Finite Group Databasebeta

← C53×#9949 #27217461

Subgroups — C53×#9949 #27217461

Order 13568.

786 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3187C211yesmaximal, characteristic
2C2×C53×#448C211yesmaximal, characteristic
3C53×#2434C211yesmaximal, characteristic, complemented
4C53×#1964C211yesmaximal, characteristic
5C53×#1787C211yesmaximal, characteristic, complemented
6C53×#1520C211yesmaximal, characteristic, complemented
7C53×#1282C211yesmaximal, characteristic, complemented
8C23×Q8×C53C2211yescharacteristic
9C22×SD16×C53—21no—
10C2×C4⋊Q8×C53—21no—
11C2×C22⋊Q8×C53C2211yescharacteristic
12C2×#121×C53C2211yescharacteristic
13C53×#483—21no—
14C53×#481—21no—
15C53×#465—22no—
16C53×#450—21no—
17C53×#448—22no—
18C53×#448—21no—
19C53×#448—22no—
20C2×#103×C53—21no—
21C2×#103×C53—22no—
22C2×#99×C53—22no—
23C2×C22⋊C8×C53C2211yescharacteristic
24C2×C22⋊C8×C53C2211yescharacteristic
25C53×#395—22no—
26C53×#391—21no—
27C53×#385—22no—
28C53×#380C2211yescharacteristic
29C4×C22⋊C4×C53C2211yescharacteristic
30C53×#328—22no—
31C22×Q8×C53—21no—
32C22×Q8×C53—21no—
33C22×Q8×C53—21no—
34C22×Q8×C53—21no—
35C22×Q8×C53—21no—
36C22×Q8×C53—44no—
37C22×Q8×C53D411yescharacteristic
38C22×Q8×C53D412yes—
39C22×D4×C53—21no—
40C22×D4×C53D411yescharacteristic
41C23×C4×C53—21no—
42C23×C4×C53C2311yescharacteristic
43C2×SD16×C53—24no—
44C2×SD16×C53—22no—
45C2×SD16×C53—44no—
46C22×C8×C53—21no—
47C22×C8×C53D412yes—
48C4⋊Q8×C53—44no—
49C22⋊Q8×C53—21no—
50C22⋊Q8×C53—21no—
51C22⋊Q8×C53—21no—
52C22⋊Q8×C53—21no—
53#121×C53—21no—
54#121×C53—21no—
55#121×C53—21no—
56#121×C53—21no—
57C2×C4⋊C4×C53—24no—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53D411yescharacteristic
61C2×C4⋊C4×C53D411yescharacteristic
62C2×C22⋊C4×C53—21no—
63C2×C22⋊C4×C53—21no—
64C2×C22⋊C4×C53—21no—
65C2×C22⋊C4×C53D412yes—
66C2×C42×C53—21no—
67#106×C53—22no—
68#103×C53—24no—
69#103×C53—22no—
70#103×C53—44no—
71#102×C53—22no—
72#99×C53—44no—
73C22⋊C8×C53—21no—
74C22⋊C8×C53—21no—
75C22⋊C8×C53—21no—
76C22⋊C8×C53D412yes—
77#95×C53—24no—
78#95×C53—21no—
79#95×C53—21no—
80C24×C53—21no—
81C24×C53C2×D411yescharacteristic
82C2×Q8×C53—24no—
83C2×Q8×C53—24no—
84C2×Q8×C53—24no—
85C2×Q8×C53—22no—
86C2×Q8×C53—21no—
87C2×Q8×C53—22no—
88C2×Q8×C53—21no—
89C2×Q8×C53—44no—
90C2×Q8×C53—42no—
91C2×Q8×C53—42no—
92C2×Q8×C53—42no—
93C2×Q8×C53—42no—
94C2×Q8×C53—44no—
95C2×Q8×C53—44no—
96C2×Q8×C53—41no—
97C2×Q8×C53—41no—
98C2×D4×C53—21no—
99C2×D4×C53—21no—
100C2×D4×C53—42no—
101C2×D4×C53—41no—
102C2×D4×C53—41no—
103C2×D4×C53—42no—
104C2×D4×C53—42no—
105C22×C4×C53—22no—
106C22×C4×C53—22no—
107C22×C4×C53—21no—
108C22×C4×C53—21no—
109C22×C4×C53—21no—
110C22×C4×C53—42no—
111C22×C4×C53—44no—
112C22×C4×C53—41no—
113C22×C4×C53—41no—
114C22×C4×C53—41no—
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53C2×D411yescharacteristic
117C22×C4×C53C2×D411yescharacteristic
118C22×C4×C53C2×D411yescharacteristic
119C22×C4×C53C2×D411yescharacteristic
120C22×C4×C53#4111yescharacteristic
121SD16×C53—48no—
122SD16×C53—82no—
123C2×C8×C53—22no—
124C2×C8×C53—22no—
125C2×C8×C53—42no—
126C2×C8×C53—42no—
127C4⋊C4×C53—21no—
128C4⋊C4×C53—21no—
129C4⋊C4×C53—21no—
130C4⋊C4×C53—21no—
131C4⋊C4×C53—48no—
132C4⋊C4×C53—42no—
133C4⋊C4×C53—41no—
134C4⋊C4×C53—41no—
135C22⋊C4×C53—22no—
136C22⋊C4×C53—41no—
137C22⋊C4×C53—41no—
138C22⋊C4×C53—41no—
139C22⋊C4×C53—41no—
140C22⋊C4×C53—42no—
141C22⋊C4×C53SD1614yescomplemented
142C42×C53—42no—
143C23×C53—41no—
144C23×C53—41no—
145C23×C53—41no—
146C23×C53—42no—
147C23×C53—81no—
148C23×C53—81no—
149C23×C53#12011yescharacteristic
150C23×C53#12111yescharacteristic
151C23×C53#12711yescharacteristic
152C23×C53C2×SD1612yes—
153C23×C53#13712yes—
154Q8×C53—24no—
155Q8×C53—24no—
156Q8×C53—42no—
157Q8×C53—42no—
158Q8×C53—44no—
159Q8×C53—44no—
160Q8×C53—42no—
161Q8×C53—81no—
162D4×C53—42no—
163D4×C53—42no—
164D4×C53—82no—
165D4×C53—81no—
166D4×C53—81no—
167C2×C4×C53—22no—
168C2×C4×C53—22no—
169C2×C4×C53—22no—
170C2×C4×C53—24no—
171C2×C4×C53—24no—
172C2×C4×C53—41no—
173C2×C4×C53—41no—
174C2×C4×C53—41no—
175C2×C4×C53—42no—
176C2×C4×C53—42no—
177C2×C4×C53—42no—
178C2×C4×C53—44no—
179C2×C4×C53—44no—
180C2×C4×C53—42no—
181C2×C4×C53—42no—
182C2×C4×C53—42no—
183C2×C4×C53—41no—
184C2×C4×C53—41no—
185C2×C4×C53—41no—
186C2×C4×C53—42no—
187C2×C4×C53—81no—
188C2×C4×C53—81no—
189C2×C4×C53—81no—
190C2×C4×C53#12011yescharacteristic
191C2×C4×C53#12011yescharacteristic
192C2×C4×C53#12111yescharacteristic
193C2×C4×C53#12111yescharacteristic
194C8×C53—42no—
195C8×C53—42no—
196#9949C5311yesmaximal, characteristic, direct product factor, Sylow
197C22×C53—21no—
198C22×C53—21no—
199C22×C53—21no—
200C22×C53—21no—
201C22×C53—21no—
202C22×C53—21no—
203C22×C53—42no—
204C22×C53—42no—
205C22×C53—81no—
206C22×C53—81no—
207C22×C53—81no—
208C22×C53—162no—
209C22×C53#39211yescharacteristic
210C22×C53#44812yes—
211C22×C53#46112yes—
212C22×C53#49211yescharacteristic
213C22×C53#49711yescharacteristic
214C4×C53—21no—
215C4×C53—21no—
216C4×C53—21no—
217C4×C53—44no—
218C4×C53—44no—
219C4×C53—81no—
220C4×C53—81no—
221C4×C53—81no—
222C4×C53—161no—
223C4×C53B(4,𝔽2)12yes—
224#3187C2×C5311yescharacteristic
225C2×#448C2×C5311yescharacteristic
226#2434C2×C5311yescharacteristic, complemented
227#1964C2×C5311yescharacteristic
228#1787C2×C5311yescharacteristic, complemented
229#1520C2×C5311yescharacteristic, complemented
230#1282C2×C5311yescharacteristic, complemented
231C2×C53—22no—
232C2×C53—21no—
233C2×C53—22no—
234C2×C53—21no—
235C2×C53—161no—
236C2×C53#200911yescharacteristic
237C2×C53#201311yescharacteristic
238C2×C53#202111yescharacteristic
239C23×Q8C22×C5311yescharacteristic
240C22×SD16—21no—
241C2×C4⋊Q8—21no—
242C2×C22⋊Q8C22×C5311yescharacteristic
243C2×#121C22×C5311yescharacteristic
244#483—21no—
245#481—21no—
246#465—22no—
247#450—21no—
248#448—21no—
249#448—22no—
250#448—22no—
251C2×#103—21no—
252C2×#103—22no—
253C2×#99—22no—
254C2×C22⋊C8C22×C5311yescharacteristic
255C2×C22⋊C8C22×C5311yescharacteristic
256#395—22no—
257#391—21no—
258#385—22no—
259#380C22×C5311yescharacteristic
260C4×C22⋊C4C22×C5311yescharacteristic
261#328—22no—
262C53#994911yescharacteristic, direct product factor, Sylow
263C22×Q8—21no—
264C22×Q8—21no—
265C22×Q8—21no—
266C22×Q8—21no—
267C22×Q8—21no—
268C22×Q8—44no—
269C22×Q8D4×C5311yescharacteristic
270C22×Q8D4×C5312yes—
271C22×D4—21no—
272C22×D4D4×C5311yescharacteristic
273C23×C4—21no—
274C23×C4C23×C5311yescharacteristic
275C2×SD16—24no—
276C2×SD16—22no—
277C2×SD16—44no—
278C22×C8—21no—
279C22×C8D4×C5312yes—
280C4⋊Q8—44no—
281C22⋊Q8—21no—
282C22⋊Q8—21no—
283C22⋊Q8—21no—
284C22⋊Q8—21no—
285#121—21no—
286#121—21no—
287#121—21no—
288#121—21no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4D4×C5311yescharacteristic
293C2×C4⋊C4D4×C5311yescharacteristic
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4D4×C5312yes—
298C2×C42—21no—
299#106—22no—
300#103—24no—
301#103—22no—
302#103—44no—
303#102—22no—
304#99—44no—
305C22⋊C8—21no—
306C22⋊C8—21no—
307C22⋊C8—21no—
308C22⋊C8D4×C5312yes—
309#95—24no—
310#95—21no—
311#95—21no—
312C24—21no—
313C24C2×D4×C5311yescharacteristic
314C2×Q8—24no—
315C2×Q8—24no—
316C2×Q8—24no—
317C2×Q8—22no—
318C2×Q8—21no—
319C2×Q8—22no—
320C2×Q8—21no—
321C2×Q8—44no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—44no—
327C2×Q8—44no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—42no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—42no—
336C2×D4—42no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—42no—
343C22×C4—44no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4C2×D4×C5311yescharacteristic
348C22×C4C2×D4×C5311yescharacteristic
349C22×C4C2×D4×C5311yescharacteristic
350C22×C4C2×D4×C5311yescharacteristic
351C22×C4C2×D4×C5311yescharacteristic
352C22×C4#41×C5311yescharacteristic
353SD16—48no—
354SD16—82no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—42no—
358C2×C8—42no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—48no—
364C4⋊C4—42no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C22⋊C4—22no—
368C22⋊C4—41no—
369C22⋊C4—41no—
370C22⋊C4—41no—
371C22⋊C4—41no—
372C22⋊C4—42no—
373C22⋊C4SD16×C5314yescomplemented
374C42—42no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—42no—
379C23—81no—
380C23—81no—
381C23#120×C5311yescharacteristic
382C23#121×C5311yescharacteristic
383C23#127×C5311yescharacteristic
384C23C2×SD16×C5312yes—
385C23#137×C5312yes—
386Q8—24no—
387Q8—24no—
388Q8—42no—
389Q8—42no—
390Q8—44no—
391Q8—44no—
392Q8—42no—
393Q8—81no—
394D4—42no—
395D4—42no—
396D4—81no—
397D4—81no—
398D4—82no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4#120×C5311yescharacteristic
423C2×C4#120×C5311yescharacteristic
424C2×C4#121×C5311yescharacteristic
425C2×C4#121×C5311yescharacteristic
426C8—42no—
427C8—42no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—42no—
435C22—42no—
436C22—81no—
437C22—81no—
438C22—81no—
439C22—162no—
440C22C53×#39211yescharacteristic
441C22C53×#44812yes—
442C22C53×#46112yes—
443C22C53×#49211yescharacteristic
444C22C53×#49711yescharacteristic
445C4—21no—
446C4—21no—
447C4—21no—
448C4—44no—
449C4—44no—
450C4—81no—
451C4—81no—
452C4—81no—
453C4—161no—
454C4C53×B(4,𝔽2)12yes—
455C2—22no—
456C2—21no—
457C2—22no—
458C2—21no—
459C2—161no—
460C2C53×#200911yescharacteristic
461C2C53×#201311yescharacteristic
462C2C53×#202111yescharacteristic
subgroupG/H
center, socleC22×C53#392
commutator, Frattini, cosocleC23×C4C23×C53
prime radicalC2×#121×C53C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
3. C53×#2434C2semidirect
5. C53×#1787C2semidirect
6. C53×#1520C2semidirect
7. C53×#1282C2semidirect
141. C22⋊C4×C53SD16semidirect
196. #9949C53direct (N × H)
226. #2434C2×C53semidirect
228. #1787C2×C53semidirect
229. #1520C2×C53semidirect
230. #1282C2×C53semidirect
262. C53#9949direct (N × H)
373. C22⋊C4SD16×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#31872
C2×C53×#4482
C53×#24342
C53×#19642
C53×#17872
C53×#15202
C53×#12822
#994953