Finite Group Databasebeta

← #25135535

Subgroups — #25135535

Order 11008.

874 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19144535C211yesmaximal, characteristic
2C2×#14583038C211yesmaximal, characteristic
3#19143253C211yesmaximal, characteristic, complemented
4#19135962C212yesmaximal, complemented
5#19135855C212yesmaximal, complemented
6#19135497C212yesmaximal, complemented
7#19135253C212yesmaximal, complemented
8#19135058C211yesmaximal, characteristic
9#19134800C211yesmaximal, characteristic, complemented
10#19134519C211yesmaximal, characteristic, complemented
11C2×C43×#419C211yesmaximal, characteristic, complemented
12C22×#10621884C2212yes—
13C2×#13063299C2211yescharacteristic
14#14583223—21no—
15#14583223—21no—
16#14583217—21no—
17#14583212—21no—
18#14583038—21no—
19#14583038—21no—
20#14583038—22no—
21#14582960—22no—
22#14582956—21no—
23#14582952—21no—
24C2×#13063260C2212yes—
25#14582942C2211yescharacteristic
26#14582942C2211yescharacteristic
27#14582926C2212yes—
28#14582903C2211yescharacteristic
29#14582903C2211yescharacteristic
30C2×#13063249C2212yes—
31C2×#13063244C2212yes—
32C2×#13063244C2212yes—
33C2×#13063241C2212yes—
34#14582863C2211yescharacteristic
35#14582863C2211yescharacteristic
36C2×C22⋊(C43⋊C8)C2211yescharacteristic
37#14582855C2212yes—
38#14582667C2212yes—
39#14582663C2212yes—
40C23×Q8×C43C2211yescharacteristic
41C22×M16×C43C2211yescharacteristic
42C2×#117×C43C2211yescharacteristic
43C43×#419—21no—
44C43×#419—21no—
45C43×#419C2212yes—
46C43×#419C2212yes—
47C2×#103×C43—21no—
48C2×#103×C43—21no—
49C22×D172C2311yescharacteristic
50C22×C22⋊D43—21no—
51C2×#10621884—24no—
52C2×#10621884—22no—
53C2×#10621884—22no—
54C2×#10621884—42no—
55C2×#10621877C2312yes—
56C2×#10621877C2311yescharacteristic
57C2×#10621877C2311yescharacteristic
58C22×C43⋊C8C2312yes—
59#13063299—21no—
60#13063299—21no—
61#13063299—21no—
62#13063299—21no—
63C2×C4⋊Dic43C2311yescharacteristic
64C2×(C2×D43)⋊C4—21no—
65#13063260—42no—
66#13063258—21no—
67#13063258—21no—
68#13063256—21no—
69#13063256—21no—
70C4⋊(C43⋊C8)—22no—
71C4⋊(C43⋊C8)—22no—
72#13063249—22no—
73#13063249—22no—
74#13063245—21no—
75#13063245—21no—
76#13063245—21no—
77#13063245—21no—
78#13063244—24no—
79#13063244—42no—
80#13063241—42no—
81C22⋊(C43⋊C8)—21no—
82C22⋊(C43⋊C8)—21no—
83C22⋊(C43⋊C8)—21no—
84C22⋊(C43⋊C8)—21no—
85C22⋊(C43⋊C8)D411yescharacteristic
86C22⋊(C43⋊C8)D411yescharacteristic
87C22⋊(C43⋊C8)D412yes—
88#13063236—22no—
89C4×C43⋊C8—22no—
90C22×Q8×C43—21no—
91C22×Q8×C43—24no—
92C22×Q8×C43—21no—
93C22×Q8×C43—21no—
94C22×Q8×C43—21no—
95C22×Q8×C43—21no—
96C22×Q8×C43—42no—
97C22×Q8×C43C2312yes—
98C23×C4×C43—21no—
99C23×C4×C43C2311yescharacteristic
100C2×M16×C43—21no—
101C2×M16×C43—42no—
102C2×M16×C43C2312yes—
103C22×C8×C43—21no—
104#117×C43—21no—
105#117×C43—21no—
106#117×C43D411yescharacteristic
107#117×C43D411yescharacteristic
108#117×C43C2311yescharacteristic
109#117×C43C2311yescharacteristic
110C2×C4⋊C4×C43D411yescharacteristic
111C2×C4⋊C4×C43D411yescharacteristic
112C2×C22⋊C4×C43—21no—
113C2×C42×C43—21no—
114#103×C43—22no—
115#103×C43—22no—
116#103×C43—22no—
117#103×C43—22no—
118C24×C43#4111yescharacteristic
119C23×D43—21no—
120C2×Q8×C43—22no—
121C2×Q8×C43—24no—
122C2×Q8×C43—24no—
123C2×Q8×C43—22no—
124C2×Q8×C43—22no—
125C2×Q8×C43—22no—
126C2×Q8×C43—22no—
127C2×Q8×C43—44no—
128C2×Q8×C43—44no—
129C2×Q8×C43—44no—
130C2×Q8×C43—42no—
131C2×Q8×C43—42no—
132C2×Q8×C43—41no—
133C2×Q8×C43—41no—
134C2×Q8×C43#4114yes—
135C22×C4×C43—22no—
136C22×C4×C43—22no—
137C22×C4×C43—21no—
138C22×C4×C43—21no—
139C22×C4×C43—22no—
140C22×C4×C43—44no—
141C22×C4×C43—41no—
142C22×C4×C43—41no—
143C22×C4×C43C2×D411yescharacteristic
144C22×C4×C43C2×D411yescharacteristic
145C22×C4×C43C2×D411yescharacteristic
146C22×C4×C43C2×D411yescharacteristic
147C22×C4×C43#4111yescharacteristic
148C22×C4×C43C2411yescharacteristic
149C2×C22⋊D43—41no—
150C2×C22⋊D43—41no—
151C2×C22⋊D43—82no—
152C22×Dic43—21no—
153C2×D172—21no—
154C2×D172—21no—
155C2×D172—22no—
156C2×D172—22no—
157M16×C43—22no—
158M16×C43—22no—
159M16×C43—42no—
160C2×C8×C43—22no—
161C2×C8×C43—42no—
162C4⋊C4×C43—21no—
163C4⋊C4×C43—21no—
164C4⋊C4×C43C2×D411yescharacteristic
165C4⋊C4×C43C2×D411yescharacteristic
166C4⋊C4×C43C2×D411yescharacteristic
167C4⋊C4×C43C2×D411yescharacteristic
168C22⋊C4×C43—21no—
169C22⋊C4×C43—21no—
170C22⋊C4×C43—21no—
171C22⋊C4×C43—21no—
172C42×C43—21no—
173C42×C43—21no—
174C42×C43—21no—
175C42×C43—21no—
176#10621884—44no—
177#10621884—44no—
178(C2×D43)⋊C4—41no—
179(C2×D43)⋊C4—41no—
180C4⋊Dic43—21no—
181C4⋊Dic43—21no—
182#10621877—22no—
183#10621877—22no—
184#10621877—22no—
185#10621877—22no—
186C2×C43⋊C8—22no—
187C2×C43⋊C8—22no—
188C2×C43⋊C8—22no—
189C2×C43⋊C8—21no—
190C2×C43⋊C8—21no—
191C2×C43⋊C8C2×D412yes—
192C2×C43⋊C8C2×D412yes—
193C23×C43—22no—
194C23×C43—22no—
195C23×C43—41no—
196C23×C43#12111yescharacteristic
197C23×C43#12111yescharacteristic
198C23×C43C2×#4111yescharacteristic
199C22×D43—41no—
200C22×D43—42no—
201C22×D43—81no—
202C22×D43—81no—
203Q8×C43—24no—
204Q8×C43—24no—
205Q8×C43—44no—
206Q8×C43—44no—
207Q8×C43—42no—
208Q8×C43—42no—
209C2×C4×C43—22no—
210C2×C4×C43—21no—
211C2×C4×C43—21no—
212C2×C4×C43—21no—
213C2×C4×C43—21no—
214C2×C4×C43—22no—
215C2×C4×C43—22no—
216C2×C4×C43—21no—
217C2×C4×C43—21no—
218C2×C4×C43—21no—
219C2×C4×C43—21no—
220C2×C4×C43—24no—
221C2×C4×C43—44no—
222C2×C4×C43—44no—
223C2×C4×C43—42no—
224C2×C4×C43—42no—
225C2×C4×C43—42no—
226C2×C4×C43—81no—
227C2×C4×C43#12111yescharacteristic
228C2×C4×C43#12111yescharacteristic
229C2×C4×C43C22×D411yescharacteristic
230C2×C4×C43C22×D411yescharacteristic
231C2×C4×C43C2×#4112yes—
232C2×C4×C43H(5,𝔽2)12yes—
233C22⋊D43—82no—
234C22⋊D43—82no—
235C2×Dic43—41no—
236C2×Dic43—81no—
237D172—41no—
238D172—41no—
239D172—42no—
240C8×C43—42no—
241C43⋊C8—22no—
242C43⋊C8—22no—
243C43⋊C8—41no—
244C43⋊C8—41no—
245C43⋊C8—42no—
246#24523—431nomaximal, Sylow
247C22×C43—21no—
248C22×C43—21no—
249C22×C43—21no—
250C22×C43—21no—
251C22×C43—21no—
252C22×C43—21no—
253C22×C43—21no—
254C22×C43—44no—
255C22×C43#47112yes—
256C22×C43C2×#12111yescharacteristic
257C22×C43#55112yes—
258C2×D43—81no—
259C2×D43—81no—
260C2×D43—81no—
261C2×D43—162no—
262C4×C43—21no—
263C4×C43—21no—
264C4×C43—44no—
265C4×C43—41no—
266C4×C43—41no—
267C4×C43—41no—
268C4×C43—41no—
269C4×C43#52511yescharacteristic
270C4×C43#53511yescharacteristic
271C4×C43#54612yes—
272Dic43—81no—
273#3094—431no—
274C2×#448—431no—
275#2924—431no—
276C2×#419D4311yescharacteristic, complemented
277#2064—432no—
278#2046—432no—
279#1977—432no—
280#1930—432no—
281#1900—431no—
282#1855—431no—
283#1784—431no—
284C2×C43—22no—
285C2×C43—22no—
286C2×C43—41no—
287C2×C43#241111yescharacteristic
288C2×C43C2×#47111yescharacteristic
289C2×C43#306911yescharacteristic
290D43—161no—
291C23×Q8C2×D4311yescharacteristic
292C22×SD16—432no—
293C22×M16C2×D4311yescharacteristic
294C2×#121—431no—
295C2×#117C2×D4311yescharacteristic
296#482—861no—
297#482—861no—
298#481—861no—
299#480—861no—
300#448—861no—
301#448—861no—
302#448—862no—
303#433—862no—
304#432—861no—
305#431—861no—
306C2×#108—432no—
307#428—431no—
308#428—431no—
309#424—432no—
310#419—21no—
311#419—21no—
312#419C2×D4312yes—
313#419C2×D4312yes—
314#418—431no—
315#418—431no—
316C2×#103—21no—
317C2×#103—21no—
318C2×#103—432no—
319C2×#101—432no—
320C2×#101—432no—
321C2×#100—432no—
322#407—431no—
323#407—431no—
324C2×C22⋊C8—431no—
325#405—432no—
326#343—432no—
327#341—432no—
328C43#2452311yescharacteristic, complemented, Sylow
329C22×Q8—21no—
330C22×Q8—24no—
331C22×Q8—21no—
332C22×Q8—21no—
333C22×Q8—21no—
334C22×Q8—21no—
335C22×Q8—42no—
336C22×Q8C22×D4312yes—
337C22×D4—431no—
338C22×D4—861no—
339C23×C4—21no—
340C23×C4C22×D4311yescharacteristic
341C2×SD16—862no—
342C2×SD16—862no—
343C2×SD16—864no—
344C2×SD16—1722no—
345C2×M16—21no—
346C2×M16—42no—
347C2×M16—431no—
348C2×M16—432no—
349C2×M16—431no—
350C2×M16C22×D4312yes—
351C22×C8—21no—
352C22×C8—432no—
353#121—861no—
354#121—861no—
355#121—861no—
356#121—861no—
357#117—21no—
358#117—21no—
359#117C22⋊D4311yescharacteristic
360#117C22⋊D4311yescharacteristic
361#117C22×D4311yescharacteristic
362#117C22×D4311yescharacteristic
363C2×C4⋊C4—431no—
364C2×C4⋊C4C22⋊D4311yescharacteristic
365C2×C4⋊C4C22⋊D4311yescharacteristic
366C2×C22⋊C4—21no—
367C2×C22⋊C4—861no—
368C2×C42—21no—
369#108—1722no—
370#107—861no—
371#107—861no—
372#106—861no—
373#106—861no—
374C4⋊C8—862no—
375C4⋊C8—862no—
376#103—22no—
377#103—22no—
378#103—22no—
379#103—22no—
380#103—862no—
381#103—862no—
382#102—861no—
383#102—861no—
384#102—861no—
385#102—861no—
386#101—864no—
387#101—1722no—
388#100—1722no—
389C22⋊C8—431no—
390C22⋊C8—431no—
391C22⋊C8—432no—
392C22⋊C8—861no—
393C22⋊C8—861no—
394C22⋊C8—861no—
395C22⋊C8—861no—
396#97—862no—
397C4×C8—862no—
398C24—861no—
399C24#1062190011yescharacteristic
400C2×Q8—22no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8—41no—
413C2×Q8—41no—
414C2×Q8#1062189814yes—
415C2×D4—861no—
416C2×D4—861no—
417C2×D4—862no—
418C2×D4—862no—
419C2×D4—3442no—
420C2×D4—1721no—
421C2×D4—1721no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—44no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—861no—
431C22×C4D4×D4311yescharacteristic
432C22×C4D4×D4311yescharacteristic
433C22×C4#1062190011yescharacteristic
434C22×C4C2×C22⋊D4311yescharacteristic
435C22×C4C2×C22⋊D4311yescharacteristic
436C22×C4C23×D4311yescharacteristic
437SD16—1724no—
438SD16—1724no—
439M16—22no—
440M16—22no—
441M16—42no—
442M16—862no—
443M16—862no—
444M16—862no—
445M16—862no—
446C2×C8—22no—
447C2×C8—42no—
448C2×C8—432no—
449C2×C8—432no—
450C2×C8—862no—
451C2×C8—862no—
452C2×C8—862no—
453C2×C8—861no—
454C2×C8—861no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—861no—
458C4⋊C4—861no—
459C4⋊C4C2×C22⋊D4311yescharacteristic
460C4⋊C4C2×C22⋊D4311yescharacteristic
461C4⋊C4C2×C22⋊D4311yescharacteristic
462C4⋊C4C2×C22⋊D4311yescharacteristic
463C22⋊C4—21no—
464C22⋊C4—21no—
465C22⋊C4—21no—
466C22⋊C4—21no—
467C22⋊C4—1721no—
468C22⋊C4—1721no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C23—22no—
474C23—22no—
475C23—41no—
476C23—3441no—
477C23—3441no—
478C23—1721no—
479C23—1722no—
480C23#1306329711yescharacteristic
481C23#1306329711yescharacteristic
482C23C2×#1062190011yescharacteristic
483Q8—24no—
484Q8—24no—
485Q8—44no—
486Q8—44no—
487Q8—42no—
488Q8—42no—
489D4—3442no—
490D4—3442no—
491D4—1721no—
492D4—1721no—
493D4—1722no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—81no—
512C2×C4—3441no—
513C2×C4—1721no—
514C2×C4#1306329711yescharacteristic
515C2×C4#1306329711yescharacteristic
516C2×C4C2×D4×D4311yescharacteristic
517C2×C4C22×C22⋊D4311yescharacteristic
518C2×C4C2×#1062189812yes—
519C2×C4#1306337012yes—
520C8—42no—
521C8—862no—
522C8—862no—
523C8—1721no—
524C8—1721no—
525C8—1722no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—44no—
534C22—3441no—
535C22—3441no—
536C22—3441no—
537C22—6882no—
538C22#1458316412yes—
539C22C2×#1306329711yescharacteristic
540C22#1458357512yes—
541C4—21no—
542C4—21no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—41no—
547C4—44no—
548C4—3441no—
549C4#1458341011yescharacteristic
550C4#1458346611yescharacteristic
551C4#1458353912yes—
552C2—22no—
553C2—22no—
554C2—41no—
555C2—6881no—
556C2#1913798011yescharacteristic
557C2C2×#1458316411yescharacteristic
558C2#1914432911yescharacteristic
subgroupG/H
centerC22C2×#13063297
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC2×C43×#419C2
socleC22×C43C2×#121
prime radicalC2×#13063299C22
hypercenterC2×#419D43
hypocenterC43#24523
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
3. #19143253C2semidirect
4. #19135962C2semidirect
5. #19135855C2semidirect
6. #19135497C2semidirect
7. #19135253C2semidirect
9. #19134800C2semidirect
10. #19134519C2semidirect
11. C2×C43×#419C2semidirect
276. C2×#419D43semidirect
328. C43#24523semidirect

Maximal subgroups

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

subgroupindex
#191445352
C2×#145830382
#191432532
#191359622
#191358552
#191354972
#191352532
#191350582
#191348002
#191345192
C2×C43×#4192
#2452343