058756312yes—
574C23#1058756312yes—
575C23C22⋊C4×D1712yes—
576C23#1058758712yes—
577C23#1058759312yes—
578C23C22×C4×D1711yescharacteristic
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—172no—
595C2×C4—172no—
596C2×C4—172no—
597C2×C4—172no—
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—341no—
607C2×C4—341no—
608C2×C4—341no—
609C2×C4—341no—
610C2×C4—341no—
611C2×C4—341no—
612C2×C4—341no—
613C2×C4—341no—
614C2×C4—341no—
615C2×C4—341no—
616C2×C4—341no—
617C2×C4—341no—
618C2×C4—684no—
619C2×C4—684no—
620C2×C4—684no—
621C2×C4—684no—
622C2×C4#1058756312yes—
623C2×C4#1058756312yes—
624C2×C4C22⋊C4×D1712yes—
625C2×C4C2×D4×D1712yes—
626C2×C4C2×#6325411yescharacteristic
627C2×C4C2×#6325411yescharacteristic
628C2×C4C2×#6325412yes—
629C8—42no—
630C8—42no—
631C8—682no—
632C8—682no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—416no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22#1176286714yescomplemented
655C22C2×C22⋊C4×D1711yescharacteristic
656C22C2×#1058756311yescharacteristic
657C22C2×#1058756311yescharacteristic
658C22C2×#1058758711yescharacteristic
659C22C2×#1058759311yescharacteristic
660C22#1176357511yescharacteristic
661C22#1176357511yescharacteristic
662C4—22no—
663C4—22no—
664C4—22no—
665C4—342no—
666C4—342no—
667C4—342no—
668C4—342no—
669C4—681no—
670C4—681no—
671C4—681no—
672C4—681no—
673C4#1176331512yes—
674C4#1176331512yes—
675C2—24no—
676C2—42no—
677C2—42no—
678C2#1453116211yescharacteristic
679C2#1453723011yescharacteristic
680C2#1453723411yescharacteristic
681C2C2×#1176286711yescharacteristic
682C2#1453741911yescharacteristic
683C2#1453743811yescharacteristic
684C2#1453744411yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D17
commutator, socleC23×C17C23×C4
FittingC17×#1779C2
prime radicalC25×C17C2×C4
cosocleC22×C4×C17C24
hypercenter#1779D17
hypocenterC17#20716
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
2. C2×#11762867C2semidirect
3. #14528853C2semidirect
4. #14528766C2semidirect
5. #14528762C2semidirect
6. #14528762C2semidirect
8. #14528552C2semidirect
9. #14528548C2semidirect
10. C22⋊C8×Dic17C2semidirect
28. C2×#10587485C22semidirect
29. C2×#10587485C22semidirect
34. C2×#10587486C22semidirect
35. C2×C8×Dic17C22semidirect
36. #11762563C22semidirect
37. #11762563C22semidirect
38. #11762563C22semidirect
39. #11762563C22semidirect
323. C22×C17#433semidirect
456. C17#20716semidirect
654. C22#11762867semidirect

Maximal subgroups

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

subgroupindex
C22×#105876122
C2×#117628672
#145288532
#145287662
#145287622
#145286942
#145285522
#145285482
C22⋊C8×Dic172
C17×#17792
#2071617