Finite Group Databasebeta

← C53×#9497 #27217009

Subgroups — C53×#9497 #27217009

Order 13568.

1184 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 1184 conjugacy classes of subgroups — too many to draw. Only its 130 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×#2403C211yesmaximal, characteristic
2C53×#2309C211yesmaximal, characteristic, complemented
3C53×#1781C212yesmaximal, complemented
4C53×#1781C211yesmaximal, characteristic, complemented
5C53×#1304C212yesmaximal, complemented
6C23×D4×C53C2211yescharacteristic
7C23×D4×C53C2211yescharacteristic
8C2×#120×C53—21no—
9C2×#120×C53C412yescomplemented
10C2×#117×C53—21no—
11C2×#99×C53—22no—
12C2×#99×C53—24no—
13C53×#392—22no—
14C53×#390—24no—
15C53×#386—21no—
16C53×#386C412yescomplemented
17C53×#379—21no—
18C53×#379C2212yes—
19C53×#379C2211yescharacteristic
20C53×#379C2212yes—
21C25×C53C2×C411yescharacteristic
22C25×C53D411yescharacteristic
23C25×C53C2311yescharacteristic
24C22×D4×C53—21no—
25C22×D4×C53—22no—
26C22×D4×C53—22no—
27C22×D4×C53—22no—
28C22×D4×C53—21no—
29C22×D4×C53—22no—
30C22×D4×C53—22no—
31C22×D4×C53—44no—
32C22×D4×C53—44no—
33C22×D4×C53C2×C412yes—
34C22×D4×C53D412yes—
35C23×C4×C53C2×C411yescharacteristic
36C23×C4×C53D411yescharacteristic
37#120×C53—44no—
38#120×C53—44no—
39#117×C53—44no—
40C2×C4⋊C4×C53—22no—
41C2×C22⋊C4×C53—22no—
42C2×C22⋊C4×C53—22no—
43C2×C22⋊C4×C53—22no—
44C2×C22⋊C4×C53—22no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—22no—
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53—22no—
49C2×C22⋊C4×C53—22no—
50C2×C22⋊C4×C53—22no—
51C2×C22⋊C4×C53—22no—
52C2×C22⋊C4×C53C2×C412yes—
53C2×C22⋊C4×C53D414yescomplemented
54C2×C42×C53—22no—
55#99×C53—48no—
56#99×C53—44no—
57#95×C53—22no—
58C24×C53—21no—
59C24×C53—21no—
60C24×C53—22no—
61C24×C53—21no—
62C24×C53—42no—
63C24×C53—42no—
64C24×C53—42no—
65C24×C53—42no—
66C24×C53—42no—
67C24×C53—42no—
68C24×C53—81no—
69C24×C53—81no—
70C24×C53—81no—
71C24×C53C22⋊C412yes—
72C24×C53C22×C411yescharacteristic
73C24×C53C2×D411yescharacteristic
74C24×C53C2×D411yescharacteristic
75C24×C53C2×D412yes—
76C24×C53#4112yes—
77C2×D4×C53—28no—
78C2×D4×C53—24no—
79C2×D4×C53—24no—
80C2×D4×C53—24no—
81C2×D4×C53—24no—
82C2×D4×C53—24no—
83C2×D4×C53—42no—
84C2×D4×C53—42no—
85C2×D4×C53—48no—
86C2×D4×C53—44no—
87C2×D4×C53—44no—
88C2×D4×C53—44no—
89C2×D4×C53—42no—
90C2×D4×C53—42no—
91C2×D4×C53—44no—
92C2×D4×C53—42no—
93C2×D4×C53—44no—
94C2×D4×C53—41no—
95C2×D4×C53—42no—
96C2×D4×C53—41no—
97C2×D4×C53—44no—
98C2×D4×C53—44no—
99C2×D4×C53—82no—
100C22×C4×C53—21no—
101C22×C4×C53—21no—
102C22×C4×C53—24no—
103C22×C4×C53—21no—
104C22×C4×C53—22no—
105C22×C4×C53—22no—
106C22×C4×C53—24no—
107C22×C4×C53—42no—
108C22×C4×C53—42no—
109C22×C4×C53C22⋊C412yes—
110C4⋊C4×C53—44no—
111C22⋊C4×C53—44no—
112C22⋊C4×C53—44no—
113C22⋊C4×C53—42no—
114C22⋊C4×C53—44no—
115C22⋊C4×C53—44no—
116C22⋊C4×C53—42no—
117C22⋊C4×C53—44no—
118C22⋊C4×C53—44no—
119C22⋊C4×C53—81no—
120C22⋊C4×C53—82no—
121C22⋊C4×C53—82no—
122C22⋊C4×C53—82no—
123C22⋊C4×C53—82no—
124C42×C53—44no—
125C23×C53—22no—
126C23×C53—22no—
127C23×C53—22no—
128C23×C53—22no—
129C23×C53—24no—
130C23×C53—22no—
131C23×C53—24no—
132C23×C53—22no—
133C23×C53—22no—
134C23×C53—22no—
135C23×C53—41no—
136C23×C53—41no—
137C23×C53—41no—
138C23×C53—41no—
139C23×C53—42no—
140C23×C53—42no—
141C23×C53—42no—
142C23×C53—42no—
143C23×C53—41no—
144C23×C53—42no—
145C23×C53—42no—
146C23×C53—42no—
147C23×C53—41no—
148C23×C53—42no—
149C23×C53—42no—
150C23×C53—42no—
151C23×C53—42no—
152C23×C53—42no—
153C23×C53—41no—
154C23×C53—42no—
155C23×C53—42no—
156C23×C53—81no—
157C23×C53—81no—
158C23×C53—82no—
159C23×C53—82no—
160C23×C53—88no—
161C23×C53—88no—
162C23×C53—88no—
163C23×C53C2×C22⋊C411yescharacteristic
164C23×C53C4×D412yes—
165C23×C53#12011yescharacteristic
166C23×C53#12112yes—
167C23×C53#12311yescharacteristic
168D4×C53—48no—
169D4×C53—48no—
170D4×C53—48no—
171D4×C53—44no—
172D4×C53—44no—
173D4×C53—88no—
174C2×C4×C53—24no—
175C2×C4×C53—22no—
176C2×C4×C53—22no—
177C2×C4×C53—22no—
178C2×C4×C53—22no—
179C2×C4×C53—24no—
180C2×C4×C53—22no—
181C2×C4×C53—41no—
182C2×C4×C53—42no—
183C2×C4×C53—41no—
184C2×C4×C53—42no—
185C2×C4×C53—42no—
186C2×C4×C53—44no—
187C2×C4×C53—44no—
188C2×C4×C53—42no—
189C2×C4×C53—44no—
190C2×C4×C53—82no—
191C2×C4×C53—82no—
192C2×C4×C53—82no—
193C2×C4×C53#9914yescomplemented
194#9497C5311yesmaximal, characteristic, direct product factor, Sylow
195C22×C53—21no—
196C22×C53—21no—
197C22×C53—21no—
198C22×C53—21no—
199C22×C53—22no—
200C22×C53—22no—
201C22×C53—21no—
202C22×C53—21no—
203C22×C53—22no—
204C22×C53—22no—
205C22×C53—22no—
206C22×C53—22no—
207C22×C53—21no—
208C22×C53—44no—
209C22×C53—44no—
210C22×C53—44no—
211C22×C53—44no—
212C22×C53—42no—
213C22×C53—42no—
214C22×C53—42no—
215C22×C53—42no—
216C22×C53—84no—
217C22×C53—82no—
218C22×C53—82no—
219C22×C53—84no—
220C22×C53—82no—
221C22×C53—84no—
222C22×C53—84no—
223C22×C53—84no—
224C22×C53—84no—
225C22×C53—81no—
226C22×C53—81no—
227C22×C53#38611yescharacteristic
228C22×C53C2×#9912yes—
229C22×C53B(4,𝔽2)12yes—
230C4×C53—24no—
231C4×C53—42no—
232C4×C53—84no—
233C4×C53—82no—
234C4×C53—162no—
235#2403C2×C5311yescharacteristic
236#2309C2×C5311yescharacteristic, complemented
237#1781C2×C5312yescomplemented
238#1781C2×C5311yescharacteristic, complemented
239#1304C2×C5312yescomplemented
240C2×C53—22no—
241C2×C53—22no—
242C2×C53—42no—
243C2×C53—42no—
244C2×C53—41no—
245C2×C53—82no—
246C2×C53—82no—
247C2×C53#188911yescharacteristic
248C2×C53#189811yescharacteristic
249C2×C53#191311yescharacteristic
250C23×D4C22×C5311yescharacteristic
251C23×D4C22×C5311yescharacteristic
252C2×#120—21no—
253C2×#120C4×C5312yescomplemented
254C2×#117—21no—
255C2×#99—22no—
256C2×#99—24no—
257#392—22no—
258#390—24no—
259#386—21no—
260#386C4×C5312yescomplemented
261#379—21no—
262#379C22×C5312yes—
263#379C22×C5311yescharacteristic
264#379C22×C5312yes—
265C53#949711yescharacteristic, direct product factor, Sylow
266C25C2×C4×C5311yescharacteristic
267C25D4×C5311yescharacteristic
268C25C23×C5311yescharacteristic
269C22×D4—21no—
270C22×D4—22no—
271C22×D4—22no—
272C22×D4—22no—
273C22×D4—21no—
274C22×D4—22no—
275C22×D4—22no—
276C22×D4—44no—
277C22×D4—44no—
278C22×D4C2×C4×C5312yes—
279C22×D4D4×C5312yes—
280C23×C4C2×C4×C5311yescharacteristic
281C23×C4D4×C5311yescharacteristic
282#120—44no—
283#120—44no—
284#117—44no—
285C2×C4⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4C2×C4×C5312yes—
298C2×C22⋊C4D4×C5314yescomplemented
299C2×C42—22no—
300#99—48no—
301#99—44no—
302#95—22no—
303C24—21no—
304C24—21no—
305C24—22no—
306C24—21no—
307C24—42no—
308C24—42no—
309C24—42no—
310C24—42no—
311C24—42no—
312C24—42no—
313C24—81no—
314C24—81no—
315C24—81no—
316C24C22⋊C4×C5312yes—
317C24C22×C4×C5311yescharacteristic
318C24C2×D4×C5311yescharacteristic
319C24C2×D4×C5311yescharacteristic
320C24C2×D4×C5312yes—
321C24#41×C5312yes—
322C2×D4—28no—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—24no—
327C2×D4—24no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—48no—
333C2×D4—42no—
334C2×D4—44no—
335C2×D4—42no—
336C2×D4—44no—
337C2×D4—42no—
338C2×D4—44no—
339C2×D4—41no—
340C2×D4—42no—
341C2×D4—41no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—82no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—24no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4C22⋊C4×C5312yes—
355C4⋊C4—44no—
356C22⋊C4—44no—
357C22⋊C4—44no—
358C22⋊C4—42no—
359C22⋊C4—44no—
360C22⋊C4—44no—
361C22⋊C4—42no—
362C22⋊C4—44no—
363C22⋊C4—44no—
364C22⋊C4—82no—
365C22⋊C4—82no—
366C22⋊C4—81no—
367C22⋊C4—82no—
368C22⋊C4—82no—
369C42—44no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—24no—
375C23—22no—
376C23—24no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—42no—
384C23—41no—
385C23—42no—
386C23—41no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—41no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—41no—
399C23—42no—
400C23—42no—
401C23—81no—
402C23—81no—
403C23—82no—
404C23—82no—
405C23—88no—
406C23—88no—
407C23—88no—
408C23C2×C22⋊C4×C5311yescharacteristic
409C23C4×D4×C5312yes—
410C23#120×C5311yescharacteristic
411C23#121×C5312yes—
412C23#123×C5311yescharacteristic
413D4—48no—
414D4—48no—
415D4—48no—
416D4—44no—
417D4—44no—
418D4—88no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4#99×C5314yescomplemented
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—22no—
444C22—22no—
445C22—21no—
446C22—22no—
447C22—22no—
448C22—21no—
449C22—21no—
450C22—22no—
451C22—22no—
452C22—44no—
453C22—44no—
454C22—44no—
455C22—44no—
456C22—42no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—84no—
461C22—82no—
462C22—82no—
463C22—84no—
464C22—82no—
465C22—84no—
466C22—84no—
467C22—84no—
468C22—84no—
469C22—81no—
470C22—81no—
471C22C53×#38611yescharacteristic
472C22C2×#99×C5312yes—
473C22C53×B(4,𝔽2)12yes—
474C4—24no—
475C4—42no—
476C4—82no—
477C4—84no—
478C4—162no—
479C2—22no—
480C2—22no—
481C2—42no—
482C2—42no—
483C2—41no—
484C2—82no—
485C2—82no—
486C2C53×#188911yescharacteristic
487C2C53×#189811yescharacteristic
488C2C53×#191311yescharacteristic
subgroupG/H
center, socleC22×C53#386
commutatorC24C22×C4×C53
Frattini, cosocleC25C23×C53
prime radicalC53×#2403C2
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C25×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#2309C2semidirect
3. C53×#1781C2semidirect
4. C53×#1781C2semidirect
5. C53×#1304C2semidirect
9. C2×#120×C53C4semidirect
16. C53×#386C4semidirect
53. C2×C22⋊C4×C53D4semidirect
193. C2×C4×C53#99semidirect
194. #9497C53direct (N × H)
236. #2309C2×C53semidirect
237. #1781C2×C53semidirect
238. #1781C2×C53semidirect
239. #1304C2×C53semidirect
253. C2×#120C4×C53semidirect
260. #386C4×C53semidirect
265. C53#9497direct (N × H)
298. C2×C22⋊C4D4×C53semidirect
438. C2×C4#99×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#24032
C53×#23092
C53×#17812
C53×#13042
#949753