Finite Group Databasebeta

← C53×#8497 #27216009

Subgroups — C53×#8497 #27216009

Order 13568.

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

normal subgroups only

This group has 820 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
1C53×#3543C211yesmaximal, characteristic
2C53×#1961C211yesmaximal, characteristic, complemented
3C53×#1555C211yesmaximal, characteristic, complemented
4C53×#1532C212yesmaximal, complemented
5C53×#1512C212yesmaximal, complemented
6C53×#560—22no—
7C53×#559C411yescharacteristic, complemented
8C53×#550C412yescomplemented
9C53×#546—22no—
10D42×C53—22no—
11D42×C53C411yescharacteristic, complemented
12C53×#540—22no—
13C53×#535—22no—
14C53×#534—22no—
15C2×#127×C53C2211yescharacteristic
16C53×#529C2212yes—
17C53×#432C2212yescomplemented
18C2×#102×C53—22no—
19C2×#97×C53C2211yescharacteristic, complemented
20C53×#382C2211yescharacteristic, complemented
21C53×#331—22no—
22C53×#330—22no—
23C53×#329—22no—
24C53×#328—22no—
25C53×#327—22no—
26C2×#41×C53—22no—
27C22×D4×C53—22no—
28C22×D4×C53—22no—
29C22×D4×C53—22no—
30C22×D4×C53D412yes—
31C22×C8×C53—21no—
32#127×C53—22no—
33#127×C53—22no—
34#127×C53C2×C411yescharacteristic
35#127×C53C2×C411yescharacteristic
36#127×C53D412yes—
37#126×C53—22no—
38#125×C53D412yes—
39#124×C53—22no—
40#124×C53D412yes—
41#123×C53—22no—
42#123×C53—24no—
43#123×C53—22no—
44#123×C53—22no—
45C22⋊Q8×C53D412yes—
46#121×C53—24no—
47#121×C53—24no—
48#121×C53—24no—
49#121×C53—22no—
50#121×C53—22no—
51#121×C53—24no—
52#121×C53D412yes—
53#120×C53—22no—
54#120×C53—24no—
55#120×C53—24no—
56#120×C53—22no—
57C4×Q8×C53C2×C412yes—
58C4×D4×C53—22no—
59C4×D4×C53—22no—
60C4×D4×C53—22no—
61C4×D4×C53C2×C412yes—
62#117×C53—22no—
63C2×C4⋊C4×C53—21no—
64C2×C22⋊C4×C53—22no—
65C2×C42×C53C2311yescharacteristic
66C4⋊C8×C53—22no—
67#102×C53—44no—
68C22⋊C8×C53—22no—
69#97×C53—22no—
70#97×C53—22no—
71#95×C53—22no—
72C24×C53—22no—
73C24×C53—22no—
74#41×C53—84no—
75C2×Q8×C53C22⋊C412yes—
76C2×D4×C53—22no—
77C2×D4×C53—24no—
78C2×D4×C53—24no—
79C2×D4×C53—24no—
80C2×D4×C53—22no—
81C2×D4×C53—22no—
82C2×D4×C53—22no—
83C2×D4×C53—24no—
84C2×D4×C53—42no—
85C2×D4×C53—42no—
86C2×D4×C53—84no—
87C2×D4×C53—84no—
88C2×D4×C53—82no—
89C2×D4×C53—82no—
90C2×D4×C53C22⋊C412yes—
91C2×D4×C53C22⋊C412yes—
92C2×D4×C53C22⋊C412yes—
93C22×C4×C53—24no—
94C22×C4×C53—22no—
95C22×C4×C53—22no—
96C22×C4×C53—41no—
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53C2×D412yes—
99C2×C8×C53—22no—
100C2×C8×C53—41no—
101C2×C8×C53—41no—
102C2×C8×C53—42no—
103C4⋊C4×C53—22no—
104C4⋊C4×C53—22no—
105C4⋊C4×C53—22no—
106C4⋊C4×C53—22no—
107C4⋊C4×C53—42no—
108C4⋊C4×C53C22⋊C412yes—
109C4⋊C4×C53C22⋊C412yes—
110C22⋊C4×C53—22no—
111C22⋊C4×C53—22no—
112C22⋊C4×C53—24no—
113C22⋊C4×C53—24no—
114C22⋊C4×C53—22no—
115C22⋊C4×C53—22no—
116C22⋊C4×C53—22no—
117C22⋊C4×C53—22no—
118C22⋊C4×C53—22no—
119C42×C53—22no—
120C42×C53C22×C411yescharacteristic
121C42×C53C2×D412yes—
122C42×C53C2×D411yescharacteristic
123C23×C53—22no—
124C23×C53—22no—
125C23×C53—22no—
126C23×C53—42no—
127C23×C53—42no—
128C23×C53—84no—
129C23×C53—82no—
130C23×C53—84no—
131C23×C53#12011yescharacteristic
132Q8×C53—42no—
133D4×C53—42no—
134D4×C53—42no—
135D4×C53—42no—
136D4×C53—44no—
137D4×C53—44no—
138D4×C53—42no—
139D4×C53—42no—
140D4×C53—82no—
141D4×C53—84no—
142D4×C53—84no—
143D4×C53—84no—
144D4×C53—84no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—22no—
148C2×C4×C53—22no—
149C2×C4×C53—22no—
150C2×C4×C53—42no—
151C2×C4×C53—44no—
152C2×C4×C53—82no—
153C2×C4×C53—84no—
154C2×C4×C53—82no—
155C2×C4×C53—81no—
156C2×C4×C53C2×C22⋊C412yes—
157C2×C4×C53C2×C22⋊C411yescharacteristic
158C2×C4×C53#12011yescharacteristic
159C2×C4×C53#12012yes—
160C8×C53—42no—
161C8×C53—82no—
162#8497C5311yesmaximal, characteristic, direct product factor, Sylow
163C22×C53—22no—
164C22×C53—42no—
165C22×C53—42no—
166C22×C53—42no—
167C22×C53—42no—
168C22×C53—44no—
169C22×C53—44no—
170C22×C53—42no—
171C22×C53—162no—
172C22×C53—162no—
173C22×C53—164no—
174C22×C53#37911yescharacteristic
175C22×C53#45311yescharacteristic
176C22×C53#45511yescharacteristic
177C4×C53—21no—
178C4×C53—21no—
179C4×C53—22no—
180C4×C53—22no—
181C4×C53—42no—
182C4×C53—42no—
183C4×C53—42no—
184C4×C53—161no—
185#3543C2×C5311yescharacteristic
186#1961C2×C5311yescharacteristic, complemented
187#1555C2×C5311yescharacteristic, complemented
188#1532C2×C5312yescomplemented
189#1512C2×C5312yescomplemented
190C2×C53—21no—
191C2×C53—21no—
192C2×C53—82no—
193C2×C53—82no—
194C2×C53—82no—
195C2×C53#179211yescharacteristic
196C2×C53#179512yes—
197#560—22no—
198#559C4×C5311yescharacteristic, complemented
199#550C4×C5312yescomplemented
200#546—22no—
201D42—22no—
202D42C4×C5311yescharacteristic, complemented
203#540—22no—
204#535—22no—
205#534—22no—
206C2×#127C22×C5311yescharacteristic
207#529C22×C5312yes—
208#432C22×C5312yescomplemented
209C2×#102—22no—
210C2×#97C22×C5311yescharacteristic, complemented
211#382C22×C5311yescharacteristic, complemented
212#331—22no—
213#330—22no—
214#329—22no—
215#328—22no—
216#327—22no—
217C53#849711yescharacteristic, direct product factor, Sylow
218C2×#41—22no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4D4×C5312yes—
223C22×C8—21no—
224#127—22no—
225#127—22no—
226#127C2×C4×C5311yescharacteristic
227#127C2×C4×C5311yescharacteristic
228#127D4×C5312yes—
229#126—22no—
230#125D4×C5312yes—
231#124—22no—
232#124D4×C5312yes—
233#123—22no—
234#123—24no—
235#123—22no—
236#123—22no—
237C22⋊Q8D4×C5312yes—
238#121—24no—
239#121—24no—
240#121—24no—
241#121—24no—
242#121—22no—
243#121—22no—
244#121D4×C5312yes—
245#120—22no—
246#120—24no—
247#120—24no—
248#120—22no—
249C4×Q8C2×C4×C5312yes—
250C4×D4—22no—
251C4×D4—22no—
252C4×D4—22no—
253C4×D4C2×C4×C5312yes—
254#117—22no—
255C2×C4⋊C4—21no—
256C2×C22⋊C4—22no—
257C2×C42C23×C5311yescharacteristic
258C4⋊C8—22no—
259#102—44no—
260C22⋊C8—22no—
261#97—22no—
262#97—22no—
263#95—22no—
264C24—22no—
265C24—22no—
266#41—84no—
267C2×Q8C22⋊C4×C5312yes—
268C2×D4—22no—
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—24no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—84no—
279C2×D4—84no—
280C2×D4—82no—
281C2×D4—82no—
282C2×D4C22⋊C4×C5312yes—
283C2×D4C22⋊C4×C5312yes—
284C2×D4C22⋊C4×C5312yes—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—41no—
289C22×C4C2×D4×C5311yescharacteristic
290C22×C4C2×D4×C5312yes—
291C2×C8—22no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—42no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—42no—
300C4⋊C4C22⋊C4×C5312yes—
301C4⋊C4C22⋊C4×C5312yes—
302C22⋊C4—22no—
303C22⋊C4—22no—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C22⋊C4—22no—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C42—22no—
312C42C22×C4×C5311yescharacteristic
313C42C2×D4×C5312yes—
314C42C2×D4×C5311yescharacteristic
315C23—22no—
316C23—22no—
317C23—22no—
318C23—42no—
319C23—42no—
320C23—84no—
321C23—82no—
322C23—84no—
323C23#120×C5311yescharacteristic
324Q8—42no—
325D4—42no—
326D4—42no—
327D4—42no—
328D4—44no—
329D4—44no—
330D4—42no—
331D4—42no—
332D4—82no—
333D4—84no—
334D4—84no—
335D4—84no—
336D4—84no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—82no—
345C2×C4—84no—
346C2×C4—82no—
347C2×C4—81no—
348C2×C4C2×C22⋊C4×C5312yes—
349C2×C4C2×C22⋊C4×C5311yescharacteristic
350C2×C4#120×C5311yescharacteristic
351C2×C4#120×C5312yes—
352C8—42no—
353C8—82no—
354C22—22no—
355C22—42no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—44no—
360C22—44no—
361C22—42no—
362C22—162no—
363C22—162no—
364C22—164no—
365C22C53×#37911yescharacteristic
366C22C53×#45311yescharacteristic
367C22C53×#45511yescharacteristic
368C4—21no—
369C4—21no—
370C4—22no—
371C4—22no—
372C4—42no—
373C4—42no—
374C4—42no—
375C4—161no—
376C2—21no—
377C2—21no—
378C2—82no—
379C2—82no—
380C2—82no—
381C2C53×#179211yescharacteristic
382C2C53×#179512yes—
subgroupG/H
center, socleC22×C53#379
commutatorC42C22×C4×C53
Frattini, cosocleC2×C42C23×C53
prime radicalC53×#3543C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#1961C2semidirect
3. C53×#1555C2semidirect
4. C53×#1532C2semidirect
5. C53×#1512C2semidirect
7. C53×#559C4semidirect
8. C53×#550C4semidirect
11. D42×C53C4semidirect
17. C53×#432C22semidirect
19. C2×#97×C53C22semidirect
20. C53×#382C22semidirect
162. #8497C53direct (N × H)
186. #1961C2×C53semidirect
187. #1555C2×C53semidirect
188. #1532C2×C53semidirect
189. #1512C2×C53semidirect
198. #559C4×C53semidirect
199. #550C4×C53semidirect
202. D42C4×C53semidirect
208. #432C22×C53semidirect
210. C2×#97C22×C53semidirect
211. #382C22×C53semidirect
217. C53#8497direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#35432
C53×#19612
C53×#15552
C53×#15322
C53×#15122
#849753