Finite Group Databasebeta

← C53×#9186 #27216698

Subgroups — C53×#9186 #27216698

Order 13568.

580 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 122 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×#2975C211yesmaximal, characteristic, complemented
2C53×#2635C211yesmaximal, characteristic
3C4×#99×C53C211yesmaximal, characteristic, complemented
4C53×#1507C212yesmaximal, complemented
5C53×#1284C212yesmaximal, complemented
6C2×#121×C53C2211yescharacteristic
7C2×C4×D4×C53C2211yescharacteristic
8C53×#436—24no—
9C53×#431—21no—
10C2×#99×C53—21no—
11C53×#407—21no—
12C53×#407C2212yes—
13C53×#400C411yescharacteristic, complemented
14C53×#399C411yescharacteristic, complemented
15C53×#394—21no—
16C53×#394C411yescharacteristic, complemented
17C53×#392—21no—
18C53×#392C411yescharacteristic, complemented
19C53×#388—21no—
20C53×#386—21no—
21C53×#381C2211yescharacteristic
22C4×C22⋊C4×C53C2212yes—
23C53×#342—21no—
24C53×#327—24no—
25C22×D4×C53—21no—
26C22×D4×C53—21no—
27C22×D4×C53C2×C411yescharacteristic
28C22×D4×C53D411yescharacteristic
29C23×C4×C53D411yescharacteristic
30C23×C4×C53C2311yescharacteristic
31C2×M16×C53—22no—
32C2×M16×C53—22no—
33#121×C53—22no—
34#121×C53—22no—
35C4×D4×C53—22no—
36C4×D4×C53—22no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53C2×C411yescharacteristic
39C2×C4⋊C4×C53C2×C411yescharacteristic
40C2×C22⋊C4×C53—21no—
41C2×C22⋊C4×C53—21no—
42C2×C22⋊C4×C53—21no—
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—22no—
47C2×C22⋊C4×C53C2×C411yescharacteristic
48C2×C22⋊C4×C53D411yescharacteristic
49C2×C42×C53—22no—
50C2×C42×C53D411yescharacteristic
51C4⋊C8×C53—22no—
52#99×C53—24no—
53C22⋊C8×C53—22no—
54C22⋊C8×C53—22no—
55C22⋊C8×C53D414yescomplemented
56C4×C8×C53—22no—
57#95×C53—21no—
58#95×C53—21no—
59#95×C53—21no—
60#95×C53—22no—
61#95×C53C2×C411yescharacteristic
62#95×C53C2×C411yescharacteristic
63C24×C53—21no—
64C24×C53C22⋊C411yescharacteristic
65C24×C53C2×D411yescharacteristic
66C2×D4×C53—22no—
67C2×D4×C53—22no—
68C2×D4×C53—22no—
69C2×D4×C53—41no—
70C2×D4×C53—42no—
71C2×D4×C53—42no—
72C2×D4×C53—81no—
73C2×D4×C53—81no—
74C2×D4×C53—81no—
75C2×D4×C53—81no—
76C2×D4×C53—81no—
77C22×C4×C53—21no—
78C22×C4×C53—21no—
79C22×C4×C53—21no—
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53—42no—
83C22×C4×C53—81no—
84C22×C4×C53—81no—
85C22×C4×C53C22⋊C411yescharacteristic
86C22×C4×C53C22⋊C411yescharacteristic
87C22×C4×C53C22⋊C411yescharacteristic
88C22×C4×C53C22×C411yescharacteristic
89C22×C4×C53C2×D412yes—
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53#4112yes—
92M16×C53—82no—
93M16×C53—82no—
94C2×C8×C53—24no—
95C4⋊C4×C53—22no—
96C4⋊C4×C53—22no—
97C4⋊C4×C53—81no—
98C22⋊C4×C53—22no—
99C22⋊C4×C53—22no—
100C22⋊C4×C53—24no—
101C22⋊C4×C53—42no—
102C22⋊C4×C53—42no—
103C22⋊C4×C53—41no—
104C22⋊C4×C53—41no—
105C22⋊C4×C53—81no—
106C22⋊C4×C53—81no—
107C22⋊C4×C53—81no—
108C22⋊C4×C53—81no—
109C42×C53—21no—
110C42×C53—21no—
111C42×C53—82no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53—41no—
124C23×C53—41no—
125C23×C53—41no—
126C23×C53—41no—
127C23×C53—41no—
128C23×C53C2×C22⋊C411yescharacteristic
129C23×C53#12112yes—
130D4×C53—44no—
131D4×C53—82no—
132D4×C53—84no—
133D4×C53—84no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—42no—
141C2×C4×C53—42no—
142C2×C4×C53—42no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—44no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—81no—
156C2×C4×C53—82no—
157C2×C4×C53—82no—
158C2×C4×C53#10412yes—
159C2×C4×C53C4×D412yes—
160C2×C4×C53#12011yescharacteristic
161C2×C4×C53#12311yescharacteristic
162C8×C53—44no—
163#9186C5311yesmaximal, characteristic, direct product factor, Sylow
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—22no—
167C22×C53—22no—
168C22×C53—22no—
169C22×C53—42no—
170C22×C53—41no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53—81no—
174C22×C53—81no—
175C22×C53—81no—
176C22×C53—81no—
177C22×C53—81no—
178C22×C53—81no—
179C22×C53—81no—
180C22×C53—81no—
181C22×C53—81no—
182C22×C53—81no—
183C22×C53#38611yescharacteristic
184C22×C53C2×#10411yescharacteristic
185C22×C53#42111yescharacteristic
186C4×C53—22no—
187C4×C53—21no—
188C4×C53—21no—
189C4×C53—22no—
190C4×C53—41no—
191C4×C53—41no—
192C4×C53—81no—
193C4×C53—81no—
194C4×C53—84no—
195C4×C53—81no—
196#2975C2×C5311yescharacteristic, complemented
197#2635C2×C5311yescharacteristic
198C4×#99C2×C5311yescharacteristic, complemented
199#1507C2×C5312yescomplemented
200#1284C2×C5312yescomplemented
201C2×C53—21no—
202C2×C53—21no—
203C2×C53—42no—
204C2×C53—81no—
205C2×C53—161no—
206C2×C53#187111yescharacteristic
207C2×C53#191011yescharacteristic
208C2×C53#191311yescharacteristic
209C2×#121C22×C5311yescharacteristic
210C2×C4×D4C22×C5311yescharacteristic
211#436—24no—
212#431—21no—
213C2×#99—21no—
214#407—21no—
215#407C22×C5312yes—
216#400C4×C5311yescharacteristic, complemented
217#399C4×C5311yescharacteristic, complemented
218#394—21no—
219#394C4×C5311yescharacteristic, complemented
220#392—21no—
221#392C4×C5311yescharacteristic, complemented
222#388—21no—
223#386—21no—
224#381C22×C5311yescharacteristic
225C4×C22⋊C4C22×C5312yes—
226#342—21no—
227#327—24no—
228C53#918611yescharacteristic, direct product factor, Sylow
229C22×D4—21no—
230C22×D4—21no—
231C22×D4C2×C4×C5311yescharacteristic
232C22×D4D4×C5311yescharacteristic
233C23×C4D4×C5311yescharacteristic
234C23×C4C23×C5311yescharacteristic
235C2×M16—22no—
236C2×M16—22no—
237#121—22no—
238#121—22no—
239C4×D4—22no—
240C4×D4—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4C2×C4×C5311yescharacteristic
243C2×C4⋊C4C2×C4×C5311yescharacteristic
244C2×C22⋊C4—22no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4C2×C4×C5311yescharacteristic
252C2×C22⋊C4D4×C5311yescharacteristic
253C2×C42—22no—
254C2×C42D4×C5311yescharacteristic
255C4⋊C8—22no—
256#99—24no—
257C22⋊C8—22no—
258C22⋊C8—22no—
259C22⋊C8D4×C5314yescomplemented
260C4×C8—22no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—22no—
265#95C2×C4×C5311yescharacteristic
266#95C2×C4×C5311yescharacteristic
267C24—21no—
268C24C22⋊C4×C5311yescharacteristic
269C24C2×D4×C5311yescharacteristic
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—41no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—81no—
277C2×D4—81no—
278C2×D4—81no—
279C2×D4—81no—
280C2×D4—81no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—42no—
287C22×C4—81no—
288C22×C4—81no—
289C22×C4C22⋊C4×C5311yescharacteristic
290C22×C4C22⋊C4×C5311yescharacteristic
291C22×C4C22⋊C4×C5311yescharacteristic
292C22×C4C22×C4×C5311yescharacteristic
293C22×C4C2×D4×C5312yes—
294C22×C4C2×D4×C5311yescharacteristic
295C22×C4#41×C5312yes—
296M16—82no—
297M16—82no—
298C2×C8—24no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—81no—
302C22⋊C4—22no—
303C22⋊C4—22no—
304C22⋊C4—24no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—81no—
310C22⋊C4—81no—
311C22⋊C4—81no—
312C22⋊C4—81no—
313C42—21no—
314C42—21no—
315C42—82no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23C2×C22⋊C4×C5311yescharacteristic
333C23#121×C5312yes—
334D4—44no—
335D4—82no—
336D4—84no—
337D4—84no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—44no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—81no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4#104×C5312yes—
363C2×C4C4×D4×C5312yes—
364C2×C4#120×C5311yescharacteristic
365C2×C4#123×C5311yescharacteristic
366C8—44no—
367C22—21no—
368C22—21no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—42no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22C53×#38611yescharacteristic
387C22C2×#104×C5311yescharacteristic
388C22C53×#42111yescharacteristic
389C4—21no—
390C4—21no—
391C4—22no—
392C4—22no—
393C4—41no—
394C4—41no—
395C4—81no—
396C4—81no—
397C4—84no—
398C4—81no—
399C2—21no—
400C2—21no—
401C2—42no—
402C2—81no—
403C2—161no—
404C2C53×#187111yescharacteristic
405C2C53×#191011yescharacteristic
406C2C53×#191311yescharacteristic
subgroupG/H
center, socleC22×C53#386
commutatorC22×C4C22×C4×C53
Frattini, cosocleC23×C4C23×C53
prime radicalC53×#2635C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C53×#2975C2semidirect
3. C4×#99×C53C2semidirect
4. C53×#1507C2semidirect
5. C53×#1284C2semidirect
13. C53×#400C4semidirect
14. C53×#399C4semidirect
16. C53×#394C4semidirect
18. C53×#392C4semidirect
55. C22⋊C8×C53D4semidirect
163. #9186C53direct (N × H)
196. #2975C2×C53semidirect
198. C4×#99C2×C53semidirect
199. #1507C2×C53semidirect
200. #1284C2×C53semidirect
216. #400C4×C53semidirect
217. #399C4×C53semidirect
219. #394C4×C53semidirect
221. #392C4×C53semidirect
228. C53#9186direct (N × H)
259. C22⋊C8D4×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#29752
C53×#26352
C4×#99×C532
C53×#15072
C53×#12842
#918653