Finite Group Databasebeta

← C53×#23895 #27231407

Subgroups — C53×#23895 #27231407

Order 13568.

2002 conjugacy classes of subgroups; only the 282 normal subgroups are shown.

normal subgroups only

This group has 2002 conjugacy classes of subgroups — too many to draw. Only its 282 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×#3484C212yesmaximal, complemented
2C53×#3026C214yesmaximal, complemented
3C53×#2021C214yesmaximal, complemented
4C53×#1913C214yesmaximal, complemented
5C53×#1781C211yesmaximal, characteristic, complemented
6C2×H(5,𝔽2)×C53C2212yescomplemented
7C23×D4×C53C2211yescharacteristic
8C53×#546—28no—
9D42×C53—28no—
10C53×#535—22no—
11C53×#534—22no—
12C53×#534C2214yes—
13C53×#534C2214yescomplemented
14C53×#525—22no—
15C2×#120×C53—22no—
16C2×#120×C53C2214yescomplemented
17C53×#518C2212yescomplemented
18C53×#458—28no—
19C53×B(4,𝔽2)—28no—
20C2×#99×C53C2218yescomplemented
21C2×#99×C53C2214yescomplemented
22C53×#379C2212yescomplemented
23C53×#342C2214yescomplemented
24C25×C53D411yescharacteristic, complemented
25C25×C53C2311yescharacteristic
26H(5,𝔽2)×C53—28no—
27H(5,𝔽2)×C53—44no—
28C2×#41×C53—22no—
29C2×#41×C53—24no—
30C22×D4×C53—22no—
31C22×D4×C53—24no—
32C22×D4×C53—24no—
33C22×D4×C53—24no—
34C22×D4×C53—28no—
35C22×D4×C53—42no—
36C22×D4×C53D412yescomplemented
37C22×D4×C53C2312yes—
38C22×D4×C53C2314yes—
39C23×C4×C53D411yescharacteristic, complemented
40#127×C53—24no—
41#124×C53—24no—
42#123×C53—24no—
43#123×C53—24no—
44#123×C53—28no—
45#123×C53D418yescomplemented
46C22⋊Q8×C53—24no—
47#121×C53—24no—
48#121×C53—28no—
49#121×C53—28no—
50#121×C53—28no—
51#121×C53—44no—
52#120×C53—28no—
53#120×C53—24no—
54#120×C53—28no—
55#120×C53—24no—
56#120×C53—24no—
57#120×C53—24no—
58#120×C53—28no—
59#120×C53D418yescomplemented
60C4×D4×C53—28no—
61C4×D4×C53—28no—
62#117×C53—22no—
63C2×C22⋊C4×C53—24no—
64C2×C22⋊C4×C53—22no—
65C2×C22⋊C4×C53—24no—
66C2×C22⋊C4×C53—42no—
67C2×C22⋊C4×C53C2314yes—
68C2×C22⋊C4×C53C2314yes—
69#99×C53—216no—
70#99×C53—28no—
71C24×C53—22no—
72C24×C53—24no—
73C24×C53—24no—
74C24×C53—22no—
75C24×C53—41no—
76C24×C53—82no—
77C24×C53—82no—
78C24×C53C2×D411yescharacteristic
79C24×C53C2×D411yescharacteristic
80C24×C53C2×D412yes—
81C24×C53C2×D414yescomplemented
82C24×C53C2411yescharacteristic
83#41×C53—416no—
84#41×C53—44no—
85#41×C53—44no—
86C2×Q8×C53—22no—
87C2×D4×C53—24no—
88C2×D4×C53—24no—
89C2×D4×C53—28no—
90C2×D4×C53—24no—
91C2×D4×C53—22no—
92C2×D4×C53—24no—
93C2×D4×C53—28no—
94C2×D4×C53—24no—
95C2×D4×C53—24no—
96C2×D4×C53—28no—
97C2×D4×C53—28no—
98C2×D4×C53—28no—
99C2×D4×C53—24no—
100C2×D4×C53—42no—
101C2×D4×C53—416no—
102C2×D4×C53—416no—
103C2×D4×C53—44no—
104C2×D4×C53—44no—
105C2×D4×C53—44no—
106C2×D4×C53—88no—
107C2×D4×C53—88no—
108C2×D4×C53C2×D418yes—
109C22×C4×C53—22no—
110C22×C4×C53—24no—
111C22×C4×C53—24no—
112C22×C4×C53—24no—
113C22×C4×C53—24no—
114C22×C4×C53—24no—
115C22×C4×C53—41no—
116C22×C4×C53C2×D412yes—
117C22×C4×C53C2×D414yescomplemented
118C4⋊C4×C53—24no—
119C4⋊C4×C53—28no—
120C4⋊C4×C53—44no—
121C22⋊C4×C53—28no—
122C22⋊C4×C53—24no—
123C22⋊C4×C53—28no—
124C22⋊C4×C53—28no—
125C22⋊C4×C53—24no—
126C22⋊C4×C53—28no—
127C22⋊C4×C53—28no—
128C22⋊C4×C53—48no—
129C22⋊C4×C53—84no—
130C22⋊C4×C53C2×D418yescomplemented
131C42×C53—24no—
132C23×C53—22no—
133C23×C53—22no—
134C23×C53—21no—
135C23×C53—22no—
136C23×C53—24no—
137C23×C53—22no—
138C23×C53—21no—
139C23×C53—28no—
140C23×C53—42no—
141C23×C53—44no—
142C23×C53—48no—
143C23×C53—48no—
144C23×C53—44no—
145C23×C53—44no—
146C23×C53—44no—
147C23×C53—82no—
148C23×C53—81no—
149C23×C53—81no—
150C23×C53—82no—
151C23×C53—84no—
152C23×C53—82no—
153C23×C53—84no—
154C23×C53—84no—
155C23×C53—84no—
156C23×C53—82no—
157C23×C53#12014yes—
158C23×C53C22×D411yescharacteristic
159C23×C53C22×D414yes—
160C23×C53H(5,𝔽2)12yes—
161Q8×C53—44no—
162D4×C53—216no—
163D4×C53—48no—
164D4×C53—44no—
165D4×C53—48no—
166D4×C53—416no—
167D4×C53—44no—
168D4×C53—44no—
169D4×C53—48no—
170D4×C53—88no—
171D4×C53—88no—
172D4×C53—164no—
173C2×C4×C53—22no—
174C2×C4×C53—22no—
175C2×C4×C53—24no—
176C2×C4×C53—28no—
177C2×C4×C53—28no—
178C2×C4×C53—44no—
179C2×C4×C53—48no—
180C2×C4×C53—44no—
181C2×C4×C53—44no—
182C2×C4×C53—84no—
183C2×C4×C53—84no—
184C2×C4×C53—84no—
185C2×C4×C53#12014yes—
186#23895C5311yesmaximal, characteristic, direct product factor, Sylow
187C22×C53—24no—
188C22×C53—22no—
189C22×C53—24no—
190C22×C53—42no—
191C22×C53—44no—
192C22×C53—41no—
193C22×C53—44no—
194C22×C53—44no—
195C22×C53—42no—
196C22×C53—41no—
197C22×C53—48no—
198C22×C53—44no—
199C22×C53—42no—
200C22×C53—41no—
201C22×C53—44no—
202C22×C53—41no—
203C22×C53—82no—
204C22×C53—88no—
205C22×C53—88no—
206C22×C53—88no—
207C22×C53—164no—
208C22×C53C2×#12012yes—
209C22×C53#53411yescharacteristic
210C22×C53D4214yes—
211C4×C53—24no—
212C4×C53—48no—
213C4×C53—44no—
214C4×C53—84no—
215#3484C2×C5312yescomplemented
216#3026C2×C5314yescomplemented
217#2021C2×C5314yescomplemented
218#1913C2×C5314yescomplemented
219#1781C2×C5311yescharacteristic, complemented
220C2×C53—22no—
221C2×C53—24no—
222C2×C53—21no—
223C2×C53—42no—
224C2×C53—44no—
225C2×C53—81no—
226C2×C53—84no—
227C2×C53#240311yescharacteristic
228C2×H(5,𝔽2)C22×C5312yescomplemented
229C23×D4C22×C5311yescharacteristic
230#546—28no—
231D42—28no—
232#535—22no—
233#534—22no—
234#534C22×C5314yes—
235#534C22×C5314yescomplemented
236#525—22no—
237C2×#120—22no—
238C2×#120C22×C5314yescomplemented
239#518C22×C5312yescomplemented
240#458—28no—
241B(4,𝔽2)—28no—
242C2×#99C22×C5318yescomplemented
243C2×#99C22×C5314yescomplemented
244#379C22×C5312yescomplemented
245#342C22×C5314yescomplemented
246C53#2389511yescharacteristic, direct product factor, Sylow
247C25D4×C5311yescharacteristic, complemented
248C25C23×C5311yescharacteristic
249H(5,𝔽2)—28no—
250H(5,𝔽2)—44no—
251C2×#41—22no—
252C2×#41—24no—
253C22×D4—22no—
254C22×D4—24no—
255C22×D4—24no—
256C22×D4—24no—
257C22×D4—28no—
258C22×D4—42no—
259C22×D4D4×C5312yescomplemented
260C22×D4C23×C5312yes—
261C22×D4C23×C5314yes—
262C23×C4D4×C5311yescharacteristic, complemented
263#127—24no—
264#124—24no—
265#123—24no—
266#123—24no—
267#123—28no—
268#123D4×C5318yescomplemented
269C22⋊Q8—24no—
270#121—24no—
271#121—28no—
272#121—28no—
273#121—28no—
274#121—44no—
275#120—28no—
276#120—24no—
277#120—24no—
278#120—24no—
279#120—24no—
280#120—28no—
281#120—28no—
282#120D4×C5318yescomplemented
283C4×D4—28no—
284C4×D4—28no—
285#117—22no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4C23×C5314yes—
291C2×C22⋊C4C23×C5314yes—
292#99—28no—
293#99—216no—
294C24—22no—
295C24—24no—
296C24—24no—
297C24—22no—
298C24—41no—
299C24—82no—
300C24—82no—
301C24C2×D4×C5311yescharacteristic
302C24C2×D4×C5311yescharacteristic
303C24C2×D4×C5312yes—
304C24C2×D4×C5314yescomplemented
305C24C24×C5311yescharacteristic
306#41—416no—
307#41—44no—
308#41—44no—
309C2×Q8—22no—
310C2×D4—24no—
311C2×D4—24no—
312C2×D4—28no—
313C2×D4—24no—
314C2×D4—22no—
315C2×D4—24no—
316C2×D4—28no—
317C2×D4—24no—
318C2×D4—24no—
319C2×D4—28no—
320C2×D4—28no—
321C2×D4—28no—
322C2×D4—24no—
323C2×D4—42no—
324C2×D4—416no—
325C2×D4—416no—
326C2×D4—44no—
327C2×D4—44no—
328C2×D4—44no—
329C2×D4—88no—
330C2×D4—88no—
331C2×D4C2×D4×C5318yes—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—41no—
339C22×C4C2×D4×C5312yes—
340C22×C4C2×D4×C5314yescomplemented
341C4⋊C4—24no—
342C4⋊C4—28no—
343C4⋊C4—44no—
344C22⋊C4—28no—
345C22⋊C4—24no—
346C22⋊C4—28no—
347C22⋊C4—28no—
348C22⋊C4—24no—
349C22⋊C4—28no—
350C22⋊C4—28no—
351C22⋊C4—48no—
352C22⋊C4—84no—
353C22⋊C4C2×D4×C5318yescomplemented
354C42—24no—
355C23—22no—
356C23—22no—
357C23—21no—
358C23—22no—
359C23—22no—
360C23—24no—
361C23—21no—
362C23—28no—
363C23—42no—
364C23—44no—
365C23—44no—
366C23—48no—
367C23—48no—
368C23—44no—
369C23—44no—
370C23—82no—
371C23—81no—
372C23—81no—
373C23—82no—
374C23—84no—
375C23—82no—
376C23—84no—
377C23—84no—
378C23—84no—
379C23—82no—
380C23#120×C5314yes—
381C23C22×D4×C5311yescharacteristic
382C23C22×D4×C5314yes—
383C23H(5,𝔽2)×C5312yes—
384Q8—44no—
385D4—216no—
386D4—48no—
387D4—44no—
388D4—48no—
389D4—416no—
390D4—44no—
391D4—44no—
392D4—48no—
393D4—88no—
394D4—88no—
395D4—164no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—28no—
399C2×C4—24no—
400C2×C4—28no—
401C2×C4—44no—
402C2×C4—48no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—84no—
406C2×C4—84no—
407C2×C4—84no—
408C2×C4#120×C5314yes—
409C22—24no—
410C22—22no—
411C22—24no—
412C22—42no—
413C22—44no—
414C22—41no—
415C22—44no—
416C22—44no—
417C22—42no—
418C22—41no—
419C22—48no—
420C22—44no—
421C22—42no—
422C22—41no—
423C22—44no—
424C22—41no—
425C22—82no—
426C22—88no—
427C22—88no—
428C22—88no—
429C22—164no—
430C22C2×#120×C5312yes—
431C22C53×#53411yescharacteristic
432C22D42×C5314yes—
433C4—24no—
434C4—48no—
435C4—44no—
436C4—84no—
437C2—22no—
438C2—24no—
439C2—21no—
440C2—42no—
441C2—44no—
442C2—81no—
443C2—84no—
444C2C53×#240311yescharacteristic
subgroupG/H
center, socleC2×C53#2403
commutator, Frattini, cosocleC24C24×C53
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C2×C53 ◁ C24×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C53×#3484C2semidirect
2. C53×#3026C2semidirect
3. C53×#2021C2semidirect
4. C53×#1913C2semidirect
5. C53×#1781C2semidirect
6. C2×H(5,𝔽2)×C53C22semidirect
13. C53×#534C22semidirect
16. C2×#120×C53C22semidirect
17. C53×#518C22semidirect
20. C2×#99×C53C22semidirect
21. C2×#99×C53C22semidirect
22. C53×#379C22semidirect
23. C53×#342C22semidirect
24. C25×C53D4semidirect
36. C22×D4×C53D4semidirect
39. C23×C4×C53D4semidirect
45. #123×C53D4semidirect
59. #120×C53D4semidirect
81. C24×C53C2×D4semidirect
117. C22×C4×C53C2×D4semidirect
130. C22⋊C4×C53C2×D4semidirect
186. #23895C53direct (N × H)
215. #3484C2×C53semidirect
216. #3026C2×C53semidirect
217. #2021C2×C53semidirect
218. #1913C2×C53semidirect
219. #1781C2×C53semidirect
228. C2×H(5,𝔽2)C22×C53semidirect
235. #534C22×C53semidirect
238. C2×#120C22×C53semidirect
239. #518C22×C53semidirect
242. C2×#99C22×C53semidirect
243. C2×#99C22×C53semidirect
244. #379C22×C53semidirect
245. #342C22×C53semidirect
246. C53#23895direct (N × H)
247. C25D4×C53semidirect
259. C22×D4D4×C53semidirect
262. C23×C4D4×C53semidirect
268. #123D4×C53semidirect
282. #120D4×C53semidirect
304. C24C2×D4×C53semidirect
340. C22×C4C2×D4×C53semidirect
353. C22⋊C4C2×D4×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#34842
C53×#30262
C53×#20212
C53×#19132
C53×#17812
#2389553