Finite Group Databasebeta

← #17979352

Subgroups — #17979352

Order 4352.

2340 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14545327C214yesmaximal, complemented
2#14545317C211yesmaximal, characteristic, complemented
3#14545309C214yesmaximal, complemented
4D4×C4⋊#595C212yesmaximal, complemented
5#14545294C214yesmaximal, complemented
6#14545274C214yesmaximal, complemented
7#14545272C214yesmaximal, complemented
8#14545265C214yesmaximal, complemented
9C4⋊Q8×#595C211yesmaximal, characteristic, complemented
10#14545241C212yesmaximal, complemented
11D17×#555C211yesmaximal, characteristic
12C2×Q8×#595C2212yes—
13C2×D4×#595C2214yes—
14C2×D4×#595C2214yes—
15C2×#10587677C2214yes—
16C22×C4⋊#595C2214yes—
17(C2×#595)⋊Q8C2212yes—
18(C2×#595)⋊Q8C2214yes—
19#11763761C2212yes—
20#11763761C2212yes—
21#11763761C2214yes—
22#11763760C2212yes—
23#11763760C2212yes—
24#11763760C2214yes—
25#11763760C2214yes—
26#11763760C2214yes—
27#11763759C2214yes—
28#11763759C2214yes—
29#11763759C2218yes—
30#11763759C2218yes—
31#11763757C2218yes—
32#11763756C2218yes—
33#11763756C2218yes—
34#11763756C2218yes—
35#11763755C2211yescharacteristic
36#11763755C2214yes—
37#11763753C2218yes—
38#11763751C2214yes—
39C4⋊C4×#595C2214yes—
40C4⋊C4×#595C2214yes—
41C4×C4⋊#595C2212yes—
42C4×C22⋊#595C2214yes—
43#11763745C2214yes—
44C42×#595C2211yescharacteristic
45#11763525C411yescharacteristic, complemented
46#11763524C412yescomplemented
47#11763523C414yescomplemented
48#11763522C414yescomplemented
49#11763521C414yescomplemented
50C4⋊Q8×D17C2211yescharacteristic
51#124×D17C2214yes—
52#121×D17C2214yes—
53C4×D4×D17C2212yes—
54#117×D17C2214yes—
55C17×#555C411yescharacteristic, complemented
56C23×#595C2318yes—
57C2×Q8×D17C2312yes—
58C2×D4×D17C2312yes—
59C2×D4×D17C2314yes—
60C22×C4×D17C2314yes—
61C2×C22⋊#595C2314yes—
62C2×C22⋊#595C2314yes—
63C2×C22⋊#595C2318yes—
64C2×C22⋊#595C2314yes—
65C2×C22⋊#595C2314yes—
66Q8×#595—44no—
67D4×#595—48no—
68D4×#595—48no—
69#10587677—48no—
70C2×C4⋊#595—48no—
71C2×C4⋊#595C2312yes—
72C2×C4⋊#595C2312yes—
73C2×C4⋊#595C2314yes—
74C2×C4⋊#595C2314yes—
75C2×C4⋊#595C2318yes—
76C2×C4⋊#595C2314yes—
77C2×C4⋊#595C2312yes—
78C2×C4⋊#595C2314yes—
79C2×C4⋊#595C2314yes—
80C2×C4×#595C2312yes—
81C2×C4×#595C2311yescharacteristic
82C2×C4×#595C2312yes—
83C2×C4×#595C2314yes—
84C2×C4×#595C2314yes—
85C2×C4×#595C2311yescharacteristic
86C2×C4×#595C2314yes—
87C17×C4⋊Q8C2×C411yescharacteristic, complemented
88C17×#124C2×C414yescomplemented
89C17×#121C2×C414yescomplemented
90C4×D4×C17C2×C412yescomplemented
91C17×#117C2×C414yescomplemented
92#10587633C2×C414yescomplemented
93#10587630C2×C412yescomplemented
94#10587625C2×C418yescomplemented
95#10587624C2×C414yescomplemented
96#10587622C2×C414yescomplemented
97D4×Dic17C2×C412yescomplemented
98#10587616C2×C414yescomplemented
99C4×C22⋊D17C2×C414yescomplemented
100#10587612C2×C414yescomplemented
101#10587581C2×C414yescomplemented
102#10587579C2×C412yescomplemented
103#10587578C2×C414yescomplemented
104#10587578C2×C414yescomplemented
105C4⋊C4×D17C2314yes—
106C4⋊C4×D17C2312yes—
107#10587574C2×C414yescomplemented
108#10587571C2×C418yescomplemented
109#10587571C2×C418yescomplemented
110#10587570C2×C418yescomplemented
111#10587567C2×C414yescomplemented
112C22⋊C4×D17C2318yes—
113C22⋊C4×D17C2314yes—
114#10587563C2×C418yescomplemented
115#10587561C2×C414yescomplemented
116C4×D68C2×C412yescomplemented
117#10587558C2×C418yescomplemented
118C42×D17C2311yescharacteristic
119C42×D17C2314yes—
120#10587555C2×C411yescharacteristic, complemented
121(C2×#595)⋊C4C2314yes—
122(C2×#595)⋊C4C2318yes—
123(C2×#595)⋊C4C2314yes—
124(C2×#595)⋊C4C2314yes—
125(C2×#595)⋊C4C2318yes—
126(C2×#595)⋊C4C2318yes—
127C23×D17C2414yes—
128C22×#595—48no—
129C22×#595—48no—
130C22×#595—48no—
131C22×#595C2411yescharacteristic
132C22×#595C2412yes—
133C22×#595C2414yes—
134C22×#595C2414yes—
135C22×#595C2414yes—
136C22×#595C2411yescharacteristic
137C2×Q8×C17C22×C412yes—
138C2×D4×C17C22×C412yes—
139C2×D4×C17C22×C414yes—
140C22×C4×C17C22×C414yes—
141C2×C22⋊D17C22×C414yes—
142C2×C22⋊D17C22×C418yes—
143C22×Dic17C22×C414yes—
144Q8×D17—42no—
145Q8×D17—42no—
146D4×D17—24no—
147D4×D17—24no—
148D4×D17—44no—
149D4×D17—44no—
150C2×D68C22×C412yes—
151C2×D68C22×C414yes—
152C2×C4×D17—28no—
153C2×C4×D17—28no—
154C2×C4×D17C22×C414yes—
155C2×C4×D17C22×C414yes—
156C2×C4×D17C2414yes—
157C2×C4×D17C2411yescharacteristic
158C2×C4×D17C2414yes—
159C2×C4×D17C2412yes—
160C2×Dic34C22×C412yes—
161C2×Dic34C22×C414yes—
162C22⋊#595—28no—
163C22⋊#595—44no—
164C22⋊#595—44no—
165C22⋊#595—48no—
166C22⋊#595—44no—
167C4⋊#595—24no—
168C4⋊#595—28no—
169C4⋊#595—24no—
170C4⋊#595—48no—
171C4⋊#595—44no—
172C4⋊#595—44no—
173C4⋊#595—44no—
174C4⋊#595—44no—
175C4⋊#595—42no—
176C4×#595—28no—
177C4×#595—22no—
178C4×#595—24no—
179C4×#595—28no—
180C4×#595—24no—
181C4×#595—44no—
182C4×#595—41no—
183C4⋊C4×C17C22×C414yes—
184C4⋊C4×C17C22×C412yes—
185C22⋊C4×C17C22×C414yes—
186C22⋊C4×C17C22×C418yes—
187C42×C17C22×C414yes—
188C42×C17C22×C411yescharacteristic
189C22⋊Dic17C22×C418yes—
190C22⋊Dic17C22×C414yes—
191#63227C22×C418yes—
192#63227C22×C418yes—
193#63227C22×C414yes—
194#63227C22×C414yes—
195C4⋊Dic17C22×C414yes—
196C4⋊Dic17C22×C412yes—
197#63225C22×C414yes—
198#63225C22×C418yes—
199C4×Dic17C22×C418yes—
200C4×Dic17C22×C414yes—
201C4×Dic17C22×C411yescharacteristic
202C4×Dic17C22×C412yes—
203#34485—171nomaximal, Sylow
204C23×C17C23×C414yes—
205C22×D17—24no—
206C22×D17—24no—
207C22×D17—24no—
208C22×D17—24no—
209C22×D17—24no—
210C22×D17—24no—
211C22×D17C23×C414yes—
212C22×D17C2511yescharacteristic
213C2×#595—22no—
214C2×#595—21no—
215C2×#595—22no—
216C2×#595—22no—
217C2×#595—24no—
218C2×#595—24no—
219C2×#595—24no—
220C2×#595—24no—
221C2×#595—28no—
222C2×#595—24no—
223C2×#595—22no—
224C2×#595—24no—
225C2×#595—24no—
226C2×#595—22no—
227C2×#595—816no—
228C2×#595H(5,𝔽2)12yes—
229Q8×C17—42no—
230D4×C17—24no—
231D4×C17—44no—
232C2×C4×C17—28no—
233C2×C4×C17C23×C414yes—
234C2×C4×C17C23×C411yescharacteristic
235C2×C4×C17C23×C414yes—
236C2×C4×C17C23×C412yes—
237C22⋊D17—28no—
238C22⋊D17—48no—
239C2×Dic17—28no—
240C2×Dic17C23×C411yescharacteristic
241C2×Dic17C23×C414yes—
242C2×Dic17C23×C414yes—
243C2×Dic17C23×C412yes—
244D68—24no—
245D68—44no—
246C4×D17—24no—
247C4×D17—21no—
248C4×D17—24no—
249C4×D17—24no—
250C4×D17—24no—
251C4×D17—21no—
252C4×D17—28no—
253C4×D17—28no—
254C4×D17H(5,𝔽2)14yes—
255C4×D172−1+414yes—
256Dic34—44no—
257Dic34—42no—
258C2×#555—171no—
259#2379—174no—
260#2369—171no—
261#2361—174no—
262D4×C4⋊C4—172no—
263#2346—174no—
264#2326—174no—
265#2324—174no—
266#2317—174no—
267C4×C4⋊Q8—171no—
268#2293—172no—
269C22×C17—24no—
270C22×C17—24no—
271C22×C17—24no—
272C22×C17C24×C411yescharacteristic
273C2×D17—24no—
274C2×D17—24no—
275C2×D17—24no—
276C2×D17—44no—
277C2×D17—44no—
278C2×D17—44no—
279C2×D17—44no—
280C2×D17C2×H(5,𝔽2)11yescharacteristic
281C2×D17C2×H(5,𝔽2)12yes—
282C2×D17C2×2−1+412yes—
283C2×D17#58511yescharacteristic
284#595—22no—
285#595—42no—
286#595—44no—
287#595—41no—
288#595—44no—
289#595—44no—
290C4×C17—24no—
291C4×C17—21no—
292C4×C17—24no—
293C4×C17#52014yes—
294Dic17—24no—
295Dic17—21no—
296Dic17—24no—
297Dic17#52014yes—
298#555—171no—
299#555—174no—
300#555—174no—
301#555—172no—
302#555—174no—
303#555#59511yescharacteristic, complemented
304C2×C4⋊Q8—171no—
305C2×#124—174no—
306C2×#121—174no—
307C2×C4×Q8—172no—
308C2×C4×D4—172no—
309C2×C4×D4—174no—
310C2×C4×D4—174no—
311C2×#117—174no—
312C2×#117—174no—
313C22×C4⋊C4—174no—
314#391—172no—
315#391—174no—
316#390—174no—
317#390—172no—
318#390—172no—
319#389—174no—
320#389—172no—
321#389—174no—
322#389—174no—
323#389—172no—
324#388—178no—
325#388—174no—
326#388—178no—
327#388—174no—
328#386—178no—
329#385—178no—
330#385—178no—
331#385—178no—
332#384—171no—
333#384—174no—
334#382—178no—
335#380—174no—
336C4×C4⋊C4—174no—
337C4×C4⋊C4—174no—
338C4×C4⋊C4—172no—
339C4×C22⋊C4—174no—
340#376—174no—
341C43—171no—
342C2×C17—44no—
343C2×C17C2×#52012yes—
344C2×C17C4×H(5,𝔽2)11yescharacteristic
345D17—44no—
346D17#354411yescharacteristic
347D17#354712yes—
348D17#354911yescharacteristic
349C22×Q8—172no—
350C22×D4—172no—
351C22×D4—174no—
352C23×C4—174no—
353C23×C4—178no—
354C4⋊Q8—172no—
355C4⋊Q8—174no—
356C4⋊Q8—171no—
357C4⋊Q8C2×#59511yescharacteristic, complemented
358#124—174no—
359#124—174no—
360#124—178no—
361#124—178no—
362#124—174no—
363#124C2×#59514yescomplemented
364#121—174no—
365#121—178no—
366#121—174no—
367#121—174no—
368#121—178no—
369#121C2×#59514yescomplemented
370C4×Q8—684no—
371C4×D4—172no—
372C4×D4—174no—
373C4×D4—172no—
374C4×D4—174no—
375C4×D4—172no—
376C4×D4—688no—
377C4×D4—688no—
378C4×D4C2×#59512yescomplemented
379#117—174no—
380#117—174no—
381#117—174no—
382#117—178no—
383#117—178no—
384#117—688no—
385#117C2×#59514yescomplemented
386C2×C4⋊C4—172no—
387C2×C4⋊C4—174no—
388C2×C4⋊C4—172no—
389C2×C4⋊C4—174no—
390C2×C4⋊C4—172no—
391C2×C4⋊C4—178no—
392C2×C4⋊C4—174no—
393C2×C4⋊C4—174no—
394C2×C4⋊C4—174no—
395C2×C4⋊C4—172no—
396C2×C4⋊C4—174no—
397C2×C4⋊C4—688no—
398C2×C22⋊C4—178no—
399C2×C22⋊C4—174no—
400C2×C22⋊C4—174no—
401C2×C22⋊C4—174no—
402C2×C22⋊C4—174no—
403C2×C22⋊C4—178no—
404C2×C22⋊C4—174no—
405C2×C42—174no—
406C2×C42—171no—
407C2×C42—172no—
408C2×C42—174no—
409C2×C42—174no—
410C2×C42—171no—
411C2×C42—172no—
412C2×C42—171no—
413C2×C42—174no—
414#95—174no—
415#95—178no—
416#95—174no—
417#95—178no—
418#95—178no—
419#95—174no—
420C17#3448511yescharacteristic, complemented, Sylow
421C24—174no—
422C2×Q8—174no—
423C2×Q8—172no—
424C2×Q8—682no—
425C2×Q8—682no—
426C2×Q8C22×#59512yes—
427C2×D4—174no—
428C2×D4—178no—
429C2×D4—172no—
430C2×D4—174no—
431C2×D4—344no—
432C2×D4—344no—
433C2×D4—684no—
434C2×D4—684no—
435C2×D4C22×#59512yes—
436C2×D4C22×#59514yes—
437C22×C4—172no—
438C22×C4—171no—
439C22×C4—174no—
440C22×C4—174no—
441C22×C4—174no—
442C22×C4—171no—
443C22×C4—174no—
444C22×C4—174no—
445C22×C4—174no—
446C22×C4—172no—
447C22×C4—174no—
448C22×C4—174no—
449C22×C4—171no—
450C22×C4—348no—
451C22×C4—348no—
452C22×C4—688no—
453C22×C4—688no—
454C22×C4—688no—
455C22×C4C22×#59514yes—
456C4⋊C4—174no—
457C4⋊C4—172no—
458C4⋊C4—178no—
459C4⋊C4—174no—
460C4⋊C4—344no—
461C4⋊C4—344no—
462C4⋊C4—348no—
463C4⋊C4—688no—
464C4⋊C4—684no—
465C4⋊C4—684no—
466C4⋊C4—684no—
467C4⋊C4—684no—
468C4⋊C4—682no—
469C4⋊C4C22×#59512yes—
470C4⋊C4C22×#59514yes—
471C22⋊C4—178no—
472C22⋊C4—174no—
473C22⋊C4—178no—
474C22⋊C4—178no—
475C22⋊C4—174no—
476C22⋊C4—174no—
477C22⋊C4—348no—
478C22⋊C4—684no—
479C22⋊C4—684no—
480C22⋊C4—684no—
481C22⋊C4—688no—
482C22⋊C4C22×#59518yes—
483C22⋊C4C22×#59514yes—
484C42—174no—
485C42—178no—
486C42—171no—
487C42—172no—
488C42—344no—
489C42—344no—
490C42—348no—
491C42—348no—
492C42—342no—
493C42—684no—
494C42—681no—
495C42C22×#59514yes—
496C42C22×#59511yescharacteristic
497C23—171no—
498C23—174no—
499C23—344no—
500C23—344no—
501C23—344no—
502C23—344no—
503C23—344no—
504C23—344no—
505C23C23×#59514yes—
506Q8—42no—
507Q8—684no—
508Q8—682no—
509D4—24no—
510D4—44no—
511D4—348no—
512D4—344no—
513D4—684no—
514D4—688no—
515C2×C4—28no—
516C2×C4—172no—
517C2×C4—174no—
518C2×C4—174no—
519C2×C4—172no—
520C2×C4—174no—
521C2×C4—171no—
522C2×C4—174no—
523C2×C4—348no—
524C2×C4—344no—
525C2×C4—341no—
526C2×C4—344no—
527C2×C4—341no—
528C2×C4—344no—
529C2×C4—344no—
530C2×C4—348no—
531C2×C4—342no—
532C2×C4—348no—
533C2×C4—344no—
534C2×C4—342no—
535C2×C4—342no—
536C2×C4—344no—
537C2×C4—341no—
538C2×C4—344no—
539C2×C4—342no—
540C2×C4—344no—
541C2×C4—342no—
542C2×C4—344no—
543C2×C4—348no—
544C2×C4—344no—
545C2×C4—344no—
546C2×C4—13616no—
547C2×C4C23×#59514yes—
548C2×C4C23×#59511yescharacteristic
549C2×C4C23×#59514yes—
550C2×C4C23×#59512yes—
551C22—24no—
552C22—24no—
553C22—24no—
554C22—172no—
555C22—172no—
556C22—171no—
557C22—171no—
558C22—344no—
559C22—344no—
560C22—344no—
561C22—684no—
562C22—684no—
563C22—684no—
564C22—684no—
565C22C24×#59511yescharacteristic
566C4—24no—
567C4—24no—
568C4—21no—
569C4—174no—
570C4—342no—
571C4—344no—
572C4—344no—
573C4—341no—
574C4—684no—
575C4—684no—
576C4—681no—
577C4—684no—
578C4—682no—
579C4#1176387814yes—
580C2—44no—
581C2—171no—
582C2—171no—
583C2—172no—
584C2—684no—
585C2C2×#1176387812yes—
586C2H(5,𝔽2)×#59511yescharacteristic
subgroupG/H
center, FrattiniC22C24×#595
commutator, socleC22×C17C24×C4
FittingC17×#555C4
prime radicalD17×#555C2
cosocleC22×D17C25
hypercenter#555#595
hypocenterC17#34485
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #14545327C2semidirect
2. #14545317C2semidirect
3. #14545309C2semidirect
4. D4×C4⋊#595C2semidirect
5. #14545294C2semidirect
6. #14545274C2semidirect
7. #14545272C2semidirect
8. #14545265C2semidirect
9. C4⋊Q8×#595C2semidirect
10. #14545241C2semidirect
45. #11763525C4semidirect
46. #11763524C4semidirect
47. #11763523C4semidirect
48. #11763522C4semidirect
49. #11763521C4semidirect
55. C17×#555C4semidirect
87. C17×C4⋊Q8C2×C4semidirect
88. C17×#124C2×C4semidirect
89. C17×#121C2×C4semidirect
90. C4×D4×C17C2×C4semidirect
91. C17×#117C2×C4semidirect
92. #10587633C2×C4semidirect
93. #10587630C2×C4semidirect
94. #10587625C2×C4semidirect
95. #10587624C2×C4semidirect
96. #10587622C2×C4semidirect
97. D4×Dic17C2×C4semidirect
98. #10587616C2×C4semidirect
99. C4×C22⋊D17C2×C4semidirect
100. #10587612C2×C4semidirect
101. #10587581C2×C4semidirect
102. #10587579C2×C4semidirect
103. #10587578C2×C4semidirect
104. #10587578C2×C4semidirect
107. #10587574C2×C4semidirect
108. #10587571C2×C4semidirect
109. #10587571C2×C4semidirect
110. #10587570C2×C4semidirect
111. #10587567C2×C4semidirect
114. #10587563C2×C4semidirect
115. #10587561C2×C4semidirect
116. C4×D68C2×C4semidirect
117. #10587558C2×C4semidirect
120. #10587555C2×C4semidirect
303. #555#595semidirect
357. C4⋊Q8C2×#595semidirect
363. #124C2×#595semidirect
369. #121C2×#595semidirect
378. C4×D4C2×#595semidirect
385. #117C2×#595semidirect
420. C17#34485semidirect

Maximal subgroups

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

subgroupindex
#145453272
#145453172
#145453092
D4×C4⋊#5952
#145452942
#145452742
#145452722
#145452652
C4⋊Q8×#5952
#145452412
D17×#5552
#3448517