Finite Group Databasebeta

← #28329120

Subgroups — #28329120

Order 13568.

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#20291258C214yesmaximal, complemented
2#20291248C211yesmaximal, characteristic, complemented
3#20291240C214yesmaximal, complemented
4D4×C4⋊#6157C212yesmaximal, complemented
5#20291225C214yesmaximal, complemented
6#20291205C214yesmaximal, complemented
7#20291203C214yesmaximal, complemented
8#20291196C214yesmaximal, complemented
9C4⋊Q8×#6157C211yesmaximal, characteristic, complemented
10#20291172C212yesmaximal, complemented
11#555×D53C211yesmaximal, characteristic
12C2×Q8×#6157C2212yes—
13C2×D4×#6157C2214yes—
14C2×D4×#6157C2214yes—
15C2×#13133296C2214yes—
16C22×C4⋊#6157C2214yes—
17(C2×#6157)⋊Q8C2212yes—
18(C2×#6157)⋊Q8C2214yes—
19#16811575C2212yes—
20#16811575C2212yes—
21#16811575C2214yes—
22#16811574C2212yes—
23#16811574C2212yes—
24#16811574C2214yes—
25#16811574C2214yes—
26#16811574C2214yes—
27#16811573C2214yes—
28#16811573C2214yes—
29#16811573C2218yes—
30#16811573C2218yes—
31#16811571C2218yes—
32#16811570C2218yes—
33#16811570C2218yes—
34#16811570C2218yes—
35#16811569C2211yescharacteristic
36#16811569C2214yes—
37#16811567C2218yes—
38#16811565C2214yes—
39C4⋊C4×#6157C2214yes—
40C4⋊C4×#6157C2214yes—
41C4×C4⋊#6157C2212yes—
42C4×C22⋊#6157C2214yes—
43#16811559C2214yes—
44C42×#6157C2211yescharacteristic
45#16811339C411yescharacteristic, complemented
46#16811338C412yescomplemented
47#16811337C414yescomplemented
48#16811336C414yescomplemented
49#16811335C414yescomplemented
50C4⋊Q8×D53C2211yescharacteristic
51#124×D53C2214yes—
52#121×D53C2214yes—
53C4×D4×D53C2212yes—
54#117×D53C2214yes—
55C53×#555C411yescharacteristic, complemented
56C23×#6157C2318yes—
57Q8×#6157—44no—
58D4×#6157—48no—
59D4×#6157—48no—
60#13133296—48no—
61C2×C4⋊#6157—48no—
62C2×C4⋊#6157C2312yes—
63C2×C4⋊#6157C2312yes—
64C2×C4⋊#6157C2314yes—
65C2×C4⋊#6157C2314yes—
66C2×C4⋊#6157C2318yes—
67C2×C4⋊#6157C2314yes—
68C2×C4⋊#6157C2312yes—
69C2×C4⋊#6157C2314yes—
70C2×C4⋊#6157C2314yes—
71C2×C22⋊#6157C2314yes—
72C2×C22⋊#6157C2314yes—
73C2×C22⋊#6157C2318yes—
74C2×C22⋊#6157C2314yes—
75C2×C22⋊#6157C2314yes—
76C2×C4×#6157C2312yes—
77C2×C4×#6157C2311yescharacteristic
78C2×C4×#6157C2312yes—
79C2×C4×#6157C2314yes—
80C2×C4×#6157C2314yes—
81C2×C4×#6157C2311yescharacteristic
82C2×C4×#6157C2314yes—
83(C2×#6157)⋊C4C2314yes—
84(C2×#6157)⋊C4C2318yes—
85(C2×#6157)⋊C4C2314yes—
86(C2×#6157)⋊C4C2314yes—
87(C2×#6157)⋊C4C2318yes—
88(C2×#6157)⋊C4C2318yes—
89C2×Q8×D53C2312yes—
90C2×D4×D53C2312yes—
91C2×D4×D53C2314yes—
92C22×C4×D53C2314yes—
93#13133217C2×C411yescharacteristic, complemented
94#13133216C2×C414yescomplemented
95#13133215C2×C412yescomplemented
96#13133206C2×C414yescomplemented
97#13133205C2×C414yescomplemented
98#13133204C2×C414yescomplemented
99#13133203C2×C418yescomplemented
100#13133203C2×C418yescomplemented
101#13133192C2×C414yescomplemented
102#13133191C2×C414yescomplemented
103#13133190C2×C418yescomplemented
104#13133189C2×C418yescomplemented
105#13133188C2×C414yescomplemented
106C4×D212C2×C412yescomplemented
107D4×Dic53C2×C412yescomplemented
108#13133179C2×C412yescomplemented
109C4×C22⋊D53C2×C414yescomplemented
110#13133177C2×C414yescomplemented
111#13133176C2×C414yescomplemented
112#13133175C2×C414yescomplemented
113#13133175C2×C414yescomplemented
114#13133174C2×C418yescomplemented
115#13133173C2×C418yescomplemented
116C4⋊C4×D53C2314yes—
117C4⋊C4×D53C2312yes—
118C22⋊C4×D53C2318yes—
119C22⋊C4×D53C2314yes—
120C42×D53C2311yescharacteristic
121C42×D53C2314yes—
122C4⋊Q8×C53C2×C411yescharacteristic, complemented
123#124×C53C2×C414yescomplemented
124#121×C53C2×C414yescomplemented
125C4×D4×C53C2×C412yescomplemented
126#117×C53C2×C414yescomplemented
127C23×D53C2414yes—
128C22×#6157—48no—
129C22×#6157—48no—
130C22×#6157—48no—
131C22×#6157C2412yes—
132C22×#6157C2414yes—
133C22×#6157C2414yes—
134C22×#6157C2414yes—
135C22×#6157C2411yescharacteristic
136C22×#6157C2411yescharacteristic
137C2×Q8×C53C22×C412yes—
138C2×D4×C53C22×C412yes—
139C2×D4×C53C22×C414yes—
140C22×C4×C53C22×C414yes—
141C2×C22⋊D53C22×C414yes—
142C2×C22⋊D53C22×C418yes—
143C22×Dic53C22×C414yes—
144Q8×D53—42no—
145Q8×D53—42no—
146D4×D53—24no—
147D4×D53—24no—
148D4×D53—44no—
149D4×D53—44no—
150C2×D212C22×C412yes—
151C2×D212C22×C414yes—
152C2×C4×D53—28no—
153C2×C4×D53—28no—
154C2×C4×D53C22×C414yes—
155C2×C4×D53C22×C414yes—
156C2×C4×D53C2414yes—
157C2×C4×D53C2411yescharacteristic
158C2×C4×D53C2414yes—
159C2×C4×D53C2412yes—
160C2×Dic106C22×C412yes—
161C2×Dic106C22×C414yes—
162C22⋊#6157—28no—
163C22⋊#6157—44no—
164C22⋊#6157—44no—
165C22⋊#6157—44no—
166C22⋊#6157—48no—
167C4⋊#6157—24no—
168C4⋊#6157—28no—
169C4⋊#6157—24no—
170C4⋊#6157—48no—
171C4⋊#6157—44no—
172C4⋊#6157—44no—
173C4⋊#6157—44no—
174C4⋊#6157—44no—
175C4⋊#6157—42no—
176C4×#6157—28no—
177C4×#6157—22no—
178C4×#6157—24no—
179C4×#6157—28no—
180C4×#6157—24no—
181C4×#6157—44no—
182C4×#6157—41no—
183C4⋊C4×C53C22×C412yes—
184C4⋊C4×C53C22×C414yes—
185C22⋊C4×C53C22×C418yes—
186C22⋊C4×C53C22×C414yes—
187C42×C53C22×C414yes—
188C42×C53C22×C411yescharacteristic
189C22⋊Dic53C22×C418yes—
190C22⋊Dic53C22×C414yes—
191#11719921C22×C418yes—
192#11719921C22×C418yes—
193#11719921C22×C414yes—
194#11719921C22×C414yes—
195C4⋊Dic53C22×C412yes—
196C4⋊Dic53C22×C414yes—
197#11719919C22×C414yes—
198#11719919C22×C418yes—
199C4×Dic53C22×C414yes—
200C4×Dic53C22×C418yes—
201C4×Dic53C22×C411yescharacteristic
202C4×Dic53C22×C412yes—
203C23×C53C23×C414yes—
204C22×D53—24no—
205C22×D53—24no—
206C22×D53—24no—
207C22×D53—24no—
208C22×D53—24no—
209C22×D53—24no—
210C22×D53C23×C414yes—
211C22×D53C2511yescharacteristic
212C2×#6157—22no—
213C2×#6157—21no—
214C2×#6157—22no—
215C2×#6157—22no—
216C2×#6157—24no—
217C2×#6157—24no—
218C2×#6157—24no—
219C2×#6157—24no—
220C2×#6157—28no—
221C2×#6157—24no—
222C2×#6157—22no—
223C2×#6157—24no—
224C2×#6157—24no—
225C2×#6157—22no—
226C2×#6157—816no—
227C2×#6157H(5,𝔽2)12yes—
228Q8×C53—42no—
229D4×C53—24no—
230D4×C53—44no—
231C2×C4×C53—28no—
232C2×C4×C53C23×C411yescharacteristic
233C2×C4×C53C23×C414yes—
234C2×C4×C53C23×C414yes—
235C2×C4×C53C23×C412yes—
236C22⋊D53—28no—
237C22⋊D53—48no—
238C2×Dic53—28no—
239C2×Dic53C23×C411yescharacteristic
240C2×Dic53C23×C414yes—
241C2×Dic53C23×C414yes—
242C2×Dic53C23×C412yes—
243D212—24no—
244D212—44no—
245C4×D53—24no—
246C4×D53—21no—
247C4×D53—24no—
248C4×D53—24no—
249C4×D53—24no—
250C4×D53—21no—
251C4×D53—28no—
252C4×D53—28no—
253C4×D53H(5,𝔽2)14yes—
254C4×D532−1+414yes—
255Dic106—44no—
256Dic106—42no—
257#34485—531nomaximal, Sylow
258C22×C53—24no—
259C22×C53—24no—
260C22×C53—24no—
261C22×C53C24×C411yescharacteristic
262C2×D53—24no—
263C2×D53—24no—
264C2×D53—24no—
265C2×D53—44no—
266C2×D53—44no—
267C2×D53—44no—
268C2×D53—44no—
269C2×D53C2×H(5,𝔽2)11yescharacteristic
270C2×D53C2×H(5,𝔽2)12yes—
271C2×D53C2×2−1+412yes—
272C2×D53#58511yescharacteristic
273#6157—22no—
274#6157—42no—
275#6157—44no—
276#6157—41no—
277#6157—44no—
278#6157—44no—
279C4×C53—24no—
280C4×C53—21no—
281C4×C53—24no—
282C4×C53#52014yes—
283Dic53—24no—
284Dic53—21no—
285Dic53—24no—
286Dic53#52014yes—
287C2×#555—531no—
288#2379—534no—
289#2369—531no—
290#2361—534no—
291D4×C4⋊C4—532no—
292#2346—534no—
293#2326—534no—
294#2324—534no—
295#2317—534no—
296C4×C4⋊Q8—531no—
297#2293—532no—
298C2×C53—44no—
299C2×C53C2×#52012yes—
300C2×C53C4×H(5,𝔽2)11yescharacteristic
301D53—44no—
302D53#354411yescharacteristic
303D53#354712yes—
304D53#354911yescharacteristic
305#555—531no—
306#555—534no—
307#555—534no—
308#555—532no—
309#555—534no—
310#555#615711yescharacteristic, complemented
311C2×C4⋊Q8—531no—
312C2×#124—534no—
313C2×#121—534no—
314C2×C4×Q8—532no—
315C2×C4×D4—532no—
316C2×C4×D4—534no—
317C2×C4×D4—534no—
318C2×#117—534no—
319C2×#117—534no—
320C22×C4⋊C4—534no—
321#391—532no—
322#391—534no—
323#390—534no—
324#390—532no—
325#390—532no—
326#389—534no—
327#389—532no—
328#389—534no—
329#389—534no—
330#389—532no—
331#388—538no—
332#388—534no—
333#388—538no—
334#388—534no—
335#386—538no—
336#385—538no—
337#385—538no—
338#385—538no—
339#384—531no—
340#384—534no—
341#382—538no—
342#380—534no—
343C4×C4⋊C4—534no—
344C4×C4⋊C4—534no—
345C4×C4⋊C4—532no—
346C4×C22⋊C4—534no—
347#376—534no—
348C43—531no—
349C53#3448511yescharacteristic, complemented, Sylow
350C22×Q8—532no—
351C22×D4—532no—
352C22×D4—534no—
353C23×C4—534no—
354C23×C4—538no—
355C4⋊Q8—532no—
356C4⋊Q8—534no—
357C4⋊Q8—531no—
358C4⋊Q8C2×#615711yescharacteristic, complemented
359#124—534no—
360#124—534no—
361#124—538no—
362#124—538no—
363#124—534no—
364#124C2×#615714yescomplemented
365#121—534no—
366#121—538no—
367#121—534no—
368#121—534no—
369#121—538no—
370#121C2×#615714yescomplemented
371C4×Q8—2124no—
372C4×D4—532no—
373C4×D4—534no—
374C4×D4—532no—
375C4×D4—534no—
376C4×D4—532no—
377C4×D4—2128no—
378C4×D4—2128no—
379C4×D4C2×#615712yescomplemented
380#117—534no—
381#117—534no—
382#117—534no—
383#117—538no—
384#117—538no—
385#117—2128no—
386#117C2×#615714yescomplemented
387C2×C4⋊C4—532no—
388C2×C4⋊C4—534no—
389C2×C4⋊C4—532no—
390C2×C4⋊C4—534no—
391C2×C4⋊C4—532no—
392C2×C4⋊C4—538no—
393C2×C4⋊C4—534no—
394C2×C4⋊C4—534no—
395C2×C4⋊C4—534no—
396C2×C4⋊C4—532no—
397C2×C4⋊C4—534no—
398C2×C4⋊C4—2128no—
399C2×C22⋊C4—538no—
400C2×C22⋊C4—534no—
401C2×C22⋊C4—534no—
402C2×C22⋊C4—534no—
403C2×C22⋊C4—534no—
404C2×C22⋊C4—538no—
405C2×C22⋊C4—534no—
406C2×C42—534no—
407C2×C42—531no—
408C2×C42—532no—
409C2×C42—534no—
410C2×C42—534no—
411C2×C42—531no—
412C2×C42—532no—
413C2×C42—531no—
414C2×C42—534no—
415#95—534no—
416#95—538no—
417#95—534no—
418#95—538no—
419#95—538no—
420#95—534no—
421C24—534no—
422C2×Q8—534no—
423C2×Q8—532no—
424C2×Q8—2122no—
425C2×Q8—2122no—
426C2×Q8C22×#615712yes—
427C2×D4—534no—
428C2×D4—538no—
429C2×D4—532no—
430C2×D4—534no—
431C2×D4—1064no—
432C2×D4—1064no—
433C2×D4—2124no—
434C2×D4—2124no—
435C2×D4C22×#615712yes—
436C2×D4C22×#615714yes—
437C22×C4—532no—
438C22×C4—531no—
439C22×C4—534no—
440C22×C4—534no—
441C22×C4—534no—
442C22×C4—531no—
443C22×C4—534no—
444C22×C4—534no—
445C22×C4—534no—
446C22×C4—532no—
447C22×C4—534no—
448C22×C4—534no—
449C22×C4—531no—
450C22×C4—1068no—
451C22×C4—1068no—
452C22×C4—2128no—
453C22×C4—2128no—
454C22×C4—2128no—
455C22×C4C22×#615714yes—
456C4⋊C4—534no—
457C4⋊C4—532no—
458C4⋊C4—538no—
459C4⋊C4—534no—
460C4⋊C4—1064no—
461C4⋊C4—1064no—
462C4⋊C4—1068no—
463C4⋊C4—2128no—
464C4⋊C4—2124no—
465C4⋊C4—2124no—
466C4⋊C4—2124no—
467C4⋊C4—2124no—
468C4⋊C4—2122no—
469C4⋊C4C22×#615712yes—
470C4⋊C4C22×#615714yes—
471C22⋊C4—538no—
472C22⋊C4—534no—
473C22⋊C4—538no—
474C22⋊C4—538no—
475C22⋊C4—534no—
476C22⋊C4—534no—
477C22⋊C4—1068no—
478C22⋊C4—2124no—
479C22⋊C4—2124no—
480C22⋊C4—2124no—
481C22⋊C4—2128no—
482C22⋊C4C22×#615718yes—
483C22⋊C4C22×#615714yes—
484C42—534no—
485C42—538no—
486C42—531no—
487C42—532no—
488C42—1064no—
489C42—1064no—
490C42—1068no—
491C42—1068no—
492C42—1062no—
493C42—2124no—
494C42—2121no—
495C42C22×#615714yes—
496C42C22×#615711yescharacteristic
497C23—531no—
498C23—534no—
499C23—1064no—
500C23—1064no—
501C23—1064no—
502C23—1064no—
503C23—1064no—
504C23—1064no—
505C23C23×#615714yes—
506Q8—42no—
507Q8—2124no—
508Q8—2122no—
509D4—24no—
510D4—44no—
511D4—1068no—
512D4—1064no—
513D4—2124no—
514D4—2128no—
515C2×C4—28no—
516C2×C4—532no—
517C2×C4—534no—
518C2×C4—534no—
519C2×C4—532no—
520C2×C4—534no—
521C2×C4—531no—
522C2×C4—534no—
523C2×C4—1068no—
524C2×C4—1064no—
525C2×C4—1061no—
526C2×C4—1064no—
527C2×C4—1061no—
528C2×C4—1064no—
529C2×C4—1064no—
530C2×C4—1068no—
531C2×C4—1062no—
532C2×C4—1068no—
533C2×C4—1064no—
534C2×C4—1062no—
535C2×C4—1062no—
536C2×C4—1064no—
537C2×C4—1061no—
538C2×C4—1064no—
539C2×C4—1062no—
540C2×C4—1064no—
541C2×C4—1062no—
542C2×C4—1064no—
543C2×C4—1068no—
544C2×C4—1064no—
545C2×C4—1064no—
546C2×C4—42416no—
547C2×C4C23×#615714yes—
548C2×C4C23×#615711yescharacteristic
549C2×C4C23×#615714yes—
550C2×C4C23×#615712yes—
551C22—24no—
552C22—24no—
553C22—24no—
554C22—532no—
555C22—532no—
556C22—531no—
557C22—531no—
558C22—1064no—
559C22—1064no—
560C22—1064no—
561C22—2124no—
562C22—2124no—
563C22—2124no—
564C22—2124no—
565C22C24×#615711yescharacteristic
566C4—24no—
567C4—24no—
568C4—21no—
569C4—534no—
570C4—1062no—
571C4—1064no—
572C4—1064no—
573C4—1061no—
574C4—2124no—
575C4—2124no—
576C4—2121no—
577C4—2124no—
578C4—2122no—
579C4#1681169214yes—
580C2—44no—
581C2—531no—
582C2—531no—
583C2—532no—
584C2—2124no—
585C2C2×#1681169212yes—
586C2H(5,𝔽2)×#615711yescharacteristic
subgroupG/H
center, FrattiniC22C24×#6157
commutator, socleC22×C53C24×C4
FittingC53×#555C4
prime radical#555×D53C2
cosocleC22×D53C25
hypercenter#555#6157
hypocenterC53#34485
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #20291258C2semidirect
2. #20291248C2semidirect
3. #20291240C2semidirect
4. D4×C4⋊#6157C2semidirect
5. #20291225C2semidirect
6. #20291205C2semidirect
7. #20291203C2semidirect
8. #20291196C2semidirect
9. C4⋊Q8×#6157C2semidirect
10. #20291172C2semidirect
45. #16811339C4semidirect
46. #16811338C4semidirect
47. #16811337C4semidirect
48. #16811336C4semidirect
49. #16811335C4semidirect
55. C53×#555C4semidirect
93. #13133217C2×C4semidirect
94. #13133216C2×C4semidirect
95. #13133215C2×C4semidirect
96. #13133206C2×C4semidirect
97. #13133205C2×C4semidirect
98. #13133204C2×C4semidirect
99. #13133203C2×C4semidirect
100. #13133203C2×C4semidirect
101. #13133192C2×C4semidirect
102. #13133191C2×C4semidirect
103. #13133190C2×C4semidirect
104. #13133189C2×C4semidirect
105. #13133188C2×C4semidirect
106. C4×D212C2×C4semidirect
107. D4×Dic53C2×C4semidirect
108. #13133179C2×C4semidirect
109. C4×C22⋊D53C2×C4semidirect
110. #13133177C2×C4semidirect
111. #13133176C2×C4semidirect
112. #13133175C2×C4semidirect
113. #13133175C2×C4semidirect
114. #13133174C2×C4semidirect
115. #13133173C2×C4semidirect
122. C4⋊Q8×C53C2×C4semidirect
123. #124×C53C2×C4semidirect
124. #121×C53C2×C4semidirect
125. C4×D4×C53C2×C4semidirect
126. #117×C53C2×C4semidirect
310. #555#6157semidirect
349. C53#34485semidirect
358. C4⋊Q8C2×#6157semidirect
364. #124C2×#6157semidirect
370. #121C2×#6157semidirect
379. C4×D4C2×#6157semidirect
386. #117C2×#6157semidirect

Maximal subgroups

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

subgroupindex
#202912582
#202912482
#202912402
D4×C4⋊#61572
#202912252
#202912052
#202912032
#202911962
C4⋊Q8×#61572
#202911722
#555×D532
#3448553