Finite Group Databasebeta

← C53×#10337 #27217849

Subgroups — C53×#10337 #27217849

Order 13568.

1274 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 1274 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2405C212yesmaximal, complemented
2C53×#2403C211yesmaximal, characteristic
3C53×#1819C211yesmaximal, characteristic, complemented
4C53×#1781C212yesmaximal, complemented
5C53×#1304C211yesmaximal, characteristic, complemented
6C23×D4×C53C2212yes—
7C2×#123×C53—22no—
8C2×#120×C53—22no—
9C22×C22⋊C4×C53C2211yescharacteristic
10C2×#99×C53—24no—
11C53×#394—24no—
12C53×#392—22no—
13C53×#386—22no—
14C53×#386—22no—
15C53×#385—22no—
16C53×#379—22no—
17C53×#379C2211yescharacteristic
18C53×#379C2211yescharacteristic
19C53×#379C2212yes—
20C53×#342—22no—
21C25×C53D412yes—
22C25×C53C2311yescharacteristic
23C22×D4×C53—22no—
24C22×D4×C53—22no—
25C22×D4×C53—22no—
26C22×D4×C53—22no—
27C22×D4×C53—22no—
28C22×D4×C53—22no—
29C22×D4×C53—22no—
30C22×D4×C53—48no—
31C23×C4×C53D412yes—
32C23×C4×C53D412yescomplemented
33#123×C53—48no—
34#120×C53—48no—
35C2×C4⋊C4×C53—24no—
36C2×C22⋊C4×C53—24no—
37C2×C22⋊C4×C53—24no—
38C2×C22⋊C4×C53—24no—
39C2×C22⋊C4×C53—21no—
40C2×C22⋊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—21no—
47C2×C22⋊C4×C53—22no—
48C2×C22⋊C4×C53D412yes—
49C2×C22⋊C4×C53D414yescomplemented
50#99×C53—48no—
51#95×C53—24no—
52#95×C53—22no—
53C24×C53—22no—
54C24×C53—22no—
55C24×C53—22no—
56C24×C53—44no—
57C24×C53—44no—
58C24×C53—42no—
59C24×C53—44no—
60C24×C53—44no—
61C24×C53C2×D412yes—
62C24×C53C2×D411yescharacteristic
63C24×C53C2×D411yescharacteristic
64C24×C53C2×D412yes—
65C24×C53#4111yescharacteristic
66C2×D4×C53—28no—
67C2×D4×C53—28no—
68C2×D4×C53—28no—
69C2×D4×C53—44no—
70C2×D4×C53—44no—
71C2×D4×C53—44no—
72C2×D4×C53—44no—
73C2×D4×C53—44no—
74C2×D4×C53—44no—
75C2×D4×C53—44no—
76C2×D4×C53—44no—
77C2×D4×C53—44no—
78C2×D4×C53—42no—
79C2×D4×C53—42no—
80C2×D4×C53—42no—
81C2×D4×C53—42no—
82C2×D4×C53—48no—
83C2×D4×C53—82no—
84C22×C4×C53—22no—
85C22×C4×C53—22no—
86C22×C4×C53—22no—
87C22×C4×C53—22no—
88C22×C4×C53—22no—
89C22×C4×C53—24no—
90C22×C4×C53—22no—
91C22×C4×C53—22no—
92C22×C4×C53—44no—
93C22×C4×C53—44no—
94C22×C4×C53—41no—
95C4⋊C4×C53—48no—
96C22⋊C4×C53—48no—
97C22⋊C4×C53—44no—
98C22⋊C4×C53—44no—
99C22⋊C4×C53—48no—
100C22⋊C4×C53—44no—
101C22⋊C4×C53—41no—
102C22⋊C4×C53—42no—
103C22⋊C4×C53—41no—
104C22⋊C4×C53—44no—
105C22⋊C4×C53—42no—
106C22⋊C4×C53—44no—
107C22⋊C4×C53—42no—
108C22⋊C4×C53—84no—
109C23×C53—24no—
110C23×C53—24no—
111C23×C53—24no—
112C23×C53—24no—
113C23×C53—22no—
114C23×C53—24no—
115C23×C53—24no—
116C23×C53—24no—
117C23×C53—42no—
118C23×C53—41no—
119C23×C53—42no—
120C23×C53—42no—
121C23×C53—42no—
122C23×C53—42no—
123C23×C53—41no—
124C23×C53—42no—
125C23×C53—42no—
126C23×C53—42no—
127C23×C53—44no—
128C23×C53—44no—
129C23×C53—42no—
130C23×C53—44no—
131C23×C53—44no—
132C23×C53—82no—
133C23×C53—82no—
134C23×C53—88no—
135C23×C53—816no—
136C23×C53#12012yes—
137C23×C53#12011yescharacteristic
138C23×C53#12111yescharacteristic
139C23×C53#12112yes—
140C23×C53#12711yescharacteristic
141D4×C53—48no—
142D4×C53—48no—
143D4×C53—48no—
144D4×C53—48no—
145D4×C53—88no—
146C2×C4×C53—24no—
147C2×C4×C53—24no—
148C2×C4×C53—24no—
149C2×C4×C53—42no—
150C2×C4×C53—42no—
151C2×C4×C53—42no—
152C2×C4×C53—42no—
153C2×C4×C53—44no—
154C2×C4×C53—44no—
155C2×C4×C53—44no—
156C2×C4×C53—44no—
157C2×C4×C53—42no—
158C2×C4×C53—44no—
159C2×C4×C53—44no—
160C2×C4×C53—42no—
161C2×C4×C53—82no—
162C2×C4×C53—82no—
163C2×C4×C53—81no—
164#10337C5311yesmaximal, characteristic, direct product factor, Sylow
165C22×C53—22no—
166C22×C53—22no—
167C22×C53—22no—
168C22×C53—22no—
169C22×C53—22no—
170C22×C53—22no—
171C22×C53—22no—
172C22×C53—21no—
173C22×C53—22no—
174C22×C53—21no—
175C22×C53—44no—
176C22×C53—48no—
177C22×C53—48no—
178C22×C53—48no—
179C22×C53—44no—
180C22×C53—82no—
181C22×C53—84no—
182C22×C53—84no—
183C22×C53—84no—
184C22×C53—82no—
185C22×C53—84no—
186C22×C53—84no—
187C22×C53—82no—
188C22×C53—82no—
189C22×C53#39211yescharacteristic
190C22×C53B(4,𝔽2)14yes—
191C22×C53#45812yes—
192C4×C53—44no—
193C4×C53—84no—
194C4×C53—82no—
195C4×C53—82no—
196C4×C53—161no—
197#2405C2×C5312yescomplemented
198#2403C2×C5311yescharacteristic
199#1819C2×C5311yescharacteristic, complemented
200#1781C2×C5312yescomplemented
201#1304C2×C5311yescharacteristic, complemented
202C2×C53—22no—
203C2×C53—24no—
204C2×C53—41no—
205C2×C53—41no—
206C2×C53—42no—
207C2×C53—84no—
208C2×C53#202112yes—
209C2×C53#202211yescharacteristic
210C23×D4C22×C5312yes—
211C2×#123—22no—
212C2×#120—22no—
213C22×C22⋊C4C22×C5311yescharacteristic
214C2×#99—24no—
215#394—24no—
216#392—22no—
217#386—22no—
218#386—22no—
219#385—22no—
220#379—22no—
221#379C22×C5311yescharacteristic
222#379C22×C5311yescharacteristic
223#379C22×C5312yes—
224#342—22no—
225C53#1033711yescharacteristic, direct product factor, Sylow
226C25D4×C5312yes—
227C25C23×C5311yescharacteristic
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4—22no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4—22no—
235C22×D4—48no—
236C23×C4D4×C5312yes—
237C23×C4D4×C5312yescomplemented
238#123—48no—
239#120—48no—
240C2×C4⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4D4×C5312yes—
254C2×C22⋊C4D4×C5314yescomplemented
255#99—48no—
256#95—24no—
257#95—22no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24—44no—
262C24—44no—
263C24—42no—
264C24—44no—
265C24—44no—
266C24C2×D4×C5312yes—
267C24C2×D4×C5311yescharacteristic
268C24C2×D4×C5311yescharacteristic
269C24C2×D4×C5312yes—
270C24#41×C5311yescharacteristic
271C2×D4—28no—
272C2×D4—28no—
273C2×D4—28no—
274C2×D4—44no—
275C2×D4—44no—
276C2×D4—44no—
277C2×D4—44no—
278C2×D4—44no—
279C2×D4—44no—
280C2×D4—44no—
281C2×D4—44no—
282C2×D4—44no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—48no—
288C2×D4—82no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4—41no—
300C4⋊C4—48no—
301C22⋊C4—48no—
302C22⋊C4—44no—
303C22⋊C4—44no—
304C22⋊C4—48no—
305C22⋊C4—44no—
306C22⋊C4—41no—
307C22⋊C4—42no—
308C22⋊C4—41no—
309C22⋊C4—44no—
310C22⋊C4—42no—
311C22⋊C4—44no—
312C22⋊C4—42no—
313C22⋊C4—84no—
314C23—24no—
315C23—24no—
316C23—24no—
317C23—24no—
318C23—22no—
319C23—24no—
320C23—24no—
321C23—24no—
322C23—42no—
323C23—41no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—41no—
329C23—44no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—44no—
334C23—42no—
335C23—44no—
336C23—44no—
337C23—82no—
338C23—82no—
339C23—88no—
340C23—816no—
341C23#120×C5312yes—
342C23#120×C5311yescharacteristic
343C23#121×C5312yes—
344C23#121×C5311yescharacteristic
345C23#127×C5311yescharacteristic
346D4—48no—
347D4—48no—
348D4—48no—
349D4—48no—
350D4—88no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—81no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—21no—
377C22—22no—
378C22—21no—
379C22—48no—
380C22—44no—
381C22—48no—
382C22—48no—
383C22—44no—
384C22—82no—
385C22—84no—
386C22—84no—
387C22—84no—
388C22—82no—
389C22—84no—
390C22—84no—
391C22—82no—
392C22—82no—
393C22C53×#39211yescharacteristic
394C22C53×B(4,𝔽2)14yes—
395C22C53×#45812yes—
396C4—44no—
397C4—82no—
398C4—84no—
399C4—82no—
400C4—161no—
401C2—22no—
402C2—24no—
403C2—41no—
404C2—42no—
405C2—41no—
406C2—84no—
407C2C53×#202112yes—
408C2C53×#202211yescharacteristic
subgroupG/H
center, socleC22×C53#392
commutator, Frattini, cosocleC25C23×C53
prime radicalC53×#2403C2
LCSG ▷ C25 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C25×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C53×#2405C2semidirect
3. C53×#1819C2semidirect
4. C53×#1781C2semidirect
5. C53×#1304C2semidirect
32. C23×C4×C53D4semidirect
49. C2×C22⋊C4×C53D4semidirect
164. #10337C53direct (N × H)
197. #2405C2×C53semidirect
199. #1819C2×C53semidirect
200. #1781C2×C53semidirect
201. #1304C2×C53semidirect
225. C53#10337direct (N × H)
237. C23×C4D4×C53semidirect
254. C2×C22⋊C4D4×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#24052
C53×#24032
C53×#18192
C53×#17812
C53×#13042
#1033753