Finite Group Databasebeta

← #17979045

Subgroups — #17979045

Order 4352.

2806 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14545293C218yesmaximal, complemented
2#14545292C218yesmaximal, complemented
3#14545264C212yesmaximal, complemented
4#14545261C214yesmaximal, complemented
5#14545228C214yesmaximal
6#14545228C212yesmaximal
7#14545222C212yesmaximal, complemented
8C22×C4⋊C4×D17C211yesmaximal, characteristic
9C2×#10587677C2214yes—
10C2×#10587677C2218yes—
11#11763760C22116yescomplemented
12#11763758C22132yescomplemented
13#11763752C2214yes—
14#11763752C2218yes—
15#11763751C2218yes—
16#11763751C2214yes—
17#11763751C2214yes—
18#11763751C2218yes—
19#11763750C2214yes—
20C4⋊C4×#595C22116yescomplemented
21C4×C22⋊#595C2218yes—
22C4×C22⋊#595C2214yes—
23C4×C22⋊#595C2214yes—
24C4×C22⋊#595C2218yes—
25C23×C4×D17C2211yescharacteristic
26C23×C4×D17C2212yes—
27C22×C4⋊Dic17C411yescharacteristic, complemented
28C22×#63225C412yescomplemented
29C2×C4⋊C4×D17C411yescharacteristic, complemented
30C2×C4⋊C4×D17C411yescharacteristic, complemented
31C2×C4⋊C4×D17C412yescomplemented
32C2×C4⋊C4×D17C414yescomplemented
33C2×C4⋊C4×D17C414yescomplemented
34C2×C4⋊C4×D17C2211yescharacteristic
35C2×C4⋊C4×D17C2212yes—
36C2×C4⋊C4×D17C2211yescharacteristic
37C2×C4⋊C4×D17C2214yes—
38C2×C4⋊C4×D17C2214yes—
39C22×C4⋊C4×C17C411yescharacteristic, complemented
40C24×D17C2311yescharacteristic
41C23×C4×C17C2×C411yescharacteristic
42C23×C4×C17C2×C412yes—
43C23×Dic17C2×C411yescharacteristic
44C23×Dic17C2×C412yes—
45C22×C4×D17—28no—
46C22×C4×D17—48no—
47C22×C4×D17C2×C411yescharacteristic
48C22×C4×D17C2×C411yescharacteristic
49C22×C4×D17C2×C412yes—
50C22×C4×D17C2×C412yes—
51C22×C4×D17C2×C414yes—
52C22×C4×D17C2×C412yes—
53C22×C4×D17C2×C412yes—
54C22×C4×D17C2×C414yes—
55C22×C4×D17C2312yes—
56C22×C4×D17C2312yes—
57C22×C4×D17C2311yescharacteristic
58C22×C4×D17C2314yes—
59C22×C4×D17C2312yes—
60C22×C4×D17C2314yes—
61C22×C4×D17C2312yes—
62C22×C4×D17C2311yescharacteristic
63C2×C22⋊#595C2318yes—
64C2×C22⋊#595C2318yes—
65C2×C22⋊#595C2318yes—
66#10587677—216no—
67#10587677—416no—
68C2×C4⋊#595C2318yes—
69C2×C4⋊#595C23116yes—
70C2×C4×#595C2318yes—
71C2×C4×#595C23116yes—
72C2×C4⋊C4×C17C2×C414yes—
73C2×C4⋊C4×C17C2×C412yes—
74C2×C4⋊C4×C17C2×C414yes—
75C2×C4⋊C4×C17C2×C411yescharacteristic
76C2×C4⋊C4×C17C2×C411yescharacteristic
77C2×C4⋊Dic17C2×C414yes—
78C2×C4⋊Dic17C2×C412yes—
79C2×C4⋊Dic17C2×C414yes—
80C2×C4⋊Dic17C2×C411yescharacteristic
81C2×C4⋊Dic17C2×C411yescharacteristic
82C2×#63225C2×C418yes—
83C2×#63225C2×C412yes—
84C2×#63225C2×C412yes—
85C2×#63225C2×C418yes—
86C2×#63225C2×C412yes—
87C2×#63225C2×C412yes—
88C4⋊C4×D17C2×C418yes—
89C4⋊C4×D17C2×C414yes—
90C4⋊C4×D17C2×C414yes—
91C4⋊C4×D17C2×C418yes—
92C4⋊C4×D17C2×C418yes—
93C4⋊C4×D17C2×C414yes—
94C4⋊C4×D17C2×C414yes—
95C4⋊C4×D17C2×C414yes—
96C4⋊C4×D17C2×C414yes—
97C4⋊C4×D17C2314yes—
98C4⋊C4×D17C2318yes—
99C4⋊C4×D17C2314yes—
100(C2×#595)⋊C4C23116yes—
101(C2×#595)⋊C4C2318yes—
102(C2×#595)⋊C4C2318yes—
103(C2×#595)⋊C4C23116yes—
104C24×C17C22×C411yescharacteristic
105C23×D17—24no—
106C23×D17—24no—
107C23×D17—24no—
108C23×D17C22×C411yescharacteristic
109C23×D17C22×C411yescharacteristic
110C23×D17C22×C411yescharacteristic
111C23×D17C2411yescharacteristic
112C23×D17C2411yescharacteristic
113C23×D17C2411yescharacteristic
114C22×#595C24116yes—
115C22×C4×C17—24no—
116C22×C4×C17—44no—
117C22×C4×C17C22×C414yes—
118C22×C4×C17C22×C411yescharacteristic
119C22×C4×C17C22×C412yes—
120C22×C4×C17C22×C412yes—
121C22×C4×C17C22×C412yes—
122C22×C4×C17C22×C412yes—
123C22×C4×C17C22×C414yes—
124C22×C4×C17C22×C411yescharacteristic
125C22×Dic17—24no—
126C22×Dic17—44no—
127C22×Dic17C22×C411yescharacteristic
128C22×Dic17C22×C412yes—
129C22×Dic17C22×C412yes—
130C22×Dic17C22×C414yes—
131C22×Dic17C22×C412yes—
132C22×Dic17C22×C412yes—
133C22×Dic17C22×C414yes—
134C22×Dic17C22×C411yescharacteristic
135C2×C4×D17—24no—
136C2×C4×D17—216no—
137C2×C4×D17—24no—
138C2×C4×D17—24no—
139C2×C4×D17—28no—
140C2×C4×D17—28no—
141C2×C4×D17—28no—
142C2×C4×D17—28no—
143C2×C4×D17—44no—
144C2×C4×D17—48no—
145C2×C4×D17—44no—
146C2×C4×D17—44no—
147C2×C4×D17—48no—
148C2×C4×D17—44no—
149C2×C4×D17—48no—
150C2×C4×D17C22×C412yes—
151C2×C4×D17C22×C414yes—
152C2×C4×D17C22×C414yes—
153C2×C4×D17C22×C412yes—
154C2×C4×D17C22×C412yes—
155C2×C4×D17C22×C412yes—
156C2×C4×D17C22×C412yes—
157C2×C4×D17C22×C414yes—
158C2×C4×D17C22×C414yes—
159C2×C4×D17C22×C414yes—
160C2×C4×D17C22×C414yes—
161C2×C4×D17C22×C412yes—
162C2×C4×D17#4118yes—
163C2×C4×D17C2414yes—
164C2×C4×D17C2414yes—
165C2×C4×D17C2412yes—
166C2×C4×D17C2412yes—
167C22⋊#595—216no—
168C22⋊#595—216no—
169C22⋊#595—216no—
170C4⋊#595—216no—
171C4⋊#595—416no—
172C4×#595—216no—
173C4×#595—416no—
174C4⋊C4×C17C22×C418yes—
175C4⋊C4×C17C22×C414yes—
176C4⋊C4×C17C22×C414yes—
177C4⋊Dic17C22×C418yes—
178C4⋊Dic17C22×C414yes—
179C4⋊Dic17C22×C414yes—
180#63225C22×C418yes—
181#63225C22×C418yes—
182#63225C22×C418yes—
183#63225C22×C418yes—
184#34178—171nomaximal, Sylow
185C23×C17—22no—
186C23×C17—22no—
187C23×C17—22no—
188C23×C17C23×C411yescharacteristic
189C23×C17C23×C411yescharacteristic
190C23×C17C23×C411yescharacteristic
191C22×D17—232no—
192C22×D17—22no—
193C22×D17—22no—
194C22×D17—22no—
195C22×D17—22no—
196C22×D17—22no—
197C22×D17—22no—
198C22×D17—22no—
199C22×D17—22no—
200C22×D17—22no—
201C22×D17—22no—
202C22×D17—22no—
203C22×D17—22no—
204C22×D17—22no—
205C22×D17—22no—
206C22×D17—22no—
207C22×D17C23×C411yescharacteristic
208C22×D17C23×C411yescharacteristic
209C22×D17C23×C411yescharacteristic
210C22×D17C2×#4114yes—
211C22×D17H(5,𝔽2)14yes—
212C22×D172−1+414yes—
213C22×D17C2511yescharacteristic
214C2×#595—216no—
215C2×#595—216no—
216C2×#595—216no—
217C2×C4×C17—22no—
218C2×C4×C17—28no—
219C2×C4×C17—24no—
220C2×C4×C17—24no—
221C2×C4×C17—42no—
222C2×C4×C17—44no—
223C2×C4×C17—42no—
224C2×C4×C17#11714yescomplemented
225C2×C4×C17C23×C414yes—
226C2×C4×C17C23×C414yes—
227C2×C4×C17C23×C412yes—
228C2×C4×C17C23×C412yes—
229C2×Dic17—22no—
230C2×Dic17—28no—
231C2×Dic17—24no—
232C2×Dic17—24no—
233C2×Dic17—42no—
234C2×Dic17—44no—
235C2×Dic17—42no—
236C2×Dic17#11714yescomplemented
237C2×Dic17C23×C412yes—
238C2×Dic17C23×C414yes—
239C2×Dic17C23×C414yes—
240C2×Dic17C23×C412yes—
241C4×D17—24no—
242C4×D17—28no—
243C4×D17—28no—
244C4×D17—24no—
245C4×D17—28no—
246C4×D17—28no—
247C4×D17—24no—
248C4×D17—24no—
249C4×D17—24no—
250C4×D17—24no—
251C4×D17—44no—
252C4×D17—44no—
253C4×D17—44no—
254C4×D17—44no—
255C4×D17#11718yescomplemented
256C4×D17C2×#4118yes—
257C23×C4⋊C4—171no—
258#2345—178no—
259#2344—178no—
260#2316—172no—
261#2313—174no—
262#2280—172no—
263#2280—174no—
264#2274—172no—
265C22×C17—28no—
266C22×C17—21no—
267C22×C17—21no—
268C22×C17—21no—
269C22×C17—21no—
270C22×C17—21no—
271C22×C17—21no—
272C22×C17C2×#11712yes—
273C22×C17#51812yes—
274C22×C17#51912yes—
275C22×C17C24×C411yescharacteristic
276C2×D17—28no—
277C2×D17—28no—
278C2×D17—28no—
279C2×D17—21no—
280C2×D17—21no—
281C2×D17—21no—
282C2×D17—21no—
283C2×D17—21no—
284C2×D17—21no—
285C2×D17—28no—
286C2×D17—28no—
287C2×D17—28no—
288C2×D17C2×#11712yes—
289C2×D17#51812yes—
290C2×D17#51912yes—
291C2×D17C22×#4112yes—
292C2×D17C2×H(5,𝔽2)12yes—
293C2×D17C2×2−1+412yes—
294#595—416no—
295C4×C17—22no—
296C4×C17—24no—
297C4×C17—24no—
298C4×C17C2×#11714yes—
299Dic17—22no—
300Dic17—24no—
301Dic17—24no—
302Dic17C2×#11714yes—
303C24×C4—171no—
304C24×C4—172no—
305C2×#117—174no—
306C2×#117—178no—
307C22×C4⋊C4—172no—
308C22×C4⋊C4—171no—
309C22×C4⋊C4—172no—
310C22×C4⋊C4—171no—
311C22×C4⋊C4—174no—
312C22×C4⋊C4—174no—
313C22×C4⋊C4—172no—
314C22×C4⋊C4—174no—
315C22×C4⋊C4—171no—
316C22×C4⋊C4—171no—
317C22×C4⋊C4—174no—
318C22×C4⋊C4—171no—
319C22×C4⋊C4#59511yescharacteristic, complemented
320#389—1716no—
321#387—1732no—
322#381—174no—
323#381—178no—
324#380—178no—
325#380—174no—
326#380—178no—
327#380—174no—
328#379—174no—
329C4×C4⋊C4—1716no—
330C4×C22⋊C4—178no—
331C4×C22⋊C4—174no—
332C4×C22⋊C4—174no—
333C4×C22⋊C4—178no—
334C2×C17—22no—
335C2×C17—22no—
336C2×C17—22no—
337C2×C17C22×#11711yescharacteristic
338C2×C17C2×#51811yescharacteristic
339C2×C17C2×#51911yescharacteristic
340D17—22no—
341D17—22no—
342D17—22no—
343D17#349214yes—
344C25—171no—
345C23×C4—171no—
346C23×C4—171no—
347C23×C4—172no—
348C23×C4—174no—
349C23×C4—171no—
350C23×C4—172no—
351C23×C4—172no—
352C23×C4—171no—
353C23×C4—172no—
354C23×C4—172no—
355C23×C4—171no—
356C23×C4—174no—
357C23×C4—172no—
358C23×C4—172no—
359C23×C4—174no—
360C23×C4—172no—
361C23×C4—174no—
362C23×C4—172no—
363C23×C4—348no—
364C23×C4—688no—
365C23×C4C2×#59511yescharacteristic
366C23×C4C2×#59512yes—
367#117—3416no—
368#117—6816no—
369C2×C4⋊C4—172no—
370C2×C4⋊C4—172no—
371C2×C4⋊C4—178no—
372C2×C4⋊C4—172no—
373C2×C4⋊C4—172no—
374C2×C4⋊C4—178no—
375C2×C4⋊C4—174no—
376C2×C4⋊C4—172no—
377C2×C4⋊C4—174no—
378C2×C4⋊C4—171no—
379C2×C4⋊C4—171no—
380C2×C4⋊C4—174no—
381C2×C4⋊C4—178no—
382C2×C4⋊C4—174no—
383C2×C4⋊C4—174no—
384C2×C4⋊C4—174no—
385C2×C4⋊C4—178no—
386C2×C4⋊C4—174no—
387C2×C4⋊C4—174no—
388C2×C4⋊C4—178no—
389C2×C4⋊C4—178no—
390C2×C4⋊C4—174no—
391C2×C4⋊C4—174no—
392C2×C4⋊C4—178no—
393C2×C4⋊C4—1716no—
394C2×C4⋊C4C2×#59514yes—
395C2×C4⋊C4C2×#59512yes—
396C2×C4⋊C4C2×#59514yes—
397C2×C4⋊C4C2×#59511yescharacteristic
398C2×C4⋊C4C2×#59511yescharacteristic
399C2×C22⋊C4—178no—
400C2×C22⋊C4—178no—
401C2×C22⋊C4—178no—
402C2×C42—178no—
403C2×C42—1716no—
404#95—1716no—
405#95—178no—
406#95—178no—
407#95—1716no—
408C17#3417811yescharacteristic, complemented, Sylow
409C24—171no—
410C24—171no—
411C24—171no—
412C24—171no—
413C24—171no—
414C24—171no—
415C24—344no—
416C24—344no—
417C24—344no—
418C24C22×#59511yescharacteristic
419C22×C4—24no—
420C22×C4—44no—
421C22×C4—171no—
422C22×C4—178no—
423C22×C4—172no—
424C22×C4—172no—
425C22×C4—172no—
426C22×C4—172no—
427C22×C4—171no—
428C22×C4—174no—
429C22×C4—172no—
430C22×C4—172no—
431C22×C4—174no—
432C22×C4—172no—
433C22×C4—172no—
434C22×C4—172no—
435C22×C4—174no—
436C22×C4—174no—
437C22×C4—172no—
438C22×C4—172no—
439C22×C4—172no—
440C22×C4—174no—
441C22×C4—174no—
442C22×C4—174no—
443C22×C4—174no—
444C22×C4—174no—
445C22×C4—174no—
446C22×C4—1716no—
447C22×C4—344no—
448C22×C4—348no—
449C22×C4—348no—
450C22×C4—348no—
451C22×C4—348no—
452C22×C4—3416no—
453C22×C4—344no—
454C22×C4—344no—
455C22×C4—344no—
456C22×C4—688no—
457C22×C4—684no—
458C22×C4—688no—
459C22×C4—684no—
460C22×C4—684no—
461C22×C4—688no—
462C22×C4—684no—
463C22×C4—684no—
464C22×C4C22×#59511yescharacteristic
465C22×C4C22×#59514yes—
466C22×C4C22×#59512yes—
467C22×C4C22×#59512yes—
468C22×C4C22×#59512yes—
469C22×C4C22×#59512yes—
470C22×C4C22×#59514yes—
471C22×C4C22×#59511yescharacteristic
472C4⋊C4—178no—
473C4⋊C4—178no—
474C4⋊C4—178no—
475C4⋊C4—178no—
476C4⋊C4—178no—
477C4⋊C4—174no—
478C4⋊C4—174no—
479C4⋊C4—3416no—
480C4⋊C4—6816no—
481C4⋊C4C22×#59518yes—
482C4⋊C4C22×#59514yes—
483C4⋊C4C22×#59514yes—
484C22⋊C4—3416no—
485C22⋊C4—3416no—
486C22⋊C4—3416no—
487C42—3416no—
488C42—6816no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—171no—
493C23—174no—
494C23—174no—
495C23—174no—
496C23—171no—
497C23—171no—
498C23—171no—
499C23—342no—
500C23—342no—
501C23—342no—
502C23—342no—
503C23—342no—
504C23—342no—
505C23—342no—
506C23—342no—
507C23—342no—
508C23—342no—
509C23—342no—
510C23—342no—
511C23—3432no—
512C23—342no—
513C23—342no—
514C23—342no—
515C23C23×#59511yescharacteristic
516C23C23×#59511yescharacteristic
517C23C23×#59511yescharacteristic
518C2×C4—22no—
519C2×C4—28no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—174no—
526C2×C4—172no—
527C2×C4—178no—
528C2×C4—178no—
529C2×C4—172no—
530C2×C4—174no—
531C2×C4—174no—
532C2×C4—344no—
533C2×C4—344no—
534C2×C4—348no—
535C2×C4—348no—
536C2×C4—344no—
537C2×C4—344no—
538C2×C4—348no—
539C2×C4—348no—
540C2×C4—344no—
541C2×C4—344no—
542C2×C4—342no—
543C2×C4—348no—
544C2×C4—344no—
545C2×C4—344no—
546C2×C4—3416no—
547C2×C4—3416no—
548C2×C4—3416no—
549C2×C4—684no—
550C2×C4—684no—
551C2×C4—684no—
552C2×C4—684no—
553C2×C4—682no—
554C2×C4—684no—
555C2×C4—682no—
556C2×C4#1058767714yescomplemented
557C2×C4C23×#59512yes—
558C2×C4C23×#59514yes—
559C2×C4C23×#59514yes—
560C2×C4C23×#59512yes—
561C22—28no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—172no—
569C22—172no—
570C22—172no—
571C22—172no—
572C22—172no—
573C22—172no—
574C22—348no—
575C22—348no—
576C22—348no—
577C22—348no—
578C22—348no—
579C22—348no—
580C22—341no—
581C22—341no—
582C22—341no—
583C22—341no—
584C22—341no—
585C22—341no—
586C22C2×#1058767712yes—
587C22#1176387612yes—
588C22#1176387712yes—
589C22C24×#59511yescharacteristic
590C4—22no—
591C4—24no—
592C4—24no—
593C4—174no—
594C4—342no—
595C4—344no—
596C4—344no—
597C4—6816no—
598C4C2×#1058767714yes—
599C2—22no—
600C2—22no—
601C2—22no—
602C2—174no—
603C2—342no—
604C2—342no—
605C2—342no—
606C2C22×#1058767711yescharacteristic
607C2C2×#1176387611yescharacteristic
608C2C2×#1176387711yescharacteristic
subgroupG/H
center, FrattiniC22C24×#595
commutator, socleC22×C17C24×C4
FittingC22×C4⋊C4×C17C4
prime radicalC24×D17C23
cosocleC22×D17C25
hypercenterC22×C4⋊C4#595
hypocenterC17#34178
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #14545293C2semidirect
2. #14545292C2semidirect
3. #14545264C2semidirect
4. #14545261C2semidirect
7. #14545222C2semidirect
11. #11763760C22semidirect
12. #11763758C22semidirect
20. C4⋊C4×#595C22semidirect
27. C22×C4⋊Dic17C4semidirect
28. C22×#63225C4semidirect
29. C2×C4⋊C4×D17C4semidirect
30. C2×C4⋊C4×D17C4semidirect
31. C2×C4⋊C4×D17C4semidirect
32. C2×C4⋊C4×D17C4semidirect
33. C2×C4⋊C4×D17C4semidirect
39. C22×C4⋊C4×C17C4semidirect
224. C2×C4×C17#117semidirect
236. C2×Dic17#117semidirect
255. C4×D17#117semidirect
319. C22×C4⋊C4#595semidirect
408. C17#34178semidirect
556. C2×C4#10587677semidirect

Maximal subgroups

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

subgroupindex
#145452932
#145452922
#145452642
#145452612
#145452282
#145452222
C22×C4⋊C4×D172
#3417817