Finite Group Databasebeta

← C53×#12759 #27220271

Subgroups — C53×#12759 #27220271

Order 13568.

1556 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 1556 conjugacy classes of subgroups — too many to draw. Only its 142 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
1C2×D42×C53C211yesmaximal, characteristic
2C2×C53×#353C211yesmaximal, characteristic, complemented
3C2×C53×AΣL(1,𝔽16)C212yesmaximal, complemented
4C53×#1889C212yesmaximal, complemented
5C53×#1887C211yesmaximal, characteristic, complemented
6C23×D4×C53—22no—
7D42×C53—22no—
8D42×C53—24no—
9D42×C53C414yescomplemented
10C2×#127×C53C2211yescharacteristic
11C2×#121×C53—22no—
12C2×#120×C53—21no—
13C2×#120×C53C2212yes—
14C53×#518C2211yescharacteristic, complemented
15C2×C4×D4×C53—21no—
16C53×#418—21no—
17C2×#100×C53C2211yescharacteristic, complemented
18C2×#99×C53C2212yescomplemented
19C53×#379—22no—
20C53×#353—22no—
21C53×AΣL(1,𝔽16)—24no—
22C53×#342—22no—
23C53×#339—21no—
24C25×C53—21no—
25C25×C53D412yes—
26C22×D4×C53—21no—
27C22×D4×C53—22no—
28C22×D4×C53—28no—
29C22×D4×C53—28no—
30C22×D4×C53—22no—
31C22×D4×C53—22no—
32C22×D4×C53—416no—
33C22×D4×C53D411yescharacteristic
34C22×D4×C53C2311yescharacteristic
35C23×C4×C53—22no—
36C2×M16×C53—21no—
37#127×C53—22no—
38#127×C53C2×C412yes—
39#127×C53D412yes—
40#121×C53—28no—
41#121×C53—28no—
42#120×C53—24no—
43#120×C53—24no—
44#120×C53—22no—
45#120×C53—22no—
46#120×C53C2×C414yes—
47#120×C53D414yes—
48C4×D4×C53—24no—
49C4×D4×C53—24no—
50C4×D4×C53—42no—
51#117×C53—21no—
52C2×C4⋊C4×C53—21no—
53C2×C22⋊C4×C53—21no—
54C2×C22⋊C4×C53—22no—
55C2×C22⋊C4×C53—22no—
56C2×C22⋊C4×C53—22no—
57C2×C22⋊C4×C53—44no—
58C2×C22⋊C4×C53D412yes—
59C2×C42×C53D411yescharacteristic
60#102×C53—42no—
61#100×C53—22no—
62#99×C53—22no—
63#99×C53—22no—
64C24×C53—22no—
65C24×C53—24no—
66C24×C53—21no—
67C24×C53—22no—
68C24×C53—42no—
69C24×C53—48no—
70C24×C53—48no—
71C24×C53—42no—
72C24×C53—81no—
73C24×C53—81no—
74C24×C53C22⋊C414yescomplemented
75C24×C53C2×D412yes—
76C2×D4×C53—24no—
77C2×D4×C53—28no—
78C2×D4×C53—28no—
79C2×D4×C53—24no—
80C2×D4×C53—24no—
81C2×D4×C53—24no—
82C2×D4×C53—24no—
83C2×D4×C53—21no—
84C2×D4×C53—41no—
85C2×D4×C53—41no—
86C2×D4×C53—48no—
87C2×D4×C53—41no—
88C2×D4×C53—41no—
89C2×D4×C53—432no—
90C2×D4×C53—432no—
91C2×D4×C53—48no—
92C2×D4×C53—48no—
93C2×D4×C53C22⋊C412yescomplemented
94C2×D4×C53C22×C411yescharacteristic
95C2×D4×C53C2×D411yescharacteristic
96C2×D4×C53C2×D412yes—
97C22×C4×C53—21no—
98C22×C4×C53—28no—
99C22×C4×C53—22no—
100C22×C4×C53—24no—
101C22×C4×C53—48no—
102C22×C4×C53—42no—
103C22×C4×C53C2×D411yescharacteristic
104M16×C53—41no—
105M16×C53—41no—
106C2×C8×C53—41no—
107C4⋊C4×C53—24no—
108C4⋊C4×C53—22no—
109C22⋊C4×C53—24no—
110C22⋊C4×C53—22no—
111C22⋊C4×C53—24no—
112C22⋊C4×C53—24no—
113C22⋊C4×C53—42no—
114C22⋊C4×C53—41no—
115C22⋊C4×C53—42no—
116C22⋊C4×C53—42no—
117C22⋊C4×C53—88no—
118C22⋊C4×C53C22⋊C414yescomplemented
119C42×C53—21no—
120C42×C53—41no—
121C42×C53C22⋊C412yescomplemented
122C23×C53—28no—
123C23×C53—24no—
124C23×C53—22no—
125C23×C53—22no—
126C23×C53—22no—
127C23×C53—44no—
128C23×C53—48no—
129C23×C53—44no—
130C23×C53—44no—
131C23×C53—44no—
132C23×C53—48no—
133C23×C53—416no—
134C23×C53—42no—
135C23×C53—42no—
136C23×C53—48no—
137C23×C53—41no—
138C23×C53—41no—
139C23×C53—82no—
140C23×C53—84no—
141C23×C53—88no—
142C23×C53—84no—
143C23×C53—88no—
144C23×C53—88no—
145C23×C53C2×C22⋊C412yes—
146C23×C53#12012yes—
147C23×C53#12011yescharacteristic
148D4×C53—216no—
149D4×C53—41no—
150D4×C53—42no—
151D4×C53—41no—
152D4×C53—42no—
153D4×C53—48no—
154D4×C53—48no—
155D4×C53—48no—
156D4×C53—48no—
157D4×C53—416no—
158D4×C53—81no—
159C2×C4×C53—24no—
160C2×C4×C53—22no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—24no—
164C2×C4×C53—416no—
165C2×C4×C53—44no—
166C2×C4×C53—42no—
167C2×C4×C53—84no—
168C2×C4×C53—82no—
169C2×C4×C53—84no—
170C2×C4×C53C2×C22⋊C411yescharacteristic
171C2×C4×C53#12011yescharacteristic
172C8×C53—81no—
173#12759C5311yesmaximal, characteristic, direct product factor, Sylow
174C22×C53—22no—
175C22×C53—28no—
176C22×C53—22no—
177C22×C53—22no—
178C22×C53—42no—
179C22×C53—42no—
180C22×C53—44no—
181C22×C53—48no—
182C22×C53—44no—
183C22×C53—48no—
184C22×C53—44no—
185C22×C53—48no—
186C22×C53—44no—
187C22×C53—42no—
188C22×C53—82no—
189C22×C53—82no—
190C22×C53—88no—
191C22×C53—84no—
192C22×C53—816no—
193C22×C53—82no—
194C22×C53#37911yescharacteristic
195C22×C53B(4,𝔽2)11yescharacteristic
196C22×C53#45811yescharacteristic
197C4×C53—24no—
198C4×C53—21no—
199C4×C53—21no—
200C4×C53—44no—
201C4×C53—82no—
202C4×C53—162no—
203C2×D42C2×C5311yescharacteristic
204C2×#353C2×C5311yescharacteristic, complemented
205C2×AΣL(1,𝔽16)C2×C5312yescomplemented
206#1889C2×C5312yescomplemented
207#1887C2×C5311yescharacteristic, complemented
208C2×C53—21no—
209C2×C53—21no—
210C2×C53—42no—
211C2×C53—42no—
212C2×C53—48no—
213C2×C53—82no—
214C2×C53#179011yescharacteristic
215C2×C53#219612yes—
216C23×D4—22no—
217D42—22no—
218D42—24no—
219D42C4×C5314yescomplemented
220C2×#127C22×C5311yescharacteristic
221C2×#121—22no—
222C2×#120—21no—
223C2×#120C22×C5312yes—
224#518C22×C5311yescharacteristic, complemented
225C2×C4×D4—21no—
226#418—21no—
227C2×#100C22×C5311yescharacteristic, complemented
228C2×#99C22×C5312yescomplemented
229#379—22no—
230#353—22no—
231AΣL(1,𝔽16)—24no—
232#342—22no—
233#339—21no—
234C53#1275911yescharacteristic, direct product factor, Sylow
235C25—21no—
236C25D4×C5312yes—
237C22×D4—21no—
238C22×D4—22no—
239C22×D4—28no—
240C22×D4—22no—
241C22×D4—22no—
242C22×D4—28no—
243C22×D4—416no—
244C22×D4D4×C5311yescharacteristic
245C22×D4C23×C5311yescharacteristic
246C23×C4—22no—
247C2×M16—21no—
248#127—22no—
249#127C2×C4×C5312yes—
250#127D4×C5312yes—
251#121—28no—
252#121—28no—
253#120—24no—
254#120—22no—
255#120—22no—
256#120—24no—
257#120C2×C4×C5314yes—
258#120D4×C5314yes—
259C4×D4—24no—
260C4×D4—24no—
261C4×D4—42no—
262#117—21no—
263C2×C4⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4D4×C5312yes—
270C2×C42D4×C5311yescharacteristic
271#102—42no—
272#100—22no—
273#99—22no—
274#99—22no—
275C24—22no—
276C24—24no—
277C24—21no—
278C24—22no—
279C24—42no—
280C24—48no—
281C24—48no—
282C24—42no—
283C24—81no—
284C24—81no—
285C24C22⋊C4×C5314yescomplemented
286C24C2×D4×C5312yes—
287C2×D4—24no—
288C2×D4—28no—
289C2×D4—28no—
290C2×D4—24no—
291C2×D4—24no—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—21no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—48no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—432no—
301C2×D4—432no—
302C2×D4—48no—
303C2×D4—48no—
304C2×D4C22⋊C4×C5312yescomplemented
305C2×D4C22×C4×C5311yescharacteristic
306C2×D4C2×D4×C5311yescharacteristic
307C2×D4C2×D4×C5312yes—
308C22×C4—21no—
309C22×C4—28no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—48no—
313C22×C4—42no—
314C22×C4C2×D4×C5311yescharacteristic
315M16—41no—
316M16—41no—
317C2×C8—41no—
318C4⋊C4—24no—
319C4⋊C4—22no—
320C22⋊C4—24no—
321C22⋊C4—22no—
322C22⋊C4—24no—
323C22⋊C4—24no—
324C22⋊C4—42no—
325C22⋊C4—41no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C22⋊C4—88no—
329C22⋊C4C22⋊C4×C5314yescomplemented
330C42—21no—
331C42—41no—
332C42C22⋊C4×C5312yescomplemented
333C23—28no—
334C23—24no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—44no—
339C23—44no—
340C23—48no—
341C23—44no—
342C23—44no—
343C23—48no—
344C23—416no—
345C23—42no—
346C23—42no—
347C23—48no—
348C23—41no—
349C23—41no—
350C23—82no—
351C23—84no—
352C23—88no—
353C23—84no—
354C23—88no—
355C23—88no—
356C23C2×C22⋊C4×C5312yes—
357C23#120×C5311yescharacteristic
358C23#120×C5312yes—
359D4—216no—
360D4—41no—
361D4—42no—
362D4—41no—
363D4—42no—
364D4—48no—
365D4—48no—
366D4—48no—
367D4—48no—
368D4—416no—
369D4—81no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—24no—
375C2×C4—416no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—84no—
379C2×C4—82no—
380C2×C4—84no—
381C2×C4C2×C22⋊C4×C5311yescharacteristic
382C2×C4#120×C5311yescharacteristic
383C8—81no—
384C22—22no—
385C22—28no—
386C22—22no—
387C22—22no—
388C22—42no—
389C22—42no—
390C22—44no—
391C22—48no—
392C22—44no—
393C22—48no—
394C22—44no—
395C22—48no—
396C22—44no—
397C22—42no—
398C22—82no—
399C22—82no—
400C22—88no—
401C22—84no—
402C22—816no—
403C22—82no—
404C22C53×#37911yescharacteristic
405C22C53×B(4,𝔽2)11yescharacteristic
406C22C53×#45811yescharacteristic
407C4—24no—
408C4—21no—
409C4—21no—
410C4—44no—
411C4—82no—
412C4—162no—
413C2—21no—
414C2—21no—
415C2—42no—
416C2—48no—
417C2—42no—
418C2—82no—
419C2C53×#179011yescharacteristic
420C2C53×#219612yes—
subgroupG/H
center, socleC22×C53B(4,𝔽2)
commutatorC2×D4C22×C4×C53
Frattini, cosocleC22×D4C23×C53
prime radicalC2×D42×C53C2
LCSG ▷ C2×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
9. D42×C53C4semidirect
14. C53×#518C22semidirect
17. C2×#100×C53C22semidirect
18. C2×#99×C53C22semidirect
74. C24×C53C22⋊C4semidirect
93. C2×D4×C53C22⋊C4semidirect
118. C22⋊C4×C53C22⋊C4semidirect
121. C42×C53C22⋊C4semidirect
173. #12759C53direct (N × H)
204. C2×#353C2×C53semidirect
205. C2×AΣL(1,𝔽16)C2×C53semidirect
206. #1889C2×C53semidirect
207. #1887C2×C53semidirect
219. D42C4×C53semidirect
224. #518C22×C53semidirect
227. C2×#100C22×C53semidirect
228. C2×#99C22×C53semidirect
234. C53#12759direct (N × H)
285. C24C22⋊C4×C53semidirect
304. C2×D4C22⋊C4×C53semidirect
329. C22⋊C4C22⋊C4×C53semidirect
332. C42C22⋊C4×C53semidirect

Maximal subgroups

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

subgroupindex
C2×D42×C532
C2×C53×#3532
C2×C53×AΣL(1,𝔽16)2
C53×#18892
C53×#18872
#1275953