Finite Group Databasebeta

← #16955711

Subgroups — #16955711

Order 4352.

2782 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587578C211yesmaximal, characteristic
2C22×#10587558C211yesmaximal, characteristic
3C22×C4⋊C4×D17C211yesmaximal, characteristic
4#14527014C211yesmaximal, characteristic, complemented
5#14527013C211yesmaximal, characteristic, complemented
6#14527013C211yesmaximal, characteristic, complemented
7#14527012C214yesmaximal, complemented
8#14527005C214yesmaximal
9C17×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×C4×D17C2212yes—
11C23×C4×D17C2211yescharacteristic
12C2×#10587578—24no—
13C2×#10587578—24no—
14C2×#10587578C418yescomplemented
15C2×#10587558—24no—
16C2×#10587558—24no—
17C2×#10587558C418yescomplemented
18C22×C4⋊Dic17C2211yescharacteristic
19C22×C4⋊Dic17C2211yescharacteristic
20C22×#63225C2212yes—
21C22×#63225C2212yes—
22C2×C4⋊C4×D17—24no—
23C2×C4⋊C4×D17—24no—
24C2×C4⋊C4×D17C418yescomplemented
25C22×#63227C2212yes—
26C22×#63227C2212yes—
27C22×C4×Dic17C2212yes—
28C22×C4×Dic17C2211yescharacteristic
29C2×#10587502C2214yes—
30C2×#10587502C2214yes—
31C2×#10587502C2214yes—
32C22×C4⋊C4×C17C2211yescharacteristic
33C22×C4⋊C4×C17C2211yescharacteristic
34C22×C42×C17C2211yescharacteristic
35C2×C17×#95C2214yes—
36C24×D17C2311yescharacteristic
37C23×C4×C17C2312yes—
38C23×C4×C17C2312yes—
39C23×C4×C17C2312yes—
40C23×C4×C17C2311yescharacteristic
41C23×Dic17C2312yes—
42C23×Dic17C2312yes—
43C23×Dic17C2312yes—
44C23×Dic17C2311yescharacteristic
45C22×C4×D17—24no—
46C22×C4×D17—24no—
47C22×C4×D17—44no—
48C22×C4×D17—44no—
49C22×C4×D17C2×C418yes—
50C22×C4×D17C2×C414yes—
51C22×C4×D17C2×C414yes—
52C22×C4×D17C2×C414yes—
53C22×C4×D17D414yes—
54C22×C4×D17D414yes—
55C22×C4×D17D414yes—
56C22×C4×D17Q814yes—
57C2×C4⋊C4×C17—22no—
58C2×C4⋊C4×C17—22no—
59C2×C4⋊C4×C17—22no—
60C2×C4⋊C4×C17—22no—
61C2×C4⋊C4×C17C2×C414yes—
62C2×C4⋊C4×C17C2×C414yescomplemented
63C2×C42×C17—22no—
64C2×C42×C17—22no—
65C2×C42×C17C2×C414yescomplemented
66C2×#63227—24no—
67C2×#63227—24no—
68C2×#63227—24no—
69C2×#63227—24no—
70C2×#63227C2×C418yes—
71C2×#63227C2×C418yes—
72C2×C4⋊Dic17—22no—
73C2×C4⋊Dic17—22no—
74C2×C4⋊Dic17—22no—
75C2×C4⋊Dic17—22no—
76C2×C4⋊Dic17C2×C414yes—
77C2×C4⋊Dic17C2×C414yescomplemented
78C2×#63225—24no—
79C2×#63225—24no—
80C2×#63225—24no—
81C2×#63225—24no—
82C2×#63225C2×C418yes—
83C2×#63225C2×C418yescomplemented
84C2×C4×Dic17—22no—
85C2×C4×Dic17—22no—
86C2×C4×Dic17—24no—
87C2×C4×Dic17—24no—
88C2×C4×Dic17C2×C414yescomplemented
89C2×C4×Dic17C2×C418yescomplemented
90#10587578—232no—
91C4⋊C4×D17—232no—
92#10587558—232no—
93C17×#95—44no—
94#10587502—44no—
95#10587502—44no—
96#10587502—44no—
97C24×C17C2411yescharacteristic
98C23×D17—22no—
99C23×D17—22no—
100C23×D17—22no—
101C23×D17—22no—
102C23×D17C22×C412yes—
103C23×D17C22×C412yes—
104C23×D17C22×C412yes—
105C23×D17C2×D412yes—
106C23×D17C2×D412yes—
107C23×D17C2×D412yes—
108C23×D17C2×Q812yes—
109C22×C4×C17—22no—
110C22×C4×C17—22no—
111C22×C4×C17—22no—
112C22×C4×C17—22no—
113C22×C4×C17—22no—
114C22×C4×C17—22no—
115C22×C4×C17—24no—
116C22×C4×C17—24no—
117C22×C4×C17—22no—
118C22×C4×C17—24no—
119C22×C4×C17—22no—
120C22×C4×C17—22no—
121C22×C4×C17—22no—
122C22×C4×C17—24no—
123C22×C4×C17C22×C414yes—
124C22×C4×C17C22×C414yes—
125C22×C4×C17C22×C414yes—
126C22×C4×C17C22×C412yes—
127C22×C4×C17C22×C412yes—
128C22×C4×C17C22×C412yes—
129C22×C4×C17C2×D412yes—
130C22×C4×C17C2×D412yes—
131C22×C4×C17C2×D412yes—
132C22×C4×C17C2×Q812yes—
133C22×Dic17—22no—
134C22×Dic17—22no—
135C22×Dic17—22no—
136C22×Dic17—22no—
137C22×Dic17—22no—
138C22×Dic17—22no—
139C22×Dic17—24no—
140C22×Dic17—24no—
141C22×Dic17—22no—
142C22×Dic17—22no—
143C22×Dic17—24no—
144C22×Dic17—22no—
145C22×Dic17—24no—
146C22×Dic17—22no—
147C22×Dic17C22×C414yes—
148C22×Dic17C22×C414yes—
149C22×Dic17C22×C414yes—
150C22×Dic17C22×C412yes—
151C22×Dic17C22×C412yes—
152C22×Dic17C22×C412yes—
153C22×Dic17C2×D412yes—
154C22×Dic17C2×D412yes—
155C22×Dic17C2×D412yes—
156C22×Dic17C2×Q812yes—
157C2×C4×D17—216no—
158C2×C4×D17—416no—
159C2×C4×D17—48no—
160C2×C4×D17—416no—
161C2×C4×D17—48no—
162C2×C4×D17C42116yes—
163C2×C4×D17C22⋊C4116yes—
164C2×C4×D17C22⋊C4116yes—
165C2×C4×D17C22⋊C4116yes—
166C2×C4×D17C4⋊C4116yes—
167C2×C4×D17C4⋊C4116yes—
168C2×C4×D17C4⋊C4116yes—
169C4⋊C4×C17—28no—
170C4⋊C4×C17—28no—
171C42×C17—28no—
172#63227—216no—
173#63227—216no—
174C4⋊Dic17—28no—
175C4⋊Dic17—28no—
176#63225—216no—
177#63225—216no—
178C4×Dic17—28no—
179C4×Dic17—216no—
180#13795—171nomaximal, Sylow
181C23×C17C23×C411yescharacteristic
182C23×C17C23×C411yescharacteristic
183C23×C17C23×C411yescharacteristic
184C23×C17C22×D411yescharacteristic
185C23×C17C22×D411yescharacteristic
186C23×C17C22×D411yescharacteristic
187C23×C17C22×Q811yescharacteristic
188C23×C17H(5,𝔽2)12yes—
189C23×C17H(5,𝔽2)12yes—
190C23×C172−1+412yes—
191C23×C172−1+412yes—
192C22×D17—24no—
193C22×D17—24no—
194C22×D17—24no—
195C22×D17—24no—
196C22×D17—24no—
197C22×D17—24no—
198C22×D17—22no—
199C22×D17—24no—
200C22×D17—22no—
201C22×D17—22no—
202C22×D17—24no—
203C22×D17—22no—
204C22×D17—24no—
205C22×D17—24no—
206C22×D17—24no—
207C22×D17—24no—
208C22×D17C2×C4214yes—
209C22×D17C2×C22⋊C414yes—
210C22×D17C2×C22⋊C414yes—
211C22×D17C2×C22⋊C414yes—
212C22×D17C2×C4⋊C414yes—
213C22×D17C2×C4⋊C414yes—
214C22×D17C2×C4⋊C414yes—
215C2×C4×C17—24no—
216C2×C4×C17—24no—
217C2×C4×C17—24no—
218C2×C4×C17—28no—
219C2×C4×C17—28no—
220C2×C4×C17—24no—
221C2×C4×C17—24no—
222C2×C4×C17—42no—
223C2×C4×C17—44no—
224C2×C4×C17—42no—
225C2×C4×C17—42no—
226C2×C4×C17—42no—
227C2×C4×C17—42no—
228C2×C4×C17—42no—
229C2×C4×C17—44no—
230C2×C4×C17—42no—
231C2×C4×C17—42no—
232C2×C4×C17C2×C4214yes—
233C2×C4×C17C2×C22⋊C414yes—
234C2×C4×C17C2×C22⋊C414yes—
235C2×C4×C17C2×C22⋊C414yes—
236C2×C4×C17C2×C4⋊C414yes—
237C2×C4×C17C2×C4⋊C414yes—
238C2×C4×C17C2×C4⋊C414yes—
239C2×Dic17—24no—
240C2×Dic17—24no—
241C2×Dic17—24no—
242C2×Dic17—28no—
243C2×Dic17—24no—
244C2×Dic17—24no—
245C2×Dic17—28no—
246C2×Dic17—42no—
247C2×Dic17—44no—
248C2×Dic17—42no—
249C2×Dic17—42no—
250C2×Dic17—42no—
251C2×Dic17—42no—
252C2×Dic17—42no—
253C2×Dic17—42no—
254C2×Dic17—42no—
255C2×Dic17—44no—
256C2×Dic17C2×C4214yes—
257C2×Dic17C2×C22⋊C414yes—
258C2×Dic17C2×C22⋊C414yes—
259C2×Dic17C2×C22⋊C414yes—
260C2×Dic17C2×C4⋊C414yes—
261C2×Dic17C2×C4⋊C414yes—
262C2×Dic17C2×C4⋊C414yes—
263C4×D17—432no—
264C4×D17#95164yescomplemented
265C22×#117—171no—
266C22×#117—171no—
267C23×C4⋊C4—171no—
268C4⋊#95—171no—
269C4⋊#95—174no—
270C4⋊#95—171no—
271C4⋊#95—171no—
272C4⋊#95D1711yescharacteristic, complemented
273C22⋊#95—174no—
274C22×C17C22×C4211yescharacteristic
275C22×C17C22×C22⋊C411yescharacteristic
276C22×C17C22×C22⋊C411yescharacteristic
277C22×C17C22×C22⋊C411yescharacteristic
278C22×C17C22×C4⋊C411yescharacteristic
279C22×C17C22×C4⋊C411yescharacteristic
280C22×C17C22×C4⋊C411yescharacteristic
281C22×C17#51812yes—
282C22×C17#51912yes—
283C22×C17#52012yes—
284C22×C17#52012yes—
285C22×C17#52012yes—
286C22×C17#52012yes—
287C22×C17#53511yescharacteristic
288C22×C17#53512yes—
289C22×C17#53511yescharacteristic
290C22×C17#53611yescharacteristic
291C22×C17#53612yes—
292C22×C17#53611yescharacteristic
293C22×C17#53712yes—
294C22×C17#53712yes—
295C22×C17#54412yes—
296C22×C17#54412yes—
297C2×D17—24no—
298C2×D17—24no—
299C2×D17—28no—
300C2×D17—24no—
301C2×D17—24no—
302C2×D17—28no—
303C2×D17—24no—
304C2×D17—28no—
305C2×D17—24no—
306C2×D17—28no—
307C2×D17C2×#9518yes—
308C4×C17—44no—
309C4×C17—44no—
310C4×C17—44no—
311C4×C17C2×#9518yes—
312Dic17—44no—
313Dic17—44no—
314Dic17—44no—
315Dic17C2×#9518yes—
316C24×C4—171no—
317C24×C4—172no—
318C2×#117—178no—
319C2×#117—178no—
320C2×#117—344no—
321C2×#117—344no—
322C2×#117—344no—
323C2×#117—344no—
324C22×C4⋊C4—172no—
325C22×C4⋊C4—172no—
326C22×C4⋊C4—171no—
327C22×C4⋊C4—171no—
328C22×C4⋊C4—178no—
329C22×C4⋊C4—344no—
330C22×C4⋊C4—344no—
331C22×C4⋊C4C2×D1711yescharacteristic
332C22×C4⋊C4C2×D1711yescharacteristic
333C22×C22⋊C4—172no—
334C22×C22⋊C4—172no—
335C22×C42—172no—
336C22×C42—171no—
337C22×C42C2×D1711yescharacteristic
338C2×#95—174no—
339C2×#95—174no—
340C2×#95—174no—
341C2×#95C2×D1714yes—
342C2×C17C22×#9511yescharacteristic
343C2×C17#227412yes—
344C2×C17#230911yescharacteristic
345C2×C17#231011yescharacteristic
346C2×C17#231112yes—
347C2×C17#231111yescharacteristic
348C2×C17#231111yescharacteristic
349C2×C17#231612yes—
350C2×C17#231711yescharacteristic
351C2×C17#231712yes—
352C2×C17#231711yescharacteristic
353D17—28no—
354C25—171no—
355C23×C4—171no—
356C23×C4—174no—
357C23×C4—174no—
358C23×C4—174no—
359C23×C4—174no—
360C23×C4—174no—
361C23×C4—174no—
362C23×C4—174no—
363C23×C4—172no—
364C23×C4—172no—
365C23×C4—172no—
366C23×C4—178no—
367C23×C4—344no—
368C23×C4—344no—
369C23×C4—684no—
370C23×C4—684no—
371C23×C4C22×D1712yes—
372C23×C4C22×D1712yes—
373C23×C4C22×D1712yes—
374C23×C4C22×D1711yescharacteristic
375#117—3432no—
376#117—3432no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—178no—
382C2×C4⋊C4—174no—
383C2×C4⋊C4—178no—
384C2×C4⋊C4—174no—
385C2×C4⋊C4—3432no—
386C2×C4⋊C4—344no—
387C2×C4⋊C4—344no—
388C2×C4⋊C4—344no—
389C2×C4⋊C4—344no—
390C2×C4⋊C4—342no—
391C2×C4⋊C4—342no—
392C2×C4⋊C4—342no—
393C2×C4⋊C4—342no—
394C2×C4⋊C4C4×D1714yes—
395C2×C4⋊C4C4×D1714yescomplemented
396C2×C22⋊C4—178no—
397C2×C22⋊C4—178no—
398C2×C22⋊C4—344no—
399C2×C22⋊C4—344no—
400C2×C22⋊C4—344no—
401C2×C22⋊C4—344no—
402C2×C42—22no—
403C2×C42—22no—
404C2×C42—178no—
405C2×C42—174no—
406C2×C42—344no—
407C2×C42—344no—
408C2×C42—342no—
409C2×C42—342no—
410C2×C42C4×D1714yescomplemented
411#95—44no—
412#95—684no—
413#95—684no—
414#95—684no—
415C17#1379511yescharacteristic, complemented, Sylow
416C24—172no—
417C24—172no—
418C24—172no—
419C24—172no—
420C24—172no—
421C24—172no—
422C24—172no—
423C24—342no—
424C24—342no—
425C24—342no—
426C24—342no—
427C24C23×D1711yescharacteristic
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—172no—
443C22×C4—172no—
444C22×C4—172no—
445C22×C4—172no—
446C22×C4—172no—
447C22×C4—172no—
448C22×C4—172no—
449C22×C4—1716no—
450C22×C4—1716no—
451C22×C4—1716no—
452C22×C4—1716no—
453C22×C4—1716no—
454C22×C4—1716no—
455C22×C4—1716no—
456C22×C4—174no—
457C22×C4—174no—
458C22×C4—174no—
459C22×C4—3416no—
460C22×C4—342no—
461C22×C4—342no—
462C22×C4—342no—
463C22×C4—342no—
464C22×C4—342no—
465C22×C4—342no—
466C22×C4—342no—
467C22×C4—344no—
468C22×C4—342no—
469C22×C4—344no—
470C22×C4—342no—
471C22×C4—344no—
472C22×C4—344no—
473C22×C4—342no—
474C22×C4—6816no—
475C22×C4—688no—
476C22×C4—6816no—
477C22×C4—688no—
478C22×C4C2×C4×D1714yes—
479C22×C4C2×C4×D1714yes—
480C22×C4C2×C4×D1714yes—
481C22×C4C2×C4×D1712yes—
482C22×C4C2×C4×D1712yes—
483C22×C4C2×C4×D1712yes—
484C22×C4D4×D1712yes—
485C22×C4D4×D1712yes—
486C22×C4D4×D1712yes—
487C22×C4Q8×D1712yes—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—3416no—
491C4⋊C4—3416no—
492C4⋊C4—348no—
493C4⋊C4—348no—
494C22⋊C4—3416no—
495C22⋊C4—3416no—
496C42—28no—
497C42—3416no—
498C42—348no—
499C23—174no—
500C23—174no—
501C23—174no—
502C23—174no—
503C23—174no—
504C23—174no—
505C23—174no—
506C23—344no—
507C23—344no—
508C23—344no—
509C23—344no—
510C23—344no—
511C23—344no—
512C23—342no—
513C23—344no—
514C23—342no—
515C23—342no—
516C23—344no—
517C23—342no—
518C23—344no—
519C23—344no—
520C23—344no—
521C23—344no—
522C23C22×C4×D1711yescharacteristic
523C23C22×C4×D1711yescharacteristic
524C23C22×C4×D1711yescharacteristic
525C23C2×D4×D1711yescharacteristic
526C23C2×D4×D1711yescharacteristic
527C23C2×D4×D1711yescharacteristic
528C23#1058769112yes—
529C23C2×Q8×D1711yescharacteristic
530C23#1058769412yes—
531C23#1058769612yes—
532C23#1058769712yes—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—174no—
551C2×C4—174no—
552C2×C4—174no—
553C2×C4—174no—
554C2×C4—174no—
555C2×C4—174no—
556C2×C4—174no—
557C2×C4—1764no—
558C2×C4—344no—
559C2×C4—344no—
560C2×C4—344no—
561C2×C4—348no—
562C2×C4—344no—
563C2×C4—344no—
564C2×C4—348no—
565C2×C4—6832no—
566C2×C4—682no—
567C2×C4—684no—
568C2×C4—682no—
569C2×C4—682no—
570C2×C4—682no—
571C2×C4—682no—
572C2×C4—682no—
573C2×C4—682no—
574C2×C4—682no—
575C2×C4—684no—
576C2×C4C42×D1714yes—
577C2×C4C22⋊C4×D1714yes—
578C2×C4C22⋊C4×D1714yes—
579C2×C4C22⋊C4×D1714yes—
580C2×C4C4⋊C4×D1714yes—
581C2×C4C4⋊C4×D1714yes—
582C2×C4C4⋊C4×D1714yes—
583C22—178no—
584C22—344no—
585C22—344no—
586C22—348no—
587C22—344no—
588C22—344no—
589C22—348no—
590C22—348no—
591C22—344no—
592C22—348no—
593C22—344no—
594C22C2×C42×D1711yescharacteristic
595C22C2×C22⋊C4×D1711yescharacteristic
596C22C2×C22⋊C4×D1711yescharacteristic
597C22C2×C22⋊C4×D1711yescharacteristic
598C22C2×C4⋊C4×D1711yescharacteristic
599C22C2×C4⋊C4×D1711yescharacteristic
600C22C2×C4⋊C4×D1711yescharacteristic
601C22#1176328112yes—
602C22#1176328512yes—
603C22#1176328812yes—
604C22#1176328912yes—
605C22#1176329012yes—
606C22#1176329112yes—
607C22#1176337211yescharacteristic
608C22#1176337512yes—
609C22#1176337711yescharacteristic
610C22#1176337911yescharacteristic
611C22#1176338212yes—
612C22#1176338411yescharacteristic
613C22#1176338812yes—
614C22#1176338912yes—
615C22#1176343612yes—
616C22#1176343912yes—
617C4—44no—
618C4—44no—
619C4—44no—
620C4—178no—
621C4—684no—
622C4—684no—
623C4—684no—
624C4#95×D1718yes—
625C2—348no—
626C2C2×#95×D1711yescharacteristic
627C2#1453113812yes—
628C2#1453136311yescharacteristic
629C2#1453136711yescharacteristic
630C2#1453137312yes—
631C2#1453137411yescharacteristic
632C2#1453137511yescharacteristic
633C2#1453139912yes—
634C2#1453140511yescharacteristic
635C2#1453140612yes—
636C2#1453140911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D17
commutatorC22×C17C22×C42
FittingC17×C4⋊#95C2
socle, cosocleC24×C17C24
prime radicalC24×D17C23
hypercenterC4⋊#95D17
hypocenterC17#13795
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
4. #14527014C2semidirect
5. #14527013C2semidirect
6. #14527013C2semidirect
7. #14527012C2semidirect
9. C17×C4⋊#95C2semidirect
14. C2×#10587578C4semidirect
17. C2×#10587558C4semidirect
24. C2×C4⋊C4×D17C4semidirect
62. C2×C4⋊C4×C17C2×C4semidirect
65. C2×C42×C17C2×C4semidirect
77. C2×C4⋊Dic17C2×C4semidirect
83. C2×#63225C2×C4semidirect
88. C2×C4×Dic17C2×C4semidirect
89. C2×C4×Dic17C2×C4semidirect
264. C4×D17#95semidirect
272. C4⋊#95D17semidirect
395. C2×C4⋊C4C4×D17semidirect
410. C2×C42C4×D17semidirect
415. C17#13795semidirect

Maximal subgroups

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

subgroupindex
C22×#105875782
C22×#105875582
C22×C4⋊C4×D172
#145270142
#145270132
#145270122
#145270052
C17×C4⋊#952
#1379517