Finite Group Databasebeta

← #17050955

Subgroups — #17050955

Order 4352.

1250 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Dic34C211yesmaximal, characteristic
2C2×#11762938C211yesmaximal, characteristic, complemented
3#14531736C211yesmaximal, characteristic, complemented
4#14529756C211yesmaximal, characteristic, complemented
5#14529755C211yesmaximal, characteristic, complemented
6#14529743C211yesmaximal, characteristic, complemented
7#14529734C211yesmaximal, characteristic, complemented
8#14529731C211yesmaximal, characteristic, complemented
9#14529730C211yesmaximal, characteristic, complemented
10#14528792C211yesmaximal, characteristic
11#14528782C211yesmaximal, characteristic
12#14528611C211yesmaximal, characteristic
13#14528609C211yesmaximal, characteristic
14#102×Dic17C211yesmaximal, characteristic
15C17×#1804C211yesmaximal, characteristic
16C23×Dic34—21no—
17D4×Dic34—24no—
18D4×Dic34C418yescomplemented
19C2×#10587555C2211yescharacteristic
20C2×#10587574C2211yescharacteristic
21C2×#10587574C2211yescharacteristic
22C2×#10587610—21no—
23C2×#10587564—21no—
24C2×#10587564—21no—
25C2×C4×Dic34C2211yescharacteristic
26C2×D4×Dic17C2211yescharacteristic
27C2×D4×Dic17C2211yescharacteristic
28#11762938—24no—
29C2×#10587485C2211yescharacteristic
30C2×#10587485C2211yescharacteristic
31C2×#10587503C2211yescharacteristic
32C2×#10587503C2211yescharacteristic
33C2×#10587474C2211yescharacteristic
34C2×#10587486C2211yescharacteristic
35C2×C8×Dic17C2211yescharacteristic
36#11762713—21no—
37#11762694—21no—
38#11762691—21no—
39#11762678C2211yescharacteristic
40#11762674C2211yescharacteristic
41C4⋊C4×Dic17C2211yescharacteristic
42C4⋊C4×Dic17C2211yescharacteristic
43C4×C4⋊Dic17C2211yescharacteristic
44#11762660—21no—
45#11762572C2211yescharacteristic
46#11762572C2211yescharacteristic
47#11762572C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762571C2211yescharacteristic
53#11762571C2211yescharacteristic
54#11762571C2211yescharacteristic
55C2×C4×D4×C17C2211yescharacteristic
56C2×C17×C4⋊C8C2211yescharacteristic
57C2×C17×#102C2211yescharacteristic
58C2×C17×#102C2211yescharacteristic
59C17×#382C2211yescharacteristic
60C17×#340C2211yescharacteristic
61C17×#340C2211yescharacteristic
62C22×D4×C17C2311yescharacteristic
63C23×C4×C17—21no—
64C23×Dic17—21no—
65C23×Dic17—21no—
66C22×Dic34—21no—
67C22×Dic34—21no—
68C22×Dic34—21no—
69C22×Dic34—28no—
70C22×Dic34—44no—
71C22×Dic34—44no—
72C22×Dic34D411yescharacteristic
73C22×C8×C17C2311yescharacteristic
74C22×C8×C17C2311yescharacteristic
75C4×D4×C17—22no—
76C4×D4×C17C2×C414yes—
77C2×C4⋊C4×C17C2311yescharacteristic
78C2×C4⋊C4×C17C2311yescharacteristic
79C2×C4⋊C4×C17C2311yescharacteristic
80C2×C22⋊C4×C17—21no—
81C2×C42×C17C2311yescharacteristic
82C2×C22⋊Dic17—21no—
83C2×C22⋊Dic17—21no—
84D4×Dic17—22no—
85D4×Dic17—22no—
86D4×Dic17C2×C414yes—
87D4×Dic17C2×C414yes—
88C2×C4⋊Dic17—21no—
89C2×C4⋊Dic17D411yescharacteristic
90C2×C4⋊Dic17C2311yescharacteristic
91C2×C4⋊Dic17C2311yescharacteristic
92C2×C4⋊Dic17C2311yescharacteristic
93C2×C4⋊Dic17C2311yescharacteristic
94#10587610—28no—
95C2×#63225—21no—
96C2×#63225—21no—
97C2×#63225—21no—
98C2×#63225—21no—
99C2×#63225D411yescharacteristic
100C2×#63225D411yescharacteristic
101C2×C4×Dic17—21no—
102C2×C4×Dic17—21no—
103C2×C4×Dic17—21no—
104C2×C4×Dic17C2311yescharacteristic
105C2×C4×Dic17C2311yescharacteristic
106C22×#3622C2311yescharacteristic
107C22×#3622C2311yescharacteristic
108#10587574—22no—
109#10587574—22no—
110#10587574C2×C414yes—
111#10587574C2×C414yes—
112#10587564—28no—
113#10587564—28no—
114#10587555—22no—
115#10587555C2×C414yes—
116C4×Dic34C2×C414yes—
117C4×Dic34D414yes—
118C17×C4⋊C8—22no—
119C17×#102—22no—
120C17×#102—22no—
121C17×#95—21no—
122C17×#95—21no—
123#10587503—22no—
124#10587503Q814yescomplemented
125#10587502—21no—
126#10587502—21no—
127#10587502—21no—
128#10587502—21no—
129#10587502—21no—
130#10587486—22no—
131#10587485—22no—
132#10587485—22no—
133C8×Dic17—22no—
134#10587474—22no—
135C24×C17—21no—
136C2×D4×C17—21no—
137C2×D4×C17—21no—
138C2×D4×C17—21no—
139C2×D4×C17C22×C412yes—
140C2×D4×C17C2×Q812yes—
141C2×D4×C17#4112yes—
142C22×C4×C17—21no—
143C22×C4×C17—24no—
144C22×C4×C17—21no—
145C22×C4×C17—21no—
146C22×C4×C17—42no—
147C22×C4×C17—42no—
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2411yescharacteristic
151C22×Dic17—21no—
152C22×Dic17—21no—
153C22×Dic17—21no—
154C22×Dic17—24no—
155C22×Dic17—24no—
156C22×Dic17—21no—
157C22×Dic17—42no—
158C22×Dic17—42no—
159C22×Dic17—42no—
160C22×Dic17—42no—
161C22×Dic17C2×D411yescharacteristic
162C22×Dic17C2×D411yescharacteristic
163C22×Dic17C2×D411yescharacteristic
164C22×Dic17C2×D411yescharacteristic
165C2×Dic34—22no—
166C2×Dic34—22no—
167C2×Dic34—24no—
168C2×Dic34—24no—
169C2×Dic34—24no—
170C2×Dic34—432no—
171C2×Dic34—42no—
172C2×Dic34—42no—
173C2×Dic34—42no—
174C2×Dic34—42no—
175C2×Dic34—42no—
176C2×Dic34—42no—
177C2×Dic34C22⋊C414yes—
178C2×C8×C17—21no—
179C2×C8×C17—21no—
180C2×C8×C17—21no—
181C2×C8×C17—21no—
182C2×C8×C17—21no—
183C2×C8×C17—21no—
184C4⋊C4×C17—21no—
185C4⋊C4×C17—21no—
186C4⋊C4×C17—21no—
187C4⋊C4×C17—21no—
188C4⋊C4×C17—21no—
189C4⋊C4×C17C22×C412yes—
190C22⋊C4×C17—24no—
191C42×C17C22×C412yes—
192C42×C17C2×D412yes—
193C22⋊Dic17—24no—
194C22⋊Dic17—24no—
195C4⋊Dic17—24no—
196C4⋊Dic17—21no—
197C4⋊Dic17—21no—
198C4⋊Dic17—21no—
199C4⋊Dic17—21no—
200C4⋊Dic17—21no—
201C4⋊Dic17C22⋊C414yes—
202C4⋊Dic17C22×C412yes—
203C4⋊Dic17C22×C412yes—
204C4⋊Dic17C2×Q812yescomplemented
205#63225—24no—
206#63225—24no—
207#63225—42no—
208#63225—81no—
209#63225C22⋊C414yes—
210#63225C22⋊C414yes—
211C4×Dic17—42no—
212C4×Dic17—81no—
213C4×Dic17—81no—
214C4×Dic17C22×C412yes—
215C4×Dic17C22×C412yes—
216C4×Dic17C2×D412yes—
217C4×Dic17C2×D412yes—
218C2×#3622—21no—
219C2×#3622—21no—
220C2×#3622—21no—
221C2×#3622—21no—
222C2×#3622—21no—
223C2×#3622C2×Q812yescomplemented
224#22523—171nomaximal, Sylow
225C23×C17—22no—
226C23×C17—22no—
227C23×C17—22no—
228C23×C17—41no—
229C23×C17—41no—
230C23×C17—41no—
231C23×C17—41no—
232C23×C17C22×D411yescharacteristic
233D4×C17—22no—
234D4×C17—22no—
235D4×C17—22no—
236D4×C17C4×Q814yes—
237C2×C4×C17—21no—
238C2×C4×C17—21no—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—22no—
242C2×C4×C17—48no—
243C2×C4×C17—41no—
244C2×C4×C17—41no—
245C2×C4×C17—41no—
246C2×C4×C17—41no—
247C2×C4×C17—41no—
248C2×C4×C17—41no—
249C2×C4×C17—41no—
250C2×C4×C17—41no—
251C2×C4×C17C2×C22⋊C412yes—
252C2×C4×C17C2×C22⋊C412yes—
253C2×C4×C17C23×C411yescharacteristic
254C2×C4×C17C22×D411yescharacteristic
255C2×C4×C17C22×Q811yescharacteristic
256C2×C4×C17C2×#4111yescharacteristic
257C2×C4×C172−1+411yescharacteristic
258C2×C4×C172−1+411yescharacteristic
259C2×Dic17—21no—
260C2×Dic17—21no—
261C2×Dic17—21no—
262C2×Dic17—21no—
263C2×Dic17—21no—
264C2×Dic17—21no—
265C2×Dic17—22no—
266C2×Dic17—22no—
267C2×Dic17—22no—
268C2×Dic17—48no—
269C2×Dic17—41no—
270C2×Dic17—41no—
271C2×Dic17—41no—
272C2×Dic17—41no—
273C2×Dic17—41no—
274C2×Dic17—41no—
275C2×Dic17—48no—
276C2×Dic17—41no—
277C2×Dic17—41no—
278C2×Dic17—41no—
279C2×Dic17C2×C22⋊C412yes—
280C2×Dic17C2×C22⋊C412yes—
281C2×Dic17C2×C22⋊C412yes—
282C2×Dic17C2×C22⋊C412yes—
283C2×Dic17AGL(1,ℤ/8)12yes—
284C2×Dic17AGL(1,ℤ/8)12yes—
285Dic34—28no—
286Dic34—48no—
287Dic34—48no—
288Dic34—48no—
289C8×C17—41no—
290C8×C17—41no—
291#3622—22no—
292#3622—41no—
293C2×D4×Q8—171no—
294C2×#446—171no—
295#2355—171no—
296#1993—171no—
297#1993—171no—
298#1990—171no—
299#1988—171no—
300#1987—171no—
301#1987—171no—
302#1804—171no—
303#1804D1711yescharacteristic
304#1802—171no—
305#1762—171no—
306#1762—171no—
307C4×#102—171no—
308C22×C17—24no—
309C22×C17—42no—
310C22×C17—42no—
311C22×C17—42no—
312C22×C17—42no—
313C22×C17—42no—
314C22×C17—42no—
315C22×C17C22×C22⋊C411yescharacteristic
316C22×C17#54811yescharacteristic
317C22×C17D4×Q811yescharacteristic
318C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
319C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
320C22×C17#57711yescharacteristic
321C22×C17#57811yescharacteristic
322C4×C17—22no—
323C4×C17—22no—
324C4×C17—42no—
325C4×C17—81no—
326C4×C17—81no—
327C4×C17C2×C4×Q811yescharacteristic
328C4×C17#51911yescharacteristic
329C4×C17#54811yescharacteristic
330C4×C17D4×Q811yescharacteristic
331Dic17—22no—
332Dic17—22no—
333Dic17—22no—
334Dic17—42no—
335Dic17—42no—
336Dic17—42no—
337Dic17—81no—
338Dic17#41814yes—
339C23×Q8—341no—
340D4×Q8—178no—
341D4×Q8—344no—
342C2×C4⋊Q8—171no—
343C2×C4⋊Q8—171no—
344C2×C4⋊Q8—171no—
345C2×C22⋊Q8—341no—
346C2×C22⋊Q8—341no—
347C2×C22⋊Q8—341no—
348C2×C4×Q8—171no—
349C2×C4×D4—171no—
350C2×C4×D4—171no—
351C2×C4×D4C2×D1711yescharacteristic
352#446—344no—
353C2×C4⋊C8—171no—
354C2×C4⋊C8—171no—
355C2×C4⋊C8C2×D1711yescharacteristic
356C2×#102—171no—
357C2×#102—171no—
358C2×#102C2×D1711yescharacteristic
359C2×#102C2×D1711yescharacteristic
360C2×#97—171no—
361C2×#97—171no—
362C2×C4×C8—171no—
363#391—341no—
364#387—341no—
365#387—341no—
366#384—171no—
367#382—171no—
368#382C2×D1711yescharacteristic
369C4×C4⋊C4—171no—
370C4×C4⋊C4—171no—
371C4×C4⋊C4—171no—
372C4×C4⋊C4—341no—
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#340C2×D1711yescharacteristic
384#340C2×D1711yescharacteristic
385C2×C17—44no—
386C2×C17Q8×C22⋊C411yescharacteristic
387C2×C17C2×#41811yescharacteristic
388C2×C17#289811yescharacteristic
389C2×C17#336711yescharacteristic
390C2×C17#337311yescharacteristic
391C2×C17#338411yescharacteristic
392C2×C17#338611yescharacteristic
393C22×Q8—171no—
394C22×Q8—341no—
395C22×Q8—341no—
396C22×Q8—348no—
397C22×Q8—341no—
398C22×Q8—684no—
399C22×Q8—684no—
400C22×D4C22×D1711yescharacteristic
401C23×C4—21no—
402C23×C4—341no—
403C23×C4—341no—
404C22×C8—171no—
405C22×C8—171no—
406C22×C8C22×D1711yescharacteristic
407C22×C8C22×D1711yescharacteristic
408C4⋊Q8—174no—
409C4⋊Q8—174no—
410C4⋊Q8—174no—
411C4⋊Q8—342no—
412C4⋊Q8—342no—
413C4⋊Q8—342no—
414C22⋊Q8—348no—
415C22⋊Q8—348no—
416C22⋊Q8—348no—
417C4×Q8—174no—
418C4×Q8—174no—
419C4×D4—22no—
420C4×D4—174no—
421C4×D4—174no—
422C4×D4—342no—
423C4×D4—342no—
424C4×D4C4×D1714yes—
425C2×C4⋊C4—171no—
426C2×C4⋊C4—171no—
427C2×C4⋊C4—171no—
428C2×C4⋊C4—171no—
429C2×C4⋊C4—171no—
430C2×C4⋊C4—171no—
431C2×C4⋊C4—171no—
432C2×C4⋊C4—341no—
433C2×C4⋊C4—341no—
434C2×C4⋊C4—341no—
435C2×C4⋊C4—341no—
436C2×C4⋊C4—341no—
437C2×C4⋊C4C22×D1711yescharacteristic
438C2×C4⋊C4C22×D1711yescharacteristic
439C2×C4⋊C4C22×D1711yescharacteristic
440C2×C22⋊C4—21no—
441C2×C22⋊C4—341no—
442C2×C22⋊C4—341no—
443C2×C42—171no—
444C2×C42—171no—
445C2×C42—341no—
446C2×C42—341no—
447C2×C42—341no—
448C2×C42C22×D1711yescharacteristic
449C4⋊C8—22no—
450C4⋊C8—342no—
451C4⋊C8—342no—
452#102—22no—
453#102—22no—
454#102—174no—
455#102—342no—
456#97—342no—
457#97—342no—
458C4×C8—342no—
459#95—21no—
460#95—21no—
461#95—341no—
462#95—341no—
463#95—341no—
464#95—341no—
465#95—341no—
466C17#2252311yescharacteristic, complemented, Sylow
467C24—21no—
468C2×Q8—174no—
469C2×Q8—342no—
470C2×Q8—342no—
471C2×Q8—344no—
472C2×Q8—344no—
473C2×Q8—344no—
474C2×Q8—682no—
475C2×Q8—682no—
476C2×Q8—682no—
477C2×Q8—682no—
478C2×Q8—682no—
479C2×Q8—682no—
480C2×Q8—6832no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4C2×C4×D1712yes—
485C2×D4#6325412yes—
486C2×D4Q8×D1712yes—
487C22×C4—21no—
488C22×C4—24no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—171no—
494C22×C4—171no—
495C22×C4—171no—
496C22×C4—171no—
497C22×C4—344no—
498C22×C4—344no—
499C22×C4—341no—
500C22×C4—341no—
501C22×C4—341no—
502C22×C4—341no—
503C22×C4—682no—
504C22×C4—682no—
505C22×C4—682no—
506C22×C4—682no—
507C22×C4D4×D1711yescharacteristic
508C22×C4D4×D1711yescharacteristic
509C22×C4C23×D1711yescharacteristic
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—172no—
517C2×C8—341no—
518C2×C8—341no—
519C2×C8—341no—
520C2×C8—341no—
521C2×C8—341no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—174no—
528C4⋊C4—174no—
529C4⋊C4—174no—
530C4⋊C4—172no—
531C4⋊C4—172no—
532C4⋊C4—172no—
533C4⋊C4—344no—
534C4⋊C4—344no—
535C4⋊C4—341no—
536C4⋊C4—341no—
537C4⋊C4—341no—
538C4⋊C4—344no—
539C4⋊C4—341no—
540C4⋊C4—341no—
541C4⋊C4—682no—
542C4⋊C4—1361no—
543C4⋊C4C2×C4×D1712yes—
544C22⋊C4—24no—
545C22⋊C4—344no—
546C22⋊C4—344no—
547C42—172no—
548C42—172no—
549C42—172no—
550C42—172no—
551C42—682no—
552C42—1361no—
553C42—1361no—
554C42C2×C4×D1712yes—
555C42D4×D1712yes—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23C2×D4×D1711yescharacteristic
564Q8—348no—
565Q8—688no—
566Q8—688no—
567Q8—688no—
568D4—22no—
569D4—22no—
570D4—22no—
571D4#1058757314yes—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—48no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—172no—
587C2×C4—172no—
588C2×C4—172no—
589C2×C4—172no—
590C2×C4—172no—
591C2×C4—172no—
592C2×C4—341no—
593C2×C4—341no—
594C2×C4—341no—
595C2×C4—341no—
596C2×C4—341no—
597C2×C4—341no—
598C2×C4—342no—
599C2×C4—342no—
600C2×C4—342no—
601C2×C4—688no—
602C2×C4—688no—
603C2×C4—681no—
604C2×C4—681no—
605C2×C4—681no—
606C2×C4—681no—
607C2×C4—681no—
608C2×C4—681no—
609C2×C4—681no—
610C2×C4—681no—
611C2×C4—681no—
612C2×C4C22⋊C4×D1712yes—
613C2×C4C22⋊C4×D1712yes—
614C2×C4C22×C4×D1711yescharacteristic
615C2×C4C2×D4×D1711yescharacteristic
616C2×C4C2×#6325411yescharacteristic
617C2×C4C2×Q8×D1711yescharacteristic
618C2×C4#1058769711yescharacteristic
619C2×C4#1058769711yescharacteristic
620C8—41no—
621C8—41no—
622C8—342no—
623C8—681no—
624C22—24no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22C2×C22⋊C4×D1711yescharacteristic
632C22#1176346511yescharacteristic
633C22#1176347211yescharacteristic
634C22AGL(1,ℤ/8)×D1711yescharacteristic
635C22AGL(1,ℤ/8)×D1711yescharacteristic
636C22#1176361611yescharacteristic
637C22#1176362311yescharacteristic
638C4—22no—
639C4—22no—
640C4—42no—
641C4—81no—
642C4—81no—
643C4—174no—
644C4—342no—
645C4—342no—
646C4—342no—
647C4—682no—
648C4—682no—
649C4—682no—
650C4—1361no—
651C4C2×#1058757311yescharacteristic
652C4#1176328611yescharacteristic
653C4#1176346511yescharacteristic
654C4#1176347211yescharacteristic
655C2—44no—
656C2#1453161611yescharacteristic
657C2D17×#41811yescharacteristic
658C2#1453731311yescharacteristic
659C2#1454149711yescharacteristic
660C2#1454157111yescharacteristic
661C2#1454168611yescharacteristic
662C2#1454170011yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1804C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
cosocleC22×C4×C17C24
hypercenter#1804D17
hypocenterC17#22523
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1804

Complements

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

NH ≅ G/Ntype
2. C2×#11762938C2semidirect
3. #14531736C2semidirect
4. #14529756C2semidirect
5. #14529755C2semidirect
6. #14529743C2semidirect
7. #14529734C2semidirect
8. #14529731C2semidirect
9. #14529730C2semidirect
18. D4×Dic34C4semidirect
124. #10587503Q8semidirect
204. C4⋊Dic17C2×Q8semidirect
223. C2×#3622C2×Q8semidirect
466. C17#22523semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic342
C2×#117629382
#145317362
#145297562
#145297552
#145297432
#145297342
#145297312
#145297302
#145287922
#145287822
#145286112
#145286092
#102×Dic172
C17×#18042
#2252317