Finite Group Databasebeta

← C53×#12202 #27219714

Subgroups — C53×#12202 #27219714

Order 13568.

976 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 976 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2029C211yesmaximal, characteristic, complemented
3C53×#1596C212yesmaximal, complemented
4C53×#1343C212yesmaximal, complemented
5C53×#1316C211yesmaximal, characteristic, complemented
6C23×D4×C53—21no—
7C53×#543C2211yescharacteristic, complemented
8C2×#120×C53—22no—
9C53×#450—22no—
10C53×#392—21no—
11C53×#392C2212yes—
12C53×#386—22no—
13C53×#379—21no—
14C53×#379C2211yescharacteristic
15C53×#342—21no—
16C53×#342C2211yescharacteristic, complemented
17C53×#328—22no—
18C53×#323C2212yescomplemented
19C25×C53—21no—
20C25×C53D411yescharacteristic
21C22×D4×C53—22no—
22C22×D4×C53—22no—
23C22×D4×C53—21no—
24C22×D4×C53—48no—
25C22×D4×C53—42no—
26C22×D4×C53—42no—
27C22×D4×C53D412yes—
28C23×C4×C53—21no—
29C2×SD16×C53—42no—
30C22⋊Q8×C53—22no—
31C22⋊Q8×C53—42no—
32#120×C53—48no—
33C2×C22⋊C4×C53—22no—
34C2×C22⋊C4×C53—21no—
35C2×C22⋊C4×C53—22no—
36C2×C22⋊C4×C53—22no—
37C2×C22⋊C4×C53—21no—
38C2×C22⋊C4×C53—42no—
39C2×C22⋊C4×C53D411yescharacteristic
40C2×C22⋊C4×C53C2311yescharacteristic
41#102×C53—42no—
42C22⋊C8×C53—22no—
43#95×C53—21no—
44#95×C53D412yes—
45C24×C53—21no—
46C24×C53—22no—
47C24×C53—21no—
48C24×C53—41no—
49C24×C53—44no—
50C24×C53—42no—
51C24×C53—44no—
52C24×C53—41no—
53C24×C53—44no—
54C24×C53—81no—
55C24×C53—81no—
56C24×C53SD1612yescomplemented
57C24×C53C2×D411yescharacteristic
58C2×Q8×C53—22no—
59C2×D4×C53—28no—
60C2×D4×C53—44no—
61C2×D4×C53—42no—
62C2×D4×C53—44no—
63C2×D4×C53—44no—
64C2×D4×C53—44no—
65C2×D4×C53—416no—
66C2×D4×C53—416no—
67C2×D4×C53—42no—
68C2×D4×C53—42no—
69C2×D4×C53—44no—
70C2×D4×C53—44no—
71C2×D4×C53—81no—
72C22×C4×C53—22no—
73C22×C4×C53—21no—
74C22×C4×C53—22no—
75C22×C4×C53—44no—
76C22×C4×C53—42no—
77C22×C4×C53—41no—
78C22×C4×C53C2×D412yes—
79SD16×C53—84no—
80C2×C8×C53—42no—
81C4⋊C4×C53—22no—
82C4⋊C4×C53—42no—
83C22⋊C4×C53—41no—
84C22⋊C4×C53—44no—
85C22⋊C4×C53—44no—
86C22⋊C4×C53—44no—
87C22⋊C4×C53—41no—
88C22⋊C4×C53—41no—
89C22⋊C4×C53—41no—
90C22⋊C4×C53—41no—
91C22⋊C4×C53—41no—
92C22⋊C4×C53—84no—
93C23×C53—24no—
94C23×C53—22no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—24no—
98C23×C53—42no—
99C23×C53—44no—
100C23×C53—42no—
101C23×C53—42no—
102C23×C53—42no—
103C23×C53—44no—
104C23×C53—42no—
105C23×C53—42no—
106C23×C53—41no—
107C23×C53—41no—
108C23×C53—44no—
109C23×C53—48no—
110C23×C53—42no—
111C23×C53—42no—
112C23×C53—42no—
113C23×C53—81no—
114C23×C53—84no—
115C23×C53—84no—
116C23×C53—82no—
117C23×C53—88no—
118C23×C53—88no—
119C23×C53—84no—
120C23×C53#12011yescharacteristic
121C23×C53C2×SD1611yescharacteristic
122C23×C53AGL(1,ℤ/8)11yescharacteristic
123Q8×C53—82no—
124D4×C53—416no—
125D4×C53—44no—
126D4×C53—48no—
127D4×C53—48no—
128D4×C53—82no—
129D4×C53—84no—
130C2×C4×C53—24no—
131C2×C4×C53—22no—
132C2×C4×C53—42no—
133C2×C4×C53—48no—
134C2×C4×C53—42no—
135C2×C4×C53—42no—
136C2×C4×C53—42no—
137C2×C4×C53—41no—
138C2×C4×C53—42no—
139C2×C4×C53—82no—
140C2×C4×C53—81no—
141C2×C4×C53—82no—
142C8×C53—82no—
143#12202C5311yesmaximal, characteristic, direct product factor, Sylow
144C22×C53—22no—
145C22×C53—24no—
146C22×C53—21no—
147C22×C53—21no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53—42no—
151C22×C53—44no—
152C22×C53—44no—
153C22×C53—44no—
154C22×C53—44no—
155C22×C53—42no—
156C22×C53—42no—
157C22×C53—44no—
158C22×C53—44no—
159C22×C53—81no—
160C22×C53—82no—
161C22×C53—82no—
162C22×C53—84no—
163C22×C53—82no—
164C22×C53—816no—
165C22×C53—82no—
166C22×C53—82no—
167C22×C53#45011yescharacteristic
168C22×C53#45511yescharacteristic
169C22×C53B(4,𝔽2)11yescharacteristic
170C4×C53—44no—
171C4×C53—42no—
172C4×C53—81no—
173C4×C53—82no—
174C4×C53—161no—
175#2403C2×C5311yescharacteristic
176#2029C2×C5311yescharacteristic, complemented
177#1596C2×C5312yescomplemented
178#1343C2×C5312yescomplemented
179#1316C2×C5311yescharacteristic, complemented
180C2×C53—21no—
181C2×C53—21no—
182C2×C53—41no—
183C2×C53—41no—
184C2×C53—44no—
185C2×C53—84no—
186C2×C53#160111yescharacteristic
187C2×C53#219612yes—
188C23×D4—21no—
189#543C22×C5311yescharacteristic, complemented
190C2×#120—22no—
191#450—22no—
192#392—21no—
193#392C22×C5312yes—
194#386—22no—
195#379—21no—
196#379C22×C5311yescharacteristic
197#342—21no—
198#342C22×C5311yescharacteristic, complemented
199#328—22no—
200#323C22×C5312yescomplemented
201C53#1220211yescharacteristic, direct product factor, Sylow
202C25—21no—
203C25D4×C5311yescharacteristic
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—21no—
207C22×D4—48no—
208C22×D4—42no—
209C22×D4—42no—
210C22×D4D4×C5312yes—
211C23×C4—21no—
212C2×SD16—42no—
213C22⋊Q8—22no—
214C22⋊Q8—42no—
215#120—48no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4D4×C5311yescharacteristic
223C2×C22⋊C4C23×C5311yescharacteristic
224#102—42no—
225C22⋊C8—22no—
226#95—21no—
227#95D4×C5312yes—
228C24—21no—
229C24—22no—
230C24—21no—
231C24—41no—
232C24—44no—
233C24—42no—
234C24—44no—
235C24—41no—
236C24—44no—
237C24—81no—
238C24—81no—
239C24SD16×C5312yescomplemented
240C24C2×D4×C5311yescharacteristic
241C2×Q8—22no—
242C2×D4—28no—
243C2×D4—44no—
244C2×D4—42no—
245C2×D4—44no—
246C2×D4—44no—
247C2×D4—44no—
248C2×D4—416no—
249C2×D4—416no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—44no—
253C2×D4—44no—
254C2×D4—81no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—44no—
259C22×C4—42no—
260C22×C4—41no—
261C22×C4C2×D4×C5312yes—
262SD16—84no—
263C2×C8—42no—
264C4⋊C4—22no—
265C4⋊C4—42no—
266C22⋊C4—41no—
267C22⋊C4—44no—
268C22⋊C4—44no—
269C22⋊C4—44no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—84no—
276C23—24no—
277C23—22no—
278C23—21no—
279C23—21no—
280C23—24no—
281C23—42no—
282C23—42no—
283C23—44no—
284C23—42no—
285C23—42no—
286C23—42no—
287C23—42no—
288C23—44no—
289C23—41no—
290C23—41no—
291C23—44no—
292C23—48no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23—81no—
297C23—84no—
298C23—84no—
299C23—82no—
300C23—88no—
301C23—88no—
302C23—84no—
303C23#120×C5311yescharacteristic
304C23C2×SD16×C5311yescharacteristic
305C23AGL(1,ℤ/8)×C5311yescharacteristic
306Q8—82no—
307D4—416no—
308D4—44no—
309D4—48no—
310D4—48no—
311D4—82no—
312D4—84no—
313C2×C4—24no—
314C2×C4—22no—
315C2×C4—42no—
316C2×C4—48no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—82no—
323C2×C4—81no—
324C2×C4—82no—
325C8—82no—
326C22—22no—
327C22—24no—
328C22—21no—
329C22—21no—
330C22—41no—
331C22—41no—
332C22—42no—
333C22—44no—
334C22—44no—
335C22—44no—
336C22—44no—
337C22—42no—
338C22—44no—
339C22—42no—
340C22—44no—
341C22—81no—
342C22—82no—
343C22—82no—
344C22—84no—
345C22—82no—
346C22—816no—
347C22—82no—
348C22—82no—
349C22C53×#45011yescharacteristic
350C22C53×#45511yescharacteristic
351C22C53×B(4,𝔽2)11yescharacteristic
352C4—44no—
353C4—42no—
354C4—81no—
355C4—82no—
356C4—161no—
357C2—21no—
358C2—21no—
359C2—41no—
360C2—44no—
361C2—41no—
362C2—84no—
363C2C53×#160111yescharacteristic
364C2C53×#219612yes—
subgroupG/H
center, socleC22×C53B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C53
prime radicalC53×#2403C2
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C53 ◁ C2×C22⋊C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#2029C2semidirect
3. C53×#1596C2semidirect
4. C53×#1343C2semidirect
5. C53×#1316C2semidirect
7. C53×#543C22semidirect
16. C53×#342C22semidirect
18. C53×#323C22semidirect
56. C24×C53SD16semidirect
143. #12202C53direct (N × H)
176. #2029C2×C53semidirect
177. #1596C2×C53semidirect
178. #1343C2×C53semidirect
179. #1316C2×C53semidirect
189. #543C22×C53semidirect
198. #342C22×C53semidirect
200. #323C22×C53semidirect
201. C53#12202direct (N × H)
239. C24SD16×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#24032
C53×#20292
C53×#15962
C53×#13432
C53×#13162
#1220253