Finite Group Databasebeta

← C53×#12248 #27219760

Subgroups — C53×#12248 #27219760

Order 13568.

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

normal subgroups only

This group has 428 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×#2609C211yesmaximal, characteristic
2C53×#2061C211yesmaximal, characteristic, complemented
3C53×#1611C211yesmaximal, characteristic, complemented
4C53×#1599C211yesmaximal, characteristic
5C53×#1346C211yesmaximal, characteristic, complemented
6C53×#1345C211yesmaximal, characteristic
7C53×#1324C211yesmaximal, characteristic, complemented
8C53×#544C2211yescharacteristic
9C2×#117×C53—21no—
10C53×#483—21no—
11C53×#450—21no—
12C53×#428—21no—
13C53×#398C2211yescharacteristic
14C53×#396—21no—
15C53×#394C2211yescharacteristic
16C53×#388—21no—
17C53×#387—21no—
18C53×#386—21no—
19C53×#385—21no—
20C53×#382C2211yescharacteristic
21C53×#343C2211yescharacteristic
22C53×#339—21no—
23C53×#328—21no—
24C53×#328—21no—
25C53×#324C2211yescharacteristic
26C53×#324C2211yescharacteristic
27C22×D4×C53D411yescharacteristic
28C23×C4×C53—21no—
29C2×SD16×C53—41no—
30C2×M16×C53—21no—
31C4⋊Q8×C53—41no—
32#125×C53—41no—
33C22⋊Q8×C53—21no—
34C22⋊Q8×C53—21no—
35#117×C53—21no—
36#117×C53—42no—
37#117×C53—42no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53D411yescharacteristic
42C2×C4⋊C4×C53D411yescharacteristic
43C2×C4⋊C4×C53C2311yescharacteristic
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53—21no—
49C2×C42×C53—21no—
50C2×C42×C53D411yescharacteristic
51#107×C53—41no—
52#106×C53—41no—
53#106×C53—41no—
54#103×C53—41no—
55#102×C53—41no—
56C22⋊C8×C53—21no—
57C22⋊C8×C53—21no—
58#95×C53—21no—
59#95×C53—21no—
60#95×C53—21no—
61#95×C53—21no—
62#95×C53D411yescharacteristic
63#95×C53D411yescharacteristic
64C24×C53—21no—
65C2×Q8×C53—21no—
66C2×Q8×C53—21no—
67C2×D4×C53—41no—
68C2×D4×C53—42no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—42no—
75C22×C4×C53—42no—
76C22×C4×C53—42no—
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80SD16×C53—82no—
81M16×C53—22no—
82M16×C53—41no—
83C2×C8×C53—41no—
84C2×C8×C53—41no—
85C2×C8×C53—41no—
86C4⋊C4×C53—21no—
87C4⋊C4×C53—21no—
88C4⋊C4×C53—21no—
89C4⋊C4×C53—21no—
90C4⋊C4×C53—41no—
91C4⋊C4×C53—42no—
92C4⋊C4×C53—42no—
93C4⋊C4×C53—41no—
94C4⋊C4×C53—41no—
95C4⋊C4×C53—41no—
96C4⋊C4×C53—41no—
97C4⋊C4×C53—41no—
98C4⋊C4×C53—41no—
99C22⋊C4×C53—42no—
100C22⋊C4×C53—42no—
101C22⋊C4×C53—42no—
102C22⋊C4×C53—41no—
103C22⋊C4×C53—41no—
104C22⋊C4×C53—41no—
105C22⋊C4×C53—81no—
106C42×C53—21no—
107C42×C53—42no—
108C42×C53—41no—
109C42×C53SD1612yescomplemented
110C23×C53—22no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53—41no—
115C23×C53—41no—
116C23×C53—41no—
117C23×C53#12011yescharacteristic
118Q8×C53—81no—
119Q8×C53—81no—
120D4×C53—42no—
121D4×C53—81no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—22no—
125C2×C4×C53—22no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—48no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53C2×SD1611yescharacteristic
147C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
148C8×C53—42no—
149C8×C53—81no—
150C8×C53—81no—
151#12248C5311yesmaximal, characteristic, direct product factor, Sylow
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—42no—
155C22×C53—42no—
156C22×C53—42no—
157C22×C53—81no—
158C22×C53—81no—
159C22×C53—81no—
160C22×C53—81no—
161C22×C53#45011yescharacteristic
162C22×C53#45611yescharacteristic
163C22×C53B(4,𝔽2)11yescharacteristic
164C4×C53—21no—
165C4×C53—21no—
166C4×C53—42no—
167C4×C53—42no—
168C4×C53—41no—
169C4×C53—41no—
170C4×C53—81no—
171C4×C53—81no—
172C4×C53—81no—
173C4×C53—81no—
174C4×C53—81no—
175#2609C2×C5311yescharacteristic
176#2061C2×C5311yescharacteristic, complemented
177#1611C2×C5311yescharacteristic, complemented
178#1599C2×C5311yescharacteristic
179#1346C2×C5311yescharacteristic, complemented
180#1345C2×C5311yescharacteristic
181#1324C2×C5311yescharacteristic, complemented
182C2×C53—21no—
183C2×C53—21no—
184C2×C53—82no—
185C2×C53#160411yescharacteristic
186C2×C53#219711yescharacteristic
187C2×C53#220411yescharacteristic
188#544C22×C5311yescharacteristic
189C2×#117—21no—
190#483—21no—
191#450—21no—
192#428—21no—
193#398C22×C5311yescharacteristic
194#396—21no—
195#394C22×C5311yescharacteristic
196#388—21no—
197#387—21no—
198#386—21no—
199#385—21no—
200#382C22×C5311yescharacteristic
201#343C22×C5311yescharacteristic
202#339—21no—
203#328—21no—
204#328—21no—
205#324C22×C5311yescharacteristic
206#324C22×C5311yescharacteristic
207C53#1224811yescharacteristic, direct product factor, Sylow
208C22×D4D4×C5311yescharacteristic
209C23×C4—21no—
210C2×SD16—41no—
211C2×M16—21no—
212C4⋊Q8—41no—
213#125—41no—
214C22⋊Q8—21no—
215C22⋊Q8—21no—
216#117—21no—
217#117—42no—
218#117—42no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4D4×C5311yescharacteristic
223C2×C4⋊C4D4×C5311yescharacteristic
224C2×C4⋊C4C23×C5311yescharacteristic
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C42—21no—
231C2×C42D4×C5311yescharacteristic
232#107—41no—
233#106—41no—
234#106—41no—
235#103—41no—
236#102—41no—
237C22⋊C8—21no—
238C22⋊C8—21no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95D4×C5311yescharacteristic
244#95D4×C5311yescharacteristic
245C24—21no—
246C2×Q8—21no—
247C2×Q8—21no—
248C2×D4—41no—
249C2×D4—42no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4C2×D4×C5311yescharacteristic
259C22×C4C2×D4×C5311yescharacteristic
260C22×C4C2×D4×C5311yescharacteristic
261SD16—82no—
262M16—22no—
263M16—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—81no—
287C42—21no—
288C42—42no—
289C42—41no—
290C42SD16×C5312yescomplemented
291C23—22no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23#120×C5311yescharacteristic
299Q8—81no—
300Q8—81no—
301D4—42no—
302D4—81no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—48no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4C2×SD16×C5311yescharacteristic
328C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
329C8—42no—
330C8—81no—
331C8—81no—
332C22—21no—
333C22—21no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22C53×#45011yescharacteristic
342C22C53×#45611yescharacteristic
343C22C53×B(4,𝔽2)11yescharacteristic
344C4—21no—
345C4—21no—
346C4—42no—
347C4—42no—
348C4—41no—
349C4—41no—
350C4—81no—
351C4—81no—
352C4—81no—
353C4—81no—
354C4—81no—
355C2—21no—
356C2—21no—
357C2—82no—
358C2C53×#160411yescharacteristic
359C2C53×#219711yescharacteristic
360C2C53×#220411yescharacteristic
subgroupG/H
center, socleC22×C53B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C53
prime radicalC22×D4×C53D4
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C53 ◁ C2×C4⋊C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#2061C2semidirect
3. C53×#1611C2semidirect
5. C53×#1346C2semidirect
7. C53×#1324C2semidirect
109. C42×C53SD16semidirect
151. #12248C53direct (N × H)
176. #2061C2×C53semidirect
177. #1611C2×C53semidirect
179. #1346C2×C53semidirect
181. #1324C2×C53semidirect
207. C53#12248direct (N × H)
290. C42SD16×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#26092
C53×#20612
C53×#16112
C53×#15992
C53×#13462
C53×#13452
C53×#13242
#1224853