Finite Group Databasebeta

← C53×#13422 #27220934

Subgroups — C53×#13422 #27220934

Order 13568.

734 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#2669C211yesmaximal, characteristic
2C2×C53×#353C211yesmaximal, characteristic, complemented
3C2×C53×AΣL(1,𝔽16)C211yesmaximal, characteristic, complemented
4C53×#1889C211yesmaximal, characteristic, complemented
5C53×#1887C211yesmaximal, characteristic, complemented
6C53×#1867C211yesmaximal, characteristic, complemented
7C53×#1390C211yesmaximal, characteristic, complemented
8C2×#127×C53C2211yescharacteristic
9C2×#121×C53—21no—
10C2×#120×C53—21no—
11C2×#120×C53C2211yescharacteristic
12C53×#518C2211yescharacteristic
13C53×#418—21no—
14C2×#100×C53C2211yescharacteristic
15C2×#99×C53C2211yescharacteristic
16C2×#99×C53C2211yescharacteristic
17C53×#394—21no—
18C53×#392—21no—
19C53×#392—21no—
20C53×#388—21no—
21C53×#386C2211yescharacteristic
22C53×#385—21no—
23C53×#379—21no—
24C53×#353—22no—
25C53×AΣL(1,𝔽16)—22no—
26C53×#342—21no—
27C53×#342—21no—
28C53×#339—21no—
29C25×C53D411yescharacteristic
30C22×D4×C53—21no—
31C22×D4×C53—21no—
32C22×D4×C53—21no—
33C22×D4×C53—21no—
34C22×D4×C53—21no—
35C22×D4×C53D411yescharacteristic
36C22×D4×C53C2311yescharacteristic
37C23×C4×C53D411yescharacteristic
38C2×M16×C53—21no—
39#127×C53—21no—
40#127×C53—21no—
41#127×C53—21no—
42#127×C53—21no—
43#121×C53—44no—
44#120×C53—21no—
45#120×C53—21no—
46#120×C53—21no—
47#120×C53—21no—
48#120×C53—44no—
49C4×D4×C53—41no—
50C4×D4×C53—41no—
51#117×C53—21no—
52C2×C4⋊C4×C53—21no—
53C2×C4⋊C4×C53—41no—
54C2×C22⋊C4×C53—21no—
55C2×C22⋊C4×C53—21no—
56C2×C22⋊C4×C53—21no—
57C2×C22⋊C4×C53—21no—
58C2×C22⋊C4×C53—21no—
59C2×C22⋊C4×C53—21no—
60C2×C22⋊C4×C53—21no—
61C2×C22⋊C4×C53—21no—
62C2×C22⋊C4×C53—21no—
63C2×C22⋊C4×C53—21no—
64C2×C22⋊C4×C53—41no—
65C2×C22⋊C4×C53—41no—
66C2×C22⋊C4×C53D411yescharacteristic
67C2×C22⋊C4×C53D411yescharacteristic
68C2×C42×C53D411yescharacteristic
69#102×C53—41no—
70#102×C53—41no—
71#100×C53—21no—
72#100×C53—21no—
73#99×C53—21no—
74#99×C53—21no—
75#99×C53—21no—
76#99×C53D412yescomplemented
77#95×C53—21no—
78#95×C53—21no—
79#95×C53—41no—
80C24×C53—21no—
81C24×C53—21no—
82C24×C53—21no—
83C24×C53—21no—
84C24×C53—41no—
85C24×C53—42no—
86C24×C53—41no—
87C24×C53—81no—
88C24×C53C2×D411yescharacteristic
89C24×C53C2×D411yescharacteristic
90C2×D4×C53—21no—
91C2×D4×C53—21no—
92C2×D4×C53—41no—
93C2×D4×C53—41no—
94C2×D4×C53—41no—
95C2×D4×C53—41no—
96C2×D4×C53—42no—
97C2×D4×C53—42no—
98C2×D4×C53—42no—
99C2×D4×C53—42no—
100C2×D4×C53—42no—
101C2×D4×C53—42no—
102C2×D4×C53—41no—
103C2×D4×C53—41no—
104C2×D4×C53—41no—
105C2×D4×C53—41no—
106C2×D4×C53—41no—
107C2×D4×C53—41no—
108C2×D4×C53—81no—
109C2×D4×C53—81no—
110C2×D4×C53—81no—
111C2×D4×C53—81no—
112C2×D4×C53—81no—
113C2×D4×C53—81no—
114C2×D4×C53C2×D411yescharacteristic
115C2×D4×C53#4111yescharacteristic
116C2×D4×C53#4111yescharacteristic
117C2×D4×C53#4111yescharacteristic
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—42no—
127C22×C4×C53—41no—
128C22×C4×C53—41no—
129C22×C4×C53—41no—
130C22×C4×C53C2×D411yescharacteristic
131M16×C53—41no—
132M16×C53—41no—
133C2×C8×C53—41no—
134C4⋊C4×C53—21no—
135C4⋊C4×C53—21no—
136C4⋊C4×C53—42no—
137C4⋊C4×C53—82no—
138C22⋊C4×C53—24no—
139C22⋊C4×C53—22no—
140C22⋊C4×C53—21no—
141C22⋊C4×C53—21no—
142C22⋊C4×C53—42no—
143C22⋊C4×C53—42no—
144C22⋊C4×C53—42no—
145C22⋊C4×C53—41no—
146C22⋊C4×C53—41no—
147C22⋊C4×C53—41no—
148C22⋊C4×C53—41no—
149C22⋊C4×C53—41no—
150C22⋊C4×C53—41no—
151C22⋊C4×C53—41no—
152C22⋊C4×C53—41no—
153C22⋊C4×C53—41no—
154C22⋊C4×C53—81no—
155C22⋊C4×C53—82no—
156C22⋊C4×C53—82no—
157C42×C53—21no—
158C42×C53—21no—
159C42×C53—41no—
160C23×C53—21no—
161C23×C53—21no—
162C23×C53—21no—
163C23×C53—21no—
164C23×C53—22no—
165C23×C53—41no—
166C23×C53—41no—
167C23×C53—41no—
168C23×C53—41no—
169C23×C53—41no—
170C23×C53—41no—
171C23×C53—41no—
172C23×C53—42no—
173C23×C53—41no—
174C23×C53—41no—
175C23×C53—41no—
176C23×C53—41no—
177C23×C53—42no—
178C23×C53—42no—
179C23×C53—41no—
180C23×C53—41no—
181C23×C53—41no—
182C23×C53—41no—
183C23×C53—41no—
184C23×C53—41no—
185C23×C53—81no—
186C23×C53—82no—
187C23×C53—81no—
188C23×C53—82no—
189C23×C53—84no—
190C23×C53—81no—
191C23×C53#12011yescharacteristic
192C23×C53#12111yescharacteristic
193C23×C53#12111yescharacteristic
194C23×C53#12311yescharacteristic
195C23×C53#12411yescharacteristic
196D4×C53—41no—
197D4×C53—41no—
198D4×C53—41no—
199D4×C53—41no—
200D4×C53—42no—
201D4×C53—81no—
202D4×C53—82no—
203D4×C53—82no—
204D4×C53—82no—
205D4×C53—82no—
206D4×C53—84no—
207D4×C53—82no—
208D4×C53—82no—
209D4×C53—84no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—42no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—42no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—81no—
229C2×C4×C53—81no—
230C2×C4×C53—81no—
231C2×C4×C53—81no—
232C2×C4×C53—81no—
233C2×C4×C53—81no—
234C2×C4×C53—81no—
235C2×C4×C53—81no—
236C2×C4×C53—81no—
237C2×C4×C53—81no—
238C2×C4×C53#12111yescharacteristic
239C2×C4×C53#12311yescharacteristic
240C8×C53—81no—
241#13422C5311yesmaximal, characteristic, direct product factor, Sylow
242C22×C53—21no—
243C22×C53—21no—
244C22×C53—21no—
245C22×C53—21no—
246C22×C53—21no—
247C22×C53—21no—
248C22×C53—41no—
249C22×C53—41no—
250C22×C53—41no—
251C22×C53—41no—
252C22×C53—42no—
253C22×C53—42no—
254C22×C53—41no—
255C22×C53—41no—
256C22×C53—41no—
257C22×C53—41no—
258C22×C53—44no—
259C22×C53—42no—
260C22×C53—81no—
261C22×C53—81no—
262C22×C53—81no—
263C22×C53—81no—
264C22×C53—81no—
265C22×C53—81no—
266C22×C53—81no—
267C22×C53—81no—
268C22×C53—82no—
269C22×C53—82no—
270C22×C53—82no—
271C22×C53—81no—
272C22×C53—81no—
273C22×C53—81no—
274C22×C53—81no—
275C22×C53—81no—
276C22×C53#39411yescharacteristic
277C22×C53B(4,𝔽2)11yescharacteristic
278C22×C53#45811yescharacteristic
279C4×C53—21no—
280C4×C53—21no—
281C4×C53—42no—
282C4×C53—42no—
283C4×C53—81no—
284C4×C53—81no—
285C4×C53—82no—
286C4×C53—81no—
287C4×C53—81no—
288C4×C53—161no—
289#2669C2×C5311yescharacteristic
290C2×#353C2×C5311yescharacteristic, complemented
291C2×AΣL(1,𝔽16)C2×C5311yescharacteristic, complemented
292#1889C2×C5311yescharacteristic, complemented
293#1887C2×C5311yescharacteristic, complemented
294#1867C2×C5311yescharacteristic, complemented
295#1390C2×C5311yescharacteristic, complemented
296C2×C53—21no—
297C2×C53—21no—
298C2×C53—41no—
299C2×C53—41no—
300C2×C53—41no—
301C2×C53—41no—
302C2×C53—82no—
303C2×C53—161no—
304C2×C53#204411yescharacteristic
305C2×C53#219611yescharacteristic
306C2×C53#220011yescharacteristic
307C2×#127C22×C5311yescharacteristic
308C2×#121—21no—
309C2×#120—21no—
310C2×#120C22×C5311yescharacteristic
311#518C22×C5311yescharacteristic
312#418—21no—
313C2×#100C22×C5311yescharacteristic
314C2×#99C22×C5311yescharacteristic
315C2×#99C22×C5311yescharacteristic
316#394—21no—
317#392—21no—
318#392—21no—
319#388—21no—
320#386C22×C5311yescharacteristic
321#385—21no—
322#379—21no—
323#353—22no—
324AΣL(1,𝔽16)—22no—
325#342—21no—
326#342—21no—
327#339—21no—
328C53#1342211yescharacteristic, direct product factor, Sylow
329C25D4×C5311yescharacteristic
330C22×D4—21no—
331C22×D4—21no—
332C22×D4—21no—
333C22×D4—21no—
334C22×D4—21no—
335C22×D4D4×C5311yescharacteristic
336C22×D4C23×C5311yescharacteristic
337C23×C4D4×C5311yescharacteristic
338C2×M16—21no—
339#127—21no—
340#127—21no—
341#127—21no—
342#127—21no—
343#121—44no—
344#120—21no—
345#120—21no—
346#120—21no—
347#120—21no—
348#120—44no—
349C4×D4—41no—
350C4×D4—41no—
351#117—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—41no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—41no—
365C2×C22⋊C4—41no—
366C2×C22⋊C4D4×C5311yescharacteristic
367C2×C22⋊C4D4×C5311yescharacteristic
368C2×C42D4×C5311yescharacteristic
369#102—41no—
370#102—41no—
371#100—21no—
372#100—21no—
373#99—21no—
374#99—21no—
375#99—21no—
376#99D4×C5312yescomplemented
377#95—21no—
378#95—21no—
379#95—41no—
380C24—21no—
381C24—21no—
382C24—21no—
383C24—21no—
384C24—41no—
385C24—42no—
386C24—41no—
387C24—81no—
388C24C2×D4×C5311yescharacteristic
389C24C2×D4×C5311yescharacteristic
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—41no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—41no—
408C2×D4—81no—
409C2×D4—81no—
410C2×D4—81no—
411C2×D4—81no—
412C2×D4—81no—
413C2×D4—81no—
414C2×D4C2×D4×C5311yescharacteristic
415C2×D4#41×C5311yescharacteristic
416C2×D4#41×C5311yescharacteristic
417C2×D4#41×C5311yescharacteristic
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—42no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4C2×D4×C5311yescharacteristic
431M16—41no—
432M16—41no—
433C2×C8—41no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—42no—
437C4⋊C4—82no—
438C22⋊C4—24no—
439C22⋊C4—22no—
440C22⋊C4—21no—
441C22⋊C4—21no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—41no—
446C22⋊C4—41no—
447C22⋊C4—41no—
448C22⋊C4—41no—
449C22⋊C4—41no—
450C22⋊C4—41no—
451C22⋊C4—41no—
452C22⋊C4—41no—
453C22⋊C4—41no—
454C22⋊C4—81no—
455C22⋊C4—82no—
456C22⋊C4—82no—
457C42—21no—
458C42—21no—
459C42—41no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—22no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—42no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—42no—
478C23—42no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—81no—
486C23—82no—
487C23—81no—
488C23—82no—
489C23—84no—
490C23—81no—
491C23#120×C5311yescharacteristic
492C23#121×C5311yescharacteristic
493C23#121×C5311yescharacteristic
494C23#123×C5311yescharacteristic
495C23#124×C5311yescharacteristic
496D4—41no—
497D4—41no—
498D4—41no—
499D4—41no—
500D4—42no—
501D4—81no—
502D4—82no—
503D4—82no—
504D4—82no—
505D4—82no—
506D4—84no—
507D4—82no—
508D4—82no—
509D4—84no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4#121×C5311yescharacteristic
539C2×C4#123×C5311yescharacteristic
540C8—81no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—42no—
552C22—42no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—44no—
558C22—42no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22C53×#39411yescharacteristic
576C22C53×B(4,𝔽2)11yescharacteristic
577C22C53×#45811yescharacteristic
578C4—21no—
579C4—21no—
580C4—42no—
581C4—42no—
582C4—81no—
583C4—81no—
584C4—82no—
585C4—81no—
586C4—81no—
587C4—161no—
588C2—21no—
589C2—21no—
590C2—41no—
591C2—41no—
592C2—41no—
593C2—41no—
594C2—82no—
595C2—161no—
596C2C53×#204411yescharacteristic
597C2C53×#219611yescharacteristic
598C2C53×#220011yescharacteristic
subgroupG/H
center, socleC22×C53B(4,𝔽2)
commutator, Frattini, cosocleC22×D4C23×C53
prime radicalC53×#2669C2
LCSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C53 ◁ C22×D4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C53×#353C2semidirect
3. C2×C53×AΣL(1,𝔽16)C2semidirect
4. C53×#1889C2semidirect
5. C53×#1887C2semidirect
6. C53×#1867C2semidirect
7. C53×#1390C2semidirect
76. #99×C53D4semidirect
241. #13422C53direct (N × H)
290. C2×#353C2×C53semidirect
291. C2×AΣL(1,𝔽16)C2×C53semidirect
292. #1889C2×C53semidirect
293. #1887C2×C53semidirect
294. #1867C2×C53semidirect
295. #1390C2×C53semidirect
328. C53#13422direct (N × H)
376. #99D4×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#26692
C2×C53×#3532
C2×C53×AΣL(1,𝔽16)2
C53×#18892
C53×#18872
C53×#18672
C53×#13902
#1342253